We calculate the new physics effects to the branching ratios of the lepton
flavor conserving decays Z -> l^+ l^- in the framework of the general two Higgs
Doublet model. We predict the upper limits for the couplings
|\bar{\xi}^{D}_{N,\mu\tau}| and |\bar{\xi}^{D}_{N,\tau\tau}| as 3\times 10^2
GeV and 1\times 10^2 GeV, respectively.Comment: 9 pages, 3 figure

A. Czarnecki

A. Ghosal

A. Posthaus

B. Pontecorvo

C. Caso

D. Buskulic

D. Decamp

E. Iltan

E. O. Iltan

F. Sanchez

H. N. Brown

J. Bernabeu

J. I. Illana

K. Abdullah

P. Abreu

P. D. Acton

R. Hawkings

T. M. Aliev

U. Stiegler

W. Bernreuther

Z. Maki

English

arXiv

Crossref

Physical Review D

Lepton flavor conserving<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mo>→</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>−</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>decays in the general two Higgs doublet model

We calculate the new physics effects to the branching ratios of the lepton flavor conserving decays Z - l(+)l(-) in the framework of the general two Higgs doublet model. We predict the upper limits for the couplings \(xi) over bar (D)(N,mutau)\ and \(xi) over bar (D)(N,tautau)\ as 3 x 10(2) GeV and 1 x 10(2) GeV, respectively

Iltan, EO

OpenMETU (Middle East Technical University)

arXiv:hep-ph/0108230v1  28 Aug 2001Lepton flavor conserving Z → l+l− decays in thegeneral two Higgs doublet modelE. O. Iltan ∗Physics Department, Middle East Technical UniversityAnkara, TurkeyAbstractWe calculate the new physics effects to the branching ratios of the lepton flavor con-serving decays Z → l+l− in the framework of the general two Higgs Doublet model.We predict the upper limits for the couplings |ξ̄DN,µτ | and |ξ̄DN,ττ | as 3 × 102 GeV and1× 102 GeV , respectively.∗E-mail address: eiltan@heraklit.physics.metu.edu.tr1 IntroductionIn the standard model (SM) of electroweak interactions lepton flavor is conserved. This con-servation can be broken with the extension of the SM, such as νSM, permitting the existenceof the massive neutrinos and the lepton mixing mechanism [1], νSM, extended with one heavyordinary Dirac neutrino or two heavy right-handed singlet Majorana neutrinos [2], Zee model[3], the general two Higgs doublet model, which contains off diagonal Yukawa couplings in thelepton sector [4].Leptonic Z-decays are among the most interesting lepton flavor conserving (LFC) and leptonflavor violating (LFV) interactions and they reached great interest since the related experimen-tal measurements are improved at present. With the Giga-Z option of the Tesla project, thereis a possibilty to increase Z bosons at resonance [5].The processes Z → l−l+ with l = e, µ, τ are the among the LFC decays and they exist inthe SM even at the tree level. The experimental predictions for the branching ratios of thesedecays are [6]BR(Z → e−e+) = 3.366± 0.0081% ,BR(Z → µ−µ+) = 3.367± 0.013% ,BR(Z → τ−τ+) = 3.360± 0.015% , (1)and the tree level SM predictions areBR(Z → e−e+) = 3.331% ,BR(Z → µ−µ+) = 3.331% ,BR(Z → τ−τ+) = 3.328% . (2)Comparision of these experimental and theoretical results shows that the main contributioncomes from the SM in the tree level and the loop contributions, even the ones beyond the SM,should lie almost in the uncertainity of the measurements of these decays.In the literature, there are various experimental and theoretical studies [7]-[15]. A methodto determine the weak electric dipole moment was developed in [9]. The vector and axialcoupling constants, vf and af , in Z-decays have been measured at LEP [11]. Furthermore, themeasurements of the weak electric dipole moments of fermions have been performed [12]. In[13], various additional types of interactions have been studied and a way to measure thesecontributions in the process Z → τ−τ+ was described. In [15], a new method to measure theelectroweak mixing angle in Z-decays to tau leptons has been proposed.1In the present work we study the LFC Z → l−l+ decays, where l = e, µ, τ , in the model IIIversion of 2HDM, which is the minimal extension of the SM. Since the SM prediction for thesedecays, even in the tree level, is almost the same as the experimental one, the contributionsbeyond the SM, which can exist at least in the loop level, should be small enough not to exceedthe experimental results. This discussion stimulates us to study the BR’s of these processeswith the addition of the contributions beyond the SM and try to predict upper limits for thenew couplings existing in the model used. In the model III, the neutral Higgs bosons h0 andA0 play the important role for the physics beyond the SM, in the calculation of the BR of theLFC decays underconsideration. This analysis shows that the predictions of the upper limitsfor the couplings |ξ̄DN,µτ | and |ξ̄DN,ττ | are ∼ 3×102GeV and ∼ 1×102GeV , respectively, however,an upper limit for the coupling |ξ̄DN,eτ | can not be found due to the small contribution of newphysics effects to the BR of the Z → e−e+ decay.The paper is organized as follows: In Section 2, we present the explicit expressions forthe branching ratios of Z → l−l+ in the framework of the model III. Section 3 is devoted todiscussion and our conclusions.2 Z → l−l+ decay in the general two Higgs Doublet model.In the SM, lepton flavor is conserved since the matter content forbids the lepton flavor violation.However, most theories beyond the SM may bring flavor changing neutral currents (FCNC) atthe tree level, unless some discrete ad hoc symmetries are imposed to eliminate them. Themodel I and II versions of 2HDM are the examples of the theories beyond where FCNC at thetree level is forbidden. In the model III version of 2HDM, the FCNC interactions at the treelevel are allowed and this makes the LFV interactions possible. Furthermore, the existence ofFCNC at the tree level brings new contributions to LFC decays. The most general Yukawainteraction for the leptonic sector in the model III isLY = ηDij l̄iLφ1EjR + ξDij l̄iLφ2EjR + h.c. , (3)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.The choice of φ1 and φ2φ1 =1√2[(0v +H0)+( √2χ+iχ0)];φ2 =1√2( √2H+H1 + iH2), (4)2with the vacuum expectation values< φ1 >=1√2(0v);< φ2 >= 0 , (5)helps us to decompose the SM particles and beyond in the tree level. In this case, the firstdoublet carries the SM particles and the other one is responsible for the particles beyond theSM. Therefore, we take H1 and H2 (see eq. (4)) as the mass eigenstates h0 and A0 respectively,since no mixing between CP-even neutral Higgs bosons h0 and the SM one, H0, occurs at thetree level. In eq. (3) ξDij are the Yukawa matrices and they have in general complex entries.Notice that in the following we replace ξD with ξDN where ”N” denotes the word ”neutral”.The general effective vertex for the interaction of on-shell Z-boson with a fermionic currentis given byΓµ = γµ(fV − fA γ5) +1mW(fM + fE γ5) σµν qν (6)where q is the momentum transfer, q2 = (p − p′)2, fV (fA) is vector (axial-vector) coupling,fM (fE) is proportional to the weak magnetic (electric dipole) moment of the fermion. Herep (−p′) is the four momentum vector of lepton (anti-lepton). For the Z → l−l+ decay, thecouplings fV and fA have contributions from the SM, fSMV and fSMA , even at the tree level,and all the couplings have contributions beyond the SM, fBeyondI , where I = V,A,M,E. Theexplicit expressions for these couplings arefSMV =igcos θWcVfSMA =igcos θWcA , (7)andfBeyondV =−ig32 cos θW π2{∫10dx cV ηVi (−1 + x) (lnLselfh0µ2LselfA0µ2)+∫10dx∫1−x0dy(12(−1 + x+ y)miml− η−i (1Lverh0 A0− 1LverA0 h0)+ cV[(−ηVi m2i + (−1 + x+ y) η+i mi ml−)1Lverh0− (ηVi m2i + (−1 + x+ y) η+i mi ml−)1LverA0+ ηVi (2− (q2 x y +m2l− (−1 + x+ y)2) (1Lverh0+1LverA0) + lnLverh0µ2LverA0µ2)])},fBeyondA =ig32 cos θW π2{∫10dx cA ηVi (−1 + x) (lnLselfh0µ2LselfA0µ2)3−∫10dx∫1−x0dy(− 2 cA ηVi lnLverh0 A0µ2LverA0 h0µ2+ cA[(ηVi m2i − (−1 + x+ y) η+i mi ml−)1Lverh0+ (ηVi m2i + (−1 + x+ y) η+i mi ml−)1LverA0+ ηVi (2− (q2 x y −m2l− (−1 + x+ y)2) (1Lverh0+1LverA0) + lnLverh0µ2LverA0µ2)])},fBeyondM =g mW256 cos θW π2∫10dx∫1−x0dy{2 η−i mi (−1 + x+ y) (1Lverh0 A0− 1LverA0 h0)+1Lverh0(η−i mi (y − x)− 4 cV (x+ y)(− η+i mi + 2 (−1 + x+ y) ηVi ml−))+1LverA0(η−i mi (x− y)− 4 cV (x+ y)(η+i mi + 2 (−1 + x+ y) ηVi ml−))},fBeyondE =g mW256 cos θW π2∫10dx∫1−x0dy{2 (1− x− y)((η+i mi + 2 (x− y) ηVi ml−) 1Lverh0 A0−(η+i mi + 2 (y − x) ηVi ml−) 1LverA0 h0+(mi(η+i (y − x) + 4 cV η−i (x+ y))+ 2 (y − x) (−1 + x+ y) ηVi ml−)1LverA0+(mi(η+i (x− y) + 4 cV η−i (x+ y))− 2 (x− y) (−1 + x+ y) ηVi ml−)1Lverh0}, (8)whereLselfh0 = m2h0 (1− x) + (m2i −m2l− (1− x)) x ,LselfA0 = Lself1, h0(mh0 → mA0) ,Lverh0 = m2h0 (1− x− y) +m2i (x+ y)− q2 x y ,Lverh0 A0 = m2h0 x+m2i (1− x− y) + (m2A0 − q2 x) y ,LverA0 = Lverh0 (mh0 → mA0) ,LverA0 h0 = Lverh0 A0(mh0 → mA0) , (9)andηVi = ξD ∗N,ilξDN,li ,η+i = ξD ∗N,ilξD ∗N,il + ξDN,liξDN,li ,η−i = ξD ∗N,ilξD ∗N,il − ξDN,liξDN,li . (10)The parameters cV and cA are cA = −14 and cV = 14 − sin2 θW . In eq. (10) the flavor changingcouplings ξDN,li represent the effective interaction between the internal lepton i, (i = e, µ, τ) and4outgoing (incoming) l− (l+) one. It is useful to redefine the coupling as ξDN,li =√4GF√2ξ̄DN,li toextract the dimensionfull part ξ̄DN,li. In general the Yukawa couplings ξ̄DN,li are complex and theycan be parametrized asξ̄DN,il = |ξ̄DN,il| eiθil , (11)with lepton flavors i, l and CP violating parameters θil. Notice that parameters θil are thesources of the lepton EDM.Finally the BR for the LFC process Z → l− l+, for the vanishing external lepton masses,can be written asBR(Z → l− l+) = mZ12 π ΓZ{|fV |2 + |fA|2 +12 cos2 θW(|fM |2 + |fE |2)} . (12)where fV = fSMV + fBeyondV , fA = fSMA + fBeyondA and fM = fBeyondM , fE = fBeyondE . Here ΓZ isthe total decay width of Z boson, namely ΓZ = 2.490± 0.007GeV .3 DiscussionFlavor conserving Z → l+l− decays are possible at the tree level in the SM model and thecontribution of one loop corrections to the tree level result is small. Our aim is to determinethe new physics effects to the BR of these decays and to predict the restrictions for the freeparameters of the model used. The model we study is the model III version of 2HDM, whichmay bring considerable contribution to the BR (Z → l+l−) beyond the SM. However, in themodel III, there are large number of free parameters, namely, the masses of charged and neutralHiggs bosons, the Yukawa couplings that can be complex in general. The Yukawa couplings inthe lepton sector are ξ̄DN,ij, i, j = e, µ, τ and it is necessary to restrict them using the presentand forthcoming experiments.The couplings ξ̄DN,ij, i, j = e, µ can be neglected compared to ξ̄DN,τ i i = e, µ, τ with theassumption that the strength of these couplings are related with the masses of leptons denotedby the indices of them. Furthermore, we assume that ξ̄DN,ij is symmetric with respect to theindices i and j. Therefore, the Yukawa couplings ξ̄DN,τ e, ξ̄DN,τ µ and ξ̄DN,τ τ play the main role inour lepton conserving decays, Z → l+l−.For ξ̄DN,µτ , the constraint coming from the experimental limits of µ lepton EDM [16],0.3× 10−19 e− cm < dµ < 7.1× 10−19 e− cm (13)(see [17] for details) or the deviation of the anomalous magnetic moment (AMM) of muon overits SM prediction [18] due to the recent experimental result of muon AMM by g-2 Collaboration5[19], can be used. The coupling ξ̄DN,eτ is restricted using the experimental upper limit of the BRof the process µ → eγ and the above constraint for ξ̄DN,µτ , since µ → eγ decay can be used tofix the Yukawa combination ξ̄DN,µτ ξ̄DN,eτ . Using the the experimental bounds of µ lepton EDMand the upper limit of the BR of the process µ → eγ |ξ̄DN,µτ | (|ξ̄DN,eτ |) has been predicted at theorder of the magnitude of 102 − 103 (10−5 − 10−3) GeV (see [17]). For |ξ̄DN,ττ | no prediction hasbeen done yet.The present work is devoted to study on the lepton flavor conserving decays Z → l−l+, wherel = e, µ, τ . The main contribution to the BR of these processes come from the SM in the treelevel. In the calculations, we neglect the one loop diagrams including the charged W± bosonsin the SM and H± bosons beyond, since the neutrinos existing in the expressions are almostmassless. Therefore, we take into account the one loop contributions including the neutralHiggs bosons h0 and A0 beyond the SM. We see that, with this additional part, the BR of theZ → l−l+ decays can be enhanced, by playing with the Yukawa couplings and the upper limits ofthese Yukawa couplings can be restricted using the existing experimental measurements. Noticethat in the theoretical calculations, we take the Yukawa couplings complex, however we usetheir magnitudes in the numerical analysis, since the BR of the processes under considerationare not sensitive to the CP violating part of these couplings. Throughout our calculations weuse the input values given in Table (1).Parameter Valueme 0 (GeV)mµ 0.106 (GeV)mτ 1.78 (GeV)mW 80.26 (GeV)mZ 91.19 (GeV)GF 1.1663710−5(GeV −2)ΓZ 2.490 (GeV )sin θW√0.2325Table 1: The values of the input parameters used in the numerical calculations.Now we would like to parametrize the BR asBR = BRSM +BRBeyond , (14)where BRSM is coming from only the SM part. In eq. (14) BRBeyond gets contributions fromthe combination of the SM and beyond, which we denote as BRMixed, and from only beyondthe SM, which we denote as BRPureBeyond.6Fig. 1 represents |ξ̄DN,eτ | dependence of the BRBeyond (Z → e− e+) for mh0 = 80GeV andmA0 = 90GeV . Here the coupling |ξ̄DN,eτ | stands in the range 10−4 − 10−3GeV respecting theabove restrictions and the BRBeyond can take the values at most at the order of the magnitudeof 10−15. This value is negligible compared to even the uncertainity in the experimental resultof the BR (Z → e− e+), ∼ 0.008%. This concludes that the new physics effects in the modelIII does not bring any contribution to the BR of the process Z → e− e+ and a new constraintfor the coupling |ξ̄DN,eτ | can not be found. Notice that the BRPureBeyond (Z → e− e+) is at theorder of the magnitude of 10−28. We also study the Higgs boson h0 mass mh0 dependence of theBRBeyond (50 < mh0 < 80GeV ) and observe that, for mh0 = 50GeV , there is an enhancementmore than a factor of two larger than the result for mh0 = 80GeV .In Fig. 2 we present |ξ̄DN,µτ | dependence of the BRBeyond (Z → µ− µ+) for mh0 = 80GeV andmA0 = 90GeV . Here the solid (dashed) line represents the SM (Beyond) contribution. It canseen that the BRBeyond can reach the SM values for the large values of the coupling, |ξ̄DN,µτ | ∼3.1× 103GeV , which lies in the above constraint region. Not to exceed the experimental resultof the process Z → µ− µ+, we need to predict an upper limit for this coupling. By taking intoaccount the uncertainity of the experimental result, namely 0.013% we get the upper limit ofthe coupling as |ξ̄DN,µτ | ∼ 3 × 102GeV . The BRPureBeyond (Z → e− e+) can also reach the SMvalue for |ξ̄DN,µτ | ∼ 3.5 × 103GeV . Furthermore, the Higgs boson h0 mass mh0 dependence ofthe BRBeyond (50 < mh0 < 80GeV ) shows that for mh0 = 50GeV , there is an enhancementmore than a factor of three larger than the result for mh0 = 80GeV .Fig. 3 is devoted to the |ξ̄DN,τi| (i = µ, τ) dependence of the BRBeyond (Z → τ− τ+) formh0 = 80GeV and mA0 = 90GeV . Here the solid (dashed, small, dotted ) line representsthe SM (Beyond). Dashed line is devoted to the |ξ̄DN,τµ| dependence for |ξ̄DN,ττ | = 103GeV ,small dashed line is to the |ξ̄DN,ττ | dependence for |ξ̄DN,τµ| = 103GeV and dotted line is to the|ξ̄DN,ττ | dependence for |ξ̄DN,τµ| = 3× 102GeV , which is the upper limit of |ξ̄DN,τµ|, obtained usingBRBeyond (Z → µ− µ+). From this figure, it is observed that BRBeyond can reach the SM valuefor the large values of the couplings, |ξ̄DN,µτ | ∼ 3.0× 103GeV and |ξ̄DN,µτ | ∼ 3.0× 103GeV . Ouraim is to get a BRBeyond so that the BR does not exceed its experimental value. To ensurethis, first, we choose the upper limit of the coupling |ξ̄DN,µτ | as 3 × 10−2GeV , respecting theBRBeyond (Z → µ− µ+) and then use the uncertainity of the experimental result for BR (Z →τ− τ+), namely 0.015%. We predict the upper limit of the coupling as |ξ̄DN,µτ | < 1 × 102GeV(see dotted line in the Fig. 3). Notice that BRPureBeyond (Z → e− e+) can reach the SM valuefor |ξ̄DN,µτ | ∼ 3.5 × 103GeV and |ξ̄DN,ττ | ∼ 3 × 103GeV . For completeness we study the mass7mh0 dependence of the BRBeyond (50 < mh0 < 80GeV ) and observe that for mh0 = 50GeV ,there is an enhancement less than a factor of two larger than the result for mh0 = 80GeV .As a summary, we study the BRBeyond’s of the lepton flavor conserving decays Z → l−l+ (l =e, µ, τ) in the model III. We observe that the one loop diagrams due to neutral Higgs bosonsh0 and A0 can give considerable contributions and its possible even to reach the tree level SMresult. This forces one to predict the upper limits for the Yukawa couplings in the lepton sector:• An upper limit for the coupling |ξ̄DN,eτ | can not be found since the new physics effects tothe BR (Z → e−e+) are exteremely small compared to the SM result.• We predict the upper limit of the coupling as |ξ̄DN,µτ | < 3× 102GeV .• We predict the upper limit of the coupling as |ξ̄DN,ττ | < 1× 102GeV .Furthermore, BRBeyond is not so much sensitive to the mass mh0 .In future, with the more reliable experimental result of the BR’s of above processes it wouldbe possible to test models beyond the SM and free parameters of these models4 AcknowledgementsReferences[1] B. Pontecorvo, Zh. Eksp. Teor. Fiz. 33 (1957) 549, Z. Maki, M. Nakagawa, and S. Sakata,Prog. Theor. Phys. 28 (1962) 870, B. Pontecorvo, Sov. Phys. JETP 26 (1968) 984.[2] J. I. Illana, M. Jack and T. Riemann, hep-ph/0001273 (2000), J. I. Illana, and T. Riemann,Phys. Rev. D63 (2001) 053004.[3] A. Ghosal, Y. Koide and H. Fusaoka, Phys. Rev. D64 (2001) 053012[4] E. Iltan, I. Turan, hep-ph/0106068 (2001).[5] R. Hawkings and K. Mönig, Eur. Phys. J. direct C8 (1999) 1.[6] Particle Data Group Collaboration, C. Caso et al., Eur. Phys. J. C3 (1998) 1.[7] CDF Collaboration, (T. Kamon for the collaboration). FERMILAB-CONF-89/246-E, Dec1989. 25pp. To be publ. in Proc. of 8th Topical Workshop on p anti-p Collider Physics,Castiglione d. Pescaia, Italy, Sep 1-5, 1989. Published in Pisa Collider Workshop 1989:0281-305 (QCD161:W64:1989).8[8] ALEPH Collaboration, D. Decamp et al. Phys.Lett. B263 (1991) 112 ; ALEPH Collabora-tion, D. Decamp et al., it Phys.Lett. B265 (1991) 430.[9] W. Bernreuther, G.W. Botz, O. Nachtmann, P. Overmann, Z.Phys. C52 (1991) 567.[10] DELPHI Collaboration, P. Abreu et al. Z. Phys. C55 (1992) 555.[11] LEP Collaboration, Phys. Lett. B276 (1992) 247.[12] OPAL Collaboration, P. D. Acton et al., Phys. Lett. B281 (1992) 405; ALEPH Collabo-ration, D. Buskulic et al. Phys.Lett. B297 (1992) 459.[13] U. Stiegler, Z. Phys. C58 (1993) 601.[14] ALEPH Collaboration, D. Buskulic et al. Z. Phys. C59 (1993) 369; U. Stiegler, Z. Phys.C57 (1993) 511; J. Bernabeu, G.A. Gonzalez-Sprinberg, J. Vidal Phys. Lett. B326 (1994)168; F. Sanchez, Phys. Lett. B384 (1996) 277.[15] A. Posthaus, P. Overmann, JHEP 9802:001 (1998).[16] K. Abdullah et.al, Phys. Rev. Lett. 65 (1990) 2347.[17] E. Iltan, Phys. Rev. D64 (2001) 013013.[18] A. Czarnecki and W. J. Marciano, Phys. Rev. D64, (2001) 013014.[19] H. N. Brown et.al, Muon g-2 Collaboration, Phys. Rev. Lett. 86, (2001) 2227.9104jDN;ej (GeV )1015BR(Z!e+e )109876543211.210.80.60.40.20Figure 1: |ξ̄DN,eτ | dependence of the BRBeyond (Z → e− e+) for mh0 = 80GeV and mA0 =90GeV .1010 2jDN;j (GeV )103BR(Z!+ )30252015105035302520151050Figure 2: |ξ̄DN,µτ | dependence of the BRBeyond (Z → µ− µ+) for mh0 = 80GeV and mA0 =90GeV . Here the solid (dashed) line represents the SM (Beyond) contribution.10 2jDN;ij (GeV )103BR(Z!+ )2520151053530252015105Figure 3: |ξ̄DN,τi| (i = µ, τ) dependence of the BRBeyond (Z → τ− τ+) for mh0 = 80GeV andmA0 = 90GeV . Here the solid line is devoted to the SM, dashed line to the |ξ̄DN,τµ| dependencefor |ξ̄DN,ττ | = 103GeV , small dashed line to the |ξ̄DN,ττ | dependence for |ξ̄DN,τµ| = 103GeV anddotted line to the |ξ̄DN,ττ | dependence for |ξ̄DN,τµ| = 3× 102GeV .11

Lepton flavor conserving Z -> l(+)l(-) decays in the general two Higgs doublet model

Lepton flavor conserving Z -> l^+ l^-$ decays in the general two Higgs
  doublet model

Lepton flavor conserving Z -> l^+ l^-$ decays in the general two Higgs doublet model

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