scholarly journals Nonminimal gradient flows in QCD-like theories

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Marco Boers
Keyword(s):  
Degrees Of Freedom ◽  
Anomalous Dimension ◽  
Gradient Flow ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Flow Equation ◽  
Gradient Flows ◽  
Leading Order ◽  
Flow Equations ◽  
Yang Mills

Abstract The Yang-Mills gradient flow for QCD-like theories is generalized by including a fermionic matter term in the gauge field flow equation. We combine this with two different flow equations for the fermionic degrees of freedom. The solutions for the different gradient flow setups are used in the perturbative computations of the vacuum expectation value of the Yang-Mills Lagrangian density and the field renormalization factor of the evolved fermions up to next-to-leading order in the coupling. We find a one-parameter family of flow systems for which there exists a renormalization scheme in which the evolved fermion anomalous dimension vanishes to all orders in perturbation theory. The fermion number dependence of different flows is studied and applications to lattice studies are anticipated.

scholarly journals Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

2021 ◽  
Author(s):  
Alexander Mielke
Keyword(s):  
Gradient Flow ◽  
Careful Analysis ◽  
Flow Equation ◽  
Uniqueness Result ◽  
Gradient System ◽  
Independent System ◽  
Exact Relation ◽  
Flow Equations ◽  
Total Variation Flow ◽  
Energetic Solutions

AbstractWe consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

THE AFFINE N = 4 YANG–MILLS THEORY

1992 ◽  
Vol 07 (11) ◽  
pp. 2469-2485
Author(s):  
A. C. CADAVID ◽  
R. J. FINKELSTEIN
Keyword(s):  
Mass Spectrum ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Effective Theory ◽  
Affine Algebra ◽  
Linear Mass ◽  
Expectation Value ◽  
Yang Mills ◽  
Exotic States ◽  
Affine Symmetry

An affine field theory may be constructed by gauging an affine algebra. The momentum integrals of the affine N = 4 Yang–Mills theory are ultraviolet finite but diverge because the sum over states is infinite. If the affine symmetry is broken by postulating a nonvanishing vacuum expectation value for that component of the scalar field lying in the L0 direction, then the theory acquires a linear mass spectrum. This broken theory is ultraviolet finite too, but the mass spectrum is unbounded. If it is also postulated that the mass spectrum has an upper bound (say, the Planck mass), then the resulting theory appears to be altogether finite. The influence of the exotic states has been estimated and, according to the proposed scenario, is negligible below energies at which gravitational interactions become important. The final effective theory has the symmetry of a compact Lie algebra augmented by the operator L0.

scholarly journals PRIVATE HIGGS AT THE LHC

2013 ◽  
Vol 28 (28) ◽  
pp. 1350149 ◽  
Author(s):  
YONI BENTOV ◽  
A. ZEE
Keyword(s):  
Dark Matter ◽  
Degrees Of Freedom ◽  
Vacuum Expectation ◽  
Mass Hierarchy ◽  
Neutrino Mixing ◽  
Leading Order ◽  
Expectation Values ◽  
Mixing Angles ◽  
Yukawa Couplings

We study the LHC phenomenology of a general class of "Private Higgs" (PH) models, in which fermions obtain their masses from their own Higgs doublets with [Formula: see text] Yukawa couplings, and the mass hierarchy is translated into a dynamical chain of vacuum expectation values. This is accomplished by introducing a number of light gauge-singlet scalars, the "darkons," some of which could play the role of dark matter. These models allow for substantial modifications to the decays of the lightest Higgs boson, for instance through mixing with TeV-scale PH fields and light darkons: in particular, one could accommodate [Formula: see text] flavor-uncorrelated deviations from the SM [Formula: see text] vertices with TeV-scale degrees of freedom. We also discuss a new implementation of the PH framework, in which the quark and neutrino mixing angles arise as one-loop corrections to the leading order picture.

scholarly journals SYMPLECTIC GEOMETRY AND HAMILTONIAN FLOW OF THE RENORMALIZATION GROUP EQUATION

1995 ◽  
Vol 10 (18) ◽  
pp. 2703-2732 ◽  
Author(s):  
BRIAN P. DOLAN
Keyword(s):  
Renormalization Group ◽  
Phase Space ◽  
Poisson Bracket ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Identity Operator ◽  
Group Equation ◽  
Renormalization Group Flow ◽  
Flow Equations ◽  
Group Flow

It is argued that renormalization group flow can be interpreted as a Hamiltonian vector flow on a phase space which consists of the couplings of the theory and their conjugate “momenta,” which are the vacuum expectation values of the corresponding composite operators. The Hamiltonian is linear in the conjugate variables and can be identified with the vacuum expectation value of the trace of the energy-momentum operator. For theories with massive couplings the identity operator plays a central role and its associated coupling gives rise to a potential in the flow equations. The evolution of any quantity, such as N-point Green functions, under renormalization group flow can be obtained from its Poisson bracket with the Hamiltonian. Ward identities can be represented as constants of the motion which act as symmetry generators on the phase space via the Poisson bracket structure.

scholarly journals ON THE LARGE N LIMIT, WILSON LOOPS, CONFINEMENT AND COMPOSITE ANTISYMMETRIC TENSOR FIELD THEORIES

2004 ◽  
Vol 19 (25) ◽  
pp. 4251-4270 ◽  
Author(s):  
CARLOS CASTRO
Keyword(s):  
Tensor Field ◽  
Degrees Of Freedom ◽  
Vacuum Expectation ◽  
Field Theories ◽  
Antisymmetric Tensor ◽  
Wilson Loops ◽  
Tensor Fields ◽  
Yang Mills ◽  
Large N ◽  
Antisymmetric Tensor Field

A novel approach to evaluate the Wilson loops associated with a SU (∞) gauge theory in terms of pure string degrees of freedom is presented. It is based on the Guendelman–Nissimov–Pacheva formulation of composite antisymmetric tensor field theories of area (volume) preserving diffeomorphisms which admit p-brane solutions and which provide a new route to scale-symmetry breaking and confinement in Yang–Mills theory. The quantum effects are discussed and we evaluate the vacuum expectation values (VEV) of the Wilson loops in the large N limit of the quenched reduced SU (N) Yang–Mills theory in terms of a path integral involving pure string degrees of freedom. The quenched approximation is necessary to avoid a crumpling of the string worldsheet giving rise to very large Hausdorff dimensions as pointed out by Olesen. The approach is also consistent with the recent results based on the AdS/CFT correspondence and dual QCD models (dual Higgs model with dual Dirac strings). More general Loop wave equations in C-spaces (Clifford manifolds) are proposed in terms of generalized holographic variables that contain the dynamics of an aggregate of closed branes (p-loops) of various dimensionalities. This allows us to construct the higher-dimensional version of Wilson loops in terms of antisymmetric tensor fields of arbitrary rank which couple to p-branes of different dimensionality.

scholarly journals Vacuum Expectation Value of the Higgs Field and Dyon Charge Quantisation from Spacetime Dependent Lagrangians

2003 ◽  
Vol 18 (31) ◽  
pp. 2207-2216
Author(s):  
Rajsekhar Bhattacharyya ◽  
Debashis Gangopadhyay
Keyword(s):  
Classical Solution ◽  
Electric Charge ◽  
Higgs Field ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Lagrangian Formalism ◽  
Expectation Value ◽  
Yang Mills ◽  
Noncommuting Coordinates

The spacetime dependent Lagrangian formalism of Refs. 1 and 2 is used to obtain a classical solution of Yang–Mills theory. This is then used to obtain an estimate of the vacuum expectation value of the Higgs field viz. ϕa = A/e, where A is a constant and e is the Yang–Mills coupling (related to the usual electric charge). The solution can also accommodate noncommuting coordinates on the boundary of the theory which may be used to construct D-brane actions. The formalism is also used to obtain the Deser–Gomberoff–Henneaux–Teitelboim results10 for dyon charge quantisation in Abelian p-form theories in dimensions D = 2(p+1) for both even and odd p.

scholarly journals Modeling of Chemical Reaction Systems with Detailed Balance Using Gradient Structures

2020 ◽  
Vol 181 (6) ◽  
pp. 2257-2303 ◽  
Author(s):  
Jan Maas ◽  
Alexander Mielke
Keyword(s):  
Chemical Reaction ◽  
Detailed Balance ◽  
Gradient Flow ◽  
Flow Equation ◽  
Gradient Flows ◽  
Reaction Systems ◽  
Chemical Reaction Systems ◽  
Detailed Balance Condition ◽  
Homogeneous Chemical ◽  
Reaction Rate Equation

AbstractWe consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary $$\Gamma $$ Γ -convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.

The ground state of an SU(2) gauge theory in a nonabelian background field

1983 ◽  
Vol 61 (10) ◽  
pp. 1442-1447 ◽  
Author(s):  
R. Parthasarathy ◽  
M. Singer ◽  
K. S. Viswanathan
Keyword(s):  
Energy Density ◽  
Ground State ◽  
Gluon Condensate ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Background Field ◽  
Yang Mills ◽  
Classical Energy ◽  
The Stability ◽  
Bag Constant

The ground state in an SU(2) Yang–Mills theory using a constant nonabelian background field is studied taking into account the gluon and quark one-loop corrections to the classical energy density. The calculated energy density is found to be lower than that of the perturbative vacuum. The local minimum of the energy density is used to calculate the vacuum expectation value of the gluon condensate and the bag constant, which is found to be in reasonable agreement with the known estimates. The stability of this approximation is briefly discussed.

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