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PERTURBATIVE QCD for BEGINNERS Instead of Introduction PERTURBATIVE QCD FOR BEGINNERS Yu.L. Dokshitzer CERN, Geneva, Switzerland and Petersburg Nuclear Physics Institute, Russia. Abstract Basic elements of the p erturbative QCD approachtomultipar- ticle pro duction in hard pro cesses are discussed. Instead of Intro duction I had a hard time writing these lectures. You will have a hard time reading them. Don't take it as the author's fault: it is rather part of the author's design. I met in Dubna young exp erimenters, eager to learn. Some of them would ask not just \How probable is it to have this-and-that", but the most clever questions suchas \Why on earth..." or \How come that..." or even \How could I estimate the fraction of such-and-suchevents". Those students are to blame for what follows. Hearing things and an easy reading can only give an illusion of understanding. To really learn the game one should try playing. The present course was designed as a dull intro duction into the realm of quarks and gluons as QCD partons. We shall discuss why and to which extentanentirely classical probabilisti c language of partons is applicable to the analysis of multiparticle pro duction in quantum eld theory in the rst place. We shall sp end quite some time with the basic ingredients of the QCD parton picture: the nature of logarithmicall y enhanced contribu- tions to high-order hard interaction cross sections; the notion of parton evolution; and the problem of coherence e ects due to quantum interference, which reveals itself in a clever choice of the \evolution time". We shall study the main building blo cks: parton splitting functions, all-order double logarithmic form factors, the running coupling. Scanning through the text, you will o ccasionally see some formula-free pages. Don't rush in there. I b elieve these lectures may help you to develop physical intuition and to learn to apply it for practically pro ducing interesting, reliable back-of-an-envelop e estimates. To get there, however, you had b etter prepare yourself to a slow reading and keep a p encil and a (A4) notepad at hand. You will nd quite a few Exercises and Problems on the way. The former are to train your hand in p erforming typical calculations; the latter will require activation of grey cells prior to a hand. Try them. Don't hurry up to the solutions supplied. Go o d luck. May the colour force b e with you. Lecture 1. BASICS OF BREMSSTRAHLUNG We start our excursion with a survey of the pro cess of soft photon radiation ac- companying hard scattering of an electron, that is the QED bremsstrahlung. Sp eaking of a hard pro cess (scattering, interaction, etc.) one means a pro cess in whicha charged 2 2 2 particle exp eriences an impact with large momentum transfer, Q jq j m . The main characters of the rst lecture will b e Electrons and Photons (with Quarks and Gluons explicitly app earing only on sp ecial o ccasions when QCD subtleties come onto the stage). It is implied that all conclusions we shall arrive at considering basic prop erties of QED bremsstrahlung hold in the QCD context as well, that is when soft gluon radiation is concerned. Consider electron scattering o an external eld V accompanied by the emission of a photon with the 4-momentum k . The radiation amplitude is describ ed by the two graphs of Fig. 1. k k p p p p 1 2 1 2 Figure 1: Feynman graphs for radiation accompanying scattering of a charge Applying Feynman rules we get for these two contributions # " ^ ^ m+^p k m +^p +k 1 2 u(p ); (1.1) V +V M = e u (p ) 1 2 2 2 2 2 m (p +k) m (p k) 2 1 where V stands for the vertex of the interaction with the external eld. The next step is to simplify the radiation amplitude by making use of the Dirac equation for the on-mass-shell fermions, u (p )(mp^ )=(mp^ )u(p )=0: (1.2) 2 2 1 1 To this end let us exchange the order ofp ^ and matrices, i (m +^p)=(mp^) +2p ; to identically rewrite the square brackets of (1.1) as ^ ^ (2p + k )V V (2p k ) 2 1 + + terms vanishing for on-mass-shell charges. (1.3) 2 2 2 2 m (p + k ) m (p k) 2 1 Using 2 2 2 2 =0) ; (p k ) m = 2p k + k = 2p k (for real photons, k i i i one arrives at " # " # ^ ^ 2p 2p V k kV 1 2 (1:3) = V + Vj + : (1.4) 2(p k ) 2(p k ) 2(p k ) 2(p k ) 1 2 1 2 Dep ending on the nature of the underlying scattering pro cess, the vertex V may a ect electron p olarizatio n states, i.e. mayhave a spinor structure. Therefore wehave carefully kept the V -factors at their prop er places according to the Feynman rules for the amplitude (1.1). Such a precaution is necessary,however, only for the term, while the rst, main, contribution to (1.4) simply provides the scattering vertex V with a scalar (with resp ect to fermion p olarizatio n space) accompanying-radiatio n factor j , p p 1 2 j (k )= : (1.5) (pk) (pk) 1 2 The reason to call j the main contribution is seen from the way the amplitude scales 0 with resp ect to the photon energy ! = k : 1 0 j = O ! ; = O ! : (1.6) This clearly makes j dominating for soft photons with ! E ;E . The origin of the 1 2 soft radiation factor (1.5) is purely classical. Let us verify this by considering scattering of a charge in the framework of the classical theory of electromagnetic radiation. 1.1 Classical Consideration From classical eld theory we know that it is the acceleration of a charge that causes electromagnetic radiation. Electromagnetic current participating in eld formation in the course of scattering consists of two terms (we suppress the charge e for simplicity) 8 3 ~ < C = ~v (~r ~v t) #(t t) ; 1 1 1 0 ~ ~ ~ C = C + C ; (1.7a) 1 2 3 : ~ C = ~v (~r ~v t) #(t t ) ; 2 2 2 0 with ~v the velo city of the initial ( nal) charge moving along the classical tra jectory 1(2) ~r = ~v t.By t we denote the moment in time when the scattering o ccurs and the velo city i 0 abruptly changes. Toachieve a Lorentz-covariant description one adds to (1.7a) an equa- tion for the propagation of the charge-density D to b e treated as the zero-comp onentof the 4-vector current C , ~ C (t; ~r ) = D (t; ~r ); C (t; ~r ) v D ; (1.7b) i i i i i with v the 4-velo cityvectors i v =(1;~v): i i ~ The emission amplitude for a eld comp onent with 4-momentum (!; k ) is prop ortional to the Fourier transform of the total current. For the two terms of the currentwehave 1 0 Z Z Z ik t 0 0 iv e 0 ~ 1 3 ix k ik (t + )i(k ~v ) 0 i C (k)= dt d re ; C (t; ~r )=v d e = 1 1 1 0 ~ 1 k (k ~v ) 1 1 (1.8) 1 0 Z Z Z ik t 0 +1 iv e 0 ~ 2 3 ix k ik (t + )i(k ~v ) 0 i C (k )= dt d re C (t; ~r )=v d e = : 2 2 2 0 ~ 0 k (k ~v ) 2 1 The solution of the Maxwell equation for the eld p otential induced by the current (1.8) reads Z Z 4 3 dk d k 0 ~ ix k i! x +i(k ~x) 2 A (x)= e e A (k ) ; (1.9) [2i(k )] C (k )= 4 3 (2 ) 2!(2 ) where A (k ) = A (k ) A (k ); (1.10a) 2 1 v i i! t 0 ~ ~ A (k ) = e ; ! = jk j ; (k ~v ) !v cos : (1.10b) i i i i ! (1 v cos ) i i Here are the angles b etween the direction of the photon momentum and that of the i corresp onding (initial/ nal ) charge. Rewriting (1.10b) in the covariant form E v p v i i i i = = ; ~ ~ (p k ) i ! (k ~v ) E (! (k ~v )) i i i we observe that the classical 4-vector \p otential" (1.10a), as exp ected, is identical to the quantum amplitude j (1.5), apart from an overall phase factor exp(i! t ). The latter is 0 irrelevant for calculating the observable cross section (see, however, section 1.10). We conclude that the classical consideration gives the correct accompanying ra- diation pattern in the soft-photon limit. This is natural b ecause in such circumstances (negligible recoil) it is legitimate to keep charges moving along their classical tra jectories, which remain unp erturb ed in the course of sending away radiation. 1.2 Radiation Cross Section To obtain the di erential cross section you haveto 1. calculate the matrix elementby pro jecting the eld-amplitude M onto the photon p olarization state , 2. square it and supply it with a prop er phase-space factor and 3. sum over : 3 3 X X d k d k 2 d / jM j = M M : (1.11) 3 3 (2 ) 2! (2 ) 2! =1;2 =1;2 The sum runs over twophysical p olarization states of the real photon, describ ed by normalized p olarization vectors orthogonal to its momentum: 0 0 (k ) (k )= ; (k ) k =0; ; =1;2: 0 ; Within these conditions the p olarization vectors maybechosen di erently.Asyou know fairly well, such an uncertainty do es not a ect physical observables (gauge invariance).
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