Questions tagged [renormalization]
This tag is for questions which relates with the renormalization, an ensemble of techniques which serves to treat the infinities which appear in quantum field theory or statistical mechanics. Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values.
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What contributes most to the mass of the proton?
i'm a bit confused about the origin of the mass of the proton (or other hadrons). It is said that it stems from the renormalization of the current quarks, which have about 1/100 of the mass of the ...
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Linearization of $\beta$-functions around Gaussian fixed points
I was reading section 8.4.4 of the book "Condensed Matter Field Theory" by Altland & Simons and I ended up with a very specific question that I couldn't resolve.
After finding the ...
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If a QFT is renormalizable, will it give better prediction at high orders?
Assuming I have a theory with a couplinh parameter with zero mass dimensions, So after calculating and adding all the counter terms I have a redifned theory, will this theory give me better prediction ...
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Expanding functions with poles in QFT Calculation
I am using Series function in Mathematica on $(1/z)(-k^2)^z$. Up to $z^0$, the function gives me $1/z + \log[-k^2]$. But in the standard textbook on QFT, it turns out the expansion should give $1/z + \...
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How to integrate a Gaussian path integral of free particle using zeta function regularization?
I am attempting to integrate this path integral in Euclidean variable $\tau $ (but this need not be the same as the $X^0$ field):
$$Z=\int _{X(0)=x}^{X(i)=x'}DX\exp \left(-\int _0^i d\tau \left[\frac{...
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Is it true that there are no known mathematically rigorous examples of interacting QFTs in 4 dimensions?
An answer to a question
(What physical processes other than scattering are accounted for by QFT? How do they fit into the general formalism?) about quantum field theory asserts
"we don't know ...
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Running coupling constants QFT
I am very confused and probably understanding the concepts of renormalization wrongly: If the running coupling constant is a real experimentally observable quantity and the change with scale has a ...
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Renormalization group equation and method of characteristics
All of this question refers to ref. 1. The equation are numbered alike.
The author claims to solve a renormalization group (RG) equation using the Method of characteristics, but there is a passage ...
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Infinite series of diagrams = one-loop RG equations？
In the RG of $\rm Fe$ based superconductor, the author wrote something like Infinite series of diagrams (RPA)= one-loop RG equations.
I wonder if it is a universal theorem.
I don't have a background ...
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Radiative corrections for the vacuum polarization
In Vanderhaeghen et al. (Phys. Rev. C 62 025501), the photon propagator modified by the vacuum polarization is
$$
\Pi(Q^2) = \frac{-e^2}{(4\pi)^2} \frac{4}{3}
\left[ \frac{1}{\epsilon_{UV}} -\gamma_E ...
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Introductory Reference for Asymptotically Safe Gravity
I have a fairly solid understanding of "classical" quantum field theory (Weinberg 1 & 2) and also of General Relativity with its various formulations. Being one that works in the field ...
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Problem in calculating the one-loop contribution to the self–energy of $\phi$ field
Consider the theory
$$
\mathcal{L}=\frac{1}{2}\left(\partial_\mu \phi\right)^2-\frac{m^2}{2} \phi^2-\frac{g}{3 !} \phi^3-\frac{\lambda}{4 !} \phi^4 .
$$
Find the expression for the self-energy and the ...
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Coupling renormalization $\lambda\phi^4$ vs QED
I have some doubts regarding the allegedly different procedures used in $\lambda\phi^4$ and QED.
First of all, I am more familiar with bare perturbation theory (no counterterms), so I would be ...
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Pauli-Villars regularization and self-energy
In the calculation of the electron self-energy in QED (one-loop level), there is a UV and IR divergent integral that needs to be regularized. A common choice for the regularization is the Pauli-...
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Peskin and Schroeder QFT Eq.(12.66), the Renormalization Group equation
I am troubled for the derivation of Eq.$(12.66)$ on Peskin and Schroeder's QFT book.
$$
\left[p \frac{\partial}{\partial p}-\beta(\lambda) \frac{\partial}{\partial \lambda}+2-2 \gamma(\lambda)\right] ...
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Why is $Z_3= Z_\xi$ in a non-abelian gauge theory?
In my lecture notes for a course on QFT it is said that, also in non-abelian gauge theories, the identity $Z_3 = Z_\xi$ holds, where those renormalization parameters belong respectively to the ...
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Conformal manifold of a supersymmetric field theory
I'm trying to understand what exactly is the conformal manifold of a theory. If I understand it right, the conformal manifold is the space of couplings.
From that point of view, it is just a subset ...
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Peskin and Schroeder Pg. 228: Expanding resummed propagator around the physical pole
I am having difficulty wrapping my head around this particular statement and equation (7.44) on pg. 228 of P&S,
I think I understand that "~" sign here is trying to denote at under $p^0 ...
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Perturbative calculation of hierarchy problem
I've been trying to understand the origin of the hierarchy problem for the Higgs mass but I've tied myself into some pretty nasty knots and I'm hoping someone can shed some light on this.
So as I see ...
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Analyzing the one-loop self-energy graph in $\phi^3$ model
Consider the $\phi^3$ model with a real scalar field $\phi(x)$ in $3+1$ dimensional Minkowski spacetime with metric $(-,+,+,+)$. Its Lagrangian density is
$$
\mathcal{L}=-\frac{1}{2} \partial_\mu \phi ...
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Justification for the derivative expansion in the Exact Renormalization Group
In the Exact Renormalization Group formalism, specifically the formalism of Wetterich, one writes down an evolution equation for the effective average action $\Gamma_k[\varphi]$, see f.ex
$$
\...
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Trouble with loop calculations
I was trying to do loop calculation. I have done chapter 6 and 7 of Peskin and Schroeder which deals with one loop correction to electron vertex function and vacuum polarization. But I don't feel very ...
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In formalizing QFT, are mathematical issues of canonical quantization approach and that of path integral approach related?
In QFT, many mathematical issues arise. Setting aside renormalization, these deal with rigorous constructions of objects underlying QFT:
i) In the canonical quantization approach, the main issue comes ...
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Why does non-perturbative QCD need to be regularized and renormalized?
The $n$-point correlation functions of QCD, which define the theory, are computed by performing functional derivatives on $Z_{QCD}[J]$, the generating functional of QCD,
$$\frac{\delta^nZ_{QCD}[J]}{\...
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Effective field theory of massive spin-1 and spin-2
I'd like to understand what could be the use of an effective field theory (EFT) of a single massive particle of spin 1 or 2, or simple modification of these (see below). By EFT, I mean the most ...
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The Callan-Symanzik equation on Peskin & Schroeder's QFT
On Peskin & Schroeder's QFT page 411, the Callan-Symanzik (CS) equation reads
$$
\left[M \frac{\partial}{\partial M}+\beta(\lambda) \frac{\partial}{\partial \lambda}+n \gamma(\lambda)\right] G^{(n)...
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1PI effective potential vs self-energy
Consider the following Lagrangian describing the interaction between a massless field $\phi$ and a massive field $\psi$:
$$
{\scr L} = \frac12(\partial\phi)^2 + \frac12 (\partial\psi)^2(1 + f(\phi/M)) ...
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Does pure Yang-Mills have a scale?
Consider pure Yang-Mills (YM) in 4 dimensions. The YM mass gap problem (as described in https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf) tells us that this is supposed to have a mass-...
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Lorentz invariance
In QED loop renormalization, Lorentz invariance is often used to express the possible momentum-dependence of the propagators.
For example, the propagator corresponding to the fermion loop is
$$ie_0^2 \...
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Radiative correction of the electron self-energy
In Mandl & Shaw's Quantum Field Theory (2nd edition p217), the radiative correction for the electron self-energy is:
$$
e_0^2 \Sigma(p) =
\frac{\tilde{e_0}^2}{16\pi^2} (p\!\!/ -4m) \left(\frac{2}{\...
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Beta function of CFT perturbed by several operators
As the title says, I am trying to derive the $\beta$ function of a CFT whose action is perturbed by several operators. The main refference I am following is https://arxiv.org/abs/1507.01960 starting ...
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Divergence of gauge kinetic coupling at the AdS boundary
This is the Einstein-Maxwell-Dilaton Gravity action:
\begin{eqnarray*}
S_{EM} = -\frac{1}{16 \pi G_5} \int \mathrm{d^5}x \sqrt{-g} \ [R - \frac{f(\phi)}{4}F_{MN}F^{MN} -\frac{1}{2}D_{M}\phi D^{M}\...
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Do irrelevant directions contribute to the IR dynamics?
Is it not a misnomer to call in a non-perturbative setting an irrelevant direction irrelevant?
I know that it comes from perturbation theory where an irrelevant direction is irrelevant for the IR ...
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Fermion mass correction always proportional to it's mass? even in case of mixing?
In QED, it is obvious that one-loop correction to the mass of the fermion ($\psi$) is proportional to its bare mass. However, it is not very clear to me whether it is general even in the case when ...
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Why does it make sense to do Dyson Resummation with first-order 1PI-diagrams?
This is somewhat related to This question.
The way I understand renormalisation (e.g of the mass) is that we consider the loop-corrections order by order in the coupling constant. If we then consider ...
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False vacua and the effective action
The effective action $\Gamma[\phi]$ of a (scalar) field theory described by the action $S[\phi]$, can be computed in the "background field method" by shifting $S[\phi + \phi_b]$, where $\...
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Renormalization-group approach to half-filling charge density wave
In Shankar's noted review paper on the renormalization group (RG) approach to many-body physics, Sec. IV deals with RG in a 1D lattice nearest-neighbour (quartically) interacting model, which leads to ...
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Beta function of $\phi^4$ from RG equation
I am currently following David Tongs lecture notes on statistical field theory (Link to the script) and I have an issue with the calculation of the beta-function from the RG equations (equation (3.45))...
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Charge renormalization using Ward identity
In Mandl & Shaw's Quantum Field Theory (p 181), the Ward identity
$$\frac{d\Sigma(p)}{dp_\mu} = \Lambda^\mu(p,p)\tag{9.60}$$
where $\Sigma(p)$ and $\Lambda^\mu (p^\prime,p)$ are respectively the ...
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Weinberg's virtual soft photon addition to Feynman diagrams
I am reading Weinberg's book on QFT, especially Chapter 13.2, talking about the effect of adding virtual soft photons to a Feynman graph in the theory of QED. Weinberg supports that adding $N$ virtual ...
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How does the renormalization process differ between actions with and without diffeomorphism invariance?
If I have an action that is not invariant under a change of coordinates. Does this effect the renormalizability of the theory?
The procedure of renormalizing a field theory essentially boils down to ...
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A diagram of P. Coleman's Introduction to Many-Body Physics
In the book Piers Coleman - Introduction to Many-Body Physics in page 548,
The renormalization of the Coulomb interaction can be understood as the screening
effect of high-frequency virtual pair ...
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Dimensional regularization yields finite result for loop integral in 3d $\phi^4$ theory [duplicate]
As an exercise I wanted to compute the mass renormalization in 3d $\phi^4$ theory (1-loop).
At 1-loop the field requires no renormalization (as in the 4d case) and the mass renormalization is ...
3
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Double slit wavefunction doesn't converge
I'm trying to plot the exact solution to wavefunction of the double slit experiment.
I'm assuming sphere waves radiating with 1/r at each point. In other words, here's the exact solution I'm solving ...
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Renormalisation of the fermionic triangle loop
I am trying to renormalise the following loop diagram in the Standard Model:
$\qquad\qquad\qquad\qquad\qquad\qquad$
Using the Feynman rules, we can write the amplitude as follows:
$$ \Gamma_f \sim - ...
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Perturbation in Euler-Heisenberg Lagrangian
If we use minimal subtraction to remove infinities, the effective Lagrangian of the background EM field is:
$$ \mathcal{L}_{EH} = -\dfrac{1}{4}F^2_{\mu\nu} - \dfrac{e^2}{32\pi^2}\int \dfrac{ds}{s}e^{...
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Vacuum bubble with the $Z$-boson
I'm interested in computing the vacuum bubble diagram of a single $Z$-boson. The propagator for a massive vector is
$$
D_{\mu\nu}(k_\mu) = \frac{i}{k^2 - m_Z^2}\left(g_{\mu\nu} - \frac{k_\mu k_\nu}{...
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Which summation should be chosen for a divergent series arising from the expression of relative mass in order to always preserve the same mass? [closed]
In all physical theories, the appearance of infinity is generally regarded as a sign that the theory is either incorrect or being applied outside its applicable domain, necessitating the search for ...
- 1
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Renormalization of Two Scalar Field Theory at Tree-Level and One-Loop Levels
I have been given a problem on renormalization, but due to my inexperience, I don't understand what to do with it. Here is the statement:
Consider a theory of two real scalar fields $\phi$ and $\Phi$ ...
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1
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Explicit form of cutoff dependent bare coupling at first order in $\phi^4$ with cutoff regularization
I would like to renormalize $\phi^4$ theory with Lagrangian
\begin{equation}
\mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m_0^2 \phi^2 - \frac{\lambda_0}{4!} \phi^4
\end{...
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