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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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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What are the six quark mass values when extrapolated to Planck energy?

Let us assume that the standard model is correct up to Planck mass. (Yes, I know, this is a big assumption.) If we continue the running of quark masses with energy (due to renormalization), what are ...
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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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Renormalization group resummation

I'm having trouble in understeanding a mathematical feature of RG, namely how it provides a way to resum the perturbation series and how that's defined mathematically. From a conceptual point of view ...
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How can we calculate pion decay constant in Chiral Perturbation Theory ?

Above diagram is an one-loop contribution to the Pion decay constant $f_\pi$. For example in this paper (Eq.7) they have written down the pion decay constant to one loop, but the calculation is not ...
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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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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: $$ ...
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Peskin's book page 334 proof of $Z_1=Z_2$ to all orders in QED perturbation theory

Peskin in his QFT page 334 argued that $Z_1=Z_2$ to all orders in QED perturbation theory, but I couldn't understand his argument: ... With a generalization of the argument given there (section ...
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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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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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RGEs of the MSSM - problems with Mathematica

I'm having some troubles with the trilinear soft couplings of the MSSM RGEs. I've used the ones written in Martin's supersymmetry primer and I run them using mathematica, if I do so without taking ...
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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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“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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Around fixed point of renormalization group

In the general formulation of renormalization group in "statistical mechanics" by P.K.Pathria, each point in parameter space is represented by a vector $\vec{K}$ and the transformed vector would be ...
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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 ...
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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 - ...
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S-matrix element for forward scattering and amputed green function

I'm studying dispersion relations applied as alternative method to perturbation theory from Weinberg's book (Vol.1) Let's consider the forward scattering in the lab frame of a massless boson of any ...
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Regularization of the Casimir effect

For starters, let me say that although the Casimir effect is standard textbook stuff, the only QFT textbook I have in reach is Weinberg and he doesn't discuss it. So the only source I currently have ...
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multi-dimensional renormalization group flow?

Suppose you have $\lambda \phi^3$ theory, and that you renormalize the 2 and 3 point one-particle irreducible graphs, $\Pi_R(p^2)$ and $\Gamma_R(p_1,p_2,p_3)$, by Taylor expanding about $p=\mu_0$ for ...
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Can the terms in the microscopic model with nonzero conformal spin generate some new term(s) under RG (renormalization group) flow?

As in the book Bosonization and Strongly Correlated Systems at page 66, it says that "We see that the original perturbation with nonzero conformal spin generates the perturbation with zero conformal ...
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Superfields and the Inconsistency of regularization by dimensional reduction

Question: How can you show the inconsistency of regularization by dimensional reduction in the $\mathcal{N}=1$ superfield approach (without reducing to components)? Background and some references: ...
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A question about the implication of UV divergence in QFT

I have a basic question about the logic of renormalization in quantum field theory (QFT). We met the ultraviolet (UV) divergence in loop corrections. The standard argument is, our current field theory ...
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Renormalization Group Invariance of Scattering Amplitude

How can one show that the scattering amplitude is renormalization group invariant using the fact that the bare Green's function $G_0^{(n)}$ is renormalization group invariant? We have: $(1) \quad ...
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Am I understanding correctly the argument that leads to the need for field and mass renormalization?

I'm studying Quantum Field Theory from Weinberg's book, and I'm to the point where he introduces the concept of renormalization. I'd like to know if I'm getting the point that Weinberg makes when ...
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Physical meaning of Ward Identity and computing vertex functions

Following the derivation of Ward Identity by Weinberg book, you get it in the form $$ (l-k)_\mu S'(k)\Gamma^\mu(k,l)S'(l) = i S'(l) - iS'(k) $$ Can anyone explain the physical meaning of this ...
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Can I use Pauli-Villars and dimensional regularization together?

There are at least two ways to compute the electron-self energy. You can use Pauli-Villars or dimensional regularization, for example. On Weinberg's book, it's chosen the first method, while on my ...
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Beta function calculation in massless minimal subtraction $\phi^4$ theory

I'm trying to understand how to calculate the beta function in massless phi^4 theory using dimensional regularisation and minimal subtraction. I'm struggling to understand: Is it possible to ...
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Why Conserved Current Should Not Need Renormalization?

May be this is trivial but I need to understand why the renormalization of conserved current is not necessary ? As for example, in this paper, they demand (2$^{nd}$ paragraph of the 2$^{nd}$ column in ...
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Intuition for parameter $\mu$ in dimensional regularization

In dimensional regularization, a dimensionless coupling $g$ is replaced by $\mu^{4-d}g$ so that it can remain dimensionless. $\mu$ is unphysical, though its choice affects the values of counterterms. ...
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Renormalization confusion

I'm starting to read about renormalization in the case of scalar field theory. I have some confusions. I will consider momentum renormalization. First, consider a theory with a coupling constant ...
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History of Renormalization Group

I want to study from an historical point of view the renormalization group (starting from statistical mechanics). Are there any historical (but also technical) books about this? Thanks.
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Symmetry of interaction lagrangian and symmetry of full lagrangian

Suppose we have lagrangian $$ \tag 1 L = \frac{\theta}{f_{\gamma}}F_{EM}\tilde{F}_{EM} +\frac{1}{2}(\partial_{\mu}\theta)^2 - \frac{1}{2}m_{\theta}^2\theta^2 + L_{SM}, $$ where $\tilde{F}_{EM}$ ...
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Why the QED coupling constant is a continuous function? [closed]

In page 316 of 'Student friendly quantum field theory', when discussing Figure 12-4, it says that the QED coupling constant is a continuous function of $\ln(p)$. But I think it's disconnected at ...
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Determinant of a propagator

Say I have a path integral $\int D \phi \exp(i S_0)$. $S_0$ is the usual free action $$S_0=\frac{1}{2}\int\phi (-\Box-m^2) \phi=\frac{1}{2}\int \phi G^{-1} \phi,$$ and at the moment I'm not ...
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How are the real-space RG transformations defined?

I'm reading Shang-keng Ma's book Modern theory of critical phenomena, and I'm a bit confused as to how the real-space RG transformations are defined. Ma basically says that these transformations are ...
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A small issue in renormalisation group formalism

In the general RG formalism, suppose $\vec{\mu}$ represents a vector in parameter space and $\vec{\mu}^*$ is the fixed point under the transformation $R$. Then for $\vec{\mu}=\vec{\mu}^*+\delta ...
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How do we know for sure a theory is non-renormalizable?

In quantum field theory, we are looking for a Lagrangian that is, amongst other, renormalizable. But how do we determine whether or not a theory is renormalizable? Is this purely done by power ...
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Renormalization and Conway/Surreal Numbers

In the final chapter of his book "An Interpretive Introduction to Quantum Field Theory", Paul Teller writes about three interpretations of renormalization in quantum field theory. In particular, ...
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Charge screening

By the charge screening effect the charge of the electron increases if the scale of energy increases. But which charge increases, the "true" charge or the bae chagre which will be normalized to the ...
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Why is the term $m\delta_2\delta_m\bar{\psi}\psi$ ignored in the QED Lagrangian?

Consider the QED Lagrangian $$\mathcal{L}=\bar{\psi}_0(i\gamma^{\mu}\partial_{\mu}-e_0\gamma^{\mu}A_{0\mu}-m_0)\psi_0-\frac{1}{4}(\partial_{\mu}A_{0\nu}-\partial_{\nu}A_{0\mu})^2$$ where the 0 ...
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What is the exact meaning that QED perturbative series is only asymptotic and eventually diverges at very high orders?

When I read paper PRB89, 235431 about the effective field theory of graphene, there is a statement that QED perturbative series is only asymptotic and eventually diverges at very high orders (e. g. ...
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Peskin Schroeder and the general solution to Callan-Symanzik Equation

I have a couple of questions regarding Peskin and Schroeder's derivation of the solution to the Callan-Symanzik equation. First of all, they claim that using ...
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SU(2) critical point and volume dependence

I am doing multi-dimensional plots of $\beta_j$ for SU(2) for infinite volume to understand the flow behavior and I was wondering, before I go too much further, if anyone knew off the top of their ...
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What happens when you apply the path integral to the Einstein-Hilbert action?

The Einstein Field Equations emerge when applying the principle of least action to the Einstein-Hilbert action, and from what I understand the path integral formulation generalizes the principle of ...
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Wilsonian Renormalization Group and Symmetries of the EFT

I have am action $S_0$ valid up to energy scale $\Lambda_0$ with renormalisable terms. I want to study the EFT at a lower scale $\Lambda \ll \Lambda_0$, by using the Wilsonian RG. It will give me an ...
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QFT: What does “finiteness” mean?

As above: what is the definition of a QFT to be "finite"? That all UV corrections are finite and there are no divergences at all? That there are divergences, but these divergences can be absorbed ...
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renormalization - free energy density scaling

By introduction to the renormalization theory we have started with the following Hamiltonian: $$ H=\int d^d x [\delta m^2+k(\nabla m)^2+h m ] $$ We have assumed that $\delta$ and $h$ scale as: $$ ...