Observables of low-lying supersymmetric vectorlike leptonic generations via loop corrections

A correlated analysis of observables arising from loop induced effects from a vectorlike generation is given. The observables include flavor changing radiative decays $\mu\to e \gamma, \tau\to \mu \gamma, \tau\to e \gamma$, electric dipole moments of the charged leptons $e,\mu, \tau$, and corrections to magnetic dipole moments of $g_\mu-2$ and $g_e-2$. In this work we give a full analysis of the corrections to these observables by taking into account both the supersymmetry loops as well as the exchange of a vectorlike leptonic generation. Thus the fermion mass matrix involves a $5\times 5$ mixing matrix while the scalar sector involves a $10\times 10$ mixing matrix including the CP violating phases from the vectorlike sector. The analysis is done under the constraint of the Higgs boson mass at the experimentally measured value. The loops considered include the exchange of $W$ and $Z$ bosons and of leptons and a mirror lepton, and the exchange of charginos and neutralinos, sleptons and mirror sleptons. The correction to the diphoton decay of the Higgs $h\to \gamma\gamma$ including the exchange of the vectorlike leptonic multiplet is also computed.


Introduction
Precision measurements can reveal small deviations from the standard model (SM) prediction and indicate the existence of new physics beyond the standard model. There are a variety of experiments which are exploring the properties of elementary particles to a high precision to this end. These include flavor changing radiative decays of the charged leptons µ → eγ, τ → µγ and τ → eγ, i.e., the MEG experiment [1], BaBar Collaboration [2] and the Belle Collaboration [3], the electric dipole moment (EDM) of the electron [4], of the muon as well as of quarks [5], and the precision measurement of the anomalous magnetic moment of the muon [6] and of the electron. In this work we explore the implications of a low-lying vectorlike generation on the leptonic processes mentioned above. Vectorlike generations exist in a variety of models including grand unified models, string models and D brane models [7,8,9]. Some of these vectorlike generations may be light. Further, vectorlike generations are anomaly free so they preserve good properties of the model as a quantum field theory. The mixings of these light vectorlike generations with the three generations of leptons can lead to contributions to the processes noted above. Several studies of the effects of vectorlike leptons in various processes already exist [10,11,12,13,14,15,16,17,18,19,20] and in non-supersymmetric context in [21,22,23,24,25,26,27]. In this analysis we perform a correlated study of the contributions of the vectorlike generation to these phenomena. The analysis involves an enlarged leptonic mass matrix which is 5 × 5 and a slepton mass-squared matrix which is 10 × 10 including the CP violating phases from the vectorlike sector. In the analysis we consider loop exchange of W and Z bosons, leptons and mirror leptons, and exchange of charginos and neutralinos along with the sleptons and mirror sleptons. The analysis is done under the constraint of the Higgs boson mass at ∼ 125 GeV, and an analysis of the contribution to the branching ratio h → γγ from the vectorlike leptonic exchange is also given.
The outline of the rest of the paper is as follows: In section 2 we give a description of the model. In section 3 we give an analysis of the flavor changing decays of the charged leptons. An analysis of the EDM of the charged leptons is given in section 4. In section 5 we give an analysis of g − 2 for the charged leptons. An analysis of the contribution of the vectorlike leptonic generation to the diphoton decay of the Higgs boson is given in section 6. A numerical analysis is given in section 7 and conclusions are given in section 8. Further details of the analysis is given in appendices A and B.

Description of the model
In this section we give details of the model used in the rest of the paper. As mentioned in section 1 the model consists of three generations of sequential leptons (e, µ, τ ) and in addition a single vectorlike generation. Thus one has four sequential families and a mirror generation. The properties of the sequential generation under SU (3) C × SU (2) L × U (1) Y are given by where the last entry on the right hand side of each ∼ is the value of the hypercharge Y defined so that Q = T 3 + Y and we have included in our analysis the singlet field ν c i , where i runs from 1−4. The mirrors are given by The main difference between the leptons and the mirrors is that while the leptons have V −A type interactions with SU (2) L × U (1) Y gauge bosons the mirrors have V + A type interactions.
We assume that the mirrors of the vectorlike generation escape acquiring mass at the GUT scale and remain light down to the electroweak scale where the superpotential of the model for the lepton part may be written in the form i 4Lν whereˆimplies superfields,ψ L ≡ψ τ L stands forψ 3L ,ψ µL stands forψ 2L andψ eL stands forψ 1L .
The mass terms for the neutrinos, mirror neutrinos, leptons and mirror leptons arise from the term where ψ and A stand for generic two-component fermion and scalar fields. After spontaneous breaking of the electroweak symmetry, ( H 1 2), we have the following set of mass terms written in the four-component spinor notation so that where the basis vectors in which the mass matrix is written is given bȳ and the mass matrix M f of neutrinos is given by We define the matrix elements (2, 2) and (5,5) of the mass matrix as m N and m ν G , respectively, so that The mass matrix is not hermitian and thus one needs biunitary transformations to diagonalize it. We define the biunitary transformation so that where ψ 1 , ψ 2 , ψ 3 , ψ 4 , ψ 5 are the mass eigenstates for the neutrinos. In the limit of no mixing we identify ψ 1 as the light tau neutrino, ψ 2 as the heavier mass mirror eigenstate, ψ 3 as the muon neutrino, ψ 4 as the electron neutrino and ψ 5 as the other heavy four-sequential generation neutrino. A similar analysis goes to the lepton mass matrix M where We introduce now the mass parameters m E and m G for the elements (2,2) and (5,5), respectively, of the mass matrix above so that CP phases that arise from the new sector are defined so that As in the neutrino mass matrix case, the charged lepton mass matrix is not hermitian and thus one needs again a biunitary transformation to diagonalize it. We define the biunitary transformation so that where τ α (α =1−5) are the mass eigenstates for the charged lepton matrix.
The mass-squared matrices of the slepton-mirror slepton and sneutrino-mirror sneutrino sectors come from three sources: the F term, the D term of the potential and the soft SUSY breaking terms. After spontaneous breaking of the electroweak symmetry the Lagrangian is given by where and L soft is given in appendix A.
Other flavor changing decays are τ → µγ and τ → eγ. Here the current experimental limits on the branching ratios of these processes from the BaBar Collaboration [2] and from the Belle Collaboration [3] are Improvement in the measurements of flavor changing processes is expected to occur at the SuperB factories [28,29,30] (for a review see [31]). Thus it is of interest to see if theoretical estimates for these branching ratios can lie close to the current experimental limits to be detectable in improved experiment. Flavor violating radiative decays have been analyzed in several previous works (see, e.g., [31,32,33,34,35,36,37]).
However, none of these works explore the class of models discussed here.
We discuss now the specifics of the model. Thus the decay µ → eγ is induced by one-loop electric and magnetic transition dipole moments, which arise from the diagrams of Fig. 1. For an incoming muon of momentum p and a resulting electron of momentum p , we define the amplitude e(p )|J α |µ(p) =ū e (p )Γ α u µ (p), where with q = p − p and where m f denotes the mass of the fermion f . The branching ratio of µ → eγ is given by where the form factors F µe 2 and F µe 3 arise from the chargino, neutralino and vector bosons contributions as follows It is also useful to define B m and B e as follows where B m is the branching ratio from the magnetic dipole operator and B e is the branching ratio from the electric dipole operator. We discuss now the individual contributions to F µe 2 and F µe 3 from supersymmetric and non-supersymmetric loops.
The chargino contribution F µe 2χ + is given by where F 3 (x) and F 4 (x) are given by and The neutralino contribution F µe 2χ 0 is given by where F 1 (x) and F 2 (x) are given by and The contributions from the W exchange F µe 2W is given by where the form factors F W (x) and G W (x) are given by and The contribution F µe 2Z from the Z exchange is given by where the form factors F Z (x) and G Z (x) are given by and The chargino contribution F µe 3χ + is given by where The neutralino contribution F µe 3χ 0 is given by where The W boson contribution F µe 3W is given by where the form factor I 1 is given by x + 1 4 And finally, the Z exchange diagram contribution F µe 3Z is given by where the form factor I 2 is given by (39), (41) and (43), are given in appendix B.
An analysis for B(τ → eγ) can be done similarly so that where the expressions for the form factors, F τ e 2 and F τ e 3 , can be obtained from Eqs. (21) and (22) by the replacements: where the expressions for the form factors F τ µ 2 and F τ µ 3 can be deduced from Eqs. (21) and (22) by the re-
The current experimental limit on the EDM of the tau lepton is [38]  Next we discuss the case when we include a vectorlike leptonic multiplet which mixes with the three generations of leptons. In this case the mass eigenstates will be linear combinations of the three generations plus the vectorlike generation which includes mirror particles. Here we discuss the contribution of the model to the lepton EDM. These contributions arise from four sources: the chargino exchange, the neutralino exchange, the W boson exchange and the Z boson exchange.
Using the interactions given in appendix B, the chargino contribution is given by where the form factor F 6 (x) is given by Eq. (38).
Using the interactions given in appendix B, the neutralino contribution is given by where the form factor F 5 (x) is given by Eq. (40).
The contributions to the lepton electric dipole moment from the W and Z exchange arise from similar loops.
Using the interactions given in appendix B the contribution arising from the W exchange diagram is given by where the form factor I 1 is given by Eq. (42).
The Z boson exchange diagram contribution is given by where the form factor I 2 is given by Eq. (44). Again, all couplings used here are given in appendix B.

Analysis of g − 2 with exchange of vectorlike leptons
The current experimental result for the muon g − 2 [5] is which is about a three sigma deviation from the standard model prediction. For the electron g e −2 experiment gives [39] ∆a e = a exp where m χ − i is the mass of chargino χ − i and mν j is the mass of sneutrinoν j and where the form factors F 3 and F 4 are given by Eqs. (26) and (27).  The contribution arising from the exchange of neutralinos, charged sleptons and charged mirror sleptons as shown in the right diagram of Fig. 3 is given by where the form factors F 1 and F 2 are given by Eqs. (29) and (30).
Next we compute the contribution from the exchange of the W and Z bosons. Thus the exchange of the W and the exchange of neutrinos and mirror neutrinos as shown in the left diagram of Fig. 4 gives where the form factors F W and G W are given by Eqs. (32) and (33).
Finally the exchange of the Z and the exchange of leptons and mirror leptons as shown in the right diagram of Fig. 4 gives where the form factors F Z and G Z are given by Eqs. (35) and (36) and m Z is the Z boson mass. The couplings that enter in Eqs. (56), (57), (58) and (59) are given in appendix B. For other works relating the muon anomalous magnetic moment to new physics see [40,41].

Leptonic vectorlike contribution to h → γγ
The observed diphoton decay of the Higgs boson shows an agreement with the standard model prediction within the limits of uncertainty which is still significant. As more data is collected and uncertainties better modeled, the signal strength, R γγ , will be measured with a larger accuracy and any new physics manifest as particles in the loop will be better probed. Thus the ATLAS and CMS collaborations [42,43] report a signal strength of In the SM, the largest contribution to h → γγ comes from the exchange of W bosons and top quarks in the loop. Thus the SM decay width of a Higgs boson of mass m h may be approximated by the expression [44] where A SM ≈ −6.49, A 1 and A 1 2 are loop functions (see Appendix of [44]), τ i = 4m 2 i /m 2 h , N c is the color number and Q t the top quark charge. The inclusion of SUSY allows for the exchange of heavier particles in the loop. In general the decay width of h → γγ in supersymmetry takes the form where α is the CP-even Higgs mixing angle, Q W is the W -boson charge, b 1 2 = 4 3 (for Dirac fermions of mass m f , number N f and charge Q f ) and b 0 = 1 3 (for charged scalars of mass m S , number N c,S and charge Q S ). The inclusion of the vectorlike leptonic generation contributes to the fermionic and scalar parts where the latter is due to the supersymmetric partners of the vectorlike leptons.
In this analysis the couplings of the Higgs boson to the first three generations are assumed negligible in comparison with the vectorlike counterparts. Hence the mixings between the vectorlike generation and the first three generations in Eq. (7) can be assumed negligible and so the lepton mass matrix from the vectorlike generation may be written as The two mass-squared eigenvalues resulting from diagonalizing the matrix of Eq. (63) are Calculating the vectorlike fermionic contribution, one finds that Considering only this fermionic contribution, we find that the Higgs diphoton rate is enhanced by a factor Now turning to the bosonic contribution which is due to the four scalar superpartners of the vectorlike leptons. The mass eigenvalues are obtained from a 4 × 4 mass-squared mixing matrix and in the basis where (M 2 4 ) 2×2 is given by and (M 2 E ) 2×2 is given by In this analysis, the scalar masses-squared,M 2 4L ,M 2 4 ,M 2 χ ,M 2 E are much larger than the vectorlike masses, |h 6 |, |h 7 | and so the 4 × 4 mass-squared matrix becomes block diagonal. Thus the two mass-squared matrices are now decoupled with superpartner˜ 41,2 for the first andẼ 1,2 for the second. The total bosonic contribution is the sum of the contributions coming from the two decoupled mass-squared matrices and can be written as Here and where, for convenience, we renamed the matrices as M 2 ≡ M 2 4 and M 2 ≡ M 2 E . Assuming σ(pp → h) obs = σ(pp → h) SM the enhancement factor R γγ is given by

Numerical Analysis
Here we present a correlated analysis of the observables discussed in the previous sections including the effect of vectorlike leptons (for other works related to vectorlike leptons see [45,46]). In the analysis we will include the CP violating phases from the vectorlike generation. SUSY CP phases are known to affect electroweak phenomena and these effects can be very significant [47,48,49,50,51,52,53,54,55,56]. In the analysis we use SUGRA model [57] with non-universal soft parameters given by m  Table 1.   Since SUSY contributions involve the exchange of scalars (sleptons and sneutrinos), the input of Table 1 suggests that such a contribution will be suppressed due to the high scalar masses (being in the several TeV range). Hence, we expect the mirror and fourth sequential generations to have a more significant contribution to the observables. The parameters in the vectorlike sector are chosen so as to be consistent with the lepton masses obtained after diagonalization. We present in Table 2 the results of the observables obtained for three benchmark points, (a), (b) and (c) of Table 1. On the right-most column, the experimental limits on the corresponding observables is summarized for comparison purpose and the computed values of the observables satisfy these bounds. Thus the branching ratios of µ → eγ and τ → µγ are below but close to their upper limits, especially for points (b) and (c) and could be probed by a small improvement in experiment. The branching ratio of τ → eγ appears to be two to three orders of magnitude smaller than its upper limit.
However, one can achieve somewhat higher values by varying the Yukawa masses m E and/or m G as we will see later. It is interesting that for the same parameter set the EDM of the electron is also close to its current limit while the EDMs of the muon and of tau are five to seven orders of magnitude smaller than the upper limits. The electron and muon anomalous magnetic moments are typically small and the contribution is not significant to explain the ∼ 3σ deviation if indeed it holds up in improved experiment. As for the diphoton rate enhancement there are discernible corrections to the branching ratio but consistent with the current limits from ATLAS and CMS, Eq. (60). Here we note that it was shown in previous works (see, e.g., [46]) that a muon g − 2 close to the experimental limit can be obtained via leptonic vectorlike exchange. To see if this is possible with the current constraints we take point (a) from Table 2 and modify the input parameters.
The results are listed in Table 3 where a muon g − 2 of O(10 −9 ) and with in the observed 3σ deviation is obtained. The rest of the observables are still in check but one of the branching ratios, namely, τ → eγ, has become very small. Also, we have obtained a four orders of magnitude increase in the muon EDM.     We discuss now in further detail the sensitivity of some of the observables on the various input parameters.
Thus in Fig. 6 we display the variation of B(µ → eγ), B(τ → eγ), B(τ → µγ) and the electron EDM, |d e | as a function of the CP phases from the vectorlike sector. It is clear that all those observables exhibit a sensitive dependence on the CP phases where the branching ratios oscillate above and below their upper limits. Also, the electron EDM shows large variations very close to the experimental upper limit. The different curves in each plot correspond to different choices of the couplings |f 3 |, |f 3 |, |f 3 |, |f 4 | and |f 4 | where larger values of the observables are obtained for larger couplings. Note that those couplings cannot take arbitrarly large values since this will spoil the lepton masses.
In Fig. 7 we show the dependence of the branching ratios of µ → eγ, τ → µγ, τ → eγ and the electron EDM on the vectorlike Yukawa masses for points (c) and (e) of Table 1 GeV while B(µ → eγ) is already below the upper limit for even smaller m E = m G . This shows how the interplay of those parameters lead to all constraints to fall in place. While it was difficult to achieve larger B(τ → eγ) values (in Table 2), it was easier to do so for point (e).   In the above we discussed the lepton flavor changing process µ → eγ but did not discuss the flavor changing processes µ → e conversion and µ → 3e. A proper treatment of these processes at the same level of care as done for the other processes treated here is outside the scope of this work. Thus, for example, for the µ → e conversion one needs computation of a set of box and penguin diagrams which would again involve our 10 × 10 scalar mass matrices in the loops. In addition µ → e conversion has much more model dependence because of nuclear physics effects. Here we give approximate results for them valid in certain limits which, however, do indicate the expected size of the branching ratios for these processes for the parameter sets in our case given in Table 2. Thus in the dipole dominance approximation, one has [60] The right hand side of Eq. (75) evaluates to ∼ 6 × 10 −3 . Using this ratio B(µ → 3e) ∼ 3.4 × 10 −15 for column 3 in Table 2. This is to be compared with the current experimental limit [61] B(µ → 3e) < 1.0 × 10 −12 at 90% C.L.
In the future one expects that experiments using Al nuclei will reach a sensitivity in the range [31,65] B(µ → e) Al < 10 −16 − 10 −18 .

Conclusion
In a large class of models such as based on grand unification, on strings and branes, one has vectorlike states some of which could be light and lie in the low energy region accessible to experiment. Their presence can affect low energy phenomena through loop corrections. In supersymmetric theories the vectorlike generations will have particles and their mirrors as well as sparticles and their mirrors. This means that in a model with three generations there will be two more particles that can appear in the mixing matrix, making the fermionic mixing matrix a 5 × 5 mixing matrix. In the slepton sector, one will have in general a 10 × 10 mixing matrix.
The analysis is done including the CP violating phases in the mixings of the vectorlike generation. In his work we have carried out a correlated study of the effects of the vectorlike generation on several observables.
These include µ → eγ, τ → µγ, τ → eγ, muon and electron magnetic moments, g µ − 2 and g e − 2, and EDMs We define the slepton mass-squared matrix M 2 τ in the basis and label the matrix elements as (M 2 τ ) ij = M 2 ij where these elements are given by We assume that the masses that enter the mass-squared matrix for the scalars are all of electroweak size.
This mass-squared matrix is hermitian and can be diagonalized with a unitary transformatioñ The mass-squared matrix in the sneutrino sector has a similar structure. In the basis (ν τ L ,Ñ L ,ν τ R ,Ñ R ,ν µL ,ν µR ,ν eL ,ν eR ,ν 4L ,ν 4R ), where the sneutrino mass squared matrix (M 2 ν ) ij = m 2 ij has elements given by Again as in the charged slepton sector we assume that all the masses are of the electroweak size so all the terms enter in the mass-squared matrix. This mass-squared matrix can be diagonalized by the unitary Appendix B Interactions that enter in the analyses of the radiative decays, of the EDMs and of the magnetic dipole moments of the leptons In this appendix we discuss the interactions in the mass diagonal basis involving charged leptons, sneutrinos and charginos. Thus we have such that where D τ L,R andD ν are the charged lepton and sneutrino diagonalizing matrices and are defined by Eq. (13) and Eq. (84), respectively and U and V are the matrices that diagonalize the chargino mass matrix M C so that [56] U Further, (κ N , κ τ , κ µ , κ e , κ 4 ) = (m N , m τ , m µ , m e , m 4 ) √ 2m W cos β , (κ E , κ ντ , κ νµ , κ νe , κ ν4 ) = (m E , m ντ , m νµ , m νe , m ν4 ) √ 2m W sin β .
where m W is the mass of the W boson and tan β = H 2 2 / H 1 1 where H 1 , H 2 are the two Higgs doublets of MSSM.
Here X are defined by where X diagonalizes the neutralino mass matrix, i.e., Further,D τ that enter in Eqs. (92) and (93) is a matrix which diagonalizes the charged slepton mass squared matrix and is defined in Eq. (81).
In addition to the supersymmetric loop diagrams, we compute the contributions arising from the exchange of W and Z bosons and leptons and mirror leptons in the loops. For the W boson exchange the interactions are given by where and Here D ν L,R are matrices of a biunitary transformation that diagonalizes the neutrino mass matrix and are defined in Eq. (9). For the Z boson exchange the interactions that enter are given by − L τ τ Z = Z ρ 5 α=1 5 β=1τ α γ ρ (C Z Lαβ P L + C Z Rαβ P R )τ β , where and with x = sin 2 θ W .