qcd and rescattering in nuclear targets lecture 3 · 2006-06-07 · june 6, 2006 1 jianwei qiu, isu...

June 6, 2006 Jianwei Qiu, ISU1

QCD and Rescatteringin Nuclear Targets

Lecture 3

Jianwei QiuIowa State University

The 21st Annual Hampton University Graduate Studies Program (HUGS 2006)

June 5-23, 2006Jefferson Lab, Newport News, Virginia


Calculations in perturbative QCD

Calculation of factorized short-distance pieces

Factorization for hadronic collisions

What do we expect when proton is replaced bya heavy nucleus in high energy collisions?

Sources of the nuclear dependence

Early surprises – magic of nuclear targetsEMC effect, Cronin effect, …


PQCD FactorizationCan pQCD calculate cross sections with identified hadrons?

Typical hadronic scale: 1/R ~ 1 fm-1 ~ ΛQCD

Energy exchange in hard collisions: Q >> ΛQCD

pQCD works at αs(Q), but not at αs(1/R)

PQCD can be useful iff quantum interferencebetween perturbative and nonperturbative scales can be neglected

phy ( ,1/ ) ( ) (1/ ) (1/ )Q QR RQ ROσ σ ϕ⊗ +∼

Measured

Short-distance

Long-distance

Power corrections

Factorization – “forgetting the past”


Parton distribution functions (PDFs)Predictive power of pQCD relies on the factorization and the universality of PDFs

+ UV CTGives the μ-dependence

PDFs as matrix elements of two parton fields:


DGLAP evolution of PDFsμ2-dependence determined by DGLAP equations

2/

2 22 ( , ) ( ', ),

'i i j jF F s FF j

xPx

x xϕ ϕμ μ α μμ

⎛ ⎞⎜= ⎟⎝

∂⊗⎠∂ ∑

Boundary condition extracted from physical x-sections

( )2QCD2

2 / /2

2 2

2

, ,( , ) ,f hh BB

q f s FFq

F x x Qx Q

CQ x Oα μμ

ϕ⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞Λ= ⊗ + ⎜ ⎟⎜ ⎟

⎝ ⎠∑

extracted parton distributions depend on the perturbatively calculated Cq and power corrections

Leading order (tree-level) Cq LO PDF’s

Next-to-Leading order Cq NLO PDF’s

Calculation of Cq at NLO and beyond depends onthe UVCT the scheme dependence of Cq

the scheme dependence of PDFs


Calculation of perturbative partsUse DIS structure function F2 as an example:

( )22QCD2

2 2,

2/2( , ) , , ,B

h B q f sq

f h FFf

x QF x Q C x Ox Q

ϕ μμ

α⎛ ⎞Λ⎛ ⎞

= ⊗ + ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑

h q→Apply the factorized formula to parton states:

( )2

22

,

2/2( , ) , , ,B

q B q f s f q Ff Fq

x QF x Q C xx

ϕ μμ

α⎛ ⎞

= ⊗⎜ ⎟⎝ ⎠

∑Feynman diagrams

Feynman diagrams

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

0th order: ( ) ( ) ( ) ( )0 02 22

02 2/( , ) ( / , / ) ,q B F qq B q FF 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

F q q F

F q q F

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( , / ) ( , ) ( , ) ,Fq q Fq q qC x Q F x Q F x Q xμ ϕ μ= − ⊗


Leading order coefficient functionProjection 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⎛ ⎞ ⎛ ⎞⎛ ⎞⋅ ⋅

= − − + − −⎜ ⎟ ⎜ ⎟⎜ ⎟⋅ ⎝ ⎠⎝ ⎠⎝ ⎠2

2 21 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

⎛ ⎞= − +⎜ ⎟

⎝ ⎠⎛ ⎞

= − +⎜ ⎟⎝ ⎠

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= −


NLO coefficient function( ) ( ) ( ) ( ) ( )1 1 02 2 2 1

22

/22( , / ) ( , ) ( , ) ,Fq q Fq q qC x Q F x Q F x Q xμ ϕ μ= − ⊗

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

⎛ ⎞− = − + −⎜ ⎟

⎝ ⎠

Feynman diagrams:

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

μν μν−

( )1,qWμν

+ UV CT

+ + +

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

Real

Virtual


Contribution from the trace of WμνLowest order in n-dimension:

( )0 2, (1 ) (1 )q qg W e xμν

μν ε δ− = − −

NLO virtual contribution:( )1 2

,

2

2 2

2

(1 ) (1 )

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

Vq q

s FF

g W e x

QC

μνμν

ε

ε δ

α π ε επ

με ε ε

− = − −

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

NLO real contribution:( )1

22

, 2

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

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

R sq

FFqg W e

Q

xxx

C

x

εμν

μνα πμ εεπ ε

ε ε εε ε ε ε

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


The “+” distribution:

( )1

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

n xx Ox x x

ε

δ ε εε

+

++

−⎛ ⎞ ⎛ ⎞= − − + + +⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠1 1( ) ( ) (1) (1 ) (1)

(1 ) 1z z

f x f x fdx dx n z fx x+

−≡ + −

− −∫ ∫

One loop contribution to the trace of Wμν:( )

21 2

, 2

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

F

sq qq qqq

QPg W e Px x ne

μνμν γ

αεπ ε μ π −

⎛ ⎞⎛ ⎞ ⎧− = − − +⎨ ⎜ ⎟⎜ ⎟⎩⎝ ⎠ ⎝ ⎠

( )2

2 (1 ) 3 1 11 ( )1 2 1 1F

n x xx nx x

C xx ++

⎡ − +⎛ ⎞ ⎛ ⎞+ + − −⎜ ⎟ ⎜ ⎟⎢ − − −⎝ ⎠ ⎝ ⎠⎣293 (1 )

2 3x xπ δ

⎫⎤⎛ ⎞ ⎪+ − − + − ⎬⎥⎜ ⎟⎝ ⎠ ⎪⎦⎭

Splitting function:21 3( ) (1 )

(1 ) 2qq Fxx xx

P C δ+

⎡ ⎤+= + −⎢ ⎥−⎣ ⎦


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

=

One loop contribution to F2 of a quark:( ) ( )

21 2 2

2C

2O

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

Eqqq

Fqq

sq P P QF x Q e x x n e x nγε

επ

π μα − ⎛ ⎞⎧⎛ ⎞= − + +⎨ ⎜ ⎟⎜ ⎟

⎝ ⎠⎩ ⎝ ⎠2 2

2 (1 ) 3 1 1 9(1 ) ( ) 3 2 (1 )1 2 1 1 2 3F

n x xx n x x xx x x

C π δ+ +

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

as 0ε⇒ ∞ →

One loop contribution to quark PDF of a quark:( )1

/UV CO

2 1 1( , ) ( ) UV-CT2F

sq q qqx P x

εμϕ

εαπ

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

Different UV-CT = different factorization scheme!


Common UV-CT terms:

MS scheme: MSUV

1UV-CT ( )2 q

sqP xα

π ε⎛ ⎞= − ⎜ ⎟⎝ ⎠

DIS scheme: choose a UV-CT, such that( ) 21 2

DIS( , / ) 0q FC x Q μ =

MS scheme: ( )MSUV

1UV-CT ( ) 1 (4 )2

Esqq x nP e γα ε π

επ−⎛ ⎞= − +⎜ ⎟

⎝ ⎠

One loop coefficient function:( ) ( ) ( ) ( ) ( )1 1 02 2 2 1

22

/22( , / ) ( , ) ( , ) ,Fq q Fq q 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

+ +

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

( )2M

21 2 2 2

S

( , / ) ( )2 qq q q

s P QC x Q e x x nαμμπ

⎧ ⎛ ⎞⎪= ⎜ ⎟⎨ ⎜ ⎟⎪ ⎝ ⎠⎩


Improvement from the fixed orderBeyond the Born term (lowest order), partonichard-parts are NOT unique, due to renormalizationof parton distributions

Once ϕ(x,μ2) is fixed in one scheme, same ϕ(x,μ2) should be used for all calculations of partonic parts

Coefficient has the2

2( )F

qqQx nPμ

⎛ ⎞⎜ ⎟⎝ ⎠

Suggests to choose the scale: 22F Qμ ∼

Coefficient has potentially large logarithms:

Resummation of the large logarithms

1 (1 )( ), , (1 ) 1

n xn xx x ++

−⎛ ⎞⎜ ⎟− −⎝ ⎠


Recover the effect of non-vanishing kT

/2 2

/44

2

22

24

/66

ˆ [1 ...] ( )

ˆ + [1 ...] ( )

ˆ + [1 ...] ( )

+ ...

phys s s

s s

s s

h i i h

ii h

ii h

xT

xT

Qx

Q

T

σ σ α α

σ α α

σ α α

= ⊗ + + + ⊗

⊗ + + + ⊗

⊗ + + + ⊗

Leading Twist

Power corrections

perturbative

Systematics of power corrections:

Factorization maynot be true for

power corrections!Need to be proved

for any given process

Sources of power corrections:Parton transverse momentum:

Target and parton masses:

Coherent multiple scattering:

22m Q

22 2 2Q Qk k⊥ ∼

( ) 2 +

2 Medium eng1 l thQ R F F +⊥⊥

⎡ ⎤⎣ ⎦


Factorization for hadronic collisionsUse Drell-Yan as an example:

( ) ( ) ( )' 'p ph Xqh + −+ → + 2 2 with q Q=

PM picture:

Long-range soft interactions before the hard collisioncould break the PDF’s universality – loss of prediction

( )' 'f xφ ( )f xφ

p

x p

'p

''x p

Soft interaction

Proof of the factorization – to show the soft interactiondoes not alter the PDFs and the factorized formula

Leading power (twist): Collins, Soper, and Sterman; BodwinNext leading power: Qiu and Sterman


Heuristic argument for factorization


Field strength is strongly contracted

( ) ( ) ( ) ( )e1 1

'

0

22 '

', '

2

lDY2

20

'' ,

, ', ' ˆ ,,',

, hh ff

f fF

f fF F

x xx x x

ddd d

d dQ Qp pp qqp

xσσ

φ φμ

μ μ= ∑ ∫ ∫

QCD at leading power has been very successful!


Power correction is very small, excellent prediction! Qiu, Zhang, PRL 2001

Success of pQCD with identified hadron


No free fitting parameter!Qiu, Zhang


D0 Run – II Upsilon data

Berger, Qiu, WangA prediction!


Proton to nucleusWhat do we expect when a proton is replaced by a heavy nucleus in high energy collisions?

Expect a simple sum of individual nucleons at high energy collision?


Magic of nuclear targets

Strong suppression at intermediate x

Strong enhancement at low x

Fermi motion of nucleons inside a nucleuscannot explain the effect


Discovery of nuclear shadowing

At small x, SLAC and EMC data show an opposite effect


1/2 3T a bk A≈ +

Fermilab data show:


E772 dataStrong xF - dependence


PHENIX vs. NA50 on J/ψ

Suppression at RHIC (√s=200 GeV) looks VERY similar to that

seen at NA50

• but ~10x collisionenergy & ~2-3x gluonenergy density

• why should it be thesame??


Systematics of PT broadening


Free nucleons vs. bound nucleonsIn a gas target:

Nucleons have a large empty space between themThey are quantum mechanically incoherent for at least the strong interactions

In a nuclear target:

Nucleons are very closed to each otherThey are quantum mechanically coherent, andbounded via a weak nuclear binding energy


Hard probe and its probing sizeHard probe – process with a large momentum transfer:

2QCD with q Q qμ ≡ Λ

Size of a hard probe is very localized and much smaller than a typical hadron at rest:

1 2 fm~ RQ

But, it might be larger than a Lorentz contracted hadron:

or equivalently1 1 1 2 0.1 2cR xm

px

Q xp mR⎛ ⎞⎜ ⎟⎝ ⎠

≡∼ ∼

If an active parton x is small enoughthe hard probe could cover several nucleons

In a Lorentz contracted large nucleus!


Coherence length in different framesUse DIS as an example – in target rest frame:virtual photon fluctuates into a q-qbar pair

If , the probe – q-qbar state of the virtual photon can interact with who hadron/nucleus coherently.

In Breit frame:coherent final-state rescattering

The conclusion is frame independent


Incoherent/independent multiple scattering

JetσJet

2 22

∝ ⊗∝

Weak quantum interference between scattering centers

Modify jet spectrum without changing the total rate

Nuclear dependence from the scattering centers’densitynumbermomentum distribution and cut-off (new scale)etc

Gyulassy, Levai, and Vitev (GLV), NPB 2001


Coherence soft multiple scatteringStrong quantum interference between scattering centers

Modify production rate as well as jet spectrum

Jet

Jetσ ∝ ∝Jet

2

Nuclear dependence from multi-parton correlationsMulti-parton correlation functions are processindependent if pQCD factorization can be appliedFourier transform from momentum to coordinate

universal matrix elements of multiple fieldsno additional scale – power suppressed

Qiu and Sterman, 2003


Single hard scatteringNon-perturbative dynamics is effectively frozen

Jetσ ∝ ∝Jet

2

Jet

2

⊗

Production rate is proportional to the PDFsNuclear dependence from nPDFs

modified DGLAP evolutioninput nPDFs for the evolutionnPDFs are universal and process independent

≈ + ...+

PDF

Mueller and Qiu, 1986

qcd and rescattering in nuclear targets lecture 3 · 2006-06-07 · june 6, 2006 1 jianwei qiu, isu...

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