Questions tagged [regularization]

In QFT, regularization is a method of addressing divergent expressions by introducing an arbitrary regulator, such as a minimal distance *ϵ* in space, or maximal energy *Λ*. While the physical divergent result is obtained in the limit in which the regulator goes away, *ϵ* → 0 or *Λ* → ∞, the regularized result is finite, allowing comparison and combination of results as functions of *ϵ, Λ*. Use for dimensional regularization as well.

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217 views

Does the angular measure matter in dimensional regularization?

In dimensional regularization, we replace a momentum integral $I= \int d^nk f(|k|)$ with the family of regularized integrals $$\mu^{n-d}\int d^dk f(|k|) = \mu^{\epsilon}\Omega_d \int p^{d-1} f(p)dp.\...
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Can dimensional regularization be viewed as a soft version of a Wilsonian cutoff?

In the Wilsonian picture of renormalization, a quantum field theory is defined to have degrees of freedom only up to an energy scale $\Lambda$. The results of low-energy experiments shouldn't change ...
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99 views

Basic cut-off regularization

I've been reading these notes on regularization by Hitoshi Murayama here, and on page 3 there's a few lines of calculations on a quick method of regularizing an integral. But I can't follow the steps ...
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Reproducing Ramond's sunset diagram calculation for $\phi^4$ theory

I am unable to reproduce the calculation of the sunset diagram for $\phi^4$ theory in Pierre Ramond's Fied Theory a Modern Primer. This is the second edition chapter 4.4. He starts with eq. (4.4.19) \...
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152 views

What cancels this tree level IR divergence?

Computing the amplitude squared for $e^-\mu^-\rightarrow e^-\mu^-$ at tree level we get \begin{equation} \frac{1}{4}\sum_\mathrm{spins}|\mathcal{M}(s,t)|^2=2e^4\frac{s^2+u^2}{t^2} \end{equation} which ...
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Help understanding loops with negative superficial degree of divergence

Consider $$\int \frac{d^4k}{(2\pi)^4} \frac{1}{k^2}\frac{1}{k^2}\frac{1}{k^2}.\tag{1}$$ We can Wick rotate $k_0 \to i k_0$: $$ i \int \frac{d^4k_E}{(2\pi)^4} \frac{1}{k_E^2}\frac{1}{k_E^2}\frac{1}...
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Free quantum field theories as fixed points of Wilsonian RG

Consider Euclidean Klein Gordon quantum field theory on the toroidal spacetime $X\simeq S^1\times \cdots\times S^1$, with action $$S(\varphi) = \int_X \varphi(\Delta+m^2)\varphi$$ and scalar field $\...
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Anomalies and short-distance divergences$.$

Let $J$ be a certain Noether current $$ J=J[\phi] $$ where $\phi$ is a field. This object is classically conserved, although in the quantum-mechanical case it may be anomalous. In the functional ...
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Integrals of the form $\int\frac{d^Dk}{(2\pi)^D} \frac{1}{k^{2n}}$ in $D=4-2\varepsilon$ dimensions?

In a massless theory we often get integrals of the form $$\int\frac{d^Dk}{(2\pi)^D} \frac{1}{k^{2n}} \tag{*}$$ where $D=4-2\varepsilon$. I have tried to calculate this in two ways in Minkowski space ...
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Wick rotations, convergence and Feynman propagators?

It is said (in e.g. Hawking, 1979, Euclidean quantum gravity) that the integral: $$ \int \mathcal{D}\phi \exp(iS[\phi])\tag{1} $$ for real fields in Minkowski space does not converge, but the Wick ...
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Anomalous triangle vertex: divergencies and symmetry argument

Consider triangle correlator of one axial-vector current $J_{\lambda 5}$ and two vector currents $J_{\mu}, J_{\nu}$ in a theory with a fermion with mass $m$: $$ \Gamma_{\lambda \mu\nu}(q,p,k) = F\bigg[...
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The role of the renormalization scale in the functional renormalization group

On p. 28 of Bertrand Delamotte's Introduction to the Nonperturbative Renormalization Group he writes $k$ [the renormalization scale] acts as an infrared regulator (for $k \neq 0$) somewhat similar ...
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“Dimensional analysis” arguments in quantum field theory

I'm uncomfortable with dimensional analysis arguments made in quantum field theory, particularly those related to renormalization. For example, in section III.2 of Zee's QFT book, it says: Consider ...
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Semiclassical propagator convergence at $t=0$

For harmonic oscillators the prefactor for the semiclassical propagator is $Fe^{iS}$ where $$F=\sqrt {m\omega/{2πi\hbar\sin(ωt)}}$$ and $$S={m\omega[(x_0^2+x_1^2)\cos(\omega t)-2x_0x_1]}/{2\sin(\omega ...
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Does every regularization/renormalization approach gives running coupling constants?

I'm studying different tools for regularization and renormalization. Until now I vaguely understand 1) the wilson approach to renormalization where one thinks of the theory as essencially effective ...
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$D$-dimensional Schrodinger's equation with a Dirac delta potential

I know that for $D\geq 2$, there is no bound state for a Dirac potential $V=-\alpha \delta(\textbf{x})$ unless we use an ultraviolet cutoff $k_{max}=1/a$. I showed this by solving the Schrodinger's ...
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What's the relation between Wilson Renormalization Group (RG) in Statistical Mechanics and QFT RG?

What's the relation between Wilson Renormalization Group(RG) in Statistical Mechanics and QFT RG? For easier to compare, I choose scalar $\phi^4$ in both cases. Wilson RG: Given $\phi^4$ model, $$Z=...
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446 views

How to derive this Matsubara sum, as presented in Wikipedia?

On the Wikipedia page for Matsubara frequencies, the following formula is presented, $$ \sum_{i\omega_n} \frac{(i\omega_n)^2}{(i\omega_n)^2 - \xi^2} = -\frac{\xi}{2}\Big(1 - 2 N_{\text{FD}}(\xi)\Big), ...
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Avoid the singularity of Coulomb matrix elements in grid-based calculations

Whenever one is doing grid-based calculations for particles interacting via the Coulomb potential, the singularity of the Coulomb potential $w(\mathbf r_1,\mathbf r_2) = \frac1{|\mathbf r_1 - \mathbf ...
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Regularization scheme independence in QFT

I know there are a few similar questions on the topic (1,2) , but I still feel they do not fully answer my questions (correct me if I am wrong!). What I am asking is a clarification on the commonly ...
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181 views

What does the cut-off $\Lambda$ stand for in the theory of QED?

The bare electron mass $m_0$, in QED, changes as $$m_0\to m=m_0+\delta m\Big(\frac{\Lambda}{E}\Big)$$ where high momentum modes from $E$ to $\Lambda$ has been integrated out. What scale does the cut-...
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265 views

Are the field renormalization factors infinite or finite?

We know that in quantum field theory we include infinities at each order of the perturbative expansion of the renormalization $Z$ factors about the coupling constant $\lambda$ to absorb the ...
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How to remove the divergent part of loop integrals when employing the cuttoff procedure?

On page 130 of "A Modern Introduction to Quantum Field Theory" by Michele Maggiore they evaluate a divergent four dimensional integral writing: "We introduce a cuttoff stating that we integrate ...
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Order 1 Correction to the Electron Mass (Peskin 7.1)

In Section 7.1 in Peskin and Schroeder, (pp. 270), the first order correction to the electron mass is calculated. They define (eq. 7.24) the physical mass $m$ of the electron as the solution to, $$[p\!...
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What is the logic behind box normalization and periodic boundary condition?

Free particle energy eigenfunctions are $A\exp{[i(Et-\textbf{p}\cdot\textbf{r})/\hbar]}$ are non-normalizable. To normalize them one introduces a procedure called 'box normalization' where one imposes ...
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Are there fundamental differences between finite and infinite systems?

Most sources on classical field theory introduce classical fields as a limit of a system with $N$ particles constrained in some way in a lattice where a continuum limit involving $N$, lattice size and ...
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612 views

Point splitting regularization for polynomials of operators

Point-splitting regularization in quantum field theory uses the fact, that UV-divergences occurring in expressions of the type $\left< \phi \left( x \right) \phi \left( x \right) \right>$ can be ...
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1answer
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What is a point-split?

I encountered the term point-split [1] several times and would like to know what this concept is all about. From my understanding, a point is splitted by adding $ε$ and $-ε$ to a local point $x$ ...
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836 views

Is the fact that the sum of all natural numbers $\sum_{n=1}^\infty n = -\frac{1}{12}$ essential to the understanding of the Casimir Force In QED?

Apparently this result is used in many areas physics including the extra dimensions of string theory, which is not the scope of the question. The result is apparently also used to understand the ...
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531 views

Possible divergence structures of a renormalizable and non-renormalizable theory

If a theory has a coupling with negative mass dimension, it will require an infinite number of counterterms. This is because the theory will have infinitely many divergence structures. To be concrete,...
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101 views

Physical understanding of the regularization of benign infinities in free field theories

Any continuum quantum field theory (QFT), free or interacting, has uncountably infinite number of degrees of freedom in spacetime. Does it have anything to do with the appearance of infinities in ...
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Why can consistent QFTs only arise from CFTs?

This is claimed by Jared Kaplan in his Lectures on AdS/CFT from the Bottom Up. He writes: It seems that all QFTs can be viewed as points along an Renormalization Flow (or RG flow, this is the ...
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How to properly truncate an infinite-dimensional Hilbert space for quantum optics simulations?

In order to solve numerically a master equation in which the Hamiltonian and the jump operators are defined in terms of the infinite-dimensional annihilation and creation operators, we have to ...
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Renormalization and Infinite Series

In the renormalization of QED, one subtracts infinities to get finite results. This is not possible in gravity because of infinitely many divergences, and infinitely many counter terms. The general ...
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540 views

Why is there no anomaly when particle mechanics is quantized?

We know that if one or more symmetries of the action of a classical field theory is violated in its quantized version the corresponding quantum theory is said to have anomaly. Is this a sole feature ...
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956 views

Dimensional Regularization and Massless Integrals

Consider the following integral: \begin{equation} \int \frac{d^3 k}{(2\pi)^3} \frac{1}{k^3} \end{equation} In dim-reg, such integrals evaluate to $0$. However, if we instead consider \begin{...
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Taylor expansion of $Ei(x)$

I'm reading a note on regularization by Muruyama, link http://hitoshi.berkeley.edu/230A/regularization.pdf On the bottom of page 2, Muruyama Taylor expanded $$ -\frac{e^{m^2/\Lambda^2}}{4\pi} \...
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Use of the delta function in the cancellation of real and virtual corrections

In light of the not so well defined integral $\int_a^b \delta(x-a) dx$ and from David Z's comment at the end of this Math.SE post, consider the following equation, which I've come across in many ...
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$\phi^{4}$ Propagator - Feynman Diagram: internal vertex that loops back to itself

In all that follows I'll be dealing with everything massless. The free, massless propagator ($\mathcal{L} = \int d^{4}x \left(\partial \phi(x) \right)^{2} $) is supposedly given by $G_{0}(x,y) = c (x-...
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The relation of two integrals

In this post about the integral $$ \int\frac{d^4k}{(2\pi)^4}\,\frac{1}{(k^2)^2}e^{ik\cdot\epsilon}=\frac{i}{(4\pi)^2}\log\frac{1}{\epsilon^2},\quad \epsilon\rightarrow 0, \tag{19.43}$$ which is (19....
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249 views

Pauli- Villars regularization in the Electron Vertex Function: Evaluation

I'm studying one loop contribution for electron vertex function form Peskin and Schroeder's book " An introduction to quantum field theory " Section: 6.3. I have some troubles with Pauli- Villars ...
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The determinant of the Dirac operator in Euclidean signature

Suppose the Dirac operator determinant in Euclidean space-time with manifold $\mathbb R^{4}$: $$ d = \text{det}(iD), \quad iD = i\gamma^\mu (\partial_\mu +A_{\mu}) $$ The Dirac operator is elliptical, ...
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How can I handle divergence that appears in many physical problem? [closed]

I came across with the following type of integration with singularity. $$\int_{s_2=0}^{s_2=\infty}\int_{s_1=0}^{s_1=s_2}\left(\frac{1}{s_2-s_1}\right)^{3/2} \,ds_1\,ds_2 \, .$$ How can I solve it?
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446 views

Why do we study anomalies with the triangle diagram?

This is a very basic question. Why do we study anomalies by means of the triangle diagram, i.e. the tree-point function of gauge/global currents and not with, for instance, a two point function? In ...
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Contributions to final states in the optical theorem

Consider the use of the optical theorem for computing the imaginary part of the total forward scattering amplitude $\mathcal M(AB \rightarrow AB)$. Then the theorem tells us to compute $$2 \text{Im} \...
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Gauge invariance, symmetries, and regularization

When regularizing integrals in a QFT with a gauge symmetry, many people state that it is important that the regulator also enjoys gauge invariance. Why is this true? What goes wrong when you use a ...
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1answer
142 views

Spinor vacuum energy

I'm reading the calculation in the book Quantum field theory in a nutshell of A. Zee of chaoter II.5 In this chapter the vacuum energy is calculated through the path integral approach. At some point ...
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327 views

How should we think of local counterterms in the context of anomalies?

Short version: effective actions, particularly ones obtained after integrating chiral fermions, are ambiguous up to the addition of local counterterms. Should we think of the counterterms as part of ...
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Can we obtain the Feynman diagrams using infinite series representation of a path integral?

While evaluating quantum amplitude of a particle using path integral approach, we deal with infinite number of paths that can usually lead to a divergent infinite series. We can then also obtain a ...
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157 views

Why do not renormalization group equations explicitly depend on cutoff?

Suppose $g$ is the parameter set and $\Lambda\equiv\Lambda_0e^{-t}$ the momentum cutoff, then usually one finds the renormalization group equations to take the form $$\frac{dg(t)}{dt}=\beta(g).$$ My ...