The Perturbative Pole Mass in QCD

by Andreas S. Kronfeld.

Fermilab report FERMILAB-PUB-98/139-T
SPIRES entry
E-print archive hep-ph/9805215

Phys. Rev. D58 (1998) 051501.

Some Background

Many physicists are probably astonished that a proof of the infrared finiteness and gauge independence of the pole mass in QCD is being written up in 1998. If you are one of them, please read the following before assuming that the results have long been known.

This paper grew out of Referee A's report on another paper of mine, hep-lat/9712024, written with Bart Mertens and Aida El-Khadra and submitted to (and published in) Physical Review D. The referee wrote

2) The authors assume that the pole mass of a quark is a well-defined concept order by order in perturbation theory. To the best of my knowledge this has not been shown in the literature. It has been demonstrated that the pole mass is infrared finite and gauge invariant to order $\alpha^2$~[C], and the corresponding finite part in the relation to the $\overline{\rm MS}$ mass has been worked out in ref.~[D].

It is quite possible that infrared problems prevent a definition of the quark's pole mass to all orders in perturbation theory....

[C] R. Tarrach, Nucl. Phys. B183 (1981) 384.
[D] N. Gray, D.J. Broadhurst, W. Grafe and K. Schilcher, Z. Phys. C 48, 673 (1990).

Let me add that other parts of the report revealed that Referee A is exceptionally well-informed on theoretical issues. It turns out, he/she also knows the literature better than most of us.

After considerable literature search I could not find a proof anywhere. During my search I found numerous authors who assert that the pole mass is, indeed, well-defined order by order in perturbation theory. Many papers cite Tarrach's paper for an all-orders proof, even though it sticks to two loops. Indeed, Tarrach is openly worried about the infrared. He writes [italics mine]

It may be evident to many theorists that the pole-mass is gauge-parameter independent in perturbative QCD, but it is less evident whether it is IR finite or not. Let us study these issues at the two loop level.
When Tarrach did his work, in 1981, there had been an effort to uncover a confining mechanism in the infrared divergences of QCD, so his concerns are a sign of the times.

In trying to trace the history of the QCD pole mass, I've noticed two folklores, which have evolved side-by-side. One, espoused by Referee A, holds that infrared divergences in QCD are so serious that nothing can be taken for granted. The other, which is nowadays probably more popular, takes for granted that the pole mass is infrared finite. (I have found no citation to a paper, even one on QED, that purports to study the problem to all orders; a remark in a footnote shows that Noboru Nakanishi knew what to do [Prog. Theor. Phys. 19 (1958) 159].)

I realize that some of you will have known the QED literature well enough to see that the generalization to QCD was straightforward. I would be happy to acknowledge unpublished work on the subject here: feel free to send me a copy of your notes. (Of course, it goes without saying that I would like to know of a detailed published reference.) At the same time, I hope that my paper serves as a useful reference, underpinning the (now publicly proven) fact that the pole mass in QCD is well defined at every order in perturbation theory.

During the time this paper was circulated as an e-print, several physicists from around the world alerted me to proofs of gauge independence of the pole mass, in QED and QCD, and of analogous quantities such as gluon damping rates at nonzero temperature. By and large, these papers do not pay close attention to infrared divergences. An exception is in Lowell Brown's text, Quantum Field Theory, which contains an elegant proof that infrared divergences and gauge dependence of the electron propagator (in QED) resides in the residue only, not the pole position. The proof is relegated to a problem and is, thus, easy to overlook. The proof assumes an Abelian gauge group, and I have not tried to generalize it.

Finally, I would also like to thank Referee A; without his/her strict report, I would not have tried to prove something that so many ``experts'' thought was done in 1981.

01 May 1998 --- Andreas Kronfeld
Modified 29 July 1998