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Basics of Qcd Perturbation Theory

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Basics of Qcd Perturbation Theory BASICS OF QCD PERTURBATION THEORY Davison E. Sop er Institute of Theoretical Science University of Oregon, Eugene, OR 97403 ABSTRACT This is an intro duction to the use of QCD p erturbation theory, em- phasizing generic features of the theory that enable one to separate short-time and long-time e ects. I also cover some imp ortant classes of applications: electron-p ositron annihilation to hadrons, deeply in- elastic scattering, and hard pro cesses in hadron-hadron collisions. Supp orted by DOE Contract DE-FG03-96ER40969. 1 Intro duction A prediction for exp eriment based on p erturbative QCD combines a particular calculation of Feynman diagrams with the use of general features of the theory. The particular calculation is easy at leading order, not so easy at next-to-leading order, and extremely dicult b eyond the next-to-leading order. This calculation of Feynman diagrams would b e a purely academic exercise if we did not use certain general features of the theory that allow the Feynman diagrams to b e related to exp eriment: the renormalization group and the running coupling; the existence of infrared safe observables; the factorization prop erty that allows us to isolate hadron structure in parton distribution functions. In these lectures, I discuss these structural features of the theory that allowa comparison of theory and exp eriment. Along the way,we will discover something ab out certain imp ortant pro cesses: + e e annihilation; deeply inelastic scattering; hard pro cesses in hadron-hadron collisions. By discussing the particular along with the general, I hop e to arm the reader with information that sp eakers at research conferences take to b e collective knowledge| knowledge that they assume the audience already knows. Now here is the disclaimer. We will not learn how to do signi cant calculations in QCD p erturbation theory. Three lectures is not enough for that. I hop e that the reader may b e inspired to pursue the sub jects discussed here 1 in more detail. A go o d source is the Handbook of Perturbative QCD by the CTEQ collab oration. More recently, Ellis, Stirling, and Webb er have written an 2 excellent book that covers most of the sub jects sketched in these lectures. For the reader wishing to gain a mastery of the theory, I can recommend the recent 3 4 5 b o oks on quantum eld theory by Brown, Sterman, Peskin and Schro eder, and 6 Weinb erg. Another go o d source, including b oth theory and phenomenology,is 7 the lectures in the 1995 TASI pro ceedings, QCD and Beyond. 2 Electron-Positron Annihilation and Jets In this section, I explore the structure of the nal state in QCD. I b egin with the + kinematics of e e ! 3 par tons, then examine the b ehavior of the cross section + for e e ! 3 par tons when two of the parton momenta b ecome collinear or one parton momentum b ecomes soft. In order to illustrate b etter what is going on, Iintro duce a theoretical to ol, null-plane co ordinates. Using this to ol, I sketch a space-time picture of the singularities that we nd in momentum space. The singularities of p erturbation theory corresp ond to long-time physics. We see that the structure of the nal state suggested by this picture conforms well with what is actually observed. I draw the distinction b etween short-time physics, for which p erturbation the- ory is useful, and long-time physics, for which the p erturbative expansion is out of control. Finally, I discuss how certain exp erimental measurements can prob e the short-time physics while avoiding sensitivity to the long-time physics. + 2.1 Kinematics of e e ! 3 Partons + Figure 1: Feynman diagram for e e ! q qg . p + Consider the pro cess e e ! q qg , as illustrated in Fig. 1. Let s b e the total energy in the c.m. frame and let q b e the virtual photon (or Z b oson) momentum, so q q = s. Let p b e the momenta of the outgoing partons (q; q; g) and let i 0 E = p b e the energies of the outgoing partons. It is useful to de ne energy i i fractions x by i 2p q E i i p = : (1) x = i s=2 s Then 0 <x : (2) i Energy conservation gives P X 2( p ) q i x = =2: (3) i s i Thus only two of the x are indep endent. i Let b e the angle b etween the momenta of partons i and j . We can relate ij these angles to the momentum fractions as follows: 2 2 2p p =(p +p ) =(qp ) =s2qp ; (4) 1 2 1 2 3 3 2E E (1 cos )=s(1 x ): (5) 1 2 12 3 Dividing this equation by s=2 and rep eating the argument for the two other pairs of partons, we obtain three relations for the angles : ij x x (1 cos ) = 2(1 x ) ; 1 2 12 3 x x (1 cos ) = 2(1 x ) ; 2 3 23 1 x x (1 cos ) = 2(1 x ) : (6) 3 1 31 2 We learn two things immediately. First, x < 1: (7) i Second, the three p ossible collinear con gurations of the partons are mapp ed into x space very simply: i ! 0 , x ! 1; 12 3 ! 0 , x ! 1; 23 1 ! 0 , x ! 1: (8) 31 2 The relations 0 x 1, together with x =2x x, imply that the i 3 1 2 allowed region for (x ;x ) is a triangle, as shown in Fig. 2. The edges x =1 of 1 2 i the allowed region corresp ond to two partons b eing collinear, as shown in Fig. 3. The corners x = 0 corresp ond to one parton momentum b eing soft (p ! 0). i i 2.2 Structure of the Cross Section One can easily calculate the cross section corresp onding to Fig. 1 and the similar amplitude in which the gluon attaches to the antiquark line. The result is 2 2 d x + x 1 s 1 2 = C ; (9) F dx dx 2 (1 x )(1 x ) 0 1 2 1 2 Figure 2: Allowed region for (x ;x ). Then x is 2 x x . 1 2 3 1 2 Figure 3: Allowed region for (x ;x ). The lab els and small pictures show the 1 2 physical con guration of the three partons corresp onding to subregions in the allowed triangle. P 2 2 + where C =4=3 and =(4 =s) Q is the total cross section for e e ! F 0 f 0 hadr ons at order . The cross section has collinear singularities: s (1 x ) ! 0 ; (2&3 col l inear ); 1 (1 x ) ! 0 ; (1&3 col l inear ) : (10) 2 There is also a singularity when the gluon is soft: x ! 0. In terms of x and x , 3 1 2 this singularity o ccurs when (1 x ) 1 const: (11) (1 x ) ! 0; (1 x ) ! 0; 1 2 (1 x ) 2 Let us write the cross section in a way that displays the collinear singularity at ! 0 and the soft singularityat E ! 0: 31 3 1 d f (E ; ) s 3 31 = C : (12) F dE d cos 2 E (1 cos ) 0 3 31 3 31 Here, f (E ; ) , a rather complicated function. The only thing that we need to 3 31 know ab out it is that it is nite for E ! 0 and for ! 0. 3 31 Now lo ok at the collinear singularity, ! 0.Ifweintegrate over the singular 31 region holding E xed, we nd that the integral is divergent: 3 Z 1 d d cos = log(1): (13) 31 dE d cos a 3 31 Similarly,ifweintegrate over the region of the soft singularity, holding xed, 31 we nd that the integral is divergent: Z a d dE = log(1): (14) 3 dE d cos 0 3 31 Evidently, p erturbation theory is telling us that we should not take the p ertur- + bative cross section to o literally. The total cross section for e e ! hadr ons is certainly nite, so this partial cross section cannot b e in nite. What we are seeing is a breakdown of p erturbation theory in the soft and collinear regions, and we should understand why. + Figure 4: Cross section for e e ! q qg , illustrating the singularity when the gluon is soft or collinear with the quark. Where do the singularities come from? Lo ok at Fig. 4 (in a physical gauge). 2 The scattering matrix element M contains a factor 1=(p + p ) where 1 3 2 (p + p ) =2p p =2E E (1 cos ): (15) 1 3 1 3 1 3 31 2 Evidently,1=(p +p ) is singular when ! 0 and when E ! 0. The collinear 1 3 31 3 singularity is somewhat softened b ecause the numerator of the Feynman diagram contains a factor prop ortional to in the collinear limit. (This is not exactly 31 obvious, but is easily seen by calculating. If you like symmetry arguments, you can derive this factor from quark helicity conservation and overall angular momentum conservation.) Wethus nd that " # 2 31 2 jMj / (16) 2 E 3 31 for E ! 0 and ! 0. Note the universal nature of these factors. 3 31 Integration over the double singular region of the momentum space for the gluon has the form Z Z 2 E dE d cos d 3 31 3 2 E dE d d: (17) 3 3 31 E 3 Combining the integration with the matrix element squared gives " # 2 Z Z 2 dE d 31 3 31 2 d E dE d d d: (18) 3 3 31 2 2 E E 3 3 31 31 Thus wehave a double logarithmic divergence in p erturbation theory for the soft and collinear region.
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