Femtoscopy of stopped protons

The longitudinal proton-proton femtoscopy (HBT) correlation function, based on the idea that in a heavy ion collision at $\sqrt{s} \lesssim 20$ GeV stopped protons are likely to be separated in configuration space, is evaluated. It shows a characteristic oscillation which appears sufficiently pronounced to be accessible in experiment. The proposed measurement is essential for estimating the baryon density in the central rapidity region, and can be also viewed as an (almost) direct verification of the Lorentz contraction of the fast-moving nucleus.


I. INTRODUCTION
The search for a possible phase structure of QCD has been a focus point in strong interaction research. Lattice QCD calculations have established that for vanishing and small net-baryon density the transition from hadrons to quarks and gluons is an analytic cross-over [1]. The situation at larger baryon density, on the other hands is less clear, since at present lattice methods cannot access this region due to the fermion sign problem. Here one has to rely on model calculations and large class of these models do indeed predict a first-order phase coexistence region which ends in a critical point (see, e.g., [2] for an overview).
In order to explore the region of large net-baryon density experimentally one studies heavy ion collisions at moderate beam energies √ s 20 GeV, where a sufficient amount of the incoming nucleons are stopped at mid-rapidity in order to achieve the necessary baryon density. Indeed, since produced baryons always come as baryon -anti-baryon pairs, the only means of producing a finite net baryon density is by stopping the nucleons of the colliding nuclei. Thus, in order to explore the QCD phase diagram at large baryon density, the question of baryon stopping is essential to understand. In fact, stopping the baryons is only a necessary condition. In addition to being at mid-rapidity in momentum space they also need to overlap in configuration space.
The mechanism by which the incoming nucleons are stopped is indeed a very interesting question [3][4][5][6][7][8]. However, independent of the specific mechanism, it seems rather unphysical that the nucleons are stopped instantaneously. Instead it will take time and space for the nucleons to decelerate. Therefore, it is rather unlikely that the stopped nucleons will end up at z 0, i.e., at the point of the collision of the two nuclei. Instead, one would expect that the nucleons from the rightgoing nucleus will end up at positions in configuration space with z > 0 and the left-going ones at z < 0 so that the stopped nucleons may actually be distributed bi-modally in configuration space. This observation was recently pointed out in [9]. Based on a simple string model, Ref. [9] found that for collision energies √ s 10 GeV the stopped nucleons actually will not overlap significantly in configuration space. Of course this observation was based on a rather simple model and it would be much better if this observation could be verified or ruled out in experiment. This is the purpose of this note, where we propose to measure longitudinal Hanbury Brown Twiss (HBT) type correlations (also known as femtoscopy [10]) of the stopped protons, i.e. protons at y cm ≈ 0 with transverse momentum not exceeding, say, 1 GeV. Since femtoscopy does not a priori distinguish between stopped and produced protons, it is important to choose a collision energy which is small enough for proton production to be negligible but sufficiently high so that the deceleration length is large enough for the stopped protons to be separated in configuration space. Thus an energy of √ s 20 GeV appears to be a good choice since at this energy the number of the anti-proton to proton ratio is still very small,p/p 0.1 [11,12].
This paper is organized as follows: In the next section we present and discuss the source function based on the same simple string model used in Ref. [9] (corrected, however, for the Fermi motion inside a nucleus). Next we will calculate the resulting femtoscopy correlation function before we close with a discussion of the various issues and limitations of this study.

II. THE SOURCE FUNCTION
The essential ingredient for femtoscopy is the underlying source of the the emitted particles, protons in our case. The source is the phase-space distribution of emission points, which are typically the points of the last interaction of the protons before they fly to the detector. Clearly a quantitative calculation of such a source function would require a sophisticated simulation. However, we believe that certain semi-quantitative aspects can be discussed without such a treatment, and it is this approach we will take in the following.
As already eluded to in the Introduction, once two nucleons collide it is very unlikely or possibly even unphysical for them to come to a stop right at the collision point. Instead they will only come to a stop after a certain distance and time.
The distance ∆z and time ∆t between the collision and the final space-time point (z, t) where and when a nucleon acquires its final rapidity y, depends on the mechanism of deceleration and thus on the model used for its description. It is in general a function of the initial and final rapidity, Y i and y as well as the typical transverse mass, M ⊥ the nucleon acquires after the collision.
For a given collision space-time point (z c , t c ) in the centre of mass frame of the two nucleons we thus have where the plus sign refers to the right-going particles and the minus sign to the left-going ones. One sees from (1) that, in order to construct the source needed for femtoscopy, we need a distribution of the collision points in space and time and a model/theory which determines ∆z and ∆t.

A. Distribution of collision points
Let us start with the collision point distribution. Here we follow Ref. [9] and assume that the distribution of nucleons inside the target and projectile nuclei can be reasonably described by a Gaussian. In this case the longitudinal (z-direction) and transverse components of the collision point distribution factorise and subsequently we will concentrate on the collision point distribution in the z-direction, which, following [9], we assume to be proportional to the overlap of the distribution of the nucleons in the left-and right-moving nuclei.
We thus have where are the positions of the centres of the nuclei at the time t. ζ 0 and −ζ 0 are positions of the centres of left-moving and right-moving nuclei at t = 0 before the nuclei have any contact with each other. This implies ζ 0 R L,R /γ and t c ≥ 0. Also, γ = cosh(Y cm ) denotes the Lorentz contraction factor for the incoming nuclei in the center of mass frame, which we are working in.
B. Distribution of nucleon emission points z and t Consider first right-movers. For the distribution of z and t we have For left-movers the formula differs by the sign of ∆z.
For identical nuclei we have where ζ 0 R/γ, Γ 2 c = R 2 /2γ 2 , and the dependence on the initial and final momentum is implicit via ∆z and ∆t.
What remains then is to determine ∆z and ∆t. For nucleons with small transverse velocities the simplest model is that of linear energy loss, as it is for instance used in the Lund model [13] or the Bremsstrahlung model [14]. Using the conditions 1 dE/dz = σ; dP/dt = σ where σ denotes the energy loss per unit length or string tension one obtains [9] ∆z Here f are the initial and final longitudinal momenta and energies These equations determine ∆z and ∆t for initial and final longitudinal momenta, P i and P f , transverse mass of the final proton, M ⊥ , and for a given rate of energy loss σ (string tension). In reality the string tension is not a constant but may fluctuate from collision to collision (e.g., depending on number of constituent quarks wounded in a given collision). Since it is unlikely, however, that a nucleon with only one or two wounded quarks may fully stop, the sample of the nucleons with the final rapidity y ≈ 0 is expected to be largely dominated by those with three wounded quarks. Thus we shall ignore fluctuations due to the string tension and take σ = 3σ 0 = 3 GeV/fm. For the transverse mass we subsequently will chose a value of M ⊥ = 1.2 GeV (we verified that the results are not sensitive to the actual value of M ⊥ ).

C. Fermi motion
In case of a nucleus-nucleus collision, the nucleons inside the target and projectile nuclei experience Fermi motion. Consequently, the initial momentum of the colliding nucleons is distributed around the nominal (mean) value of the nucleus-nucleus collision. This broadens the emission source in the longitudinal spatial direction, and thus affects the femtoscopy signal. We have where G F (P i − P i ) is the distribution of the actual initial momentum P i of the nucleon around the average P i . We shall take it in the form where k F is the Fermi momentum. 2 Note that due to the Lorentz boost the width of the distribution, Γ F scales with the Lorentz factor γ, Γ F ∼ γ. This increases substantially the width of this distribution in the energy region of interest.

III. THE HBT CORRELATION FUNCTION
The femtoscopic longitudinal correlation function we are seeking for is given by [10,15] where δq z is the difference of the longitudinal momenta of the two protons, δq 0 is the difference of their energies and is the Fourier transform of the density. Since we are working with Gaussians, the Fourier transforms are straightforward. We have where for the Fourier transform in t it was essential to use the condition ζ 0 R/γ which allowed to integrate over time from −∞.
Thus the final result for the correlation function, including Fermi motion, is where where we have omitted a common phase which does not play any role in the correlation function. Here we have assumed that the momentum dependence of the shift ∆z, Eq. (8), may be approximated by where E i is the mean (nominal) energy of the incident nuclei, and ∆Z denote the shift in space for the mean energy. We further assumed that the velocity of the nuclei in the c.m. frame is close to unity, V 1. For the energies under consideration, √ s > 10 GeV, these should be very good approximations and indeed we find that the resulting correlation function agree with the full result within a few per mil.

IV. RESULTS
The correlation function (15) evaluated for δq 0 = 0 (corresponding to protons with equal and opposite rapidities) is shown in Fig. 1. Here we used for the radius R = 7 fm and M ⊥,1 = M ⊥,2 = 1.2 GeV. In panel (a) we show the predicted femtoscopy correlation function for a collision energy of √ s = 20 GeV and in panel (b) for √ s = 14 GeV. The black dashed lines represent the result for the model discussed above. The blue solid lines are the results, where we doubled the width Γ c of the collision point distribution in order to allow for additional smearing (induced, e.g., by repulsive pp interaction at short distances) not taken into account in our model.
In Fig. 2 we show the corresponding time-integrated source distribution, Here we used the same approximation as before, Eq. (17). We see that the separation of the stopped protons exhibited in the source distribution manifests itself as extra oscillation in the femtoscopy correlation function. For a collision energy of √ s = 20 GeV the signal is clearly visible for both the model result as well the more conservative result, where we doubled the width Γ c . At √ s = 14 GeV the signal is much weaker, however.

V. CONCLUSION AND REMARKS
In conclusion, we have presented a calculation of the longitudinal femtoscopy correlation function of stopped protons based on the observation/idea that in a heavy ion collision at 10 GeV √ s 20 GeV such protons are likely to be separated in configuration space. The resulting correlation function shows extra oscillations which appear sufficiently pronounced to be accessible in experiment. Clearly such a measurement, if feasible, would be most desirable. It will provide useful information about the longitudinal configuration space distribution of the nucleons in a heavy ion collision, and, more importantly, it will provide essential constraints on the mechanism by which baryon number is transported to mid-rapidity.
Some remarks are in order.
• The observation of the suggested extra oscillations will not only confirm the idea that the nucleons do not stop immediately after collision. It should also allow to measure the effective distance at which the energy is deposited in the produced particles. Indeed, as seen from Eq. (16), Φ F (and thus also C F ) explicitly depends on ∆Z, the average distance required to stop a proton.
• Even if the oscillations are not seen, the measurement will determine the (longitudinal) size of the volume from which the protons at y cm ≈ 0 are emitted. This should allow to estimate the actual density of protons in configuration space, the quantity essential for the studies of this system. One also obtains the upper limit on the distance the nucleons travel before attaining the rapidity y ≈ 0, thus improving our understanding of the process of the energy loss by the leading particles in a high energy collision.
• The definition of the longitudinal correlation function requires that the vector δ q points in the z-direction, i.e. δq ⊥ = 0. In our approximation of the nuclear densities as Gaussians this restriction is not important, as the longitudinal and transverse degrees of freedom factorise.
To increase statistics, one may thus integrate over transverse momenta. Since the Lund model is best justified at small transverse velocities, and since the Gaussian form is only an approximation, it seems reasonable, however, to restrict measurements to protons with transverse momenta not exceeding, say, 1 GeV.
• In our work we have neither taken into account corrections due to the Coulomb interaction nor the effects of the proton-proton final state s-wave scattering. These will have to be taken into account in the data analysis.
• Our calculation ignored entirely possible correlations between the outgoing protons due to quark mixing at very short distances [16]. Introducing such correlations may result in the correlation function being positive in some region of δq z . As shown in [16], however, this effect is small and should not modify our conclusions.
• Finally, let us add that our results rely strongly on the idea that the longitudinal distribution of nucleons inside moving nucleus are Lorentz-contracted and that this contraction survives during the collision. The proposed measurement should thus provide an interesting test of this commonly used assumption (for the recent discussion of the measurements of Lorentz contraction, see [17]).