The quark-lepton unification : LHC data and neutrino masses

The recent discovery of nonzero $\theta_{13}$ (equal to Cabbibo angle $\theta_C$ up to a factor of $\sqrt{2}$), the masses of supersymmetric particles $\gapp$ TeV from LHC data, and the sum of three active neutrino masses $\sum_i m_{\nu_i}\lapp 1$ eV from the study of large scale structure of the universe motivate to study whether quark and lepton mixing have the same origin at the grand unification scale. We find that both results from neutrino experiments and LHC are complementary in quark-lepton unified model. A new constraint on SUSY parameters appears from electroweak symmetry breaking with a new correlation between the lower bounds on sparticle masses and the upper bound on $\sum_i m_{\nu_i}$. In addition, we find that only $\mu>0$ (which is favored by $(g-2)$ of muon) is allowed and $m_{\tilde q, \tilde \l} \gapp$ TeV if $\sum_i m_{\nu_i} \lapp 1$ eV. On the other hand, a small change in lower limit on $\theta_{13}$ from zero leads to a large increase in lower limits on sparticles masses ($\gapp 2$ TeV), which are also the bounds if recently discovered boson at LHC with mass around 125 GeV is the Higgs boson.


Introduction
The standard model (SM) [1] of elementary particles is now a completely successful unified theory with an answer to the origin of masses of quarks, leptons and gauge bosons if the recently discovered new boson at Large Hadron Collider (LHC) at a mass around 125 GeV [2] is the standard model Higgs boson. The present data on neutrino masses indicate see-saw scale equal or very close to the grand unification scale. Again, recent result after Daya Bay and RENO experiments [3] θ 13 ≈ θ C (equal up to a factor of √ 2) possibly be a hint of a connection between leptonic mixing and quark mixing. The connection, particularly, the quark-lepton unification has been first enunciated in the grand unified theories (GUTs) with an additional family symmetry in [4] and then worked out in a series of papers [5]. The family symmetry (e.g., S 4 , A 4 and SO (3), etc.) dictates the organizing principle for the structure of Yukawa matrix (to generate the observed mass pattern in both lepton and quark sector at weak scale).
The quark masses originate from electroweak symmetry breaking (EWSB), while neutrino masses have different origin around the GUT scale -the see-saw mechanism. All observed quark mixing angles are very small, while in neutrino sector 1-2 and 2-3 mixing angles are large and 1-3 mixing angle is small. The large magnifications of solar and atmospheric mixing angles E-mail address: abhijit.samanta@gmail.com. through renormalization group evolution (RGE) from a high scale around the GUT scale to weak scale are possible [6] for quasidegenerate neutrinos. Here, the required quasi-degenerate hierarchical (normal) neutrinos can be generated in GUT models with type II see-saw with an additional family symmetry [5] (discussed later).
The intermediate scale between GUT scale and electroweak (EW) scale is relevant for see-saw mechanisms, but not necessary at all for quark masses and their mixing. The lower limits of the see-saw scales are increased as the neutrino masses becomes smaller and smaller. The improvement on sky survey data are showing more stronger lower limit on the sum of neutrino masses ∼ 0.6 eV or more smaller, which leads to see-saw scales equal or very close to the GUT scale. If the see-saw scale is at the GUT scale, the RG evolved neutrino masses at the EW scale are fitted well over the allowed ranges obtained from sky survey data [7] as well as the mass squared differences from neutrino oscillation experiments. Moreover, this range can also be accessible in future double beta decay experiments [8] and in KATRIN experiment [9].
Here, we consider the hypothesis of equal quark and lepton mixings and hierarchical (normal) neutrino masses at GUT scale as in [4]. In this scenario we find a correlation between the upper limit on the sum of the active neutrino masses ( i m ν i ) and lower bound on tan β as the quark-lepton unification fixes the lower limit on tan β (discussed later) for a given upper limit on i m ν i . Once lower limit on tan β is fixed, lower bounds on sparticle masses are fixed from EWSB condition. It determines the stable the electroweak symmetry breaking minima and fixes μ 2 : If μ 2 < 0, Higgsino mass is imaginary and the EWSB minima becomes unstable. The RGE of Higgs mass parameters m 2 H u and m 2 H u strongly depends on the sparticle masses and tan β. This leads to a correlation between lower limit on tan β and lower bounds on sparticle masses at weak scale. For i m ν i 1 eV (constrained from large scale structure of universe), sparticle masses are 1 TeV (consistent with LHC bounds) and only μ > 0 is allowed (supported by (g − 2) of muon).
Again, a small change in lower limit on θ 13 from zero (≈ θ C / √ 2 after Daya Bay and RENO experiments) leads to a large increase in lower limits on sparticles masses ( 2 TeV) from quark-lepton unification, which is also the case if recently discovered boson at LHC with mass around 125 GeV is the Higgs boson. We present our result for constrained minimal supersymmetric extension of the SM (CMSSM) [10]. In our study the running of neutrino parameters are exact as they are coupled with the running of minimal supersymmetric standard model (MSSM) parameters using the ISASUGRA program of the ISAJET package (V7.81) [11].

Models for degenerate neutrino masses at high scale
The most natural way to understand the smallness of neutrino mass is the see-saw mechanism. Here, the neutrino mass matrix is generated by the effective dimension 5 Weinberg operator [12]. In conventional type I see-saw [13], the SM is extended with additional heavy right handed neutrinos (which are not connected with the left handed fermions) and the mass matrix can be written as where, M N is the mass matrix of right handed neutrinos and M D is the Dirac neutrino mass matrix. Here, the neutrino masses are expected to be hierarchical in the similar way to the quark masses. However, in case of type II see-saw [14] the SM model is extended by a charged Higgs triplet and the neutrino mass matrix can be expressed as Now, the most general neutrino mass matrix can be written as The Yukawa coupling matrix Y depends on high scale physics and it is unconstrained by SM data. One can therefore choose it to be unit matrix. If one considers the neutrino mass matrix dominated by the first term, then the neutrino masses are quasi-degenerate in conjunction with the lepton mixing angles close to quark mixing angles. The realization of above type II see-saw scenario has been shown to be achieved in GUT models with gauge group SU(2) L × SU(2) R × SU(4) C and with an additional S 4 global symmetry in [4].
This is not an adhoc assumption, while the gauge group can be a subgroup of number of GUTs like SO(10), E 6 , SO (18), etc. Here, we have not considered the running of the parameters between two scales for decoupling of heavy fields as the values of neutrino parameters are considered as the input at the lowest scale of decoupling of Heavy fields and this lowest scale is assumed as the GUT scale.

Renormalization group evolution
The solution of the coupled RGEs for neutrino parameters along with SUSY parameters are obtained by an iterative cyclic process (weak-to-GUT and then GUT-to-weak) with GUT boundary conditions following CMSSM [10]. The neutrino parameters are also set at GUT scale. The Higgsino mass μ and the soft Higgs bilinear term B are fixed from radiative electroweak symmetry breaking (REWSB). This has been done by the following steps. First, we set the gauge couplings and Yukawa couplings at EW scale and run only these couplings up to GUT scale (where g 1 and g 2 meet) setting other required mass parameters at SUSY breaking scale with approximate values. Now, we set the GUT boundary conditions  From the RGE it is clear that large value of y τ (which requires large tan β), quasi-degenerate neutrino masses, and normal hierarchical mass pattern are needed to generate large radiative magnification of the 1-2 and 2-3 mixing angles at electroweak scale from quark-lepton unified mixing angles at the GUT scale.

Result
The sparticle masses are complicated function of soft SUSY breaking parameters which are obtained at weak scale through running of their coupled RGE from GUT scale. We find the allowed parameter space (and lower limits on masses) by scanning randomly over the following ranges of the parameters; common scalar mass (m 0 ): 0.05-3 TeV, common gaugino mass  We set the experimental bounds obtained from LEP data: m h > 114.5 GeV, mχ± > 103 GeV [16] as these masses can be dominated by the value of μ. If we withdraw the LEP bounds, the relatively lower sparticle masses are allowed, but the correlation of the bounds (discussed later) with i m ν i remains. We consider only the points in the parameter space that can produce neutrino oscillation parameters at weak scale within the 3σ range obtained from global-fit [17]: sin 2 θ 23 0.52 +0.12 −0.13 ; sin 2 θ 12 0.312 +0.048 −0.042 , and sin 2 θ 13 0.039 (the case for nonzero bound on θ 13 has been discussed later). It is expected that the addition of threshold corrections to the mass squared differences [18] may restrict the parameter space as it restricts the choices of m ν i s for a given i m ν i . We have studied the unification with and without threshold corrections varying the ranges of mixing angles and mass squared differences. But, these are not very significant in this scenario with normal slepton mass hierarchy and no significant change in result is observed as we have used large allowed ranges for all parameters in our scanning (both neutrino parameters as well as SUSY parameters), where the parameters are chosen randomly. We present the plots for m 2 21 : 5-10 × 10 −5 eV 2 and m 2 32 : 2-3 × 10 −3 eV 2 , respectively, considering the threshold corrections. Obviously, more stronger constraints are obtained for narrower ranges of oscillation parameters (discussed later). The generation of neutrino mixing angles at EW scale in the ranges allowed by global-fit of neutrino oscillation data needs large radiative magnifications and demands very high value of y τ . As the ranges of the mixing angles at present are very narrow, it almost fixes y τ and consequently determines the lower bound on tan β for a given m ν i at the EW scale. As i m ν i is lowered, higher value of y τ is required and it demands more larger value of tan β. This is shown in the first plot of Fig. 1. The solar and atmospheric mass squared differences are different by two order of magnitude as well as the magnification for solar angle is ∼ 3 and for atmospheric angle is ∼ 20. To accommodate all parameters in the experimentally allowed ranges for a given neutrino mass scale the Majorana phases are constrained in very narrow regions (see Fig. 1). This can be understood from Eq. (5a) and from Eq. (5c).
The upper bound on tan β is either fixed from REWSB or from the LEP bounds on mχ± or m h (which are lowered for smaller μ values). As tan β increases m 2 H d decreases through RG evolution; and it can even be negative. This leads to smaller μ at larger tan β. At more higher tan β, μ 2 becomes negative as the minima of Higgs potential become unstable and then REWSB becomes impossible. In case of μ < 0, the loop correction to m 2 H d leads to a more lower value compared to μ > 0 and we find an upper limit on tan β 55 from REWSB. This restricts the increase in y τ and consequently leads to a lower bound on i m ν i ≈ 1 eV, which is strongly disfavored by the present cosmological data [7]. On the other hand, for μ > 0 one can increase tan β up to 65 leading to a decrease in i m ν i ≈ 0.6 eV, which is very highly favored by sky survey data [7]. The lower bounds on sparticle masses depend only on upper limit on i m ν i as other parameters are scanned over their whole ranges. All these bounds follow the recent LHC results [2]; and again, the neutrino mass limit i m ν i 1 eV is strongly favored by the sky survey data.
In Fig. 2  At this large value of tan β,τ 1 can be the lightest supersymmetric particle (LSP) in some cases depending on the choices of other parameters, mainly A 0 and sign(μ). Another consequences of such high values of tan β with positive sign of μ are successful explanation of g − 2 of muon [19]. Again, for positive μ, t − b − τ unification is also possible [20].
The present global-fit of neutrino data [21] after Daya Bay and RENO experiments [3] gives sin 2 θ 13 > 0.017 and θ 13 ≈ θ C / √ 2. We have found that a small change in its lower limit leads to a large increase in lower limits on sparticles masses ( 2 TeV) from quarklepton unification, which is the case if recently discovered boson at LHC with mass around 125 GeV is the Higgs boson (see Fig. 2). However, since the RGE considered for CKM parameters are approximate, we have not represented the bounds for different lower bounds on θ 13 . This would be more precise and reliable when one considers exact running of CKM parameters. As the allowed range of tan β (which is determined mainly to fix y τ within an interval) is very narrow there appears a definite pattern in the differences between two sparticle masses (see Fig. 2). In case of other supersymmetry breaking scenarios one can also expect similarly strong bounds on sparticle masses from the quark-lepton unification as one always needs large tan β within a narrow range. But, the differences in the sparticle masses will then have different definite pattern due to different GUT boundary conditions. The difference between the patterns becomes more prominent when the allowed range of tan β is very narrow. This may make the possibility to distinguish different models.

Conclusion
The quark-lepton unification not only satisfies and/or predicts all experimental results available till now, but also shows that both results from neutrino experiments and LHC are complementary. The quark-lepton unification leads to a strong constraint on the parameter space along with very strong correlations between the upper limit on i m ν i and the lower limits on sparticle masses.
This arises due to the fact that there exists a lower limit on tan β for a given i m ν i when one demands quasi-degenerate neutrino masses at the GUT scale (which can be generated in GUT models with type II see-saw scenario with an additional family symmetry). As i m ν i decreases lower limit on tan β increases. For a given high value of tan β there appears very strong lower bounds of sparticle masses form EWSB. As tan β increases lower bounds of sparticle masses increase significantly. We find that tan β 55 is not allowed for μ < 0 (as μ 2 becomes negative and EWSB minima is unstable) and it constrains i m ν i 1 eV. For i m ν i 1 eV (constraint from large scale structure of universe) only μ > 0 (which is favored by (g − 2) of muon) is allowed and there exists strong lower bounds on sparticle masses TeV (which are also the bounds from LHC). A small change in lower limit of θ 13 from zero (θ 13 ≈ θ C / √ 2 after Daya Bay and RENO results) leads to a large increase in lower limits on sparticles masses ( 2 TeV) from quark-lepton unification, which is also the case if recently discovered boson at LHC with mass around 125 GeV is the Higgs boson.