Relativistic corrections to e + e cJ + in NRQCD Vladyslav - PowerPoint PPT Presentation

Relativistic corrections to e + e − → χ cJ + γ in NRQCD Vladyslav Shtabovenko 1 in collaboration with N. Brambilla 1 , W. Chen 2 , Y. Jia 2 and A. Vairo 1 1 Technische Universität München, Germany 2 Institute of High Energy Physics Beijing, China C OLD Q UANTUM C OFFEE S EMINAR 16 TH OF M AY , 2017, H EIDELBERG Physik-Department T30f V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017 Electromagnetic χ cJ production 1 / 40

Outline Theoretical framework and motivation 1 Heavy quarkonia Nonrelativistic QCD Relativistic corrections to the exclusive production e + e − → χ cJ + γ 2 Overview of the existing results Contributions from the higher Fock state | Q ¯ Qg � NRQCD-factorized production cross-sections Perturbative matching between QCD and NRQCD Final results Automatic nonrelativistic calculation with F EYN C ALC 3 Nature of the problem FeynCalc FeynCalc 9.3 FeynOnium Summary and Outlook 4 V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017 Electromagnetic χ cJ production 2 / 40

Theoretical framework and motivation Heavy quarkonia c ◮ Bound states of a heavy quark and a heavy b antiquark of the same flavor ◮ Heavy quarks: charm, bottom and top c b ◮ The top quark decays too fast to form a bound state ◮ Q ¯ c ) and bottomonia ( b ¯ Q -bound states: charmonia ( c ¯ b ) ◮ Heavy quarkonia are an ideal laboratory to test our understanding of QCD ◮ Nonrelativistic system ◮ Rich phenomenology ◮ Creation/annihilation of Q ¯ Q -pairs in short-distance processes ◮ Formation of Q ¯ Q -bound states in long-distance (nonperturbative) processes V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017 Electromagnetic χ cJ production 3 / 40

Theoretical framework and motivation Heavy quarkonia ◮ Production of heavy quarkonia is governed by an interplay of perturbative and non-perturbative effects: ◮ The formation of a heavy quark pair from a collision process can be evaluated in perturbation theory. ◮ However, we should expand not only in α s but also in the relative heavy quark velocity v ! ◮ The evolution of a heavy quark pair to a physical quarkonium requires non-perturbative input. ◮ We need a way to disentangle those effects from each other ⇒ Factorization. ◮ Even the perturbative part alone is not simple due to the presence of different entangled scales ⇒ Multiscale problem in a non-relativistic system with a hierarchy of scales. ◮ The nonperturbative part requires a clear field theoretical definition (e. g. for the evaluation on the lattice) V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017 Electromagnetic χ cJ production 4 / 40

Theoretical framework and motivation Heavy quarkonia ◮ First heavy quarkonia ( J /ψ and ψ ( 2 s ) ) were discovered over 40 years ago [Aubert et al., 1974] , [Augustin et al., 1974] ◮ Crucial for the establishment of QCD as the correct theory of strong interactions (November revolution of 1974, evidence for the existence of the charm quark, . . . ) ◮ Early attempts to develop a theoretical description of heavy quarkonia: phenomenological models ◮ Spectra from potential models ◮ Color singlet model [Einhorn & Ellis, 1975, Ellis et al., 1976, Carlson & Suaya, 1976] ◮ Color evaporation model [Fritzsch, 1977, Halzen, 1977, Halzen & Matsuda, 1978, Gluck et al., 1978] ◮ Common short-comings of model-based approaches ◮ Require tuning to data ◮ Relation to the full QCD is unclear ◮ Predictions fail when the experimental precision increases ◮ No way to calculate higher order corrections systematically ◮ Effective Field Theory (EFT) methods [ Weinberg, 1979 , Wilson, 1974 ] : the modern way to treat Q ¯ Q -bound states V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017 Electromagnetic χ cJ production 5 / 40

Theoretical framework and motivation Heavy quarkonia ◮ The general idea is to construct a QFT that approximates the given high energy theory at energies E ≪ Λ ◮ using the most appropriate degrees of freedom and ◮ providing the simplest description of the relevant physics ◮ The EFT approach is, in general, applicable to systems with several well-sepearated dynamical scales Λ ≫ Λ 1 ≫ Λ 2 ≫ . . . ◮ The main steps to construct an EFT [Pich, 1998] ◮ Identify the relevant scales, symmetries and degrees of freedom ◮ The most general EFT Lagrangian: expansion in the small ratios of the relevant scales, contains all operators O i compatible with the symmetries. c i � L EFT = Λ d i − 4 O i i ◮ Introduce power-counting rules ⇒ Systematics, predictive power ◮ If possible, determine matching coefficients c i from comparing suitable quantities in the high energy theory and in the EFT ⇒ Matching V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017 Electromagnetic χ cJ production 6 / 40

Theoretical framework and motivation Nonrelativistic QCD ◮ Relevant dynamical scales of a heavy quarkonium are ≫ mv 2 ≫ mv m ���� ���� ���� hard soft ultrasoft with v 2 c ∼ 0 . 3 , v 2 b ∼ 0 . 1 ◮ ⇒ relativistic corrections are very important for charmonia! ◮ The formation of a Q ¯ Q -pair occurs within a distance 1 / m (short distance process) ◮ The formation of a heavy quarkonium happens over distances of order 1 / ( mv ) or larger in the quarkonium rest frame (long distance process). ◮ A suitable EFT for studying quarkonium production is Non-Relativistic QCD (NRQCD) [Caswell & Lepage, 1986, Bodwin et al., 1995] ◮ Starting from the full QCD, all scales above mv are integrated out. ◮ We can always do this perturbatively, since m ≫ Λ QCD ◮ The effects of the high-energy contributions are encoded in the matching coefficients c n ( α s ( m ) , µ ) multiplying NRQCD operators � O n ( µ ) � . c n ( α s ( m ) ,µ ) ◮ L NRQCD = � � O n ( µ ) � is an expansion in α s and v . m dn − 4 n ◮ ∞ -number of operators with increasing mass dimension. ◮ Contributions to a process at the given accuracy estimated by velocity scaling rules. V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017 Electromagnetic χ cJ production 7 / 40

Theoretical framework and motivation Nonrelativistic QCD ◮ NRQCD is not a model, it precisely reproduces the full QCD at energies E ≪ m order by order in 1 / m . ◮ Non-perturbative contributions go inside long distance matrix elements (LDME) � O n ( µ ) � . ◮ LDMEs must be extracted from experiment or calculated on the lattice, but they do not depend on the short-distance process (universality). ◮ Predictive power of NRQCD: Extract LDMEs from one measurement and use them for predictions in a different measurement. ◮ Matching condition on the level of cross-sections for production F n ( α s ( m ) , µ ) � � � 0 |O Q ¯ ! σ ( Q ¯ Q Q ) = n ( µ ) | 0 � | pert. NRQCD . � pert. QCD m d n − 4 n ◮ Once the matching coefficients F n ( α s ( m ) , µ ) are determined in perturbative matching, we can write down the NRQCD factorized production cross-section F n ( α s ( m ) , µ ) � � 0 |O H σ ( H ) = n ( µ ) | 0 � . m d n − 4 n V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017 Electromagnetic χ cJ production 8 / 40

Theoretical framework and motivation Nonrelativistic QCD ◮ NRQCD tells us that a Q ¯ Q -pair evolving into quarkonium does not necessarily has to be in the color singlet configuration. ◮ Fock-state expansion of a heavy quarkonium | H � ∼ a 0 | Q ¯ Q � + a 1 | Q ¯ Qg � + a 2 | Q ¯ Qgg � + . . . ◮ Higher order Fock states with Q ¯ Q -pairs in the color octet (CO) configuration are suppressed by power of v . ◮ Nevertheless, they must be taken into account when higher order relativistic or radiative corrections are computed! ◮ The presence of the CO mechanism is an important feature that distinguishes NRQCD from other approaches. ◮ Studying the importance of the CO contributions for phenomenology is an important test for the validity of NRQCD. V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017 Electromagnetic χ cJ production 9 / 40

Relativistic corrections to the exclusive production e + e − → χ cJ + γ Overview of the existing results ◮ Electromagnetic spin triplet P -wave quarkonium production in e + e − -annihilation: virtual photon decays into a hard ( k p ∼ m ) on-shell photon and χ cJ : γ e − e + χ cJ ◮ No experimental data available, good perspectives for this measurement will exist at Belle II in Japan ◮ An early study [Chung et al., 2008] based on O ( α 0 s v 0 ) results predicted cross-sections that might be measurable at B-factories (here for √ s = 10 . 6 GeV ): ◮ σ ( e + e − → χ c 0 + γ ) = 1 . 3 fb ◮ σ ( e + e − → χ c 1 + γ ) = 13 . 7 fb ◮ σ ( e + e − → χ c 2 + γ ) = 5 . 3 fb ◮ Subsequently, corrections of order O ( α s v 0 ) ( [Sang & Chen, 2010b] , [Li et al., 2009] ), O ( α 0 s v 2 ) ( [Li et al., 2013, Chao et al., 2013] ) and finally O ( α s v 2 ) ( [Xu et al., 2014] ) were obtained as well. ◮ C.f. also treatment in the light cone formalism [Braguta, 2010] , [Wang & Yang, 2014] V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017 Electromagnetic χ cJ production 10 / 40