Questions tagged [renormalization]

This tag is for questions which relates with the renormalization, an ensemble of techniques which serves to treat the infinities which appear in quantum field theory or statistical mechanics. Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values.

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0answers
70 views

Why the divergent part of the 1-loop correction for the photon propagator *cannot* depend on the fermion mass $m$?

Last week my QFT II professor claimed that the divergent part of the diagram of the one-loop correction to the photon propagator by means of a fermion cannot depend on the fermion's mass. I haven't ...
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Vacuum polarisation in QED - why is it significant to renormalisation?

I have followed along for the derivation of the amplitude of the 2-photon vacuum polarisation and the book says the result is important for the renormalisation of QED, why is this?
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1answer
42 views

Homogeneous solutions in the renormalization-group pertrubation method

I am trying to get a handle on the renormalization-group approach to perturbation theory. I understand the overarching approach, but I'm stumbling on the mechanics of early steps when actually doing ...
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1answer
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A relationship between the proof of a renormalizability and gauge fixing conditions?

I already know that QCD is renormalizable in several gauges, including the $\xi$ gauge and the background field gauge. That is, the divergence of the quantum effective action is limited by symmetry, ...
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"Absorbing" UV divergences

What does it mean when someone writes that a finite number of parameters "absorb" the UV divergences of a perturbatively renormalizable theory? Is there a physical interpretation to this?
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What renormalisation scheme do we use for Standard Model masses?

On this Wikipedia page, the free parameters of the Standard Model are listed. The renormalisation scheme is given for the quark masses and gauge couplings. However, for the other parameters, the ...
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1answer
64 views

What justifies regularization with a high-momentum cutoff?

Before renormalizing a perturbative series, during the regularization step when we insert a high-momentum UV cutoff, what justifies this step given that it's only formal and does not have a physical ...
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Perturbative renormalization as re-parametrization

I've read that the source of UV divergences in perturbative renormalization is due to a bad choice of parametrization. But since we have some freedom over how to reparametrize, and some of them are &...
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Wouldn't a simple scalar field fix the non-renormalizability of gravity?

It is well known that quadratic gravity is renormalizable. On the other hand it is possible to transform the partition function of Einstein-Hilbert + free minimally coupled complex scalar field into a ...
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21 views

Epstein-Glaser renormalization and perturbative renormalization

I am trying to understand the differences between perturbative renormalization and Epstein-Glaser renormalization, which conceives of renormalization as the extension of distributions. What is ...
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54 views

Beta function for $U(N)$ Yang-MIlls?

What is the one-loop beta function $\beta(g)$ for $U(N)$ pure Yang-Mills? I expect it to behave rather differently than $SU(N)$, since when $N=1$ we have electrodynamics, for which $\beta(e)=0$. As a ...
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Multiplying distributions for QFT

My understanding is that UV divergences arise due to improperly handling the product of distributions. In what sense is it "improper"? And how does its proper handling relate to the notion ...
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1answer
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In Srednicki's book, when calculating loop corrections to the propagator, why doesn't he include both diagram topologies at second order?

This might be a somewhat basic question, so apologies in advance for that. I've only recently started learning QFT, and so I'd really like to make sure I understand this. In Srednicki's textbook, in ...
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$\beta$-function in non-Abelian scalar QCD

I study non-Abelian scalar QCD in $d = 4 - \epsilon$ dimensions (using dimensional reduction). That is, I consider complex scalars, $\Phi^i$ with $i \in \{1, ..., N\}$, that satisfies $U(N)$-symmetry, ...
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1answer
50 views

Symmetry protects against symmetry-breaking counterterms in renormalization

My lecturer said that when we renormalize a theory (any theory, not necessarily a renormalizable one) we can do so by adding counterterms to the original Lagrangian $\mathscr{L}_B$, turning it into $\...
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Radius of convergence of beta function of fine structure constant

I'm looking at the beta function of the fine structure constant $\alpha=\frac{e_R^2}{4\pi}$ \begin{equation} \beta(\alpha)=\mu \frac{d\alpha}{d\mu}=-2 \alpha \left[ \frac{\epsilon}{2}+ \left(\frac{\...
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Why doesn't the gluon fusion channel for Higgs production diverge?

The heavy quark triangle integral has a superficial degree of divergence of $D=1$, so one would naively expect it to diverge (more than logarithmically, in fact) in 4 dimensions. It happens, though, ...
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How does beta-function describes coupling constant? [duplicate]

I can't understand what does Beta-function (in QFT), that describes coupling constant, actually says. It's said that $$ \beta (g)= \frac{\partial g}{\partial log( \mu )}. $$ But how do we determine ...
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1answer
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Strings in background fields: how to prove that $\beta^G_{\mu\nu}=\beta^B_{\mu\nu}=0 \implies \nabla_\nu \beta^\phi = 0$

In many String Theory texts (e.g. Polchinski), when discussing the bosonic string in presence of background fields $G_{\mu\nu}$, $B_{\mu\nu}$ and $\phi$, respectively symmetric and antisymmetric ...
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For real emission graphs can we put the additional photons also in the initial state?

To compute the NLO for e.g. $e^+e^- \rightarrow \mu^+\mu^-(+\gamma)$ in the Schwartz book, we need to compute the real emission graphs for $e^+e^- \rightarrow \mu^+\mu^-\gamma$ to cancel the IR ...
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1answer
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Functional derivatives on position and momentum spaces

I'll first give some context for the problem I'm having, but the essence of it seems to be related to only what is in the title. I've been working with the Wetterich equation for the Functional ...
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46 views

Connection between Symmetry Breaking and RG flow

I was wondering if this interpretation of how RG flows connects to the conventional symmetry breaking description of phase transitions is correct. Say I am an experimentalist and I move through ...
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Renormalization group applied to a simple QCD problem

Consider page 551 of Peskin's book on QFT, where he is treating the process $e^{+}e^{-} \to \text{Hadrons}$. I have 3 misunderstandings regarding some lines of thought effected there: 1) Peskin argues ...
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Why do the critical exponents at the Gaussian fixed point coincide with mean field theory?

In the Ising model, we know that in dimensions higher than the upper critical dimension, $d_u=4$, the critical exponents can be found from mean field theory. We also know that for the same dimensions, ...
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How do I calculate the Altarelli-Parisi splitting function for $g \rightarrow q\bar{q}$?

I'm trying to calculate the Altarelli-Parisi splitting function in the collinear limit for a gluon splitting into a quark-antiquark pair, but I keep getting stuck. Let $p$ and $k$ be the momenta for ...
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1answer
167 views

Perspective on the renormalization group

From reading the renormalization group description from high energy theory texts like peskin & schroeder, one may be tempted to think it has to do with regulating infinities. However, my ...
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Confusion about solution of the Callan-Symanzik Equation

In the QFT textbook by Peksin and Schroeder, they apply a hydrodynamic-bacteriological analogy to derive the solution of the Callan-Simanzik equation. Yet I'm confused with the integration boundary on ...
3
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1answer
66 views

Beta-function for mass

I'm trying to find the renormalization group fixed points for a free, massive, scalar theory \begin{equation} S = \int_{\mathbb{R}^d}d^dx \left( \frac{(\partial_\mu\phi)^2}{2} + \frac{m^2\phi^2}{2} \...
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What are the conditions for RG flows to have strange attractors?

this question served as a great reference for RG flows that can end up with more complicated dynamics than fixed points, such as limit cycles. As far back as K.G. Wilson's 1971 original RG paper, ...
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Could the divergences in loop Feynman diagrams be resolved by applying an exponentially decreasing probability eg fluctuation theorem?

Since the divergence originates from having to integrate over an infinite range of intermediate energies, surely the bigger the temporary energy delta, the less likely it will be, and the shorter the ...
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1answer
814 views

What does mathematical consistency in QFT mean?

My question is more naive than Is QFT mathematically self-consistent? Just when people talk about the mathematical consistency of QFT, what does consistency mean? Do people want to fit QFT into ZFC? ...
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1answer
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Why would the validity of UV/IR decoupling imply the explicit constructability of black hole remnants?

A paper written in 2020 by Harlow and Shaghoulian (reference 1) proposes a connection between unitary black hole evaporation and the non-existence of global symmetries in quantum gravity. In passing (...
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1answer
110 views

QFT and divergences: what makes the finite part be regularization-independent?

It seems that the "finite part" of divergent loop integrals are the same, irrelevant of the regularization scheme used to regulate the integrals - why is this? Consider the following ...
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Does choice of renormalisation scheme affect the consequences of Haag's theorem?

So Haag's theorem means that the interaction and Hamiltonian picture are not equivalent. The reason seems to be that renormalization mixes interactions and free particles (ie self energy of a free ...
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0answers
47 views

1-loop diagrams in Scalar Yang-Mills

Disclaimer: I've been calculating the renormalization constants $Z_i$ for the ScalarQED seen as the abelian limit of the Scalar Yang-Mills, and I know that I've made some mistakes because I find the ...
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0answers
28 views

Is the set of every renormalization group countable and finite?

Is the set of every renormalization group countable and finite? Suppose A is a renormalization group, and the elements of it compose of the set B. Is B the set countable and finite?
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1answer
61 views

Does the Slavnov-Taylor identity still hold for scalar Yang-Mills?

I want to renormalize the minimally-coupled scalar Yang-Mills theory: $$\mathcal{L}_{YM\phi}=(D_\mu\phi)^\dagger(D^\mu\phi)-\frac{1}{4}F_{\mu\nu}^a{F^{\mu\nu}}^a-\frac{1}{2\xi}(\partial_\mu {A^\mu}^a)^...
2
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1answer
74 views

1-loop renormalization of general scalar in general dimension

I'm interested in studying the theory $$S=\int d^dx\left(\frac{1}{2}(\partial\phi)^2-\frac{g}{(2k)!}\phi^{2k}\right)$$ in $d=2, 3, 4$ and for $k=2, 3, 4\dots$ and I'm having trouble with the the 1-...
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2answers
101 views

Functional Renormalization Group and Dirac Fermions — Yukawa Theory

I've been practicing with FRG techniques and I wanted to obtain the usual beta functions for Yukawa theory using the Wetterich equation. However, this has been more troublesome than I expected. If I'm ...
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0answers
51 views

What, exactly, is the difference between a 'gauge coupling constant' and ones like the Fine Structure Constant?

Several times in my reading I have come across mentions of gauge coupling constants, usually denoted $g_1$, $g_2$ and $g_3$, which have different values than the Gravitational Coupling Constant $G$ or ...
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1answer
77 views

Renormalization of $\phi^2(x)$

In order for observables like scattering amplitudes to be finite, one must implement a renormalization scheme for parameters like the mass $m$, the coupling $\lambda$, and the field $\phi$. Question: ...
6
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1answer
209 views

Universality and quantum gravity

Here is apparently a quote by Jacques Distler from his blog https://golem.ph.utexas.edu/~distler/blog/archives/000612.html: “I started off by recounting the tale of Howard Georgi, back in 1982, ...
2
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1answer
85 views

Do Tadpoles Contribute to Self-energy?

In evaluating contributions to the two-point function in say $\phi^3$ theory to: $$\langle 0|\phi(x)\phi(y)e^{-i\int d^4z\frac{\lambda}{3!}\phi^3(z)}|0\rangle,$$ at $\mathcal{O}(\lambda^2)$, one of ...
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0answers
57 views

Connection Between Renormalization Group and Phase Transitions

I have a couple of questions on the relation of RG and phase transitions. I've heard in many sources that the theory of most transitions (excluding novel phase transitions like Quantum Critical ...
3
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1answer
106 views

Does every Renormalization Group fixed point correspond to a phase of the system?

(Sorry if this has been asked before but I can't find a similar question on here) My bachelors thesis had some connections to renormalization group theory. I covered some theory briefly and used the ...
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1answer
71 views

$\phi^4$ theory in 5 dimensions

$\phi^4$ theory is not perturbatively renormalizable in 5 dimensions. I have come across literature where renormalizability is discussed w.r.t $N$, for fields obeying $O(N)$ symmetry. But it is not ...
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How does one usually deal with this kind of loop processes?

Now, I ask this because of the propagator of the inner line of the lepton going through a loop. Let's assume the initial Z has a momentum $p$, and the outgoing leptons have a momentum $p_1$ (for the ...
2
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1answer
103 views

Physical intuition for bare quantities to be infinite

I don't know much about interacting field theory, as far as I understand, in the interacting field theories, when doing renormalization, one usually assume the bare quantities are infinite to cancel ...
3
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0answers
93 views

How to compute the beta-functions given the euclidean action for two scalar fields?

Given the Euclidean action for two scalar fields in $d$ dimensions: \begin{equation} S_E = \int d^dx\frac{1}{2}((\nabla\phi_1)^2 + \nabla\phi_2)^2) + \frac{\lambda}{4!}(\phi^4_1 + \phi^4_2) + \frac{2\...
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2answers
92 views

Single-vertex tadpole self-loop in QED

I'm starting to learn the basic Feynman rules of QED, and applying them to basic processes to first order in $\alpha=\frac{e^2}{4\pi}$. This includes things like Møller-, Bhabha- and Compton ...

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