Double parton sca/ering for perturba3ve transverse momenta Maarten - - PowerPoint PPT Presentation

Double parton sca/ering for perturba3ve transverse momenta

Maarten Buffing

QCD evolution workshop 2016

May 31, 2016

In collaboration with Markus Diehl and Tomas Kasemets

Content - outline

• Brief motivation/introduction
• Soft factors
• Color for DPDFs/DTMDs
• Evolution equations

– Writing them down for DTMDs – Solving them

• Matching: cross section contributions for large y
• Conclusions

2

Motivation

• DPDs: double parton distribution functions
• Factorization: stick to singlets in final states

– Double Drell-Yan – Higgs + W/Z

• For perturbative qT → significant predictive

results

• Motivation and goals

– Formulate description to handle soft factors – Write down evolution equations – Solve evolution equations – Matching equations for DPDFs/DTMDs Diehl, Ostermeier, Schäfer, JHEP 1203 (2012) 089

3

Short-distance expansion

• Differences compared to TMDs

– Two hard processes involved – Two coefficient functions per DTMD – Positions z1 and z2 (compare with bT for the TMD case) – Additional distance y

• Consider the limit

– |z1|, |z2| much smaller than 1/Λ – |z1|, |z2|≪ y, with y fixed

• Gives separate matching factors

with

Fus(xi, zi, y) = Cf(x0

1, z1) ⊗ x1 Cf(x0 2, z2) ⊗ x2 Fus(x0 i, y)

C(x0) ⊗

x F(x0) =

Z 1

x

dx0 x0 C(x0)F ⇣ x x0 ⌘

Figure: modified from Diehl, Ostermeier, Schäfer, JHEP03 (2012) 089

4

Soft factors

• Wilson line structure from

factorization formula.

• Nontrivial color complications.

Collinear and soft factors carry color indices.

• Wilson line self-interactions

drop out in cross section.

Collins, Foundations of perturbative QCD, (2011); Aybat, Rogers, PRD 83 (2011) 114042; Diehl, Gaunt, Ostermeier, Plößl, Schäfer, JHEP01 (2016) 076

Figure: Diehl, Gaunt, Ostermeier, Plößl, Schäfer, JHEP01 (2016) 076

5

Soft factors

• TMDs
• Soft functions for the single TMD related to K through
• Soft function not matrix valued
• Square root construction for TMD (see Collins’ book)
• DTMDs
• For DPDs: matrix valued functions (working hypothesis)
• Soft function matrix valued
• Square root construction extended to matrix expressions

Collins, Foundations of perturbative QCD, (2011); Aybat, Rogers, PRD 83 (2011) 114042

K(z; µ) = 1 2  ∂ ∂yA log S(z; yA, −∞) − ∂ ∂yB log S(z; +∞, yB)

• S(z1, z2, y, yA, yB) = exp

h (yA − yB)K(z1, z2, y) i

6

Soft factors

• Subtracted DPD distributions are defined as

with Fus vector in color space and S a matrix.

• Matrix equivalent of square root construction

using composition law and a similar expression for left moving particles.

• Wilson line self-interactions drop out in F.

Collins, Foundations of perturbative QCD, (2011); Aybat, Rogers, PRD 83 (2011) 114042

S−1(vL, vC) = S1/2(−vC, vR)S−1/2(vL, vR)S−1/2(vL, vC)

7

(technical details) Fqq(vc) = lim

v2

L→0 S−1

qq (vL, vC)Fus,qq(vL)

S(vA, −vB)S(vB, vC) = S(vA, vC)

Soft factors

• Wilson line structure for double Drell-Yan

with Wilson lines and similarly for the adjoint representation.

• We will need uncontracted color indices in the middle.

Wij(z, v) = P exp  −igta Z 0

−∞

dλvAa(z + λv) z+=z−=0

ij

8

Soft factors

• Uncontracted indices in the middle
• Soft factor for DTMDs factorizes

in small-distance expansion as

• Wilson lines in S(y) pairwise at the

same transverse position.

• We require a simplification of the

color indices.

9

S(z1, z2, y) = Cs(z1) Cs(z2) S(y)

Color structure

• Recall full Wilson line structure
• Hard scattering couples four parton

lines, insert color projectors

• Examples of color projectors

– Quarks: – For gluons: more possibilities – Mixed quark-gluon projectors also exist

• Highly nontrivial whether color structure can be factorized.

pj1j0

1 k1k0 1

1

= 1 Nc δj1j0

1δk1k0 1

pj1j0

1 k1k0 1

8

= 2ta

j1j0

1ta

k1k0

1

10

Color structure

• Recall full Wilson line structure
• Hard scattering couples four parton

lines, insert color projectors

• Color trick (in collinear situation: WW† = 1)

For proof: use color Fierz identity

11

Color structure

• Color trick (in collinear situation: WW† = 1)
• For proof: use color Fierz identity:
• Trick also works for adjoint Wilson lines. Use color Fierz identity and

W ab = 2 Tr  taWtbW †  2ta

ii0ta jj0 = δij0δi0j − 1

Nc δii0δjj0

12

Color structure

• Color trick (in collinear situation: WW† = 1)
• Dynamical and not just some color algebra
• With same trick show that S(y) is color diagonal.

13

Implications for soft factor

• Color projection of fields at infinity rather than ξ + = ξ - = 0.
• Allows for relating most general soft function with open indices in the

middle with soft function with contracted indices in the middle.

• For collinear factorization case only!

⇐ ⇒

Related

14

Renormalization and rapidity evolution

• Short-distance expansion

– The two hard processes are separated

• Evolution equations for DTMDs

– Two renormalization scales: µ1 and µ2

• Soft factor recap

– Working hypothesis – Soft factor becomes

• For phenomenology: only four independent collinear soft functions

S(z1, z2, y, yA, yB) = exp h (yA − yB)K(z1, z2, y) i

RR0S(z1, z2, y) = RCs(z1)RCs(z2)RRS(y)δRR0

15

Renormalization and rapidity evolution

• TMDs
• DTMDs
• DTMD renormalizations are independent, since they are separated.

Collins, Foundations of perturbative QCD, (2011); Aybat, Rogers, PRD 83 (2011) 114042

∂ ∂log ζ F(x, z, µ, ζ) = 1 2K(z; µ)F(x, z, µ, ζ) ∂ ∂log µF(x, z; µ, ζ) = γF (µ, ζ)F(x, z; µ, ζ) ∂ ∂log µ1

RF(xi, zi, y; µi, ζ) = γF (µ1, x1ζ/x2)RF(xi, zi, y; µi, ζ)

∂ ∂log µ2

RF(xi, zi, y; µi, ζ) = γF (µ2, x2ζ/x1)RF(xi, zi, y; µi, ζ)

∂ ∂log ζ

RF(xi, zi, y, µi, ζ) = 1

2 X

R0 RR0K(zi, y; µi)R0F(xi, y, µi, ζ)

16

Evolution: TMDs vs DTMDs

PDF/TMDs

• Soft function not matrix valued
• Just the position of one parton
• Renormalization scale µ
• Rapidity evolution scale ζ
• One coefficient function per

TMD DPDF/DTMDs

• Soft function matrix valued
• Positions of two partons and the

distance y

• Renormalization scales µ1, µ2
• Rapidity evolution scale ζ

– ζ dependence also for collinear distri- bution if R ≠ 1.

• Two coefficient functions per

DTMD

17

DTMD evolution

• The evolution of DTMDs is in the short-distance matching given by
• From additive structure of the Collins-Soper evolution kernel we have

the sum for the two contributions for the µ1 and µ2 dependences.

• K(z1,z2,y)-kernel splits in three separate contributions: K(z1,µ01), K(z2,µ02)

and J(y,µ01,µ02) when collinear soft function becomes diagonal.

RF(xi, zi, y; µ1, µ2, ζ)

= exp ⇢Z µ1

µ01

dµ µ  γF (µ, µ2) − γK(µ) log p x1ζ/x2 µ

• + RK(z1, µ01) log

p x1ζ/x2 µ01 + Z µ2

µ02

dµ µ  γF (µ, µ2) − γK(µ) log p x2ζ/x1 µ

• + RK(z2, µ02) log

p x2ζ/x1 µ02 + RJ(y, µ01, µ02) log √ζ √ζ0

• × RC(x0

1, z1; µ01, µ2 01) ⊗ x1 RC(x0 2, z2; µ02, µ2 02) ⊗ x2 RF(x0 i, y; µ01, µ02, ζ0)

18

Cross section contribution

• Cross section contribution given by
• The z1, z2 and y contributions nicely factorize.

Wlarge y = X

R

exp ⇢Z µ1

µ01

dµ µ  γF (µ, µ2) − γK(µ) log Q2

1

µ2

• + RK(z1, µ01) log Q2

1

µ2

01

+ Z µ2

µ02

dµ µ  γF (µ, µ2) − γK(µ) log Q2

2

µ2

• + RK(z2, µ02) log Q2

2

µ2

02

• × RC(x0

1, z1; µ01, µ2 01) ⊗ x1 RC(x0 2, z2; µ02, µ2 02) ⊗ x2

× RC(x0

1, z1; µ01, µ2 01) ⊗ x1 RC(x0 2, z2; µ02, µ2 02) ⊗ x2

× h Φ(νy) i2 exp 

RJ(y, µ0i) log

p Q2

1Q2 2

ζ0

• RF(xi, y; µ0i, ζ0)RF(xi, y; µ0i, ζ0)

19

Cross section contribution

• Cross section contribution given by
• There is ζ – dependence for color non-singlet DPDFs.

Wlarge y = X

R

exp ⇢Z µ1

µ01

dµ µ  γF (µ, µ2) − γK(µ) log Q2

1

µ2

• + RK(z1, µ01) log Q2

1

µ2

01

+ Z µ2

µ02

dµ µ  γF (µ, µ2) − γK(µ) log Q2

2

µ2

• + RK(z2, µ02) log Q2

2

µ2

02

• × RC(x0

1, z1; µ01, µ2 01) ⊗ x1 RC(x0 2, z2; µ02, µ2 02) ⊗ x2

× RC(x0

1, z1; µ01, µ2 01) ⊗ x1 RC(x0 2, z2; µ02, µ2 02) ⊗ x2

× h Φ(νy) i2 exp 

RJ(y, µ0i) log

p Q2

1Q2 2

ζ0

• RF(xi, y; µ0i, ζ0)RF(xi, y; µ0i, ζ0)

20

Polarizations (work in progress)

• Including parton labels in equations for DTMDs and cross section. E.g.
• Parton labels like a1 not only q, q and g, but also δq, Δq, δg, Δg, etc.
• Splitting kernels from PDF/TMDs can largely be recycled

Collins, Foundations of perturbative QCD, (2011); Aybat, Rogers, PRD 83 (2011) 114042; Bacchetta, Prokudin, NPB 875 (2013) 536; Echevarría, Kasemets, Mulders, Pisano, JHEP 1507 (2015) 158; MGAB, Diehl, Kasemets, work in progress.

_

RFa1a2(xi, zi, y; µ1, µ2, ζ)

= X

b1b2

exp ⇢ Z µ1

µ01

dµ µ  γF,a1(µ, µ2)−γK,a1(µ) log p x1ζ/x2 µ

• +RKa1(z1, µ01) log

p x1ζ/x2 µ01 + Z µ2

µ02

dµ µ  γF,a2(µ, µ2)−γK,a2(µ) log p x2ζ/x1 µ

• +RKa2(z2, µ02) log

p x2ζ/x1 µ02 + RJ(y, µ01, µ02) log √ζ √ζ0

• ×RCa1b1(x0

1, z1; µ01, µ2 01)⊗ x1 RCa2b2(x0 2, z2; µ02, µ2 02)⊗ x2 RFb1b2(x0 i, y; µ01, µ02, ζ0)

Cg/g(x0, z; µ0, µ2

0)

Cg/δg(x0, z; µ0, µ2

0)

Cq/g(x0, z; µ0, µ2

0)

etc.

21

Conclusions

• We use short-distance expansion

– |z1|, |z2| much smaller than 1/Λ – |z1|, |z2|≪ y

although part of our results are also valid outside this region.

• Description for soft function

– Separation in a y-dependent contribution and two pieces depending on either z1 or z2. – We have shown the correct way to deal with color.

• Matching equations for DPDs

– Evolution equations for DTMDs. – Expression for matching at level of individual DTMDs/DPDFs and cross section.

• Work in progress: explicit expressions for matching of all polarization-

modes.

22

Backup slides

23

Properties of the DTMD soft factor

• Parity and boost
• Parity and time reversal
• Hermitian conjugation
• Charge conjugation

and

• Composition law
• Independent collinear soft matrix elements (singlet configurations are 1)

Sa1a2(z1, z2, y, vA, vB) = Sa1a2(z1, z2, y, vB, vA) Sa1a2(z1, z2, y, vA, vB) = Sa1a2(z1, z2, y, −vA, −vB) Sa1a2(z1, z2, y, vA, vB) = S†

a1a2(z1, z2, y, vA, vB)

S∗

qq = Sqq

S∗

gg = Sgg

Sa1a2(vA, −vB)Sa1a2(vB, vC) = Sa1a2(vA, vC)

88S = 88Sqq = AASqg = SSSqg = AASgq = SSSgq = AASgg = SSSgg DDS = DDSgg 27 27S = 27 27Sgg

24

Solving evolution equations for DTMDs

DTMDs: µ1 and µ2 scale evolution

• µ1 scale evolution governed by an equation of the form

and similarly for µ2.

• For the starting values:
• Starting scales µ10 and µ20 for µ1 and µ2.
• We define the ζ value as the geometric mean
• We get the result
• Note the additive structure

∂ ∂log µ1

RF(xi, zi, y; µi, ζ) = γF (µ1, x1ζ/x2)RF(xi, zi, y; µi, ζ) RF(xi, zi, y; µi, ζ) = RF(xi, zi, y; µ0i, ζ)

× exp ⇢ Z µ1

µ01

dµ µ γF (µ, x1ζ/x2) + Z µ2

µ02

dµ µ γF (µ, x2ζ/x1)

• 25

Solving evolution equations for DTMDs

DTMDs: µ1 and µ2 scale evolution and ζ evolution

• ζ evolution governed by
• Solving for rapidity dependence, we then get the result
• The K-kernel splitting in three separate contributions is crucial, but
• nly true in limit where we can do the DTMD → DPDF matching.

∂ ∂log ζ

RF(xi, zi, y, µi, ζ) = 1

2 X

R0 RR0K(zi, y; µi)R0F(xi, y, µi, ζ) RF(xi, zi, y; µi, ζ) = RF(xi, zi, y; µ0i, ζ)

× exp ⇢ Z µ1

µ01

dµ µ  γF (µ, x1ζ/x2) − γK(µ) log p x1ζ/x2 µ

• +

Z µ2

µ02

dµ µ  γF (µ, x2ζ/x1) − γK(µ) log p x2ζ/x1 µ

• +

h

RK(z1, µ01)+ RK(z2, µ02)+ RJ(y, µ0i)

i log √ζ √ζ0

• 26

Coefficient functions

• Consider the limit

– |z1|, |z2| much smaller than 1/Λ – |z1|, |z2|≪ y, with y fixed

• We calculate the coefficient functions for a value of z at O(αs).

– Collinear contribution given by – Soft function contribution given by

• The expression for the coefficient function at order at O(αs) is then

given by

RCf,ab(x, z, vL) = δabδ(1 − x) + αs RC(1) f,ab(x, z, vL) + O(α2 s) RCs,a(z, vL, vC) = 1 + αs RC(1) s,a(z, vL, vC) + O(α2 s) RCab(x, z, . . . /ζ . . .) = δabδ(1 − x)

+αs lim

v2

L→0



RC(1) f,ab(x, z, vL)−δabδ(1−x)RC(1) s,a(z, vL, vC)

• +O(α2

s)

27