Renormalization is an ensemble of techniques which serves to treat the infinities which appear in quantum field theory or statistical mechanics.

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Evaluate $1$-loop contribution to the $4$-point Green's function

I am trying to evaluate the following integral \begin{equation} I = \int \frac{d^d p_\text{E}}{(2 \pi)^d} \frac{1}{(p_\text{E}^2+m^2)((q_\text{E}-p_\text{E})^2 + m^2)} \tag{1} \end{equation} where ...
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How do we know for sure a theory is non-renormalizable?

In quantum field theory, we are looking for a Lagrangian that is, amongst other, renormalizable. But how do we determine whether or not a theory is renormalizable? Is this purely done by power ...
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Observable which dependes on the cutoff

In arXiv:0710.4330v1 Balitsky calculate the eikonal scattering of dipole composed of quark anti-quark, $Tr(U_{x}U^{\dagger}_{y})$, to NLO accuracy. The result he found is: Where $\mu$ is the ...
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Why is $\vert \phi \vert ^2$ infinite in QFT?

I've read here¹ that for a scalar field $\phi$, the square $\vert \phi \vert ^2$ is infinite (which gives an infinite contribution to mass), more precisely: the square of the field – a quantity ...
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134 views

Mass corrections to fermions proportional to the mass?

In this post regarding quantum corrections to a massless fermion field, the answerer stated that quantum corrections to the mass will always be proportional to the mass (at least in QED). This point ...
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The phrase “Trace Anomaly” seems to be used in two different ways. What's the relation between the two?

I've seen the phrase "Trace Anomaly" refer to two seemingly different concepts, though I assume they must be related in some way I'm not seeing. The first way I've seen it used is in the manner, for ...
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Could quarks be free in higher-dimensional space than 3D?

Reading this answer, I now wonder: if quarks are confined by $r^2$ potential, could their potential allow infinite motion in higher-dimensional space? To understand why I thought this might be ...
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A question about the implication of UV divergence in QFT

I have a basic question about the logic of renormalization in quantum field theory (QFT). We met the ultraviolet (UV) divergence in loop corrections. The standard argument is, our current field theory ...
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105 views

Quick-and-dirty way to integrate out heavy fields

I understand the roughly understand the process of integrating out heavy degrees of freedom of a Lagrangian, namely, taking the action and performing the path integral over the high momentum modes. ...
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60 views

Equality of electric charges of all leptons

What does it precisely mean the often repeated statement that the electric charges of all leptons are the same. Let's consider QED with two leptons: electron and muon. The interaction part of the ...
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109 views

Neglecting mass at asymptotic spacelike momenta

What is the rational/reason for neglecting masses at asymptotic non-exceptional space-like momenta. I have come across this as a first fix for being able to extract information from the ...
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69 views

Some questions about the large-N Gross-Neveu-Yukawa model

Consider the following action with a fermionic field $\psi$ and a scalar field $\sigma$, $S = \int d^dx \{ -\bar{\psi}(\gamma^\mu \partial_\mu +\sigma )\psi + \Lambda^{d-4}[ \frac{(\partial_\mu ...
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No non-trivial UV asymptotically free and IR free

How it could be proven that a non-trivial theory cannot be both asymptotically free and IR free (g=0 both in the UV and IR with some interpolating function in between)? This is of course contrary to ...
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113 views

Regularization and renomalization in the lightcone quantization of bosonic string

This question relates to this link. But I still don't understand it >_< In Polchinski's string theory vol I, p. 22, there is a divergence term (when $\epsilon \rightarrow 0$) in the zero point ...
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257 views

Divergent sum in lightcone quantization of bosonic string theory

I had the following question regarding lightcone quantization of bosonic strings - The normal ordering requirement of quantization gives us this infinite sum $\sum_{n=1}^\infty n$. This is regularized ...
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176 views

Divergent bare parameters/couplings: what is the physical meaning of it? Do this have any relation with wilson's renormalization group approach?

I understand that bare parameters in the Lagrangian are different from the physical one that you measure in an experiment. I'm wondering if the fact that they are divergent has any physical meaning? ...
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The sum of positive integers equals minus one twelfth [duplicate]

I was watching a lecture online from the american physicist Lawrence Krauss, when he made an off the cuff remark about the sum of all the positive integers being equal to one twelfth. My question is ...
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How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”? [duplicate]

How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics? I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis ...
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53 views

Assymptotic freedom significance

So I have read a bit on this, and get the idea and mathematical machinery leading up to this. I get that it sheds light on the relationship between coupling strengths and length scales. Can someone ...
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Quantum corrections to massless fermionic field

in QED the corrections to electron propagator change the bare electron mass from $m_0$ to $m=m_0+δm=m_0+∑(\not{p}=m)$ (Peskin, formula 7.27). This is the consequence of the fact, that the quantum ...
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QED coupling constant at one loop

On page 257 in Peskin's QFT book a qualitative sketch of the QED coupling is given (see the picture below). Why should I expect such a behavior from QED? The QED beta function is ...
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Renormalization beta-function

I want to know the other forms of beta-function that make manifest certain properties of renormalization group, for instance dependence on poles/residue and more. If possible can you state a ...
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86 views

independence of the bare parameters on μ for beta function

So I know re-normalization has bean "beaten to death". I want to understand something a bit specific which might seem trivial. Independence of the bare parameters on $\mu$ and relevance to the beta ...
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583 views

Divergent Series

Why is it that divergent series make sense? Specifically, by basic calculus a sum such as $1 - 1 + 1 ...$ describes a divergent series (where divergent := non-convergent sequence of partial sums) ...
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Regularization of the Casimir effect

For starters, let me say that although the Casimir effect is standard textbook stuff, the only QFT textbook I have in reach is Weinberg and he doesn't discuss it. So the only source I currently have ...
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How to show that tensor gravity is nonrenormalizable?

Let's have the tensor gravity theory, which represented the massless spin-2 field: $$ L = -\frac{1}{32 \pi G}\left( \frac{1}{2}(\partial_{\alpha}h_{\nu \beta}) \partial^{\alpha}\bar {h}^{\nu \beta} - ...
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Derivative with respect to ${\not}{p}$

When studying renormalization of QED in standard textbooks, we typically encounter derivatives with respect to ${\not}{p}=p^\mu \gamma_\mu$, i.e., $\partial/\partial{\not}p$. As far as I understand, ...
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Basic question about the S-Matrix, Unitarity and Effective Field Theory

Consider scattering some particles in a state collectively denoted by $i$ to a final state denote by $f$. The scattering amplitude, S-matrix is then defined by: $S_{fi}\equiv \langle ...
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Peskin's book page 334 proof of $Z_1=Z_2$ to all orders in QED perturbation theory

Peskin in his QFT page 334 argued that $Z_1=Z_2$ to all orders in QED perturbation theory, but I couldn't understand his argument: ... With a generalization of the argument given there (section ...
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68 views

Explicit calculation of bosonic string Weyl invariance at one loop

I have been trying to do all the calculations in the Green, Schwarz and Witten Superstring Theory textbook. At the end of chapter 3, the author did one-loop calculation for Weyl invariance for the ...
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Semantic problem about renormalizability

This post relates to this previous one. My question is, what is the actual meaning of a theory being renormalizable? There might be at-least two possibilities (correct me if I am wrong) ...
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146 views

One-loop beta functions of the Standard Model

For my master's research on energy scale independent combinations of renormalization group equations in supersymmetric theories, I need an overview of all the one-loop beta functions of the Standard ...
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Renormalization of Composite Chiral Superfields

On page 24 of these lecture notes http://arxiv.org/abs/hep-th/0309149 it is stated that products of chiral superfields do not suffer from short distance singularities. In other words, if I want to ...
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Anomalous dimension for bare actions with a standard kinetic term

In this paper on p42, it is explained that when starting with a bare action that contains a standard kinetic term, this kinetic term attains a correction in the course of the RG flow which can be ...
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Running chargino/neutralino masses in MSSM

Consider the plot below, showing the running of different masses due to renormalization for a certain point of the (c)MSSM. I am able to exactly reproduce the plot, including the running of M1, M2, ...
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Amplitudes in renormalized perturbation theory

This question arose while reading Peskin and Schroeder, specifically, it arose in regards to the sum of diagrams above their Eq. (10.20) on pg. 326. The context is $\phi ^4$ theory and they are using ...
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Link between anomalous dimensions and fractal dimensions

I just realized that anomalous dimensions in quantum/statistical field theory is not that different from fractal dimensions of objects. They both describe how quantitaive objects transform under a ...
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174 views

Why does the counterterm's propagator have inverse units of the propagator? $\phi^4$-theory

According to Peskin & Schroeder (page 325), the Feynman rule for the counterterm ------(x)----- for $$ \frac12 ...
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Why do three dimensional gauge theories flow to conformal theories in the infrared?

What is meant with the fact that Super Yang-Mills flows to a conformal field theory in the infrared? Also, is this a general fact or does this depend on the fact of considering a certain class of ...
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128 views

Misunderstanding Wick Ordering

In M. Salmhofer's "Renormalization, An Introduction" Wick ordering is defined as follows: Let $C = C_\Gamma$ be a nonnegative symmetric operator on $\mathbb{C}^\Gamma$. For $J: \Gamma \to ...
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Why isn't renormalization called dark physics?

In QED there is extra energy that has to be gotten rid of to match observations. Kind of the opposite to GR/ND where you have to add extra energy/matter to match observations. Why isn't ...
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Evaluating propagator without the epsilon trick

Consider the Klein–Gordon equation and its propagator: $$G(x,y) = \frac{1}{(2\pi)^4}\int d^4 p \frac{e^{-i p.(x-y)}}{p^2 - m^2} \; .$$ I'd like to see a method of evaluating explicit form of $G$ ...
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121 views

IR non-renormalizable theory

can be a theory with an infinite number of divergent integrals of the form $$ \int \frac{d^{p}k}{k^{m}} $$ for m=1 , 2 , 3 , 4 ,...... so the theory would be IR non renormalizable and you would need ...
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Beta function from exact renormalization group equation

I'm trying to calculate beta functions from exact (or functional) renormalization group equation (I mean Wilson-Polchinski RGE). I've got the equations but I don't know exactly how to use the ...
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1answer
140 views

Renormalizability of the Polyakov Action

I was told today that the Polyakov action for a $p$-brane is (superficially) re-normalizable iff $p\leq 1$. Of course, when I went to check for myself, I screwed up my power-counting, and I'm having ...
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Why are only logarithmic divergence relevant for the Callan-Symanzik equation? Intuitive understanding?

I may be wrong, but it seems that only logarithmic divergences need to be retained when using the Callan-Symanzik equation, finding running couplings, etc. Why is this the case? Is there some simple ...
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If Renormalization Scale is Arbitrary, Why Do We Care about Running Couplings?

For the bounty please verify the following reasoning [copied from comment below] Ah right, so the idea is that overall observable quantities must be independent of the renormalization scale. But at ...
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What happens when you apply the path integral to the Einstein-Hilbert action?

The Einstein Field Equations emerge when applying the principle of least action to the Einstein-Hilbert action, and from what I understand the path integral formulation generalizes the principle of ...
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Why do we expect our theories to be independent of cutoffs?

Final edit: I think I pretty much understand now (touch wood)! But there's one thing I don't get. What's the physical reason for expecting the correlation functions to be independent of the cutoff? ...
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Is the effective Lagrangian the bare Lagrangian?

In standard (non-Wilsonian) renormalization we split the bare Lagrangian $\mathcal{L}_0$ into a physical Lagrangian $\mathcal{L}_p$ with measurable couplings and masses counterterms ...