This paper presents a complete algebraic proof of the renormalizability of the gauge invariant $d=4$ operator $F_{μν}^2(x)$ to all orders of perturbation theory in pure Yang-Mills gauge theory, whereby working in the Landau gauge. This renormalization is far from being trivial as mixing occurs with other $d=4$ gauge variant operators, which we identify explicitly. We determine the mixing matrix $Z$ to all orders in perturbation theory by using only algebraic arguments and consequently we can uncover a renormalization group invariant by using the anomalous dimension matrix $Γ$ derived from $Z$. We also present a future plan for calculating the mass of the lightest scalar glueball with the help of the framework we have set up.
A method for computing the low-energy non-perturbative properties of SUSY GFT, starting from the microscopic lagrangian model, is presented. The method relies on covariant SUSY Feynman graph techniques, adapted to low energy, and Renormalization-Group-improved perturbation theory. We apply the method to calculate the glueball superpotential in N=1 SU(2) SYM and obtain a potential of the Veneziano-Yankielowicz type.
Using techniques developed in a previous paper three-point functions in field theories described by holographic renormalization group flows are computed. We consider a system of one active scalar and one inert scalar coupled to gravity. For the GPPZ flow, their dual operators create states that are interpreted as glueballs of the N=1 SYM theory, which lies at the infrared end of the renormalization group flow. The scattering amplitudes for three-glueball processes are calculated providing precise predictions for glueball decays in N=1 SYM theory. Numerical results for low-lying glueballs are i…
This paper presents a complete algebraic analysis of the renormalizability of the $d=4$ operator $F^2_{μν}$ in the Gribov-Zwanziger (GZ) formalism as well as in the Refined Gribov-Zwanziger (RGZ) version. The GZ formalism offers a way to deal with gauge copies in the Landau gauge. We explicitly show that $F^2_{μν}$ mixes with other $d=4$ gauge variant operators, and we determine the mixing matrix $Z$ to all orders, thereby only using algebraic arguments. The mixing matrix allows us to uncover a renormalization group invariant including the operator $F^2_{μν}$. With this renormalization group i…
This proceeding gives an overview of the renormalization of $F^2_{μν}$ using the Faddeev-Popov action and the more complicate Gribov-Zwanziger action, which deals with Gribov copies. We show that using the Faddeev-Popov action, $F^2_{μν}$ mixes with other $d=4$ operators. However, due to the BRST invariance of the action, this mixing is not relevant at the level of the correlator, $\Braket {F^2_{μν}(x) F^2_{αβ}(y)}$. In contrast, when turning to the Gribov-Zwanziger action, the mixing of $F^2_{μν}$ with other $d=4$ operator does have consequences at the level of the correlator. This is due to …
The renormalization group approach towards the string representation of non abelian gauge theories translates, in terms of the string sigma model beta function equations, the renormalization group evolution of the gauge coupling constant and Zamolodchikov`s $c$ function. Tachyon stability, glueball mass gap, renormalization group evolution of the $c$ function and the area law for the Wilson loop are studied for a critical bosonic string vacuum corresponding to a non abelian gauge theory in four dimensional space-time. We prove that the same intrinsic geometry for the string vacuum is universal…
We present variational estimates for the low-lying energies of a simple matrix model that approximates $SU(3)$ Yang-Mills theory on a three-sphere of radius $R$. By fixing the ground state energy, we obtain the (integrated) renormalization group (RG) equation for the Yang-Mills coupling $g$ as a function of $R$. This RG equation allows to estimate the masses of other glueball states, which we find to be in excellent agreement with lattice simulations.
This Ph.D. thesis presents a study of the Gribov-Zwanziger framework: from propagators to glueballs. The chapters 2 and 3 are meant as an introduction and only require a basic knowledge of quantum field theory. Chapter 2 explains the techniques behind algebraic renormalization, which shall be widely used throughout this thesis, while chapter 3 tries to give a pedagogic overview of the Gribov-Zwanziger framework as this is not available yet in the literature. The subsequent chapters contain own research. First in chapter 4, we shall dig a bit deeper in the Gribov-Zwanziger framework, by explori…
Solving by a canonical string theory, of closed strings for the glueballs and open strings for the mesons, the 't Hooft large-$N$ expansion of QCD is a long-standing problem that resisted all the attempts despite the advent of the celebrated gauge/gravity duality in the framework of string theory. We demonstrate that in the canonical string framework such a solution does not actually exist because an inconsistency arises between the renormalization properties of the QCD S matrix at large $N$ -- a consequence of the asymptotic freedom (AF) -- and the open/closed duality of the would-be string s…
Within a self-consistent proper-time Renormalization Group (RG) approach we investigate an effective QCD trace anomaly realization with dilatons and determine the finite-temperature behavior of the gluon condensate. Fixing the effective model at vanishing temperature to the glueball mass and the bag constant a possible gluonic phase transition is explored in detail. Within the RG framework the full non-truncated dilaton potential analysis is compared with a truncated potential version.
We point out that perturbation theory in conjunction with the renormalization group (RG) puts a severe constraint on the structure of the large-N non-perturbative glueball propagators in SU(N) pure YM, in QCD and in n=1 SUSY QCD with massless quarks, or in any confining asymptotically-free gauge theory massless in perturbation theory. For the scalar and pseudoscalar glueball propagators in pure YM and QCD with massless quarks we check in detail the RG-improved estimate to the order of the leading and next-to-leading logarithms by means of a remarkable three-loop computation by Chetyrkin et al.…
The recently developed gauge-invariant formalism for the treatment of fluctuations in holographic renormalization group (RG) flows overcomes most of the previously encountered technical difficulties. I summarize the formalism and present its application to the GPPZ flow, where scattering amplitudes between glueball states have been calculated and a set of selection rules been found.
This master's thesis gives a thorough and pedagogical introduction to the Dijkgraaf-Vafa conjecture which tells us how to calculate the exact effective glueball superpotential in a wide range of $\mathcal{N}=\textrm{1}$ supersymmetric gauge theories in four space-time dimensions using a related matrix model. The introduction is purely field theoretical and reviews all the concepts needed to understand the conjecture. Furthermore, examples of the use of the conjecture are given. Especially, we find the one-cut solution of the matrix model and use this to obtain exact superpotentials in the case…
We work out the interplay between lowest-order perturbative computations in the 't Hooft coupling, $g^2=g^2_{YM} N$, operator mixing, renormalization-group (RG) improved ultraviolet (UV) asymptotics of leading-order (LO) nonplanar/planar contributions to $2$-point correlators, and nonperturbative large-$N$ expansion of perturbatively massless QCD-like theories. As concrete examples, we compute to the lowest perturbative order in $SU(N)$ YM theory the ratios, $r_i$, of LO-nonplanar to planar contributions to the $2$-point correlators in the orthogonal basis in the coordinate representation of t…
We consider the N=1 super Yang-Mills theory with gauge group U(Nc) or SU(Nc) and one adjoint Higgs field with an arbitrary polynomial superpotential. We provide a purely field theoretic derivation of the exact effective superpotential W(S) for the glueball superfield S in the confining vacua. We show that the result matches with the Dijkgraaf-Vafa matrix model proposal. The proof brings to light a deep relationship between non-renormalization theorems first discussed by Intriligator, Leigh and Seiberg, and the fact that W(S) is given by a sum over planar diagrams.
We derive and solve a subset of the fluctuation equations about two domain wall solutions of D=5, N=8 gauged supergravity. One solution is dual to D=4, N=4 SYM theory perturbed by an N=1, SO(3)-invariant mass term and the other to a Coulomb branch deformation. In the first case we study all SO(3)-singlet fields. These are assembled into bulk multiplets dual to the stress tensor multiplet and to the N=1 chiral multiplets Tr Phi^2 and Tr W^2, the former playing the role of anomaly multiplet. Each of these three multiplets has a distinct spectrum of "glueball" states. This behavior is contrasted …
A long-standing conjecture on the structure of renormalized, gauge invariant, integrated operators of arbitrary dimension in Yang-Mills theory is established. The general solution of the consistency condition for anomalies with sources included is also derived. This is achieved by computing explicitely the cohomology of the full unrestricted BRST operator in the space of local polynomial functionals with ghost number equal to zero or one. The argument does not use power counting and is purely cohomological. It relies crucially on standard properties of the antifield formalism.
Resembling the renormalization group from ultraviolet to infrared, we construct a dynamical holographic model in the graviton-dilaton-scalar framework, where the dilaton background field $Φ$ and scalar field $X$ are responsible for the gluodynamics and chiral dynamics, respectively. At the UV boundary, the dilaton field is dual to the dimension-4 gluon operator, and the scalar field is dual to the dimension-3 quark-antiquark operator. The metric structure at IR is automatically deformed by the nonperturbative gluon condensation and chiral condensation in the vacuum. The produced scalar gluebal…
The renormalization properties of two local BRST invariant composite operators, $(O,V_μ)$, corresponding respectively to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are scrutinized in the $U(1)$ Higgs model by means of the algebraic renormalization setup. Their renormalization $Z$'s factors are explicitly evaluated at one-loop order in the $\overline{\text{MS}}$ scheme by taking into due account the mixing with other gauge invariant operators. In particular, it turns out that the operator $V_μ$ mixes with the gauge invariant quantity $\partial_ν…
We study the Connes-Kreimer Hopf algebra of renormalization in the case of gauge theories. We show that the Ward identities and the Slavnov-Taylor identities (in the abelian and non-abelian case respectively) are compatible with the Hopf algebra structure, in that they generate a Hopf ideal. Consequently, the quotient Hopf algebra is well-defined and has those identities built in. This provides a purely combinatorial and rigorous proof of compatibility of the Slavnov-Taylor identities with renormalization.
In close analogy to the Bloch-Feshbach formalism known from the theory of nuclear dynamics, I develop a mathematical framework that allows one to understand renormalization in terms of purely algebraic operations (projections, dilatations) in Hilbert space. This algebraic approach is put to the test in the study of the low-energy dynamics of interacting quantum gases, and proves to be efficient in deriving such diverse results as the renormalization group equation for an interacting Bose gas, the $β$ function of $ϕ^4$ theory, the screening of fermion-fermion interactions or the BCS instability…
QCD chiral phase diagram from weak functional renormalization group
Center for Theoretical Physics and College of Physics, Jilin University（吉林大学理论物理中心与物理学院）；Department of Physics and Astronomy, Kwansei Gakuin University（关西学院大学物理天文学系）
We present a perturbative construction of interacting quantum field theories on any smooth globally hyperbolic manifold. We develop a purely local version of the Stueckelberg-Bogoliubov-Epstein-Glaser method of renormalization using techniques from microlocal analysis. As byproducts, we describe a perturbative construction of local algebras of observables, present a new definition of Wick polynomials as operator-valued distributions on a natural domain, and we find a general method for the extension of distributions which were defined on the complement of some surfaces.
In the present paper, which is a mathematical follow--up of [16] taking inspiration from [11], we present an abstract formulation of exact renormalization group (RG) in the framework of Batalin--Vilkovisky (BV) algebra theory. In the first part, we work out a general algebraic and geometrical theory of BV algebras, canonical maps, flows and flow stabilizers. In the second part, relying on this formalism, we build a BV algebraic theory of the RG. In line with the graded geometric outlook of our approach, we adjoin the RG scale with an odd parameter and analyse in depth the implications of the r…
The Hadamard renormalization procedure is applied to a free, massive Dirac field $ψ$ on a 2 dimensional Lorentzian spacetime. This yields the state-independent divergent terms in the Hadamard bispinor $G^{(1)}(x, x') = \frac{1}{2} \left\langle \left[ \barψ(x'), ψ(x) \right] \right\rangle$ as $x$ and $x'$ are brought together along the unique geodesic connecting them. Subtracting these divergent terms within the limit assigns $G^{(1)}(x, x')$, and thus any operator expressed in terms of it, a finite value at the coincident point $x' = x$. In this limit, one obtains a quadratic operator instead …
We study asymptotically free SU($N_c$) gauge theories with ${\cal N}=1$ supersymmetry, including the purely gluonic theory and theories with $N_f$ copies of a pair of massless chiral superfields in the respective representations $R$ and $\bar R$ of SU($N_c$). The cases in which $R$ is the fundamental representation and the symmetric and antisymmetric rank-2 tensor representation are considered. We calculate Padé approximants to the beta functions for these theories in the $\overline{\rm DR}$ scheme up to four-loop order for the gluonic theory and up to three-loop order for the theories with ma…
The formulation of an exact functional renormalization group equation for Quantum Einstein Gravity necessitates that the underlying effective average action depends on two metrics, a dynamical metric giving the vacuum expectation value of the quantum field, and a background metric supplying the coarse graining scale. The central requirement of "background independence" is met by leaving the background metric completely arbitrary. This bimetric structure entails that the effective average action may contain three classes of interactions: those built from the dynamical metric only, terms which a…
The Renormalization Group (RG) is a set of methods that have been instrumental in tackling problems involving an infinite number of degrees of freedom. What all these methods have in common -- which is what explains their success -- is that they allow a systematic search for those degrees of freedom that happen to be relevant to the phenomena in question. In the standard approaches the RG transformations are implemented by either coarse graining or by changes of variables. When these transformations are infinitesimal the formalism can be described as a continuous dynamical flow in a fictitious…
Stochastic dynamics for group field theories II: Methods for nonequilibrium renormalization group
Université Paris Saclay（巴黎萨克雷大学）；Cea（法国原子能和替代能源委员会）；Faculté des Sciences et Techniques (ICMPA-UNESCO Chair) Université d’Abomey-Calavi（阿博米-卡拉维大学科学与技术学院）
The two-dimensional renormalization group acting as the Ricci flow $Λ\frac{\partial}{\partialΛ} g_{μν} = R_{μν}$ produces a specific 1+3 dimensional space-time metric which describes an expanding universe that starts with a big bang $a \sim t^{\scriptscriptstyle 1/\sqrt3}$ then decelerates until $z=0.2$ then accelerates until ending at $t_{\max}=1.6\,t_{H}$ with a big blowup $a \sim (t_{\max}-t)^{\scriptscriptstyle -1/\sqrt3}$. The only free parameters are the overall time scale and the value of the present time $t_{0}$. These are fixed by the Hubble constant $H_{0}=t_{H}^{-1}$ and the present…
The Pauli--Villars regularization procedure confirms and sharpens the conclusions reached previously by covariant point splitting. The divergences in the stress tensor of a quantized scalar field interacting with a static scalar potential are isolated into a three-parameter local, covariant functional of the background potential. These divergences can be naturally absorbed into coupling constants of the potential, regarded as a dynamical object in its own right; here this is demonstrated in detail for two different models of the field-potential coupling. here is a residual dependence on the lo…
We study the competition of spin- and charge-density waves and their quantum multicritical behavior for the semimetal-insulator transitions of low-dimensional Dirac fermions. Employing the effective Gross-Neveu-Yukawa theory with two order parameters as a model for graphene and a growing number of other two-dimensional Dirac materials allows us to describe the physics near the multicritical point at which the semimetallic and the spin- and charge-density-wave phases meet. With the help of a functional renormalization group approach, we are able to reveal a complex structure of fixed points, th…
The standard nonperturbative approaches of renormalization group for tensor models are generally focused on a purely local potential approximation (i.e. involving only generalized traces and product of them) and are showed to strongly violate the modified Ward identities. This paper as a continuation of our recent contribution [Physical Review D 101, 106015 (2020)], intended to investigate the approximation schemes compatibles with Ward identities and constraints between $2n$-points observables in the large $N$-limit. We consider separately two different approximations: In the first one, we tr…
These notes provide a concise introduction to important applications of the renormalization group (RG) in statistical physics. After reviewing the scaling approach and Ginzburg-Landau theory for critical phenomena, Wilson's momentum shell RG method is presented, and the critical exponents for the scalar Phi^4 model are determined to first order in an eps expansion about d_c = 4. Subsequently, the technically more versatile field-theoretic formulation of the perturbational RG for static critical phenomena is described. It is explained how the emergence of scale invariance connects UV divergence…
We study some of the implications for the perturbative renormalization program when augmented with the Borel-Ecalle resummation. We show the emergence of a new kind of non-perturbative fixed point for the scalar $ϕ^4$ model, representing an ultraviolet self-completion by transseries. We argue that this completion is purely non-Wilsonian and it depends on one arbitrary constant stemming from the transseries solution of the renormalization group equation. On the other hand, if no fixed points are demanded through the adjustment of this arbitrary constant, we end up with an effective theory in wh…
We develop a new formulation of the functional renormalization group (RG) for interacting fermions. Our approach unifies the purely fermionic formulation based on the Grassmannian functional integral, which has been used in recent years by many authors, with the traditional Wilsonian RG approach to quantum systems pioneered by Hertz [Phys. Rev. B 14, 1165 (1976)], which attempts to describe the infrared behavior of the system in terms of an effective bosonic theory associated with the soft modes of the underlying fermionic problem. In our approach, we decouple the interaction by means of a sui…
