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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How to understand singularities in physics?

The question is probably two-folded and I will try not to make it too vague, but nonetheless the question remains general. First fold: In most physical laws, that we have analytic mathematical ...
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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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Is QFT mathematically self-consistent?

After recently going through a short program of self-study in quantum mechanics, I was surprised to find a quote attributed to Feynman essentially saying he was extremely bothered by the computational ...
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Gauge choice after Spontaneous Symmetry Breaking

After the spontaneous breakdown of local symmetry in presence of gauge fields (Higgs Mechanism), we can always choose a gauge where the Goldstone bosons are eaten up by the gauge field (also called ...
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Naturalness arguments and dimensional regularization?

How do issues of naturalness arise when regularizing QFT using dimensional regularization? I can only recall ever seeing naturalness arguments (hierarchy problem, cosmological constant problem, etc.) ...
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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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In QFT how do you write down the most general interactions?

This past year I took a QFT class and I now feel comfortable solving scattering problems, but I am still a bit perplexed by how physicists write down a Lagrangian in the first place. In particular, ...
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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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What is IR CFT and UV CFT?

What is IR CFT and UV CFT? In many physics related materials, they often mention IR, and UV. I think it is related with regularization (I remember in QFT, there is UV cutoff in some regularization ...
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Infinity of running couplings

A Landau pole - an infinity occurring in the running of coupling constants in QFT is a known phenomena. How does the Landau pole energy scale behave if we increase the order of our calculation, (more ...
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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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Why Zeta regularization is not valid for multiple-loops?

Why zeta regularization only valid at one-loop? I mean there are zeta regularizations for multiple zeta sums. Also we could use the zeta regularization iteratively on each variable to obtain finite ...
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Cut-off Regularisation and Renormalisation in Scalar Field Theory, Deriving the Cutoff Independent Physical Mass

I'm having trouble reproducing Equation 42: \begin{equation}\tag{1} m^{2}_{\text{phys}}= m^{2}_{r} + m^{2}_{r} \tilde{\lambda} \text{log} \left( \dfrac{m^{2}_{r}}{\mu^{2}} \right) \end{equation} ...
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Why are the eigenvalues of a linearized RG transformation real?

The RG transformation $R_\ell$ maps a set of coupling constants $[K]$ of a model Hamiltonian to a new set of coupling constants $[K']=R_\ell[K]$ of a coarse-grained model where the length scale is ...
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Regarding Non-renormalizatibility of GR

I've been doing some reading trying to get to a better understanding of some renormalization issues with the Einstein-Hilbert action. But, something odd came into mind that I'm hoping some users may ...
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What does it mean to renormalize an effective field theory?

This is in reference to slide 19 of this talk "As always in Effective Field Theory, the theory becomes predictive when there are more observables than parameters" Can one explain what this exactly ...
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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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What is meant by the phrase “this operator does not renormalize this other operator”, and how can understand it using diagrammatic arguments?

I am trying to understand some sentences in a paper. In section two the following theory of a (complex) massless scalar coupled to a $U(1)$ gauge boson is introduced ...
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Why holographic renormalization?

Why is there a need to perform holographic renormalization for the normal $AdS_5\times S^5$/CFT$_4$ correspondence if the brane theory is conformal? Since the flow along the AdS direction $r$ is ...
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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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Protection of the electron mass by chiral symmetry

In many textbooks it is said that mass renormalization of the electron mass is only logarithmic $\delta m \sim m\, log(\Lambda/m)$ because it is protected by the chiral symmetry. I understand that in ...
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What is the connection between Conformal Field Theory and Renormalization group in QFT?

As I know, the fundamental concept of QFT is Renormalization Group and RG flow. It is defined by making 2 steps: We introduce cutting-off and then integrating over "fast" fields $\widetilde{\phi}$, ...
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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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Why is $R^2$ gravity not unitary?

I have often heard that $R^2$ gravity (as studied by Stelle) is renormalisable but not unitary. My question is: what is it that causes the theory to suffer from problems with unitarity? My naive ...
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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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How do fermion and scalar masses run with energy? Is the difference in their running the core of the hierarchy problem?

Do fermion and scalar running masses run in the same way? Specifically, what are the qualitative differences in the mass beta functions for, say, scalar $\lambda\phi^4$ field theory and the fermion ...
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The Euler equations as a RNG fixed point

In this paper at the at the beginning of the last paragraph on p.2 it is said, that the Euler equations, which are an infinite Reynolds number limit of the Navier-Stokes equations, arise as an RNG ...
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Why don't we have logarithms or exponentials of the fields in the Lagrangians?

All tbe Lagrangian densities I have seen have always been polynomials of the fields. Is this a coincidence or is there a reason forbid, say, Lagrangians with logarithms or exponentials of the fields?
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Questions regarding $D=4 $ ${\cal N}=4$ supersymmetric Yang-Mills

I have some questions regarding the $D=4 $ ${\cal N}=4$ super-Yang-Mills theory (the one with a really long action which can be acquired by compactifying the 10-dimensional ${\cal N}=1$ theory). I ...
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Field redefinitions and new counterterms

My question was motivated by my attempt to answer this question. Suppose we are given an action and we make a change of variables such that the theory is non-renormalizable. Does the new theory then ...
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Deriving Feynman rules from Renormalized Lagrangian

In the context of Renormalized Pertubation Theory Peskin Schröder says: The Lagrangian $$ \mathcal{L}=\frac{1}{2} (\partial_\mu\phi_r)^2-\frac{1}{2}m^2\phi_r^2-\frac{\lambda}{4!}\phi_r^4 + \frac{1}{2} ...
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Is the Higgs bare mass larger than the physical mass?

The Higgs boson propagator can be written $$\frac{1}{p^2-m^2+\Sigma(p^2)}$$. If we take $p^2=m_P^2$ the physical mass, we get $m_P^2=m^2-\Sigma(m_P^2)$. Now, if $\Sigma\sim \Lambda^2$, we get ...
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Scalar field divergent mass correction interpretation question (hierarchy problem)

Simple power counting tells you that a scalar field coupled to some fermions at one-loop picks up a correction to the mass of the order $\Lambda^2$. Based on this people say things like "it's natural ...
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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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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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Drawing the RG flow diagram

In real-space renormalization group how does one find the complete RG flow exactly, (not schematically)? I understand it needs to be done on a computer. For example, I have the ising model on a ...
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conformal anomaly of free scalar in 2D

I'm trying to calculate the conformal anomaly $c$ of a free scalar on a 2-sphere. I've seen other, indirect ways to do this, but since this is a free theory I feel like it should be possible to see ...
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Does perturbation theory break down for quantum gravity?

Perturbation theory presumes we have a valid family of models over some continuous (infinitely differentiable, in fact) range for some parameters, i.e. coupling constants. We have some special values ...
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Defining a CFT using beta-functions

Won't it be correct to define a CFT as a QFT such that the beta-function of all the couplings vanish? But couldn't it be possible that the beta-function of a dimensionful coupling vanishes but it ...
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Why is GR renormalizable to one loop?

I have read in a few places that GR is renormalizable at one loop. (hep-th/9809169 for example, second sentence, although they don't seem to develop this point at all). Is this do to some hidden ...
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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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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 of the R-charge?

In general I would like to know as to known or what is/are the standard references about R-charge renormalization in supersymmetric theories. When does it do so and what is expected or known to be ...
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Why do we need to prove the gauge invariance of QED (or all of the gauge theories) on the Feynman diagrams language?

Let's have the QED lagrangian. It has explicit gauge invariance, so, by the naive thinking, all of the EM processes must satisfy the property of gauge invariance. So why do we need to recheck of gauge ...
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Significance of massive states in string theory

A free superstring has an infinite tower of states with increasing mass. The massless states correspond to the fields of the corresponding SUGRA. In "Quantum Fields and Strings: A Course 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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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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Zeta regularization and Renormalization group

Is there a physical method to prove for example when the zeta regularization of a series $$ 1+2^{k}+3^{k}+............= \zeta (-k) $$ gives the correct result: Casimir effect, vacuum energy and when ...
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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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Can Divergences in Nonrenormalizable Theories Always Be Absorbed by (An Infinite Number of) Counterterms?

For example, consider the $\phi^3$ theory in $d=8$, with Lagrangian: $\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-\frac{1}{3!}\lambda_{3}\phi^{3}$. In 8 ...