qcd in collisions with polarized beams€¦ · the goal: to understand qcd and the strong...

QCD in Collisions

with

Polarized Beams

Jianwei Qiu Brookhaven National Laboratory

Stony Brook University

Annual Hua-Da School on QCD: EIC physics China Central Normal University (CCNU), Wuhan, China, May 23 – June 3, 2016

q  The Goal:

To understand QCD and the strong interaction dynamics, and to explore hadron structure and its properties by studying high energy collisions with polarized beams

q  The Plan (approximately):

Electron-Ion Collider

Connecting QCD quarks and gluons to observed hadrons and leptons

Fundamentals of QCD factorization and evolution

Three lectures

Hard scattering processes with transversely polarized beams

Two lectures

Hard scattering processes with longitudinally polarized beams

Three lectures

The plan for my eight lectures

See also talks by Yuan and Xiao

Summary of lecture one

q Cross section with identified hadron(s) is NOT completely calculable in QCD perturbation theory

q QCD Factorization – neglecting quantum interference between dynamics at hard partonic scattering and those at hadronic scales – approximation

q Predictive power of QCD factorization relies on the universality of PDFs (or TMDs, GPDs, …), the calculations of perturbative coefficient functions – hard parts

q EIC is a ultimate QCD machine: 1) to discover and explore the quark/gluon structure and

properties of hadrons and nuclei, 2) to search for hints and clues of color confinement, and 3) to measure the color fluctuation and color neutralization

q  EIC is a tomographic machine for nucleons and nuclei with a resolution better than 1/10 fm

How to connect QCD quarks and gluons

to observed hadrons and leptons?

Fundamentals of QCD factorization

and evolution

QCD factorization – approximation

q  Creation of identified hadron(s):

⇡

O✓ hk2i

Q2

�↵◆+

k2 = 0

Perturbative!

Non-perturbative, but, universal

Factorization: factorized into a product of “probabilities” !

q  Scattering amplitude:

( ) ( ) ( )', '; ' , q u k ie u kλ µ λλ λ σ γ⎡ ⎤⎣ ⎦=Μ −

( )'2

i gq

µµ⎛ ⎞−⎜ ⎟

⎝ ⎠

( )' 0 ,emX eJ pµ σ

*

*

q  Cross section:

( )( ) ( )

2 3 32DIS

3 3, ', 1

1 1 ', '; ,2 2 2 2 2 2 '

Xi

X i i

d l d kd qs E Eλ λ σ

σ λ λ σπ π=

⎡ ⎤⎛ ⎞= Μ ⎢ ⎥⎜ ⎟⎝ ⎠ ⎢ ⎥⎣ ⎦

∑ ∑ ∏ ( )4 4

1

2 'X

iil k p kπ δ

=

⎛ ⎞+ − −⎜ ⎟

⎝ ⎠∑

( ) ( )2DIS

3 2

1 1 , '''

,2

dE W qd k s

k pQ

L kµνµν

σ ⎛ ⎞= ⎜ ⎟

⎝ ⎠

q  Leptonic tensor:

( )2

' ' '2( , ')

2eL k k k k k k k k gµν µ ν ν µ µν

π= + − ⋅– known from QED

μ

μ’ μ’

μ

Example: Inclusive lepton-hadron DIS

q  Hadronic tensor:

q  Symmetries:

( ) ( )

( ) ( ) ( )( )

( )

2 21 22 2 2

222 2

1

1

. .+

.

, ,

, ,.

B B

p B B

q q p q p qW g p q p qq p

S SS

q q q

p q q piM q

p

F x Q F x Q

g xq

g x Qp q

Q

µ νµν µν µ µ ν ν

σ σµνρσ σρε

⎛ ⎞ ⎛ ⎞⎛ ⎞⋅ ⋅= − − + − −⎜ ⎟ ⎜ ⎟⎜ ⎟⋅ ⎝ ⎠

⎡ ⎤⎢ ⎥⎢ ⎥⎣

⎝⎠

−+

⎦

⎠⎝

q  Structure functions – infrared sensitive:

( ) ( ) ( ) ( )2 2 2 21 2 1 2, , , , , , ,B B B BF x Q F x Q g x Q g x Q

4 †1( , , ) e , ( ) (0) ,4

iq zS S SW q p d z p J z J pµν µ νπ⋅= ∫

²  Parity invariance (EM current) ²  Time-reversal invariance

²  Current conservation

*

sysmetric for spin avg.

real

0

W W

W W

q W q W

µν νµ

µν µν

µ νµν µν

=

=

= =

No QCD parton dynamics used in above derivation!

DIS structure functions

xB =Q

2

2p · q

Q2 = �q2

Long-lived parton states

q  Feynman diagram representation:

W µν ∝ …

q  Perturbative pinched poles:

42 2

0

1 1 perturbati1T( , vely( , ) ) Hr

Qd k k kk i k iε ε⎛ ⎞⎛ ⎞ ⇒⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∞+ −∫

q  Perturbative factorization: 2 2

2T

Tk kk xx

p n kp n

µ µ µ µ+= + +

⋅

22 2

0

2 2H( , 0) 1 1 ( 1T , )T dkdx d k ki ik

Qr

kkx ε ε

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

=+ −∫ ∫

Short-distance

Nonperturbative matrix element

q  Collinear approximation, if Q ∼ xp ⋅n ≫ kT , k 2

Parton’s transverse momentum is integrated into parton distributions, and provides a scale of power corrections

Scheme dependence

≈2

2TkOQ⎛

+⎞

⎜ ⎟⎝ ⎠

⊗ +UVCT

– Lowest order:

Same as an elastic x-section

q  Corrections:

q  DIS limit: 2 whil, , e fixedBQ xν →∞

Feynman’s parton model and Bjorken scaling

Spin-½ parton!

Collinear factorization – further approximation

Parton distribution functions (PDFs)

q  PDFs as matrix elements of two parton fields: – combine the amplitude & its complex-conjugate

But, it is NOT gauge invariant!

can be a hadron, or a nucleus, or a parton state!

ZO(µ2)

– need a gauge link:

ZO(µ2)

– corresponding diagram in momentum space: Z

d

4k

(2⇡)4�(x� k

+/p

+)

μ-dependence

Universality – process independence – predictive power

+UVCT(µ2)

Gauge link – 1st order in coupling “g”

q  Longitudinal gluon:

q  Left diagram:

= g

Zdy

�1

2⇡n ·Aa(y�1 )t

a

Zdx1

1

�x1 + i✏

eix1p+(y�

1 �y

�)

�M = �ig

Z 1

y�dy�1 n ·A(y�1 )M

Zdx1

Zp

+dy

�1

2⇡eix1p

+(y�1 �y

�)n ·Aa(y�1 )

�M(�igt

a)� · pp

+

i((x� x1 � x

B

)� · p+ (Q2/2x

B

p

+)� · n)(x� x1 � x

B

)Q2/x

B

+ i✏

q  Right diagram: Zdx1

Zp

+dy

�1

2⇡eix1p

+y

�1n ·Aa(y�1 )

��i((x+ x1 � x

B

)� · p+ (Q2/2x

B

p

+)� · n)(x+ x1 � x

B

)Q2/x

B

� i✏

(+igt

a)� · pp

+M

= g

Zdy

�1

2⇡n ·Aa(y�1 )t

a

Zdx1

1

x1 � i✏

eix1p+y

�1

�M = ig

Z 1

0dy�1 n ·A(y�1 )M

q  Total contribution: �ig

Z 1

0

�Z 1

y�

�dy�1 n ·A(y�1 )MLO

O(g)-term of the gauge link!

q  NLO partonic diagram to structure functions:

2 212

10

Q dkk

−

∝ ∫ Dominated by

k12 0

tAB →∞

Diagram has both long- and short-distance physics

q  Factorization, separation of short- from long-distance:

QCD high order corrections

q  QCD corrections: pinch singularities in 4id k∫

+ + + …

q  Logarithmic contributions into parton distributions:

⊗ + +…+UVCT

( )22

22QCD2

2 2( , ) , , ,fB

FFf

B fsx QF x Q C x Ox Q

α ϕ µµ

⎛ ⎞Λ⎛ ⎞= ⊗ + ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑

q  Factorization scale: 2Fµ

To separate the collinear from non-collinear contribution

Recall: renormalization scale to separate local from non-local contribution

QCD high order corrections

q  Use DIS structure function F2 as an example:

( )22QCD2

2 2,

2/2( , ) , , ,B

h B q f sq f

f hx QF x Q C x Ox Q

α ϕ µµ

⎛ ⎞Λ⎛ ⎞= ⊗ + ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑

h q→²  Apply the factorized formula to parton states:

( )2

22

,

2/2( , ) , , ,B

q B q f sq

ff

qx QF x Q C xx

ϕ µµ

α⎛ ⎞

= ⊗⎜ ⎟⎝ ⎠

∑Feynman diagrams

Feynman diagrams

²  Express both SFs and PDFs in terms of powers of αs:

0th order: ( ) ( ) ( ) ( )02 2/

0 02 22 ( , ) ( / , / ) ,q qq B q BF x Q C x x Q xµ ϕ µ= ⊗

( ) ( )0 02( ) ( )q qC x F x= ( ) ( ) ( )0

/ 1q q qqx xδ δϕ = −

1th order: ( ) ( ) ( ) ( )( ) ( ) ( )

1 12 02 222

0 2

/

12 2/

( , ) ( / , / ) ,

( / , / ) ,

q B q B

q B

q q

q q

F x Q C x x Q x

C x x Q x

µ ϕ µ

µ ϕ µ

= ⊗

+ ⊗

( ) ( ) ( ) ( ) ( )1 1 02 2 2 12 2/2 2( , / ) ( , ) ( , ) ,q q qq qC x Q F x Q F x Q xµ ϕ µ= − ⊗

How to calculate the perturbative parts?

q  Change the state without changing the operator:

– given by Feynman diagrams

q  Lowest order quark distribution:

²  From the operator definition:

PDFs of a parton

q  Leading order in αs quark distribution:

²  Expand to (gs)2 – logarithmic divergent:

UV and CO divergence

q  Projection operators for SFs:

( ) ( )2 21 22 2 2

1, ,q q p q p qW g F x Q p q p q F x Qq p q q qµ ν

µν µν µ µ ν ν

⎛ ⎞ ⎛ ⎞⎛ ⎞⋅ ⋅= − − + − −⎜ ⎟ ⎜ ⎟⎜ ⎟⋅ ⎝ ⎠⎝ ⎠⎝ ⎠

22 2

1 2

22 2

2 2

1 4( , ) ( , )2

12( , ) ( , )

xF x Q g p p W x QQ

xF x Q x g p p W x QQ

µν µ νµν

µν µ νµν

⎛ ⎞= − +⎜ ⎟

⎝ ⎠

⎛ ⎞= − +⎜ ⎟

⎝ ⎠

q  0th order: ( )

( ) ( ) ( )

0(0)2 ,

22

2

1( ) 4

1Tr 2 ( )4 2

(1 )

q q

q

q

F x xg W xg

exg p p q p q

e x x

µν µνµν

µνµ ν

π

γ γ γ γ πδπ

δ

= =

⎡ ⎤= ⋅ ⋅ + +⎢ ⎥⎣ ⎦

= −( )0 2( ) (1 )q qC x e x xδ= −

Partonic cross sections

q  Complex n-dimensional space:

(2) Calculate IRS quantities here

(3) Take εè 0 for IRS quantities only

Re(n)

Im(n)

4 6

UV-finite, IR divergent

UV-finite, IR-finite

Theory cannot be renormalized!

(1) Start from here: UV renormalization a renormalized theory

How does dimensional regularization work?

( ) ( ) ( ) ( ) ( )1 1 02 2 2 12 2/2 2( , / ) ( , ) ( , ) ,q q qq qC x Q F x Q F x Q xµ ϕ µ= − ⊗

q  Projection operators in n-dimension: 4 2g g nµνµν ε= ≡ −

( )2

2 2

41 (3 2 ) xF x g p p WQ

µν µ νµνε ε

⎛ ⎞− = − + −⎜ ⎟

⎝ ⎠

q  Calculation: ( ) ( )1 1, ,and q qg W p p Wµν µ ν

µν µν−

q  Feynman diagrams:

( )1,qWµν

+ UV CT

+ + +

+ + c.c. + + c.c.

Real

Virtual

NLO coefficient function – complete example

q  Lowest order in n-dimension:

( )0 2, (1 ) (1 )q qg W e xµν

µν ε δ− = − −

q  NLO virtual contribution:

( )1 2,

2 2

2 2

(1 ) (1 )

4 (1 ) (1 ) 1 3 1 * 4(1 2 ) 2

Vq q

sF

g W e x

QC

µνµν

ε

ε δ

α π ε επ ε ε ε

µ

− = − −

⎡ ⎤ Γ + Γ −⎛ ⎞ ⎡ ⎤− + +⎢ ⎥⎜ ⎟ ⎢ ⎥Γ − ⎣ ⎦⎝ ⎠ ⎣ ⎦

q  NLO real contribution:

( )2

1 2, 2

4 (1 )(1 )2 (1 2 )

1 2 1 1 2 * 11 1 2 2(1 2 )(1 ) 1 2

R sq q Fg W e

Q

xxx

C

x

ε

µνµν

α π εε

π ε

ε ε εε ε ε ε

µ⎡ ⎤ Γ +⎛ ⎞− = − − ⎢ ⎥⎜ ⎟ Γ −⎝ ⎠ ⎣ ⎦

⎧ ⎫− ⎡ ⎤ −⎛ ⎞⎛ ⎞− − + + +⎨ ⎬⎜ ⎟⎜ ⎟⎢ ⎥− − − − −⎝ ⎠⎝ ⎠⎣ ⎦⎩ ⎭

Contribution from the trace of Wμν

q  The “+” distribution:

( )1

21 1 1 (1 )(1 )1 (1 ) 1

n xx Ox x x

ε

δ ε εε

+

++

−⎛ ⎞ ⎛ ⎞= − − + + +⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠

l

1 1( ) ( ) (1) (1 ) (1)(1 ) 1z z

f x f x fdx dx n z fx x+

−≡ + −

− −∫ ∫ l

q  One loop contribution to the trace of Wμν:

( )2

1 2, 2

1(1 ) ( ) ( )2 (4 )Es

qq qqq q P Qg W e x x ne

Pµνµν γ

αε

π ε µ π −

⎛ ⎞⎛ ⎞⎧− = − − +⎨ ⎜ ⎟⎜ ⎟⎩⎝ ⎠ ⎝ ⎠

l

( )2

2 (1 ) 3 1 11 ( )1 2 1 1Fn x xx n

x xC x

x ++

⎡ − +⎛ ⎞ ⎛ ⎞+ + − −⎜ ⎟ ⎜ ⎟⎢ − − −⎝ ⎠ ⎝ ⎠⎣

l l

293 (1 )2 3

x xπδ

⎫⎤⎛ ⎞ ⎪+ − − + − ⎬⎥⎜ ⎟

⎝ ⎠ ⎪⎦⎭q  Splitting function:

21 3( ) (1 )(1 ) 2qq F

xx xx

P C δ+

⎡ ⎤+= + −⎢ ⎥−⎣ ⎦

q  One loop contribution to pμpν Wμν:

( )1, 0Vqp p Wµ ν

µν = ( )2

1 2, 2 4R sq q F

Qp p W ex

Cµ νµν

απ

=

q  One loop contribution to F2 of a quark:

( ) ( )2

1 2 22 2

CO

1( , ) ( ) 1 (4 ) ( )2

Eqq q

sq qq

QF x Q e x x n eP x nPγεε

απ

π µ− ⎛ ⎞⎧⎛ ⎞= − + +⎨ ⎜ ⎟⎜ ⎟

⎝ ⎠⎩ ⎝ ⎠l l

2 22 (1 ) 3 1 1 9(1 ) ( ) 3 2 (1 )

1 2 1 1 2 3Fn x xx n x x x

x x xC π

δ+ +

⎫⎡ ⎤⎛ ⎞− + ⎪⎛ ⎞ ⎛ ⎞+ + − − + + − + − ⎬⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪⎣ ⎦⎭

l l

as 0ε⇒ ∞ →

Different UV-CT = different factorization scheme!

q  One loop contribution to quark PDF of a quark:

( )1/

V CO

2

U

1 1( , ) ( ) UV-CT2s

q q qqPx xαµ

εϕ

π ε⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + − +⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎩ ⎭

– in the dimensional regularization

q  Common UV-CT terms:

²  MS scheme: MSUV

1UV-CT ( )2 qs

qP xαπ ε

⎛ ⎞= − ⎜ ⎟⎝ ⎠

²  DIS scheme: choose a UV-CT, such that ( )1 2 2

DIS( , / ) 0qC x Q µ =

²  MS scheme: ( )MSUV

1UV-CT ( ) 1 (4 )2

Esqq x nP e γα

ε πεπ

−⎛ ⎞= − +⎜ ⎟⎝ ⎠

l

q  One loop coefficient function:

( ) ( ) ( ) ( ) ( )1 1 02 2 2 12 2/2 2( , / ) ( , ) ( , ) ,q q qq qC x Q F x Q F x Q xµ ϕ µ= − ⊗

2 22 (1 ) 3 1 1 9(1 ) ( ) 3 2 (1 )

1 2 1 1 2 3Fn x xx n x x x

x x xC π

δ+ +

⎫⎡ ⎤⎛ ⎞− + ⎪⎛ ⎞ ⎛ ⎞+ + − − + + − + − ⎬⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪⎣ ⎦⎭

l l

( )2

1 2 2 22MS

( , / ) ( )2 qq q qs P QC x Q e x x nα

µπ µ

⎧ ⎛ ⎞⎪= ⎜ ⎟⎨ ⎜ ⎟⎪ ⎝ ⎠⎩

l

q  Physical cross sections should not depend on the factorization scale

22

22 ( , ) 0BFF

d F x Qd

µµ

=

µF2 ddµF

2C f

xBx,Q

2

µF2,αs

!

"##

$

%&&

'

())

*

+,,f

∑ ⊗ϕ f x,µF2( )+ C f

xBx,Q

2

µF2,αs

!

"##

$

%&&

f∑ ⊗µF

2 ddµF

2ϕ f x,µF

2( ) = 0Evolution (differential-integral) equation for PDFs

DGLAP evolution equation: 2

/2 2

2 ( , ) ( ', ),'i i j jF F s F

F j

xPx

x xµ ϕ µ α ϕ µµ

⎛ ⎞⎜= ⎟⎝ ⎠

∂⊗

∂ ∑

q  PDFs and coefficient functions share the same logarithms

PDFs:

Coefficient functions:

( ) ( )2 2 2 20 QCDlog or logF Fµ µ µ Λ

( ) ( )2 2 2 2log or logFQ Qµ µ

Dependence on factorization scale

F2(xB , Q2) =

X

f

Cf (xB/x,Q2/µ

2F ,↵s)�f (x, µ

2F )

Calculation of evolution kernels

q  Evolution kernels are process independent

²  Parton distribution functions are universal

²  Could be derived in many different ways

“Gain” “Loss” Change

q  Extract from calculating parton PDFs’ scale dependence

P

kk

P p p

k k

PP

p-kp

kpPP

pk

p-k

PP

+1

2+

1

2

Collins, Qiu, 1989

²  Same is true for gluon evolution, and mixing flavor terms

q One can also extract the kernels from the CO divergence of partonic cross sections

Scaling and scaling violation

Q2-dependence is a prediction of pQCD calculation

σ totDIS ∼ ⊗

1 OQR⎛ ⎞

+ ⎜ ⎟⎝ ⎠

From one hadron to two hadrons

q  One hadron: e p

Hard-part Probe

Parton-distribution Structure

Power corrections Approximation

q  Two hadrons:

σ totDY ∼ ⊗

1 OQR⎛ ⎞

+ ⎜ ⎟⎝ ⎠

s

Predictive power: Universal Parton Distributions

q

Drell-Yan process – two hadrons

q  Drell-Yan mechanism: S.D. Drell and T.-M. Yan Phys. Rev. Lett. 25, 316 (1970)

q2 ⌘ Q2 � ⇤2QCD ⇠ 1/fm2

with

q  Original Drell-Yan formula:

2 2

2

2

⌦ ⌦

No color yet!

Right shape – But – not normalization

Rapidity:

Lepton pair – from decay of a virtual photon, or in general, a massive boson, e.g., W, Z, H0, … (called Drell-Yan like processes)

Drell-Yan process in QCD

q  Spin decomposition – cut diagram notation:

( all � structure: �↵ �↵�5, , �↵�(or �5�↵�

), I, �5

( all � structure: �↵ �↵�5, , �↵�(or �5�↵�

), I, �5

q  Parity-Time reversal invariance:

hp,�~s|O( , Aµ)|p,�~si = hp,~s|PT O†( , Aµ)T �1P�1|p,~si

q  Factorized cross section:

�(Q,~s)± �(Q,�~s) / hp,~s|O( , Aµ)|p,~si± hp,�~s|O( , Aµ)|p,�~si

q  Good operators:

hp,~s|PT O†( , Aµ)T �1P�1|p,~si = ±hp,~s|O( , Aµ)|p,~si“+” for spin-averaged cross section PDFs:

hp,~s| (0)�+ (y�)|p,~si hp,~s|F+i(0)F+j |p,~si(�gij),

p,~s p,~s

Drell-Yan process in QCD – LO

q  Spin-averaged cross section – Lowest order:

q  Lowest order partonic cross section:

p,~s1

2� · p =

1

2

X

s

us(p)us(p)

1

2p+�

+�(x� p

+1 /p

+)dx

p1 p1 s = (p1 + p2)2 = Q2

q  Drell-Yan cross section:

Drell-Yan process in QCD – NLO

q  Real contribution:

q  Virtual contribution:

q  NLO contribution:

Absorbed into PDFs – scheme dependence

Drell-Yan process in QCD – factorization

q  Beyond the lowest order: ²  Soft-gluon interaction takes

place all the time ²  Long-range gluon interaction

before the hard collision

Break the Universality of PDFs Loss the predictive power

q  Factorization – power suppression of soft gluon interaction:

Drell-Yan process in QCD – factorization

q  Factorization – approximation:

²  Suppression of quantum interference between short-distance (1/Q) and long-distance (fm ~ 1/ΛQCD) physics

Need “long-lived” active parton states linking the two

Perturbatively pinched at p2a = 0

Active parton is effectively on-shell for the hard collision

²  Maintain the universality of PDFs:

Long-range soft gluon interaction has to be power suppressed

²  Infrared safe of partonic parts:

Cancelation of IR behavior Absorb all CO divergences into PDFs

Collins, Soper, Sterman, 1988

Drell-Yan process in QCD – factorization

q  Leading singular integration regions (pinch surface):

Hard: all lines off-shell by Q

Collinear: ²  lines collinear to A and B ²  One “physical parton”

per hadron

Soft: all components are soft

q  Collinear gluons:

²  Collinear gluons have the

polarization vector:

²  The sum of the effect can be

represented by the eikonal lines,

which are needed to make the PDFs gauge invariant!

Drell-Yan process in QCD – factorization

q  Trouble with soft gluons:

²  Soft gluon exchanged between a spectator quark of hadron B and

the active quark of hadron A could rotate the quark’s color and

keep it from annihilating with the antiquark of hadron B

k±

k±²  The soft gluon approximations (with the eikonal lines) need not

too small. But, could be trapped in “too small” region due to the

pinch from spectator interaction: k± ⇠ M2/Q ⌧ k? ⇠ M

Need to show that soft-gluon interactions are power suppressed

Drell-Yan process in QCD – factorization

q  Most difficult part of factorization:

0?

0? y?

y?

²  Sum over all final states to remove all poles in one-half plane

– no more pinch poles

²  Deform the k± integration out of the trapped soft region

²  Eikonal approximation soft gluons to eikonal lines

– gauge links

²  Collinear factorization: Unitarity soft factor = 1

All identified leading integration regions are factorizable!

Factorized Drell-Yan cross section

q  TMD factorization ( ):

The soft factor, , is universal, could be absorbed into the definition of TMD parton distribution

q  Collinear factorization ( ):

q? ⌧ Q

q? ⇠ Q

+O(1/Q)

q  Spin dependence:

The factorization arguments are independent of the spin states of the colliding hadrons

same formula with polarized PDFs for γ*,W/Z, H0…

PT–distribution (PT << M) – two scales

q  Z0-PT distribution in pp collisions:

PT as low as [0,2.5] GeV bin (or about 1.25 GeV)

PT–distribution (PT << M) – two scales

q  Interesting region – where the most data are:

q  Fixed order pQCD calculation is not stable!

/ 1

q2T! 1+

q  Large logarithmic contribution from gluon shower:

↵s ln

2

✓M2

Z

q2T

◆�n

Resummation is necessary!

PT << MZ ~ 91 GeV Two observed, but, very different scales

See Yuan’s talk

PT–distribution (PT >> M) – two hard scales

q  PT-distribution – factorizable if M >> ΛQCD:

d�AB

dydp

2T dQ

2=

X

a,b

Zdxa fa/A(xa)

Zdxb fb/B(xb)

d�ab

dydp

2T dQ

2(xa, xb,↵s)

How big is the logarithmic contribution?

q  Improved factorization:

Beger et al. 2015

PT–distribution (PT >> M) – two hard scales

q  Fragmentation functions of elementary particles:

q  Evolution equations:

q  Evolution kernels:

If , reorganization of perturbative

expansion to remove all logarithms of hard parts

Q � ⇤QCD

PT–distribution (PT >> M) – two hard scales

Fragmentation logs are under control!

q PQCD factorization approach is mature, and has been extremely successful in predicting and interpreting high energy scattering data with momentum transfer > 2 GeV

Summary of lecture two

q NLO calculations are available for most observables, Many new techniques have been developed in recent years for NNLO or higher order calculations (not discussed here), NNLO are becoming available for the search of new physics

q  Leading power/twist pQCD “Factorization + Resummation” allow to have precision tests of QCD theory in the asymptotic regime, and to control the background so well to discover potential “new physics” beyond SM

See Yuan’s lectures

What about the power corrections, richer in dynamics?

Backup slides

A complete example – “Drell-Yan”

²  Cross section with single hard scale:

q Heavy boson production in hadronic collisions:

�AB!V (MV ) =X

ff 0

ZdxA f(xA, µ

2)

ZdxB f(xB , µ

2) �ff 0!V (xA, xB ,↵s(µ);MV )

²  Cross section with two different hard scales:

pT ⇠ MV

d�AB!V

dydp2T(pT ⇠ MV )

d�AB!V

dy(MV ) �AB!V (MV ), ,

– Fixed order pQCD calculation

d�AB!V

dydp2T(pT ⌧ MV ) – Resummation of double logarithms:

↵ns ln2n(M2

V /p2T )

d�AB!V

dydp2T(pT � MV )

– Resummation of single logarithms:

↵ns lnn(p2T /M

2V )

Same discussions apply to production of Higgs, and other heavy particles

A(PA) +B(PB) ! V [�⇤,W/Z,H0, ...](p) +X

Total cross section – single hard scale

q  Partonic hard parts:

(Hamberg, van Neerven, Matsuura; Harlander, Kilgore 1991)

q  NNLO total x-section : �(AB ! W,Z)

²  Scale dependence:

a few percent

²  NNLO K-factor is about

0.98 for LHC data, 1.04

for Tevatron data

Rapidity distribution – single hard scale

q  NNLO differential cross-section: Anastasiou, Dixon, Melnikov, Petriello, 2003-05

Rapidity distribution – single hard scale

q  NNLO differential cross-section: Anastasiou, Dixon, Melnikov, Petriello, 2003-05

Determination of mass and width

q  W mass & width: , CTEQ SS2012

Charge asymmetry – single hard scale

q  Charged lepton asymmetry:

Ach(ye) =d�

W+

/dye � d�

W�/dye

d�

W+/dye + d�

W�/dye

�! d(xB ,MW )/u(xB ,MW )� d(xA,MW )/u(xA,MW )

d(xB ,MW )/u(xB ,MW ) + d(xA,MW )/u(xA,MW )

y ! ymax

The Ach data distinguish between the PDF models, reduce the PDF uncertainty

Tevatron data

D0 – W charge asymmetry

Charge asymmetry – single hard scale

q  Charged lepton asymmetry:

Ach(ye) =d�

W+

/dye � d�

W�/dye

d�

W+/dye + d�

W�/dye

�! d(xB ,MW )/u(xB ,MW )� d(xA,MW )/u(xA,MW )

d(xB ,MW )/u(xB ,MW ) + d(xA,MW )/u(xA,MW )

y ! ymax

Sensitive both to d/u at x > 0.1 and u/d at x ~ 0.01

Flavor asymmetry – single hard scale

q  Flavor asymmetry of the sea:

Could QCD allow ubar(x) > dbar(x)?