Note on gauge and gravitational anomalies of discrete ZNZ_N symmetries

In this note, we discuss the consistency conditions which a discrete $Z_N$ symmetry should satisfy in order that it is not violated by gauge and gravitational instantons. As examples, we enlist all the $Z_N$ ${\cal R}$-symmetries as well as non-${\cal R}$ $Z_N$ symmetries (N=2,3,4) in the minimally supersymmetric standard model (MSSM) that are free from gauge and gravitational anomalies. We show that there exists non-anomalous discrete symmetries that forbid Baryon number violation up to dimension 6 level (in superspace). We also observe that there exists no non-anomalous $Z_3$ ${\cal R}$-symmetry in the MSSM. Furthermore, we point out that in a theory with one Majorana spin 3/2 gravitino, a large class of $Z_4$ ${\cal R}$-symmetries are violated in the presence of Eguchi-Hanson (EH) gravitational instanton. This is also in general true for higher $Z_N$ ${\cal R}$-symmetries. We also notice that in 4 dimensional ${\cal N}=1$ supergravity, the global $U(1)$ ${\cal R}$-symmetry is always violated by the EH instanton irrespective of the matter content of the theory.Comment: Version to appear in JHE

Role of Tensor operators in RKR_K and RK∗R_{K^*}

The recent LHCb measurement of $R_{K^*}$ in two $q^2$ bins, when combined with the earlier measurement of $R_K$, strongly suggests lepton flavour non-universal new physics in semi-leptonic $B$ meson decays. Motivated by these intriguing hints of new physics, several authors have considered vector, axial vector, scalar and pseudo scalar operators as possible explanations of these measurements. However, tensor operators have widely been neglected in this context. In this paper, we consider the effect of tensor operators in $R_K$ and $R_{K^*}$. We find that, unlike other local operators, tensor operators can comfortably produce both of $R_{K^*} ^{\rm low}$ and $R_{K^*} ^{\rm central}$ close to their experimental central values. However, a simultaneous explanation of $R_K$ is not possible with only Tensor operators, and other vector or axial vector operators are needed. In fact, we find that combination of vector and tensor operators can provide simultaneous explanations of all the anomalies comfortably at the $1 \sigma$ level, a scenario which is hard to achieve with only vector or axial vector operators. We also comment on the compatibility of the various new physics solutions with the measurements of the inclusive decay $B_d \to X_s \ell^+ \ell^-$.Comment: Version to appear in PL

Unification with Vector-like fermions and signals at LHC

We look for minimal extensions of Standard Model with vector like fermions leading to precision unification of gauge couplings. Constraints from proton decay, Higgs stability and perturbativity are considered. The simplest models contain several copies of vector fermions in two different (incomplete) representations. Some of these models encompass Type III seesaw mechanism for neutrino masses whereas some others have a dark matter candidate. In all the models, at least one of the candidates has non-trivial representation under $SU(3)_{color}$. In the limit of vanishing Yukawa couplings, new QCD bound states are formed, which can be probed at LHC. The present limits based on results from 13 TeV already probe these particles for masses around a TeV. Similar models can be constructed with three or four vector representations, examples of which are presented.Comment: 48 pages; v3:major corrections with discussion on threshold corrections, version accepted for publication in JHE