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In charge renormalization equation, $e=e_{0}^{2}\left[1-e_{0}^{2}A\right]$, how can an infinite $e_{0}$ and $A$ give finite $e$ in any limit?

In Griffiths elementary particle book (chapter 7, 'Quantum electrodynamics', equation 7.188), one gets the following equation for the vacuum polarization calculated to one loop correction. $$\frac{e_{...
2
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1answer
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 ...
3
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1answer
182 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-...
2
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1answer
147 views

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\!...
3
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2answers
838 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 ...
3
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1answer
1k views

What is a logarithmic divergence?

I am reading about renormalisation in QED and I come across the term logarithmic divergence several times. Can somebody explain to me about it in simple terms?
1
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0answers
157 views

Radiative correction to the charge form factor $F_1$ in QED

In QED, one can calculate the correction to the form factor $F_2$. To the lowest order, $F_1=1$ and $F_2=0$. At one loop, it is found that $F_2(0)$ receives a non-zero finite correction which is ...
3
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1answer
206 views

What are the observable effects of finite pieces of the loop corrections in QED?

I'm lost amidst the calculation of regularization and renormalization process in QED. In addition to the divergent piece in the in the self-energy correction (similarly in vacuum polarization ...
4
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2answers
184 views

A derivative about chiral current in Peskin's book

In Peskin's book (an introduction to QFT), Page 655, the axial vector current is defined as follows, \begin{eqnarray*} j^{\mu5} & = & \text{symm }\lim_{\epsilon\rightarrow0}\bigg\{\bar{\psi}(x+...
0
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1answer
223 views

Divergent diagrams in QED

I was reading about how to choose divergent diagrams in QED by using the concept of Superficial degree of divergence. We have an empirical relation $$ D= 4-E_b -\frac{3}{2}E_f $$ where $E_b$ is number ...
1
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1answer
109 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 $$ i\Pi^{\mu\...
2
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1answer
130 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 $$\frac{-8e^2}{(4\pi)^{d/2}}\int_0^1dx\,x(1-x)\frac{\Gamma(2-\...
0
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1answer
110 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 ...
1
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1answer
521 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} + i\...