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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Instantons in Minkowski spacetime? or only valid in Euclidean spacetime?

In the usual description of the instanton of nonabelian gauge theory in $D=4$ spacetime, we always (or just usually?) choose the $D=4$ Euclidean spacetime see for example https://en.wikipedia.org/wiki/...
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How to Wick rotate the Yang-Mills instanton winding number?

How to Wick rotate the instanton number of Yang-Mills theory? (Related to the earlier question Wick rotate the Yang-Mills $SU(N)$ gauge theory's field strength?) My question is particularly about ...
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Wick rotate the Yang-Mills $SU(N)$ gauge theory's field strength? [duplicate]

How do we Wick rotate the Yang-Mills $SU(N)$ gauge theory's field strength? Say in 3 space and 1 time dimensions? Suppose we start with a Lorentz signature with coordinates $(x_0, x_1, x_2, x_3)$, ...
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If having a finite action for Maxwell or Yang-Mills theory, $A_μ$ goes to a pure gauge configuration?

I had an empirical understanding that --- If we like to have a finite action for Maxwell or Yang-Mills theory, so that the field strengh $F_{μν}$ must go to zero at space-time infinity, meaning that $...
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Determinant of differential operator as exponential of Wess-Zumino-Witten action

I am currently reading this paper (Mass Gap and Confinement in (2+1)-Dimensional Yang-Mills Theory, Dimitra Karabali) and between equations (6) and (7) the following identity is used: \begin{equation*}...
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Fock-Schwinger gauge in pure Yang-Mills theory and coordinate dependence of equations of motion

Let $(\Omega^\bullet (\mathbb{R}^n,\mathfrak{g}),d_A)$ be the Yang-Mills cochain complex on $\mathbb{R}^n$, where $d_A$ is the gauge covariant derivative, $d_A \circ d_A=0$. I was wondering: if we ...
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Can Montonen-Olive duality be used for studying $\mathcal{N}=4$ SYM at strong coupling? If not, why not?

It's all in the title. To be more complete, the following is stated in the preamble of the Wikipedia article about S-duality: One of the earliest known examples of S-duality in quantum field theory ...
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Trace properties of the gauge potential in non-Abelian gauge theory

I want to proof equation 69.18 in Srednicki's book "Quantum field theory", which reads: \begin{equation} A_\mu^a(x)=2\text{Tr}[A_\mu(x)T^a].\tag{69.18} \end{equation} $A_\mu(x)$ is the non-...
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Equation of motion of the curvature form $F$ in Yang-Mills theory

Following 4.2.1 in this document (Muharrem Küskü, The Free Maxwell Field in Curved Spacetime, 2001), I tried to adapt the method used (in particular equations 4.21 and 4.32) to Yang-Mills theory ...
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Abelian theories with more than one charge

I have a question about the non-abelian character of QCD. In order to write a gauge-invariant Lagrangian, there must be a term with the strength tensor $X^{\mu\nu}_{a}X_{\mu\nu}^{a}$ where $$ X^a_{\mu\...
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Normalization relation of the generators of the Lie-algebra of the Yang-Mills gauge group

At the introduction to Yang-Mills-theory and its gauge group typically a $SU(N)$-group, the generators $t_A$ of the corresponding Lie-algebra are supposed to fulfill the following normalisation: $$Tr(...
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Higher form symmetries and Yang Mills

I have been reading about higher-form symmetries, particularly how they are applied to non-abelian gauge theories. I have come across the claim that pure $SU(N)$ Yang Mills (i.e. with no quarks) ...
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Why less gauge fixing conditions in Faddeev Popov method?

According to Dirac theory of constraints systems, to study the dynamics of gauge invariant observables, we can fix the gauge freedom by fixing it using gauge fixing conditions which are equal to ...
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Eqs of motion under dual field strength tensor

With a $SU(2)$-invariant Lagrangian of the form $$ {\cal L} = -\frac{1}{2g^2}tr\{F_{\mu\nu}F^{\mu\nu}\} - \frac{\theta}{2g^2}tr\{F_{\mu\nu}\widetilde{F}^{\mu\nu}\},\quad \widetilde{F}^{\mu\nu} = \frac{...
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Assumed form for beta function

In a classical paper on Hierarchy of Interactions in Unified Gauge Theories, Georgi et al define the renormalization group equation $$ \mu \frac{\partial g(\mu)}{\partial \mu } =\beta(g(\mu)). $$ He ...
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Does the structure constant of Yang-Mills field change sign under time reversal?

The time reversal of Abelian (electromagnetic) field strength is pretty straight forward. The electric field $F_{0i}$ is even under time reversion. The magnetic field $F_{ij}$ is odd under time ...
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Why does putting field theory on $S^4$ deform the scalar kinetic term?

I was reading the famous localization paper by Pestun, where he proves, among other things, that $\mathcal{N}=4$ super Yang-Mills (SYM) on $S^4$ localizes to a Gaussian matrix model. However, I was ...
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Can a gauge transformation eliminate singularity of gauge potential?

Suppose I have a gauge potential $A$ which goes to infinity at some point $x_0$. Can I use a gauge transformation \begin{equation} A'=U^{-1}AU+U^{-1}dU,~~~U=\exp\{-i\alpha^a(x)T^a\} \end{equation} to ...
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Slavnov-Taylor identities and the Ward identity

Suppose we have a vertex $\Gamma$ that satisfies the Slavnov-Taylor identity: $$ p^{\mu} q^{v} \Delta_{\sigma \lambda}^{\mathrm{tr}}(r) \Gamma_{\mu \nu \lambda}(p, q, r) =\frac{1}{\widetilde{Z}\left(p^...
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Positive Definiteness of Killing Form in Gauge Theory

This question is related to requirement that the gauge group of a gauge theory be a direct product of compact simple groups and $U(1)$ factors but is not the same as, for example, this question (...
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Misunderstanding the dimension of QCD

From my point of view, the definition of the tension tensor is contradictory. $$F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu} +ig[A_{\mu},A_{\nu}]$$ $$[A_{\mu}]=\frac{1}{cm \times g};[F_{\...
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How to compute gauge potential $A$ from the field strength $F$?

Let $F=dA+A \wedge A$ be the field strength that solves vaccum Yang-Mills equation. The question is: how to recover the gauge potential $A$? Is there any standard way? or any theorem stating the ...
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How is the Theta angle of $SU(5)$ grand unified theory related to the three Theta angles in the $U(1) \times SU(2) \times SU(3)$ standard model?

There are three Theta angles in the $U(1) \times SU(2) \times SU(3)$ standard model: call them $$U(1): \theta' F \tilde{F}$$ $$SU(2):\theta'' F \tilde{F}$$ $$SU(3):\theta''' F \tilde{F}$$ But there is ...
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How to decompose the spinor kinetic term of $D=10$ SYM in terms of lower dimensional parts?

The kinetic term for the spinors in $D=10$ SYM is $\lambda \gamma^\mu \partial_\mu \lambda$, where $\lambda$ is a 16 component Majorana-Weyl spinor and $\gamma^\mu$ is a 16 by 16 matrix satisfying $\{\...
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Understanding the QED Lagrangian using Yang Mills formalism

In QED the Lagrangian is $$ \mathcal{L} = \bar{\psi}(i \not \partial - m ) \psi - \frac{1}{4} F_{\mu \nu} F^{\mu \nu} - e \bar{ \psi} \gamma^\mu \psi A_\mu $$ which is the sum of a Dirac term, the ...
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What are the physical degrees of freedom in Yang-Mills theories?

I am pretty familiar with the Lagrangian formulation of quantum electrodynamics and perturbation theory techniques; however, I am hoping to move into QCD and other Yang-Mills Theories. As I do, I am ...
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Renormalization of the coupling constant in $\mathcal{N}=1$ SYM

I have been watching the lecture https://youtu.be/lrikIt9MXpQ from the school LACES 2020 by INFN. The $\mathcal{N}=1$ SYM is investigated, with the action: $$ \mathcal{L} = \frac{1}{32\pi} \mathrm{Im} ...
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One-loop exactness of self-dual Yang-Mills theory

The self-dual Yang-Mills theory (gauge group $G$) with the action: $$ \mathcal{S} = \int_{M} \text{Tr} (B^{+} \wedge F) $$ where $B^{+}$ is a self-dual field, transforming in the adjoint ...
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Black holes analogies in Yang-Mills theory

Theory of gravity is very similar to Yang-Mills theory. In General relativity there are black holes. Is some analogies of black holes in Yang-Mills theory?
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$SO(3)$-invariant Lagrangian and null kinetic term for gauge fields

Let's say we have a Yang-Mills $SO(3)$ theory coupled to a real scalar field $\phi$. Then the Lagrangian can be written as $$ {\cal L} = \frac{1}{2}(D_\mu \phi)^T D_\mu \phi + \mu^2 \phi^T \phi - \...
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Equations of motion involving terms with four vectors

So I am trying to find equations of motion for the Lagrangian associated with a non-Abelian Gauge theory for $SU(N)$, and while I was doing the math, I was a bit confused the indices. So I have $\...
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Different features of Gravity and Yang-Mills

I am reading a famous paper by S.Hawking - "Quantum gravity and path integrals" https://doi.org/10.1103/PhysRevD.18.1747. On the third page left column there is a statement, after the ...
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Derivative of a field strength tensor wrt field potential in YM gauge

I'm currently following this article to cosntruct a gauge invariant energy stress tensor for pure Yang-Mills gauge: $$ \mathcal{L} = -\frac{1}{4}F_{\mu\nu}^aF_{\mu\nu}^a, \qquad F_{\mu\nu}^a = \...
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Conservation law of colour current in Yang-Mills theories

In a Yang-Mills theory where the fermion fields transform under $\Psi \rightarrow e^{-\theta^A t_A} \Psi$ with $t_A$ generators of a Lie-algebra fulfilling $[t_A,t_B]=f^A_{BC}t_C$ a Noether current $...
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Most general Gauge Lie group in a Yang-Mills theory

Mathematicians have done a complete classification of all possible Lie groups. Is there a set of conditions that allows us to identify which Lie groups from the classification can possibly act as a ...
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Interchanging positions of Gell-Mann matrices with Dirac matrices, Pauli matrices

The anti-commutation relations for Gamma matrices $\big\{\gamma ^\mu , \gamma ^\nu \big\} = 2g ^{\mu \nu} $ can be used for interchanging positions of the respective matrices in a given expression, ...
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The planar limit, self-duality and their relation to two dimensions

In the lecture notes by Beisert on integrability, it is stated that integrability is a property mainly in two-dimensional field theories, with some higher-dimensional examples. As higher-dimensional ...
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Why does $A_\mu$ undergo an adjoint representation matrix transformation?

This question pertains to the following passage from Weinberg's second volume on QFT. It appears on page 4, section 15.1. To make the Lagrangian invariant, we need a field $A^\alpha_\mu$, whose ...
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Renormalization of composite operators in gauge theories

I am trying to understand, how composite operators in gauge theory are renormalized. In this paper the authors consider the renormalization of Yang-Mills stress-energy tensor: $$ \mathcal{O}_{\mu \nu}^...
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Deriving the Momentum Conjugate to $\mathbf{A}$ in Canonical Quantisation of Yang-Mills

In David Tong's lecture notes on Gauge Theory, in the section "Canonical Quantisation of Yang-Mills", the momentum conjugate to the field $\mathbf{A}$ is computed for the following ...
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Faddeev-Popov Gauge Fixing Procedure

I want to know that does $F^{a}[A_{\mu}] = 0$ condition used in Faddeev-Popov Quantization has unique solution $A_{\mu}$ or is it $F^{a}[A^{\theta}_{\mu}] = 0$ should have unique $\theta$ as solution ...
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Why Yang-Mills Potential Is A Linear Combination of Pauli Matrices?

What is meant in the following paragraph (this is Yang-Mills original 1954 paper "Conservation of Isotopic Spin and Isotopic Gauge Invariance") Where Equation (3) is the transformation of B ...
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Yang-Mills massive ghosts

Is there any procedure to add a mass to Faddeev-Popov Lagrangian density of a pure Yang-Mills theory, other than just add it from nowhere?
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Are the fermions in 4D Euclidean $\mathcal{N}=4$ SYM Majorana? If not, what are they?

It is stated in this paper (as well as in many other) that the fermions of $4$D Euclidean $\mathcal{N}=4$ Super Yang-Mills (SYM) are Majorana fermions (see eq. (42) and (43)). However it is stated in ...
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How does the Lorentz group fit into the Standard Model?

I'm trying to get a better sense of how the various group theory applications in physics fit together and I have some outstanding issues in my understanding: The gauge group of the standard model is $...
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Confusion about supersymmetric Ward identities for $\mathcal{N}=4$ super Yang-Mills theory

I'm trying to understand Eq. 2.6 in this paper. I understand the idea and derivation of the SUSY Ward identity itself and I know how to apply it in the $\mathcal{N}=1$ case. What confuses me here is ...
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Can Maxwell's equations be generalized to all fields?

For having studied both classical and quantum optics, I regard Maxwell's equations as the grand "cheat sheet" from which (almost) all optical/photonic phenomena can be derived. Yet, I also ...
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Center symmetry on wavefunction

Assume we know the wavefunctions of a SU(N) Chern-Simons (or YM) on a 3-mfld $M$, perhaps using holomorphic quantization. How do the center symmetry transformations act on the wavefunctions? Is it ...
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QCD energy scale $\Lambda_{\rm MS} $, $\Lambda_{\rm QCD}$, …?

Why there seems to be different conventions of QCD energy scales? Is that due to the running coupling? For example in Wikipedia https://en.wikipedia.org/wiki/Coupling_constant#QCD_scale: $$ \Lambda_{\...
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How do generators of the Lie algebra correspond to gauge fields?

I’m tackling physics recreationally from a pure math perspective. Right now I’m looking at just the outline of gauge theory. The Wikipedia article explains that gauge fields correspond to generators ...

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