zhongbo kang - indico2.riken.jp · comments on phenomenology! summary ... xd v! p a = x a p ... jet...

Introduction on spin-dependent fragmentation and evolution of TMDs

Workshop on Fragmentation Functions and QCD 2012RIKEN, Wako, Japan, November 9–11, 2012

Zhongbo KangTheoretical Division, Group T-2Los Alamos National Laboratory

n Introductionn Spin-dependent fragmentation functionsn Transverse momentum dependent distributions

n TMD factorizationn Evolution of TMDs: basic idean Evolution of Sivers function

n Difference between SIDIS and DY regarding sign changen Evolution of Collins function

n Comments on phenomenologyn Summary

Nov 10, 2012 Zhongbo Kang, LANL

Outline:

2


Parton distribution and fragmentation function

n One of the major goal of hadron physics is to understand hadron structure:n how quarks and gluons distributed inside the hadronn how quarks and gluons are formed into hadron

n One of the major tool to extract these information is through the high energy scattering experiments, by relying on QCD factorization

n PDFs and FFs are closely related to each othern The better extraction of one could lead to better understanding of the other

3

N

(E, p)

hπ

π

q

e

+

(E, p )’ ’

π

u

du

*γ

dσ ∼ φ(x)⊗ σ̂ ⊗D(z)

PDF: parton distribution function

FF: fragmentation function


The status of collinear PDFs and FFs

n The collinear PDFs and FFs are widely studied

4

3

What Do We Know About Glue in Matter?

• Scaling violation: dF2/dlnQ2 and

linear DGLAP Evolution !

G(x,Q2)!

Deep Inelastic Scattering : d

2" ep#eX

dxdQ2

=4$%e.m.

2

xQ4

1& y +y

2

2

'

( )

*

+ , F2(x,Q2) &

y2

2FL (x,Q2)

-

. /

0

1 2

Gluons dominate low-x wave function

)20

1( !xG

)20

1( !xS

vxu

vxd

!

pa = xaPADSS parametrization


pQCD with PDFs and FFs: they generally work very welln Jet cross section (p+p→jet+X): only PDFs

n Involving hadronn both PDFs and FFs

5

Jets (p. 4)

Introduction

Background KnowledgeJets from scattering of partons

Jets are unavoidable at hadroncolliders, e.g. from parton scat-tering

p

p

jet

jet

[GeV/c]JETTp

0 100 200 300 400 500 600 700

[n

b/(

Ge

V/c

)]JE

T

T d

pJE

T /

dy

!2

d

-1410

-1110

-810

-510

-210

10

410

710

1010 D=0.7TK

)-1

CDF data ( L = 1.0 fb

Systematic uncertainties

NLO: JETRAD CTEQ6.1M

corrected to hadron level

0µ / 2 = JET

T = max pFµ = Rµ

PDF uncertainties

)-6 10"|<2.1 (JET

1.6<|y

)-3 10"|<1.6 (JET

1.1<|y

|<1.1JET

0.7<|y

)3

10"|<0.7 (JET

0.1<|y

)6 10"|<0.1 (JET

|y

Jet cross section: data and theory agree over many orders of magnitude !probe of underlying interaction

Tevatron

1

10

210

310

410

510

610

710

810

0.2

0.6

1.0

1.4

1.8

1/2! d"

/(d#

dp

T)

[pb

/Ge

V]

da

ta /

th

eo

ry

[GeV/c]pT

10 20 30 40 500

2

(a)

(b)

Combined MB

Combined HT

NLO QCD (Vogelsang)

Systematic UncertaintyTheory Scale Uncertainty

p+p g jet + X GeVmidpoint-coner =0.40.2<#<0.8

=200s

cone

STAR

STAR, PRL 97 (2006), 252001

RHIC

Inclusive jet pT spectrum

Hard Probes 2010 Hermine K. Wöhri : CMS results in pp collisions

(GeV)T

p20 30 100 200 1000

(p

b/G

eV

)T

/dyd

p!

2d

-110

10

310

510

710

910

11101024)"|y|<0.5 (

256)"|y|<1.0 (#0.5

64)"|y|<1.5 (#1.016)"|y|<2.0 (#1.5

4)"|y|<2.5 (#2.0

1)"|y|<3.0 (#2.5

NLO pQCD+NP

Exp. uncertainty

= 7 TeVs-1CMS preliminary, 60 nb

R=0.5 PFT

Anti-k

(GeV)T

p20 30 100 200 1000

(p

b/G

eV

)T

/dyd

p!

2d

-110

10

310

510

710

910

1110 coverage of the full

pT range combining

triggers with different thresholds

experimental systematic

uncertainties dominated

by jet energy scale and resolution, and by the

luminosity measurement

17

!! Extending the high pT limit beyond Tevatron reach

!! Accessing the low pT part using different jet reconstruction algorithms

!! Good agreement with NLO predictions

(GeV)T

p20 30 100 200 1000

Trig

ge

r E

ffic

ien

cy

0

0.2

0.4

0.6

0.8

1

MinBias

Jet6u

Jet15u

Jet30u


|y| < 0.5

R=0.5 PFTAnti-k

37 56 84

(GeV)T

p20 30 100 200 1000

Trig

ge

r E

ffic

ien

cy

0

0.2

0.4

0.6

0.8

1

(GeV)T

p20 30 100 200 1000

Da

ta /

Th

eo

ry

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

|y| < 0.5PF

Theory uncertainty

Exp. uncertainty

ansatz


(GeV)T

p20 30 100 200 1000

Da

ta /

Th

eo

ry

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

PAS QCD-10-011

J. Weng’s talk

G. Martinez’s talk

K.S. Grogg’s poster

LHC


Going beyond the current picture: two ways - I

n Beyond the collinear PDFs and FFsn probe also parton the transverse momentumn since parton kt is always much smaller than the longitudinal component in high

energy experiments, one typically need “transverse spin” to correlate with the parton kt

n kt-dependent PDFs and FFs are usually involving spin vector, either the hadron spin or the parton spin

6

parton distribution function Fragmentation function


Three important TMDs will be addressed

n Transversity: survive kt-integrationn a distribution of transversely polarized quark in a transversly polarized hadron

n Sivers function: change sign from SIDIS to DY

n Collins function: universal function

7

fq/h↑(x,k⊥, �S) ≡ fq/h(x, k⊥) +12∆Nfq/h↑(x, k⊥) �S · p̂× k̂⊥

q

∆(z, p⊥) =1

2

�D(z, p2⊥)/n+H

⊥1 (z, p2⊥)σ

µν p⊥µnν

M

�

n Collins effect observed by BELLE Collaboration: cos(Φ1+Φ2)


Current status for Collins fragmentation function

8PRL 96, 232002 (2006)PRD 78, 032011 (2008)

-0.05

0

0.05

0.1

0.15

0.2

-0.05

0

0.05

0.1

0.15

0.2

0.2 0.4 0.6 0.8

A12

0.2<z1<0.3

AUL

AUC

0.3<z1<0.5

z2

A12

0.5<z1<0.7

z2

0.7<z1<1

0.2 0.4 0.6 0.8

See also update: R. Seidl, PRD 86 (2012) 039905


Also observed by BarBar

n Collins effect from BarBar

n Compare with Belle: generally consistent

9

some discrepancy for AUC

should be now resolvedR. Seidl, PRD86 (2012) 039905

Garzia, talk at QCD-N’12see more on Muller’s talk


Extraction of Collins in e+e- could help extract transversity

n Collins extraction from e+e-

n Combine this with SIDIS data

10

S. Melis, 2012

n Dihadron fragmentation function

n DFF

n IFF


Going beyond the current picture: two ways - II

11

∆ ∼ γ−D1(z1, z2,M

2h) +

σ−αRTα

2|RT |H

�1 (z1, z2,M

2h)

H�1 (z1, z2,M

2h) ∼ Sq × (P1⊥ − P2⊥)

D1(z1, z2,M2h)

Bianconi-Boffi-Jakob-Radici, 99


They are better theoretical objects

n They follow a simple collinear DGLAP type evolution equationn follow the DGLAP evolution for spin-averaged collinear

fragmentation functionn follow the DGLAP evolution for the quark transversity

n Usual collinear factorization could be used to describe the experimental datan SIDIS: hadron pair, transversity × IFFn e+e-: two hadron pair, IFF × IFFn pp:

n hadron pair, transversity × IFFn two hadron pair, IFF × IFF

12

D1(z1, z2,M2h)

H�1 (z1, z2,M

2h)

Ceccopieri-Radici-Bacchetta, 07


Data on IFF

n Belle

13

PRL, arXiv: 1104.2425


Extract IFF from these datan First extraction has been made by Courtoy-Bacchetta-Radici-Bianconi

n DFF and IFF

14

PRD, arXiv: 1202.0323


Extract transversity from dihadron fragmentation function

n A first extraction of transversity (a combination) from dihadron fragmentation function (based on SIDIS data) has been made by Bacchetta-Courty-Radiccin It is in a reasonable agreement with the transversity extracted from Collins

effect

15

PRL, arXiv: 1104.3855


IFF in proton+proton

n STAR also has measurements now, it will be good to use these data to extract transversity

16

1104.3855

Vossen, RHIC/AGS User’s meeting, 2012


Evolution effect

n Our experimental data span very different energies: n SIDIS is performed at Q2~1-3 GeV2, Belle and BarBar at Q2~100 GeV2

n Future RHIC experiment can also access very high Q2~16-81 GeV2 for DYn Our kt-extended parton model has made great success in recent yearsn It is the time now to go beyond these parton model result, and understand the

evolution of the relevant TMDs

n So far we talked about evolution of DFF and IFF, saying that they are following the usual DGLAP evolution equation (with splitting kernel either same as unpolarized PDF or transversity)

n What does evolution means?n How they are connected to higher order corrections?

17


Cross section depends on where you factorize?

n Start with collinear factorization, cross section can be written as follows

n In the QCD formalism, there is a factorization scale dependence µ2, where does it come from? How does the PDFs change (evolve) with respect to this scale?n If in the hard part, one choose µ~Q, then we need PDFs at all different scales

as experimental measurements are done at different Qs

18

σ(x,Q2) = σ̂(x,Q2/µ2)⊗ φa/p(x, µ2)


Recall: DIS again

n Deep Inelastic Scattering (DIS)

n Hadronic tensor in perturbative expansion

n Leading order factorization: parton model

19

All the interesting physics (QCD dynamics) is contained in

=e

e

P

q

X µ

µqqσ

2

= ×

Lµν Wµν

(leptonic tensor) (hadronic tensor)Wµν

= + + ...Wµν

q

k1

µ

P

qk

P

q q

k

µ

≈q q

µ

P

k

P

q q

k

µ

xp

+ O

�k2

T

Q2,

k2

Q2

�


QCD dynamics beyond leading order

n Radiative corrections

20

v intermediate quark is on-shell

v gluon radiation takes place long before the photon-quark interaction a part of PDF

tAB → ∞k21 ∼ 0

Partonic diagram has both long- and short-distance physics

⇒�

d4k1i

k21+i�

−i

k21−i�

⇒∞

?

Collinear divergence!!! (from )k21 ∼ 0

A

B

P

kq

1µ

qk

q

k1

µ

P

qk

≈


Factorization: separation of short- from long-distance

21

LO + evolution

NLO

q

k1

µ

P

qk

q

k1

µ

P

qk

q

k1

µ

P

qk

q qµ

q1

µq

k

P

k

k1k

P

� ∼Q2

0dk2

1

� ∼Q2

µ2dk2

1

� µ2

0dk2

1

� µ2

0dk2

1

k21 ≈ 0

� ∼Q2

µ2dk2

1

� k21

0dk2

C(0)⊗φ(1)

C(1)⊗φ(0)

= +

=

+

n Systematically remove all the long-distance physics into PDFs


DGLAP evolution = resummation of single logs

n Evolution = Resum all the gluon radiation

n By solving the evolution equation, one resums all the single logarithms of

n Same idea for DFF and IFF22

P

kk1

k

P

kP

= + + + ...φ(x, µ2)

DGLAP Equation

P

k-P

kk1

k

P+ + ...=φ(x, µ2)

∂

∂ lnµ2φi(x, µ2) =

�

j

Pij(x

x� )⊗ φj(x�, µ2)

Evolution kernel splitting function

�αs ln

µ2

Λ2

�n


Go from Collinear PDFs to TMDs

n Evolution of collinear PDFs follow the usual DGLAP-type evolution equation, which is equivalent to resum the single-logarithmic contributions to all order

n Evolution of TMDs follow Collins-Soper-type evolution equation, which is equivalent to resum the double-logarithmic contributions to all order, which is usually more difficult

23

�αs ln

Q2

µ2

�n

�αs ln

2 Q2

q2T

�n


Evolution of TMDs: basic idea

n These double logarithms in the cross section level come from the double logarithm in the kt-dependent distribution and fragmentation functionsn A generic idea: parton distribution/fragmentation function is

associated with a hadron which has a large longitudinal component P+, with these two scales in the TMDs (kt, P+), its perturbative tail will have double logarithms

n It is these logarithms which will translate into the cross section level

n In the true calculation, these P+ comes in as a Lorentz invariant

n Thus one studies the variable dependence of the TMDs, this is the evolution of TMDsn technique part: is introduced to regularize the rapidity divergence

24

φ(x, k2⊥)

�αs ln

2 P+2

k2⊥

�n

�αs ln

2 Q2

q2T

�n

ζ = 4(P · v)/v2

ζ

ζ


Evolution is simpler in b-space: SIDIS cross section

n in momentum space:

n in b-space:

n b-space the cross section is a simple product of TMDs and hard-coefficient functions, thus the evolution is simpler in b-space

25


Evolution of TMDs

n Since one now needs to resum double logarithms, typically it involves two steps:n Energy evolution of the unpolarized PDFs

n Since it contains double logarithms, the kernel still contains single logarithms

n Solving these two equations, equivalently one resums the double logsn First for the evolution equation of K and G

n Then feed the solution back to the energy evolution equation

26

µd

dµK(µ, b) = −γK = −µ

d

dµG(µ, ζ)

Idilbi-Ji-Ma-Yuan, 2004

ζ∂

∂ζq(x, b, µ, ζ) = (K(µ, b) +G(µ, ζ)) q(x, b, µ, ζ)


Solve evolution equations

n Once all the logs are resummed, the rest of b-dependent PDFs and FFs can be expanded as collinear PDFs and FFs

n All the large logarithms are resummed to the Sudakov exponential term

27

dσ

dxBdydzhd2Ph⊥= σ0FUU

FUU =

�d2b

(2π)2ei �q⊥·�bWUU (b,Q, xB , zh)

WUU (b,Q, xB , zh) = e−S(b,Q)�

q

e2q�Cq/i ⊗ fi/A

�(xB , µ =

c

b)

×�DB/j ⊗ C̃j/q

�(zh, µ =

c

b)

S(b,Q) =

� Q2

c2/b2

dµ2

µ2

�A ln(Q2/µ2) +B

�A =

�

n=1

A(n)�αs

π

�n

A(1) = CF B(1) = −3

2CF


Evolution of Sivers functionn The Collins-Soper energy evolution is really for the whole correlator

n So for Sivers function, it really is that evolves as a wholen in b-space, it is (also the b-derivative of Sivers function)

n it follows the same energy evolution equation

n Thus one should get a very similar resummation formalism

28

kα⊥f⊥1T (x, k

2⊥)

f̃ (⊥α)1T (x, b, µ, ζ) =

1

M

�d2k⊥e

−i�k⊥·�b⊥kα⊥ f⊥1T (x, k⊥, µ, ζ)

ζ∂

∂ζf̃ (⊥α)1T (x, b, µ, ζ) = [K(µ, b) +G(µ, ζ)] f̃ (⊥α)

1T (x, b, µ, ζ)

Kang-Xiao-Yuan, PRL, 2011


The Sivers effect for SIDIS

n The resummation formalism (consistent with experimental convention)

n spin-dependent structure function

n only soft-gluonic pole Qiu-Sterman function appears in this part

29

dσ

dxBdydzhd2Ph⊥= σ0

�FUU + |s⊥| sin(φh − φs)F

sin(φh−φs)UT

�

F sin(φh−φs)UT = − 1

4π

� ∞

0db b2J1(q⊥b)WUT (b,Q, xB , zh)

WUT (b,Q, xB , zh) = e−S(b,Q)�

q

(∆CTq/i ⊗ Ti,F )(xB , µ =

c

b)

×(DB/j ⊗ C̃j/q)(zh, µ =c

b)

(∆CTq/i ⊗ Ti,F )(xB , µ) =

� 1

xB

dx

x∆CT

q/i(xB

x, µ)Ti,F (x, x, µ)

Qiu-Sterman function


The evolution of Collins function

n Recall: the evolution equation is really for the correlator, immediately, you know the evolution for Collins functionn Again it is which should be evolving as a wholen In the b-space, define

n It follows the same evolution just like unpolarized fragmentation functionn quark transversity follows the same evolution for the unpolarized PDFs f1

n One thus also has the same type of resummation formalism for the Collins effect

30

H̃(⊥α)1 (z, b, µ, ζ) =

1

M

�d2p⊥e

−i�p⊥·�b⊥pα⊥ H⊥1 (z, p⊥, µ, ζ)

pα⊥H

⊥1 (z, p2⊥)


Comments on SIDIS and DY

n The only difference comes from so-called coefficient functionn leading order

n at next-leading-order: well-known difference due to Q2>0 (<0)

n Thus in the full perturbative QCD region, Sivers between SIDIS and DY is not just a sign: it is interesting to study the consequence

31

∆CT (0)i/j (z, µ =

c

b) = δijδ(1− z)

∆CT (1)i/j (z, µ =

c

b) = δij

�− 1

4Nc+

CF

2(π2

2− 4)δ(1− z)

�

∆CT (1)i/j (z, µ =

c

b) = −δij

�− 1

4Nc+

CF

2(−4)δ(1− z)

�

∆CT (0)i/j (z, µ =

c

b) = −δijδ(1− z)

DY

SIDIS

DY

SIDIS


Phenomenological study: what’s the difficulty?n For small qt region, we could use the resumed formalism. Don’t need

to worry about the perturbative tail from collinear twist-3 contribution

n Only at small b-region (corresponds to large momentum), one can calculate the relevant coefficients perturbatively.

n However, in order to Fourier transform back to qt-space, we need the whole b-region. Since large b-region will be non-perturbative, we need a non-perturbative input. This part should be universal if QCD factorization holds for the process.

32

WUU (b,Q, xA, xB) = e−S(b,Q)�

q

e2q(Cq/i ⊗ fi/A)(xA, µ =c

b)

×(Cq̄/j ⊗ fj/B)(xB , µ =c

b)

dσ

dQ2dyd2q⊥=

σ0

2π

� ∞

0db bJ0(q⊥b)WUU (b,Q, xA, xB)


The parametrization for the non-perturbative function

n Different approaches for the non-perturbative functions in “DY”

n Parametrize the full b-space function

n function form (through extrapolation): n fitted form directly from experiments:

33

dσ

dQ2dyd2q⊥=

σ0

2π

� ∞

0db bJ0(q⊥b)WUU (b,Q, xA, xB)

WUU (b,Q, xA, xB) = W pertUU (b,Q, xA, xB)F

NP (b,Q, xA, xB)

Qiu-Zhang, 2001

W pertUU (b,Q, xA, xB) = e−S(b,Q)

�

q

e2q(Cq/i ⊗ fi/A)(xA, µ =c

b)

×(Cq̄/j ⊗ fj/B)(xB , µ =c

b)

Brock-Landry-Nadolsky-Yuan, 2003

WUU (b,Q, xA, xB) = W pertUU (b∗, Q, xA, xB)F

NP (b,Q, xA, xB) b∗ =b�

1 + (b/bmax)2

FNP (b,Q, xA, xB) = exp

�−�g1(1 + g3 ln(100xAxB)) + g2 ln

�Q

2Q0

��b2�


Works perfectly fine for Tevatron and LHC

n Z boson production at Tevatron and LHCn the non-perturbative function is not suitable for low energy (highly biased by

high energy data)

34

10-3

10-2

10-1

0 10 20 30 40 50 60 70 80 90pT (GeV)

1/ d

/dp T

(GeV

-1)

10-3

10-2

10-1

1

10

0 10 20 30 40 50 60 70 80 90pT (GeV)

d/d

p T (p

b/G

eV)

CDF LHC

Kang-Qiu, 2012, to appear


Works very well for HERA data: SIDIS

n z-flow distribution

n The non-perturbative function fitted from HERA only works for small-x data, and do not apply for large x (Sivers data)n note this function in SIDIS is different from that in DY, as SIDIS non-

perturbative function comes from PDF and FF, while DY only comes from PDFn In the current Sivers analysis, this functional form is not adopted. It will be a

very good cross-check to see if the current form can describe HERA data!!!

35

Nadolsky, et.al., 1999, 2001


Works directly with TMDs

n Seem to work fine

n See also work by Anselmino, et.al.n With TMD evolution, it seems to describe the data slightly better

36

Aybat-Prokudin-Rogers, 2011

�Q2� � 3.6 GeV

�Q2� � 2.4 GeV

Anselmino-Boglione-Melis, 2012


Naively apply this functional form ...

n It leads to dramatic effect, which might not be true

n One needs a refit, which concentrates on the low energy data

37

-0.1

-0.08

-0.06

-0.04

-0.02

0

-3 -2 -1 0 1 2 3

s=200 GeV0<qT<1 GeV4<Q<9 GeV

warning: test-only

y

AN

S. Melis, 2012


Summary

n There are tremendous progress recently in measuring/extracting spin-dependent fragmentation function, in turn to better extract other TMD parton distribution functions, particularly quark transversity

n QCD evolution is very important in the future for better understanding of the spin asymmetries

n QCD evolution for all TMDs in principle has already become available

n We just need more time/better data to pin down our theoretical parameters, in order to apply them more accurately in the phenomenology

38


Summary

n There are tremendous progress recently in measuring/extracting spin-dependent fragmentation function, in turn to better extract other TMD parton distribution functions, particularly quark transversity

n QCD evolution is very important in the future for better understanding of the spin asymmetries

n QCD evolution for all TMDs in principle has already become available

n We just need more time/better data to pin down our theoretical parameters, in order to apply them more accurately in the phenomenology

38

Thank you

zhongbo kang - indico2.riken.jp · comments on phenomenology! summary ... xd v! p a = x a p ... jet...

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