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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Confusion on the renormalizability and the dimension of coupling constant (From Srednicki's book)

I am trying to understand the renormalizability and the dimension of couplings from section 18 of Srednicki's QFT book. In section 18, it mentions if we can use finitely many new terms (counter-term) ...
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Normalization of a spin-2 state

I'm wondering what is the correct normalization for a spin-2 state. Imagine you have a quantum state $|\psi^{ab}\rangle$ with $a,b=1,2,3$. If this is traceless and symmetric it should have the right ...
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Proof that Wilsonian renormalization only generates terms consistent with the symmetry of the action

In the Wilsonian approach to renormalization it's easy to see that integrating out high momentum dofs in the path integral generates an infinite number of terms in the renormalized action. It's often ...
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Perturbative RG approach

I'm currently trying to understand Shankar's review (1994) on perturbative RG, and I'm slightly confused above the field theory approach of the $\phi^4$ theory (section II.D). Indeed, what Shankar ...
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Could higher-dimensional quantum field theories be obtained as continuum limits of lattice models?

Traditional lore, based mostly on lagrangians and perturbation theory, said that nontrivial quantum field theories exist only in $d$-dimensional spacetime with $d\leq 4$. Now the lore is changing: at ...
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Classification of couplings in continuum renormalisation

Wilsonian and continuum renormalization are distinct ideas. In Wilsonian renormalisation, we have a theory with a lagrangian, $\mathcal{L}$, defined up to an momentum scale $\Lambda$ with couplings $ \...
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98 views

Dependence by the renormalization scheme in the beta function coefficents in QCD

Given the coupling costant $\alpha_s$ of QCD and it's RGE equation $\frac{d\ln \alpha_s(\mu^2)}{d\ln\mu^2}=\beta(\alpha_s)$, with $\beta(\alpha_s)=-\beta_0\alpha_s-\beta_1\alpha_s^2 -\beta_2\alpha_s^3+...
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1answer
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Renormalization conditions for QED self-energies

I'm having trouble understanding renormalization conditions: what I know is that they are the conditions required so that at some point called "renormalization point" (which is usually just ...
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59 views

Transverse and longitudinal photon propagator

I'm studying QED renormalization and the Ward-Takahashi identity, and I'm having trouble understanding two things about the longitudinal and transverse parts of the photon propagator. What I ...
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Doesn't vector bosons enter in the formula for the calculation of beta function coefficients?

It is a well established result that, for a popular set of conventions, the lowest order $\beta$ function coefficient is calculated as $$ b=-\frac{11}{3}C_2(G)+\frac{4}{3}\kappa S_2(F)+\frac{1}{6}\eta ...
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Should counterterms enter the computation of amplitudes in loop level?

https://imgur.com/gallery/F5bWzSt I am currently studying the one-loop renormalization structure for $\phi^4$ theory and I can’t seem to understand what’s stopping me from making one loop diagrams (I ...
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$\phi^4$ theory in higher dimensions

For a scalar, the Lagrangian $$L = \partial_{\mu}\phi\partial^{\mu}\phi - m^2\phi^2 + \lambda \phi^4$$ seems to be particularly suited for 4 dimensional space-time. In four dimensions, the coupling ...
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What are clover fermions?

I've seen the term been used quite a lot when reading about lattice gauge theory calculations. So far what I've gathered is the following, from this source [1]. Lorentz invariance of the action is ...
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Doing chemistry with atoms or "Renormalization group on chemistry"

When I studied molecular mechanics we saw all of this in terms of electrons with classical nucleus. Is there a formulation in term of atoms-as-particles (ignoring internal structure, but maybe taking ...
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The 1-loop anomalous dimension of massless quark field for $SU(N)$ gauge theory with $n_f$ quark flavours

Considering $SU(N)$ gauge theory with $n_f$ massless quarks I want to find the anomalous dimension to order of 1-loop of the massless quark field, that defined by: $$\gamma_q(g^{(R)})=\frac{1}{2Z_q}\...
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Dimensionless vs. dimensional RG-Flow equations

When one writes down RG-Flow equations for any theory, at some point one encounters statements like "It is useful to properly rescale the above exact flow equations and rewrite them in ...
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Finding type of deformation for a given interaction to a Gaussian action in 3 dimensional Euclidian space

Consider the following 3 dimensional Euclidian Gaussian action: $$S_0=\int d^3x \left[ \frac{1}{2}(\partial_\mu\phi)^2 +\sum_{a=1}^N\bar \psi_a(\gamma^\mu \partial_\mu+\frac{1}{\sqrt N}\sigma)\psi_a \...
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RG flow of coupled two scalar fields

Considering an action in $1\le d<4$ dimensions that including deformation of a Gaussian action: $$S=S_0+\int d^dx[g_1 \mu^{\epsilon_1}\cdot\phi_1^4(x)+g_2 \mu^{\epsilon_2}\cdot\phi_2^4(x)+g_3\mu^{\...
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The interpretation of the quantum field

In QM we have always been told that for each quantum mechanical field there is an associated particle. This works in the free theory where from canonical quantisation we promote a field to a field ...
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Which quantities are expandable in $1/M$ in effective (quantum) field theories?

According to this Wikipedia article on Effective field theory, the effective field theories used in QFT can be seen as an expansion in $1/M$, where $M$ is a characteristic mass scale of a certain ...
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How AdS/CFT solves the non-renormalizability of gravity?

According to AdS/CFT, all gravity calculation in AdS can be mapped to CFT calculations on the boundary. Then what are the loop divergences in AdS gravity be mapped to the CFT side? Or put it ...
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Relationship between the bare charge and the elementary charge of the electron

I'm trying understand the philosophy of the renormalization. Sorry if my question is trivial, but I haven't found this in any book so far. Bare charge should be the charge of the particle without any ...
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Can one disregard tadpole diagrams even if an operator is being inserted?

I understand that the renormalization condition $\langle0|\phi|0\rangle=0$ permits one to disregard tadpole diagrams in most calculations due to the leg of the tadpole carrying zero momentum. If an ...
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113 views

Correlation function under RG flow

I got stuck in understanding how the correlation function changes under the RG flow. Consider that the correlation function of a scalar field $\phi(x)$ in $d$ dimension is that : \begin{equation} \...
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2answers
91 views

Why propagator pole is associated to the mass?

We say that the pole of the all-orders photon propagator, $$\frac{1}{q^2[1+\Pi(q^2)]}$$ doesn't shift if $\Pi(q^2=0)$ is regular. This amounts to say that the photon remains massless to all orders in ...
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Scaling dimension in statistical field theory

I got stuck in understanding the scaling dimension in statistical field theory. Currently I am reading the statistical field theory written by Prof. David Tong. In his note(p.63), it states that the ...
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Understanding classical massive real scalar $\phi^4$ Callan-Symanzik equation

Considering classical massive real scalar field theory by the action: $S[\phi] = \frac{1}{2}\int d^4x[\partial_\mu\phi\partial^\mu\phi-m^2\phi^2-\frac{g}{12}\phi^4] $ we assume the theory is ...
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1answer
57 views

RG flow diagram plotting

I want to be able to plot a flow diagram with a given recursion relation. For example, I have the follow recursion relation: \begin{align*} \frac{dT}{d\ell} &= 2T{y_0}^2 a^2 \\ \frac{d ...
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1answer
76 views

Is my interpretation of the fixed points of the renormalization group correct?

I would like to know wether or not I understood the meaning of the fixed points of the RG concerning the phase diagram of a system. This is how I understand it: Since a RG transformation leaves the ...
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58 views

Loop integral in $d$ dimensions

I am studying large $N$ Quantum Field Theory and I am having a hard time calculating the expansion of the simple loop integral of eq.(13.123) of Peskin and Schroeder. $$ \int\frac{d^dk}{(2\pi)^d}\frac{...
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Effective potential [closed]

Given that after renormalization, we can write: $$\Im\left(\mathcal{M} \right) = \Im\left(-\frac{-\lambda^2}{32 \pi^2} \int_{0}^{1} dx \ln(m^2 + p^2x(1-x)) \right)$$ The $(m^2 + p^2x(1-x))$ needs to ...
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80 views

Renormalization in strongly coupled theories without Lagrangian description

In a weakly coupled theories, it is quite transparent how the flow works since we have a path integral and then we integrate out fields to see the behaviour at different length scales. But if I don't ...
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Renormalization Group approaches to many-body problems: An algebraic perspective

This is related to a question I asked a few months ago here. I haven't looked at it again until recently because I was busy with something very different, but now I am coming back at it and wanted to ...
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In the derivation of LSZ formula, why do we need $\langle k| \phi(0)|0 \rangle =1$? (Srednicki's book)

In the section 5 of the book, it says The LSZ formula is valid provided that the field obeys $$\langle 0|\phi(x)|0\rangle=0, \langle k|\phi(x)|0\rangle=1.$$ The second one is needed to ensure one-...
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5d nonabelian gauge theory or 5d QCD explicit $\beta$-function

Question: Are there some derived $\beta$-function formula $$ \beta(g) = \frac{\partial g}{\partial \log(\mu)} $$ for nonabelian gauge theory in the 5-dimensional spacetime at some energy scale $\mu$ ...
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Why can we neglect the beta term in the Callan-Symanzik equation of scalar field theory

In Peskin p413, He writes In any renormalizable massless scalar field theory, the two-point Green's function has the generic form $$ \begin{aligned} G^{(2)}(p) &= \longrightarrow+(\text { loop ...
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Equivalence between wilsonian and non-wilsonian RGE on QFT

My current way of viewing Wilsonian RGE applied to QFT: (1) We start with a lagrangian that accurately models dynamics up to a scale $\Lambda_0$. (2) We fix $\Lambda_0$ as a cutoff to regulate the ...
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27 views

Wilsonian renormalization group effective interactions

My current way of viewing wilsonian RGE applied to QFT: (1) We start with a lagrangian that accurately models dynamics up to a scale $\Lambda_0$. (2) We fix $\Lambda_0$ as a cutoff to regulate the ...
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59 views

When does perturbative renormalization group (RG) fail?

In the typical perturbative RG scheme, one looks for fixed points by computing the beta function after truncating the interaction strength (say $\alpha$) to some desired order. In particular, we're ...
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Resources on calculation of beta function for $\mathrm{O}(N)$ model

Is there any useful introductory material about the calculation of the beta function in $\phi^4$ scalar theory and $O(N)$ model? I would also like to generalize this to the large-$N$ limit.
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The cross on Feynman diagrams

In figure (a), in the upper part of the loop, What is the meaning of the cross between smuon and the node of selectron and photino?
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Expansion of the $S$-operator and Normal Ordering

To phrase my question I will use the example, which we used as an exercise in my introducory QFT lecture. We considered a theory of a real scalar field $\Phi$ and a complex scalar field $\phi$. The ...
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Similarity between 1PI effective action and Wilsonian effective action

A similar question about the difference between 1PI and Wilsonian effective actions was asked and answered here. Now I ask, when are they the same? Particularly, Seiberg says here (Pg 6, Sec 2.3) that ...
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Anomalous Dimension and Infinitesimal Transformation

In the theory of RG transformations I ended up with the following equation $$\left( \frac{Z(\mu)}{Z(\mu / s)} s^{d-2}\right)^{1/2} = 1+\left(\frac{1}{2}(d-2)+\gamma_{\phi}\right)\delta s$$ where $Z(\...
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Nonrenormalizable but quantizable theory: gravity?

In p.8 of Michio Kaku book Introduction to Superstrings and M-Theory-Springer (1998), he said The gravitational force. Gravity research was totally uncoupled from research in the other interactions. ...
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1answer
79 views

Do we have to fix parameters by experiment when using the renormalization group?

In traditional renormalization, renormalized masses have to be fixed by experiment before going on to make other predictions. Do renormalization group methods, like Wilson's, require fixing parameters ...
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The difference between field-theoretic renormalization group and Wilsonian renormalization group methods

I know that they deliver the same answers if we are in the vicinity of a fixed point (but may have different beta functions otherwise), and that field-theoretic approaches are not as general, but I am ...
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Asymptotic Series in QFT: What to do when all "trustworthy" terms are known?

In my Introduction to QFT lecture, we quantized a Klein-Gordon Field and as a toy model we looked at $\phi^3$ theory. For this toy model we expanded the $S = U(-\infty, \infty)$ operator in a series (...
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Eigenvalues of the linearized renormalization group flow

In Wilsonian renormalization we linearize the flow around the fixed point as $\mathbf{k}'(\mathbf{k}^*+\Delta\mathbf{k})=\mathbf{k}^*+\mathbf{R}\Delta\mathbf{k}+\ldots$ where the $\mathbf{k}$, $\...
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Predictions using perturbative renormalization

If physical quantities such as renormalized masses and couplings are fixed by experiment, what exactly are physicists using calculative perturbative renormalization techniques for?

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