Z. Phys. C Particles and Fields 37, 635-636 (1988) Zeitschriff P a r t i c ~ for l:Ynysik C

and F Ls �9 Springer-Verlag 1988

Radiative corrections for baryonic correlators

M. Jamin Institut f/Jr Theoretische Physik, Universitfit, Philosophenweg 16, D-6900 Heidelberg, Federal Republic of Germany

Received 6 December 1987

Abstract. We reanalyse the first order radiative cor- rections of the bare loop and of the quark condensate for nucleon and delta two-point function. It turned out, that there were changes in the next to leading order for the logarithms.

1 Introduction

Inspired by our work on mixing and factorization of four quark operators [1], we started a reanalysis of nucleon and delta sum rules, with the emphasis on the influence of factorization on the latter. Al- though the first order radiative corrections for the bare loop and the quark condensate had already been evaluated [2] we included a recalculation, since for this contributions there appeared differences with [3], which were also mentioned in [4].

The influence of the four quark operators is still under investigation and will be presented in a later publication. Nevertheless we give here the corrected results for the former contributions, to clarify the situ- ation. The notation used here is the same as in [2], which is our main reference, since we base our work on the method outlined there.

2 Calculation o f the two-point functions

We start with the two-point function II,(q) for a bar- yonic current J,(x) which is given by

lls(q)=i(27r)4SdDxeiqX(f2[ T{ :JB(x)~(0):}[O) (2.1)

where If2) is the physical vacuum. The relevant cur- rents for nucleon and delta are given in [2]. Applica- tion of Wick's theorem to the time ordered product leads to the diagrams for the bare loop (perturbation theory) and the quark condensate if one pair of quark fields is uncontracted. For a pictorial representation of some diagrams, the reader is also refered to [2].

In the calculation of the bare loop graphs no spe- cial difficulties are encountered, but one type of graphs which contribute to the quark condensate leads to infrared divergencies which we had to deal with. In this case one can keep the quark masses finite, as was explained in [2], to get an explicit cancelation of the logarithmic divergencies, which appear now in the quark masses, and the corresponding counter terms for the quark operator. The computation of the Feynman-integrals was done with the aid of some useful formulas found in [5, 6].

With the help of the Lehmann representation, the two-point function can be decomposed into

H ~(q) = O" AB(q 2 ) q- I . BR(q 2) + . . . (2.2)

where the dots denote other operator contributions, i.e. the gluon condensate and higher order operators. On dimensional grounds one can see that the function AB(q 2) arises from the perturbative part whereas the function BB(q 2) corresponds to the quark condensate. The results for A(q 2) and B(q 2) for nucleon and delta are given below:

AN(q2) = --218 (q2) 2 I n ( 7 q 2 2 ) ( 5 a 2 + 5 b 2 + 2 a b )

{ (/)} 71as 1 c~,ln �9 1-{ 12 ~z 2 =

~z 2 --q:

2

1 __q2

7+g

A~(q2)= - - -

636

82 B~(qZ)=3 ~ (qq) q2 In

�9 -fl 5 as 2 a s _ q 2

3 Conclusion

The results ob ta ined agree nei ther with those of refer- ence [2] nor with [3]. Compared to reference [-2] there are only al terat ions in the next to leading order terms for the radiat ive corrections. Al though the nu- merical analysis has no t yet been completed we as- sume, that the differences will no t play an impor t an t role for the sum rules since the radiat ive correct ions give only mino r cont r ibut ions . Fo r a detailed investi- gat ion we refer to a for thcoming publ ica t ion were

the mixing of the four quark operators under factori- za t ion will be included as well.

Acknowledgements. The author wishes to thank H.G. Dosch and M. Kremer for helpful and enlightening discussions on the subject. Moreover we are deeply indebted to F.V. Tkachov and L. Surgu- ladze for kindly checking our results.

References

1. M. Jamin, M. Kremer: Nucl. Phys. B277 (1986) 349 2. Y. Chung, H.G. Dosch, M. Kremer, D. Schall: Z. Phys. C25

(1984) 151 3. N.V. Krasnikov, A.A. Pivovarov, N.N. Tavkhelidze: Z. Phys. C 19

(1983) 301 4. L.J. Reinders, H.R. Rubinstein, S. Yazaki: Phys. Rep. 127.1 (1985) 5. S. Narison: Phys. Rep. 84.4 (1982) 6. P. Pascual, R. Tarrach: Renormalization for the Practitioner,

Lect. Notes in Physics, No. 194. Berlin, Heidelberg, New York: Springer 1984