Next-to-leading order corrections to the valon model

Transcription

1 PRAMANA c Indian Academy of Sciences Vol. 86, No. 1 journal of January 216 physics pp Next-to-leading order corrections to the valon model G R BOROUN E ESFANDYARI Physics Department, Razi University, Kermanshah 67149, Iran Corresponding author. grboroun@gmail.com; boroun@razi.ac.ir MS received 17 January 214; revised 31 October 214; accepted 21 November 214 DOI: 1.17/s ; epublication: 25 July 215 Abstract. A seminumerical solution to the valon model at next-to-leading order (NLO) in the Laguerre polynomials is presented. We used the valon model to generate the structure of proton with respect to the Laguerre polynomials method. The results are compared with H1 data other parametrizations. Keywords. Valon model; structure function; Laguerre polynomials. PACS Nos 13.6.Hb; x In the valon model [1 7], a valon is defined to be a dressed valance quark in quantum chromodynamics (QCD) with a cloud of gluons sea quarks antiquarks. In this model, the proton is considered as a bound state of three valons (UUD). They contribute independently in an inclusive hard collision with a Q 2 -dependence that can be calculated in QCD at high Q 2. The proton structure function F p 2 (x, Q2 ) is related to the valon structure function f v ( x y,q2 ) for each valon v which can be shown as F p 2 (x, Q2 ) = v 1 x G v/n (y)f v ( x y,q2 ) dy, (1) where the summation in this equation is over the three valons G v/n (y) indicates the probability for the v valon to have a momentum fraction y in the proton. The valon distributions are assumed to have a simple relation in a hadron as G UUD (y 1,y 2,y 3 ) = g(y 1 y 2 ) α y β 3 δ(y 1 + y 2 + y 3 ). (2) The U D valon distributions can be defined by G U (y) = 1 y 1 y y2 dy 2 dy 3 G UUD (y 1,y 2,y 3 ) = gb(α + 1,β + 1)y α (1 y) α+β+1 (3) Pramana J. Phys., Vol. 86, No. 1, January

2 G R Boroun E Esfyari G D (y) = 1 y 1 y y1 dy 1 dy 2 G UUD (y 1,y 2,y 3 ) = gb(α + 1,α+ 1)y β (1 y) 2α+1, (4) where g(= (B(α + 1,β+ 1)B(α + 1,β+ α + 2)) 1 ) is given by the normalization factor B(i,j) is the Euler beta function with α =.65 β =.35 [1 7]. After doing the inverse Mellin transformation, the valon distributions have the forms G U (y) = 7.98y.65 (1 y) 2 (5) G D (y) = 6.1y.35 (1 y) 2.3. (6) The valon structure functions can be given in terms of the flavour singlet (S) nonsinglet (NS) terms as F U 2 (z, Q2 ) = 2 9 z[gs (z, Q 2 ) + G NS (z, Q 2 )] (7) F D 2 (z, Q2 ) = 1 9 z[2gs (z, Q 2 ) G NS (z, Q 2 )], (8) where N f G S (z, Q 2 ) = (G qi /v + G i=1 N f qi /v) (9) G NS (z, Q 2 ) = (G qi /v G qi /v). (1) i=1 In the moment representation we have M 2 (n, Q 2 ) = M α (n, Q 2 ) = 1 1 x n 2 F 2 (x, Q 2 )dx (11) x n 1 G α (x, Q 2 )dx (12) where α = v/n, S, NS. Therefore, the proton moment is given as M p (n, Q 2 ) = M v/n (n)m v (n, Q 2 ). (13) v Here, the valon momentum distributions are defined as B(α + n, α + β + 2) U(n) M U/p (n) = B(α + 1,α+ β + 2), B(β + n, 2α + 2) D(n) M D/p (n) = B(α + 1, 2α + 2), (14) 78 Pramana J. Phys., Vol. 86, No. 1, January 216

3 NLO corrections to the valon model also the moments of parton distributions are M u/v (n, s) = 2U(n)M NS (n, s), M d/v (n, s) = D(n)M NS (n, s), M g (n, s) =[2U(n) + D(n)]M gq (n, s), M sea (n, s) = 1 [2U(n) + D(n)][M S (n, s) M NS (n, s)]. (15) 2N f The evolution parameter s is defined as s = ln ln Q2 / 2 ln Q 2, (16) / 2 where Q 2 are the QCD cut-off parameter initial scale parameter, respectively. In the leading-order analysis (LO), the moments of quark (S NS) quark-to-gluon distribution inside the proton are defined as M NS (n, s) = exp( d NS s), (17) M S (n, s) = 1 2 (1 + ρ)exp( d +s) (1 ρ)exp( d s) (18) M gq (n, s) = 1 d gq [exp( d + s) exp( d s)]. (19) Here, the anomalous dimensions are [1,2] where d = γ n 2β, (2) γns n = α s 4π γ qq (),n (21) γij n = α s 4π γ (),n ij, i,j = q, g, (22) where γ ij is the dependence to the splitting function. The running coupling constant at LO is α s (Q 2 4π ) = β ln ( Q 2 / 2), (23) where β = (33 2N f )/3, the other parameters are given as ρ = (d NS d gg ), (24) = d + d =[(d NS d gg ) 2 + 4d gq d qg ] 1/2, (25) Pramana J. Phys., Vol. 86, No. 1, January

4 G R Boroun E Esfyari d NS = 1 2 n 1 3πb n(n + 1) + 4 1, (26) j j=2 d gq = n + n 2 3πb n(n 2 1), (27) d qg = N f 2 + n + n 2 2πb n(n + 1)(n + 2), (28) d gg = 3 1 πb n(n + 1) + 1 (n + 1)(n + 2) N n f 18 1, (29) j d ± = 1 2 [d NS + d gg ± ], (3) j=2 where b = β /4π. Using the next-to-leading order analysis (NLO), the running coupling constant is ( α s (Q 2 4π ) = β ln ( Q 2 / 2) 1 β 1 ln ln(q 2 / 2 ) ) β 2, (31) ln(q2 / 2 ) where β 1 = 12 38N f, 3 the moments of NS quark are proportional to [3,8 1] [ M NS = 1 + α s(q 2 ) α s (Q 2 ) ( γ (1),n NS 4π 2β So the evolutions of the γ, s up to NLO are given by γ n NS = α s 4π γ (),n qq + ( αs 4π ( αs β 1γ qq (),n 2β 2 )] (αs (Q 2 ) α s (Q 2 ) ) γ qq (),n 2β. (32) ) 2 γ (1),n NS (33) ) 2 γ (1),n ij, i,j = q, g, (34) γij n = α s 4π γ (),n ij + 4π also the moments of the S quark quark-to-gluon distribution inside the proton are [3,8 1] ( ) M S (n, Q 2 M g (n, Q 2 = ( αs (Q 2 ) λn 2β ) [P n α s (Q 2 ) 1 α s (Q 2 ) α s(q 2 ) 2β 4π P n γ n P n α s(q 2 ) α s(q 2 ( ) αs (Q 2 ) 4π 4π α s (Q 2 ) P n γ n P n + 2β + λ n + λ n 8 Pramana J. Phys., Vol. 86, No. 1, January 216 ) λn + λn 2β ]) (1) + {Q + }. (35)

5 NLO corrections to the valon model Here γ n = γ (1),n (β 1 /β )γ (),n (1) is the unit matrix in eq. (35). The term {Q + } implies that all subscripts are exchanged. Also, the other coefficients are given by [5,1] λ n ± = 1 (),n [γ qq + γ gg (),n ± (γ gg (),n γ qq (),n ) γ qg (),n γ gq (),n (36) P± n n =±γ λ n λ n + λ n. (37) For our calculation at LO up to NLO, the initial scale is equal to Q 2 =.2 GeV2 the QCD cut-off parameters are LO =.255 GeV NLO =.217 GeV, respectively. The parton distributions can be determined for a single value of s (or Q 2 ), by using a fit to moments with respect to a sum of beta functions. In turn, the moment of parton distributions are given by zq v (z, Q 2 ) = N v z a v (1 z) b v, v = u, d, (38) zq sea (z, Q 2 ) = a s z b s (1 z) c s (1 + e s z.5 + d s z) (39) zg(z, Q 2 ) = a g z b g (1 z) c g (1 + d g z.5 ). (4) According to the H1 data [11], the parameters a i,b i,... are further considered to be functions of s, as can be seen in the Appendix. In figures 1 4, we show the shape of the distributions in eqs (38) (4) for the valence, sea quarks gluon distributions at Q 2 = 45 GeV 2. We have computed the predictions for distributions inside the proton in the kinematic range where it has been measured using the experimental data have compared our predictions with the GJR parametrizations [12]. As we can see in these figures, the agreements at NLO are good. Figure 1. Distribution function for u valence quark in U valon at Q 2 = 45 GeV 2. Dot dash curves are our results at LO NLO, respectively as compared with the GJR parametrization [12]. Pramana J. Phys., Vol. 86, No. 1, January

6 G R Boroun E Esfyari Figure 2. Distribution function for d valence quark in D valon at Q 2 = 45 GeV 2. Dot dash curves are our results at LO NLO, respectively as compared with the GJR parametrization [12]. Figure 3. Distribution function for sea quarks per valon at Q 2 = 45 GeV 2.Dot dash curves are our results at LO NLO, respectively as compared with the GJR parametrization [12]. Figure 4. Distribution function for gluons per valon at Q 2 = 45 GeV 2. Dot dash curves are our results at LO NLO, respectively as compared with the GJR parametrization [12]. 82 Pramana J. Phys., Vol. 86, No. 1, January 216

7 NLO corrections to the valon model To obtain the proton structure function in valon model with respect to the Laguerre polynomials one needs to use an elegant fast numerical method at LO up to NLO. Therefore, we concentrate on the Laguerre polynomials in our determinations. In the Laguerre method [13,14], the Laguerre polynomials are defined as (n + 1)L n+1 (x) = (2n + 1 x)l n (x) nl n 1 (x), (41) where the orthogonality condition is defined as follows: e x L n (x )L m (x )dx = δ n,m. (42) The general integrable function f(e x ) is transformed into the sum as follows f(e x ) = N f(n)l n (x ), (43) where f(n)= e x L n (x )f (e x )dx. (44) In what follows we use the variable transformations, x = e x, y = e y to obtain the valonic structure function form to the Laguerre polynomials form. Then we combine exp each term of this equation on Laguerre polynomials according to eqs (43) (44) use the properties as x dy L n (x y )L m (y ) = L n+m (x ) L n+m+1 (x ), (45) The equation obtained determines F p 2 (x, Q2 ) in terms of the Laguerre polynomials at LO up to NLO according to the parton distributions as F p 2 (n, Q2 ) = n G v/p (m)[f2 v (n m, Q2 ) v m= F2 v (n m 1,Q2 )], (46) where F v 2 (n, Q2 ) = G v/p (m) = dx e x F v 2 (e x,q 2 )L n (x ) (47) dy e y G v/p (e y )L m (y ). (48) Here F2 v (x) is defined according to eqs (7) (8) along with eqs (38) (4) their coefficients at LO NLO (Appendix), also G v/p (y) is defined according Pramana J. Phys., Vol. 86, No. 1, January

8 G R Boroun E Esfyari to eqs (5) (6), respectively. Therefore, the proton structure function in valon model is defined by solving the recursion relation as N ( F p 2 (x, Q2 ) = F p 2 (n, Q2 )L n Ln 1 ), (49) x n= where F p 2 (n, Q2 ) is the proton structure function with respect to the Laguerre model is defined by eqs (46) (48) as the coefficients in these equations are obtained with respect to the valon model at LO NLO. This result is completely general gives the expression for the proton structure function with respect to the Laguerre polynomials model. Here we can exp the integrable functions till a finite order N = 3, as we can converge these series in the numerical determinations. We computed the predictions for all detail of the proton structure function in the kinematic range where it has been measured by H1 Collaboration [11] compared with GJR parametrization [12]. Our numerical predictions are presented as functions of x at Q 2 = 45 GeV 2. The result is presented in figure 5 along with a comparison with the H1 data with the results obtained with the help of other stard parametrizations. We compared the results at LO NLO with predictions of F2 P in perturbative QCD where the input densities are given by GJR parametrizations [12]. The agreement between the Laguerre polynomials method for the proton structure function in valon model at NLO data at low- high-x are remarkably good. The good agreement indicates that the Laguerre polynomials method in valon model for the proton structure function at NLO has a good asymptotic behaviour is compatible with both the data the other stard models at x values. This model has this advantage that we get a very elegant solution for the proton structure function. In summary, we have used the Laguerre polynomials method to describe the proton structure function in valon model at LO NLO. The proton structure can be determined in terms of the valon distributions valon structure functions with respect to the Figure 5. Proton structure function in valon model with respect to the Laguerre polynomials at Q 2 = 45 GeV 2. Dot dash curves are our results at LO NLO, respectively as compared with the GJR parametrization [12] H1 data that are associated with total errors [11]. 84 Pramana J. Phys., Vol. 86, No. 1, January 216

9 NLO corrections to the valon model Laguerre polynomials. To confirm the method results at NLO, the calculated values are compared with the H1 data on the proton structure function. It is shown that, there is a good agreement at NLO with the experimental H1 data for F p 2, if one considers the total errors, it is consistent with a higher order QCD calculations of F p 2 which essentially show an increase as x decreases. Also at NLO, this model gives a good description of the parton distributions at low- high-x values. Appendix The functional form of the free parameters of eqs (38) (4) are given by the following forms in terms of s at LO NLO analyses. Coefficients for u valence in U valon are: At LO a u = s.166s 2, b u = s 1.622s 2, N u = s 2.755s 2. (A.1) At NLO a u = s +.584s 2, b u = s +.89s 2, N u = s +.517s 2. (A.2) Coefficients for d valence in D valon are: At LO a d = s +.418s 2, b d = s s 2, N d = s s 2. (A.3) At NLO a d = s +.432s 2, b d = s +.949s 2, N d = s s 2. (A.4) Coefficients for sea quarks in each valon are: At LO a s = s s s 3, b s = s s s 3, c s = s s 2, d s = s.887s 2, e s = s s s 3. (A.5) Pramana J. Phys., Vol. 86, No. 1, January

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