Ratio of the structure functions and the color dipole model bound

doi:10.1016/j.nuclphysa.2019.07.011

Nuclear Physics A

Volume 990, October 2019, Pages 244-258

https://doi.org/10.1016/j.nuclphysa.2019.07.011 Get rights and content

Abstract

We observe that the DGLAP evolution equations at NNLO analysis predicts a ratio of the structure functions in region of small Bjorken variable x. The ratio $F_{L} (x, Q^{2}) / F_{2} (x, Q^{2})$ is obtained and compared with the prediction of the dipole model and HERA data. In particular we show that this ratio is lower than dipole model bound at high- $Q^{2}$ values and it is higher at low- $Q^{2}$ values. Then the effect of adding a higher twist term to the description of the ratio $F_{L} (x, Q^{2}) / F_{2} (x, Q^{2})$ for $Q^{2} < 20 {GeV}^{2}$ is investigated. Also the bounds are discussed by including charm distribution on $F_{L} / F_{2}$ . We discuss, furthermore, how this ratio can be determined the proton structure function with respect to the reduced cross section at high-y values.

Introduction

Measurements of the inclusive deep inelastic scattering (DIS) cross section have been pivotal in the development of the understanding of strong interaction dynamics [1], [2], [3], [4], [5]. The cross section in this measurement depends on two structure function $F_{2}$ and $F_{L}$ , which depend on the kinematic variables x and $Q^{2}$ . The structure functions obtained from these experiments have helped develop the description of hadrons. Hadrons are composite objects from the quarks and gluons at low and high-x values. The longitudinal structure function $F_{L} (x, Q^{2})$ comes as $F_{L} (x, Q^{2}) = F_{2} (x, Q^{2}) - 2 x F_{1} (x, Q^{2})$ , where $F_{2} (x, Q^{2})$ is the transverse structure function and it can be expressed at leading order by a sum of the quark-antiquark momentum distributions $x q_{i} (x)$ weighted with the square of the quark electric charges $e_{i}$ : $F_{2} = x \sum_{i} e_{i}^{2} (q + \overline{q})$ . The longitudinal structure function $F_{L}$ is directly dependent on the gluon distribution and it is proportional to the running coupling constant $α_{s}$ .

In the one-photon exchange approximation the neutral current reduced cross section is defined as $σ_{r} (x, Q^{2}) = F_{2} (x, Q^{2}) [1 - \frac{y^{2}}{Y_{+}} \frac{F_{L} (x, Q^{2})}{F_{2} (x, Q^{2})}],$ where $Y_{+} = 1 + {(1 - y)}^{2}$ , $y = Q^{2} / x s$ is the inelasticity and s is the center-of-mass squared energy of incoming electrons and protons respectively. The transverse and longitudinal structure functions, $F_{2} (x, Q^{2})$ and $F_{L} (x, Q^{2})$ , are related to the transverse and longitudinal virtual photon absorption cross sections, $σ_{T}$ and $σ_{L}$ . It is convenient to define the structure functions as follows $F_{2} (x, Q^{2}) = \frac{Q^{2}}{4 π^{2} α_{e m}} (1 - x) [σ_{T} (x, Q^{2}) + σ_{L} (x, Q^{2})], F_{L} (x, Q^{2}) = \frac{Q^{2}}{4 π^{2} α_{e m}} (1 - x) σ_{L} (x, Q^{2}) .$ Where the contribution of $F_{L}$ to reduced cross section (Eq. (1)) is significant only at high value of the inelasticity y, in spite of the fact that data on $F_{L}$ are generally difficult to extract from the cross section measurements.

In the first approximation of the parton model, the longitudinal structure function is equal identically zero but in actual DIS experiments should be nonzero since it arises from gluon corrections. Therefore $F_{L} (x, Q^{2})$ behavior depends on values of $Q^{2}$ . This behavior in the dipole picture [6] for DIS $F_{L}$ is nonzero. In the dipole model a strict bound for the ratio of $\frac{F_{L} (x, Q^{2})}{F_{2} (x, Q^{2})}$ is defined as [7], [8] $\frac{F_{L} (x, Q^{2})}{F_{2} (x, Q^{2})} \leq 0.27 .$ Based on the dipole formulation of the $γ^{⁎} p$ scattering [9], the standard formulae for $F_{2}$ and $F_{L}$ are defined by $F_{2} (x, Q^{2}) = \frac{Q^{2}}{4 π^{2} α_{e m}} (1 - x) \sum_{q} \int d^{2} r [w_{T}^{(q)} (r, Q^{2}) + w_{L}^{(q)} (r, Q^{2})] {\hat{σ}}^{(q)} (r, ξ), F_{L} (x, Q^{2}) = \frac{Q^{2}}{4 π^{2} α_{e m}} (1 - x) \sum_{q} \int d^{2} r w_{L}^{(q)} (r, Q^{2}) {\hat{σ}}^{(q)} (r, ξ),$ where $w_{T, L}^{(q)}$ are the probability densities for the virtual photon splitting into a $q \overline{q}$ pair and $\hat{σ}$ is the dipole cross section which describes the interaction of the dipole with the proton. This cross section depends on r where it is the transverse separation of the quarks in the quark-antiquark pair, and ξ is an energy variable in this formalism.

The bound for the ratio $\frac{F_{L} (x, Q^{2})}{F_{2} (x, Q^{2})}$ defined [10], [11] $g (Q, r, m_{q}) = \frac{w_{L}^{(q)} (r, Q^{2})}{w_{T}^{(q)} (r, Q^{2}) + {w^{(q)}}_{L} (r, Q^{2})},$ where $m_{q}$ is the mass of the quark q. It was shown in literatures that for all $Q \geq 0$ , $r \geq 0$ and $m_{q} \geq 0$ the bound (5) for the ratio $F_{L} / F_{2}$ is valid.

The paper is organized as follows. In section 2 we describe a formalism for the solution of DGLAP evolution equations [12] at NNLO analysis. We suggest an evolution method for the ratio $\frac{G (x, Q^{2})}{{F_{2}}^{s} (x, Q^{2})}$ in this section. Then the ratio $\frac{F_{L}}{F_{2}}$ obtained from the Altarelli-Martinelli equation [13] compared with HERA data and with the color dipole model bound. The results and discussion of our predictions are presented in section 3. The relation between the structure function from the DGLAP evolution equations with the color dipole model (CDM) is discussed in this section. Then allows one to draw conclusions about the role of higher twist effects in the ratio of structure functions. An influence of heavy quark contribution to the ratio $F_{L} / F_{2}$ is discussed in section 4. We conclude in section 5.

Section snippets

The ratio $\frac{G (x, Q^{2})}{F_{2}^{s} (x, Q^{2})}$

The DGLAP evolution equations for the singlet and gluon density in the standard form are given by $\frac{d}{d l n Q^{2}} [\begin{matrix} q_{s} (x, Q^{2}) \\ g (x, Q^{2}) \end{matrix}] = [\begin{matrix} P_{q q} & P_{q g} \\ P_{g q} & P_{g g} \end{matrix}] \otimes [\begin{matrix} q_{s} (x, Q^{2}) \\ g (x, Q^{2}) \end{matrix}]$ which emphasized that quark and gluon densities are coupled. The convolution express the possibility that a parton i with momentum fraction x may originate from the branching of a parent parton j of the higher momentum fraction y ( $P_{i j}$ is the splitting function). The symbol ⊗ indicates convolution over the variable x as $P_{i j} \otimes f_{j} = \int_{x}^{1} \frac{d y}{y} P_{i j} (\frac{x}{y}) f_{j} (y,$

Result and discussion

In this paper, we obtain the ratio $G (x, Q^{2}) / F_{2}^{s} (x, Q^{2})$ and $F_{L} (x, Q^{2}) / F_{2} (x, Q^{2})$ and the proton structure function at NNLO analysis respectively. The analysis is performed in the range $10^{- 5} \leq x \leq 10^{- 2}$ and $1.5 \leq Q^{2} \leq 150 {GeV}^{2}$ . In Fig. 1 the ratio $G / F_{2}^{s}$ extracted with respect to τ variable where $τ = \frac{Q^{2}}{Q_{0}^{2}} {(\frac{x}{x_{0}})}^{- λ}$ . Here $Q_{0}^{2} = 1 {GeV}^{2}$ , $x_{0} = 3.0 \times 10^{- 4}$ and $λ = λ_{s}$ . The effective exponents for gluon and singlet distributions are defined with an exponent of $λ_{g} = 0.424$ and $λ_{s} = 0.327$ respectively [18]. These values are compatible with

Heavy flavor contribution

As our further research activities we hope to study the ratio of structure functions to get analytical solutions for heavy quark contributions of the structure functions. When the virtual photon interacts indirectly with a gluon in the proton then a heavy quark pair produced via the direct boson-gluon fusion processes. At low-x this behavior is related to the growth of gluon distribution via the $g \to q_{H} {\overline{q}}_{H} (H = c, b)$ transition [27], [28]. Then the perturbative predictions for $F_{L} (x, Q^{2})$ at the $N_{f} = 3$

Conclusion

In this paper we have found that there is in general an analytical relation between the gluon distribution function and singlet structure function at low x region into the effective exponents. The ratio of the structure functions, into the DGLAP evolution equations at small x at NNLO analysis, is studied and compared with EMNS bound in this region. Results are comparable with the experimental data and they are lower than EMNS bound at high- $Q^{2}$ values. Our results are very close to the bounds for

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Ratio of the structure functions and the color dipole model bound

Abstract

Introduction

Section snippets

The ratio G(x,Q2)F2s(x,Q2)

Result and discussion

Heavy flavor contribution

Conclusion

Phys. Lett. B

Eur. Phys. J. C

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Phys. Rev. C

Eur. Phys. J. A

Eur. Phys. J. C

Eur. Phys. J. C

Eur. Phys. J. C

Phys. Rev. C

Z. Phys. C

Z. Phys. C

Phys. Rev. D

Phys. Rev. D

Phys. Lett. B

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Acta Phys. Pol. B

The ratio $\frac{G (x, Q^{2})}{F_{2}^{s} (x, Q^{2})}$