Charged Lepton Flavor Violation $\mu\rightarrow e\gamma$ in $\mu-\tau$ Symmetric SUSY SO(10) mSUGRA, NUHM, NUGM, and NUSM theories and LHC

Charged Lepton Flavor Violation (cLFV) processes like $ \mu \rightarrow e \gamma $ are rare decay processes, that are another signature of physics beyond Standard Model (BSM). These processes have been studied in various models, that could explain neutrino oscillations and mixings. In this work, we present bounds on cLFV decay $ \mu \rightarrow e \gamma $ in a $ \mu $-$ \tau $ symmetric SUSY SO(10) theory, using type I seesaw mechanism. The updated constraints on BR($ \mu \rightarrow e \gamma $) from MEG experiment, recently measured value of Higgs mass at LHC and value of $\theta_{13}$ from reactor data have been used. We present our results in mSUGRA, NUHM, NUGM and NUSM models, and sensitivity to test these theories at next run of LHC is also discussed.

scale is slightly lower than the GUT scale [4,[16][17][18][19][20]. Such studies in different see saw mechanisms have been carried out in [4,[16][17][18][19][20][21]. In [4], such studies were done in scenario when neutrino masses and mixings arise due to type I See Saw mechanism of SUSY SO(10) theory. In this work the Dirac neutrino Yukawa couplings were of the type-Y ν = Y u Similar studies were done in [22] in type II See Saw scenario. Charged Lepton Flavor Violation in SUSY type II seesaw [23] models have also been studied earlier in [18,[19][20].
In this work we carry out studies on cLFV decay (µ → eγ) using type I see saw mechanism in µ-τ symmetric SUSY SO(10) theories [24], and hence chek the sensitivity to test the observation of sparticles at next run of LHC [15], in mSUGRA, NUHM, and NUGM [25] models. Such studies in NUGM models are done for the first time in this work.
It may be noted that µ-τ symmetric SUSY SO(10) theory provides good fit to observed neutrino oscillations and mixings. The analysis have been done for tan β = 10, and M GUT = 2 × 10 16 GeV. The form of Dirac neutrino Yukawa couplings is used from [24]. The value of Higgs mass as measured at LHC [15] and global fit values of reactor mixing angle θ 13 as measured at Daya Bay, Reno [26] have been used in this work. Also, after the improved constraints on BR(µ → eγ) at MEG experiment [27], this is the first study in type I See Saw scenario. Such studies in type II See Saw have been carried out in [22] also, using CKM or PMNS like Dirac neutrino Yukawa couplings. In [4], such studies were done using type I see saw formula, using older value of BR(µ → eγ) [27].
It is well known that SUSY can be broken by soft terms of type −A 0 , m 0 , M 1/2 , where A 0 is the universal trilinear coupling, m 0 is the universal scalar mass, and M 1/2 is the universal gaugino mass. Strict universality between Higgs and matter fields of mSUGRA models can be relaxed in NUHM (Non Uniersal Higgs Mass) [28] models. As shown in our results in Sec.IV in mSUGRA, the spectrum of M 1/2 and m 0 is found to lie towards heavy side, as allowed by MEG constraints on BR(µ → eγ), though in NUHM, lighter spectra is possible (due to partial cancellations in flavor violating term). So it motivated us to investigate cLFV decay µ → eγ in NUGM (Non Universal Gaugino Mass Models) [25]. Non Universality of gaugino masses can be realised in various scenarios, including grand unification [29]. In these models, gaugino masses are non universal at GUT scales, unlike in mSUGRA/NUHM models. From [25] we have used for SO(10) theory.Here, M 1 , M 2 and M 3 are the gaugino masses at GUT scale. In NUGM, an increase in allowed SUSY soft parameter space is observed, as compared to mSUGRA and NUHM that lies within BR(µ → eγ) limits of MEG 2013. The BR(µ → eγ) is found to increase with increase of m 0 here, which is opposite to mSUGRA and NUHM. In NUGM model, the |A 0 | is found to shift towards large value side, as compared to mSUGRA and NUHM models.
From above it is seen that signatures of cLFV could be tested at next run of LHC, if SUSY sparticles are observed within few TeV range, as discussed in more detail in next sections. It is worth mentioning here that, during 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 sparticles, which could imply a tuning of EW symmetry at a few percent level [30][31][32][33][34][35]. And hence some alternatives to low scale SUSY theories have been proposed. Some of them are − minisplit SUSY [36] and maximally natural SUSY [37]. 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 the range (10−10 5 ) TeV. In maximally natural SUSY, the 4D theories arise from 5D SUSY theory, with Scherk-Schwarz SUSY breaking at a Kaluza-Klein scale ∼ 1 R of several TeV [37]. Charged Lepton Flavor Violation in these models would be studied in our future works. Table I: Expected present and future sensitivities from the current generation experiments on various LFV processes [22] The paper has been organised as follows. In section II, we give connections of cLFV with type I See Saw mechanism in µ − τ symmetric SO(10) theories. In section III, the values of various parameters used in our analysis has been presented. We have used software SuSeFLAV [38] to compute BR(µ → eγ). Section IV contains our results and their analysis. Section V summarises the work.

II. CHARGED LFV µ → eγ DECAY IN SUSY SO(10) IN CONNECTION WITH TYPE I SEE SAW
Neutrino oscillations and mixings are now a proved phenomenon, and through a neutrino oscillation, a cLFV process could be induced as Here W means a vertex involving a W boson. The process requires neutrino mass insertion at two points. In type I See Saw 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 an intermediate scale leads to the running and generate flavor violating entries in the left-handed slepton mass matrix at the weak scale [4]. These entries in the Leading Log Estimates in mSUGRA are [39] here M X is the GUT scale, M R k is the scale of the k th heavy RH majorana neutrino, m 0 and A 0 are universal soft mass and trilinear terms at the high scale. Y ν are the Dirac neutrino Yukawa couplings. The fermion masses can be generated by renormalisable Yukawa couplings of the 10⊕126⊕1 20 representation of scalars of SO(10) GUTs. We have used the Dirac neutrino Yukawa couplings Y ν at the high scale in SO(10) GUTs in our work from [24] .
M D is the Dirac neutrino mass matrix. The flavor violating off-diagonal entries at the weak scale in eq. (3) are then completely determined by using Y ν from eq. (4).
Possibly the finest way to understand the lepton flavour violating entries in the SO(10) SUSY GUT framework is in terms of the low energy parameters. We employ the so called Mass Insertion (MI) [40] notation to represent the various flavour violating entries of the slepton mass matrix. These flavour violating entries are zero at the high scale, where SUSY breaking soft scalar masses are universal. At the weak scale, the universality is broken by the RG evolution and the 6 × 6 slepton squared-masses matrix M 2 l takes the following form where the flavour violation is parameterized in terms of the quantity δ ij = ∆ij m 2 l . Herem 2 l is the geometric mean of the slepton squared masses [41], and ∆ i =j are flavour non diagonal entries of the slepton mass matrix induced at the weak scale due to RG evolution. The mass insertions are branched into the LL/LR/RL/RR types [42], according to the chirality of the corresponding SM fermions.

A. LL Insertions From The Running
To calculate the δs from the RGEs, we use the leading log approximation. Assuming the soft masses to be flavour universal at the input scale, off diagonal entries in the LL sector are induced by right handed neutrinos running in the loops. To use the leading log expression (eq. (3)) we need the mass of the heaviest right handed neutrino, which we have used from [24] by diagonalising matrix M R , and found to be ∼ 10 16 GeV. The induced off-diagonal entries relevant to l i → l j +γ are of the order of (putting A 0 to 0)  The branching ratio of a charged LFV decay where M SUSY is the SUSY breaking scale. In NUHM models, the term Here, m Hu is the soft mass terms of the up type Higgs at the high scale. We consider the NUHM1 case (at the GUT scale) Moreover, there can be a relative sign difference between the universal mass terms for the matter fields and the Higgs mass terms at the GUT scale. This can clearly leads to cancellations for Or enhancements for compared to mSUGRA in the flavor violating entries at the weak scale.

III. CALCULATION OF BR(µ → eγ) IN MSUGRA, NUHM AND NUGM
In this section we present our calculations and results on the charged LFV constraints in µ-τ symmetric SO (10) SUSY theory, using type I Seesaw mechanism with mSUGRA, NUHM and NUGM like boundary conditions through detailed numerical analysis. For mSUGRA we scan the soft parameter space in the following ranges.
We perform random scans for the following range of parameters in NUGM model with non universal and opposite sign gaugino masses at M GUT , with the sfermion masses assumed to be universal maintaining the ratio between the non universal gaugino masses [25].
Here m 0 is the universal SSB mass parameter for sfermions, and M 1 , M 2 , and M 3 denote the gaugino masses for U (1) Y , SU (2) L and SU (3) C respectively. A 0 is the trilinear scalar interaction coupling, tanβ is the ratio of the MSSM Higgs vacuum expectation values (VEVs).
We have done the numerical analysis using the publicly available package SuSeFLAV [38]. We also study cLFV for   Table 1.

IV. ANALYSIS AND DISCUSSION ON RESULTS
In this section, we will present analysis and discussion on results obtained in section III.

A. Complete Universality -cMSSM (mSUGRA)
In mSUGRA at the high scale, the parameters of the model are m 0 , A 0 and unified gaugino mass M 1/2 . In addition to these, there is the Higgs potential parameter µ and the undetermined ratio of the Higgs VEVs, tanβ. The entire supersymmetric mass spectrum is determined once these parameters are given. We find that, the updated MEG limit [22] together with a large θ 13 [26] puts significant constraints on SUSY parameter space in mSUGRA. As can be seen from fig 1a, only small part of the paramater space survives for tan β = 10 in mSUGRA allowed by future MEG limit for BR(µ → eγ). This leads to the conclusion that the parameter space M 1/2 ≥ 1 TeV is allowed by present MEG bounds on BR(µ → eγ), while future MEG limit excludes small M 1/2 space ≤ 3. From the studies in mSUGRA and NUHM model in above subsections, we see that the SUSY parameter space, as allowed by future MEG bounds on BR(µ → e + γ) shifts to heavier side. And hence, we are motivated to do such studies in NUGM models. In this section we discuss the scenario with non universal and opposite sign gaugino masses at M GUT , with the sfermion masses assumed to be universal. We perform random scans for ranges of the parameters given in eq. (14). We concentrate on the specific model 24 of [25] with the gaugino masses having the ratios M 1 : M 2 : M 3 = −1/2 : −3/2 : 1. In fact the non-universality of the gaugino masses is by no means a peculiar phenomenon, rather it is realized in various scenarios, including some approaches to grand unification [29].  From fig. 3a, we find that the branching ratio Log[BR(µ → e + γ)] increases with increase in scalar masses (in contrast to mSUGRA and NUHM). This could be due to some strong cancellations occuring because of the particular ratios of gaugino masses in NUGM model.
As can be seen, large part of the paramater space survives for tanβ = 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 0 ≥ 1.5 TeV is allowed. Squark masses m 0 ≥ 1.5 TeV corresponding to 126 GeV Higgs are mostly favoured which would be  fig. 4f. The patches in the plot are due to cancellation in the entries of the left handed slepton mass matrices δ LL i =j between the soft universal mass terms. We find that in CMSSM/mSUGRA like models, the present experimental limit on BR(µ → eγ) disfavors the soft SUSY breaking parameters m 0 ≤ 6 TeV and M 1/2 ≤ 2 TeV if the Dirac neutrino Yukawas are used from [24]. LFV constraint on SUSY spectrum is relaxed if NUHM model is considered and we find that interesting cancellation in the magnitude of charged LFVs arise if the universality condition is relaxed for the soft mass of up type Higgs m 2 Hu . As a result of this, as compared to mSUGRA, relatively soft parameter space is allowed in NUHM, by BR(µ → eγ) bounds. In mSUGRA if the seesaw scale is slightly lower than the GUT scale, mixings among the sleptons of different generation get induced at the seesaw scale through (i) renormalization group evolution (RGE) effects and (ii) lepton flavor violating Yukawa couplings. As a result, slepton mass matrices no longer remain diagonal at the seesaw scale.
At the weak scale, the off-diagonal entries in the slepton mass matrices generate large rate of LFV decays. These effects have been studied in the literature in all three variants of the seesaw mechanisms [4,[18][19][20].
In Tables III and IV we have summarised the comparison of our study with [4]. The new results in NUGM which we find in our work are the following: 1. Lighter m 0 is also allowed as compared to mSUGRA.

A wider SUSY parameter space is allowed.
3. A 0 range in this work is shown in the Table V. 4. BR(µ → eγ) increases with increase of masses.  Fig 4b,4d,4f shows the allowed space for different parameters, that is allowed by MEG 2013 bound.

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 type I see saw mechanism, in mSUGRA, NUHM and NUGM models. We have used the value of Higgs mass   as measured at LHC, latest global data on the reactor mixing angle θ 13 for neutrinos, and latest constraints on BR(µ → eγ) as projected by MEG [14]. We find that in mSUGRA very heavy M 1/2 region is allowed by future MEG bound of BR(µ → eγ), though in NUHM case a low M 1/2 is also allowed. Hence we further studied the non universal gaugino mass model (NUGM). In mSUGRA, the m 0 values as allowed by MEG 2013 bound, shifts toward heavier spectrum, as compared to allowed m 0 of [4] (which was allowed by a less stringent bound of MEG 2011). As compared to mSUGRA, in NUHM, a wider parameter range is allowed. For Higgs mass central value 125.4 GeV, our analysis allows a slightly lower value of m 0 than [4], both in mSUGRA and NUHM (as can be seen from Tables III and IV). In NUGM, these calculations are presented for the first time here in this work. We find that NUGM allows in general, a wider parameter space, as compared to both mSUGRA and NUHM. Here BR(µ → eγ) is found to increase with increase in m 0 which could be due to particular ratios of gaugino masses. In NUGM, we find that allowed values of |A 0 | are shifted towards heavier side (compared to mSUGRA and NUHM). Hence any observation of heavy particles at next run of LHC, could help us understand to discriminate among these models, in reference to constraints put by cLFV decays. This in turn could contribute towards a better understanding of theories beyond standard model   TeV -12 < A 0 < -6 -13 < A 0 < -7 -15 < A 0 <-10