Overview of TMD Factorization and Evolution(corrected)

John Collins (Penn State)

• TMD factorization/evolution

• How should non-perturbative part of evolution kernel behave?

• Tool for diagnosis and comparison of formalisms/fits:

– Introduce scheme-independent L(bT) function– Examples

QCD Evolution workshop, May 12, 2014

Basic parton model inspiration: Case of Drell-Yan at qT Q

Ap

Bp

PB

PA

• Lorentz contracted high-energy hadrons

• qT(leptons) =∑

kT(quarks)

• Use parton distribution in x and kT

• But parton model needs to be substantially modified in QCD

Symptom of the QCD complications: qT distribution broadens

0.01

0.1

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

E605 Data

Data

Normalized LY-G Fit

Normalized DWS-G Fit

Normalized BLNY Fit

qT (GeV)

Ed d

qy

3

30

03

σp

bG

eV

nu

cleo

nat

-2

=

.

Q = 7-8GeV

Q = 13.5-18GeV

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

0 5 1 0 1 5 2 0

CDF Z Run 1

Data

Normalized LY-G Fit

Normalized DWS-G Fit

Normalized BLNY Fit

qT (GeV)

d dq

Tσp

b

GeV

Q = 91 GeV

Q: 7–18 GeV,√s = 38.8 GeV Q = mZ,

√s = 1800 GeV

(Plot of E dσ/d3q) (Plot of dσ/dqT: has qT factor.)

(Adapted from Landry et al., PRD 67,073016 (2003))

Steps to derive factorization,given typical structure of graphs + momentum regions:

PB

PA Fourier trans. of 〈p|ψ WL ψ|p〉

• Approximations at leading power

• + Ward identities =⇒ Wilson lines for “misattached” glue (Feynman gauge)

• + contour deformation + “unitarity cancellation”, etc

• =⇒ initial-state Wilson lines for DY

• Factorization of contributions of different regions, including central/soft

• Reorganize: Construct subtractions, define TMD pdfs, with glue restricted tocorrect hemisphere, etc. Soft factors somewhere.

=⇒ Broadening from pert. and non-pert. glue into increasing rapidity range.

Full TMD factorization (modernized Collins-Soper form)

dσ

d4q dΩ=

2

s

∑j

dσj(Q,µ)

dΩ

∫eiqT·bT fj/A(xA, bT; ζA, µ) f/B(xB, bT; ζB, µ) d2bT

+ poln. terms + high-qT term + power-suppressed

where can set ζA = ζB = Q2, µ = Q.

Evolution:∂ ln ff/H(x, bT; ζ;µ)

∂ ln√ζ

= K(bT;µ)

dK

d lnµ= −γK(αs(µ))

d ln ff/H(x, bT; ζ;µ)

d lnµ= γf(αs(µ); 1)− 1

2γK(αs(µ)) ln

ζ

µ2

Small-bT:

ff/H(x, bT; ζ;µ) =∑j

∫ 1+

x−Cf/j(x/x, bT; ζ, µ, αs(µ)) fj/H(x;µ)

dx

x+ O[(mbT)p]

Key to predictivity: Universality etc derived from QCD

• All process use the same TMD pdfs (and fragmentation functions)

• Except:

– Scale dependence: Use evolution– Reversed Wilson lines in TMD pdfs between DY and SIDIS– Hence sign reversal for Sivers function etc

• Same evolution kernel K (color triplet) in all cases, including all polarized cases(Sivers, Boer-Mulders, etc)

• But breakdown of TMD factorization in HH → HH +X

• Non-perturbative information:

– Ordinary pdfs– Large bT TMD pdfs: “intrinsic transverse momentum”– Large bT of evolution kernel K(bT, µ): recoil against radiation per unit rapidity

Formalisms used: They don’t all appear compatible

Parton model: QCD complications ignored

Original CSS: non-light-like axial gauge; soft factor

Ji–Ma–Yuan: non-light-like Wilson lines; soft factor; parameter ρ

New CSS: clean up, Wilson lines mostly light-like;

absorb (square roots of) soft factor in TMD pdfs

Becher–Neubert: SCET, but without actual finite TMD pdfs

Echevarrıa–Idilbi–Scimemi: SCET

Mantry–Petriello: SCET

Boer, Sun-Yuan: Approximations on CSS

Disagreement on non-perturbative contribution to evolution (K(bT) at large bT), oreven whether it exists.

Geography of evolution of cross section

qT

0.01

0.1

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

E605 Data

Data

Normalized LY-G Fit

Normalized DWS-G Fit

Normalized BLNY Fit

qT (GeV)

Ed d

qy

3

30

03

σp

bG

eV

nu

cleo

nat

-2

=

.

Q = 7-8GeV

Q = 13.5-18GeV

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

0 5 1 0 1 5 2 0

CDF Z Run 1

Data

Normalized LY-G Fit

Normalized DWS-G Fit

Normalized BLNY Fit

qT (GeV)

d dq

Tσpb

GeV

Q = 91 GeV

bT

0.25 0.5 0.75 1 1.25 1.5 1.75 2

b @GeV-1D

0

10

20

30

40

Nfit-

1b

WHb

,Q,x

A,x

BL@

fbG

eVD

p + Cu ® Μ+Μ- + X;!!!

s=38.8 GeV; Q=11 GeV; y=0

bmax=1.5 GeV-1, C3=b0, Nfit=1.19

bmax=1.5 GeV-1, C3=2b0, Nfit=1.05

bmax=0.5 GeV-1, C3=b0, Nfit=1.09

Qiu-Zhang , bmax=0.3 GeV-1, Nfit=1

0.2 0.4 0.6 0.8 1 1.2 1.4

b @GeV-1D

0

0.1

0.2

0.3

0.4

0.5

0.6

bWHb

,Q,x

A,x

BL@

nbG

eVD

p + p-

® Z0+ X;

!!!s=1.96 TeV; Q=MZ; y=0

bmax=1.5 GeV-1, C3=b0

bmax=1.5 GeV-1, C3=2b0

bmax=0.5 GeV-1, C3=b0

Q: 7–18 GeV,√s = 38.8 GeV Q = mZ,

√s = 1800 GeV

(Adapted from Landry et al., PRD 67,073016 (2003), Konychev & Nadolsky, PLB 633, 710 (2006))

Evolution of cross section (a la CSS)

0.25 0.5 0.75 1 1.25 1.5 1.75 2

b !GeV!1"

0

10

20

30

40

Nfit!

1b

W"#b

,Q,x

A,x

B$!

fb%G

eV"

p # Cu $ Μ#Μ! # X;&'''

s&38.8 GeV; Q&11 GeV; y&0

bmax&1.5 GeV!1, C3&b0, Nfit&1.19

bmax&1.5 GeV!1, C3&2b0, Nfit&1.05

bmax&0.5 GeV!1, C3&b0, Nfit&1.09

Qiu!Zhang , bmax&0.3 GeV!1, Nfit&1

Q = 11 GeV Q = 91GeV

dσ

d4q= norm.×

∫eiqT·bTW (bT, s, xA, xB) d2bT

∂W

∂ lnQ2

∣∣∣∣∣fixed xA, xB

=∂W

∂ ln s

∣∣∣∣∣fixed xA, xB

=(K(bT, µ) +G(Q,µ)

)W

• Universal K

• Perturbative: G, K at small bT, with RG

• Non-perturbative K at large bT

Different results for evolution at large bT

With CSS prescription: K(bT, µ) = K(b∗, µ)− gK(bT; bmax) fits at Q up to mZ give:

gK(bT) =0.68+0.01

−0.02

2b2T (BLNY, bmax = 0.5 GeV−1 = 0.1 fm)

gK(bT) =0.158± 0.023

2b2T (KN, bmax = 1.5 GeV−1 = 0.3 fm)

But this implies wrong behavior at large bT, smaller Q:

With this parameterization

W = . . . e−b2[coeff(x)+const ln(Q

2/Q

20)]

2 5 10 20 50-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Blue: BLNY, Red: KN

(Sun & Yuan, PRD 88, 114012 (2013))

Tool to compare different methods: The L function(JCC & Rogers, in preparation)

• Shape change of transverse momentum distribution comes only frombT-dependence of K

• So define scheme independent

L(bT) = − ∂

∂ ln b2T

∂

∂ lnQ2 ln W (bT, Q, xA, xB)CSS= − ∂

∂ ln b2TK(bT, µ)

• QCD predicts it is

– independent of Q, xA, xB– independent of light-quark flavor– RG invariant– perturbatively calculable at small bT

– non-perturbative at large bT

Relation of L function to properties of cross section

If L were constant, then

W (bT, Q) = normalization× W (bT, Q0)×(

1

b2T

)L ln(Q2/Q

20)

L is positive: W decreases at large bT and increases at small bT when Q increases.

Of course, L is not actually constant.

Comparing different results using the L function(Preliminary)

0 1 2 3 4 5bT !GeV

!1"

0.2

0.4

0.6

0.8

1.0

L!bT "

!"#$%&'%

()*+%,%-./%01234%

'#%&'%

()*+%,%4./%01234%

5%6789*:%;<))1=9<%%

%%%%%%%%(91*>8?&%5%6@?A?1)1?=%

"B%C0%D)E9@F1GH%

#@%(I%=*)8?&%

0 1 2 3 4 5bT !GeV

!1"

0.2

0.4

0.6

0.8

1.0

L!bT "

!"#$!#

%&'()&*'##

+#,#-#./0#

%&'()&*'##

+#,#-1#./0#

2#3456*7#89::/;69##

########<6/*=5'$# 2#3>'?'/:/';#

@A#B.#C:D6>E/FG#

">#<H#;*:5'$#

Q Typical bT

2 GeV 3 GeV−1

10 GeV 1.2 GeV−1

mZ 0.5 GeV−1

SY = Sun & Yuan (PRD 88, 114012 (2013)):

LSY = CFαs(Q)

πDepends on Q: contrary to QCD

Implications

• Important to determine actual non-perturbative part of TMD evolution (i.e.,K(bT) at large bT).

• Older fits (e.g., KN) OK for bT up to around 3 GeV−1 = 0.6 fm.

• But their extrapolation to larger bT makes K(bT) too large.

• Use L(bT) to diagnose the issues: It’s a universal scheme independent function inQCD.

• What does this mean physically? . . .

Physical meaning of non-perturbative K(bT)

• Overall principle: Emission of glue is uniform in rapidity

• Old idea/intuition:

– In one unit of rapidity emit Gaussian (??) distribution of kT:

e−k2T/k

20 T

1

πk20 T

– Exponential convolution =⇒ W (bT, Q) = W (bT, Q0)e−b2T ln(Q

2/Q

20)k

20 T/4

– Gives K(bT)NP = −b2Tk20 T/4

• New proposal

– Per unit rapidity: a probability of no emission, and a probability of emitting aparticle (or more)

– So

K(bT)NP = FT of c[−δ(2)(kT) + e−k

2T/k

20 T/(πk2

0 T)]

= c[−1 + e−b

2Tk

20 T/4

]– ¿Change to exponential at large bT instead of Gaussian? (Normal for Euclidean

correlation function)