scholarly journals Six-dimensional ultraviolet completion of the ℂℙ(N) σ model at two loops

2020 ◽  
Vol 35 (22) ◽  
pp. 2050188
Author(s):  
J. A. Gracey
Keyword(s):  
Renormalization Group ◽  
Quantum Electrodynamics ◽  
Matter Field ◽  
Coupling Constants ◽  
Universal Theory ◽  
Gauge Parameter ◽  
Loop Analysis ◽  
Four Dimensions ◽  
Anomalous Dimensions ◽  
Group Functions

We extend the recent one-loop analysis of the ultraviolet completion of the [Formula: see text] nonlinear [Formula: see text] model in six dimensions to two-loop order in the [Formula: see text] scheme for an arbitrary covariant gauge. In particular we compute the anomalous dimensions of the fields and [Formula: see text]-functions of the four coupling constants. We note that like Quantum Electrodynamics (QED) in four dimensions the matter field anomalous dimension only depends on the gauge parameter at one loop. As a nontrivial check we verify that the critical exponents derived from these renormalization group functions at the Wilson–Fisher fixed point are consistent with the [Formula: see text] expansion of the respective large [Formula: see text] exponents of the underlying universal theory. Using the Ward–Takahashi identity we deduce the three-loop [Formula: see text] renormalization group functions for the six-dimensional ultraviolet completeness of scalar QED.

scholarly journals The all-loop perturbative derivation of the NSVZ $$\beta $$-function and the NSVZ scheme in the non-Abelian case by summing singular contributions

2020 ◽  
Vol 80 (9) ◽  
Author(s):  
K. V. Stepanyantz
Keyword(s):  
Renormalization Group ◽  
Beta Function ◽  
Covariant Derivatives ◽  
Anomalous Dimensions ◽  
Higher Covariant Derivative ◽  
Higher Covariant Derivatives ◽  
Loop Approximation ◽  
Abelian Case ◽  
Group Functions ◽  
Supersymmetric Gauge

AbstractThe perturbative all-loop derivation of the NSVZ $$\beta $$ β -function for $${{\mathcal {N}}}=1$$ N = 1 supersymmetric gauge theories regularized by higher covariant derivatives is finalized by calculating the sum of singularities produced by quantum superfields. These singularities originate from integrals of double total derivatives and determine all contributions to the $$\beta $$ β -function starting from the two-loop approximation. Their sum is expressed in terms of the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This allows obtaining the NSVZ equation in the form of a relation between the $$\beta $$ β -function and these anomalous dimensions for the renormalization group functions defined in terms of the bare couplings. It holds for an arbitrary renormalization prescription supplementing the higher covariant derivative regularization. For the renormalization group functions defined in terms of the renormalized couplings we prove that in all loops one of the NSVZ schemes is given by the HD + MSL prescription.

scholarly journals A GEOMETRICAL INTERPRETATION OF RENORMALIZATION GROUP FLOW

1994 ◽  
Vol 09 (08) ◽  
pp. 1261-1286 ◽  
Author(s):  
BRIAN P. DOLAN
Keyword(s):  
Renormalization Group ◽  
Euclidean Space ◽  
Point Of View ◽  
Dimensional Euclidean Space ◽  
Lie Derivative ◽  
Coupling Constant ◽  
Four Dimensions ◽  
Anomalous Dimensions ◽  
Definition Of ◽  
Group Flow

The renormalization group (RG) equation in D-dimensional Euclidean space, RD, is analyzed from a geometrical point of view. A general form of the RG equation is derived which is applicable to composite operators as well as tensor operators (on RD) which may depend on the Euclidean metric. It is argued that physical N-point amplitudes should be interpreted as rank N covariant tensors on the space of couplings, [Formula: see text], and that the RG equation can be viewed as an equation for Lie transport on [Formula: see text] with respect to the vector field generated by the β functions of the theory. In one sense it is nothing more than the definition of a Lie derivative. The source of the anomalous dimensions can be interpreted as being due to the change of the basis vectors on [Formula: see text] under Lie transport. The RG equation acts as a bridge between Euclidean space and coupling constant space in that the effect on amplitudes of a diffeomorphism of RD (that of dilations) is completely equivalent to a diffeomorphism of [Formula: see text] generated by the β functions of the theory. A form of the RG equation for operators is also given. These ideas are developed in detail for the example of massive λφ4 theory in four dimensions.

scholarly journals RUNNING COUPLING CONSTANTS OF FERMIONS WITH MASSES IN QUANTUM ELECTRODYNAMICS AND QUANTUM CHROMODYNAMICS

2001 ◽  
Vol 16 (16) ◽  
pp. 2873-2894 ◽  
Author(s):  
GUANG-JIONG NI ◽  
GUO-HONG YANG ◽  
RONG-TANG FU ◽  
HAIBIN WANG
Keyword(s):  
Renormalization Group ◽  
Quantum Chromodynamics ◽  
Momentum Transfer ◽  
Quantum Electrodynamics ◽  
Renormalization Group Equation ◽  
Energy Scale ◽  
Coupling Constants ◽  
Group Equation ◽  
Running Coupling ◽  
Renormalization Method

Based on a simple but effective regularization-renormalization method (RRM), the running coupling constants (RCC) of fermions with masses in quantum electrodynamics (QED) and quantum chromodynamics (QCD) are calculated by renormalization group equation (RGE). Starting at Q=0 (Q being the momentum transfer), the RCC in QED increases with the increase of Q whereas the RCCs for different flavors of quarks with masses in QCD are different and they increase with the decrease of Q to reach a maximum at low Q for each flavor of quark and then decreases to zero at Q→0. Thus a constraint on the mass of light quarks, the hadronization energy scale of quark–antiquark pairs are derived.

Zeta-function regularization of quantum field theory

1990 ◽  
Vol 68 (7-8) ◽  
pp. 620-629 ◽  
Author(s):  
A. Y. Shiekh
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Zeta Function ◽  
Analytic Continuation ◽  
Quantum Field ◽  
Four Dimensions ◽  
Group Functions

Analytic continuation leads to the finite renormalization of a quantum field theory. This is illustrated in a determination of the two loop renormalization group functions for [Formula: see text] in four dimensions.

scholarly journals Renormalization Group evolution from on-shell SMEFT

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Minyuan Jiang ◽  
Teng Ma ◽  
Jing Shu
Keyword(s):  
Field Theory ◽  
Renormalization Group ◽  
Standard Model ◽  
Effective Field Theory ◽  
Effective Field ◽  
High Dimensional ◽  
Anomalous Dimensions ◽  
The Standard Model ◽  
Group Evolution

Abstract We describe the on-shell method to derive the Renormalization Group (RG) evolution of Wilson coefficients of high dimensional operators at one loop, which is a necessary part in the on-shell construction of the Standard Model Effective Field Theory (SMEFT), and exceptionally efficient based on the amplitude basis in hand. The UV divergence is obtained by firstly calculating the coefficients of scalar bubble integrals by unitary cuts, then subtracting the IR divergence in the massless bubbles, which can be easily read from the collinear factors we obtained for the Standard Model fields. Examples of deriving the anomalous dimensions at dimension six are presented in a pedagogical manner. We also give the results of contributions from the dimension-8 H4D4 operators to the running of V+V−H2 operators, as well as the running of B+B−H2D2n from H4D2n+4 for general n.

scholarly journals THE CASE OF ASYMPTOTIC SUPERSYMMETRY

2013 ◽  
Vol 28 (14) ◽  
pp. 1350053 ◽  
Author(s):  
BRUCE L. SÁNCHEZ-VEGA ◽  
ILYA L. SHAPIRO
Keyword(s):  
Fixed Point ◽  
Renormalization Group ◽  
Gauge Coupling ◽  
High Energy ◽  
Systematic Investigation ◽  
Coupling Constants ◽  
Scalar Coupling ◽  
Asymptotic State ◽  
Stable Fixed Point ◽  
Scalar Couplings

We start systematic investigation for the possibility to have supersymmetry (SUSY) as an asymptotic state of the gauge theory in the high energy (UV) limit, due to the renormalization group running of coupling constants of the theory. The answer on whether this situation takes place or not, can be resolved by dealing with the running of the ratios between Yukawa and scalar couplings to the gauge coupling. The behavior of these ratios does not depend too much on whether gauge coupling is asymptotically free (AF) or not. It can be shown that the UV stable fixed point for the Yukawa coupling is not supersymmetric. Taking this into account, one can break down SUSY only in the scalar coupling sector. We consider two simplest examples of such breaking, namely N = 1 supersymmetric QED and QCD. In one of the cases one can construct an example of SUSY being restored in the UV regime.

Determine the critical fermion flavor in three-dimensional QED using nonlocal gauge

2014 ◽  
Vol 29 (33) ◽  
pp. 1450159
Author(s):  
Hua Jiang ◽  
Yong-Long Wang ◽  
Wei-Tao Lu ◽  
Chuan-Cong Wang
Keyword(s):  
Symmetry Breaking ◽  
Critical Point ◽  
Chiral Symmetry ◽  
Gauge Boson ◽  
Quantum Electrodynamics ◽  
Chiral Symmetry Breaking ◽  
Three Dimensional ◽  
Gauge Parameter ◽  
Dynamical Chiral Symmetry Breaking

We determine the critical fermion flavor for dynamical chiral symmetry breaking in three-dimensional quantum electrodynamics using nonlocal gauge (gauge parameter depends on the momentum or coordinate). The coupled Dyson–Schwinger equations of the fermion and gauge boson propagators are considered in the vicinity of the critical point. Illustrated by using the transverse vertex proposed by Bashir et al., we show that: for a variety of the transverse vertex, the critical flavor is still 128/3π2, the same as using the bare vertex.

Infrared Yang–Mills theory: A renormalization group perspective

2016 ◽  
Vol 25 (07) ◽  
pp. 1642002 ◽  
Author(s):  
Axel Weber ◽  
Pietro Dall’Olio ◽  
Francisco Astorga
Keyword(s):  
Fixed Point ◽  
Renormalization Group ◽  
Analytical Approach ◽  
Landau Gauge ◽  
Momentum Dependence ◽  
Lattice Data ◽  
Four Dimensions ◽  
Yang Mills ◽  
Epsilon Expansion ◽  
Renormalization Group Equations

We describe a technically very simple analytical approach to the deep infrared regime of Yang–Mills theory in the Landau gauge via Callan–Symanzik renormalization group equations in an epsilon expansion. This approach recovers all the solutions for the infrared gluon and ghost propagators previously found by solving the Dyson–Schwinger equations of the theory and singles out the solution with decoupling behavior, confirmed by lattice calculations, as the only one corresponding to an infrared attractive fixed point (for space-time dimensions above two). For the case of four dimensions, we describe the crossover of the system from the ultraviolet to the infrared fixed point and determine the complete momentum dependence of the propagators. The results for different renormalization schemes are compared to the lattice data.

Renormalization Group

2020 ◽  
pp. 289-318
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Critical Phenomena ◽  
Degrees Of Freedom ◽  
Gaussian Model ◽  
Coupling Constants ◽  
Quantum Field ◽  
Important Notion ◽  
Effective Hamiltonians

Chapter 8 introduces the key ideas of the renormalization group, including how they provide a theoretical scheme and a proper language to face critical phenomena. It covers the scaling transformations of a system and their implementations in the space of the coupling constants and reducing the degrees of freedom. From this analysis, the reader is led to the important notion of relevant, irrelevant and marginal operators and then to the universality of the critical phenomena. Furthermore, the chapter also covers (as regards the RG) transformation laws, effective Hamiltonians, the Gaussian model, the Ising model, operators of quantum field theory, universal ratios, critical exponents and β‎-functions.

Sign in / Sign up

Export Citation Format

Share Document