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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What is the significance of the branch cut in renormalization group logarithms?

What is the physical significance of the branch cut in renormalization group logarithms? (Is this just an avatar of the optical theorem, or is there something to be understood about these logarithms ...
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Do exact beta functions exist in (super)gravity theories and string theory?

An exact beta function exists for Super-Yang-Mills theories in 4D without matter - the so-called NSVZ beta function. Does a similar exact beta-function exist in gravity or supergravity theories? In ...
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Renormalization of worldsheet energy-momentum tensor

At the end of section 2.3, Polchinski (in his volume 1) derives the energy-momentum tensor for free massless scalars on worldsheet. He adds a footnote that "the only possible ambiguity introduced by ...
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Renormalizing composite operators

Consider the QED Lagrangian, \begin{equation} {\cal L} = \bar{\psi} ^{(0)} ( i \partial_\mu \gamma^\mu - m ) \psi ^{(0)} - e A _\mu ^{(0)} \bar{\psi} ^{(0)} \gamma ^\mu \psi ^{(0)} - \frac{1}{4} ...
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Anomalously broken conformal symmetry

I'm trying to understand an argument made by Bardeen in On Naturalness in the Standard Model. The argument is about quadratic divergences in Standard Model. My notation is that the SM Higgs potential ...
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Anomalous Dimensions of Gauge Interactions

Peskin and Schroeder mention a few times that the anomalous dimension of a gauge interaction operator is zero. The justification for this is that the charge operator shouldn't get modified under ...
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Superfields and the Inconsistency of regularization by dimensional reduction

Question: How can you show the inconsistency of regularization by dimensional reduction in the $\mathcal{N}=1$ superfield approach (without reducing to components)? Background and some references: ...
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298 views

Renormalization Group and Ising with d=1 and D=1

I have a question about the results of RG on Ising model. I know it's possible to obtain two couple of relations $K'(K)$, $q(K')$ $K(K')$, $q(K)$ between the coupling costants. My problem arise ...
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Which values of the Riemann zeta funtion at negative arguments come up in physics?

For my bachelor's thesis, I am investigating Divergent Series. Apart from the mathematical theory behind them (which I find fascinating), I am also interested in their applications in physics. ...
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Some conceptual questions on the renormalization group

I recently followed courses on QFT and the subject of renormalization was discussed in some detail, largely following Peskin and Schroeder's book. However, the chapter on the renomalization group is ...
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Dimensional regularization and the finite part

Let be a dimensional regularized integral $$ \int d^{4-\epsilon}kF(k,m,s)= \frac{2}{\epsilon}+\frac{m^{2}}{3}(\gamma +log(4\pi)-\frac{1}{\epsilon}))$$ then formally if we elmiinate the divergent ...
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Renormalizing with external momenta set to zero

I've often seen in textbooks that authors renormalize diagrams by setting external momentum to zero. Under what conditions is this justified? An example of this is done in Manohar and Wise's book on ...
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Proof of renormalizability based on analyzing the symmetry of effective action: isn't regulator also important?

In QFT Vol2 written by Weinberg(Chap 16-17), or very much similarly in Adel Bilal's notes(Chap 7), a powerful way of proving renormalizability is presented: Analyze the symmetries of the quantum ...
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What exactly is regularization in QFT?

The question. Does there exist a mathematicaly precise, commonly accepted definition of the term "regularization procedure" in perturbative quantum field theory? If so, what is it? Motivation and ...
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Does the 4/3 problem of classical electromagnetism remain in quantum mechanics?

In Volume II Chapter 28 of the Feymann Lectures on Physics, Feynman discusses the infamous 4/3 problem of classical electromagnetism. Suppose you have a charged particle of radius $a$ and charge $q$ ...
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Power counting with a cutoff

In Effective Field Theory video lectures found here, the professor explained power counting in effective field theories and the difficulties of power counting associated with loop diagrams. He then ...
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Quantum field theories with asymptotic freedom

QCD is the best-known example of theories with negtive beta function, i.e., coupling constant decreases when increasing energy scale. I have two questions about it: (1) Are there other theories with ...
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Does dynamical critical phenomena results into renormalized time?

I am trying to construct dynamical landau theory for self-assembly. From my constructed landau energy functional i derive dynamical equations using model-A of hohenberg and halperin. I solve those ...
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Trace of Fermion Loops in Effective Field Theories

I'd like to know whether we need to take the trace of fermion loops in effective theory in the same way that we need to do so for renormalizable theories. At first thought, it seems obvious that ...
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549 views

Why is mass renormalization insufficient to explain electron mass?

In the Standard Model, I understand that the mass of the electron is assume to arise from two effects: A bare mass given by Yukawa interaction with the Higgs field, and A mass correction from mass ...
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Renormalizing IR and UV divergences

In lectures on effective field theory the professor wanted to find the correction to the four point vertex in massless $\phi^4$ theory by calculating the diagram, $\hspace{6cm}$ We consider the zero ...
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Are “confinement” and “asymptotic freedom” two sides of the same coin?

On Wikipedia it says that the two peculiar properties of quantum chromodynamics (QCD) are: confinement and asymptotic freedom. Asymptotic freedom is the idea that at low energies we cannot use ...
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In what sense is the renormalization group equation a group?

The renormalization group equation is given by: \begin{equation} \left[\mu \frac{\partial}{\partial \mu} + \beta \frac{\partial}{\partial g} + m \gamma_{m^2} \frac{\partial}{\partial m} - n \gamma_d ...
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How does the renormalization scale $\mu$ cancel in all finite observables?

In dimensional regularization, we must shift the dimensionless coupling $g$ by the renormalization scale $\mu$ (which has unit mass dimension): \begin{equation} g \rightarrow \mu^{4-d} g \tag{1} ...
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Renormalization and the Hierarchy Problem

The hierarchy problem is roughly: A scalar particle such as the Higgs receives quadratically divergent corrections, that have to cancel out delicately with the bare mass to give the observed Higgs ...
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Field Strength Renorm in Peskin&Schroeder

On page 237 in PS we have (the unnumbered equation after eq. 7.58) $$\mathcal{P} \sim \frac{iZ}{p^2-m^2-iZ\,\mathrm{Im}M^2(p^2)}$$ but after deriving it myself I obtained $$\mathcal{P} \sim ...
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Renormalizing QED with on-shell fermions

When renormalizing QED, we calculate the 1 loop correction to the fermion-fermion-photon vertex using the diagram, $\hskip2in$ When doing the calculation we typically let the photon go off-shell ...
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How do we measure the physical field of a particle?

In the renormalization procedure of quantum field theories, say $\lambda \phi^4$ theory for simplicity, we use the physical mass $m$, the physical coupling constant $\lambda$ and the physical field ...
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Reference for the renormalization of a scalar field's mass

There are a couple of interesting lectures by Leonard Susskind online, and in the first lecture on Supersymmetry & Grand Unification he explains renormalization. His example is the mass ...
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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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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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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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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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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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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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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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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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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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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 ...