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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Why Is Abelian Gauge Theory So Special?

I have a perhaps stupid question about Maxwell equations. Let $G$ be a generic Lie group. We consider a $G$-gauge theory. Let $A$ be the associated connection $1$-form, and $F=dA+A\wedge A$ be the ...
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Global Part of Non-Abelian Gauge Transformation

I have a perhaps stupid question about Noether's theorem. In Abelian gauge theory, say $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\Psi}(iD\!\!\!\!/-m)\Psi, \tag{1.0} $$ where $D_{\mu}=\...
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A Question about Yang-Mills Equation

The non-homogeneous part of the Yang-Mills equations is given by $$D\star F=\star J,$$ where $D=d+A$ is the covariant derivative, $\star$ is the Hodge star and $J$ is the source current. Under a ...
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Is Brandt-Neri-Coleman stability analysis valid?

My question is related to the problem of stability of magnetic monopoles in Yang-Mills-Higgs theories. I have read "The Magnetic Monopole 50 years later" from Coleman and, in section 3.5, he discusses ...
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Supersymmetry Generator Definition for ${\cal N }= 1$

I am studying SYM $\mathcal{N}$ = 1 in D = 10, and using the bimodular representations for the 32x32 gamma matrices $\Gamma^a$. This means that I work with the off-diagonal 16x16 matrices, which I ...
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In what sense $Z_\mu^0$ is orthogonal to $A_\mu$?

I am reading Standard model. Please explain in what sense the $Z$-boson $$Z_\mu^0=(g^2+g^{\prime 2})^{-1/2}(g A^3_\mu-g^\prime B_\mu)$$ is an orthogonal linear combination of the photon $$A_\mu=(g^2+...
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Is it possible to have a complex gauge field?

Beyond the obvious fact that the particles in the standard model described by gauge fields do not have an anti-particle pair, is there a reason why a complex gauge field is typically not considered? ...
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A specific derivation of Yang-Mills equations of motion

I am not happy about the derivation of Yang-Mills equations of motion (YM eom) given here @Prahar https://physics.stackexchange.com/a/312681/42982: @Prahar said: Yang-Mills action is $$ S = \int ...
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Winding number in 4D & $SU(2)$ group

In the book Quantum field theory by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such ...
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Inconsistency between $d_A = d + A \wedge$ and $d_A = d(..) + [A,..]$?

I am confused by something basic stated in this https://physics.stackexchange.com/a/429947/42982 by @ACuriousMind and some fact I knew of. Here $d_A$ is covariant derivative. $d_A A=F$ --- @...
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A question on supersymmetry variation of the Wilson loop in $\mathcal{N}=4$ SYM

The Wilson loop in $\mathcal{N}=4$ SYM is $$W=\frac{1}{N}tr P \exp \int ds (i A_\mu(x) \dot{x}^\mu+\Phi_i(x)\theta^i|\dot{x}|).\tag{2.3}$$ In order to check whether this operator is supersymmetric I ...
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Legal values of spin-1 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, ..?

For the spin-1/ boson field $A_\mu$, we may choose it to be a vector which needs to be real $\mathbb{R}$ usually for photon field. The field strength $F= dA$ is also real. Same for the nonabelian ...
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Reference request for Gribov ambiguity

I was hoping to find a reference (book or article) with a good introduction to the Gribov Ambiguity in non-abelian gauge theories. I’ve looked through QFT books by Schwartz and Srednicki, Rubakov’s ...
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Gribov's phenomenon

In the well known textbook by Itzykson-Zuber "Quantum Field Theory" there is a discussion of the Gribov phenomenon in non-abelian gauge theories (see Section 12-2-1). To my taste, the discussion ...
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Gauge invariance on Yang-Mills Lagrangian

How do I verify the invariance on Yang-Mills' Lagrangian: $$L = -\frac{1}{4} \sum_{a} \left(\partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A_\mu^bA_\nu^c \right)^2$$ under the transformation:...
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$* d * $ operator — Digest the (differential/geometry) meaning

I like to digest better: the $* d * $ operator in Maxwell differential form equation the $* D * $ operator in Yang-Mills differential form equation We already knew that in Maxwell differential ...
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Simplify Yang-Mills Equation of Motion in the 1-form gauge field $A$

We know the Yang-Mills theory Equation of Motion (eom) without source $$ * D * F = * (d (* F ) + [A, (* F )])= 0. $$ My question is that what are the most simple form we can boil down this ...
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Classical Yang-Mills equation of motion with both electric and magnetic sources?

We know the classical Maxwell equation of motion (eom) with both electric and magnetic source can be written as: (1) Explicit form or more schematically as: (2) Differential form $$ d * F = * J_e $$...
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$U(N)$ & $SU(N)$ : What's the conceptual difference in Gauge Theory?

I know the mathematical difference that one means $ absolutevalue(det) = 1$ and one means det = 1 (rotation) and that ones the subgroup of the other and so on. But: has a local/gauged $SU(3)$ ...
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Mistake or Rewriting of Yang-Mills in Nakahara

I am familiar with Yang-Mills equation of motion E.O.M. (without matter or source fields) in differential form. $$ D * F =0 $$ and Bianchi identity $$ D F=0 $$ where $F= dA + A \wedge A$ and $D=d + [...
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Questions about BRST symmetry [closed]

For a course about the standard model, I am writing a paper on BRST symmetry. For this I am mainly following the material developed in chapter 16.4 of Peskin and Schroeder. I am mostly done, however ...
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Why aren't gravitons spin 1?

Expressing the metric as $g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$, assuming $h_{mu \nu} \ll 1$ we can write the Einstein Hilbert action to leading order in $h_{\mu \nu}$ and quantize the ...
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Relating the Yang-Mills field-strength to the Maxwell tensor in $SU(2)$ gauge theory

I'm studying topological monopoles in a $SU(2)$ Yang-Mills theory with spontaneous symmetry breaking, through the book "Topological Solitons", by Manton and Sutcliffe. In section 8.2, the authors ...
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Gauge group of Electroweak theory

I am doing a question that asks me to identify the gauge groups of a Lagrangian with the field strength tensors $$\bf{F}_{\mu \nu} = \partial_{\mu}\bf{W}_{\nu} - \partial_{\nu} \bf{W}_{\mu} - g\bf{...
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Asymptotically free/flat

What does the expression: "...the theory becomes asymptotically free/conformal" mean? If it means that the spacetime $M$ on which the fields are defined is e invariant under conformal ...
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Fermionic contribution to central charge in $\mathcal{N}=2$ Super Yang-Mills?

I am trying to replicate the calculation of the central charge for $\mathcal{N}=2$ Super Yang-Mills, by following Weinberg's textbook in section 27.9. He calculates it by finding how one supercurrent ...
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“Hidden” theta-term in Hamiltonian formulation of Yang-Mills theory

I've read in David Tong's lecture notes on gauge theory that the Hamiltonian of Yang-Mills theory does not depend on the angular parameter $\theta$, because it can be absorbed in the electric field: $...
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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}...