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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Renormalization in non-relativistic quantum mechanics

I read many articles about renormalization in the Internet, but as I currently don't know much of QFT (currently just studying classical field theory and QM), and as all this looks quite interesting, ...
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Renormalization, integrating out high momenta Wilson way

In equation $(12.5)$ in Peskin and Schroeder, they write out the generating function but leave out all quadratic terms of the form $\phi\hat{\phi}$ arguing that they vanish since Fourier ...
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What is the exact renormalization regularization for divergent harmonic serise?

given the harmonic series $$ \sum_{n=0}^{\infty}\frac{1}{n+a} $$ what is the correct option for the regularization ? a) $ \sum_{n=0}^{\infty}\frac{1}{n+a}= -\Psi (a) $ Digamma function b) $ ...
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Zamolodchikov's c-theorem paper

I am reading the 1986 paper [1] where Zamolodchikov proves the c-theorem and I would like to understand how equations (7a), (7b) and (8) are derived from the Callan-Symanzik equation. For ...
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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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Convergence of light by light scattering amplitude

Perhaps I'm too exhausted to see the answer of why the photon-photon scattering should contain no divergences. In Peskin and Schroeder page 320 we find that because of the Ward identity the ...
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Are critical exponents below and above the critical point always same?

The scaling relations don't distinguish the the critical exponents below and above the critical value. In the mean field level, I understand these critical exponents are same whatever one approaches ...
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List of known universality classes

I am working with RG and have a pretty good idea of how it works. However I have noticed that even though the idea of universality class is very general and makes it possible to classify critical ...
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Is the stability matrix of a linearised RG flow always diagonalisable?

This is a follow up on "Why are the eigenvalues of a linearized RG transformation real?". My question is simple: Is there some physical (or mathematical) reason for the stability matrix of ...
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Correlation functions scale as powers of external momenta

In the 1st paragraph of section 12.2 on page 406 of Peskin and Schroeder, it says "the correlation functions turn out to scale as powers of their external momenta, with power laws that do not appear ...
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119 views

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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Beta function of the non-linear sigma model

In chapter 7.1.1. inTong's notes about String Theory could someone sketch how can I show the statements that he nmakes around eq. 7.5 That the addition of the counterterm can be absorbed by ...
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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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Unitarity and renormalizability

What is the difference between the unitarity of the theory and its renormalizability? Can we say that renormalizable theory is unitary after renormalization? The questions have arisen after I have ...
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1answer
71 views

Typical form of the beta function of the renormalization group

Why in "typical" cases (according to some non-English text I read), does the $\beta$-function have the form $$ \beta (g) = ag^{2} + bg^{3} + O(g^{4})\ ? $$ I.e., why are there no linear or logarithmic ...
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How to get the relation for dependence of anomalous dimension on regularization?

Here is the anomalous dimension: $$ \gamma_{\Gamma}(t, g) = \left[\frac{\partial }{\partial t}\ln \left(Z_{\Gamma}(t , g) \right)\right]_{t = 1}, $$ where $Z_{\Gamma}$ is renormalization factor which ...
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1answer
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Asymptotic freedom in QCD

From renormalization group equation $$ t \frac{d \bar{g}(t , g)}{dt} = \beta (\bar{g}(t , g)), \quad \bar{g}(1 , g) = g $$ (here $t$ is momentum scale factor, $g$ is initial coupling constant and ...
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Constants of infinity [duplicate]

Many times in physics when we analyze a physical system mathematicly we get divergences, but when those divergences has no dependence on any actual physical quantity of interest we tend to disregard ...
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A Conceptual Problem With the Field Equations of General Relativity

I have two questions: Suppose that we have an amount of energy in the form of a perfect fluid in the right hand side of Einstein field equations (energy momentum tensor), this will lead to a ...
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Lack of scale in Schrödinger equation for square-inverse potential

I see that if we set our potential in schrodinger equation to be a inverse-square dependence we don't have a typical unit of length as we have for hydrogen atom. $$-{\hbar^2\over 2m}\nabla^2\psi + ...
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140 views

Is there a way to obtain an RG flow equation for Quantum spin systems using MERA

We restrict ourselves to ground states of translationally invariant 1d quantum systems. I understand that there is the scale invariant MERA(multiscale entanglement renormalization ansatz) which ...
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365 views

Question about infinite sum in quantum field

I read from some books of number theory that $$\sum_{n=1}^{\infty}\frac{1}{n^s} = -\frac{1}{12}\text{,when } s=-1.$$ Now is there such a result $$\sum_{n=1}^{\infty}\frac{1}{n^s} = \pi \text{,when } ...
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Effective Field Theory (EFT) decoupling top

The decoupling theorem of Appelquist-Carazzone says that if you want to decouple a particle, the low energy resulting theory need to be renormalizable. You can't do that for the top, because you break ...
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Logarithmic discretization in Anderson´s model

Is there some motivation for the construction of Ladder operator that compound the recursive halmitonian of the Anderson model for numerical renormalization contained is this paper?
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Toy models of asymptotic safety?

Are there some toy model QFTs where the asymptotic safety scenario is realized?
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1answer
188 views

Fermion Self-Interaction

I'm trying to think of a theory with a Fermion self-interaction, similar to the $\phi^4$ theory. The first difficulty is of course that such a theory would have a non-renormalizable mass dimension: ...
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138 views

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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$U(1)$ abelian/axial/chiral anomaly in 4D

I am reading $U(1)$ abelian/axial/chiral anomaly in 3+1 dimensions using the path integral method (Fujikawa). Am I wrong in assuming that the anomaly can be cancelled by introducing a counter term in ...
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145 views

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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1answer
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How non-abelian gauge coupling runs below confinement or QCD scale?

I know the famous beta function of asymptotic free, but that seems describe the running coupling beyond confinement/QCD scale so that a perturbative analysis can apply. But how coupling runs below ...
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126 views

Where does this delta of zero come from?

It is common when evaluating the partition function for a $O(N)$ non-linear sigma model to enforce the confinement to the $N$-sphere with a delta functional, so that $$ Z ~=~ \int d[\pi] d[\sigma] ~ ...
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Massless $\phi^3$ theory in $d=6$ dimensions

I am asked to calculate renormalization for a massless $\phi^3$ theory in $d=6$ dimensional space using dimensional regularization. I'm having trouble finding the three-point vertex correction as one ...
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Another Power Counting/ mass dimension question

Are the mass dimension of the Dirac field different from those of the Klein-Gordon field, or is this just another issue of "cannonical normalization?" For instance if $\mathcal{L}_{KG}=\int ...
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O(N) sigma model at large N

I would like to better understand the main principles of large-N expansion in quantum field theory. To this end I decided to consider simple toy-model with lagrangian (from Wikipedia) $ \mathcal{L} = ...
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Renormalization of diagrams in QFT [duplicate]

Can any one suggest a good reference for studying renormalization of disjoint, nested and overlapping divergences in Feynman diagrams (for example, $\Phi^4$ theory)?
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One more time about LSZ-theorem

This question is the continuation of this one. For simplicity, let's use $(1)$ from the linked question (it is called n-point Green function and in particle case coincides with internal diagram), $$ ...
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LSZ reduction theorem derivation in Weinberg QFT

When deriving LSZ reduction theorem Weinberg in his QFT book have assumed n-point generalized Green functions, $$ G(q_{1},...,q_{n}) = \int d^{4}x_{1}...d^{4}x_{n}e^{-i\prod_{i =1}^{n}q_{j}x_{j}} ...
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Renormalizability of standard model

I'm wonder what precisely is meant by the renormalizability of the standard model. I can imagine two possibilities: The renormalizability of all of the interaction described by the Lagrangian before ...
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One question about renormalization

The idea of renormalization of "naked" perturbation theory is in principal possibility of addition counterterms which reduce infinity when calculating matrix elements. But I have met such concepts as ...
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291 views

Massless $\lambda \phi^4$ QFT

The $\lambda \phi^4$ quantum filed theory is the textbook example (which probably cannot be constructed nonperturbatively; I'm purely interested in perturbation theory). However, usually one treats ...
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power counting and (superficial) non-renormalizability

Comment: This stuff is new to me so it doesn't entirely make sense (yet). Question: As I understand from Peskin and Schroeder chap 10 if you have a theory with interaction terms $\lambda \phi^n$ in ...
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Perturbative vs. non-perturbative approaches to a well-defined Yang-Mills theory in 4 dimensions

Another question regarding the Yang-Mills Existence and Mass Gap problem (http://www.claymath.org/sites/default/files/yangmills.pdf). Does the problem require that the "construction" of a four ...
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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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Naive unification of scalar QFT and GR is possible?

I am thinking on the Klein-Gordon equation with curved (non-diagonal) metrics. Is it possible? Doesn't have it some inherent contradiction? If yes, what? If no, what is this combined formula?
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How to cancel infinite mass corrections for quantities without counterterms?

I'm trying to understand how infinite mass corrections are cancelled for a particle that is massless at tree level. In short the problem is that we have infinite diagrams, but we don't have a ...
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Finite corrections to the Higgs mass in SUSY

I am worried here about specific supersymmetric scenarios in which an energy scale above the TeV is introduces. An example would be the Seesaw extension of the Minimal Supersymmetric Standard Model. ...
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318 views

Difference between regularization and renormalization?

In my studies on quantum field theory we have come up with the concepts of regularization and renormalization. I'm a little confused about these two. In my understanding regularization is a way to ...
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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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Do divergent parts cancel out between the 1-loop contribution to the vertex and the self-energy on the electron legs of the vertex diagram?

I regret I don't know how to draw Feynman diagrams here, so I refer to the standard book Peskin-Schroeder. At the beginning of section 7.5 on page 244 of this book several Feynman diagrams are shown, ...
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Anomalies from a Renormaization Group Equation (RGE)

This is an approach to anomalies which seems unfamiliar to me.. Firstly what is this function $W$ which seems to satisfy the equation, $\frac{\partial W }{\partial g^{\mu \nu} } = \langle T_{\mu ...