I 1. Introduction

Leptonic moments, CP phases and the Higgs boson mass constraint


Amin Aboubrahim***Email:a.aboubrahim@neu.edu, Tarek IbrahimEmail:tibrahim@zewailcity.edu.eg and Pran NathEmail:p.nath@neu.edu

Department of Physics, Northeastern University, Boston, MA 02115-5000, USA

University of Science and Technology, Zewail City of Science and Technology,

6th of October City, Giza 12588, Egypt§§§Permanent address: Department of Physics, Faculty of Science, University of Alexandria, Alexandria 21511, Egypt


Abstract

Higgs boson mass measurement at GeV points to a high scale for SUSY specifically the scalar masses. If all the scalars are heavy, supersymmetric contribution to the leptonic moments will be significantly reduced. On the other hand the Brookhaven experiment indicates a deviation from the standard model prediction. Here we analyze the leptonic moments in an extended MSSM model with inclusion of a vector like leptonic generation which brings in new sources of CP violation. In this work we consider the contributions to the leptonic moments arising from the exchange of charginos and neutralinos, sleptons and mirror sleptons, and from the exchange of and bosons and of leptons and mirror leptons. We focus specifically on the moments for the muon and the electron where sensitive measurements exist. Here it is shown that one can get consistency with the current data on under the Higgs boson mass constraint. Dependence of the moments on CP phases from the extended sector are analyzed and it is shown that they are sensitively dependent on the phases from the new sector. It is shown that the corrections to the leptonic moments arising from the extended MSSM sector will be non-vanishing even if the SUSY scale extends into the PeV region.

Keywords: Leptonic moments, CP phases, Higgs mass, PeV scale.
PACS numbers: 12.60.-i, 14.60.Fg

I 1. Introduction

The observation by ATLAS Aad:2012tfa () and by CMS  Chatrchyan:2012xdj () of the Higgs boson with a mass of GeV has put very stringent constraints on low scale supersymmetry. Since the tree level mass of the Higgs boson lies below , a large loop correction from the supersymmetric sector is needed which in turn implies a high scale for the weak scale supersymmetry and specifically for the scalar masses. A large SUSY scale also has direct implications for the of the muon. Thus the current experimental result gives for the muon  Beringer:1900zz ()

(1)

which is about a three sigma deviation from the standard model prediction. Similarly for the electron the experimental determination of is very accurate and the uncertainty is rather small, i.e., one has Giudice:2012ms ()

(2)

This result relies on a QED calculation up to four loops. Thus along with Eq. (1), Eq. (2) also acts as a constraint on the standard model extensions. Supersymmetric theories with low weak scale mass can make corrections to which could be as large as the standard model electroweak corrections and even larger and have strong CP phase dependence Ibrahim:1999hh (); Ibrahim:1999aj (); Ibrahim:2001ym () (for early work see Yuan:1984ww ()). These arise largely from the chargino and sneutrino exchange diagram with the neutralino and smuon exchange diagram making a relatively small contribution. However, if the scalar masses are large, the supersymmetric exchange contributions will be small due to the largeness of the sneutrino and the smuon masses.

In this work we give an analysis of the for the muon and for the electron in an extended MSSM model with a vector like leptonic generation. We note that vector like multiplets are anomaly free and they appear in a variety of settings which include grand unified models, strings and D brane models Ibrahim:2008gg (); vectorlike (); Babu:2008ge (); Liu:2009cc (); Martin:2009bg (). Further, it is known that has a sharp dependence on CP phases Ibrahim:1999hh (); Ibrahim:1999aj (); Ibrahim:2001ym (). For this reason we investigate also the dependence of the muon and the electron on the CP phases in the extended MSSM model. Here we are particularly interested in the dependence on the CP phases that arise from the new sector involving vector like leptons. We note that the CP phases are constrained in this case by the electric dipole moment of the electron which currently has the value cm Baron:2013eja () while the upper limit on the muon EDM is cm Beringer:1900zz () and is rather weak. As discussed in several works even with large phases the EDMs can be suppressed either by mass suppression Nath:1991dn (); Kizukuri:1992nj () or via the cancellation mechanism Ibrahim:1998je (); Ibrahim:1997gj (); Falk:1998pu (); Ibrahim:1998je (); Brhlik:1998zn (); Ibrahim:1999af (). Several analyses of the vector like extensions of MSSM already exist in the literature Ibrahim:2010va (); Ibrahim:2010hv (); Ibrahim:2011im (); Ibrahim:2012ds (); Aboubrahim:2013gfa (); Dermisek:2013gta (); Nickel:2015dna (); Aboubrahim:2015zpa (); Ibrahim:2015hva (); Ibrahim:2016rcb ().

The outline of the rest of the paper is as follows: In section 2 we give an analytical computation for the contribution of the vectorlike lepton generation to of the muon and of the electron. In section (3) we give a numerical analysis of the contributions arising from MSSM and from the extended MSSM with a vector like leptonic generation. Conclusions are given in section 4. Details of the extended MSSM model with a vector like leptonic generation are given in the Appendix. The explanation of the muon anomaly with vector like leptons was considered previously in  Dermisek:2013gta () within a non-supersymmetric framework. Our analysis is within a supersymmetric framework where we carry out a simultaneous fit to both the muon as well as the electron anomaly. Further, we explore the implications of the CP phases arising from the new sector.

Ii 2. Analysis of and with exchange of vector like leptons

The extended MSSM with a vector like leptonic generation is discussed in detail in the Appendix. Using the formalism described there we compute the contribution to the anomalous magnetic moment of a charged lepton . We discuss now in detail the various contributions. The contribution arising from the exchange of the charginos, sneutrinos and mirror sneutrinos as shown in the left diagram in Fig. 1 is given by

(3)

where is the mass of chargino and is the mass of sneutrino and where the form factors and are given by

(4)

and

(5)

The couplings appearing in Eq. (3) are given by

(6)
(7)

where and are the charged lepton and sneutrino diagonalizing matrices and are defined by Eq. (47) and Eq.(57) and and are the matrices that diagonalize the chargino mass matrix so that Ibrahim:2007fb ()

(8)

Further,

(9)
(10)

where is the mass of the boson and where are the two Higgs doublets of MSSM.

Figure 1: The diagrams that contribute to the leptonic () magnetic dipole moment via exchange of charginos (), sneutrinos and mirror sneutrinos () (left diagram) inside the loop and from the exchange of neutralinos (), sleptons and mirror sleptons () (right diagram) inside the loop.
Figure 2: The W loop (the left diagram) involving the exchange of sequential and vectorlike neutrinos and the Z loop (the right diagram) involving the exchange of sequential and vectorlike charged leptons that contribute to the magnetic dipole moment of the charged lepton .

The contribution arising from the exchange of neutralinos, charged sleptons and charged mirror sleptons as shown in the right diagram in Fig. 1 is given by

(11)

where the form factors are

(12)

and

(13)

The couplings that enter in Eq. 11 are given by

(14)
(15)

where

(16)
(17)

and

(18)
(19)

and where

(20)
(21)

Here are defined by

(22)
(23)

where diagonalizes the neutralino mass matrix, i.e.,

(24)

Further, that enter in Eqs. (14) and (15) is a matrix which diagonalizes the charged slepton mass squared matrix and is defined in Eq. (53.).

Next we compute the contribution from the exchange of the and bosons. Thus the exchange of the and the exchange of neutrinos and mirror neutrinos as shown in the left diagram of Fig. 2 gives

(25)

where the form factors are given by

(26)

and

(27)

The couplings that enter in Eq. (25) are given by

(28)
(29)

Here are matrices of a bi-unitary transformation that diagonalizes the neutrino mass matrix and are defined in Eq. (43).

Finally the exchange of the and the exchange of leptons and mirror leptons as shown in the right diagram of Fig. 2 gives

(30)

where

(31)

and

(32)

and is the boson mass. The couplings that enter in Eq. (30) are given by

(33)

and

(34)

Iii 3. Estimates of and

We begin by discussing the prediction for and for MSSM when the scalar masses are large lying in the several TeV region. In Tables 1 and 2 we exhibit the results for two benchmark points where we assume universality and take the scalar masses and the trilinear couplings to be all equal. Table 1 exhibits the result of the computation for where individual contributions arising from the chargino exchange, neutralino exchange, exchange and exchange are listed. The entries exhibit the contributions over and above what one expects from the standard model and so the entries for the and exchanges show a null value. Thus the entire contribution in this case arises from the chargino and the neutralino exchange and their sum gives a value which is two orders of magnitude smaller than the experimental result of Eq. (1). A very similar analysis is given in Table 2 for where again the contribution to arises from the exchange of charginos and neutralinos and their sum is which is three orders of magnitude smaller than the result of Eq. (2). Thus with a high scale of the scalar masses one cannot explain the results of Eq. (1) and Eq. (2).

We turn now to the analysis within the extended MSSM with a vector like leptonic generation. As in the analysis within MSSM here also we assume the universality of the soft parameters so that we set and in the computation of the charged slepton mass squared matrix. Similarly we assume and for the computation of the sneutrino mass squared matrix (see Appendix). The contributions from the chargino exchange, the neutralino exchange, and the and exchange are listed in Table 1 and Table 2 for two benchmark points. In this case the boson and the boson exchange contributions are non-vanishing and the contributions listed are those over and above what one expects in the standard model. As in the MSSM case here also one finds that the contributions from the chargino exchange and from the neutralino exchange fall significantly below the experimental results of Eq. (1) and Eq. (2). However, in this case including the contributions from the exchange and from the boson exchange one finds that consistency with Eq. (1) and Eq. (2) is achieved. At the same time one has the Higgs boson mass in the model for both benchmarks (a) and (b) at GeV consistent with the experimental measurements by ATLAS Aad:2012tfa () and by CMS  Chatrchyan:2012xdj (). Here the loop correction that gives mass to the Higgs boson comes from the MSSM sector while the extra vector like leptonic generation makes a negligible contribution.

In the analysis of and the exchange of both the sequential leptons and the mirrors play a role with the mirror exchange being the more dominant. The analysis requires diagonalization of a mass matrix in the charged lepton-charged mirror lepton sector and diagonalization of a mass matrix in the neutrino-mirror neutrino sector. Parameter choices are made to ensure that the eigenvalues in the charged lepton sector give the desired experimental values for , and along with two additional masses, one for the sequential fourth generation lepton and the other for the mirror charged lepton. Their values are listed in Table 3 for the case of two benchmark points (a) and (b). A similar analysis holds for the neutrino-mirror neutrino sector where we get two additional eigenvalues, one for the fourth generation neutrino and the other for the mirror neutrino. Their values are also listed in Table 3 for two benchmark points. The analysis also requires diagonalization of a matrix in the charged slepton and charged mirror slepton sector, as well as diagonalization of a matrix in the sneutrino and the mirror sneutrino sector.

(a) (b)
Contribution MSSM Vectorlike MSSM Vectorlike
Chargino
Neutralino
W Boson
Z Boson
Total
Table 1: The contribution of the vectorlike multiplet vs the contribution from the MSSM sector to the anomalous magnetic moments of the muon for two illustrative benchmark points (a) and (b). They are: (a) , , , , , , and (b) , , , , , , . Other parameters have the values , , , , , , , , , , , , , , , , , , . For the MSSM analysis the following parameters were used for both cases (a) and (b): The scalar masses are taken to be universal with and the trilinear coupling is taken to be universal . Other inputs are: , , , , . All masses are in GeV and phases in rad.
(a) (b)
Contribution MSSM Vectorlike MSSM Vectorlike
Chargino
Neutralino
W Boson
Z Boson
Total
Table 2: The contribution of the vectorlike multiplet vs the contribution from the MSSM sector to the anomalous magnetic moments of the electron for two illustrative benchmark points (a) and (b) as given in table 1.
Mass Spectrum (GeV)
Particles (a) (b)
Mirror Neutrino 208 207
Fourth Sequential Neutrino 816 395
Mirror Lepton 253 349
Fourth Sequential Lepton 545 226
Table 3: The mass of the heavy particles obtained after diagonalizing the lepton and neutrino mass matrices for benchmark points (a) and (b) of Table 1.

We discuss now some further features of the analysis which includes the vector like leptonic generation. In Figure 3 we show the variation of as a function of the mass of the mirror lepton as given by Eq. (45), for four values. A remarkable feature of this graph is the dependence on it exhibits. Notice that for a fixed , decreases for increasing values of as varies from to . Now we recall that the Yukawa coupling of a charged lepton has a dependence and as a consequence the contribution of the charged lepton to becomes larger for larger which is a well known result. However, the Yukawa coupling of the mirror lepton goes like  Ibrahim:2008gg () and so decreases for larger values of . This feature explains the dependence in Figure  3. It also shows that the and exchange contributions in this case are being controlled by exchange of the mirror particles. A very similar dependence on is exhibited by .

The anomalous magnetic moments are quite sensitive to CP phases as first demonstrated in the analysis of  Ibrahim:1999hh (); Ibrahim:1999aj (); Ibrahim:2001ym () for the case of CP phases that arise in supergravity  Ibrahim:1999hh (); Ibrahim:2001ym () and more generally for the case of MSSM Ibrahim:1999aj (). In those analyses it was also found that large CP phases could be made consistent with the experimental constraints on the EDMs by the cancellation mechanism Ibrahim:1997gj (); Falk:1998pu (); Ibrahim:1998je (); Brhlik:1998zn (); Ibrahim:1999af (). In the present analysis the contribution from the MSSM sector is suppressed and the dominant contribution arises from the and exchanges. For the case of three generations this sector does not have any CP phases in the leptonic sector. However, the extended MSSM with a vector like leptonic multiplet allows for CP phases which cannot be removed by field redefinitions. It is of interest then to discuss the dependence of and on the CP phases that arise in the extended MSSM. We discuss now the dependence of and on such phases. In Fig. (4) we exhibit the dependence of and on , which is the phase of (see Appendix). A sharp dependence on is seen for both and for . A very similar sensitivity to the CP phase which is the phase of (see Appendix) is exhibited in Fig. (5). To explore further the sensitivity of and of to parameters in the vector like sector we exhibit in Fig. (6) the dependence of and on which is the co-efficient of the term in the superpotential (see Eq. (37)). One can see in Fig. (6) the strong dependence of and on . In the analyses given so far both and have very significant dependence on the parameters arising from inclusion of the vector like sector. However, there are parameters which affect and differently. This is the case for . Here as seen in the left panel of Fig. (7) is a sensitive function of but not so for the case for (not exhibited) because of its much larger size. Finally we note that even if the SUSY scale lies in the PeV region, the contributions from the and exchange arising from Fig. (2) survive while the diagrams of Fig. (1) give a vanishingly small contribution. This is illustrated in the right panel of Fig. (7).

Figure 3: as a function of when = 20, 25, 30, 35. Other parameters are , , , , , , , , , , , , , , , , , , , , , , , . All masses are in GeV and phases in rad.
Figure 4: (left panel) and (right panel) as a function of in the range when . Other parameters are , , , , , , , , , , , , , , , , , , , , , , , , , . All masses are in GeV and phases in rad.
Figure 5: (left panel) and (right panel) as a function of in the range when . Other parameters are , ,