Femtoscopic scales in p+p and p+ Pb collisions in view of the uncertainty principle
V.M. Shapoval, P. Braun-Munzinger, Iu.A. Karpenko, Yu.M. Sinyukov
aa r X i v : . [ h e p - ph ] J u l Femtoscopic scales in p + p and p + Pb collisions in view of theuncertainty principle
V.M. Shapoval a , P. Braun-Munzinger b,c , Iu.A. Karpenko a,c and Yu.M. Sinyukov a ( a ) Bogolyubov Institute for Theoretical Physics,Metrolohichna str. 14b, 03680 Kiev, Ukraine ( b ) ExtreMe Matter Institute EMMI,GSI Helmholtz Zentrum f¨ur Schwerionenforschung, D-64291 Darmstadt, Germany ( c ) Frankfurt Institute for Advanced Studies,Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany
Abstract
A method for quantum corrections of Hanbury-Brown/Twiss (HBT) interferometric radii pro-duced by semi-classical event generators is proposed. These corrections account for the basicindistinguishability and mutual coherence of closely located emitters caused by the uncertaintyprinciple. A detailed analysis is presented for pion interferometry in p + p collisions at LHC energy( √ s = 7 TeV). A prediction is also presented of pion interferometric radii for p +Pb collisions at √ s = 5 .
02 TeV. The hydrodynamic/hydrokinetic model with UrQMD cascade as ’afterburner’ isutilized for this aim. It is found that quantum corrections to the interferometry radii improve sig-nificantly the event generator results which typically overestimate the experimental radii of smallsystems. A successful description of the interferometry structure of p + p collisions within thecorrected hydrodynamic model requires the study of the problem of thermalization mechanism,still a fundamental issue for ultrarelativistic A + A collisions, also for high multiplicity p + p and p +Pb events. PACS numbers: 13.85.Hd, 25.75.Gz
Keywords: correlation femtoscopy, HBT radii, proton-proton collisions, proton-nucleus colli-sions, LHC, uncertainty principle, coherence
I. INTRODUCTION
The quantum-statistical enhancement of the pairs of identical pions produced with closemomenta was observed first in ¯ p + p collisions in 1959 [1]. It took more than a decadeto develop the method of pion interferometry based on the discovered phenomenon. Thiswas done at the beginning of the 1970s by Kopylov and Podgoretsky [2]. Their theoreticalanalysis assumed the radiating source as consisting of independent incoherent emitters. Infact, such a representation is used for a long time for the analysis of the space-time structureof particle sources created in ¯ p + p , p + p , e + + e − and A + A collisions. The concept ofindependent emitters was applied to a further development of the interferometric method,in particular, to account for momentum-position correlations of the emitted particles [3–6]that, in turn, has resulted in a general interpretation of the measured radii as the homogene-ity lengths in the Wigner functions [7–9]. This concept is important for a study of A + A collision processes within the hydrodynamic approach. Also a detailed analysis of the par-ticle final state (Coulomb) interactions brings the significant contribution to the traditionalmethod of correlation femtoscopy [10, 11].In a recent paper [12] the correlation analysis is taken beyond the model of independentparticle emitters. It is found that the uncertainty principle leads to (partial) indistinguisha-bility of closely located emitters that fundamentally impedes their full independence andincoherence. The partial coherence of emitted particles is because of the quantum nature ofparticle emission and happens even if there is no specific mechanism to produce a coherentcomponent of the source radiation. This effect leads to a reduction of the interferometryradii and suppression of the Bose-Einstein correlation functions. The effect is significantonly for small sources with typical sizes less than 2 fm. We shall apply this approach [12]to the analysis of data in p + p collisions at the LHC energy of √ s = 7 TeV, where themeasured interferometry radii are just within the above scale. A simple estimate will bedone also for p +Pb, where the radii are larger and such corrections are less important.A first attempt of the systematic theoretical analysis of the pion interferometry of p + p √ s = 0 . p + p collisions, at least, for high multiplicityevents. Then, to reproduce high multiplicity, the initially very small p + p system has tobe superdense at early times. This leads to very large collective velocity gradients, and sothe homogeneity lengths should be fairly small. However, as we shall demonstrate, evenat the maximally possible velocity gradients at the given multiplicity, one gets again anoverestimate of the interferometry radii in p + p collisions. The similar result is obtained inhydrodynamics in Ref. [15] . Therefore, one can conclude that the problem of theoreticaldescription of the interferometry radii in p + p collisions may be a general one for differenttypes of event generators associated with various particle production mechanisms. Herewe try to correct the results on interferometry from event generators using for this aimthe quantum effects accounting for partial indistinguishability and mutual coherence of theclosely located emitters due to the uncertainty principle [12].In this Letter we employ the hydrokinetic model (HKM) [17, 18] in its hybrid form[19] where the UrQMD hadronic cascade is considered as the semi-classical event generatorat the post freeze-out (“afterburner”) stage of the hydrodynamic/hydrokinetic evolution.We analyze two aspects of the analysis of p + p collisions. The main one is: whetherquantum corrections can help to describe the experimental data. If yes, it gives hope thatit can be successfully applied for any event generator associated with another mechanismsof the particle production. The second aspect is more sophisticated: whether the typicalhybrid models developed for A + A collisions (here hybrid = hydrodynamic/hydrokinetic+ hadronic cascade) with correspondingly modified initial conditions and with the above-mentioned quantum corrections can be real candidates to describe the bulk observables in p + p collisions at LHC energies. For this aim we study the space-time structure of p + p The results for p + p [16] obtained using EPOS 2.05 + hydro, need to be clarified since that version ofEPOS underestimates the transverse energy per unit of rapidity [16]. p T -behavior of the HBT radii. It is worth noting that a satisfactory descriptionof the corresponding experimental data challenges the theoretical picture of p + p collisions,however supporting the Landau pioneer suggestion [20] to use relativistic hydrodynamictheory for the hadron collisions with high multiplicity. Certain arguments in the favor ofthis suggestion are presented, for example, in [21, 22], where multiparticle production innuclear collisions is related to that in hadronic ones within the model based on dissipatingenergy of participants and their types, which includes Landau relativistic hydrodynamicsand constituent quark picture. II. HYDROKINETIC MODEL: DESCRIPTION AND RESULTS FOR p + p COL-LISIONS
The hydrokinetic model [17–19] of A + A collisions consists of several ingredients describ-ing different stages of the evolution of matter in such processes. At the first stage of system’sevolution the matter is supposed to be chemically and thermally equilibrated and its expan-sion is described within perfect (2+1)D boost-invariant relativistic hydrodynamics with thelattice QCD-inspired equation of state in the quark-gluon phase [23] matched with a chemi-cally equilibrated hadron-resonance gas via crossover-type transition. The hadron-resonancegas consists of 329 well-established hadron states made of u,d,s-quarks, including σ -meson( f (600)). With such an equilibrated evolution the system reaches the chemical freeze-outisotherm with the temperature T ch = 165 MeV. At the second stage with T < T ch , thehydrodynamically expanding hadron system gradually looses its (local) thermal and chemi-cal equilibrium and particles continuously escape from the system. This stage is describedwithin the hydrokinetic approach [17, 18] to the problem of dynamical decoupling. In hHKMmodel [19] the hydrokinetic stage is matching with hadron cascade UrQMD one [25] at theisochronic hypersurface σ : t = const (with T σ ( r = 0) = T ch ), that guarantees the correctnessof the matching (see [17–19] for details). The analysis provided in Ref. [19] shows a fairlysmall difference of the one- and two-particle spectra obtained in hHKM and in the caseof the direct matching of hydrodynamics and UrQMD cascade at the chemical freeze-outhypersurface. Thus, in this Letter we utilize just the latter simplified “hybrid” variant for According to Particle Data Group compilation [24]. p + p collisions at √ s = 7TeV aiming to get the minimal interferometry radii/volume at the given multiplicity bin.As it is known [17] the maximal average velocity gradient, and so the minimal homogeneitylengths can be reached for a Gaussian-like initial energy density profile. For the same aimwe use the minimal transverse scale in ultra-high energy p + p collision, close to the size ofgluon spots [26] in a proton moving with a speed v ≈ c . In detail, the initial boost-invarianttube for p + p collisions has a Gaussian energy density distribution in the transverse plane ǫ i ( r ) with width (rms) R = 0.3 fm [26] and, following Ref. [19], we attribute it to aninitial proper time τ = 0.1 fm/c. At this time there is no initial transverse collective flow.The maximal initial energy density is defined by all charged particle multiplicity bin. Themaximum initial energy density, ǫ i ( r = 0), is determined in HKM, for selected experimentalbins in multiplicity, by fitting of the mean charged particle multiplicity in those bins.The correlation function for bosons in the UrQMD event generator is calculated accordingto C ( q ) = P i = j δ ∆ ( q − p i + p j )(1 + cos( p j − p i )( x j − x i )) P i = j δ ∆ ( q − p i + p j ) (1)where δ ∆ ( x ) = 1 if | x | < ∆ p/ p being the bin size in histograms.The method (1) accounts for the smoothness approximation [27]. The output UrQMD 3Dcorrelation histograms in the LCMS for different relative momenta q = p − p are fittedwith Gaussians at each p T = | p T + p T | bin C ( q ) = 1 + λ · exp( − R out q out − R side q side − R long q long ) . (2)The interferometry radii R out ( p T ), R side ( p T ), R long ( p T ) and the suppression parameter λ areextracted from this fit.In Fig.1 we demonstrate the results from hydrokinetic model for the pion interferometricradii, comparing them with the ones measured by the ALICE Collaboration at the LHC [28]in p + p collisions at the energy √ s = 7 TeV. As one can see there is a significant systematicoverestimate of the predicted interferometry volume V int = R out R side R long in p + p collisioneven at the minimal homogeneity lengths possible for the given multiplicity classes. This isconsistent with the results of the first paper devoted to the same topic “Pion interferometrytesting the validity of hydrodynamical models” [29]. In what follows we shall try to improve5he results of the semi-classical HKM event generator by means of the quantum correctionsto them [12]. III. THE QUANTUM CORRECTIONS TO THE HYDROKINETIC RESULTS
In [12] it is shown that, for small systems formed in particle collisions (e.g. pp , e + e − ) wherethe observed interferometry radii are about 1–2 fm or smaller, the uncertainty principledoesn’t allow one to distinguish completely between individual emission points. Also thephases of closely emitted wave packets are mutually coherent. All that is taken into accountin the formalism of partially coherent phases in the amplitudes of closely spaced individualemitters. The measure of distinguishability and partial coherence is then the overlap integralof the two emitted wave packets. In thermal systems the role of the corresponding coherencelength is played by the thermal de Broglie wavelength that defines also the size of a singleemitter. The Monte-Carlo method (1) cannot account for such effects since it deals withclassical particles and point-like emitters (points of the particle’s last collision). The classicalprobabilities are summarized according to the event generator method (1), while in thequantum approach a superposition of partially coherent amplitudes, associated with differentpossible emission points, serves as the input for further calculations [12]. Such an approachleads to a reduction of the interferometry radii as compared to Eq. (1). In addition, theascription of the factor 1 + cos( x − x )( p − p ) to the weight of the pion pair in (1)is not correct for very closely located points x and x because there is no Bose-Einsteinenhancement if the two identical bosons are emitted from the same point [12, 30]. Theeffect is small for large systems with large number of independent emitters. For smallsystems, however, it can be significant and one has to exclude unphysical contributions(“double counting” [12]) in the two-particle emission amplitude. Such corrections lead toa suppression of the Bose-Einstein correlations that is manifested in a reduction of theobserved correlation function intercept compared with one in the standard method (1).The results of Ref. [12] are presented in the non-relativistic approximation related to therest frame of the source moving with four-velocity u µ . In the hydrodynamic/hydrokineticapproach the role of such a source at a given pair’s half-momentum bin near some value p is played by the fluid element or piece of the matter with the size equal to the homogeneitylength λ ( p ) [7]. These lengths are extracted from the HKM simulations, namely, from the6 /d ch dN , f m i n t V (0.2,0.3) GeV/c T ALICE pp @ 7 TeV, p = 0.1 fm/c τ HKM model 3D fit, = 2.05 γ HKM model 3D fit corrected, = 1 γ HKM model 3D fit corrected,
FIG. 1. The pion interferometry volume dependency on the charged particles multiplicity at p T = 0 . − . interferometry radii defined by the Gaussian fits to the correlation functions obtained inHKM. All the pairs in procedure (1) are considered in the longitudinally co-moving system(LCMS) that in the boost-invariant approximation automatically selects the longitudinalrest frame of the source and longitudinal homogeneity length in this frame (it is Lorentz-dilated as compared to one in the global system [5]). The femtoscopy analysis is typicallyrelated to a fixed p T bin and so one needs also to determine the transverse source size inthe transverse rest frame. The corresponding Lorentz transformations do not change the side -homogeneity length; as for the out -direction we proceed in the way proposed in Ref. [5].Remaining within the Gaussian approximation, realized for expanding inhomogeneoussystems in the saddle point method [7, 8], let us fix some p = ( p T ,
0) in the basic LCMSreference system ( basic -RS) and select the transversely moving reference systems (markedby the sign tilde ) where the emission density distribution, related to the space-time centerof this local source ˜ x i ( p ), ˜ t ( p ), can be well approximated as the following ρ ( x, t ) ∝ e − P i ˜ x i / λ i ( p ) − ˜ t / T ( p ) (3)7hen in tilde -RS the correlation function has the form (2) where R long = λ long , R side = λ side , R out → ˜ R out = ˜ λ out + ˜ p out ˜ p ˜ T with ˜ T defining the duration of emission in this tilde -RS. Thepair’s half-momentum p corresponds to the concrete experimental bin taken in the basic -RS,the difference of the particle momenta q components in selected tilde -RS are ˜ q out , ˜ q side = q side ,˜ q long = q long . Therefore, only q out and correspondingly R out , including λ out and T , are reallytransformed in (2) at the Lorentz boosts along the transverse momentum of the pair. Thecorrelation function C ( p, q ) is the Lorentz invariant, therefore ˜ R out ( p )˜ q out = inv.To relate the interferometry radius in the rest frame of the source (marked by the asterisk)to the one in basic -RS one should express both values through the radius in tilde -RS usingthe invariance property similar as it is done in [5]. Then one can get R ∗ out ( p ) = R out ( p ) cosh y T cosh( y T − η T ) , R ∗ side = R side , R ∗ long = R long (4) λ ∗ out = λ out cosh y T cosh( y T − η T ) , p ∗ out p ∗ T ∗ = T sinh y T cosh( y T − η T )Here η T is a rapidity of the source in transverse direction, y T = ( y T + y T ) / basic -RSaccording to (4). Note that y ∗ T = y T − η T , and if the rapidity of the pair is equal to therapidity of the source, y ∗ T = 0, then in this particular case the radius in the rest frame isLorentz-dilated by the factor γ . Generally, the reference system where the pair’s momentumis zero does not coincide with the rest frame of the source that emits the pair. Therefore, thedirect application of these formulas is not an easy task for the rather complicated emissionstructure in a hypothetical hydrodynamic/hydrokinetic model of p + p collisions. In Fig.2one can see the structure of the chemical freeze-out hypersurface with the maximum valueof collective velocity labeled for high multiplicity p + p events in comparison with the onesfor central Pb+Pb collisions at LHC . Of course, the details of the transformation will bedifferent for a string-based event generator, therefore we present the analysis for the radiitransformation just in the two limiting cases R ∗ out = R out and R ∗ out = γR out ( γ = cosh y T ).We provide the quantum corrections at each p T bin in the rest frame of the correspondingsource using Eq. (4) and then come back again to the basic -RS. To preserve the previous Note, that the initial maximal energy densities are close in both these processes and the peculiarities ofthe freeze-out hypersurface and velocity profile in p + p case are caused by the very large gradients ofinitial density because of the small initial transverse size. fm T r , f m / c τ T = 165 MeV switching hypersurface in HKM = 17.9 η /d ch ALICE pp @ 7 TeV, dNT = 165 MeV switching hypersurface in HKMLHC PbPb @ 2.76 TeV = 0.893 max v = 0.727 max v FIG. 2. The chemical freeze-out hypersurface in HKM in the transverse plane for √ s = 7 TeV p + p collisions at dN ch dη = 17 . τ, r T coordinates by the factor 1/3 for √ s = 2 .
76 TeV Pb+Pb collisions (blue line). The maximal initialenergy densities are close in both cases. The maximal velocities are marked in the correspondingpoints on the curves. notations one can suppose that the source rest frame coincides with tilde -RS. In what followsthe tilde and asterisk marks are omitted and all values are related to the source rest frame.To account that due to the uncertainty principle the emitters (strictly speaking emittedwave packets) have finite sizes h (∆ x ) i ∼ k − ( k is the momentum variance of the particleradiation) when defining the lengths of coherence, one should at first consider the amplitudeof the radiation processes and only then make statistical averaging over phases of the wavepackets using the overlap integral as the coherence measure [12].Following [12] we present the quantum state ψ x i ( p, t ) corresponding to a boson with mass m emitted at the time t i from the point x i as a wave packet with momentum variance k which then propagates freely: ψ x i ( p, t ) = e ipx i − iEt e iϕ ( x i ) ˜ f ( p ) (5)9here ϕ ( x i ) is some phase and ˜ f defines the primary momentum spectrum f ( p ) that wetake in the Gaussian form, f ( p ) = ˜ f ( p ) = 1(2 πk ) / e − p k , (6)with the variance k = mT . The effective temperature of particle emission in the local restframes in HKM, T , is close to the chemical freeze-out temperature T ch .The amplitude of the single-particle radiation from some 4-volume can be written atvery large times t ∞ as a superposition of the wave functions ψ x i ( p ) with coefficients b ρ ( x i ) = p ρ ( x i ) that leads in the case of completely random phases to the emitter distribution (3)in the local rest frames of the sources: A ( p, t ) = c Z d x i ψ x i ( p, t ) b ρ ( x i ) , (7)where c is the normalization constant.In the paper [12] the two-particle state is considered as a product of the single-particleamplitudes, thus suggesting the maximal possible distinguishability and independence ofdifferent emitters compatible with the uncertainty principle for momentum & position andenergy-momentum & time measurements. The latter is accounted for by the averaging ofsuch a two-particle amplitude over partially coherent phases between different emitters withoverlap integral measure [12, 31] .With that said the single- and two-particle spectra, averaged over the ensemble of emissionevents with partially correlated phases ϕ ( x ) are W ( p ) = c Z d xd x ′ e ip ( x − x ′ ) b ρ ( x ) b ρ ( x ′ ) f ( p ) h e i ( ϕ ( x ) − ϕ ( x ′ )) i W ( p , p ) = c Z d x d x d x ′ d x ′ e i ( p x + p x − p x ′ − p x ′ ) ·· f ( p ) f ( p ) b ρ ( x ) b ρ ( x ) b ρ ( x ′ ) b ρ ( x ′ ) h e i ( ϕ ( x )+ ϕ ( x ) − ϕ ( x ′ ) − ϕ ( x ′ )) i . (8)The phase averages are associated with corresponding overlap integrals [12] h e i ( ϕ ( x ) − ϕ ( x ′ )) i = G xx ′ = I xx ′ = (cid:12)(cid:12)(cid:12)(cid:12)Z d r ψ x ( t, r ) ψ ∗ x ′ ( t, r ) (cid:12)(cid:12)(cid:12)(cid:12) , (9) h e i ( ϕ ( x )+ ϕ ( x ) − ϕ ( x ′ ) − ϕ ( x ′ )) i = G x x ′ G x x ′ + G x x ′ G x x ′ − G x x ′ G x x ′ G x x (10) Such a ’minimal’ consideration of the uncertainty principle only does not exclude, of course, an existenceof the concrete mechanisms of correlation and coherence between emitters; note, however, that such morecomplicated picture might lead to the results different from these that are set forth here. ψ x i ( t, r ) = π ) / R f ( p ) e − i p ( r − x i ) e − i p m ( t i − t ) d p are the wave functions of singlebosonic states in coordinate representation.Then the correlation function C ( p , q ) can be expressed through the homogeneity lengthsin the local rest frame R L ≡ λ ∗ long ( p ), R S ≡ λ ∗ side ( p ), R O ≡ λ ∗ out ( p ) that are expressedthrough the HBT radii obtained from the Gaussian fit (2) of the HKM correlation functionsand transformation law (4) as described above. C ( p , q ) = W ( p , p ) W ( p ) W ( p ) == 1 + e − q O R O k R O k R O − q S R S k R S k R S − q L R L k R L k R L − ( q · p )2 T m k T k T − C d ( p , q ) , (11)where k = k / (1 + αk T /m ), parameter α ( k R ) is defined from the model numerically(it is the order of unity for R ∼ C d ( p , q ) = e − q Ok R O ( k R O )( k R O )( k R O +8 k R O ) − q Sk R S ( k R S )( k R S )( k R S +8 k R S ) − q Lk R L ( k R L )( k R L )( k R L +8 k R L ) ·· e − k T p · q )2 ( k T ) m ( p T )( k T k T ) F ( k R i , k T ) ,F ( k R i , k T ) = (cid:18) k k (cid:19) / (cid:18) k T k T + 8 k T k R O k R O + 8 k R O ·· k R S k R S + 8 k R S k R L k R L + 8 k R L (cid:19) / (12)corresponds to the elimination of the double counting.Now we can see that the apparent interferometry radii extracted from the Gaussian fitsto the correlation function (11) are reduced as compared to those obtained in the standardapproach.Particularly, if we neglect the double counting effects, truncate the subtracted term C d ( p , q ) in (11), and fit the correlation function with the Gaussian (2), we obtain thefemtoscopic radii R out , R side , R long related to the standard ones R out,st , R side,st , R long,st asfollows R out R out,st = (cid:18) R O k R O k R O + T v out k T k T (cid:19) / (cid:0) R O + T v out (cid:1) R side R side,st = 4 k R S k R S (13) R long R long,st = 4 k R L k R L v out = p ∗ out /p ∗ ≪ A + A collisions, k R ≫ k T ≫ T = a ( R O + R S + R L ) / R O (and T ) through R i,st . The latter are connected with ones taken in the basic -RS according to transformationlaws (4). The value a is a free model parameter. Then we put these extracted values intothe expression (11) for the correlation function and perform its fitting with the Gaussian (2).This gives us finally the interferometry radii R out , R side and R long in view of the uncertaintyprinciple. The radii are presented then in the basic -RS using the transformations inverseto (4).The correlation function is the ratio of the two- and one-particle spectra. It is found [12]that quantum corrections to this ratio are not so sensitive to different forms of the wavepackets as the spectra itself. In particular, the effective temperature of the corrected trans-verse spectra depends on whether the parameter of mean particle momentum is includedor not into the wave packet formalism. If yes, the corrected effective temperature for smallsources R ∼ T = k /m , whilefor the wave packets in the form (5) it is lower [12]. Besides of this, in the non-relativisticapproximation one can describe only very soft part of the spectra. That is why we focusin the Letter on the corrections to the Bose-Einstein correlation functions where in the restframe of the source the total and relative momenta of the boson pairs are fairly small. IV. THE RESULTS FOR p + p AND p + Pb COLLISIONS, AND DISCUSSION
The initial conditions for HKM are described in Section 2. The HKM event generatorprovides us with the interferometry radii in basic -RS. To find the corresponding homogeneitylengths in the rest frame of the source according to (4) we use, as discussed in Section 3,the two limiting cases: the transverse boost to the rest frame of the pair from the basicLCMS system, or no transformation at all. For the former case it is defined by the p T binand for p T = 0 . − . y T = γ = 2 .
05. The parameter a connecting T with R i increases linearly with multiplicity from 0.8 to 1.0, primary momentum spectrum dispersion k = 0 .
16 GeV/c (T=0.18 GeV), pair mean transverse momentum in the source rest frame12 /3 η /d ch dN , f m ou t R (0.2,0.3) GeV/c T ALICE pp @ 7 TeV, pHKM modelHKM model corrected> = 0.15 GeV/c T* 15 GeV/c. The α parameter is set to linearly decrease with multiplicity from 0.8to 0.6. As for the γ = 1 case, the parameter a decreases linearly with multiplicity from 1.1to 0.9, k = 0 . 18 GeV/c and p ∗ T = 0 . 13 GeV/c. The α parameter decreases linearly withmultiplicity from 1.35 to 0.9 which is close to the theoretical results [12]. In Fig. 1 alongwith the experimental and pure HKM results we present the multiplicity dependence of thequantum corrected interferometry volume at p T = 0 . 25 GeV/c. The solid line representsthe corrected values calculated under the assumption that the R out interferometry radii,observed in basic -RS, are Lorentz-contracted by a factor γ = 2 . 05 for the chosen p T = 0 . γ = 1. As one can see the accounting for the uncertaintyprinciple allows one to describe the overall multiplicity dependence of the interferometricradii. Figure 3 represents the dependence on multiplicity of individual radius parameters.The suppression of the Bose-Einstein correlations for small sources with closely located13mitters takes place even without specific coherence mechanism and resonance contributions.To see this effect the double counting in the correlation function should be eliminated asEq. (11) demonstrates. Then the additional suppression parameter λ c oh < λ = λ c oh λ HKM . The result of ourcalculations gives λ coh = 0.9 – 0.95 for not very small multiplicities.In addition to the correlation analysis of p + p collisions, let us make the simplest estimatesand try to predict the HBT radii for p +Pb collisions at the LHC energy √ s = 5 . 02 GeV.We ignore the possible asymmetry of the hydrodynamic tube in the longitudinal directionand present our prediction within hHKM for centrality c = 0 − 20 % with dN ch /dη = 35.The results are calculated for the two initial radii with rms equal to 0.9 fm and 1.5 fm andalso for the two initial times: τ = 0.1 fm/c and 0.25 fm/c. It turns out that the latterfactor is not essential if we keep fixed final multiplicity: only the longitudinal radii are 3–4% higher at τ = 0.25 fm/c than at 0.1 fm/c. The transverse radii practically coincide.Therefore, we finally demonstrate only the case τ = 0.1 fm/c. The initial transverse sizesof the system, created in the p +Pb collision, are taken from Ref. [32]: “In the conventionalwounded nucleon model it is assumed that the sources are located in the transverse planein the centers of the participating nucleons. This amounts to rather large initial transversesizes in the p − Pb system, R = 1 . N N system is also admissible, which leads to a more compact initial distribution, R = 0 . p + p system, with γ = 1 to the case of larger sizes typical for the p +Pb collisions. At that the k and p ∗ T valuesare left the same as for the p + p case, whereas α and a are chosen to be smaller. For the R = 0 . α = 0 . a = 0 . R = 1 . α = 0 . a = 0 . p + p and p +Pb colli-sions we cannot bypass the scaling hypothesis issue [33], that suggests a universal lineardependence of the HBT volume on the particle multiplicity. It means that the observedinterferometry volume depends roughly only on the multiplicity of particles produced incollision, but not on the geometrical characteristics of the collision process. At the sametime, as it was found in the theoretical analysis in Ref. [34], the interferometry volumeshould depend not only on the multiplicity, but also on the initial size of colliding sys-14ems. In more detail, the intensity of the transverse flow depends on the initial geometricalsize R g of the system: roughly, if the pressure is p = c ǫ , then the transverse acceleration a = ∇ x T p/ǫ ∝ p ( x T = 0) / ( R g ǫ ) = c /R g . The interferometry radii R T , that are associatedwith the homogeneity lengths, depend on the velocity gradient and geometrical size, andfor non-relativistic transverse expansion can be approximately expressed through R g , theaveraged transverse velocity h| v T |i and inverse of the temperature β at some final moment τ [8, 35, 36]: R T = R g ( τ ) q π h| v T |i βm T ≈ R g (cid:18) τ c R g ) − βm T τ c π ( R g ) (cid:19) (14)The result (14) for the HBT radii depends obviously on R g and, despite its roughness,demonstrates the possible mechanism of compensation of the growing (in time) geometricalradii of an expanding fireball in the femtoscopy measurements. For some dynamical modelsof expanding fireballs [37] the interferometry radii, measured at the final time of system’sdecoupling, are fully coincided with the initial geometrical ones, no matter how large themultiplicity is. The reason for such a behavior is explained in Ref. [17]: if there is nodissipation in the expanding system, namely, the evolution corresponds to a solution of theBoltzmann equation with F gain ( t, x ) = F loss ( t, x ), then the spectra and correlation functionsare coincided with the initial ones. The detail study of hydrodynamically expanding systemsis provided in Ref. [34]. It is found that at the boost-invariant isentropic and chemicallyfrozen evolution the interferometry volume, if it were possible to measure the interferometryradii at some evolution time τ , is approximately constant: V int ( τ ) ≃ C ( √ s ) dN/dy ( τ ) h f i τ T eff ( τ ) (15)where h f i is the averaged phase-space density [38] which is found to be approximatelyconserved during the hydrodynamic evolution under above conditions as well as dNdy [34].As for the effective temperature of the hadron spectra, T eff ( τ ) = T ( τ ) + m h v T ( τ ) i , onecan see that when the system’s temperature T drops, the mean v T increases, therefore T eff does not change much during the evolution (it slightly decreases with time for pions andincreases for protons). Hence V int , if it has been measured at some evolution time τ , will alsoapproximately conserve. Of course, the real evolution is neither isentropic, nor chemicallyfrozen, includes also QGP stage, but significant dependence of the femtoscopy scales on theinitial system size is preserved anyway. 15 /d ch dN , f m i n t V (0.2,0.3) GeV/c T ALICE pp @ 7 TeV, pAGS, SPS, RHIC, LHC Au+Au and Pb+Pb, c=0--5%data compilation from arXiv:1012.4035 [nucl-ex]HKM modelHKM model corrected FIG. 4. The interferometry volume dependency on charged particles multiplicity. The curvefragments in the middle correspond to the HKM prediction for p +Pb collision at the LHC energy √ s = 5 . 02 GeV. The upper one is related to the initial transverse system size R = 1 . R = 0 . p + p and A + A central collisions respectively, compared to the experimental data at AGS, SPS,RHIC and LHC, taken from papers [28], [40] – [47]. The pp volumes are calculated as a product R out R side R long of respective experimental radii. The blue lines correspond to pure HKM results,whereas the quantum corrections to them are presented by the red lines. Fig. 4 shows the dependency V int ( h dN ch /dη i ) for the case of p + p collisions at the LHC, √ s = 7 TeV, and for the most central (only!) collisions of nuclei having similar sizes, Pb+Pband Au+Au, at the SPS, RHIC and LHC. We have also added on the plot our predictionfor the interferometry volume of p Pb system, that has an initial size larger than that forthe pp system. As one can see, the different groups of points corresponding to p + p , p +Pband A + A events cannot be fitted by the same straight line. This apparently confirmsthe result obtained in [34] that the interferometry volume is a function of both variables:the multiplicity and the initial size of colliding system. The latter depends on the atomicnumber A of colliding objects and the collision centrality c .16 GeV/c T p , f m ou t R >=9.2 η /d ch ALICE pp @ 7 TeV, V. CONCLUSIONS One can conclude that quantum corrections to pion interferometry radii in p + p col-lisions at the LHC can significantly improve the (semi-classical) event generator resultsthat typically give an overestimate of the experimental interferometry radii and volumes.The corrections account for the basic (partial) indistinguishability and mutual coherence ofclosely located emitters because of the uncertainty principle [12]. The additional suppressionof the Bose-Einstein correlation function also appears. The effects become important forsmall sources, 1–2 fm or smaller. Such systems cannot be completely random and so requirea modification of the standard theoretical approach for correlation analysis. The predictedinterferometric radii for p +Pb collisions need some small corrections only for its minimal18 GeV/c T p , f m ou t R HKM model, R=0.9 fmHKM model, R=1.5 fmHKM model corrected, R=0.9 fm, a = 0.7HKM model corrected, R=1.5 fm, a = 0.6> = 0.13 GeV/c T* 02 GeV, h dN ch /dη i = 35. values corresponding to the initial transverse size of p Pb system 0.9 fm.More sophisticated result of this study is a good applicability of the hydrodynam-ics/hydrokinetics with the quantum corrections for description of HBT radii not only in A + A collisions but also, at least for large multiplicities, in p + p events. These radii arewell reproduced for not too large p T . Whether it means the validity of the hydrodynamicmechanism for the bulk matter production in the LHC p + p collisions is still an openquestion. It is also related to the problem of early thermalization in the processes of heavyion collisions; the nature of this phenomenon is still a fundamental theoretical issue. VI. ACKNOWLEDGMENT Yu.M.S. is grateful to B. Kopeliovich, R. Lednick´y, L.V. Malinina, E.E. Zabrodin forfruitful discussions and to the ExtreMe Matter Institute EMMI for visiting professor position.Iu.A.K. acknowledges the financial support by the ExtreMe Matter Institute EMMI and19essian LOEWE initiative. The research was carried out within the scope of the EUREA:European Ultra Relativistic Energies Agreement (European Research Group: “Heavy ions atultrarelativistic energies”), and is supported by the State Fund for Fundamental Researchesof Ukraine (Agreement F33/24-2013). [1] G.Goldhaber, S.Goldhaber, W. Lee, A. Pais, Phys. Rev. (1960) 325.[2] G.I. Kopylov, M.I. Podgoretsky, Sov. J. Nucl. Phys.: (1972) 219; (1972) 219 (1973)336; (1974) 215.[3] S. Pratt, Phys. Rev. D (1986)1314.[4] A.N. Makhlin, Yu.M. Sinyukov, Sov. J. Nucl. Phys. (1987) 345;A. N. Makhlin, Yu. M. Sinyukov, Z. Phys. C (1988) 69.[5] Yu. M. Sinyukov, Nucl. Phys. A (1989) 151.[6] Y. Hama, S.S. Padula, Phys. Rev. D (1988) 3237.[7] Yu.M. Sinyukov, Nucl.Phys. A (1994) 589c;Yu.M. Sinyukov, in: Hot Hadronic Matter: Theory and Experiment, eds. J. Letessier, H.H.Gutbrod and J. Rafelski (Plenum, New York) 1995, 309.[8] S.V. Akkelin, Yu.M. Sinyukov, Phys. Lett. B (1995) 525; S.V. Akkelin, Yu.M. Sinyukov,Z. Phys. C (1996) 501.[9] Yu.M. Sinyukov, S.V. Akkelin, Iu.A. Karpenko. Act. Phys. Pol. (2009) 1025.[10] Yu.M.Sinyukov, R.Lednick´y, J.Pluta, B.Erazmus, S.V.Akkelin. Phys. Lett. B (1998) 248.[11] Gordon Baym, Peter Braun-Munzinger, Nucl, Phys. A (1996) 286c.[12] Yu. M. Sinyukov, V. M. Shapoval, Phys.Rev. D (2013) 094024, arXiv:1209.1747.[13] M.S. Nilsson, L.V. Bravina, E.E. Zabrodin, L.V. Malinina,J. Bleibel, U. Mainz. Phys.Rev. D (2011) 054006.[14] Q. Li, G. Graef, M. Bleicher, arXiv: 1209.0042 [hep-ph], 2012.[15] P. Bozek, Acta Phys. Pol. B41 (2010) 837.[16] K. Werner et al, Phys.Rev. C (2011) 044915.[17] Yu.M. Sinyukov, S.V. Akkelin, and Y. Hama, Phys. Rev. Lett. (2002) 052301.[18] S.V. Akkelin, Y. Hama, Iu.A. Karpenko, Yu.M. Sinyukov. Phys. Rev. C (2008) 034906.Iu.A. Karpenko, Yu.M. Sinyukov. Phys. Rev. C (2010) 054903. 19] Iu.A. Karpenko, Yu.M. Sinyukov, K. Werner. Phys.Rev. C (2013) 024914.[20] L.D. Landau, Izv. Akad. Nauk SSSR, Ser. Fiz. (1953) 51.[21] Edward K.G. Sarkisyan, Alexander S. Sakharov, Eur. Phys. J. C (2010) 533-541.[22] Edward K.G. Sarkisyan, Alexander S. Sakharov, AIP Conf. Proc. (2006) 35-41,arXiv:hep-ph/0510191.[23] M. Laine, Y. Schroder, Phys. Rev. D (2006) 085009.[24] K. Nakamura et al. (Particle Data Group), J. Phys. G , 075021 (2010) and 2011 partialupdate for the 2012 edition.[25] S.A. Bass et al., Prog. Part. Nucl. Phys. (1998) 255; Prog. Part. Nucl. Phys. (1998)225; M. Bleicher et al., J. Phys. G (1999) 1859.[26] B.Z. Kopeliovich , A. Schafer, and A. V. Tarasov, Phys. Rev. D (2000) 054022;E. Shuryak and I. Zahed, Phys. Rev. D (2004) 014011.[27] S. Pratt, Phys. Rev. C (1997) 1095.[28] K. Aamodt, et al. (ALICE Collaboration), Phys. Rev. D (2011) 112004.[29] B. Lorstad, Yu.M. Sinyukov, Phys. Lett. B ( 1991 ) 159.[30] R. Lednick´y, V.L. Lyuboshits, M.I. Podgoretsky. J.Nucl Phys. (1983) 251.[31] Yu.M.Sinyukov, A.Yu.Tolstykh. Z.Phys.C – Particles and Fields (1994) 593.[32] P. Bozek and P. Broniowski arXiv:1301.3314 (2013).[33] M. Lisa, S. Pratt, R. Soltz, U. Wiedemann. Ann.Rev.Nucl.Part.Sci. (2005) 357;M. Lisa Braz.J.Phys. (2004) 064901;S.V. Akkelin, Yu.M. Sinyukov, Phys. Rev. C (2006) 034908.[35] T. Csorgo and B. Lorstad, Lund Preprint LUNFD6 (NFPL-7082), 1994.[36] S.V. Akkelin, P. Braun-Munzinger, Yu.M. Sinyukov, Nucl. Phys. A (2002) 439.[37] P. Csizmadia, T. Csorgo, and B. Lukacs, Phys. Lett. B (1998) 21.[38] G.F. Bertsch, Phys. Rev. Lett. (1994) 2349; (1996) 789.[39] A. Bzdak, B. Schenke, P. Tribedy, R. Venugopalan, arXiv:1304.3403 (2013).[40] K. Aamodt, et al (ALICE Collaboration). Phys.Lett. B (2011) 328.[41] Dariusz Antonczyk. Acta Phys. Polon. B (2009) 1137.[42] S.V. Afanasiev et al, NA49 Collaboration. Phys. Rev. C (2002) 054902.[43] C. Alt et al, NA49 Collaboration. Phys.Rev. C (2008) 064908. 44] J. Adams et al. (STAR Collaboration). Phys.Rev.Lett. (2004) 112301.[45] J. Adams et al. (STAR Collaboration). Phys.Rev. C (2004), 044906.[46] S.S. Adler et al. (PHENIX Collaboration). Phys.Rev. C (2004) 034909.[47] S.S. Adler et al. (PHENIX Collaboration). Phys.Rev.Lett. (2004) 152302.(2004) 152302.