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4
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0answers
35 views

How to handle the infrared divergence of massless $\phi^4$ in scattering

For massless $\phi^4$ theory, if exterior momentums are going to zero, then this diagram will be $$\int \frac{dk^4}{k^4}$$ will suffer from infrared divergence. Because the infrared divergence, ...
1
vote
1answer
59 views

How to choose the proper loop correction?

I review my QFT lecture notes and I am having hard times to figure out the significance of Ward identity in vacuum polarization. In class, we calculated one loop correction stated as $$ ...
2
votes
1answer
104 views

Two math methods apply the same loop integral lead different results! Why?

I tried to adopt the cut-off regulator to calculate a simple one-loop Feynman diagram in $\phi^4$-theory with two different math tricks. But in the end, I got two different results and was wondering ...
2
votes
1answer
99 views

Doubts with basic renormalization

When we renormalize to obtain the physical mass, the $\Lambda$ dependence of the physical mass is removed by introducing the counterterms in the Lagrangian. So whether we put ...
0
votes
0answers
48 views

What is primitive divergence?

As in the title, what is primitive divergence? How is it distinguished from normal divergence? As a followup, what is a primitive divergent graph in a theory? Some simple examples?
2
votes
0answers
122 views

Why does regularization work in this Bessel function integral?

I encountered some days before an integral representation for a modified Bessel function and should differentiate it. But in this representation : $$K(\omega,a)=\int_0^{\infty} \frac{ds}{s} ...
1
vote
1answer
78 views

Renormalization Using Momentum Cut-off Regularization, What Are The Subtraction Schemes Used?

In most of the books on QFT, the author talks about various methods of regularization but in the end chooses the dimensional regularization and MS-bar scheme when discussing the final renormalization, ...
0
votes
0answers
23 views

Infinite bare quantities and dressed quantities confusion

I'm getting very confused. Taking the example of the mass of the Z-boson. Constructing the GWS model using gauge symmetry breaking one finds a lagrangian which is a function of the Z-boson mass: ...
1
vote
1answer
70 views

Pauli Villars Regularization

Consider the t-channel diagram of phi-4 one loop diagrams. Evaluated it is, with loop momenta p, $\frac{\lambda^2}{2}\displaystyle\int\frac{d^4p}{(2\pi)^4}\frac{1}{(p+q)^2-m^2}\frac{1}{p^2-m^2}$ If ...
4
votes
1answer
78 views

Point splitting technique in Pesking and Schroeder

One of the cornerstones of point splitting technique of calculating chiral anomaly (Peskin and Schroeder 19.1, p.655) is a symmetric limit $\epsilon \rightarrow 0$. And this is the point that I don't ...
0
votes
0answers
35 views

Fermionic oscillator and Hurwitz zeta function

Good resources for calculating Partition function of the fermionic oscillator using the Hurwitz zeta function? I liked the way Nakahara explain this, but some parts are really tricky for me (e.g. the ...
1
vote
1answer
39 views

Why IR divergences cancel by cross sections of next-to-leading diagrams?

I was reading QFT & Standard Model by Schwartz, Chapter 20 which is about IR divergences. He says that IR divergences only cancel cross sections for processes involving different initial or ...
1
vote
1answer
39 views

Dirac Current Spectral Representation

I'm reading Strocchi's book on The Non-Perturbative Foundations of Quantum Field Theory. In the chapter concerning point-splitting regularization, where the free Dirac current is defined as follows ...
2
votes
1answer
38 views

How do logarithms show up in the one loop calculation of the vacuum polarization in QED?

I am following Peskin with the computation of the vacuum polarization in QED and there is one thing I do not see. Equation (7.90) reads ...
1
vote
0answers
41 views

Dependence of finite part of loop integral on regularization

Recently I've calculated some process in which arise triangle loop with running two $W$ bosons and one massless fermion. The expression for integral is following: $$ I_{\alpha \beta}(r, q) = \int ...
1
vote
0answers
44 views

Connection between the cosmological constant $\Lambda$ and the cutoff scale $\Lambda$

I'm trying to understand the connection between the $\Lambda$ from cosmology and the $\Lambda$ from QFT. Cosmology: The cosmological constant enters the Einstein equations. In the special case of the ...
6
votes
5answers
496 views

How to understand singularities in physics?

The question is probably two-folded and I will try not to make it too vague, but nonetheless the question remains general. First fold: In most physical laws, that we have analytic mathematical ...
3
votes
1answer
73 views

Length path integral

Let's consider a 2-dimensional Euclidean plane. The length between two points $a$ and $b$ can be defined in the following way: $$ (ab) := \inf_{\gamma} \,\int_0^1 d\tau \,\sqrt{\delta_{ab} ...
0
votes
0answers
81 views

Divergent harmonic series

Unlike the zeta regularization for the series $ \sum_{n=1}^{\infty}n^{k} $ which is the exact value for the generalized harmonic series? I mean the series, $$ \sum_{n=0}^{\infty}\frac{1}{(n+a)} $$ ...
4
votes
1answer
104 views

Which cardinality of infinities are subtracted in the renormalisation of quantum field theory?

In quantum field theory, e.g. in quantum electrodynamics, renormalisation is used to make sense of an infinite number of virtual particles. This, crudely, involves the subtraction of infinities. But ...
0
votes
0answers
70 views

Pauli-Villars regularization in QED and appearance of massive spin-1 particle

To regularize QED, which is defined by the following Lagrangian, \begin{equation} \mathcal{L}=-\frac{1}{4}F_{\mu\nu}^2+\bar\psi(i\partial_\mu\gamma^\mu-eA_\mu\gamma^\mu-m)\psi \end{equation} One ...
1
vote
0answers
70 views

Shifting the integration variable in loop integrals

We know that, in four dimensions, shifting the integration variables is valid only for convergent and logarithmically divergent integrals. If we employ a hard cutoff $\Lambda$, is it permissible to ...
-1
votes
1answer
74 views

Why renormalizable theory is useful?

Why renormalizable theory is useful? I want to know detail reason for above question. At a glance, I know following things. In quantum field theory, $i.e$ computing self-energy(or self-interaction) ...
4
votes
2answers
519 views

Does the need for renormalization in QFT vanish once you use a more fundamental theory (e.g., string theory)?

It is often explained that renormalization arises in QFT because QFT is a low-energy effective theory that needs to be replaced by a more fundamental theory at higher energies/smaller distances. While ...
0
votes
0answers
74 views

Why string theorist use the following result? [duplicate]

$1+2+3.......$so on $ = -1/12.$ I have seen a few proofs of this result. And I hope most of you are familiar with them. Why string theorist use this ambiguous result in string theory, when assigning ...
2
votes
2answers
164 views

If path integrals aren't well-defined, how can they have any physical meaning?

I am confused about a particular point about the nature of path integration. According to what I've read, what we really mean when we say functional integration is \begin{equation} ...
2
votes
1answer
88 views

What is the relation between zeta and cut-off regularization of the Casimir effect?

In the literature there are at least two methods to derive Casimir effect: original one by Casimir himself: take the quantized energy between plates minus the free space energy, then regularize, ...
4
votes
0answers
106 views

$d=2$ pole argument of quadratic divergences in Peskin & Schroeder's book

Q1: My question is, in the context of dimensional regularisation(DREG, in dimension $d$), why do they mention the absence of $d=2$ pole in the gauge theory cases[1], whereas the $d=2$ pole is not ...
6
votes
1answer
177 views

Cut-off Regularisation and Renormalisation in Scalar Field Theory, Deriving the Cutoff Independent Physical Mass

I'm having trouble reproducing Equation 42: \begin{equation}\tag{1} m^{2}_{\text{phys}}= m^{2}_{r} + m^{2}_{r} \tilde{\lambda} \text{log} \left( \dfrac{m^{2}_{r}}{\mu^{2}} \right) \end{equation} ...
1
vote
1answer
69 views

Introducing cut-off in a renormalisation procedure for quantum mechanics

I've been reading a paper on renormalisation theory as applied to a simple one-particle Coulombic system with a short-range potential. In the process of renormalisation, the authors introduce an ...
3
votes
1answer
163 views

Why use a particular regularization for $\int_0^\infty \mathrm{d}x\,e^{i p x}$?

There are many badly defined integrals in physics. I want to discuss one of them which I see very often. $$\int_0^\infty \mathrm{d}x\,e^{i p x}$$ I have seen this integral in many physical problems. ...
4
votes
2answers
234 views

The integral is zero! $\int \frac{\mathrm{d}^d k}{(2\pi)^d} = 0$

In using dimensional regularization in QFT calculations, one comes across integrals over propagators, they might look like $(d = \text{dimension of spacetime}, n = \text{a number})$ ...
0
votes
0answers
29 views

Zeta regularization for a series and an integral

Given the UV divergence $$ \int_{0}^{\infty}dx $$ in order to regularize it, I introduce a regulator $|x|^{-s}$, so I have $$ \int_{0}^{\infty}x^{-s}dx $$ My problem and doubt is how can I relate ...
0
votes
0answers
47 views

Which renormalisation techniques are available for 3+1 QED?

I hope my question is not too naive, but I would like to know what are the available renormalisation techniques for 3+1 QED. I have read a bit about Pauli-Villars, but I am wondering if there are ...
4
votes
1answer
199 views

Second derivative of dirac delta expression

I have come across the expression $$ \int f(x) \delta(x-a) \delta''(x-a) \mathrm dx$$ where the prime represents the derivative. Usually with derivatives of the delta distribution I'd partially ...
1
vote
1answer
104 views

Regularization of infrared divergences

Let's have diagrams in QED when we don't use Feynman gauge. Then the bare photon propagator will look like $$ \tag 1 D_{\mu \nu}(p) = -\frac{g_{\mu \nu} - \frac{p_{\mu}p_{\nu}}{p^{2}}}{p^{2} + ...
1
vote
0answers
37 views

is the renormalization unique? [closed]

mean let be a theory A in which the divergent integrals appear $ \int_{0}^{\infty}dx $ and $ \int_{0}^{\infty}xdx $ and let be another physical theory with 3 types of divergences $ ...
1
vote
0answers
37 views

What is the exact renormalization regularization for divergent harmonic serise?

given the harmonic series $$ \sum_{n=0}^{\infty}\frac{1}{n+a} $$ what is the correct option for the regularization ? a) $ \sum_{n=0}^{\infty}\frac{1}{n+a}= -\Psi (a) $ Digamma function b) $ ...
6
votes
2answers
161 views

What is IR CFT and UV CFT?

What is IR CFT and UV CFT? In many physics related materials, they often mention IR, and UV. I think it is related with regularization (I remember in QFT, there is UV cutoff in some regularization ...
10
votes
2answers
187 views

Massless integrals in dim-reg

Consider the massless divergent integral $$ \int dk^4 \frac{1}{k^2}, $$ which occurs in QFT. We can't regularize this integral with dim-reg; the continuation from the massive to the massless case is ...
2
votes
0answers
147 views

Anomalies in QFT books

Why in most QFT books when author discusses of non-invariance of measure of path integral (massless fermions interact with gauge fields) $$ \int D\bar{\Psi} D\Psi \to |\Psi \to U\Psi , \quad ...
2
votes
1answer
72 views

Typical form of the beta function of the renormalization group

Why in "typical" cases (according to some non-English text I read), does the $\beta$-function have the form $$ \beta (g) = ag^{2} + bg^{3} + O(g^{4})\ ? $$ I.e., why are there no linear or logarithmic ...
0
votes
0answers
35 views

Constants of infinity [duplicate]

Many times in physics when we analyze a physical system mathematicly we get divergences, but when those divergences has no dependence on any actual physical quantity of interest we tend to disregard ...
2
votes
2answers
129 views

A step in zeta function regularization

I'm just wondering about the mathematical step $$\sum_{n=1}^\infty n\exp[-\epsilon n\sqrt x]=\frac1{\epsilon^2 x}-\frac1{12}+\mathcal O(\epsilon).$$ Why is this equality so? I see that ...
3
votes
2answers
367 views

Question about infinite sum in quantum field

I read from some books of number theory that $$\sum_{n=1}^{\infty}\frac{1}{n^s} = -\frac{1}{12}\text{,when } s=-1.$$ Now is there such a result $$\sum_{n=1}^{\infty}\frac{1}{n^s} = \pi \text{,when } ...
2
votes
1answer
111 views

Interacting Lagrangian - Coupling constant and cutoff factor

I have a general question concerning a given interacting Lagrangian: $$\mathfrak{L}_I = \frac{g}{\Lambda^2} \bar{\chi} \ \gamma^\mu \gamma_5 \ \chi \ \partial^\nu F_{\mu\nu}$$ where $F_{\mu\nu}$ is ...
8
votes
1answer
441 views

Difference between regularization and renormalization?

In my studies on quantum field theory we have come up with the concepts of regularization and renormalization. I'm a little confused about these two. In my understanding regularization is a way to ...
1
vote
0answers
54 views

Apparent elimination of overlapping divergences

The integral, $$ \iint_{\mathbb{R}^{2+}}\frac{xy}{1+x+y} \mathrm{d}y \, \mathrm{d}x$$ possesses an overlapping divergence when $ x \to \infty $ and $ y \to \infty $. However, under a change of ...
1
vote
0answers
29 views

What properties does the conductor making the plates for Casimir effect have?

Reading this explanation, I've understood that the divergence in computation of Casimir force on two parallel conducting plates is because of an unphysical model of ideal conductor, which makes EM ...
2
votes
0answers
57 views

renormalization by differentiation how does it work?

i mean let be the integral $$ \int_{0}^{\infty} \frac{p^{3}}{(p^{2}+m^{2})^{2}} $$ logartihmic divergence but if a apply differentiation with respect to $ m^{2} $ i get $$ \int_{0}^{\infty} ...