An Analytical Study of the Nonsinglet Spin Structure Function 
                        <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.2715pt" id="M1" height="21.6166pt" version="1.1" viewBox="-0.0657574 -16.3451 61.9629 21.6166" width="61.9629pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-104" d="M546 430L539 434C529 434 505 438 495 440C473 444 450 448 430 448C352 448 265 412 213 366C145 306 96 203 96 103C96 22 135 -12 160 -12C190 -12 238 14 262 32C310 68 368 120 411 184H413C403 117 396 75 384 21C353 -118 325 -158 291 -184C270 -200 241 -205 208 -205C133 -205 90 -164 74 -110C70 -98 58 -100 49 -107C34 -119 23 -140 23 -155C23 -190 74 -261 166 -261C219 -261 280 -233 314 -208C383 -157 446 -79 470 81C491 223 529 388 546 430ZM456 386C452 357 433 283 420 252C402 216 366 174 325 129C288 88 239 56 212 56C192 56 182 77 182 120C182 165 199 242 226 292C256 348 281 377 311 389C327 395 353 402 375 402C408 402 436 394 456 386Z"/></g><g transform="matrix(.013,0,0,-0.013,10.177,-7.899)"><path id="g190-79" d="M719 650H483V622C553 619 576 606 581 560C585 532 588 487 588 394V148H585L167 650H20V622C67 618 88 610 109 585C127 562 129 557 129 484V264C129 173 126 124 123 93C118 44 92 31 35 28V0H272V28C204 32 181 44 176 95C173 124 170 172 170 264V515H172L598 -9H629V394C629 487 631 532 635 563C639 606 663 619 719 622V650Z"/></g><g transform="matrix(.013,0,0,-0.013,19.514,-7.899)"><path id="g190-84" d="M409 504C401 567 396 607 392 642C354 654 312 665 266 665C137 665 60 583 60 487C60 374 161 325 225 290C300 250 355 215 355 141C355 68 311 21 235 21C131 21 86 122 71 183L41 176C48 128 61 42 68 21C78 16 93 8 118 0C142 -7 175 -15 216 -15C349 -15 438 69 438 174C438 287 344 333 265 374C186 414 138 449 138 522C138 576 172 631 249 631C336 631 363 562 380 499L409 504Z"/></g><g transform="matrix(.013,0,0,-0.013,9.728,5.206)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g><g transform="matrix(.018,0,0,-0.018,26.127,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.018,0,0,-0.018,32.315,0)"><path id="g113-121" d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"/></g><g transform="matrix(.018,0,0,-0.018,42.313,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.018,0,0,-0.018,49.388,0)"><path id="g113-117" d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"/></g><g transform="matrix(.018,0,0,-0.018,55.486,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> Up to NLO in the DGLAP Approach at Small <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2811508pt" id="M2" height="8.47856pt" version="1.1" viewBox="-0.0657574 -8.19741 10.1659 8.47856" width="10.1659pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><use xlink:href="#g113-121"/></g></svg>

Borah, Neelakshi N. K.; Choudhury, D. K.

doi:https://doi.org/10.1155/2014/379823

Advances in High Energy Physics

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Research Article | Open Access

Volume 2014 | Article ID 379823 | https://doi.org/10.1155/2014/379823

An Analytical Study of the Nonsinglet Spin Structure Function Up to NLO in the DGLAP Approach at Small

Neelakshi N. K. Borah¹and D. K. Choudhury^1,2,3

Academic Editor: George Siopsis

Received16 Jul 2014

Accepted28 Oct 2014

Published23 Nov 2014

Abstract

A next-to-leading order QCD calculation of nonsinglet spin structure function at small is presented using the analytical methods: Lagrange’s method and method of characteristics. The compatibility of these analytical approaches is tested by comparing the analytical solutions with the available polarized global fits.

1. Introduction

Study of flavour nonsinglet and singlet evolution equations with next-to-leading order corrections in helps us to understand accurately the spin content of the nucleon. The spin-dependent DGLAP [1–4] evolution equations provide us the basic framework to study the polarized quark and gluon structure functions which finally give us polarized proton and neutron structure functions. Apart from the discussions about the numerical solutions [5–17] of DGLAP equations, analytical approaches towards these evolution equations at small are also available in literature [18–23] with reasonable phenomenological success.

There are many QCD working groups continuously upgrading the QCD parameterization for the polarized global fits [24–32]. NNPDF [24, 29] is a new approach to PDF fitting based on Monte Carlo sampling and Neural Networks. HOPPET is an -space evolution code which provides polarized PDFs for longitudinally polarised evolution up to NLO [30]. These Parton Distribution Function (PDF) evolution programs are used by the QCD working group to set some benchmark results in recent QCD NLO analysis.

In this work we extend our present analytical analysis up to NLO, obtain analytical solutions of spin-dependent DGLAP evolution equations at small , and calculate evolution for nonsinglet structure function, as well as making a comparative study of the two analytical methods.

The structure of the paper is as follows: Section 1 is the Introduction, Section 2 describes the formalism part, and in Section 3, we discuss our findings and show the results, and Section 4 contains the conclusion.

2. Formalism

2.1. Approximation of DGLAP Equation at Small

The polarized nonsinglet structure function evolves independently of the polarized singlet and gluon distribution in DGLAP framework. The evolution equation for is [33] Here is the polarized splitting kernel [34–36], is the NLO running coupling constant, and . The quark-quark splitting function can be expressed as where , are LO and NLO quark-quark splitting functions, respectively.

Introducing a variable and expanding the argument on the r.h.s. of (1) in a Taylor series as well as neglecting the higher order terms we get two approximate relations, respectively, The levels of approximation of (3) and (4) are as discussed in [37]. Using both (3) and (4) in (1) separately and putting the expressions for NLO polarized splitting function at small [34–36, 38], we get two reduced forms of (1) as a function of and . We can express them as with . Here we introduce the running coupling constant up to NLO as with The functions , , , and are, respectively,

Case 1. For the PDE derived using (3) (),

Case 2. For the PDE derived using (4) (),
Unlike in LO, (5) cannot be solved analytically. Hence, as in [39, 40], we introduce an assumption which linearizes as where is a numerical parameter to be obtained from range under consideration as has been done in [39, 40]. We will make a detailed study of this parameter later in this present work.
We now solve (5) analytically by Lagrange’s method [41] and then by method of characteristics [42, 43].

2.2. Solution by Lagrange’s Method

To obtain solutions of the PDE (5), we recast it in the form with the forms of , , and () given as Two independent solutions and () (say) are to be obtained for this auxiliary system, so that the genenral solution of (11) can be written as being an arbitrary function of and (). It yields for both Cases 1 and 2 () that

We define and as, with , where , . In (15), with . Demanding the linearity of the solution for we get the possible form of as where and are constants to be determined using the appropriate boundary conditions. Using relations equation (18) and physically plausible boundary conditions we get the solution for at NLO as follows.

Case 1. Using relation equation (18), We can put (19) in the form as where The term gives us the measure of NLO effect on polarized structure function at small for the solution equation (21).

Case 2. Here we use (4). In this case also we get the following.
Using (18), with As in Case 1, we express (23) in a form as with giving us the measure of NLO effect at small for the solution equation (25).
Thus we get two analytical solutions of (5) by Lagrange’s method for at NLO at small , given by (21) and (25).

2.3. Approximate Analytical Forms of and at Small

To obtain the analytical forms of (22) and (26) we need the analytical forms of and . To that end we need explicit corresponding forms of , , , and . It is possible only in the very small limit. In the very small region the analytical form of , as defined in (16), can be obtained as which after integration yields Here with Under a similar small approximation as for , the takes the form where To obtain an analytical form of we need an additional approximation as used in deriving equation (27), which yields

This yields, , where Here is Euler’s constant [44, 45] and has the value . Taking first three terms in the series expansion of the integral [44] and regularizing the value of at , we get Using above equation takes the form

2.4. Solution by Method of Characteristics

To use this method, it is convenient if (5) can be rewritten as Further the running coupling constant as defined by (6) is reexpressed as with In the method of characteristics, the original set of variables are changed to a new set of variables and the PDE becomes an ordinary differential equation with respect to either or . Now along the characteristic curve the PDE (38) becomes an ODE and takes the form with where and () are as defined in our earlier section.

Case 1. Using (3) (), where In obtaining (44), we used “ultra small ” limit. Integrating (41) along the characteristic curve and then going back to using (44), the solution for at NLO comes out as

Case 2. Using (4) (), solutions of the characteristic equations for the PDE (38) are where Thus the solution of the equation for charateristic curve leads us to the solution for at NLO (using (4)) as Since the expressions of ( and ) are different (44) and (50), in this case too we have two solutions (46) and (52), corresponding to the level of approximations equations (3) and (4).
In the next section we will discuss the relative merits of the four solutions.

3. Results and Discussion

3.1. The Parameter

We have defined the running coupling constant as given in (10). Here is a parameter to be determined numerically for the particular range under study. There are many illustrated values for this numerical parameter available in literature, some of which are phenomenologically justified [39, 40, 46]. It is reasonable to identify as the average value of the coupling constant for the particular range under study [39]. Taking GeV², for all the ranges within the perturbative region, CCFR range yields [39].

Again within the range and , and for E665 as well as and for NMC, the valid range of is found to be [40]. It is also observed that, in the range , as per requirement of the range of data compared, choice of a suitable value of (), can minimize the error [46].

However such approach does not yield any definite NLO effect to be compared on LO. In stead it leads to an NLO analysis with an additional parameter fitted from data. In this work we will rather find a theoretical limit on in the relative range of compatible with the perturbative expectation.

It is to be noted that the approximation for linearising as per equation (10) is exactly true only for very small variation of with . That is, this approach is applicable only in a very limited range of . If a comparison of the prediction of the model is done in a large range the assumption is expected to break down. So the best way to fit the is to consider experimentally accessible range and find the upper and lower limits of and to consider its average value.

In the experimentally available range for for HERMES [47] is found to be in the range . So taking the average value of both the upper and lower limit of we derive the value of as .

3.2. Constraints on the Analytical Solutions

The analytical solutions at small using Lagrange’s method have two more restrictions.

(i) should be positive in (28).

(ii) Allowed region should satisfy (33) for any given .

For , since and , (28) yields , which is outside the the expected small region. In fact for any positive value of there is no small solution for . This implies that a physically plausible analytical solution by Lagrange’s method at NLO equation (5) with () at small does not exist. Equation (21) is therfore not pursued further.

In case of , on the other hand we use (33) with the value of and obtain the limiting value of to be , up to which the analytical solution (25) is valid. We also take it to be the regularized value of in (37).

(iii) Due to nonintegral exponent of factor in the standard PDF forms [25–28, 48], in (46) becomes in general complex. Hence these cannot be used in the solution given by (46) which was obtained by method of characteristics. Hence (46) is no more considered for comparative study.

We are left with (25) and (52) to pursue the comparative analysis of our analytical methods.

3.3. Comparison with Exact Results

The formalism developed above is valid at small , , which yields that should be . On the other hand (25) has specific small range of validity, , while (52) does not have such defining limits. We therefore study the two solutions within , while comparing with exact results.

We compare our results for ((25) and (52)) with AAC03, GRSV01, LSS10, BB10, and recent Khorramian polarized NLO global fits [25–28, 48] at two different values. In this calculation, we choose GeV for and and GeV for and . The number of active flavors is fixed by the number of quarks with taking GeV as in [27].

In Figure 1, (25) and (52) are shown separately with the AAC03, GRSV01, and LSS10 global fits at GeV². We observe that, within our valid small range, (25) compares better as (52) evolves rapidly as we approach .

Figure 2 shows a similar comparison of our solutions at GeV² with the available NLO exact solutions by BB10 and Khorramian group [28, 48]. It is observed that in the small region both the analytical solutions evolve with a good agreement with the theoretical prediction by Blumlein and Bottcher. It is to be noted that a preference of one solution over the other is not possible phenomenologically from Figure 2. Again our analytical models are not consistent with the global fit developed by Khorramian group [48].

In Figure 3, we therefore plot both solutions together and address if any of these two solutions fares better in the range , which is the common range of validity for both. From it, we observe that at both , within the range , the solution given by (25) is more consistent with the exact theoretical predictions, than the solution by (52), thus giving Lagrange’s method an edge over method of characteristics as a more appropriate method for obtaining small analytical solutions in polarized NLO case also.

In Figures 1–3, we have taken the minimum cut-off of range , since resummation effects might be important in such very small region [49] below and DGLAP equations fail to describe these resummation effects.

4. Conclusion

In this comparative study, we have obtained analytical solutions for the polarized nonsinglet structure function at NLO, using the two analytical methods: Lagrange’s method and method of characteristics. Due to physical constraints these two methods lead us to only two suitable solutions for at NLO, valid for small . In this particular work, we have compared our analytical solutions only with the polarized global fits to test the consistency and plotted our solutions against for two different values, approximately in the range . Within this range, it is observed that Lagrange’s method is more consistent over the method of characteristics. Though we considered two levels of approximations as given in (3) and (4), our analysis indicates that only one approximation equation (4) leads us to physically plausible analytical solutions at small , although theoritically former one (3) is preferred. Instead of various numerical methods, our method too proves out to be workable alternative to study these polarized evolution equations at small .

A new insight in the structure functions drawn from this analytical approach is this: the dependence of the exponents of in (25) and (52) plays a decisive role in selecting the kinematic range of phenomenological validity of the solutions in certain range. Again as both solutions (25) and (52) show identical behaviour numerically at range, we can infer that their exponents, and , are almost equal in that considered kinematic range. Moreover using recent works available [50, 51] the comaparative study of these analytical methods can be extended up to NNLO.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Copyright

Copyright © 2014 Neelakshi N. K. Borah and D. K. Choudhury. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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