Bose-Einstein correlations of identical charged-pion pairs produced in hadronic Z decays are analyzed in terms of various parametrizations. The tau-model with a one-sided L\'evy proper-time distribution provides a good description, enabling the source function to be reconstructed.Bose-Einstein correlations of identical charged-pion pairs produced in hadronic Z decays are analyzed in terms of various parametrizations. The tau-model with a one-sided L\'evy proper-time distribution provides a good description, enabling the source function to be reconstructed

Csorgo, T.

Kittel, W.

Metzger, W.J.

Novak, T.

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Vol. 40 (2009) ACTA PHYSICA POLONICA B No 4INITIAL CONDITIONS, TIME EVOLUTIONAND BE CORRELATIONS IN e+e− ANNIHILATION∗T. Novák†, T. Csörgő‡, W. Kittel, W.J. Metzgeron behalf of the L3 CollaborationRadboud UniversityHeyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands(Received February 4, 2009)Bose–Einstein correlations of identical charged-pion pairs produced inhadronic Z decays are analyzed in terms of various parametrizations. Theτ -model with a one-sided Lévy proper-time distribution provides a gooddescription, enabling the source function to be reconstructed.PACS numbers: 13.38.Dg, 13.66.Bc1. IntroductionIn particle and nuclear physics intensity interferometry provides a directexperimental method for the determination of sizes, shapes and lifetimes ofparticle-emitting sources (for recent reviews see [1–3]). In particular, bosoninterferometry provides a powerful tool for the investigation of the space-time structure of particle production processes, since Bose–Einstein correla-tions (BEC) of two identical bosons reflect both geometrical and dynamicalproperties of the particle radiating source.For our analysis we use a sample of about 500 thousand two-jet events,selected by the Durham algorithm [6] with ycut = 0.006, from e+e−annihi-lation data collected by L3 at a center-of-mass energy of 91.2 GeV.2. Parametrizations of BECThe two-particle correlation function is defined as:R2(p1, p2) =ρ2(p1, p2)ρ1(p1)ρ1(p2), (1)∗ Presented by T. Novák at the IV Workshop on Particle Correlations and Femtoscopy,Kraków, Poland, September 11–14, 2008.† Present address: Károly Róbert College, Mátrai 36, 3200, Gyöngyös, Hungary.‡ Visitor from Budapest, Hungary, sponsored by the Scientific Exchange between Hun-gary (OTKA) and The Netherlands (NWO), project B64-27/N25186.(919)920 T. Novák et al.where ρ2(p1, p2) is the two-particle invariant momentum distribution, ρ1(pi)the single-particle invariant momentum distributions and pi the four-momen-tum of particle i. Since we are only interested in BEC, the product of singleparticle densities is replaced by the so-called reference sample, ρ0(p1, p2),the two-particle density that would occur in the absence of Bose–Einsteininterference. Here we use mixed events as a reference sample [7].After some assumptions [1, 2], this two-particle correlation function isrelated to the Fourier transformed source distribution. In this caseR2(p1, p2) = 1 + |f˜(Q)|2 , (2)where Q is the invariant four-momentum difference, Q =√−(p1 − p2)2 andf˜(Q) is the Fourier transform of the density of the source, f(x).2.1. Gaussian distributed sourceThe simplest assumption is that the source has a symmetric Gaussiandistribution, in which case f˜(Q) = exp(iµQ− (RQ)22)andR2(Q) = γ[1 + λ exp(−(RQ)2)](1 + δQ) , (3)where the parameter γ is a constant of normalization, λ is an interceptor incoherence factor, which measures the strength of the correlation, and(1 + δQ) is introduced to parametrize possible long-range correlations notadequately accounted for in the reference sample.A fit of Eq. (3) to the data results in an unacceptably low confidencelevel [7] from which we conclude that the shape of the source deviates froma Gaussian. The fit is particularly bad at low Q values.2.2. Lévy distributed sourceAdopting Nolan’s S(α, β = 0, γ, δ; 1) convention [8] for the symmetricLévy stable distribution with rescaling of the scale parameter γ to R and thelocation parameter δ to x0, the Fourier transform (characteristic function)f˜(Q) has the following general form:f˜(Q) = exp(iQx0 − |RQ|α) . (4)The index of stability, α, satisfies the inequality 0 < α ≤ 2. The case α = 2corresponds to a Gaussian source distribution. For more details see [8].Then R2 has the following, relatively simple form [9]:R2(Q) = γ [1 + λ exp (−(RQ)α)] (1 + δQ) . (5)After fitting Eq. (5) to the data it is clear that the correlation function is farfrom Gaussian: α ≈ 1.3. The confidence level, although improved comparedto the fit of Eq. (3), is still unacceptably low [7].Initial Conditions, Time Evolution and BE Correlations in e+e− . . . 9213. The τ -modelA model of strongly correlated phase-space was developed [12] to ex-plain the experimentally found invariant relative momentum Q dependenceof Bose–Einstein correlations in e+e− reactions. This model also predictsa specific transverse mass dependence of R2, that we subject to an experi-mental test here. In this model, it is assumed that the average productionpoint xµ of particles with a given momentum kµ is given byxµ(kµ) = dkµ . (6)In the case of two-jet events, d = τ/mt, where τ =√t2 − k2z is the lon-gitudinal proper-time and mt =√m2 + p2t is the transverse mass. Thesecond assumption is that the distribution of xµ(kµ) about its average,δ∆(xµ(kµ) − xµ(kµ)), is narrower than the proper-time distribution. Thenthe emission function of the τ -model isS(x, k) =∞∫0dτH(τ)δ∆(x− dk)N1(k) , (7)whereH(τ) is the longitudinal proper-time distribution, the factor δ∆(x−dk)describes the strength of the correlations between coordinate space and mo-mentum space variables and N1(k) is the experimentally measurable single-particle spectrum. In the plane-wave approximation the Yano–Koonin for-mula [13] gives the following two-pion multiplicity distribution:ρ2(k1, k2) =∫d4x1d4x2S(x1, k1)S(x2, k2) (1 + cos [(k1 − k2)(x1 − x2)]) .(8)Approximating the δ∆ function by a Dirac delta function, the argument ofthe cosine becomes(k1 − k2)(x¯1 − x¯2) = −0.5(d1 + d2)Q2 . (9)Then the two-particle Bose–Einstein correlation function is obtained asR2(k1, k2) = 1 + λReH˜2(Q22mt), (10)where H˜(ω) =∫dτH(τ) exp(iωτ) is the Fourier transform of H(τ). Thusan invariant relative momentum dependent BEC appears.Guided by the result of the previous section, we use a one-sided Lévydistribution for the longitudinal proper-time density. The corresponding922 T. Novák et al.BEC function has an analytic form [9–11]:R2(Q2,mt)=γ[1+λ cos(τ0Q2mt+A(∆τQ2mt)α)exp(−(∆τQ2mt)α)]B ,(11)where the parameter τ0 is the proper-time of the onset of particle production,∆τ is a measure of the width of the proper-time distribution, A = tan(αpi4)and B = (1 + δQ).Assuming that particle production starts immediately and defining an ef-fective radius, R, [11] R2 simplifies toR2(Q) = γ[1 + λ cos[(RaQ)2α]exp(−(RQ)2α)](1 + δQ) , (12)where Ra is related to R by R2αa = tan(αpi/2)R2α.The fit of Eq. (12) to the data is statistically acceptable [7]. The dataare well described by the fit. For Q between 0.5 GeV and 1.5 GeV the datapoints go below the level of the long-range correlations extrapolated to lowerQ values. These data points indicate an anti-correlation in the Q ≈ 1 GeVregion. This property of the data is well reproduced by the fitted curve,which also goes below unity as a result of the cosine term in Eq. (12), whichcomes from the asymmetric Lévy assumption.After fitting Eq. (11) for various mt intervals we find that the quality ofthe fits is statistically acceptable and the fitted values of the model param-eters are stable and within errors the same in all investigated mt intervals,confirming the mt-dependence predicted by the τ -model. The τ -model witha one-sided Levy proper-time distribution describes the data with parame-ters τ0 = 0 fm, α ≈ 0.43± 0.03 and ∆τ ≈ 1.8± 0.4 fm (the difference in mtof the two pions is required to be less than 0.2 GeV).4. Reconstruction of the emission functionIn order to reconstruct the space-time picture of the emitting process weassume that the emission function can be factorized in the following way:S(r, z, t) = I(r)G(η)H(τ) , (13)where I(r) is the single-particle transverse distribution, G(η) is the space-time rapidity distribution of particle production, which approximately coin-cides with the single-particle rapidity distribution, and H(τ) is the observedproper-time distribution.With these assumptions one can reconstruct the longitudinal part of theemission function integrated over the transverse distribution. It is plottedas a function of t and z in Fig. 1. It exhibits the typical boomerang shapeInitial Conditions, Time Evolution and BE Correlations in e+e− . . . 923-20-1001020z  (fm)01020t  (fm)012 S(z,t)  (fm   ) -2-20-1001020 z  (fm)01020t  (fm)012 S(z,t)  (fm   ) -2Fig. 1. Two views of the longitudinal part of the source function normalized to theaverage number of pions per event.τ                   = 0.05 (fm)-0.4-0.200.20.4 x  (fm)-0.4-0.200.20.4y  (fm)00.050.15        τ     S(x,y,  ) (fm   )  -3 τ                  = 0.1 (fm)-0.4-0.200.20.4 x  (fm)-0.4-0.200.20.4y  (fm)00.050.15        τ     S(x,y,  ) (fm   )  -3 τ                   = 0.15 (fm)-0.4-0.200.20.4 x  (fm)-0.4-0.200.20.4y  (fm)00.050.15        τ     S(x,y,  ) (fm   )  -3 τ                  = 0.2 (fm)-0.4-0.200.20.4 x  (fm)-0.4-0.200.20.4y  (fm)00.050.15        τ     S(x,y,  ) (fm   )  -3 τ                   = 0.25 (fm)-0.4-0.200.20.4 x  (fm)-0.4-0.200.20.4y  (fm)00.050.15        τ     S(x,y,  ) (fm   )  -3 τ                  = 0.3 (fm)-0.4-0.200.20.4 x  (fm)-0.4-0.200.20.4y  (fm)00.050.15        τ     S(x,y,  ) (fm   )  -3 Fig. 2. The source function normalized to the average number of pions per eventfor various proper-times.with a maximum at low t and z but with tails reaching out to very large tand z values.The transverse profile, which follows from Eq. (7), is given byd4ndτd3r=m3tτ3H(τ)N1(k =mtrτ). (14)924 T. Novák et al.This equation describes the particle production in coordinate space as a func-tion of the proper-time τ . It describes the expansion of the source as theproper-time increases. The particle production probability is proportionalto the proper-time distribution H(τ). Fig. 2 shows the transverse part ofthe emission function for various proper-times. Particle production startsimmediately, increases rapidly and decreases slowly. A ring-like structure,similar to the expanding, ring-like wave created by a pebble in a pond, isreconstructed from L3 data. An animated gif file that shows this effect isavailable from [16].This research was supported by the OTKA grants NK73143 and TO49466,as well as by the exchange program of the Hungarian Academy of Sciencesand the Polish Academy of Arts and Sciences.REFERENCES[1] W. Kittel, Acta Phys. Pol. B 32, 3927 (2001).[2] T. Csörgő, Heavy Ion Physics 15, 1 (2002).[3] T. Csörgő, http://arxiv.org/abs/0901.0766[4] M. Althoﬀ et al. [TASSO Collaboration], Z. Phys. C30, 355 (1986).[5] T. Csörgő, B. Lörstadt, Phys. Rev. C54, 1390 (1996).[6] S. Cantani et al., Phys. Lett. B279, 432 (1991).[7] T. Novák, Ph.D. thesis, Radboud University Nijmegen (2008),http://webdoc.ubn.ru.nl/mono/n/novak_t/bosecoine.pdf[8] J.P. Nolan, Stable Distributions: Models for Heavy Tailed Data,http://academic2american.edu/˜jpnolan/stable/CHAP1.PDF[9] T. Csörgő, S. Hegyi, W.A. Zajc, Eur. Phys. J. C36, 67 (2004); T. Csörgő,hep-ph/0301164.[10] T. Csörgő, private communication.[11] T. Csörgő, W. Kittel, W.J. Metzger, T. Novák Phys. Lett. B663, 214 (2008).[12] T. Csörgő, J. Zimányi, Nucl. Phys. A517, 588 (1990).[13] F.B. Yano, S.E. Koonin, Phys. Lett. B78, 556 (1978).[14] N.M. Agababyan at al. [NA22 Collaboration], Phys. Lett. B422, 359 (1998).[15] A. Ster, T. Csörgő, B. Lörstad, Nucl. Phys. A661, 419 (1999).[16] http://www.hef.kun.nl/˜novakt/movie/movie.gif

Initial conditions, time evolution and BE correlations in e+e- annihilation

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