Sivers Differential Cross Section for $\pi+X$ Production via $p^{\uparrow}+p$ Collisions, and the Quark Sivers Function

TL;DR

This paper estimates the Sivers differential cross section for pion production in polarized proton collisions and derives the Sivers function using experimental data and quark distribution functions.

Contribution

It introduces a method to estimate the Sivers differential cross section and extract the Sivers function from experimental data involving polarized proton collisions.

Findings

Estimated the Sivers differential cross section from experimental data.

Derived the Sivers function using quark distribution functions.

Provided insights for future experiments to determine the Collins fragmentation function.

Abstract

We recently estimated pion+X production via proton collisions with a polarized proton target to enable the Collins fragmentation function to be determined via future experiments. In the present article we first estimate the Silvers differential cross section obtianed from experiments with the target proton having both positive and negative polarizations, and then estimate the Sivers Function using quark distribution functions.

Equations

$$A({\varphi }_h,{\varphi }_s)$$

$$\frac{d{\sigma }^{Sivers}}{dy}$$

$$\frac{d{\sigma }_{p^{\uparrow }p\to \pi X}}{dy}$$

$$y$$

$$x(y)$$

$$\Delta f_g(x)$$

$$\Delta f_d(x)$$

$$\Delta f_u(x)$$

$$\Delta f_{\bar{u}}(x)$$

$$\Delta f_{\bar{d}}(x)$$

$$D_{q^{\uparrow }\to q\bar{q}}$$

$$\frac{d{\sigma }^{Sivers}}{dy}$$

$$D^{Sivers}$$

$$(1/\Delta f_g(x)\Delta f_g(a/x))\times [\Delta f_g(x)\Delta f_g(a/x)+\Delta f_d(x)\Delta f_d(a/x)+$$

$$\Delta f_{\bar{d}}(x)\Delta f_{\bar{d}}(a/x)+\Delta f_u(x)\Delta f_u(a/x(y),2m)+$$

$$\Delta f_{\bar{u}}(x)\Delta f_{\bar{u}}(a/x))]\simeq 1.4$$

$$\frac{d{\sigma }^{Sivers}}{dy}$$

$$f_{q{p}^{\uparrow }}(z,{\vec{k}}_T)-f_{q{p}^{\uparrow }}(z,-{\vec{k}}_T)$$

$$f_S(z)$$

$$f_{q{p}^{\uparrow }}(z)$$

$$f_S(z)$$

Full text

PACS Indices:12.38.Aw,13.60.Le,14.40.Lb,14.40Nd

1 Introduction

Recently we estimated the differential cross section for a proton colliding with a polarized proton target to produce a pion and a hadron $X$, $\frac{d\sigma_{p^{\uparrow}p\rightarrow\pi X}}{dy}$, with $y$ the rapidity[1], with a factor $D_{q^{\uparrow}\rightarrow q\bar{q}}$, the fragmentation probability for a polarized quark to fragment into a $q\bar{q}$[2]. From this cross section, and the fragmentation probability to produce a hadron from a transversely polarized quark, one can extract the Collins fragmentation function[3].

The present article is motivated in part by the Letter of Intent by the E1039 collaboration[4], with spokespersons from the Los Alamos National Laboratory, to measure the Sivers Function[5] via Drell-Yan experiments. A Drell-Yan process[6] occurs in hadron-hadron scattering with a quark from one hadron and an antiquark from the other hadron annihilating to create a boson. A number of Deep Inelastic Scattering experiments[7, 8, 9, 10] have measured non-zero values for the Sivers Function, but the E1039 Collaboration plan to make the first measurement of the Sivers Function of sea quarks.

In the following section the Sivers Function obtained from proton collision experiments with a polarized proton target is discussed and the Sivers differential cross section is estimaterd. In the next section the Sivers Function in terms of the quark distribution function in polarized protons is defined and used for our estimate of the Sivers Function.

2 The Sivers Function and the Sivers Cross Section

The Sivers and Collins Functions in terms of experimental cross sections are defined, e.g., in Refs[7, 10]. The Sivers function was defined in terms of quark distribution functions with polarized protons in Ref.[11].

2.1 Sivers and Collins Functions in terms of experiments

For proton collisions with a polarized proton target producing a pion and a hadron (X), which is similar to the differential cross section estimated in Ref[1] for the Collins Fragmentation Function, except that both up and down proton polarizations are needed. The Sivers and Collins Functions are given in terms of the assymmetry $A(\phi,\phi_{s})$, see Refs.[7, 10], where $\sigma_{p^{\uparrow}p\rightarrow\pi X},\sigma_{p^{\downarrow}p\rightarrow\pi X}$ are cross sections for the production of $\pi$ + X via p-p scattering, with the target proton polarization up, down,

[TABLE]

where ($\phi_{h},\phi_{s}$) are the azimuthal angles of the (hadron momentum, target spin) relative to the scattering plane, $S_{T}$ is the component of the target proton spin orthogonal to the scattering plane, $A_{C}$ is the Collins moment, and $A_{S}$ is the Sivers moment. Since the cross sections $\sigma_{p^{\uparrow}p\rightarrow\pi X}(\phi_{h},\phi_{s}),\sigma_{p^{\downarrow}p\rightarrow\pi X}(\phi_{h},\phi_{s})$ are very difficult for experiments to measure we estimate the Sivers differential cross section in the next subsection section and the Sivers Function in the following section.

2.2 Sivers Differential Cross Section

From Eq(1) we define the Sivers Differential Cross Section $\frac{d\sigma^{Sivers}}{dy}$ as

[TABLE]

with From Ref[1] Eq(11) the differential cross section used to evaluate the Sivers differential cross section is, with $Aqq$ a known constant,

[TABLE]

with rapidity $y$

[TABLE]

with $E,p_{z}$ the proton energy and momentum.

From Ref.[12] the distribution functions[13] for polarized $p-p$ collisions for $Q=$ 3 GeV and E=200 GeV

[TABLE]

with $\Delta f_{g}$ the gluon distribution function and $\Delta f_{q},\Delta f_{\bar{u}}$ the quark and anti-quark distribution functions.

[TABLE]

with $D_{q^{\uparrow}\rightarrow q\bar{q}}(z)$ evaluated from Ref.[2]

From Eqs(2,3) the differential cross section $\frac{d\sigma^{Sivers}}{dy}$, $D^{Sivers}=D_{q^{\uparrow}\rightarrow q\bar{q}}-D_{q^{\downarrow}\rightarrow q\bar{q}}$, is

[TABLE]

[TABLE]

Note also that for $-1\leq y\leq 1$

[TABLE]

Therefore, from Eqs(8,2.2)

[TABLE]

Note that in Ref[1] $\frac{d\sigma_{p^{\uparrow}p\rightarrow\pi X}}{dy}$ was calculated for E=200 Gev.

Therefore from Figure 2 in Ref[1] one finds $\frac{d\sigma^{Sivers}}{dy}$ by simply multiplying the differential cross section in Figure 2 in Ref[1] by 3.54

$d\sigma^{Sivers}/dy$ for E=200 GeV is shown in figure 1

3 Sivers Function in terms of Quark Distribution

Functions

The Sivers Function was defined in terms of quark distribution functions in Refs[11, 14].

The Sivers Function $f_{S}(z,k_{T}^{2})$ is given by[14]

[TABLE]

where $\vec{P}$ is the proton momentum, $\vec{k}_{T}$ is quark momentum perpendicular to $\vec{P}$, $\vec{S}$ is the normalized proton spin, with $S^{2}=-1.0$, and $M$ is the proton mass.

With the choice of frame the target rest frame, the Sivers Function can be derived from the quark distribution function with polarized protons:

[TABLE]

Therefore, an approximate estimate of the Sivers Function can be obtained from the quark fragmentation function $D_{q^{\uparrow}\rightarrow q\bar{q}}(z)$ from Ref[1], which was obtained from Ref[2] .

3.1 Estimate of the Sivers Function in terms of Quark

Distribution Functions

We make an approximent estimate of the Sivers function $f_{S}(z)$ using $f_{qp^{\uparrow}}(z)\simeq D_{q^{\uparrow}\rightarrow q\bar{q}}(z)$ The estimate of $D_{q^{\uparrow}\rightarrow q\bar{q}}(z)$ from the Eq(31) in Ref[2] was discussed in Ref[1].

[TABLE]

with $0\leq z\leq 1$. Therefore from Eqs(12,13)

[TABLE]

For E=$\sqrt{P^{2}+M^{2}}$=200 GeV and the proton mass M$\simeq$ 1 GeV, $P/2M\simeq 100$, giving

[TABLE]

The Sivers function, $f_{S}(z)$, is shown in Figure 2.

4 Conclusions

We have estimated the Sivers Differential Cross Section. As shown in Figure 1 it should be possible to measure $\frac{d\sigma^{Sivers}}{dy}$ . We have also estimated the Sivers Function, shown in Figure 2.

**Acknowledgements

**This work was supported in part by the DOE contracts W-7405-ENG-36 and DE-FG02-97ER41014. The author LSK was a visitor at Los Alamos National Laboratory, Group P25. The authors declare that there is no conflict of interest regarding the publication of this paper.