# Analytic derivation of the next-to-leading order proton structure function based on the Laplace transformation

###### Abstract

An analytical solution based on the Laplace transformation technique for the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi DGLAP evolution equations is presented at next-to-leading order accuracy in perturbative QCD. This technique is also applied to extract the analytical solution for the proton structure function, , in the Laplace -space. We present the results for the separate parton distributions for all parton species, including valence quark densities, the anti-quark and strange sea parton distribution functions (PDFs), and the gluon distribution. We successfully compare the obtained parton distribution functions and the proton structure function with the results from GJR08 [Eur. Phys. J C 53 (2008) 355-366] and KKT12 [J. Phys. G 40 (2013) 045002] parametrization models as well as the -space results using QCDnum code. Our calculations show a very good agreement with the available theoretical models as well as the deep inelastic scattering (DIS) experimental data throughout the small and large values of . The use of our analytical solution to extract the parton densities and the proton structure function is discussed in detail to justify the analysis method considering the accuracy and speed of calculations. Overall, the accuracy we obtain from the analytical solution using the inverse Laplace transform technique is found to be better than 1 part in 10 to 10. We also present a detailed QCD analysis of non-singlet structure functions using all available DIS data to perform global QCD fits. In this regard we employ the Jacobi polynomial approach to convert the results from Laplace space to Bjorken space. The extracted valence quark densities are also presented and compared to the JR14, MMHT14, NNPDF and CJ15 PDFs sets. We evaluate the numerical effects of target mass corrections (TMCs) and higher twist (HT) terms on various structure functions, and compare fits to data with and without these corrections.

###### pacs:

12.39.-x, 14.65.Bt, 12.38.-t, 12.38.Bx###### Contents

- I Introduction
- II Theoretical formalism
- III Singlet solution in Laplace space at the next-to-leading order approximation
- IV Non-singlet solution in Laplace space at the next-to-leading order approximation
- V Proton structure function in Laplace space
- VI The results of Laplace transformation technique
- VII Jacobi polynomials technique for the DIS analysis
- VIII Summary and conclusion

## I Introduction

Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations Dokshitzer:1977sg (); Gribov:1972ri (); Lipatov:1974qm (); Altarelli:1977zs () are a set of an integro differential equations which can be used to evolve the parton distribution functions (PDFs) to an arbitrary energy scale, Q. The solutions of the DGLAP evolution equations will provide us the gluon, valence quark and sea quark distributions inside the nucleon. Consequently these equations can be used widely as fundamental tools to extract the deep inelastic scattering (DIS) structure functions (SFs) of the proton, neutron, and deuteron to enrich our current information about the structure of hadrons. The standard procedure to obtain the dependence of the gluon and quark distributions is to solve numerically the DGLAP equations and compare the solutions with the data in order to fit the PDFs to some initial factorization scale, typically less than the square of the -quark mass Q (2 GeV). The initial distributions for the gluon and quark are usually determined in a global QCD analysis including a wide variety of DIS data from HERA Abramowicz:2016ztw (); Abt:2016zth (); Abramowicz:2015mha (); Aaron:2009kv (); Aaron:2009bp (); Aaron:2009aa () and COMPASS Adolph:2015saz (), hadron collisions at Tevatron Aaltonen:2008eq (); Abazov:2008ae (); Abulencia:2007ez (); Abbott:2000ew () fixed-target experiments over a large range of and Q, as well as data from CHORUS and NuTeV Onengut:2005kv (); Tzanov:2005kr (), and also the data for the longitudinal structure function Collaboration:2010ry (). Finally using the coupled integro-differential DGLAP evolution equations one can find the PDFs at higher energy scale, Q. For the most recent studies on global QCD analysis, see for instance Harland-Lang:2014zoa (); Khanpour:2012tk (); Alekhin:2012ig (); ::2014uva (); Buckley:2014ana (); Ball:2014uwa (); Martin:2009iq (); JimenezDelgado:2008hf ().

Some analytical solutions of the DGLAP evolution equations using the Laplace transform technique, initiated by Block et al., have been reported in recent years Block:2010du (); Block:2011xb (); Block:2010fk (); Block:2009en (); Block:2010ti (); Block:2007pg (); Block:2008xc (); Zarei:2015jvh (); AtashbarTehrani:2013qea (); Boroun:2015cta (); Boroun:2014dka () with considerable phenomenological success. In this paper, a detailed analysis has been performed, using repeated Laplace transforms, in order to find an analytical solutions of the DGLAP evolution equations at next-to-leading order (NLO) approximations. We also analytically calculate the individual gluon, singlet and non-singlet quark distributions from the initial distributions inside the nucleon. We present our results for the valence quark distributions and , the anti quark distributions and , the strange sea distribution , and finally the gluon distribution . Using the Laplace transform technique, we also extract the analytical solutions for the proton structure function as the sum of flavor singlet , and flavor non singlet distributions. The obtained results indicate an excellent agreement with the DIS data as well as those obtained by other methods such as the fit to the structure function performed by KKT12 Khanpour:2012tk () and GJR08 Gluck:2007ck ().

In the present work, we also demonstrate once more the compatibility of the Laplace transform technique and the Jacobi polynomial expansion approach at the next-to-leading order and extract the valence quark densities as well as the values of the parameter from the QCD fit to the recent DIS data. The effect of target mass corrections (TMCs), which are important especially in the high- and low-Q regions, and the contribution from higher twist (HT) terms are also considered in the analysis. To quantify the size of these corrections, we evaluate the structure functions at next-to-leading order in QCD, and compare the results with the DIS data used in our PDF fits.

The present paper is organized as follows: In Sec. II, we provide a brief discussion of the theoretical formalism of the proton structure function at the NLO approximation of QCD. A detailed formalism to establish an analysis method for the solution of DGLAP evolution using the repeated Laplace transforms for the singlet sector have been presented in Sec. III. In Sec. IV, we also review the method of the analytical solution of DGLAP evolution equations based on Laplace transformation techniques for the non singlet sector. In Sec. V, we utilize this method to calculate the proton structure function by Laplace transformation. We attempt a detailed comparison of our next-to-leading order results with recent results from the literature in Sec. VI. We also discuss in detail the use of our analytical solution to justify the analysis method in terms of accuracy and speed. A completed comparison between the obtained results and available DIS data is also presented in this section. The application of the Laplace transformation techniques and Jacobi polynomial expansion machinery at the next-to-leading order are described in detail in Sec. VII. The method of the QCD analysis including the PDF parametrization, statistical procedures, and data selection are also presented in this section. The numerical effects of target mass corrections (TMCs) and higher twist terms (HT) on various structure functions are also discussed. Finally, we give our summary and conclusions in Sec. VIII. In Appendix A, we render the results for the different splitting functions in the Laplace transformed space, and Appendix B includes the analytical expression for the coefficient functions of the singlet and gluon distribution in space.

## Ii Theoretical formalism

The present DIS and hadron collider data provide the best determination of quark and gluon distributions in a wide range of Abramowicz:2015mha (); Aaron:2009bp (); Aaron:2009aa (). In this article we will be concerned specifically with the proton structure function at next-to-leading order accuracy in perturbative QCD. In the common renormalization scheme the structure function, extracted from the DIS process, can be written as the sum of a flavour singlet , and a flavour non-singlet distributions in which we will have,

(1) | |||||

here and represent the gluon and quark distribution functions respectively. The stands for the usual flavour non-singlet combination, , and stand for the flavour-singlet quark distribution,

where denotes the number of active massless quark flavours. In Equation (1) the symbol denotes the convolution integral which turns into a simple multiplication in Mellin space and represents the average squared charge. and are the common next-to-leading order Wilson coefficient functions Vermaseren:2005qc (). The analytical expression for the additional next-to-leading order gluonic coefficient function can be found in Ref. Vermaseren:2005qc (). As we already mentioned the gluon and quark distribution functions at the initial state can be determined by fit to the precise experimental data over a large numerical range for and Q. The individual quark and gluon distributions are parametrized with the pre-determined shapes as a standard functional form. This function is given in terms of and a chosen value for the input scale Q. The gluon distribution is a far more difficult case for PDF parametrizations to obtain precise information due to the small constraints provided by the recent data Khanpour:2012tk (); Martin:2009iq ().

In the following, we will present our analytic method based on the newly developed Laplace transform technique to determine the non singlet and singlet and structure functions using the input distributions , and at Q = 2 GeV. We use the KKT12 Khanpour:2012tk () and GJR08 Gluck:2007ck () input parton distributions to determine the individual parton distribution functions at an arbitrary Q > Q, which can be obtained, using the DGLAP evolution equations. Having the parton distribution functions and using the inverse Laplace transform, one can extract the proton structure function as a function of at any desired Q value.

## Iii Singlet solution in Laplace space at the next-to-leading order approximation

For the most important high energy processes the next-to-leading order approximation is the standard one which we also consider it in our analysis. The DGLAP evolution equations can describe the perturbative evolution of the singlet and gluon distribution functions. The coupled DGLAP evolution equations at the next-to-leading order approximation, using the convolution symbol , can be written as Block:2007pg (); Block:2008xc ()

(2) |

(3) |

where is the running coupling constant and the splitting functions and are the Altarelli-Parisi splitting kernels at one and two loop corrections respectively as Altarelli:1977zs (); Curci:1980uw (); Furmanski:1980cm (),

(4) |

In the evolution equations, we take N = 4 for and N = 5 for and adjust the QCD parameter at each heavy quark mass threshold, and . Consequently the renormalized coupling constant can be run continuously when the N changes at the and mass thresholds Botje:2010ay ().

We are now in a position to briefly review the method of extracting the parton distribution functions via analytical solution of DGLAP evolution equation using the Laplace transformation technique. By considering the variable changes and , one can rewrite the evolution equations presented in Eqs.(III) and (III) in terms of the convolution integrals and with respect to and variables as Block:2010du (); Zarei:2015jvh ()

where the Q dependence of above evolution equations is expressed entirely thorough the variable as . Note that we used the notation and . The above convolution integrals show that using one-loop and two-loop kernels where the and are a combination of quark or gluon , one can obtain the singlet and gluon sectors of distributions.

Defining the Laplace transforms and and using this fact that the Laplace transform of a convolution factors is simply the ordinary product of the Laplace transform of the factors, which have been presented in Block:2010du (); Block:2011xb (), the Laplace transforms of Eqs.(III), and (III) convert to ordinary first-order differential equations in Laplace space with respect to variable . Therefore we will arrive at

(7) | |||||

(8) | |||||

whose the leading-order splitting functions for the structure function , presented in Altarelli:1977zs (); Floratos:1981hs () in Mellin space, are given by and at Laplace space by

(9) |

(10) |

(11) | |||||

and

(12) |

where the N is the number of active quark flavors, is the Euler’s constant and is the digamma function. The next-to-leading order splitting functions and are too lengthy to be include here and we present them in Appendix A. One can easily determine these next-to-leading order splitting functions in Laplace space using the next-to-leading order results derived in Ref. Altarelli:1977zs (); Curci:1980uw (); Furmanski:1980cm (). The leading-order solution of the coupled ordinary first order differential equations in Eqs.(7) and (8) in terms of the initial distributions are straightforward. Considering the initial distributions for the gluon, , and singlet distributions, , at the input scale Q GeV, the evolved solutions in the Laplace space are given by Block:2010du (); Block:2011xb (),

(13) |

The inverse Laplace transform of coefficients in the above equations are defined as kernels and the input distributions by and . Then the following decoupled solutions with respect to and Q variables and in terms of the convolutions integrals can be written as,

(14) | |||||

(15) |

Considering the , one can finally arrive at the solutions of the DGLAP evolution equations with respect to and Q variables. As we mentioned earlier, the dependence of the distributions functions and are specified by variable. Clearly knowledge of the initial distributions and at is needed to obtained the distributions at any arbitrary energy scale

Now we intend to extend our calculations to the next-to-leading order approximation for gluon and singlet sectors of unpolarized parton distributions. In this case, to decouple and to solve DGLAP evolutions in Eqs.(7) and (8) we need an extra Laplace transformation from space to space. The will be a parameter in this new space. In the rest of the calculation, the is replaced for brevity by . Therefore the solution of the first-order differential equations in Eqs.(7) and (8) can be converted to

We can consider a very simple parametrization for as . Generally to do a more precise calculation at the next-to-leading order approximation, one can consider the following expression for the as Zarei:2015jvh ()

(18) |

This expansion involves excellent accuracy to a few parts in . Using defined in the above equation and the conventions which were presented in Block:2010du (); Block:2011xb (), the following simplified notations for the splitting functions in space can be introduced by:

(19) |

Equations.(III) and (III) can be solved simultaneously to get the desired coupled algebraic equations for singlet and gluon distributions arriving at,

The simplified solutions of above equations can be obtained by setting in Eq.(18). For , the Eqs.(III) and (III) lead us to,

(22) |

(23) |

One can easily solve these equations and extract the and distributions. The results are clearly based on the input quarks and gluon distribution functions at Q. Using the Laplace transform technique, it is possible to go back from space to space, leading to the desired and expressions. The complete solutions of Eqs. (III) and (III) can be obtained via iteration processes. The iteration can be continued to any required order but we will restrict ourselves to getting a sufficient convergence of the solutions. Our results show that the second order of iterations is sufficient to get a reasonable convergence. Using the iterative solution of Eqs. (III) and (III) and the inverse Laplace transform technique to get back from space to space, the following expressions for the singlet and gluon distributions can be obtained Block:2010du (); Block:2011xb (); AtashbarTehrani:2013qea ():

The analytical expressions for the next-to-leading order approximation of coefficients , , and up to the desired steps of iteration are given in Appendix B. Using Laplace inversion in Eq. (III) from to space, we can arrive to the decoupled solutions (, ) space as the result of convolution defined by the Eqs. (III) and (15).

As a brief description, we have used the Laplace transform algorithm presented in Refs. Block:2009en (); Block:2010ti () for the numerical inversion of Laplace transformations and convolutions to obtain the required parton distribution functions. The analytical result at the LO approximation is given by Eq. (III). Employing the iterative numerical method through Eq. (III) -(23), up to desired order to achieve a sufficient convergence, will yield us the analytical expressions for the patron densities in space at the NLO approximation given by Eq. (III). To return the distributions to the space we need to convolution integral, Eqs. (III) and (15) in both LO and the NLO approximations. The Q dependence of the solutions are determined by the variable and recalling that , the solutions can be transformed back into the usual space. Consequently, one can obtain the singlet and gluon distributions as and respectively.

We have used the numerical Laplace transform algorithm presented in Refs. Block:2009en (); Block:2010ti () for the numerical inversion of Laplace transformations and convolutions to obtain the parton distribution functions and structure function in and space.

## Iv Non-singlet solution in Laplace space at the next-to-leading order approximation

Here we wish to extend our calculations to the next-to-leading order approximation for the non singlet sector of the parton distributions. For the non singlet distribution , one can schematically write the logarithmic derivative of as a convolution of non-singlet distribution with the non-singlet splitting functions, and Altarelli:1977zs (); Curci:1980uw (); Furmanski:1980cm (). Therefore the next-to-leading order contributions for the can be written as:

Again changing to the required variable, , and going to the Laplace space , we arrive at the simple solution as,

Going to Laplace space, we can obtain the first-order differential equations in Laplace space with respect to the variable for the non-singlet distributions :

(27) |

The above equation has a very simplified solution,

(28) |

where is contains the next-to-leading order contributions of the splitting functions at space, defined as

(29) |

The evaluation of is straightforward but too lengthy to present here. The analytical results for the unpolarized splitting functions in the transformed Laplace space at the next-to-leading order approximation are given in Appendix A. The Q dependence of the evolution equations is represented by at the leading order approximation and by at the next-to-leading order approximation which the latter one defined as Block:2010du (); Block:2011xb (); AtashbarTehrani:2013qea (),

Since all parts of the current analysis are done at the next-to-leading order approximation, we should use the variable as well. However to simplify in notation, the variable is used insteadly through out the whole paper.

Similar to the singlet case, any non-singlet solution, , can be obtained using the non-singlet kernel which is defined by,

(31) |

Using again the appropriate change of variable, , the solution of Eq.(31) can be converted to the usual space. The iterative numerical method of Laplace transformations at the NLO approximation is followed by the convolutions, based on Eqs. (III) and (15). For the numerical inversion of Laplace transformations and convolutions to obtain the appropriate PDFs and SF in and space, we again used the numerical inversion routine presented in Refs. Block:2009en (); Block:2010ti ().

## V Proton structure function in Laplace space

We perform here an next-to-leading order analytical analysis for the proton structure function using the Laplace transform technique. The result for singlet, gluons and non singlet parton distributions which we obtained in previous sections are used to extract the nucleon structure function. The next-to-leading order proton structure function for massless quarks can be written as Dokshitzer:1977sg (); Gribov:1972ri (); Lipatov:1974qm (); Altarelli:1977zs ()

(32) | |||||

where and are the next-to-leading order quarks and gluon Wilson coefficients, and , , and are the quark, anti-quark and gluon distributions, respectively. We exactly follow the method that we introduced before to solve the DGLAP evolution equations analytically to drive the proton structure function at the next-to-leading order approximation first in Laplace space and then in Bjorken space. As we already mentioned, only the initial knowledge of singlet , gluon , and non singlet distributions is required to solve the DGLAP evolution equations via the Laplace transform technique.

For our numerical investigation, we use the KKT12 Khanpour:2012tk () and GJR08 Gluck:2007ck () parton distribution functions at Q = 2 GeV. The valence quark distributions and , the anti-quark distributions and , the strange sea distribution and the gluon distribution of the KKT12 and GJR08 models are generically parameterized via the following standard functional form

(33) |

subject to the constraints that , , and the total momentum sum rule

(34) |

After changing to the variable and using the Laplace transform , one can easily obtain Eq.(33) in Laplace space,

(35) | |||||

We use the following standard parametrizations in Laplace space at the input scale Q=2 GeV for all parton types , obtained from GJR08 set of the free parton distribution functions Gluck:2007ck ():

(36) | |||||

(38) | |||||

(39) | |||||

(40) |

where is the common Euler beta function. The strange quark distribution function is assumed to be symmetric ( ) and it is proportional to the isoscalar light quark sea which parameterized as

(41) |

where in practice is a constant fixed to = Khanpour:2012tk (); Gluck:2007ck ().

The proton structure function in Laplace space, up to the next-to-leading order approximation, can be written as

(42) |

where the flavour singlet and gluon contribution read

(43) | |||||

(44) |

Finally the non-singlet contribution for three active (light) flavours is given by

where the and are the common next-to-leading order approximation of Wilson coefficients functions, derived in Laplace space by and ,

Once again the Q