Finite size of hadrons and Bose-Einstein correlations in $pp$ collisions at 7 TeV

Space-time correlations between produced particles, induced by the composite nature of hadrons, imply specific changes in the properties of the correlation functions for identical particles. The expected magnitude of these effects is evaluated using the recently published blast-wave model analysis of the data for $pp$ collisions at $\sqrt{s}$ = 7 TeV.

1.It has been recently pointed out [1] that since hadrons produced in high-energy collisions are not point-like objects, they cannot be uncorrelated.Indeed, being composite, hadrons cannot occupy too close space-time points (because at small distance the constituents of hadrons mix and there are no separate hadrons to interfere).Since the HBT experiment measures the quantum interference between wave functions of hadrons, it cannot see hadrons which are too close to each other.Consequently, the distribution function of the pair of hadrons must vanish at small distances between them.
This implies of course a correlation in space-time.As this correlation is the necessary consequence of the composite structure of hadrons (and thus it is a general property of the system) it is interesting to investigate to what extent it modifies the accepted ideas about the quantum interference which are, usually, derived under the assumption that such correlations can be neglected [2].
It was already shown in [1] that such space-time correlations may be responsible for the observation that the two-pion Bose-Einstein correlation function takes values below unity [3][4][5], at variance with the well-known theorem valid when the correlations are ignored [2].
In the present paper the investigation of this phenomenon is continued, using the recently published [6] analysis of the data on HBT radii, measured by the ALICE collaboration [7].This allows to estimate quantitatively the magnitude of the effect and to give predictions for its size in all three directions long, side and out, commonly used in discussion of the quantum interference [2].
In the next section the consequences of the space-time correlations for the HBT correlation functions are explained.In Sections 3 and 4 the blast-wave model used for the quantitative estimate of the effect is presented.The results are presented and summarized in the last two sections.
2. In absence of correlations between produced hadrons, the two-particle source function is the simple product where w(p, x) is the single-particle source function (Wigner function).Consequently, the Bose-Einstein correlation function between the momenta of two identical particles is given by [2] C(p where The data from the L3 collaboration [3] and from the CMS colllaboration [5] show that the correlation function C(p 1 , p 2 ) takes values below unity, contrary to Eq. (3).Thus the particles must be correlated and we propose that this effect is due to the composite nature of hadrons.
To implement these space-time correlations, we replace formula (1) for the two-particle source function by where D(x 1 − x 2 ) is the cut-off function that satisfies the constraint D(x 1 − x 2 = 0) = 1 and tends to 0 at larger distances (above, let us say, 1 fm).Then, the HBT correlation function becomes where the uncorrelated part C noncorr (P 12 , Q) is given by (3), while the correction due to space-time correlations reads where One sees that the contribution from the correlation part is negative.Moreover, since it obtains contributions from a small region of space-time, its dependence on Q is much less steep than that of the uncorrelated part.Consequently, at Q large enough C(P 12 , Q) may easily fall below one.
To describe the actual measurements one has to take into account that particles produced very far from the center (e.g.those arising from long-lived resonances) form a "halo" and do not contribute to the HBT correlations [8].Thus we have where p is the probability that both particles originate from the "core".
In the ALICE experiment [7] Ĉobs was, in addition, normalized to 1 at some Q 0 where the influence of quantum interference is expected to be negligible.Thus we finally have to consider the function Introducing the (measured) intercept parameter λ by the condition one obtains This allows to evaluate the measured correlation function in terms of the measured intercept parameter λ and the evaluated correlation function C(P 12 , Q).

3.
To have an idea on the magnitude of the effect we discuss, we have used the blast-wave model described in detail in [6,9].In this model, at freeze-out, hadrons are created at a fixed (longitudinal) proper time The single-particle source function (in the longitudinal c.m.s.system) becomes w(p, x) = k 0 cosh ηe −U cosh η+V cos φ f (r)rdrdφdη (15 where k 0 = m 2 + k 2 ⊥ , whereas η, φ and r are space-time rapidity, azimuthal angle and transverse distance from the symmetry axis 1 .We have also introduced the notation with T = 1/β being the freeze-out temperature.Finally, θ describes the transverse flow by the relation with ω being a parameter.The function f (r) describes the transverse profile of the source.It was shown in [6] that the model is flexible enough to describe the HBT radii measured by the ALICE collaboration [7].The function f (r) was taken in the form corresponding to a "shell" of the width √ 2 δ and radius R. Thus the model contains 5 free parameters: T, ω, τ f , R and δ, which may depend on the multiplicity of the event.Their values, giving a good description of the HBT radii measured in [7], are given in [6].
4. It remains to select the cut-off function D(x 1 , x 2 ).Since very little is known about its shape, it seems reasonable to start with the simplest possibility, i.e the Gaussian where d(x 1 , x 2 ) is the distance between the two pions and ∆ is the characteristic distance beyond which pions do not interact anymore.Since we treat particles as extended objects produced on the hyperbola (14), the longitudinal distance between the two hadrons located at the space-time rapidities η 1 , η 2 should be calculated along this curve, which yields In the frame where η 1 + η 2 = 0 we also have t 1 = t 2 and thus the total distance between particles is Since this expression is invariant under boost in the longitudinal direction, it is also valid in the LCMS system, and thus we finally have ----  5. Using (19), ( 22) and the source function of the model described in Sec. 3, with the parameters taken from [6] 2 , we have evaluated corrections to the HBT correlation functions (6) for all intervals of the particle multiplicity and transverse momentum (as measured in [7]), and for all three directions of the vector Q.The cut-off distance ∆ was taken to be 1 fm.Some of the results are shown if Figs. 1-3.
One sees that for the long direction the correlation function falls below 1 at all multiplicities and transverse momenta of the pair.The depth of the minimum in the long direction varies from ∼ 0.02 to ∼ 0.01 (below 1) when the HBT radius R long increases from ∼ 0.8 to ∼ 2 fm. 1 All irrelevant constants are cancelled in the definition of w(p, x).In the side and out directions the results are strongly dependent on the value of the transverse momentum of the pair.At k ⊥ ≤ 300 MeV for the side and k ⊥ ≤ 400 MeV for the out direction the correlation functions are always larger than 1 in the investigated region.In the side direction the correlation function shows a clear structure: a minimum followed by a maximum (particularly at low multiplicities).At larger k ⊥ the minimum below 1 shows up in both cases.
In the side direction the minimum at k ⊥ ≥ 300 MeV is similar to that found in the long direction.It is about twice deeper in the out direction (above 400 MeV).In both cases the minimum is deeper when the multiplicity increases.Also in this case the change is controlled by the corresponding HBT radii.
6.In summary, we have estimated to what extent the space-time correlations implied by the excluded volume effect modify the HBT correlation functions.
Our conclusions can be formulated as follows: (i) The space-time correlations induced by the finite size of hadrons lead to a rich structure of the HBT correlation functions, depending on (i) the measurement direction, (ii) multiplicity and (iii) the transverse momentum of the pair.
(ii) The difference between the long and the two other directions at small k ⊥ is particularly striking.
(iii) At large k ⊥ the minimum below 1 shows up in every direction.It is about twice deeper for out than for the long and side directions.
Finally, it should be emphasized that our results depend on the validity of the blast-wave model [6] used for description of ALICE data [7].Thus the experimental verification of the presented results could be an important contribution to the discussion about the validity of the blast-wave picture in such small systems as those produced in pp collisions at the LHC energies.

Figure 1 :
Figure 1: (Color online) Correlation function C obs for the long direction in the interval 0.2 GeV ≤ Q ≤ 0.8 GeV (normalized to 1 at Q= 1GeV).The dashed lines describe the results for k ⊥ = 163 MeV and the two multiplicity classes: Nc = 12-16 and Nc = 52-151.The solid lines describe the results for k ⊥ = 547 MeV and the same two multiplicity classes.

Figure 2 :
Figure 2: (Color online) The same as Fig. 1 but for the side direction.

2 Figure 3 :
Figure 3: (Color online) The same as Fig. 1 but for the out direction.