- Email: [email protected]
Compositeness effects in the anomalous weak-magnetic moment of leptons
26 December 1996
PHYSICS
ELSEVIER
LElTERS
B
Physics Letters B 389 ( 1996)707-7 12
Compositeness effects in the anomalous weak-magnetic moment of leptons M.C. Gonzalez-Garciaa,
S.F. Novaes b
a Theory Division, CERN, CH-1211 Geneva 23, Switzerland b Instituto de Fisica Teo’rica. Universidade Estadual Paulista, Run Pamplona 14.5, CEP 01405-900 Srio Pa&o, Brazil
Received 24 July 1996 Editor: R. Gatto
Abstract
We investigate the effects induced by excited leptons, at the one-loop level, in the anomalous magnetic and weak-magnetic form factors of the leptons. In particular, we compute their contributions to the weak-magnetic moment of the 7 lepton, which can be measured on the Z peak, and we compare it with the contributions to g, - 2, measured at low energies.
The standard model of electroweak interactions (SM), in spite of its remarkable agreement with the present experimental data at the 2 pole [ 11, leaves some important questions unanswered. In particular, the reason why fermion generations repeat and the understanding of the complex pattern of quark and lepton masses are not furnished by the model. With the proliferation of fermion flavours, it is natural to ask whether these particles are truly elementary or composite states. The idea of composite models assumes the existence of an underlying substructure, characterized by a mass scale A, with the fermions sharing some of the constituents [ 21. As a consequence, excited states of each known lepton should show up at some energy scale, and the SM should be seen as the low-energy limit of a more fundamental theory. Several experimental collaborations have been searching for excited leptonic states [ 31 without any direct evidence for the existence of these particles. Their analyses are based on a model proposed some years ago [ 41, where the couplings of the excited leptons are described by an effective SU(2) x U( 1) invariant Lagrangian. On the other hand, an important source of indirect information about new particles and interactions is provided by the precise measurement of the electroweak parameters. Virtual effects of these new states can alter the SM predictions for some of these parameters, and the comparison with the experimental data can impose bounds on their masses and couplings. Bounds have been derived from the contribution to the anomalous magnetic moment of leptons [5] and to the Z observables at LEP [6]. In this work we investigate the effects induced by excited leptons, at the one-loop level, in the anomalous weak-magnetic form factors of the leptons, at an arbitrary energy scale. In particular, we study the contribution to the weak-magnetic moment of the 7 lepton that can be measured on the Z peak [ 71. Our results show that for universal couplings the existing limits from g, - 2 strongly constrain the possibility of observing this effect on the anomalous weak-magnetic moment of the 7 lepton at LEP, given the expected experimental sensitivity. 0370-2963/96/$12.00 Copyright 0 1996Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(96)01332-9
M.C. Gonzalez-Garcia, S.F. Novaes/Physics
708
Letters B 389 (1996) 707-712
We consider excited fermionic states with spin and isospin 3, and we assume that the excited fermions acquire their masses before the SU( 2) x U( 1) breaking, so that both left-handed and right-handed states belong to weak isodoublets. The most general dimension-six effective Lagrangian [4] that describes the coupling of the excited-usual fermions, which is SU(2) x U( 1) invariant and CP-conserving, can be written as &f
= -
c
C,P@“(
1 - ys) fJpVy - i c
DwfpcY”( v=Y,z
v=Yzw
1 - ys) fW,V,
+ h.c.,
(1)
where F = N, E represent the excited states, and f = Y, e, the usual light fermions of the first generation. A pure left-handed structure is assumed for these couplings in order to comply with the strong bounds coming from the measurement of the anomalous magnetic moment of leptons [ 51. In the same way, the coupling of gauge bosons to excited leptons can be written as CFF = -
c
F( AwFypVg
+ K~~ac”“~,Vv) F .
(2)
v=y,zw The definition of the coupling constants A~F, Cwf, Dwf, and K~F can be found elsewhere [ 61. They depend on the weak mixing angle, Bw, and on f2 (k2) and fl (kl), the weight factors associated to the SU( 2) and U( 1) coupling constants, and the dimensionful constants are proportional to 1/A, where A is the compositeness scale. The matrix element of a boson (Vt ) current has the general form:
.P = e iif(p1 )
2sine
Uf(P2) lcosew~~ [Fv(s2> - &(q2>r5] + $$‘(q2W’qv
W
9
(3)
where VI = y or Z, and q = pi +p2. The terms FV and FA are present at tree level in the SM, e.g. for the Z boson, FU@?= -T{ + 2Qf sin’ Bw, and Fiw = -T[, The contribution of the excited leptons to these form factors at V the one-loop level has been evaluated in Ref. [ 61. The anomalous weak-magnetic form factor, a;, is generated only at one-loop both in the SM and in models with excited fermions. In the latter case, there are twelve one-loop Feynman diagrams involving excited fermions that contribute to the anomalous electroweak-magnetic moment of leptons, which are shown in Fig. 1. For each of these contributions, we define the amplitudes S~(q2,M2,M~),i=l ,..., 12, where V2 is the virtual vector boson with mass Mv, running in the loops, and we can write the excited lepton contribution to a: as
ay(q2>= iy sV,-ff(q2),
(4)
with ~v,+f+f-(q2)
=qqZ,M2,0)
+g+,(q2,MZ,o)
+s?(q2,M2,M$)
+s~(q2,M2,M$)
+ Sf+,(q2,M2&)
+S~+,(q2,M2,M$)
+ $+g(q2,
+ S7w,s(q2, M2, M2,)
+$V(42,M2,M$) + s;+,(q? + g+,,(q2,
+ $+,2(&
M2,0) M2,0)
+ S,z,,dq2,
M2, M2,) .
M2&) M2, M;)
+ Sr+&2,
M2, M2,)
(5)
We have neglected the fermion masses in the evaluation of the integrals @ (i.e. rr$ << M2, MC), and in this limit SF+,(q2, M2, M$) = 0.
M.C. Gonzalez-Garcia, S.F. Novaes/Physics
Letters B 389 (1996) 707-712
709
cu A=200
Gi-
101
E-
100
E-
lo-
f
GeV
z
(47
1
11111111111,,,,-1 k”
-2
1
2
0.8
1
5 V f
f
103
T
T CI)
102 02
f
f
i
T
(S)
xz101
100
0.2 &
Fig, 1. The contribution of the excited leptons to the anomalous electroweak magnetic moments.
0.4
0.6
(TeV)
Fig. 2. (a) Accessible region oi parameters for Ia,”( Mi ) [ 1 10m4 (above the curves) in the If/* versus k plane for fixed values of M = A. (b) Accessible region (above the curves) in the lf[* versus M = A plane for fixed values of k.
The loop contributions of the excited leptons were evaluated in D = 4 - 2~ dimensions using the dimension regularization method [ 81, which is a gauge-invariant regularization procedure, and we adopted the unitary gauge to perform the calculations. The results in D dimensions were obtained with the aid of the Mathematics package FeynCalc [ 91, and the poles at D = 4 (E = 0) and D = 2 (E = 1) were identified with the logarithmic and quadratic dependence on the scale A [ lo]. Our results for Sy (q2,M2, Mc2) are rather lengthy. We show here only approximate expressions, which are valid in the limit q2,M$ << M2, at first order in RQ = q2/M2 and Rv = M$/M2:
sy --L C&f 14473 N
-
RQ(~SAKFF
S2v:s - -!24&
6&,~~(20 + 9Rv) +
•I- ~~K&FF
%FfcKFf
Ml
-
72
~~Kv,FF
M(3 + Rv)
(AWF + K~,FF M) log $
1,
3+6Rv+12RvlogRv+4RQ(I+210gRv)+610g$]
710
M.C. Gonzalez-Garcia, SF. Novaes/Physics
39+6Rv+ $2 2 -i-42
144T2
79 -
y2Ffgflw
c VzFf
KFf
sv2 9+10- -
12RvlogRv-
+ 2Rv)
,
16RQlogRv+610g-
+
120Rv - 4210g
2Rv+3RvlogRv-3Rvlog-+-
(&+&)
ic&,)2 cV,FfcVzFf
- 6 Av,FF(~
~RQ
Letters B 389 (19%) 707-712
&,FF(
1.5 +
14Rv) + &,FF
log
$
A2 1+ 12Rv - 610g M2 where g& and gh are the vector and axial coupling
A2
M2
M2
,
M(22 + 21Rv)
1
+
A2
+
4;
(A&w
+ KvzFF M)
,
,
of the vector bosons
to the usual fermions:
g”? = -e,
- 1)/(4cosfYw), g”y= 0; & = & = g/(2&); for f = v, & = & = g/(4 cos&~); for f = e, & = g(4sin2& = -g/(4cos 0~). The coupling g&w refers to the triple vector boson vertex: gvm = g sin Bw, and and gz gzww = gcOsew. We should notice that g+s is infrared-divergent for q2 # 0. However, we have explicitly checked that this divergence cancels against the one coming from real photon emission, and the final result is therefore infrared-finite. The approximate final results for the anomalous magnetic (u’f) and weak-magnetic (af ) moments, assuming and fl = f;! = f and kl = k2 = k is,
M2 = A2 > M&, u
f”m;
sin2 ew c0s2 ew
[
f’m$
f
961r
M2
I3
37+74c0s2ew+(24+39cos2ew)k
LY .z=---
a:=Gx
37+2c0s2ew(27-74c0s2ew) i
+6(4-
sin3 ew cos3 ew
‘
13c0s4ew)k 1.
(7)
Our results for the anomalous magnetic moment u; are in agreement with those of Ref. [5], for k = 0. We now turn to the attainable values for the weak-magnetic moment of the r lepton at LEP energies. In Ref. [ 71, Bernabeu et al. compute the SM contribution to this observable and discuss the attainable sensitivity at LEP. They propose to measure the weak-magnetic moment through the analysis of the angular asymmetry of the semileptonic 7 decay products, which carries information about the weak-magnetic moment of the parent lepton. They assume that the r direction is fully reconstructed and they deduce a sensitivity of the order of Ia,“( 5 10-4. In Fig. 2, we show the accessible region in the parameter space f, k and A. For the sake of simplicity, we have assumed that A = M. As seen in this figure only models with strong coupling, i.e. f N G/e, and compositeness scale A 5 200 GeV could lead to a value for the anomalous weak-magnetic moment of the r large enough to be observed at LEP. If we assume that the couplings to the excited fermions are universal, i.e. if fi, ki, M and A are the same for the three generations, the attainable value for the r lepton weak-magnetic moment is already constrained by the existing limits from the anomalous electromagnetic moment of the muon measured at low energies. Nowadays, the most precise determination of the anomalous magnetic moment of the muon a: = (gcL - 2)/2 comes from a CERN experiment, i.e. a: = 11659 230 (84) x lo-r0 [ 1 I]. This result should be compared with the existing
M.C. Gonzalez-Garcia, S.F. Novaes/Physics
.
10-Z
w
\
_
711
h=lTeV ---
2
0
-1
-2
Letters B 389 (1996) 707-712
k
:
II
_
II Ii
10-4 r ,( :: II II
102
----
A=200 GeV 7 A=1 TeV
101
x
N
N
T=i-
$
100
I-
N
10-l
- aJ
10-2
0.2
A,
0.4
0.6
0.8
1
(TeV)
Fig. 3. (a) Excluded region of parameters from IaLl 5 8 x 10m9 (above the curves) in the lfl* versus k plane for fixed values of M = A. (b) Corresponding limits in the lfl* versus M = A plane
-2
-1
0
1
2
k Fig. 4. Attainable values of Ia: (Mi) 1for universal excited lepton couplings after imposing the constraints from g, - 2.
for fixed values of k. of the QED [ 121, electroweak [ 131, and hadronic [ 141 contributions, which are known The main theoretical uncertainty comes from the hadronic contributions which is of the order of 20 x lo-to. Therefore the present limit on the non-standard contributions to the anomalous magnetic moment of the muon is theoretical
calculations
with
precision.
high
Isag 5 8 x 10-9. The proposed AGS experiment at the Brookhaven National Laboratory [ 151 will be able to measure the anomalous magnetic moment of the muon with an accuracy of about f4 x 10-t’. In Fig. 3 we show the present limits from Eq. (8) in the parameter space f, k and A. Our results for k = 0 are in agreement with those from Ref. [ 51. However, the presence of an anomalous magnetic moment term at tree level in the coupling between a pair of excited fermions (see Eq. (2) ) alters the attainable bounds on f. As seen in Fig. 3a, there is a value k = /q 21 -1.56 (-1.72), for A = 0.2 ( 1.0) TeV, for which the limit on the coupling strength f becomes very weak. This comes as a consequence of the cancellation of the leading terms in Eq. (7). The exact dependence of ka on A is due to higher-order terms, which are not displayed in Eq. (7). Taking these results into account, we plot in Fig. 4 the attainable values for the r anomalous weak-magnetic moment, assuming universal couplings, after imposing the constraints from g, - 2 measurements. We can see that only for a narrow band of k values around ko can ]u,” (M$) 1 be large enough to be observed at LEE.
712
M.C. Gonzalez-Garcia, S.F. Novaes/Physics
Leners B 389 (1996) 707-712
Summarizing, we have investigated the effects induced by excited leptons at the one-loop level in the anomalous magnetic and weak-magnetic form factors of the leptons at an arbitrary scale. Using a general effective Lagrangian approach to describe the couplings of the excited leptons, we have computed their contributions to the weak-magnetic moment of the T, which can be measured on the Z peak, and we compare it with the contributions to g, - 2 measured at low energies. Our results show that although for universal couplings, the existing limits from gp - 2 strongly constrain the possibility of observing the anomalous weak-magnetic moment of the T lepton at LEP, there is a very narrow region of parameters in which the observation is still possible. One must notice, however, that in this region the model becomes strongly coupled. References [ 1] The LEP Collaborations ALEPH, DELPHI, L3, OPAL, and the LEP Electroweak Working Group, contributions to the 1995 Europhysics Conference on High Energy Physics (EPS-HEP), Brussels, Belgium, and to the 17th International Symposium on Lepton-Photon Interactions, Beijing, China, Report No. CERN-PPE/95-172 ( 1995). [ 21 For a review, see for instance: H. Harari, Phys. Reports 104 (1984) 159; H. Terazawa, Proceedings of the XXII International Conference on High Energy Physics, Leipzig ( 1984). eds. A. Meyer and E. Wieczorek, p. 63; W. Buchmtiller, Acta Phys. Austriaca, Suppl. XXVII (1985) 517; M.E. Peskin, Proceedings of the 1985 International Symposium on Lepton and Photon Interactions at High Energies, Kyoto ( 1985) p. 714, eds. M. Konuma and K. Takahashi. [ 31 See for instance: ALEPH Collaboration, D. Decamp et al., Phys. Reports 216 (1992) 253; L3 Collaboration, 0. Adriani et al., Phys. Reports 236 (1993) 1; Hl Collaboration, I. Abt et al., Nucl. Phys. B 396 (1993) 3; ZEUS Collaboration, M. Derrick et al., Z. Phys. C 65 (1994) 627. [4] N. Cabibbo, L. Maiani and Y. Srivastava, Phys. Lett. B 139 ( 1984) 459; A. De Rtijula, L. Maiani and R. Petronzio, Phys. Lett. B 140 (1984) 253; J. Kuhn and P Zerwas, Phys. Lett. B 147 (1984) 189; K. Hagiwara, S. Komamiya and D. Zeppenfeld, Z. Phys. C 29 (1985) 115; E Boudjema, A. Djouadi and J.L. Kneur, Z. Phys. C 57 (1993) 425. [5] S.J. Brodsky and S.D. Drell, Phys. Rev. D 22 (1980) 2236; EM. Renard, Phys. Lett. B 116 ( 1982) 264; P. Metry, SE. Moubarik, M. Perrottet and FM. Renard, Z. Phys. C 46 (1990) 229; R. Escribano and E. Masso, UAB-FT-385 and hep-phI9607218. [6] M.C. Gonzalez-Garcia and SF Novaes, CERN-TH/96-123, IFT-POl3/96 and hep-ph/9608309, Nucl. Phys. B, to appear. [7] J. Bemabeu, GA. Gonzalez-Sprinberg, M. Tung and J. Vidal, Nucl. Phys. B 436 (1995) 474. [ 81 G. ‘t Hooft and M. Veltman, Nucl. Phys. B 44 (1972) 189; C.G. Bollini and J.J. Giambiagi, Nuovo Cim. B 12 (1972) 20. [9] R. Mertig, M. Bohm and A. Denner, Comput. Phys. Commun. 64 (1991) 345. [lo] K. Hagiwara, S. lshihara, R. Szalapski and D. Zeppenfeld, Phys. Lett. B 283 (1992) 353; Phys. Rev. D 48 (1993) 2182. [ 111 J. Bailey et al., Nucl. Phys. B 150 ( 1979) 1; E.R. Cohen and B.N. Taylor, Rev. Mod. Phys. 59 (1987) 1121. [ 121 For a review of the QED calculations see: T. Kinoshita and W.J. Marciano, Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore, 1990) p. 419, and references therein. [ 131 G. Altarelli, N. Cabibbo and L. Maiani, Phys. Len. B 40 ( 1972) 415; I. Bars and M. Yoshimura, Phys. Rev. D 6 (1972) 374; K. Fujikawa, B.W. Lee and A.I. Sanda, Phys. Rev. D 6 (1972) 2923; R. Jackiw and S. Weinberg, Phys. Rev. D 5 ( 1972) 2473; A. Czamecki, B. Krause and W.J. Marciano, Phys. Rev. Lett. 76 (1996) 3267; S. Peris, M. Perrottet and E. de Rafael, Phys. Lett. B 355 ( 1995) 523. [ 141 J. Calmet, S. Narison, M. Perrottet and E. de Rafael, Phys. Lett. B 61 (1976) 283; Rev. Mod. Phys. 49 (1977) 21; T. Kinoshita, B. Nitic and Y. Okamoto, Phys. Rev. D 31 (1985) 2108; M. Hayakawa, T. Kinoshita and A.I. Sanda, Phys. Rev. Lett. 75 ( 1995) 790; J. Bijnens, E. Pallante and J. Prades, Phys. Rev. Lett. 75 (1995) 1447. [ 151 B.L. Roberts, Z. Phys. C 56 (1992) SlOl.
PHYSICS
ELSEVIER
LElTERS
B
Physics Letters B 389 ( 1996)707-7 12
Compositeness effects in the anomalous weak-magnetic moment of leptons M.C. Gonzalez-Garciaa,
S.F. Novaes b
a Theory Division, CERN, CH-1211 Geneva 23, Switzerland b Instituto de Fisica Teo’rica. Universidade Estadual Paulista, Run Pamplona 14.5, CEP 01405-900 Srio Pa&o, Brazil
Received 24 July 1996 Editor: R. Gatto
Abstract
We investigate the effects induced by excited leptons, at the one-loop level, in the anomalous magnetic and weak-magnetic form factors of the leptons. In particular, we compute their contributions to the weak-magnetic moment of the 7 lepton, which can be measured on the Z peak, and we compare it with the contributions to g, - 2, measured at low energies.
The standard model of electroweak interactions (SM), in spite of its remarkable agreement with the present experimental data at the 2 pole [ 11, leaves some important questions unanswered. In particular, the reason why fermion generations repeat and the understanding of the complex pattern of quark and lepton masses are not furnished by the model. With the proliferation of fermion flavours, it is natural to ask whether these particles are truly elementary or composite states. The idea of composite models assumes the existence of an underlying substructure, characterized by a mass scale A, with the fermions sharing some of the constituents [ 21. As a consequence, excited states of each known lepton should show up at some energy scale, and the SM should be seen as the low-energy limit of a more fundamental theory. Several experimental collaborations have been searching for excited leptonic states [ 31 without any direct evidence for the existence of these particles. Their analyses are based on a model proposed some years ago [ 41, where the couplings of the excited leptons are described by an effective SU(2) x U( 1) invariant Lagrangian. On the other hand, an important source of indirect information about new particles and interactions is provided by the precise measurement of the electroweak parameters. Virtual effects of these new states can alter the SM predictions for some of these parameters, and the comparison with the experimental data can impose bounds on their masses and couplings. Bounds have been derived from the contribution to the anomalous magnetic moment of leptons [5] and to the Z observables at LEP [6]. In this work we investigate the effects induced by excited leptons, at the one-loop level, in the anomalous weak-magnetic form factors of the leptons, at an arbitrary energy scale. In particular, we study the contribution to the weak-magnetic moment of the 7 lepton that can be measured on the Z peak [ 71. Our results show that for universal couplings the existing limits from g, - 2 strongly constrain the possibility of observing this effect on the anomalous weak-magnetic moment of the 7 lepton at LEP, given the expected experimental sensitivity. 0370-2963/96/$12.00 Copyright 0 1996Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(96)01332-9
M.C. Gonzalez-Garcia, S.F. Novaes/Physics
708
Letters B 389 (1996) 707-712
We consider excited fermionic states with spin and isospin 3, and we assume that the excited fermions acquire their masses before the SU( 2) x U( 1) breaking, so that both left-handed and right-handed states belong to weak isodoublets. The most general dimension-six effective Lagrangian [4] that describes the coupling of the excited-usual fermions, which is SU(2) x U( 1) invariant and CP-conserving, can be written as &f
= -
c
C,P@“(
1 - ys) fJpVy - i c
DwfpcY”( v=Y,z
v=Yzw
1 - ys) fW,V,
+ h.c.,
(1)
where F = N, E represent the excited states, and f = Y, e, the usual light fermions of the first generation. A pure left-handed structure is assumed for these couplings in order to comply with the strong bounds coming from the measurement of the anomalous magnetic moment of leptons [ 51. In the same way, the coupling of gauge bosons to excited leptons can be written as CFF = -
c
F( AwFypVg
+ K~~ac”“~,Vv) F .
(2)
v=y,zw The definition of the coupling constants A~F, Cwf, Dwf, and K~F can be found elsewhere [ 61. They depend on the weak mixing angle, Bw, and on f2 (k2) and fl (kl), the weight factors associated to the SU( 2) and U( 1) coupling constants, and the dimensionful constants are proportional to 1/A, where A is the compositeness scale. The matrix element of a boson (Vt ) current has the general form:
.P = e iif(p1 )
2sine
Uf(P2) lcosew~~ [Fv(s2> - &(q2>r5] + $$‘(q2W’qv
W
9
(3)
where VI = y or Z, and q = pi +p2. The terms FV and FA are present at tree level in the SM, e.g. for the Z boson, FU@?= -T{ + 2Qf sin’ Bw, and Fiw = -T[, The contribution of the excited leptons to these form factors at V the one-loop level has been evaluated in Ref. [ 61. The anomalous weak-magnetic form factor, a;, is generated only at one-loop both in the SM and in models with excited fermions. In the latter case, there are twelve one-loop Feynman diagrams involving excited fermions that contribute to the anomalous electroweak-magnetic moment of leptons, which are shown in Fig. 1. For each of these contributions, we define the amplitudes S~(q2,M2,M~),i=l ,..., 12, where V2 is the virtual vector boson with mass Mv, running in the loops, and we can write the excited lepton contribution to a: as
ay(q2>= iy sV,-ff(q2),
(4)
with ~v,+f+f-(q2)
=qqZ,M2,0)
+g+,(q2,MZ,o)
+s?(q2,M2,M$)
+s~(q2,M2,M$)
+ Sf+,(q2,M2&)
+S~+,(q2,M2,M$)
+ $+g(q2,
+ S7w,s(q2, M2, M2,)
+$V(42,M2,M$) + s;+,(q? + g+,,(q2,
+ $+,2(&
M2,0) M2,0)
+ S,z,,dq2,
M2, M2,) .
M2&) M2, M;)
+ Sr+&2,
M2, M2,)
(5)
We have neglected the fermion masses in the evaluation of the integrals @ (i.e. rr$ << M2, MC), and in this limit SF+,(q2, M2, M$) = 0.
M.C. Gonzalez-Garcia, S.F. Novaes/Physics
Letters B 389 (1996) 707-712
709
cu A=200
Gi-
101
E-
100
E-
lo-
f
GeV
z
(47
1
11111111111,,,,-1 k”
-2
1
2
0.8
1
5 V f
f
103
T
T CI)
102 02
f
f
i
T
(S)
xz101
100
0.2 &
Fig, 1. The contribution of the excited leptons to the anomalous electroweak magnetic moments.
0.4
0.6
(TeV)
Fig. 2. (a) Accessible region oi parameters for Ia,”( Mi ) [ 1 10m4 (above the curves) in the If/* versus k plane for fixed values of M = A. (b) Accessible region (above the curves) in the lf[* versus M = A plane for fixed values of k.
The loop contributions of the excited leptons were evaluated in D = 4 - 2~ dimensions using the dimension regularization method [ 81, which is a gauge-invariant regularization procedure, and we adopted the unitary gauge to perform the calculations. The results in D dimensions were obtained with the aid of the Mathematics package FeynCalc [ 91, and the poles at D = 4 (E = 0) and D = 2 (E = 1) were identified with the logarithmic and quadratic dependence on the scale A [ lo]. Our results for Sy (q2,M2, Mc2) are rather lengthy. We show here only approximate expressions, which are valid in the limit q2,M$ << M2, at first order in RQ = q2/M2 and Rv = M$/M2:
sy --L C&f 14473 N
-
RQ(~SAKFF
S2v:s - -!24&
6&,~~(20 + 9Rv) +
•I- ~~K&FF
%FfcKFf
Ml
-
72
~~Kv,FF
M(3 + Rv)
(AWF + K~,FF M) log $
1,
3+6Rv+12RvlogRv+4RQ(I+210gRv)+610g$]
710
M.C. Gonzalez-Garcia, SF. Novaes/Physics
39+6Rv+ $2 2 -i-42
144T2
79 -
y2Ffgflw
c VzFf
KFf
sv2 9+10- -
12RvlogRv-
+ 2Rv)
,
16RQlogRv+610g-
+
120Rv - 4210g
2Rv+3RvlogRv-3Rvlog-+-
(&+&)
ic&,)2 cV,FfcVzFf
- 6 Av,FF(~
~RQ
Letters B 389 (19%) 707-712
&,FF(
1.5 +
14Rv) + &,FF
log
$
A2 1+ 12Rv - 610g M2 where g& and gh are the vector and axial coupling
A2
M2
M2
,
M(22 + 21Rv)
1
+
A2
+
4;
(A&w
+ KvzFF M)
,
,
of the vector bosons
to the usual fermions:
g”? = -e,
- 1)/(4cosfYw), g”y= 0; & = & = g/(2&); for f = v, & = & = g/(4 cos&~); for f = e, & = g(4sin2& = -g/(4cos 0~). The coupling g&w refers to the triple vector boson vertex: gvm = g sin Bw, and and gz gzww = gcOsew. We should notice that g+s is infrared-divergent for q2 # 0. However, we have explicitly checked that this divergence cancels against the one coming from real photon emission, and the final result is therefore infrared-finite. The approximate final results for the anomalous magnetic (u’f) and weak-magnetic (af ) moments, assuming and fl = f;! = f and kl = k2 = k is,
M2 = A2 > M&, u
f”m;
sin2 ew c0s2 ew
[
f’m$
f
961r
M2
I3
37+74c0s2ew+(24+39cos2ew)k
LY .z=---
a:=Gx
37+2c0s2ew(27-74c0s2ew) i
+6(4-
sin3 ew cos3 ew
‘
13c0s4ew)k 1.
(7)
Our results for the anomalous magnetic moment u; are in agreement with those of Ref. [5], for k = 0. We now turn to the attainable values for the weak-magnetic moment of the r lepton at LEP energies. In Ref. [ 71, Bernabeu et al. compute the SM contribution to this observable and discuss the attainable sensitivity at LEP. They propose to measure the weak-magnetic moment through the analysis of the angular asymmetry of the semileptonic 7 decay products, which carries information about the weak-magnetic moment of the parent lepton. They assume that the r direction is fully reconstructed and they deduce a sensitivity of the order of Ia,“( 5 10-4. In Fig. 2, we show the accessible region in the parameter space f, k and A. For the sake of simplicity, we have assumed that A = M. As seen in this figure only models with strong coupling, i.e. f N G/e, and compositeness scale A 5 200 GeV could lead to a value for the anomalous weak-magnetic moment of the r large enough to be observed at LEP. If we assume that the couplings to the excited fermions are universal, i.e. if fi, ki, M and A are the same for the three generations, the attainable value for the r lepton weak-magnetic moment is already constrained by the existing limits from the anomalous electromagnetic moment of the muon measured at low energies. Nowadays, the most precise determination of the anomalous magnetic moment of the muon a: = (gcL - 2)/2 comes from a CERN experiment, i.e. a: = 11659 230 (84) x lo-r0 [ 1 I]. This result should be compared with the existing
M.C. Gonzalez-Garcia, S.F. Novaes/Physics
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Letters B 389 (1996) 707-712
k
:
II
_
II Ii
10-4 r ,( :: II II
102
----
A=200 GeV 7 A=1 TeV
101
x
N
N
T=i-
$
100
I-
N
10-l
- aJ
10-2
0.2
A,
0.4
0.6
0.8
1
(TeV)
Fig. 3. (a) Excluded region of parameters from IaLl 5 8 x 10m9 (above the curves) in the lfl* versus k plane for fixed values of M = A. (b) Corresponding limits in the lfl* versus M = A plane
-2
-1
0
1
2
k Fig. 4. Attainable values of Ia: (Mi) 1for universal excited lepton couplings after imposing the constraints from g, - 2.
for fixed values of k. of the QED [ 121, electroweak [ 131, and hadronic [ 141 contributions, which are known The main theoretical uncertainty comes from the hadronic contributions which is of the order of 20 x lo-to. Therefore the present limit on the non-standard contributions to the anomalous magnetic moment of the muon is theoretical
calculations
with
precision.
high
Isag 5 8 x 10-9. The proposed AGS experiment at the Brookhaven National Laboratory [ 151 will be able to measure the anomalous magnetic moment of the muon with an accuracy of about f4 x 10-t’. In Fig. 3 we show the present limits from Eq. (8) in the parameter space f, k and A. Our results for k = 0 are in agreement with those from Ref. [ 51. However, the presence of an anomalous magnetic moment term at tree level in the coupling between a pair of excited fermions (see Eq. (2) ) alters the attainable bounds on f. As seen in Fig. 3a, there is a value k = /q 21 -1.56 (-1.72), for A = 0.2 ( 1.0) TeV, for which the limit on the coupling strength f becomes very weak. This comes as a consequence of the cancellation of the leading terms in Eq. (7). The exact dependence of ka on A is due to higher-order terms, which are not displayed in Eq. (7). Taking these results into account, we plot in Fig. 4 the attainable values for the r anomalous weak-magnetic moment, assuming universal couplings, after imposing the constraints from g, - 2 measurements. We can see that only for a narrow band of k values around ko can ]u,” (M$) 1 be large enough to be observed at LEE.
712
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Leners B 389 (1996) 707-712
Summarizing, we have investigated the effects induced by excited leptons at the one-loop level in the anomalous magnetic and weak-magnetic form factors of the leptons at an arbitrary scale. Using a general effective Lagrangian approach to describe the couplings of the excited leptons, we have computed their contributions to the weak-magnetic moment of the T, which can be measured on the Z peak, and we compare it with the contributions to g, - 2 measured at low energies. Our results show that although for universal couplings, the existing limits from gp - 2 strongly constrain the possibility of observing the anomalous weak-magnetic moment of the T lepton at LEP, there is a very narrow region of parameters in which the observation is still possible. One must notice, however, that in this region the model becomes strongly coupled. References [ 1] The LEP Collaborations ALEPH, DELPHI, L3, OPAL, and the LEP Electroweak Working Group, contributions to the 1995 Europhysics Conference on High Energy Physics (EPS-HEP), Brussels, Belgium, and to the 17th International Symposium on Lepton-Photon Interactions, Beijing, China, Report No. CERN-PPE/95-172 ( 1995). [ 21 For a review, see for instance: H. Harari, Phys. Reports 104 (1984) 159; H. Terazawa, Proceedings of the XXII International Conference on High Energy Physics, Leipzig ( 1984). eds. A. Meyer and E. Wieczorek, p. 63; W. Buchmtiller, Acta Phys. Austriaca, Suppl. XXVII (1985) 517; M.E. Peskin, Proceedings of the 1985 International Symposium on Lepton and Photon Interactions at High Energies, Kyoto ( 1985) p. 714, eds. M. Konuma and K. Takahashi. [ 31 See for instance: ALEPH Collaboration, D. Decamp et al., Phys. Reports 216 (1992) 253; L3 Collaboration, 0. Adriani et al., Phys. Reports 236 (1993) 1; Hl Collaboration, I. Abt et al., Nucl. Phys. B 396 (1993) 3; ZEUS Collaboration, M. Derrick et al., Z. Phys. C 65 (1994) 627. [4] N. Cabibbo, L. Maiani and Y. Srivastava, Phys. Lett. B 139 ( 1984) 459; A. De Rtijula, L. Maiani and R. Petronzio, Phys. Lett. B 140 (1984) 253; J. Kuhn and P Zerwas, Phys. Lett. B 147 (1984) 189; K. Hagiwara, S. Komamiya and D. Zeppenfeld, Z. Phys. C 29 (1985) 115; E Boudjema, A. Djouadi and J.L. Kneur, Z. Phys. C 57 (1993) 425. [5] S.J. Brodsky and S.D. Drell, Phys. Rev. D 22 (1980) 2236; EM. Renard, Phys. Lett. B 116 ( 1982) 264; P. Metry, SE. Moubarik, M. Perrottet and FM. Renard, Z. Phys. C 46 (1990) 229; R. Escribano and E. Masso, UAB-FT-385 and hep-phI9607218. [6] M.C. Gonzalez-Garcia and SF Novaes, CERN-TH/96-123, IFT-POl3/96 and hep-ph/9608309, Nucl. Phys. B, to appear. [7] J. Bemabeu, GA. Gonzalez-Sprinberg, M. Tung and J. Vidal, Nucl. Phys. B 436 (1995) 474. [ 81 G. ‘t Hooft and M. Veltman, Nucl. Phys. B 44 (1972) 189; C.G. Bollini and J.J. Giambiagi, Nuovo Cim. B 12 (1972) 20. [9] R. Mertig, M. Bohm and A. Denner, Comput. Phys. Commun. 64 (1991) 345. [lo] K. Hagiwara, S. lshihara, R. Szalapski and D. Zeppenfeld, Phys. Lett. B 283 (1992) 353; Phys. Rev. D 48 (1993) 2182. [ 111 J. Bailey et al., Nucl. Phys. B 150 ( 1979) 1; E.R. Cohen and B.N. Taylor, Rev. Mod. Phys. 59 (1987) 1121. [ 121 For a review of the QED calculations see: T. Kinoshita and W.J. Marciano, Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore, 1990) p. 419, and references therein. [ 131 G. Altarelli, N. Cabibbo and L. Maiani, Phys. Len. B 40 ( 1972) 415; I. Bars and M. Yoshimura, Phys. Rev. D 6 (1972) 374; K. Fujikawa, B.W. Lee and A.I. Sanda, Phys. Rev. D 6 (1972) 2923; R. Jackiw and S. Weinberg, Phys. Rev. D 5 ( 1972) 2473; A. Czamecki, B. Krause and W.J. Marciano, Phys. Rev. Lett. 76 (1996) 3267; S. Peris, M. Perrottet and E. de Rafael, Phys. Lett. B 355 ( 1995) 523. [ 141 J. Calmet, S. Narison, M. Perrottet and E. de Rafael, Phys. Lett. B 61 (1976) 283; Rev. Mod. Phys. 49 (1977) 21; T. Kinoshita, B. Nitic and Y. Okamoto, Phys. Rev. D 31 (1985) 2108; M. Hayakawa, T. Kinoshita and A.I. Sanda, Phys. Rev. Lett. 75 ( 1995) 790; J. Bijnens, E. Pallante and J. Prades, Phys. Rev. Lett. 75 (1995) 1447. [ 151 B.L. Roberts, Z. Phys. C 56 (1992) SlOl.