PRODUCTION AT HADRON COLLIDERS

Hee Sok Chung Argonne National Laboratory

Based on Geoffrey T. Bodwin, HSC, U-Rae Kim, Jungil Lee, PRL113, 022001 (2014)

Geoffrey T. Bodwin, HSC, U-Rae Kim, Jungil Lee, Yan-Qing Ma, Kuang-Ta Chao, in preparation

OUTLINE

• Leading-power fragmentation in quarkonium production

• Cross section and polarization of

• direct

• and

• prompt

• Summary

2

HEAVY QUARKONIUM

3

• Bound states of a heavy quark and a heavy antiquark : e.g. , , , , , , , ...

•

• which allow nonrelativistic description : for charmonia, for bottomonia

• Typical energy scales , Ideal for studying interplay between perturbative and nonperturbative physics

J/ ⌘c 0 �cJhc ⌥(nS) �bJ⌘b

2mb > 2mc � ⇤QCD

mJ/ ⇡ m⌘c ⇡ 2mc, m⌥(1S) ⇡ m⌘b ⇡ 2mb,

v2 ⇡ 0.3 v2 ⇡ 0.1

m > mv > mv2 ⇡ ⇤QCD

S-w

ave

P-w

ave

Spin SingletSpin Triplet

J/ ⌘c 0,

�cJ hc

S-w

ave

P-w

ave

Spin SingletSpin Triplet

⌥(nS)

�bJ

⌘b

hb

Charmonia Bottomonia

HEAVY QUARKONIUM • Quark model assignments of some heavy quarkonia

• The -differential cross section has been measured at hadron colliders like RHIC, Tevatron and the LHC.

• is usually identified from its leptonic decay.

• Large contributions from B hadron decays are subtracted to yield the “prompt” cross section, which includes contributions from direct production and from decays of heavier charmonia

INCLUSIVE PRODUCTION

5

INCLUSIVE PRODUCTION

6

• Color-singlet model (CSM) : A pair with same spin, color and C P T is created in the hard process, which evolves in to the . The universal rate from to (color-singlet long-distance matrix element) is known from models, lattice measurements, and fits to experiments.

p p

cc̄

J/

color-singlet LDME

cc̄

cc̄

J/

J/

INCLUSIVE PRODUCTION

7

• CSM is incomplete :

• A color-octet (CO) pair can evolve into a color singlet meson by emitting soft gluons.

• In the effective theory nonrelativistic QCD, CO LDMEs are suppressed by powers of .

• In many cases, color-octet channels are necessary : • For S-wave vector quarkonia ( ),

CSM severely underestimates the cross section.• For production or decay of P-wave quarkonia

( ), CS channel contains IR divergences that can be cancelled only when the CO channels are included.

cc̄

J/ , (2S),⌥(nS)

�cJ ,�bJ

INCLUSIVE PRODUCTION

8

• CSM prediction vs. measurement at Tevatron

• LO CSM ( ) is inconsistent with both shape and normalization.

• Radiative corrections are larger than LO and has different shape ( ), but still not large enough 10 -3

10 -2

10 -1

1

10

5 10 15 20

BR(J/ μ+μ-) d (pp_ J/ +X)/dpT (nb/GeV)s =1.8 TeV; | | < 0.6

pT (GeV)

LO colour-singletcolour-singlet frag.

⇠ 1/p4T

⇠ 1/p8T

INCLUSIVE PRODUCTION • NRQCD can be used to describe the physics of scales

smaller than the quarkonium mass.

• NRQCD factorization conjecture for production of

• Short-distance cross sections are essentially the production cross section of that can be computed using perturbative QCD

• The LDMEs are nonperturbative quantities that correspond to the rate for the to evolve into

• LDMEs have known scaling with 9

Bodwin, Braaten, and Lepage, PRD51, 1125 (1995)

Short-distance cross section LDME

QQ̄

QQ̄

INCLUSIVE PRODUCTION • NRQCD factorization conjecture for production of

• Usually truncated at relative order : , , , channels for

• The short-distance cross sections have been computed to NLO in by three groups :

• It is not known how to calculate color-octet LDMEs, and are usually extracted from measurements

10

Bodwin, Braaten, and Lepage, PRD51, 1125 (1995)

Short-distance cross section LDME

Kuang-Ta Chao’s group (PKU) : Ma, Wang, Chao, Shao, Wang, ZhangBernd Kniehl’s group (Hamburg) : Butenschön, KniehlJianxiong Wang’s group (IHEP) : Gong, Wan, Wang, Zhang

INCLUSIVE PRODUCTION • In order to extract CO LDMEs from measured cross

sections we need to determine the short-distance cross sections as functions of

• NLO corrections give large K-factors that rise with ; this casts doubt on the reliability of perturbation theory

11 Ma, Wang, Chao, PRL106, 042002 (2011)

POLARIZATION PUZZLE

• NRQCD at LO in predicts transverse polarization at large

• Disagrees with measurement

• NLO corrections are large in the and channels

• NRQCD at NLO still predicts transverse polarization

12

(GeV/c)Tp5 10 15 20 25 30

α

-1-0.8-0.6-0.4-0.20

0.20.40.60.81

CDF DataNRQCD

-factorizationTk

(a)

CDF, PRL99, 132001 (2007)Braaten, Kniehl, and Lee, PRD62, 094005 (2000)

CMS, PLB727, 381 (2013)Butenschoen and Kniehl, PRL108, 172002 (2012)

: Transverse

: Unpolarized

: Longitudinal

LP FRAGMENTATION • Large NLO corrections arise because new channels

that fall off more slowly with open up at NLO

• The leading power (LP) in ( ) is given by single-parton fragmentation

• Corrections to LP fragmentation go as 13

Collins and Soper, NPB194, 445 (1982)Nayak, Qiu, and Sterman, PRD72, 114012 (2005)

Fragmentation Functions

Parton productioncross sections

z : fraction of momentum transferred from parton k to hadron

i, j, k run over quarks, antiquarks, and gluon

d�

dp2T

[ij ! cc̄+X]

=XXX

k

ZZZ 1

0dz

d�

dp2T

[ij ! k+X]D[k ! cc̄+X] +O(m2c/p

6T )

FRAGMENTATION FUNCTIONS • Fragmentation functions (FFs) for production of

can be computed using perturbative QCD

• A gluon can produce a pair in state directly : gluon FF for this channel starts at order , involves a delta function at

• A gluon can produce a in state by emitting a soft gluon : gluon FF for this channel starts at order , involves distributions singular at

• A gluon can produce a in state by emitting a gluon : gluon FF for this channel starts at order , does not involve divergence at order

14

cc̄

Collins and Soper, NPB194, 445 (1982)

cc̄

cc̄

cc̄

z = 1

z = 1

FRAGMENTATION FUNCTIONS • For the and channels, gluon

polarization is transferred to the pair, and therefore the is mostly transverse.

• For the channel, the is unpolarized because it is isotropic.

15

cc̄cc̄

cc̄

LP FRAGMENTATION • LP fragmentation explains the large, -dependent

K-factors that appear in fixed-order calculations

• channel is already at LP at LO : NLO correction is small

• and channels do not receive an LP contribution until NLO : NLO corrections are large

16

• LP fragmentation reproduces the fixed-order calculation at NLO accuracy at large The difference is suppressed by

• The slow convergence in channel is because the fragmentation contribution is small (no function or plus distribution from IR divergence)

LP FRAGMENTATION

17

LP+NLO • We combine the LP fragmentation contributions with

fixed-order NLO calculations to include corrections of relative order

• We take in order to suppress possible non-factorizing contributions

18

LP fragmentation resummed leading logs

LP fragmentation to NLO accuracy

fixed-order calculation to NLO

corrections of relative order

LP+NLO • Alternatively, one can consider the LP fragmentation

to supplement the fixed-order NLO calculation

19

LP fragmentation resummed leading logs

LP fragmentation to NLO accuracy

fixed-order calculation to NLO

Additional fragmentation contributions

channel and channels

LP CONTRIBUTIONS THAT WE COMPUTE

• We resum the leading logarithms in to all orders in

• Corrections to LP contributions give “normal” K-factors ( )

20

LO NLO NNLO

NLO NNLO

NNLO

Fragmentation functions

Pa

rto

n p

rod

uct

ion

cro

ss s

ecti

on

s

Gribov and Lipatov, Yad. Fiz. 15, 781 (1972) / Lipatov, Yad. Fiz. 20, 181 (1974)Dokshitzer, Zh. Eksp. Teor. Fiz. 73, 1216 (1977) / Altarelli and Parisi, NPB126, 298 (1977)

- NLO NNLO

- NNLO

-

Fragmentation functions

Not yet available Leading logarithms onlyAvailable

. 2

RESUMMATION OF LEADING LOGARITHMS• The leading logarithms can be resummed to all orders

by solving the LO DGLAP equation

• This equation is diagonalized in Mellin space; the inverse transform can then be carried out numerically

• Because the FFs are singular at the endpoint, the inversion is divergent at ; special attention is needed for contribution at

21

z = 1

z ⇡ 1

d

d logµ2f

✓DS

Dg

◆=

↵s(µf )

2⇡

✓Pqq 2nfPgq

Pqg Pgg

◆⌦✓DS

Dg

◆

DS =P

q(Dq +Dq̄)

RESUMMATION OF LEADING LOGARITHMS• We split the integral :

• is chosen so that near

22

z

N �̂(z) ⇡ �̂(1)zN

Z 1

0dz �̂(z)D(z) =

Z 1�✏

0dz �̂(z)D(z) +

Z 1

1�✏dz �̂(z)D(z)

⇡Z 1�✏

0dz �̂(z)D(z) + �̂(z = 1)

Z 1

1�✏dz zND(z)

z ⇡ 1Z 1

1�✏dz zND(z) =

Z 1

0dz zND(z)�

Z 1�✏

0dz zND(z)

Well defined in Mellin space (Mellin transform of D(z))

Well behaved numerically

LP+NLO • The additional fragmentation contributions have

important effects on the shapes in the channel

• Large corrections to the shape of the channel because the LO and NLO contributions cancel at about

23

PRODUCTION • We obtain good fits to

the cross section measurements by CDF and CMS

• was used in the fit

• 25% theoretical uncertainty from varying fragmentation, factorization and renormalization scales

24

CDF, PRD71, 032001 (2005)CMS, JHEP02, 011 (2012)Bodwin, HSC, Kim, Lee, PRL113, 022001 (2014)

PRODUCTION • The data falls off faster

than and channels

• The fit constrains the and channels to cancel

• channel dominates the cross section

• This possibility was first suggested by Chao et al.

25

Chao, Ma, Shao, Wang, Zhang, PRL108, 242004 (2012)

hOJ/ (1S[8]0 )i = 0.099± 0.022 GeV3

hOJ/ (3S[8]1 )i = 0.011± 0.010 GeV3

hOJ/ (3P [8]0 )i = 0.011± 0.010 GeV5

POLARIZATION • Because of dominance,

is almost unpolarized

• FIRST PREDICTION OF UNPOLARIZED IN NRQCD

• This is in good agreement with CMS data and much improved agreement with CDF Run II data

• Caveat : feeddown ignored26

CMS, PLB727, 381 (2013)CDF, PRL 85, 2886 (2000), PRL99, 132001 (2007)

Bodwin, HSC, Kim, Lee, PRL113, 022001 (2014)

PROMPT PRODUCTION • can also be

produced from decays of and

• LDMEs from fit to CMS and CDF cross section data

• LDMEs from fit to ATLAS cross section data

• 30% theoretical uncertainty from scale variation

27

PRELIMINARY

PRELIMINARY

CDF, PRD80, 031103 (2009)CMS, JHEP02, 011 (2012)CMS-PAS-BPH-14-001

ATLAS, JHEP1407, 154 (2014)

POLARIZATION • We predict that the

is slightly transverse at the Tevatron

• We predict that the is slightly transverse at the LHC Agrees with CMS data within errors

28

PRELIMINARY

PRELIMINARY

CDF, PRL85, 2886 (2000)CDF, PRL99, 132001 (2007)

CMS, PLB727, 381 (2013)

PRODUCTION • and channels

contribute at leading order in

• We obtain good fits to ATLAS data

• The matrix element obtained from fit agrees with the potential model calculation

29

PRELIMINARY

→Suggests that NRQCD factorization works

ATLAS, JHEP1407, 154 (2014)

Potential model

Eichten and Quigg, PRD 52, 1726 (1995)

Our fit

POLARIZATION OF FROM DECAY

• We predict that the from decay is slightly transverse at LHC

• We assume transition in (higher-order transitions have little effect)

30

PRELIMINARY

Faccioli, Lourenco, Seixas, and Wohri, PRD83, 096001 (2011)

PROMPT PRODUCTION • After including

feeddown contributions, we again obtain good fits

• Again, was used in the fit

31

PRELIMINARY

PRELIMINARY

Fractions of direct and feeddown contributions

CDF, PRD71, 032001 (2005)CMS, JHEP02, 011 (2012)CMS-PAS-BPH-14-001

PROMPT PRODUCTION

32

PRELIMINARY

• The direct cross section falls off faster than and channels

• The fit constrains the and channels to cancel

• channel dominates the direct cross section

PROMPT POLARIZATION • Direct and

from feeddown is slightly transverse

• PROMPT HAS SMALL POLARIZATION

• This is in reasonably good agreement with CMS data

33

PRELIMINARY

PRELIMINARY

CMS, PLB727, 381 (2013)

CDF, PRL85, 2886 (2000), PRL99, 132001 (2007)

SUMMARY• We present new LP fragmentation contributions that

have a significant effect on calculations of production in hadron colliders

• When we include LP fragmentation contributions, we predict the to have near-zero polarization at high at hadron colliders

• This is the first prediction of small polarization at high in NRQCD

• Work on higher-order corrections, as well as other quarkonium states is in progress

34

BACKUP

GLUON FRAGMENTATION INTO • Lowest-order diagrams of a gluon producing CO

36

GLUON FRAGMENTATION INTO • Computation of fragmentation functions involve

Eikonal lines, and their interaction with gluons

37

PREDICTIONS AT NLO

• Used cross section measurements at HERA and Tevatron to fix LDMEs, predicts transverse polarization at large

• H1 and ZEUS data are at small • Does not include feeddown

38

Butenschoen and Kniehl, Mod.Phys.Lett.A28, 1350027 (2013)

Bernd Kniehl’s group

PREDICTIONS AT NLO

• Used CDF and LHCb cross section data to fit LDMEs

• Includes feeddown

• Prediction still more transverse than measurement

39

Jianxiong Wang’s group

Gong, Wan, Wang, Zhang, PRL110, 042002 (2013)

PREDICTIONS AT NLO

40Shao, Han, Ma, Meng, Zhang, Chao, hep-ph/1411.3300 (2014)

Kuang-Ta Chao’s group

• Used CDF, ATLAS, CMS, LHCb cross section data to fit LDMEs

• Included feeddown

• Assumed positivity of all LDMEs, although LDME has strong factorization scale dependence

• Prediction still more transverse than data

PREDICTIONS AT NLO

41

• Used CDF, ATLAS, CMS, LHCb cross section data to fit LDMEs

• Included feeddown

• Assumed positivity of all LDMEs, although LDME has strong factorization scale dependence

• Prediction still more transverse than data

Shao, Han, Ma, Meng, Zhang, Chao, hep-ph/1411.3300 (2014)

Kuang-Ta Chao’s group