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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Is there a block spin renormalization group scheme that preserves Kramers-Wannier duality?

Block spin renormalization group (RG) (or real space RG) is an approach to studying statistical mechanics models of spins on the lattice. In particular, I am interested in the 2D square lattice model ...
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Wilsonian Renormalisation — Peskin & Schroder Sect. 12.1

I'm working my way through Peskin & Schroeder, but some of the details of the calculations done in their introduction to the renormalisation group are slipping past me. For concreteness, the ...
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50 views

Problem with RG beta function in toy model

In the article "A Hint of Renormalization" (https://arxiv.org/abs/hep-th/0212049), the author considers a toy model in which a physical quantity (say, a scattering amplitude) $F$, depending on some ...
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73 views

Wick renormalization

I'm trying to understand the Wick renormalization in the framework of the Ito integral. I saw the Wick theorem as presented on Wikipedia in a QFT course and I would like to understand how that is ...
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489 views

Why Does Renormalized Perturbation Theory Work?

I've read about renormalization of $\phi^4$ theory, ie. $\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-m^2\phi^2-\frac{\lambda}{4!}\phi^4\,,$ particularly from Ryder's book. But I am ...
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Bound states in two and three dimensional delta potential in non relativistic QM

I would like to find bound state energies in let's say 2D delta function potential. So my eigenvalue equation is: $$(-\frac{1}{2}\Delta - g\delta(r)) \psi = -B \psi$$ and by the means of Fourier ...
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In QED/Yang Mills, why do fermions contribute 4 times as much as scalars to vacuum polarization?

Consider a Yang-Mills theory in $4D$ over a gauge group $G$ $$ \mathcal{L} = - \frac{1}{4} F^{a\mu\nu}F_{\mu\nu}^a + \bar \psi i D_\mu \gamma^\mu \psi + (D_\mu \phi)^\dagger D^\mu \phi $$ where $\...
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1answer
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Renormalization group invariant objects of a quantum field theory

Consider an arbitrary QFT with $g_b$ as the bare coupling constant. After dimensional regularization, is $g_b \mu^\epsilon$ a renormalization group invariant object of the theory? In other words, is ...
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41 views

renormalization and set theory

I am in high school, thus most of what I know about the topics I ask comes from popular science books, which risks asking some dumb questions, because I rarely understand the math behind. It is my ...
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The relation between anomalous dimensions and renormalization constants

I am trying to understand the general strategy and technical details of calculating $\beta$-function at higher orders. $\beta$-function is the anomalous dimension of the coupling constant and there is ...
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3answers
308 views

Is Bohmian Mechanics incompatible with loop corrections?

For those who continue to be unsatisfied with Quantum Mechanics (QM), Bohmian Mechanics (BM) is an alternative worth considering. It is sometimes claimed that BM is equivalent to QM, but Lubos Motl ...
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Equality of renormalized coupling constant

I want to show, that the renormalized coupling constants of a SU(N) Yang-Mills field with fermions included, are all equal. In the most textbooks it is written, that this could been shown by the Ward-...
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Logarithms in Renormalization

I am learning renormalization in Quantum field theory and following mainly Schwartz (Quantum field theory and standard model) for it. While explaining Renormalization group equations it says it mainly ...
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1answer
82 views

How to understand the idea of functional renormalization group?

I have been looking at how to use the functional RG method in many-body systems, but I don't quiet get the idea of it, it look different from Wilson's RG approach (eg. why shall we integrate out the ...
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295 views

How can dimensional regularization “analytically continue” from a discrete set?

The procedure of dimensional regularization for UV-divergent integrals is generally described as first evaluating the integral in dimensions low enough for it to converge, then "analytically ...
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Superficial degree of divergence for scalar theories

I have a few questions regarding the derivation of the degree of divergence for feynman diagrams. The result is $$D = [g_E] - \sum_{n=3}^{\infty} V_n [g_n]$$ (following notation in Srednicki, $P118$) ...
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1answer
100 views

What does it mean by “infinities” when dealing with QFT? [closed]

I found this PDF online here while browsing Nobel Prize winner contributions, which explains a bit about renormalization (a concept for which Kenneth G. Wilson won the Nobel). However I was somewhat ...
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Renormalization Point for Coulomb Potential?

In Introduction to Quantum Field Theory by Matthew Schwartz at page 177 he explains that we use the renormalization point $p_0=0$ in order to derive Eq. 17.54: \begin{equation} \tilde{V}(p)= \...
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45 views

Structure of Mass Renormalisation

I'm currently working on the renormalisation part in Peskin, Schroeder QFT. There it is stated that non-logarithmic UV divergences give a mass renormalisation and thus are forbidden, e.g. for the ...
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Decimation of a triangular lattice [closed]

Consider the network of spins shown below. The Hamiltonian is given by $$H = - \sum_{\langle i j k \rangle} [J \sigma_i \sigma_j \sigma_k + J_0]$$ with $J,J_o \geq 0$ and $\langle i j k \rangle$ ...
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Divergent diagrams in QED

I was reading about how to choose divergent diagrams in QED by using the concept of Superficial degree of divergence. We have an empirical relation $$ D= 4-E_b -\frac{3}{2}E_f $$ where $E_b$ is number ...
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Is a lagrangian with a background field interaction renormalizable ? If yes, when?

Consider the Lagrangian, $$ L = -\partial_{\mu} \chi \partial_{\nu} \chi^{\dagger} - m^2 \chi \chi^{\dagger} + g\chi \chi^{\dagger}\phi,$$ where $\phi$ is a background field and $\chi$ is a complex ...
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1answer
67 views

Cutoff-dependent “inverse propagator” for renormalization

In Zee's QFT in a Nutshell, when introducing mass renormalization, he calculates the "inverse propagator" for a $\phi^4$ scalar field theory to order $\lambda^2$ by considering the two diagrams shown: ...
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Why would renormalization be necessary without divergent integrals? [duplicate]

Weinberg uses the LSZ reduction formula to introduce field renormalization,and on page 441, he says: As this discussion should make clear: the renormalization of masses and fields has nothing to ...
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Resummation of large logarithms in effective field theory

I am trying to learn about the ideas of large-logarithms in effective field theory from a set of notes. However the following paragraph is confusing me: This doesn't seem quite right to me. Surely ...
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51 views

Running coupling, effective potential and the stability of vacuum

Consider the potential $$V(\phi)=\frac{1}{2}\mu^2\phi^2+\lambda\phi^4$$ where $\phi=\phi(t,\textbf{x})$ is a real scalar field. Let, $\mu^2<0$ and $\lambda>0$ then the potential is bounded from ...
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What is the problem with quantizing GR in the Effective Field Theory approach?

In the modern view due to Wilson, the cut-off $\Lambda$ is an intrinsic property of a theory and renormalization just means that the theory is invariant under scale transformations below $\Lambda$. ...
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66 views

1-Loop Mass Splitting of vector-like Fermions

In this paper the author argues that for a vector-like fermion doublet, with degenerate mass $M$ at tree level, we always have a mass splitting between the charged component of the doublet $L$ and the ...
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56 views

Questions about the beta function in QFT

When someone defines $\beta(g)=\mu\frac{dg}{d\mu},\quad (1)$ he is implicitly assuming that the result of the rhs of this equation can be written only in terms of $g$ instead of $\mu$, which is not ...
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67 views

Why is quantising gravity so difficult? [duplicate]

Since gravity is so similar with the Yang-Mills theory, the Christoffel connection is the gauge potential, the Riemann curvature is the field strength, then why is quantising gravity so difficult when ...
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What should I think of a diverging beta function (in Renormalisation Group flow)?

I have written a set of RG flow equations using Functional Renormalisation Group methods. I am looking at the flow of a well known problem with an additional original coupling. I did not do anything ...
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77 views

Meaning of counterterms and quantum corrections

When one talks about the “classical Lagrangian” of a field, does one mean the tree-level Lagrangian with physical masses and physical couplings? If yes, does it therefore mean that the bare Lagrangian ...
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81 views

Dimensional Regularization for $\phi 4$ theory

When using dimensional Regularization for $\phi 4$ theory to calculate the running of the mass, why there is no quadratic divergence as expected? For details, cf https://workspace.imperial.ac.uk/...
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Numerically extracting free energy in renormalization procedure

I have some doubts about how to apply real space renormalization numerically. I understand the theoretical concept, and how we require $Z'=Z$, being $Z'$ the partition function of the renormalized ...
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129 views

What is “the scale at which a theory is defined”?

I'm trying to learn the renormalization group, but I am confused about renormalization schemes. The general idea of RG is that physical predictions are independent of "the scale at which a theory is ...
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How do people go about looking for asymptotic safety in quantum gravity?

Do we have (proposed?) methods to look for fixed points in the renormlaization group flow of the Einstein-Hilbert action? My understanding of the RG is still somewhat sketchy at this point and I am ...
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59 views

Computing the pole mass from a given $\overline{MS}$ mass?

Given a Yukawa coupling as a function of scale $\mu$ and a vev, therefore $m_R(μ)=Y(μ)⟨ϕ⟩$, how can I compute the corresponding pole mass $m_p$? Relations I was able to find are (page 39) $$m_p=m_R−Σ(...
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1answer
86 views

Renormalisation group equation for Green's functions

The renormalization group equations for the $n$-point Green’s function $$\Gamma(n) = \langle \psi_{x_1} \dots \psi_{x_n}\rangle $$ in a four-dimensional massless field theory are $$\mu \frac{d}{d \mu}...
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Mass of an imaginative stable top quark at low energies? [duplicate]

Masses are not fixed quantities, but change with energy because of the renormalization group running. These can be calculated in a given renormalization scheme, for example, the MS scheme: Imagine ...
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Top quark mass $m_t$ at energy scales $\mu < m_t$?

Edit - Maybe formulated differently: Does it make sense to talk about the top mass at energies below $m_t$, although in all processes the corresponding energy scale is above $m_t$, because of the rest ...
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1answer
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On the c-theorem

I have been reading a few papers on CFT and AdS/CFT regarding the c-theorem and I have a few questions regarding c-theorems: a) Why is it that the c-theorem is usually considered for only unitary ...
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Continuous renormalisation

In Quantum Field Theory and the Standard Model, M.D. Schwartz talks about how Wilsonian renormalisation relates to continuous renormalisation. He states that continuous renormalisation looks a ...
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W and Z boson masses' running neglected?

Since the Lagrangian mass term of $W$ boson involves the bare coupling $g$, it cannot be the measured mass. Then the measured mass will "run" with momentum transferred. But everywhere I look the "...
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Understanding the $\phi^4$ phase diagram

I'm having trouble making sense of this phase diagram. The model is a $V(\phi)=g_2 \phi^2+g_4\phi^4$ scalar field theory. Here's what I think I understand: the capital letters represent different ...
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62 views

Callan-Symanzik equation for the QCD scattering cross section of the $e^{-}e^{+}\to q\bar{q}$ process

In Peskin and Schroeder (Section 17.2) it is stated without derivation that the scattering cross section for the $e^{-}e^{+}\to q\bar{q}$ process obeys the following Callan-Symanzik equation: $$ \left[...
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Energy scale dependence of coupling constants

I am trying to understand the meaning of the renormalization group equation and what i have understood is that, since observable (or physical?) quantities must not depend on arbitrary energy scales, ...
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Superficial degree of divergence on Weinberg

Reading volume 1 of Weinberg's QFT book, chapter 12, page 505 he says that if you consider a diagram with degree of divergence $D\geq{}0$, its contribution can written as a polynomial of order $D$ in ...
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Field strength renormalization: going outside of Fock space? [closed]

When one talks about field strength renormalization, one defines the renormalized field $\psi^R(x)$ in the following way (I'm using the notation from Matthew Schwartz's book): $\psi^R(x) \equiv \frac{...
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“Irreversibility” of the RG flow

In his remarkable work, Zamolodchikov proved a theorem regarding two dimensional QFT Renormalization Group (RG) flow, describing a monotonically decreasing function in the flow parameter which is ...
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Linear term in six dimensional $\phi^3$-theory

In our current QFT homework we are given the following Lagrangian in six spacetime dimensions. It is $$ \mathcal L = \frac12 [\partial \phi]^2 - c_0 \phi - \frac{m_0^2}2\phi^2 - \frac{g_0}{...