International Journal of Theoretical Physics

, Volume 57, Issue 2, pp 495–505 | Cite as

Decoupling of the Leading Order DGLAP Evolution Equation with Spin Dependent Structure Functions



We propose an analytical solution for DGLAP evolution equations with polarized splitting functions at the Leading Order (LO) approximation based on the Laplace transform method. It is shown that the DGLAP evolution equations can be decoupled completely into two second order differential equations which then are solved analytically by using the initial conditions \(\delta F^{\mathrm {S}}(x,Q^{2})=\mathcal {F}[\partial \delta F^{\mathrm {S}}_{0}(x), \delta F^{\mathrm {S}}_{0}(x)]\) and \({\delta G}(x,Q^{2})=\mathcal {G}[\partial \delta G_{0}(x), \delta G_{0}(x)]\). We used this method to obtain the polarized structure function of the proton as well as the polarized gluon distribution function inside the proton and compared the numerical results with experimental data of COMPASS, HERMES, and AAC’08 Collaborations. It was found that there is a good agreement between our predictions and the experiments.


DGLAP equations Structure function Laplace transform method 


  1. 1.
    Airapetian, A., [HERMES Collaboration], et al.: Precise determination of the spin structure function g 1 of the proton, deuteron, and neutron. Phys. Rev. D 75, 012007 (2007)ADSCrossRefGoogle Scholar
  2. 2.
    Khorraminan, A.N., Atashbar Tehrani, S., Olness, F., Taheri Monfared, S., Arbabifar, F.: Non-singlet spin-dependent structure functions. Nucl. Phys. B (Proc. Suppl.) 65, 207–208 (2010)Google Scholar
  3. 3.
    Alekseev, M.G., [COMPASS Collaboration], et al.: The spin-dependent structure function of the proton \({g^{p}_{1}}\) and a test of the Bjorken sum rule. Phys. Lett. B 690, 466 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    Blümlein, J.: The theory of deeply inelastic scattering. Prog. Part. Nucl. Phys. 69, 2884 (2013)CrossRefGoogle Scholar
  5. 5.
    Taghavi-Shahri, F., Mirjalili, A., Yazdanpanah, M.M.: A new approach to calculate the gluon polarization. Eur. Phys. J. C 71, 1590 (2011)ADSCrossRefGoogle Scholar
  6. 6.
    Khorramian, A.N., Atashbar Tehrani, S., Taheri Monfared, S., Arbabifar, F., Olness, F.I.: Polarized deeply inelastic scattering (DIS) structure functions for nucleons and nuclei. Phys. Rev. D 83, 054017 (2011)ADSCrossRefGoogle Scholar
  7. 7.
    Airapetian, A., [The HERMES Collaboration], et al.: Measurement of the proton spin structure function \({g^{p}_{1}}\) with a pure hydrogen target. Phys. Lett. B 442, 484 (1998)ADSCrossRefGoogle Scholar
  8. 8.
    Adeva, B., [Spin Muon Collaboration], et al.: Spin asymmetries a 1 and structure functions g 1 of the proton and the deuteron from polarized high energy muon scattering. Phys. Rev. D 58, 112001 (1998)ADSCrossRefGoogle Scholar
  9. 9.
    Adeva, B., [Spin Muon Collaboration], et al.: Spin asymmetries a 1 of the proton and the deuteron in the low x and low q 2 region from polarized high energy muon scattering. Phys. Rev. D 60, 072004 (1999)ADSCrossRefGoogle Scholar
  10. 10.
    Alekseev, M., [COMPASS Collaboration], et al.: The polarised valence quark distribution from semi-inclusive DIS. Phys. Lett. B 660, 458 (2008)ADSCrossRefGoogle Scholar
  11. 11.
    Deshpande, A.: Spin physics at RHIC: present and future. Pramana - J. Phys. 61, 859 (2003)ADSCrossRefGoogle Scholar
  12. 12.
    Bazilevsky, A., [RHIC spin collaboration]: The RHIC spin program overview. J. Phys.: Conf. Ser. 678, 012059 (2016)Google Scholar
  13. 13.
    Dokshitzer, Y.L.: Calculation of the structure functions for deep inelastic scattering and e + e annihilation by perturbation theory in Quantum Chromodynamics. Sov. Phys. JETP. 46, 641 (1977)ADSGoogle Scholar
  14. 14.
    Altarelli, G., Parisi, G.: Asymptotic freedom in parton language. Nucl. Phys. B 126, 298 (1997)ADSCrossRefGoogle Scholar
  15. 15.
    Gribov, V.N., Lipatov, L.N.: Deep inelastic e p scattering in perturbation theory. Sov. J. Nucl. Phys. 15, 438 (1972)Google Scholar
  16. 16.
    Nadolsky, P.M., Lai, H.L., Cao, Q.H., Joey Huston, J., Pumplin, J., Stump, D., Tung, W.K., Yuan, C.-P.: Implications of CTEQ global analysis for collider observables. Phys. Rev. D 78, 013004 (2008)ADSCrossRefGoogle Scholar
  17. 17.
    Martin, A.D., Stirling, W., Thorne, R., Watt, G.: Parton distributions for the LHC. Eur. Phys. J. C 63, 189 (2009)ADSCrossRefMATHGoogle Scholar
  18. 18.
    Botje, M.: A QCD analysis of HERA and fixed target structure function data. Eur. Phys. J. C 14, 285 (2000)ADSCrossRefGoogle Scholar
  19. 19.
    Cafarella, A., Corianò, C., Guzzi, M.: NNLO Logarithmic expansions and exact solutions of the DGLAP equations from x-space: new algorithms for precision studies at the LHC. Nucl. Phys. B 748, 253 (2006)ADSCrossRefGoogle Scholar
  20. 20.
    Cafarella, A., Corianò, C., Guzzi, M.: Precision studies of the NNLO DGLAP evolution at the LHC with Candia. Comput. Phys. Commun. 179, 665–684 (2008)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Salam, G.P., Rojo, J.: A higher order perturbative parton evolution toolkit (HOPPET). Comput. Phys. Commun. 180, 120 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    Vogt, A.: Efficient evolution of unpolarized and polarized parton distributions with QCD-pegasus. Comput. Phys. Commun. 170, 65 (2005)ADSCrossRefGoogle Scholar
  23. 23.
    Sarma, J.K., Choudhury, D.K., Medhi, G.K.: x-distribution of deuteron structure function at low-x. Phys. Lett. B 403, 139 (1997)ADSCrossRefGoogle Scholar
  24. 24.
    Ratcliffe, P.G.: A matrix approach to numerical solution of the DGLAP evolution equations. Phys. Rev. D 63, 116004 (2001)ADSCrossRefGoogle Scholar
  25. 25.
    Ghasempour Nesheli, A., Mirjalili, A., Yazdanpanah, M.M.: Numerical solution of DGLAP equations using Laguerre polynomials expansion and Monte Carlo method. SpringerPlus 5, 1672 (2016)CrossRefGoogle Scholar
  26. 26.
    Boroun, G.R., Amiri, M.: Laguerre polynomials method in the valon model. Phys. Scr. 88, 035102 (2013)ADSCrossRefMATHGoogle Scholar
  27. 27.
    Rezaei, B., Boroun, G.R.: Analytical solution of the longitudinal structure function in the leading and next-to-leading-order analysis at low x with respect to Laguerre polynomials method. Nucl. Phys. A 857, 42 (2011)ADSCrossRefGoogle Scholar
  28. 28.
    Atashbar Tehrani, S., Khorramian, A.N.: The Jacobi polynomials QCD analysis for the polarized structure function. J. High Energy Phys. 07, 048 (2007)CrossRefGoogle Scholar
  29. 29.
    Khanpour, H., Taheri Monfared, S., Atashbar Tehrani, S.: Nucleon spin structure functions at NNLO in the presence of target mass corrections and higher twist effects. Phys. Rev. D 95, 074006 (2017)ADSCrossRefGoogle Scholar
  30. 30.
    Taghavi-Shahri, F., Khanpour, H., Atashbar Tehrani, S., Alizadeh Yazdi, Z.: Next-to-next-to-leading order QCD analysis of spin-dependent parton distribution functions and their uncertainties: Jacobi polynomials approach. Phys. Rev. D 93, 114024 (2016)ADSCrossRefGoogle Scholar
  31. 31.
    Block, M.M., Durand, L., McKay, D.W.: Analytic derivation of the leading-order gluon distribution function G(x, q 2) = x g(x, q 2) from the proton structure function \({f^{2}_{p}} (x, q^{2})\). Phys. Rev. D 77, 094003 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    Atashbar Tehrani, S., Taghavi-Shahri, S., Mirjalili, A., Yazdanpanah, M.M.: NLO Analytical solution to the polarized parton distributions, based on the Laplace transformation. Phys. Rev. D 87, 114012 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    Boroun, G.R.: Solutions of independent DGLAP evolution equations for the gluon distribution and singlet structure functions in the next-to-leading order at low x. JETP 106, 700 (2008)ADSCrossRefGoogle Scholar
  34. 34.
    Baishya, R., Sarma, J.K.: Method of characteristics and solution of DGLAP evolution equation in leading and next to leading order at small x. Phys. Rev. D 74, 107702 (2006)ADSCrossRefGoogle Scholar
  35. 35.
    Baishya, R., Sarma, J.K.: Semi numerical solution of non-singlet Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equation up to next-to-next-to-leading order at small x. Eur. Phys. J. C 60, 585 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    Baishya, R., Sarma, J.K.: Solution of non-singlet DGLAP evolution equation in leading order and next to leading order at small-x. Indian J. Phys. 83, 1333 (2009)ADSCrossRefGoogle Scholar
  37. 37.
    Devee1, M., Baishy, R., Sarma, J.K.: Evolution of singlet structure functions from DGLAP equation at next-to-next-to-leading order at small-x. Eur. Phys. J. C 72, 2036 (2012)ADSCrossRefGoogle Scholar
  38. 38.
    Baishya, R., Jamil, U., Sarma, J.K.: Evolution of spin-dependent structure functions from DGLAP equations in leading order and next to leading order. Phys. Rev. D 79, 034030 (2009)ADSCrossRefGoogle Scholar
  39. 39.
    Barone, V., Predazzi, E.: High-Energy Particle Diffraction. Springer, Berlin (2002)CrossRefMATHGoogle Scholar
  40. 40.
    Boroun, G.R.: Analysis of the logarithmic slope of f 2 from the Regge gluon density behavior at small x. JETP 111, 567 (2010)ADSCrossRefGoogle Scholar
  41. 41.
    Boroun, G.R., Rezaei, B., Sarma, J.K.: A phenomenological solution small x to the longitudinal structure function dynamical behavior. Int. J. Mod. Phys. A 29, 1450189 (2014)ADSCrossRefGoogle Scholar
  42. 42.
    Rezaei, B., Boroun, G.R.: Analytic approach to the approximate solution of the independent DGLAP evolution equations with respect to the hard-Pomeron behavior. JETP 112, 380 (2011)ADSCrossRefGoogle Scholar
  43. 43.
    Block, M.M.: A new numerical method for obtaining gluon distribution function G(x, q 2) = x g(x, q 2), from the proton structure function \(f^{p}_{2}(x, q^{2})\). Eur. Phys. J. C 65, 1 (2010)ADSCrossRefGoogle Scholar
  44. 44.
    Block, M.M., Durand, L., Ha, P., McKay, D.W.: Applications of the leading-order Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations to the combined HERA data on deep inelastic scattering. Phys. Rev. D 84, 094010 (2011)ADSCrossRefGoogle Scholar
  45. 45.
    Boroun, G.R., Rezaei, B.: Decoupling of the DGLAP evolution equations at next-to-next-to-leading order (NNLO) at low-x. Eur. Phys. J. C 73, 2412 (2013)ADSCrossRefGoogle Scholar
  46. 46.
    Boroun, G.R., Zarrin, S., Teimoury, F.: Decoupling of the DGLAP evolution equations by Laplace method. Eur. Phys. J. Plus 130, 214 (2015)CrossRefGoogle Scholar
  47. 47.
    Ellis, R.K., Stirling, W.J., Webber, B.R.: QCD And Collider Physics. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  48. 48.
    Hirai, M., [Asymmetry Analysis collaboration(AAC)], et al.: Determination of gluon polarization from deep inelastic scattering and collider data. Nucl. Phys. B 813, 106 (2009)ADSCrossRefGoogle Scholar
  49. 49.
    Leader, E., Sidorov, A.V., Stamenov, D.B.: Impact of CLAS and COMPASS data on polarized parton densities and higher twist. Phys. Rev. D 75, 074027 (2007)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of PhysicsRazi UniversityKermanshahIran

Personalised recommendations