Questions tagged [renormalization]

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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Does the action remain dimensionless after the renormalization?

After the renormalization procedure, fields will gain an anomalous dimension, $\gamma$, which means that their scaling dimension will be different from what we would guess from the dimensional ...
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Why do theories of nature prefer to be renormalizable and not super-renormalizable?

It seems to me (correct me if I am wrong) that all theories in the Standard Model are exactly renormalizable, as opposed to non-renormalizable or super-renormalizable. In a sense, we could say that ...
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Interpretation of Feynman propagator for massive scalar field in position space

I've always treated propagator in the momentum representation so when it diverges, we are on-shell. But what is the interpretation of light-cone divergences in position space? If it is something we ...
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How is $\Lambda_{\textrm{QCD}}$ relevant in the non-perturbative regime?

The famous $\Lambda_{\textrm{QCD}}$ parameter enters through the one-loop running of the QCD coupling, through a relation similar to the following: $$\alpha_S(Q^2)=\frac{\alpha_S(Q^2_0)}{1+b\ln(Q^2/Q^...
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The non-singular term in the transformation of the free energy per lattice site (Or: Is Cardy wrong about the RG transformation)

In Scaling and renormalization in statistical physics by Cardy on page 44 he asserts that the free energy per lattice site defined by $f(\{K\})\equiv-N^{-1} \ln Z$ transforms as $$f(\{K\})=g(\{K\})+...
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Rescaling of effective hamiltonian coupling constants in the Wilsonain renormalization group

I am confused about an aspect of coupling constant rescaling in the Wilsonian renormalization group procedure. (I'm following Kardar's "Statistical Physics of Fields, Ch5). I think I understand the ...
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Is there a difference between renormalization and renormalization group?

Is there a difference between renormalization and renormalization group? In his book 'Scaling and Renormalization in Statistical Mechanics', John Cardy states the following about the term ...
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Question about Renormalization of Ghosts in QCD in MS scheme

During a calculation of the Renormalization constant of the ghosts in QCD I stumbled over the following question: When I calculate the self-energy of the Faddeev-Popov ghosts in $SU(N)$ non-abelian ...
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Electrical conductivity of complex scalar field

I want to compute the conductivity of a complex scalar field in both the symmetric and symmetry broken phases. To define the current operator one needs to couple the matter field to EM gauge field ...
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Pauli Term contribution to muon' s anomalous magnetic moment

I'm trying to compute the contribution of a non-renormalizable term in the QED lagrangian to the muon's anomalous magnetic moment. The term in question is: $\frac{ie}{\Lambda}\bar{\psi}\sigma^{\mu\nu}\...
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Renormalization group in statistical mechanics: (1) rescaling of parameters and (2) calculating the free energy

I have some questions about the momentum space renormalization group procedure as described in the textbook "Statistical Mechanics of Fields" by Kardar (Ch5). The first is about the rescaling of ...
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Vertex correction in QED

I've been working through the chapters in Schwartz on the renormalisation of QED, and I have some confusion to do with the form of the Vertex correction. By my understanding, the correlation function ...
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Numerical renormalization of 2D Ising lattice

I'm trying to make some toy computations on the $2D$ Ising model on a square lattice. I want to apply a renormalization transformation, and try to estimate observables on the renormalized lattice ...
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Doesn't the massless $g\phi^4$ theory bound to have an infrared fixed point?

A free, massless scalar theory, $\mathcal{L}_1=\frac{1}{2}(\partial\phi)^2$, is scale-invariant both classically and quantum mechanically. However, a $g\phi^4$ theory, $\mathcal{L}_2=\frac{1}{2}(\...
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$\phi^3$ theories in 2+1 dimensions

I quite often see papers considering a $\phi^4$ theory in three spacetime dimensions, but rarely do I see papers with $\phi^3$ terms. I understand that these kinds of interactions terms can have ...
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References on infraparticles in QED

I recently became interested in a notion of infraparticles as "true" scattering states in, for example, QED. It is well known that S-matrix elements in QED suffer from infrared divergences due to ...
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Computation of beta function for $N$ scalar fields and introduction of deformations

Consider the Euclidean field theory with $N$ real scalar fields $\phi_{i}$ with Lagrangian density: \begin{equation} L=\frac{1}{2}\partial_{\mu}\phi_{i}\partial^{\mu}\phi_{i}+\frac{1}{2}m^{2}\phi_{...
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A bit of confusion with central idea of “running” coupling constants

An effective quantum field theory of a single scalar field $\phi$ is described by an action, $S(\phi,\{g_n\})$ where $\{g_n\}$ denote the coupling constants of the theory. The corresponding path-...
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Defining upper critical dimension

Considering the usual Landau functional of the form: $$ \beta L[\phi] = \int d^D r [\frac{1}{2} |\nabla \phi(r)|^2 + \frac{r_0}{2} |\phi(r)|^2 + \frac{u_0}{4} |\phi(r)|^4 ] $$ In searching for the ...
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QFT-$\phi^6$ Theory-Counter Terms in 3D [duplicate]

I recently asked a previous question about renoramalisation in $\phi^6$ theory for which there was a great answer but I'm still confused about counter terms in the Lagrangian. I'm mostly confident in $...
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2D CFT and the $c$-theorem

Does the $c$-theorem imply that in two dimensions all RG-Fixpoints are CFTs? Or does it just imply that the monotonously decreasing function $c$ coincides with the central charge if the fixpoint is a ...
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Wilson's approach to renormalisation according to Peskin & Schroeder

Although Peskin & Schroeder treats Wilson's approach to renormalisation theory in some depth, I don't get one of its main points. According to P&S (p.401): Imagine that we wish to compute ...
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Quantum Field Theory - Interacting Scalar Fields

Is there an infinite number of interacting theories? Or is there a limit? For example, I know about $\phi^6$ theory, which is non-renormalisable in 4D spacetime, but I've never really gone beyond $\...
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An identity in Zamolodchikov's $c$ theorem paper

I have seen this mentioned in various papers/lecture notes/Stack Exchange questions [1-4]: $$\Theta = \beta^i(g) \Phi_i.$$ Here $\Theta$ is the trace of the stress tensor, $g=\{g^i\}$ is the set of ...
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Finding the critical temperature in the zero field Ising model via renormalisation

Inspired by images such as these, I would think that one can find the critical temperature in the zero-field Ising model computationally as follows: Start with any initial temperature $T$ and ...
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Confusion about dimensional regularization

I am recently trying to understand dimensional regularization in the context of quantum field theory. So to solve an integral $$ \int_{\mathbb R^d} \frac{\text d ^d p}{(2 \pi)^d} \frac{1}{(p^2 + m^2)^...
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Renormalization of mass: does it change sign from high temperature to low temperature?

Consider a Landau Ginzburg theory for ferromagnets with Hamiltonian $$H=\int d^{D} x \frac{1}{2}(\nabla\phi(x))^{2} + \frac{1}{2} \mu^{2} \phi^{2}(x) + \frac{\lambda}{4!}\phi^{4}(x)$$ I can compute ...
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Terms that appear in the RG flow with regards to symmetries of the Lagrangian

Consider a theory with Lagrangian $\mathcal{L}$ with a global $U(1)$ symmetry, and consider adding a deformation $$\mathcal{L}_g = g \mathcal{O}(x)$$ which breaks the $U(1)$ symmetry explicitly. ...
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Infrared divergences: total energy of soft photons

One way to deal with IR-divergences in QED is sum over soft photons, which can't be detected due to finite sensitivity of detector: only photons with energy bigger that resolution of detector $E>E_{...
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Resources for One-Loop Calculation in QED

I am recently trying to study the calculation on one-loop diagrams in QED. Since the most resources i found where rather cryptic since they where very very general i wanted to ask whether or not there ...
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Does lattice field theory also have infinity like perturbation before regularizations?

In loop corrections, the integrals are often infinity before regularization. The reason is attributed to the applicability of field theory. Does lattice approach also have infinity? If yes, where can ...
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How to get the solution of the QCD renormalization group equation at higher orders

In Peskin & Schroeder Exercise 17.1, it is asked to find the solution at higher order of the renormalization group equation with $\beta$-function at $$ \beta (g) = -\frac{b_0}{(4\pi)^2} g^3 - \...
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What are the scaling variables in the RG approach?

I am reading Cardy's "Scaling and Renormalization in Statistical Physics" and I am a little confused about a concept he introduces in section $3.8$. He starts off with a theory which has some ...
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Why is gravity the only force not quantizable only at high energies?

There are a lot of questions on this site about the quantization of gravity. Now what I do understand is, that we can quantize gravity, but only at low energies, and this is really different from all ...
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Is the starting distribution in the solution of DGLAP IR-bare?

On p.27 of this paper by John Collins, he says that when defining PDFs in terms of partonic number operators, one acquires an IR-divergent bare PDF (eq. 52). The residue of the IR-divergent term is ...
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Why are the renormalisation constants the same in LSZ and renormalisation?

To keep it simple I'll phrase everything in terms of scalar fields. We seem to have three constants called $Z$: When we do LSZ reduction we say, as $t\rightarrow-\infty$ then $\phi\rightarrow \sqrt{...
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Picture of poor man's scaling for AFM/FM interaction in Kondo problem

Poor man's scaling in Kondo problem For the Kondo model: $$H=-t\sum_{i,j}c_i^\dagger c_j+JS\cdot \sigma(0)$$ which only including itinerant electrons with the band-width $ W \in[-D,D]$, and $S$ is ...
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Why don't counterterms appear in one-loop correction for 1PI effective action?

In Zee's "QFT in a Nutshell: Second Edition", section IV.3, the author calculates the 1PI effective potential for a single real scalar field. The full Lagrangian is given by equation (1): $$\mathcal{...
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In renormalization group theory, why fixed point gives self-similar probability distribution

I got this question when learning Kardar’s course “Statistical Physics of Fields”. The description of this question can be found in Chapter 4 of Kardar's book. In his lectures, the renormalization ...
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Photon mass Infrared divergence regulization in the one-loop electron self-energy in QED

So basically I'm trying to calculate the one-loop mass and field strength counterterms from the electron's self-energy in QED using Pauli-Villars regularization (i.e. some heavy particle of mass $\...
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Fine structure constant and the renormalisation group

I read "The fine structure constant, alpha (α), describes how electromagnetic radiation affects charged particles. It has the numerical value 0.007297351" Given it relates to the electric charge ...
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In a QCD FO calculation, which divergences need to be reshuffled into the PDFs using factorization?

I am trying to understand how divergences of different sources are brought under control in fixed order QCD calculations of observables, specifically at NLO, for example of an inclusive hadronic cross ...
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The form of QED Vertex Correction

In chapter 6 of Peskin-Schroeder's text Introduction to Quantum Field Theory it is argued that the form of vertex correction for QED can only have the following form (since we have only the constants, ...
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Fixed Point of RG Flow of $\phi^3$ theory in 6 dimensions

I was calculating RG Flow equations for $\phi^3$ theory in 6 dimensions. The partition function and lagrangian are given below, $$Z = \int D\phi\ e^{-\int^{\Lambda} d^6x \mathcal{L}[\phi] }$$ $$\...
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Why do two vertex need to be located at the same position for momentum shell RG of $\phi^4$ theory?

When we evaluate the momentum shell RG for $\phi^4$ theory (assuming in Euclidean space): $$S[\phi]_{E}=\int d^{D} x\left[\frac{1}{2}(\partial \phi)^{2}+\frac{1}{2} r \phi^{2}+\frac{1}{4 !} g \phi^{4}\...
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What do the fixed points of a RG equation mean and what are its importance?

Can somebody explain to me what the fixed points of a renormalization group mean? What is their physical significance in the sense that why do we study them and what do we get to know from them?
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Renormalization in effective field theory [closed]

I have tried to solve these two questions. I have read different papers and textbooks in order to get an answer but all effort proved abortive. Someone should kindly assist in these equations.
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When is Schwartz's method for “integrating out” a field valid?

In Schwartz's QFT book, heavy fields are often "integrated out" by simply solving their equations of motion formally (i.e. allowing things like $\Box^{-1}$) and plugging them back into the Lagrangian. ...
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Where can I find Hollowood's lecture notes on cutoffs and continuum limits?

I'm trying to find a copy of Tim Hollowood's "Cutoffs & Continuum Limits: A Wilsonian Approach to Field Theory". These are unpublished lecture notes that I've seen dated to 1998 or so, and have ...

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