[PPT] - Double parton scattering: factorisation, evolution and matching M. PowerPoint Presentation

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Double parton scattering: factorisation, evolution and matching

M. Diehl

Deutsches Elektronen-Synchroton DESY

REF (Resummation, Evolution, Factorization) Madrid, 13 to 16 Nov. 2017

DESY

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Hadron-hadron collisions

◮ standard description based on factorisation formulae

cross sect = parton distributions × parton-level cross sect

◮ net transverse momentum pT of hard-scattering products:

pT integrated cross sect collinear factorisation
pT ≪ hard scale of interaction TMD factorisation

◮ particles resulting from interactions between spectator partons unobserved

M. Diehl

Double parton scattering: factorisation, evolution and matching 2

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Hadron-hadron collisions

◮ standard description based on factorisation formulae

cross sect = parton distributions × parton-level cross sect

◮ net transverse momentum pT of hard-scattering products:

pT integrated cross sect collinear factorisation
pT ≪ hard scale of interaction TMD factorisation

◮ particles resulting from interactions between spectator partons unobserved ◮ spectator interactions can be soft underlying event

r hard multiparton interactions

◮ here: double parton scattering with factorisation formula

cross sect = double parton distributions × parton-level cross sections

M. Diehl

Double parton scattering: factorisation, evolution and matching 3

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Single vs. double parton scattering (SPS vs. DPS)

◮ example: prod’n of two gauge bosons, transverse momenta q1 and q2

q2 q1

single scattering: |q1| and |q2| ∼ hard scale Q |q1 + q2| ≪ Q

q2 q1

double scattering: both |q1| and |q2| ≪ Q

◮ for transv. momenta ∼ Λ ≪ Q :

dσSPS d2q1 d2q2 ∼ dσDPS d2q1 d2q2 ∼ 1 Q4 Λ2

but single scattering populates larger phase space :

σSPS ∼ 1 Q2 ≫ σDPS ∼ Λ2 Q4

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Double parton scattering: factorisation, evolution and matching 4

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Single vs. double parton scattering (SPS vs. DPS)

◮ example: prod’n of two gauge bosons, transverse momenta q1 and q2

q2 q1

single scattering: |q1| and |q2| ∼ hard scale Q |q1 + q2| ≪ Q

q2 q1

double scattering: both |q1| and |q2| ≪ Q

◮ for small parton mom. fractions x

double scattering enhanced by parton luminosity

◮ depending on process: enhancement or suppression

from parton type (quarks vs. gluons), coupling constants, etc.

M. Diehl

Double parton scattering: factorisation, evolution and matching 5

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

A numerical example

gauge boson pair production

W + W +

single scattering: qq → qq + W +W + suppressed by α2

s

W + W +

integrated cross section

J Gaunt et al, arXiv:1003.3953

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Double parton scattering: factorisation, evolution and matching 6

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Drell-Yan: factorisation for qT ≪ Q

S H H B A

⇒

H H B A S

◮ fast-moving longitudinal gluons coupling to hard scattering

include in Wilson lines in parton density

◮ soft gluon exchange between left- and right-moving partons

include in soft factors = vevs of Wilson lines

needs: eikonal approximation, Ward identities, Glauber cancellation

essential for establishing factorisation
permits resummation of Sudakov logarithms

TMD factorisation

Collins, Soper, Sterman 1980s; Collins 2011

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Double parton scattering: factorisation, evolution and matching 7

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Drell-Yan: factorisation for qT ≪ Q

S H H B A

⇒

H H B A S

absorb soft factor into parton densities

σ = ˆ σBSA = ˆ σ(B √ S)( √ SA) = ˆ σfB fA

S requires a rapidity cutoff for the gluons:

right-moving gluons fA, left-moving ones fB

separation at central rapidity Y (or equivalent variable)

ζ = 2(xp+

Ae−Y )2

¯ ζ = 2(¯ xp−

B e+Y )2

ζ ¯ ζ = Q4

resum Sudakov logarithms log(qT /Q) via evolution equations

d d log ζ fA(ζ)

and

d d log ¯ ζ fB(¯

ζ)

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Double parton scattering: factorisation, evolution and matching 8

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Drell-Yan: factorisation for qT ≪ Q

W (z) = P exp

−igta
−∞

dλ vAa(λv+z)

WR(z/2)

W †

R(−z/2)

W †

L(z/2)

WL(−z/2)

H H B A S

◮ transverse variables

z Fourier conjugate to q:

dσ/d2q ∝

d2z eizqfA(x, z; ζ)fB(¯

x, z; ¯ ζ)

soft factor S =

1 Nc

tr W †

L( z 2) WR( z 2) W † R(− z 2) WL(− z 2)

collinear factorisation: in
d2q (dσ/d2q) have z = 0

⇒ S = 1 soft gluon exchanges cancel in sum over all graphs no Sudakov logarithms

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Double parton scattering: factorisation, evolution and matching 9

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Double parton scattering

◮ can generalise previous treatment from single to

double Drell-Yan and other DPS processes

M Buffing, T Kasemets, MD 2017

S H1 H2 H2 H1 B A

⇒

H1 H2 H1 H2 B A S

◮ basic steps can be repeated:

collinear gluons Wilson lines in DPDs
soft gluons soft factor

MD, D Ostermeier, A Sch¨ afer 2011; MD, J Gaunt, P Pl¨

ßl, A Sch¨

afer 2015

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Double parton scattering: colour complications

◮ DPDs have several colour combinations of partons j k k′ j′

z1/2 + y z2/2 −z2/2 −z1/2 + y

x1 x2 x1 x2

colour projection operators
singlet: P jj′,kk′

1

= δjj′δkk′/3 as in usual PDFs

octet: P jj′,kk′

8

= 2tjj′

a tkk′ a

for gluons: 8A, 8S, 10, 10, 27

◮ corresponding combinations in soft factor

soft factor → matrix in colour space
for colour octet (and other non-singlets):

WRtaW †

R = 1 when at same position

⇒ S = 1 Sudakov factors even in collinear factoris’n

M Mekhfi 1988; A Manohar, W Waalewijn 2012

ta ta z1/2 + y z2/2 −z2/2 −z1/2 + y

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Double parton scattering: factorisation, evolution and matching 11

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Coloured final states

◮ processes with coloured final states (jets etc)

collinear factorisation only with measured small qT no TMD factorisation even for single scattering

P Mulders, T C Rogers 2010

H1 H2 H1 H2 B A S

soft factor with more open

colour indices

to be contracted with hard

scattering

for large distance y

non-perturbative

◮ looks grim for phenomenology . . .

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Double parton scattering: factorisation, evolution and matching 12

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Simplification for collinear factorisation

◮ projector identity for Wilson

lines at same position

◮ includes all interactions ◮ also for adjoint Wilson lines

(gluons) and mixed case

P jj′,k′k

R

= P ii′,j′j

R

W(z) W †(z) j j′ k k′ W(z) W †(z) i i′ j j′

◮ use this to show

S for jet production etc. same as for Drell-Yan:

=

S(y) is diagonal in colour:

RR′S(y) ∝ δRR′ with R = 1, 8, . . .

and octet 88S(y) is same for quarks and gluons

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Collinear factorisation

◮ in collinear factorisation simple colour structure

σDPS ∼

R Rˆ

σ1

Rˆ

σ2

d2y RFB(y) RFA(y)

with RFA = √

RRS RA and RFB likewise

◮ evolution of RF(x1, x2, y; µ1, µ2, ζ) with Collins-Soper type equation:

2∂ ∂ log ζ RF = RJ(y; µ1, µ2) RF ∂ ∂ log µ1 RJ = − RγJ(µ1)

can choose separate factorisation scales µ1, µ2 for hard scatters
for colour singlet have 1J = 0
for colour octet:

8J(y) = kernel for rapidity evolution of single gluon TMD

A Vladimirov 2016

◮ solution has form

RF(x1, x2, y; µ1, µ2, ζ) = e− RE(x1,x2,y;µ1,µ2,ζ) R

F(x1, x2, y; µ1, µ2) where R F follows DGLAP equations in µ1 and µ2 with kernels RP(µ)

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Double parton scattering: factorisation, evolution and matching 14

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

TMD factorisation

◮

RR′S = S(z1, z2, y; Y ) nontrivial matrix in colour space

◮ rapidity evolution of S understood at perturbative two-loop level

A Vladimirov 2016

◮ assume that general structure valid beyond two loops:

∂ ∂Y S(Y ) =

K S(Y ) for Y ≫ 1

work towards an all-order proof: A Vladimirov 2017

◮ define FA = s A (s = matrix equivalent of

√ S)

◮ cross section σ ∝ ˆ

σ1 ˆ σ2

R

RFB RFA

H1 H2 H1 H2 B A S

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

TMD factorisation: evolution

◮ evolution of RF(x1, x2, z1, z2, y; µ1, µ2, ζ)

j k k′ j′

z1/2 + y z2/2 −z2/2 −z1/2 + y

x1 x2 x1 x2

∂ ∂ log ζ F = K(z1, z2, y; µ1, µ2) F ∂ log µ1 K = 1

1 γK(µ1)

∂ log µ1 F = γF (µ1, x1ζ/x2) ∂ ∂ log ζ γF = γK

γF and γK same as for single-parton TMDs

where have Collins-Soper kernel K(z, µ)

write K = 1

1 [K(z1, µ1) + K(z2, µ2)] + M ⇒ M indep’t of µ1,2

◮ solution:

F(xi, zi, y; µ1, µ2, ζ) = e−E(z1;µ1,x1ζ/x2)−E(z2;µ2,x2ζ/x1) × e M(zi,y) log(ζ/ζ0) F(xi, zi, y; µ0, µ0, ζ0)

E(z; µ, ζ) = Sudakov exponent for single-parton TMD

contains double logarithm, is colour independent

matrix exponential of M gives single logarithms
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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Transverse momenta Λ ≪ qT ≪ Q: matching

f(x) f(x, z)

dσ/d2q ∝

d2z eizqfA(x, z; ζ) fB(¯

x, z; ¯ ζ)

◮ recall single DY: cross section dominated by |z| ∼ 1/qT ≫ 1/Λ

match TMD on PDF: fa(x, z; µ, ζ) =

b Cab(x, z, µ, ζ) ⊗ x fb(x; µ)

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Transverse momenta Λ ≪ qT ≪ Q: matching

f(x) f(x, z)

y + 1

2z1 1 2z2 −1 2z2 y − 1 2z1

◮ recall single DY: cross section dominated by |z| ∼ 1/qT ≫ 1/Λ

match TMD on PDF: fa(x, z; µ, ζ) =

b Cab(x, z, µ, ζ) ⊗ x fb(x; µ)

◮ double DY: dominated by |z1|, |z2| ∼ 1/qT ◮ for |y| ∼ Λ have matching

RFa1a2(xi, zi, y; µi, ζ) =

b1b2

RCa1b1(x1, z1, µ1, x1ζ/x2)

⊗

x1 RCa2b2(x2, z2, µ2, x2ζ/x1) ⊗ x2 RFb1b2(xi, y; µi, ζ)

colour singlet coefficients 1C same as for single TMDs

Collins-Soper kernel simplifies:

RR′Ka1a2(zi, y; µi) = δRR′

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

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

A second regime for DPD matching

◮ region of small |y| ∼ 1/qT ∼ |zi|

match on collinear distributions (all fields at same transv. position) F(xi, zi, y) = Fint + Fspl

+ twist-three contribution, negligible for low x

y + 1

2z1 1 2z2 −1 2z2 y − 1 2z1

y + 1

2z1 1 2z2

−1

2z2

y − 1

2z1

Fint = G + Ctw4(zi, y; µi) ⊗ G ∼ Λ2 Fspl ∼ y+ y2

+

y− y2

−

Pspl · f(x1 + x2) ∼ q2

T

G = twist 4 distribution f = PDF , y± = y ± 1

2 (z1 + z2)

Ctw4 ∝ αs (unknown) Pspl ∝ αs (known) in cross section get scaling ∼            α2

s q2 T

from Fspl × Fspl (1v1) αs Λ2 from Fspl × Fint (1v2) Λ4/q2

T

from Fint × Fint (2v2) Λ2 from |y| ∼ 1/Λ

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

A second regime for DPD matching

◮ region of small |y| ∼ 1/qT ∼ |zi|

match on collinear distributions (all fields at same transv. position) F(xi, zi, y) = Fint + Fspl

+ twist-three contribution, negligible for low x

y + 1

2z1 1 2z2 −1 2z2 y − 1 2z1

y + 1

2z1 1 2z2

−1

2z2

y − 1

2z1

Fint = G + Ctw4(zi, y; µi) ⊗ G ∼ Λ2 Fspl ∼ y+ y2

+

y− y2

−

Pspl · f(x1 + x2) ∼ q2

T

G = twist 4 distribution f = PDF , y± = y ± 1

2 (z1 + z2)

Ctw4 ∝ αs (unknown) Pspl ∝ αs (known)

◮ combine approximations for small and large y

with subtraction terms to remove double counting

adapt subtraction formalism of Collins 2011

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Introduction DPS: Colour Evolution and cross section TMD matching Summary

Summary

◮ factorisation for double parton scattering:

largely established at similar level as for SPS

◮ two aspects ≫ complicated than in SPS:

colour structure, especially in soft gluon exchange
presence of additional distance scale y

◮ significant simplifications for collinear factorisation

due to projector identity for Wilson lines

no cross talk between different colour representations R
all R except for colour singlet are Sudakov suppressed

◮ rapidity evolution for TMD factorisation → matrix in colour space

but: Sudakov double logarithms = two copies of SPS

◮ significant simplifications for Λ ≪ qT ≪ Q

match double TMDs on collinear DPDs, PDFs, and twist-four distributions

◮ many aspects to be studied quantitatively → impact on phenomenology

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