probe triple partons interaction through three quarkonia ... · quarkonium production and double...

Introduction The frame of Calculation Numerical Result Summary

Probe triple partons interaction throughthree quarkonia associated production at

LHC

Yu-Jie Zhang

Beihang Univeisity

Hadron 2019 @ GXNU

Based on Hua-Sheng Shao, YJZ,PRL122(2019)192002/Arxiv:1902.04949

August 18, 20193 Quarkonium at LHC


Outline

1 IntroductionMultiparton scatteringQuarkonium productions

2 The frame of Calculation

3 Numerical ResultNumerical Result of J/ψ + J/ψ + J/ψ + XNumerical Result for Υ+ J/ψ + φ

4 Summary

3 Quarkonium at LHC


Introduction

3 Quarkonium at LHC


Multiparton scattering

SPS, DPS, and TPS

SPS, DPS, and TPS1 Single parton scattering (SPS) / Double-parton scattering

(DPS): involve one / two partons in each hadron colliding.2 Triple-parton scattering (TPS): involve three partons in

each hadron colliding. TPS are absent due to their morecomplicated final states and much fewer yields.

J/ψ

J/ψ

J/ψ

p

p

J/ψ

J/ψ

J/ψ

p

p

J/ψ

J/ψ

J/ψ

p

p

Figure: SPS/DPS/TPS triple J/ψ production at SPPC3 Quarkonium at LHC



MPI

MPI1 The cross sections of MPI are either strongly model

dependent or assuming no correlation between MPI.2 We assume no correlation between MPI.3 The DPS studies at the LHC and Tevatron suggest that no

correlation assumption is a rather good approximation.4 A N-parton scattering (NPS) cross-section (1708.07519)

σNPSf1···fN

=mN!

∏Ni=1 σ

SPSfi

(σeff,N)N−1 , (1)

3 Quarkonium at LHC



NPS

NPS1 A N-parton scattering (NPS) cross-section (1708.07519)

σNPSf1···fN

=mN!

∏Ni=1 σ

SPSfi

(σeff,N)N−1 , (2)

2 The factor mN! : the indistinguishable final state symmetry.

3 σSPSfi

: the SPS cross section of producing final state fi .

4 σeff,N: the effective cross section, which should bedetermined by experiments.

5 The DPS and TPS cases correspond to N = 2 and N = 3.6 Ref. 1612.05582 derives σeff,3 = (0.82 ± 0.11)× σeff,2.

3 Quarkonium at LHC



σeff,2 and heavy quarkonium (1710.06315, 1811.07474)

σeff,2

1 σeff,2 ∼10 mb: extracted from the quarkonium data.2 σeff,2 ∼15 mb: extracted from the weak gauge boson data.3 However, it is still far from being conclusive in view of the

remaining large uncertainties.

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1 2�

3 4

5 6§

7 89

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1. CDF 4j 1.8 TeV 325 nb-1

2. LHCb J�Ψ+C mesons H4 channelsL 7 TeV 355 pb-1

3. ATLAS W+jj 7 TeV 36 pb-1

4. CMS same sign WW Honly Σeff lower limitL 8 TeV 19.7 fb-1

5. CMS W+jj 7 TeV 5 fb-1

6. D0 Γ+3j H2 channelsL 1.96 TeV 8.7 fb-1

7. ATLAS 4j 7 TeV 37.3 pb-1

8. D0 2Γ+2j 1.96 TeV 8.7 fb-1

9. D0 J�Ψ+J�Ψ 1.96 TeV 8.1 fb-1

10. LHCb J�Ψ+J�Ψ HDifferent SPS modelsL 13 TeV 5 fb-1

syst. err. 25%

syst. err. 15%

æ æ

WÄ jj

æ æ

ZÄ jj

æ æ æ æ

W±ÄW±A B C D E F G H

A. Fit DrelpT 13 TeV 300 fb-1

B. Fit DrelpT 100 TeV 300 fb-1

C. Fit DlrelpT 13 TeV 300 fb-1

D. Fit DlrelpT 100 TeV 300 fb-1

E. Fit DΗl l 13 TeV 300 fb-1

F. Fit DΗl l 13 TeV 3000 fb-1

G. Fit DΗl l 100 TeV 300 fb-1

H. Fit DΗl l 100 TeV 3000 fb-1

HbL

æ æ æ æ

0

5

10

15

20

25

30

35

40

45

50

0

5

10

15

20

25

30

35

40

45

50

Σef

f@m

bD

3 Quarkonium at LHC



σDPSeff , Atlas, 1811.11094

DPS and W ,Z

Experim

ent

(energ

y,

final sta

te,

year)

0 5 10 15 20 25 30

[mb]eff

σ

ATLAS

AFS (√

s = 63 GeV, 4 jets, 1986)UA2 (

√

s = 630 GeV, 4 jets, 1991)CDF (

√

s = 1.8 TeV, 4 jets, 1993)CDF (

√

s = 1.8 TeV, γ+ 3 jets, 1997)DØ (

√

s = 1.96 TeV, γ+ 3 jets, 2010)

LHCb (√

s = 7 TeV, J/ψΛ+c , 2012)

LHCb (√

s = 7 TeV, J/ψD+s , 2012)

LHCb (√

s = 7 TeV, J/ψD+, 2012)

LHCb (√

s = 7 TeV, J/ψD0, 2012)ATLAS (

√

s = 7 TeV, W+ 2 jets, 2013)CMS (

√

s = 7 TeV, W+ 2 jets, 2014)DØ (

√

s = 1.96 TeV, γ+ b/c + 2 jets, 2014)DØ (

√

s = 1.96 TeV, γ+ 3 jets, 2014)DØ (

√

s = 1.96 TeV, J/ψ + J/ψ, 2014)ATLAS (

√

s = 8 TeV, Z + J/ψ, 2015)

LHCb (√

s = 7&8 TeV, Υ(1S)D0,+, 2015)DØ (

√

s = 1.96 TeV, J/ψ +Υ, 2016)DØ (

√

s = 1.96 TeV, 2γ+ 2 jets, 2016)ATLAS (

√

s = 7 TeV, 4 jets, 2016)ATLAS (

√

s = 8 TeV, J/ψ + J/ψ, 2017)CMS (

√

s = 8 TeV, Υ+Υ, 2017)LHCb (

√

s = 13 TeV, J/ψ + J/ψ, 2017)

CMS (√

s = 8 TeV, W±W±, 2018)ATLAS (

√

s = 8 TeV, 4 leptons, 2018)

Figure: DPS3 Quarkonium at LHC



dphi @ D0, Shao, YJZ, 1605.03061

Br(

J/ψ

→µ

+µ

- )Br(

Υ→

µ+µ

- )×d

σ/d

|∆φ| [fb]

|∆φ(J/ψ,Υ)|

prompt J/ψ+Υ(1S,2S,3S) production at 1.96 TeV Tevatron

SPS: kT smearing ⟨kT⟩=3 GeV

COM: Set I LDMEs

σeff=2.2 mb

σeff=5 mb

σeff=15 mb

DR

EW-INTER

DRLI

EW-INTER

COMDPS

D0 Data

10-5

10-4

10-3

10-2

10-1

100

101

102

103

0 1 2 3

HELAC-Onia 2.0

3 Quarkonium at LHC



Double parton scattering Picture (Enterria, Snigirev, 1708.07519)

DPS

PDPSpp→ab = PSPS

pp→a × PSPSpp→b

=σ(pp → a + X )

σinel(pp)× σ(pp → b + X )

σinel(pp)(3)

Then σnPSeff ∼ σinel(pp). But σinel(pp) ∼ 30 − 50 mb and

σDPSeff ∼ 10 − 15 mb.

3 Quarkonium at LHC



TPS

TPS (1612.05582, 1703.07163, 1710.1152)

TPS theoretical studies in literature are limited to openheavy-flavor productions so far. The complete study includingSPS and DPS is not available.

c

c̄

c

c̄

c

c̄x′3

x′2

x′1

x1 x2

x3

p

p

Figure: TPS of pPb → cc̄ + cc̄ + cc̄ (PRL118, 122001).

3 Quarkonium at LHC


Quarkonium productions


NLO J/ψ at LHCb, Chao/Wang/Kniehl, 1506.03981

) [GeV/c]ψ(J/T

p0 5 10 15

θλ

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

NLO NRQCD(1)NLO NRQCD(2)NLO NRQCD(3)

NLO CS

2.5 < y < 4.0

= 7 TeVsLHCb

3 Quarkonium at LHC



CO LDMEs, 1212.2037

Butenschoen, Gong, Wang, Chao, Ma, Shao, Wang, Zhang52

Kniehl18 Wan, Zhang53 default set set 2 set 3

〈OJ/ψ(3S[1]1 )〉 1.32 GeV3 1.16 GeV3 1.16 GeV3 1.16 GeV3 1.16 GeV3

〈OJ/ψ(1S[8]0 )〉 0.0497 GeV3 0.097 GeV3 0.089 GeV3 0 0.11 GeV3

〈OJ/ψ(3S[8]1 )〉 0.0022 GeV3 −0.0046 GeV3 0.0030 GeV3 0.014 GeV3 0

〈OJ/ψ(3P[8]0 )〉 −0.0161 GeV5 −0.0214 GeV5 0.0126 GeV5 0.054 GeV5 0

〈Oψ′

(3S[1]1 )〉 0.758 GeV3

〈Oψ′

(1S[8]0 )〉 −0.0001 GeV3

〈Oψ′

(3S[8]1 )〉 0.0034 GeV3

〈Oψ′

(3P[8]0 )〉 0.0095 GeV5

〈Oχ0 (3P[1]0 )〉 0.107 GeV5

〈Oχ0 (3S[8]1 )〉 0.0022 GeV3

3 Quarkonium at LHC



Double J/ψ, Lansberg, Shao, 1410.8822

Energy and quarkonium cuts σexp. σSPS,prompt

LOσ

SPS,prompt

NLO(⋆) σDPS,prompt χ2

LHCb√

s = 7 TeV, Pψ1,2

T< 10 GeV,

2 < yψ < 5 [34]

18 ± 5.3 pb 41+51−24

pb 46+58−27

31+11−6.3

(+24−15

) pb 0.5 − 1.2

D0

√s = 1.96 TeV, P

ψ1,2

T> 4 GeV, SPS: 70 ± 23 fb 53+57

−27fb 170+340

−110fb – –

|ηψ| < 2.0 [12] (+ µ cuts in caption) DPS: 59 ± 23 fb – – 44+16−9.1

(+7.5−5.1

) fb 0.06 − 0.5

CMS√

s = 7 TeV, Pψ1,2

T> 6.5 → 4.5 GeV

depending on |yψ1,2| ∈ [0, 2.2] (see the

caption) [35]

5.25 ± 0.52 pb 0.35+0.26−0.17

pb 1.5+2.2−0.87

pb 0.69+0.24−0.14

(+0.039−0.027

) pb 1.09 − 1.14

ATLAS√

s = 7 TeV, Pψ1,2

T> 5 GeV and |yψ1,2

| <2.1 (+ µ cuts in the caption) [48]

– 6.4+4.3−2.6

fb 36+49−20

fb 19+6.8−4.0

(+2.2−1.6

) fb N/A

3 Quarkonium at LHC



Double J/ψ at CMS, Sun, Han, Chao, 1404.4042

3 Quarkonium at LHC



Double J/ψ at CMS, Lansberg, Shao, Yamanaka, YJZ 1906.10049d

σ/d

M [

nb/G

eV

]

M(J/ψ+J/ψ) [GeV]

Prompt J/ψ+J/ψ production at √s=7 TeV LHC

NLO* SPS

NLO*+LI SPS

CMS

10-6

10-5

10-4

10-3

10-2

10-1

10 20 30 40 50 60 70 80

HELAC-Onia 2.0

dσ

/d∆

y [

nb]

|∆y(J/ψ,J/ψ)|

Prompt J/ψ+J/ψ production at √s=7 TeV LHC

NLO* SPS

NLO*+LI SPS

CMS

10-5

10-4

10-3

10-2

10-1

100

0 0.5 1 1.5 2 2.5 3 3.5 4

HELAC-Onia 2.0

3 Quarkonium at LHC



Quarkonium production and double parton scattering

Many quarkonium associated production processes seemsto be dominant by Double-Parton Scattering (DPS).

1 J/ψ + W and J/ψ + Z , (ATLAS, arXiv:1401.2831,1412.6428)

3 Quarkonium at LHC






2 J/ψ + charm and Υ+ charm (LHCb, arXiv:1205.0975,1510.05949)

3 Quarkonium at LHC







3 J/ψ + J/ψ (D0, arXiv:1406.2380; CMS, arXiv:1406.0484)

3 Quarkonium at LHC







3 J/ψ + J/ψ (D0, arXiv:1406.2380; CMS, arXiv:1406.0484)4 Υ+ J/ψ (D0, arXiv:1511.02428)

3 Quarkonium at LHC



Quarkonium associated production

Quarkonium associated production at hadron colliders1 σ(J/ψ + cc̄)@α4

s : Artoisenet, Lansberg, Maltoni, 0703129.

3 Quarkonium at LHC





s : Artoisenet, Lansberg, Maltoni, 0703129.2 σ(Υ + 3jets)@α5

s : Artoisenet, Campbell, Lansberg,Maltoni, Tramontano, 0806.3282.

3 Quarkonium at LHC







3 σ(Bccc + c̄ + c̄ + c̄)@α6s : Chen, Wu, 1106.0193.

3 Quarkonium at LHC







3 σ(Bccc + c̄ + c̄ + c̄)@α6s : Chen, Wu, 1106.0193.

4 σ(J/ψ + J/ψ + J/ψ + g)@α7s : Shao, YJZ, 1902.04949 ....

3 Quarkonium at LHC


The frame of Calculation

3 Quarkonium at LHC


Total triple J/ψ hadroproduction cross sections

DPS and TPS cross sectionsWe will use the following concrete formula:

σDPS(pp → J/ψJ/ψJ/ψ + X )

=σSPS(pp → J/ψJ/ψ + X )σSPS(pp → J/ψ + X )

σeff,2,

σTPS(pp → J/ψJ/ψJ/ψ + X )

=16

[

σSPS(pp → J/ψ + X )]3

(σeff,3)2 (4)

to calculate DPS and TPS cross sections.

In total, there are three different SPS cross sections, i.e., thoseof one, two and three J/ψ production, to be computed.3 Quarkonium at LHC


SPS cross sections

Hadron and Parton level cross sections

σ(h1h2 → JJJ ) =∑

a,b

fa/h1⊗ fb/h2

⊗σ̂(ab → C + J + J + X ). (5)

Parton level cross section

d σ̂(ab→JJJ ) =∑

n1,n2,n2

σ̂(ab → cc̄[n1]cc̄[n2]cc̄[n3] + X )

〈OJ (n1)〉〈OJ (n2)〉〈OJ (n3)〉 (6)

3 Quarkonium at LHC


One of 28774 Feynman Diagrams of SPS

J/ψ

J/ψ

J/ψ

p

p

3 Quarkonium at LHC


Feynman Diagrams of DPS

J/ψ

J/ψ

J/ψ

p

p

3 Quarkonium at LHC


Feynman Diagrams of TPS

J/ψ

J/ψ

J/ψ

p

p

3 Quarkonium at LHC


Numerical Result

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X

SPS cross sections with pT (J/ψ) > 2GeV

σSPS Order 14 TeV 100 TeV

J/ψ +X α3s 72± 1 µb 300 ± 8 µb

J/ψJ/ψ +X α4s 67± 2 nb 343 ± 13 nb

J/ψJ/ψJ/ψ +X α7s 1± 0.6 pb 4.2± 3.2 pb

σ(J/ψJ/ψJ/ψ +X) Order 14 TeV CMS @ 14 TeV 100 TeV

TPS α9s 620 ± 20 pb 45± 1.5 pb 45000 ± 1500 pb

DPS α7s 480 ± 20 pb 35± 1.5 nb 10000 ± 1000 pb

SPS α7s 1± 0.6 pb 0.07± 0.04 pb 4.2± 3.2 pb

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X

Triple J/ψ production at 100 TeV SPPC

σ(M

(3J/ψ

)>M

min

)×B

r3 [

fb]

Mmin [GeV]

Prompt 3-J/ψ production at √s=100 TeV

σeff,2=10 mb,σeff,3=10 mb

6 fb-1

300 fb-1

3 ab-1

10 ab-1

100 events

SPS

DPS

TPS

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

0 10 20 30 40 50 60 70 80 90 100

HELAC-Onia 2.0

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X


σ(P

T>

PT

,min

)×B

r3 [

fb]

PT,min [GeV]



kT smearing ⟨kT⟩=3 GeV

|yJ/ψ|<2.4

300 fb-1

3 ab-1

10 ab-1

100 events

SPS

DPS

TPS

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

0 1 2 3 4 5 6 7 8 9 10

HELAC-Onia 2.0

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X


σ(M

(3J/ψ

)>M

min

)×B

r3 [

fb]

Mmin [GeV]



6 fb-1

300 fb-1

3 ab-1

10 ab-1

100 events

SPS

DPS

TPS

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

0 10 20 30 40 50 60 70 80 90 100

HELAC-Onia 2.0

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X


σ(P

T>

PT

,min

)×B

r3 [

fb]

PT,min [GeV]




|yJ/ψ|<2.4

300 fb-1

3 ab-1

10 ab-1

100 events

SPS

DPS

TPS

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

0 1 2 3 4 5 6 7 8 9 10

HELAC-Onia 2.0

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X

J/ψJ/ψJ/ψ + X at CMS/Atlas

Search TPS at CMS/Atlas

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X


Search TPS at CMS/Atlas1 Integrated luminosity of CMS/Atlas is about 160 fb−1 at√

s = 13 TeV.

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X



s = 13 TeV.2 Br [J/ψ → µ+µ−] = 0.059.

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X



s = 13 TeV.2 Br [J/ψ → µ+µ−] = 0.059.3 Number of events for 3J/ψ(µ+µ−) with pT (J/ψ) > 2 GeV

is about 2700 ± 72 , which is 1500 ± 50 from TPS and1200 ± 50 from DPS.

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X



s = 13 TeV.2 Br [J/ψ → µ+µ−] = 0.059.3 Number of events for 3J/ψ(µ+µ−) with pT (J/ψ) > 2 GeV

is about 2700 ± 72 , which is 1500 ± 50 from TPS and1200 ± 50 from DPS.

4 We can introduce cut of phase space to distinguish SPS,DPS, and TPS contributions.

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X

Triple J/ψ production at LHC

σ(M

(3J/ψ

)>M

min

)×B

r3 [

fb]

Mmin [GeV]

Prompt 3-J/ψ production at √s=13 TeV LHC


6 fb-1

23 fb-1

300 fb-1

3 ab-1

100 events

SPS

DPS

TPS

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

0 10 20 30 40 50 60 70 80 90 100

HELAC-Onia 2.0

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X


σ(|

∆y|>

|∆y| m

in)×

Br3

[fb

]

|∆y|min




6 fb-1

23 fb-1

300 fb-1

3 ab-1

100 events

SPS

DPS

TPS

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

0 0.5 1 1.5 2 2.5 3 3.5

HELAC-Onia 2.0

3 Quarkonium at LHC


J/ψ + J/ψ + J/ψ + X


σ(P

T>

PT

,min

)×B

r3 [

fb]

PT,min [GeV]




|yJ/ψ|<2.4

150 fb-1

300 fb-1

3 ab-1

100 events

SPS

DPS

TPS

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

0 1 2 3 4 5 6 7 8 9 10

HELAC-Onia 2.0

3 Quarkonium at LHC


Numerical Result for Υ + J/ψ + φ

Numerical Result for Υ+ J/ψ + φ

Υ, J/ψ, φ at LHCb

1 σ(Υ) ∼ 0.20 µb (1804.09214), σ(J/ψ) ∼ 15 µb(1509.00771), and σ(φ) ∼ 3 mb (1107.3935).

3 Quarkonium at LHC






2 We have gotten σSPS [Υ + J/ψ] ∼ 10 pb in 1605.03061.

3 Quarkonium at LHC






2 We have gotten σSPS [Υ + J/ψ] ∼ 10 pb in 1605.03061.3 SPS Υ+ J/ψ + φ: O(α9

s), very small.

3 Quarkonium at LHC







s), very small.

4 DPS Υ+ J/ψ + φ: about 3 × σSPS[Υ + J/ψ] σ[φ]σDPS

eff∼ 10 pb

for pT (φ) > 2 GeV and σDPSeff ∼ 10 mb.

3 Quarkonium at LHC







s), very small.


eff∼ 10 pb


5 TPS Υ+ J/ψ + φ: about σ[Υ]σ[J/ψ]σ[φ](σTPS

eff )2 ∼ 100 pb for

σTPSeff ∼ 10 mb.

3 Quarkonium at LHC



Search for TPS in Υ+ J/ψ + φ at LHCb

Estimate the number of events1 Br [Υ(J/ψ) → µ+µ−] = 0.024(0.06) and

Br [φ→ K+K−] = 0.5.

3 Quarkonium at LHC





Br [φ→ K+K−] = 0.5.2 Integrated luminosity of LHCb is about 6 fb−1 at√

s = 13 TeV.

3 Quarkonium at LHC






s = 13 TeV.3 Number of events for Υ(µ+µ−) + J/ψ(µ+µ−) + φ(K+K−)

with pT (φ) > 2 GeV is about 440.

3 Quarkonium at LHC







with pT (φ) > 2 GeV is about 440.4 We can introduce cut to suppress SPS and DPS

contributions.

3 Quarkonium at LHC




Υ+ J/ψ + φ at Atlas/CMS

1 σ(Υ) ∼ 0.6 µb (1211.7255), σ(J/ψ) ∼ 50 µb (1104.3038),and σ(φ) ∼ 6 mb (1402.6162).

3 Quarkonium at LHC






2 We have gotten σSPS [Υ + J/ψ] ∼ 100 pb in 1605.03061.

3 Quarkonium at LHC







s), very small.

3 Quarkonium at LHC







s), very small.


eff∼ 60 pb


3 Quarkonium at LHC







s), very small.


eff∼ 60 pb


5 TPS Υ+ J/ψ + φ: about σ[Υ]σ[J/ψ]σ[φ](σTPS

eff )2 ∼ 2 nb for

σTPSeff ∼ 10 mb.

3 Quarkonium at LHC



Search for TPS in Υ+ J/ψ + φ at Atlas/CMS


Br [φ→ K+K−] = 0.5.

3 Quarkonium at LHC





Br [φ→ K+K−] = 0.5.2 Integrated luminosity of Atlas/CMS is about 160 fb−1 at√

s = 13 TeV.

3 Quarkonium at LHC







is about 2 × 105.

3 Quarkonium at LHC







is about 2 × 105.4 We can introduce cut to suppress SPS and DPS

contributions.

3 Quarkonium at LHC


Summary

We have performed the first analysis of simultaneousproduction of J/ψ + J/ψ + J/ψ and Υ+ J/ψ + φ fromSPS/DPS/TPS contributions at LHC.

Our work shows that it is in fact most probably dominated byTPS and DPS contributions.

Finally, we show that J/ψ + J/ψ + J/ψ and Υ+ J/ψ + φproduction at LHC may be studied by experimenters.

3 Quarkonium at LHC


3 Quarkonium at LHC


Multi parton scattering

The inclusive cross section to produce n hard particles inhadronic colliders is a convolution of generalized n-partondistribution functions (PDF) and elementary partonic crosssections summed over all involved partons,

σNPShh′→a1...an

=(m

n!

)

∑

i1,..,in,i ′1,..,i′

n

∫

Γi1...inh (x1, .., xn;b1, ..,bn;Q2

1 , ..,Q2n)

× σ̂i1i ′1a1

(x1, x′1,Q

21) · · · σ̂

in i ′nan (xn, x ′

n,Q2n) (7)

× Γi ′1...i

′

nh′ (x ′

1, . . . , x′n;b1 − b, . . . ,bn − b;Q2

1 , . . . ,Q2n)

× dx1 . . . dxn dx ′1, . . . ,dx ′

n d2b1, . . . ,d2bn d2b.

3 Quarkonium at LHC


The n-parton distribution function (1708.07519)

It encodes all the 3D structure information of the hadron.1 Assumption 1: the n-PDF are factored in terms of

longitudinal and transverse components,

Γi1...inh = Di1...in

h (x1, .., xn;Q21 , ..,Q

2n)f (b1)...f (bn) (8)

2 We can get hadron-hadron overlap functionT (b) =

∫

f (b1)f (b1 − b)d2b1, where 1 =∫

T (b)d2b.3 Assumption 2: the longitudinal components reduce to the

product of independent single PDF

Di1...inh (x1, .., xn;Q2

1 , ..,Q2n) = Di1

h (x1;Q21) · · ·Din

h (xn;Q2n) (9)

3 Quarkonium at LHC


The cross sections and σnPSeff (Enterria, Snigirev, 1708.07519)

The cross sections of n-particle associated production

Then we can get

σnPShh′→a1...an

=(m

n!

) σSPShh′→a1

· · · σSPShh′→an

(

σnPSeff

)n−1 , (10)

σnPSeff

(

1

σnPSeff

)n−1

=

∫

d2b T n(b) (11)

3 Quarkonium at LHC

probe triple partons interaction through three quarkonia ... · quarkonium production and double...

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