Studies of Renormalization-Group Flows in Quantum Field Theories A Dissertation presented by Gongjun Choi to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics Stony Brook University May 2018 Copyright by Gongjun Choi 2018 Stony Brook University The Graduate School Gongjun Choi We, the dissertation committe for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation Robert Shrock - Dissertation Advisor Professor, Department of Physics and Astronomy Chang Kee Jung - Chairperson of Defense Professor, Department of Physics and Astronomy Rouven Essig - Committee Member Associate Professor, Department of Physics and Astronomy Hooman Davoudiasl - Committee Member Physicist, Department of Physics, Brookhaven National Laboratory This dissertation is accepted by the Graduate School Charles Taber Dean of the Graduate School ii Abstract of the Dissertation Studies of Renormalization-Group Flows in Quantum Field Theories by Gongjun Choi Doctor of Philosophy in Physics Stony Brook University 2018 A transformation of the scheme used for regularization and renormaliza- tion in a quantum field theory is a map that connects coupling spaces of a theory in different schemes of this type. Especially in quantum chromody- namics, there have been active studies inventing schemes aimed at trying to reduce higher-order corrections in perturbative calculations, which work well for the coupling very close to zero. However, the problem becomes differ- ent provided one aims at studying a scheme transformation for values of the coupling away from zero, as is relevant for a vanishing of the beta function away from the origin. It turns out that many existing scheme transforma- tions that work well for the coupling near zero cannot be applied so simply for the coupling away from zero (but still in the perturbative regime, i.e. g2=4π = α ≤ 1), and may cause violation of unitarity and perturbativity in a given theory. Motivated by this, we study the construction of scheme trans- formations that are applicable in the vicinity of a zero of the beta function away from the origin [1, 2]. We construct and apply scheme transformations at an infrared (IR) zero of the beta function, αIR;zero, in an vectorial, asymp- totically free SU(N) gauge theory with Nf fermions in a representation R to study the scheme dependence of αIR;zero. We show that the shift of αIR;zero with increasing loop order tends to decrease, providing a quantitative mea- iii sure of the size of the scheme-dependence in the calculation of αIR;zero up to four-loop order at not only small but also moderate coupling away from zero. As the second part of the thesis work, we study the renormalization-group evolution of a N = 1 supersymmetric SU(N) gauge theory by investigating β functions in different schemes [3]. We take two particular schemes and com- pute Pad´eapproximants to probe the degree of similarity of the respective β functions in these schemes. Finally, we explore the renormalization-group evolution of the Gross- Neveu model with N-component fermions by examining its β function up to four-loop order [4]. As part of this study, we apply scheme transforma- tions to this β function and also calculate and analyze Pad´eapproximants to the β function. From this analysis, we show that there is no evidence for presence of an infrared zero in the finite-N Gross-Neveu model. iv Dedication Page To Shinhae v Contents 1 Introduction 1 1.1 Regularization . 2 1.2 Renormalization and its Scheme . 4 1.3 Beta Function of Non-Abelian Gauge Theory . 5 1.4 Scheme Transformations . 12 2 Construction and Application of Renormalization Scheme Transformation 17 2.1 The SLr Scheme Transformation . 18 2.2 The Rational Scheme Transformation S[p;q] . 23 2.2.1 The SQr Scheme Transformation . 24 2.2.2 S[1;1] Scheme Transformation . 27 2.2.3 S[1;2] Scheme Transformation . 29 2.3 Scheme Transformations from an Integral Formalism . 30 2.3.1 Transformation with an Algebraic J(y) . 32 2.3.2 Transformation with a Transcendental J(y) . 33 2.3.3 Scheme Transformations for which J(y) = (d=dy) ln h(y) 34 2.3.4 Scheme Transformations for which J(y) = (d=dy)eφ(y) . 38 3 Renormalization-Group Evolution of N = 1 Supersymmetric Gauge Theories 43 3.1 Beta Function and Exact Results . 44 3.1.1 Beta Function . 44 3.1.2 NSVZ Beta Function . 46 3.1.3 General Result on IR Phase Properties . 47 3.2 N = 1 Supersymmetric Yang-Mills Theory . 47 3.3 Supersymmetric SU(Nc) Quantum Chromodynamics . 56 3.3.1 Some General Properties . 56 3.3.2 Calculations of Pad´eApproximants . 57 3.3.3 Analysis for Interval r 2 Ir;NACP . 60 3.3.4 Analysis for 0 < r < 3=2 . 70 3.4 Chiral Superfields in Symmetric and Antisymmetric Rank-2 Tensor Representations . 72 vi 3.4.1 Beta Function and IR Zeros . 72 3.4.2 Pad´eApproximants . 76 4 Renormalization-Group Evolution of the Finite-N Gross-Neveu Model 79 4.1 Some Relevant Background on the Gross-Neveu Model . 80 4.2 Beta Function for General N ................... 82 4.3 Analysis with Pad´eApproximants . 88 4.4 Analysis Using Scheme Transformations . 93 4.5 Comparison with Results in the LN Limit and Behavior for d > 2................................ 95 5 Conclusions 98 vii 1 Introduction In this dissertation, we will study some topics in quantum field theory (QFT), including the dependence of a β function calculated to finite loop order, and the corresponding renormalization-group flows of some models, on the scheme used for regularization and renormalization. We will call this simply a scheme in the following. The original research discussed in this thesis was performed in collaboration with Prof. R. Shrock and published in the four papers [1]- [4], the last of which also involved Prof. T. Ryttov (of Southern Denmark University) as a coauthor. Early papers on the renormalization group (RG) include [5]. The specific models we will go through for the study of these topics are • SU(Nc) gauge theory with Nf massless Dirac fermions •N = 1 supersymmetric SU(Nc) gauge theory with Nf massless chiral superfields • Gross-Neveu model with finite N component massless fermions These theories have the common feature of asymptotic freedom due to the negativity of the one-loop β function for a range of the number of fermion matter fields, Nf , that each theory has. Since we will be investigating the in- frared behavior of these asymptotically free theories, our restriction to mass- less fermions incurs no loss of generality. The reason for this is that if a fermion had a nonzero mass m0, it would be integrated out of the effective field theory at scales µ < m0, and hence would not affect the IR limit µ ! 0 considered here. As we will see later, a finite `-loop order β function with ` ≥ 3 depends on a choice of scheme used for regularization and renormal- ization, and thus one may wonder how this fact affects physical observables in perturbative QFT. To be specific, the possible questions that arise due to subtlety lying behind renormalization scheme dependence could be • How could one map coupling spaces of different schemes in a quantum field theory? • How much is a finite-loop calculation of a physical observable sensitive to change of scheme? • How is a renormalization-group flow of quantum field theory affected by change of scheme? 1 We shall investigate these questions by the method of construction of various scheme transformations and applying those to a physical observable of the theories presented above, with reference to an infrared zero of the associ- ated β functions. To this end, in this introduction section, we review the basic procedure of regularization and renormalization and the way how one defines a scheme in Sec.1.1 and Sec.1.2. Then in Sec.1.3, we briefly review the β function of a non-Abelian gauge theory and define an infrared zero of the β function. In addition, we study several possible infrared phases of an asymptotically free theory, which depend on the value of the infrared zero of β function. Finally in Sec.1.4, we study the basic formalism of scheme trans- formations mapping coupling spaces of different renormalization schemes. 1.1 Regularization In computing physical quantities within the framework of QFT, there arise certain divergences, although we know the final answer for a physical quan- tity must be finite. The expectation is that the divergences may occur in the intermediate steps of the computation of physical observables. In the com- putational procedure, it is necessary to isolate a meaningful finite part from a divergent expression and the technique for the isolation is called regulariza- tion. As an explicit example of regularization procedure, we illustrate here the order O(g2) fermion loop correction to the gluon propagator in quantum chromodynamics (QCD). Using the fermion propagator Figure 1: The Feynman diagram for one-loop correction to the gluon propa- gator. Curly external lines and solid internal lines correspond to gluon and fermion, respectively. 2 6 p + m iδ (1) ij p2 − m2 + i µ a and fermion-fermion-gauge vertex −igγ Tij with i and j color indices, one can evaluate the Feynman diagram in Fig.1 and it reads Z d4k Tr[γµ(6 k + m)γν(6 k+ 6 p + m)] −iΠµν(p2) = −g2Tr(T T ) (2) ab a b (2π)4 (k2 − m2)([k + p]2 − m2) where Ta is generator of the gauge group SU(3) and our metric is (+; −; −; −). The denominator of the integrand can be expressed with the use of the iden- tity 1 Z 1 dα = 2 (3) A · B 0 [A(1 − α) + Bα] 2 2 2 2 µν 2 with A = k − m and B = [k + p] − m .