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\begin{document}
\centerline{ 
\Large Hadron spin-flip amplitude: an analysis of the new
$A_N$ data from RHIC }  
\vskip 1cm
\centerline{\large O.V. Selyugin\footnote{on leave from Bogoliubov
 Laboratory of Theoretical Physics, JINR, 141980, Dubna, Moscow Region,
  Russia}$^,$\footnote{selugin@qcd.theo.phys.ulg.ac.be} and J.-R. Cudell
\footnote{J.R.Cudell@ulg.ac.be}}

\centerline{
Institut de Physique, B\^at. B5a, Universit\'e de Li\`ege, Sart Tilman, B4000
  Li\`ege, Belgium }
\vskip 1cm

\centerline{Abstract}
\begin{quote}
\small Through a direct analysis of the scattering amplitude,
we show that the preliminary measurement of $A_N$ obtained by the E950 
Collaboration at different energies are mainly sensitive
to the spin-flip part of the amplitude in which
the proton scatters with the $^{12}C$ nucleus as a whole. 
The imaginary part of this amplitude is negative,
and the real part positive. We give predictions for 
$p_L = 600$~GeV/$c$, which  depend mainly on
the size of the real part of the amplitude. 
\end{quote}
\vskip 1cm
\newcommand{\ahsf}{${\cal A}^H_{sf}$}
\newcommand{\ahnf}{${\cal A}^h_{nf}$}
\newcommand{\acsf}{${\cal A}^{em}_{sf}$}
\newcommand{\acnf}{${\cal A}^{em}_{nf}$}
Diffractive polarized experiments open a new window on the
spin properties of QCD at large distances. In particular,
the recent data from RHIC and HERA indicate 
that, even at high energy, the hadronic amplitude has 
a significant spin-flip contribution, ${\cal A}^h_{sf}$,
which remains proportional to the spin-non-flip part, \ahnf, 
as energy is increased \cite{akch,sel-pl}.  
In other words, the pomeron coupling to the proton 
and/or to the nuclei has a non-trivial spin structure.
%This is expected from perturbative \cite{ter} as well 
%as non-perturbative \cite{bsw,zpc} estimates.

The new RHIC fixed-target data, from E950, consists 
in measurements of the analysing power 
\be A_N(t)={\ua-\da\over \ua+\da}\ee
for momentum transfer $0 \leq |t| \leq 0.05$~GeV$^2$,
for a polarized $p$ beam hitting a (spin-0) $^{12}C$. In this 
region of $t$, 
the electromagnetic amplitude is of the same order of magnitude 
as the hadronic amplitude, and the interference of the imaginary part 
of \ahnf with the spin-flip part of the 
electromagnetic amplitude \acsf leads to a peak in the analyzing 
power $A_N$,
usually referred to as the Coulomb-Nuclear Interference (CNI) effect
\cite{schwinger,bib14,lead}. This effect was observed in the data from
\ci{akex}, but the errors were too big to draw any conclusion 
on the hadron spin-flip amplitude.

The first RHIC measurements at $p_L = 22 $~GeV/$c$ \cite{an22}  
in $p ^{12}C$ scattering indicated however that
$A_N$ changes sign already at very small momentum transfer.
Such a behaviour cannot be described by the CNI effect alone. Indeed,
fits to the data \cite{kt} give  
for 
\ba r_5&=& \lim\limits_{t \to 0}{ \ \tilde{ \cal A}^h}_{sf}/{Im(\cal A}^h_{nf})
\equiv R+iI:\label{r5}\\
     R&=&0.088 \pm 0.058;
     \ \ \ I=-0.161 \pm 0.226  
\ea
As usual, $
 \tilde{\A}_{sf}(s,t) \equiv  2\ m_p\ \A_{sf}(s,t)/ \sqrt{|t|}
$ is the  ``reduced'' spin-flip amplitude
factoring out trivial kinematical factors.
The large error on $Im(r_5)$ unfortunately 
leads to a high uncertainty on the size of 
the hadronic spin-flip amplitude. 

Before analysing the the new (preliminary) data at $p_L=24$ and $100$ 
GeV/$c$ \cite{an100}, we need to define the ingredients of the theoretical
description. 
%In general, the shape of the analyzing power, $A_N$, and
%the position of its maximum depend on the parameters of 
%the elastic scattering amplitude ${\cal A}_e(s,t)$: $\sigma_{tot}(s)$,  
%$\rho(s,t)={Im({\cal A}_e)\over Re({\cal A}_e)}$, 
%the Coulomb-nucleon interference phase  $\varphi_{CN}(s,t)$
%and the elastic slope  $B(s,t)$.
%The latter was calculated
%in the entire diffraction domain taking into account the form factors
%of the  nucleons \cite{prd-sum}.
Isoscalar targets such as $^{12}C$ simplify the calculation as they 
suppress the contribution of the isovector reggeons $\rho$ and $a_2$, 
by some power of the atomic number. Also, as $^{12}C$ is spin 0,
there are only two independent helicity amplitudes: proton spin flip 
and proton spin non flip.
However, nuclear targets lead to large theoretical uncertainties
because of the difficulties linked to nuclear structure, and 
because of the lack of high-energy proton-nucleus scattering 
experiments (see, for example, \cite{harr}).
Given these problems, we shall not rely on theoretical models
(such as the Glauber formalism) but rather parametrise the scattering 
amplitude directly from data, and take the interference terms fully 
into account.

The elastic and total cross sections and the analysing power 
$A_N$ for $p ^{12}C$ scattering are given by
\ba
d\sigma/dt \ &=& \   2 \pi \left(|\A_{nf}|^2+ |\A_{sf}|^2\right), 
 \nonumber\\
\sigma_{tot}&=& 4 \pi Im(\A_{nf}),\label{an}\\
A_N \ d\sigma/dt \ &=&  \ - 2\pi 
                   Im[ \A_{nf} \A_{sf}^{*})]. \nonumber
\ea
 
Each term includes a hadronic and an electromagnetic contribution:
$
 \A_i(s,t) = \p^h_{i}(s,t) 
        + \p_{i}^{em}(t) e^{i\delta}, (i=nf, sf),
$
where $\p^h_{i}(s,t)$ describes the strong interaction of $p ^{12}C$,
and $\p_{i}^{em}(t)$  the  electromagnetic interaction.
$\alpha_{em}$ is the electromagnetic fine structure constant, and
the Coulomb-hadron phase $\delta$ is given by
$\delta=Z\alpha_{em} \varphi_{CN}$ 
with $Z$ the charge of the nucleus, and $\varphi_{CN}$
the Coulomb-nuclear phase \cite{prd-sum}.
The electromagnetic part of the scattering amplitude can be written 
as
\ba
\A^{em}_{nf}&=& {2 \alpha_{em} \ Z\over t} \ F^{^{12}C}_{em} F^{p}_{em1},
 \\ \nonumber
\A^{em}_{sf}&=& - {\alpha_{em} \ Z\over m_p \sqrt{|t|}} \ 
                           F^{^{12}C}_{em} F^{p}_{em2},
\ea
where 
$ F^{p}_{em1}$ and $ F^{p}_{em2}$ are
the electromagnetic form factors of the proton, and
$F^{^{12}C}_{em}$ that of $^{12}C$.
We use 
\ba 
F^{p}_{em1}&=& {4 m_p^2-t (\kappa_p+1)\over (4 m_p^2-t)(1-t/0.71)^2} , \\ 
F^{p}_{em2}&=& {4 m_p^2\kappa_p\over (4 m_p^2-t)(1-t/0.71)^2} , 
\ea
 where $m_p$ is the mass of the proton and $\kappa_p$ its anomalous 
magnetic moment.
We obtain $F^{^{12}C}_{em}$
from the electromagnetic density of the nucleus 
\ba
D(r) = D_0 \ \left[1+ \tilde{\alpha}\left({r\over a}\right)^2
\right]e^{-\left({r\over a}\right)^2}.
\ea
$\tilde{\alpha}=1.07$ and $a=1.7$ fm give the best description
 of the data \cite{jansen} in the small-$|t|$ region,
and produce a zero of $F^{^{12}C}_{em}$ at $|t| = 0.130 $~GeV$^2$. 
We also calculated $F^{^{12}C}_{em}$ by integration of the nuclear 
form factor given by a sum of Gaussians \cite{data35} and obtained 
practically the same result with the zero now at  $|t| = 0.133$ 
GeV$^2$. 


The parts of the scattering amplitudes due to strong interaction 
are assumed to be well approximated by falling exponentials in the 
small-$t$ region. The slope parameter $B(s,t)/2$ is then the derivative 
of the logarithm of the amplitude with respect to $t$.  
If one considers only one contribution to the amplitude, 
this coincides with 
the slope of the differential cross section. In a more
complicated case,
there is no direct correspondance with the
cross section because of interference terms.   

The hadron spin-non flip amplitude  consists of
two parts, describing respectively the scattering of the 
proton on separate nucleons in the nucleus, and 
the scattering of the proton on the nucleus as a whole: 
\ba
{\cal A}^{h}_{nf}(s,t) \ &=& \ \A^{pN}_{nf}(s,t) \ + \   \A^{pA}_{nf}(s,t),\label{full}\\
\A^{pN}_{nf}(s,t) &=& (1+\rho^{pN})  
{\sigma_{tot}^{pN}(s)\over 4\pi} \exp\left({B^{pN}\over 2}t\right),
\\ 
\A^{pA}_{nf}(s,t) &=& (1+\rho^{pA}) 
{\sigma_{tot}^{pA}(s)\over 4\pi} \exp\left({B^{pA}\over 2}t\right) .
\label{f-h}
\ea
The size and energy dependence of $\rho^{pN}$ and $B^{pN}$
are assumed to be the same as in the $pp$ case
\cite{selex}:    
\ba
\rho^{pN}(s)& =& 6.8/p_{L}^{0.742} -  6.6/p_{L}^{0.599}+0.124,
\\ \nonumber
B^{pN}(s)& =& 11.13 -  6.21/\sqrt{p_{L}} - 0.3 \ \ln{p_{L}}.
\ea
For  $\sigma_{tot}^{pN}(s)$, we use the fact that $\sigma_{tot}^{pN}\approx
2\sigma_{tot}^{pp}$ and take the best form obtained
 in \cite{prd-comp}, which works well in this energy region:
\ba
\sigma_{tot}^{pN}(s) & \approx &2\ \sigma_{tot}^{pp}(s)\label{fsigg}\\
&=&86~ (s/s_1 )^{-0.46} \ - 66~(s/s_1 )^{-0.545} \nonumber\\
   &&  + 71 + 0.614~\ln^2{\left(s/s_0\right)},
\ea
 with all coefficients in mb, $s_{1}=1$~GeV$^2$ and
 $s_{0}=29$~GeV$^2$. 
The factor $2$ in (\ref{fsigg})
reflects a complicated hadron-hadron interaction in the nucleus
and is determined from the comparison of our calculation of 
$d \sigma / dt$ with the data of $p ^{12}C$ 
scattering \cite{shiz} in the region 
$0.1\leq |t|\leq 0.2$~GeV$^2$, where the term 
$ \A^{pN}_{nf}(s,t)$
 of the scattering amplitude gives the main contribution.

For the determination of $ \A^{pA}_{nf}(s,t)$, we rely
on the data obtained by the SELEX Collaboration.
This experiment on $pC$ scattering at $p_L = 600$~GeV/$c$ 
gives us 
  $\sigma^{pC}_{tot} \ = \ 341 $~mb;  $B^{pC}(t \approx 0.02{\ \rm GeV}^2)
 =  62$~GeV$^{-2}$.

To obtain the values of these parameters for other energies, 
we make the following 
assumptions on their energy dependence:
some analyses \cite{karol} and the data \cite{murthy}
show that the ratio $R_{C/p}$ of $\sigma_{tot}(p ^{12}C)$ to $\sigma_{tot}(pp)$
decreases very slowly in the region $5 \leq p_L \leq  600 $~GeV/$c$.
We take its energy
dependence, according to the data~\cite{shiz}, as 
$ R_{C/p} = 9.5 \ (1 - 0.015 \ln{s})$.
From this we obtain $\sigma_{tot}^{pA}(s)\equiv\sigma_{tot}^{pC}(s) 
- \sigma_{tot}^{pN}(s)
$.
We assume that the slope slowly rises with $\ln{s}$ in a way 
similar to the $pp$ case, and normalise it so that the full amplitude (\ref{full}) has a slope of  $62$~GeV$^{-2}$ at $p_L= 600$~GeV/c and
 $|t|=0.02$~GeV$^2$. This gives $ B^{pA}= 70 \ (1 +0.05 \ \ln{s})$.

We do not know the energy dependence of $\rho^{pA}$, 
but because the $\rho$ and $a_2$ trajectories are suppressed, and
because they contribute negatively, it
must be larger than in the $pp$ case, where it is about $-0.1$ in 
this energy region.
In fact, 
the data from \cite{selex,shiz} indicate that $\rho^{pA}$ is positive. We also
know that at very high energy, $\rho^{pA}$ should be of the order of
$\rho^{pp}$, which is about 0.1. 
We thus assume that at RHIC energies, it is of the order of 0.05, 
and that it changes
logarithmically with $s$, similarly to the $pp$ case.
We also allow for an extra term proportional 
to $\rho_{pp}$
with a linear suppression in $A$. This gives us two variants:  
\ba
\rho^{pA}& =& 0.05/(1- 0.05 \ \ln{s}) + \rho_{pp}/A, \label{fro} \\ 
\rho^{pA}& =& 0.05/(1- 0.05 \ \ln{s}). \label{sro}
\ea

 At small transfer momenta  $|t| \leq 0.03$~GeV$^2$, the ratio of the effective hadronic form factor ${\cal A}_{nf}^h(s,t)/{\cal A}_{nf}^h(s,0)$ to 
 the effective electromagnetic form factor ${\cal A}_{nf}^{em}(s,t)/{\cal A}_{nf}^{em}(s,0)$ is roughly equal to 1
and grows slowly  to $1.25$ at $|t|  = 0.05$~GeV$^2$.  

The hadron spin-flip amplitude of  $p ^{12}C$ scattering is 
mainly due to 
the interaction of the proton with the nucleus as a whole. 
The spin-dependent scattering of the proton on one separate
nucleon of $^{12}C$ is averaged to zero by the nuclear
wave function.  Scattering on multiple nucleons is suppressed
because these must resum to spin zero and because
the anomalous magnetic moments
of the proton and of the neutron have opposite signs.
We show in Figs.~2 and 3 the (small) additional effect of a $pN$ contribution set to 
10\% of the $pp$ spin non flip:
\ba
{\tilde\A^{pN}_{sf}}(s,t)={{\A^{pN}_{nf}}(s,t)/ 10}    
.\label{sfpN}  
\ea     

We parametrise the remainder spin-flip part of $p ^{12}C$ scattering as    
\ba
{\A^{h}_{sf}}(s,t)&=&   (k_2\rho^{pA}+i k_1) 
  { \sqrt{|t|}\sigma_{tot}^{pA}(s)\over 4\pi}\nonumber\\ 
&\phantom{=}& \times \exp\left({ B^{pA}\over 2}t\right).  
\ea     
We have assumed here that 
the spin-flip and the spin-non-flip amplitude have the same slope.
One could of course allow for more freedom and take different slopes,
but the data are not yet precise enough to test this.
%Fig. 1
\begin{figure}
\epsfysize=6.cm
\epsfxsize=8.5cm
\vglue -1cm
\centerline{\epsfbox{c12bhxa.ps}}
\vglue -1cm
\centerline{\epsfbox{c12bhxb.ps}}
\caption{ 
The analysing power $A_N $ (in \%) at $p_{L} = 24$~GeV/$c$ and
$100 $~GeV/$c$, compared with the data 
\cite{an22,an100} (only statistical errors are shown). 
The two scenarions (\ref{fro}) and \ref{sro} lead respectively to
the upper and lower curves (these are indistinguishable at 24~GeV/$c$).
The dot-dashed curves correspond to the addition of a small
spin-flip contribution from nucleons (\ref{sfpN}).}
\end{figure}

From the full scattering amplitude, the analyzing power is given by  
\ba
  A_N\frac{d\sigma}{dt} =
         - 4 \pi [Im(\A_{nf})Re(\A_{sf})-Re(\A_{nf})Im(\A_{sf})],\nonumber 
\ea 
each term having electromagnetic and hadronic contributions.
We can now calculate the form of the analyzing power $A_N$ at
small momentum transfer with different 
coefficients $k_1$ and $k_2$ chosen to obtain the best description
of $A_N$ at $p_L=24 $~GeV/$c$ and $p_L = 100 $~GeV/$c$. 
Of course, we only aim at a qualitative description
as the data are only preliminary and as they are
normalized to those
at $p_L = 22 $~GeV/$c$ \cite{an22}.

The preliminary data show that $A_N$ decreases
very fast after its maximum and is almost zero in a large
region of momentum transfer. This behaviour can be explained
only if one assumes a negative contribution of the interference 
between different parts of the hadron amplitude, that    
changes slowly with energy. 
%Fig. 3
\begin{figure}
\epsfysize=6.cm
\epsfxsize=9.cm
\vglue -1.cm
\centerline{\epsfbox{c12bhxc.ps}}
\caption{
  The  predictions 
for   $A_N $ (in \%) at $p_{L} = 600 $~GeV/$c$. The curves are as in Fig.~1.}
\end{figure}

 The data at  $p_L = 100 $~GeV/$c$ decrease faster
than those at $p_L = 24 $~GeV/$c$, and the zero 
of $A_N$ moves to lower values of $|t|$.
This change of sign is independent from the normalization 
of the data. It would be very interesting to
obtain new data with higher accuracy and at higher energies in order to
distinguish between the two scenarios (\ref{fro}) and (\ref{sro}).

The best descriptions 
of the data, shown in Fig.~1, lead to the values of $R$ and $I$ given in Table I. \\ 
%\begin{itemize}
%\item 
%in the case (\ref{fro}) \\
%$$ I = - 0.15, \ \ R_1 = 0.1 \ \ \ R_2 = 0.14\ \ \ R_3 = 0.16; $$ 
%\item 
%in the case (\ref{sro}), \\
%$$ I = - 0.15 \ \ R_1 = 0.1 \ \ \ R_2 = 0.09 \ \ R_3 = 0.085 $$
%\end{itemize}
\begin{table}
\begin{tabular}{|c|c|c|c|}
\hline
$p_L$ (GeV)& ~ form of $\rho$ ~ &~ ~ ~ ~ I~ ~ ~ ~ &~ ~ ~ ~ R~ ~ ~ ~ \\
\hline
24& (\ref{fro}) & -0.146 & 0.098 \\
  & (\ref{sro}) & -0.146 & 0.092\\
100& (\ref{fro})& -0.145 & 0.137\\
& (\ref{sro})   & -0.145 & 0.086\\
600& (\ref{fro})& -0.144 & 0.162\\
& (\ref{sro})   & -0.143 & 0.080\\
\hline
\end{tabular}
\caption{Energy dependence of $I$ and $R$ from our calculations.}
\end{table}
 The two sets of parameters $k_1$ and $k_2$,  chosen to obtain  
equivalent descriptions of the data at $p_L = 24$~GeV/c (Fig. 1), 
produce quite different 
 curves
 at higher energies, as can be seen in Figs. 1 and 2, 
 for $p_L = 100 $~GeV$/c$ 
 and $ 600 $~GeV$/c$.
  The difference between the two variants with different 
  energy dependence of $\rho$
  grows and reaches $1 \% $ at $|t| = 0.03 $~GeV$^2 $ and
 $p_L = 600$~GeV/c.   
Around
the maximum of the Coulomb-hadron interference, 
this difference 
is very small, but it grows for $|t| \ > 0.02 $~GeV$^2$.
More importantly, the two variants lead to values of $A_N$ 
with opposite signs. This characteristic is obviously independent
on the normalisation of the data.
So, a determination of the size and energy dependence  
of the spin-flip amplitude requires
a good knowledge of $\rho^{pA}$.

Both variants give the same size and negative sign 
for the imaginary part 
of the spin-flip amplitude, as shown in Fig.~3. 
As mentioned above, such an amplitude gives
an additional positive contribution to the CNI-effect at the maximum.
Its size is mostly determined by the magnitude of $A_N$ at small $|t|$.
The fast change of sign of $A_N$ is explained by the 
interference of different parts of the hadronic amplitude.

Hence, the shape and energy dependence of the analyzing power 
depend mostly on the size and energy dependence of 
$\rho_{pC}$. If we choose  another size and energy dependence, 
we can obtain a different shape for $A_N$ and different 
magnitudes for $k_1$ and $k_2$. However all conclusions
will stand and  $I=Im(r_5)$ will remain negative. Note that a positive
$Im(r_5)$ would lead to an increase with energy of the value of $p_L$
at which $A_N$ has a zero.

\begin{figure}
%Fig. 3
\epsfysize=8.cm
\epsfxsize=8.cm
\vglue -1.5cm
\centerline{\epsfbox{rir5fro.ps}}
%\vspace{-1.cm}
%\centerline{\epsfbox{imr5fro.ps}}
\caption{
  The $t$ dependence of $r_5$ at energies 
     $ p_{L} = 24 $~GeV/$c$ (solid line), 
  at $ p_{L} = 100 $~GeV/$c$ (long-dashed line) and 
  at  $ p_{L} = 600 $~GeV/$c$ (short-dashed line), without the 
  contribution (\ref{sfpN}).}
\end{figure}
Accurate measurements of the analyzing power in the  Coulomb-hadron
interference region can reveal the structure of the hadron 
spin-flip amplitude, and give us further information on the
the hadron interaction potential at large distances, as
used {\it e.g.} in the peripheral dynamic model \ci{zpc}.
Our analysis shows the existence of a hadronic spin-flip amplitude,
which cannot be neglected even at RHIC energies. 
The amplitude corresponds to the interaction of the proton with 
the Carbon nucleus as a whole.     
The ratio of the reduced spin-flip 
amplitude to the spin-non-flip amplitude is approximately 
$15 \%$. 
The hadron spin-flip amplitude gives an additional positive 
contribution to the maximum of the CNI effect with a small 
energy dependence.  It is very important to carry out the 
experiment at higher energies, for example $p_L = 600$~GeV/$c$,
and in some larger region of momentum transfer, so that we can distinguish
between the various possibilities outlined in this letter.

{\noindent\it Acknowledgments} O.V.S. is a Visiting Fellow of the 
Fonds National pour la 
Recherche Scientifique, Belgium. We thank V.~Kanavets and D.~Svirida
for their comments and discussions.
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\end{document}


