Z. Phys. C 64, 495-497 (1994) ZEITSCHRIFT FORPHYSIK C 9 Springer-Verlag 1994 Dimensional regularization at finite temperature Yasushi Fujimoto*'**, M. Loewe, J.C. Rojas Facultad de Fisica, Pontificia Universidad Catdlica de Chile, Casilla 306, Santiago 22, Chile Received: 13 September 1993 / In revised form: 16 May 1994 Abstract. It is shown that dimensional regularization, when applied to massless theory at finite temperature, properly regularizes both ultraviolet and infrared diver- gences so that the temperature ,dependent ultraviolet di- vergences cancell with each other. It is also shown that a set of highly singular diagrams turns out to be finite in dimensional regularization. 1 Introduction Dimensional regularization [1] has been a great tool in regularizing divergences and performing higher order cal- culations with a relative ease in zero temperature (T= 0) perturbations. The present work is concerned with the application of dimensional regularization to finite tem- perature (T:# 0) cases. Our particular interest is in mass- less cases where perturbations are expected to breakdown due to infrared singularities. To have a control over the divergences we make use of dimensional regularization in accordance with T= 0 case. To be explicit, we work on the massless r theory in four dimensions and calculate three loop vacuum graphs to see how the tem- perature dependent divergences are regularized in dimen- sional regularization. 2 Massless (t~ 4) theory at three loops Consider a three loop diagram in Fig. 1. Petals are easy to calculate. Each petal yields ,,~ T 2. For the central ring we have, * Also at, Department of Pure and Applied Sciences,University of Tokyo-Komaba 3-8-1 Komaba, Meguro-Ku, Tokyo, Japan ** Present address: Institute of Particle Physics, Hua-Zhong Nor- real University, Wuhan, People's republic of China . d"k I i 1 i i ie)2)ea'k~ 1 + ((k2+ie) 2 (k 22 -1 = I(T= O) + I(T. 0), (n=4-e,k2=k2-k2). (1) For our later convenience, T= 0 part may be devided into two parts so that one part contains only the ultra- violet divergence and the other the infrared divergence. dnk ~ -ira2 I(T=0)= --j" ~ ((k2+ie)2(k2_m2+ie) i 1 (k2 + ie)(kS_m2 + ie) _ 1 C) (4/'C) 2 I( -~2+ln(m2)+e + (2-1n (mZ)- C)I = 0 (C : a constant). (2) In the end one recovers a well known result. Although, in the above, terms of higher orders in e have been dropped, this are identically zero. Next we calculate T. 0 part which contains an infrared divergence. I(r=t:O) d3-"k ~ (1 1 )(kZ=k:) -- I (27r)3-e c~k2 e zk-1 -4r~2 1-B r ,+lnfl2+ C' , (C': a constant), (3) ~ ~4 ~ 2