Double Parton Distributions

Gunnar Bali, Peter Bruns, Luca Castagnini, Markus Diehl,Jonathan Gaunt, Benjamin Gläßle, Andreas Schäfer,

Christian Zimmermann

QCD

PDF4LHC Oxford, March 24, 2017

Outline

Why DPDs?What we computeSome results

Gunnar Bali (Regensburg) Double Parton Distributions QCD 2 / 11

Double hard scattering: DPDs

Needed to describe what is not from primary hard scattering at the LHC(“underlying event”)Example W+W+ production. qq → qqW+W+. Double parton scatteringgives an important contribution! [G Gaunt et al, arXiv:1003.3953]

Also (considering gluons) cccc over cc production ∼ 100 times higherthan theory predictions [LHCb, arXiv:1205.0975]!

1st DPD moment: AudVV (y2, py = 0) =

∫ 1

−1dζ

∫∫ 1

−1dx1dx2 Fud (x1, x2, ζ, y2)

Typically convolution of DPD into two generalized form factors is assumed.

Gunnar Bali (Regensburg) Double Parton Distributions QCD 3 / 11

Even the GFFs are not well known!Event generators usually approximate multiple parton cross sections asproducts of uncorrelated single parton reactions.Little difference is made whether the partons come from the same or fromdifferent protons and an effective σeff parameter [G Calucci,D Treleani, hep-ph/9902479] (and a symmetry factor S = 1, 2) is used:

dσDPS =dσSPSdSPS

σeff

DPD moments can be extracted from four point functions on the lattice.Example (two currents O1 and O†2 at equal time):

〈π+p | [uγµd ]︸ ︷︷ ︸O†

2

(y) [uγνd ]︸ ︷︷ ︸O1

(0)|π−p 〉

Note that for non-flavour changing currents this corresponds to thehadronic tensor in (Euclidean!) coordinate space, discussed earlier today.Above, two Wick contractions are possible (C1 and A)

Gunnar Bali (Regensburg) Double Parton Distributions QCD 4 / 11

LiteratureSome previous work on γ4-γ4 (and 1-1) correlation functions:[C Alexandrou, G Koutsou, arXiv:0809.2056](using 2-hand/stochastic sandwich trick)[C Alexandrou, P De Forcrand, A Tsapalis, hep-lat/0206026,hep-lat/0307009][R Gupta, D Daniel, J Grandy, hep-lat/9304009][M C Chu, M Lissia, J W Negele, NPB 360 (1991) 31][W Wilcox, K-F Liu, PRD34 (1986) 3882]

Suggestion to directly compute the Compton amplitude on the lattice:[U Aglietti et al, hep-ph/9804416]

Note that this is also related to LEET (Large Energy Effective Theory)[M J Dugan, B Grinstein, PLB 255 (1991) 583]:

〈p|q(y)Γ2

replace by Pei∫ y0

dy′µAµ(y′)︷ ︸︸ ︷

q(y)q(0) Γ1q(0)|p〉Gunnar Bali (Regensburg) Double Parton Distributions QCD 5 / 11

Quark line diagrams to be consideredIsospins of the initial/final states and currents determine diagrams.

4pt GraphsObtain six independent Wick contractions:

u ud d

π+p (0) π+

p (t)O1(τ,0)

O2(τ,y)

C1

u q u

d

π+p (0) π+

p (t)O1(τ,0)

O2(τ,y)

C2

u

dd

uπ+

p (0) π+p (t)O1(τ,0)

O2(τ,y)

A

q

ud d

π+p (0) π+

p (t)O1(τ,0)

O2(τ,y)

S1qq

ud

π+p (0) π+

p (t)O2(τ,y)

O1(τ,0)S2q

u

q

d

π+p (0) π+

p (t)O1(τ,0)

O2(τ,y)

D

Choose t = 15a, expect plateau at 6a . τ . 9a⇒ fit or average

Christian Zimmermann | Lattice 2016 Double Parton Distributions

C1, A: “Two-hand-trick”, C2, S1: sequential one-end with HPE all-to-all,S2: one-end and non-stoch 2-point, D: one-end and HPE all-to-all (noisy).

Gunnar Bali (Regensburg) Double Parton Distributions QCD 6 / 11

p = 0 pseudoscalar-pseudoscalar and scalar-scalar elementsMπ ≈ 290MeV, lattice extent: L = 40a, a ≈ 0.071 fm.

-15

-10

-5

0

5

10

15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

𝑀P

P(|𝑦

|)[10

−2G

eV4]

|𝑦|[fm]

Results for 𝑀PP (𝐿 = 40)

S1C1C2A

10−4

10−3

10−2

10−1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

|𝑀P

P(|𝑦

|)|[G

eV4]

|𝑦|[fm]

Results for 𝑀PP (logscale, 𝐿 = 40)

S1C1C2A

-15

-10

-5

0

5

10

15

20

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

𝑀S

S(|𝑦

|)[10

−2G

eV4]

|𝑦|[fm]

Results for 𝑀SS (𝐿 = 40)

S1C1C2A

10−4

10−3

10−2

10−1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

|𝑀S

S(|𝑦

|)|[G

eV4]

|𝑦|[fm]

Results for 𝑀SS (logscale, 𝐿 = 40)

S1C1C2A

Gunnar Bali (Regensburg) Double Parton Distributions QCD 7 / 11

p = 0 vector-vector and axial-axial elementsMπ ≈ 290MeV, lattice extent: L = 40a, a ≈ 0.071 fm.

-1.5

-1

-0.5

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

𝑀V

0V

0(|𝑦

|)[10

−2G

eV4]

|𝑦|[fm]

Results for 𝑀V0V0(𝐿 = 40)

S1C2C1A

10−6

10−5

10−4

10−3

10−2

10−1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

|𝑀V

0V

0(|𝑦

|)|[G

eV4]

|𝑦|[fm]

Results for 𝑀V0V0(logscale, 𝐿 = 40)

S1C2C1A

-1

-0.5

0

0.5

1

1.5

2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

𝑀A

0A

0(|𝑦

|)[10

−2G

eV4]

|𝑦|[fm]

Results for 𝑀A0A0(𝐿 = 40)

S1C2C1A

10−6

10−5

10−4

10−3

10−2

10−1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

|𝑀A

0A

0(|𝑦

|)|[G

eV4]

|𝑦|[fm]

Results for 𝑀A0A0(logscale, 𝐿 = 40)

S1C2C1A

Gunnar Bali (Regensburg) Double Parton Distributions QCD 8 / 11

Invariant functions

Current-current correlators are moments of DPDs.(Details in [M Diehl, D Ostermeier, A Schäfer, arXiv:1111.0910])They can be decomposed into Lorentz structures times invariant functionsA(p · y = −py, y2 = −y2). Below: ASS(0,−y2)

−20

−15

−10

−5

0

5

10

15

20

25

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

𝐴SS

(|𝑦|)[

10−

3 GeV

2 ]

|𝑦|[fm]

Results for 𝐴SS (𝐿 = 40)

C1C2A10−5

10−4

10−3

10−2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

|𝐴SS

(|𝑦|)|

[GeV

2 ]

|𝑦|[fm]

Results for 𝐴SS (logscale, 𝐿 = 40)

C1C2A

Can one write A as a convolution of two formfactors?

Gunnar Bali (Regensburg) Double Parton Distributions QCD 9 / 11

Comparison with the convolution of EM formfactors

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

𝐴u

dS

S(|𝑦

|)[10

−2G

eV2]

|𝑦|[fm]

Comparison for 𝐴udSS: 3pt∗3pt to 4pt (𝐿 = 40)

1u�u�

4pt data3pt∗3pt error (u� = 1)

3pt∗3pt convolution (u� = 1)3pt∗3pt error (u� = 2)

3pt∗3pt convolution (u� = 2)

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

𝐴u

dV

V(|𝑦

|)[10

−3G

eV2]

|𝑦|[fm]

Comparison for 𝐴udVV: 3pt∗3pt to 4pt (𝐿 = 40)

1u�u�

4pt data3pt∗3pt error(u� = 1)

3pt∗3pt convolution (u� = 1)3pt∗3pt error (u� free)

3pt∗3pt convolution (u� free)

Seems to work in vector channel for |y| & 0.4 fm(Small |y|: lattice artefacts. Large |y|: finite size effects.)In general, there are considerable violations.

Gunnar Bali (Regensburg) Double Parton Distributions QCD 10 / 11

Summary

We experiment with new ideas. The pion is an ideal test case.(Generalized) current-current matrix elements have manyapplications: Relation to structure functions at large-x , multiple hardscattering etc.The renormalization straight-forward and easier than when trading ina quark propagator for a Wilson line.The necessary techniques are developed.Only after all practical and theoretical questions are under control forthe pion, we will move on to the proton.

Gunnar Bali (Regensburg) Double Parton Distributions QCD 11 / 11