close

Вход

Забыли?

вход по аккаунту

?

Search for 140 microeV Pseudoscalar and Vector Dark Matter Using Microwave Cavities

код для вставкиСкачать
A bstract
Search for 140 m icroeV Pseudoscalar and Vector
Dark M atter U sing Microwave Cavities
Ana T. Malagon
2014
The question of what constitutes dark m atter remains unanswered despite evidence
th at has firmly established that galaxies are primarily composed of non-baryonic,
non-luminous m atter th at must have a significant non-relativistic component. This
cold dark m atter plays an important role in structure formation and is included in
the standard cosmological model, which estimates that dark m atter composes 23%
of the total energy density in the universe. There are many postulated particles that
are theorized to be the constituents of cold dark matter; however, none have been
observed experimentally. One strongly motivated particle that could be cold dark
m atter is the axion, a light pseudoscalar boson with a two photon vertex. Other
particles with similar properties, called axion-like particles (ALPs). can also be good
dark m atter candidates, and experiments searching for axions can also place lim­
its on ALPs' coupling versus mass parameter space. Searches for axions have been
experimentally challenging, as they interact very feebly with ordinary m atter. Ex­
perimental techniques to detect dark m atter axions rely on a multiphoton radiative
transition; in the presence of a strong magnetic field axions can convert to photons.
If a microwave cavity is placed in the interaction region, when the produced photon’s
frequency is resonant with a mode of the cavity, the signal power is enhanced further.
As the axion mass is not known, the resonant frequency of the cavity must be swept
to search for possible converted photons. This method has been used to constrain the
axion-photon interaction strength for masses between 1-3 //eV. with plans to search
up to 12 peV. However, prior to the work in this dissertation, the microwave cavity
method had not been applied to look for axions of higher mass in the 0.1-1 meV range.
It is im portant to search for axions in this mass range in order to cover all possible
parameter space, as the axion mass is constrained to lie roughly between 1 peV and
1 meV. We present here the first microwave cavity search for dark m atter ALPs with
mass m a ~ 140 /ieV. The experiment measured the power in the Ka-band frequency
range from the TM 020 mode of a cryogenically cooled cavity in a 7 Tesla background
magnetic field. High Electron Mobility Transistor (HEMT) amplifiers decreased the
system noise tem perature to approximately 20 Kelvin. We took data for six months
and swept the microwave cavity resonant frequency from 33.9-34.5 GHz. correspond­
ing to an axion mass range of 140.2-142.7 peV. We did not observe any statistically
significant signals, and thus were able to place an upper bound on the axion to two
photon coupling of gail < 8.75 x 10“ 11 1/GeV, marginally improving on the previous
best limit obtained from the CAST experiment. W ith the same data set we were
also able to set new limits on dark m atter "hidden photon’--photon interactions of
X < 5 x 10“ 10, significantly improving upon previous bounds. Hidden photons are
postulated massive vector bosons that would only interact with Standard Model pho­
tons, and arise from new gauge extensions to the Standard Model. In conclusion, as
with the first generation of microwave cavity experiments that searched for axions in
the peV mass range, there are several technical challenges that must be overcome in
order to reach the sensitivity to observe or exclude canonical axion models. However,
as we do not know the axion mass a priori, it is valuable to develop experiments
and techniques to search for them in their entire possible mass range. If observed,
axion (and ALP) dark m atter would not only represent an important advance in our
knowledge of dark m atter but also provide clues about processes at high energy scales
inaccessible to collider experiments.
Search for 140 m icroeV Pseudoscalar and
V ector Dark M atter U sing M icrowave
C avities
A D issertation
Presented to the Faculty of the G rad u ate School
of
Yale University
in C andidacy for the Degree of
D octor of Philosophy
by
A na T. M alagon
D issertation Advisor: O. K eith Baker
December 2014
UMI Number: 3582272
All rights reserved
INFORMATION TO ALL USERS
The quality of this reproduction is dependent upon the quality of the copy submitted.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
Di!ss0?t&iori Publishing
UMI 3582272
Published by ProQuest LLC 2015. Copyright in the Dissertation held by the Author.
Microform Edition © ProQuest LLC.
All rights reserved. This work is protected against
unauthorized copying under Title 17, United States Code.
ProQuest LLC
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, Ml 48106-1346
Copyright (c) 2014 by Ana T. Malagon
All rights reserved.
Contents
A cknow ledgm ents
1
2
3
4
5
ix
Introd u ction
1
1.1
Dark M a tte r ...................................................................................................
1
1.2
O u tlin e.............................................................................................................
6
Background
8
2.1
A n o m alies......................................................................................................
8
2.2
Strong CP P ro b le m ......................................................................................
11
2.3
Solutions to the Strong CP Problem
......................................................
14
2.4
Peccei Quinn M echanism ............................................................................
14
Dark M atter A xions and H idden P h oton s
21
3.1
Non-Thermal Production of Dark M a tte r ...............................................
22
3.2
Hidden Photons as Dark M a tte r ................................................................
25
Param eter Space
27
4.1
Axion and ALP B o u n d s ............................................................................
27
4.2
Hidden Photon B o u n d s................................................................................
37
Signal Pow er
45
5.1
45
Dark M atter Signal P ro p e rtie s...................................................................
iii
6
7
8
5.2
Axion Signal P o w e r......................................................................................
49
5.3
Hidden Photon Signal P o w er......................................................................
55
E xperim ent
58
6.1
The M a g n e t...................................................................................................
58
6.2
Cryogenic S y s te m s ......................................................................................
60
6.3
Insert
62
6.4
The Cavity
...................................................................................................
64
6.5
Cryogenic A m p lifiers...................................................................................
73
6.6
Room Temperature Electronics
................................................................
78
.............................................................................................................
D a ta A nalysis
84
7.1
D ata D e s c rip tio n .........................................................................................
84
7.2
Baseline R e m o v a l..........................................................................................
87
7.3
Combining Power S p e c t r a .........................................................................
94
7.4
T h r e s h o ld ......................................................................................................
97
E xclusion Lim it
101
8.1
Axion B o u n d s .................................................................................................
101
8.2
Hidden Photon B o u n d s................................................................................
103
9
C onclusions
104
A
C avity A ssem bly D raw ings
108
B
D ead Spot
110
iv
List of Figures
1.1
Direct detection WIMP exclusion bounds on the coupling versus WIMP
inass....................................................................................................................
4
2.1
Triangle loop diagram of ir —>7 7 ................................................................
10
2.2
Triangle loop diagram of axions to gluons.................................................
11
2.3
Spontaneous and explicit symmetry b reak in g .........................................
15
4.1
Primakoff e f f e c t.............................................................................................
29
4.2
ALP parameter s p a c e ...................................................................................
33
4.3
LSW experimental setup for pseudoscalar searches................................
34
4.4
Dark m atter ALP b o u n d s ..........................................................................
38
4.5
Dark m atter ALP parameter s p a c e .........................................................
39
4.6
Hidden photon parameter s p a c e ................................................................
40
4.7
Longitudinal mode limits 011 hidden p h o to n s .........................................
43
4.8
Longitudinal mode limits from microwave cavity LSW searches . . . .
44
6.1
Magnet
..........................................................................................................
59
6.2
Top half of the cryostat, with inset showing full cryostat......................
61
6.3
Sketch of cavity and waveguides in cryostat..............................................
63
6.4
Cavity b o d y ...................................................................................................
64
6.5
TM 020 form factor versus fre q u e n c y ..........................................................
67
6.6
Tuning (warm): frequency versus rod insertion d e p t h .........................
68
v
6.7
Cavity field lines
.........................................................................................
69
6.8
Reflection measurements of c a v ity ............................................................
71
6.9
Cavity assem bly.............................................................................................
72
6.10 Gain of first amplifier as measured by vendor.........................................
74
6.11 Noise tem perature
......................................................................................
75
6.12 O utput power density versus LO frequency for a 50 ohm r u n .............
77
6.13 Schematic of e x p e rim e n t............................................................................
78
6.14 Response of room tem perature electronics c h a i n ...................................
80
6.15 Loaded Q versus frequency.........................................................................
83
7.1
D ata acquisition schedule.............................................................................
85
7.2
Power spectral shape c h a n g e s ...................................................................
88
7.3
Temperature stable run.................................................................................
89
7.4
Spectra after baseline re m o v a l...................................................................
91
7.5
Power of test tone and leakage b i n ............................................................
92
7.6
Histogram of power flu c tu a tio n s................................................................
93
7.7
Standard error of the mean versus number ofaverages............................
94
7.8
Lorentzian response w eig h tin g ...................................................................
96
7.9
Histogram of power fluctuations in the total co-added power spectrum
97
7.10 Threshold for co-added power sp ectru m ...................................................
98
7.11 Persistent sig n a l.............................................................................................
100
8.1
Exclusion limit from two bin a n a ly s is ......................................................
102
8.2
Exclusion limit for hidden photons.............................................................
103
A .l
Bellows-sub-assembly...................................................................................
108
A.2 Cavity body and w aveguides......................................................................
109
A.3 Cavity b o d y ....................................................................................................
109
vi
B.l Cavity response without shim.......................................................................
Ill
B.2 Cavity response with s h i m ..........................................................................
112
vii
List of Tables
4.1
Summary of astrophysical bounds on the axionmass
............................
31
6.1
Mode co m p ariso n ..........................................................................................
66
6.2
Power levels at different points in the receiverc h a i n ...............................
80
7.1
Candidates
99
...................................................................................................
vm
Acknowledgments
I would like to thank my advisor O. Keith Baker for giving me the opportunity to work
on this project. The experiment was a group effort; Penny Slocum designed the room
tem perature receiver chain, wrote the interface for the data acquisition system, and
separately analyzed the data for cross-checking. Andrew Martin built the insert and
made it vacuum compatible, diagnosed the cavity ‘‘dead spot'’ and fixed the problem
as well as doing a great deal of troubleshooting during data runs. The members of the
Beam Lab, Yong Jiang and Sergey Shchelkunov, were very generous with equipment
and their time. Andy Szymkowiak also offered valuable assistance and equipment.
Thank you to Tom Hurteau for machining the cavity and helping test the cavity
prototypes. Also, my thanks to Will Emmett for helpful discussions on cryogenic and
vacuum engineering and for the engineering drawings of the system. The magnet is
on loan from Kurt Zilm.
Thanks to Camille, Tomomi, and Nicole for being wonderful people and cheering
me up when I needed it; Ben for fun seminars reading axion papers and puzzling out
QFT mysteries; and Craig for machine shop advice and help. My thanks to Michael
Betz for helpful discussions on cavity design and data analysis. I am also grateful to
Eustace for his advice on HFSS and life.
Finally. I am very grateful to rny family for encouraging me throughout my time
at grad school and providing much-needed support. Thank you to my mother and
sister for reading this dissertation many times.
Chapter 1
Introduction
1.1
Dark M atter
One of the most basic open questions in physics today concerns the nature of dark
m a tte r . T h e re is s tro n g o b se rv a tio n a l evid en ce t h a t non-luminous g ra v ita tio n a lly in­
teracting m atter exists,as first discussed by Zwicky in 1933 [1], who observed a much
higher mass to light ratio in the Coma cluster than expected from the luminous m atter
alone. The evidence for dark m atter has mounted since then: classic evidence comes
from the rotation speed of gas in spiral galaxies, which decreases with radius as one
would expect from a galaxy embedded in a diffuse dark m atter halo [2]. Arguably
the most important evidence comes from measurements of the Cosmic Microwave
Background (CMB) power spectrum, as measured by the WMAP [3] and Planck mis­
sions [4]. Fitting the observed power spectrum to the standard cosmological ACDM
model results in values for the baryon and m atter density of the universe:
ttbaryonh2 = 0.0226 ± .0006
( 1 . 1)
il,„att,
(1.2)
r h 2
= 0.135 ± .009
1
where h =
k^ g ec is the reduced Hubble constant. The baryon density is clearly
less than the m atter density - this conclusively shows that most ofthe m atter in the
universe is non-baryonic. The baryon density can also be estimated from Big Bang
Nucleosynthesis (BBN) and measurements of the primordial deuterium abundance in
gas clouds at high redshift [5]. The result is consistent with the CMB estimate:
t t b a r y o n h
2 = 0.0216 ± 0.0020.
(1.3)
The total m atter density can also be measured with a different technique, using the
power spectrum of the galaxy-galaxy correlation function and looking for imprints
from baryon acoustic oscillations [6 ]. The estimated m atter density from this method
is:
a , lat t e
r h 2
= 0 .1 3 0 ± 0 . 0 1 0
(1 .4 )
which also agrees well with the value from CMB.
The bulk of the dark m atter must be non-relativistic ("cold dark m atter"). O th­
erwise the dark m atter particles will be moving so fast that they wash out small scale
structure, leading to "top-down” structure formation, where large superclusters form
first and then fragment to form galaxies [7]. This contradicts the observational data
from large redshift surveys showing that the universe we live in was formed from the
accretion of small structures into larger ones [8 ]. Simulations of cold dark m atter
(CDM) [9] agree much more closely with the observed structure formation.
Therefore the identity of the dark m atter particle is constrained by the following
properties:
• its dominant component must be non-relativistic
• it must be non-baryonic
2
• as no radiation is observed from the dark m atter, the particles must be neutral
and interact very feebly with Standard Model particles
• it must be stable (or nearly so) so that the dark m atter particles produced in
the early universe would still be present in large abundances today.
There are many proposed candidates for dark matter; to date none have been
experimentally observed. Arguably the two most strongly motivated candidates are
Weakly Interacting Massive Particles (WIMPs) and axions.
W IM P s
WIMPs have been the most widely studied as dark m atter candidates; the term
refers to neutral particles that have electroweak scale interactions with m atter. The
theoretical motivation for WIMPs comes from theories of supersyinmetry (SUSY).
SUSY theories double the degrees of freedom by adding new partners for all the
known particles with opposite spin-statistics [10]. This elegantly solves a serious
problem with the Higgs mechanism, which is that the Higgs boson has a mass which
is quadratically divergent - what is known as the gauge hierarchy problem.
The lightest particle of various SUSY theories generally has the correct mass
and interaction strength to be a cold dark m atter candidate [11]. By analyzing the
thermalization of such particles as the universe cools, one finds that the present-day
abundance of WIMPs would be approximately equal to the observed dark m atter
density [12]: this startling coincidence is known as the "WIMP miracle". Various
experiments have searched for WIMPs with masses in the 10 GeV —1 TeV mass range,
either directly through the energy deposited when WIMPs scatter off nuclei [13],
indirectly through the energy resulting from their annihilation [14]. or from searches
in colliders [15]. All experiments have returned null results and the constraints on
WIMPs and SUSY theories [16] are now quite stringent. The most recent limits from
direct detection experiments at the time of writing are shown in Figure 1.1 (from [17].
3
10 45 5
C o g e n t N eutrino ScaH enng on CaWO,
1
2
3
4
5
6 7 8 910
20
30
WIMP mass [GeV c2]
Figure 1.1: Direct detection WIMP exclusion bounds on the coupling versus WIMP
mass. The gray area on the lower left shows the region where backgrounds will be
dominated by coherent neutrino scattering, which will be the limiting factor for the
sensitivity of these experiments.
As no evidence for WIMPs has appeared, interest has grown in investigating other
dark m atter candidates.
A xions and A L Ps
Axions are another strongly motivated dark m atter candidate. The axion arises from
an extension to the Standard Model proposed by Peccei and Quinn in 1977 [18] to
explain the conservation of charge parity (CP) symmetry in Quantum Chromody­
namics (QCD), which is known as the strong CP problem. This extension involves
a new global symmetry th at has an anomalous interaction with QCD, leading all
CP-violating terms in the QCD Lagrangian to dynamically go to zero. Weinberg [19]
and Wilczek [20] independently noticed that the pseudo-Nambu-Goldstone boson of
the broken symmetry was a low-energy observable: the axion. One consequence of
the Peccei-Quinn mechanism is that the mass and coupling strength of the axion are
proportional to each other and inversely proportional to the energy breaking scale,
so limits on the coupling and mass of sub-eV particles can act as probes of very high
energy scales.
In other extensions to the Standard Model, the pseudo-Nambu-Goldstone bosons
of continuous global symmetries are called axion-like particles (ALPs) if they have
a coupling to two photons, a —v 7 7 , which is the dominant decay mode if the ALP
mass m a is lighter than the electron mass. Familons and majorons are two concrete
examples of ALPs: familons come from the breaking of a global family symmetry [21]
- these symmetries are introduced to explain the structure of the fermion masses
and their mixings. Majorons arise from broken lepton-number symmetries [22]. In
string theory ALPs genericallv arise in all compactifications [23]. The two photon
vertex is the interaction most experiments use to search for axions and ALPs. For
these more general ALP models, the coupling and mass are no longer proportional, so
there are two free parameters in the theory. As well, because ALPs do not necessarily
arise from anomalous interactions with QCD they would not solve the strong CP
problem. However the techniques used to search for axions also apply to ALPs, with
experimental and theoretical bounds applying to both as well. From here on out, I
use the term pseudoscalar or ALP when referring to general axion-like particles that
have a coupling to two photons; I will refer to the axion arising specifically from the
Peccei-Quinn mechanism as the QCD axion.
Finally, another proposed particle that arises from additional 1/(1) symmetries in
the Standard Model is the hidden photon [24], a massive vector boson which would
interact only with the standard photon through kinetic mixing. Hidden photons are
of particular interest in relation to axions because similar techniques can be used to
search for both. These vector bosons are also known as dark photons or paraphotons.
They arise in many models of string theories [25], with wide ranges of the kinetic
5
mixing parameter predicted; for a review see Ref [26]. If they are long-lived and can
be produced non-thermally, their feeble interactions make them good dark m atter
candidates [27].
At present there is no clear evidence for any of these particles, thus it is worthwhile
investigating all dark m atter candidates. Moreover, dark m atter could be formed
from a combination of any or all of these particles. This work focuses on a dark
m atter axion-like particle search. The search, done as part of the Yale Microwave
Cavity Experiment (YMCE), looked for dark m atter pseudoscalars using the process
7 *a —>7 , with the off-shell photon provided by a static magnetic field. This project
is part of YMCE's work in constraining exotic particles in the 140 jueV mass region
using cryogenically cooled microwave cavities, with previous projects being a search
for dark m atter scalars [28] and another project which looked for vector bosons using
a light-shining-through-wall technique [29]. We use the data from the dark m atter
axion search to also set limits on hidden photon dark m atter. This mass region is so
far unexplored by other microwave cavity experiments, as it is more challenging to
reach the sensitivities needed to detect the QCD axion.
1.2
Outline
This dissertation will describe the pilot run of the Yale Microwave Cavity Experiment
(YMCE) to look for dark m atter axions in the mass range 140.2-142.7 ^eV. From
the data taken, the analysis excludes axion-like particles with two-photon coupling
9a~f'y < 8.75 x 10“ 11 1/GeV, and sets an upper bound on the hidden photon mixing
parameter with photons: y < 5 x 10-10. The experiment used a tunable microwave
cavity immersed in a strong magnetic field to look for dark m atter axions converting to
photons as well as dark m atter hidden photons converting to photons independently
of the magnetic field. The detection principle will be described further in Chapter 5.
The outline of the dissertation is as follows:
6
C hapter 2: Background describes the motivation for the QCD axion and the
symmetry breaking process by which it arises in order to provide a case study of
pseudoscalars arising from broken symmetries.
C hapter 3: Dark M atter A xions and H idden P h oton s describes the mechanism
by which axions and hidden photons can be generated in the early universe in the
right abundance to match the dark m atter density observed.
C hapter 4: Param eter Space goes over the current bounds on axion and hidden
photon coupling and mass, and describes where microwave cavity searches fit into
this field.
C hapter 5: Signal Pow er explains the sensitivity we can achieve using microwave
cavities in strong magnetic fields and how the expression for signal power determines
what we optimize in the experiment
C hapter 6: E xperim ent goes through the components of the experiment, design
of the microwave cavity, and calibrations of the system.
C hapter 7: D a ta A nalysis details the full analysis chain that takes the raw time
domain voltage measurements and turns them into average power spectra. Baseline
estimation, cuts, and corrections to the data are described.
C hapter 8: E xclusion Lim its presents the upper bound on the axion to two
photon coupling for the mass range investigated. Also described are limits on the
hidden photon coupling to photons.
C hapter 9: C onclusions concludes by discussing lessons learned and offering sug­
gestions for future work.
7
Chapter 2
Background
We now discuss the theoretical background for axions and hidden photons in more de­
tail. Anomalies are a concept of fundamental importance in particle physics and play
an im portant role in the physics surrounding the strong CP problem and the PecceiQuinn solution. We briefly sketch the main idea of anomalous symmetry breaking in
Section 2.1. Section 2.2 describes the strong CP problem, with possible solutions in
Section 2.3. Section 2.4 goes over the Peccei Quinn solution to this problem, which
has the axion as an observable consequence. We also review two of the commonly
discussed axion models in the literature, the KSVZ and DFSZ axion models, sum­
marize the arguments against the Peccei Quinn solution, and conclude by discussing
the generalization of the axion solution to ALPs.
2.1
Anom alies
In the context of the strong CP problem and the Peccei-Quinn solution, anomalies
explain how the axion gains mass and ultimately interacts with two photons, the
process a —v 7 7 mentioned previously. This transition is called Primakoff emission as
it is directly analogous to the original Primakoff process [30] in which a neutral pion
decays to two photons 1
7T —►7 7 .
(2 .1 )
An anomaly occurs when a classical symmetry is not conserved upon quantization.
Let us take a global axial (chiral) symmetry as a working example,
U a ( 1) ,
which
transforms fermion fields as
0 -»■ eiaTl%)
( 2 .2 )
where a is the infinitesimal parameter of the rotation. If the Lagrangian of the system
is invariant under the Ua {1) rotations, the axial current j^{x) is conserved,
fi(x) =
(2-3)
d-f; (x) = 0
(2.4)
with a corresponding conserved charge.
However, a problem arises upon quanti­
zation, specifically when calculating the vacuum expectation value of axial-vector
interactions. Ref [33] works through the calculations for the case of Abelian and
non-Abelian fields, and we refer the reader to this reference for more details. The end
result isth at aterm isintroduced to the divergence ofthe current,
so then it is no
longer conserved. In the case of a chiral anomaly with QCD, the term added to the
axial current divergence is (in the case of massless quarks):
Na
= -jfG G
(2.5)
1. The Primakoff effect arises due to an axial anomaly of a global t/.i(l) symm etry in QED. It was
the observable decay of the pion which first led Bell, Jackiw, [31] and independently Adler [32] to
study anomalies, as the decay would have been forbidden if the axial current was conserved, taken
to be the case at the time.
9
where G is the color field strength, G ab = \zabcdG cd the dual, and a s the strong force
fine structure constant. N is the number of fermions that have a charge under the
symmetry.
To lowest order in perturbation theory, anomalies are represented in Feynman
diagram form by triangle graphs, as shown in Figure 2.1 for the pion decay; diagram
from [33]. The propagation of the virtual fermions in the current shows that the
current is not conserved.
Y
r
Figure 2.1: Triangle loop diagram of 7r -* 7 7 .
As discussed later in Section 2.4, the Peccei-Quinn mechanism introduces a new
symmetry
U ( 1 ) pq
which has an anomalous symmetry breaking with QCD. It is this
anomaly which produces the interaction of the axion field with gluons, shown in
Figure 2.2 (diagram from [34]). If the virtual fermions in the triangle loop carry
electric charge, then the axion has an anomaly with QED and the analogous graph
with the gluons replaced by photons is possible, which produces the a -¥ 77 process.
10
g.
(~6~6~6~6~6Tf0~0~6~0~6T5TT' g
“0"0"0"0'0'O ‘07TClT0"0"C's g
Figure 2.2: Triangle loop diagram of axions to gluons.
Another contribution to this process comes from axion-meson mixing, from which the
mass m a for the axion [35] can be calculated to be of order
( 2 .6 )
mafpQ « m v f n
where
/ pq
is the spontaneous symmetry breaking scale of the
U ( 1 ) pq
symmetry,
f„ % 93 MeV is the pion decay constant and m n = 135 MeV the pion mass.
Having introduced the concept of anomalous symmetry breaking, we now discuss
the strong CP problem and its most widely accepted resolution, the Peccei-Quinn
mechanism.
2.2
Strong CP Problem
CP symmetry, or charge parity symmetry, refers to the symmetry of physical equa­
tions under reversal of spatial reflection and charge conjugation. If the physics of a
system is not symmetric under charge and parity reversal, the CP violation would
be manifested as a visible distinction between particles and antiparticles [36]. CP
violation was first observed in neutral kaon systems [37], showing that the CP is
not a good symmetry of the electroweak force. The strong force also has a source
of CP violation, the ‘0-terin’, which appears in the QCD effective lagrangian as a
11
consequence of the complex structure of the QCD vacuum [38] and has the form:
C
= 9— GG
47T
(2.7)
where 9 is a phase between 0 and 2n parameterizing the amount of CP violation.
To see th at this term is parity and time violating, construct the color electric and
magnetic field equivalents from the tensor. Recall that for the electromagnetic field
tensor F F = E 2 — B 2 and F F = E • B.
Therefore for the gluon field tensor,
GG — Ec ■Br.2 where Ec and B c are the equivalent color electric and magnetic
fields. Under parity and time reversals the fields transform as
E ^ -E
( 2 .8 )
E ^ E
B ^-B .
(2.9)
From this we see that the scalar product E c ■B c reverses sign under parity or time
reversal. By the CPT theorem [39], T-violation is equivalent to CP-violation, so the
#-term is CP violating unless 9 = 0. However 9 is the sum of two independent terms
so there is no reason to think they would cancel:
9=
9q
c d
+ arg det A4.
(2 . 10 )
9q c d is a phase th at comes from the fact that QCD has a topologically non-trivial
vacuum structure which is not invariant under chiral transformations. This com­
plex structure has physical significance and is needed to correctly predict the rj mass
and decay width [38]. The QCD vacuum is a superposition of various vacua con2. We note th at the analogous CP violating term s do not arise in electroweak theory (they would
look like dFftl,F ltu, where Ftl„ is the electromagnetic field strength). This is because each 0-term is
a total four -divergence, so only contributes surface term s to the action. Since the field strength F,tv
fall off faster than 1 /r 2, the surface term s are zero as r —> oo. However in QCD. because the gluon
fields are self-interacting the term GG is not zero as r —> oo, and so the surface term s are non-zero.
12
nected by tunneling events called instantons; physically
Oq
c d
can be thought of as
characterizing the density of instantons [40]. M is the quark mass matrix from the
electroweak sector. The phase arg det M. enters into the expression for 9, when, in
order to render the quark mass matrix real and diagonal, the matrix is rotated by
axial transformations; due to the anomalous interaction with QCD a term of the form
(arg det M ) G G
(2 . 11 )
arises [41].
Observable effects result from the CP violating #-term, such as a permanent neu­
tron electric dipole moment dn of order [41]
\9\ ~ 1(T 16|0| e • cm
|d J ~
(2 . 12 )
m n \m nJ
where rnq is the mass of the lightest quark, m n the neutron mass, and e the electric
charge. No particle has yet been observed to have a permanent electric dipole moment,
including the neutron, and measurements of the neutron EDM [42] place upper bounds
at the level:
dn < 2.9 x 10-26 e • cm,
(2.13)
which constrains the QCD phase to be
9 < 1CT10.
(2.14)
The strong CP problem is the question of why 9 is so small, when a “natural" value
for this periodic variable, in the sense defined by t'Hooft [43]. would be of order unity.
13
2.3
Solutions to the Strong CP Problem
If any of the quarks are massless, then there is no strong CP problem, as arg det M
becomes undefined and the remaining phase angle
appropriate transformation, so that
9
Oq
c d
can be rotated away by an
no longer has physical significance. However,
lattice calculations of the up and down quark mass ratio disfavors the massless up
quark hypothesis [44,45]. There are also arguments for why neither the up or down
quark mass can be nonzero as they influence the baryon and meson masses [46], which
leads to bounds on the running mass at the scale of 1 GeV:
m d{1 GeV) > m u( 1 GeV) ~ 5 MeV.
(2.15)
Ref [47] argues th at the bounds in Eq. 2.15 are actually placing limits on an effective
quark mass induced by instanton effects, while the intrinsic up quark mass is zero. If
this is so, the strong CP problem is solved. However, the following question would
be what mechanism causes the up quark mass to be zero.
Another explanation for the strong CP problem imposes CP as a symmetry on
the Lagrangian. This sets 9 — 0 to lowest order, with corrections from spontaneous
breaking leading to calculable, but small non-zero values of
9
- for details of these
theories, known as soft CP violation models, see [48]. These theories generally intro­
duce new particles at high energies, such as new Higgs bosons, and do not provide
low-energy observables, so are difficult to test experimentally [50].
The most widely accepted solution to the strong CP problem is the Peccei-Quinn
mechanism, wrhich we now discuss.
2.4
Peccei Quinn M echanism
In 1977 Peccei and Quinn proposed extending the standard model symmetry group
with a new global chiral symmetry
U ( 1 ) pq
[18]. In order to achieve this new symmetry
14
a complex scalar field $ is added to the standard model Lagrangian
c =
C sm +
- H(|<h|) +
Cint.
(2.16)
Lint is a model-dependent, interaction term parameterizing the coupling of the scalar
field to fermions. V is the potential, which is the standard quartic potential as for
the Higgs mechanism, shown on the left in Figure 2.3 (from [51]).
Figure 2.3: Spontaneous symmetry breaking shown in the left picture: the axion is the
Goldstone boson associated with the azimuthal degree of freedom. Explicit symmetry
breaking as shown in the right figure can be visualized as a tipping of the potential,
which forces the axion field to pick a particular value.
At energy scale f p q the scalar field acquires a non-zero vacuum expectation value
($) = J pq / \ / 2. The excitations around the ground state may be written in terms of
fields a and a:
$ = ( $ ) + a{x)eia{x)/fl’v
(2.17)
The radial mode a has mass proportional to fp q (see discussion in [52]) and is thus
extremely heavy, so cannot be detected experimentally. However the angular modes
a can be quantized as particles and these massless Goldstone bosons are axions.
15
Under the PQ transformation the axion field shifts as:
a —v a, + ctf pq
(2.18)
which would imply th at only terms involving the derivative of the axion field can
appear in the PQ-invariant Lagrangian.
However, due to a chiral anomaly with
QCD, the PQ symmetry is no longer conserved and a term also arises of the form
a ( x) ^ Q Q
(2 19)
f p q / N 4tr
with a model-dependent dimensionless term £ which depends on how the fermions
in the theory behave under PQ rotations. When only one fermion has PQ charge
(N = 1 ), the effect of the anomaly can be visualized as a tipping of the potential,
as shown on the right in Figure 2.3, which causes the axion field expectation value,
previously free to choose any value in (0 ,27r), to settle at one particular value. From
here on out we absorb the factor N into the energy scale and write it as /„ = / p
q / N .
The term in Equation 2.19 appears as a periodic potential after integrating out
the gluons and quarks to look at the effective physics below the confinement scale
Aqcd <ftnd is parameterized as3:
V(a) ~ \ Q
i CD cos ( a / f a - 0)
(2.20)
Peccei and Quinn calculated the vacuum expectation value of the axion field with
the anomalous symmetry breaking and found
(a) = - e k
(2 .2 1 )
3. Technically the r; meson also appears in this potential but the effects are suppressed by /„ . See
Ref [53].
16
canceling the 6-term exactly [54], This solves the strong CP problem elegantly and
has a testable low-energy prediction - the axion.
In Section 2.1 we noted that the anomaly produced axion-meson mixing, gener­
ating a mass for the axion. Another way to see this is to take the potential term (Eq
2.19) and find the mass from the curvature of this potential:
m l = (%V) oc y d a(GG)
(2.22)
Ja
which, taking AqCD as the scale of (GG) [41], gives the mass from dimensional argu­
ments as
ma ~
Ja
(2.23)
Original A xion M odel
In the original Peccei-Quinn model, all known fermions had PQ charge and the scalar
field was made by introducing two new Higgs doublets, which had a corresponding
energy scale of the electroweak strength:
Jew
= (v/2G /,) ~ 1/2 « 250 GeV.
(2.24)
From current algebra techniques the axion mass could be estimated [35] as roughly
m a ~ 100 keV and was conclusively ruled out by non-observations in a variety of
collider and beam dump measurements [55-58].
New models were constructed that decoupled f a from the electroweak scale, mak­
ing lighter axions possible. However, because the interaction strength goes as £ / / a,
these lighter axions were also much more feebly interacting, and the new theories
were dubbed “invisible-axion” models as it was deemed impossible to experimentally
detect these light particles. This turned out to be unnecessarily pessimistic and ex­
17
periments are now probing the parameter space predicted by these axion models, as
will be described in Chapter 4. We briefly summarize the properties of invisible-axion
models commonly discussed in the literature: the DFSZ and KSVZ models.
KSVZ
The Kim, Vainshtein, Shfiman, Zakharov (KSVZ) model [59,60], extends the Peccei
Quinn mechanism by introducing a new heavy quark and complex scalar field, as well
as a discrete symmetry. The axion then couples directly to this heavy quark and
through the quark to ordinary m atter. The known fermions and leptons do not have
PQ charge.
DFSZ
The Dine. Fischler, Srednicki, Zhitniskii (DFSZ) model [61.62], adds a single heavy
scalar field to the standard model in addition to two Higgs doublets. In this model
ordinary fermions and leptons have PQ charge and so interact with the axion. The
DFSZ model creates the U (l) symmetry in a way that can be easily embedded in
theories with larger symmetries.
A xion -P h oton Interaction
Despite the different assumptions of these models, they predict similar values for
the axion-photon interaction strength. The coupling to m atter is different for the
two models, as the KSVZ model has no tree-level coupling to electrons, but here we
only show the axion coupling to photons, as that is the interaction we will use to
experimentally search for axions.
The coupling constant is given by [63]:
9- =-^a«V ira(f-L92±H
18
<2’25)
where E characterizes the number of electrically charged fermions that couple to the
axion (see [52] for details). For the DFSZ model E / N is 8/3. for the KSVZ model it
is 0. For a summary of the axion interactions with electrons, protons, and neutrons
for these models see [34].
A rgum ents against th e P eccei-Q uinn M echanism
The argument th at all global symmetries should be violated by quantum gravity ef­
fects [64] has led to claims that the Peccei-Quinn mechanism becomes untenable,
producing values of 9 above the experimental upper bounds [65]. As well, if the PQ
symmetry breaking happens after inflation, the domain wall problem arises. This
refers to the fact that domains of different CP phase are produced after sponta­
neous symmetry breaking. Calculations of the energy stored in the walls between
each domain show that the stored energy exceeds the critical energy density of the
universe [49]. As this clearly cannot happen, the spontaneous symmetry breaking
must take place before inflation, so that we would live in one domain today. The
requirement that the PQ symmetry breaking occur before inflation pushes f a nearer
the GUT scale, making the axion lighter and therefore harder to detect. Even if the
PQ symmetry breaking scale occurs after inflation, there are ways to get around the
domain wall problem, such as a soft breaking of the PQ symmetry [6 6 ].
A L Ps
Having gone through the Peccei-Quinn mechanism, one can generalize the process of
spontaneous and anomalous symmetry breaking to other continuous global symme­
tries. the pseudo-Nambu-Goldstone bosons of which are called axion-like particles. If
the anomalous symmetry breaking occurs due to dynamics with an associated char­
acteristic scale A, the ALP mass is suppressed as m 0 ~ A2/ w h e r e f 0 is the energy
scale of the spontaneous symmetry breaking. For ALPs to arise from anomalous in-
19
teractions with symmetries beyond the Standard Model, one must have A > TeV (so
for f,p = f a,
m a) or the symmetries come from a hidden sector. Axions, ALPs
and other low mass particles therefore act as low energy probes of high energy physics
and are tied to fundamental considerations of symmetry violations.
In 1983 a seminal paper by Pierre Sikivie outlined a technique to search for axions
using the two photon interaction [67]. This is now the main method by which experi­
mentalists attem pt to detect axions. The technique used in this experiment relies on
the two photon interaction and the assumption that axions are the dark m atter in
the Milky Way galactic halo. Axions have been considered as potential dark m atter
candidates due to their feeble couplings and stability. However, in order for axions to
be good cold dark m atter candidates, they must also have the correct abundance and
be non-relativistic. The primordial production of a population of axions and ALPs is
the topic of Chapter 3.
20
Chapter 3
Dark M atter Axions and Hidden
Photons
Thermal axions are not good cold dark m atter candidates. To see this, we briefly re­
view the thermal production of particles in the early universe. To start with, particles
created in the hot, early universe will no longer be produced after the tem perature of
the universe drops below the mass of the particle, T < m. If these particles do not
decay, the only way to decrease their abundance is through annihilations with each
other. As the universe expands, it becomes harder for the particles to interact so
the abundance becomes fixed and they '‘freeze-out’7. Dark m atter particles produced
this way are called “thermal relics” . As the thermal velocities of light particles are
still relativistic when they freeze-out, species such as neutrinos and axions would be
classified as hot dark matter. Axions can also be produced non-thermally, however,
and it is this process which creates the cold dark m atter population. Section 3.1 de­
scribes the different processes which can produce dark m atter axions with the correct
abundance, focusing on the misalignment mechanism. The misalignment mechanism
can be generalized to produce string axions [68 ], general ALPs [69], and hidden pho­
tons [27]. Section 3.2 outlines the argument for how hidden photons could be dark
21
m atter candidates as well using the misalignment mechanism.
3.1
N on-Therm al Production o f Dark M atter
M isalignm ent M echanism
The misalignment mechanism [70, 71] creates a present-day energy density of cold
axions from oscillations of the axion field about the minimum of the potential. The
present day abundance of these primordial axions will be dictated by the initial value
of the axion field, o0, and subsequent evolution of the field in the expanding universe.
The evolution of the axion field obeys the equations of motion in the FriedmannRobertson-Walker metric. Gradients are neglected (dta = 0) as the expansion of the
universe will make spatial variations unimportant. Then the effective action due to
the axion is
where a is the axion field. R(t) the cosmological scale factor and the potential is
approximated by the quadratic mass term.
The decay width of the axion Ta is
negligible. The equation of motion for the zero-momentum mode is
+ R h n i ( T ) a = 0,
(3.2)
which can be rewritten in terms of the Hubble parameter H = R j R as
a + 3Ha + m'l(T)a = 0.
(3.3)
Eq. 3.3 is the familiar expression for a damped harmonic oscillator where the fric­
tion term is given by the Hubble parameter. At high temperatures, T
AQ C D -
the instanton effects which produce the anomaly are suppressed severely [72] so the
22
potential is nearly flat and the axion is effectively massless. The field will then be
stuck at its initial value:
a = n0,
where a0 is the initial
(3-4)
misaligned valueof the field, which can be between 0 and 2 n f a.
The total energy density in the field is given by the potential energy, so pa —
Thus we have a
gas
at rest with number density na = ^ m aaQ.As the
of axions
tem perature decreases, the axion mass will increase, and when
m a(T0) = 3//(To)
(3.5)
at tem perature Tq, the axion field will begin to oscillate with frequency m a(T). The
energy density stored in the field is then
= Im .tW 2
( 3 .6)
so th at the present axion density is
Pa = Pa{to)
rna ( R ( t 0) Y
7TT —5 ■
rna(t0) V R J
(3 J )
Calculations of the present day axion energy density due to misalignment [73] yield
(3.8)
where §0 = a0/ / o is the initial misalignment angle. There is a factor of 10 uncertainty
in this estimate from theoretical and cosmological uncertainties. If the PQ symmetry
spontaneous breaking happens after inflation, then
there are many domains with
different initial angles 90.Taking the root mean square average ofall values between
23
(0, 7r) gives 90,rmS — tt/^ 3 . Putting in 9Q^rms to Equation 3.8 provides an upper bound
on f a from the constraint that
must be less than the critical energy density:
f a < 1012 GeV,
(3.9)
which can be translated into a bound on the axion mass of
m a > 1(T6 eV.
(3.10)
However if the PQ symmetry breaks before inflation, then the entire observable uni­
verse is in one domain and i9() is an unknown random value, so the above estimate
no longer holds.
90
is limited by quantum fluctuations of the axion field induced by
inflation, which are known as isocurvature perturbations. The fluctuations in the ax­
ion to photon ratio in turn lead to tem perature fluctuations of the cosmic microwave
background; for details see [74], The inflation generated fluctuations are bounded by
89
~ Hi/2ixfa,
(3.11)
where Hi is the Hubble scale at the end of inflation. Therefore a minimum dark
m atter density due to axions is set for a specified Hj.
A xion Strings
Other non-thermal processes can produce cold axions.
The
U ( 1 ) pq
symmetry is
analogous to the symmetry of a superconductor. Therefore it may also possess the
analogy to vortices, namely a configuration where in traversing a closed path the
phase of the PQ field <&varies over 2im while the magnitude remains the same, where
n
must be an integer for <I> to remain single-valued. For the
U(1)pq
symmetry, these
vortices are "axion strings’' [75,76]; strings with |n| > 1 lose energy by radiating
24
axions [77].
Axion decay from other topological defects, such as domain walls, also contributes
to the total energy density. Adding the effects of all these contributions to the energy
density yields a bound on the energy scale of f a < 1.2 —2.3 x 1010 GeV. which is two
orders of magnitude lower than the bound given earlier from the misalignment mech­
anism [78]. There are large uncertainties in this estimate, and different calculations
in the literature of the contribution from strings give values ranging from 1-100 times
the misalignment contribution [79,80]. However if the spontaneous symmetry break­
ing happens before inflation the defects would dilute away and no longer contribute.
For a review of axion creation from topological defects, see [81].
3.2
H idden P hotons as Dark M atter
Before discussing the hidden photon in connection with the misalignment mechanism,
we take a moment to review the field equations and point out some general properties
of these massive vector bosons.
The hidden photon arises from the addition of f/(l) gauge symmetries to the
standard model. For sub-eV hidden photon masses, the dominant interaction with
standard model particles will be through gauge kinetic mixing with the photon [82].
The effective low-energy Lagrangian for this vector boson is then written as
£ = - -1{ F ^ F , V + F ' ^ F ' ^ + 2x F ' ^ F ^ ) +
where A M is the photon field and A ' the hidden photon field.
(3.12)
is the ordinary
electromagnetic current, and the hidden photon field tensor is defined in the usual
way: F '„ = d^A^ - dvA'^. W ith a transformation A —» .4 - \ .4', A' —►A ' , the
interactions between the photon and hidden photon can be made explicit:
i
^
C =
+ r ( A , - X A'„).
(3.13)
The hidden photon interaction with the electromagnetic current is suppressed by
If the hidden photon mass goes to zero, the couplings of the hidden photon to elec­
tromagnetic currents can be rotated away and then has no physical significance. The
vector boson mass can come about either non-dynamically (i.e. from a Stuckelberg
mechanism) or from the spontaneous breaking of a symmetry U ( l)y by a new Higgs
field. The two mechanisms for generating the mass my cause different effects for small
masses. For the Stuckelberg case all processes with the hidden photon are suppressed
by m y, while there is no dependence on mass for the Higgs case.
The misalignment mechanism for vector particles proceeds analogously to the case
for axions. Each spatial component of the field A' satisfies the damped harmonic
oscillator equation (Eq 3.3). The only difference is that as the hidden photon is a
vector, there is a particular polarization associated with each domain [27]. The hidden
photon must not thermalize if it is to remain non-relativistic. Ref [27] examines the
possible interaction processes (Compton-like scattering cpe —> j e is dominant) and
concludes th at their effects are negligible. Ref [83] finds an enhanced interaction
rate for Compton-like scattering when m y = m7. However the most important
bounds for constraining the allowed region for hidden photon dark m atter come from
considerations of cosmological observables, as will be discussed in Chapter 4.
26
Chapter 4
Parameter Space
This chapter reviews the limits on axion and hidden photon parameter space. Cosmo­
logical and astrophysical observables are powerful tools in constraining the possible
couplings and masses of these particles. Section 4.1 reviews the axion bounds, begin­
ning with the first unsuccessful laboratory searches for high mass (~ 100 keV) axions.
describing the astrophysical limits on gail, and then detailing the current state of ex­
perimental searches. The cosmological bounds for dark m atter axions were covered in
the previous chapter but are summarized here as well to show the regions of allowed
ALP dark m atter that current and future direct searches can feasibly explore. Section
4.2 goes through the cosmological, astrophysical, and experimental bounds on hidden
photons, focusing on the available parameter space for dark m atter hidden photons.
4.1
A xion and ALP Bounds
The arguments which lead to bounds on the coupling strength gail are applicable to
both axions and ALPs. These bounds can be translated to limits on the QCD axion
mass and where relevant, these results are listed as well.
H igh-E nergy Laboratory Searches
27
The first laboratory searches for the original ~ 100 keV axion took place shortly after
the Peccei-Quinn axion was postulated. Measurements of meson decays,
J/\k —> ya
T —>ya
K + -» 7r+a
(4-1)
nuclear de-excitations,
N* ->• N a
(4.2)
a —>■yy, e+e~
(4.3)
and beam dump experiments,
found no evidence for an axion coupled to fermions or nucleons up to / > 104 GeV
(rna < 0.6 keV) [84,85]. There is still available parameter space for heavy ALPs with
weak couplings, however cosmological arguments severely constrain the possibilities
for this regime.
A strophysical and C osm ological Lim its
Astrophysical observables provide strong constraints for weakly interacting particles,
primarily through energy loss arguments. First considered by [86 ], the argument is
the following: weakly interacting particles produced in a star provide an additional
source of energy loss; they freely stream out of the star by virtue of their feeble
interactions, transporting energy away and accelerating the star's cooling rate. If
the observed stellar lifetime is consistent with expectations, this limits the coupling
strength of these particles. Neutrinos are a prime example of an existing weakly
interacting particle that provides an energy loss channel in stars, in fact becoming
the dominant cooling mechanism for hot and dense stars [52].
28
P t v o io n
A x io n
3
Coulomb-field
Ze
Figure 4.1: Primakoff effect; a photon can scatter off the Coulomb field of an electron
or nucleus to produce an axion.
The bestconstraint on ga~r, comes from stars which have reached the helium
burning phase,calledhorizontal branch (HB) stars. The stellar lifetime of these stars
can be inferred by comparing the the number of HB stars, N h b • to the number of
stars on the red giant branch (RGB). N rgb -, hi fifteen globular clusters:
R = R h b / N rgb = tnB/tRGB
(4.4)
and this ratio was consistent with expectations [87]. The Primakoff effect is the
dominant production process of pseudoscalars in HB stars (see Figure 4.1, from [88 ])
and by limiting this process, one finds the constraint
gail < 0.6 x lO" 10 1/GeV.
(4.5)
There is a factor of two uncertainty in the limit so the value
gail < 1(T 10 1/GeV
(4.6)
is commonly taken as the actual bound, and is known as the globular cluster limit [89].
Two recent analyses strengthen this limit from different arguments. Ref [90] argues
29
that the existence of Cepheid variable stars constrains
gall < 0.8 x 10” 10 1/GeV
(4.7)
while Ref [91] takes into account the helium mass fraction in evaluating the ratio R
to obtain strengthened bounds of
gall < 0.66 x 10~ 10 1/ GeV
(4.8)
This can be translated into a bound on the axion mass with model dependence for
the various DFSZ and KSVZ parameters.
Another strong bound comes from white dwarf cooling.
Measurements of the
white dwarf luminosity function constrain the axion-electric coupling to be [92]
gaee < 1.3 x H T 13 1/GeV,
(4.9)
which limits the QCD axion mass to be m a < 10-2 eV for DFSZ models, taking the
PQ charges to be of order unity. KSVZ (hadronic) models have no tree-level couplings
to electrons and are thus unconstrained by this observable.
The strongest bound on the axion mass that can be placed for both DFSZ and
hadronic models comes from supernova SN1987A. In the core collapse of the origi­
nal star, the gravitational binding energy was radiated away by MeV neutrinos in
approximately ten seconds [93]. The neutrino burst was observed at underground
detectors [94] and both the duration of the burst and number of events detected were
consistent with the standard picture of neutrino physics in supernovae [89]. The pre­
dominant process of ALP emission in the star would be ALP-nucleon bremsstrahhing;
30
in order to not be in conflict with the data the coupling must be less than [52]
3
X
10- 10 < gaNN < 3 x 10- 7 1/GeV.
For a discussion of the uncertainties inherent in these limits see[89].
(4.10)
The upper
cutoff coniesabout because strongly interacting pseudoscalars becometrapped
in the
supernova core and are unable to freely radiate energy, becoming ineffective as sources
of energy loss [95]. Although these particles would not affect the observed neutrino
energies, they would excite oxygen nuclei in the detectors and cause the emission
of MeV photons, thus increasing the number of detected events [96]. The energy
trapping argument leads to bounds of
9 x l(r7 < gaNN < 1 x 1(T3 1/GeV,
(4.11)
which translates to an exclusion range for the axion mass between
0.01 eV < m a < 20 keV
(4.12)
with a small window of 3-8 eV which has been excluded by telescope searches looking
for thermal relic axions [97,98], as well as the globular cluster limit. A summary of
the excluded ranges for the axion mass (from energy loss arguments only) in different
models is shown in Table 4.1.
System
Sun
Red Giants
HB stars
White Dwarfs
SN 1987A
Process
7 7 * -> a
ey —» ae
7 7 * -* a
eZ
aeZ
N N -> N N a
DFSZ
2.7 eV - 10 keV
9 meV- 100 keV
7 meV - 10 keV
5 meV - few keV
16 meV - 2 eV
Hadronic
17 eV - 10 keV
2 eV - 100 keV
0.7 eV - 10 keV
no limit
16 meV - 2 eV
Table 4.1: Summary of astrophysical bounds on the axion mass
31
All of the limits on the coupling strength are valid as long as the pseudoscalar
mass is less than the tem perature in the stellar interior, typically 10 keV for the
systems considered. This is the reason Table 4.1 has an upper cutoff for the exclusion
ranges shown.
Cosmological observables place extremely strong bounds on gail for masses greater
than 1 eV. The arguments rely on the fact that a thermal population of ALPs could be
created in the early universe; they can thermalize and decay via the Primakoff effect
and the resulting photons would be injected into the primordial plasma. Depending on
the energy of these photons and the ALP lifetime, the existence of light pseudsocalars
would impact deuterium abundance, anisotropies of the cosmic microwave background
(CMB), and the extragalactic background. See Ref [53,85,99] for reviews.
Figure 4.2 (from [53]) shows the current bounds on ALP mass m 0 and photon inter­
action strength ga r r Not shown here are recent bounds on 100 GeV ALPs from LHC
measurements at the level gayi < 10" 4 1/GeV [100]. The ALPS. CAST+SUMICO,
PVLAS, and Haloscope bounds are from experimental searches, which we now discuss.
E xperim ental Lim its
Laboratory experiments searching for signatures of light pseudoscalars are distin­
guished primarily by their assumptions about possible axion sources. Solar helio­
scopes look for pseudoscalars streaming from the sun that then convert to photons
in strong laboratory magnetic fields. As the relativistic particles produced in the sun
have a thermal distribution, the limits obtained from solar experiments on the axion
mass are broadband. The CAST experiment is the most sensitive of these helioscope
experiments, with limits of
gary < 8.8 x 10" 11 1/GeV
32
(4.13)
Y >+SIIV
f' +c r- in’
PVLAS
'A ' I ' I ''
ALPS
CAST+SUMICO
V»
.*T
r r t .n ^ i - c r c v .'I .he - n r . t r c
Log,:i('m.-/eV)
Figure 4.2: Bounds on ALP-photon coupling strength versus ALP mass. The high
energy laboratory limits are depicted in blue; collected exclusion results from cos­
mological observables are in red. The globular cluster limit, labeled HB. and the
supernova limits (SN) rule out a wide segment of the QCD axion model space, de­
noted by the dashed band. Light-shining-through-wall searches are in yellow, and
solar axion searches in dark green. Haloscope searches for dark m atter axions are
shown in dark red.
33
for m a < 2 eV [101]. An improved helioscope IAXO has been proposed with projected
sensitivity at the level gail < 10 ' 12 1/GeV [102]. The new setup would achieve this
sensitivity through stronger magnetic fields and a larger interaction volume.
"Light-shining-through-walF (LSW) searches attem pt to produce axions with a
laser (or microwave source) in a strong magnetic field, which provides the off-shell
photon in the process
(4.14)
7 7 -¥ a.
If pseudoscalars are produced, they will go through barriers that stop photons as they
have feeble interactions with m atter. W ith a magnetic field on the other side of the
barrier there is a small probability that the ALPs will then decay back into photons,
which can be detected. A schematic of the LSW setup is shown in Figure 4.3 (taken
from [51]).
Wo
M a g re t
Magnet
B0
r ~ w
~
; ■u
[j_a?»er 2 h “rv/>y>k'"
r _
i________
Figure 4.3: LSW experimental setup for pseudoscalar searches
This process suffers in sensitivity because the overall probability of the productionregeneration process is suppressed by a factor g4 B 4. The reach of LSW experiments
can be improved by placing resonant cavities on either side of the barrier to allow
for coherent conversion and reconversion.
Experiments with lasers [103 -105] and
microwaves [106] have placed bounds on the ALP-photon interaction strength (for
34
masses less than 1 meV) of
9a-y~i < 1 0
1 /GeV.
(4.15)
ALPs can also induce changes in polarization for a linearly polarized laser beam in a
strong magnetic field. The best constraints are provided by PVLAS experiment [107],
with results comparable to the LSW searches.
Experiments looking for dark m atter pseudoscalars, known as haloscope experi­
ments [108,109], (this work) currently have the greatest sensitivity to the ALP-photon
coupling constant . This is because there is an enormous number density of these dark
m atter particles. n a « 1012 cm -3 if ALPs are cold dark m atter, so that the probability
of detecting a converted photon is within experimental capabilities despite the weak
interaction rate. As these particles would be non-relativistic, experiments look for a
nearly monochromatic signal at the particle mass m a that would result from an ALP
converting to a photon in a background magnetic field. To enhance the output signal
to noise ratio a resonant cavity is used, and cryogens are employed to decrease the
competing thermal noise. The ADMX experiment has so far excluded KSVZ model
axions in the mass range [108]
1.9 < m a < 3.53 /reV.
(4.16)
At the time of writing. ADMX is upgrading their setup in order to run with a lower
system noise tem perature and plans to search for axions up to 10 //eV. ADMX-HF
will search for axions starting at « 20 //eV with the capability to scan from 16 —33
geV with sensitivity 1.5 x the KSVZ limit in the next three years [110]. As the experi­
ments are narrowband, it is desirable to have many experiments operating at different
frequencies in order to cover the available parameter space more quickly. Microwave
cavity experiments are the only demonstrated method to reach axion model band
35
sensitivities in the //eV range, and although it is more challenging to reach the QCD
axion model band at higher frequencies with this technique (see Chapter 5). it is still
the most relevant method of doing a direct search for axions in the 1 /rev - 1 meV
range.
N on-P rim akoff Searches
Although most direct searches try to exploit the a —>7 7 transition, there are other ex­
periments th at do not rely on this interaction. Pseudoscalars can mediate T-violating
macroscopic forces such as long-range forces between spin-polarized bodies [111]. Tor­
sion pendulum experiments [112,113] and spin precession searches [114] have found
110 evidence for such ALP-mediated forces, and place extremely strong bounds on the
available parameter space. Finally, a recent idea has been proposed to look for dark
m atter axions with m a < 10~9 eV by searching for oscillating electric dipole moments 1
induced by the dark m atter pseudoscalar field [116]. This approach is complementary
to the haloscope experiments, which are limited by the experimental size to probing
masses m a > 1(D6 eV.
Lim its on Dark M atter A L Ps
The limits discussed so far. with the exception of the haloscope experiments, have been
independent of whether ALPs are cold dark m atter. The SN1987A limit constrains
the ALP mass to be greater than 0.01 eV; astrophysical and helioscope bounds place
an upper bound on the coupling of 0.66 x 10-1° 1/GeV. As discussed in the previous
chapter, the misalignment mechanism can produce ALPs in sufficient quantities to
equal the critical density of the universe for masses near 10-6 eV. According to some
1. It is interesting to note th at the strength of the induced oscillating EDM is independent of
f a [115]. This is in contrast to the axion photon coupling ga~n , which scales as g,ni x / “ L making
searches for low mass (high /„) axions more challenging since the axion-photon interaction becomes
weaker.
36
estimates, axion production from string decay could contribute another two orders
of magnitude to the energy density from the misalignment mechanism, which would
place a lower bound on the axion mass closer to 10“ 4 eV. If the Peccei Quinn scale
breaks before inflation, then the estimates from the misalignment mechanism and
string decay no longer hold. However, the initial value of the displacement angle
is constrained by isocurvature perturbations together with the scale of inflation and
cannot be too small, as this would be in conflict with observations. Figure 4.4 (from
[53]) sketches out various regimes for dark m atter ALPs.
Figure 4.5 (from [117]) shows the reach of proposed experiments. There is a gap
for axion masses in the 100 //.eV to 1 meV range that is not covered by any proposed
experiment. It is therefore im portant to build experiments which can explore this
mass range with the capability to probe the QCD axion model band. The work in this
dissertation was the first haloscope search at 140 //eV: the results from this search
experimentally confirm the improved bounds from the re-analysis of the globular
cluster limit. We now discuss the available parameter space for hidden photons.
4.2
Hidden P hoton Bounds
Many of the same arguments that constrain ALPs can be applied to hidden photons
as well, such as energy losses in stars and distortions of the CMB. The bounds which
are most relevant to cold dark m atter hidden photons come from cosmological observ­
ables, the production of hidden photon longitudinal modes in the sun, and haloscope
experiments.
C osm ological O bservables
Figure 4.6 (from [83]) gives an overview of the low mass hidden photon parameter
space. Not shown are recent bounds based on analyzing the contribution of longi­
tudinal modes to hidden photon production in the sun. For now we focus on the
37
4
6
Log10m* [eV]
Figure 4.4: Dark m atter ALP bounds. A minimal upper bound can be constructed by
estimating that the ALP field must begin oscillating by the time of matter-radiation
equality (the allowed region for dark m atter ALPs under this analysis is shown in
pink). If the axion mass receives no thermal corrections then the region of allowed
dark m atter is shown in red (labeled by mo = mi). A more in-depth analysis takes
into account the evolution of the mass with temperature, which requires calculation
of instanton effects: these bands are labeled by the m \ / m Q = ( A / T ) 3 region. The
gray region is where the ALP lifetime would be less than the age of the universe,
making the pseudoscalars unviable as dark m atter candidates.
38
Excluded
Figure 4.5: Dark m atter ALP parameter space. Proposed upgrades to current exper­
iments are shown in green. Microwave cavity searches are represented by ADMX and
can probe the parameter space for low mass dark m atter axions.
39
bounds for the allowed hidden photon dark m atter (see Ref [83] for a comprehensive
treatm ent of hidden photon dark matter).
Coulomb
ALPS
Solar
Lifetime
Allowed
HPCDM
Haloscope
Searches
-4
-1
Log10mr-[eV]
Figure 4.6: Hidden photon parameter space, with the allowed regions for dark m atter
hidden photons shown in pink. The other regions are exclusion limits from experi­
mental searches (ALPs. CAST) or energy loss arguments (HB, Solar Lifetime).
• r 2 > 1: As the density fluctuations of the CMB give an estimate the dark m atter
density which is consistent with observations, the lifetime of hidden photons is
constrained - they must not decay into photons at any epoch in which these
photons could distort the observed CMB fluctuations. This condition imposes
an upper bound for extremely low-mass hidden photons.
• CMB distortions: hidden photons absorbed in the primordial plasma would
heat the thermal bath, affecting the blackbody distribution although only at
low frequencies where processes such as inverse bremmstrahlung and Compton
40
scattering can efficiently thermalize the particles. Numerical estimates are dif­
ficult, but in the limit of small distortions [118] and FIRAS constraints [119],
the bound shown in the figure above is reached.
• N ej f hidden photon oscillations would also affect the photon to baryon ratio
through the increase of tem perature in the thermal bath. The effective number
of relativistic species N ej f parameterizes the amount of invisible energy density
at decoupling and is constrained very tightly [120 ], which translates into a limit
on the amount of possible tem perature increase from hidden photon decays.
• X-ray background: hidden photons can decay into three photons via an electron
loop, a suppressed process, but one which can be bounded by the argument that
the population of photons from 7 ' —> 7 should be less than the diffuse X-ray
background.
L ongitudinal M odes
Energy loss arguments can constrain the hidden photon coupling and are shown by
the Solar Lifetime and HB exclusion regions in Figure 4.6. However, these estimates
were made from calculations of the hidden photon flux that did not include the
contribution from longitudinal modes [121 ], which turn out to dominate the total
flux in the small mass limit, m y -C u/p where ujp is the plasma frequency [122 ],
For a hidden photon made with energy oj where u>
rriy, there are two transverse
polarization directions, and one longitudinal mode. The longitudinal portion scales
as x 2rny/a;2, and the transverse portion scales as y 2m y /o /. Hidden photon emission
stops near 300 eV as the plasma frequency is limited by the solar core temperature.
Helioscope experiments can detect the hidden photon flux from the sun: in addition,
the XenonlO experiment [123] can strongly constrain the hidden photon flux as it is
sensitive to low energy ionization signals, which can be caused by hidden photons
absorbed by atoms which then ionize [124]. See Figure 4.7 (taken from [124]) for the
41
strengthened bounds assuming a Stuckelberg mechanism generates the hidden photon
mass.
Previous work by YMCE constrained the hidden photon coupling to photons in
the 140 peV using a LSW technique. In th at project, the noise from one cryogenically cooled cavity was recorded while driving a second cavity placed next to the first
cavity but electromagnetically shielded from the cooled resonator. Both cavities had
the same resonant frequency and were monitored in the T E on mode. Detection of
excess power in the shielded and cooled cavity would have been a possible signature
of hidden photons being produced in the driven cavity, traversing the shielding and
reconverting to photons. This technique is analogous to the ALP LSW searches, with
the only difference being that no magnetic field is required. The first observation that
ALP LSW searches could be adapted for hidden photon searches was from Ref [125];
later proposals developed the technique for microwave cavities [126]. The CROWS
experiment (green region in the plot) was the first LSW experiment that used mi­
crowave cavities to perform both an ALPs measurement (with a magnetic field) and a
hidden photon measurement [106]. It operated at around 2 GHz and did not employ
cryogenic cooling, but rather long measurement times to reduce the noise floor. Fig­
ure 4.7 shows th at the longitudinal mode solar limits place optical LSW experiments
(ALPS-I) in excluded parameter space, as well as the YMCE search. The majority of
the region covered by proposed upgrades to the optical searches is ruled out as well.
This would seem to make laboratory LSW experiments unattractive as methods to
look for hidden photons. However a new analysis of hidden photon LSW searches
suggests that longitudinal modes can be detected in microwave cavity experiments,
enhancing the low-mass sensitivity [127]. To benefit from the longitudinal contri­
bution. these experiments must have the electric fields in the two cavities aligned,
meaning that viable configurations must use stacked cavities in the TM modes.
Figure 4.8 (from [127]) show's the increased sensitivity in the lowr-mass region for
42
Iiiiii^ ! I iiim^—i'll!rr;
YMCE
ALPS-I
1 (T 6
CROWS
IO " 7
ss
bC
ALPS-II
(proj.)
IO " 8
CAST (T) !
.s
.2
io~9
a
o
+3
a
a
3
IO" 10
io -n
io-12
SC: Sun
(L)
(T)
Xenon 10
i o - 13
10~14
10
15
11.liui
----—
.......
-----
“■“I 1IUUl 1i iniuL
IQ -6 1 0 -5 10-4
1Q
■Im l
■
I i m i ll
3 1CT2 io-1
1
I
i i m u l
I
101
i i m l J
I
102
i i l i m
103
m y (eV)
Figure 4.7: Longitudinal mode limits on hidden photons. The dark blue region is ex­
cluded by previous work done by our group using a light-shining-through-wall method.
The green region are exclusion results from a lower frequency LSW microwave cavity
search. The longitudinal mode limits from the sun (black) and from low background
detectors (red) exclude much of the region where optical LSW experiments are sen­
sitive.
43
the CROWS experiment, which had an experimental setup that could detect longitu­
dinal mode contributions, and outlines proposed limits from future experiments that
will take advantage of superconducting cavities and band-gap resonators to increase
the sensitivity [128]. Optical LSW experiments cannot benefit from this longitudinal
mode contribution as it requires the two regions of production and detection to be
sensitive to near field radiation, a condition satisfied only by the microwave cavity
setup.
frequency
10-13L
-
.Earth
>... .
i
.
t
.
*— .-----------»
*
—i----------•----------- *— •— *------— *— -----*------— J
i r lk i<r15 io-14 n r u io- 12 io- '1 io-‘° io-’ io-* io~7 ur‘
io*5 i<r4 nr3
* r .[eVl
Figure 4.8: Longitudinal mode limits from current microwrave cavity LSW searches,
and limits that could be set by future microwave cavity searches using superconduct­
ing resonators.
While LSW hidden photon experiments are important for exploring new parameter
space for masses 1 meV and below, they are not able to make significant inroads into
the allowed dark m atter region for hidden photons.
To explore this regime with
direct searches as well as the allowed region for dark m atter ALPs it is necessary to
use haloscope techniques, as described in the next chapter.
44
Chapter 5
Signal Power
In this chapter the expected power Pout is derived for signals induced by dark m atter
axions or hidden photons in a microwave cavity. Section 5.1 describes the properties
of these cold dark m atter bosons relevant for the expected signal power; Section 5.2
goes through the derivation of the expected signal power for axions using a field
equation approach and discusses the scaling of this expression with the axion mass.
Section 5.3 concludes by highlighting the differences for the hidden photon signal
power derivation.
5.1
Dark M atter Signal Properties
The dark m atter halo distribution is assumed to be an isothermal sphere, meaning
a self gravitating system with no other interactions. This distribution is chosen for
its simplicity although it predicts a higher halo density near galaxy centers than
that observed1. The isothermal sphere model also allows for an easily calculated
velocity dispersion. By the Virial Theorem the velocity of light bosons moving in the
1. The observed densities can be better fit with other distributions such as the Einasto profile [129].
45
gravitational potential of the galaxy is approximately
v ~ 270 km/sec =>• v ~ 103c,
with a Maxwellian distribution. This distribution is boosted bythe
(5.1)
rotation of the
Earth around the galaxy center; for details see [73]. In addition the signal peak is also
broadened by the E arth ’s orbital velocity around the sun and rotational velocity - the
contribution from these effects is less than 4% [130]. The axion (or hidden photon)
is expected to have a fractional energy dispersion of order
^
~ nr6
(5.2)
Ea
which can also be given in terms of an axion quality factor Qa = 106. Therefore
an axion converting to a microwave photon with frequency 34 GHz would have an
expected signal width of roughly 34 kHz. A higher quality factor, or narrower dis­
persion, can result if cold flows of non-virialized axions exist [131] or a dark disk
of axions is assumed [132]. An analysis looking for these narrow peaks would have
enhanced sensitivity compared to virialized axion searches simply because the noise
level is proportional to the bandwidth - thus narrower bandwidth searches have a
higher signal to noise ratio. A dark disk analysis [133] and cold flow search [134]
for axions were performed by the ADMX experiment along with a virialized axion
search. While these models provide enhanced sensitivity, observations do not sup­
port the hypothesis of a dark disk [135] and one must make assumptions about the
directionality in cold flow searches, which introduces additional model dependence to
any exclusion limit. These fine resolution searches also require a significant increase
in computing time and power. Given the additional assumptions of these models,
the practical difficulties of the corresponding analysis, and the lack of observational
evidence, we chose to use the more robust model of an isothermal sphere distribution
46
instead.
Estimates of the local density of dark m atter in the Milky Way yield
pa = 0.3 GeV/cm3
(5.3)
with a factor of two uncertainty [136,137]. Taking this energy density to be composed
predominantly of axions (or hidden photons), the corresponding number density is
n a = 3 x 1012 cm~3
(5.4)
for a boson of mass rna — 1CT4 eV. These dark m atter particles have a long deBroglie
wavelength of
A„ =
m av
^ 1m
(5.5)
which is significantly larger than the dimensions of the experiment, the size of which
is approximately equal to the wavelength A7 of the converted photon:
A7 ~ — = 1 cm
ma
(ma = 10~4 eV corresponds to a frequency of 30 GHz). The inequality \ a »
(5.6)
A7
implies that over the volume of the interaction region, the field due to the dark
m atter boson is coherent and spatially constant, with oscillations at frequency m a.
In the axion case, the interaction of the axion field with the static magnetic field
induces an electric field with an amplitude proportional gB\ the electric field follows
the oscillations of the axion field. For the hidden photon case the magnetic field is
not needed, and instead mixing of the hidden photon and photon states induces an
observable electric field. For a cavity with a mode resonant at m a the electric field
47
will build up coherently.
The experimental parameters that are controllable are the magnetic field B q, cav­
ity quality factor Q , and to some extent the volume V. The dark m atter density p is
given by astrophysical observations and a form factor C (or G for hidden photons) is
fixed once the cavity mode is selected. The minimum detectable signal power will be
determined by the noise floor of the experiment, and the spectral resolution set by
the expected width of the signal A ua. The thermal noise power in bandwidth B is
expressed as a tem perature times a bandwidth: Pth = kBTthB, with the fluctuations
of this noise power, <5P„, given by the radiometer equation [138] as:
(5.7)
where Tsys is the system noise temperature, kB is Boltzmann’s constant, and r the
integration time. The noise power fluctuations are reduced by decreasing Tsys in
cooling the cavity to liquid helium temperatures and by integrating for r = 1 hour.
The Signal to Noise ratio (SNR) then for a given integration time is:
(5.8)
where Tout = Poutl k BA u a is the output equivalent temperature. Equation 5.8 shows
th at in order to have a high SNR for a given signal power at fixed mass, it is most
effective to decrease the system noise tem perature Tsys. In this experiment the system
noise temperature, which is the sum of the physical tem perature and the intrinsic
amplifier noise temperature, is dominated by the electronic noise of the first amplifier
of approximately 15 K (see Chapter 6), so it is not useful to cool the system to
tem peratures lower than those of liquid helium.
Having established the expected signal width and amplitude for light dark m atter
bosons, we now derive the expression for the output signal power Pout from the field
48
equations. For an analogous derivation that treats the cavity as an RLC circuit with
a driving axion voltage see [139].
5.2
A xion Signal Power
The axion interaction with photons adds a term to the low-energy effective Lagrangian
of the form:
Ci = ^g allaFilvF >lu = gailaE ■B
(5.9)
where F ^ is the electromagnetic field tensor, and F^, — \ ( abnuFab its dual. The
source-free Maxwell's equations are modified by this axion interaction term in the
following way:
V - E = ga ilB - V a
(5.10)
V x E + dtB = 0
(5.11)
V x B - d tE = ga i l{E x V a - B d ta)
(5.12)
V -£ = 0
(5.13)
with an accompanying equation of motion for the axion:
(df — V 2 + rn2)a = gailE ■B.
(5.14)
In this experiment the external (static) magnetic field B0 is much stronger than the
cavity magnetic field, so B can be replaced by B0 and the terms proportional to E on
the right hand side of Equation 5.12 can be neglected. Then the modified Maxwell's
49
equations are identical to the original equations in the presence of a polarized medium
P = - g a ilB Qa,
(5.15)
which yields
V -E = - V - P
(5.16)
V x B - d tE = dtP.
(5.17)
Assuming a spatially uniform axion field a ( x ,t) = a(t), the equations of motion
for the electric field come about by taking the time derivative of Eq 5.17:
V 2E - d 2E = gailBQd2a.
(5.18)
By introducing a term to account for losses in the cavity, parameterized by Q and
the cavity resonance frequency
the equation for the evolution of electric fields in
a driven cavity with losses is produced:
V 2E + ^ d tE - d 2E = gailB 0d2a.
(5.19)
The total electric field E(x) inside the cavity can be decomposed in the basis of the
cavity modes, each labeled by index i:
J
E(x, t) = 'Lia i(t)Ei{x),
d V tr lE ^ x )? = N t
(5.20)
where iV* are normalization factors and e,. the relative permittivity inthe cavity. The
50
coefficients Qi(t) satisfy:
V 2q* + y ^ d tai - dfoti = bi(t)
v
(5.21)
where the drive coefficient 6,(t) is
ga~nB0d?a(t)
t
i{ ) ~
[ dV E ; (x ) ■::
Ni
( 5 . 22 )
and in Fourier space:
bM
=
g-^
a(Ul) J d V E ; ( x )
•
z.
( 5 . 23 )
Solving for the coefficients a, in Fourier space yields:
77y
= - 5 -----------Uq — ILULOo/ Q
(5-24)
The steady-state energy stored in cavity mode i is
Ui = ( k ( < ) |2) J d V e r\Et(x)\2
(5.25)
and the time average of the coefficients can be rewritten in terms of their frequency
representation, following the discussion in [140]:
/
DO
|ati(u)\2daj.
(5.26)
■OO
Therefore, from Equation 5.24.
<|ffl' (' )|2> * / _ „ M
-
L
<5'27)
The halo density can be written in terms of the amplitude of the field (averaged over
51
oscillations) and the mass m a: p = m^{a2), so for a single frequency axion signal
a(t) — a0e~luJt centered at u = u;0, the integral in Equation 5.27 evaluates to -f7^l(lQ 2.
This holds as long as the axion quality factor Qa is higher than the cavity quality
factor; in the case that Qa < Q we no longer capture the entire signal in the cavity
bandwidth and the expression for the output power must be adjusted accordingly.
The stored energy in mode i is therefore given by
Ut = & 2 C V Q
mia
(5.28)
where C is a form factor expressing the alignment of the electric field with the external
magnetic field:
c _ \Jv E t( x ) - z d V \ 2
(5.29)
V j v er \ Ei ( x ) \ 2d V
The general expression for power in mode i can be expressed in terms of the quality
factor Q and the average energy stored Up
(5.30)
where k is the coupling of the cavity to the detector. On resonance the power evaluates
to
Pout = ( g Bo ) 2— m a x ( Q , . Q a )K C ,
where the loaded quality factor is Qi =
k Q.
(5.31)
For typical parameters of this experiment ,
the output power expected for an axion coupling to two photons with strength gail =
52
8 x 10"u 1/GeV is:
2
= 1.9 x 10-24 w g s )
( T^
)
( £ j
x
G x KHM/Gev) (o.3 GeV/cm3) (34 G H z)
(5-32)
The axion form factor is only non-zero when the scalar product of the cavity
electric field and the external magnetic field is non-zero. This means, for a non-zero
form factor, we must operate with transverse magnetic modes which have an axial
electric field. The TMono modes have a non-zero form factor and C scales as
Cono <x 4 xL
(5-33)
for cylindrical cavities, where x 0n is the nth root of the first order Bessel function
Jo(r). This decreases approximately as n2 with increasing mode index, so the form
factor goes down quickly for higher order modes.
Scaling
The expected power sensitivity from Equation 5.31 can be rewritten by explicitly
grouping together all terms with dependence on the axion mass m a:
P°ut =
[V m a\CQi
(5-34)
where the interaction strength dependence on m a is separated by writing
®
Z7T
9cm = i r -
^
l C Q C \
0
1
(0.35)
for c-y a dimensionless parameter of order unity in the axion models. As shown above
the form factor is largest for the lowest order mode of the cavity. The TM0io mode
53
resonant frequency is:
"
=S s
(536)
which means the cavity diameter R scales as the wavelength of the converted photon
or equivalently as the inverse of the axion mass: A7 ~ u i” 1. For a cavity length that
scales as L = 0 ( R ), the volume is proportional to V oc ra“3 so the output power goes
as
Pout <x Vrna ~ m ~2.
(5.37)
The noise power depends on the system noise tem perature and bandwidth. For a
linear amplifier the noise tem perature is bounded below by the Standard Quantum
Limit, which gives [141-143]
k b T sql = hu.
(5.38)
At 34 GHz the standard quantum limit is at 1.65 K. Combining the quantum limit
with the
factth at the measurement bandwidth is proportional to the axion frequency,
Nua ~ Qaua. the noise power then depends on the axion mass as
Pn = TsysAva (X m \.
(5.39)
Recall that the signal to noise ratio in bandwidth /\u a for time r is
SNR =
*n
(5.40)
so the scanning rate, given as the ratio of the cavity bandwidth m a/Qi to the inte-
54
gration time, becomes:
djmg/Qt)
B4
(5.41)
dr
This is a steeply increasing power of m a in the denominator. Therefore, although
higher mass axions have stronger couplings, the sensitivity of microwave experiments
to these higher mass particles is much weaker. The limiting factors come from the
decrease of volume as V ex. m~3, and increase of amplifier noise tem perature with fre­
quency. Several ideas have been proposed to increase the sensitivity for higher mass
cavity experiments, such as using hybrid superconducting resonators [110], chaining
multiple cavities together to increase the detected power [144], designing photonic
band-gap structures [108], using open resonators [145], and detecting signals with
nonlinear amplifiers to evade the standard quantum limit [146], While there has been
no demonstration yet of these ideas for Ka-band frequencies, there is strong interest
in developing technologies to explore this frequency range.
5.3
H idden P hoton Signal Power
The steps to derive the signal power for hidden photons are almost identical to the
axion case; the only difference is the form of the driving force. The hidden photon
interaction with photons appears in the Lagrangian as the term
Cx —
+ xA'^)
where J71 is the charged current of the standard photon,
(5.42)
the photon field strength,
and A' the hidden photon field strength.
The equations of motion for the massive vector boson are known as Proca's equa-
tions and can be expressed as:
d2 - V 2 + m2, U ' = xj
(5.43)
which are similar to the electromagnetic equations (Equation 5.44) except for the
addition of the mass and the suppression of the charge density p and current density
j by x-
(5.44)
The driving coefficient for the hidden photon case is
(5.45)
with the resulting expression for the power on resonance due to hidden photons being
Pout = K{xm y ) 2— Q VG
(5.46)
where G, the hidden photon form factor, is
c
\fv E,(x)-hdVf
V f t er | £ , ( i ) | W ’
(5.47)
for h the polarization direction of the hidden photon vector potential A ' .
Note th at the hidden photon form factor G is similar to the axion form factor C
with the direction of the hidden photon polarization n replacing the magnetic field
axis. If we rewrite h — z cos 6 then G = C cos2 0.
A conservative estimate of the form factor assuming all directions for the polar­
ization are equally likely and that the real value of cos2($) is larger than the estimate
with 95% probability yields cos2($) = 0.0025 [83]. Then in laboratory units the
■56
expected power would be
= 2.4 x IO-24 w (
( — V -- I x
0.0025 A 9 x 103 J V16 cm3/
2 /
\
/
\
\ 5 x IO"10J VO-3 GeV/cm3J \ M GHZ/
-1
. (5.48)
This concludes the background and theory portion of this work. The remainder
of the dissertation focuses on the experimental apparatus, measurements taken, and
analysis done to produce an exclusion limit on the electromagnetic coupling of dark
m atter axions and hidden photons.
57
Chapter 6
Experiment
The experiment consists of three major components: the magnet, the cavity and cryo­
genic systems, and the room tem perature electronics chain. The magnet is described
in Section 6.1 and the separate cryogenic system that houses the cavity assembly in
Section 6.2. The cryogenic components of the experiment are collectively referred to
as the insert and are described in Section 6.3. The work I did in designing and testing
the microwave cavity is presented in Section 6.4. To detect an axion signal from the
cavity, cryogenically cooled low noise amplifiers are used; the performance of these
amplifiers is described in Section 6.5. Finally the characterization and design of the
room tem perature electronics is described in Section 6.6.
6.1
The M agnet
The magnet is a superconducting solenoid from Oxford Instruments, made of niobium
titanium alloy with copper windings and a warm bore diameter of 89 mm. The magnet
was cooled in the summer of 2011 and has run in persistent mode at 7 Tesla up to the
time of writing. Periodic fills of liquid cryogens are required but otherwise no other
maintenance is needed. Figure 6.1 shows the solenoid and cryogenic systems for the
magnet.
58
To* TitCIHAL L;nh
— n
p ic
Tc***!**!. TVat*
Li«Ui£ Hfkiun
Llttu iB
' Ni t HoSEN
ao-TTo^
ThCHMAl l i n k '
Figure 6.1: Magnet
59
6.2
Cryogenic System s
A separate gas-flow cryostat from Cryo Industries houses the cavity and cryogenic
amplifier assembly. This cryostat is situated in the bore of the magnet, with an inner
diameter of 39.88 mm (1.57 inches), and is offset from the bore center so that there is
enough space for a second cavity to be placed in the bore but outside of the cryostat.
This was useful in conducting a previous light-shining-through-wall experiment to
look for hidden photons [29]. Figure 6.2 shows a cutaway of the cryostat, which has
three liter reservoirs for liquid cryogens placed at the top; a capillary tube allows liquid
helium to flow from the reservoir to the bottom of the cryostat, where it passes over a
block containing a heating element. The liquid helium boils off and the resulting gas
flows up to the top of the cryostat, cooling the entire space. For the work presented
in this dissertation, the valve which controlled the liquid helium flow rate into the
capillary tube had to be adjusted manually. Automating the flow valve adjustment
is desirable but due to time limitations was not implemented.
The helium consumption rate depended to a great extent on the strength of the
vacuum in the cryostat insulation layer; this also greatly affected the tem perature
stability. The helium consumption would generally increase from 30 L/week to 70
L/week after two weeks of continuous running without pumping out the cryostat
insulation layer. During the data acquisition period we pumped out the cryostat
vacuum insulation layer every weekend in order to prevent tem perature instability
during data runs which was associated with a high helium consumption rate.
A typical cool-down procedure began by pumping out the cryostat space, done
overnight. The next day the capillary tube would be filled with helium gas to clear
any air or nitrogen; after that the liquid nitrogen reservoirs were filled and then the
liquid helium reservoirs. Cooling the bottom of the cryostat from room tem perature
to liquid helium temperatures took approximately three hours and two helium fills.
60
Figure 6.2: Top half of the cryostat, with inset showing full cryostat.
6.3
Insert
For a sketch of the components placed inside the cryostat, see Figure 6.3 (not to scale).
The main components are (1) waveguides, which transm it the signal in and out of
the cryostat, (2) cryogenic amplifiers, and (3) the cavity and tuning mechanism. We
give a brief overview of the entire system and discuss the particular characteristics of
the cavity and amplifiers further in later sections.
In this experiment, waveguides are used to transmit RF signals to and from the
cavity. Waveguides were used as they have significantly lower losses than cables at
34 GHz. The loss of WR28 copper waveguide is 0.576 dB /m at 32 GHz; for coax
cable, common ratings show losses of 7 dB /m at 20 GHz, and the losses increase with
frequency.
The waveguides are primarily copper, with copper-clad stainless steel portions near
the top of the cryostat to reduce the heat load into the system. WR28 waveguide
has a cutoff frequency of f c = 21.5 GHz, below which it will not propagate signals.
The nominal frequency range for the lowest order waveguide mode is from 26.5 to 40
GHz.
Mylar radiation baffles are also placed around the waveguides to reduce the radi­
ation heat load. Two flexible 3 inch sections of waveguide connect to feedthroughs at
the top of the cryostat, where a further section of waveguide takes the signal from the
cryostat to the room tem perature electronics. The waveguide and cavity system is
vacuum sealed and a pump (separate from the pump on the cryostat space) is active
during cool-downs. It was im portant to have the waveguide region pumped on dur­
ing cool-downs, as an interm ittent vacuum leak was sometimes observed (we suspect
the leak was due to the stress from repeated thermal cycling). The leak could allow
helium gas to enter the waveguide region and then condense into liquid, which could
then flow into the cavity space and shift the resonance frequency. Taking data with
62
w a v e g u id e
HEMT
amplifiers
c a v ity
m agnet
He vapor
vaporizer
Figure 6.3: Sketch of cavity and waveguides in cryostat.
63
liquid helium in the cavity was not a viable option, as the resonance frequency was
seen to change by an 0 (1 ) fraction of the cavity bandwidth every few seconds when
the cavity had liquid - this was possibly caused by helium boiling inside the cavity,
creating an unsteady state.
Besides the waveguide, there are two other feedthroughs into the cryostat: one
is the tuning rod for adjusting the cavity resonance, and the other is an electrical
stock for tem perature sensors and the cryogenic amplifier power leads. One tem­
perature sensor is connected to the bottom plate of the cavity, another to the first
cryogenic amplifier, and additional sensors monitor the tem perature at higher points
in the cryostat. The tem perature sensors were produced and calibrated by Lakeshore
Cryotronics.
We now discuss the design of the cavity in greater detail.
6.4
The Cavity
The cavity is machined from oxygen free high conductivity (OFHC) copper, and has
two parts: a cylindrical body with inner diameter 15.24 mm and height 8.64 mm.
and a bottom end cap (see Figure 6.4). All inner surfaces were polished to achieve
a surface roughness of < 2 //in. An indium seal was placed between the bottom cap
and body.
"O
°
'*
: ?\ r . r *
1A V'7> O
..
.
,
^.
A -
O
. j-
-
O. 1
■
Figure 6.4: Cavity body
The experiment monitored the power in the transverse magnetic TM 020 mode,
64
which has electromagnetic fields of the form:
where
xq2 =
E = E0J0(x02r /R )e ~ iujtz
(6.1)
B = - i y / r rEoJ2(xQ2r/R)e-lujt4>
(6.2)
5.52 is the second root of the first Bessel function Jq and R is the cavity
radius. The oscillating electric field creates currents th at run vertically; hence it is
im portant to have good contact between the cap and the cavity body. The bottom
cap therefore has a knife edge on the surface to ensure good electrical contact.
In Chapter 5 the form factor was shown to be a decreasing function of the mode
index
n
for the TM0„o modes. This would imply th at it is favorable to work in the
lowest order TM0io mode. However, the full expression for the signal power depends
not only on the form factor, but also the volume and quality factor. Even though the
form factor is smaller for higher order modes, for two modes at the same frequency
the volume will be larger for the higher order mode. In addition, the theoretical
quality factor for a cylindrical TMmnp mode is a function of the cavity geometry and
is equal to [147]:
^ .... A [Xmn + { p ^ r / L f Y 12
Q
S
2tt(1 + 2R / L )
{
’
where R is the radius, L the length, S the skin depth and A the wavelength corre­
sponding to the cavity resonance. For the TM 0no modes, this simplifies to
9 = 7 2 ,0 T iR IL Y
i6A)
Since x 0n ex n, the quality factor increases for higher n. However, the density of modes
also increases with higher n. which makes tuning more complicated. Table 6.1 shows
the expected contribution to the signal power in terms of the volume, form factor.
65
and Q for the two lowest order modes (with the same aspect ratio of 2R / L = 1.76
and theoretical Q evaluated at 34 GHz for a copper cavity at room temperature).
The two modes give the same expected contribution - in practice the quality facTable 6.1: Mode comparison
Mode
TMoio
TM 020
V (cm3)
0.70
1.6
C
0.69
0.13
VCQ (cm3)
Q
0.52
1.08 x 104
2.5 x 104
0.52
tor may be lower than the numbers listed in the table due to surface imperfections,
but the conclusion is that one achieves similar sensitivity working in either mode. We
chose to work in the higher mode for reasons related to the tuning, as described below.
Tuning
The cavity resonance frequency is adjusted by vertically inserting a dielectric rod
into the cavity. This perturbs the fields, with the net effect being that the resonance
frequency decreases. This is the same technique as used by the first dark m atter axion
searches using microwave cavities [148,149]. This tuning method is straightforward
but also degrades the form factor as the rod insertion depth increases, due to mode
localization [150], This limits the tuning range one can access with a single rod. The
ADMX experiment achieves a wide tuning range by using a combination of metal and
dielectric posts and moving them sideways [151]; for the frequency range covered in
this work, the vertical insertion method did not measurably degrade the form factor
(see Figure 6.5) and so was deemed an acceptable solution.
We use a 1.6 mm diameter rod made of alumina ceramic with a relative permittiv­
ity of e ^ 9.3; the loss tangent is specified to be less than 10-5 at Ka-band frequencies.
The dielectric can act as a waveguide and transm it signals out of the cavity, so it is
important to make the rod small enough such that the cutoff frequency is above the
66
in
coO
C
oN
C
1
o
o
O
CO
in
sco
33.8
34.0
34.2
34.4
frequency (GHz)
Figure 6.5: TM 020 form factor versus frequency
resonant frequency of the cavity in the mode of interest. For our dimensions, the
cutoff frequency is 36.22 GHz. The relative frequency shift induced by the presence
of the dielectric depends on the ratio of the dielectric volume to the cavity volume:
Suj
r 2l
— oc fyyyp
u)
R ZL
(6.5)
where r is the rod radius and I the insertion depth. In order to be able to tune the
cavity by an 0( 1) fraction of the cavity bandwidth without needing extremely fine
control over the rod's vertical motion, it was easier to work with a larger cavity in
the TMoao mode at 34 GHz instead of a small cavity in the TM0io mode. Figure 6.6
shows the TM 020 resonant frequency as a function of the rod insertion depth for the
cavity we used in the experiment. There were no other modes in the frequency range
33.9-34.5 GHz; the closest mode was the T E U2 mode near 36 GHz.
A perture C oupling
There are two apertures used to couple power in and out of the cavity. The first is
a 1.27 mm diameter hole which is very weakly coupled (< 50 dB). The second is a
67
34.6
34 4
•
•
•
N
£
CD
34 2
oI
34
33.8
0. 015
0. 03
0. 045
0. 06
R o d insertion D epth (in)
Figure 6.6: Tuning (warm): frequency versus rod insertion depth
strongly coupled aperture and signals from this port go to the first stage amplifiers.
Both apertures use inductive coupling by picking up overlap in the magnetic field of
the waveguide T E 10 mode and the azimuthal magnetic field of the cavity, shown in
Figure 6.7.
The weak coupling port is used to send in power from a vector network analyzer
so th at the cavity resonance and quality factor can be determined in situ from trans­
mission measurements. The strongly coupled port was designed to be near critical
coupling when the cavity is at liquid helium temperatures. Critical coupling occurs
when power losses through the aperture equal the ohmic losses in the cavity (« = 0.5).
As we used waveguides to couple to the cavity, it is not straightforward to adjust the
coupling after assembly as is done with antenna couplings. Therefore we simulated
the cavity response for different dimensions of the aperture size using Ansoft HFSS,
taking into account the change in dimensions and conductivity upon cooling to 4 K.
Critical coupling was achieved in the simulations for a racetrack shaped hole with
length 3.048 nun and radius 0.762 mm, positioned at a height where the waveguide
68
(a)
TM020 magnetic field lines.
(b) Waveguide magnetic field lines
Figure 6.7: Cavity field lines
magnetic fields were strongest. Two cavities were built and tested. The first had a
lower quality factor due to poor surface finish and was overcoupled, so was not used.
The second cavity had a high quality factor and the coupling of the strong port was
in agreement with the predictions from simulations. The smaller aperture was much
more weakly coupled than expected and had a loss of —65 dB. This turned out to be
such weak coupling th at during the actual data runs, there was a band of frequencies
where the cavity resonance was no longer visible from transmission measurements
because the signal power was too weak due to the attenuation from this port; see
Appendix B for details.
The ohmic losses are determined by the skin depth; at room tem perature this is
given by:
<5 = (2/u;//cr) 1^2
(6 .6)
where p is the relative permeability and a is the bulk conductivity. As the temper­
ature is reduced the skin depth decreases due to an increase in conductivity cr. The
mean free path of electrons in the metal also increases, and when the mean free path
69
is greater than the skin depth, the losses must be determined using the equations for
the anomalous skin depth [152]:
x
/ v^3c m evp v 1/3
( 6 -7 )
where rae is the electron mass, e the electron charge, vp the Fermi velocity, and n the
electron number density. In this regime the losses are independent of temperature.
At 34 GHz the anomalous skin effect regime becomes dominant at around 100 K,
therefore the measurements of the coupling coefficient at 77 K should be representative
of the values for lower temperatures. Figure 6.8 shows reflection measurements made
for the second cavity both at room tem perature and at 77 K, plotted in terms of the
logarithmic power reflection Su versus frequency. The coupling is given in terms of
the coefficient (3, where
k
=
such that critical coupling occurs for (3 = 1. The
values of (3 can be calculated from the reflection measurement [153] to yield a value
of (3 = 1.15 for the 77 K case, with a 15% variation over the frequency range scanned.
The unloaded Q was measured to be 1.9 x 104 and 1 x 104, respectively for the 77
K and room tem perature cases. This is in reasonable agreement with the theoretical
values for the unloaded quality factor of 3.2 x 104 and 2.5 x 104, for a cold and warm
cavity. The loaded quality factor, estimated from the 3 dB bandwidth of the cavity
response, is seen to increase from 6455 at room temperature to 9422 at cryogenic
temperatures.
Tuning M echanism
The dielectric rod insertion depth is adjusted by turning a screw at the top of the cryo­
stat. This rotates a long G-10 rod which is attached to a fitting on a 100 threads/inch
bushing. As the attachment moves up and down through the rotation on the threads,
this pushes a pivot arm which is placed on a bridge above the cavity. The motion of
70
Cavity at 2 9 0 K
0
-2
-4
6
Ii
it
-8
:
fres = 34.402 GHz
Ql = 6455
P = 0.68
f
34 38
3 4 .3 9
3 4 .4 0
34 4 1
34 .42
F req u en cy (Gh'z)
(a)
Cavity at 77 K
)
)
0
Qo = 19 866
fres = 34.517 GHz
Ql = 9223
34 52
F r e q u e n c e (GHz)
(b)
Figure 6.8: Reflection measurements of the cavity at room tem perature and 77 K. We
extract the coupling from these measurements to see that at cryogenic temperatures,
the cavity strong port is close to critical coupling.
71
the pivot arm then pushes the dielectric rod up or down. To keep the cavity assembly
vacuum tight, the dielectric rod is encased in a beryllium copper bellows which has
two vacuum seals, one to the pivot arm and the other to the top of the cavity. The
bellows is also surrounded by a tube to limit the motion to the vertical direction. The
range of the lever arm is 16.9 degrees for a total of 4.7 mm of rod travel. Figure 6.9
shows the entire cavity assembly with the tuning mechanism.
16.90°
?GR 0 . 1 8 5 I N O ROC TRAVEL
O
Figure 6.9: Cavity assembly, showing the cavity body, dielectric rod, and encasing
bellows and support structure. A lever arm translates the off-center motion of a G-10
rod into a vertical force on the tuning rod.
72
6.5
Cryogenic Amplifiers
The first stage amplifiers are High Electron Mobility Transistors (HEMTs) ordered
from JPL [154] which are placed in the cryostat approximately 10 cm above the cavity.
There are two HEMTs cascaded together with a 3 dB attenuator between them for a
total gain of 56.7 dB. The first of the HEMTs was measured by the vendor to have a
minimal noise tem perature of 35 K at 34.3 GHz for a physical tem perature of 15 K
and has a gain of 27.5 ± .2 dB. Figure 6.12 shows the gain as measured by the vendor
from 32-36 GHz; however these amplifiers can operate down to 11 GHz. The Friis
formula for the total noise tem perature TN of a system with components that have
noise tem peratures Ti, T2, . .. and gains Gi, G2l ■. ■goes as:
TN = Tl + T 2/ G l + T 3/ G lG2 +
( 6 .8 )
which shows that the noise tem perature of the first component is the most important
term in determining the overall noise temperature. The second cryogenic amplifier
has a noise tem perature of 100 K at a physical tem perature of 15 K, which adds 0.1 K
to the total noise temperature; all noise contributions from later stages in the receiver
are negligible.
N oise T em perature
The noise tem perature of the first cryogenic amplifier was measured directly using
the Y-factor method. In this measurement a 50 Q RF termination was connected to
the input of the first amplifier, and the noise power measured for the termination at
4 K and 27 K. A 64 mm piece of stainless steel co-axial cable was put between the
load and amplifier to thermally isolate the HEMT. The output noise power is given
by the relation:
Pout = kBG B ( T + Ta)
73
(6.9)
Vendor Measured Gain for 1st Amp
33.4
33.2
33.0
32.8
32.6
32.4
32.2
32.0
31.8
31.6
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
Freq (Hz)
Figure 6.10: Gain of first amplifier as measured by vendor
where k s is Boltzmann's constant, G the gain of the system, B the bandwidth, Ta
the noise tem perature of the first amplifier and T the physical temperature.
By
comparing the noise power for the two physical temperatures we were able to extract
the amplifier noise temperature, shown in Figure 6.11 as a function of frequency. The
measured data points were taken over a bandwidth of 3 MHz, and so a smoothed line
was plotted over the data to show the average value of the noise tem perature in the
frequency range 33.9 to 34.5 GHz of:
Ta ^ 15K.
(6.10)
The electrical loss of the stainless steel segment was measured to be 0.5 dB and in­
cluded in the calculations. It is crucial to measure the noise tem perature accurately
as this value is used when determining the system gain, which then allows us to cal­
culate the absolute power at the output to the cavity from the measured power at the
end of the receiver chain. The uncertainties in measuring the noise tem perature come
74
A m p lifie r Noise Tem perature
a 20-
&
Q.
•I
E
& 15-
33.9
S V
• •
- -
y -fa cto r m eas
data sm oothed
• —•
W einreb meas.
34.0
34.1
•
rv ^ * t f V - '
34.2
34.3
34.4
34.5
Frequency (GHz)
Figure 6.11: Noise tem perature of the first amplifier as measured with the Y-factor
method. The measured data points (black) were taken over a bandwidth of 3 MHz.
The smoothed data (red) shows the average noise temperature is near 15 K. The noise
tem perature of the first amplifier was measured by the vendor (blue) to be near 35
K for a physical tem perature of 15 K.
75
from various factors: there are experimental uncertainties such as imperfect thermal
contact between the tem perature sensors and the load, or not knowing the exact loss
of the stainless steel segment at cryogenic temperatures. There are also effects from
the nontrivial impedance mismatch between the cryogenic amplifier and waveguide to
coax adaptor, discussed in the next section. Taking into account these uncertainties
at the 30% level gives a conservative value for the amplifier noise tem perature of 20
K - this value was used in the calculation of the system gain.
Im pedance M atching
The input return loss is specified by the vendor to be greater than 9 dB at 26 GHz,
meaning that up to 12.5% of the power could be reflected at the input to the am­
plifier. There is an additional source of impedance mismatch due to the waveguide
to coax adaptor. Two waveguide-coax adaptors from Maury Microwave were used to
connect the HEMTs to waveguides, stated to have a VSWR at room tem perature of
0.15. This translates to a power reflection coefficient of 4%.
S ystem G ain
The total gain of the system is determined directly by measuring the output power
at the end of the receiver when a 50 ft termination is placed at the input to the first
cryogenic amplifier and the system is cooled. This was done at 9 K. and the output
noise power density as a function of the first local oscillator frequency is shown in
Figure 6.12. both with the RF switch in and taken out of the chain. This was done
as early data sets were acquired without the RF switch in place. Using the fact that
the system noise tem perature is the sum of the load tem perature and amplifier noise
temperature, and taking conservative values for the noise tem perature as discussed
in the previous section, the gain of the system is found to be 86.7 dB (with the RF
switch).
76
2.8
50 O hm Runs
le -1 0
£
sw itch
f -4 no sw itch
2.6
5
£ 2.4
C
Q)
q
-) -N
2.2
0>
5o
Q_
4—>
3
Q.
3
2.0
o
cu 1.8
a*
0L_)
F
1.6
. - - 4
f
1.4
33.9
34.0
34.1
34.2
34.3
------ *
34.4
34.5
RF Freq (G Hz)
Figure 6.12: O utput power density versus LO frequency for a 50 ohm run. Given an
amplifier noise tem perature of 20 K, we can extract the gain to be 86.7 dB with the
microwave switch in place.
77
6.6
R oom Tem perature Electronics
The output signal from the first stage amplifiers is fed out of the cryostat through
waveguide and mixed down to baseband frequencies by a triple heterodyne receiver.
See Figure 6.13 for a schematic of the mixing scheme. Not shown in the schematic is
an RF switch before the first mixer which allows the output signal from the amplifiers
to be fed either in to the receiver chain or the network analyzer for a transmission
measurement. The first mixer1 has a tunable local oscillator2 and mixes the RF signal
Mixer
4 .0 9 2 GHz
Mixer
^92
HEMT
amplifiers
Mixer
2 MHz
magnet
ADC
cavity +■
tuning rod
DISK
Figure 6.13: Schematic of experiment
to 4.092 GHz. Bandpass filters are placed before and after each mixer to suppress
image frequencies and harmonics. Image frequencies occur at cu* = 2ujio ~ ^
rf
and
get mixed to the same intermediate frequency as the RF signal. An amplifier3 boosts
1. M ITEQ M2640W1
2. Anrit.su MG3694C signal generator
3. M ITEQ AFS33-04000420-20-10P-GW-44
78
the mixed signal by 60 dB and then a second mixer (with bandpass filters again)
mixes the signal down to 592 MHz. There is an additional stage of amplification4 and
then the signal goes to an IQ mixer5, which mixes frequencies dowm to 2 MHz and
separates the output into its in-phase (I) and quadrature (Q) components. The I and
Q channels then go through 4 MHz lowpass filters, a voltage pre-amplifier6, and finally
are digitized by a National Instruments PCI-5114 digitizer with a 10 MHz sampling
rate and 8 bit resolution. The digitizer and all mixers are locked to a common 10
MHz reference clock.
The local oscillator frequencies were chosen to minimize harmonics near the signal
frequency, and the amplification chosen so as not to saturate any component in the
receiver chain. The mixers and amplifiers are commercially available from MITEQ,
and the filters are from Spectrum Microwave. Table 6.2 shows the power levels at
each point in the receiver chain. The measured powers are in reasonable agreement
with what we expect; as we cannot measure the loss of the waveguides directly and
there are non-zero reflections between the different receiver components, we do not
expect exact agreement between the expected and measured values.
The measured power spectral density (PSD) of the receiver chain with room tem­
perature noise at the input is shown in Figure 6.14; the shape of the spectrum is from
the 4 MHz low-pass filters and the spike at 0 Hz is from DC and 1/f noise.
As the digitizer measured voltage samples from the I and Q channels at a sampling
rate of 10 MHz, this generated a large volume (37 GB per channel) of data for
the typical one hour measurements taken during data runs. The time domain data
was streamed to external hard drives during data runs; afterwards an offline script
processed the raw data into power spectra. We used the FFTW libraries [155]. which
4. M ITEQ AFS3-00100400-13-10P-4
5. M ITEQ IR0502LC1Q
6. SRS SR445A
79
Component
Expected
Power
Density
(dBm/Hz)
Measured
Power
Density
(dBm/Hz)
Cavity
Cryogenic Amplifiers w / 10 dB atten.
RF Filter and waveguide to electronics
1st mixer and cable
1st amplifier and filters
2nd mixer
16 dB attenuation
2nd amplifier and filters
IQ mixers and filters
-185
-144.3
-147.7
-162.7
-103.7
-110
-126.2
-96.8
-105.3
-150
-161.4
-107.6
-111
-127.3
-98
-104.4
Table 6.2: Power levels measured at different points in the receiver chain agree with
expected values. The receiver chain was designed so that the power at each stage is
well below the saturation level of the electronics.
4 0 le -1 1
4 MHz F ilter (Room Tem p E lectronics Chain On)
3.0
£ 2.5
§
E
Q
in
Q_
1.0
0.5
-6
-4
-2
0
2
4
6
Frequency (M Hz)
Figure 6.14: Response of room tem perature electronics chain with a room tem perature
terminator at the input to the receiver. The main structure is due to the 4 MHz lowpass filters and DC noise.
80
employ 0 ( n log n) algorithms to compute the discrete Fourier transform efficiently,
defined as:
N-1
X k = J 2 ^ n e ~ l27rnk/N
k = 0,...,N-l
(6.11)
n= 0
where x n is the n th time domain sample and X k the A;th frequency bin. Although
the algorithm works best on sample sizes that are powers of two or multiples of small
prime numbers, no significant improvement was noticed between evaluating a 256
point spectrum and a 294 spectrum, so we used the 294 spectrum as this gives us a
resolution bandwidth (with square windowing7) of
resBW = 4 =
iV
294
- 34013.6 Hz
(6.12)
which is close to the expected axion signal width of 34 kHz. The total power spectrum
is the squared sum of the I and Q voltage transforms divided by the characteristic
impedance of the system:
Each spectra is saved to a hie which consists of two vectors corresponding to the
baseband frequency and power density, respectively. We then analyze these spectra
to identify potential axion-induced peaks, which is the main topic of the next chapter.
Before discussing the analysis, we conclude by describing the run protocol used for
the d ata acquisition.
7. Using other windowing functions would reduce the ‘'scalloping loss” : a decrease in signal ampli­
tude if the signal frequency does not match the center frequency of a bin, causing signal leakage into
surrounding bins due to the nature of the Fourier Transform. However these windows also increase
the resolution bandwidth, so require longer measurement times to achieve the same signal to noise
ratio as the square window.
81
R un Procedure
Before acquiring data, the dielectric rod is adjusted to tune the cavity resonance to the
desired frequency. The loaded quality factor, center frequency, and cavity bandwidth
are measured with a vector network analyzer8. A one hour time trace of the I and
Q voltage channels is then acquired and saved to external hard drives in little-endian
binary format as signed 8-bit integers.
A Labview program continuously records
the physical temperatures of the cavity and first amplifier during data acquisition.
The cavity is then tuned by 3 MHz and the process repeated. Figure 6.15 shows
the loaded quality factor measured at different cavity frequencies. The low quality
factors near 34.4 GHz were associated with a “dead spot". where due to the extremely
weak coupling of the small aperture the power sent into the cavity was too low to
make a proper transmission measurement for certain frequencies (see Appendix B for
a discussion of the problem and its resolution). After fixing this, data was retaken
for those frequencies, which showed a higher quality factor.
8. PNA E8364C
82
loaded Q fo r D ata Runs
10000
2
7000
6000
5000
4000
33.7
33.8
33.9
34.0
34.1
34.2
34.3
C avity Freq (GHz)
Figure 6.15: [Loaded Q versus frequency. Measurements below 33.9 GHz were taken
when the cavity volume was full of liquid helium and were not used in the analysis.
Runs with Q below 6000 corresponded to distorted measurements near the “dead
spot'’ and were re-taken after a shim was put in place in the calibration weak port.
83
Chapter 7
D ata Analysis
The aim of the data analysis for this experiment was to search for an excess of power
above the thermal noise from a cavity in the TM 020 mode.
In the absence of a
statistically significant excess, an exclusion limit was placed on the strength of ALP
coupling to photons. The data sets are summarized in Section 7.1. Baseline removal
and noise statistics are discussed in Section 7.2. The procedure for weighting the data
and combing overlapping spectra is laid out in Section 7.3; these combined spectra
are used to look for excursions. Section 7.4 shows how potential axion candidates
were selected. A second set of data was taken at all frequencies where statistically
significant peaks were observed; no candidates persisted from these rescans.
7.1
D ata D escription
Measurements were taken from November 19, 2014 to June 31, 2014 for 500 resonance
frequencies between 33.9 and 34.5 GHz. The physical temperature of the cavity was
between 4 and 10 degrees Kelvin for the data taken, with a typical observation time
for each cavity setting of 1.06 hours. Twenty eight measurements were taken for
longer times to study the system noise floor. For data taken between Jan 29. 2014
and March 13, 2014, a test tone signal was injected into the cavity at 3 MHz below the
84
cavity center frequency, A test tone was used to ensure the local oscillator frequencies
were not drifting over the long integration times and to study the effect of the signal
processing on these 1-bin peaks. Figure 7.1 shows the timeline of data runs, with
labels for changes to the system, such as adding a microwave switch to autom ate the
network analyzer measurements and a heater feedback control loop to better stabilize
the temperature.
34.6
F requency C o ve ra g e
34.4
n
x
34.2
o
u 34.0 k
<u
CT
Sf
u_ 33.8
>,
4-*
3 33.6
insert out;
cavity weak
coupling
altered
te s t tone
applied
33.4
33.2
D ate
Figure 7.1: Data acquisition schedule
As mentioned in the previous chapter, the time-domain data is converted into a
double-sided power spectra with 294 bins and 34013.6 Hz per bin. 500,000 individual
spectra (equivalent to 14.7 seconds integration time) are averaged together at a time
and saved to file; for the standard observation length of 1.06 hrs. 261 such averaged
spectra could be produced for each data set. The reason we did not average all the
individual spectra together at once is that temperature and pressure changes during
the run caused frequency drifts, as discussed below.
85
Pow er Spectral C hanges
The averaged spectra were inspected to see if the structure of the ith trace changed
significantly from the shape of the first trace, for i — 1, . . . , 261. Any traces exhibiting
a significant change in structure were discarded. We found th at a change in the cavity
tem perature of more than 0.5 K over the course of a measurement was enough to alter
the observed power spectra shape. When the frequency of the cavity was measured
after such runs with the network analyzer, it had usually shifted by an 0(1 ) fraction
of the cavity bandwidth. To avoid these frequency drifts it was therefore essential
th at the tem perature be stable on hour long time scales. The frequency stability of
the cavity was also tied to whether a vacuum leak was present in the system, as this
would change the pressure of helium gas inside the cavity. The first microwave cavity
axion search [148] found the following relation between the helium pressure and cavity
frequency:
30 kHz/GHz/psi.
(7.1)
A feedback loop was added to the tem perature monitoring system which adjusted the
power sent to a heater in the cryostat in an effort to stabilize the temperature. This
was only somewhat effective, as small adjustments to the helium flow valve caused
large differences in the cooling rate, so for each new cool-down the feedback loop
parameters had to be re-evaluated for a new flow valve setting. In the end. it was
most effective to pump out the cryostat vacuum jacket before each cool-down (which
reduced the helium consumption rate).
The low helium consumption cool-downs
usually did not have good tem perature stability on the first day when the cryostat
went from room tem perature to 4 K, but once the cryostat had been cold for a day,
it was possible to keep the temperatures stable to within < 0.02 K for twelve hours
86
at a time before the helium reservoir would need to be re-filled.
Figure 7.2 shows representative plots of power spectral changes by plotting five
traces (i = 1, n/4 , n / 2 , 3n/4, n) for datasets each with n average spectra. Figure
7.2a is from a dataset where the tem perature gradually decreased by 2 K over a
twenty minute measurement. The shape of the power spectrum remains the same
but the magnitude of the noise power decreases. A test tone signal at -1 MHz is
also visible. Figure 7.2b shows a more drastic example; in the middle of this run the
cavity tem perature dropped from 6 K to below 4 K, which was a sign that liquid
helium had entered the system. A plot of overlaid spectra for an hour run with good
tem perature stability, shown in Figure 7.3, does not exhibit any changes in the shape
of the response for different spectra. The overall power level is higher in the stable run
as the microwave switch was not yet installed in the receiver chain for that dataset.
Tracking the cavity resonance by periodically switching to the network analyzer
to measure the cavity frequency and then automating the first local oscillator would
have been a way to avoid discarding data from frequency drifts. This automation was
not implemented because of time constraints, but would greatly increase the amount
of usable data.
7.2
Baseline Removal
The first and last thirty bins of the spectra are cut out of each trace kept as well
as the twenty bins around the center to remove the DC spike and 1/f noise. The
baseline of each trace is then estimated; this baseline is subtracted from the power
spectra to obtain the spectrum of excess noise power SP. The baseline refers to the
background curve of the power spectra, which is in general not flat and is a function
of electronic noise, the filter response, and the cavity-amplifier interaction. The shape
of the cavity-amplifier response can be theoretically predicted either by analyzing the
equivalent circuit diagram of the cavity as a RLC circuit with a noisy amplifier [156],
87
3.0
02-04-14-03-57
le -1 0
2.5
—
43
64
—
2.0
85
1.5
a
0.5
0.0
-6
-4
-2
0
2
Freq O ffset (MHz) from 34.411 GHz
(a) Gradual power spectral change.
3.0 ,l e
h?
0 1 -1 4 -1 6 -1 4 -5 3
66
131
196
261
Q.
0.5
0.0
Freq O ffset (MHz) from 33.981 GHz
(b) Cavity filled with liquid helium.
Figure 7.2: Power spectral shape changes. A gradual decrease in tem perature of 2 K
over a twenty minute period caused the observed change in power spectra shown in
(a): sharp changes in tem perature were accompanied by large frequency shifts which
placed the cavity resonance outside of the receiver passband (b).
88
4.0
1 2 -1 9 -2 1 -5 8 -1 2
le -1 0
3.5
—
66
—
131
196
3.0
—
261
2.5
2.0
1.5
Q.
1.0
0.5
0.0
-6
-4
-2
0
2
Freq O ffset (MHz) from 34.135 GHz
Figure 7.3: Temperature stable run.
or from a noise wave model [139]. Although the theoretical response fit the overall
shape of the data, it did not account for a small periodic structure at the level of
0 ( 10~'?) which was seen in the power spectra for long integration times and believed
to be due to the room tem perature electronics1. Estimating the baseline empirically
by doing a smooth of the power spectrum was found to be more effective at structure
while still preserving the amplitude of 1-bin peaks. The data was smoothed using
a Savitzky-Golay filter [158], which is a type of moving-average filter th at performs
a least squares fit of a polynomial of order m to n data points at a time, taking
the center of the polynomial fit as the smoothed data point. By having higher-order
polynomials one can fit data with curvature of order m: at the same time, a wider
set of data n is then needed to accurately fit the data. The optimal choice of rn
and n is then a compromise between fitting higher order (nonlinear) structure while
not unduly suppressing peaks of interest or otherwise distorting the baseline-removed
1. This structure was also observed in the first generation ADMX runs [157].
89
trace. For the data examined we found th at a polynomial order of 4 and window size
of 11 points for the Savitzky-Golay filter were satisfactory parameters as the resultant
baseline-subtracted traces had no obvious residual structure and narrow peaks were
still visible.
Figure 7.4 shows two plots of a dataset (the plots are centered on the region near
the cavity resonance) after removing the baseline with a moving average filter (Figure
7.4a) and with a Savitzky-Golay filter (Figure 7.4b). For the moving average estima­
tion a residual structure is evident. There is also a wide peak near 3.5 MHz which is
an artifact of the electronics. When the baseline is estimated with the Savitzky-Golay
filter the baseline-subtracted trace has much less residual structure but the peak near
1.5 MHz is still identifiable.
Further C uts
After removing the baseline using the Savitzky-Golay filter, an additional cut is made
of the first and last seven bins. Removing these points guards against an inaccurate
estimation of the baseline at the end points as a result of insufficient data. Further
cuts were found to be necessary for spectra with test tone signals. The first local
oscillator was usually set so that the cavity resonance would be mixed down to 2
MHz; the test tone signals were injected at 3 MHz below the cavity resonance (so
mixed down to -1 MHz in the baseband). As the data analysis procedure only looked
for axion signals at the baseband frequencies near the cavity center (0.5-3.5 MHz in
the baseband), the test tone was initially thought to not affect this region. However
after averaging many traces together a persistent signal was observed at 1 MHz in
power spectra. Figure 7.5 shows the excess noise power after baseline subtraction
seen at ±1 MHz for test tone runs (plotted as a function of the cavity resonance); the
noise power level in surrounding bins is at the level of lO” 14 mW /Hz, so the 1 MHz
power levels are quite high in comparison.
90
01 11 0 8 24-55
-
-
-
X
I
E
53
0.5
5
o
Q.
-a
at
0.0
AO
in
( i ' l l
-0 .5
-
1.0
0.5
1.0
1.5
2.0
3.0
Freq Offset (MHz) from 33.973 GHz
3.5
4.0
(a) Systematic residual structure seen when baseline is estim ated using moving average filtering.
0 1 - 1 1 - 0 8 - 2 4 -5 5
le-14
M
X
3
£
ir
-2
-4
-6
0.5
1.0
2.5
3.0
1.5
2.0
Freq Offset (MHz) from 33.973 GHz
(b) Residual when baseline is estim ated using Savitzky-Golay filtering.
Figure 7.4: Spectra after baseline removal. The systematic structure is reduced more
effectively when the baseline is estimated with Savitzky-Golay filtering as opposed to
the moving average filtering.
91
This suggests that there was leakage at the 0.5% level from the test tone signal
into the 1 MHz bin. As this bin is in the region investigated for an axion signal, five
bins were cut around 1 MHz from all datasets that had a test tone, and measurements
retaken to have d ata at those frequencies.
Test Tone Power o f -60 dBm
10 '
icr
i*
fM
x
Q.
10
•
t
5 10 -11
£
<u
o5
Q-
•
12
4-*
D
■ ■
o
10
I
13
•
i t
■
■
te s t to n e
■ leakage
10
14
33.90
33.95
34.00
34.05
34.10
34.15
34.20
34.25
Cavity Frequency (GHz)
Figure 7.5: Power in the ±1 MHz baseband bins for different cavity settings. The
test tone was mixed down to -1 MHz, but leakage into the 1 MHz bin was consistently
observed. The fluctuations in surrounding bins were at the level of (9(10~14).
N oise S tatistics
The frequency bins outside of the 3 dB bandwidth of the cavity are considered to have
purely background noise, as an axion signal would be much suppressed for these offresonant bins. 65 of these background bins were tracked over 261 traces (after baseline
correction) to evaluate the distribution of the excess noise power. Figure 7.6 shows
the measured distribution with a Gaussian fit. The radiometer equation predicts that
92
the width of this Gaussian should be given by the system noise tem perature Tsys and
the number of averages N in each trace:
a = Tsys/ V N = 27 K /V 5 x 105 - 0.038 K,
(7.2)
and from the measured width of aP — 2.45 x 10^13 mW /Hz, dividing by the system
gain and Boltzmann’s constant yields cr™eas — 0.039 K, consistent with the expected
value. As the statistics of the measured noise are seen to agree with the expected
Gaussian noise, an initial uncertainty is assigned to each bin j of magnitude Oj —
P j j \ f N . where P, is the value of the smoothed output power for the jt h bin.
, -le l2
0 1 - 1 6 - 1 1 - 3 0 - 2 8 : ;i = 1.368581, - 1 7 , a = 2 .4 4 6 7 8 4 r-13
1.8 I----------------!----------------1---------------- 1---------------- 1------1.6
data
-
1.4
1.2
_ 1.0
c
o
u 0.8
0.6
0.4
0.2
0.0
-
1.0
-0 .5
0.0
0.5
1.0
1.5
le-12
Figure 7.6: Histogram of power fluctuations shows a Gaussian distribution, with
standard deviation a = 0.039 K after conversion to thermodynamic temperatures.
The fluctuations of the noise power versus number of averages should also obey
the radiometer equation by decreasing as the square root of the number of aver­
ages 1/ x/iV; this holds until some systematic noise floor is reached, which limits the
93
ultimate sensitivity of the experiment. Figure 7.7 shows the logarithm of the com­
puted standard error as a function of the number of averaged traces (after removing
the baseline) for a five hour dataset when the temperatures were stable. For this
integration time a systematic noise floor has not yet been reached.
rsj
T.
§
j=
Sj
\A
C
2
03
3
U
3
c
o
>
<
D
Tk3.
fU
X
J
C
0
3
■w
10
a
LD
N u m b e r of A verages
Figure 7.7: Standard error of the mean versus number of averages. No systematic
noise floor is seen for a five hour integration time.
7.3
Combining Power Spectra
Lorentzian C orrection
A potential axion signal will be attenuated by the Lorentzian response of the cavity
if it does not occur exactly on resonance; to remove this factor we weight the bins
94
with frequency / by the inverse response 1/ h(f ), where:
A(/) i + 4(/-/„)2/r2’
( 7.3 )
/o is the cavity resonance and T the bandwidth of the response. This weighting is
applied to the uncertainty <jj of each bin as well. After this weighting, the signal
height will be independent of the cavity resonance position. Figure 7.8b shows the
effect of dividing out the cavity Lorentzian. The entire spectrum is shown including
the background bins to show the effect for far off-resonance bins.
C o-adding Pow er Spectra
To combine bins at the same frequency from different datasets (indexed by r), a
weighted average of the power values is performed with weights wr given by the
inverse uncertainty squared:
1 1
A = Z r~ r
(7.4)
(7.5)
<7r2
where A normalizes the weights to unity. This weighting gives the maximum likeli­
hood of the mean for values with non-uniform uncertainties and is the same method
used in the ADMX experiment for data analysis [139,157]. The physical intuition
behind doing the weighted arithmetic mean is that it favors bins which have smaller
uncertainties. The uncertainty for combined bins at frequency / is a j and is given by
2 v'' 2 2
O f = h r w r o r.
(7.6)
The logarithmic histogram of the power fluctuations in the co-added power spec­
trum is compared with a simulation done by adding Gaussian noise to the baseline
95
05 - 16 - 12 - 15-04
le-1 4
N
r
!
0
_1____________ 1____________I____________ L.
-4
-
-3
2
-
1
0
1
2
Freq Offset (MHz) from 3 4 .2 6 5 GHz
(a) Baseline subtracted trace.
05-16-12-15-04
le-1 3
-
4
-
3
-
2
-
1
0
1
2
3
Freq Offset (MHz) from 3 4 .2 6 5 GHz
(b) Lorentzian response removed.
Figure 7.8: Lorentzian response weighting. Far off-resonance bins are included to
show the response of the lorentzian correction.
96
curves and running it through the data analysis pipeline in the same manner as the
real data. Both the real data and simulation have a Gaussian distribution; however
the distribution of the actual data is slightly narrower than th at of the simulation.
The same effect was seen in [157] and deemed to be due to the baseline removal.
H isto g ram o f C o -ad ded Power S p ectru m
3 d ata
3 s im u lated
—
'
4
C
3O
-0.8
-0.6
-0.4
-0.2
0.0
0.2
r>T(m W /Hz)
0.4
0.6
0.8
1.0
le -1 3
Figure 7.9: Histogram of power fluctuations in the total co-added power spectrum.
Running Gaussian noise through the analysis pipeline produces the distribution in
green; it is slightly wider than the observed distribution (blue).
7.4
Threshold
A threshold is defined for each bin with frequency / in the co-added spectrum to be
at 3 (7/ and rescans are done at all points where the excess power in the bin is above
the threshold. There were thirty candidates that exceeded the threshold; the number
of candidates n expected for Gaussian noise for a threshold of p sigma and looking at
which for the 18345 bins in our spectrum gives n = 25. Figure 7.10 shows the co-added
spectrum with the threshold marked in red.
30 ca n d id a te s
1 33.8
33.9
34.0
34.1
34.2
34.3
34.4
34.5
34.6
Frequency (GHz)
Figure 7.10: Co-added power spectrum in blue: 3cr threshold in red. There are 30
candidates that surpass the threshold.
If the bin of interest in the rescan does not pass the threshold, we conclude that
the candidate was due to noise. There were three candidates that persisted after the
first round of rescans. Two of these had peaks at 2.27 MHz and the -2.27 MHz bin
in these power spectra also showed an excess of similar magnitude. The appearance
of a peak in both the ±2.27 MHz bins with similar amplitude is suggestive of pickup
in the cables carrying the I and Q signals. Figure 7.11 shows the full spectrum of
the original run in which a candidate appeared and the rescan, both of which show a
peak at ±2.27 MHz. A second round of rescans was done - in order to avoid the 2.27
MHz signal at the bin of interest runs were taken with the first local oscillator shifted
98
Table 7.1: Candidates
Frequency (GHz)
33.922278828
33.9821087504
34.032823028
34.0330951368
34.0335713272
34.0352720072
34.0439794888
34.0585713232
34.0614284656
34.0632652
34.0662583968
34.1129930832
34.1426189288
34.1448298128
34.1449318536
34.1458502208
34.1463944384
34.1475168872
34.1725508968
34.1742515768
34.2082651768
34.2452719736
34.2590814952
34.3315644768
34.3332651568
34.338265156
34.3485712768
34.3699998448
34.5108501624
34.5175508416
34.5192515216
PSD (mW/Hz)
2.6433062220
4.01889850759e-14
3.9813823045e-14
3.57069525293e-14
2.44207326219e-14
1.88003796249e-14
4.02448855563e-14
2.25485631512e-14
3.28436642257e-14
3.16572898809e-14
2.87516371617e-14
1.59198873493e-14
4.16770641435e-14
2.30593607493e-14
2.03901772311e-14
1.92768292095e-14
1.70908654746e-14
2.22833996376e-14
2.90983734422e-14
7.20190044285e-14
3.08782413537e-14
2.28537708519e-14
2.38090356895e-14
1.33971266806e-14
8.30755776663e-15
1.99348483186e-14
2.06439988028e-14
2.39983850695e-14
3.96664566965e-14
1.57719161032e-14
1.30334171873e-14
99
Threshold
2.56177873618e-14
3.75779859252e-14
3.22466144484e-14
3.19619691831e-14
1.94152529304e-14
1.82006596004e-14
2.21003268442e-14
2.0484659532 le -14
2.84855312922e-14
2.2479112676e-14
2.33616496463e-14
1.3809913528e-14
3.78242771001e-14
1.91897857856e-14
1.85352212317e-14
1.62473638403e-14
1.69355141309e-14
2.1356105071e-14
2.24914699629e-14
2.09515421745e-14
2.68277290415e-14
1.82106093843e-14
2.05178958959e-14
9.12957448161e-15
5.97072478179e-15
1.88348838013e-14
1.91898996061e-14
2.32990970718e-14
3.1224174159e-14
1.56830081055e-14
1.2418264003le-14
by 200 kHz, with the cavity frequency shifted by 1 MHz from the original setting,
and with both the cavity and first local oscillator shifted. The peaks did not persist
in the bin of interest after the second round of rescans nor were they seen at 2.27
MHz in the baseband of the second rescans.
le - 14__________
0 5 -1 5 -2 3 -5 0 -0 2 _______
F re q O f f s e t
(MHz)
fro m
.
u,
l e _ 13
34.061 GHz
.
0 7 -0 9 -0 1 -0 6 -5 8
Fr« !
(a) Signal seen in original runs
-
(MHz) from 34.061 GHz
(b) Rescan at the same setting
Figure 7.11: Environmental signal. When investigating the peaks which persisted
after the first round of rescans, a spike was observed in the negative frequency bin as
well. This suggested that the signal arose after the mixers. The first round of rescans
also showed the signal in both the positive and corresponding negative frequency bin.
A second round of rescans with the first local oscillator detuned no longer showed an
excess in the bin corresponding to the RF frequency of interest.
From this we can set an exclusion limit on the strength of the interaction a —¥ 7 7
as well as the kinetic mixing parameter for hidden photon - photon oscillations \ . as
outlined in Chapter 8 .
100
Chapter 8
Exclusion Limit
As no statistically significant signals persisted after rescans, we set an exclusion limit
based on the defined threshold. The procedure is the same for the ALP and hidden
photon case; we discuss the procedure and results for the ALP case in Section 8.1
and translate this to results for the hidden photon case in Section 8.2.
8.1
A xion Bounds
The confidence level was determined by finding the probability of a signal with power
per passing the 3 cr threshold:
( 8 . 1)
so a signal with power 4.3cr would be detected with 90% confidence. As two scans
were taken, the original runs and the rescans, it is more accurate to obtain the total
confidence of 90% by requiring that the signal pass the cut in each run with probability
95% (.952 ~ .9). In this case the analysis would be sensitive to a signal power at the
level 4.75er for an overall confidence of 90%.
After determining the signal power for a given confidence level, the next step to
deriving an exclusion limit was to correct for a loss in signal amplitude due to possible
101
spectral leakage. This leakage is a consequence of the discrete Fourier transform
operating on finite time ml30easurements, with the effect being that some fraction
of the signal power will leak into the surrounding bins if the frequency of a signal
does not equal the center frequency of the closest bin (for an analysis of of spectral
leakage with square windowing, see [134,139]). In the worst case 40.5% of the power
is lost for a narrow signal on the boundary between two bins. A two bin analysis was
slightly more sensitive than a 1-bin search, as in the worst case 81% of the power in
a narrow peak remained in any given two-bin.
The final step of the analysis was to convert the power level of 4.75cr (corrected
for spectral leakage) and convert this to an expected ALP-photon coupling gail using
Equation 5.32. The exclusion limit of this work for a two bin analysis (90% total
C.L.) is shown in Figure 8.1 in blue; the CAST exclusion limit is in gray for reference.
Exclusion Limit: 2 bin search
--------------- 1--------------0.0 I--------------- 1--------------- 1--------------- 1--------------- ■
33.9
34.0
34.1
34.2
34.3
34.4
34.5
F r e q u e n c y (GHz)
Figure 8.1: Exclusion limit from this work for a two bin analysis shown in blue. For
comparison the limit from the CAST experiment is in gray.
102
8.2
Hidden P hoton Bounds
We can use the same data set to restrict the coupling of hidden photons with masses
140.2 — 142.7 /reV. The limit is shown in Figure 8.2, labeled DM YMCE, with the
estimate th at the average value of cos2 9 was 0.0025.
- K—| . . . j r "| t—|—i—|—i—...... r.. ji"pb
Excluded
10
7
4
!
2
lo g m oY ]
Figure 8.2: Exclusion limit for hidden photons.
103
Chapter 9
Conclusions
Sum m ary
The nature of dark m atter remains a mystery. In addition, the mechanism causing CP
invariance in the strong force is not understood. Chapter 2 showed that axions are a
compelling solution to the strong CP problem; the general mechanism of anomalous
symmetry breaking can provide low-mass bosons associated with new symmetries.
These light particles can be produced non-thermally in the early universe and thus
make good cold dark m atter candidates (Chapter 3). There is a rich phenomenology
associated with light particles that couple to photons, and Chapter 4 reviewed the
bounds from cosmology and astrophysics on the interaction strength with photons, as
well as limits from direct search techniques. The signal power for the haloscope search
method was derived in Chapter 5, showing that the sensitivity decreased for high mass
resonant searches. Chapters 6 and 7 were the heart of this work and detailed the ex­
periment and data analysis. Chapter 8 showed the resulting exclusion limits. The
work done in this dissertation was the first demonstration of the haloscope technique
in the ICC4 eV mass range, showing that the small table-top microwave cavity setup
was able to improve upon the CAST limits in a narrow frequency band and confirm
improved globular cluster limits.
104
Lessons Learned
YMCE's pilot run covered 600 MHz (1.7% fractional coverage) in seven months of
data taking. Several improvements could be made to increase the amount of useful
data and optimize the run procedure. First of all, portions of the experiment were
manually operated, such as tuning the cavity resonance and adjusting the first local
oscillator.
Automating these steps would decrease the chances of operator error.
As well, for the initial analysis the raw time domain data was saved and processed
later - having a real time analysis would be greatly beneficial. This would not only
speed up the d ata reduction but allow for immediate feedback if frequency drifts
occurred or mistakes were made in the receiver settings. Temperature stability was
a serious issue (discussed in detail in Chapter 7), especially during the first half of
the data taking, and as a result a significant portion of the data taken had to be
discarded. Stability improved somewhat after we began pumping out the cryostat
vacuum regularly; however using a different cryostat with larger reservoirs would
help, as each refill disturbed the cryogenic system.
Minor modifications to the current haloscope setup could increase the sensitivity
by roughly a factor of three; these adjustments could be conceivably implemented
for a second generation run of the experiment. The cavity geometry was designed
with a conservative aspect ratio to keep other modes well separated from the TM 020
resonance (see Chapter 6 ); by using a longer cavity and thus increasing the overall
volume, one would obtain a higher signal to noise ratio. Working at lower frequencies
(11-20 GHz) would also allow for a larger volume while still being complementary to
other haloscope experiments. This would require changing the signal waveguides to
coax cable and using a different first stage mixer. All together these modifications
would give modest gains to the experimental sensitivity due to reduction in the noise
tem perature of the amplifiers at the lower frequency, the larger volume, and the
105
slightly increased axion number density p / m a at the smaller mass.
While the haloscope experiments are important as a means of probing dark m at­
ter ALP and hidden photon parameter space, an order of magnitude increase in
sensitivity would require technology 1 which is at present not developed for Ka-band
frequencies. In contrast, a microwave light-shining-through-wall search for hidden
photons (see Section 4.1) holds promise as a means of probing a large region of un­
explored parameter space and achieving an order of magnitude improvement over
previous results. The expected signal power increases with the source power sent
to the generation cavity, where the conversion of photons to hidden photons would
occur. By increasing this input microwave source power and making an extremely
narrowband (O(pHz) frequency resolution) measurement of the noise power from the
detection cavity, the sensitivity could be improved to surpass the solar longitudinal
limits (Section 4.2). As this setup does not require a magnet, could be operated at
room temperature, and does not require the cavity to be tuned, operating the exper­
iment is considerably simpler. However to reach the allowed region for dark m atter
hidden photons, superconducting cavities would most likely be needed to boost the
sensitivity. This would complicate the setup once more by requiring cryogenic cooling.
C onclusions
In conclusion, microwave cavity experiments provide a unique way of probing the
peV - meV parameter space of axions, ALPs, and hidden photons, and are the only
demonstrated experimental technique to reach the allowed dark m atter regions for
these particles. Haloscope experiments have extremely good sensitivity compared
to other detection strategies but are narrowband, so multiple experiments must be
performed to cover the entire parameter space for dark m atter particles. The work
1.
Proposed ideas to increase the quality factor or decrease the noise tem perature are mentioned
in C hapter 5.
106
presented here used amplifier technology that was state of the art for this frequency re­
gion. However improvements are necessary to probe the axion model band, which has
couplings at 10-4 eV roughly three orders of magnitude weaker than those excluded
in this dissertation. Just as in the ADMX experiment, where research was done to
produce RF-frequency quantum amplifiers that significantly lowered the noise floor
from previous HEMT-based runs, lower noise amplifiers will be required at Ka-band
frequencies to improve the sensitivity of these high-frequency haloscope searches. This
by itself will not be enough to reach the axion model band. Thus it is im portant to
explore all strategies for optimizing the relevant experimental parameters (cavity ge­
ometry, amplifier noise, quality factor, etc.) in order to develop QCD axion-sensitive
haloscope experiments near 10-4 eV.
107
Appendix A
Cavity Assembly Drawings
*>RESS-FlT
MIL-MAX SOCKET INTO ASSY FO LL O W IN G SOLDER
SECTION A -A
SOLDER
SOLDER
Figure A.l: Bellows-sub-assembly
108
SOLDER
Arv •//
SOLDER
Figure A.2: Cavity body and waveguides
'6 o IN
O
5z
i
o
>
>
t
Figure A.3: Cavity body
109
Appendix B
Dead Spot
Between 33.368 and 33.398 GHz the cavity resonance disappears. We initially believed
this to be due to a mode crossing, even though no other modes are visible, and though
it might be due to a mode that usually didn’t couple to the waveguide being excited.
However, the cause turned out to be the extremely weak coupling of the calibration
port suppressing the mode at that frequency.
However after adding the metal shim in the bottom of the waveguide, the cavity
resonance re-appears in this region.
110
SI? i
II,
M ? OfUMB
76 n,!!'.
66.00
•
68.00
•
7 0
•
0 0
7 8 ,0 0
-80,00
82.00 •
84.00
Stop 34.4105 GHz
C hi: Start 34.3900 GHz
(a) The cavity response is suppressed.
S I ? I iiijM ?.i.)i'0i]B
/G.OitH,
66 00
68.00
•
70.00 •
7 6 .0 0
■ 7 8 .0 0
► -------------------------------------------
^
4
#
- - - -- - -- - -- - ----------------
80.00
82.00 •
8 6 .0 0
1
C l.
1
A v i|
Stop 34.3780 GHz
-Chi: Start 34.3580 GHz
(b) Resonance severely suppressed. Feb 05, 2014.
Figure B.l: Cavity response without shim.
I ll
30 00
/'i-'t'f I
■20 j o ? d f j
32.00
34.00
-36.00
38 00 ►
40.00
44.00
46.00
48.00 i Cli 1 Avijj
1
20
Slop 34 3800 Gl i t
■Ch1: Start 34.3600 GHz —
S
1
I
-ii | M
1 O f il l i l f
3/
* I I .tt:H M r l *
4 i ! R
|
28.00
:
I lllj
'v4.3f)-1 !0 ;
,
HW:
c. toiler.
G:
1 j s :>:
Ml.: 4:
t
.X.
30.00
14
S
.. . . . . . . . . . . . . . . . . . . . . ! /
32.00
i
34.00
\
i
/'■ ]
r/
36.00
%
\
.4
/*
\
J
h
38 001
f
J
4000
? ii?o 7 ,iH
4'>364 Mi l /
34.364 '.it 1.
/'./ u
•2, l 30? <H'
\
/I
.
gh
j
\
K
1
’X
1 "'
_
y
4 2 .0 0
v _
:
j r
|
44.00
46.00
:
I Gh 1 AvtJ
:
i
;
:
:
;>()
Stop 34 3740 Gl 1/
-Ch1: SrHit 34 3M 0 GM/ —
Figure B.2: Cavity response with shim
112
Bibliography
[1] F. Zwicky. APJ 8 6 , 217 (1937).
[2] V. C. Rubin, W. K. J. Ford, and N. . Thonnard, APJ 238, 471 (1980).
[3] D. N. Spergel et al., A PJ 148, 175 (2003), astro-ph/0302209.
[4] Planck Collaboration, P. Ade et al., ArXiv e-prints (2013), 1303.5076.
[5] R. H. Cyburt, Phys. Rev. D 70, 023505 (2004), astro-ph/0401091.
[6 ] SDSS Collaboration, D. J. Eisenstein et al., Astrophys.J. 633, 560 (2005),
astro-ph/0501171.
[7] C. S. Frenk and S. D. M. White, Annalen der Physik 524, 507 (2012 ). 1210.0544.
[8 ] J. H. Oort, Ann. Rev. of Astronomy and Astrophysics 21, 373 (1983).
[9] M. Davis. G. Efstathiou, C. S. Frenk, and S. D. M. White, APJ 292, 371
(1985).
[10] S. P. Martin,
Perspectives On Supersymmetry. Series: Advanced Series on
Directions in High Energy Physics 18, 1 (1998), hep-ph/9709356.
[11] K. Nakamura and P. D. Group, Journal of Physics G: Nuclear and Particle
Physics 37, 075021 (2010).
113
[12] M. Kamionkowski, WIMP and Axion Dark M atter, in High Energy Physics
and Cosmology, 1997 Summer School, edited by E. Gava et al., p. 394, 1998,
hep-ph/9710467.
[13] (LUX Collaboration), D. Akerib et al., Phys. Rev. Lett. 112, 091303 (2014).
[14] T. R. Slatyer, N. Padmanabhan, and D. P. Finkbeiner, Phys. Rev. D 80, 043526
(2009).
[15] A. Rajaraman, W. Shepherd, T. M. Tait, and A. M. Wijangco, Phys. Rev.
D 84, 095013 (2011), 1108.1196.
[16] ATLAS, CMS collaborations, S. Lowette et al,
ArXiv e-prints (2012 ),
1205.4053.
[17] CRESST Collaboration et al, ArXiv e-prints (2014), 1407.3146.
[18] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).
[19] S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
[20] F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
[21] H. Murayama, G. Raffelt, C. Hagmann, K. van Bibber, and L. Rosenberg. The
European Physical Journal C - Particles and Fields 15, 298 (2000).
[22] G. Gelrnini and M. Roncadelli, Physics Letters B 99, 411 (1981).
[23] P. Svrcek and E. W itten, JHEP 0606, 051 (2006), hep-th/0605206.
[24] L. Okun. Sov. Phys. JE T P 56 (1982).
[25] S. Abel, M. Goodsell, J. Jaeckel, V. Khoze, and A. Ringwald, Journal of High
Energy Physics 2008, 124 (2008).
[26] J. Jaeckel, ArXiv e-prints (2013), 1303.1821.
114
[27] A. E. Nelson and J. Scholtz, Phys. Rev. D84, 103501 (2011), 1105.2812.
[28] P. Slocum et al., Search for a scalar axion-like particle at 34 ghz, in Conf. Proc.
DESY-PROC-2012-04, 2012 .
[29] P. Slocum et al., NIM (submitted for publication).
[30] H. Primakoff, Phys. Rev. 81, 899 (1951).
[31] J. Bell and R. Jackiw, II Nuovo Cimento A 60, 47 (1969).
[32] S. L. Adler, Phys. Rev. 177, 2426 (1969).
[33] R. A. Bertlmann,
Anomalies in quantum field theory (Oxford; New York:
Clarendon Press, 1996).
[34] J. K. Vogel and K. C. Knigsmann, Searching for Solar Axions in the eV-Mass
Region with the CCD Detector at C A ST, PhD thesis, Freiburg U.. Freiburg,
Germany, 2009, Presented on 20 May 2009.
[35] W. A. Bardeen and S.-H. Tye, Physics Letters B 74. 229 (1978).
[36] K. Gottfried and V. Weisskopf, Concepts of Particle Physics (Ebsco Publishing,
1986).
[37] J. Christenson, J. Cronin, V. Fitch, and R. Turlay, Phys. Rev. Lett. 13, 138
(1964).
[38] G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976).
[39] G. Luders, Annals of Physics 2, 1 (1957).
[40] A. Belavin, A. Polyakov, A. Schwartz, and Y. Tyupkin, Physics Letters B 59,
85 (1975).
[41] C. Jarlskog, editor, CP Violation (World Scientific, Singapore. 1988).
115
[42] C. Baker et al., Phys. Rev. Lett. 97. 131801 (2006), hep-ex/0602020.
[43] G. Hooft,
Recent developments in gauge theoriesNATO ASI series: Physics
(Plenum Press, 1980).
[44] D. R. Nelson, Partially Quenched Chiral Perturbation Theory and a Massless
Up Quark: A Lattice Calculation of the Light-Quark-Mass Ratio, PhD thesis,
Ohio State University, 2002 .
[45] D. R. Nelson, G. T. Fleming, and G. W. Kilcup, Phys. Rev. Lett. 90, 021601
(2003).
[46] J. Gasser and H. Leutwyler, Physics Reports 87, 77 (1982).
[47] K. Choi, C. W. Kim, and W. K. Sze, Phys. Rev. Lett. 61, 794 (1988).
[48] S. M.
Barr and A. Zee, Phys. Rev. Lett. 55, 2253
(1985).
[49] Y. B. Zel’dovich, I. Kobzarev, and L. Okun, Sov. Phys. JE T P 40 (1975).
[50] I. Bigi and A. Sanda, CP Violation (Cambridge University Press, 2000).
[51] A. S. Chou, ArXiv e-prints (2009), 1009.4718.
[52] G. G. Raffelt, Stars as Laboratories for Fundamental Physics (University of
Chicago Press, 1995).
[53] D. Cadamuro.
Cosmological limits on axions and axion-like particles, PhD
thesis, PhD Thesis, 2012, 2012.
[54] R. D. Peccei and H. R. Quinn, Phys. Rev. D 16, 1791 (1977).
[55] D. Antreasyan et al., Physics Letters B 251, 204 (1990).
[56] A. Zehnder, K. Gabathuler. and J.-L. Vuilleumier, Physics Letters B 110, 419
(1982).
116
[57] F. De Boer et al., Phys. Lett. B180, 4 (1986).
[58] I. Antoniadis and T. N. Truong, Phys. Lett. B109, 67(1982).
[59] J. E. Kim, Phys. Rev. Lett. 43, 103 (1979).
[60] M. Shifman, A. Vainshtein, and V. Zakharov, Nuclear Physics B 166, 493
(1980).
[61] M. Dine, W. Fischler, and M. Srednicki, Physics Letters B 104, 199 (1981).
[62] A.
Zhitnitsky, Sov. J. Nucl. Phys. 31, 260 (1980).
[63] D.
B. Kaplan, Nuclear Physics B 260, 215 (1985).
[64] H. M. Georgi, L. J. Hall, and M. B. Wise, Nuclear Physics B 192, 409 (1981).
[65] M. Kamionkowski and J. March-Russell,
Phys.Lett. B282, 137 (1992), hep-
th/9202003.
[66 ] P. Sikivie, Phys. Rev. Lett. 48, 1156 (1982).
[67] P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983).
[68 ] D. J. E. Marsh. Phys. Rev. D 83, 123526 (2011), 1102.4851.
[69] E. Masso, F. Rota, and G. Zsembinszki,Phys. Rev. D 70, 115009
(2004).
hep-ph/0404289.
[70] L. Abbott and P. Sikivie, Physics Letters B 120, 133(1983).
[71] J.
Preskill, M. B. Wise, and F. WTilczek, Physics Letters B 120, 127 (1983).
[72] D. J. Gross. R. D. Pisarski, and L. G. Yaffe, Rev.Mod. Phys. 53, 43 (1981).
[73] M. S. Turner. Phys. Rept,. 197, 67 (1990).
117
[74] D. Seckel and M. S. Turner, Phys. Rev. D 32, 3178 (1985).
[75] A. Vilenkin and A. E. Everett, Phys. Rev. Lett. 48, 1867 (1982).
[76] R. Davis, Physics Letters B 180, 225 (1986).
[77] A. Vilenkin and T. Vachaspati, Phys. Rev. D 35, 1138 (1987).
[78] T. Hiramatsu, M. Kawasaki, K. Saikawa, and T. Sekiguchi, Phys. Rev. D 85,
105020 (2012).
[79] D. Harari and P. Sikivie, Physics Letters B 195, 361 (1987).
[80] R. A. Battye and E. P. S. Shellard, Recent Perspectives on Axion Cosmol­
ogy, in Dark matter in Astro- and Particle Physics, edited by H. V. KlapdorKleingrothaus and Y. Ramachers, p. 554, 1997, astro-ph/9706014.
[81] A. Vilenkin and E. Shellard,
Cosmic Strings and Other Topological Defects
(Cambridge University Press, 2000).
[82] B. Holdom, Phys. Lett. B166, 196 (1986).
[83] P. Arias et al., JCAP 1206, 013 (2012), 1201.5902.
[84] E. Masso and R. Toldra, Phys. Rev. D52, 1755 (1995), hep-ph/9503293.
[85] E. Masso and R. Toldra, Phys. Rev. D 55, 7967 (1997).
[86 ] K. Sato and H. Sato, Progress of Theoretical Physics 54, 1564 (1975).
[87] A. Buzzoni, F. F. Pecci, R. Buonanno, and C. E. Corsi, Astronomy and Astro­
physics 128. 94 (1983).
[88 ] G. G. Raffelt and D. S. P. Dearborn, Phys. Rev. D 36, 2211 (1987).
118
[89] G. G. Raffelt, Astrophysical Axion Bounds, in Axions, edited by M. Kuster,
G. Raffelt, and B. Beltran, , Lecture Notes in Physics, Berlin Springer Verlag
Vol. 741, p. 51, 2008, hep-ph/0611350.
[90] A. Friedland, M. Giannotti, and M. Wise, Physical Review Letters 110, 061101
(2013), 1210.1271.
[91] A. Ayala, I. Dominguez, M. Giannotti, A. Mirizzi, and O. Straniero, Arxiv
e-prints (2014), 1406.6053.
[92] A. H. Corsico, O. G. Benvenuto, L. G. Althaus, J. Isern, and E. Garcia-Berro,
New Astro. 6, 197 (2001), ast,ro-ph/0104103.
[93] R. M. Bionta et al., Phys. Rev. Lett. 58, 1494 (1987).
[94] K. Hirata et al., Physical Review Letters 58, 1490 (1987), cited By (since
1996)621.
[95] A. Burrows, M. T. Ressell, and M. S. Turner, Phys. Rev. D 42, 3297 (1990).
[96] J. Engel, D. Seckel, and A. C. Hayes, Phys. Rev. Lett. 65, 960 (1990).
[97] M. T. Ressell, Phys. Rev. D 44, 3001 (1991).
[98] D. Grin et a l, Phys. Rev. D. 75. 105018 (2007), astro-ph/0611502.
[99] D. Cadamuro and J. Redondo,
Journal of Physics: Conference Series 375.
022002 (2012).
[100] J. Jaeckel, M. Jankowiak, and M. Spannowsky, Physics of the Dark Universe
2, 111 (2013), 1212.3620.
[101] CAST Collaboration, M. Arik et al., Phys. Rev. Lett. 107, 261302 (2011).
[102] J. K. Vogel et a l, ArXiv e-prints (2013), 1302.3273.
119
[103] K. Ehret et al., Physics Letters B 689, 149 (2010).
[104] M. Sulc et al., Nuc. Instr. and Methods A 718, 530 (2013), Proceedings of the
12th Pisa Meeting on Advanced Detectors La Biodola, Isola d ’Elba, Italy, May
20-26, 2012 .
[105] GammeV (T-969) Collaboration, A. S. Chou et al.,
Phys. Rev. Lett. 100,
080402 (2008), 0710.3783.
[106] M. Betz, F. Caspers, M. Gasior, M. Thurnm, and S. Rieger, Phys. Rev. D88,
075014 (2013), 1310.8098.
[107] E. Zavattini et al, Phys. Rev. D. 77, 032006 (2008), 0706.3419.
[108] S. J. Asztalos et al, Phys. Rev. Lett. 104, 041301 (2010).
[109] K. Yamamoto and et al., The Rydberg-Atom-Cavity Axion Search, in Dark
Matter in Astro- and Particle Physics, edited by H. V. Klapdor-Kleingrothaus,
p. 638, 2001, hep-ph/0101200.
[110] K. van Bibber and G. Carosi, ArXiV e-print (2013), 1304.7803.
[111] J. E. Moody and F. Wilczek, Phys. Rev. D 30, 130 (1984).
[112] S. A. Hoedl, F. Fleischer, E. G. Adelberger, and B. R. Heckel, Phys. Rev. Lett.
106, 041801 (2011).
[113] B. R. Heckel, W. A. Terrano, and E. G. Adelberger, Phys. Rev. Lett. 111 .
151802 (2013).
[114] K. Tullney et al., Phys. Rev. Lett. I l l , 100801 (2013). 1303.6612.
[115] P. W. Graham and S. Rajendran, Phys. Rev. D 8 8 , 035023 (2013). 1306.6088.
120
[116] D. Budker, P. W. Graham, M. Ledbetter, S. Rajendran, and A. 0 . Sushkov,
Phys. Rev. X 4, 021030 (2014).
117 J. Redondo and B. Dobrich, ArXiv e-prints (2013), 1311.5341.
118 L. Danese and G. de Zotti, Astron. and Astrophys. 107, 39 (1982).
119 D. J. Fixsen et al., APJ 473, 576 (1996), astro-ph/9605054.
120 S. Hannestad, A. Mirizzi, and G. Raffelt, JCAP 7, 2 (2005), hep-ph/0504059.
121 J. Redondo, JCAP 7, 8 (2008), 0801.1527.
122 H. An, M. Pospelov, and J. Pradler, Phys. Rev. Lett. I l l , 041302 (2013),
1304.3461.
123 R. Essig, A. Manalaysay, J. Mardon, P. Sorensen, and T. Volansky, Physical
Review Letters 109. 021301 (2012), 1206.2644.
124 H. An, M. Pospelov, and J. Pradler, Phys.Lett. B725, 190 (2013), 1302.3884.
125 M. Ahlers, H. Gies, J. Jaeckel, J. Redondo, and A. Ringwald, Phys. Rev. D.
76. 115005 (2007). 0706.2836.
126 J. Jaeckel and A. Ringwald. Physics Letters B 659, 509 (2008), 0707.2063.
127 P. W. Graham, J. Mardon, S. Rajendran, and Y. Zhao, ArXiv e-prints (2014),
1407.4806.
128 R. Seviour, I. Bailey, N. Woollett, and P. Williams,
Journal of Physics G
Nuclear Physics 41, 035005 (2014).
129 J. Einasto and U. Haud, AAP 223, 89 (1989).
130 L. J. Rosenberg and K. A. van Bibber, Physics Reports 325, 1 (2000).
121
[131] P. Sikivie, Physics Letters B 695, 22 (2011), 1003.2426.
[132] C. W. Purcell, J. S. Bullock, and M. Kaplinghat, A PJ 703, 2275 (2009).
[133] M. Hotz, A SQUID Based R F Cavity Search for Dark Matter Axions, PhD
thesis, University of Washington, 2012.
[134] L. Duffy, High Resolution Search for Dark Matter Axions in Milky Way Halo
Substructure, PhD thesis, University of Florida, Gainesville, 2012.
[135] C. Moni Bidin, G. Carraro, R. A. Mendez, and W. F. van Altena, A PJ 724,
L122 (2010), 1011.1289.
[136] J. I. Read, Journal of Physics G Nuclear Physics 41, 063101 (2014), 1404.1938.
[137] J. Bovy and S. Tremaine, APJ 756, 89 (2012), 1205.4033.
[138] R. H. Dicke, Review of Scientific Instruments 17, 268 (1946).
[139] E. J. Daw, A search for halo axions, PhD thesis, MIT, 1998.
[140] L. Krauss, J. Moody, F. Wilczek, and D. E. Morris, Phys. Rev. Lett. 55, 1797
(1985).
[141] H. A. Haus and J. A. Mullen, Phys. Rev. 128, 2407 (1962).
[142] C. M. Caves, Phys. Rev. D 26, 1817 (1982).
[143] A. A. Clerk, M. H. Devoret. S. M. Girvin, F. M arquardt, and R. J. Schoelkopf,
Rev. Mod. Phys. 82, 1155 (2010).
[144] D. S. Kinion, First Results from a Multiple-Microwave-Cavity Search for DarkMatter Axions. PhD thesis, UC Davis, 2001.
[145] G. Rybka and A. Wagner, ArXiv e-prints (2014), 1403.3121.
122
[146] S. K. Lamoreaux, K. A. van Bibber, K. W. Lehnert, and G. Carosi, Phys. Rev.
D 8 8 , 035020 (2013).
[147] C. Poole, Electron Spin Resonance (Dover Publications, 1996).
[148] J. T. Rogers, Limits on the electromagnetic coupling and density of galactic
axions, PhD thesis, Rochester Univ., Rochester, 1987.
[149] C. A. Hagmann,
A search for cosmic axions,
PhD thesis, Florida Univ.,
Gainesville., 1990.
[150] C. Hagmann, P. Sikivie, N. Sullivan, D. Tanner, and S. Cho, Rev. Sci. Instrum.
61, 1076 (1990).
[151] S. Asztalos et al., Phys. Rev. D 64, 092003 (2001).
[152] A. A. Abrikosov, Fundamentals of the Theory of Metals (Amsterdam, Elsevier
Science Pub, 1988).
[153] D. Kajfez, IEEE transactions on microwave theory and techniques 42, 1149
(1994).
[154] A. Akgiray et al., Microwave Theory and Techniques, IEEE Transactions on
61, 3285 (2013).
[155] M. Frigo and J. S.G., Proceedings of the IEEE (2005).
[156] B. E. Moskowitz and J. Rogers, Nuclear Instr. and Methods A 264, 445 (1988).
[157] D. B. Yu, An improved R F cavity search for halo axions, PhD thesis, MIT,
2004.
[158] A. Savitzky and M. J. E. Golay, Analytical Chemistry 36, 1627 (1964).
123
Документ
Категория
Без категории
Просмотров
0
Размер файла
5 099 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа