Physical limits in the color-dipole model bounds

Abstract

The ratio of the cross-sections for the transverse and longitudinal virtual photon polarizations, $\sigma ^{\gamma ^{*}\mathrm{p}}_\mathrm{L/T}$, at high photon–hadron energy scattering is studied. The relationship between the gluon distributions obtained using the color-dipole model and standard gluons obtained from the Dokshitzer–Gribov–Lipatov–Altarelli-Parisi (DGLAP) evolution and the Altarelli–Martinelli equations is investigated. It is shown that the color-dipole bounds are dependent on the gluon distribution behavior. This behavior is considered by expansion and Laplace-transform methods. Numerical calculations at leading order (LO) and next-to-leading order (NLO) approximations and a comparison with the color-dipole model (CDM) bounds indicate the range of validity of this method at small dipole sizes, $r\sim 1/Q{\ll }1/Q_\mathrm{s}$.

Appendices

Appendix A: Details on the derivation of (26)

In this appendix, we provide a brief exposition of the derivation Eq. (26) in Ref. [77]. The evolution equations for the parton distributions are defined by

$$\begin{aligned} \frac{\partial }{\partial {\ln }Q^{2}}f_{i}(x,Q^{2})=P_{ij}(x,Q^{2}) {\otimes }f_{j}\left( x,Q^{2}\right) , \end{aligned}$$

(58)

where $f_{i}(x,Q^{2})$ stands for the number distributions of partons in a hadron and $\otimes$ stands for the Mellin convolution. Equation (58) represents a system of $2n_{f}+1$ coupled integro-differential equations. The splitting functions $P_{ij}(x,Q^{2})$ for $N^{m}LO$ approximation are defined by $P^{N^{m}\mathrm{LO}}_{ij}(x,Q^{2})=\sum _{k=0}^{m}a_{s}^{k+1}(Q^{2})P_{ij}^{k}(x)$ with $a_{s}(Q^{2})=\alpha _{s}(Q^{2})/4\pi$. The flavor-singlet quark density is defined $f_\mathrm{s}=\sum _{i=1}^{n_{f}}\left[ f_{i}+{\overline{f}}_{i}\right]$. The LO evolution equation for $F_{2}$ at low x for four flavors is defined by

$$\begin{aligned} \frac{\partial F_{2}(x,Q^{2})}{\partial {\ln }Q^{2}}=\frac{10\alpha _{s}}{9\pi } \int _{0}^{1-x}P_\mathrm{qg}(z) G\left( \frac{x}{1-z},Q^{2}\right) \mathrm{d}z. \end{aligned}$$

(59)

Here the fact is used that at low values of x the quark density can be neglected and the nonsinglet contribution can be ignored. The authors of [77] used the expansion of the gluon distribution at an arbitrary point $z=a$ as at the limit $x{\rightarrow }0$, the equation obtained is

$$\begin{aligned} \frac{\partial F_{2}(x,Q^{2})}{\partial {\ln {Q^{2}}}}\simeq \frac{5\alpha _\mathrm{s}(Q^{2})}{9\pi }\frac{2}{3}G\left( \frac{x}{1-a}\left( \frac{3}{2}-a\right) ,Q^{2}\right) . \end{aligned}$$

(60)

Therefore the gluon distribution can be expressed by

$$\begin{aligned} G(x,Q^{2})=\frac{9\pi }{5\alpha _\mathrm{s}(Q^{2})}\frac{3}{2}\frac{\partial F_{2}\left( x\frac{1-a}{\frac{3}{2}-a},Q^{2}\right) }{\partial {\ln {Q^{2}}}}. \end{aligned}$$

(61)

The result of comparing them with GRV94(LO) [90] showed that the better choices have been in the range $0.5{\le }a{\le }0.8$ and with Ryskin et al. [91] this corresponds to $a=0.75$.

Appendix B: Details on the derivation of (32)

The gluon density used in this analysis obeys the following Laplace-transform method [92,93,94,95,96,97,98,99,100,101,102], as the coordinate transformation introduced by $\upsilon {\equiv }{\ln }(1/x)$. Further, Eq. (31) is rewritten

$$\begin{aligned} \widehat{{\mathcal {F}}}(\upsilon ,Q^{2})=\int _{0}^{\upsilon }{\widehat{G}}(w,Q^{2}){\widehat{H}}(\upsilon -w){\mathrm{d}w}, \end{aligned}$$

(62)

where ${\widehat{H}}(\upsilon ){\equiv }e^{-\upsilon }{\widehat{K}}_\mathrm{qg}(\upsilon )$ and ${\widehat{f}}(\upsilon ,Q^{2}){\equiv }f(e^{-\upsilon },Q^{2})$ as ${\widehat{K}}_\mathrm{qg}(\upsilon )=1-2e^{-\upsilon }+2e^{-2\upsilon }$. The Laplace transform of the right-hand of Eq. (62) is defined by

$$\begin{aligned} {\mathcal {L}}\bigg [\int _{0}^{\upsilon }{\widehat{G}}(w,Q^{2}){\widehat{H}}(\upsilon -w){\mathrm{d}w};s\bigg ]=g(s){\times }h(s) \end{aligned}$$

(63)

where $h(s){\equiv }{\mathcal {L}}[{\widehat{H}}(\upsilon );s]=\int _{0}^{\infty }{\widehat{H}}(\upsilon )e^{-s\upsilon }\mathrm{d}\upsilon$ with the condition ${\widehat{H}}(\upsilon )=0$ for $\upsilon <0$.

The gluon distribution function in $\upsilon$-space is obtained in terms of the inverse transform of a product to the convolution of the original functions as

$$\begin{aligned} {\widehat{G}}(\upsilon ,Q^{2})= & {} {\mathcal {L}}^{-1}\left[ f(s,Q^{2}{\times }h^{-1}(s);\upsilon \right] \nonumber \\= & {} \int _{0}^{\upsilon }\widehat{{\mathcal {F}}}(w,Q^{2}){\widehat{J}}(\upsilon -w){\mathrm{d}w}, \end{aligned}$$

(64)

where

(65)

Therefore the explicit solution for the gluon distribution in $\upsilon$-space is defined in terms of the integral

(66)

Appendix C: Details of the parameterization of $F_{L}(x,Q^{2})$

The authors in Ref. [47] obtained two analytical relations for the longitudinal structure function at LO and NLO approximations in terms of the effective parameters of the parameterization of the proton structure function. The results show that the obtained method provides a reliable longitudinal structure function at the HERA domain and also the structure functions $F_\mathrm{L}(x,Q^{2})$ manifestly obey the Froissart boundary conditions. The structure functions, $F_{2}(x,Q^{2})$ and $F_\mathrm{L}(x,Q^{2})$, and their derivatives to $\ln Q^{2}$, are defined with respect to the singlet and gluon distribution functions $xf_\mathrm{a}(x,Q^{2})$ as

$$\begin{aligned}&F_{k\{=2,L\}}(x,Q^{2})=<e^{2}>\sum _{a=s,g}\left[ B_{k,a}(x){\otimes }xf_{a}(x,Q^{2})\right] \nonumber \\&\quad \mathrm {and}\nonumber \\&\frac{\partial }{\partial {\ln }Q^{2}}xf_{a}(x,Q^{2})=-\frac{1}{2}\sum _{a,b=s,g}P_{ab}(x){\otimes }xf_{b}(x,Q^{2}). \end{aligned}$$

(67)

The quantities $B_{k,a}(x)$ and $P_{ab}(x)$ are the Wilson coefficient and splitting functions, respectively. The high order corrections to the coefficient functions can be found in Ref. [47]. With respect to the Mellin transform method, the leading order longitudinal structure function is obtained at low x by the following form:

$$\begin{aligned} F_\mathrm{L}^{\mathrm {LO}}(x,Q^{2})= & {} (1- x)^{n}\sum _{m=0}^{2}C_{m}(Q^{2})L^{m}, \end{aligned}$$

(68)

where

$$\begin{aligned} C_{2}= & {} {\widehat{A}}_{2}+\frac{8}{3}a_{s}(Q^{2})DA_{2}\nonumber \\ C_{1}= & {} {\widehat{A}}_{1}+\frac{1}{2}{\widehat{A}}_{2}+\frac{8}{3}a_{s}(Q^{2})D\left[ A_{1}+\left( 4\zeta _{2}-\frac{7}{2}\right) A_{2}\right] \nonumber \\ C_{0}= & {} {\widehat{A}}_{0}+\frac{1}{4}{\widehat{A}}_{2}-\frac{7}{8}{\widehat{A}}_{2}+\frac{8}{3}a_{s}(Q^{2})D\bigg [A_{0} +\left( 2\zeta _{2}-\frac{7}{4}\right) A_{1}\nonumber \\&+\left( \zeta _{2}-4\zeta _{3}-\frac{17}{8}\right) A_{2}\bigg ], \end{aligned}$$

(69)

and

$$\begin{aligned} {\widehat{A}}_{2}= & {} {\widetilde{A}}_{2}\nonumber \\ {\widehat{A}}_{1}= & {} {\widetilde{A}}_{1}+2DA_{2}\frac{\mu ^{2}}{\mu ^{2}+Q^{2}}\nonumber \\ {\widehat{A}}_{0}= & {} {\widetilde{A}}_{0}+DA_{1}\frac{\mu ^{2}}{\mu ^{2}+Q^{2}}\nonumber \\ {\widetilde{A}}_{i}= & {} {\widetilde{D}}A_{i}+D{\overline{A}}_{i}\frac{Q^{2}}{Q^{2}+\mu ^{2}}\nonumber \\ {\widetilde{D}}= & {} \frac{M^{2}Q^{2}\bigg [(2-\lambda )Q^{2}+\lambda M^{2}\bigg ]}{\bigg [Q^{2}+M^{2}\bigg ]^{3}}\nonumber \\ {\overline{A}}_{m}= & {} a_{m1}+2a_{m2}L_{2},~~a_{02}=0. \end{aligned}$$

(70)

The standard representation for QCD running coupling constant in the LO and NLO approximations are described by

$$\begin{aligned} a_\mathrm{s}^{\mathrm {LO}}(Q^{2})= & {} \frac{1}{\beta _{0}\ln (Q^{2}/\Lambda ^{2})}, \nonumber \\ a_\mathrm{s}^{\mathrm {NLO}}(Q^{2})= & {} \frac{1}{\beta _{0}\ln (Q^{2}/\Lambda ^{2})}-\frac{\beta _{1}{\ln }\ln (Q^{2}/\Lambda ^{2})}{\beta _{0}[\beta _{0}\ln (Q^{2}/\Lambda ^{2})]^{2}}, \end{aligned}$$

(71)

for which the QCD parameter $\Lambda$ at LO and NLO approximations has been extracted with $\Lambda ^\mathrm{LO}(n_\mathrm{f}=4)=136.8~\mathrm {MeV}$ and $\Lambda ^\mathrm{NLO}(n_\mathrm{f}=4)=284.0~\mathrm {MeV}$, respectively. The QCD $\beta$-functions are

$$\begin{aligned} \beta _{0}=\frac{1}{3}(11C_{A}-2n_{f}),~\beta _{1}=\frac{1}{3}(34C_{A}^{2}-2n_\mathrm{f} (5C_{A}+3C_{F})) \end{aligned}$$

where $C_{F}$ and $C_{A}$ are the Casimir operators in the SU$(N_\mathrm{c})$ color group.

The NLO longitudinal structure function at small x is defined by

(72)

where the coefficient functions read

$$\begin{aligned} {\widehat{B}}^{(1)}_\mathrm{L,s}= & {} 8C_{F}\bigg [\frac{25}{9}n_\mathrm{f}-\frac{449}{72}C_{F}+(2C_{F}-C_{A})\nonumber \\&\left( \zeta _{3}+2\zeta _{2}-\frac{59}{72}\right) \bigg ],\nonumber \\ {\overline{B}}^{(1)}_\mathrm{L,s}= & {} \frac{20}{3}C_{F}(3C_{A}-2n_\mathrm{f})\nonumber \\ {\widehat{\delta }}^{(1)}_\mathrm{sg}= & {} \frac{26}{3}C_{A},\nonumber \\ {\overline{\delta }}^{(1)}_\mathrm{sg}= & {} 3C_{F}-\frac{347}{18}C_{A},\nonumber \\ {\widehat{R}}^{(1)}_\mathrm{L,g}= & {} -\frac{4}{3}C_{A},\nonumber \\ {\overline{R}}^{(1)}_\mathrm{L,g}= & {} -5C_{F}-\frac{4}{9}C_{A},\nonumber \\ L_{A}= & {} L+\frac{A_{1}}{2A_{2}},\nonumber \\ L_{C}= & {} L+\frac{C_{1}}{2C_{2}},\nonumber \\ L= & {} \ln (1/x)+L_{1},\nonumber \\ L_{1}= & {} {\ln }\frac{Q^{2}}{Q^{2}+\mu ^{2}}. \end{aligned}$$

(73)