II. Non-commuting Matrix Solution of DGLAP; F 2 p,d Data Leading to Partons Directly without Parameterization
aa r X i v : . [ h e p - ph ] S e p II. Non-commuting Matrix Solution of DGLAP; F p,d Data Leading to PartonsDirectly without Parameterization
M. Goshtasbpour ,and M. Zandi Dept. of Physics, Shahid Beheshti University, G. C., Evin 19834, Tehran, Iran. II. Physikalisches Institut Georg August Universitt Gttingen, Germany.
Dominant present path for determination of parton distribution functions (pdfs) from data isbased on pre-assumed form of parametric pdfs. Here, an alternative direct, or non-parametricmethod of pdf extraction is spelled out. As the main task, least square estimates of the centralvalues of pdfs are obtained at a chosen Q , and at x i , i = 1 , ..., n of the analyzed F p,d datapoints. In the process, numerically singular system of LO PQCD weighted linear combination ofdecomposition equations of the data points, each at a given ( x i , Q ij ) , j = 1 , ..., n i , obtained froma respective χ , together with the equations of ZM VFN constraints, are solved. In each dataequation, the corresponding data points are decomposed into their pdf components, evolved fromthe set of unknown pdfs at ( x i , Q ) , i = 1 , ..., n . A similar evolution is done in the constraints. Asa complementary task, the constrained discretization of Bjorken x , required for the commutingsolution of DGLAP, [1], is relaxed, and a non-commuting solution on the more natural set of exact x -points of the data is developed. PACS numbers: 12.38.Bx
INTRODUCTION
Decomposition equation of each data point serves as aweighted component of data equations on parton distri-bution functions (pdfs), and each ZM VFN constraint isa theoretical equation on pdfs, at different Q of the dataand constraints, at a given order in PQCD. The integro-differential DGLAP evolution equations of the same or-der in perturbation may then be solved and used to bringall the pdfs to the unique Q of the unknowns in the re-sulting numerically singular system of linear equationsto be solved (via SVD). Here, in the second paper of theseries, the process is realized, for the simplest example,on F p,d data at leading order (LO), with possible widerimplications. NON-COMMUTING SOLUTIONS OF ALLDGLAP EQUATIONS
In contrast to commuting solutions of DGLAP equa-tions, [1], with commuting banded (bdd) lower triangular(l.t.) splitting function matrices, here, more general non-commuting solutions of DGLAP equations, with non-commuting, non-bdd l.t. splitting function matrices areconsidered because of their advantage in data analysis. n Dimensional x -Space of Data Points and each pdfSet Refering to the corresponding section in [1], commut-ing solutions of DGLAP constrain the discrete x -basis,for the linear algebra, to be of the form x i = x i , i = 1 , · · · , n, (1)in order to have commuting banded (bdd) lower trian-gular (l.t.) splitting function matrices. Discrete x in (1)are not the exact x -points of the data. So, eventually, fordata analysis, interpolation techniques should be used.The non-commuting numerical solutions, to be pre-sented in this section, can simply use the exact Bjorken x = { x i , i = 1 , ..., n } of the data set. On either discrete x -set, a discrete set of n basis vectors for each pdf setis defineable. Thus, the splitting functions operating onthis basis (space) can be calculated as n × n matrices. Flavor Evolution Equation
At LO, where equality of splitting functions (neglect-ing quark masses), leaves only four different ones, theevolution equation can be written in the simple form of( m + 1) coupled equations of m independent quarks andantiquarks coupled through gluon distribution, (2). ∂∂t q ( x, t )... q m ( x, t ) g ( x, t ) = Z x dyy P qq ( xy ) 0 · · · P qg ( xy )0 . . . . . . ... ...... . . . . . . 0 ...0 · · · P qq ( xy ) P qg ( xy ) P gq ( xy ) · · · · · · P gq ( xy ) P gg ( xy ) . q ( y, t )... q ( m ) ( y, t ) g ( y, t ) (2)where dtdLn ( Q ) = α s ( Q )2 π . (3)For n f flavors, However, usually, a linear combination,may reduce m and modify coefficient of P qg , e.g., m → n f − c and b pdfs; or m → n f in the F p,d data analysis, in the present paper,due to use of q total for all the flavors. Flavor evolution DGLAP (2) is considered as an al-ternative to non-singlet, singlet (
N S − S ) division ofDGLAP already discussed in [1]. Thus, both have thesame number of independent parton distributions. Here,FIG. 1 to FIG. 6 show the ( n F + 1) resulting pdfs of our F p,d data analysis.Similar to the case of singlet , [1], as all the kernels are t independent, we expect the solution for finite Q intervalto be of the form: f F ( t ) = e ( t − t ) P F f F ( t ) , (4)where f F , is a ( m + 1) column vector of quarks and gluonand P F is the kernel of (2) respectively.The singlet and the flavor kernels have some essen-tial similarities that help in finding the analytic commut-ing solution of the flavor DGLAP for finite Q interval.The result may be presented in a future paper. For thepresent, we can have the numerical non-commuting solu-tions of FIG. 1 to FIG. 6. Numerical Non-commuting Solutions for Finite Q Interval for all forms of DGLAP Equations
Here, based on the exact Bjorken x i of data points, e.g.for the structure functions, numerical non-commuting so-lution of the DGLAP equations is presented. DGLAPequations are of the generic form: df K ( x, t ) dt = Z x dyy P K ( xy ) f K ( y, t ) , (5) where K can be N S, S, or F . Having the matrix forms ofthe splitting functions kernels, section (2), matrix formsof the DGLAP equations are: df Ki ( t ) dt = i X j =1 P Kij f Kj ( t ) ⇔ d f K ( t ) dt = P K . f K ( t ) . (6)Independence of the kernels from the variable t , leads thefollowing solution to equation (6 ), for the finite evolutionfrom t to t . f K ( t ) = e ( t − t ) P K . f K ( t ) ≡ E K ( t − t ) . f K ( t ) . (7)The final ( t − t ) dependent E K matrix constitutes theessential solution of DGLAP equations for finite evolu-tion.To determine the matrix form of E K ( t − t ) = e ( t − t ) P K , the choice of (1) is mandatory to have com-muting, banded triangular, splitting functions matrices,for a finite expansion of the exponential in the analyti-cal commuting solution of (7), [1, 2]. But experimentaldata points do not exactly match the x points of (1). So,eventually, interpolation techniques should be used.However, as hinted below equation (1), such restric-tions is not necessary. Exponential of the non-banded(non-commuting) form of the LO kernels P NS , P S and P F can be numerically calculated, for the non-commutingsolution of (7), at least within our limited data analysis.Using numerical algorithms to approximate the exponen-tial in (7) brings freedom from banded triangular matri-ces, so one could use experimental x points directly todiscretize the x space.In this paper, we go directly to the non-commutativesolutions, based on the sequence of x points of data,aimed at F p,d data analysis for finding N S, S, and g pdfs. The question of comparison with flavor solutionscan (may?) be trivial, considering prevailing linearity, or(massless) SU ( n F ) , n F INTODUCTION TO DATA ANALYSIS OFSTRUCTURE FUNCTIONS F p,d LO decomposition equation (factorization) of F is: F = x (1 + α s π C q ) ⊗ ˆ F q + x α s π C g ⊗ ( X F e F ) g, (8)with the quark structure function in term of n F = 5flavors N S − S , with usual definitions for p and d ˆ F q/p = 190 (22Σ + 3 q (24) − q (15) + 5 q (8) + 15 q (3) ) , ˆ F q/d = ˆ F q/p + ˆ F q/n q (24) − q (15) + 5 q (8) ) . (9)The simplest choice we have made by writing (9), unre-alistically with SU F (5) symmetry, needs explanation.Within a VFN point of view, if it was not for bringingin some minimal mass effects of heavy quarks c and b ,QED DIS would be limited to three light quarks. Whyso? Because zero-mass assumption for the three lightestquarks is well justified, m q << Q , Q in the wholerange of data and simulation. There is the possibility ofdifferentiating square of electric charge, e q at γ ∗ quarkvertex in proton or deutron DIS, so u total stands apart.Isospin symmetry differentiates d total and s total . In otherwords, detection of electric charge squared weight of q total may leave us two quarks in form of a singlet q S , anda non-singlet q NS , and isospin symmetry differentiatesdata and the set of variables q ∝ F p − d ) from q NS which now becomes q , the SU F (3) octet. In this sense,could we say there is something of an SU F (3) symmetry?It is trivial that F p,d have no information on the non-singlet valence quarks. Thus, pdfs are extracted from F p,d as total: sea and valence are not separable.Bringing in minimal mass effects of heavy quarks, c and b , to the measured data points, F p ( x ij , Q j ) and F d ( x ij , Q j ), in different regions of Q , correspondingto 4, and 5 flavors, allows unveiling of the 4th and 5thflavors. SU F (3) −→ SU F (5) symmetry under the as-sumption of having five zero-mass flavors symmetrically,as unrealistic or highly errored, as a simple ZM VFNscheme may be.Under SU F (5), differentiation of q and q once againleaves q NS = q as it was. Thus, two additional setsof nonsinglet degrees of freedom are brought in q n − with addition of the nth quark as a heavy flavor as Q increases. Finally, the number of extractable pdf centralvalue sets increases to n F + 1. The additional one pdf isthe gluon of gluon vertex.Simulation of our direct method is a testing groundfor this point of view, its correction and refinement. In-vestigation of deviation from SU F (5) symmetry in thewell-known regions of Q , is left out to be dealt withsoon.Minimal mass effects, at masses of c and of b quarks,for us, are continuity constraints c ( Q = m c ) = 0, and b ( Q = m b ) = 0 at Bjorken x = { x i , i = 1 , ..., n } of thedata set, at the boundary between regions of Q where itis assumed that n F respectively crosses 3 to 4, and 4 to5 flavors. In other words, constraints equate the singlet,Σ, and the ( n F − N S ) of the SU ( n F ) fla-vor symmetry of the region above each boundary; thus,dropping the flavor number, n F , by 1 at and below theboundary. The constraint equations, together with con-sistent evolution, keep the continuity of the pdfs whilethe flavor numbers change at the boundary.Along with continuity of pdfs, we have continuity forthe coupling constant as a changing Q crosses massivequark m b .Positions of Q = m b is in practice very different from Q = m c . Q = m c lies asymptotically outside the Q range, as equations of BCDMS data, [3], are sim-ulated here. It is only for b that a break takes placewhen evolution between the data and our simulationcrosses Q = m b . Then, solving DGLAP for finite evo-lution [ Q j , Q ] is divided into two stages of [ Q j , m b ] and[ m b , Q ]. Furthermore, there are two sets of inputs, thefirst set has n f = 4 flavors at Q j , the second set has( n f = 5) flavors at m b . In addition, the second set ofinputs is the first set of outputs with an extra input pdffor the heavy quark, b ( Q = m b ) ≡
0, our minimal effectof mass of b ! In order to carry out the comparison withMSTW, we take their value, m b = 4 . Gev [4].In the direct, non-parametric, data analysis of DIS,a singular system of linear equations of data and VFNconstraints is encountered. Direct pdf variables of thismethod are a few times more (next section), also in [5],than the number of parameters of the parametric dataanalysis. A very simple idealization, a ZM-VFN schemeemployed via the constraint equations, brought the firstsuccessful removal of singularity via SVD. It is presentedin its simplicity here, before further realistic elaborationsof the effects of heavy quarks masses. In a coming step,a critique, including that of the massless coefficient func-tions, (10) and (12 ), having large errors, is to be done.
Example Calculation of Coefficient FunctionMatrices
The kernel of the convolutions, in DIS data analysis,is either the splitting functions for DGLAP, or the coef-ficient functions of the hadronic structure function, e.g.(8). Construction of the matrix coefficient functions, forDIS, follows closely the path of construction of splittingfunctions, resulting in either commuting, banded lowertriangular, or noncommuting matrices, respectively, onlydepending upon whether equation (1) is followed fordiscretization of x or not. The two possibilities lead toanalytical, [1], or more general numerical approachs.With examples on splitting functions, derivation of P qq was spelled out in [1]. In a second example calculation,the matrix form of the LO M S coefficient functions C q ( x )and C g ( x ), suited for our ZM VFN scheme, are derivedhere, beginning with [6]: C q ( z ) = 43 [2( ln(1 − z )1 − z ) + −
32 ( 11 − z ) + − (1 + z ) × ln(1 − z ) − z − z ln z + 3 + 2 z − ( π δ (1 − z )] . (10)Using definition of ”+” regularization, integrating C ⊗ ˆ F q , (8), by parts, in which dv =
83 ln(1 − y )1 − y dy , and dv = [ − (1 + y ) ln(1 − y ) − y − y ln y + 3 + 2 y ] dy ; the matrixof coefficients of n − tuple q may be read as: C q : ( C q ) ii = a + 1 x i − x i − x i − Z x i { v ( x i y ) + v ( x i y ) } dy ( C q ) ik = 1 x k − x k − x k − Z x k dy { v ( x i y )+ v ( x i y ) } − x k +1 − x k x k Z x k +1 dy { v ( x i y ) + v ( x i y ) } . (11)where a = v (1) − v (0) − π / − C g ( x ), beginning with [6]: C g ( z ) = 12 [((1 − z ) + z ) ln( 1 − zz ) − z + 8 z − , (12)we will get the matrix form: C g : ( C g ) ii = v (1) + 1 x i − x i − x i − Z x i v ( x i y ) dy ( C g ) ik = 1 x k − x k − x k − Z x k v ( x i y ) dy − x k +1 − x k x k Z x k +1 v ( x i y ) dy (13)where dv = C g ( y ) dy . EXTRACTION OF LEAST SQUARE ESTIMATESOF THE CENTRAL VALUES OF PDFS FROM F p,d DATA AND THEIR COMPARISON WITHMSTW PDFS
At this stage, we are ready, with a bare minimum re-quired for SVD to work, to get the central values of pdfsfrom F p,d data. For the numerical solution, each F p,d data point at a given ( x i , Q i j ) of an experiment, e.gBCDMS [3], is decomposed into its partonic componentsvia (8) and (9), evolved by our solutions of DGLAP, froma set of unknown pdfs, u k , k = 1 , ..., m = ( n F +1) × n − x i , Q ) , i = 1 , ..., n , with a chosen Q .The linear system of m data equations and (2 n − A ij u j = b i , i = 1 , ..., m + (2 n − , j = 1 , ..., m, (14)where the data equations come from minimization of χ : χ = X ij ( F L ij − F R ij ) σ ij , (15) with respect to the unknowns, ∂χ ∂u k = 0 , k = 1 , ..., m. (16) F L ij is the value of data point on the left side of (8), with σ ij its quadratically calculated total error, and F R ij itsLO decomposition corresponding to the right side of (8,9). In (15), for data sets such as BCDMS [3], a sum overproton and deutron data points is understood.The cut on data is set, in principle, to separate the Q regions of PQCD and higher twists at the invariantmass squared W = 20 Gev , similar to MSTW LO cuts,[4, 5]. Number of proton and deutron data points usedare 153 and 146 respectively, thus less than 4% of thetotal BCDMS [3] data is cut out. Ability to utilize correctcuts is a major improvement of the present version of thepaper. It will be further discussed in [5].Least square estimates of the central values of unknownpdfs u k , k = 1 , ..., m , at the chosen Q , are obtained bysolving the numerically singular system of linear equa-tions (14) of the data, and ZM VFN constraints. Sin-gular value decomposition (SVD) is the essential tool forbringing out the physically acceptable solutions from thecontext of singularity. In [5] there will be extended ex-position of the workings of SVD. Here, is how we learnedto use SVD.Matrix of the coefficients A in (14) is singular. SVDhelps separating and managing the singularity, namelythe null subspace of the linear space of the singular ma-trix [7]. The magic of SVD here is to pinpoint numeri-cally too small, deletable, eigenvalues, corresponding toa deletable set of eigenvectors of the numerical null sub-space.Operationally, i.e, in the process of trial motion up ordown an indicator of the scale of ordered eigenvalues inthe simulation program, there is a single physical crite-rion for uncovering the border of null subspace, or theplace (the indicator value) of its largest eigenvalue: sud-den appearance of well patterned, physically acceptable,set of solutions of the linear system, here the LS esti-mates of every pdf set, which takes place along with thedeletion of the corresponding null subspace.** We’ll beusing this concept as the first characterization or ”qual-itative, intuitive, physical definition” of null subspace ofa singular matrix, developed quantitatively in [5]. Results
The results are presented in the first six figures, withsimilar graphic symbols (described in the 1st legend), attwo values of Q = 37 . Gev , at which so-lutions are computed independently, and then comparedwith MSTW. The first Q is chosen near the center ofpopulation of the data points, in the non-asymptotic areaof our ZM VFN in conflict with GM VFN of MSTW; thesecond is in the deep asymptotic area. FIG. 1 shows theSU(5) singlet, the most exact of the directly extractedpdf points from F p,d data point. FIG. 2 shows the fiveflavor gluon, the least exact of the extracted pdfs points. MSTW at 1000 Gev results at 1000 Gev MSTW & errors at 37.5 Gev results & errors at 37.5 Gev xq s FIG. 1. Points of the SU(5) singlet pdf at the x -points ofthe BCDMS F p,d data from which they are extracted, incomparison with MSTW’s. The choices of Q = 37 . Gev of the graphs are arbitrary. The first is chosennear the center of population of the data points, in the non-asymptotic area of our ZM VFN in conflict with GM VFN ofMSTW. The second is in the deep asymptotic area. The erroranalysis is done in [5]. This is the most exact of the directlyextracted pdf points from F p,d data point. Symbols similar to FIG.1 xg FIG. 2. Gluon; graphic symbols and caption similar to thoseof FIG. 1. This is the least exact of the extracted pdfs points.Partially, smoothened at Q = 1000 Gev due to larger inter-vals of evolution. Choosing Q = 1000 well in the asymptotic regionwill not improve the obsevable lower large- x , and highersmall- x mismatch with respect to MSTW, but smoothensout the gluon and the N S pdfs (except q because of itsindependence) due to larger intervals of evolution, takingplace while setting up equations (16). before obtainingthe SVD solutions. xq FIG. 3. NS pdf xq . Graphic symbols and caption similarto those of FIG. 1. There is an observable large- x small- x mismatch with MSTW as described in the text, common toall NS pdfs. xq FIG. 4. Independendent (in the sense of text in ”Discus-sion”) NS pdf xq . Caption similar to that of FIG. 3. Discussion and prospects
The lower large- x , and higher small- x mismatch withrespect to MSTW, pronounced in all N S pdfs, seems tocome out of an x-dependent convolution found in factor-izatins. Thus we may have a clue where to search forit. but no errors for resultsSymbols similar to FIG.1, xq FIG. 5. NS pdf xq . Caption similar to that of FIG. 3. Errorsand their discussion is left to [5]. but no errors for resultsSymbols similar to FIG.1, xq FIG. 6. NS pdf xq . Caption similar to that of FIG. 3.Errors and their discussion is left to [5]. First of all, we intend to remedy the very simpleZM VFN constraints (scheme), entering abruptly at theboundaries Q = m c,b . This is to be done after thepresent paper.Here attention is drawn to the following computationaltest of the theoretical necessity of havig our c quark con-straints as a possible minimum mass effect for having( n F + 1) pdf central values.At Q = m c = 1 . Gev , constraints may not ap-pear to be as crutial as those at Q = m b from a prac-tical point of view. We are in the asymptotic regionfor c pdf, when working above the minimum of BCDMSdata at Q = 7 . Gev . At m c , even these constraintsare overlapping with the higher twist boundary set at W = Q (1 /x −
1) = 20
Gev . However, even in thatasymptotic Q region, without c constraints, SVD cannotproduce physical results, FIG. 7 and FIG. 8, magically(at this stage of understanding) pointing out a deep defi-ciency in such a practical point of view. Anchorage of thepdf c ( m c ) = 0 sets essential physical constraints towardsasymptotic use of our ZM VFN, whether evolution takes - H u + d + s L xq MSTW
FIG. 7. At Q = 1000 Gev , MSTW (solid line), and our NS pdf q ( 11 black dots) and its two, positive ( u + d + s ) andnegative ( − c ), components (10* dots in color each), withboth series of c and b constraints. SVD produces acceptablephysical results. - - - xc x H u + d + s L xq MSTW
FIG. 8. Here, everything is the same as in FIG. 7, but withoutthe series of c constraints. q is the most adversely affectedpdf from lack of ZM VFN constraints at Q = m c . Here,SVD cannot produce physical results . place in the asymptotic region for c pdf or not.Operating. At, m b = 22 . Gev , in the middleof the finite ∆ Q = [7 . , Gev interval of BCDMSdata, constraints are not only unavoidable in the abovesense, but also are practically used in process of evolutionthrough Q = m b .The path towards global data analysis, begining withelectroweak and going to hard scattering, to deter-mine (2 n F + 1) , n F → N S com-muting solution of DGLAP. Most importantly, A HigherOrder Perturbative Parton Evolution Toolkit, HOPPET,[11], which we became familiar with only very recently,is exactly of the same family as ours, and of great valuein this respect.
ACKNOWLEDGEMENTS
Thanks to P. G. Ratcliffe for inception of the idea. [1] Mehrdad Goshtasbpour and Seyed Ali Shafiei, arXiv:1303:3985[2] M. Goshtasbpour and P. G. Ratcliffe, 14th InternationalSpin Physics Symposium, Osaka, Japan, Oct. 16-21,2000, P. 879; P. G. Ratcliffe,
Phys. Rev. D , 116004(2001), arXiv: hep-ph/ 0012376; S.A. Shafiei, A Studyof DIS and a Novel Solution to DGLAP Evolution Equa-tion , M.S. Thesis, Shahid Beheshti University (2002).[3] Benvenuti et al, BCDMS, Phys. Lett. B223, 485 (1989).[4] A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt
Eur.Phys.J. C Singular Value Decomposition, Hes-sian Errors, and Linear Algebra of Non-parametric Ex-traction of Partons from DIS.
To be submitted[6] R.K. Ellis, W.J. Stirling, B.R. Webber,
QCD and Col-lider Physics , Cambridge university press (1996).[7] Press, W.H., et al., Numerical Recipes, Cambridge Uni-versity Press, 1992.[8] Arneodo, et al., NMC,
Phys. Lett. B , 107 (1995).[9] Mehrdad Goshtasbpour and Seyed Ali Shafiei, 15th In-ternational Spin Physics Symposium, Long Island, N. Y., Sept. 9-14, 2002, AIP Conf. Proc. , 299-302 (2003).Also in ”Upton/Danvers 2002, Spin 2002 ” 299-302.[10] Mohammad Zandi, A New Numerical Matix Solution forDGLAP Evolution Equation of PQCD and Application ,M.S. Thesis, Shahid Beheshti University (2010).[11] Gavin P. Salam and Joan Rojo,
Comp. Phys. Comm. , 120 (2009), arXiv:0804:3755. * Subtraction of one, from the number of unknownsand the corresponding constraint equations, is an indi-cation of lack of information or data on a particular pdfvariable, namely, b ( x = . m b = 22 . Gev , and at x = .
07, BCDMS’shighest Q = 19 < . Gev . This leads to a miss-ing point at x = .
07 in graphs of the results wherever b ( x = .
07) is involved.**A note on the history of this point. The 3rd refer-ence of [2], has a good overlap with this paper. There,we went as far as decomposing F of NMC [8] with theCommutative Matrix solutions of DGLAP with the bestfit of Bjorken xx