Perturbative QCD

Chul Kim

Open KIAS, Pyeong-Chang Summer Institute 2013

Seoultech

Pyeong-Chang, Alpensia Resort, July 9, 2013

Part II : Structure functions for DY One loop correction to current

Drell-Yan Process

P1

P2

N2

N1

p1

p2

q l�

l+

}p0 = p1 + p2 � q

■ Scattering Cross Section

q

q2 = Q2

q f

q f

l�

l+iM = hl+l�X|(iQ f e)q f gµq f · (ie)lgµl|N1N2i

�iq2

|M|2 =e4Q2

f

4Q4 hN1N2|J†µ|XihX|Jn|N1N2iLµn

Lµn(k1, k2) =14 Â

s,s0us(k1)g

nvs0(k2)vs0(k2)gµus(k1)

=14

Trk/1gnk/2gµ = kµ1 kn

2 � kn1kµ

2 � gµnk1 · k2

s(N1N2 ! l+l�X) =12s Â

X

Z d3k1(2p)3

d3k2(2p)3

12k0

1 · 2k02(2p)4d(P1 + P2 � q � pX)Â

f|M|2

=32p2a2

Q4s Âf

Q2f Wµn(P1, P2, q)

Z d3k1(2p)3

d3k2(2p)3

Lµn(k1, k2)

2k01 · 2k0

2

=32p2a2

Q4s Âf

Q2f

Zd4qWµn(P1, P2, q)

Z d3k1(2p)3

d3k2(2p)3

Lµn(k1, k2)

2k01 · 2k0

2d(q � k1 � k2).

s = (P1 + P2)2

Z d3k1(2p)3

d3k2(2p)3

Lµn(k1, k2)

2k01 · 2k0

2d(q � k1 � k2) = �p

6(�q2gµn + qµqn)

�!Z

dQ2d(q2 � Q2)⇣�p

6Q2gµn

⌘.

Homework:

qµWµn = qnWµn = 0

● Hadronic tensor

Wµn(P1, P2, q) = ÂX(2p)4d(P1 + P2 � q � pX)hN1N2|J†

µ|XihX|Jn|N1N2i

● Structure function ds

dQ2 = Âf

Q2f4pa2

3Q2s

Z d4q(2p)4 d(q2 � Q2)(�gµn)Wµn(P1, P2, q) =

s0s

FDY(t, Q2)

FDY(t, Q2) = �Nc ÂX

Zd4qd(q2 � Q2)d(P1 + P2 � q � pX)hN1N2|J†

µ|XihX|Jµ|N1N2i

= �Nc

Z d4q(2p)4 d(q2 � Q2)

Zd4ze�iq·zhN1N2|J†

µ(z)Jµ(0)|N1N2i

s0 = Âf

Q2f

4pa2

3NcQ2LO x-section at parton level:

will perform short distance expansion● Matching of EM current

Jµ = qgµq ! qnWngµ?W†

n qn

n/qn = O(l), n/qn = O(l)

Homework

Derivation of FT TheoremFDY(t) = �Ncd(q2 � Q2)hN1N2|qnWng

µ?W†

n qn · qnWng?µ W†

n qn|N1N2i

= �Nc

Zdy1dy2d(q2 � Q2)hN1N2|qnWng

µ?W†

n qn

⇥hqnWnd(y2 +

n · P†

n · P2)ig?

µ

hd(y1 �

n · Pn · P1

)W†n qn

i|N1N2i

● PDF projection

hN|hd(y � n · P

n · P

)Yn

ia

a

⇣Y

n

⌘b

b

|Ni = n · P

2N

c

dab⇣

n/2

⌘

ab

f

q/N

(x), Yn = W†n qn

hN|⇣

Yn

⌘a

a

hYnd(y +

n · P†

n · P)ib

b|Ni = n · P

2Ncdab

⇣n/2

⌘

abfq/N(y)

hN1N2|qnWngµ?W†

n qn ·hqnWnd(y2 +

n · P†

n · P2)ig?

µ

hd(y1 �

n · Pn · P1

)W†n qn

i|N1N2i

=s

4N2c

Nc fq/N1(y1) fq/N2(y2)Trn/2

gµ?

n/2

g?µ = � s

Ncfq/N1(y1) fq/N2(y2)

q = P1 + P2 � pX = (P1 � pXn) + (P2 � pXn) + pXH = p1 + p2 � pXH

■ LO factorization theorem FDY(t) = s

Zdy1dy2 fq/N1(y1) fq/N2(y2)d(q2 � Q2)

=Z dy1

y1

dy2y2

fq/N1(y1) fq/N2(y2)d(1 � z) =Z 1

t

dzz

d(1 � z)Lqq(t

z)

Lqq(t) =Z 1

t

dy1y1

fq/N1(y) fq/N2(t

zy1)

● Kinematic variables

s = (p1 + p2)2 = n · p1n · p2 = y1y2s

P1

P2

N2

N1

p1

p2

q l�

l+

}p0 = p1 + p2 � q

q2 � Q2 = s � 2pXH · (p1 + p2) + p2XH

� Q2

pXn

pXn

pXH

z ⌘ Q2

s, t ⌘ Q2

s

At LO,

pXH = 0 ! q2 � Q2 = s(1 � z)

■ Factorization theorem

FDY(t) =Z 1

t

dzz

HDY(z, Q2, µ)Lqq(t

z, µ)

F(0)DY(t) = d(1 � t)

※ LO structure function at parton level

H(0)DY = d(1 � z)

f (0)q/q(y) = f (0)q/q(y) = d(1 � y)

● At the higher order - The hard function should be IR-finite- All the IR divergence should reside in PDFs- Structure fn should be scale invariant

µ ⇠ Q

µ ⇠ LQCD

RG evolution

HDY(z, Q2, µ)

fq/N1 , fq/N2

One loop virtual computation in full QCD

p

p0

k

p+ k

p0 + k

MVqgµq = �ig2CFµ2#MS

Z dDk(2p)D

1k2(k + 2p · k)(k + 2p0 · k)

qgn(k/ + p/0)gµ(k/ + p/)gnq

Z dDk(2p)D

1k2(k + 2p · k)(k + 2p0 · k)

= 2Z 1

0da

Z 1

1�adb

Z dDl(2p)D

1(l2 � D)3

■ One loop correction to current

l = k + ap + bp0

D = abM2, M2 ⌘ 2p · p0 = 2p · q

qgn(k/ + p/0)gµ(k/ + p/)gnq =h (D � 2)2

Dl2 + 2M2(1 � a � b + (1 � #)ab)

iqgµq

● Numerator

● Loop integrals

gµgµ = D, gµgagµ = (2 � D)ga, gµg5gµ = �Dg5,

gµgagbgµ = 4gab + (D � 4)gagb,

gµgagbgdgµ = �2gdgbga + (4 � D)gagbgd.

In D-dimension, after Wick rotation such as l0 = il0

E, l2 = �l2

E, we have

Z dDlE(2p)D =

Z dWD(2p)D

Z •

0

dlElD�1

E ,

ZdWD =

2pD/2

G(D/2). (1)

Therefore Feynman integrals becomes

Z dDlE(2p)D

l2mE

(l2

E + D)n =1

(4p)D/2

G(m + D/2)G(D/2)

G(n � m � D/2)G(n)

⇣1

D

⌘n�D/2�m.

(2)

Usually D = 4� 2e, and Gamma function can be expanded in powers of e such

as

G(1 + e) = eG(e) = 1 � gEe +1

2

(g2

E +p2

6

)e2 + · · · . (3)

● Result at one loop

MV =asCF4p

⇣µ2eg

M2

⌘# Z 1

0da

Z 1

1�adb

h2(1 � 2#)G(#)D0�#

� 2G(1 + #)D0�1�#(1 � a � b + (1 � #)ab)i

=asCF2p

h 12 #UV

� 1#2

IR� 1

#IR(2 + ln

µ2

M2 )� 4 +p2

12� 3

2ln

µ2

M2 � 12

ln2 µ2

M2

i.

Cancelled by field strength renormalization of quark fields

- Double pole (logarithm) results from Sudakov effect

- For totally inclusive process, all the IR divergences cancel, but....

D0 = D/M2 = ab

- Can be matched onto EFT (SCET), Remaining finite terms gives the hard corrections