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 • Bound states of a heavy quark and a heavy antiquark : e.g. , , , , , , , ... Υ ( nS ) ψ 0 J/ ψ h c η c η b χ cJ χ bJ • 2 m b > 2 m c � Λ QCD m J/ ψ ≈ m η c ≈ 2 m c , m Υ (1 S ) ≈ m η b ≈ 2 m b , • which allow nonrelativistic description : for charmonia, for bottomonia v 2 ≈ 0 . 3 v 2 ≈ 0 . 1 • Typical energy scales , m > mv > mv 2 ≈ Λ QCD Ideal for studying interplay between perturbative and nonperturbative physics 3
HEAVY QUARKONIUM • Quark model assignments of some heavy quarkonia Spin Triplet Spin Singlet Spin Triplet Spin Singlet S-wave S-wave J/ ψ ψ 0 Υ ( nS ) , η c η b P-wave P-wave h c h b χ cJ χ bJ Charmonia Bottomonia
INCLUSIVE PRODUCTION • 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 5
INCLUSIVE PRODUCTION • Color-singlet model (CSM) : A pair with same spin, color and C P T c ¯ c is created in the hard process, which evolves in to the . J/ ψ J/ ψ The universal rate from to J/ ψ c ¯ c (color-singlet long-distance matrix c ¯ color-singlet c element) is known from models, lattice LDME measurements, and fits to experiments. p p 6
INCLUSIVE PRODUCTION • CSM is incomplete : • A color-octet (CO) pair can evolve into a color c ¯ c 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 ( ), J/ ψ , ψ (2 S ) , Υ ( nS ) CSM severely underestimates the cross section. • For production or decay of P-wave quarkonia ( ), CS channel contains IR divergences that χ cJ , χ bJ can be cancelled only when the CO channels are included. 7
INCLUSIVE PRODUCTION • CSM prediction vs. measurement at Tevatron • LO CSM ( ) is ∼ 1 /p 8 10 T _ μ + μ - ) d (pp BR(J/ J/ +X)/dp T (nb/GeV) inconsistent with both s =1.8 TeV; | | < 0.6 shape and normalization. 1 LO colour-singlet colour-singlet frag. • Radiative corrections -1 10 are larger than LO and has different shape -2 10 ( ), but still not ∼ 1 /p 4 T large enough -3 10 5 10 15 20 p T (GeV) 8
INCLUSIVE PRODUCTION • NRQCD can be used to describe the physics of scales smaller than the quarkonium mass. • NRQCD factorization conjecture for production of Bodwin, Braaten, and Lepage, PRD51, 1125 (1995) Short-distance cross section LDME • Short-distance cross sections are essentially the production cross section of that can be computed Q ¯ Q using perturbative QCD • The LDMEs are nonperturbative quantities that correspond to the rate for the to evolve into Q ¯ Q • LDMEs have known scaling with 9
INCLUSIVE PRODUCTION • NRQCD factorization conjecture for production of Bodwin, Braaten, and Lepage, PRD51, 1125 (1995) Short-distance cross section LDME • Usually truncated at relative order : , , , channels for • The short-distance cross sections have been computed to NLO in by three groups : Kuang-Ta Chao’s group (PKU) : Ma, Wang, Chao, Shao, Wang, Zhang Bernd Kniehl’s group (Hamburg) : Butenschön, Kniehl Jianxiong Wang’s group (IHEP) : Gong, Wan, Wang, Zhang • It is not known how to calculate color-octet LDMEs, and are usually extracted from measurements 10
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 Ma, Wang, Chao, PRL106, 042002 (2011) 11
POLARIZATION 1 (a) 0.8 PUZZLE 0.6 0.4 CDF Data NRQCD 0.2 k -factorization T α 0 : Transverse -0.2 -0.4 -0.6 : Unpolarized -0.8 -1 5 10 15 20 25 30 : Longitudinal p (GeV/c) T CDF, PRL99, 132001 (2007) • NRQCD at LO in predicts Braaten, Kniehl, and Lee, PRD62, 094005 (2000) transverse polarization at large • Disagrees with measurement • NLO corrections are large in the and channels • NRQCD at NLO still predicts transverse polarization CMS, PLB727, 381 (2013) Butenschoen and Kniehl, PRL108, 172002 (2012) 12
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 Collins and Soper, NPB194, 445 (1982) single-parton fragmentation Nayak, Qiu, and Sterman, PRD72, 114012 (2005) d σ [ ij → c ¯ c + X ] dp 2 Parton production Fragmentation T cross sections Functions Z 1 dz d σ Z Z X X X c + X ] + O ( m 2 c /p 6 [ ij → k + X ] D [ k → c ¯ T ) = dp 2 0 T k z : fraction of momentum transferred from parton k to hadron i, j, k run over quarks, antiquarks, and gluon • Corrections to LP fragmentation go as 13
FRAGMENTATION FUNCTIONS • Fragmentation functions (FFs) for production of c ¯ c can be computed using perturbative QCD Collins and Soper, NPB194, 445 (1982) • A gluon can produce a pair in state directly : c ¯ c gluon FF for this channel starts at order , involves a delta function at z = 1 • A gluon can produce a in state by emitting a c ¯ c soft gluon : gluon FF for this channel starts at order , involves distributions singular at z = 1 • A gluon can produce a in state by emitting a c ¯ c gluon : gluon FF for this channel starts at order , does not involve divergence at order 14
FRAGMENTATION FUNCTIONS • For the and channels, gluon polarization is transferred to the pair, and c ¯ c therefore the is mostly transverse. c ¯ c • For the channel, the is unpolarized because c ¯ c it is isotropic. 15
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 • 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) 17
LP+NLO • We combine the LP fragmentation contributions with fixed-order NLO calculations to include corrections of relative order fixed-order LP fragmentation calculation to NLO to NLO accuracy LP fragmentation resummed leading logs corrections of relative order • We take in order to suppress possible non-factorizing contributions 18
LP+NLO • Alternatively, one can consider the LP fragmentation to supplement the fixed-order NLO calculation LP fragmentation LP fragmentation resummed leading logs to NLO accuracy fixed-order calculation to NLO Additional fragmentation contributions 19
LP CONTRIBUTIONS THAT WE COMPUTE channel and channels Fragmentation functions Fragmentation functions Parton production cross sections - NLO NNLO LO NLO NNLO - NNLO NLO NNLO - NNLO Available Not yet available Leading logarithms only • We resum the leading logarithms in to all orders in 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) • Corrections to LP contributions give “normal” K- factors ( ) . 2 20
RESUMMATION OF LEADING LOGARITHMS • The leading logarithms can be resummed to all orders by solving the LO DGLAP equation ✓ D S ◆ ✓ P qq ◆ ✓ D S ◆ = α s ( µ f ) d 2 n f P gq ⊗ d log µ 2 D g P qg P gg D g 2 π f D S = P q ( D q + D ¯ q ) • 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 z = 1 needed for contribution at z ≈ 1 21
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