Muon g-2 and lepton flavor violation in a two Higgs doublets model for the fourth generation

# Muon g-2 and lepton flavor violation in a two Higgs doublets model for the fourth generation

Shaouly Bar-Shalom Physics Department, Technion-Institute of Technology, Haifa 32000, Israel    Soumitra Nandi Physique des Particules, Université de Montréal, C.P. 6128, succ. centre-ville, Montréal, QC, Canada H3C 3J7 Theoretische Elementarteilchenphysik, Department Physik, Universität Siegen, D-57068 Siegen, Germany    Amarjit Soni Theory Group, Brookhaven National Laboratory, Upton, NY 11973, USA
July 29, 2019
###### Abstract

In the minimal Standard Model (SM) with four generations (the so called SM4) and in “standard” two Higgs doublets model (2HDM) setups, e.g., the type II 2HDM with four fermion generations, the contribution of the 4th family heavy leptons to the muon magnetic moment is suppressed and cannot accommodate the measured access with respect to the SM prediction. We show that in a 2HDM for the 4th generation (the 4G2HDM), which we view as a low energy effective theory for dynamical electroweak symmetry breaking, with one of the Higgs doublets coupling only to the 4th family leptons and quarks (thus effectively addressing their large masses), the loop exchanges of the heavy 4th generation neutrino can account for the measured value of the muon anomalous magnetic moment. We also discuss the sensitivity of the lepton flavor violating decays and and of the decay to the new couplings which control the muon g-2 in our model.

preprint: SI-HEP-2011-18

## I Introduction

Particle magnetic moments provide an important and valuable test of QED and of the Standard Model (SM). In the case of the muon and the electron magnetic moments, both the experimental measurements and the SM predictions are very precisely known. However, due to its larger mass, the muon magnetic moment is considered more sensitive to massive virtual particles and hence to new physics (NP).

In the SM, the total contributions to the muon () can be divided into three parts: the QED, the electroweak (EW) and the hadronic contributions. While the QED qed () and EW ew () contributions are well understood, the main theoretical uncertainties lies with the hadronic part which are difficult to control qcd (). The hadronic loop contributions cannot be calculated from first principles, so that one relies on a dispersion relation approach 45844 (). At present the available data are used to calculate the leading-order (LO) and higher-order vacuum polarization contributions to ; the estimated contributions are given by arXiv:0908.4300 (); hep-ph/0611102 ()

On the other hand, the hadronic light-by-light contribution cannot be calculated from data, hence, its evaluation relies on specific models. The latest determination of this term is nyffeler ()

Including all these corrections, the complete SM prediction is given by

 aSMμ=116591834(2)(41)(26)×10−11 , (3)

whereas the current experimentally measured value is pdg ()

 aexpμ=116592089(54)(33)×10−11 . (4)

The SM prediction, therefore, differs from the the experimentally measured value by (see also gminus2 ())

 anewμ=aexpμ−aSMμ=(255±80)⋅10−11 , (5)

which allows some room for new physics. For the purpose of this work we are going to assume that the discrepancy in Eq. 5 is due to NP, although we are aware that the estimates of the hadronic contributions have appreciable uncertainties that may provide part of the discrepancy.

In most extensions of the SM, new charged or neutral states, can contribute to the muon anomalous magnetic moment (AMM) at the one-loop (lowest) level. For example, the AMM plays an important role in constraining the supersymmetric (SUSY) parameter space, where, as in the SM, the leading SUSY contribution to arises at one-loop, and is found to be enhanced for large . In particular, as was shown in muong2-susy (), SUSY can address the observed muon discrepancy for and (Higgsino mass parameter), with typical SUSY masses, of the particles involved in the loops, in the range .

Model independent analysis show that (for details see gminus2 ()), for small enough couplings, scalar exchange diagrams could account for the observed AMM with a scalar mass in the range , whereas pseudoscalar and axial-vector one-loop exchanges contribute with the wrong sign and the one-loop vector exchange contributions are too small.

In this paper we will consider the AMM in a new 2HDM framework with a heavy 4th generation family. Indeed, we will show that the access (with respect to the SM prediction) shown in Eq. 5 can be explained by one-loop exchanges of the heavy 4th generation neutrino () in a model with two Higgs doublets that we have constructed in 4G2HDM () and named the 4G2HDM. These new class of two Higgs doublet models were proposed in 4G2HDM () as viable low energy effective frameworks for models of 4th generation condensation. In particular, a theory with new heavy fermionic states is inevitably cutoff at the near by TeV-scale, where one thus expects some form of strong dynamics and/or compositeness to occur. Thus, as was noted already 20 years ago luty (), the low-energy (i.e., sub-TeV) dynamics of such a scenario may be more naturally embedded in multi-Higgs theories, where the new composite scalars are viewed as manifestations of the several possible bound states of the fundamental heavy fermions. Besides, our 4G2HDM can naturally (albeit effectively) accommodate the large (EW-scale) mass of the heavy 4th generation neutrino, which otherwise remains a cause of concern in theories with a 4th family of fermions.

We recall that an additional fourth generation of fermions cannot be ruled out by any symmetry argument, and is not excluded by EW precision data ewpt (). It is also interesting to note, that already the simplest 4th generation extension of the SM, the so called SM4, has the potential to address some of the current open questions in particle physics, such as the observed baryon asymmetry gh08 (), the Higgs naturalness problem Hashimoto:2009ty (), the fermion mass hierarchy problem Hung:2009ia () etc…. The SM4 can also accommodate the emerging possible hints for new flavor physics SAGMN08 (); SAGMN10 (); ajb10B (); gh10 (); Nandi:2010zx (); lenz_fourth1 (). However, the SM4 as such cannot explain the observed muon g-2 discrepancy, see e.g., hou_muong2 (). In fact, even “standard” 2HDM frameworks (like the type II 2HDM that underlies the minimal SUSY model) with an additional 4th generation of heavy fermions, was shown to fail in explaining the measured AMM hou_muong2 ().

In section II we calculate the AMM in the 4G2HDM framework. In sections III and IV we consider the constraints on AMM from the lepton flavor violating (LFV) decays and and from , respectively, and in section V we summarize our results.

## Ii Muon g−2 in the 4G2HDM

At the tree level the muon magnetic moment is predicted by the Dirac equation to be with . The effective vertex of a photon with a charged fermion can in general be written as

 ¯u(p′)eΓμu(p)=¯u(p′)e[γμF1(q2)+iσμνqν2mfF2(q2)]u(p), (6)

where, to lowest order, and . While remains unity at all orders due to charge conservation, quantum corrections yield . Thus, since , it follows that .

In our 4G2HDM 4G2HDM () the one-loop contribution to the AMM can be subdivided as

 aμ=[aμ]SM4W+[aμ]4G2HDMH, (7)

where contains the charged and neutral Higgs contributions coming from the one-loop diagrams in Fig. 1 (see below; the diagrams with and in the loop dominate), whereas the SM4-like contribution, , comes from the one-loop diagram with in the loop and is given by jlev ()

 [aμ]SM4W|U24|2=GFm2μ4√2π2A(xν′) , (8)

where is the 24 element of the CKM-like PMNS leptonic matrix, and the loop function is given by

 A(xi)=3x3ilogxi(xi−1)4+4x3i−45x2i+33xi−106(xi−1)3. (9)

For values of in the range one finds , so that for (as expected) the simple SM4 cannot accommodate the observed discrepancy in .

Let us recapitulate the salient features of the 4G2HDM setups introduced in 4G2HDM (). In these models one of the Higgs fields (the “heavier” field) couples only to heavy fermionic states, while the second Higgs field (the “lighter” field) is responsible for the mass generation of all other (lighter) fermions. Applying this principle to the 4th generation leptonic sector we have

 LY=−¯EL(ΦℓYe⋅(I−I)+ΦhYe⋅I)eR−¯EL(~ΦℓYν⋅(I−I)+ΦhYν⋅I)νR+h.c. , (10)

where are left(right)-handed fermion fields, is the left-handed lepton doublet and are general Yukawa matrices in flavor space. Also, and are the two Higgs doublets, is the identity matrix and . The Yukawa texture of (10) can be realized in terms of a -symmetry under which the fields transform as follows: , , , (for ), (for ), and , .

From the point of view of the leptonic sector, the Yukawa interaction in (10) is the natural underlying setup that can effectively accommodate the heavy masses of the 4th generation leptons, by coupling them to the heavy Higgs doublet. This setup might also be an effective underlying description of more elaborate constructions in models of warped extra dimensions gustavo3 ().

The Yukawa interactions between the physical Higgs bosons and the leptonic states are then given by (see 4G2HDM ())

 L(hℓiℓj) = g2mW¯ℓi{mℓisαcβδij−fhβ⋅[mℓiΣℓijR+mℓjΣℓ∗jiL]}ℓjh , (11) L(Hℓiℓj) = g2mW¯ℓi{−mℓicαcβδij+fHβ⋅[mℓiΣℓijR+mℓjΣℓ∗jiL]}ℓjH , (12) L(Aℓiℓj) = −iIℓgmW¯ℓi{mℓitanβγ5δij−fβ⋅[mℓiΣℓijR−mℓjΣℓ∗jiL]}ℓjA , (13) L(H+νiej) = g√2mW¯νi{[mejtanβ⋅Uji−mekfβ⋅UkiΣekj]R (14) +[−mνitanβ⋅Uji+mνkfβ⋅Σν∗kiUjk]L}ejH+ ,

with

 fβ≡tanβ+cotβ , fhβ≡cαsβ+sαcβ , fHβ ≡ cαcβ−sαsβ , (15)

and is the ratio between the two VEVs. Also, is the charged Higgs, are the physical neutral Higgs states ( and are the lighter and heavier CP-even neutral states, respectively, and is the neutral CP-odd state), and or with weak isospin and , respectively. Also, and is the leptonic CKM-like PMNS matrix. Finally, are new mixing matrices in the charged(neutral)-leptonic sectors, obtained after diagonalizing the lepton mass matrices

 Σeij=L⋆R,4iLR,4j , Σνij=N⋆R,4iNR,4j , (16)

where are the rotation (unitary) matrices of the right-handed charged and neutral leptons, respectively. Notice that and depend only on the elements of 4th rows of and , respectively, which we will treat as unknowns, i.e., by expressing physical observables in terms of and or, equivalently in terms of and .

Following jlev (), let us redefine the Higgs Yukawa interactions as

 L(Hℓiℓj)≡¯ℓi[SHℓiℓj+PHℓiℓjγ5]ℓjH , (17)

with or and or . Then, neglecting terms of order for and terms of order for , the above scalar and pseudoscalar couplings, , and , ( or 3), which mix the 4th generation leptons with the light leptons, are given in our 4G2HDM by

 SH0ℓiτ′=−PH0ℓiτ′=g4m′τmWFH0Σν44δΣi , SH−ℓiν′=g2√2mτ′mWfβU∗44Σν44[mν′mτ′(1−tβfβΣν44)δUi−δΣi] , PH−ℓiν′=−g2√2mτ′mWfβU∗44Σν44[mν′mτ′(1−tβfβΣν44)δUi+δΣi] , (18)

where or , , , (see Eq. 15) and

 δUi≡U∗i4U∗44 , δΣi≡Σe∗4iΣν44 , (19)

which are the small quantities that parameterize the amount of mixing between the 4th generation leptons and the light leptons of the 1st, 2nd and the 3rd generations. In what follows we will take all quantities in Eq. 18 to be real and always set and (for limits on in the 4G2HDM see 4G2HDM ()). We note that and the branching ratios for the LFV decays are proportional to (see Eqs. 22 and 34), so that there is no enhancement for .

Using Eq. 18, the charged and neutral Higgs contributions to [with and in the loop ( or ), respectively, see diagrams in Fig. 1] are given by (see also jlev ())

 [aμ]4G2HDMH±≈m2μ8π2∫10dxx(x−1){x(∣∣SH−μν′∣∣2+∣∣PH−μν′∣∣2)+mν′mμ(∣∣SH−μν′∣∣2−∣∣PH−μν′∣∣2)}m2H−x+m2ν′(1−x), (20)
 (21)

Note that, for the neutral Higgs case, the term proportional to vanishes since (see Eq. 18). Therefore, the dominant contribution, by far, to comes from the charged Higgs exchange, in particular, from the second term (proportional to ) in the numerator of Eq. 20, where

 ∣∣SH−μν′∣∣2−∣∣PH−μν′∣∣2=−g22mν′mτ′m2Wf2β|U44|2|Σν44|2⋅Re{(1−tβfβΣν44)δU2δ∗Σ2} , (22)

so that is proportional to the product .

In Fig. 2 we plot as a function of the product (assuming its real) for several values of and and fixing ( depends linearly on , see Eq. 22). Depending on the mass , we find that is typically required to accommodate the measured value of .

In what follows we will consider the constraints from the lepton flavor violating decays and from the decay , both of which are sensitive to the quantities and , as will be explained below.

## Iii Constraints from Lepton flavour violation

LFV decays such as and , which are absent in the SM, are often found useful for constraining NP models that can potentially contribute to the AMM, as such processes do not suffer from hadronic uncertainties. The current experimental 90%CL upper bounds on these LFV decays are pdg (); meg ()

 Br(τ→μγ)<4.4×10−8  ,  Br(μ→eγ)<2.4×10−12 . (23)

Let us define the amplitude for the transition as

 M(ℓi→ℓjγ)=¯uℓj(p′)[iσμνqν(A+Bγ5)]uℓi(p)ϵμ∗, (24)

where is the photon polarization. The decay width is then given by

 Γ(ℓi→ℓjγ)=m3ℓi8π⎛⎝1−m2ℓjm2ℓi⎞⎠⎡⎣⎛⎝1+m2ℓjm2ℓi⎞⎠(|A|2+|B|2)+4mℓjmℓi(|A|2−|B|2)⎤⎦ . (25)

Here again, the new 4G2HDM amplitude can be divided as

 M(ℓi→ℓjγ)4G2HDM≡MSM4W(ℓi→ℓjγ)+M4G2HDMH+(ℓi→ℓjγ)+M4G2HDMH0(ℓi→ℓjγ) , (26)

where are the SM4-like W-exchange contribution which is obtained from the diagram (right) of Fig. 1 with replaced by plus the diagrams which contain the self-energy corrections to the external fermion line or . In particular, using the definition in Eq. 24 and taking the limit , the net contribution to with internal in the loop is given by beneke_buras ()

 ASM4W=BSM4W=eGFmℓi4√2π2Uj4U∗i4F(xν′) , (27)

where and is given by

 F(xi)=xi(1−6xi+3x2i+2x3i−6x2ilogxi)4(1−xi)4 . (28)

Here also, we find that is much smaller than the charged and neutral Higgs amplitudes, and (calculated from the diagrams in Fig. 1), for which we obtain

 A4G2HDMH− = e32π2{(mℓi+mℓj)(SH−ℓiν′∗SH−ℓjν′+PH−ℓiν′∗PH−ℓjν′)IH+1+2mν′(SH−ℓiν′∗SH−ℓjν′−PH−ℓiν′∗PH−ℓjν′)IH+2} , (29) B4G2HDMH− = e32π2{(mℓi−mℓj)(SH−ℓiν′∗PH−ℓjν′+SH−ℓjν′PH−ℓiν′∗)IH+1+2mν′(SH−ℓiν′∗PH−ℓjν′−SH−ℓjν′PH−ℓiν′∗)IH+2} , (30) A4G2HDMH0 = e32π2{(mℓi+mℓj)(SH0ℓiτ′∗SH0ℓjτ′+PH0ℓiτ′∗PH0ℓjτ′)IH01+2mτ′(SH0ℓiτ′∗SH0ℓjτ′−PH0ℓiτ′∗PH0ℓjτ′)IH02} , (31) B4G2HDMH0 = e32π2{(mℓi−mℓj)(SH0ℓiτ′∗PH0ℓjτ′+SH0ℓjτ′PH0ℓiτ′∗)IH01+2mτ′(SH0ℓiτ′∗PH0ℓjτ′−SH0ℓjτ′PH0ℓiτ′∗)IH02} , (32)

where the loop integrals , , and are given by (taking ):

 IH+1 ≈ ∫10dxx2(x−1)m2H−x+m2ν′(1−x) , IH+2 ≈ ∫10dxx(x−1)m2H−x+m2ν′(1−x) , IH01 ≈ ∫10dxx2(1−x)m2τ′x+m2H0(1−x) , IH02 ≈ ∫10dxx2m2τ′x+m2H0(1−x) .

The dominant terms in Eqs. 29-32 are the ones proportional to from the charged Higgs exchange contribution,

 (SH−ℓiν′∗SH−ℓjν′−PH−ℓiν′∗PH−ℓjν′) = −g24mν′mτ′m2Wf2β|U44|2|Σν44|2(1−tβfβΣν44)(δ∗UiδΣj+δUjδ∗Σi) , (SH−ℓiν′∗PH−ℓjν′−SH−ℓiν′∗PH−ℓjν′) = −g24mν′mτ′m2Wf2β|U44|2|Σν44|2(1−tβfβΣν44)(δ∗UiδΣj−δUjδ∗Σi) , (34)

since the terms proportional to in the neutral Higgs exchanges vanish due to (see Eq. 18).

We thus find that in our 4G2HDM, the decays and are sensitive to and through the products and , respectively, so that, in principle, one can avoid constraints on the quantities and if and are sufficiently small.

In Figs. 3 we plot as a function of and , for GeV, GeV and fixing . We see that for e.g. GeV and for values of and of [for which the product reproduces the measured (see Fig. 2)], and are required to be smaller than , implying that and .

In Fig. 4 we plot as a function of and , and in Fig. 5 we give a scatter plot of the allowed values in the plane, for which (i.e., below its 90%CL bound). In both plots we use GeV, GeV and we fix . The individual couplings and are randomly chosen to always be within values that reproduce the measured (see Fig. 2). We see that the products and are required to be at most , in order to be consistent with the current bounds on .