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

код для вставкиСкачатьA 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. 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