Z. Phys. A 348, 123-127 (1994) ZEITSCHRIFT FURPHYSIKA 9 Springer-Verlag 1994 Higher twist and higher order contributions to the pion electromagnetic form factor G.A. Miller ~, J. Pasupathy 2 1Department of Physics, FM-15, University of Washington, Seattle, WA 98195, USA 2 Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560 012, India Received: 7 January 1994 (; .m4,2 ~2] Abstract. The 0 and 0 (~2) corrections to \tm~ + ma) ~2 ] the leading term of the perturbative QCD calculation of the pion electromagnetic form factor are examined nu- merically. Both sets of terms provide significant correc- tions for values of Q2 between 1 and 15 GeV2/c 2. PACS: 13.40.Fn; 14.40.Ag; 12.38.Bx 1. Introduction Computing the pion electromagnetic form factor F~ (Q2) was one of the early challenges to perturbative quantum chromodynamics QCD calculations [1-4] of exclusive processes. Such calculations have been extended by using non-asymptotic pion wave functions [5] and criticized [6, 7] for depending too much on the end-point regions of the wave function. Most recently the effects of Sudakov evolution [8] have been incorporated [9, 10] in the leading twist two contribution to F~ (Q2). The results [10] are within a factor of two or so of the data. We describe this history in a bit more detail and define our notation in Sect. 2. Our purpose here is to compute two sorts of correc- tions to that calculation. Geshkenbein and Terent'ev [ 12] identified a term that supplies a power correction of ( (m,+ma)2Q2 j. This term arises from the twist-3 pion wave function and is especially important because of the small values of the current quark masses m and m a. It is known that this term is actually divergent unless Sudakov effects are included. Thus we evaluate this power correction incorporating the effects of Sudakov suppres- sion; see Sect. 3. But this is not enough. Mueller [13] has argued that the higher twist contributions are meaningful only if the leading twist term has been calculated to a sufficiently high order in c~ s (Q2). Therefore we examine the corrections of order es 2 . These were evaluated in [14-17]; with Braaten and Tse [17] resolving some tech- nical differences between the earlier calculations. Al- though the earlier calculations contained numerical re- sults, it seems that no numerical evaluation of the Braa- ten-Tse formulae has appeared in the literature. We pro- vide such a calculation in Sect. 4. The power corrections and higher order corrections are closely related because power corrections arise from the infrared regons of the integrations needed to perform the higher order calculations [ 13]. Thus the higher twist and higher order correction terms can not be simply added. Our approach is to make separate examinations of each of the above mentioned correction terms. The results are summarized and discussed in Sect. 5. 2. History This section is concerned with a brief review of the early history and also establishes some notation. The classic prediction [1-4] is that the electromagnetic form factor of the pion tends to the limit 8 Jrc~ s (Q2)f~ lira F~(Q2)=--F~(Q2) - Q2 Q2~co (1) where f= = 133 MeV is the pion decay constant, Q2 is the negative of the square of the transferred four momentum, and es (Q2) is the QCD running coupling constant. In leading order, es (Q2) = 4 ~c/[(11 - } ny) log (Q2/A2)], with ni as the number of flavors (here 3). Bebek et al. [11] have summarized the form factor in the interval 0.15 GeV2< Q2< 10 GeV 2. An adequate fit to the data was obtained by using the representation 1 F~ (Q2) = l + Q2/(0.462 + 0.024 GeV2) " (2) If one assumes that (1) is applicable for Q2~ 10 GeV 2, then the use of A -- 200 MeV/c shows that the theoretical prediction is smaller than the experimental value by about