QCD Reummation for Heavy Quarkonium Production in High Energy - PowerPoint PPT Presentation

QCD Reummation for Heavy Quarkonium Production in High Energy Collisions Zhongbo Kang Iowa State University PHENO 2008 SYMPOSIUM Madison, Wisconsin, Apr 28-30, 2008 based on work with J. -W. Qiu 1 Apr 29 , 2008 Zhongbo Kang, ISU

Success of NRQCD  NRQCD approach for quarkonium production � � ij → ( Q ¯ � O H � � Braaten, Bodwin, Lepage 1995 σ ( pp → H + X ) = dx 1 dx 2 φ i/p ( x 1 ) φ j/p ( x 2 )ˆ Q ) n n � σ _ i,j,n ij → ( Q ¯ : production of QQ state with quantum number n, calculable in pQCD � � ˆ Q ) n σ as a expansion of α s � O H n � : can be expanded in powers of v 2  Comparison with Tevatron data based on LO formula 10 1 _ " J/ ! +X)/dp T (nb/GeV) _ "! (2S)+X)/dp T (nb/GeV) BR(J/ !" µ + µ - ) d # (pp BR( ! (2S) " µ + µ - ) d # (pp $ s =1.8 TeV; | % | < 0.6 $ s =1.8 TeV; | % | < 0.6 -1 1 10 total total colour-octet 1 S 0 + 3 P J colour-octet 1 S 0 + 3 P J colour-octet 3 S 1 colour-octet 3 S 1 LO colour-singlet LO colour-singlet colour-singlet frag. colour-singlet frag. -1 -2 10 10 -2 -3 10 10 -3 -4 10 10 5 10 15 20 5 10 15 20 p T (GeV) p T (GeV) Apr 29 , 2008 Zhongbo Kang, ISU 2

NLO contributions Color-singlet contribution for J/ ψ and Upsilon production at Tevatron NNLO P. Artoisenet, F. Maltoni, et.al. 2007 NLO LO associate LO direct Large uncertainty band ⇒ strong scale dependence Large NLO, NNLO contribution ⇒ how perturbative series converge? Apr 29 , 2008 Zhongbo Kang, ISU 3

Scale dependence of the cross section  Scale dependence of the ttbar cross section at NLO With NLO correction included, scale- dependence is strongly reduced Linear scale Bonciani, Catani, Mangano, Nason, NPB529 (1998) 424  Scale dependence is still large for J/ ψ at NLO: large NLO corrections e + e − → J/ ψ + c ¯ c σ ( fb ) m = 1 . 4 ∼ 1 . 5GeV Λ = 0 . 338GeV √ s = 10 . 6GeV | R S (0) | 2 = 1 . 01GeV 3 Next-to-leading order 600 Leading order 400 Log scale 200 5.5 µ ( GeV ) 1.5 2.5 3.5 4.5 Zhang and Chao PRL98, 092003(2007) Campbell, Maltoni, Tramontano, PRL98(2007) 252002 Apr 29 , 2008 Zhongbo Kang, ISU 4

Why NLO contribution is LARGE?  LO • Scale dependence from φ ( x, µ ) 2 α 3 s ( µ ) (2 m ) 4 1 • P T dependence α 3 P s P 8 P 8 T T T  NLO: new channel NLO: high power α s (µ) , low power in P T (2 m ) 2 α 4 s P 6 T P T  NLO to existing LO channels 1 α 4 s P 4 P T T Apr 29 , 2008 Zhongbo Kang, ISU 5

Large logarithmic contributions  NNLO � P 2 � �� 1 α 4 T α s ln · s P 4 m 2 T P T  To have a stable perturbative expansion, one need resum all the large logarithms: resummation  Same large log contribution for color-octet channels Apr 29 , 2008 Zhongbo Kang, ISU 6

New factorized formula with QCD resummation  Fragmentation contributions E. Braaten, et.al., 1993 � σ F ( pp → H + X ) = � dx 1 dx 2 dz φ i/p ( x 1 ) φ j/p ( x 2 )ˆ σ [ ij → k ] D k → H ( z ) i,j,k D k → H (z) resums all the logarithms. This is the dominant contribution when P T 2 >>m 2 ❖ Q: What is the relation between fragmentation contribution and fixed order results in NRQCD? P 2 T ∼ m 2 : σ ≈ σ P ert calculated by fixed order NRQCD. Logarithms are not important P 2 T ≫ m 2 : σ ≈ σ F Logarithms dominate / resummed ❖ How to transform smoothly between these two regimes? ❖ How to avoid double counting beyond LO?  We propose a new factorized formula: σ = σ Dir + σ F resum all the fragmentation logs No logs σ Dir = σ P ert − σ Asym separation between Direct and Fragmentation contribution depends on the definition of fragmentation function D(z, µ 2 ) Apr 29 , 2008 Zhongbo Kang, ISU 7

Fragmentation function D q → J/ ψ (z f ,µ 2 )  Operator definition for D q → J/ ψ (z f ,µ 2 ) P z 2 d 4 k 4 k + δ ( z f − P + � f γ + T ( k, P ) D k → H ( z f , µ 2 ) = � � = k + )Tr T(k, P) (2 π ) 4 k 2 ≤ µ 2 k  Calculation of leading order fragmentation function: D (0)q → J/ ψ (z f ,µ 2 ) � ( z f − 1) 2 + 1 α 2 � z f µ 2 1 − 4 m 2 � � �� D (0) s 36 m 3 � O 8 ( 3 S 1 ) � · q → J/ ψ ( z f , µ ) = ln − z f 4 m 2 z f µ 2 z f  Evolution equation of D q →ψ (z f ,µ 2 ) : inhomogeneous term � 1 � z f � µ 2 d dµ 2 D q → J/ ψ ( z f , µ ) = γ q → J/ ψ ( z f , µ ) + α s d ξ D q → J/ ψ ( ξ , µ ) + · · · ξ P q → q 2 π ξ z f � ( z f − 1) 2 + 1 α 2 − 4 m 2 µ 2 − 4 m 2 � � � s 36 m 3 � O 8 ( 3 S 1 ) � γ q → J/ ψ ( z f , µ ) = θ µ 2 z f z f Apr 29 , 2008 Zhongbo Kang, ISU 8

Case study: e + e - → J/ ψ +qq  NRQCD perturbative results 2 z = 2 E J/ ψ ξ = 4 m 2 √ s � � E 2 s J/ ψ ln + � z 2 − 4 ξ z L = zm 2 �� ( z − 1) 2 + 1 d σ P ert α 2 � O 8 ( 3 S 1 ) � = σ 0 · 2 + 2 ξ 2 − z + ξ 2 2 � ln z + z L � s √ s − 2 z L 18 m 3 z − z L dE J/ ψ z z z  How to identify the logarithms before the full calculations σ Asym k//P 2 P ˆ P k p k 3 ⊗ ≈ k d σ Asym z f = P + q → J/ ψ ( z f , µ 2 , 4 m 2 ) dz f k + = 1 ≈ σ 0 · D (0) 2 [ z + z L ] dE J/ ψ dE J/ ψ Apr 29 , 2008 Zhongbo Kang, ISU 9

Smooth transition  Direct contribution σ Dir = σ P ert − σ Asym = σ P ert − 2 σ 0 · D (0) q → J/ ψ ( z, µ 2 ) -3 x 10 d σ Dir α 2 � O 8 ( 3 S 1 ) � σ 0 · 2 s = √ s 0.25 18 m 3 dE J/ ψ ! s/2 # 0 d # /dE " �� ( z − 1) 2 + 1 + 2 ξ 2 − z + ξ 2 2 � ln z + z L σ ! s=91GeV × − 2 z L 0.2 σ Dir z − z L z z z � ( z f − 1) 2 + 1 � z f µ 2 1 − 4 m 2 � � ��� − z f 0.15 σ F ln − z f 4 m 2 z f µ 2 z L z f 0.1 µ = 2 E J/ ψ  Full cross section 0.05 σ = σ Dir + σ F 0 5 10 15 20 25 30 35 40 -3 E " x 10 with evolved fragmentation function 0.25 ! s/2 # 0 d # /dE " ⇒ log resummed ! s=91GeV 0.2 ∼ d σ Dir σ d σ ❖ when E J/ ψ ~m σ P ert dE J/ ψ dE J/ ψ 0.15 σ F d σ F d σ ❖ when E J/ ψ >>m ∼ 0.1 dE J/ ψ dE J/ ψ  Compare to lowest order NRQCD calculation 0.05 0 Apr 29 , 2008 Zhongbo Kang, ISU 5 10 15 20 25 30 35 40 10 E "

Hadronic collisions - in progress σ = σ Dir + σ F D (0) q → H ( z, µ 2 )  Direct contribution: σ Dir = σ P ert − σ Asym 2 2 2 2 + ⊗ + · · · P − P T T LO NLO  Fragmentation contribution: 2 2 σ F = ⊗ D g → H + ⊗ D Q → H + ... Stay tuned Apr 29 , 2008 Zhongbo Kang, ISU 11

Summary  We proposed a QCD resummed factorization formula for heavy quarkonium production  We reorganized the perturbative series of NRQCD calculation  New formula is reliable for a wide range of collision energy Apr 29 , 2008 Zhongbo Kang, ISU 12