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.

Filter by
Sorted by
Tagged with
1
vote
2answers
53 views

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 ...
2
votes
0answers
14 views

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 ...
2
votes
2answers
105 views

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 ...
3
votes
1answer
50 views

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 ...
6
votes
1answer
141 views

2-dimensional QFTs invariant under area-preserving diffeomorphisms

In introductory textbooks & lecture notes on conformal field theory, it is usually stated that solving the highly nontrivial dimensional quantum field theory in 2 dimensions is possible due to the ...
1
vote
1answer
75 views

Yang-Mills Action for Non-Trivial Bundle

Suppose we have a principal $G$ bundle $(P,M,π)$ where $M$ is a 4-dimenational manifold and $G$ a Lie group (and $\mathfrak{g}$ its Lie algebra).The Yang Mills action is a functional of the gauge ...
6
votes
1answer
152 views

$CP$ Invariance of Yang-Mills Vacua in Electroweak Theory

It is well know that quantum Yang-Mills theory has a periodic vacuum structure. Consider electroweak theory. For a single generation of fermions, the theory is CP invariant. I would like to know if ...
3
votes
1answer
76 views

BRST quantization: Explicit computation request

Following Green, Schwarz and Witten's book on Superstrings, the BRST charge is given by $$Q_B = c^i K_i-\frac{1}{2}f_{ij}^{~~~k}c^ic^jb_k\tag{3.2.4}$$ with $$[K_i, K_j] = f_{ij}^{~~~k}K_k,\tag{3.2.1}...
0
votes
0answers
17 views

BPS Wilson loop operators and supersymmetries

In recent papers the circular Wilson loop in $\mathcal{N}=4$ SYM is always called a 1/2 BPS operator. So, my initial idea was that a 1/2-BPS operator was an operator that preserves half of the ...
1
vote
1answer
34 views

Abelian and non-Abelian holonomies

I read the article Geometric Manipulation of Trapped Ions for Quantum Computation, and it mentioned “Abelian and non-Abelian geometric operations (holonomies)”. I know what is holonomy, and what is ...
1
vote
1answer
51 views

The relation between Chern-Simons Theory and Yang-Mills Theory

So from this page, I know that there is a relation between Chern-Simons Theory and Yang-Mills Theory, but I have difficulty proving the identities in the document. I was going to prove $$\partial_\...
3
votes
2answers
493 views

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 ...
0
votes
0answers
43 views

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}=\...
3
votes
1answer
61 views

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 ...
0
votes
0answers
35 views

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 ...
1
vote
0answers
24 views

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 ...
0
votes
0answers
38 views

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+...
1
vote
0answers
56 views

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? ...
0
votes
0answers
47 views

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 ...
1
vote
1answer
48 views

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 ...
2
votes
4answers
136 views

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. $d_A A=F$ --- @...
3
votes
1answer
65 views

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 ...
0
votes
0answers
27 views

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 ...
1
vote
1answer
27 views

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 ...
1
vote
0answers
33 views

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 ...
0
votes
1answer
80 views

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:...
1
vote
0answers
36 views

$* 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 ...
0
votes
0answers
28 views

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 ...
0
votes
0answers
39 views

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 $$ d * F = * J_e $$...
2
votes
1answer
87 views

$U(N)$ & $SU(N)$ : 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 $SU(3)$ ...
0
votes
1answer
54 views

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 + [...
2
votes
1answer
61 views

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 ...
0
votes
0answers
87 views

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 ...
3
votes
1answer
231 views

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 ...
1
vote
1answer
43 views

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{...
0
votes
1answer
40 views

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 ...
2
votes
0answers
32 views

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 ...
5
votes
1answer
199 views

“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: $...
3
votes
1answer
63 views

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 ...
3
votes
1answer
142 views

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 ...
0
votes
2answers
84 views

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 ...
1
vote
1answer
57 views

“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 ...
3
votes
0answers
110 views

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 ...
0
votes
0answers
22 views

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 ...
3
votes
0answers
108 views

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 ...
0
votes
1answer
122 views

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,...
3
votes
2answers
134 views

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 ...
0
votes
0answers
109 views

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 $W(x)$ ...
1
vote
0answers
24 views

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?
2
votes
2answers
189 views

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 ...