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Questions tagged [yang-mills]

Yang–Mills theory is a QFT, a *gauge theory* normally symmetric under a compact non-Abelian Lie group relying on (originally massless) gauge vector fields. YM theories describe the strong and electroweak interactions of elementary particle physics, the Standard Model.

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Understanding the field strength tensor $F_{\mu\nu}$ as a commutator

As far as I understand, one way to define the field-strength tensor is by using the commutator of covariant derivatives as follows: $$-igT^aF^a_{\mu\nu} = [D_\mu, D_\nu]$$ where $T^a$ is a basis for ...
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Gauss law, gauge and global symmetries

I am reading Witten's paper on the confinement/deconfinement phase transition in $\mathcal{N}=4$ $\mathrm{SU}(N)$ SYM theory. I am a bit stuck at section "Confinement" at Finite Volume, page ...
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Super Hilbert space of SYM

I hope this message finds you well. I am currently try to understand explicitly, at least in some sense, $d=4$, ${\cal N}=4$ super Yang-Mills theory. What is the explicit construction of the super ...
d'Alembert's user avatar
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Derivation of the BRST invariance in QCD

I am trying to follow the proof for the BRST invariance in QCD in the following pdf file: https://scipp.ucsc.edu/~haber/ph222/BRST.pdf (section 3, from end of page 6) I can understand the derivations ...
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A list of failed attempts towards a proof of confinement [closed]

Can one give a list of failed or open attempts (not necessarily Supersymmetric) towards a proof of confinement in 4d regarding YM or QCD?
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How to expand $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ in $SU(2)$?

I would like to calculate the following expression: $$(D_\mu\Phi)^\dagger(D^\mu\Phi)$$ where $$D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$$ and $A_\mu^a$ are the components of a real $SU(...
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Gauge transformation rule for $dA$, where $A$ is the gauge field

Let $G$ be a non-Abelian simple compact gauge group and $\{ t^\alpha\}$ be a normalized set of generators for its Lie algebra $\mathfrak{g}$. Let $C^{\alpha \beta}_\gamma$ be the coupling constant for ...
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Writing gauge transformation of the gauge fields explicitly in terms of coordinates

Let $G$ be a non-Abelian simple compact gauge group and $\{ t^\alpha\}$ be a normalized set of generators for its Lie algebra $\mathfrak{g}$. Let $C^{\alpha \beta}_\gamma$ be the coupling constant for ...
Keith's user avatar
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Is tetrad postualte independent of gauge field?

Here is what I know, $g_{\mu \nu} = e^{a}_{\mu} e^{b}_{\nu} \eta_{ab}$ and the tetrad postulate is \begin{equation} D_{\rho} e^{a}_{\nu} = \partial_{\rho} e^{a}_{\mu} - \Gamma^{\lambda}_{\rho \nu}e^{a}...
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Derivation of the Noether current (Gauss law operator) in anomalous chiral gauge theory

I am reading Fujikawa-Suzuki's Path Integrals and Quantum Anomalies, §6.3. The Lagrangian I am looking at is \begin{equation} \mathcal{L}=-\frac{1}{4g^2}\left(\partial_\mu L_\nu^a-\partial_{\nu}L_\mu^...
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Why the Slavnov operator is self-adjoint? [duplicate]

In the context of BRST we can define the Slavnov operator $\Delta_{BRST}$ which generates BRST transformations. My lecture notes claim that $\Delta_{BRST}$ is self-adjoint, but I don't see why.
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The commutation relations of photon and gluon?

In QED, the photon field has the following commutation relations: \begin{equation} [A^{\mu}(t,\vec{x}),A^{\nu}(t,\vec{y})]=0, \tag{1} \end{equation} where $A^{\mu}(t,\vec{x})$ is the photon filed. ...
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2+1-dimensional $SU(N)$ Yang-Mills Theory

In recent years, there has been significant progress and growing interest in conducting quantum simulations of field theories using quantum devices. This typically involves formulating a Hamiltonian ...
Quantization's user avatar
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Konishi operator anomalous dimension [closed]

The Konishi operators are operators in the ${\cal N}=4$ SYM theory and are given by: $$ K = \sum _{i=1}^6Tr\ (\phi^i\phi^i) $$ The 2 point function of this operator is: $$ \big\langle K(x)K(y)\big\...
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How to do Variational Principle in QFT? ($SU(2)$-Yang-Mills)

I am currently familiarizing myself with QFT and have a question about this article. My understanding is that the Lagrangian density in (2) couples my gauge fields to the Higgs field. And with ...
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1 answer
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Asymptotic Freedom QCD

I'm trying to understand the derivation of asymptotic freedom with the renormalisation group equations. I'm reading Taizo Muta's book on QCD. What I don't understand is how he obtains the last ...
Gogoman96 X's user avatar
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Proving conservation of supercurrent

I am trying to prove that the supercurrent $J^\mu = \gamma^{\nu \rho} F^A_{\nu \rho} \gamma^\mu \lambda^A $ is conserved in ${\cal N}=1$ SUSY Yang-Mills theory ( basically trying to reproduce equation ...
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Motivation for pure Yang-Mills Lagrangian

The Lagrangian for pure Yang-Mills theory is given by $$-\frac14 F^{a\mu\nu}F^a_{\mu\nu} \tag{1}$$ where $$F^a_{\mu\nu} = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A^b_\mu A_\nu^c.\tag{2}$$...
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Extracting a gauge-invariant variable from a given Wilson line? (NOT Wilson loop)

Let $W[x_i,x_f]$ be the Wilson line as defined here. Under a local gauge transform $g(x)$, it transforms as \begin{equation} W[x_i,x_f] \to g(x_f)W[x_i,x_f] g^{-1}(x_i) \end{equation} which is shown ...
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4 votes
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Interpretation of self-interacting terms in the expansion of a pure YM Lagrangian?

Let $A^{\alpha}_\mu$ be the gauge field of a Yang-Mills theory where $\alpha$ is the gauge index of generators for some Lie algebra with structure constant $C_{\alpha \beta}^\gamma$ and $\mu$ is the ...
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Higgs mechanism and mass gap in the Standard Model - asking for some clarification

Higgs mechanism is known to give "mass" to gauge bosons, especially in electroweak theory where the gauge group is given by $SU(2) \times U(1)$. However, as in this PE post or the statement ...
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Noether current for Yang-Mills theory in the absence of scalar field

The theory with an arbitrary compact gauge group $G$ is given. And global transformations are valid (see below) $$ A_{\mu}\mapsto{A^{'}_{\mu}={\omega}A_{\mu}\omega^{-1}} $$ also $\omega \in G$ and it ...
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Non-abelian Yang-Mills in 1+1 dimensions

Abelian electrodynamics in 1+1 dimensions is solvable, in the sense that we can find the space of solutions for the equation of motions $\partial_\mu F^{\mu\nu}=0$. To see this, one first notice that ...
BVquantization's user avatar
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Visualization of a gauge field with non-null winding number

In QCD you may add the term $\mathcal{L}_{\theta} = \theta\dfrac{g^2}{16\pi^2} \text{Tr}F\tilde{F}$, which turns out to be a total derivative. Now, it can be proven that the action of this lagrangian ...
Gabriel Ybarra Marcaida's user avatar
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Meaning of “transforms like the adjoint” in the context of Yang Mills Theory and connection to Lie Algebra

In Srednicki Chapter 69, we say something transforms like the adjoint if its transformation under the $SU(N)$ group action is $$W\rightarrow UWU^\dagger$$ The Field strength and the covariant ...
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What makes electric and magnetic fields in Yang-Mills theories gauge co-variant?

Specifically in QCD, why is it so?
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Fermions coupled to BF theory and asymptotic freedom

Suppose we couple $N$ colors of fermions to an $SU(N)$ gauge field $A$, but instead of a Yang-Mills action, there is a BF theory that restricts the gauge field to be flat $dA+A\wedge A\equiv F=0$ (by ...
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Global properties of the gauge group

In this very good P.E. answer, it is explained precisely what it means for a quantum system/theory to have a symmetry group $G$ (where $G$ is a Lie group): going back to first principles, it means ...
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Why semi-simple and compact Gauge Group in YM Theory? [duplicate]

I'm studying the Yang-Mills theory, with the Action: $$ S=-\frac{1}{2}\int\mathrm{tr}_{\rho}(\mathcal{F}\wedge\star\mathcal{F}) $$ where $\mathcal{F}:=\mathrm{d} \mathcal{A}+\frac{1}{2}[\mathcal{A},\...
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Wilson loop is not an element of $\mathrm{SU}(3)$ in color deconfinement

The center symmetry in QCD comes from the $$a\ \mathcal{P}\mathrm{exp}\left(ig_s \int_C dx^\mu \ A_\mu(x)\right) a^{-1} = \mathcal{P}\mathrm{exp}\left(ig_s \int_C dx^\mu \ A_\mu(x)\right),$$ where $C$ ...
Joao Vitor's user avatar
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2 answers
115 views

Gauge boson interacting with derivative of the Higgs field

Consider the $\mathrm{SU}(2)\times \mathrm U(1)_Y$ Lagrangian $$ \mathcal L=-\frac{1}{4} (W^a_{\mu\nu})^2 -\frac{1}{4} B_{\mu\nu}^2 + (D_\mu H)^\dagger (D^\mu H) +m^2 H^\dagger H -\lambda (H^\...
Liu Zhiyu's user avatar
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Help understanding gauge symmetry and principal bundles

i'm diving into gauge theories and i'm having a hard time understanding the concept of Gauge symmetry. What i understand is: gauge symmetry is the invariance of a field theory under a certain family ...
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Hamiltonian form for Yang-Mills theory

For trying to convert a Lagrangian system to Hamiltonian form, I've seen references to two different methods for dealing with constraints; the Dirac-Bergmann method (see reference here) or the Faddeev-...
Matt Dickau's user avatar
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1 answer
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Yang-Mills mass gap caused by gluonballs or because dark matter WIMPs?

Yang-Mills quantum field theory predicts the existence of the lightest massive Bosonic (i.e. integer spin) particle. This massive Boson will be much lighter than the $W$ and $Z$ Boson and therefore ...
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Gauge transformation and Kaluza-Klein metric

The Kaluza-Klein metric, by reduction, can be written as a $(4+m) \times (4+m)$ symmetric matrix, where $m$ is the dimension of the additional spacetime (if we decompose $M_D = M_4 \times M_m$). It ...
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2 votes
1 answer
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Silly confusion about gauge invariance in supersymmetric Lagrangians - in particular, in the ${\cal N}=1$ superfield formulation of ${\cal N}=4$ SYM

Hoping to resolve a simple confusion I have about supersymmetric gauge theory, one which I ran into while trying to understand the ${\cal N}=1$ superfield formulation of ${\cal N}=4$ supersymmetric ...
Cyrus R.O.'s user avatar
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How to derive the gauge invariance of Yang-Mills action with external source?

In the Faddeev-Popov procedure of path integral of $$ Z[J] = \int [DA] e^{iS(A,J)}, \quad S(A,J)= \int d^4x [-\frac{1}{4}F^{a\mu\nu}F_{a\mu\nu} + J^{a\mu}A_{a\mu} ] $$ we have used that $S(A,J)$ is ...
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Wilson lines with Chan-Paton factors in string theory

In the context of compactifying the open string with Chan-Paton factors, Polchinski (Volume I Section 8.6) considers a toy example with a point particle of charge $q$ which has the action $$ S = \int ...
Adrien Martina's user avatar
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Infinitesimal transformation of the Yang-Mills field

I am trying to obtain the infinitesimal transformation for the Yang-Mills field $A_{\mu}$. I want to show that $$ A^{\prime a}_\mu=A_\mu^a-\partial_\mu \theta^a-g_s f^{a b c} \theta^b A_\mu^c $$ For ...
David Lazaro's user avatar
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A reference request on fundamental modular domains in the context of Gribov ambiguity

I see that there are some references in the post PE on the Gribov ambiguity. However, resolution of this ambiguity, as stated in wiki, is to find the fundamental modular region (FMR). I looked into ...
1 vote
1 answer
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Quantum effective action for Yang-Mills theories

During a course I came across a formula for the quantum effective action of a Yang-Mills theory in euclidean space and it appears like this (some indices may be dropped but I hope that won't be a ...
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0 answers
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About "linear" Yang-Mills theories

Is it sensible to say about "linear" YM theories with tiny YM fields, thus removing the nonlinear YM self-interaction terms from the Lagrangian which therefore only contains the linear YM ...
Ian Darius's user avatar
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1 answer
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Is it possible to solve numerically the classical Yang-Mills for a generic source?

The classical Yang-Mills equation in the presence of a source $J^\nu(x)$ can be written as $$ \partial_\mu F^{\mu \nu} - i g [A_\mu, F^{\mu \nu}] = J^\nu (x), $$ where $F^{\mu \nu} = \partial^\mu A^\...
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Why we only have one gauge coupling constant for $\rm SU(2)$ gauge symmetry?

I know that in the case of a Yang-Mills $\rm SU(2)$ gauge symmetry, the covariant derivative is written as: $$D^{\mu} = \partial^{\mu} + i\frac{g}{2}W^{\mu}_{a}\tau^{a}$$ With $g$ is the coupling ...
ياسين المهتدي's user avatar
1 vote
1 answer
88 views

Perturbative expansion and renormalization of non-abelian Yang-Mills theory solely in terms of gauge-invariant quantities?

In standard QFT, each term in the perturbative expansion for a gauge theory is not necessarily gauge-invariant. Only the whole sum of Feynman diagrams is guaranteed so. However, at least for QED, ...
Keith's user avatar
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Is there an experimental set up that would produce a "macroscopic" a weak or strong nuclear force fields?

I was wondering if there is an experimental set up that would produce something equivalent to a classical electromagnetic field for the weak and strong nuclear forces. I know that the those forces are ...
Bryan D's user avatar
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Asking for correct ways to do "power counting" for gauge theories

I am looking into Weinberg "Quantum Theory of Fields" Volume 1 Ch. 12-1. There, he discusses the general rules of power counting. He defines the UV asymptotics of the propagator $\Delta_f$ ...
Keith's user avatar
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Is there any physical reason behind the choice of Lie group in a Yang-Mills theory?

A Yang-Mills theory can be constructed for any Lie group that is compact and semisimple. The motivation behind this is discussed in this question. Is there any physical reason we choose $SU(3)$ or $U(...
CBBAM's user avatar
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1 vote
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Subleading correction to the gluon propagator in large $N$ expansion

I was reading Callan, Coote and Gross' paper on 2-dimensional QCD, where they show that the model that 't Hooft proposes in his work indeed produces quark confinement. In section VIII, they analyze ...
Marcosko's user avatar
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Possible cases of matter fields for $SU(2)$ theory which retains asymptotic freedom?

Let us assume $4$ spacetime dimensions. QCD, the $SU(3)$ gauge theory with quarks as the matter fields, have the asymptotic freedom property as long as there are 16 quark flavors of mass below the ...
Keith's user avatar
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