Femtoscopy within a hydrodynamic approach based on flux tube initial conditions

In pp scattering at LHC energies, large numbers of elementary scatterings will contribute significantly, and the corresponding high multiplicity events will be of particular interest. Elementary scatterings are parton ladders, identified with color flux tubes. In high multiplicity events, many of these flux tubes are produced in the same space region, creating high energy densities. We argue that there is good reason to employ the successful procedure used for heavy ion collisions: matter is assumed to thermalize quickly, so that the energy from the flux tubes can be taken as initial condition for a hydro-dynamic expansion. This scenario gets spectacular support from very recent results on Bose-Einstein correlations in pp scattering at 900 GeV at the LHC.

1 EPOS is a multiple scattering model in the spirit of the Gribov-Regge approach [1]. Here, one does not refer to simply multiple hard scatterings, the elemen tary processes correspond to complete parton ladders, which means hard scatterings plus initial state radia tion. In this case, this elementary process carries an important fraction of the available energy, and there fore we treat very carefully the question of energy shar ing in the multiple scattering process. Open and closed ladders have to be considered, in order to have a con sistent quantum mechanical treatment. The corre sponding graphs are squared, and we employ cutting rule techniques and Markov chains to obtain finally partial cross sections. The cut parton ladders are iden tified with longitudinal color fields or flux tubes, treated via relativistic string theory.
In case of very high energy pp collisions (at the LHC) or heavy ion scatterings already at RHIC, many flux tubes overlap and produce high energy densities. Let us consider the energy density at an early time in an AuAu scattering at RHIC, as obtained from an EPOS simulation [1]. The energy density shows a very bumpy structure concerning the x-y dependence (transverse coordinates), whereas the variation with the space time rapidity s is small. There are in partic ular peaks in the x-y plane, which show up at the same position at different values of η s . So we have sub flux 1 The article is published in the original. tubes which exhibit a long range structure in the longi tudinal variable η s , with a typical transverse width of the order of a fermi. This is exactly the width we obtain if we compute the initial energy density in protonproton scattering at the LHC. This means, if a hydro dynamic treatment is justified for Au-Au collisions at RHIC, it is equally justified for pp scattering at the LHC, provided the energy densities are high enough, which seems to be the case.
We are therefore going to employ a new tool for treating very high energy hadronic interactions including a hydrodynamic evolution (even in pp), as explained in [1]. In [1], we test the approach by inves tigating all soft observables of heavy ion physics, in case of AuAu scattering at 200 GeV. Let us consider proton proton scattering now. In [2], we compare three different scenarios: the full calculations, includ ing hydro evolution and hadronic cascade (full), the calculation without hadronic cascade (no casc), and the calculation without hydro and without cascade (base). Concerning particle spectra, we refer to [2], where we compare the corresponding calculations with experimental data, for pp scattering at 900 GeV.
The space time evolution of the "full" hydrody namic approach will be completely different com pared to the "base" approach, where particles are directly produced from breaking strings, as can be seen from Fig. 1, where we plot the distribution of forma tion points of π + as a function of the radial distance r = (in the pp center of mass system (cms)) for the three scenarios. Only particles with space time rapidities around zero are considered. All calculations in this paper refer to high multiplicity events in pp scattering at 900 GeV, with a mean dn/dη(0) equal to 12.9. The "base calculation" (dotted line) gives as expected a steeply falling distribution as a function of r. In the two cases involving a hydrodynamic evolu tion, particle production is significantly delayed, even more in the case of the full calculation, with hadronic cascade. The bump in the two latter scenarios is due to particles being produced from the fluid, the small p t contribution is due to corona particles. This particular space time behavior of the hydrody namic expansions should clearly affect Bose-Einstein correlations-what we are going to investigate in the following. There is a long history of so called femto scopic methods [4], where the study of two particle correlations provides information about the source function S(P, r'), being the probability of emitting a pair with total momentum P and relative distance r'. Under certain assumptions, the source function is related to the measurable two particle correlation function CF(P, q) as CF(P, q) = ∫d 3 r'S(P, r')|Ψ(q', r')| 2 , with q being the relative momentum, and where Ψ is the out going two particle wave function, with q' and r' being relative momentum and distance in the pair center of mass system. The source function S can be obtained from our simulations, concerning the pair wave func tion, we follow [5], some details are given in [1].
Here, we investigate π + πcorrelations. We calcu late the correlation function with Bose-Einstein (BE) quantum statistics included, but no Coulomb correc tions. Weak decays are not carried out. We compute correlation functions for different k T interval defined as (in MeV): KT1 = [100, 250], KT3 = [400, 550], KT5 = [700, 1000], where k T of the pair is defined as k T = (|p t (pion 1) + p t (pion 2)|)/2. In Fig. 2a, we show the results for KT1. We compare the three different scenarios: "full calculation" (solid line), "calculation without hadronic cascade" (dashed), and "calculation without hydro and without cascade" (dotted), and data from ALICE [6]. The data are actually not Cou lomb corrected, because the effect is estimated to be small compared to the statistical errors. We consider here the high multiplicity class, with dn/dη(0) = 11.2, close to the value of 12.9 from our simulated high multiplicity events. We compare with the real data (not polluted with simulations), normalized via mixed events, and we do the same with our simulations. Despite the limited statistics, in particular at large k T , we see very clearly that the "full" scenario, including hydro evolution and hadronic cascade, seems to fit the data much better than the two other ones. Usually people like to extract radii from these distributions, so    when we make a fit of the form CF -1 = λexp(-R|q|), in the |q| range from 0.05 to 0.70. We obtain the radii given in the figure. So the radii are very different, vary ing from 0.69 fm (base approach) to 1.80 fm (full model), which is understandable from Fig. 1. We pre fer an exponential fit rather than a Gaussian, simply because the former one works, the latter one does not. We do not want to give a precise meaning to R, it sim ply characterizes the distribution. Normalizing by mixed events is something one can easily do experimentally (this is why we compare with these data), but it is clear that one has still unwanted correlations, like those due to energy momentum conservation, which is not an issue in mixed events. Doing simulations, life is easier. We can take simula tions without Bose-Einstein correlations as base line, rather than mixed events. This is referred to as "real/bare" normalization (to be distinguished from the "real/mixed" case discussed earlier). The corre sponding results are shown in Fig. 2b, the solid line (full calculation) is now completely horizontal away from the peak region, the radius from the exponential fit is 2.11 fm instead of 1.83 fm for the "mixed" nor malization. For the other k T regions, the situation is similar, the final results for all the three k T regions for the full calculation are shown in Fig. 3a, together with the radii from the exponential fit: they are almost identical, around 2 fm.
We get to the same conclusion as outlined in [6]: the radii are k T independent, contrary to what has been observed in AuAu scattering. How can it be that our hydrodynamic scenario gives a strong k T depen dence in AuAu, but not in pp? To answer this question, we compute the "true" correlation function (real/bare normalization) for the calculation without hydro and without cascade (just string decay). The results are shown in Fig. 3b. Surprisingly, here we get a strong k T dependence of the radii, but the "wrong" way: we have 0.64 for KT1 and 1.63 fm for KT5! Actually such behavior is quite normal: the distribution is broader for high p t particles, because high p t resonances live longer and can move further out before decaying. This effect is in principle also present in AuAu scattering, but it is much more visible for the small pp system. So in pp we have two competing effects: • radii increase with k T , due to the bigger size of the source of the high p t particles compared to the low p t ones, • radii decrease with k T , as in AuAu (see [1]), in case of collective flow, due to the p-x correlation.
This p-x correlation exists indeed for the case of hydrodynamic evolutions, and is much smaller in the basic scenario. So in the hydro scenarios, the two com peting effects roughly cancel, the radii are k T inde pendent.
To summarize: we employ a hydrodynamic approach to pp scattering at 900 GeV. A very interest ing application is Bose-Einstein correlations. We have shown that, as in heavy ion scattering, the hydrody namic expansion leads to momentum-space correla tions, which clearly affect the correlation functions. To see the signal is nontrivial due to the fact that in addition to the x-p correlations (which leads to decreasing radii with k T ), there is a second effect which works the other way round: the single particle source size is p t dependent, which is an important effect in pp, not so in heavy ion scattering. In this sense we can interpret the k T independence of the radii as a real flow effect. Our simulation does not only reproduce the k T independence, but also the whole correlation functions, which is not at all reproduced from the "base scenario" without hydro and without cascade. So the correlation data provide a very strong evidence for a collective hydrodynamic expansion in pp scattering at the LHC.
The research was carried out within the scope of the EUREA: European Ultra Relativistic Energies Agreement (European Research Group: "Heavy Ions at Ultrarelativistic Energies"). Iu. K. acknowledges partial support by the National Academy of Sciences of Ukraine (Agreement 2012) and by the State Fund for Fundamental Researches of Ukraine (Agreement 2012). T. P. and K. W. acknowledge partial support by a PICS (CNRS) with KIT (Karlsruhe). K. M. acknowledges partial support by the RFBR CNRS grants no. 08 02 92496 NTsNIL_a and no. 10 02 9311 l NTsNIL_a.