In collaboration with V. Shapoval, Iu.Karpenko Yu.M. Sinyukov, BITP, Kiev WPCF-2012 September 09 – FIAS Frankfurt

First attempts to apply hydro for both types of high energy collisions: A+A and p+p Interferometry radii R T, τ from R L, T f.o Experiment. data for Slope of pion p T spectra T f.o. μ

T=0.170 GeV, τ=1.5 – 2.1 fm/c The simplest estimate for p+p collisions at Freeze-out at τ : transversally, longitudinally: boost-invar. Experiment (ALICE):  Disagreement with the data: Transverse flow smaller homogeneity length (?)

Fast forward Twenty Years Later The modern dynamical models are too complicated to use analytics. To check validity of the model of space-time evolution of the matter in hadron and nuclear collisions one should do the following:  Select initial time from which the model can be applied (for thermal or hybrid ones it is 1~2 fm/c; pre-thermal models are related to the interval: 0.1 – 1÷1.5 fm/c).  Select initial profile of the energy (or entropy) density (e.g., MC Glauber-like one, CGC, EPOS, etc…).  Select parameters of IC (e.g., maximal initial energy density) from the condition of description of and observed transverse spectra of pions, kaons, protons…  Select EoS from, e.g., lattice QCD.  Then the model, fixed as above, has to describe observed femtoscopy scales (interferometry radii), otherwise it does not catch the real evolution of the system and, so, cannot pretend to be the space-time model of the reactions of hadronic or nuclear collisions.

The simulations of p+p 7 TeV within HKM model gives the minimal interferometry volume by the factor more then the experiment.

Yu.S., V. Shapoval: arXiv (this Tuesday)

The idea of the correlation femtoscopy is based on an impossibility to distinguish between registered particles emitted from different points because of their identity. R ab detector 12 p1p1 p2p2 x1x1 x2x2 x3x3 x1x1 x2x2 x3x3 xaxa xbxb t= |q i | D Momentum representation + symmetrization Probabilities: q 1/R i

Probabilities of one- and two- identical bosons emitted independently from distinguishable/orthogonal quantum states with points of emission x 1 and x 2 (x 1 - x 2 = ∆x) Double accounting! HBT (Glauber, Feynman, 1965)

Both cases (independent- and fully non-independent emitters) can be described in the formalism of the partially coherent phases (Yu.S., A. Tolstykh, Z.Phys. 1991): If the states with different emission points and are not independent at all (full coherence), the phases are equal :

The distance between the centers of emitters is larger than their sizes related to the widths of the emitted wave packets 1/Δp. 1/Δp Spectrum: Criterion : overlapping of the wave packages: Phase correlations Wave function for emitters Uncertainty principle and distinguishability of emitters Distinguishable emitters The states are orthogonal Indistinguishable emitters The states are not orthogonal

The measurement of the particle momenta p has accuracy depending on the duration of the measurement Δt: Δp ~ 1/Δt [Landau, Lifshitz, v. IV]. So one can measure the time of particle emission without noticeable violation of the momentum spectra with accuracy not better than 1/ Δp The 4-points phase correlator for maximally possible chaotic and independent emitters permitted by uncertainty principle is decomposed into the sum of products of the two- point correlators G ij and contains also term that eliminate the double accounting. Overlapping integral Femtoscopy for independent distinguishable emitters (standard model) Femtoscopy for partially indistinguishable and so partially non-independent emitters (for small systems where the uncertainty principle is important) The milestones

single-particle amplitude wave-function of emitter emitter’s spectrumspatiotemporal distr. of emitters overlapping integral approximation for two-point phase average two-particle amplitude The correlation femtoscopy in the Gaussian approximation

The accuracy of the approximation for the overlapping integral The comparison of the correlation function without subtraction of the double accounting with corresponding analytical approximations. The alpha- parameters, which give a good agreement with the exact results, are presented for different system sizes. The momentum dispersion k=m=0.14 GeV, p_T=0, T=R.

The results for the Gaussian part of B.-E. correlation functions R_st means interferometry radii in the (standard) model of independent distinguishable emitters. In the region of the source sizes fm α = The reduction of the interferometry radii:

The behavior of the two-particle Bose-Einstein correlation function ( side- projection) where the uncertainty principle and correction for double accounting are utilized. The momentum dispersion k=m=0.14 GeV, p_T=0, T=R. The Bose-Einstein correlation function for small systems

 The effects of the reduction of the interferometry radii and suppression of the correlation function are caused by not complete distinguishabity of the emitting points of the source because of the uncertainty principle: Δp ~ 1/Δt, Δx ~ 1/Δp  The elimination of the double accounting leads also to the reduction of the intercept of the correlation function when the system size shrinks. So, there is the positive correlation between observed interferometry radii and the intercept when the system size is changing.  The effects are practically absent for A+A collisions where effective sizes (homogeneity lengths) more then 3 fm. RESUME

Interferometry volume vs multiplicity in HKM after corrections for partial indistinguishability of the emitters

Femtoscopy scales vs multiplicity in the HKM after corrections

Femtoscopy scales vs p_T in the HKM after corrections

 Our preliminary results on p+p collisions at the LHC energy demonstrates that the uncertainty principle may play an important role for such small systems and allows one to explain the observed overall femtoscopy scale (interverometry volume) and its dependence on multiplicity.  An analysis of the p_T – dependence of the femtoscopic scales corrected for uncertainty principle does not exclude the possibility of the hydrodynamic interpretation of p+p collisions at the LHC energies.  The comparison of vs for pp and AA collisions conforms probably the result of Akkelin, Yu.S. : PRC (2004); PRC (2006) that the interferometry volume depends not only on multiplicity but also on the initial size of colliding systems.