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Bounds on R-Parity Violating Parameters from Fermion EDM's


CUMQ/HEP 97 February 1, 2008

Bounds on R-Parity Violating Parameters from Fermion EDM’s
arXiv:hep-ph/9706510v1 25 Jun 1997
M. Franka,1 and H. Hamidianb,2
a

Department of Physics, Concordia University, 1455 De Maisonneuve Blvd. W. Montreal, Quebec, Canada, H3G 1M8 b Department of Physics, Stockholm University, Box 6730, S-113 85 Stockholm, Sweden

Abstract
We study one-loop contributions to the fermion electric dipole moments in the Minimal Supersymmetric Standard Model with explicit R-parity violating interactions. We obtain new individual bounds on R-parity violating Yukawa couplings and put more stringent limits on certain parameters than those obtained previously.

1 2

e-mail mfrank@vax2.concordia.ca e-mail hamidian@vanosf.physto.se

Introduction
The Standard Model (SM) of electroweak interactions is constructed in such a way that it automatically conserves both the baryon number B and the lepton ?avor number L. These (accidental) global symmetries of the SM result from the particle content and the SU(2)L × U(1)Y gauge invariance of the theory and naturally explain the non-observation of B- and L-violating processes, as well as the stability of the nucleon. However, in supersymmetric (SUSY) extensions of the SM these features no longer follow. In fact, by promoting the SM ?elds to super?elds, additional gauge- and Lorentz-invariant terms will be generated which violate B and L conservation. For example, in the SM the Higgs doublet and the are di?erent; whereas in the SUSY extensions of the SM the distinction between the Higgs doublet and the lepton doublet disappears and one would naturally expect lepton number violation. In order to forbid B- and L-violating interactions in SUSY extensions of the SM a parity quantum number de?ned as R = (?1)3B+L+2S , where S is the spin, is assigned to each component ?eld and invariance under R transformation is imposed [1]. Although the assignment of the ad hoc R-parity to super?elds in the SUSY extensions of the SM reproduces the global B and L U(1) symmetries of the SM, it is by no means the only symmetry which allows the construction of phenomenologically viable SUSY extensions of the SM [2]. In fact, from a phenomenological point of view, it is most important to ensure that there are no interaction terms in the Lagrangian which lead to rapid proton decay and, in this respect, other discrete symmetries can be used which are even more e?ective than R-parity. This is simply due the fact that R-parity only forbids dimension-four Band L-violating operators in the Lagrangian, while dimension-?ve operators can still remain dangerous, even if suppressed with cut-o?’s as large as the Planck mass (see Aulakh et al. in Ref.[1] for an interesting alternative to R-parity which forbids dimension-four and dimension?ve B-violating interactions while allowing L-violating operators of the same dimensions in the Lagrangian). Since there is no fundamental theoretical basis for imposing R-parity conservation on the minimal supersymmetric extension of the Standard Model (MSSM), it is worthwhile to investigate the phenomenological constraints on theoretically allowed R parity breaking couplings in the MSSM. Unlike the SM in which all the elementary matter ?elds are fermions and, consequently, B lepton doublet have the same SU(2) × U(1) quantum numbers, but the spins of the particles

1

and L quantum numbers are separately conserved, the MSSM has a particle content which contains scalar leptons and quarks, thus allowing separate B and L violating interaction terms in the Lagrangian. Using only the MSSM super?elds, the most general renormalizable R-parity violating superpotential can be written as:
′ ′′ ? ? ? ? ? W = λijk Li Lj Ek + λijk Li Qj Dk + λijk Ui Dj Dk ,

(1)

where i, j, k are generation indices and we have rotated away a term of the form ?ij Li Hj . ? ? In (1) L and Q denote the lepton and quark doublet super?elds respectively, and Ui , Dj and
′′ ? Dk are singlet SU(2) super?elds. The couplings λijk and λijk are antisymmetric with respect

to the interchange of SU(2) ?avor indices: λijk = ?λjik and λijk = ?λikj .

′′

′′

To avoid rapid proton decay, it is not possible for both λ, λ′ type and λ′′ type couplings

to have nonzero values. We shall assume here—as is often done in the literature—that only the lepton number is violated (see [3] for restrictions on the λ′′ type couplings). Viewing the SM as a low-energy e?ective theory, one often searches for potential contributions arising from the physics beyond the SM. In this manner, numerous studies on R-parity violating decays have either resulted in separate bounds on the λ, λ′′ couplings, or on their products. Some of the most important studies involve limits coming from proton stability, n–? oscillations, νe -Majorana mass, neutrinoless double β decays, charged current univern sality, ν? –e deep inelastic scattering, atomic parity violation, e–?–τ universality, K + -decays, τ -decays, D-decays and precision measurements of LEP electroweak observables [4]. In addition, when the assumption of R-parity conservation is relaxed, the superpartner spectrum for the MSSM is expected to be dramatically di?erent from the one with R-conservation, the most important consequence being that the lightest supersymmetric particle can decay. The strongest bounds so far on the λ and λ′ couplings come from cosmological considerations on the survival of the cosmic ?B [5]. They are usually obtained assuming the L-violating interactions to be constantly out of equilibrium until the weak scale, so that they where the L-violating interactions are allowed to survive at the weak scale to give rise to lepton-number violations. Lepton-number violating interactions in the context of R-parity breaking have been studied recently and more stingent bounds than those reported earlier have been found [6]. 2 cannot wash out any (B ? L) asymmetry previously generated. They are relaxed in the case

The most precise low-energy measurements in leptonic physics are the lepton ?avora? , and the EDM of the electron. These are all very sensitive probes and are often used violating decays such as ? → eγ, ? → eee, the anomalous magnetic moment of the muon

to explore physics beyond the SM. In particular, the electric dipole moment (EDM) of the leptons (especially the electron) and the neutron are strictly bound by experiments with the currently available upper limits given by[7], de d? dτ for the leptons, and dn < 1.1 × 10?25 e cm (3) = (3 ± 8) × 10?27 e cm, = (3.7 ± 3.4) × 10?19 e cm, < (3.7 ± 3.4) × 10?17 e cm, (2)

for the neutron. In this letter we investigate bounds on R-parity violating interactions by using the available experimental upper limits on the EDM’s of the fermions. As we shall discuss below, the individual bounds that we put on certain R-parity violating couplings are more stringent than those found prior to this work. We shall also brie?y discuss the signi?cance of the EDM bounds compared to the ones obtained by using the—also accurately measured—values of the lepton anomalous magnetic moments.

R-Parity Violating Interactions and Fermion EDM’s
We shall begin with a brief review of the fermion EDM’s and then proceed to evaluate the leading order contributions that arise by including R-parity violating interactions in the MSSM. The electric dipole moment of an elementary fermion is de?ned through its electromagnetic form factor F3 (q 2 ) found from the (current) matrix element f (p′ )|J? (0)|f (p) = u(p′ )Γ? (q)u(p), ? where q = p′ ? p and Γ? (q) = F1 (q 2 )γ? + F2 (q 2 )iσ?ν q ν /2m + FA (q 2 )(γ? γ5 q 2 ? 2mγ5 q? ) + F3 (q 2 )σ?ν γ5 q ν /2m, (5) 3 (4)

with m the mass of the fermion. The EDM of the fermion ?eld f is then given by df = ?F3 (0)/2m, corresponding to the e?ective dipole interaction i ? LI = ? df fσ?ν γ5 f F ?ν 2 in the static limit. Since a non-vanishing df in the SM results in fermion chirality ?ip, it requires both CP violation and SU(2)L symmetry breaking. Even if one allows for CP -violation in the leptonic sector of the SM, the lepton EDM’s vanish to one-loop order due to the cancellation of all the CP -violating phases. Two-loop calculations for the electron [8] and for quarks [9] also yield a zero EDM. In the MSSM, however, there are many more sources of CP violation than in the SM. In addition to the usual Kobayashi-Maskawa phase δ from the quark mixing matrix, there are phases arising from complex parameters in the superpotential and in the soft supersymmetry breaking terms. The phases of particular interest to us are those coming from the so-called A-terms, Au,d = |Au,d | exp (iφAu ,d ) [10]. The CP -violating e?ects arise from the squark mass matrix which has the following form: LMu = (?? u? ) uL ? R ? A? mu ?2 + m2 u L u 2 Au mu ?R + m2 , u uL ? uR ? (7) (6)

,

(8)

and similarly for LMd , where the mass parameters |Au |, ?L and ?R are expected to be of the ? order of the W -boson mass MW . The ?elds uL , uR can be transformed into mass eigenstates ? ? u1 , u2 , ? ? 1 uL = exp(? iφAu )(cos θ u1 + sin θ u2 ), ? ? ? 2 1 ? ? uR = exp( iφAu )(cos θ u2 ? sin θ u1 ), ? 2 where the mixing angle θ is given by tan 2θ = 2|Au |mu /(?2 ? ?2 ). L R are 1 2 M1,2 = ?2 + ?2 + 2m2 ± [(?2 ? ?2 )2 + 4m2 |Au |2 ]1/2 . R u L R u 2 L 4 (11) (10)

(9)

and the physical masses, M1,2 , corresponding to the eigenvalues of the mass matrix in (8)

The lepton EDM’s at one-loop order are generated by the interactions in Fig.1 (with similar Feynman diagrams for the muon and the tau) and resemble those in the MSSM with charginos or neutralinos in the loop. The contributions to the lepton EDM’s are then given by 4e mdk |Auj | sin θ cos θ sin(φAu )f3 (xdk ) 3 m3 ? f

dei = ? |λ′ijk |2 ? |λ′ijk |2

2e mdk |Auj | sin θ cos θ sin(φAu )f4 (xdk ) 3 m3 ? f 2e muj ? |λ′ijk |2 |Adk | sin θ cos θ sin(φAd )f3 (xuj ) 3 m3 ? f 4e muj |Adk | sin θ cos θ sin(φAd )f4 (xuj ), ? |λ′ijk |2 3 m3 ? f

(12)

? where xu,d = (mu,d /mf?)2 , with f the scalar quark in the loop, and the loop integrals are expressed in a familiar form in terms of the functions 1 1+x+ 2(1 ? x)2 1 f4 (x) = 3?x+ 2(1 ? x)2 f3 (x) = 2x ln x 1?x 2 ln x . 1?x

(13)

In order to simplify, we will assume degenerate squark masses, ?L ≈ ?R ≈ |Au,d | = O(MW ),
2 and expand only to the leading order in mu,d |Au,d |/M1,2. Comparing the above expressions

for the fermion EDM’s with the usual ones obtained in the MSSM [11], it is not di?cult to see that there are two sources of enhancement: one coming from the absence of the electroweak coupling constant, αew , responsible for an enhancement of O(102 ), and another from potentially large fermionic masses in the loop. Indeed, in this scenario it is possible to obtain a contribution to the electron EDM proportional to the mass of the top quark, in contrast to the usual one proportional to me . (Note that even if the EDM is proportional to the mass

de , d? , and dτ numerically is not completely straightforward since no model-independent

of the up quark, there will still be an enhancement of O(20).) Unfortunately estimating

experimental information is available on squark masses and mixing angles. We shall assume, √ without great loss of generality, that cos θ = sin θ = 1/ 2. We shall also assume, φAu = φAd and |Auj ,dk | ≈ mf? = O(MW ), which is in agreement with the naturalness of the MSSM. 5

Putting all these together, Eq. (12) becomes mf? |Au,d | 1 )?3 ( ) sin(φA ) dei = ? |λ′ijk |2 ( 3 100GeV 100GeV × mdk 4f3 (xuj ) + 2f4 (xuj ) + muj [2f3 (xdk ) + 4f4 (xdk )] × 10?21 e cm. (14) We would like to comment that in addition to these contributions, in theories with massive neutrinos one could have contributions coming from the Feynman diagrams such as those in Fig.2. These are not included here since we restrict ourselves to the particle spectrum of the MSSM. Including these contributions in other SUSY extensions of the SM could provide restrictions on the λijk parameters, however they would all depend rather sensitively on the neutrino mass. Any estimate of the lepton EDM’s must be correlated and further restricted by estimates of the neutron EDM. In the R-parity-conserving MSSM the neutron EDM severely restricts the masses of the squarks, barring accidental cancellations between the supersymmetric phases in the squark and gluino matrices. With the introduction of R-parity violating interactions, the terms contributing to the quark EDM’s are Yukawa-type only, such as those shown in Fig.3 for the up quark. Taking all the scalar quark masses to be the same, and taking mu ? md = 10 MeV , the up- and down- quark EDM’s are equal and the neutron EDM is dn = EDM is |dn | < 1.2 × 10 conserving MSSM. The limits that we obtain on the individual λ′ijk parameters from the electron (λ′1jk ), muon (λ′2jk ) EDM’s are given in Table 1. Unfortunately, the weak constraint on the tau EDM is insu?cient to adequately restrict all the λ′3jk coe?cients. Two cases are considered in Table 1: (i) light slepton spectrum (mf? = 100 GeV ) and (ii) heavy slepton spectrum (mf? = 1 T eV ). In both cases we ?nd strong bounds from the electron EDM and weaker ones from the ? or τ EDM’s. The strongest limits appear to be the ones on λ1j3 because of with other estimates on the λ′ parameters. The di?erent scenarios bridge the gap of any other possible assumptions on the superparticle spectrum. For instance, if instead of assuming a 6 e?ects of O(mt ). The di?erence between a light and a heavy squark spectrum is in agreement
4 dd 3 ?25

e cm the limits obtained on the λijk parameters are weaker than

?

1 3

du ? dd . Since the experimental limit on the neutron

those obtained from the electron EDM, in contrast with the existing situation in R-parity

degenerate squark mass spectrum, we assume a universal scalar mass spectrum at the GUT scale and then evolve masses at the electroweak scale as in [12], our assumptions resemble very closely the case in which tan β ≤ 10 . In this case the squarks of the ?rst two families and ?R have very similar masses and only the stop is heavier. That would a?ect mostly the b λi3k coe?cients; the results will be of the same order of magnitude with slightly di?erent numerical factors. If we assume, as has been sometimes done in the literature, that the λ′ijk are ?avor blind, i.e. λ′ijk = λ′ , we will be able to severely restrict all the R-parity violating couplings, as shown in Figures 4 and 5 respectively for the light and heavy squark mass scenarios. In both cases the restrictions on the λ′ijk couplings are more stringent than previously found limits. The EDM’s do not, unfortunately, provide any limits on the λijk or λ′′ couplings, ijk so one would have to rely on limits obtained by considering other phenomena, such as those mentioned in the introduction. We shall now brie?y comment on possible restrictions coming from the anomalous magand its measured deviation from the SM prediction lies within a range of ?2 × 10?8 ≤ ?aexp ≤ 2.6 × 10?8 . The one-loop contributions to the muon magnetic moment also come ? from Feynman diagrams similar to those in Fig.1 and are given by a? = F2 (0)/e, netic moment of the muon, a? . Its experimental value is aexp = 1165922(9) × 10?9 [7] ?

(15)

where F2 (0) is the static limit of the electromagnetic form factor de?ned in (5). Unfortunately, the restrictions that a? would place on the R-parity violating couplings λ and λ′ are extremely weak compared to those coming from the EDM’s. The EDM of the electron is expected to be known to about ?ve orders of magnitude more accurately than its anomalous factor yield ?[(g ? 2)/2] = 1 × 10?11 , which corresponds to ?(F2 (0)/2me ) = 2 × 10?22 e cm. magnetic moment within a few years. Indeed the precise measurements of the electron (g ?2)

The same is true for the anomalous magnetic moment of the muon [13]. In addition, for the calculation of the form factor F2 (q 2 ) for the electron (muon) the helicity ?ip occurs on the external fermion line, giving rise to an amplitude proportional to the electron (muon) mass. These two factors combined, give weaker bounds on the R-parity violating Yukawa couplings than previously found, despite the fact that the anomalous magnetic moment does not require CP violation. 7

To conclude, we have studied one-loop contributions to the lepton and neutron EDM’s in the MSSM with explicit R-parity violating interactions. We have used the—very accurately measured—experimental values of the lepton EDM’s to severely restrict the magnitude of the R-parity violating Yukawa couplings. This analysis has allowed us to obtain new bounds on individual λ′ijk couplings, rather than their combinations with other parameters. Acknowledgement This work was supported by NSERC and the Swedish Natural Science Research Council. M.F. would like to thank the Field Theory and Particle Physics Group of Stockholm University (where this work was initiated) for their hospitality.

References
[1] C.S. Aulakh and R.N. Mohapatra, Phys. Lett. B119, 136 (1982); L.J. Hall and M. Suzuki, Nucl. Phys. B231, 419 (1984); J. Ellis et al., Phys. Lett. B150, 142 (1985); G.G. Ross and J.W.F. Valle, Phys. Lett. B151, 375 (1985); S. Dawson, Nucl. Phys. B261, 297 (1985); R. Barbieri and A. Masiero, Nucl. Phys. B267, 679 (1986). [2] L.E. Iba? ez and G.G. Ross, Nucl. Phys. B368, 3 (1992). n [3] D. Chang and W-Y. Keung , hep-ph/9608313 (1996). A.Y. Smirnov and F. Vissani, Phys. Lett. B380, 317 (1996). [4] For an up-to-date review and further references, see G. Bhattacharyya in Proceedings of the 4-th International Conference on Supersymmetry, (SUSY 96), Nucl. Phys. Proc. Suppl. 52A 83 (1997). [5] B. Campbell, S. Davidson, J. Ellis and K.A. Olive Phys. Lett. B256, 457 (1991). [6] M. Chaichian, K. Huitu Phys. Lett. B384, 157 (1996). [7] Particle Data Group, Phys. Rev. D 54, 1 (1996). [8] J.F. Donoghue, Phys. Rev. D 18, 1632 (1978). [9] E.P. Shabalin, Yad. Fiz. 28, 151 [Sov. J. Nucl. Phys. 28, 75 (1978)]. 8

[10] For a complete analysis of phases in supersymmetric theories see M. Dugan, B. Grinstein, L.J.Hall Nucl. Phys. B255, 416 (1985). [11] Y. Kizukuri and N. Oshimo Phys. Rev. D 46, 3025 (1992). [12] S.P. Martin, P.Ramond Phys. Rev. D 48, 5365 (1993). [13] W. Bernreuther, M. Suzuki Rev. Mod. Phys. 63, 313 (1991). [14] K. Agashe and M. Graesser, hep-ph/9510439. [15] V. Barger, G.F. Giudice and T.H. Tan, Phys. Rev. D 40, 2987 (1989). [16] R.M. Godbole, R.P. Roy and X. Tata, Nucl. Phys. B401, 67 (1993).

9

Figure Captions:
Figure 1: Graphs contributing to the electron EDM that put limits on the value of the λ′1jk couplings. Similar graphs contribute to the ?- and τ -EDM’s and put limits on λ′2jk and λ′3jk respectively. Figure 2: Graphs that could contribute to the electron EDM in theories with massive neutrinos. Similar graphs contribute to the ?- and τ -EDM’s. These graphs could restrict the values of the λijk couplings. Figure 3: Graphs contributing to up quark EDM’s and therefore to the neutron EDM. For the down quark the same graphs contribute, with d ? u and e ? ec . Figure 4: The electron EDM as a function of a universal coupling λ′ijk = λ′ (horizontal axis) for the light squark scenario , mf? = 100 GeV . We take the following values for the quark masses masses: mu = md = 10 MeV , ms = 300 MeV , mc = 1.5 GeV , mb = 4.5 GeV and mt = 175 GeV [7]. Figure 5: The electron EDM as a function of a universal coupling λ′ijk = λ′ (horizontal axis) for the heavy squark scenario , mf? = 1 T eV . We take the following values for the quark masses masses: mu = md = 10 MeV , ms = 300 MeV , mc = 1.5 GeV , mb = 4.5 GeV and mt = 175 GeV [7].

Table Captions:
Table 1: Bounds on the R-parity violating parameters, λ′ijk , from the electron (λ′1jk ) and muon (λ′2jk ) EDM’s, compared with previous bounds obtained from: (a) K + -decay [14]; (b) Atomic parity violation and eD asymmetry [15]; (c) t-decay [14]; (d) νe -Majorana mass [16]; (e) ν? deep-inelastic scattering [15]. 10

Table 1 |λ′ijk |2 ≤ |λ′111 |2 |λ′112 |2 |λ′113 |2 |λ′121 |2 |λ′122 |2 |λ′123 |2 |λ′131 |2 |λ′132 |2 |λ′133 |2 |λ′211 |2 |λ′212 |2 |λ′213 |2 |λ′221 |2 |λ′222 |2 |λ′223 |2 |λ′231 |2 |λ′232 |2 |λ′233 |2 mf? = 100 GeV 3 × 10?9 5 × 10?10 9 × 10?11 7.5 × 10?11 6.6 × 10?11 4 × 10?11 2.6 × 10?12 2.6 × 10?12 2.5 × 10?12 3 × 10?1 5 × 10?2 9 × 10?3 7.5 × 10?3 6.6 × 10?3 4 × 10?3 2.6 × 10?4 2.6 × 10?4 2.5 × 10?4 mf? = 1 T eV 2.4 × 10?7 3 × 10?8 4 × 10?9 4.3 × 10?9 4 × 10?9 2 × 10?9 2.4 × 10?11 2.4 × 10?11 2.3 × 10?11 24 3 4 × 10?1 4.3 × 10?1 4 × 10?1 2 × 10?1 2.4 × 10?3 2.4 × 10?3 2.3 × 10?3 Previous Limits for mf? = 100 GeV 1.44 × 10?4 (a) 1.44 × 10?4 1.44 × 10?4 (a) 1.44 × 10?4 (a) 1.44 × 10?4 (a) 1.44 × 10?4 (a) ?4 1.44 × 10 (a), 0.0676 (b) 1.44 × 10?4 (a), 0.16 (c) 1.44 × 10?4 (a), 10?6 (d) 1.44 × 10?4 (a) 1.44 × 10?4 (a) 1.44 × 10?4 (a) 1.44 × 10?4 (a) 1.44 × 10?4 (a) 1.44 × 10?4 (a) 1.44 × 10?4 (a), 0.0484 (e) 1.44 × 10?4 (a), 0.16 (c) 1.44 × 10?4 (a), 0.16 (c)

11

γ ~c u eL d γ uc eL ~ d uc eR eL ~c u eR eL ~ d

γ ~ d u
c

eL

γ d ~c u d eR

Figure 1

γ ~ e eL ν ~ e eR
Figure 2

γ e eL ~ ν e eR

γ ~ ec uL d γ d uL ~c e d uR uL ~c e uR u L ~ d

γ d ec γ ec ~ d ec uR uR

Figure 3

EDM x 10^27

10

8

6

4

2

-7 5. 10 1. 10

-6 1.5 10

-6 2. 10

-6 2.5 10

-6 3. 10

-6

Figure 4

EDM x 10^27 12

10

8

6

4

2

-6 5. 10

0.00001

0.000015

0.00002

0.000025

0.00003

Figure 5


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