Quarkonia production at the LHC in NRQCD with k T -factorization Sergey Baranov P.N.Lebedev Institute of Physics, Moscow Artem Lipatov D.V.Skobeltsyn Institute of Nuclear Physics, MSU, Moscow P L A N O F T H E T A L K 0. Introduction: experimental observables 1. First acquaintance with k t -factorization 2. Implementing the Quarkonium physics 3. Numerical results 4. Conclusions
Sergey Baranov, 12th Quarkonium Group Workshop, Beijing, Nov.2017 EXPERIMENTAL OBSERVABLES – Differential cross sections for charm and bottom families – Differential cross sections for ground states and excited states – Direct to indirect production ratios (feed-down from χ c and χ b ) – P -wave production ratios σ ( χ c 1 ) /σ ( χ c 2 ) , σ ( χ b 1 ) /σ ( χ b 2 ) – Polarization THEORETICAL APPROACHES Several approaches are competing: Color-Singlet versus Color-Octet model; both may be extended to NLO or tree-level NNLO*; both may be incorporated with collinear or k T -factorization. This talk is devoted to k T -factorization. – Deep theory: a method to calculate high-order contributions (ladder-type diagrams enhanced with “large logarithms”). – Practice: making use of so called k T -dependent parton densities. (Unusual properties: nonzero k T and longitudinal polarization.) 2
Sergey Baranov, 12th Quarkonium Group Workshop, Beijing, Nov.2017 First acquaintance with k T -factorization parton off-shellness and nonzero k T QED QCD Weizs¨ acker-Williams approximation Conventional Parton Model (collinear on-shell photons) (collinear gluon density) 2 π [1 + (1 − x 2 )] log α s x G ( x, µ 2 ) F γ ( x ) = 4 m 2 Equivalent Photon approximation Unintegrated gluon density F γ ( x, Q 2 ) = α Q 2 [1 + (1 − x 2 )] 1 F ( x, k 2 t , µ 2 ) 2 π � F ( x, k 2 Q 2 ≈ k 2 t , µ 2 ) dk 2 t = x G ( x, µ 2 ) t / (1 − x ) Photon spin density matrix Gluon spin density matrix L µν ≈ p µ p ν ǫ µ ǫ ν ∗ = k µ t / | k T | 2 t k ν use k = xp + k t , then do gauge shift so called nonsense polarization ǫ → ǫ − k/x with longitudinal components Looks like Equivalent Photon Approximation extended to strong interactions. 3
Sergey Baranov, 12th Quarkonium Group Workshop, Beijing, Nov.2017 The underlying theory: Initial State Radiation cascade Every elementary emission gives α s · 1 /x · 1 /q 2 x = longitudinal momentum fraction q 2 = gluon virtuality Integration over the phase space yields α s · ln x · ln q 2 so called large logarithms, the reason to focus on this type of diagrams Random walk in the k T -plane: ... � k T i − 1 � ≪ � k T i � ≪ � k T i +1 � ... ... � x i − 1 � > � x i � > � x i +1 � ... Technical method of summation: integro-differential QCD equations BFKL E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45, 199 (1977); Ya. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978); or CCFM S.Catani, F.Fiorani, G.Marchesini, Phys.Lett.B 234, 339 (1990); Nucl.Phys. B336, 18 (1990); G.Marchesini, Nucl.Phys. B445, 49 (1995); M.Ciafaloni, Nucl.Phys. B296, 49 (1998); CCFM is more convenient for programming because of strict angular ordering ...θ i − 1 < θ i − 1 < θ i +1 ... ⇒ A step-by-step solution. 4
Sergey Baranov, 12th Quarkonium Group Workshop, Beijing, Nov.2017 k t -factorization is: s [ln(1 /x )] n [ln( Q 2 )] n up A method to collect contributions of the type α n to infinetly high order. Sometimes it may be better than conventional calculations to a fixed order. (None of the methods is compete.) The evolution cascade is part of the hard interaction process; it affects both the kinematics (initial k T ) and the polarization (off-shell spin density matrix). The corrections always have the same (ladder) structure, irrespective of the ‘central’ part of hard interaction, and can be conveniently absorbed into redefined parton densities F ( x, k 2 t , µ 2 ) ⇒ k T -dependent = “unintegrated” distribution functions Advantages: With the LO matrix elements for ‘central’ subprocess we get access to effects requiring complicated next-to-leading order calculations in the collinear scheme. Many important results have been obtained in the k t -factorization approach much earlier than in the collinear case. Includes effects of soft resummation and makes predictions applica- ble even to small p t region. 5
Sergey Baranov, 12th Quarkonium Group Workshop, Beijing, Nov.2017 Implementing the Quarkonium production Color-Singlet mechanism Perturbative production of a heavy quark pair within QCD; g = k µ ǫ µ standard rules except gluon polarization vectors: T / | k T | E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45, 199 (1977); Ya. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978); L.V. Gribov, E.M. Levin, M. G. Ryskin, Phys. Rep. 100, 1 (1983). Spin projection operators to guarantee the proper quantum numbers: P ( 3 S 1 ) = � ǫ V ( � p Q + m Q ) / (2 m Q ) for Spin-triplet states P ( 1 S 0 ) = γ 5 ( � p Q + m Q ) / (2 m Q ) for Spin-singlet states Probability to form a bound state is determined by the wave function: for S -wave states | R S (0) | 2 is known from leptonic decay widths; P (0) | 2 is taken from potential models. for P -wave states | R ′ E. J. Eichten, C. Quigg, Phys. Rev. D 52, 1726 (1995) If L � = 0 and S � = 0 we use the Clebsch-Gordan coefficients to reexpress the | L, S � states in terms of | J, J z � states, namely, the χ 0 , χ 1 , χ 2 mesons. 6
Sergey Baranov, 12th Quarkonium Group Workshop, Beijing, Nov.2017 What was wrong with Color-Singlet Model? – Wrong p t dependence of the cross sections ⇒ an indication that something important is missing. Partly corrected by k T -factorization. – Quarkonium formation is assumed to be completed already at the perturbative stage; but what if the Q ¯ Q pair is produced in the color octet state? Treating soft gluons in a perturbative manner is wrong: soft gluons cannot resolve the Q ¯ Q pair into quarks. Need another language to describe the emission of gluons by the entire Q ¯ Q system, not by individual quarks. Come to NRQCD. What was wrong with Color-Octet Model (NRQCD)? – Assumes that soft gluons can change the color and other quantum numbers of a Q ¯ Q system without changing the energy-momentum. An obvious conflict with confinement that prohibits radiation of in- finitely soft colored quanta. Need to consider not infinitely small energy-momentum exchange. Not only a kinematric correction! – Long-distance matrix elements (LDMEs) for 3 S [8] 1 → J/ψ transitions are treated as spin-blind numbers. Need to replace them with ampli- tudes showing well defined spin structure. 7
Sergey Baranov, 12th Quarkonium Group Workshop, Beijing, Nov.2017 Modified Color-Octet mechanism – Step 1: use perturbative QCD to create a heavy quark pair Q ¯ Q in the hard gluon-gluon fusion subprocess. – Step 2: use multipole expansion for soft gluon radiation. Another perturbation theory where the small parameter is the relative quark velocity (or the size of the Q ¯ Q system over the gluon wavelength) Both steps are combined into a single amplitude: Q ¯ Q spin density matrix is contracted with E1 transition amplitudes (same as for real χ c decays). Color-Electric Dipole transitions A ( χ c 0 ( p ) → J/ψ ( p − k ) + γ ( k )) ∝ k µ p µ ε ν ( J/ψ ) ε ( γ ) ν A ( χ c 1 ( p ) → J/ψ ( p − k ) + γ ( k )) ∝ ǫ µναβ k µ ε ( χ c 1 ) ε ( γ ) ε ( J/ψ ) β ν α A ( χ c 2 ( p ) → J/ψ ( p − k ) + γ ( k )) ∝ p µ ε αβ [ k µ ε ( γ ) ( χ c 2 ) ε ( J/ψ ) − k β ε ( γ ) µ ] β α A.V.Batunin, S.R.Slabospitsky, Phys.Lett.B 188, 269 (1987) P.Cho, M.Wise, S.Trivedi, Phys. Rev. D 51, R2039 (1995) One or two subsequent transitions to convert a color octet into J/ψ : 3 P [8] 3 S [8] 3 P [8] 3 P [8] → J/ψ + g or → J + g, → J/ψ + g, J = 0 , 1 , 2 . J 1 J 8
Sergey Baranov, 12th Quarkonium Group Workshop, Beijing, Nov.2017 J/ψ from 3 P [8] polarization J k T -factorization LO collinear factorization Dashed = 3 P 2 ; dash-doted = 3 P 1 ; doted = 3 P 0 Approximate cancellation between 3 P [8] and 3 P [8] channels. 1 2 9
Sergey Baranov, 12th Quarkonium Group Workshop, Beijing, Nov.2017 J/ψ from 3 S [8] polarization 1 Pure 3 P [8] Interfering channels states J 2 2 + 3 P [8] 2 + 3 P [8] � 3 P [8] � � Dash = 3 P 2 ; dash-dot = 3 P 1 ; dot = 3 P 0 ; Solid= � � 2 √ � Solid = 3 S [8] spin preserved. normalized to 2 J + 1 1 10
Sergey Baranov, 12th Quarkonium Group Workshop, Beijing, Nov.2017 Model versus CMS and LHCb data Helicity frame ψ (2 S ) Collins-Soper frame ψ (2 S ) 1 1 λ θ λ θ |y| < 0.6 |y| < 0.6 0.6 < |y| < 1.2 0.6 < |y| < 1.2 1.2 < |y| < 1.5 1.2 < |y| < 1.5 0.5 0.5 0 0 -0.5 -0.5 -1 -1 15 20 25 30 35 40 45 50 15 20 25 30 35 40 45 50 p T [GeV] p T [GeV] 1 1 λ θ λ θ 2 < y < 2.5 2.5 < y < 3 3 < y < 3.5 3.5 < y < 4 0.5 0.5 4 < y < 4.5 0 0 2 < y < 2.5 2.5 < y < 3 -0.5 -0.5 3 < y < 3.5 3.5 < y < 4 4 < y < 4.5 -1 -1 4 6 8 10 12 14 4 6 8 10 12 14 p T [GeV] p T [GeV] 11
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