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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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Are theta vacua topologically protected?

In discussions of Yang-Mills instantons it is often stated that one should sum in the path integral over all contributions of fluctuations around all the topologically distinct vacua labelled by ...
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Why doesn't the $\theta$ Angle Renormalize?

The $\theta$ term for Yang-Mills takes the form $$L_{\theta}=\frac{\theta}{64\pi^2}\varepsilon^{\mu\nu\rho\sigma}F^a_{{\mu\nu}}F^a_{\rho\sigma}$$ A fact that I have heard is that $\theta$ does not ...
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Showing the form of the covariant derivative of $\phi$, if $\phi$ transforms as the adjoint representation of $SU(n)$

I want to show that if $\phi$ transforms as the adjoint representation of SU(n), its covariant derivative is given by $\textbf{D}_\mu \phi = \partial_\mu \phi + i [\textbf{A}_\mu, \phi]$. (Exercise in ...
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“Pure” Yang-Mills and the absence of light matter

I am researching various models of Neutral Naturalness which involve the addition of an additional gauge group whose matter content is uncharged under SM color. Many of these theories state that their ...
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Compactification of space in Hamiltonian formulation of Yang-Mills theory

I am reading David Tong's lecture notes on Gauge Theory where he talks about Hilbert space interpretation of Yang-Mills theories in Section 2.2 of Chapter 2. When discussing the gauge dependence of ...
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Does the Yang Mills Mass Gap represent “Absolute Bottom”?

The question of the existence of a Yang Mills mass gap is a complex and technical one. In this paper Philip Gibbs asks "Is fundamentality then a relative concept with no absolute bottom, or is there ...
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A Naive Question about Gauge Theory

I am suffering from a question I encountered from the lecture notes of gauge theory by David Tong. The problem comes from page 67 on the gauge fixing in back-ground gauge method. In David Tong's ...
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Why quarks in the fundamental and gluons in the adjoint?

I have been told that in gauge theories “fermionic matter goes in the fundamental rep of $SU(N)$, while gauge fields go in the adjoint rep”. I understand how this works, and for instance, in QCD,...
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Triviality of Yang Mills in $d>4$?

It has been proved that the $\phi^4$ theory is trivial in spacetime dimensions $d>4$. By trivial I mean that the field $\phi$ is a generalized free field, or in other words, it's only nonzero ...
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Wilson loop and Polyakov loop

As I understand, the Wilson line is the operator $W(x) = P\exp(i\int_{xi}^{xf} A.dx)$, where $P$ is path ordering. The Polyakov loop $P(x)$ on the other hand is the trace of the Wilson loop $W(x)$ ...
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Prerequisites on Yang Mills Theory [duplicate]

I was wondering of what kind of subject are necessary to study the Yang - Mills theory from basic university level. Can you suggest some good book for subject?
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Invariance of Yang-Mills Lagrangian under charge conjugation

The Yang-Mills Lagrangian gauge invariant under an $SU(N)$ tranformation can be written as $${\cal L} = -\frac{1}{4}F_{\mu\nu}^i F^{i\ \mu\nu} \tag1$$ (Sum over $i$ implicit) This Lagrangian ...
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Why do like charges repel in a spin-1 gauge theory, while they attract in spin-2 theories?

A presenter recently said this at a colloquium, but they didnt explain why. I know a fair bit of GR and Yang-Mills theory, so dont shy away from the details (unless they are cumbersome and don't aid ...
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Gauge-covariance of the Yang-Mills field strength $F_{\mu\nu}^a$

Accordingly to Yang-Mills theories, after the introduction of a covariant derivative such that $$D_\mu = \partial_\mu - igA_\mu, \tag1$$ you can built the kinetic term for the gauge potential $A_\...
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Notation and concepts of Yang Mills Theory

I am studying loop quantum gravity using the book by Pullin and Gambini. I am having some trouble understanding and getting past the chapter on Yang Mills theory, mainly because I am confused about ...
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Checking modularity-like transformation property

Assume $M$ is a 4 manifold. Let $Z_v$ be partition function of fixed magnetic flux $v$ with all instanton configuration summed over where $v\in H^2(M,Z/nZ)$. $\tau$ denotes complex parameter on upper ...
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Differentiating D3 brane worldvolume theories with NS5 brane and NS5 antibrane boundary conditions

In 'Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory', Gaiotto and Witten derive boundary conditions for the worldvolume theory of the D3 brane. In particular the boundary conditions (...
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Is color charge quantized?

I was reading this stackexchange question, and found the answer to my question not totally answered. Clearly there is color and anti-color in analogy to electric charge, and color charge clearly ...
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Why is Lattice QCD done in Euclidian 4-space?

This could be a really naive question (and honestly I just don't want to dig through ArXiV review papers on lattice QCD), but my question is simple: Why exactly is Lattice QCD done in $\mathbb{R}^4$ ...
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Gauge fixing while preserving supersymmetry

In supersymmetric gauge theories, the vector potential is a part of a vector supermultiplet which is represented by a real superfield $V$. Expanded out in components, the Lagrangian for such a field ...
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Why is color confinement a difficult problem?

Assuming color force follows a constant rule of force instead of an inverse square rule of force. And that red, green and blue are all attracted to each other. Why is color confinement considered a ...
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Another way to write the Einstein-Hilbert action?

Let's take a look at the equation for the Riemann tensor in terms of an arbitrary 1-form: $$\nabla_{\mu}\nabla_{\nu}A_{\alpha}-\nabla_{\nu}\nabla_{\mu}A_{\alpha}=R_{\mu\nu\alpha}^{\quad\:\delta}A_{\...
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Why can we write lagrangian for gauge theory without the traces?

I understand that trace is needed in order to preserve gauge invariance of the lagrangian equation by using the cycling property. But I fail to see why the following equation holds true: $$-\frac{1}{2}...
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What is the relation between the Gribov problem and color confinement?

I have heard that the Gribov problem is in some way related to color confinement (For instance: Gribov copies and confinement). Although I understand what both the Gribov and confinement problems are, ...
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Understanding Measure in Path integrals

I know this is still a current topic of study, so my question is less about mathematical underpinnings, and more about the transition from the "sum-over-all-possible-intermediate-points" measure $$\...
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Where do the color indices come back in $SU(3)$ Yang-Mills Quantization?

Can the partition function of $SU(3)$ (the Generic Partition function for a yang-mills theory found on the linked wiki page below), be split into a sum of 8 functional integrals for each gauge field? ...
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Feynman rules out of the Lagrangian

Accordingly to chapter 10, section 10.6 Feynman Rules of 'Introduction to Elementary Particles' by David Griffiths, there is a way to extract the vertex and propagators just by inspection of the ...
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Is it enough to assume $F_{\mu\nu}\to 0$ at infinity but not $A_\mu$ to derive the equation of motion?

Suppose the the Lagrangian $\mathscr{L}$ of the free electromagnetic field is augmented with the term $$F_{\mu\nu}\tilde{F}^{\mu\nu}=\partial_{\mu}(\epsilon^{\nu\nu\lambda\rho}A_\nu F_{\lambda\rho}).$$...
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Isn't there a unique vacuum of the Yang-Mills quantum theory?

The theta vacua$^1$ of a Yang-Mills quantum theory are given by $$|\theta\rangle=\sum\limits_{n=-\infty}^{\infty}e^{in\theta}|n\rangle.$$ In Srednicki's Quantum Field Theory, he claims that the ...
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Why are two different gauge transformations of $A_\mu=0$ in $U(1)$ gauge thoery equivalent?

Two inequivalent gauge transformations of $\mathbb{A}_\mu=0$, described by $U$ and $\tilde{U}$ of a pure $SU(N)$ Yang-Mills theory as $$\mathbb{A}_\mu=\frac{i}{g} U\partial_\mu U^\dagger~\text{and}~\...
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Commutator of Gauge Transformations for Yang-Mills Theory

Following the conventions of "Quantum Field Theory and the Standard Model" by Schwartz, we have that for Yang-Mills Theory, an infinitesimal gauge transformation acts like $$\delta_{\alpha} A = d\...
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Conserved currents in Yang-Mills theory: gluon current vs. quark current

In Yang-Mills theory there are two currents we can construct. There is the well-known quark current related to the global $SU(3)_C$ symmetry, $$j{}^{\mu\,A}_\text{quark} = \overline{\psi}{}^i \gamma{}^...
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What is the difficulty in extending geometrodynamics to non-abelian fields?

In an attempt to widen my own horizons I've decided to educate myself in Wheeler's Geometrodynamics. In the so-called "already unified theory" one can essentially reproduce an electromagnetic field ...
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Covariant Derivative in QCD: How does it act on gluons?

Let the covariant derivative be $$D_\mu = \partial_\mu + \text ig\ A_\mu^a t^a,\quad a=1,\ldots,8$$where $g$ is the bare QCD coupling, $A_\mu^a$ are the eight gluon fields and $t^a=\tfrac{1}{2}\lambda^...
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What is the Noether charge associated with the the color $SU(3)$ symmetry of QCD?

A version of the Noether's theorem applies to local gauge symmetries. What is the Noether's charge associated with a non-abelian gauge symmetry such as the color $SU(3)$ and how is that derived? I ...
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Infrared divergencies in Yang-Mills theory

I'm trying to better understand the nature of infrared divergencies in YM theory; for now, I'm only interested in soft divergences. The usual explanation one is given about the origin of IR ...
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Can Yang-Mills field strength be defined as covariant derivative squared?

In Yang-Mills theory the field strength tensor $F_{\mu \nu}$ can be calculated as $$ \begin{equation} F_{\mu\nu} \equiv \frac{i}{g} [D_\mu,D_\nu] = \partial_\mu A_\nu - \partial_\nu A_\mu -ig[A_\mu,A_\...
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Local charge current for gauge field and conservation of charge

Motivation: It is a well known fact that the gravitational field (in General Relativity and direct generalizations of it) has no local energy-momentum density. Usually there are two reasons stated, ...
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Are the nonphysical degrees of freedom in Yang-Mills theory analogous to the worldsheet metric in the Polyakov formalism?

The Polyakov string action on a flat background (in the Euclidean signature) $$S_{P}[X,\gamma]\propto\int_{\Sigma}\mathrm{d}^2\sigma\,\sqrt{\text{det}\gamma}\,\gamma^{ab}\delta_{\mu\nu}\partial_{a}X^{...
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What are chromoelectric and chromomagnetic fields?

Are they the normal elctric and magnetic fields from Maxwell fields? Or are they just the corresponding components from $G_{\mu\nu}^a$ (the gluon fields), say chromoelectric fields are simply $G_{0i}^...
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Is Yang-Mills theory as a topological quantum field theory a good alternative?

Suppose we know a classical solution for gauge Boson fields $A_{\mu,c}$ and Fermion fields $\psi_c$ and now we want to consider ist Quantum fluctuations. These fluctuations arise from loop corrections,...
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What does that means? “QCD is a non-linear and non-trivial field theory?”

I know QCD is represented by the $SU(3)$ group and is non-abelian. Then, as a consequence QCD is a non-linear and non-trivial field theory. I would like to know why? and what does that means?
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Which of the Wightman axioms are not incorporated by four dimensional quantum Yang-Mills?

I am trying to understand the quantum Yang-Mills existence problem but the best I have seen so far is the statement that there is no known interacting relativist field theory in four dimensions which ...
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$SU(5)$ representation and higher anti-symmetric traces

In Zee QFT book v2 p.411 eq.16-17, he shows the SU(5) gauge theory anomaly cancellation by the following: The 1st line in fundamental of SU(5) $$ tr(T^3)=3(+2)^3+2(-3)^3=30, $$ is easy to follow, ...
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Why not regard all large gauge transformations as genuine ones?

A large gauge transformation is a gauge transformation that is not connected to the identity. When quantizing a gauge theory, we must take configurations related by ordinary gauge transformations to ...
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Updating link variables in lattice $SU(N)$ gauge theory

I'm currently writing a basic program in python to simulate a 1 + 1 dimensional yang mills gauge theory with symmetry group $SU(2)$. On the lattice you work with link variables, which are $SU(N)$ ...
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What justifies compactifying space and spacetime, in the context of instantons?

When studying Yang-Mills instantons, there are two instances where one compactifies a space. When classifying vacuum states, one demands $A_\mu(\mathbf{x})$ to become a constant as $\mathbf{x} \to \...
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A doubt in Yang-Mills procedure

My question is this: I saw the next relation in a Yang Mills theory paper: $$L_{i}P^{\mu \nu}_{i}=P^{\mu \nu}$$ With $L_{i}$ a generator of su(2) and for any $P^{\mu \nu}_{i}$. But I can't ...
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Proof of the expression of field strength in a non-abelian theory

I want to prove that the expression for the field strength in a non-abelian theory is \begin{equation} G_{μν}Ψ=[D_{μ},D_{ν}]Ψ=(∂_{μ}A_{ν}-∂_{ν}A_{μ}+[A_{μ},A_{ν}])Ψ \end{equation} where the covariant ...
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Is there a name for symmetry in which fermions and bosons are in identical adjoint representations?

In a Yang-Mills theory, with gauge group $G$, if the Fermions are in an adjoint representation then for every Fermion with "charge" $Q$ there is a boson with charge $Q$. i.e. there is no difference ...