Perturbative QCD. Chul Kim. Seoultech. Part I : Introduction to QCD Structure functions for DIS

Transcription

1 Perturbative QCD Part I : Introduction to QCD Structure functions for DIS Chul Kim Seoultech Open KIAS, Pyeong-Chang Summer Institute 2013 Pyeong-Chang, Alpensia Resort, July 8, 2013

2 QCD QCD Lagrangian 1 L QCD = q(id/ m)q 4 Gµn,a Gµn a D µ = µ igt a Aµ, a Gµn a = µ An a n Aµ a + gf abc AµA a b n QCD Beta function : calculated as a negative value d d ln µ g(µ) =b(g(µ)) b(g) = g  k=0 b k as 4p k+1 = b0 g 3 16p 2 b 1 g 5 (16p 2 ) 2 + b 0 = 11N c 3 2n f, b 1 = 34 3 N2 c 10 3 N cn f 2C F n f

3 Coupling Constant a s (µ) = g2 (µ) 4p = 4p b 0 ln µ2 L 2 QCD h 1 b 1 ln ln µ2 L 2 QCD b 2 0 ln µ2 L 2 QCD i a s (µ! L QCD )! Confinement : long distance interactions a s (µ! )! 0 Asymptotically Free : short distance interactions

4 Operator Product Expansion (OPE) ho 1 (x)o 2 (0)i = Â n C n 12 (x, µ)ho n(µ)i C12 n (x, µ) : Complex function including Wilson coefficient Expanded by the short distance x ho n (µ)i : Includes all the information on the long distance interactions EX) Hadronic tensor for DIS W µn = 1 2p Z d 4 ze iq z hn J µ(z)j n (0) Ni z! 0 : Short distance expansion

5 QCD Factorization Theorem Systematically separate the short and long distance interactions EX) Factorization theorem of DIS structure function F 1 (x) = Z 1 x dz z H(Q2, z, µ) f q/p ( x z, µ) - Describe the short distance interactions - Corresponding to Wilson coefficient - Can be computed by perturbation - Describe the long distance interactions - Corresponding to the matrix element of the nonlocal operator - Cannot be computed, instead fit to experiments - Structure function has no renormalization scale variance Perturbative QCD has a predictive power

6 Structure Function for DIS

7 Deep inelastic scattering l k k 0 q L µn = 1 2 Â s,s0 ū s (k)g µ u s 0(k 0 ) ū s 0(k 0 )g n u s (k) = 1 2 Trk/gµ k/ 0 g n = 2(k µ k 0n + k n k 0µ g µn k k 0 ) P p } p0 = p + q N s(ln! lx) = 1 2s = p s Z Z d 3 k 0 (2p) 3 1 d 3 k 0 (2p) 3 1 2k 0 0 2k 0 0 Â X (2p) 4 d(k + P k 0 p X )hn J µ XihX J n Ni e2 Q 2 f Q 4 e 2 Q 2 f Q 4 L µn(k, k 0 )W µn (q, P) Lµn

8 Hadronic tensor and the structure functions W µn (q, P) = 1 2p  X = 1 Z 2p (2p) 4 d(q + P d 4 ze iq z hn J µ(z)j n (0) Ni p X )hn J µ XihX J n Ni, = ( g µn + q µq n q 2 )F 1 +(P µ q µ P q q 2 )(P n q n P q q 2 )F 2 Breit frame : q µ = Q( n µ + n µ )/2 n 2 = n 2 = 0, n n = 2 - Incoming hadron: - n-collinear momentum P µ = n P nµ 2 + m2 n P n µ 2 n µ =(1, 0, 0, 1), n µ =(1, 0, 0, 1) p µ = n p nµ 2 + pµ? + n p nµ 2 = O(Q)+O(Ql)+O(Ql2 ) - Bjorken variable: x = Q2 2P q Q n P q µ W µn = q n W µn = 0 - Final state momentum: p 2 X =(P + q)2 = m 2 + 2P q Q 2 Q 2 1 x x 0 apple x apple 1

9 Electromagnetic current µ q q H J µ = qg µ q! q H g µ? q n n/q n = O(l) q n Hadronic tensor in the Breit frame W µn = g µn? F 1 + n µ 2 + nµ 2 n n 2 + nn 2 F L Callan-Gross relation Q 2 F L = F 2 F 1! 0 4x (At leading twist) Suppressed part

10 Wilson lines q H W n q n n-collinear momenta Effective theory description J µ = qg µ q! q H g µ? q n! q H g µ? W n q n h W n (x) =P exp ig Gauge-invariant combination Z x dsn A n (sn µ i )

11 Derivation of the collinear Wilson lines When n-collinear gluons radiate from quark other than n-collinear (a) (b) p + k p + k 1 + k 2 p k p + k 1 p k 2 k 1 p k 1 k 2 M a = (p/ + k/)a/ gg q = (p + k) 2 2p A gg 2p k q = M b1 = g 2 n A(k 2 )n A(k G 1 ) q n k 1 n (k 1 + k 2 ) M b2 = g 2 n A(k G 1 )n A(k 2 ) q n k 2 n (k 1 + k 2 ) n An p gg n kn p q = G( g n A n k )q A µ = n A nµ 2 + Aµ? + n A nµ 2 = O(Q)+O(Ql)+O(Ql2 ) W n = 1 g 1 n P+ ie n A n + g 2 1 n P+ ie n A n 1 n P+ ie n A n +

12 Expression in the coordinate space n A n (n q) = Z d ze in q z n A n ( z) g 1 n P+ ie n A n( x) = = g 2p = ig g 2p Z dn q Z Z d z Z x d zn A n ( z) e in q x n q + ie n A n(n q) dn q e in q( x z) n q + ie n A n( z) 2pi Q( x z) Homework: show it. g 2 1 n P+ ie n A 1 n( x) n P+ ie n A n( x) = ( ig)2 2! P Z x d zn A n ( z) Z x dȳn A n (ȳ) h W n ( x) =P exp ig h W n( x) = P exp ig Z x i d zn A n ( z) Z x i W nw n = W nw n = 1 d zn A n ( z)

13 Factorization of the structure Fn. F 1 (x,,q 2 ) = g µn? 2 W µn = 4p 3  d(q + P X = 4p 3 Z 1 x dyâ X d(q + P p X )hn J?µ XihX Jµ? Ni p X )hn q n W n g µ? q H XihX q H g? µ d(y n P n P )W n q n Ni Separation of the final state l  X =  X H  X n, p X = p XH + p Xn, Xi = X H i X n i P = p + p Xn, q + P p X = q + p p XH k k 0 q incoming parton s momentum P p } p0 = p + q : p XH N p Xn

14 Parton Distribution Function (PDF) f q/n (y, µ) = 1 n P  n/ hn(p) q n W n 2 X nihx n d(y X n n P n P )W n q n N(P)i n/ = hn(p) q n W n 2 d(yn P n P)W n q n N(P)i Z dn z = 4p e ixn Pn z/2 hn(p) q n ( n z h n z i 2 ) 2,0 q n (0) N(P)i h [ z,0] = W n ( z)w n = P exp ig h = P exp ig Z z PDF at parton level - LO result 0 i d z 0 n A n ( z 0 ) Z z d z 0 n A n ( z )i 0 P exp gauge invariant n/ f q/q (y) =hq(p) q n W n 2 d(yn p n P)W n q n q(p)i f (0) q/q (y) = 1 2  ū s (p) n/ s 2 d(yn p n p)u s(p) = 1 2 = d(1 y) h ig  X n ihx n = 1 X n Z 0 i d z 0 n A n ( z 0 ) 1 n p d(1 y)trp/ n/ 2

15 PDF projection hn h d(y n P n P )Y n i a a Ȳn b Ni = n P d ab n/ b 2N c 2 Factorization theorem Z 1 F 1 (x, Q 2 ) = 4p 3 n P dy f 2 q/n (y)  d(q + p p XH ) x X H 1 n/ a a h0 N c 2 gµ? q H X HihX H q H g µ? 0i a a = Z 1 LO hard function x dy y f q/n(y, µ)h( x y, Q2, µ) = Z 1 x ab f q/n(x), Y n = W n q n dz z H(z, Q2, µ) f q/n ( x z, µ) H( x y, µ) = 2p3 n p  d(q + p p XH ) 1 n/ a h0 N X c 2 gµ? H q X HihX H q H g µ? a H H (0) ( x y, µ) = 2p3 n p Z d 3 k 1 (2p) 3 d(q + p k)tr n/ 2k 0 2 gµ? k/g? µ = n pn kd(k 2 ) k=q+p = n pqd(n pq Q 2 )=yn Pd(yn P Q) = yd(y x) =d(1 x y ) a a 0i

16 F 1 (x, Q 2 )= Z 1 x dz z H(z, Q2, µ) f q/n ( x z, µ) H(z, Q 2, µ) RG evolution µ Q At the higher order - The hard function should be IR-finite - All the IR divergence should reside in PDF - All the IR divergence should reside in PDF f q/n ( x z, µ) µ L QCD Structure function should scale invariant d d ln µ F 1 = 0! d d ln µ H f q/n + H d d ln µ f q/n