Coulomb-Nuclear Interference in Polarized pA Scattering

Abstract

We made the first attempt to understand the observed unusual t dependence of single-spin asymmetry observed in the HJET experiment at RHIC. Usually, the interaction of hadrons is presented as a long-range Coulomb interaction and a short-range strong interaction with Coulomb corrections. Such a division gives rise to a Coulomb phase of the hadronic term. Conversely, here we consider short-range hadronic interaction as a correction to the long-range electromagnetic term, i.e., we treat it as an absorptive correction. This significantly affects the Coulomb-nuclear interference, which is a source of single-spin azimuthal asymmetry at small angles.

1. Introduction

Elastic scattering is usually characterized by spin non-flip $f_{nf}$ and spin-flip $f_{sf}$ amplitudes, which determine the differential elastic cross-sections and single-spin azimuthal asymmetry $A_N\left(t\right)$,

$$\begin{array}{ccc}\hfill \frac{d{\sigma }_{el}}{dt}& =& \pi \left(|f_{nf}{|}^2+{\left|f_{sf}\right|}^2\right),\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill A_N\frac{d{\sigma }_{el}}{dt}& =& -2\pi \mathrm{Im}\left(f_{nf}{f}_{sf}^{*}\right),\hfill \end{array}$$

(2)

where t is 4-momentum transfer squared. The latter contains interference of the helicity amplitudes, which makes polarization effects sensitive to the hadron interaction dynamics [1,2].

According to Equation (2), an important condition for azimuthal asymmetry is the existence of a phase shift between the spin amplitudes. However, to the best of our knowledge, the phases of $f_{nf}$ and $f_{sf}$ at high energies are similar. For example, in the Regge pole model, $f_{sf}^h$ and $f_{nf}^h$ have the same phase given by the signature factor. To maximize $A_N\left(t\right)$, one should combine hadronic and electromagnetic amplitudes [3]. While the former is predominantly imaginary, the latter is nearly real. Apparently, a sizable effect is expected at small momentum transfer squared t, where the Coulomb and hadronic amplitudes are of the same order. The t dependence of asymmetry in $pp$ scattering is described in a simplified approximation [3] by

$$A_N^{pp}\left(t\right)=A_N^{pp}\left(t_p\right)\frac{4y^{3/2}}{3y^2+1},$$

(3)

where $y=-t/t_p^{pp}$ and

$$\begin{array}{ccc}\hfill t_p^{pp}& =& \frac{8\sqrt{3}\pi {\alpha }_{em}}{{\sigma }_{tot}^{pp}};\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill A_N^{pp}\left(t_p\right)& =& \frac{\sqrt{3t_p}}{4m_p}({\mu }_p-1).\hfill \end{array}$$

(5)

Here, ${\alpha }_{em}=1/137$ is the fine structure constant; ${\mu }_p=2.79$ is the proton magnetic moment. The asymmetry $A_N^{pp}\left(t\right)$ reaches a maximum (5) of about 4–5%, at $t=t_p\approx 2\times {10}^{-3}{\mathrm{GeV}}^2$. For the sake of simplicity, we assume here, like in [3], a pure non-flip and imaginary hadronic amplitude, no Coulomb phase, etc. In what follows, we present most of these simplifications.

The CNI asymmetry Equation (3) predicted in [3] was confirmed by measurements in [4,5]. It is worth mentioning at least two important practical applications of the CNI effect. First, it is a unique opportunity to measure the spin-flip component of the hadronic amplitude at high energies [6,7].

Second, an important application of CNI-generated azimuthal asymmetry is the possibility to measure the polarization of the beam, in particular at RHIC. The usage of CNI as a polarimeter has been intensively studied in [8,9,10,11,12,13,14,15,16,17,18,19,20].

Similar relations can be applied to proton–nucleus elastic scattering,

$$t_p^{pA}=K_A{t}_p^{pp},$$

(6)

with

$$K_A=\frac{Z{\sigma }_{tot}^{pp}}{{\sigma }_{tot}^{pA}}.$$

(7)

Correspondingly, the maximal value of $A_N$ at $t=t_p^{pA}$ reads,

$$A_N^{pA}\left(t_p^{pA}\right)=\sqrt{K_A}{A}_N^{pp}\left(t_p\right).$$

(8)

A simple estimate ${\sigma }_{tot}^{pA}$∼$A^{2/3}{\sigma }_{tot}^{pp}$ leads to non-dramatic modification of the CNI asymmetry, with about a 20% increase in $A_N^{pA}\left(t_p^{pA}\right)$ even for heavy nuclei, e.g., gold.

The energy of $100\mathrm{GeV}$ in the Lab frame is not sufficiently high to suppress iso-vector Reggeons, which have quite a large (dominating) spin-flip component. This is why it is difficult to disentangle large Reggeon and small Pomeron spin-flip terms. On iso-scalar nuclei, e.g., carbon, copper, etc., iso-vector Reggeons are completely excluded, otherwise they are suppressed by a small factor $(A-2Z)/A$.

2. Born Approximation

The elastic $pA$ amplitude is fully described by two helicity amplitudes $f_{nf,sf}$ defined in [2], each having hadronic and electromagnetic parts, $f_i\left(q_T\right)=f_i^h\left(q_T\right)+f_i^{em}\left(q_T\right)$, where for small-angle elastic scattering $t\equiv -q^2\approx -q_T^2$, they are comparable.

Hadronic Born amplitudes can be represented as

$${\left.f_{nf}^{pA\left(h\right)}\left(q_T\right)\right|}_B=\frac{{\sigma }_{tot}^{pA}}{4\pi }{F}_A^h\left(q_T^2\right),$$

(9)

$${\left.f_{sf}^{pA\left(h\right)}\left(q_T\right)\right|}_B=r_5^{pA}\frac{q_T}{m_N}\frac{{\sigma }_{tot}^{pA}}{4\pi }\mathrm{Im}{F}_A^h\left(q_T^2\right),$$

(10)

where $r_5^{pA}$ is a nuclear analog of $r_5^{pp}$ defined in [2]

$$r_5^{pp}=\frac{m_N{f}_{sf}\left(q_T\right)}{q_T\mathrm{Im}{f}_{nf}\left(q_T\right)}.$$

(11)

For $pp$ elastic scattering, $r_5^{pp}$ was fitted to data in [5,7].

The hadronic nuclear form factor $F_A^h\left(q_T^2\right)$ in Equations (9) and (10) can be evaluated within the Glauber approximation [21],

$$F_A^h\left(q_T\right)=\frac{2i}{{\sigma }_{tot}^{pA}}\int d^{2}b{e}^{i{\overrightarrow{q}}_T·\overrightarrow{b}}\left[1-{\left(1-\frac{1}{2A}{\sigma }_{tot}^{pp}(1-i{\rho }_{pp}\left(s\right))T_A^h\left(b\right)\right)}^A\right],$$

(12)

as well as the total nuclear cross-section,

$${\sigma }_{tot}^{pA}=2\mathrm{Im}\int d^{2}bi\left[1-{\left(1-\frac{1}{2A}{\sigma }_{tot}^{pp}\left(s\right)(1-i{\rho }_{pp}\left(s\right))T_A^h\left(b\right)\right)}^A\right].$$

(13)

Correspondingly, the Born electromagnetic amplitudes read

$${\left.f_{nf}^{pA\left(em\right)}\left(q_T\right)\right|}_B=-2Z{\alpha }_{em}\frac{1}{q_T^2+{\lambda }^2}{F}_A^{em}\left(q_T\right),$$

(14)

$${\left.f_{sf}^{pA\left(em\right)}\left(q_T\right)\right|}_B=-Z{\alpha }_{em}\frac{1}{m_N{q}_T}({\mu }_p-1)F_A^{em}\left(q_T\right),$$

(15)

where the small fictitious photon mass $\lambda$ is introduced to avoid infrared divergence. The final results are checked for stability at $\lambda \to 0$.

The nuclear electromagnetic form factor in Equations (14) and (15) has the form

$$F_A^{em}\left(q_T\right)=\frac{1}{Z}\int d^{2}b{e}^{i{\overrightarrow{q}}_T·\overrightarrow{b}}{T}_Z\left(b\right),$$

(16)

The nuclear thickness functions are defined as

$$T_A\left(b\right)={\int }_{-\infty }^{\infty }dz\rho (b,z),$$

(17)

where $\rho (b,z)$ is the nuclear density distribution function, and $T_Z\left(b\right)=T_A\left(b\right)Z/A$.

Following [21,22], next we replace the nuclear thickness function with a more accurate effective thickness function convoluted with $NN$ elastic amplitude

$$T_A\left(b\right)\Rightarrow T_A^h\left(b\right),$$

(18)

where

$$T_A^h\left(b\right)=\frac{2}{{\sigma }_{tot}^{hN}}\int d^{2}s\mathrm{Re}{\Gamma }^{hN}\left(s\right)T_A(\overrightarrow{b}-\overrightarrow{s});$$

(19)

$$\mathrm{Re}{\Gamma }^{hN}\left(s\right)=\frac{{\sigma }_{tot}^{hN}}{4\pi B_{hN}}exp\left(-\frac{s^2}{2B_{hN}}\right),$$

(20)

where $B_{hN}$ is the slope of the differential $hN$ elastic cross-section. This effective nuclear thickness function can be simplified to

$$\begin{array}{ccc}\hfill T_A^h\left(b\right)& =& \frac{2}{{\sigma }_{tot}^{hN}}\int d^{2}s\frac{{\sigma }_{tot}^{hN}}{4\pi B_{hN}}exp\left(-\frac{s^2}{2B_{hN}}\right)T_A(\overrightarrow{b}-\overrightarrow{s})\hfill \\ & =& \frac{1}{2\pi B_{hN}}\int dsd\varphi sexp\left(-\frac{s^2}{2B_{hN}}\right)T_A\left(\sqrt{b^2+s^2-2bscos\varphi }\right).\hfill \end{array}$$

(21)

3. Hadronic vs. Electromagnetic Amplitudes

The long-range Coulomb forces affect the strong-interaction amplitude by generating a phase shift, known as the Coulomb phase [23,24,25]. The interplay of Coulomb and hadronic interaction mechanisms is illustrated in Figure 1, following the consideration of this problem in [23].

In [23] the last two graphs, (N) and (CN) were combined and treated as a Coulomb-modified strong-interaction amplitude. The modification was approximated by giving an extra phase factor to the hadronic amplitude. In the literature, this factor is called the Coulomb phase [23,24,25].

Alternatively, one can combine (C) and (CN) and obtain a hadronic correction to the Coulomb amplitude. In what follows, we call this absorptive correction [26,27].

As usual, multiple interaction amplitudes, depicted in Figure 1, are easily calculated in impact parameter representation, where the result is just a product of multiple amplitudes. Thus, we switch from $q_T$ to b dependent amplitudes, and simultaneously, from the Born approximation to the eikonal optical model.

$$\begin{array}{ccc}\hfill {\gamma }_{nf}^{pA\left(h\right)}\left(b\right)& =& \frac{i}{2\pi }\int d^2{q}_T{e}^{-i{\overrightarrow{q}}_T·\overrightarrow{b}}{f}_{nf}^{pA\left(h\right)}\left(q_T\right)\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\gamma }_{sf}^{pA\left(h\right)}\left(b\right)& =& \frac{i}{2\pi }\int d^2{q}_T{e}^{-i{\overrightarrow{q}}_T·\overrightarrow{b}}{f}_{sf}^{pA\left(h\right)}\left(q_T\right)\hfill \end{array}$$

(23)

These hadronic eikonal phases were used in the Glauber model expressions (12) and (13).

Adding higher order terms to the Born Coulomb amplitudes (22) and (23), as is explained in [27], we arrive at the eikonal form for the electromagnetic amplitudes

$$f_{nf}^C\left(q_T\right)=\frac{i}{2\pi }\int d^{2}b{e}^{i{\overrightarrow{q}}_T·\overrightarrow{b}}\left(1-e^{i{\chi }_{nf}^C\left(b\right)}\right),$$

(24)

$$f_{sf}^C\left(q_T\right)=\frac{1}{2\pi }\int d^{2}b{e}^{i{\overrightarrow{q}}_T·\overrightarrow{b}}{\chi }_C^{sf}\left(b\right)e^{i{\chi }_{nf}^C\left(b\right)},$$

(25)

with Coulomb eikonal phases, given by the Born amplitudes in impact parameter representation.

$$\begin{array}{ccc}\hfill {\chi }_{nf}^C\left(b\right)& =& \frac{-2Z{\alpha }_{em}}{2\pi }\int d^2{q}_T{e}^{-i{\overrightarrow{q}}_T·\overrightarrow{b}}\frac{F_A^{em}\left(q_T\right)}{q_T^2+{\lambda }^2}\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill {\chi }_{sf}^C\left(b\right)& =& \frac{-Z{\alpha }_{em}({\mu }_p-1)}{2\pi m_N}\int d^2{q}_T{e}^{-i{\overrightarrow{q}}_T·\overrightarrow{b}}\frac{F_A^{em}\left(q_T\right)}{q_T^2+{\lambda }^2}\frac{{\overrightarrow{q}}_T·\overrightarrow{b}}{b}.\hfill \end{array}$$

(27)

Combining the Coulomb (C) with Coulomb-nuclear (CN) mechanisms depicted in Figure 1, one obtains the Coulomb terms, Equations (24) or (25), which acquire an absorption factor, given by the standard Glauber eikonal approximation [28],

$$\begin{array}{ccc}\hfill S\left(b\right)& =& 1-{\gamma }_{nf}^{pA\left(h\right)}\left(b\right)={\left(1-\frac{1}{2A}{\sigma }_{tot}^{pp}\left(s\right)(1-i{\rho }_{pp}\left(s\right))T_A^h\left(b\right)\right)}^A\hfill \\ & \approx & exp\left[-\frac{1}{2}{\sigma }_{tot}^{pp}\left(s\right)T_A^h\left(b\right)\right].\hfill \end{array}$$

(28)

As usual, the complicated multi-loop integrations in momentum representation are essentially simplified to a multiplicative combination in impact parameters. This is why the correction $S\left(b\right)$ of Equation (28) enters the final expressions as a factor. Thus, the absorption-corrected amplitudes $\tilde{f}$ take the form

$${\tilde{f}}_{nf}^C\left(q_T\right)=\frac{i}{2\pi }\int d^{2}b{e}^{i{\overrightarrow{q}}_T·\overrightarrow{b}}\left(1-e^{i{\chi }_{nf}^C\left(b\right)}\right)S\left(b\right),$$

(29)

$${\tilde{f}}_{sf}^C\left(q_T\right)=\frac{1}{2\pi }\int d^{2}b{e}^{i{\overrightarrow{q}}_T·\overrightarrow{b}}{\chi }_C^{sf}\left(b\right)e^{i{\chi }_{nf}^C\left(b\right)}S\left(b\right),$$

(30)

Eventually, we arrive at the full amplitudes.

$$f_{nf,sf}\left(q_T\right)={\tilde{f}}_{nf,sf}^C\left(q_T\right)+f_{nf,sf}^N\left(q_T\right).$$

(31)

4. Results vs. Data

Now we are in a position to calculate the proton–nucleus total cross-section and single-spin asymmetry, given by Equations (1) and (2). The results are compared with the data in Figure 2, Figure 3, Figure 4 and Figure 5.

First of all, we performed calculations within the Born approximation, Equations (9), (10), (14) and (15). The hadronic spin-flip component was set to zero. The results are depicted with black dotted curves in Figure 2 and Figure 5. The magnitude of $A_N$ substantially exceeds the data.

Then, we relied on the eikonal form of higher order terms, Equations (24) and (25), keeping $r_5=0$. The results are depicted with blue double-dot-dashed curves. The effect of eikonalization turns out to be rather mild, and the discrepancy with the data remains significant.

The next step is the introduction of absorptive corrections, which significantly reduce the values of $A_N\left(t\right)$ as is demonstrated by the red solid curves, calculated with Equations (29) and (30) ($r_5$ is still zero).

Eventually, we can adjust the single unknown parameter, $r_5^{pA}$, for each nuclear target. The results of the fit are presented in

Table 1. Results of fit for $r_5^{pA}$ to data on $A_N\left(t\right)$ for different nuclei [29,30,31].

Experiment	Re ${\mathit{r}}_5$ fit	Im ${\mathit{r}}_5$ fit
p-C @ 100 GeV	$-0.071\pm 0.003$	$-0.063\pm 0.005$
p-Al @ 100 GeV	$-0.080\pm 0.001$	$-0.101\pm 0.002$
p-Ru @ 100 GeV	$-0.068\pm 0.001$	$-0.469\pm 0.002$
p-Au @ 100 GeV	$-0.027\pm 0.008$	$-0.526\pm 0.006$

.

The found values of $r_5^{pA}$ are pretty close to the values for the Pomeron spin component found by fitting to the $pp$ data in [7].

With values of $r_5$ in Table 1 and amplitude Equations (29) and (30), we plot the green dashed curves, which we treat as our final results.

Notice that the nuclear data are quite sensitive to the value of $r_5^{pA}$; this is why the CNI method was proposed [6] as a unique way for measuring the hadronic spin-flip amplitudes at high energies.

5. Conclusions

Concluding, we performed the first calculation of single-spin asymmetry in polarized proton–nucleus elastic scattering in the CNI region. We achieved a reasonable agreement with the data, in spite of the rather complicated theoretical construction. The remarkable feature of the nuclear form factor (12) is a change in the sign of the imaginary part of the elastic hadronic non-flip amplitude corresponding to the first zero of the Bessel function $J_1\left(t\right)$. Since the non-flip hadronic amplitude significantly exceeds the spin-flip part, they become of the same order right before and after the former changes sign. The asymmetry $A_N\left(t\right)$ Equation (2) reaches maximum at $f_{nf}=f_{sf}$, so $A_N\left(t\right)$ should abruptly vary between positive and negative maxima in the vicinity of the Bessel zero. Comparison with the data in Figure 4 and Figure 5 confirms such behavior; indeed, the measured $A_N\left(t\right)$ exposes two maxima with opposite signs with positions close to predicted.

However, the magnitude of these maxima is exaggerated in our calculations, so there is still room for improvements. Here, nuclear effects were evaluated within the Glauber approximation, which is subject to Gribov inelastic shadowing corrections [32,33]. Their calculation requires substantial modeling, including knowledge of the proton wave function, interaction mechanism, etc. This needs a detailed study, as was performed in [22]. We leave this issue for future development.