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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Integrating out massive degrees of freedom of Super Yang-Mills Action

From this paper, I want to integrate out the massive degrees of freedom. The total action $S$ is given by $$S = S_{Y} + S_{A} + S_{Fermi} + S_{ghost} $$ where the terms are given in equations (2.9),(...
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Yang-Mills connections for Palatini $f(R)$ Theory

It is known that for some modified theories of gravity such as Palatini $f(R)$ Theory, we have two kinds of the connections: Affine connection and the Christoffel symbol, which are related to each ...
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Equations of motion of classical chromodynamics with Yang-Mills theory

I am currently reading a paper about classical chromodynamics: https://arxiv.org/abs/hep-th/0607203 However I have problems understanding equation (2) and (4) (2): \begin{equation} F_{\mu \nu}= \...
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Supersymmetric localisation of 2D super YM on $S^2$

I wanna to understand how to calculate partition function for pure abelian Yang-Mills theory. To do this, I need follow some usual step's (I follow Benini, Localization in supersymmetric field ...
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Why is the standard model gauge group $SU(3) \times SU(2) \times U(1)$ and not $U(3) \times U(2) \times U(1)$?

The bilinears in the Lagrangian are invariant under the full group. Where does the restriction to unit determinant come from?
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Consistency condition for Yang-Mills on a Torus

So I was recently studying 't Hooft's paper on self-dual solutions to Yang-Mills on $\mathbb{T}^4$. So the basic idea is that you consider a box with periodic boundary conditions and then you impose ...
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Why are magnetic monopoles hard to find (if exist)?

I understand the Yang-Mill perspective of $U(1)$-gauge theory. In that, you can easily write down the field of a Dirac magnetic monopole. What interests me is the fact that it's so hard to find (if ...
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Finding the Feynman rules and invariant amplitude of a non-abelian gauge theory [closed]

I am having trouble with finding the invariant amplitude from a given Lagrangian. I am given a Lagrangian of a non-abilian gauge field and a complex scalar field. It is given by, $$ \mathscr{L}=\...
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What are the status and the latest papers on Yang-Mills existence and the mass gap?

The Clay Millennium problem is still open. What is the recent status? Is there anybody publishing about it? On http://www.claymath.org/millennium-problems/yang%E2%80%93mills-and-mass-gap the last ...
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Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates

In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented. Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...
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Is it possible to derive a non-mass gap from the Yang-Mills action? [duplicate]

The action: $$J=∫Tr(F∧⋆F).$$ Is it possible to find the non-mass gap property from the Yang mills action? If so, how?
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What is quark 'confinement' in the context of Yang-Mills Theory?

In the context of Yang Mills Theory, what is quark confinement? Please try to explain as simple as possible (obviously without being too general)
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Why hasn't Peter Higgs solved the Yang-Mills prize problem?

The Higgs mechanism gets rid of the mass gap problem, and it's been experimentally proven, so why is there still a problem? Why are the million dollars still up for grabs?
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Why does Yang Mills Theory only work for massless particles? [duplicate]

Why does the non-abelian element of the Yang Mills theory (SU(3), SU(2), etc.), inherently imply a 'non mass gap' and long range forces? Please explain as simple as possible, I haven't seen a clear ...
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Physical content of $[D_\mu, D_\nu]=ieQF_{\mu\nu}$ and $[D_\mu, D_\nu]=igT^a G^a_{\mu\nu}$

For the abelian QED theory, $$[D_\mu, D_\nu]=ieQF_{\mu\nu}$$ where $D_\mu=\partial_\mu+ieQA_\mu$ is the gauge covariant derivative in QED and $F_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu$ is ...
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Missing factor in Dimensionally reduced Yang Mills Ghost Field

I'm trying to calculate the ghost field in the background field gauge for the dimensionally reduced Yang Mills action in this paper. I am using the expression from Srednicki's book, chapter 78. The ...
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Why is the self-energy for quarks in $d=2$ Large $N$ QCD only order $g^2$?

In an interesting article by 't Hooft , he is able to find the exact quark propagator, in the large $N$ limit of QCD. He finds that the full 1PI self-energy is given by: $$\Gamma(p)=-\frac{g^2}{2\pi} \...
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‘t Hooft loops, yang mills, and “free energy”

When reading through the N=4 SYM s-duality literature, one will encounter some interesting properties i.e. twists (on boundary conditions), non-trivial electric and magnetic fluxes, the so called “...
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Besides instantons and large-$N$ what are some other general non-perturbative methods for quantum field theory?

Besides large-$N$, instantons, lattice QFT, what are some other non-perturbative methods that help us better understand QFTs like the large distance dynamics of Yang Mills and QCD?
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What are the $A_{\mu}{}^a$ fields in Yang-Mills theory?

At some point of the demonstration of Yang Mills theory we assume an ansatz that $A_{\mu}=t^a A_{\mu}{}^a$ where $a=1, \ldots,n^2-1$ and the $t^a$ are the generators of the $SU(n)$ symmetry in order ...
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What happens when somebody solved mass gap problem like ultraviolet catastrophe? [closed]

Just curious if somebody came along and proposed a good mathematical solution to the millennium problem but it is not intuitive at first, what happens then? this person can rigorously proof the ...
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When one says that Yang Mills theory is a non-abelian field theory, what specific gauges and gauge transformations are implied in this statement?

What specific gauges and gauge transformations are implied when one states that the order of such gauge groups are vital? Can this please be explained as simple as possible (:
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Finding ghost terms for super Yang-Mills theory in background field gauge

I want to find the ghost terms (2.16) for the action in this paper. The gauge field action is given by $$ \begin{align}S_{A} =& i \int d\tau \Big(\frac{1}{2}A_{1}(\partial_{\tau}^2 - r^2)A_{1} + \...
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Commutator of covariant derivatives to get the curvature/field strength

For notation and convention, please see Gauge theory formalism and Generalizing the covariant derivate for gauge theory. The covariant derivative can be used to construct curvatures (called field ...
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Intuition about ADHM construction

I'm trying to understand reasons, why self-dual Yang-Mills equation can be reduced to algebraic equations. It's seem like a miracle. In article Construction of Instanton and Monopole Solutions and ...
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What is path ordering

Consider the Wilson loop for Yang-Mills theories $$ W_C=P\exp\left(ig \oint_C A_{\mu}^a(z)T^adz^{\mu} \right) $$ where $A_{\mu}^a$ are the gauge fields, $T^a$ are the generators and $g$ is the gauge ...
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Understanding the prefactor $\frac{\theta g^2}{32\pi^2}$ of the $F\tilde{F}$ term in Yang-Mills theories

The most general Yang-Mills (YM) action consistent with Lorentz invariance, gauge invariance and renormalizability should contain a term $$\kappa F_{\mu\nu a}\tilde{F}^{\mu\nu a}\tag{1}$$ where $\...
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How can we tell a theory is confining?

Physically, I understand what it means for a theory to be confining. The elementary particles are not observable, but only composite particles are. The classic example is QCD, where quarks are ...
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A basic question about how to apply the gauge covariant derivative in Yang-Mills theory

I am sorry if this question is too stupid... We know that Yang-Mills equation (without source) can be written as $$D^\mu F_{\mu\nu}=0,\tag{1}$$ where $$D^{\mu}=\partial^\mu-ig A^{\mu}$$ and $$A^\mu=A^...
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How to prove that these integrals are the same with a Fourier transform?

In the paper BMN Correlators and Operator Mixing in $\mathcal{N}$=4 Super Yang-Mills Theory, they claim in Appendix $A.2$ (p.26, eq. $(A.7)$) that the following relation holds: $$F_{12,34} = \frac{(\...
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Are Yang-Mills Fields sections of associated bundles to the orthonormal frame bundle?

Let $\pi: P \to M$ be a principal bundle and $\omega$ a connection on it. Given a section $\sigma: M \to P$ we define Yang-Mills fields by $$A=\sigma^*\omega$$ Now since under Lorentz ...
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What is the way to express Yang-Mills symmetry groups without gauges?

Given a Yang-Mills theory such as $SU(3)$ which has 8 gluons. After we gauge-fix this theory, it no longer has $SU(3)$ guage symmetry. Yet, we still use the group constants and the 8 types of gluons ...
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Comparator operator in QFT

In Peskin and Shroeder, for a local $U(1)$ transformation, the comparator operator is expanded as: \begin{equation} U(x+\epsilon n, x) = 1 -ie\epsilon n^{\mu}A_{\mu} + \mathcal{O}(\epsilon^2) \tag{15....
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Intuition for Asymptotic Freedom

In QED, the $\beta$-function has a positive sign. This means that the coupling increases at higher energies, or equivalently, smaller length scales. This picture is made intuitively clear by the ...
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Difference between sign conventions in the action of $\mathcal{N}=4$ SYM

In the paper called Wilson Loops in N=4 Supersymmetric Yang-Mills Theory, the authors define the action for the $\mathcal{N}=4$ Supersymmetric Yang-Mills (SYM) theory including the following term: $$...
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Are there winding-number vacua in Weinberg-Salam (Or are they a gauge artifact)?

In pure SU(2) Yang-Mills the vacua van be grouped in homotopy classes labeled by their winding number. Instantons connect these giving rise to the theta-vacuum. I’m studying the SU(2) sphaleron in ...
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Yang-Mills Feynman rules

Good morning/evening. In Peskin & Schroeder chapter 16 on gauge invariance, the gauge boson self interaction vertex rules are given. For three gauge bosons, the relevant interaction term in the ...
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How to prove that a given correlation function is protected?

I would be interested in proving that $2$-point functions made of $1/2$-BPS operators are protected in $\mathcal{N}=4$ SYM (Supersymmetric Yang-Mills), i.e. that the correlator $\langle \mathcal{O}_2(...
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Question about the Lie group $SU(3) \times SU(2) \times U(1)$ and the concept of manifold

I don't know if this question is a duplicate, so I'll delete if is. Well, I'm in the very beginning of the study of contemporary topics such as gauge theories, I would say that I'm in a "science ...
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Does gauge invariance of scalars/fermions in the adjoint representation induce the existence of Wilson loop and (then) covariant derivative?

First, for the known case of $U(N)$ gauge invariance we have scalars (it works for fermions too) transforming as (fundamental representation) $$ \phi(x)\to V(x)\phi(x), \ \ V(x)\in U(N) $$ So then we ...
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How I can get the numerical factor in the relation between string coupling and YM coupling?

I'm trying to understand some references about Wilson loops being used to test AdS/CFT. Some of them are Nadav Drukker, David J. Gross: An Exact Prediction of N=4 SUSYM Theory for String Theory ...
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Feynman rules of $\mathcal{N}=4$ supersymmetric Yang-Mills in Euclidean space

I am trying to derive the Feynman rules for $\mathcal{N}=4$ supersymmetric Yang-Mills. The (Euclidean) action that I start with comes from this paper (Wilson Loops in $\mathcal{N}=4$ Supersymmetric ...
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Transformation of Yang-Mills fields

I am trying to recover the transformation properties of a Yang-Mills field and I'm not sure if I am wrong or if I am misunderstanding what is meant by a Yang-Mills field. Suppose I had a principle $G$...
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Does the temporal gauge condition uniquely determine a gauge in case of non-Abelian gauge theory?

For a $U(1)$-gauge theory, we can fix $A_0 = 0$ by choosing a temporal gauge. Can we do the same for all of the gauge components of the $SU(2)$ gauge field, i.e., $W^a_0 = 0$ for $a \in \{1,2,3\}$? ...
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What happens if I Wick contract a trace operator internally?

In theories such as $\cal{N}=4$ supersymmetric Yang-Mills, we often consider operators such as $\cal{O}(x_1)=$Tr$(\phi(x_1)\phi(x_1))$, with $\phi$ the scalar field(s) of the theory. Then we go on ...
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Supercurrent conservation for super-Yang-Mills in D=3,4,6,10 dimensions

I am following the book by Freedman and Van-Proeyen and this question is related to exercise 6.3. The supercurrent of a super Yang-Mills theory is given by $\mathcal{J}^{\mu} = \gamma^{\nu \rho} F^...
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How does the underlying symmetry of QCD imply the allowance of a 4-gluon vertex?

Quantum chromodynamics allows for a four-gluon vertex such as this, in a diagram Such a vertex would never be allowed in quantum electrodynamics, which has an underlying U(1) gauge symmetry. I know ...
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Time reversal symmetry of the Faddeev-Popov determinant

I am studying the Faddeev-Popov procedure to quantize a non-Abelian gauge theory, and I got confused by the status of the time reversal symmetry. People have different definitions of the time reversal ...
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2answers
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Yang-Mills Bianchi identity in tensor notation vs form notation

I've seen the Yang-Mills Bianchi identity written as both $$0 = dF^a + f^{abc} A^b \wedge F^c$$ and, in tensor notation, as $$\epsilon^{\mu\nu\lambda\sigma}D_{\nu} F^a_{\lambda\sigma} = 0.$$ Here ...
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Fierz identity for symplectic group

For the fundamental representation of $SU(N)$, there is a Fierz identity: $$ \sum_iT^i_{ab}T^i_{cd}=\frac{1}{2}\left(\delta_{ad}\delta_{bc}-\frac{1}{N}\delta_{ab}\delta_{cd}\right) $$ where $T^i$ is ...

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