Cancellation of infrared divergence

[kof9595995]

Infrared contribution of vertex correction gives an infinity, the resolution is to add infrared bremsstrahlung contributions as well, I can follow the math, but I'm not so convinced by the justification of this resolution given by Peskin(and Weinberg, and whatever I can find on internet). Basically they argue that, below a certain energy bremsstrahlung cannot be distinguished from vertex correction, because real detectors all have a threshold. I don't quite buy it: what in principle can prevent us making the detecting threshold be 0?I don't t think it's prevented by QED in theory. So the argument seems to be based on the limitation of current technology, not from any theoretical first prinples.

 

[strangerep]

Infrared contribution of vertex correction gives an infinity, the resolution is to add infrared bremsstrahlung contributions as well, I can follow the math, but I'm not so convinced by the justification of this resolution given by Peskin(and Weinberg, and whatever I can find on internet). Basically they argue that, below a certain energy bremsstrahlung cannot be distinguished from vertex correction, because real detectors all have a threshold. I don't quite buy it: what in principle can prevent us making the detecting threshold be 0?I don't t think it's prevented by QED in theory. So the argument seems to be based on the limitation of current technology, not from any theoretical first principles.

Over the decades, lots of people have remained dissatisfied with this approach.

There's another technique for dealing with IR problems in QED. (Chung; Kibble; Kulish & Faddeev; Bagan, Lavelle, McMullan; and many others.)

The problem's origin is that the Coulomb 1/r interaction doesn't vanish quickly enough at infinity, so the usual assumption that asymptotic states correspond to free states is not correct in this case. The other techniques involve finding a correct space of asymptotic states -- in this case dressing the electrons with a coherent photon field.

Try Google Scholar for the Bagan, Lavelle, McMullan papers to get a reasonably modern perspective on this.

 

[kof9595995]

After reviewing Peskin chap 6 more carefully, I think Peskin's text is a bit misleading.It seems that it's not a big problem if we don't include bremsstrahlung. For pure loop corrections without external photons, the infrared contribution is actually zero as long as we sum the perturbation to all orders, c.f. eqn (6.79)&(6.80). This is physically acceptable, because it means the amplitude is 0 for a scattering without any emission of photon. So, quite on the contrary to what he claimed earlier in the chapter, the effect of adding soft bremsstrahlung contribution is not to cancel the infrared divergence, but to make the measured cross-section non-zero.
To this point I thought my question is resolved(or am I still wrong?), but your reply raised me more questions, what's the reason that "Over the decades, lots of people have remained dissatisfied with this approach."? And how can we identify infrared divergence with the problem of asymptotic states?

 

[strangerep]

After reviewing Peskin chap 6 more carefully, I think Peskin's text is a bit misleading.It seems that it's not a big problem if we don't include bremsstrahlung. For pure loop corrections without external photons, the infrared contribution is actually zero as long as we sum the perturbation to all orders, c.f. eqn (6.79)&(6.80). This is physically acceptable, because it means the amplitude is 0 for a scattering without any emission of photon. So, quite on the contrary to what he claimed earlier in the chapter, the effect of adding soft bremsstrahlung contribution is not to cancel the infrared divergence, but to make the measured cross-section non-zero.
To this point I thought my question is resolved(or am I still wrong?), but your reply raised me more questions, what's the reason that "Over the decades, lots of people have remained dissatisfied with this approach."?

Look at P&S's paragraph between eqs(6.80) abd (6.81). Note the words "...integrate the squared matrix element over the photon's phase space" (my emphasis). And later, just after eq(6.83): "...gives our final result for the measured cross section..." (my emphasis again).

The reason "lots of people have remained dissatisfied with this approach" is because it only gives cross-sections, deprecating the S-matrix. The "lots of people" wished for some other approach that could sensibly preserve the primary role of the S-matrix.

And how can we identify infrared divergence with the problem of asymptotic states?

That's a rather long story to explain fully. For starters, do you understand the details of how the ordinary Coulomb potential causes problems in nonrelativistic scattering theory before the 1/r potential doesn't decay quickly enough?

Following is a bibliography if you *really* want to get into this. Start with the Bagan+Lavelle+McMullan papers which include an overview. Kulish+Faddeev includes a short instructive calculation for the nonrelativistic Coulomb case, although their method in the relativistic case is flawed, as pointed out by Bagan+Lavelle+McMullan. Dollard's paper gives more rigorous detail on the nonrelativistic Coulomb problem. Chung's paper is the seminal one. The stuff in the supplement of the Jauch+Rohrlich textbook gives a readable overview.

\bibitem{BagLavMcMul-1}
E. Bagan, M. Lavelle, D. McMullan,
"Charges from Dressed Matter: Construction",~\\
(Available as hep-ph/9909257.) \\
Abstract: {\em A crucial element of scattering theory and the LSZ reduction
formula is the assumption that the coupling vanishes at large times. This is
known not to hold for the theories of the Standard Model and in general such
asymptotic dynamics is not well understood. We give a description of
asymptotic dynamics in field theories which incorporates the important
features of weak convergence and physical boundary conditions. Applications to
theories with three and four point interactions are presented and the results
are shown to be completely consistent with the results of perturbation
theory.} \\
(Also includes summary of Kulish-Fadeev method. Some content overlap with
\cite{HorLabMcM-1}.)

\bibitem{BagLavMcMul-2}
E. Bagan, M. Lavelle, D. McMullan,\\
"Charges from Dressed Matter: Physics \& Renormalisation",~\\
(Available as hep-ph/9909262.)\\
Abstract: {\em Gauge theories are characterised by long range interactions.
Neglecting these interactions at large times, and identifying the Lagrangian
matter fields with the asymptotic physical fields, leads to the infra-red
problem. In this paper we study the perturbative applications of a
construction of physical charges in QED, where the matter fields are combined
with the associated electromagnetic clouds. This has been formally shown, in a
companion paper, to include these asymptotic interactions. It is explicitly
demonstrated that the on-shell Greens functions and S-matrix elements
describing these charged fields have, to all orders in the coupling, the pole
structure associated with particle propagation and scattering. We show in
detail that the renormalisation procedure may be carried out straightfor-
wardly. It is shown that standard infrared finite predictions of QED are not
altered and it is speculated that the good infrared properties of our
construction may open the way to the calculation of previously uncalculable
properties. Finally extensions of this approach to QCD are briefly discussed.}

\bibitem{Chu}
V. Chung,
"Infrared Divergences in Quantum Electrodynamics", ~\\
Phys. Rev., vol 140, (1965), B1110.
(Reprinted in \cite{KlaSkag}.)

\bibitem{Dol}
J. D. Dollard,
"Asymptotic Convergence and the Coulomb Interaction",~\\
J. Math. Phys., vol, 5, no. 6, (1964), 729-738.

\bibitem{HorLabMcM-1}
R. Horan, M. Lavelle, D. McMullan,~
"Asymptotic Dynamics in QFT",~\\
Arxiv preprint hep-th/9909044.\\
Abstract: {\em A crucial element of scattering theory and the LSZ reduction
formula is the assumption that the coupling vanishes at large times. This is
known not to hold for the theories of the Standard Model and in general such
asymptotic dynamics is not well understood. We give a description of
asymptotic dynamics in field theories which incorporates the important
features of weak convergence and physical boundary conditions. Applications to
theories with three and four point interactions are presented and the results
are shown to be completely consistent with the results of perturbation
theory.}\\
(Also includes summary of Kulish-Fadeev method. Some content overlap with
\cite{BagLavMcMul-1}.)

\bibitem{HorLabMcM-2}
R. Horan, M. Lavelle, D. McMullan,~\\
"Asymptotic Dynamics in QFT -- When does the coupling switch off?",~\\
Arxiv preprint hep-th/0002206.\\
Abstract: {\em We discuss the approach to asymptotic dynamics due to Kulish
and Faddeev. We show that there are problems in applying this method to
theories with four point interactions. The source of the difficulties is
identified and a more general method is constructed. This is then applied to
various theories including some where the coupling does switch off at large
times and some where it does not.}\\
(Shorter paper. Some content overlap with \cite{BagLavMcMul-1},
\cite{HorLabMcM-1}.)

\bibitem{JauRoh}
Jauch \& Rohrlich
"The Theory of Photons \& Electrons" (2nd Edition),~\\
Springer-Verlag, 1980, ISBN 0387072950.

\bibitem{Kib1}
T.W.B. Kibble, ~\\
"Coherent Soft-Photon States \& Infrared Divergences. I. Classical Currents",~\\
J. Math. Phys., vol 9, no. 2, (1968), p. 315.

\bibitem{Kib2}
T.W.B. Kibble, ~\\
"Coherent Soft-Photon States \& Infrared Divergences. II.
Mass-Shell Singularities of Green's Functions",~\\
Phys. Rev., vol 173, no. 5, (1968), p. 1527.

\bibitem{Kib3}
T.W.B. Kibble, ~\\
"Coherent Soft-Photon States \& Infrared Divergences.
III. Asymptotic States and Reduction Formulas.",~\\
Phys. Rev., vol 174, no. 5, (1968), p. 1882.

\bibitem{Kib4}
T.W.B. Kibble, ~\\
"Coherent Soft-Photon States \& Infrared Divergences.
IV. The Scattering Operator.",~\\
Phys. Rev., vol 175, no. 5, (1968), p. 1624.

\bibitem{KlaSkag}
J. R. Klauder \& B. Skagerstam, ~\\
"Coherent States -- Applications in Physics \& Mathematical Physics",~\\
World Scientific, 1985, ISBN 9971-966-52-2

\bibitem{KulFad}
P.P. Kulish \& L.D. Faddeev, ~\\
"Asymptotic Conditions and Infrared Divergences in Quantum Electrodynamics",~\\
Theor. Math. Phys., vol 4, (1970), p. 745

 

[kof9595995]

Look at P&S's paragraph between eqs(6.80) abd (6.81). Note the words "...integrate the squared matrix element over the photon's phase space" (my emphasis). And later, just after eq(6.83): "...gives our final result for the measured cross section..." (my emphasis again).

The reason "lots of people have remained dissatisfied with this approach" is because it only gives cross-sections, deprecating the S-matrix. The "lots of people" wished for some other approach that could sensibly preserve the primary role of the S-matrix.

So you mean if we calculate S-matrix(still of pure loop corrections+bremsstrahlung) instead of cross section the divergence will not go away? Seems the only difference is Peskin added bremmstrahlung photons incoherently to get the cross section, while a more convincing calculation would be to add the amplitudes of loop correction and bremsstrahlung, then square it. Is this what you are talking about?

That's a rather long story to explain fully. For starters, do you understand the details of how the ordinary Coulomb potential causes problems in nonrelativistic scattering theory before the 1/r potential doesn't decay quickly enough?

Following is a bibliography if you *really* want to get into this ...

Thanks, these look quite formidable though.

 

[strangerep]

So you mean if we calculate S-matrix(still of pure loop corrections+bremsstrahlung) instead of cross section the divergence will not go away? Seems the only difference is Peskin added bremmstrahlung photons incoherently to get the cross section, while a more convincing calculation would be to add the amplitudes of loop correction and bremsstrahlung, then square it. Is this what you are talking about?

Not really. It involves using different electron states at $t = \pm\infty$ (composite of bare electron and coherent photon states) and re-doing the scattering calculations. One finds that the IR divergences do not occur.

I.e., using asymptotic in-out states that are more physically-correct, one avoids the IR divergences.

 

[kof9595995]

Not really. It involves using different electron states at $t = \pm\infty$ (composite of bare electron and coherent photon states) and re-doing the scattering calculations. One finds that the IR divergences do not occur.

I.e., using asymptotic in-out states that are more physically-correct, one avoids the IR divergences.

This is not what I intended to ask. I still don't see why people are unsatisfied with the treatment in Peskin. You said:

Look at P&S's paragraph between eqs(6.80) abd (6.81). Note the words "...integrate the squared matrix element over the photon's phase space" (my emphasis). And later, just after eq(6.83): "...gives our final result for the measured cross section..." (my emphasis again).

The reason "lots of people have remained dissatisfied with this approach" is because it only gives cross-sections, deprecating the S-matrix. The "lots of people" wished for some other approach that could sensibly preserve the primary role of the S-matrix.

The original intention of Peskin is to get rid of the infinity of the loop correction, and he achieved this when he summed the loops to all order and obtained (6.80), which is a amplitude not a squared amplitude. So I assume you must be talking about (6.81)~(6.83), but I don't consider it as a problem, because firstly the key goal is already acheived(get rid of loop ddivergences); secondly, although (6.82) shows that the cross-section of emitting soft photons is infinity, this is not very disturbing(at least not as disturbing as the loop divergences) since this just means we get infinite photons with infinitely low energies; thirdly it might be possible to replace (6.81)~(6.83) by an amplitude calculation(as a reminder this is what I assumed you are unsatisfied with, correct me if I misunderstood you).

 

[strangerep]

The original intention of Peskin is to get rid of the infinity of the loop correction, and he achieved this when he summed the loops to all order and obtained (6.80), which is a amplitude not a squared amplitude.

No, (6.80) is not the whole story - it only covers virtual photons (see the LHS diagram in that equation).

So I assume you must be talking about (6.81)~(6.83), but I don't consider it as a problem, because firstly the key goal is already achieved (get rid of loop divergences);

Like I said, the key goal is not already achieved at that point. Take a look a bit further back near the top of p203 where P&S says:

Infrared divergences arise from photons with "soft" momenta: real photons with energy less than some cutoff $E_\ell$, and virtual photons with (after Wick rotation) $k^2 < E_\ell^2$. [...]

secondly, although (6.82) shows that the cross-section of emitting soft photons is infinity, this is not very disturbing(at least not as disturbing as the loop divergences) since this just means we get infinite photons with infinitely low energies;

Yes, but this is an artifact caused by the way the standard treatment ignores the fact that the EM interaction does not vanish satisfactorily between infinitely-separated charged particles. This invalidates the usual adiabatic hypothesis in QFT that the in-out states correspond to free states.

thirdly it might be possible to replace (6.81)~(6.83) by an amplitude calculation

This is exactly what lots of people have tried (and failed) to achieve over the decades, and why the other approach (involving electrons dressed with a coherent photon factor) were developed.

 

[kof9595995]

I see, that clarifies all the matter. Thanks!