- Email: [email protected]
Weak electric dipole moments of heavy fermions in the MSSM
23 April 1998
Physics Letters B 425 Ž1998. 322–328
Weak electric dipole moments of heavy fermions in the MSSM W. Hollik a
a,1
, J.I. Illana
a,2
, S. Rigolin
a,b,3
, D. Stockinger ¨
a,4
Institut fur ¨ Theoretische Physik, UniÕersitat ¨ Karlsruhe, D-76128 Karlsruhe, FR Germany b Dipartimento di Fisica, UniÕersita` di PadoÕa and INFN, I-35131 Padua, Italy Received 28 November 1997; revised 9 February 1998 Editor: P.V. Landshoff
Abstract A minimal supersymmetric version of the Standard Model with complex parameters allows contributions to the weak-electric dipole moments of fermions at the one-loop level. Assuming generation-diagonal trilinear soft-susy-breaking terms and the usual GUT constraint, a set of CP-violating physical phases can be introduced. In this paper the general expressions for the one-loop contribution to the WEDM in a generic renormalizable theory are given and the size of the WEDM of the t lepton and the b quark in such a supersymmetric model is discussed. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction In the electroweak Standard Model ŽSM. there is only one source of CP violation, the d CK M phase of the Cabibbo–Kobayashi–Maskawa mixing matrix for quarks w1x. 5 The only place where CP violation has been currently measured, the neutral K system, fixes the value of this phase but does not constitute itself a test for the origin of CP violation w2x. On the other hand, if the baryon asymmetry of the universe has
1
E-mail address: [email protected]. E-mail address: [email protected]. 3 E-mail address: [email protected]. 4 E-mail address: [email protected]. 5 The QCD Lagrangian includes an additional source of CP violation, the u QC D , but we will not consider it here. Extreme fine tuning is needed in order that its contribution to the neutron EDM does not exceed the present experimental upper bound. Various mechanisms beyond the SM have been proposed to solve this problem w2x. 2
been dynamically generated, CP must be violated. The SM cannot account for the size of the observed asymmetry w3x. Many extensions of the SM contain new CP-violating phases, in particular, the supersymmetric models w4x. It has also been shown that the Minimal Supersymmetric Standard Model ŽMSSM. w5x can provide the correct size of baryon asymmetry in some range of parameters if the CPviolating phases are not suppressed w6x. One needs soft-breaking terms to introduce physical phases in the MSSM, different from the d CK M w7x. We assume that the soft-breaking terms preserve R-parity. Other possibilities for CP violation can arise in R-parity violating models Žcf. e.g. w8x in the context of R-parity violating scalar interactions.. For simplicity, we restrict ourselves to generation-diagonal trilinear soft-breaking terms to prevent FCNC. Doing this we ignore CP-violating effects that have been already considered in the literature w9x. In our analysis we do not make any additional assumption, except for the unification of the soft-breaking gaug-
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 2 4 7 - 0
W. Hollik et al.r Physics Letters B 425 (1998) 322–328
ino masses at the GUT scale. However we do not assume unification of the scalar mass parameters or trilinear mass parameters. In such a constrained framework the following SUSY parameters can be complex: the Higgs–Higgsino mass parameter m ; the gaugino mass parameters M1 , M2 and M3 ; the bilin2 ear mixing mass parameter m12 and the trilinear soft supersymmetry breaking parameters At , A t and A b Žand accordingly for the other two generations.. Not all of these phases are physical. Namely, the MSSM has two additional UŽ1. symmetries for vanishing m and soft-breaking terms: the Peccei–Quinn and the R-symmetry. For non vanishing m and soft-breaking terms these symmetries can be used to absorb two of the phases by redefinition of the fields w10x. In addition, the GUT relationship between M1 , M2 and M3 leads to only one common phase for the gaugino mass parameters. Hence, we are left with four independent CP-violating physical phases Žonly two assuming universality in the sfermion sector.. The most significant effect of the CP-violating phases in the phenomenology is their contribution to electric dipole moments ŽEDMs. w11x. Unlike the SM, where the contribution to the EDM of fermions arises beyond two loops w12x, the MSSM can give a contribution already at the one-loop level. The measurement of the neutron EDM w13x constrains the phases and the supersymmetric spectrum in a way that may demand fine tuning Žsupersymmetric CP problem.: phases of O Ž10y2 . w4x, or SUSY particles very heavy Žseveral TeV w14x.. This problem could be solved if soft supersymmetry breaking terms are universal and the genuine SUSY CP phases vanish Žthe Yukawa matrices are then the only source of CP violation, like in the SM.. It has been argued that one could still keep the SUSY phases of O Ž1. and the SUSY spectrum not very heavy and satisfy the experimental bounds due to cancellations among the different components of the neutron EDM w15x. Furthermore it has been recently shown w16x that, even without such cancellations and in the context of non universal soft-breaking terms, the current experimental limits on the neutron EDM can be met with almost no fine tuning on the CP-violating phases Ževen for the first generation ones., at the only price of argŽ m . of O Ž10y2 .. In our analysis we do not assume universality and keep all the SUSY phases as free parameters.
323
As a generalization of the electromagnetic dipole moments of fermions, one can define weak dipole moments ŽWDMs., corresponding to couplings with a Z boson instead of a photon. The most general Lorentz structure of the vertex function that couples a Z boson and two on-shell fermions Žwith outgoing momenta q and q . can be written in terms of form factors Fi Ž s ' Ž q q q . 2 . as
½ ž
GmZ f f s ie gm
ž
FV y
y FA y
Õf 2 sW cW
af 2 sW cW
/
/
g5
q Ž q y q . m w FM q FE g 5 x y Ž q q q . m w FS q FP g 5 x 4 ,
Ž 1.
where Õ f ' Ž I3f y 2 sW2 Q f ., a f ' I3f . The form factors FM and FE are related to the anomalous weak magnetic and electric dipole moments of the fermion f with mass m f as follows: AWMDM' aWf s y2 m f FM Ž MZ2 . , 2 WEDM' d W f s ie FE Ž MZ . .
The FM Ž FE . form factors are the coefficients of the chirality-flipping term of the CP-conserving ŽCPviolating. effective Lagrangian describing Z-fermion couplings. Therefore, they are expected to get contributions proportional to some positive power of the mass of the fermions involved. This allows the construction of observables which can be probed experimentally most suitably by heavy fermions. Hence, for on-shell Z bosons, where the dipole form factors are gauge independent, the b quark and t lepton are the most promising candidates. In this work we concentrate on the analysis of the one-loop contribution of the MSSM with complex parameters and conserved R-parity to the WEDM of the t lepton and the b quark. 6
6
The AWMDM has been considered in Refs. w18x where real supersymmetric couplings were used.
W. Hollik et al.r Physics Letters B 425 (1998) 322–328
324
dW f
2. The WEDM All the possible one-loop contributions to the WEDM can be classified in terms of six classes of triangle diagrams ŽFig. 1.. The vertices are labelled by generic couplings, according to the following interaction Lagrangian, for vectors VmŽ k . s Am , Zm , Wm , Wm†, general fermions C k and general scalars F k :
qieGjk Z mF E F k q eCf Ž S jk y Pjk g 5 . C kF j
½
Ž 2.
The expressions for the WEDM are evaluated in the ’t Hooft–Feynman gauge Žall the would-be-Goldstone bosons must be included. and are written in terms of three-point one-loop tensor integrals C and vertex coefficients: 7 dW f e
a
Ž I. s
4p
½
4 m f Ý Im VjkZ Ž Vf Žj l .AŽflk.) q AŽflj. Vf Žkl .) . jkl
q A Zjk Ž Vf Žj l . Vf Žkl .) q AŽflj. AŽflk.) . = 2Cqy y Cy 2 1 Ž k , j,l . y4 Ý m k Im VjkZ Ž Vf Žj l .AŽflk.) y AŽflj. Vf Žkl .) . jkl
y A Zjk Ž Vf Žj l . Vf Žkl .) y AŽflj. AŽflk.) . = 2Cq 1 y C 0 Ž k , j,l . 4 , dW f e
a
Ž II . s y q
½
4p
Ž 3.
2 m f Ý Im J Ž Vf Žl j.AŽfkl .)
AŽf j.l Vf Žl k .)
jkl
.
4Cqy y Cy 2 1 Ž k , j,l .
q6 Ý m l Im J Ž Vf Žl j.AŽfkl .) y AŽf j.l Vf Žl k .) . jkl
=Cq 1
Ž k , j,l . 4 ,
½
y2 m f Ý Im VjkZ Ž Pl j Sl)k q Sl j Pl)k .
A Zjk
jkl
q Sl j Sl)k q Pl j Pl)k = 2Cqy y Cy k , j,l 2 1 Z q2 m k Im Vjk Pl j Sl)k y Sl j Pl)k
Ž
.
Ž
Ý
.
Ž
.
jkl
q A Zjk Ž Sl j Sl)k y Pl j Pl)k .
½
qeVmŽ k .Cjg m Ž VjlŽ k . y AŽjlk .g 5 . C l
qeK jk Z m VmŽ k .F j q h.c. 4 .
4p
y = Cq Ž 5. 1 q C1 Ž k , j,l . 4 , W df a 2 m f Ý Im Gjk Ž S jl Pk)l q Pjl Sk)l . Ž IV . s e 4p jkl
L s ieJ Ž Wmn† W m Z n y W mn Wm†Zn q Z mn Wm† Wn .
†l j m
e
a
Ž III . s
Ž 4.
7 Equivalent expressions for classes III and IV can be found in Ref. w19x where a different set of generic couplings and tensor integrals is employed.
= 2Cqy y Cy 2 1 Ž k , j,l . y2 Ý m l Im Ž Gjk S jl Pk)l . jkl
= 2Cq 1 y C 0 Ž k , j,l . 4 , W df a Ž V q VI . s y Ý 2Im K jk Ž VfŽlk . Pjl) e 4p jkl q AŽfkl .S jl) .
Ž 6.
y Cq 1 q C1 Ž k , j,l . .
Ž 7. The arguments of the tensor integrals refer to C Ž k, j,l . ' C Žyq,q, Mk , M j , Ml . of Ref. w20x. The tensor integrals are defined in such a way that for qy equal external fermion masses Cy are 1 and C 2 antisymmetric under the interchange of k and j, whereas C0 and Cq 1 are symmetric. The contribution of diagrams of class I and II vanishes as they can only involve SM fermions in the loop ŽMSSM preserves R-parity. whose couplings to gauge bosons are either real Ž Z-exchange. or self-conjugated ŽWexchange.. The gluonic contribution in class I contains only real couplings. For the class V and VI diagrams, the only contribution to the WEDM might occur when a pseudoscalar Higgs boson is involved in the loop, but there is no coupling of two neutral gauge bosons to a pseudoscalar and hence they also vanish. One can easily check that the Higgs sector for both the SM and the MSSM to class III and IV diagrams does not contribute to the WEDM, consistently with the CP-conserving character of both the SM and MSSM Higgs sectors. In a general 2HDM a CP-violating contribution is possible w17x. These considerations lead to the well known result that the SM one-loop contribution to the WEDM is zero. The MSSM contribution comes from charginos, neutrali-
W. Hollik et al.r Physics Letters B 425 (1998) 322–328
325
Fig. 1. The one-loop Zff diagrams.
nos, gluinos and sfermions via diagrams of class III and IV. Finally, notice that all the contributions are proportional to one of the fermion masses involved, as expected from the chirality flipping character of the dipole moments. 8
We make a full scan of the SUSY parameter space and determine the values of the CP-violating phases that yield the maximum effect on the WEDM. The result of this analysis is described below. 3.1. Chargino and scalar neutrino contribution to dtW
3. The WEDM of t lepton and the b quark The conventions for couplings and mixings in the MSSM are the ones in Ref. w5,21x except for the complex character of the m parameter and the trilinear soft supersymmetry breaking parameters At , A t and A b . For convenience, we deal with the following CP-violating phases: 9 wm ' argŽ m ., w f˜ ' f . Ž argŽ m LR f s t , t, b . with m tLR ' A t y m ) cot b and t,b m LR ' At , b y m ) tan b . We assume a common squark mass parameter m q˜ ' m Q˜ s mU˜ s m D˜ as well as a common slepton mass parameter m l˜' m L˜ s m E˜ . We take real gaugino mass parameters constrained by the GUT relations: as M1 s 53 tan2u W M2 , M3 s sW2 M2 . Ž 8. a A ‘‘natural’’ scale for the EDMs is a ‘‘magneton’’ defined by m f ' er2 m f s 1.7 = 10y 15 Ž0.7 = 10y1 5 . ecm for the t lepton Ž b quark.. In the plots the dimensionless quantity d W f rm f is displayed. 8
For the diagrams of class V and VI the chirality flipping occurs at the scalar–fermion vertex and the fermion mass is embedded in S and P, which are in this case Yukawa couplings. 9 Such a choice leads to a dependence on wm of chargino and neutralinos masses. Conversely the sfermion masses are independent on w f˜.
There is only one phase, wm , involved in the chargino contribution as there is no mixing in the scalar neutrino sector. The result grows with tan b . It also depends on the common slepton mass Žwhose effect consists of dumping the result through the tensor integrals. and the < m < and M2 mass parameters. A value wm s pr2 enhances the WEDM. Taking M2 s < m < s 250 GeV, ReŽ dtW wcharginosx. s 0.18 Ž5.52. = 10y6 mt for tan b s 1.6 Ž50. and m l˜s 250 GeV. There is no contribution to the imaginary part assuming the present bounds on the chargino masses. 3.2. Neutralino and t˜ slepton contribution to dtW Now both wm and wt˜ contribute. Assuming that < mtL R < is of the order of < m
The size of < mtL R < is critical for the t sleptons to have a physical mass.
326
W. Hollik et al.r Physics Letters B 425 (1998) 322–328
w x. Fig. 2. The boundaries of the different shaded areas are contour lines in the plane w f˜y wm showing Ža. ReŽ d W b charginos in units of y6 w x. 10y6 m b for low tan b and Žb. ReŽ d W m b for high tan b . M2 s < m < s m q˜ s 250 GeV and < m tL R < s < m
y0.01 Žy0.25. = 10y6 mt for tan b s 1.6 Ž50.. For presently non excluded masses of the neutralinos there can be a contribution to the imaginary part, of the order of 10y6 mt .
3.3. Chargino and t˜ squark contribution to d bW Two CP-violating phases are involved in this contribution: wm and w t˜. In Fig. 2Ža. the dependence
w x. Fig. 3. The boundaries of the different shaded areas are contour lines in the plane M2 y ImŽ m . showing ReŽ d W b charginos in units of 10y6 m b for Ža. low and Žb. high tan b with
W. Hollik et al.r Physics Letters B 425 (1998) 322–328
on these phases is shown, for M2 s < m < s m q˜ s 250 GeV, < m tL R < s < m
327
4. Conclusions
The two relevant CP-violating phases for this case are: wm and w b˜ . As before, the most important effect from w b˜ arises when the off-diagonal term is larger, which in this case corresponds to high tan b , as the trilinear soft breaking parameter is taken to be of the order of < m
Unlike the SM, an R-parity preserving MSSM version with complex parameters contains enough freedom to provide a contribution to the ŽW.EDMs to one loop: considering generation-diagonal trilinear soft-susy-breaking terms, to reduce undesired FCNC, and the GUT constraint between the gauginos mass parameters, at most two Žthree. CP-violating physical phases are available for lepton Žquark. WEDMs, apart from the d CK M . In this work, the one-loop analytical expressions for the ŽW.EDM of fermions in any renormalizable theory are given in terms of a set of generic couplings. Moreover, a full scan of the MSSM parameter space has been performed in search for the maximum effect on the WEDM of the t lepton and the b quark. The Higgs sector does not contribute and chargino diagrams are more important than neutralino ones. Gluinos are also involved in the b case and compete in importance with charginos. In the most favourable configuration of CP-violating phases and for values of the rest of the parameters still not excluded by experiments, these WEDMs can be as much as twelve orders of magnitude larger than the SM predictions, although still far from experimental reach:
3.5. Gluino contribution to d bW
3.4. Neutralino and b˜ squark contribution to d bW
The gluino contribution is affected only by m bL R , m q˜ and the gaugino mass M3 . Therefore the maximum value occurs for w b˜ s pr2. The mixing in the b˜ sector is determined by m bL R and intervenes in the contribution due to chirality flipping in the gluino internal line Žthe contribution proportional to M3 .. The contribution to the AWMDM is enhanced by the largest values of < m bL R < compatible with an experimentally not excluded mass for the lightest b˜ squark. For zero gluino mass, only the term proportional to the mass of the b quark provides a contribution. As we increase the gluino mass, the term proportional to M3 dominates, especially for large < m bL R <, being again suppressed at high M3 due to the gluino decoupling. Thus for w b˜ s pr2 one gets ReŽ d bW wgluinosx. s 0.26 Ž9.31. = 10y6 m b for low Žhigh. tan b and < m bL R < s < m
Acknowledgements We thank T. Gajdosik for discussions on Ref. w19x and T. Hahn for valuable help in the preparation of
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the figures. J.I.I. is supported by the Fundacion ´ Ramon ´ Areces and partially by the spanish CICYT under contract AEN96-1672. S.R. is supported by the Fondazione Ing. A. Gini and by the italian MURST. References w1x N. Cabibbo, Phys. Rev. Lett. 10 Ž1963. 531; M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49 Ž1973. 652. w2x For recent reviews on CP violation see e.g.: K. Gronau, D. London, Phys. Rev. D 55 Ž1997. 2845; Y. Grossman, Y. Nir, R. Rattazzi, hep-phr9701231; Y. Nir, hep-phr9709301; X.G. He, hep-phr9710551. w3x G.R. Farrar, M.E. Shaposhnikov, Phys. Rev. D 50 Ž1994. 774; M.B. Gavela et al., Nucl. Phys. B 430 Ž1994. 382; P. Huet, E. Sather, Phys. Rev. D 51 Ž1995. 379; K. Kajantie, M. Laine, K. Rummukainen, M. Shaposhnikov, Phys. Rev. Lett. 77 Ž1996. 2887. w4x W. Buchmuller, D. Wyler, Phys. Lett. B 121 Ž1983. 321; J. ¨ Polchinski, M.B. Wise, Phys. Lett. B 125 Ž1983. 393; F. del Aguila, M.B. Gavela, J.A. Grifols, A. Mendez, Phys. Lett. B ´ 126 Ž1983. 71; M. Dugan, B. Grinstein, L.J. Hall, Nucl. Phys. B 255 Ž1985. 413. w5x H.E. Haber, G.L. Kane, Phys. Rep. 117 Ž1985. 75; H.P. Nilles, Phys. Rep. 110 Ž1984. 1. w6x P. Huet, A.E. Nelson, Phys. Rev. D 53 Ž1996. 4578; M. Aoki, N. Oshimo, A. Sugamoto, hep-phr9612225; hepphr9706287; hep-phr9706500; M. Carena, M. Quiros, ´ A. Riotto, I. Vilja, C.E.M. Wagner, Nucl. Phys. B 503 Ž1997. 387; G.M. Cline, M. Joyce, K. Kaimulaine, hep-phr9708393. w7x S.A. Abel, J.M. Frere, ´ Phys. Rev. D 55 Ž1997. 1623. w8x S.A. Abel, Phys. Lett. B 410 Ž1997. 173. w9x F. Gabbiani, A. Masiero, Nucl. Phys. B 322 Ž1989. 235; G.S. Hagelin, S. Kelley, T. Tanaka, Nucl. Phys. B 415 Ž1994. 293; F. Gabbiani, E. Gabrielli, A. Masiero, L. Silvestrini, Nucl. Phys. B 477 Ž1996. 321. w10x S. Dimopoulus, S. Thomas, Nucl. Phys. B 465 Ž1996. 23.
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Physics Letters B 425 Ž1998. 322–328
Weak electric dipole moments of heavy fermions in the MSSM W. Hollik a
a,1
, J.I. Illana
a,2
, S. Rigolin
a,b,3
, D. Stockinger ¨
a,4
Institut fur ¨ Theoretische Physik, UniÕersitat ¨ Karlsruhe, D-76128 Karlsruhe, FR Germany b Dipartimento di Fisica, UniÕersita` di PadoÕa and INFN, I-35131 Padua, Italy Received 28 November 1997; revised 9 February 1998 Editor: P.V. Landshoff
Abstract A minimal supersymmetric version of the Standard Model with complex parameters allows contributions to the weak-electric dipole moments of fermions at the one-loop level. Assuming generation-diagonal trilinear soft-susy-breaking terms and the usual GUT constraint, a set of CP-violating physical phases can be introduced. In this paper the general expressions for the one-loop contribution to the WEDM in a generic renormalizable theory are given and the size of the WEDM of the t lepton and the b quark in such a supersymmetric model is discussed. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction In the electroweak Standard Model ŽSM. there is only one source of CP violation, the d CK M phase of the Cabibbo–Kobayashi–Maskawa mixing matrix for quarks w1x. 5 The only place where CP violation has been currently measured, the neutral K system, fixes the value of this phase but does not constitute itself a test for the origin of CP violation w2x. On the other hand, if the baryon asymmetry of the universe has
1
E-mail address: [email protected]. E-mail address: [email protected]. 3 E-mail address: [email protected]. 4 E-mail address: [email protected]. 5 The QCD Lagrangian includes an additional source of CP violation, the u QC D , but we will not consider it here. Extreme fine tuning is needed in order that its contribution to the neutron EDM does not exceed the present experimental upper bound. Various mechanisms beyond the SM have been proposed to solve this problem w2x. 2
been dynamically generated, CP must be violated. The SM cannot account for the size of the observed asymmetry w3x. Many extensions of the SM contain new CP-violating phases, in particular, the supersymmetric models w4x. It has also been shown that the Minimal Supersymmetric Standard Model ŽMSSM. w5x can provide the correct size of baryon asymmetry in some range of parameters if the CPviolating phases are not suppressed w6x. One needs soft-breaking terms to introduce physical phases in the MSSM, different from the d CK M w7x. We assume that the soft-breaking terms preserve R-parity. Other possibilities for CP violation can arise in R-parity violating models Žcf. e.g. w8x in the context of R-parity violating scalar interactions.. For simplicity, we restrict ourselves to generation-diagonal trilinear soft-breaking terms to prevent FCNC. Doing this we ignore CP-violating effects that have been already considered in the literature w9x. In our analysis we do not make any additional assumption, except for the unification of the soft-breaking gaug-
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 2 4 7 - 0
W. Hollik et al.r Physics Letters B 425 (1998) 322–328
ino masses at the GUT scale. However we do not assume unification of the scalar mass parameters or trilinear mass parameters. In such a constrained framework the following SUSY parameters can be complex: the Higgs–Higgsino mass parameter m ; the gaugino mass parameters M1 , M2 and M3 ; the bilin2 ear mixing mass parameter m12 and the trilinear soft supersymmetry breaking parameters At , A t and A b Žand accordingly for the other two generations.. Not all of these phases are physical. Namely, the MSSM has two additional UŽ1. symmetries for vanishing m and soft-breaking terms: the Peccei–Quinn and the R-symmetry. For non vanishing m and soft-breaking terms these symmetries can be used to absorb two of the phases by redefinition of the fields w10x. In addition, the GUT relationship between M1 , M2 and M3 leads to only one common phase for the gaugino mass parameters. Hence, we are left with four independent CP-violating physical phases Žonly two assuming universality in the sfermion sector.. The most significant effect of the CP-violating phases in the phenomenology is their contribution to electric dipole moments ŽEDMs. w11x. Unlike the SM, where the contribution to the EDM of fermions arises beyond two loops w12x, the MSSM can give a contribution already at the one-loop level. The measurement of the neutron EDM w13x constrains the phases and the supersymmetric spectrum in a way that may demand fine tuning Žsupersymmetric CP problem.: phases of O Ž10y2 . w4x, or SUSY particles very heavy Žseveral TeV w14x.. This problem could be solved if soft supersymmetry breaking terms are universal and the genuine SUSY CP phases vanish Žthe Yukawa matrices are then the only source of CP violation, like in the SM.. It has been argued that one could still keep the SUSY phases of O Ž1. and the SUSY spectrum not very heavy and satisfy the experimental bounds due to cancellations among the different components of the neutron EDM w15x. Furthermore it has been recently shown w16x that, even without such cancellations and in the context of non universal soft-breaking terms, the current experimental limits on the neutron EDM can be met with almost no fine tuning on the CP-violating phases Ževen for the first generation ones., at the only price of argŽ m . of O Ž10y2 .. In our analysis we do not assume universality and keep all the SUSY phases as free parameters.
323
As a generalization of the electromagnetic dipole moments of fermions, one can define weak dipole moments ŽWDMs., corresponding to couplings with a Z boson instead of a photon. The most general Lorentz structure of the vertex function that couples a Z boson and two on-shell fermions Žwith outgoing momenta q and q . can be written in terms of form factors Fi Ž s ' Ž q q q . 2 . as
½ ž
GmZ f f s ie gm
ž
FV y
y FA y
Õf 2 sW cW
af 2 sW cW
/
/
g5
q Ž q y q . m w FM q FE g 5 x y Ž q q q . m w FS q FP g 5 x 4 ,
Ž 1.
where Õ f ' Ž I3f y 2 sW2 Q f ., a f ' I3f . The form factors FM and FE are related to the anomalous weak magnetic and electric dipole moments of the fermion f with mass m f as follows: AWMDM' aWf s y2 m f FM Ž MZ2 . , 2 WEDM' d W f s ie FE Ž MZ . .
The FM Ž FE . form factors are the coefficients of the chirality-flipping term of the CP-conserving ŽCPviolating. effective Lagrangian describing Z-fermion couplings. Therefore, they are expected to get contributions proportional to some positive power of the mass of the fermions involved. This allows the construction of observables which can be probed experimentally most suitably by heavy fermions. Hence, for on-shell Z bosons, where the dipole form factors are gauge independent, the b quark and t lepton are the most promising candidates. In this work we concentrate on the analysis of the one-loop contribution of the MSSM with complex parameters and conserved R-parity to the WEDM of the t lepton and the b quark. 6
6
The AWMDM has been considered in Refs. w18x where real supersymmetric couplings were used.
W. Hollik et al.r Physics Letters B 425 (1998) 322–328
324
dW f
2. The WEDM All the possible one-loop contributions to the WEDM can be classified in terms of six classes of triangle diagrams ŽFig. 1.. The vertices are labelled by generic couplings, according to the following interaction Lagrangian, for vectors VmŽ k . s Am , Zm , Wm , Wm†, general fermions C k and general scalars F k :
qieGjk Z mF E F k q eCf Ž S jk y Pjk g 5 . C kF j
½
Ž 2.
The expressions for the WEDM are evaluated in the ’t Hooft–Feynman gauge Žall the would-be-Goldstone bosons must be included. and are written in terms of three-point one-loop tensor integrals C and vertex coefficients: 7 dW f e
a
Ž I. s
4p
½
4 m f Ý Im VjkZ Ž Vf Žj l .AŽflk.) q AŽflj. Vf Žkl .) . jkl
q A Zjk Ž Vf Žj l . Vf Žkl .) q AŽflj. AŽflk.) . = 2Cqy y Cy 2 1 Ž k , j,l . y4 Ý m k Im VjkZ Ž Vf Žj l .AŽflk.) y AŽflj. Vf Žkl .) . jkl
y A Zjk Ž Vf Žj l . Vf Žkl .) y AŽflj. AŽflk.) . = 2Cq 1 y C 0 Ž k , j,l . 4 , dW f e
a
Ž II . s y q
½
4p
Ž 3.
2 m f Ý Im J Ž Vf Žl j.AŽfkl .)
AŽf j.l Vf Žl k .)
jkl
.
4Cqy y Cy 2 1 Ž k , j,l .
q6 Ý m l Im J Ž Vf Žl j.AŽfkl .) y AŽf j.l Vf Žl k .) . jkl
=Cq 1
Ž k , j,l . 4 ,
½
y2 m f Ý Im VjkZ Ž Pl j Sl)k q Sl j Pl)k .
A Zjk
jkl
q Sl j Sl)k q Pl j Pl)k = 2Cqy y Cy k , j,l 2 1 Z q2 m k Im Vjk Pl j Sl)k y Sl j Pl)k
Ž
.
Ž
Ý
.
Ž
.
jkl
q A Zjk Ž Sl j Sl)k y Pl j Pl)k .
½
qeVmŽ k .Cjg m Ž VjlŽ k . y AŽjlk .g 5 . C l
qeK jk Z m VmŽ k .F j q h.c. 4 .
4p
y = Cq Ž 5. 1 q C1 Ž k , j,l . 4 , W df a 2 m f Ý Im Gjk Ž S jl Pk)l q Pjl Sk)l . Ž IV . s e 4p jkl
L s ieJ Ž Wmn† W m Z n y W mn Wm†Zn q Z mn Wm† Wn .
†l j m
e
a
Ž III . s
Ž 4.
7 Equivalent expressions for classes III and IV can be found in Ref. w19x where a different set of generic couplings and tensor integrals is employed.
= 2Cqy y Cy 2 1 Ž k , j,l . y2 Ý m l Im Ž Gjk S jl Pk)l . jkl
= 2Cq 1 y C 0 Ž k , j,l . 4 , W df a Ž V q VI . s y Ý 2Im K jk Ž VfŽlk . Pjl) e 4p jkl q AŽfkl .S jl) .
Ž 6.
y Cq 1 q C1 Ž k , j,l . .
Ž 7. The arguments of the tensor integrals refer to C Ž k, j,l . ' C Žyq,q, Mk , M j , Ml . of Ref. w20x. The tensor integrals are defined in such a way that for qy equal external fermion masses Cy are 1 and C 2 antisymmetric under the interchange of k and j, whereas C0 and Cq 1 are symmetric. The contribution of diagrams of class I and II vanishes as they can only involve SM fermions in the loop ŽMSSM preserves R-parity. whose couplings to gauge bosons are either real Ž Z-exchange. or self-conjugated ŽWexchange.. The gluonic contribution in class I contains only real couplings. For the class V and VI diagrams, the only contribution to the WEDM might occur when a pseudoscalar Higgs boson is involved in the loop, but there is no coupling of two neutral gauge bosons to a pseudoscalar and hence they also vanish. One can easily check that the Higgs sector for both the SM and the MSSM to class III and IV diagrams does not contribute to the WEDM, consistently with the CP-conserving character of both the SM and MSSM Higgs sectors. In a general 2HDM a CP-violating contribution is possible w17x. These considerations lead to the well known result that the SM one-loop contribution to the WEDM is zero. The MSSM contribution comes from charginos, neutrali-
W. Hollik et al.r Physics Letters B 425 (1998) 322–328
325
Fig. 1. The one-loop Zff diagrams.
nos, gluinos and sfermions via diagrams of class III and IV. Finally, notice that all the contributions are proportional to one of the fermion masses involved, as expected from the chirality flipping character of the dipole moments. 8
We make a full scan of the SUSY parameter space and determine the values of the CP-violating phases that yield the maximum effect on the WEDM. The result of this analysis is described below. 3.1. Chargino and scalar neutrino contribution to dtW
3. The WEDM of t lepton and the b quark The conventions for couplings and mixings in the MSSM are the ones in Ref. w5,21x except for the complex character of the m parameter and the trilinear soft supersymmetry breaking parameters At , A t and A b . For convenience, we deal with the following CP-violating phases: 9 wm ' argŽ m ., w f˜ ' f . Ž argŽ m LR f s t , t, b . with m tLR ' A t y m ) cot b and t,b m LR ' At , b y m ) tan b . We assume a common squark mass parameter m q˜ ' m Q˜ s mU˜ s m D˜ as well as a common slepton mass parameter m l˜' m L˜ s m E˜ . We take real gaugino mass parameters constrained by the GUT relations: as M1 s 53 tan2u W M2 , M3 s sW2 M2 . Ž 8. a A ‘‘natural’’ scale for the EDMs is a ‘‘magneton’’ defined by m f ' er2 m f s 1.7 = 10y 15 Ž0.7 = 10y1 5 . ecm for the t lepton Ž b quark.. In the plots the dimensionless quantity d W f rm f is displayed. 8
For the diagrams of class V and VI the chirality flipping occurs at the scalar–fermion vertex and the fermion mass is embedded in S and P, which are in this case Yukawa couplings. 9 Such a choice leads to a dependence on wm of chargino and neutralinos masses. Conversely the sfermion masses are independent on w f˜.
There is only one phase, wm , involved in the chargino contribution as there is no mixing in the scalar neutrino sector. The result grows with tan b . It also depends on the common slepton mass Žwhose effect consists of dumping the result through the tensor integrals. and the < m < and M2 mass parameters. A value wm s pr2 enhances the WEDM. Taking M2 s < m < s 250 GeV, ReŽ dtW wcharginosx. s 0.18 Ž5.52. = 10y6 mt for tan b s 1.6 Ž50. and m l˜s 250 GeV. There is no contribution to the imaginary part assuming the present bounds on the chargino masses. 3.2. Neutralino and t˜ slepton contribution to dtW Now both wm and wt˜ contribute. Assuming that < mtL R < is of the order of < m
The size of < mtL R < is critical for the t sleptons to have a physical mass.
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w x. Fig. 2. The boundaries of the different shaded areas are contour lines in the plane w f˜y wm showing Ža. ReŽ d W b charginos in units of y6 w x. 10y6 m b for low tan b and Žb. ReŽ d W m b for high tan b . M2 s < m < s m q˜ s 250 GeV and < m tL R < s < m
y0.01 Žy0.25. = 10y6 mt for tan b s 1.6 Ž50.. For presently non excluded masses of the neutralinos there can be a contribution to the imaginary part, of the order of 10y6 mt .
3.3. Chargino and t˜ squark contribution to d bW Two CP-violating phases are involved in this contribution: wm and w t˜. In Fig. 2Ža. the dependence
w x. Fig. 3. The boundaries of the different shaded areas are contour lines in the plane M2 y ImŽ m . showing ReŽ d W b charginos in units of 10y6 m b for Ža. low and Žb. high tan b with
W. Hollik et al.r Physics Letters B 425 (1998) 322–328
on these phases is shown, for M2 s < m < s m q˜ s 250 GeV, < m tL R < s < m
327
4. Conclusions
The two relevant CP-violating phases for this case are: wm and w b˜ . As before, the most important effect from w b˜ arises when the off-diagonal term is larger, which in this case corresponds to high tan b , as the trilinear soft breaking parameter is taken to be of the order of < m
Unlike the SM, an R-parity preserving MSSM version with complex parameters contains enough freedom to provide a contribution to the ŽW.EDMs to one loop: considering generation-diagonal trilinear soft-susy-breaking terms, to reduce undesired FCNC, and the GUT constraint between the gauginos mass parameters, at most two Žthree. CP-violating physical phases are available for lepton Žquark. WEDMs, apart from the d CK M . In this work, the one-loop analytical expressions for the ŽW.EDM of fermions in any renormalizable theory are given in terms of a set of generic couplings. Moreover, a full scan of the MSSM parameter space has been performed in search for the maximum effect on the WEDM of the t lepton and the b quark. The Higgs sector does not contribute and chargino diagrams are more important than neutralino ones. Gluinos are also involved in the b case and compete in importance with charginos. In the most favourable configuration of CP-violating phases and for values of the rest of the parameters still not excluded by experiments, these WEDMs can be as much as twelve orders of magnitude larger than the SM predictions, although still far from experimental reach:
3.5. Gluino contribution to d bW
3.4. Neutralino and b˜ squark contribution to d bW
The gluino contribution is affected only by m bL R , m q˜ and the gaugino mass M3 . Therefore the maximum value occurs for w b˜ s pr2. The mixing in the b˜ sector is determined by m bL R and intervenes in the contribution due to chirality flipping in the gluino internal line Žthe contribution proportional to M3 .. The contribution to the AWMDM is enhanced by the largest values of < m bL R < compatible with an experimentally not excluded mass for the lightest b˜ squark. For zero gluino mass, only the term proportional to the mass of the b quark provides a contribution. As we increase the gluino mass, the term proportional to M3 dominates, especially for large < m bL R <, being again suppressed at high M3 due to the gluino decoupling. Thus for w b˜ s pr2 one gets ReŽ d bW wgluinosx. s 0.26 Ž9.31. = 10y6 m b for low Žhigh. tan b and < m bL R < s < m
Acknowledgements We thank T. Gajdosik for discussions on Ref. w19x and T. Hahn for valuable help in the preparation of
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