All Questions
Tagged with quantum-chromodynamics renormalization
76
questions
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Asymptotic Freedom QCD
I'm trying to understand the derivation of asymptotic freedom with the renormalisation group equations. I'm reading Taizo Muta's book on QCD. What I don't understand is how he obtains the last ...
- 15
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0
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35
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Evidence of more generations in the QCD beta function
We know that the beta function for QCD is
$$
\beta = -\left(11 - \dfrac{2N_f}{3}\right),
$$
where $N_f$ is the number of fermions in the theory. We have $\beta_{\text{SM}} = -7$.
Now, my question is, ...
- 1,611
2
votes
1
answer
105
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How does the on-shell (OS) scheme work if we assume mass to be zero?
When calculating the self-energy correction of a massless quark up to one loop, I get
$$i\Sigma(p)=i\frac{\alpha_s}{4\pi}C_F/\!\!\!{p}\left[\frac{1}{\varepsilon_{\text{UV}}}-\gamma+\ln(4\pi)+1+\ln(\...
- 172
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1
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Possible cases of matter fields for $SU(2)$ theory which retains asymptotic freedom?
Let us assume $4$ spacetime dimensions.
QCD, the $SU(3)$ gauge theory with quarks as the matter fields, have the asymptotic freedom property as long as there are 16 quark flavors of mass below the ...
- 1,669
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0
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67
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Deriving Euler-Lagrange Equations in Light-Front Quantization from the Heisenberg Equation
I'm delving into light-front quantization, with a focus on understanding the roles of good and bad fermions. Using Collins' formulation in Foundations of Perturbative QCD, we define the projectors as:
...
- 11
2
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1
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256
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Why does non-perturbative QCD need to be regularized and renormalized?
The $n$-point correlation functions of QCD, which define the theory, are computed by performing functional derivatives on $Z_{QCD}[J]$, the generating functional of QCD,
$$\frac{\delta^nZ_{QCD}[J]}{\...
- 343
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0
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72
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Renormalizability of massless Gross-Neveu theory
We have the following Lagrangian density:
$$ \mathcal{L} = \bar{\psi}_i i \gamma^\mu \partial_\mu \psi_i
+ \frac{g^2}{2} \left( \bar{\psi}_i \psi_i \right)^2 $$
which corresponds to the two-...
- 1,124
1
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1
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163
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Asymptotic Freedom and the Continuum Limit
Suppose I have a family of effective field theories, parametrized by a cut-off $\Lambda$ and a coupling constant $g$ and specified in terms of generating functionals $Z_{\Lambda,g}(J)$. Asymptotic ...
- 155
3
votes
1
answer
120
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Renormalization of quark bilinears
I'm looking at the one-loop corrections to the amputated quark two-point functions ($\Gamma_i$) with insertions of quark bilinears (indexed by $i\in\{S,P,V,A,T\}$) with off-shell legs in Euclidean QCD....
- 71
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244
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One-loop renormalization of the gauge coupling
Quoting Yuji Tachikawa, chapter 3 of "${\cal N}=2$ Supersymmetric Dynamics for Pedestrians":
Recall the one-loop renormalization of the gauge coupling in a general Lagrangian field theory $$...
user242231
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1
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Expansion at first order ${\cal O}(\alpha_s)$ in counterterms for the QCD vertex renormalization at 1-loop
What is the meaning of the expansion at first order ${\cal O}(\alpha_s)$ in $\delta_2$ and $\delta_3$ at the second step in the last line? These quantities are not "small" - on the contrary,...
- 152
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0
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279
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Tadpole diagram in one-loop QCD gluon self-energy renormalization
I am trying to study QCD renormalization at 1-loop order. So, when I take into account the gluon self-energy corrections, the book I am studying from says that there are three diagrams who contribute ...
- 21
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Why is there still disagreement over the mass of the bottom (or beauty) quark, but none of the others?
Wikipedia (among other places) lists two values for the alleged mass of the B quark, 4.18 and 4.65 GeV.
Only one of the two possible masses listed has a link to another Wiki page explaining the ...
- 4,509
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1
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256
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What are all the 1PI gluon 3-point function Feynman diagrams at 1-loop?
As an exercise in renormalization, I want to calculate the divergent part of the gluon 3-point function and gluon 4-point function matrix element. What are all the 1PI one-loop Feynman diagrams?
- 2,599
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280
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Is there a simple explanation of why coupling constants run with $\log(E)$?
The inverse coupling constants run with $\log(E)$, where $E$ is the energy or four-momentum.
Some coupling constants increase, some decrease with $\log(E)$.
Is there a simple argument that explains ...
user85598
3
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1
answer
304
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Renormalization group applied to a simple QCD problem
Consider page 551 of Peskin & Schroeder's book on QFT, where he is treating the process $e^{+}e^{-} \to \text{Hadrons}$. I have 3 misunderstandings regarding some lines of thought effected there:
...
- 1,742
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135
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How do I calculate the Altarelli-Parisi splitting function for $g \rightarrow q\bar{q}$?
I'm trying to calculate the Altarelli-Parisi splitting function in the collinear limit for a gluon splitting into a quark-antiquark pair, but I keep getting stuck. Let $p$ and $k$ be the momenta for ...
- 43
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1
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258
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What makes the (non-abelian) strong interaction so special that it leads to confinement?
The strong interaction has a coupling constant of $\alpha_s(91GeV)\approx 0.1$ whereas the weak interaction has a much lower coupling constant $\alpha_w \approx 10^{-6}$. Both theories are non-abelian ...
- 9,778
2
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1
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576
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Dependence by the renormalization scheme in the beta function coefficents in QCD
Given the coupling costant $\alpha_s$ of QCD and it's RGE equation $\frac{d\ln \alpha_s(\mu^2)}{d\ln\mu^2}=\beta(\alpha_s)$, with $\beta(\alpha_s)=-\beta_0\alpha_s-\beta_1\alpha_s^2 -\beta_2\alpha_s^3+...
- 823
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1
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320
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What are clover fermions?
I've seen the term been used quite a lot when reading about lattice gauge theory calculations. So far what I've gathered is the following, from this source [1].
Lorentz invariance of the action is ...
- 5,371
3
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124
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The 1-loop anomalous dimension of massless quark field for $SU(N)$ gauge theory with $n_f$ quark flavours
Considering $SU(N)$ gauge theory with $n_f$ massless quarks
I want to find the anomalous dimension to order of 1-loop of the massless quark field, that defined by: $$\gamma_q(g^{(R)})=\frac{1}{2Z_q}\...
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0
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71
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5d nonabelian gauge theory or 5d QCD explicit $\beta$-function
Question: Are there some derived $\beta$-function formula
$$
\beta(g) = \frac{\partial g}{\partial \log(\mu)}
$$ for nonabelian gauge theory in the 5-dimensional spacetime at some energy scale $\mu$ ...
- 7,848
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0
answers
59
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What is mass factorisation?
I know that for some observables in QCD, e.g. inclusive DIS cross sections, one can use factorisation by writing said observable as a convolution of hard scattering coefficients with PDFs. But what is ...
- 1,187
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0
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121
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Negative running coupling constant in QCD?
Considering the one-loop running coupling constant in QCD:
$$\alpha_s(\mu) = \frac{a_s(\mu_0)}{1 + \frac{33-2f}{6 \pi}ln\left( \frac{\mu}{\mu_0}\right)}$$
where f is the number of quarks flavours with ...
- 585
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0
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179
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QCD quark self-energy, is the propagator momentum in the right direction?
For the quark self-energy diagram the amplitude is giving by:
The fermion propagator is given by $\frac{i}{\not p + l}$ in the lecture. It is not supposed to be $\frac{i}{\not p - l}$? It's seems ...
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1
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230
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Is $\mu$ the renormalization or factorization scale in the DGLAP equations?
The DGLAP equations read
$$\frac{\partial f_i(x,\mu^2)}{\partial\ln\mu^2}=\sum_j\int^1_x\frac{dz}{z}P_{ij}(z,\alpha_s(\mu^2))f_j\left(\frac{x}{z},\mu^2\right),$$
where the $f_i$ are the parton ...
- 1,187
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373
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Calculating strong coupling constant $\alpha_s$
I'm trying to find out how to calculate the value of strong coupling constant. but haven't found any exact answer, all I get is reference to perturbation theory.
is there any exact defined formula to ...
- 141
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42
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How to expand the QCD loops in terms of infrared scale for matching with heavy quark effective field theory
According to Manohar:"argument applies almost without change to a practical example,the derivation of the HQET Lagrangian to one-loop."
on page 37 of the following paper on effective field ...
- 1,268
5
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3
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740
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How is $\Lambda_{\textrm{QCD}}$ relevant in the non-perturbative regime?
The famous $\Lambda_{\textrm{QCD}}$ parameter enters through the one-loop running of the QCD coupling, through a relation similar to the following:
$$\alpha_S(Q^2)=\frac{\alpha_S(Q^2_0)}{1+b\ln(Q^2/Q^...
- 5,371
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0
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199
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Question about Renormalization of Ghosts in QCD in MS scheme
During a calculation of the Renormalization constant of the ghosts in QCD I stumbled over the following question:
When I calculate the self-energy of the Faddeev-Popov ghosts in $SU(N)$ non-abelian ...
- 854
3
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0
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86
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Is the starting distribution in the solution of DGLAP IR-bare?
On p.27 of this paper by John Collins, he says that when defining PDFs in terms of partonic number operators, one acquires an IR-divergent bare PDF (eq. 52). The residue of the IR-divergent term is ...
- 1,187
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1
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157
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How do you reconcile quark masses with notion of confinement?
In trying to understand exactly what confinement means, I have been reading 't Hooft s original paper on 2D QCD at large $N$. In the paper he shows that the quark propagator pole is moved to infinity, ...
- 744
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1
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375
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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 ...
- 3,514
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0
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288
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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 ...
user240990
3
votes
2
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681
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1
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1
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305
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What does QCD look like in higher dimensions?
It was pointed out as a comment on my question on atomic physics in higher dimensions that that question implicitly rests on an assumption that QCD, and thus the structure of atomic nuclei, is pretty ...
- 2,196
3
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0
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324
views
A Question about Wave-Function Renormalization Factor in SQCD
Here, I have a question about the one-loop computation of the wave-function renormalization factor in SQCD.
According to Seiberg duality, the following electric $\mathrm{SQCD}_{e}$
\begin{gather}
...
- 2,923
5
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0
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180
views
Instantons, renormalization, and the Schwinger Model
Instantons in QCD contribute to the up, down, and strange quark masses (see, e.g., Georgi and McArthur (1981) or Kaplan and Manohar (1986)). Some papers claim that this contribution is equivalent to ...
- 1,783
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Would Color Confinement apply in higher dimensions?
As I understand it color confinement comes from the fact that as the distance between two color charges increases the color potential energy increases, instead of decreasing, and the energy needed to ...
- 2,001
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1
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232
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Why does literature list the strong coupling at the scale of the Z-boson's mass?
In the 2004 edition of the book "QCD as a Theory of Hadrons" by S. Narison, the author provides a value for the strong coupling at a scale of the mass of the Z boson,
$$ \alpha_s (M_Z) = 0.1181 \pm ...
- 2,648
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0
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177
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Question about DGLAP evolution equation
I am reading chapter 32.2 of Schwartz's QFT book, where he defines the renormalized PDFs $f_i(x, \mu)$. This leads to an equation (32.48), which relates PDFs at different scales $\mu, \mu_1$:
$f_i(x,\...
- 1,129
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0
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119
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Renormalization of interaction parameter in QCD
In my book of QCD, when talking about renormalization, the author mentions the difference between QED and QCD:
In QED, the interaction paramenter $\alpha$ of renormalization factor $\mu$ and ...
- 269
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0
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106
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How to get group U(1) from SU(N)?
I have read that the unitary group is somehow given by the direct product $U(N)=U(1)*SU(N)$ and it follows that for $N$ going to zero we get just $U(1)$.
How it can be possible? What does it mean $SU(...
- 479
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514
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Can the On-Shell Scheme be used for QCD?
I am reading a textbook on QFT and it mentions that the on-shell scheme cannot be used for QCD.
Could someone explain the on-shell renormalization scheme in a bit more detail, plus why QCD cannot be ...
- 1,410
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2
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530
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What is relevant for Yang-Mills theory below $d = 4$?
There are two ways to write the Lagrangian for Yang-Mills, differing by the scaling of the Yang-Mills field. Fancy theorists tend to write
$$S = \int d^dx \, \frac{1}{4e^2} \, \text{tr}(F^2)$$
while ...
- 103k
4
votes
0
answers
512
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Beta function for non-Abelian gauge theory with both fermionic and bosonic matter fields
Consider a non-Abelian gauge theory with both fermionic $\psi$ and bosonic $\phi$ matter fields in 4D, which may be called a QCD-Higgs model,
$$L=-\frac{1}{4 g^2}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}i\gamma^...
- 11.9k
2
votes
2
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77
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Error estimation of $\alpha_s$
I have calculated the strong coupling constant $\alpha_s$ using and approximate solution of the Renormalization Group Equation
$$\mu_R^2\frac{d\alpha_s}{d\mu_R^2}=-(b_0 \alpha_s^2 + b_1\alpha_s^3 + ...
- 21
11
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1
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What is meant by “factorization scale factor” in QCD calculations?
What is “factorization scale factor” $\mu_F$ and how is it different from “renormalization scale factor” $\mu_R$ in QCD calculations?
When are both of them equal, such that $\mu_R=\mu_F=\mu_0=\sqrt{m^...
- 573
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442
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Is Yang-Mills theory confining in any dimensions?
What is the current understanding of Yang-Mills theory (pure non-Abelian gauge theory without matter field) in the infrared limit? (To avoid the subtlety of renormalizability, we may restrict our ...
- 11.9k
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1
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369
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Connection between QCD Lagrangian and the strong running coupling constant
I know the QCD Lagrangian as well as the running coupling constant for the strong force.
But how are they connected? The Lagrangian should contain the coupling constant, shouldn't it?
- 1,507