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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Yang-Mills theories, confinement and chiral symmetry breaking

I was thinking about hadrons in general Yang-Mills theories and I have some doubts that I'd like to discuss with you. Suppose that we have a Yang-Mills theory that, like QCD, tend to bind quarks ...
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Reason for not using Higgs mechanism to solve mass gap problem

I am curious as to precisely why one can't introduce masses for gluons in Yang-Mills by a Higgs-type mechanism as in electroweak theory. Is it because then one would end up with an unwanted massless ...
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Question from Terning's book

In Chapter 7 of Terning's book (Modern Supersymmetry), the first example considered is that of an SO(3) gauge theory, a complex scalar in the triplet representation of SO(3) and a potential term: ...
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Nahm's theorem and related with SYM ;other theories

To study $S$-duality in more detail, i tried to read, Electric-magnetic duality and the Geometric Langlands Program. In section 2.1, there is a comment about 10d SYM. Excerpt from the paper above, ...
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Unitarity of the S-matrix and Feynman Diagrams

There are several questions on the unitarity of the S matrix, but unfortunately non of them answers directly the following question. The S matrix is unitary and that can be proven by the fact that ...
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What exactly is the “diagonal embedding” in the supersymmetric topological twist?

Consider $\mathcal{N}=2$ pure SYM theory. If we want to put the theory in a 4-manifold we take its topological twist. The global symmetry group $$G= SU(2)_{+} \times SU(2)_{-} \times SU(2)_I \times ...
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108 views

How can I understand instantons as sheaves?

In specific, instantons are considered or interpeted as torsion free coherent sheaves. Why is that the case? Is there a nice way to understand this relation and of course also understand how the two ...
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163 views

Gauge theory for mathematicians?

I'm looking for a textbook or set of lecture notes on gauge theory for mathematicians that assumes only minimal background in physics. I'd prefer a text that uses more sophisticated mathematical ...
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84 views

Completeness Relations of Polarization Vectors in QCD

What are the completeness relations of the polarization vectors of (external) particles in QCD amplitude calculation? (I assume the polarization vectors depend on the gauge and even so still have some ...
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92 views

Yang and Mills' (and others') justification for local gauge invariance

In most physics textbooks, local gauge invariance is simply postulated---you start with a global symmetry, e.g. the global phase, then allow it to depend on the spacetime point, make the necessary ...
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28 views

The BPS mass formula for 3d N=4

Is the mass formula (or equivalently the central charge formula) known for BPS particles in 3d $\mathcal{N}=4$ susy gauge theories? In particular, what is its dependence on the Fayet-Iliopoulos ...
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What are the global symmetries of $\mathcal{N}=2$ SYM before twisting and after twisting?

I am confused with the global symmetries of $\mathcal{N}=2$ SYM. On one hand I know that the theory has a $U(2)_R = SU(2)_R \times U(1)_R$ symmetry. Now, there exists also a $U(1)_B$ global symmetry. ...
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Vector potential in gauge transformation

While applying Gauge transformation, $\psi\prime = U \psi$ , where $ U= e^{i q \lambda(x)}$ , transformation law for "Vector Potential" comes out to be : $$ A_{\mu}\prime= ...
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Are correlators constructed out of Wilson loops singular in pure Yang-Mills?

If I have some gauge invariant function of two Wilson loops (such as $\left<\text{Tr}W_1 \text{Tr}W_2\right>$) does the expectation value diverge when the loops coincide the same way ...
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215 views

Einstein-Yang-Mills Connections

I am playing around with coupling a classical $SU(2)$ Yang-Mills theory to Einstein's equations. Assuming spherical symmetry, the $SU(2)$ connection can be written \begin{equation} A = ...
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Is it true that particles propagate on geodesics in Yang-Mills theory?

I mean free particles. Sorry for the inaccurate wording, I'm new in this field of physics.
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How are quadruple gluon vertices related to $SU(2)$ and $SU(3)$?

I once read that the non-commutativity of the Lie Groups $SU(2)$ and $SU(3)$ is the reason that the weak and strong interactions have Feynman diagrams with quadruple vertices, where four gauge bosons ...
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How does the Higgs field relate to the Yang-Mills fields and gauge theories in general?

I asked this in astronomy How does the Higgs field relate to the Yang-Mills fields and gauge theories in general? but they suggested I ask it here. It is very confusing. Is there an easy ...
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230 views

Invariance of supersymmetric Yang-Mills theory under supersymmetry

I was following Brink, Scherk and Schwartz, "Supersymmetric Yang-Mills theories". The variation of the Lagrangian w.r.t a supersymmetry transformation can be reduced to $$ \delta L = -igf_{a b c} ...
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82 views

Why is QCD hard to solve if I know the beta functions?

Why is it still hard to solve QCD if we know the beta functions of the coupling? Aren't only the loops causing problems? And am I not able to write every possible interaction exact at tree-level with ...
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Why is $U(1)$ special when defining global charges?

For gauge groups like $SU(2)$ and $SU(3)$ etc. we know that observable states such as mesons or baryons must be charge neutral. However, for a $U(1)$ gauge group we can have charged initial states in ...
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88 views

Intro to Super Yang Mills theory

I'm looking to start learning Super Yang Mills theory. Currently I have studied Peskin and Schroeder up to the Renormalization Group, but don't know supersymmetry yet. I know some Conformal Field ...
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166 views

Geometric interpretation of quantum Yang-Mills field

In most books\articles review geometric interpretation of classical Yang-Mills field in terms of principal bundle, connections...etc. What are geometric interpretation of quantum Yang-Mills field? ...
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163 views

Commutator of Gauge Covariant derivatives

What is the physical meaning of $$ [D_{\mu}, D_{\nu}] ~\propto~ F_{\mu, \nu}, $$ where $D_{\mu}$ is the gauge covariant derivative and $F_{\mu,\nu}$ is the field strength? Is it just a definition? ...
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Translational versus dilatational zero modes?

Why are the zero modes of the SU(2) Yang Mills instanton referred to as translational or dilatational zero modes? Is this standard terminology?
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Clarification: Why the gauge symmetry of pure Yang-Mills is $PU(n)$ and not $SU(n)$? [closed]

I am quoting the following from the Wikipedia article on the projective unitary group: In the pure Yang–Mills $SU(n)$ gauge theory, which is a gauge theory with only gluons and no fundamental ...
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177 views

How to compute this loop integral? [closed]

I have a gauge boson that splits into two scalars and the loop is closed by a gauge boson as shown in the picture. The incoming boson has $\mu$ index while the boson that runs in the loop has momentum ...
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Questions regarding $D=4 $ ${\cal N}=4$ supersymmetric Yang-Mills

I have some questions regarding the $D=4 $ ${\cal N}=4$ super-Yang-Mills theory (the one with a really long action which can be acquired by compactifying the 10-dimensional ${\cal N}=1$ theory). I ...
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90 views

Field renormalization of scalar Yang-Mills

In most books, one can find the field renormalization $Z_3$ in Yang-Mills with fermionic matter in the fundamental. In the $\overline{MS}$ scheme, tt is given by $$ Z_3 = 1 + \frac{g^2}{16\pi^2 ...
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63 views

Is there a well-defined partition function of 4d Yang-Mills?

So I've looked everywhere to find a resource on 4 dimensional Yang-Mills partition functions, but have only managed to find examples using supersymmetry. Is there a resource describing the partition ...
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61 views

What makes Lattice Yang-Mills hard?

I've been reading up on non-perturbative Yang-Mills, and have found the following equation: $$Z[\gamma, g^2, G]=\int \! \prod e^{-S}\mathrm{d}U_i$$ Now I don't know much about computational physics, ...
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156 views

What are Maximally Helicity Violating (MHV) Amplitudes?

Definition of MHV amplitudes on Wikipedia: In theoretical particle physics, maximally helicity violating amplitudes are amplitudes with n external gauge bosons, where n-2 gauge bosons have a ...
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90 views

Calculation of the Poisson bracket of a (Classical) Yang-Mills generator

This question might be too technical or minute, but I believe someone can give me the right advise. What I want to calculate is a Poisson bracket algebra of classical YM gauge generators, ...
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67 views

Deconfinement at high T $\leftrightarrow$ spontaneous breaking of the center of the gauge group

I am reading Witten's "Anti-de Sitter Space, Thermal Phase Transitions, And Confinement In Gauge Theories" (see here), in which he connects the confinement-deconfinement transition in $\mathcal{N}=4$ ...
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145 views

Yang-Mills constraints and Poisson brackets

Let's have constraints for Yang-Mills theory: $$ \varphi_{a} = \partial_{i}\pi^{i}_{a} - f_{abc}\pi^{b}_{i}A^{c}_{i}. $$ I have read the statement that $$ \tag 1 [\varphi_{a}(\mathbf x), ...
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243 views

Are the Yang-Mills equation and its generalization gauge invariant?

I have derived the Yang-Mills equation and its generalization coupled to a current of a scalar field $\phi$ by extremalizing the action describing a $\mathrm{SU}(2)$ scalar field gauge theory: ...
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Why are non-Abelian gauge theories Lorentz invariant quantum mechanically?

I seem to be missing something regarding why Yang-Mills theories are Lorentz invariant quantum mechanically. Start by considering QED. If we just study the physics of a massless $U(1)$ gauge field ...
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103 views

A Variation on Laplace's equation (context: Yang-Mills N-Instantons, Rajaraman's book)

Statement of the problem I need to solve the equation \begin{align} 0 = \frac{1}{\phi} \partial_{\sigma}\partial_{\sigma} \phi \hspace{20mm} (1) \end{align} where $\phi$ is a scalar field and ...
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Quantum Yang-Mills Theory and AdS/CFT

I just read the first chapter of Becker-Becker-Schwarz. To quote: A remarkable discovery made in the late 1990s is the exact equivalence (or duality) of conformally invariant quantum field ...
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Is there a method which quantizes non-abelian gauge theories without path integrals formalism?

In the most QFT books there is a method of quantization of non-abelian theories through path integral methods. But I want to learn also the other methods without using of this formalism. Does anyone ...
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Mass term in the Lagrangian

I have read that the mass term appearing in the electroweak Lagrangian stops it (the Lagrangian) from becoming gauge invariance. Can someone explain where and why this term is creating the problem?
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On self-duality of N=4 super Yang Mills theories

I am looking at S-duality a bit, and was wondering if anyone had the answer to the following question. It is known that the supersymmetric Yang-Mills theory in 4 space dimensions is self-dual with ...
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How to show isotropy of $SU(2)$ Yang Mills stress energy tensor?

When I vary the action of the YM Lagrangian density $$L = -\frac{1}{4} F^a_{\mu \nu}F^{\mu \nu}_a + J_a^\mu A^a_\mu$$ with respect to the metric, I obtain: $$T_{\mu \nu} = \frac{-2}{\sqrt{|g|}} ...
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76 views

Yang Mills theory and SU(N) groups [duplicate]

Trying to get a better understanding of the relation between a SU(N) Yang Mill theory and its number of "color" space. Most of the description I've found so far are either way to complex/specific. ...
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279 views

Lagrangian depends on second derivative of field

In case of the gauge-fixed Faddeev-Popov Lagrangian: $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}\,^{a}F^{\mu\nu ...
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Yang-Mills Lagrangian invariant under BRST

In equation 16.47 in Peskin & Schroeder, it is claimed that $$ -\frac{1}{2}g^2f^{abc}f^{cde}\left(A_{\mu}\,^{b}c^{d}c^{e}+A_{\mu}\,^{d}c^{e}c^{b}+A_{\mu}\,^{e}c^{b}c^{d}\right) ~=~ 0 \tag{16.47}$$ ...
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Gauge Invariance of Yang Mills Lagrangian

I am trying to show the invariance of the following Yang Mills Lagrangian: $$L= -\frac{1}{4} F^a_{\mu \nu} F_a^{\mu\nu} + J_a^\mu A_\mu^a$$ under the following gauge transformation ($\theta$ being a ...
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Deriving field equation in Yang Mills theory

Trying to show that $$D_\mu\vec{F^{\mu \nu}} = \partial_{\mu}\vec{F^{\mu \nu}} + g \vec{A_\mu} \times \vec{F^{\mu \nu}} = 4 \pi \vec{J^\nu},$$ or (correct me if I'm wrong) $$ \partial_{\mu} F^{\mu ...
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Perturbative vs. non-perturbative approaches to a well-defined Yang-Mills theory in 4 dimensions

Another question regarding the Yang-Mills Existence and Mass Gap problem (http://www.claymath.org/sites/default/files/yangmills.pdf). Does the problem require that the "construction" of a four ...
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Rigorous QFT on a Torus

The problem description for the Yang-Mills Existence and Mass Gap problem (http://www.claymath.org/sites/default/files/yangmills.pdf) says in its "Mathematical Perspective" section that Some ...