AA Based on work with Maarten Bu ffi ng and Markus Diehl MPI@LHC - - - PowerPoint PPT Presentation

Matching and resummation in double parton scattering

Tomas Kasemets

Nikhef / VU

MPI@LHC - San Cristóbal de las Casas, November 29, 2016

√ AA

Based on work with Maarten Buffing and Markus Diehl

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

• Total cross section
• DPS populates final state phase space in a different way than SPS



 
 DPS same power as SPS

• Makes small transverse momentum region a very interesting region for

DPS

• Any factorization theorem for this region, must include both single and

double parton scattering

q1 q2

DPS differential in transverse momenta

2

q1 q2

σDPS/σSPS ∼ Λ2 Q2 |q1|, |q2| ∼ Λ << Q : dσSP S d2q1d2q2 ∼ dσDP S d2q1d2q2 ∼ 1 Q4Λ2

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

• TMD Drell-Yan cross section (unpolarized) 



 
 with
 TMDs: 
 and born x hard-matching

• TMDs defined as combination of soft and collinear to cancel rapidity

divergencies

• Depends on two scales, UV and rapidity regularization.

Lessons from TMD factorization and pT resummation

∂ ∂ log µ fa(x, z; µ, ζ) = γF,a(µ, ζ) fa(x, z; µ, ζ) ∂ ∂ log ζ γF,a(µ, ζ) = −1 2 γK,a(µ) . ∂ ∂ log ζ fq(x, z; ζ, µ) = 1 2 Kq(z) fq(x, z; ζ, µ) . ∂ ∂ log µ Ka(z; µ) = −γK,a(µ) dσ dxd¯ xd2q = X

q

ˆ σq¯

q(q2, µ2)

Z d2z (2π)2 e−iqz Wq¯

q(x, ¯

x, z; µ) Wq¯

q = fq(x, z; µ, ζ)f¯ q(¯

x, z; µ, ¯ ζ) γK = Γcusp

3

Collins, 2011; Echevarria, Idilbi, Scimemi, 2011; 
 Echevarria, TK, Mulders, Pisano, 2015

ˆ σq¯

q(q2, µ2) = ˆ

σ0

q¯ qCH(q2, µ2)

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

• For perturbatively small , can match TMDs onto PDFs
• Solving the evolution equations gives the evolved TMDs
• Can be supplemented by non-perturbative 


transverse momentum dependence etc.

• Perturbative input alone gives pT resummed 


cross section

• High scale processes, e.g. Higgs:


small dependence on non-pert. input

4

TMD factorization and pT resummation

z

0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 dσ/dqT qT [GeV] NNLL bc = 1.5 GeV−1 √s = 13 TeV 0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 dσ/dqT qT [GeV] NNLL bc = 1.5 GeV−1 √s = 13 TeV 0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 dσ/dqT qT [GeV] NNLL bc = 1.5 GeV−1 √s = 13 TeV 0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 dσ/dqT qT [GeV] NNLL bc = 1.5 GeV−1 √s = 13 TeV 0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 dσ/dqT qT [GeV] NNLL bc = 1.5 GeV−1 √s = 13 TeV 0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 dσ/dqT qT [GeV] NNLL bc = 1.5 GeV−1 √s = 13 TeV 0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 dσ/dqT qT [GeV] NNLL bc = 1.5 GeV−1 √s = 13 TeV 0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 dσ/dqT qT [GeV] NNLL bc = 1.5 GeV−1 √s = 13 TeV λf = λh = λQ = 0 λf = λh = λQ = 0.5

Echevarria, TK, Mulders, Pisano, 2015

fa(x, z; ζ, µ) = X

b

Cab(x0, z; ζ, µ) ⊗

x fb(x0; µ) ,

fa(x, z; µ, ζ) = X

b

Cab(x, z; µ0, µ2

0) ⊗ x fb(x0; µ0, ζ0)

× exp ⇢Z µ1

µ01

dµ µ  γF,a(µ, µ2) − γK,a(µ) log √ζ µ

• + 1Ka(z, µ0) log

√ζ µ0

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

Goal of project:

• Set up the theoretical (DTMD) framework, within QCD
• As few assumptions as possible
• As much perturbative input as possible, to enhance predictive power
• Provide the basis, correctly including and treating the different effects.
• Once set up in place, can introduce modeling and approximations to

connect with experiments

• Additional difficulties compared to TMDs for SPS
• Different regions which require different matchings
• Color (and polarization) structure
• etc.
• Compared to the pocket formula, it represents the other end of DPS

research

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talk by Markus Diehl

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

Soft and collinear functions

• DPS cross section proportional to
• We define rapidity divergency free DTMDs as
• Collinear matrix element


• perators dressed by Wilson lines (adjoint rep.)
• Soft function, matrix in color space



 perturbative calculation at NNLO

6

√ AA

Fus,gg(x1, x2, z1, z2, y) ⇠ Z dz−

1

2π dz−

2

2π dy−e−i(x1z−

1 +x2z− 2 )p+

⇥ hp| Og(0, z2) Og(y, z1) |pi ,


 Diehl, Schäfer, Ostermeier, 2011

Ogi(y, zi) = gT µν W†G+ν WG+µ

• z+

i =y+=0,

S ∼ ⌦

• W W† W W†W W† W W†

↵

talk by Markus Diehl

Vladimirov, 2016

F T

us,gg(YR) sT −1(YR − YC) s−1(YC − YL) Fus,gg(YL)

= F T

gg(YC) Fgg(YC)

Fgg(YC) = lim

YL→−∞ s−1(YC − YL) Fus,gg(YL) ,

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

DTMD cross section

• For color singlet production (photon, z, Higgs etc.) at



 
 
 
 
 with:

• removes UV region . Choose . 


dependence cancelled by subtraction

• Double TMDs (DTMDs) depend on:


color label, parton and polarization label 
 momentum fractions, transverse distances
 UV renormalization scales, rapidity regularization scale,

7

y± = y ± 1

2(z1 − z2)

x1,2 = y, z1,2 = µ1,2 = ζ = ζ ¯ ζ = Q2

1Q2 2

|q1,2| ⇠ qT ⌧ Q R = 1, 8, ... a1,2, b1,2 = Φ(νy±) ν ∼ Q Φ y± ⌧ 1/ν

RFa1a2(xi, zi, y; µi, ζ)

dσDPS dx1 dx2 d¯ x1 d¯ x2 d2q1 d2q2 = 1 C X

a1,a2,b1,b2

ˆ σa1b1(q2

1, µ2 1) ˆ

σa2b2(q2

2, µ2 2)

× Z d2z1 (2π)2 d2z2 (2π)2 d2y e−iq1z1−iq2z2 Wa1a2b1b2(¯ xi, xi, zi, y; µi, ν) W = Φ(νy+) Φ(νy−) X

R RFb1b2(¯

xi, zi, y; µi, ¯ ζ) RFa1a2(xi, zi, y; µi, ζ)

talk by Jo Gaunt

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

Scale evolution

• UV and rapidity scale


• Complicated functions (3 transverse vectors!), little predictive power
• When : 



 
 
 then region of perturbative dominates result

• But what about the size of 


— can be either small or large

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Z d2z1 (2π)2 d2z2 (2π)2 d2y e−iq1z1−iq2z2 Wa1a2b1b2(¯ xi, xi, zi, y; µ1, µ2, ν) Λ ⌧ qT ⌧ Q |q1| ∼ |q2| ∼ |q1 ± q2| ∼ qT |zi| ∼ 1/qT y |y| ∼ 1/qT |y| ∼ 1/Λ ∂ ∂ log µ1

RFa1a2(xi, zi, y; µi, ζ) = γF,a1(µ1, x1ζ/x2) RFa1a2

∂ ∂ log ζ

RFa1a2(xi, zi, y; µi, ζ) = 1

2

RR0Ka1a2(z1, z2, y) R0Fa1a2

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

Region of large y

• Scalings
• Match DTMDs onto the DPDFs
• Mixing between quark and gluon distributions
• Combine and into subtracted DTMD possible since


(independent of parton type)

• We calculate soft function and matching coefficients at one-loop order


(all parton types, polarizations and color representations, CSS and SCET)

• Coefficients equal to TMDs — PDFs matching coeffs. appart from: 


1) Color factors for non-singlet
 2) Different vector dependence, since DTMDs and DPDs are parametrized in terms of same distance between partons
 3) additional polarizations possible

9

Λ ⌧ qT ⌧ Q

RRSqq(y) = RRSgg(y)

|zi| ∼ 1 qT , |y| ∼ 1 Λ

RFa1a2(xi, zi, y) =

X

b1b2 RCf,a1b1(x0 1, z1)⊗ x1 RCf,a2b2(x0 2, z2)⊗ x2 RFb1b2(x0 i, y) RRSqq RFus,a1b1

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

Region of large y

• Rapidity evolution kernel simplifies considerably

• Diagonal in color, distance dependence separated


usual Collins-Soper kernel

• remains for DPDFs (rapidity scale evolution for collinear func.)
• Solution to evolution equations:

10 1Ka1(z1; µ1) RFa1a2(xi, zi, y; µi, ζ)

= X

b1b2 R⇥

Ca1b1(z1) ⊗

x1 Ca2b2(z2)⊗ x2Fb1b2(x0 i, y; µ0i, ζ0)

⇤ × exp ⇢Z µ1

µ01

dµ µ  γF,a1 − γK,a1 log p x1ζ/x2 µ

• + RKa1(z1) log

p x1ζ/x2 µ01 + Z µ2

µ02

dµ µ  γF,a2 − γK,a2 log p x2ζ/x1 µ

• + RKa2(z2) log

p x2ζ/x1 µ02 + RJ(y) log √ζ √ζ0

• RJ(y; µi)

RR0Ka1a2(zi, y; µi) = δRR0

⇥RKa1(z1; µ1) + RKa2(z2; µ2) + RJ(y; µi) ⇤

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

Region of large y

• Cross section for large y:


• Non-perturbative input: — collinear DPDFs, only one transverse distance

and several model calculations available

• at large scales, color singlet distributions dominate
• ideal future, measured distributions — still a long way to go

11

Wlarge y = X

c1c2d1d2, R

⇥ Φ(νy) ⇤2 exp 

RJ(y, µ0i) log

p q2

1 q2 2

ζ0

• × exp

⇢Z µ1

µ01

dµ µ  γF,a1(µ, µ2) − γK,a1(µ) log q2

1

µ2

• + RKa1(z1, µ01) log q2

1

µ2

01

+ Z µ2

µ02

dµ µ  γF,a2(µ, µ2) − γK,a2(µ) log q2

2

µ2

• + RKa2(z2, µ02) log q2

2

µ2

02

• × R⇥

Cb1d1(z1) ⊗

¯ x1 Cb2d2(z2) ⊗ ¯ x2Fc1c2(xi, y; µ0i, ζ0)

⇤ × R⇥ Ca1c1(z1) ⊗

x1 Ca2c2(z2) ⊗ x2Fd1d2(¯

xi, y; µ0i, ζ0) ⇤

talk by Matteo Rinaldi

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

Region of small y

• Scaling:
• Soft function perturbatively calculable
• Expand on collinear distributions (all fields at same position)


• Size of the contributions


12 RR0Sa1a2(zi, y) = RR0Cs,a1a2(zi, y) RF = RFsplit + RFintr

y ∼ 1/qT ∼ zi Z d2yW(zi, y)

• small y

∼ 8 > < > : α2

sq2 T

from Fsplit × Fsplit (1vs1) αsΛ2 from Fsplit × Fintr (1vs2) Λ4/q2

T

from Fintr × Fintr (2vs2) Fintr = G + C ⊗ G ∼ Λ2, G = twist 4, C ∝ αs Ftw3,

• nly chiral odd, discard

Fspl ∼ y+ y2

+

y− y2

−

T · f(x1 + x2) ∼ q2

T , f = PDF, T ∝ αs

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

Region of small y

• DPS cross section contribution
• Non-perturbative input: PDF 


and twist-four collinear (all color representations)

• Twist-four contribution can (at leading order) be modeled through DPDFs

(part of both DTMDs and DPDFs in this region can be matched onto twist four distributions)

13

Wsmall y = exp ⇢Z µ1

µ0

dµ µ ⇥ γF,a1 − γK,a1 log q2

1

µ2 ⇤ + 1Ka1(z1, µ0) log p q2

1 q2 2

ζ0 + Z µ2

µ0

dµ µ ⇥ γF,a2 − γK,a2 log q2

2

µ2 ⇤ + 1Ka2(z2, µ0) log p q2

1 q2 2

ζ0

• ×

X

RR0

⇥RFspl+int, b1b2(¯ xi, zi, y; µ0i, ζ0) ⇤RR0exp ⇥ Ma1a2(zi, y) log p q2

1 q2 2

ζ0 ⇤ × ⇥R0Fspl+int, a1a2(xi, zi, y; µ0i, ζ0) ⇤

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

Combine regions

• Contributions from the two regions:
• Combine large and small y:

• Collins subtraction formalism
• r , equal up to differences in scale choice


(beyond accuracy for suitable choices)

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Wsubt = Wlarge y

• |y|⌧1/Λ

Wsubt = Wsmall y

• |zi|⌧|y|

W = Wlarge y − Wsubt + Wsmall y Wlarge y = ⇥ Φ(νy) ⇤2 X

R

exp ⇢

RS(z1) + RS(z1)

• exp



RJ(y, µ0i) log

p q2

1 q2 2

ζ0

• × R⇥

C(z1) ⊗

¯ x1 C(z2)⊗ ¯ x2F(¯

xi, y; µ0i, ζ0) ⇤ R⇥ C(z1) ⊗

x1 C(z2)⊗ x2F(xi, y; µ0i, ζ0)

⇤ Wsmall y = Φ(νy+)Φ(νy−)exp ⇢

1S(z1) + 1S(z2)

X

RR0

⇥RFspl+int(¯ xi, zi, y; µ0i, ζ0) ⇤ × RR0exp ⇥ M(zi, y) log p q2

1 q2 2

ζ0 ⇤ ⇥R0Fspl+int(xi, zi, y; µ0i, ζ0) ⇤

talk by Jo Gaunt

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

What can we do with this

• Describe the DPS cross section for double color singlet production

differential in the transverse moment of each of the singlets

• Region where DPS is large and at same power as single parton scattering
• Example processes:
• Double same-sign W (where DPS is enhanced over SPS)
• Other diboson produciton, ZZ, HW, etc
• Color singlet quarkonia
• Perturbative input alone gives transverse momentum resummed cross

section (same non-perturbative input as collinear DPS)

• Provides a framework where we know what we neglect in

phenomenology, and where we can study the common approximations

• Can serve as input to refine the modeling and improve Monte-Carlo
• generators. Can ZZ results serve as validation for DPS modeling?

15

MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets

Summary

• Large fraction of DPS at low/intermediate transverse momenta
• Transverse momentum dependent framework necessary
• Development of DTMD framework: definitions of DTMDs, their

evolution and matching in different regimes

• Provide maximal perturbative information: significantly limits the new

non-perturbative (unknown) information required compared to integrated cross section.

• A lot of potential to:
• do phenomenology
• connect with experiments
• provide useful input to MC generators
• suggestions welcome!

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