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Double parton distributions in the pion in the Nambu–Jona-Lasinio model

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Two-parton correlations in the pion, a non perturbative information encoded in double parton distribution functions, are investigated in the Nambu–Jona-Lasinio model. It is found that double parton distribution functions expose novel dynamical information on the structure of the pion, not accessible through one-body parton distributions, as it happens in several estimates for the proton target and in a previous evaluation for the pion, in a light-cone framework. Expressions and predictions are given for double parton distributions corresponding to leading-twist Dirac operators in the quark vertices, and to different regularization methods for the Nambu–Jona-Lasinio model. These results are particularly relevant in view of forthcoming lattice data.


  1. [1]

    P. Bartalini and J.R. Gaunt, Multiple Parton Interactions at the LHC, Adv. Ser. Direct. High Energy Phys.29 (2018) 1 [INSPIRE].

    Article  Google Scholar 

  2. [2]

    ATLAS collaboration, Measurement of hard double-parton interactions in W (→ lν) + 2 jet events at \( \sqrt{s}=7 \)TeV with the ATLAS detector, New J. Phys.15 (2013) 033038 [arXiv:1301.6872] [INSPIRE].

  3. [3]

    N. Paver and D. Treleani, Multi-Quark Scattering and Large pT Jet Production in Hadronic Collisions, Nuovo Cim.A 70 (1982) 215 [INSPIRE].

    ADS  Article  Google Scholar 

  4. [4]

    M. Diehl, D. Ostermeier and A. Schafer, Elements of a theory for multiparton interactions in QCD, JHEP03 (2012) 089 [Erratum ibid.1603 (2016) 001] [arXiv:1111.0910] [INSPIRE].

  5. [5]

    M. Guidal, H. Moutarde and M. Vanderhaeghen, Generalized Parton Distributions in the valence region from Deeply Virtual Compton Scattering, Rept. Prog. Phys.76 (2013) 066202 [arXiv:1303.6600] [INSPIRE].

    ADS  Article  Google Scholar 

  6. [6]

    R. Dupré, M. Guidal and M. Vanderhaeghen, Tomographic image of the proton, Phys. Rev.D 95 (2017) 011501 [arXiv:1606.07821] [INSPIRE].

    ADS  Google Scholar 

  7. [7]

    B. Blok, Yu. Dokshitser, L. Frankfurt and M. Strikman, pQCD physics of multiparton interactions, Eur. Phys. J.C 72 (2012) 1963 [arXiv:1106.5533] [INSPIRE].

    ADS  Article  Google Scholar 

  8. [8]

    B. Blok, Yu. Dokshitzer, L. Frankfurt and M. Strikman, Perturbative QCD correlations in multi-parton collisions, Eur. Phys. J.C 74 (2014) 2926 [arXiv:1306.3763] [INSPIRE].

    ADS  Article  Google Scholar 

  9. [9]

    A. Del Fabbro and D. Treleani, Scale factor in double parton collisions and parton densities in transverse space, Phys. Rev.D 63 (2001) 057901 [hep-ph/0005273] [INSPIRE].

    ADS  Google Scholar 

  10. [10]

    M. Rinaldi and F.A. Ceccopieri, Hadronic structure from double parton scattering, Phys. Rev.D 97 (2018) 071501 [arXiv:1801.04760] [INSPIRE].

    ADS  Google Scholar 

  11. [11]

    M. Rinaldi and F.A. Ceccopieri, Double parton scattering and the proton transverse structure at the LHC, JHEP09 (2019) 097 [arXiv:1812.04286] [INSPIRE].

    ADS  Article  Google Scholar 

  12. [12]

    T. Kasemets and S. Scopetta, Parton correlations in double parton scattering, Adv. Ser. Direct. High Energy Phys.29 (2018) 49 [arXiv:1712.02884] [INSPIRE].

    Article  Google Scholar 

  13. [13]

    H.-M. Chang, A.V. Manohar and W.J. Waalewijn, Double Parton Correlations in the Bag Model, Phys. Rev.D 87 (2013) 034009 [arXiv:1211.3132] [INSPIRE].

    ADS  Google Scholar 

  14. [14]

    M. Rinaldi, S. Scopetta and V. Vento, Double parton correlations in constituent quark models, Phys. Rev.D 87 (2013) 114021 [arXiv:1302.6462] [INSPIRE].

    ADS  Google Scholar 

  15. [15]

    M. Rinaldi, S. Scopetta, M. Traini and V. Vento, Double parton correlations and constituent quark models: a Light Front approach to the valence sector, JHEP12 (2014) 028 [arXiv:1409.1500] [INSPIRE].

    ADS  Article  Google Scholar 

  16. [16]

    T. Kasemets and A. Mukherjee, quark-gluon double parton distributions in the light-front dressed quark model, Phys. Rev.D 94 (2016) 074029 [arXiv:1606.05686] [INSPIRE].

    ADS  Google Scholar 

  17. [17]

    M. Rinaldi, S. Scopetta, M. Traini and V. Vento, Double parton scattering: a study of the effective cross section within a Light-Front quark model, Phys. Lett.B 752 (2016) 40 [arXiv:1506.05742] [INSPIRE].

    ADS  Article  Google Scholar 

  18. [18]

    M. Rinaldi, S. Scopetta, M.C. Traini and V. Vento, Correlations in Double Parton Distributions: Perturbative and Non-Perturbative effects, JHEP10 (2016) 063 [arXiv:1608.02521] [INSPIRE].

    ADS  Article  Google Scholar 

  19. [19]

    M. Traini, M. Rinaldi, S. Scopetta and V. Vento, The effective cross section for double parton scattering within a holographic AdS/QCD approach, Phys. Lett.B 768 (2017) 270 [arXiv:1609.07242] [INSPIRE].

    ADS  Article  Google Scholar 

  20. [20]

    R. Kirschner, Generalized Lipatov-Altarelli-Parisi Equations and Jet Calculus Rules, Phys. Lett.B 84 (1979) 266 [INSPIRE].

    ADS  Article  Google Scholar 

  21. [21]

    V.P. Shelest, A.M. Snigirev and G.M. Zinovev, The Multiparton Distribution Equations in QCD, Phys. Lett.B 113 (1982) 325 [INSPIRE].

    ADS  Article  Google Scholar 

  22. [22]

    M. Diehl and J.R. Gaunt, Double parton scattering theory overview, Adv. Ser. Direct. High Energy Phys.29 (2018) 7 [arXiv:1710.04408] [INSPIRE].

    Article  Google Scholar 

  23. [23]

    RQCD collaboration, Double Parton Distributions of the Pion, PoS(LATTICE2016)152 (2016) [arXiv:1701.05479] [INSPIRE].

  24. [24]

    G.S. Bali et al., Two-current correlations in the pion on the lattice, JHEP12 (2018) 061 [arXiv:1807.03073] [INSPIRE].

    ADS  Article  Google Scholar 

  25. [25]

    M. Rinaldi, S. Scopetta, M. Traini and V. Vento, A model calculation of double parton distribution functions of the pion, Eur. Phys. J.C 78 (2018) 781 [arXiv:1806.10112] [INSPIRE].

    ADS  Article  Google Scholar 

  26. [26]

    S.P. Klevansky, The Nambu–Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys.64 (1992) 649 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  27. [27]

    R.M. Davidson and E. Ruiz Arriola, Parton distributions functions of pion, kaon and eta pseudoscalar mesons in the NJLS model, Acta Phys. Polon.B 33 (2002) 1791 [hep-ph/0110291] [INSPIRE].

    ADS  Google Scholar 

  28. [28]

    L. Theussl, S. Noguera and V. Vento, Generalized parton distributions of the pion in a Bethe-Salpeter approach, Eur. Phys. J.A 20 (2004) 483 [nucl-th/0211036] [INSPIRE].

    ADS  Google Scholar 

  29. [29]

    E. Ruiz Arriola and W. Broniowski, Pion light cone wave function and pion distribution amplitude in the Nambu–Jona-Lasinio model, Phys. Rev.D 66 (2002) 094016 [hep-ph/0207266] [INSPIRE].

    ADS  Google Scholar 

  30. [30]

    A. Courtoy and S. Noguera, Enhancement effects in exclusive pi pi and rho pi production in gamma* gamma scattering, Phys. Lett.B 675 (2009) 38 [arXiv:0811.0550] [INSPIRE].

    ADS  Article  Google Scholar 

  31. [31]

    A. Courtoy and S. Noguera, The Pion-photon transition distribution amplitudes in the Nambu–Jona Lasinio model, Phys. Rev.D 76 (2007) 094026 [arXiv:0707.3366] [INSPIRE].

    ADS  Google Scholar 

  32. [32]

    A. Courtoy, Generalized Parton Distributions of Pions. Spin Structure of Hadrons, Ph.D. Thesis, Valencia University, Valencia Spain (2010) [arXiv:1010.2974] [INSPIRE].

  33. [33]

    S. Noguera and S. Scopetta, The eta-photon transition form factor, Phys. Rev.D 85 (2012) 054004 [arXiv:1110.6402] [INSPIRE].

    ADS  Google Scholar 

  34. [34]

    H. Weigel, E. Ruiz Arriola and L.P. Gamberg, Hadron structure functions in a chiral quark model: Regularization, scaling and sum rules, Nucl. Phys.B 560 (1999) 383 [hep-ph/9905329] [INSPIRE].

    ADS  Article  Google Scholar 

  35. [35]

    S. Noguera and S. Scopetta, Pion transverse momentum dependent parton distributions in the Nambu and Jona-Lasinio model, JHEP11 (2015) 102 [arXiv:1508.01061] [INSPIRE].

    ADS  Article  Google Scholar 

  36. [36]

    W. Broniowski and E. Ruiz Arriola, Partonic quasidistributions of the proton and pion from transverse-momentum distributions, Phys. Rev.D 97 (2018) 034031 [arXiv:1711.03377] [INSPIRE].

    ADS  Google Scholar 

  37. [37]

    F.A. Ceccopieri, A. Courtoy, S. Noguera and S. Scopetta, Pion nucleus Drell-Yan process and parton transverse momentum in the pion, Eur. Phys. J.C 78 (2018) 644 [arXiv:1801.07682] [INSPIRE].

    ADS  Article  Google Scholar 

  38. [38]

    J.R. Gaunt and W.J. Stirling, Double Parton Distributions Incorporating Perturbative QCD Evolution and Momentum and Quark Number Sum Rules, JHEP03 (2010) 005 [arXiv:0910.4347] [INSPIRE].

    ADS  Article  Google Scholar 

  39. [39]

    A.V. Radyushkin, Nonforward parton distributions, Phys. Rev.D 56 (1997) 5524 [hep-ph/9704207] [INSPIRE].

    ADS  Google Scholar 

  40. [40]

    X.-D. Ji, Off forward parton distributions, J. Phys.G 24 (1998) 1181 [hep-ph/9807358] [INSPIRE].

    ADS  Article  Google Scholar 

  41. [41]

    M. Diehl and T. Kasemets, Positivity bounds on double parton distributions, JHEP05 (2013) 150 [arXiv:1303.0842] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  42. [42]

    A. Courtoy, S. Noguera and S. Scopetta, Two current correlations in the pion in the Nambu–Jona-Lasinio model, in progress.

  43. [43]

    C. Zimmermann, private communication.

  44. [44]

    M. Burkardt, Light front quantization, Adv. Nucl. Phys.23 (1996) 1 [hep-ph/9505259] [INSPIRE].

    Google Scholar 

  45. [45]

    M. Burkardt, Much ado about nothing: Vacuum and renormalization on the light front, in QCD, light cone physics and hadron phenomenology. Proceedings of 10th Nuclear Summer School and Symposium, NuSS’97, Seoul Korea (1997), pg. 170 [hep-ph/9709421] [INSPIRE].

  46. [46]

    K. Itakura and S. Maedan, Dynamical chiral symmetry breaking on the light front. 2. The Nambu–Jona-Lasinio model, Phys. Rev.D 62 (2000) 105016 [hep-ph/0004081] [INSPIRE].

    ADS  Google Scholar 

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Correspondence to Sergio Scopetta.

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ArXiv ePrint: 1909.09530

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Courtoy, A., Noguera, S. & Scopetta, S. Double parton distributions in the pion in the Nambu–Jona-Lasinio model. J. High Energ. Phys. 2019, 45 (2019).

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