All Questions
38
questions
1
vote
1
answer
83
views
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
1
vote
0
answers
39
views
Loop Calculations of A Spontaneous Broken gauge theory with fermions
Let me first rephrase the background. Consider adding a massless fermion to the spontaneously broken $U(1)$ gauge theory through a chiral interaction:
$$
\mathcal{L}=\bar{\psi}_{L}i \gamma_{\mu}D^{\mu}...
- 81
2
votes
1
answer
147
views
Derivative interactions in the Wilsonian renormalisation Group
I am currently working through some basic renormalisation group problems, and have come to one about derivative interactions. It has been a while since I have studied QFT formally so bear with me ...
- 90
-1
votes
1
answer
130
views
Massless Sunset Diagram $\phi^4$ [closed]
I should compute an explicit calculation for the sunset diagram in massless $\phi^4$ theory.
The integral is $$-\lambda^2 \frac{1}{6} (\mu)^{2(4-d)}\int \frac{d^dk_1}{(2\pi)^d} \int \frac{d^dk_2}{(2\...
- 7
2
votes
1
answer
146
views
2-loop correction to exact 3-point vertex in a complex scalar field theory with cubed interaction
I am a graduate student with 1 quarter of relativistic QFT at the level of Srednicki (covered up to Chapter 30 this Fall). This question is not in any book that I know off and it wasn't assigned as ...
- 1,713
1
vote
0
answers
52
views
External leg correction to 3-point QED Green's function
I am trying to calculate the following diagram to solve the Callan-Symanzik equation for the three-point Green's function (two massless fermions and a photon).
The counterterm to the photon ...
- 1,075
1
vote
1
answer
115
views
Calculation of $ \gamma(\lambda) $ in massless renormalizable scalar field theory
In Peskin & Schroeder p.413 and 414, the Callan-Symanzik equation for a 2-point Green's function is used to calculate $ \gamma(\lambda) $ for a massless renormalizable scalar field theory. The two-...
- 1,075
0
votes
0
answers
158
views
Analyzing the one-loop self-energy graph in $\phi^3$ model
Consider the $\phi^3$ model with a real scalar field $\phi(x)$ in $3+1$ dimensional Minkowski spacetime with metric $(-,+,+,+)$. Its Lagrangian density is
$$
\mathcal{L}=-\frac{1}{2} \partial_\mu \phi ...
- 153
1
vote
0
answers
102
views
Beta function from renormalized coupling
While trying to derive a specific beta function for a CFT a stumbled upon the following. I have some bare coupling $g_b$ and introduce a renormalized coupling $g$ as
$$g_b=\mu^\epsilon(g+\frac{zg^2}{\...
- 61
3
votes
0
answers
50
views
Divergent verticies in mesonic scalar theory [closed]
Considering the following Lagrangian density:
$$ \mathcal{L} = - \frac{1}{2} ( \partial_{\mu} \phi \partial^{\mu} \phi + m^2 \phi^2) + \bar{\psi} (i \gamma^{\mu} \partial_{\mu} - m) \psi + g \bar{\psi}...
- 1,124
1
vote
3
answers
414
views
Yukawa decay at one-loop
I am trying to calculate the amplitude for a decay $\phi \to e^+e^-$ under a Yukawa interaction $\mathcal{L}_I = -g\phi \bar{\psi}\psi$ to one-loop order (with massless fermions for simplicity).
If I'...
- 23
1
vote
0
answers
116
views
From which interaction term does the self-energy diagram of $\phi^4$ theory come?
In 4D, let us start with the normal-ordered product of free neutral scalar fields $:\phi^4:$.
Then, we can in fact write $$:\phi^4:=\sum_{i=0}^4 V_i$$ where each $V_i$ is an operator-valued ...
- 1,669
1
vote
1
answer
124
views
On-shell renormalization (Schwartz Quantum Field Theory Equation (18.48))
I have a question about how, in section 18.3.2 in Schwartz's quantum field theory, he goes from equation (18.47) to (18.48) using Pauli-Villars regularization. It comes down to showing that to leading ...
- 213
2
votes
0
answers
45
views
How to show that the low energy effective bosonic sigma model is consistent : $\beta^G=\beta^B=0 \Rightarrow \beta^\phi=cnst$
I am totally stuck at an exercise regarding the consistency of nonlinear sigma model...
Let
\begin{equation}
\beta^G_{ab}=R_{ab}+2D_a D_b \phi-\frac{1}{4}H_{acd}H^{bcd}\text{, }\beta^H_{ab}=-\frac{1}{...
- 1,669
4
votes
0
answers
110
views
Effective potential [closed]
Given that after renormalization, we can write:
$$\Im\left(\mathcal{M} \right) = \Im\left(-\frac{-\lambda^2}{32 \pi^2} \int_{0}^{1} dx \ln(m^2 + p^2x(1-x)) \right)$$
The $(m^2 + p^2x(1-x))$ needs to ...
- 1,213