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Effective Flavor-changing Weak Neutral Current in the Standard Theory and Z-Boson Decay.

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Annalen der Physik. 7. Folge, Band 40, Heft 6, 1983, S. 334-346
J. A. Bsrth, Leipzig
Effective Flavor-changing Weak Neutral Current
in the Standard Theory and 2-Boson Decay
G. MA”
Zentralinstitut fur solar-terrestrische Physik der Akademie der Wissenschaften der DDR, Berlin
T. RIEMANN
Institut fur Hochenergiephysik der Akademie der Wissenschaften der DDR, Berlin-Zeuthen’)
Abstract. We have studied the effective Zfi f,-vertex in the Glashow-Salam-Weinbergtheory
arising from 1-loop corrections, allowing for arbitrary fermion masses miin the loop. We use our
results to study the branching ratio of the 2-boson decay into two different fermions.
This is of phenomenological interest if there exist yet unknown fermions with large mass which
could show up at present and designed accelerators only through radiative corrections.
Effektiver flavor-andernder schwacher neutraler Strom in der Standard-Theorio
und Z-Boson-Zerfall
-
Inhaltsubersicht. Wir untersuchen den durch 1-loop-Korrekturen entstehenden Zf,f.-Vertex
in der Glashow-Salam-Weinberg-Theorie.
Im Loop sind beliebige Fermionmassen zugelassen. Die
Resultate werden benutzt, um das Verzweigungsverhiiltnisfiir den 2-Boson-Zerfall in zwei unterschiedliche Fermionen zu untersuchen. Diese Zerfallsmode ist von phanomenologischem Interesse
bei dcr Suche nach unbekannten Fermionen sehr grol3er Masse, die an derzeitigen und geplanten
Beschleunigern nur durch Strahlungskorrekturen nachweisbar sein werden.
1. Introduction
I n recent years one distinguished theory of electroweak particle interactions was obtained -the GLASHOW-SALAM-WEINBERG
theory [11. One of the interestingfeatureswithin this
theory is that of the non-diagonal weak neutral currents. Being flavor-diagonal in tree
approximation, the weak neutral current obtains a small flavor-violating component
from loop corrections in the presence of symmetry breaking. Earlier studies of the corresponding phenomenological consequences were limited t o the low energy region. I n
refs. [2, 31, e.g., the effective Zd-coupling has been studied in the limit of vanishing
external momenta. Here we allow for a non-vanishing momentum of the weak neutral
current, qi
0. Furthermore, in the effective Zf,f,-coupling we do not specify whether
f, are up- or down-type leptons or quarks in the left handed weak isospin dublets.
Already a rough estimate shows the dependence of the form factor V on the mass
splitting between the fermions f i with mass mi in the loop. With N families of fermions,
+
N
V - 93
1)
2 PziPR V(m?/M&).
i=l
Now: Joint Institute for Nuclear Research, Dubna, USSR.
(1.1)
335
Effective Flavor-changing Weak Neutral Current in the Standard Theory
Unitarity of the KOBAYASHI-MASKAWA
mixing matrix Pii [4] forces V to vanish if
not the masses mi are different2). This reflects the possibility to define a separate,
conserved fermion number for each family if all the up- or all the down-type fermion
masses are equal thus preventing family mixing and, in turn, flavor-violation by the
neutral current ,).
Eq. (1.1) implies rather small predictions for all presently known fermions (mi Hw).
But it opens the possibility to observe effects of rather heavy fermions (mi 3 M,)
which cannot be produced as free particles at present or designed accelerators. These
rather massive fermions could, however, be virtually exchanged in the loop diagrams
describing the Z&fz-vertexenhancing the couplings of this vertex so strongly that the
vertex and, in this indirect way, the massive fermions may be observable.
The article is organized as follows: I n Sect. 2 we shortly discuss renormalization of
the vertex. In Sect. 3 we present the vertex for arbitrary q i and mi in terms of the
3-point functions C,, Cii of 'T HOOFT
and VELTMAN[9, 101 as functions of I, and Q
of the virtual fermion.
Sect. 4 contains an application of the results of Sect. 3-the decay Z -+
f2.
After t h e Discussion we define in App. A the lagrangian and present in App. B some
results on the 3-point functions C,, Cii which may be useful in other contexts too.
We also relate them to the functions I, introduced in ref. [5] for the study of muon
number violation.
<
&+
2. Renormalization
Although there is no independent 2-factor of the Zf,f,-vertex, it gets an infinite
contribution from fermion wave function renormalization.In the on mass shell renormalization program [ll, 121 g and 4 = sinz Ow are fixed as follows:
a, = M ~ / W W&,, = 1 - ozl,g2& = ez = 4 ~ 0 1 .
(2.1)
Diagrams with self-energy insertions on externaI legs give a vanishing contribution
to the matrix element under consideration. Nevertheless, th6 self-energy diagrams of
Fig. 1 must be calculated since they define the non-diagonal part of the lepton wave
function renormalization; and it is this procedure which induces and completely fixes
a counter term for the ZKf2-vertex.
'I
Fig. 1. Self-energy diagrams defining non-diagonal fermion wave function renormalization
+
+
a) The same phenomenon occurs in the processes ,u+ e
y, vl +. v2 y and related ones as
studied, e.g., in refs. [5, 61.
3) VELTMAN'B
observation [7] of the influence of mass splitting within a fermion doublet on the
charged to neutral current ratio in neutrino scattering is another example of large massea showing
up in low energy physics, but there it is due to breaking of weak isospin. All these phenomena demonstrated the limited validity of the decoupling theorem [S] in theories with spontaneously broken
gauge invariance.
G . NANN
and T.RIEMANN
336
For the left-handed fermion fields fz,8we introduce a matrix of 8-factors:
-.
= (zL):p fL,t ( 8 , t = 1, . 9 N ) .
(2.2)
The same must be done for the right-handed fields. Since mass eigenstates are already
introduced, mass renormalization may be treated by
mt = (2L);1’2
m:Zm,8(ZR);1’z (no sum over 8 . )
The following conditions on the regularized self-energies (including counter term contributions) define the 8-factors introduced:
z;,(p) u, = 0 I p s , -m’t
f%,8
9
( 8 , t = 1, . ..,N )
(2.3)
a, &(p) = 0
-m;
(u and a are on mass shell wave functions of fermions.)
The lengthy procedure of solving (2.3) [13] simplifies considerably in the approximation ma = 0 which will be assumed to be valid throughout the paper. Then, ZR,sl- dat
and with no loss of generality one may choose ZL,st= 2z,ra.
Corresponding to the
diagrams of Pig. 1 we get for the nondiagonal elements of the 2-factor matrix
Ips=
where A i= m:/M& and the 2-point function B, is defined in eqs. (Bl-2) of App. B.
I f inserted into the originally diagonal lagrangian, among other contributions the
matrix 8, induces the following counter term of order g3:
Y,, = BZ,,, f,’y”(l
+ 7’)
f:Z‘P
+ h.c.,
(2.5)
% (v a ) (zL)y~
BZ,,, = 2%
with Z, from eq. (2.4) and v, a being the vector and axial vector couplings of the weak
neutral current to f, and f t . The expression eq. (2.5) is the counter term for the flavorchanging weak neutral current we searched for. We stress once again that it is completely
determined by fermion wave function renormalization.
+
3. The Amplitude
Neglecting masses of external (on mass shell) fermions the Z&f,-vertex may be
described by two form factors. With the conventions and Feynman rules of refs. [13,10]
and App. A the vertex is
p
= ix2
$3
-y P ( 17
4C @
+
(3.1)
Fig. 2. Dingrams contributing to the effective Zf,f,-vertes
Effective Flavor-changingWeak Sciitr,tl Current in the Standard Theory
337
I n the (:lashow-Salam-Weinberg theory the vector and axial vector form factors are
equal. Contributions to them come from t h e diagrams of Fig. 2. Using dimensional
regularization, one gets in the 't Hooft-Fepnman gauge
(3.2)
a ) 2ACo]
+ 1" ,(?;+a)The first curly bracket is calculated from the first diagram in Fig. 2, and so on. Frirthermore,
c g = oil,
c
=
-yi/Mf,,,
:7
=
Z(13 - 2 Q 4 ) ,
u = 213.
A i= m:/M&,
(3.4)
(33)
(I,, Q are weak isospin and electric charge of ferinion f i ) . The dimensionless 3-point
functions C = C( l , O, d ) and d = C(d, I, 1)and B, are defined following refs. [9, 101
and discussed in App. B.
The functions C2,, 6, and B, are divergent. While V ( A i ,a) is free of divergencies
proportional to A;, i t has a constant divergent contribution. This constant divergenc.v
vanishes as a result of the (nondiagonal) unitary sum when V(di,.a)is used to calcnlatc
the form factors (3.2)*).
We have compared our numerical results in detail with those of MA and PRAMUI)ITA
[3] for a = 0. This is a good check of (3.3) which expresses the vertex in terms of loop
functions C, 6, Bl.I n numerical computer calculations we have used the analytic
expressions (B. 1-7) of App. B.
The reduction of 3-point functions to C,, 6, and complex logarithms has the advantage of avoiding roughly cancelling large contributions in intermediate steps which
plague other calculational schemes. Furthermore, we made an independent computer
numeric integration of the 3-point functions and the form factor (3.4) and got complete
agreement with the analytic results.
The dependence of V(d ,a) on /I is shown in Fig. 3 for the special choice a = a, =
M i / M & = 1.25 ( M , = 94 GeV, M , % 84 GeV). As long as A < a/4, the vertex has
an imaginary part. For applications the large A behaviour is of special interest. Using
4 ) Concerning renormalization one further comment may be useful. In Sect. 2 we have renormalized using Z-factors. Alternatively, one could add diagrams containing the square root of
full propagators at external fermion legs. This rule leads to the addition of half the non-diagonal
self-energydiagrams to t,he vertex as ha8 been done, e.g., in ref. 131 for the case of zero momentum
transfrr u = 0.
G . MANN and T. RIEMANN
338
eqs. (B. 10-16) of App. B one gets the following approximation:
+ [ - - - -I3-1b +
1
18
8
+ [ - -+
3
-0.-
) 4]
-+-u
(
l
: 8:
2
3
A(y) = ?J arctan (1/2y),
!J
= (1/0 - 1/4)’’!!.
(3.7)
Eq. (3.6) implies a dependence of observable quantities on the fourth power of the large
ferinion masses in the loop 5).
On the 2-boson mass shell the numerical value is
V(O
1, 1.25) 21 13[d
+ 2.84,5 l n d - 4.4!12] + &[-0.04861.
(3.8)
The large A hehaviours of the four vertices (rip- and down-type leptons and quarks)
are rather siniilar to each other; up to an overall sign (of relevance in interference
effects) it is only the numerically unimportant dependence on the charge Q which makes
a difference. So we have drawn in Fig. 3 only two of the vertices. The situation changes
for small A where the vertices split up. For later use we need Y ( 0 ,a) which may be
calculated from (3.3), (B. 17-26). Nunierically,
V ( 0 ,1.25) = (2.498
+ i 2.037) + (-0.234
13
- i 0.411) Q .
(3.9)
The (T has only a non-leading influence on I’ in (3.t;). One may get a further impression
of the smooth dependence of the vertices on (J from the coinparison of (3.8) with
i’(4
I, a = 0 ) .
The general expression (3.3) yields in this limit:
<
V ( A , 0)
=
13[d(0- 10) I l
+ 8L + 61.
(3.10)
This function is also shown in Fig. 3.
1
For I3 = Tit is in accordance with ref. [3].
The functions I,, have been introduced for the study of (low energy) muon number
et al. [ b ]and are discussed together with L in -4pp. B. The form
violation by ALTARELLI
factor V ( A , 0) has no dependence on sin2 @,, or Q as may he seen also from the Ward
identity [el. Especially,
?‘(A 9 I , 0) rr, 13(0+ 3 I n d
- 3)
(3.11)
(which should be compared to (3.8)) and
l*(d < 1,O) rr, 13[2A In A
+ 641.
(3.12)
The dependence on /lIn A for small A is absent if CT =+= 0.
In the next section we will apply our results to the nondiagonal decay of the %boson into fermions.
5) Three diagrams contain terms of order A In A which canccl each other. The remaining leading
term in (3.6) comes from the Higgs diagrams including the counter term.
339
Effective Flavor-changing Weak Neutral Current in the Standard Theory
100
10
1
0.1
10
Fig. 3. Form factor V(d, u), (3.3), of the
effective Zf,f,-vertex as function of the
fermion mass mi. External fermions are
charged leptons (dashed curve) and downtype quarks (solid curve)
100
1000
mi [ GeV 1
Fig. 4. Dependence of the branching ratio
B(f,, f2), (4.4), on the fermion mass mi. External fermions are charged leptons (r = 1;
dashed curve) and down-type quarks ( r = 3;
solid curve)
4. Non-Diagonal 2-Boson Decay
The next generation of accelerators will allow the production of a large number of
2-bosons. This raises interest in all its decay modes, including rather rare ones.
The non-diagonal 2-boson decay is possible through radiative corrections containing
virtual ferinions of necessarily different mass. This allows the search for these fermions
even if they are too heavy to be produced directly. Of course they must have lighter
partners f, with m, < M , and Kobayashi-Maskawa mixing. Then a decay of Z into f,
and f,, is a signal of the existence of the f i . Here we will study this option in more detail
using the results of Chapter 3.
G. M ~ N and
N T.RIEMANN
340
I n order to define a branching ratio from the width (2+ f2
a norinalization :
+
f l ) we
have to choose
The sum over the two charge configurations makes B(f,, f z ) insensitive to CP violation;
neglecting this, the two contributions to the numerator of (4.1) are equal. With (3.1),
the branching ratio becomes
(4.8)
( r = l ( 3 ) for leptons (quarks)). Using sin20rV2: 0.8 and (3.2), one inay estimate the
order of ma.gnittide :
I?,
I N
(4.3)
Another factor of about 2/(11.3 N ) must be taken into account if one wants to relate
(Z-t fl
to the anticipated total decay rate [14]. The resulting number has to be
confronted with the proposed possibility to produce millions of 2-boson decays per
experiment a t a future accelerator like LEl'or SLC [14]. It is evident that there remains
a chance to observe non-diagonal 2-boson decay only if the form factor V(Ai,ax) is
larger (better: much larger) than 1. But this is not enough. What really is necessary is
the splitting between the form factors V ( A i ,az) of different generations.
Constant contributions are killed by the (non-diagonal) unitary sum.
The main features of the problem inay be demonstrated in the following example
choosing N = 3, fl = b-quark, f 2 = q u a r k [lj]. The following compact expression
arises for the branching ratio (4.2):
+ 7,)
Here we have used that
V ( A u ,a) = V(0,a)
+ const - A ,
'v
V(0,o)
(4.5)
(and so for Ac). The expression for V ( 0 ,a) is given in (3.9).
I n Fig. 4 we demonstrate the dependence of B(b, s) of (4.4) on the mass of the tquark. We also show the corresponding function for external charged leptons. The other
two cases are rather similar as is evident from the preceding discussion. The range of
interest is between m(t)N 20 GeV (experimental lower limit. [16] and m(t) O(SO0 GeV).
The upper limit stems from the analysis of partial wave unitary bounds and applicability of perturbation theory [17]. I n Fig. 4 we have divided out froin B(b, s) the
influence of the unknown Kobayashi-Maskawa matrix elements [18].
Furthermore, another factor of roughly 2/(11.3N) must be included to get the total
rate. For comparison we have also drawn (4.4)for a = 0 in which case there is no threshold behaviour. Apart from this the curve approaches that for a = az already at
moderate A . This behaviour is an indication of the relative unimportance of external
masses for the physical effect we are interested in. Of similar restricted numerical relevance should be the inclusion of external fernlion masses.
The spirit of the approach has been demonstrated. To go one step further in the
analysis of possible Z-boson decay one had t o speculate on relevant parameter sets and
possible further fernlion generations, so we will stop here [19].
N
341
Effective Flavor-changingWeak Neutral Current in the Standard Theory
Conclusions
We have generalized earlier work in two directions. We allowed for 0
0 which
made possible the investigation of non-diagonal 2-boson decays. Our results demonstrate
that the search for flavor-changing 2-boson decays could give information on the
existence of rather massive fermions. Of course, the sensitivity of 2-boson decays with
a t best millions of events is much siiialler than that of, e.g., the rare muon decay
p +e
y and corresponding light quark physics, but i t is sensitive also t o other particle
mixings and with this in mind of its own value.
Secondly, we generalized earlier work in that we allowed for non-diagonal couplings
of 2-boson and neutrinos, charged leptons, up- and down-type quarks6).
Of course, massive fermions may show up in the loop effect studied here ifthey have
couplings to particles with mass smaller than M,. I n view of this our technically motivated restriction to vanishing external fermion masses (m, < M,) is rather restrictive
even if the results are not too strongly affected numerically. .
Nuinerical calculations of the 2-boson decay (0 =az) along the line of our Chapter 4
but allowing for arbitrary quark masses have been presented in preprints we got knowledge of in preparing the present article. CAPDEQUI-PEYRANERE
and TALON[20] coinputed the cross section of e+e--annihilation into two different massive quarks; they
investigated several parameter sets but did not analyze the large A limit. AXELROD
[all mainly investigated the decay Z+ t + ij (generally, into charge 213 quarks) and
got for this choice small rates. Finally, GANAPATHIet al. [ 2 2 ] and CLEMENTSe t al. [ 2 3 ]
studied the general case of 2 + + Q allowing rather large ferniion masses in the loop
arriving a t similar conclusions as those contained in our Sect. 4.
A c k n o w l e d g e m e n t s . We thank D. Yu. BARDW,C.-J. BIEBL, S. M. BILENKY,
P. CH.CHRISTOVA,D. EBERT,R . YECCEI and E. WIECZOREK
for useful comments and
discussions. One of us (T. R.) thanks S. SCHLENSTEDT
for introducing him to the use of
a computer.
+
+
a
-4ppendix A : Charged Current Interactions
We use the lagrangian of ref. [lo]. Fermion masses are introduced by nondiagonal
Yukawa couplings of SU(2)x U(1) eigenstates t o the Higgs dublet. These couplings
imply the following charged Higgs boson-fermion interactions some of them are given in
[lo] :
y;P=-
2 1/2
+ y5) P,ifyw”-
y@(l
+ %z
2)/2
[-Mw (1 + y s )
md
mzL
- -(1- f ) ]
1
Here j d ( f u ) is a ferniion with I3 = 2
Maskawa mixing matrix.
Mw
F8ifp-
+ h.c.
(+ T1 ) ,P,; are elements of the Kobayashi-
Appendix B: The Vertex Integrals
Our definition of one-loop functions agrees with that of ’T HOOFTand VELTMANN
[9] (see also [lo]). The only exception is the normalization-our functions are diniensionless. They depend on the dimensionless variables ni(Mwni = pi) and cri(M&cri = mf).
6) We should mention here that it is not totally excluded by experiment (including cosmology)
t,o have rather heavy (unstable)neutral leptons so that we decided to handle qnarks and leptons with
equal right here.
342
%point function B1(d;al,az)
J dnq q"/N2
- i ~ [)( q p1l2+ rn; - i ~ ] .
in2piB1(n: ;ul) ~
N2 = (a2 + m4
2=
)
)
+
We need t,his function in the limit $ = 0:
1
+ B;,
B l ( O ; d ,1) = - 7 8
2
1 A
Bi(O;A,l)=-2 1-A
S = 2/(4 - n ) - Y E
1
- I n n - In (M&/p2).
%point functions C(z!, ng;al, as, a3)
in2(Co;CY;C"v)= M $ J d n q { l ; q"; ql"q"}/N,,
N , = (q2
+ mf - [ ( q + p1)' + mz - i ~ []( q + p2)' + mg - i ~ ] (B.3)
)
+p v c
"C + P;P;c22 + (PYP; + ZGP3 c2, + JG6/"'C2*.
it)
c" = p",c,,
C"" =
2 12)
1Pl 21
The dimensionless functions C,, Cii depend on the arguiiients {-(nl- n2)2,n;,n;;
bl, G2' GZ}.
Here they are t o be calculated for arbitrary -(PI - P ~ ) ~ / M
=$u but vanishing
masses of external fermions. The remaining two sets of scalar 3-point functions depend
on two different sets of parameters :
{ b , O , O ; l , d , d ) and { a , O , O ; A , 1).
I n short: ( l , A , d ) and ( A , 1,l).
We may drop the actual calculation of them. It has been shown in ref. [ 9 ] how to
express Co in terms of Spence functions @ ( Z ) :
+ co, + c o 3 ; ac,i
co = -co1
=
@(A)
(b),
- @ x -1
x
2.
x -x.
d
The definition of variable9 X~ has been taken over from ref. [ 9 ] too. Their values in the
limit of two vanishing external monenta squared are listed in Table 1.
Table 1. Variables being used to calculate the two types of 3-point)functions
1 - (1- A ) / o
1/(1- A )
-iE
Effective Fliivor-changing Weak Neutral Current in the Standard Theory
343
All the other 3-point functions Cii inay be expressed by C, and complex logarithms
also depending on the xi. Of course, performing the integrations one carefully must do
the necessary subtractions as demonstrated for C, in [I)].
The result may be expressed
using 3 auxiliary functions
1 - xi
-2.
Gi = (1- xi)ln-x;ln U = -1 - G,
G2 G,;
x, - 2i
z, - zi
1
1-xi
1-xi
H=--H,+H,+H,:
H a. -- - - - In 1 - x, x, - xi x, - xi
1 -xi
-xi
In xo zo - zi x, - xi
+ +
3-point functions tor u = 0
Much simpler results map be obtained if CT = 0 as, e.g., in [3] or in the study of the
rare muon dacey u
, +e
y [5]. I n this case the Feynman integrals allow one trivial
integration leaving the integrals I, [ 5 ] , L :
+
1
L(u,~)=
jdxxIn(az+b).
0
(B-8)
We found it convenient to calculate the I, using I, and the recursion relation
UI,,
+ bInp1 = -.n1
The 3-point functions are then (c = 0, arbitrary A ) :
c, = I,,
c11 - C 12 --- + ,
C,,
c
j
24
= C,, =
1
2C2, =
---b--L,
1
1
- 4
2
(B.9)
1
I,,
G. M ~ N N
and T.RIEMANN
344
whereli=u(d,1,1)= -u(l,d,A)= l - J , b = h ( d ,
Furthermore, to arrive at (3.10) we made use of
-
2dZ,
-
41 - I
-
1; - I, - AIT
l , l ) = d , b ( l , d , A ) = 1(B.lO)
= 0,
1
+-=
2
(B.11)
0.
Vertex functions tor A < 1
Using the general expressions (B.4- 7) it is rather simple to study further specialized
situations. Here we write down vertex funct,ions in the limit of a large fermion inass d
in the loop. We neglect all terms and omit all vertex functions which are not larger than
O(l/d) when d 1 in the context of this paper (coinpare eq. (3.3)):
(B.12)
1
C23(1,A, A ) = 1SA’
(B.13)
(B.14)
1
C o ( A , l , l ) = -1n3
A
1
+[1 - 4A(y)]-,
(B.13)
._1
1
c~~(A,I,I)=--~~A
4
+ [- 1-2 +
3
1
2
B;(O;d,1)=--nil
-
3
4
(B.1G)
-
1:
1
3 (4 - a) A(!))
a
--d
72
+-
1
+ -1n
:I
- 9
1
1 -2A ’
(B.17)
A(y) and y are defined in (3.7).
Vertex functions for A
=0
1
-d(Col,0, 0) = z 2 / 6 - @,[l/(a I)] - _In2 (a
2
-+
+ 11+ i z In (a + I ) ,
(B.18)
aC,,(l, 0,O) = oc,, = c, - 1
-d2C2,(1, 0, 0)
=
+ 111 a
1
(a + 2 ) c, - -a
a
-
- 7.3,
(€3.19)
2 -c 2(ln a - i z ) ,
(B.20)
+ I) C, + 30 + 2 - (a -t 2) (111 0 - i n )
kC;,(l,O, 0) = - 2 ( ~
1
B;(O, 0 , l ) = - -,
4
7
(B.21)
(B.22)
Effc ctive Flavor-changing Weak Neutral Current in the Standard Theory
b(C00, 1, 1)= n2/G - @(l- a)
345
+ 2 Re @
(B.23)
d , , ( O , 1, 1) = -[Go - B
+ I],
(B.24)
a2C2,(0, I, I) = 2
(B.25)
20c;,(O,
1, 1) = - C,
B
= ByCarctan (2y)
-
I3 + 1 -
3
2
1
+ nay - 2ayarctan(2y) , (B.26)
+ arctan [‘Ly(a - I)/(Y- 41).
(B.27)
As long as 0 < CT < 4, only the functions C(1, A , A ) develop an imaginary part (for
A < Atl,, = --d/4).C,, Cll, 5, contain terms of order d In d if 0 < d < 1,but these cancel
when they come together in the form factor eq. (3.3) [12]. Also, the combined limit
d + 0, a+ 0 is well-defined for the form factor although this is not the case for all
the single vertex integrals.
References
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[23] CLEJIENTS,
Bei der Redaktion eingegangen am 12. September 1983.
Anschr. d. Vcrf.: Dr. G. MANN
Zentralinstitut fiir solnr-tcrrestrischc Physik
der AdW der DDR
Observatorium fiir Radioastronomie
DDR-1,501Tremsdorf
Dr. T. RIEMANN
Institut fur Hochenergiephysik
der AdW der DDR
DDR-1615 Zeuthen
Platanenallee G
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