Supersymmetric QCD on a Spacetime Lattice
May, 2024Citations per year
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Abstract: (Cyprus U.)
In this thesis, we outline numerous motivations to delve into the study of supersymmetric theories using lattice techniques. Nevertheless, various well-known obstacles emerge from the breaking of supersymmetry in a lattice-regulated theory, such as the requirement for fine-tuning of the theory's bare Lagrangian. Our approach to address these issues involves mandating that all symmetries of the action, which are broken on the lattice, must be fully restored as the continuum limit is approached. We also propose some ways to reduce the amount of the parameters that need fine-tuning in order to render the numerical lattice calculations easier. For the first project, we investigate the fine-tuning of the Yukawa (gluino-quark-squark interactions) and quartic (four-squark interactions) couplings of N=1 supersymmetric QCD, discretized on a Euclidean lattice. We use perturbation theory at one-loop level and to the lowest order in the lattice spacing. The Modified Minimal Subtraction Scheme (\bar{MS}) is employed; by its definition, this scheme requires perturbative calculations, in the continuum and/or on the lattice. On the lattice, we utilize the Wilson formulation for gluon, quark and gluino fields; for squark fields we use na\"ive discretization. All Green's functions and renormalization factors are analytic expressions depending on the number of colors, Nc, the number of flavors, Nf, and the gauge parameter, α, which are left unspecified. Knowledge of these renormalization factors is necessary in order to relate numerical results, coming from nonperturbative studies, to the renormalized, ``physical" Green's functions of the theory.The sheer difficulties of this study lie in the fact that different components of squark fields mix among themselves at the quantum level and the action's symmetries, such as parity and charge conjugation, allow an additional Yukawa coupling. Consequently, for an appropriate fine-tuning of the Yukawa terms, these mixings must be taken into account in the renormalization conditions. Note that while we provide the results of the renormalization factors for the Yukawa couplings in both dimensional and lattice regularization, we present the outcomes of the renormalization factors for the quartic couplings only in dimensional regularization. The computations regarding the renormalization of the quartic couplings on the lattice are underway.For the second project, we perform calculations to determine the renormalization of the four-quark operators in the framework of QCD. We employ a Gauge Invariant Renormalization Scheme (GIRS), which can be advantageous compared to other schemes, especially in nonperturbative lattice investigations, and the Modified Minimal Subtraction Scheme (\bar{MS}). From our perturbative computations we extract the elements of the conversion matrices between these two renormalization schemes at the next leading order. A formidable issue in the study of the four-quark operators is the fact that operators with different Dirac matrices mix among themselves upon renormalization. Furthermore, computations in GIRS, at a given order in perturbation theory, involve diagrams with more than one loop. Note that we focus on both Parity Conserving and Parity Violating four-quark operators with ΔF=2.The extraction of the elements of the conversion matrices entails the calculation of two-point Green's functions, which involve products of two four-quark operators or one four-quark operator and one bilinear operator, as well as three-point Green's functions which involve one four-quark and two bilinear operators; all operators are situated at distinct spacetime points. Moreover, we concentrate on both Parity Conserving and Parity Violating four-quark operators. The significance of our results lies in their potential to refine our understanding of QCD phenomena, offering valuable insights into the precision of Cabibbo–Kobayashi–Maskawa (CKM) matrix elements and shedding light on the nonperturbative treatment of complex mixing patterns associated with four-quark operators.References(0)
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