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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
June 6, 2006 Jianwei Qiu, ISU2
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, …
June 6, 2006 Jianwei Qiu, ISU3
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”
June 6, 2006 Jianwei Qiu, ISU4
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:
June 6, 2006 Jianwei Qiu, ISU5
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
June 6, 2006 Jianwei Qiu, ISU6
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= − ⊗
June 6, 2006 Jianwei Qiu, ISU7
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= −
June 6, 2006 Jianwei Qiu, ISU8
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
June 6, 2006 Jianwei Qiu, ISU9
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 −⎝ ⎠ ⎣ ⎦⎧ ⎫− ⎡ ⎤ −⎛ ⎞⎛ ⎞− − + + +⎨ ⎬⎜ ⎟⎜ ⎟⎢ ⎥− − − − −⎝ ⎠⎝ ⎠⎣ ⎦⎩ ⎭
June 6, 2006 Jianwei Qiu, ISU10
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+
⎡ ⎤+= + −⎢ ⎥−⎣ ⎦
June 6, 2006 Jianwei Qiu, ISU11
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!
June 6, 2006 Jianwei Qiu, ISU12
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
⎧ ⎛ ⎞⎪= ⎜ ⎟⎨ ⎜ ⎟⎪ ⎝ ⎠⎩
June 6, 2006 Jianwei Qiu, ISU13
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 ++
−⎛ ⎞⎜ ⎟− −⎝ ⎠
June 6, 2006 Jianwei Qiu, ISU14
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 +⊥⊥
⎡ ⎤⎣ ⎦
June 6, 2006 Jianwei Qiu, ISU15
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
June 6, 2006 Jianwei Qiu, ISU16
Heuristic argument for factorization
June 6, 2006 Jianwei Qiu, ISU17
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!
June 6, 2006 Jianwei Qiu, ISU18
Power correction is very small, excellent prediction! Qiu, Zhang, PRL 2001
Success of pQCD with identified hadron
June 6, 2006 Jianwei Qiu, ISU19
No free fitting parameter!Qiu, Zhang
June 6, 2006 Jianwei Qiu, ISU20
D0 Run – II Upsilon data
Berger, Qiu, WangA prediction!
June 6, 2006 Jianwei Qiu, ISU21
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?
June 6, 2006 Jianwei Qiu, ISU22
Magic of nuclear targets
Strong suppression at intermediate x
Strong enhancement at low x
Fermi motion of nucleons inside a nucleuscannot explain the effect
June 6, 2006 Jianwei Qiu, ISU23
Discovery of nuclear shadowing
At small x, SLAC and EMC data show an opposite effect
June 6, 2006 Jianwei Qiu, ISU24
June 6, 2006 Jianwei Qiu, ISU25
1/2 3T a bk A≈ +
Fermilab data show:
June 6, 2006 Jianwei Qiu, ISU26
E772 dataStrong xF - dependence
June 6, 2006 Jianwei Qiu, ISU27
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??
June 6, 2006 Jianwei Qiu, ISU28
June 6, 2006 Jianwei Qiu, ISU29
Systematics of PT broadening
June 6, 2006 Jianwei Qiu, ISU30
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
June 6, 2006 Jianwei Qiu, ISU31
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!
June 6, 2006 Jianwei Qiu, ISU32
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
June 6, 2006 Jianwei Qiu, ISU33
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
June 6, 2006 Jianwei Qiu, ISU34
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
June 6, 2006 Jianwei Qiu, ISU35
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
≈ + ...+
Mueller and Qiu, 1986
June 6, 2006 Jianwei Qiu, ISU36
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