On long-range pionic Bose-Einstein correlations -- Including analyses of OPAL, L3 and CMS BECs
aa r X i v : . [ h e p - ph ] F e b On long-range pionic Bose-Einstein correlations– Including analyses of OPAL, L3 and CMS BECs –
Takuya Mizoguchi and Minoru Biyajima National Institute of Technology, Toba College, Toba 517-8501, Japan Department of Physics, Shinshu University, Matsumoto 390-8621, Japan
February 23, 2021
Abstract
Long-range correlation plays an important role in analyses of pionic Bose-Einstein corre-lations (BECs). In many cases, such correlations are phenomenologically introduced. In thisinvestigation, we propose an analytic form. By making use of the form, we analyze the OPALBEC and the L3 BEC at Z -pole and the CMS BEC at 0.9 and 7 TeV using our formulasand the τ -model. The parameters estimated by both approaches are found to be consistent.Utilizing the Fourier transform in four-dimensional Euclidean space, a number of pion-pairdensity distributions are also studied. The following conventional formula is utilized as a standard tool in many analyses of pionic Bose-Einstein correlation (BEC) [1–10]:CF I = [1 . λE BE ( R, Q )] · LRC , (1)where E BE is the exchange function between two identical pions and Q = − p − ( p − p ) is themagnitude of the momentum transition squared between them. Typically, E BE is given by theGaussian distribution and/or the exponential function. The degree of coherence is expressed by λ .The following long-range correlation (LRC) is frequently used:LRC ( δ ) = C (1 + δQ ) (2)In this present paper, we first pay attention to the BEC created at the Z -pole by the OPALcollaboration, because they used a second kind of LRC, which was given asLRC ( δ, ε ) = C (1 + δQ + εQ ) (3)As is seen in Table 1 and Fig. 1, when LRC ( δ, ε ) (i.e., Eq. (3)) is utilized in the analysis, weobtain a better χ value than that using Eq. (2). This fact may suggest us that Eq. (3) reflectssome physical meaning.In Section 2, we consider the analytic form of the LRC. In Section 3, we analyze the CMSBEC created at 0.9 and 7 TeV by the CMS Collaboration using Eqs. (3) and (4) and an analytic1able 1: Analysis of OPAL data using Eqs. (1)–(2). “G” denotes the Gaussian distribution.LRC R (fm)(G) λ c δ (GeV − ) ε (GeV − ) χ / ndf( δ ) 1.12 ± ± ± ± δ , ε ) 0.95 ± ± ± ± − ± N ( + : − ) / N B G Q (GeV)Excluded from fitOPAL BEC at Z -pole χ / ndf = / CF I ( G )× LRC (δ) N ( + : − ) / N B G Q (GeV)Excluded from fitOPAL BEC at Z -pole χ / ndf = / CF I ( G )× LRC (δ , ε) Figure 1: Analysis of the OPAL BEC at the Z -pole using by Eq. (1) with Eqs. (2) and (3).form mentioned in the next section. In Section 4, we analyze the OPAL BEC at the Z -pole usinga new conventional formula introduced in Section 3. In Section 5, by making use of the Fouriertransform, we show the density distribution of pion-pairs in a four-dimensional Euclidean space,where ξ = ( x + y + z + ( ct ) ) / is introduced. Finally, in Section 6, concluding remarks anddiscussions are presented. It is remarkable that Eq. (3) utilized by the OPAL collaboration works well; given that it is aphenomenological form that depends upon the parameters δ and ε , its asymptotic behavior is asfollows: Eq. (3) Q → ∞−−−−−−−−−−−−→ − large value . In Table 1, at Q = 0 . c is 0 . ± .
01, which is much lower than C ∼ = 1 . Q → ∞ , we propose the following analytic form:LRC ( α, β, n ) = C [1 + αQ n exp( − βQ )] , (4)where α and β are parameters. As n = 1,Eq. (4) ( Q → ∞−−−−−−−−−−−−→ C ≃ . ,Q → −−−−−−−−−−−−→ C (1 . αQ − αβQ ) . (5)2his is the same form as Eq. (3); moreover, to make certain a case with n = 2 is investigated. Inaddition to Eq. (5), the following correspondences are expected: α > · · · reproducing Eqs. (2) and (3) ,α ≈ · · · no any effect ,α < · · · because of negative contribution to BEC, this sign implies the subtractionof non-BE effect: the contamination between different hadron pairs, resonanceseffect, and/or the energy conservation.By using Eq. (4) with smaller χ values, we are able to determine some physical informationcontained within the LRCs. It should be noted that the OPAL collaboration reported two kinds of data, i.e., an “ordinary”data ensemble and a “corrected” data ensemble that had been renormalized using the Monte Carlocalculation N (2+: 2 − ) /N BG ,N (2+: 2 − )MC /N BGMC = N (2+: 2 − ) /N (2+: 2 − )MC N BG /N BGMC . (a) First of all, we analyze the OPAL BEC at the Z -pole. Our results by means of Eq. (4) with n = 1 and 2 are displayed in Table 2 and Fig. 2. As α > ( δ, ε ) in Table 1. For α <
0, improvements about χ ’s are seen in Table 2. Thisfact probably means that the OPAL BEC at the Z -pole contains non-BE effects, which must besubtracted from the data (see Section 6).Table 2: Analysis of the OPAL data using Eqs. (1) and (4). Values set with *) indicate that | α | = 6 .
24 GeV − is larger than | α | = 1 .
61 GeV − for n = 1.LRC R (fm)(G) λ c α (GeV − n ) β (GeV − ) χ / ndf α > n = 1 .
0) 0.94 ± ± ± ± ± n = 2 .
0) 0.96 ± ± ± ± ± α < n = 1 .
0) 0.94 ± ± ± − ± ± n = 2 .
0) 1.25 ± ± ± − ± ± (b) Next, we analyze the second data ensemble by means of Eqs. (1) and (4). Our results areshown in Fig. 3 and Table 3. 3 .60.70.80.91 0 0.5 1 1.5 2 L RC Q (GeV/c)OPAL at Z -poleLRC (δ) LRC (δ , ε) LRC (α < , β ,n= LRC (α < , β ,n= Figure 2: LRC in the OPAL BEC at the Z -pole ( α <
0) (see Tables 1 and 2). N M C ( + : − ) / N M CB G Q (GeV)Excluded from fitOPAL BEC at Z -pole χ / ndf = . / CF I ( G )× LRC (α , β ,n= Figure 3: Analysis of the OPAL BEC at the Z -pole renormalized using the Monte Carlo calculationbased on Eqs. (1) and (4).Table 3: Analysis of the OPAL BEC at the Z -pole renormalized using the Monte Carlo calculationbased on Eqs. (1) and (4).LRC R (fm)(G) λ c α (GeV − n ) β (GeV − ) χ / ndf α > n = 1 .
0) 0.91 ± ± ± ± ± n = 2 .
0) 0.91 ± ± ± ± ± α < n = 1 .
0) 0.89 ± ± ± − ± ± n = 2 .
0) 0.92 ± ± ± − ± ± Because the L3 collaboration reported BEC data for 2-jet ( q ¯ q jet) and 3-jet ( q ¯ qg jet) cases, we areinterested in analyzing those data. Such data can be categorized into the same kinds of ensembleswith two and 3three jets, respectively. Thus, we may analyze them using the CF I with LRC ( α, β, n ) .Our results are shown in Fig. 4 and Table 4. To compare them with those obtained using the4 -model [11–15], F τ ( e + e − ) = (cid:8) λ cos (cid:2) ( R a Q ) α τ (cid:3) exp (cid:2) − ( RQ ) α τ (cid:3)(cid:9) × LRC ( δ ) , (6)with R α τ a = tan( α τ π/ R α τ , we analyze them. As seen in Fig. 5, the effective degree of coherencein the τ -model is oscillating. The results from Eq. (6) are also presented in Table 4. N ( + : − ) / N B G Q [GeV]L3 BEC Z -pole 2-jet χ / n.d.f. = . / CF I ( E )× LRC (α , β ,n= N ( + : − ) / N B G Q [GeV]L3 BEC Z -pole 3-jet χ / n.d.f. = . / CF I ( E )× LRC (α , β ,n= Figure 4: Analysis of the L3 BEC at the Z -pole using Eqs. (1) and (4).Table 4: Analysis of the L3 BEC at the Z -pole using Eqs. (1) and (4) and the τ -model.CF I with LRC ( α, β, n =2) event R (fm)(E) λ c α (GeV − ) β (GeV − ) χ / ndf2-jet 0.87 ± ± ± − ± ± ± ± ± − ± ± τ -model with LRC ( δ ) event R (fm) λ c α τ δ (GeV − ) χ /ndf2-jet 0.78 ± ± ± ± ± ± ± ± ± ± -0.500.51 0 1 2 3 4 Q [GeV]L3 BEC Z -pole 2-jet, τ -model λ cos (( QR a ) τ )× c (1 + δ Q )λ× c (1 + δ Q ) -0.200.20.40.6 0 1 2 3 4 Q [GeV]‘dip’ cos (( QR a ) τ ) exp (−( QR ) τ ) Figure 5: “Effective degree of coherence” and “dip structure” in the τ -model.5 Analysis of CMS BEC
We analyze the CMS BEC at 0.9 and 7 TeV using the following formula:CF II = [(1 . λ E BE ( R , Q ) + λ E BE ( R , Q )] · LRC , (7)where the second λ ( λ ) and the second exchange function ( E BE ) are introduced to describe theBEC data at the LHC. A detailed derivation and analysis with Eq. (3) (i.e., LRC ( δ ) ) are presentedin Refs. [7, 9, 10]. For our purposes, the analytic LRC ( δ, ε ) is also necessary in the CF II . Our resultsare shown in Figs. 6 and 7, and Table 5.In Table 5, we also show the results obtained using the τ -model formula, which is appropriatefor LHC collisions. This means that there are two formulas, Eq. (6) for e + e − collisions and Eq. (8)for LHC collisions. Thus, to analyze the CMS BEC at 0.9 and 7 TeV, the authors of [5, 11, 12, 16]employ the following equation: F τ = n . λ cos h ( R Q ) + tan (cid:16) α τ π (cid:17) ( RQ ) α τ i exp( − ( RQ ) α τ ) o × LRC ( δ ) , (8)where R is the free parameter.As seen in Table 5 and Fig. 7, three LRCs are grouped together and the estimated parametersare almost the same.The three columns in Table 5 indicate that (in the center column) is almost the same as the setobtain using the τ -model, provided that LRC ( α, β, n =2) is adopted. Comparing the second set withthe first one with LRC ( δ, ε ) , we find that the estimated R s in the first set are somewhat smallerthan R those in the second set. The situation is the opposite for R . This is probably attributableto the sets’ normalization factors, c = 0 .
91 or 0.93. N ( + : − ) / N B G Q [GeV]CMS 0.9 TeVCF II × LRC (α , β ,n= LRC (α , β ,n= (1 + λ E BE )× LRC (α , β ,n= c (1 + λ E BE ) N ( + : − ) / N B G Q [GeV]CMS 7 TeVCF II × LRC (α , β ,n= LRC (α , β ,n= (1 + λ E BE )× LRC (α , β ,n= c (1 + λ E BE ) Figure 6: Analysis of the CMS BEC at 0.9 and 7 TeV.6able 5: The estimated parameters for the CMS BEC at 0.9 and 7 TeV.CF II × LRC ( δ, ε ) CF II × LRC ( α, β, n =2) F τ ( τ -model)0.9 TeV R (fm) R (fm) R (fm) R = 2 . λ = 0 .
82 (E) R = 3 . λ = 0 .
71 (E) R = 2 .
98 fm R = 0 . λ = 0 .
15 (G) R = 0 . λ = 0 .
10 (G) R = 0 . λ + λ = 0 . λ + λ = 0 . λ = 1 . α τ = 0 . c = 0 . c = 1 . c = 0 . δ = 0 .
13 fm α = − .
07 fm ε = − .
035 fm β = 0 .
65 fm χ / ndf = 209 / χ / ndf = 209 / χ / ndf = 240 / R (fm) R (fm) R (fm) R = 3 .
15 fm, λ = 0 .
83 (E) R = 3 .
42 fm, λ = 0 .
75 (E) R = 3 .
46 fm R = 0 .
53 fm, λ = 0 .
12 (G) R = 0 .
15 fm, λ = 0 .
08 (G) R = 0 .
22 fm λ + λ = 0 . λ + λ = 0 . λ = 1 . α τ = 0 . c = 0 . c = 1 . c = 0 . δ = 0 .
020 fm α = − .
08 fm ε = − .
001 fm β = 0 .
72 fm χ / ndf = 208 / χ / ndf = 207 / χ / ndf = 289 / II × LRC ( δ ) is utilized in the analysis of the CMS BEC at 7 TeV,the following estimated parameters are obtained [9]: R = 3 .
88 fm, λ = 0 .
84 (E), R = 0 .
71 fm, λ = 0 .
12 (G), and χ = 540.Adopting LRC ( α, β, n =2) , we obtain a better χ value, as mentioned above. L RC Q (GeV/c) CMS 7 TeVLRC (δ) LRC (δ , ε) LRC (α < , β ,n= LRC (α < , β ,n= Figure 7: LRC in the CMS BEC at 7 TeV ( α <
Analysis of OPAL BEC renormalized by Monte Carlo at Z -pole using Eq. (7) Because the OPAL BEC at the Z -pole is not separable into “2-jet” and “3-jet” events, we ap-ply Eq. (7) to this case. Because the OPAL BEC prefers Gaussian distributions, we choose acombination of G+G.Our results are shown in Table 6 and Fig. 8. In Table 6, we observe that R (= 1 .
39 fm)increases as | α | = 2 .
41 GeV − increases; we conclude that these quantities are linked to each other.Table 6: Analysis of the OPAL BEC at the Z -pole using Eqs. (4) and (7). “G” indicates theGaussian distribution.data R (fm)(G) λ R (fm)(G) λ c α (GeV − ) χ / ndf β (GeV − ) N (2+: 2 − ) N BG ± ± ± ± ± − ± ± N (2+: 2 − )MC N BGMC ± ± ± ± ± − ± ± N M C ( + : − ) / N M CB G Q (GeV)Excluded from fitOPAL BEC at Z -pole χ / ndf = . / CF II ( G+G )× LRC (α , β ,n= LRC (α , β ,n= Figure 8: Analysis of the OPAL BEC at the Z -pole using Eqs. (4) and (7). We are able to calculate the pion-pairs density distribution in Euclidean space via the Fourieranalysis [17–20]: ρ BE ( ξ, R ) = 1(2 π ) ξ Z ∞ Q E BE ( Q, R ) J ( Qξ ) dQ, (9)where ξ = p x + y + z + ( ct ) , and J ( Qξ ) is the modified Bessel function. The variable ξ isdisplayed in Fig. 9. 8igure 9: Geometrical picture of the BEC. ξ denotes the distance between two production points( r , t ) and ( r , t ) in the Euclidean space-time. p and p are momenta of identical pion pair.For the LRC, by substituting the expression (LRC − .
0) = αQ n e − βQ into Eq. (9), we obtain ρ (LRC − ( ξ, α, β, n = 2) = 1(2 π ) ξ Z ∞ Q [LRC( Q, α, β, n = 2) − . J ( Qξ ) dQ = α (2 π ) ξ Γ(6)( β + ξ ) / P − β p β + ξ ! , (10)where P − ( x ) is the associated Legendre function and Γ( x ) is the gamma function.For n = 1, we have the following formula: ρ (LRC − ( ξ, α, β, n = 1) = α (2 π ) ξ Γ(5)( β + ξ ) P − β p β + ξ ! . (11)The pion-pairs density distributions are shown in Table 7 and Figs. 10 and 11. The suffixes E andG indicate the exponential functions and Gaussian distributions, respectively. The contributions ofthe Gaussian distributions may contain contamination between different hadron-pairs, resonances,and/or energy conservation among produced hadrons.Table 7: Correlation functions, source functions, and pion-pairs density distribution. E BE ( R, Q ) ρ ( ξ ) number of pairs density distributionsexp( − R Q ) 116 π R exp (cid:18) − ξ R (cid:19) π ξ ρ ( ξ )exp( − RQ ) 34 π R ξ/R ) ) / (2 π ξ : phase space)9 π ξ ρ (f m − ) ξ (fm)CMS BEC 7 TeV R = . fm ) , λ = . R = . fm ) , λ = . ×2π ξ ρ E (ξ ,R )λ ×2π ξ ρ G (ξ ,R ) -0.3-0.2-0.100.1 0 1 2 3 4 5 6 α × π ξ ρ L RC − (f m − ) ξ (fm)CMS BEC 7 TeV α = −0 . fm β = . fm ξ = √4 / = . fm |S N |=S P = . S N S P N ( + : − ) / N B G ( ξ ) ξ (fm)CMS BEC 7 TeV . + λ ×2π ξ ρ E (ξ ,R )1 . + λ ×2π ξ ρ G (ξ ,R ) α× π ξ ρ (LRC-1) ( ξ , β )sum = . + ξ (λ ρ E + λ ρ G + αρ ( LRC −1) ) Figure 10: Pion-pairs density distributions of the CMS BEC at 0.9 and 7 TeV in four-dimensionalEuclidean space, where ξ denotes the distance between two pion-production points. The Contri-butions of the crossed terms ( ρ G × ρ (LRC − and ρ E × ρ (LRC − ) are invisible. Maximal points inthe distributions are shifted by the phase space 2 π ξ : R G → √ R G and R E → p / R E .10 π ξ ρ (f m − ) ξ (fm)OPAL BEC at Z -pole R = . fm ) , λ = . R = . fm ) , λ = . ×2π ξ ρ G (ξ ,R )λ ×2π ξ ρ G (ξ ,R ) -0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.2 0 1 2 3 4 5 6 7 8 α × π ξ ρ ( L RC − ) (f m − ) ξ (fm)OPAL BEC at Z -pole α = −0 . fm β = . fm ξ = = . fm |S N |=S P = . S N S P N M C ( + : − ) / N M CB G ( ξ ) ξ (fm)OPAL BEC at Z -pole . + λ ×2π ξ ρ G (ξ ,R )1 . + λ ×2π ξ ρ G (ξ ,R )1 . + α×2π ξ ρ ( LRC −1) (ξ , β) sum = . + ξ (λ ρ E + λ ρ G + αρ ( LRC −1) ) Figure 11: Pion-pairs density distributions of the OPAL BEC at the Z -pole in four-dimensionalEuclidean space, where ξ denotes the distance between two pion-production points. The contri-butions of crossed terms ( ρ G × ρ (LRC − and ρ E × ρ (LRC − ) are invisible. Maximal points in thedistributions are shifted by the phase space 2 π ξ : R G → √ R G . The following has been concluded:
C1)
In many analyses of BEC, the phenomenological forms of LRC have been adopted. In thispaper, we propose an analytic form, i.e., Eq. (4), which is probably reproducing producing Eqs. (2)and (3).
C2)
As seen in Table 3, four kinds of analyses for the OPAL BEC at the Z -pole show almostthe same values of R s(G) ( ∼ χ ( ∼ α ) contribution at n = 1. C3)
When analyzing the L3 and CMS BECs, we can compare our results obtained using CF I and CF II with those using Eq. (4) with the same quantities estimated using the τ -model. As seenin Tables 4 and 5, it can be said that they are almost the same, provided that Eq. (4) is utilized for11F I and CF II . On the contrary, the results from CF II × LRC ( δ ) are improved using LRC ( α, β, n =2) .See the right column of Table 5. C4)
As seen in Tables 2, 4, 5, and 7, the exchange functions (i.e., the Gaussian distribution andthe exponential function) and the power number ( n ) of the LRC ( α, β, n ) are related to one another(see Table 8). In other words, the deformations of exchange functions due to LRC ( α, β, n =2) arenecessary for analysis of BECs. C5)
As seen in Figs. 10 and 11, the pion-pairs density distributions in the regions with ξ ≤ Z -pole and an inverse power law with s = 5 / pp collisions at the LHC. This suggests thatthe interaction region in the pp collisions at the LHC is larger than that at the Z -pole. D1)
To describe the distributions in Minkowski space [21], we need distributions on energydifferences (∆ E = ( p − p )). D2)
A study of the Fourier transform in the Levy stochastic process [22, 23] is necessary foradvanced investigations.
D3)
For presently unclear reasons, the OPAL BEC preferred the Gaussian distribution to theexponential function, whereas the L3 BEC with separable data (2- and 3-jet cases) takes theexponential function.
D4)
As seen in Table 5, R = 0 .
15 fm when estimated using CF II × LRC ( α, β, n =2) and R = 0 . τ -model, which are shown respectively. It is not clear why thesevalues are so similar.Table 8: Empirical relationship between exchange the functions and LRC ( α, β, n ) . E BE ( R, Q ) LRC ( α, β, n ) dataGaussian distribution ←→ n = 1 OPALExponential function ←→ n = 2 L3, CMS Acknowledgments.
We are thankful to the organizer of 2020 Zimanyi Winter School and variouscomments presented there. Concerning L3 BEC data, we are indebted to W. J. Metzger andM. Csanad for their kindness. M. Biyajima thanks his colleagues at the Department of Physics ofShinshu University for their kindness.
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