I Infrared Divergences in Vertex and Self Energy diagrams

[Elmo]

TL;DR

I am unsure how IR divergences show up in these amplitudes.

and Link Removed

 

[malawi_glenn]

What are your thoughts so far?

Is the solid black line a charged fermion, or are you dealing with scalar QED?

 

[Elmo]

What are your thoughts so far?

Is the solid black line a charged fermion, or are you dealing with scalar QED?

sorry about the late reply.
Yes this solid line is meant to be a charged fermion. This can be taken to mean a gluon loop on quarks for example or even a self energy diagram in HQET.
I have elaborated my question in the attached pdf file "Q" .

 

[malawi_glenn]

sorry about the late reply.
Yes this solid line is meant to be a charged fermion. This can be taken to mean a gluon loop on quarks for example or even a self energy diagram in HQET.
I have elaborated my question in the attached pdf file "Q" .

Insert that here on the forum. It supports latex

 

[Elmo]

Insert that here on the forum. It supports latex

ah man sorry ,for whatever reason whenever I preview my latex typing in this chat box it never actually renders correctly. SO i use pdf

 

[malawi_glenn]

ah man sorry ,for whatever reason whenever I preview my latex typing in this chat box it never actually renders correctly. SO i use pdf

Post the source code and I can read and fix it.

 

[Elmo]

Post the source code and I can read and fix it.

oh thank you !

[Moderator's note: I have used magic moderator powers to edit the below and remove the invalid LaTeX codes.]

To the best of my knowledge the two above diagrams are both UV and IR divergent.

Some textbooks ( like Peskin eqtn 7.19 ) introduce a photon (or gluon) mass $\mu$ to regulate an IR divergence giving

$$
\int_{0}^{1}dx(2m-x \displaystyle{\not}{p})\ln\left(\frac{x\Lambda^2}{(1-x)m^2+x\mu^2-x(1-x)p^2}\right)
$$

(This was done in PV regularization).

Firstly I am not sure if this integral can be done in Mathematica using

" Integrate[,{}]//Normal " but if it can then indeed we get some $\ln(\mu)$ which is divergent as $\mu\rightarrow 0$.

Yet I have seen other sources like Schwartz ( eqtn 18.12) or an MIT ocw lecture (for HQET self energy) dispense with the artificial mass which gives a UV divergence as before but an IR divergence is not apparent to me, if there is one. So I am not sure what is the purpose of an artificial mass.

Also I don't know why IR divergences have to be cancelled at the level of cross sections by adding in soft real emission diagrams. Can they not be subtracted off from the diagram itself ?

 

[malawi_glenn]

Use

Also read https://www.physicsforums.com/help/latexhelp/ and https://www.physicsforums.com/threads/preview-button-not-working-as-expected.1016875/

If it is the first post in a thread that contain LaTeX, you might need to hit "refresh" on your browser to make it render.

 

[PeterDonis]

for whatever reason whenever I preview my latex typing in this chat box it never actually renders correctly

Whatever you are using to compose your LaTeX is assuming you are writing a document and using LaTeX pagebreak, begin document, and end document codes. You're not writing a document in posts here. You won't be able to just cut and paste the LaTeX from whatever source you used to write the PDF; you'll have to then remove the LaTeX codes that are only valid in a document context, not here.

 

[PeterDonis]

You won't be able to just cut and paste the LaTeX from whatever source you used to write the PDF; you'll have to then remove the LaTeX codes that are only valid in a document context, not here.

I have now done this with post #7; it should render properly now.

 

[PeterDonis]

For reference, here is the LaTeX code for the equation in post #7:

$$
\int_{0}^{1}dx(2m-x \displaystyle{\not}{p})\ln\left(\frac{x\Lambda^2}{(1-x)m^2+x\mu^2-x(1-x)p^2}\right)
$$

Also note that here, inline LaTeX uses double pound signs ##, not single dollar signs. That is the only LaTeX markup in the rest of post #7.

 

[malawi_glenn]

Can they not be subtracted off from the diagram itself ?

What do you mean by this?

Have you studied Peskin & Schroeder chapter 6.5?

 

[Elmo]

What do you mean by this?

Have you studied Peskin & Schroeder chapter 6.5?

I was under the impression that you add a soft leading order real emission diagram to the vertex correction diagram and the IR divergences in the soft real emission amplitude mod square and the vertex-real emission cross term ,they mutually cancel as in here :
$| \mathcal{M}_{real emission}|^2 +2Re[\mathcal{M}_{real emission}.\mathcal{M}_{vertex}]$
Counterterm diagrams being added to loop diagrams is done to cancel UV divergences as far as I am aware.

 

[malawi_glenn]

Counterterm diagrams being added to loop diagrams is done to cancel UV divergences as far as I am aware.

One add counter terms to the Lagrangian.

But what I meant is that you wrote "subtracted off the diagram" when you mentioned IR-divergences.
You mean why one does not add counter terms for IR-divergences as well? I am just trying to understand your question.

If you have not studied chapter 6.5 in P&S, do it now :)

 

[Elmo]

well yes one adds counterterms to the lagrangian from which one can construct counterterm daigrams (for UV divergences) but yes I meant that why does one not add counterterms to subtract off IR divergences as well,why do they resort to subtracting off divergences from cross sections rather, as has been done in P&S and a number of other texts ?
Sure ill have a proper read of P&S 6.5
But that is only part of my question.
I had first asked about the manifestation of the IR divergences from the above equation (from my original post) and the fact that some sources simply do not use the regulator gluon mass.
And also whether or not it is correct to solve the above integral ( which is from the self energy diagram ) using Mathematica Integrate[]//Normal command ? (the integral still evaluates fine whether or not you have a regulator mass but if you don't add regulator mass then you don't get $\ln(\mu)$ which is clearly the culprit of IR divergence.

 

[vanhees71]

The reason, why you don't add counterterms for IR divergences is that IR divergences are cured by resummations of soft-photon ladder diagrams, i.e., they occur, because you have to reorganize your perturbative calculation, because due to the denominators from the propagators involving massless particles you have infinitely many diagrams contributing to a given order of the coupling constant.

The physics behind this is that plane waves for charged particles are not the right asymptotic states if a massless gauge field as in electrodynamics is involved.

 

[Elmo]

The reason, why you don't add counterterms for IR divergences is that IR divergences are cured by resummations of soft-photon ladder diagrams, i.e., they occur, because you have to reorganize your perturbative calculation, because due to the denominators from the propagators involving massless particles you have infinitely many diagrams contributing to a given order of the coupling constant.

The physics behind this is that plane waves for charged particles are not the right asymptotic states if a massless gauge field as in electrodynamics is involved.

ah thanks.
Although what about the part of what form do these divergences take ? That some books do these integrals with and some without the regulator mass.

 

[vanhees71]

I think, as long as the matter particles are massive, you don't need an IR regulator in QED as long as you use minimal subtraction (using dimensional regularization) or any other mass-independent renormalization scheme. The on-shell scheme introduces artificial IR divergences. In any case you get rid of the IR divergences by the appropriate resummation of the soft-photon ladders. The usual procedure is to use the usual naive perturbation theory and apply the arguments by Bloch and Nordsieck. For the non-Abelian case it's the Kinoshita-Lee-Nauenberg theorem.

Another more physical approach is to use infraparticle asymptotic states, taking into account the "photon cloud" (i.e., the electromagnetic field of a point charge) properly. For a pedagogical introduction to this topic, see

P. Kulish and L. Faddeev, Asymptotic conditions and infrared
divergences in quantum electrodynamics, Theor. Math. Phys.
4, 745 (1970), https://doi.org/10.1007/BF01066485

 

[Elmo]

Thanks a lot everyone for your replies. IR divergences are bit of a new topic to me and in the recent days I had been reading up on it.
From what I have understood so far :
IR divergences come in loop integrals if any of the propagator denominators do not have a mass term. Those divergences are regulated by putting an artificial mass to the massless particles and it is this mass which will show up as the divergence, in the limit it tends to zero.
And if I understand correctly, the IR divergences can be alternatively regulated in DR by taking different signs for $\epsilon$.
Also do please confirm that if an IR regulator mass is not used and the integral solved by DR, will the $1/\epsilon$ terms definitely contain both UV and IR divergences ?
and that using a regulator mass separates the UV from the IR divergences, making them show up as distinct entities ?