We calculate the electric dipole moment of electron using the experimental
result of muon electric dipole moment and upper limit of the BR(\mu -->
e\gamma) in the framework of the general two Higgs doublet model. Our
prediction is 10^{-32} e-cm, which lies in the experimental current limits.
Further, we obtain constraints for the Yukawa couplings \bar{\xi}^{D}_{N,\tau
e} and \bar{\xi}^{D}_{N,\tau\mu}. Finally we present an expression which
connects the BR(\tau\to \mu\gamma) and the electric dipole moment of
\tau-lepton and study the relation between these physical quantities.Comment: 8 pages, 4 Figures (required epsf sty

D. Chang

D. E. Groom

E. D. Commins

E. O. Iltan

G. C. Branco

G. Couture

J. Bailey

J. Liu

K. Abdullah

K. S. Babu

M. L. Brooks

R. Barbieri

R. Diaz

S. Ahmed

S. M. Barr

S. Weinberg

T. P. Cheng

T. V. Duong

Y. Okada

English

arXiv

Crossref

Physical Review D

Electric dipole moments of charged leptons and lepton flavor violating interactions in the general two Higgs doublet model

We calculate the electric dipole moment of the electron using the experimental result of the muon electric dipole moment and upper limit of the BR(mu-->e gamma) in the framework of the general two Higgs doubler model. Our prediction is 10(-32) e cm, which lies in the experimental current limits. Further, we obtain constraints for the Yukawa couplings )over bar>(D)(N,taue) and )over bar>(D)(N,tau mu). Finally, we present an expression which connects the BR(tau-->mu gamma) and the electric dipole moment of the tau lepton and study the relation between these physical quantities

Iltan, EO

OpenMETU (Middle East Technical University)

arXiv:hep-ph/0101017v2  17 May 2001Electric Dipole moments of charged leptonsand lepton flavor violating interactions in thegeneral two Higgs Doublet modelE. O. Iltan ∗Physics Department, Middle East Technical UniversityAnkara, TurkeyAbstractWe calculate the electric dipole moment of electron using the experimental result ofmuon electric dipole moment and upper limit of the BR(µ → eγ) in the framework ofthe general two Higgs doublet model. Our prediction is 10−32 e − cm, which lies in theexperimental current limits. Further, we obtain constraints for the Yukawa couplingsξ̄DN,τe and ξ̄DN,τµ. Finally we present an expression which connects the BR(τ → µγ) andthe electric dipole moment of τ -lepton and study the relation between these physicalquantities.∗E-mail address: eiltan@heraklit.physics.metu.edu.tr1 IntroductionLepton Flavor Violating (LFV) interactions are one of the important source of physics beyondthe Standard model (SM) and have reached great interest with the improvement of experimentalmeasurements. The processes µ → eγ and τ → µγ are the examples of LFV interactions and thecurrent limits for their branching ratios (BR) are 1.2×10−11 [1] and 1.1×10−6 [2] respectively.In such decays, the assumption of the non-existence of Cabibbo-Kobayashi-Maskawa (CKM)type matrix in the leptonic sector forbids the charged Flavor Changing (FC) interactions andtherefore the physics beyond the SM plays the main role, where the general two Higgs doubletmodel (2HDM), so called model III, is one of the candidate. In this model, LFV interactionscan exist at loop level with the help of the internal neutral higgs bosons h0 and A0. The Yukawacouplings appearing in these loops are free parameters and their strength can be deteminedby the experimental data. In the literature, there are several studies on LFV interactionsin different models. Such interactions are studied in a model independent way in [3], in theframework of model III 2HDM [4], in supersymmetric models [5, 6, 7, 8, 9, 10, 11].CP violating effects also provide comprehensive information in the determination of freeparameters of the various theoretical models. In extensions of the SM, new source of CP-violating phases occur. The exchanges of neutral Higgs bosons in the 2HDM [12] or the chargedsector in multi Higgs doublet models (more than two Higgs doublets) [13] induce CP violatingeffects. In the model III 2HDM, one of the source for the CP violation is the complex Yukawacoupling. Non-zero Electric Dipole Moments (EDM) of the elementary particles are the signof such violation. These are interesting from the experimental point of view since there areimprovements in the experimental limits of charged lepton EDM. EDM of electron, muonand tau have been measured and the present limits are de = 1.8 ± 1.2 ± 1.0 × 10−27 [14],dµ = (3.7± 3.4)× 10−19 [15] and dτ = 3.1× 10−16 [16].Dipole moments of leptons have been studied in the literature extensively. In [17], it isemphasized that the dominant contribution to the EDM of lighter leptons comes from two loopdiagrams that involve one power of the Higgs Yukawa couplings. In this work, the CP-violationis assumed to be mediated by neutral Higgs scalars [18] and the EDM of electron is predictedat the order of the magnitude of 10−26 e − cm. Further EDM of leptons have been analyzedin supersymmetric models [6, 11, 19]. In the recent work [20] EDM of leptons are studiedby scaling them with corresponding lepton masses and the EDM of electron is predicted as10−27 e− cm.In our work, we study EDM of leptons e, µ, τ and LFV processes µ → eγ, τ → µγ in the1general 2HDM (model III). The source of EDM of a particle is the CP violated interactionand it can come from the complex Yukawa couplings. In the model III, it is possible to get aconsiderable EDM at one-loop level. Further, LFV processes can exist also at one-loop levelwith internal mediating neutral particles h0 and A0 since there is no CKM type matrix andtherefore no charged FC interaction in the leptonic sector according to our assumption.The paper is organized as follows: In Section 2, we present EDMs of e, µ, τ leptons and theexpressions for the decay width of processes µ → eγ and τ → µγ in the framework of the modelIII. Section 3 is devoted to discussion and our conclusions.2 Electric dipole moments of charged leptons and LFVinteractions in the general two Higgs Doublet model.The 2HDM type III permits flavor changing neutral currents (FCNC) at tree level and makesCP violating interactions possible with the choice of complex Yukawa couplings. The Yukawainteraction for the leptonic sector in the model III isLY = ηDij l̄iLφ1EjR + ξDij l̄iLφ2EjR + h.c. , (1)where i, j are family indices of leptons, L and R denote chiral projections L(R) = 1/2(1∓γ5), φifor i = 1, 2, are the two scalar doublets, liL and EjR are lepton doublets and singlets respectively.Here φ1 and φ2 are chosen asφ1 =1√2[(0v +H0)+( √2χ+iχ0)];φ2 =1√2( √2H+H1 + iH2), (2)and the vacuum expectation values are< φ1 >=1√2(0v);< φ2 >= 0 . (3)With this choice, the SM particles can be collected in the first doublet and the new particlesin the second one. The part which produce FCNC at tree level isLY,FC = ξDij l̄iLφ2EjR + h.c. . (4)Here the Yukawa matrices ξDij have in general complex entries. Note that in the following wereplace ξD with ξDN where ”N” denotes the word ”neutral”. The complex Yukawa couplings arethe source of CP violation and EDM of particles are created by the CP violated interactions.The effective EDM interaction for a lepton is defined asLEDM = idl l̄ γ5 σµν l F µν , (5)2where Fµν is the electromagnetic field tensor and ”dl” is EDM of the lepton. Here, ”dl” is a realnumber by hermiticity. Concentrating only on neutral currents in the model III, the neutralHiggs bosons h0, A0 can induce CP violating interactions which can create EDM at loop level.Note that, we take H1 and H2 in eq. (2) as the mass eigenstates h0 and A0 respectively sinceno mixing between CP-even neutral Higgs bosons h0 and the SM one, H0, occurs at tree level.Due to possible small mixing at loop level we also study the mixing effects on the physicalparameters in the Discussion part.Now, we give the necessary 1-loop diagrams due to neutral Higgs particles in Fig. 1. Since,in the on-shell renormalization scheme, the self energy∑(p) can be written as∑(p) = (p̂−ml) ¯∑(p)(p̂−ml) , (6)diagrams a, b in Fig. 1 vanish when l-lepton is on-shell. However the vertex diagram c in Fig.1 gives non-zero contribution. The most general Lorentz-invariant form of the coupling of acharged lepton to a photon of 4-momentum qν can be written asΓµ = G1(q2) γµ + +G2(q2) σµν qν+ G3(q2) σµνγ5 qν (7)where qν is photon 4-vector and q2 dependent form factors G1(q2) and G2(q2) are proportional tothe charge and anamolous magnetic moment of l-lepton respectively. Non-zero value of G3(q2)is responsible for the CP violation and it is proportional to EDM of l-lepton. By extractingthis part of the vertex, l-lepton EDM ”dl” (l = e, µ, τ) (see eq.(5)) can be calculated as a sumof contributions coming from neutral Higgs bosons h0 and A0,dl = −iGF√2e32π2Qτmτ((ξ̄D ∗N,lτ)2 − (ξ̄DN,τl)2) (F1(yh0)− F1(yA0)) , (8)for l = e, µ anddτ = −iGF√2e32π2{Qτmτ((ξ̄D ∗N,ττ )2 − (ξ̄DN,ττ)2) (F2(rh0)− F2(rA0))− Qµmµm2τ((ξ̄D ∗N,µτ)2 − (ξ̄DN,τµ)2) (rh0 ln (zh0)− rA0 ln (zA0))}, (9)where the functions F1(w), F2(w) and F3(w) areF1(w) =w (3− 4w + w2 + 2 lnw)(−1 + w)3 ,F2(w) = w lnw +2 (−2 + w)w ln 12(√w −√w − 4)√w (w − 4). (10)3Here yH =m2τm2H, rH =1yHand zH =m2µm2H, Qτ and Qµ are charges of τ and µ leptons respectively.In eqs. (8) and (9) ξ̄DN,ij is defined as ξDN,ij =√4GF√2ξ̄DN,ij. In eq. (8) we take into accountonly internal τ -lepton contribution since, in our assumption, the Yukawa couplings ξ̄DN,ij, i, j =e, µ, are small compared to ξ̄DN,τ i i = e, µ, τ due to the possible proportionality of the Yukawacouplings to the masses of leptons underconsideration in the vertices. In eq. (9) we take alsothe internal µ-lepton contribution besides internal τ -lepton one and ignore the one coming fromthe internal e-lepton respecting our assumption (see Discussion part). Note that, we make ourcalculations in arbitrary q2 and take q2 = 0 at the end.Using the parametrizationξ̄DN,τl = |ξ̄DN,τl| eiθl , (11)the Yukawa factors in eqs. (8) and (9) can be written as((ξ̄D ∗N,lτ)2 − (ξ̄DN,τl)2) = −2 isin 2θl |ξ̄DN,τl|2 (12)where l = e, µ, τ . Here θl are CP violating parameters which are the sources of the leptonEDM.Now, let us consider the lepton number violating process µ → eγ which is a good candidatein the determination of the Yukawa couplings and new physics beyond the SM. Since we takeinto account only the neutral Higgs contributions in the lepton sector of the model III, thecontribution comes from neutral Higgs bosons h0 and A0 (see Fig. 1). In the on-shell renor-malization scheme the self energy diagrams are cancelled and the vertex diagram (Fig. 1-c)contributes. Taking only τ lepton for the internal line, the decay width Γ reads asΓ(µ → eγ) = c1(|A1|2 + |A2|2) , (13)whereA1 = Qτ18mµmτξ̄DN,τe ξ̄DN,τµ (F1(yh0)− F1(yA0)) ,A2 = Qτ18mµmτξ̄D∗N,eτ ξ̄D∗N,µτ (F1(yh0)− F1(yA0)) , (14)c1 =G2Fαemm5µ32π4and the function F1(w) is given in eq. (10). Here the amplitudes A1 and A2have right and left chirality respectively. In eq. (13) we ignore the contributions coming frominternal µ and e leptons respecting our assumption on the Yukawa couplings (see Discussion).Another LFV process is τ → µγ and it is rich from the theoretical point of view. The decaywidth of this process can be calculated using the same procedure and reads asΓ(τ → µγ) = c2(|B1|2 + |B2|2) , (15)4whereB1 = Qτ148mµmτξ̄DN,τµ{ξ̄D∗N,ττ(G1(yh0) + G1(yA0)) + 6ξ̄DN,ττ(F1(yh0)− F1(yA0)},B2 = Qτ148mµmτξ̄D∗N,µτ{ξ̄DN,ττ(G1(yh0) + G1(yA0)) + 6ξ̄D∗N,ττ(F1(yh0)− F1(yA0)}, (16)and c2 =G2Fαemm5τ32π4. Here the amplitudes B1 and B2 have right and left chirality respectively.The function G1(w) is given byG1(w) =w (2 + 3w − 6w2 + w3 + 6w lnw)(−1 + w)4 .Note that, in the case of the possible loop induced mixing of the neutral Higgs bosons h0 andthe SM one H0, the functions F1(yh0), F2(rh0), ln(zh0) and G1(yh0) in eqs. (8, 9, 14, 16) shouldbe multiplied by cos2 α where α is the small mixing angle.3 DiscussionThe Yukawa couplings ξ̄DN,ij, i, j = e, µ, τ play the main role in the determination of the leptonEDM and the physical quantities for LFV interactions, in the model III. These couplings arecomplex in general and they are free parameters which can be fixed by present and forthcomingexperiments. The experimental results on the EDM of leptons e, µ, τ and LFV process µ → eγare our starting point of the predictions on Yukawa couplings related with leptons. Note that,in our predictons, we assume that the Yukawa couplings ξ̄DN,ij, i, j = e, µ, are small comparedto ξ̄DN,τ i i = e, µ, τ since the strength of these couplings are related with the masses of leptonsdenoted by the indices of them, similar to the Cheng-Sher scenerio [21]. Further, we assumethat ξ̄DN,ij is symmetric with respect to the indices i and j.Since non-zero EDM can be obtained in case of complex couplings, there exist a CP violatingparameter θl coming from the parametrization eq. (11). At this stage, we find a restriction forξ̄DN,τµ using eq. (8) and the experimental result of µ EDM [22],0.3× 10−19 e− cm < dµ < 7.1× 10−19 e− cm . (17)Fig. 2 shows mA0 dependence of ξ̄DN,τµ for sinθµ = 0.5 and mh0 = 70GeV . Here the couplingis restricted in the region between solid lines. As shown in the figure, ξ̄DN,τµ is at the order ofthe magnitude 102 and take smaller values if mA0 takes larger values compared to mh0 in thetheory. If neutral Higgs masses are almost degenerate, namely mA0 = mh0 , ξ̄DN,τµ should bevery large since the contributions of h0 and A0 have opposite signs and the same functional5dependences. Therefore to be able control the numerical value of the coupling ξ̄DN,τµ, the massesshould not be degenerate.Now, we use the upper limit of the BR of the process µ → eγ to restrict the Yukawacombination ξ̄DN,τµ ξ̄DN,τe (see eq. (14)). Here we take into account only the internal τ -leptoncontribution. Using the restrictions for ξ̄DN,τµ and ξ̄DN,τµ ξ̄DN,τe we find a constraint region forξ̄DN,τe. In Fig. 3 we plot ξ̄DN,τe as a function of mA0 for sinθµ = 0.5 and mh0 = 70GeV . Herethe coupling is restricted in the region between solid lines. As shown in the figure ξ̄DN,τe is atthe order of the magnitude 10−4 and becomes smaller if the mass of the particle A0 is largercompared to mh0 in the theory, similar to the coupling ξ̄DN,µτ . Fig. 4 represente the effect ofpossible mixing between CP even neutral Higgs bosons h0 and H0 on the coupling ξ̄DN,τe. Whenthe mixing parameter sin α changes in the range 0 − 0.1, the upper limit of ξ̄DN,τe is affectedat the order of the magnitude 3% for the fixed values of sinθµ = 0.5 , mh0 = 70GeV andmA0 = 80GeV . The lower limit of the coupling is almost non-sensitive to the mixing as shownin the figure.At this stage we are ready to predict the electron EDM, de. Using eq. (8) and the restrictionfor ξ̄DN,τe, de is plotted with respect to sin θe in Fig. 5 for fixed values of sinθµ = 0.5, mh0 =70GeV and mA0 = 80GeV . For the intermediate values of sinθe, de is at the order of themagnitude 10−32 which lies in the experimental restriction region [14]de = 1.8± 1.2± 1.0× 10−27 e− cm (18)We also present mh0 dependence of de for fixed values of sinθe = 0.5 and mA0 = 80GeV inFig. 6. This figure shows that de is bounded by the solid lines and decreases with increasingvalues of mh0 . For completeness, we study h0-H0 mixing effect on de. Fig. 7 shows that de isnot sensitive to mixing of two CP-even bosons in the given range of sin α.Finally, we would like to predict the behaviour of dτ with respect to BR(τ → µγ). Sincewe take into account internal τ and µ leptons in the calculation of dτ , it has a functionaldependence (see eq. (9))dτ = f1 + f2|ξ̄DN,ττ |2 , (19)where f1 contains the coupling ξ̄DN,τµ and its complex conjugate, which can be fixed by theconstraint plotted in Fig. 2. On the other hand BR(τ → µγ) has in the formBR(τ → µγ) = g1|ξ̄DN,τµ|2 |ξ̄DN,ττ |2 , (20)6where |ξ̄DN,τµ|2 is again fixed by the constraint we have. As a result, we connect dτ andBR(τ → µγ) by an expression in the formdτ = f1 + g2BR(τ → µγ) . (21)Here the lower limit of f1 is at the order of the magnitude of 10−15 for the fixed values ofmh0 = 70GeV , mA0 = 80GeV and sinθµ = 0.5. Therefore, even internal µ-lepton causes toexceed the upper limit of experimental value of dτ [16]dτ = 3.1× 10−16 e− cm , (22)In eq. (21), the part g2BR(τ → µγ) is due to the internal τ -lepton contribution. Here g2 isat the order of the magnitude of 10−14. Therefore it is not possible to get a strict bound forBR(τ → µγ) using the eqs. (21) and the experimental result eq. (22).In our work we choose the 2HDM type III and assume that only FCNC interactions exist attree level. First we obtain constraints for the couplings ξ̄DN,τe and ξ̄DN,τµ. The previous one is atthe order of the magnitude of 10−4 and the last one is 102. Numerical values of these couplingsdecrease when the mass difference of neutral Higgs bosons h0 and A0 increases. Further, wepredict the de as 10−32 e − cm using the experimental result of dµ and the upper limit ofBR(µ → eγ). This lies in the experimental restriction region of de, however it is far fromthe present limits. Finally, we calculated the BR(τ → µγ) and dτ and connect these physicalquantities in the form of a linear expression (eq. (21)). This expression does not provide astrict bound for BR(τ → µγ) and the constraint region for the coupling ξ̄DN,ττ .In future, with the reliable experimental result of BR(τ → µγ) and obtaining more strictbounds for BR(µ → eγ), de, dµ and dτ it would be possible to test models and free parametersunderconsideration more accurately.4 AcknowledgementI would like to thank Prof. T. M. Aliev for useful discussions.References[1] M. L. Brooks et. al., MEGA Collaboration, Phys. Rev. Lett. 83, 1521 (1999).[2] S. Ahmed et.al., CLEO Collaboration, Phys. Rev. D61, 071101 (2000).[3] D. Chang, W. S. Hou and W. Y. Keung, Phys. Rev. D48, 217 (1993).7[4] R. Diaz, R. Martinez and J-Alexis Rodriguez, hep-ph/0010149 (2000).[5] R. Barbieri and L. J. Hall, Phys. Lett. B338, 212 (1994).[6] R. Barbieri, L. J. Hall and A. Strumia, Nucl. Phys. B445, 219 (1995).[7] R. Barbieri, L. J. Hall and A. Strumia, Nucl. Phys. B449, 437 (1995).[8] P. Ciafaloni, A. Romanino and A. Strumia, IFUP-YH-42-95.[9] T. V. Duong, B. Dutta and E. Keith, Phys. Lett. B378, 128 (1996).[10] G. Couture, et. al., Eur. Phys. J. C7, 135 (1999).[11] Y. Okada, K. Okumara and Y. Shimizu, Phys. Rev. D61, 094001 (2000).[12] G. C. Branco and M. N. Rebelo, Phys. Lett. B160, 117 (1985); J. Liu and L. Wolfenstein,Nucl. Phys. B289, 1 (1987); S. Weinberg, Phys. Rev. D42, 860 (1990).[13] S. Weinberg, Phys. Rev. Lett. 36, 294 (1976).[14] E. D. Commins et.al, Phys. Rev. A 50, (1994) 2960.[15] J. Bailey et al, Journ. Phys. G4, (1978) 345;[16] Particle Data Group, D. E. Groom et.al., European Phys. Journ. C15, 1 (2000).[17] S. M. Barr and A. Zee, Phys. Rev.Lett. 65, 21 (1990).[18] S. Weinberg, Phys. Rev. Lett. 63, 2333 (1989).[19] K. S. Babu, B. Dutta and R. N. Mohapatra, Phys. Rev. Lett. 85, 5064 (2000).[20] K. S. Babu, S. M. Barr and I. Dolsner, hep-ph/0012303 (2000).[21] T. P. Cheng and M. Sher, Phy. Rev. D35, 3383 (1987).[22] K. Abdullah et.al, Phys. Rev. Lett. 65, (1990) 2347.8h A00 , γl l( a ) ( b )( c )Figure 1: One loop diagrams contribute to EDM of l-lepton and LFV interactions (if externalleptons l have different flavors) due to the neutral Higgs bosons h0 and A0 in the 2HDM. Dashedlines represent the electromagnetic field, h0 and A0 fields.9mA0(GeV )10 4DN;(GeV)15014013012011010090800.20.180.160.140.120.10.080.060.040.020Figure 2: ξ̄DN,τµ as a function of mA0 for mh0 = 70GeV and sin θµ = 0.5. Here the coupling isrestricted in the region between solid lines.10mA0(GeV )104DN;e(GeV)15014013012011010090803.532.521.510.5Figure 3: The same as Fig. 2 but for ξ̄DN,τe.sin104DN;e(GeV)0.10.080.060.040.02043.532.521.510.50Figure 4: ξ̄DN,τe as a function of sin α for mh0 = 70GeV , mA0 = 80GeV and sin θµ = 0.5. Herethe coupling is restricted in the region between solid lines.11sin1032de(e cm)0.80.70.60.50.40.30.20.12.521.510.50Figure 5: de as a function of sin θe for mh0 = 70GeV and mA0 = 80GeV . Here de is restrictedin the region bounded by solid lines.mh0GeV1032de(e cm)75706560555045401.61.41.210.80.60.40.20Figure 6: The same as Fig. 5 but de as a function of mh0 for sin θe = 0.5.12sin1032de(e cm)0.10.080.060.040.0201.61.41.210.80.60.40.20Figure 7: de as a function of sin α for mh0 = 70GeV , mA0 = 80GeV and sin θe = 0.5. Here deis restricted in the region bounded by solid lines.13

Electric Dipole moments of charged leptons and lepton flavor violating
  interactions in the general two Higgs Doublet model

Electric Dipole moments of charged leptons and lepton flavor violating interactions in the general two Higgs Doublet model

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