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We review unitarity and crossing constraints on scattering amplitudes for particles with spin in four dimensional quantum field theories. As an application we study two to two scattering of neutral spin 1/2 fermions in detail. Assuming Mandelstam analyticity of its scattering amplitude, we use the numerical S-matrix bootstrap method to estimate various non-perturbative bounds on quartic and cubic (Yukawa) couplings.

We review unitarity and crossing constraints on scattering amplitudes for particles with spin in four dimensional quantum field theories. As an application we study two to two scattering of neutral spin 1/2 fermions in detail. Assuming Mandelstam analyticity of its scattering amplitude, we use the numerical S-matrix bootstrap method to estimate various non-perturbative bounds on quartic and cubic (Yukawa) couplings.

We demonstrate the use of several code implementations of the Mellin-Barnes method available in the public domain to derive analytic expressions for the sunset diagrams that arise in the two-loop contribution to the pion mass and decay constant in three-flavoured chiral perturbation theory. We also provide results for all possible two-mass configurations of the sunset integral, and derive a new one-dimensional integral representation for the one mass sunset integral with arbitrary external momentum. Thoroughly annotated Mathematica notebooks are provided as ancillary files, which may serve as pedagogical supplements to the methods described in this paper.

In supersymmetric E 6 the masses of the third family quarks and charged lepton, t − b − τ , as well as the masses of the vector-like quarks and leptons, D −D and L−L, may arise from the coupling 27 3 ×27 3 ×27 H , where 27 3 and 27 H denote the third family matter and Higgs multiplets respectively. We assume that the SO(10) singlet component in 27 H acquires a TeV scale VEV which spontaneously breaks U (1) ψ and provides masses to the vector-like particles in 27 3 , while the MSSM doublets in 27 H provide masses to t, b and τ . Imposing Yukawa coupling unification h t = h b = h τ = h D = h L at M GU T and employing the ATLAS and CMS constraints on the Z ψ boson mass, we estimate the lower bounds on the third family vector-like particles D −D and L −L masses to be around 5.85 TeV and 2.9 TeV respectively. These bounds apply in the supersymmetric limit.

A wide variety of unified models predict asymptotic relations at M GU T between the b quark and τ lepton Yukawa couplings. Within the framework of supersymmetric SU(4) × SU(2) L × SU(2) R , we explore regions of the parameter space that are compatible with b-τ quasi-Yukawa unification and the higgsinos being the lightest supersymmetric particles ( 1 TeV). Among the colored sparticles, the stop weighs more than 1.5 TeV or so, whereas the squarks of the first two families are signifcantly heavier, approaching 10 TeV in some cases. The gluino mass is estimated to lie in the 2-4 TeV range which raises the important question: Will the LHC find the gluino?

We present ψ ′ MSSM, a model based on a U (1) ψ ′ extension of the minimal supersymmetric standard model. The gauge symmetry U (1) ψ ′ , also known as U (1)N , is a linear combination of the U (1)χ and U (1) ψ subgroups of E6. The model predicts the existence of three sterile neutrinos with masses 0.1 eV, if the U (1) ψ ′ breaking scale is of order 10 TeV. Their contribution to the effective number of neutrinos at nucleosynthesis is ∆Nν ≃ 0.29. The model can provide a variety of possible cold dark matter candidates including the lightest sterile sneutrino. If the U (1) ψ ′ breaking scale is increased to 10 3 TeV, the sterile neutrinos, which are stable on account of a Z2 symmetry, become viable warm dark matter candidates. The observed value of the standard model Higgs boson mass can be obtained with relatively light stop quarks thanks to the D-term contribution from U (1) ψ ′ . The model predicts diquark and diphoton resonances which may be found at an updated LHC. The well-known µ problem is resolved and the observed baryon asymmetry of the universe can be generated via leptogenesis. The breaking of U (1) ψ ′ produces superconducting strings that may be present in our galaxy. A U (1) R symmetry plays a key role in keeping the proton stable and providing the light sterile neutrinos.

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