Charged Lepton Flavor Violation μ→eγ in μ−τ Symmetric SUSY SO(10) mSUGRA, NUHM, NUGM, and NUSM theories and LHC
aa r X i v : . [ h e p - ph ] D ec Charged Lepton Flavor Violation µ → eγ in µ - τ Symmetric SUSY SO(10) mSUGRA,NUHM, NUGM and NUSM Theories and LHC
Kalpana Bora, Gayatri Ghosh
Department of Physics, Gauhati University, Guwahati 781014, India
Charged Lepton Flavor Violation (cLFV) processes like µ → eγ are rare decay processes that areanother signature of physics beyond standard model. These processes have been studied in variousmodels that could explain neutrino oscillations and mixings. In this work, we present bounds on thecLFV decay µ → eγ in a µ - τ symmetric SUSY SO(10) theory, using the type I seesaw mechanism.The updated constraints on BR( µ → eγ ) from the MEG experiment, the recently measured value ofHiggs mass at LHC, and the value of θ from reactor data have been used. We present our resultsin mSUGRA, NUHM, NUGM, and NUSM models, and the sensitivity to test these theories at thenext run of LHC is also discussed. I. INTRODUCTION
The flavor changing neutral currents (FCNCs) are forbidden in the standard model (SM) of particle physics, at treelevel. They are allowed beyond tree level, but highly suppressed by the GIM mechanism [1]. Flavor mixing in thestandard model quark sector is well established, through processes like K − ¯ K oscillations, B d − ¯ B d mixing etc. Thephenomenon of neutrino oscillations, already proved by experiments, require one to go beyond the standard model.These neutrino oscillations, and hence mixings, are also expected to induce flavor violations in the charged leptonicsector. Theoretically, such cLFV processes could be induced in different theories with BSM particles such as SUSYGUT [2], SUSY seesaw [3–5], Little Higgs model [6], and models with extra dimensions [7]. In this work we considercLFV decay µ → eγ , getting contributions from neutrino oscillations and mixings.Many processes involving cLFV decays could be possible such as µ → e , τ → µ or τ → e transitions. For µ → eee decay, an improvement of up to four orders of magnitude is expected [8], and similarly for µ → e conversions in atomicnuclei improvements are expected [9–13]. Improvements for µ → eγ decay at the next phase of the MEG experimentis expected to reach BR ( µ → e + γ ) ≤ × − [14, 15]. In this work, we have only considered the decay µ → e + γ ,asthis is best constrained by experiments. Such experimental searches and theoretical studies on cLFV can help usconstrain the new physics or BSM theories that could be present just above the electroweak scale, or within the reachof the next run of LHC. It is worth mentioning that in the next run of LHC, the center of mass energies are expectedto go to 14 TeV [16–18].SUSY GUTs naturally give rise to tiny neutrino masses via seesaw mechanisms in which significant contributionsto cLFVs could come from flavor violations among heavy sleptons. The lepton flavor violation effects could becomesignificant due to radiative corrections to Dirac neutrino Yukawa couplings (DNY), which might arise if the seesawscale is slightly lower than the GUT scale [4],[19–34]. Such studies addressing different seesaw mechanisms have beencarried out in [4],[19–35]. In [4], such studies were done in the scenario when neutrino masses and mixings arise dueto the type I seesaw mechanism of SUSY SO(10) theory. In this work the Dirac neutrino Yukawa couplings were ofthe type - Y ν = Y u and Y ν = Y diagu U P MNS , where Y u = V CKM Y diagu V † CKM . Similar studies were done in [36] in thetype II seesaw scenario. Charged Lepton Flavor Violation in the SUSY type II seesaw [37–40] models have also beenstudied earlier [26–34].In this work we carry out studies on cLFV decay ( µ → eγ ) using the type I seesaw mechanism in µ - τ symmetricSUSY SO(10) theories [41, 50], and hence we check the sensitivity to test the observation of sparticles at the nextrun of LHC [16–18], in mSUGRA, non-universal Higgs mass (NUHM), non-universal gaugino mass (NUGM) [43], andNUSM [44] models. Such studies in NUGM models were done earlier in [45]. It may be noted that µ - τ symmetricSUSY SO(10) theory provides a good fit to the observed neutrino oscillations and mixings. In [4], such studies weredone using the type I seesaw formula, using an older value of BR( µ → eγ ) [46]. We have used the form of Diracneutrino Yukawa couplings from [41], for tan β = 10, and M GUT = 2 × GeV. The value of the Higgs mass asmeasured at LHC [16] and global fit values of the reactor mixing angle θ as measured at Daya Bay, Reno [47] havebeen used in this work. In the global data, the octant of the atmospheric angle θ [48, 49] still needs attention. Somestudies on LFV in SO(10) GUTs have also been presented in [50, 51].The minimal supergravity model (mSUGRA) is a well motivated model [52–55] ; for a review, see [56–59]; forreviews of the minimal supersymmetric standard model, see [60, 61] . In mSUGRA, SUSY is broken in the hiddensector and is communicated to the visible sector MSSM fields via gravitational interactions. The generation of gauginomasses [62–66] in mSUGRA (N = 1 supergravity) involves two scales − the spontaneous SUGRA breaking scale inthe hidden sector through the singlet chiral superfield and the other one is the GUT breaking scale through thenon-singlet chiral superfield [52–59] . In principle these two scales can be different. But in a minimalistic viewpoint,they are usually assumed to be the same [52–59]. This leads to a common mass m for all the scalars, a commonmass M / for all the gauginos and a common trilinear SUSY breaking term A at the GUT scale, M GUT ≃ × GeV.Next, we would like to discuss the universal sfermion masses, assumed in the mSUGRA, NUHM, and NUGMmodels. SO(10) symmetric soft terms essentially mean boundary conditions close to NUHM. We are working in theframework of SO(10) theories, in which all the matter fields and the right-handed neutrino are present in the same16-dimensional representation, and, hence, all the matter fields will have the same mass at the high scale. However,the Higgs fields can have a different mass, as they are not present in the same representation as the matter field.Thus, the boundary terms for the SO(10) theory are consistent with NUHM and mSUGRA (in mSUGRA, all theHiggs fields will be in the same representation). Deviation from NUHM boundary conditions would typically signal adeviation from the SO(10) boundary conditions. Similarly, it should be noted that NUGM boundary conditions arealso derived from SO(10) models. If the hidden sector has representations that are not singlets under SO(10), onecan expect non-trivial gaugino mass boundary conditions. So, to summarize, both NUHM and NUGM are boundaryconditions which are a result of assuming SO(10) symmetric boundary conditions at the GUT scale in two differentways. One can, of course, assume completely non-SO(10) symmetric soft terms at the high scale, but then it wouldnot be compatible with the present framework. Moreover, as can be seen from results presented in Sect. IV, lowenergy flavor phenomenology is not much affected by these different boundary conditions at the high scales. In thisanalysis we also carry out cLFV ( µ → eγ ) studies in non-universal scalar masses, the NUSM model [44] where the firsttwo generations of scalar masses and the third generation of sleptons are very massive. Low energy flavor changingneutral current processes (FCNCs) get a contribution due to this non-universality through SUSY loops. But therequirement of radiative breaking of electroweak symmetry REWSB forbids the scalar masses from being too massive.This circumstance is evaded by allowing third generation squark masses and the Higgs scalar mass parameters tobe small [44]. This smallness also serves to keep the naturalness problem within control. We show the variation ofBR( µ → eγ ) with m − m m + m , where m and m are the masses of the first and second generation sfermions, respectively,in Fig.6. It can be seen that the branching ratio of the cLFV decay µ → eγ is not affected much by these completelynon-universal SO(10) symmetric mass terms at the GUT scale. It is well known that SUSY can be broken by softterms of type − A , m , M / , where A is the universal trilinear coupling, m is the universal scalar mass, and M / is the universal gaugino mass. Strict universality between Higgs and matter fields of mSUGRA models can be relaxedin NUHM [67] models. As shown in our results in Sect. IV in mSUGRA, the spectrum of M / and m is found to lietoward the heavy side, as allowed by MEG constraints on BR( µ → eγ ), though in NUHM, lighter spectra are possible(due to partial cancelations in the flavor violating term). So it motivated us to investigate cLFV decay µ → eγ inNUGM [43]. Non-universality of gaugino masses can be realized in various scenarios, including grand unification [68].In these models, gaugino masses are non-universal at GUT scales, unlike in mSUGRA/NUHM models. From [43] wehave used M : M : M = − / − / M , M and M are the gaugino masses at the GUT scale. In NUGM, an increase inthe allowed SUSY soft parameter space is observed, as compared to mSUGRA and NUHM, which lies within theBR( µ → eγ ) limits of MEG 2013. The BR( µ → eγ ) is found to increase with the increase of m here, which is oppositeto mSUGRA and NUHM. This could be explained due to cancelations between chargino and neutralino contributions[45, 69]. In the NUGM model, the | A | is found to shift toward the large value side, as compared to mSUGRA andNUHM models. As shown in our results in Sect.IV, we find that in the NUSM model the gaugino masses are verylarge, so as to allow very large scalar masses. As long as the third generation squark masses and the Higgs scalarmass parameters are small, the fine tuning problem of naturalness does not get worse. In order to have a Higgs massaround 125.9 GeV, the first two generations of squark and slepton masses as well as the third generation of sleptonmasses lies around 12.516 TeV.From above it is seen that the signatures of cLFV could be tested at the next run of LHC, if the SUSY sparticles areobserved within a few TeV, as discussed in more detail in the next sections. It is worth mentioning here that, duringthe last run of LHC, no SUSY partner of SM has been observed, and this could point to a high scale SUSY theory.The LHC has stringent limits on the sparticles, which could imply a tuning of EW symmetry at a few percent level[70–75]. And hence some alternatives to low scale SUSY theories have been proposed. Some of them are minisplitSUSY [76] and maximally natural SUSY [77]. In the former the scalar sparticles are heavier than the sfermions(gauginos and higgsinos), so that sfermions could be observed at LHC. Scalar sparticles could be anywhere in therange (10 − ) TeV. In maximally natural SUSY, the 4D theories arise from 5D SUSY theory, with ScherkSchwarzSUSY breaking at a KaluzaKlein scale ∼ R of several TeV [77]. Some aspects of LFV in such theories have beenstudied in [78]. Charged lepton flavor violation in these models will be studied in our future work. LFV Processes Present Bound Near Future Sensitivity Of Ongoing Experiments BR ( µ → eγ ) 5 . × − [14] 6 × − [15] BR ( τ → eγ ) 3 . × − [79] 10 − − − [80] BR ( τ → µγ ) 4 . × − [79] 10 − − − [80] BR ( µ → eee ) 1 . × − [81] 10 − [82] BR ( τ → eee ) 2 . × − [83] 10 − − − [80] BR ( τ → µµµ ) 2 . × − [83] 10 − − − [80] Table I:
Present experimental limits and future sensitivities for some LFV processes.
The paper is organized as follows. In Sect. II, we give connections of cLFV with the type I seesaw mechanism insymmetric SO(10) theories. In Sect.III, the values of various parameters used in our analysis have been presented.Values of different LFV observables, from [79–83], are listed in Table I. We have used software SuSeFLAV [84] tocompute BR( µ → eγ ). Section IV contains our results and their analysis. Section V summarizes the work. II. CHARGED LFV µ → eγ DECAY IN µ − τ SYMMETRIC SUSY SO(10) THEORYA. cLFV
Neutrino oscillations and mixings are now a proved phenomenon, and through neutrino oscillations, a cLFV processcould be induced as l i W −→ ν l i → ν l j W −→ l j (2)Here W means a vertex involving a W boson. The process requires neutrino mass insertion at two points. In the type Iseesaw mechanism, ∆L = 2 Majorana neutrino masses arise from tree level exchange of a heavy right-handed neutrino.The SUSY SO(10) theory naturally incorporates the seesaw mechanism. The presence of heavy RH neutrinos at anintermediate scale leads to the running and generates flavor violating entries in the left-handed slepton mass matrixat the weak scale [4]. The lepton flavor violating entries in the SO(10) SUSY GUT framework can be understood interms of the low energy parameters. These entries in the leading log approxi- mation in mSUGRA are [85]. (cid:0) m L (cid:1) i = j = − m o + A o π X k ( Y ⋆ν ) ik ( Y ν ) jk log (cid:18) M X M R k (cid:19) (3)Here M X is the GUT scale, M R k is the scale of the k th heavy RH majorana neutrino, m and A are universalsoft mass and trilinear terms at the high scale. Y ν are the Dirac neutrino Yukawa couplings. The flavor violation isparameterized in terms of the quantity δ ij = ∆ ij m l . Here m l is the geometric mean of the slepton squared masses [23],and ∆ i = j are flavor non-diagonal entries of the slepton mass matrix induced at the weak scale due to RG evolution. Themass insertions are branched into the LL/LR/RL/RR types [86], according to the chirality of the corresponding SMfermions. The fermion masses can be generated by renormalizable Yukawa couplings of the 10 ⊕ ⊕
120 representationof scalars of SO(10) GUTs. We have used the Dirac neutrino Yukawa couplings Y ν at the high scale in µ - τ symmetricSO(10) GUTs in our work from [41], Y ν = 1 υsinβ M D (4) M D is the Dirac neutrino mass matrix. The flavor violating off-diagonal entries at the weak scale in Eq. (3) arethen completely determined by using Y ν from Eq. (4). To calculate the δ s from the RGEs, we use the leading logapproximation. Assuming the soft masses to be flavor universal at the input scale, off-diagonal entries in the LLsector are induced by right-handed neutrinos running in the loops. To use the leading log expression (Eq.3) we needthe mass of the heaviest right-handed neutrino, which we have used from [41] by diagonalizing the matrix M R , andit is found to be ∼ GeV. The induced off-diagonal entries relevant to l i → l j + γ are of the order of (putting A to 0), ( δ LL ) µe = − π ( Y ⋆ν ) ( Y ν ) ln (cid:18) M X M R (cid:19) (5)( δ LL ) τµ = − π ( Y ⋆ν ) ( Y ν ) ln (cid:18) M X M R (cid:19) (6)( δ LL ) τe = − π ( Y ⋆ν ) ( Y ν ) ln (cid:18) M X M R (cid:19) (7) LFV contributions For µ - τ symmetric case δ . × − δ . × − δ . × − Table II:
Values (dominant) of δ ij that enter Eq. (5,6,7) for µ - τ symmetric theory. The branching ratio of a charged LFV decay l i → l j is [4],BR ( l i → l j + γ ) ≈ α | δ LLij | G F M SUSY tan β BR ( l i → l j ν i ˜ ν j ) (8)where M SUSY is the SUSY breaking scale. In NUHM models, the term ( − m o + A o ) of the mSUGRA models in Eq.(3) is replaced by ( − m o + A o + m H u ). Here, m H u is the soft mass terms of the up type Higgs at the high scale. Weconsider the NUHM1 case (at the GUT scale) m H u = m H d (9)Moreover, there can be a relative sign difference between the universal mass terms for the matter fields and the Higgsmass terms at the GUT scale. This can clearly lead to cancellations for m H u ≈ − m (10)Or enhancements for m H u ≥ m (11)compared to mSUGRA in the flavor violating entries at the weak scale. B. µ − τ symmetry Neutrino mixings observed in various oscillation experiments can be explained through the structure of both theneutrino and the charged lepton mass matrices. In a basis where the charged leptons are mass eigenstates, the µ ↔ τ interchange symmetry has proved useful in understanding the experimentally observed near-maximal value of ν µ ↔ ν τ mixing angle ( θ ≃ π ). In µ − τ symmetry the mass matrix remains invariant under the interchange of the 23 sector.The mass matrix becomes M ν = x a aa b ca c b (12)At high scales the µ ↔ τ symmetry can be assumed to be an exact symmetry. But at low scales the µ ↔ τ symmetryis effective only in the neutrino Yukawa couplings but not in the charged lepton sectors, since m τ > m µ . Animmediate consequence of this class of theories is that θ = π and θ = 0. Recently evidence of θ = 0 from reactorexperiments [87–89] has been found. This reduces µ − τ symmetry to an approximate symmetry. A small, explicit,tiny breaking of the µ − τ symmetry, to explain the reactor angle θ , has been studied in [41] . This can be done byadding a 120-dimensional Higgs to the 10+126 representation of Higgs. Yukawa interactions of the model are givenby the lagrangian − L Y = 16 i [ H ij
10 + F ij
126 + G ij j + h.c (13)with H ij = H ji ; F ij = F ji ; G ij = − G ji . It may be noted that some results on neutrino masses and mixings usingupdated values of running quark and lepton masses in SUSY SO(10) have also been presented in [90]. Though weconsider 3-flavor neutrino scenario, 4-flavor neutrinos with sterile neutrinos as fourth flavor are also possible [91]. Wehave not considered the CP violation phase [92], in this work. III. CALCULATION OF BR( µ → eγ ) IN MSUGRA, NUHM, NUGM, AND NUSM In this section we present our calculations and results on the charged LFV constraints in µ - τ symmetric SO(10)SUSY theory, with the type I seesaw mechanism using the NUHM, mSUGRA, NUGM, and NUSM like boundaryconditions through detailed numerical analysis. We scan the soft parameter space for mSUGRA in the followingranges: tanβ = 10 m h ∈ [122 . , .
5] GeV∆ m H ∈ m ∈ [0 ,
7] TeV M / ∈ [0 . , .
5] TeV A ∈ [ − m , +3 m ] sgn ( µ ) ∈ {− , + } (14)We perform random scans for the following range of parameters in NUGM model with non-universal and oppositesign gaugino masses at M GUT , with the sfermion masses assumed to be universal maintaining the ratio between thenon-universal gaugino masses [43] m h ∈ [122 . , .
5] GeV m ∈ [0 ,
7] TeV M ∈ [ − . , − .
8] TeV M ∈ [ − . , − .
4] TeV M ∈ [ . , .
6] TeV tanβ = 10 A ∈ [ − m , +3 m ] (15)Here m is the universal soft SUSY breaking mass parameter for sfermions, and M , M , and M denote the gauginomasses for U (1) Y , SU (2) L and SU (3) C respectively. A is the trilinear scalar interaction coupling, tan β is the ratioof the MSSM Higgs vacuum expectation values (VEVs).We have done the numerical analysis using the publicly available package SuSeFLAV [84]. We also study cLFV forthe non-universal Higgs model without completely universal soft masses at a high scale. The ranges of the scan ofvarious SUSY parameters, used by us, in NUHM are: m h ∈ [122 . , .
5] GeV30 GeV ≤ m ≤ ≤ M / ≤ . − . ≤ m H u ≤ +8 . − . ≤ m H d ≤ +8 . −
18 TeV ≤ A ≤ +18 TeV (16)The ∆ LLi = j due to non-universal Higgs and m h ≥
125 GeV puts a strong constraint on SUSY parameter space. Also,because of partial cancelations in the entries of ∆
LLi = j in the NUHM case, a large region of parameter space can beexplored by MEG. We also perform random scans for the following range of parameters in NUSM model [44] andgenerate the SUSY particle spectrum. The ranges of the SUSY parameters used at GUT scale are: tanβ = 10 m ∈ [0 ,
16] TeV M / ∈ [0 ,
6] TeV A = 0 TeV m H u = m H d = 0 TeV (17)The masses of the heavy neutrinos used in our calculations are - M R = 10 GeV, M R = 10 GeV, and M R =10 GeV.. For ∆ m sol , ∆ m atm and θ , we use the central values from the recent global fit of neutrino data [47].The present limits on different LFV observables are summarized in Table I. In Table II we have given the dominantvalues of δ ij that enter Eqs. (5), (6), and (7). IV. ANALYSIS AND DISCUSSION OF RESULTS
In this section, we will present analysis and discussion of results obtained in Sect. III.
A. Complete universality: cMSSM (mSUGRA)
In mSUGRA at the high scale, the parameters of the model are m , A , and the unified gaugino mass M / . Inaddition to these, there is the Higgs potential parameter µ and the undetermined ratio of the Higgs VEVs, tan β .The entire supersymmetric mass spectrum is determined once these parameters are given. We find that the updatedMEG limit [36] together with a large θ [47] puts significant constraints on the SUSY parameter space in mSUGRA.As can be seen from Fig. 1a, only a small part of the paramater space survives for tan β = 10 in mSUGRA allowedby the future MEG limit for BR( µ → eγ ). This leads to the conclusion that the parameter space M / ≥ µ → eγ ), while the future MEG limit excludes a small M / space ≤ m ≥ M / or M / ≥ m . In Fig. 1c, d we plot the lightest Higgs mass, m h , as a function of m , M / in the mSUGRA case. Wesee that for the range of the Higgs mass as given by the data at LHC, i.e. i.e 122.5 GeV ≤ m h ≤ m ≥ µ → eγ ). The space M / ≥ µ → eee . In SUSY (with conserved R-parity)0the dominant contribution to this process arises from the same dipole operator responsible for µ → eγ (for R-parityin SUSY theories, see Refs. [93–100]). Such a prediction is consistent with our results shown in Fig. 1e for tanβ = 10. In SUSY with conserved R-parity the two processes µ → eγ and µ → eee are correlated. This correlation isclearly seen in Fig. 1e as as BR( µ → e γ ) ∼ α em BR( µ → e ). Here α em is the electromagnetic dipole operator. Theasymmetry in the value of A can be seen in Fig. 1f. B. Non Universal Higgs Model (NUHM1)
Next, we present our results obtained in NUHM1 case. In Fig. 2a we have shown M / vs. log[BR( µ → e + γ )and the Fig. 2b in the right panel shows m [GeV] vs. M / [GeV]. Different horizontal lines in Fig. 2a correspondto present and future bounds on BR( µ → e + γ ). We can see from Fig. 2a, b that even in the presence of partialcancelations, most of the NUHM1 parameter space is going to be explored by present and future bounds of MEG.In Fig. 2c, d, the SUSY parameter space M / − m h and m − m h is presented, as allowed by present MEG bounds.For a Higgs mass around 126 GeV, almost all values of M / are allowed in the range (1002500 GeV). Similarly for m h around 126 GeV, the region 3 TeV ≤ m ≤ δ LLi = j due to cancelations between m H u and m , a large region of soft parameter space is allowed which would be easily accessible at the next run of LHCsatisfying the current cLFV constraints. Figure 2e shows A vs. m h [GeV]. A is slightly more symmetric comparedto mSUGRA. C. NON UNIVERSAL GAUGINO MASS MODELS (NUGM)
From the studies in mSUGRA and NUHM model in the above subsections, we see that the SUSY parameter space,as allowed by future MEG bounds on BR( µ → e + γ ) shifts to the heavier side. Hence, we are motivated to dosuch studies in NUGM models. In this section we discuss the scenario with non-universal and opposite sign gauginomasses at M GUT , with the sfermion masses assumed to be universal. We perform random scans for ranges of theparameters given in Eq. (15). We concentrate on the specific model 24 of [43] with the gaugino masses having theratios M : M : M = − / − / µ → e + γ )] increases with increase in scalar masses (in contrast to mSUGRAand NUHM). This could be due to some strong cancelations occurring because of the particular ratios of gauginomasses in NUGM model as discussed earlier. In Fig. 3b, the SUSY parameter space m − M as allowed by the MEG2013 bound on BR( µ → eγ ) is presented. We find that M ≥ m ;while for low M ≤ m are favored. The region below the curve line is excluded by SUSY.1 Figure 1:
The results of our calculations are presented for mSUGRA case. In a , different horizontal lines represent thepresent (MEG 2013) and future MEG bounds for BR( µ → e + γ ). b-d SUSY parameter space allowed by MEG 2013 bound.
In Fig. 4a, c, e we present the constraints from BR ( µ → e + γ ) on NUGM parameter space for tan β = 10. As canbe seen, a large part of the paramater space survives for tan β = 10 in NUGM, as compared to NUHM and mSUGRA.From Fig. 4b we find that for Higgs mass m h around 125.9 GeV, the whole parameter space m ≥ m ≥ BR ( µ → e + γ ) ≤ . × − ). From Fig. 4d, we see that for2 Figure 2:
The results of our calculations are presented for NUHM case. In a , different horizontal lines represent the present(MEG 2013) and future MEG bounds for BR( µ → e + γ ). b-e shows the allowed space for different parameters, that isallowed by MEG 2013 bound. a Higgs mass around 126 GeV, M lies between − . ≤ M ≤ − A [ GeV ] Vs m h [ GeV ] is shown in Fig. 4f. The patches in the plot are due to cancelation in the entries of theleft-handed slepton mass matrices δ LLi = j between the soft universal mass terms.3 Figure 3: a we show the plot m [GeV] vs. log [BR( µ → e + γ )], b represents parameter space of m and M , for NUGMmodel. Different horizontal lines represent present and future bounds on BR( µ → eγ ). D. Non Universal Scalar Mass Models (NUSM)
The parameters of NUSM model are given by [101],tan β , M / , A , sgn( µ ), and m .The parameters play exactly the same role as those in mSUGRA, except for a significant difference in the scalarsector. The masses of the first two generations of scalars (squarks and sleptons) and the third generations of sleptonsare designated as m at the GUT scale. Here m is allowed to span up to a very large value of up to tens of TeVs.However, the Higgs scalars and the third family of squarks are assumed to have vanishing mass values at M GUT .In this analysis the mass parameters for the third generation of squarks and Higgs scalars are set to zero. We limitourselves to a vanishing A in our analysis [44]. We present our results obtained with the non-universal scalar massesat M GUT in Fig. 5. In Fig. 5a we show the SUSY parameter space as allowed by the present and future MEGbounds on BR ( µ → eγ ). For the present MEG bound on log BR ( µ → eγ ), the allowed region of M / parameterspace becomes constrained with a lower limit of 400 GeV. Figure 5b in the right panel shows m [GeV] Vs M / [GeV] as allowed by the MEG 2013 bound on BR( µ → eγ ) (The values of m (GeV) along the x-axis in Fig. 5b, care multiplied by 10 ). We find that the M / ≥ m ≥ m ≥ M / lies between 4 TeV ≤ M / ≤ µ → eγ and µ → e are correlated ascan be seen from Fig. 5e.The variation of BR( µ → eγ ) with m − m m + m is shown in Fig. 6, where m and m are the masses of the first and secondgeneration sfermions, respectively. The range of m − m m + m is taken to be from -0.1 to 0.1. The value of log[ BR ( µ → eγ )]varies from 14.8 to 14.9 for the given interval of m − m m + m , and this change is quite insignificant. We find that the lowenergy flavor phenomenology is not much affected by these completelynon-universal SO(10) symmetric mass terms atthe GUT scale.We find that in CMSSM/mSUGRA like models, the present experimental limit on BR ( µ → eγ ) disfavors the softSUSY breaking parameters m ≤ m ≤ µ - τ Table III:
Masses in this table are comparison between [4] and this work for NUHM.
Range of parameters allowed by Range of parameters allowed byfor BR ( µ → eγ ) < . × − BR ( µ → eγ ) < . × − MEG 2013 (from this work) MEG 2012 (L. Cabbibibi et al.[4]) .Figure 2a: .Only M / ≥ M / space allowed .Figure 2b: (MEG 2013) . m ≥ m ≥ M / , M / ≥ M / ≥
500 GeV for small m wider space is allowed in this work. .Figure 2d: . m ≥ m ≥ m h = 125 . m h = 125 . .Figure 2e: .Almost same as in ours.-13 TeV < A < -7 TeVfor m h = 125 . m H u . As a result of this, as compared to mSUGRA, a relativelysoft parameter space is allowed in NUHM, by the BR( µ → eγ ) bounds. In mSUGRA if the seesaw scale is lowerthan the GUT scale, mixings take place among the sleptons of a different generation at the seesaw scale through (i)renormalization group evolution (RGE) effects and (ii) lepton flavor violating Yukawa couplings. As a result, theslepton mass matrices no longer remain diagonal at the seesaw scale. At the weak scale, the off-diagonal entries inthe slepton mass matrices generate LFV decays. These effects have been studied in the literature in all three variantsof the seesaw mechanisms [26–30].In Tables III and IV we have summarized the comparison of our study with [4]. In Tables V and VI we havehighlighted the comparative study of our analysis between NUGM and NUSM. The new results in NUGM which wefind in our work are the following:1. Lighter m is also allowed as compared to mSUGRA.2. A wider SUSY parameter space is allowed.3. The A range in this work is shown in the Table 5.4. BR( µ → eγ ) increases with increase of masses.5 Table IV:
Masses in this table are comparison between NUGM and NUSM of this work.
Range of parameters allowed by Range of parameters allowed byfor BR ( µ → eγ ) < . × − for BR ( µ → eγ ) < . × − in NUGM in NUSM .Figure 4e: .Figure 5a: constrained region of M / almost whole M space is allowed, parameter space is allowed,100 GeV ≤ M ≤ M / ≥
400 GeV. .Figure 3b: .Figure 5b: M / ≥ M ≥ m , m ≥ m ≤ M ≤ M / .for small m ≤ .Figure 4b: .Figure 5c: m ≥ m ≥
850 GeV for for m h = 125 GeV. m h = 125 GeV. .Figure 4d: .Figure 5d: 4 TeV ≤ M / ≤ | M | ≥
900 GeV for m h = 125 GeV.for m h =125 GeV. Table V:
Masses in this table are comparison between [4] and this work for mSUGRA.
Range of parameters allowed by Range of parameters allowed byfor BR ( µ → eγ ) < . × − BR ( µ → eγ ) < . × − MEG 2013 (from this work) MEG 2012 (L. Cabbibibi et al.[4]) .Figure 1a: .For MEG 2011, M / ≥ M / ≥ M / ≥ M / ≥ µ → eγ ) < − by future MEG bound [14] .Figure 1b (MEG 2013): . For MEG 2011 of BR ( µ → eγ ), m ≥ M / m ≥ M / ,M / ≥ m M / ≥ m M shifts to slightly heavier side .Figure 1c: . m ≥ m h = 125 . m ≥ m h = 125 . .Figure 1f: . -11 TeV < A < -6 TeV for-12 TeV < A < -6 TeV m h = 125 . m h = 125 . (a) (b)(c) (d)(e) (f) Figure 4:
The results of our calculations are presented for NUGM case. In a,c,e different horizontal lines show the present(MEG 2013) and future MEG bounds for BR( µ → e + γ ). b,d,f The space for different parameters that is allowed by MEG2013 bound. Table VI:
Comparison of A between mSUGRA, NUHM and NUGM (this work) A (mSUGRA) A (NUHM) A (NUGM)(TeV) (TeV) (TeV) − < A < − − < A < − − < A < − V. CONCLUSION
To conclude, in this work we have studied the rare cLFV decay µ → eγ in µ − τ symmetric SUSY SO(10) theories,using the type I seesaw mechanism, in mSUGRA, NUHM, NUGM and NUSM models. We have used the value ofthe Higgs mass as measured at LHC, the latest global data on the reactor mixing angle θ for neutrinos, and thelatest constraints on BR( µ → eγ ) as projected by MEG [14, 15]. We find that in mSUGRA a very heavy M / region is allowed by the future MEG bound of BR( µ → eγ ), though in the NUHM case a low M / is also allowed.Hence we further studied the non-universal gaugino mass model (NUGM). In mSUGRA, the m values as allowedby MEG 2013 bound, shift toward a heavier spectrum, as compared to allowed m of [4](which was allowed by a lessstringent bound of MEG 2011). As compared to mSUGRA, in NUHM, a wider parameter range is allowed. For aHiggs mass central value 125.9 GeV, our analysis allows a slightly lower value of m than [4], both in mSUGRA andNUHM (as can be seen from Tables III and IV). We find that NUGM allows, in general, a wider parameter space, ascompared to both mSUGRA and NUHM. Here BR( µ → eγ ) is found to increase with increase in m , which couldbe due to the particular ratios of gaugino masses. In NUGM, we find that the allowed values of | A | are shiftedtowards the heavier side (compared to mSUGRA and NUHM). In NUSM, the allowed M / parameter space at lowenergies becomes constrained as compared to the other three models. For a Higgs boson mass around 125 GeV, M / lies between 4 TeV ≤ M / ≤ m ≥ µ → eγ does notchange significantly with variation of first and second generation sfermion masses at the GUT scale, in the completelynon-universal NUSM model.The results presented in this work can influence the experimental signatures for the production of SUSY particlesand can motivate a special detector set up to guarantee that the largest possible class of supersymmetric models leadto observable signatures at the present and future run of LHC. Hence any observation of heavy particles at the nextrun of LHC could help us understand and discriminate among these models, in reference to constraints put by cLFVdecays. This in turn could contribute towards a better understanding of theories beyond the standard model. VI. ACKNOWLEDGEMENTS
KB and GG would like to thank Katen Patel for discussions, and for bringing this problem to their attention.They also thank Sudhir Vempati for critically reading the manuscript and fruitful discussions and suggestions atIISc Bangalore. They sincerely thank the referee for very constructive and helpful comments and suggestions. KBwould like to thank Gauhati University, and ICTP, Italy, for providing support to visit ICTP, where part of this workwas carried out. She also thanks John March Russel, for discussions on Maximally SUSY theories, at ICTP. GG8acknowledges support from CHEP, IISc, Bangalore, to visit the Institute, where a major part of this work has beendone. She would also like to thank UGC, India, for providing an RFSMS fellowship to her.
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The results of our calculations are presented for NUGM case. In a,c,e different horizontal lines show the present(MEG 2013) and future MEG bounds for BR( µ → e + γ ). b,d,f The space for different parameters that is allowed by MEG2013 bound. Figure 6:
Variation of log[BR( µ → e + γ )] as a function of m − m m + m is shown. The interval of m − m2
Variation of log[BR( µ → e + γ )] as a function of m − m m + m is shown. The interval of m − m2 m + m2