CCoulomb interaction for L ´evy sources ∗ B´alint Kurgyis E ¨otv ¨os Lor´and University, Hungary, H-1117 Budapest, P´azm´any P. s. 1/A ,e-mail: [email protected]
July 21, 2020
Abstract
During the study of Bose-Einstein correlations in heavy ion collisions one has to takeinto account the final state interactions, amongst them the Coulomb interaction playing aprominent role for charged particles. In some cases measurements have shown that thecorrelation function can be best described by L´evy sources, and three dimensional mea-surements have indicated the possibility of deviation from spherical symmetry. Therefore,one would like to study the Coulomb interaction for non-spherical L´evy sources. We re-sort to numerical methods which are most commonly used in order to take into accountthe Coulomb interaction such measurements. Here, we utilize the Metropolis-Hastings al-gorithm. The symmetric L´evy distribution that describes the source can be characterizedby three L´evy scale parameters and the L´evy exponent. We investigate the roles of theseparameters in the correlation function. We show the results for the Bose-Einstein correla-tion functions for ellipsoidal L´evy sources with Coulomb interaction. We also compare ourresults with previous ways to treat the Coulomb interaction in the presence of L´evy sources.
The investigation of Bose-Einstein-correlations or HBT measurements offer a way to get infor-mation about heavy-ion collisions on a femtoscopic scale. These analyses can yield informationabout the space-time geometry of the collision, particle production mechanisms and could indi-cate critical phenomena [1, 2, 3, 4].For the study of Bose-Einstein correlation function one usually makes an assumption forthe source function. Recent measurements indicate that there are cases when there is a longrange component to the source and the most suitable choice is a L´evy-type source [4, 5, 6].Additionally, the available statistics can be more often utilized and the measurement simplifiedby spherical symmetry i.e. when the correlation function is measured as a function of only onemomentum variable, the length of momentum difference. However, three-dimensional mea-surements can yield further information about the collision, thus it is desirable to perform thisalso whenever possible [3, 7].These measurements are very often carried out with pairs of identical charged particles, e.g.pions or kaons. As such, one has to take into account the Coulomb repulsion between the out-going particles. This final state interaction is handled by a Coulomb correction in experimental ∗ Presented by B. Kurgyis at 17th International Scientific Days, 5 June 2020, Gy¨ongy¨os, Hungary a r X i v : . [ nu c l - t h ] J u l nalyses [8, 9]. At the moment, the Coulomb correction is at hand for spherically symmetricL´evy HBT measurements [10]. Our goal is twofold: first, to investigate the Coulomb correc-tion for three-dimensional L´evy sources and determine a sound method to use in experimentalworks. Second, we are looking at the implications of using different coordinate frames forthe measurements and calculations, namely longitudinally comoving system (LCMS) and paircenter of mass system (PCMS). The n -particle correlation functions are defined as C n = N n ( k , · · · k n ) (cid:81) ni =1 N ( k i ) , (1)where N n is the n -particle invariant momentum distribution which we can write up using the n -particle wave-function ψ n ( x , · · · x n , k , · · · k n ) and the source function S ( x, k ) as N n ( k , · · · k n ) = (cid:90) | ψ n ( x , · · · x n , k , · · · k n ) | n (cid:89) i =1 S ( x i , k i )d x i . (2)Instead of the single-particle source function it is useful to introduce the pair-distribution D ( ρ, K ) ,which is the autoconvolution of the source function in the first variable and with k = K .The two-particle correlation function then can be expressed with properly normalized wave-function and source the following way if we assume that the particles are of similar momentum( k ≈ k ): C ( q, K ) = (cid:90) | ψ ( q, K, ρ, R ) | D ( ρ, K )d ρ d R, (3)where we introduced relative ( q, ρ ) and average ( K, R ) quantities.
For the source function we assume that we have a symmetric L´evy distribution [11]: S ( r, K ) = L (4 D ) ( r µ , α ( K ) , R σν ( K )) = (cid:90) d q (2 π ) e iq µ r µ e − | q σ R σν q ν | α/ , (4)where α is the L´evy-exponent and R σν is a 2-index symmetric tensor containing the squaresof the L´evy-scale parameters; these parameters carry the momentum dependence of the source.The autoconvolution of such a L´evy-distribution is itself a L´evy-distribution but with scaleparameters R ’ = 2 /α R . By choosing a coordinate frame and making some assumptionswe constrain the form of the R σν matrix. Most measurements are done in the LCMS system[3, 4], thus we will use the assumption that our source can be described by a spatially threedimensional symmetric L´evy (only diagonal terms) and that the freeze-out is simultaneous forparticles with the same average momentum (no temporal part). Therefore, the R σν tensor hasthe following form: R σν = R out R side
00 0 0 R long , (5)2here we used the out, side, long to indicate that we are using Bertsch-Pratt coordinates [12,13]. We can then simplify the four-dimensional L´evy distribution as a product of a Dirac deltafunction and a three-dimensional symmetric L´evy distribution: L (4 D ) = δ ( t L ) L (3 D ) ( (cid:126)r L , α, R out , R side , R long ) , (6) L (3 D ) ( (cid:126)r L , α, R out , R side , R long ) = (cid:90) d q (2 π ) e − i(cid:126)q(cid:126)r L e −| q out R out + q side R side + q long R long | α/ , (7)where the L superscript indicates that these coordinates are in LCMS. Without final state inter-actions we can easily get the form of the two-particle correlation function in the LCMS with theabove mentioned source [11]: C (0)2 ( (cid:126)q, α, R out , R side , R long ) = 1 + e −| q out R out + q side R side + q long R long | α/ . (8) To take into account the Coulomb interaction one has to use the Coulomb interacting two par-ticle wave function. We get that as the solution of the two-particle Schr¨odinger equation withrepulsive Coulomb force . It can be solved in the PCMS [8, 14], and the fully symmetric wavefunction for identical bosons is ψ ( (cid:126)R P , (cid:126)r P , (cid:126)K P , (cid:126)k P ) = N √ e − i (cid:126)K (cid:126)R (cid:2) e i(cid:126)k(cid:126)r F ( − iη, , i ( kr − (cid:126)k(cid:126)r ))++ e − i(cid:126)k(cid:126)r F ( − iη, , i ( kr + (cid:126)k(cid:126)r )) (cid:3) , (9)where we used (cid:126)k = (cid:126)q/ , k = | (cid:126)k | and the following notations: η = mc α hck , where α is the fine-structure constant, (10) N = e − πη Γ(1 + iη ) , (11)with Γ( z ) being the gamma function and F ( a, b, z ) is the confluent hypergeometric function. Toevaluate the two-particle correlation function we need the norm squared of the wave function,with that the (cid:126)R and (cid:126)K dependence is lost: | ψ ( (cid:126)r P , (cid:126)k P ) | = 2 πηe πη − (cid:2) | F ( − iη, , i ( kr + (cid:126)k(cid:126)r )) | ++ e i(cid:126)k(cid:126)r F ( − iη, , i ( kr − (cid:126)k(cid:126)r )) F ( iη, , − i ( kr + (cid:126)k(cid:126)r )) (cid:3) . (12)To get the two-particle correlation function one has to evaluate a d r integral over the wholespace-time, this could be performed in any coordinate frame. We have several options to ex-plore:1. Let us assume, that the R σν , thus the source is the same in PCMS and LCMS, this is anapproximation of (cid:126)K ≈ . However this is a rather strong approximation and one of thegoals of HBT measurements is to explore the average momentum (or transverse mass)dependence of the parameters that describe the source.3. There are two objects, one in PCMS (the wave function) and the other in LCMS (thesource function). We could try to transform the wave function from PCMS to LCMSand then use the simple form of source function and get the result in LCMS coordinates.However, the two-particle wave function of eq. 12 is not a relativistic expression, thus werefrain from trying to come up with the right transformation of this object.3. The third option is to evaluate the integral in PCMS as the two-particle Coulomb wavefuntion is only known in PCMS. This means that the L´evy-source has to be transformedfrom LCMS to PCMS.Below we proceed with the third option listed above. We introduce some further notations:the mass of the particles m (e.g. pion mass), the average transverse momentum in LCMS K T ,the transverse mass m T = (cid:112) m + K T and the β T = K T /m T factor. The Lorentz-boost fromLCMS to PCMS is then Λ νµ = 1 m m T − K T − K T m T m
00 0 0 m . (13)The L´evy distribution then transforms as a scalar from LCMS to PCMS, which means that wehave to evaluate eq. 6 at the coordinates r (cid:48) = Λ − r , where the transformation is the following: (cid:18) t L (cid:126)r L (cid:19) = 1 m m T t P + K T r P out K T t P + m T r P out mr P side mr P long . (14)The temporal integral then can be easily evaluated and then we are left with the expression(where (cid:126)k = (cid:126)q ) C ( C )2 ( (cid:126)q ) = (cid:90) d r | ψ ( (cid:126)k, (cid:126)r ) | L (3 D ) (cid:18)(cid:113) − β T r out , r side , r long , α, R out , R side , R long (cid:19) , (15)where we dropped the P superscripts for simplicity but every momentum and spatial coordinateis in PCMS. We can further work on the expression for the three-dimensional L´evy-distributionand obtain the following relationship: L (3 D ) (cid:18)(cid:113) − β T r out , r side , r long , α, R out , R side , R long (cid:19) ∼∼ L (3 D ) (cid:18) (cid:126)r, α, R out / (cid:113) − β T , R side , R long (cid:19) . (16)Then the integral we would like to calculate is C ( C )2 ( (cid:126)q, α, R , R , R ) = (cid:90) d r | ψ ( (cid:126)k, (cid:126)r ) | L (3 D ) ( (cid:126)r, α, R , R , R ) , (17)where R = R out / (cid:112) − β T , R = R side , R = R long . This expression can be evaluated numeri-cally, we utilize the Metropolis-Hastings algorithm.4 .2 Numerical simulations The Metropolis-Hastings algorithm can be used to evaluate integrals of the form I = (cid:90) Ω d xf ( x ) · g ( x ) , (18)where f ( x ) can be thought of as a probability distribution and g ( x ) is the function of interest [15,16]. In our case the three dimensional symmetric L´evy distribution is the probability distributionand the function of interest is from eq. 12: f ( x )d x := L (3 D ) ( (cid:126)r, α, R , R , R ) d r, (19) g ( x ) := | ψ ( (cid:126)k, (cid:126)r ) | . (20)We can utilize two transformations. First, with the reflection relations of the confluent hyperge-ometric functions the second term in eq. 12: e i(cid:126)k(cid:126)r F ( − iη, , i ( kr − (cid:126)k(cid:126)r )) F ( iη, , − i ( kr + (cid:126)k(cid:126)r )) == F (1 + iη, , − i ( kr − (cid:126)k(cid:126)r )) F (1 − iη, , − i ( kr + (cid:126)k(cid:126)r )) . (21)Additionally, we can transform the 3D symmetric L´evy distribution: L (3 D ) ( (cid:126)r, α, R , R , R ) = L (1 D ) ( s ( (cid:126)r ) , α, R R R , (22) s ( (cid:126)r ) = (cid:115) r out R + r side R + r long R . (23)The integral was performed using spherical coordinates on the domain Ω = [0 , r max ] × [0 , π ] × [0 , π ] , with an r max chosen so that the integral of the L´evy distribution ( I = (cid:82) L ) to be amaximum of less than ( I ≥ . ). First we are going to look at the comparison of our three dimensional calculations and otheravailable, spherically symmetric calculations for L´evy sources. Then, we are going to investi-gate the implications of the fact that most measurements are in LCMS, and the source is assumedto be spherical there for one-dimensional analyses, but the integral of eq. 17 is in PCMS.
The three dimensonal calculation is rather time consuming and its numerical precision could bealso problematic for implementing it for experimental analyses. Our approach here was, that wefixed a set a parameters ( α, R , R , R ) and evaluated the integral at points in momentumspace. This gives us a fine enough resolution in momentum space for purposes of comparison.First let us compare the two-particle correlation functions in PCMS. On fig. 1 we can see theBose-Einstein correlation functions with Coulomb interaction (Full BEC) and without any finalstate interaction (Free BEC) from our 3D calculation, from 1D calculation with quadratic andarithmetic average scale parameters and the angle averaged values of the 3D calculation. Inthe spherical case, on the left hand side plot, everything is the same as we would expect; but5 [MeV/c] 0 10 20 30 40 50 60 70 80 90 10000.20.40.60.811.21.41.61.82 Correlation functions in PCMS = 1.4, R1 = 7, R2 = 7, R3 = 7 a
1D FullBEC with quadratic mean R1D FreeBEC with quadratic mean R1D FullBEC with arithmetic mean R1D FreeBEC with arithmetic mean R3D FullBEC 3D FreeBEC Angle averaged 3D FullBECAngle averaged 3D FreeBEC
Correlation functions in PCMS q [MeV/c] 0 10 20 30 40 50 60 70 80 90 10000.20.40.60.811.21.41.61.82
Correlation functions in PCMS = 1.4, R1 = 12, R2 = 2, R3 = 2 a
1D FullBEC with quadratic mean R1D FreeBEC with quadratic mean R1D FullBEC with arithmetic mean R1D FreeBEC with arithmetic mean R3D FullBEC 3D FreeBEC Angle averaged 3D FullBECAngle averaged 3D FreeBEC
Correlation functions in PCMS
Figure 1: On the left hand side the two-particle correlation functions are shown in a sphericalcase for the three dimensional calculation in comparison with one dimensional calculations inpresence of Coulomb interaction in without final state interactions. On the right hand side a non-spherical three dimensional calculation is shown alongside with one dimensional calculationswith quadratic and arithmetic mean scale parameters. q [MeV/c] 0 10 20 30 40 50 60 70 80 90 10000.20.40.60.811.21.41.61.82
Coulomb correction in PCMS = 1.4, R1 = 12, R2 = 2, R3 = 2 a )/3 +R3 +R2 (R11D Coul. corr., R= )/3 +R3 +R2 (R1Full BEC w/ 1D Coul., R=1D Coul. corr., R=(R1+R2+R3)/3Full BEC w/ 1D Coul., R=(R1+R2+R3)/33D Coul. corr.Full BEC w/ 3D Coul.3D Pure BECAngle averaged 3D BEC Coulomb correction in PCMS q [MeV/c] 0 10 20 30 40 50 60 70 80 90 10000.20.40.60.811.21.41.61.82
Coulomb correction in PCMS = 1.4, R1 = 12, R2 = 7, R3 = 2 a )/3 +R3 +R2 (R11D Coul. corr., R= )/3 +R3 +R2 (R1Full BEC w/ 1D Coul., R=1D Coul. corr., R=(R1+R2+R3)/3Full BEC w/ 1D Coul., R=(R1+R2+R3)/33D Coul. corr.Full BEC w/ 3D Coul.3D Pure BECAngle averaged 3D BEC Coulomb correction in PCMS
Figure 2: The Coulomb corrections and the Coulomb corrected three-dimensional two-particlecorrelation function is shown in two non-spherical cases.on the right hand side plot, when we have a non-spherical source for the 3D calculation wecan see that there is large difference between the correlation functions, both in the Coulombinteracting and in the free case. However, we are interested in the question whether we coulduse the 1D calculation for the purposes of Coulomb correction only, viz. the ratio of the fulland free BEC functions ( K = C ( C )2 /C (0)2 ). We can see the comparison of Coulomb correctionson fig. 2 with two sets of parameters, both non-spherical. The Full BEC functions are herethe Coulomb corrected three-dimensional correlation functions (FullBEC = K · C (0)2 , D ). Theone dimensional Coulomb corrections are evaluated at | (cid:126)q | in PCMS, thus at q inv and at an av-erage R for R , R and R . Although the correlation functions were quite different, we cansee that the Coulomb corrections are very much the same. Now, we would just like to pointout the fact that one-dimensional and three-dimensional Coulomb corrections are very similar,therefore in an experimental analysis it is sufficient to use a one-dimensional Coulomb correc-tion, with the right parameter values. The error caused by the spherical Coulomb correctioncould be estimated, but it is not in the scope of this paper to give a quantitative limit on thisuncertainty. The application of the Coulomb correction in three-dimensional analyses is quitestraightforward: if the measurement is in LCMS and we have momenta q L = ( q L out , q L side , q L long ) and L´evy scale parameters R out , R side , R long for particles with an average transverse momentum6f K T , which gives us β T . Then using the assumption that the Coulomb correction transformsas a scalar we evaluate the Coulomb correction (which was calculated in PCMS) at momenta q P = ( (cid:112) − β T q L out , q L side , q L long ) and scale parameters R = R out / (cid:112) − β T , R = R side and R = R long . Accordingly, we use q inv = (cid:113) (1 − β T ) q L out + q L side + q L long and some average of R , R and R if we use a 1D Coulomb correction, for example the quadratic average: R PCMS = (cid:115) R out − β T + R side + R long . (24)Therefore the Coulomb-correction that can be applied in a three-dimensional measurement isthe following: K = C ( C )2 , D ( q inv , R PCMS , α )1 + exp ( −| q inv R PCMS | α ) , (25)where C ( C )2 , D is the result from the integral of Eq. 17. in a spherical case with radius of R PCMS according to Eq. 24. and at momentum q inv which can be calculated for every point in a three-dimensional measurement in LCMS. Below we investigate the implications of our calculations for one-dimensional HBT measure-ments. When we perform a one-dimensional measurement in LCMS we assume that the sourcehere is spherical, thus R = R out = R side = R long and we have a single momentum vari-able q LCMS = (cid:113) q L out + q L side + q L long , but the Coulomb correction is calculated in PCMS with R , R , R . This means, that a spherical source in LCMS would imply a non-spherical ( R = R/ (cid:112) − β T , R = R = R ) source in PCMS and the need for a three-dimensional Coulomb-correction. However, we have seen above that the non-spherical Coulomb-correction can bewell approximated with a spherical Coulomb-correction if we use the right average R, viz. in-stead of R LCMS = R we have to use R PCMS = (cid:115) − β T − β T R, (26)if we use a quadratic average R. Another problem stems from the fact that we can not reconstruct q inv from q LCMS . An obvious solution would be to measure all momentum variables instead ofjust the length of the momentum difference, but then the advantages of the 1D measurementover the 3D would be lost. We can try to overcome this obstacle in some other ways, one solidapproximation could be the following: we measure an A ( q LCMS , q inv ) distribution of particlepairs and then we use this to obtain a weighted Coulomb-correction: K weighted ( q LCMS ) = (cid:82) A ( q LCMS , q inv ) K ( q inv )d q inv (cid:82) A ( q LCMS , q inv )d q inv . (27)On fig. 3 we can see the Coulomb correction and the corrected three-dimensional two-particlecorrelation functions for K T = 0 . GeV/ c in LCMS. The parameters are chosen so, that inthe LCMS we have an approximately spherically symmetric source ( R out = 2 . fm, R side = R long = 2 fm). We can see that there is a clear difference between the two one-dimensional cor-rections, one with an LCMS average R, the other with an average in accordance with eq. 26. In7 ) long = q side = q out q [MeV/c] ( q0 20 40 60 80 100 120 140 160 180 200 22000.20.40.60.811.21.41.61.82 = 0.8 T Coulomb correction in LCMS at K = 1.4, R1_lcms = 2.0624, R2 = 2, R3 = 2 a R^2 matrix is diagonal in LCMS without temporal part1D Coul. corr. with lcms average RFull BEC w/ 1D Coul. with lcms average R1D Coul. corr. with pcms averageFull BEC w/ 1D Coul. with pcms average) / o
3D Coul. corr. with RFull BEC w/ 3D Coul. in LCMS3D Pure BEC LCMSAngle averaged 3D Coul. corrAngle averaged 3D BEC = 0.8 T Coulomb correction in LCMS at K [MeV/c] ( qother = 0 ) out q0 20 40 60 80 100 120 140 160 180 200 22000.20.40.60.811.21.41.61.82 = 0.8 T Coulomb correction in LCMS at K = 1.4, R1_lcms = 2.0624, R2 = 2, R3 = 2 a R^2 matrix is diagonal in LCMS without temporal part1D Coul. corr. with lcms average RFull BEC w/ 1D Coul. with lcms average R1D Coul. corr. with pcms averageFull BEC w/ 1D Coul. with pcms average) / o
3D Coul. corr. with RFull BEC w/ 3D Coul. in LCMS3D Pure BEC LCMSAngle averaged 3D Coul. corrAngle averaged 3D BEC = 0.8 T Coulomb correction in LCMS at K
Figure 3: The Coulomb corrections and the Coulomb corrected three-dimensional two-particlecorrelation function is shown in LCMS, when the source is spherical in LCMS but not for thecalculation. On the left hand side we take the three-dimensional Coulomb correction along adiagonal line and on the right hand side along the q out axis.the low- q region there is some difference between the angle averaged, the one-dimensionaland the three dimensional Coulomb-corrections, also the numerical precision of the three-dimensional calculation make if difficult to decide between the options. However, we canclearly see that from q > MeV/ c the angle averaged and the three-dimensional Coulomb cor-rection are in good agreement with the one-dimensional Coulomb-correction with the average Rof eq. 26, and there is consistent difference from the other one. The fact that the angle averagedcase is most similar to the one dimensional with the transformed average R of eq. 26 indicatesthat it is best to use the latter for one-dimensional measurements. On the left hand side, thethree-dimensional correlation function is taken at a diagonal line in LCMS ( q out = q side = q long )and on the right hand side along the out axis. For the one-dimensional Coulomb correctionwe did not rely on a weighted average, as we could calculate q inv . Let us now list the possi-ble approaches to deal with the Coulomb interaction in one-dimensional measurements that arecarried out in LCMS. We only list the options that make use of a one-dimensional calculationfor the integral of eq. 17, in these cases the factor of ref. [10] can be used. A more simplis-tic solution would be to use the Gamow-factor, where the source size is neglected. The mostsophisticated approach would be to use the angle averaged Coulomb correction from a three-dimensional calculation, but this would be an overly intricate solution. The possibilities formaking use of a one-dimensional Coulomb integral calculation are the following, in increasingsophistication:1. Simply use C ( C )2 ( q LCMS , R
LCMS ) , which means that we formally substitute q LCMS = q inv and R PCMS = R LCMS .2. Take into account the fact that q inv (cid:54) = q LCMS but neglect the same for the scale parameters,and use the weighting method of eq. 27. However, not for the Coulomb-correction but forthe correlation function instead. Thus use C , weighted ( q LCMS , R
LCMS ) for the fitting: C ( q LCMS , R
LCMS ) = (cid:82) A ( q LCMS , q inv ) C ( q inv , R LCMS )d q inv (cid:82) A ( q LCMS , q inv )d q inv . (28)3. The same approach as above, use R LCMS for the Coulomb correction, and use a weightedaverage but for the Coulomb-correction this time. This approach is more sensible if8e consider fig. 1, where we saw that the correlation functions can look rather differ-ent even if on fig. 2 the Coulomb corrections look very much the same. Now we use K weighted ( q LCMS , R
LCMS ) · C (0)2 ( q LCMS , R
LCMS ) for fitting.4. One improvement to the above mentioned methods is to take into account the transforma-tion of scale parameters, so use the average of eq. 26. The simpler version is the same asno. 3. above, when we weigh the correlation function and use C , weighted ( q LCMS , R
PCMS ) for fitting. Here however, we lose the explicit form of C (0)2 in LCMS which is known.5. The most sophisticated option would be to use R PCMS only for the Coulomb correction,and use the weighting of eq. 27. The function used for fitting is now K weighted ( q LCMS , R
PCMS ) · C (0)2 ( q LCMS , R
LCMS ) .6. Finally, an approach that is easier to implement than the previous ones making use ofa distribution A ( q LCMS , q inv ) , is to make an approximation for the q LCMS - q inv relation-ship that is appropriate for the Coulomb correction. One could be motivated by the lefthand side plot of fig. 3, as the one-dimensional Coulomb correction with R PCMS and theangle averaged three-dimensional calculation are in a relatively good agreement. Therelationship q inv = (cid:112) − β T / q LCMS could be used, as it would hold for the diagonalline q out = q side = q long line. Therefore the function we could use for fitting would be K ( (cid:112) − β T / q LCMS , R
PCMS ) · C (0)2 ( q LCMS , R
LCMS ) .Additionally, either the a distribution of particle pairs from same events (usually denoted with A ) or some background distribution, that has no quantum-statistical effects ( B ) could be usedfor weighting C and K [4]. Here, one could argue in the favor of the latter, however it is notexpected to make a significant difference. The soundest approach for one dimensional analysesis no. 5. We have investigated the Coulomb interaction for HBT measurements in presence of L´evysources. Our results can be applied to three-dimensional and one-dimensional measurementsalike. The results hold for Gaussian or Cauchy sources as well because these are special casesof the L´evy source ( α = 2 for Gaussian and α = 1 for Cauchy). We have learned that a one-dimensional Coulomb correction can be reasonably well applied for three-dimensional mea-surements if we use the the average scale parameter of eq. 24 and use q inv as the momentumvariable for the Coulomb correction. For one dimensional measurements in LCMS we saw thatwe should use the average scale parameter of eq. 26 and we should also evaluate the Coulombcorrection at q inv as we calculated this in PCMS, which in practice can be estimated with aweighted Coulomb correction according to the option no. 5 in the previous section. The abovedetailed treatment of Coulomb interaction in heavy-ion collisions could be readily applied toexperimental measurements. Acknowledgements
B. K. expresses gratitude for the support of Hungarian NKIFH grant No. FK-123842. Theauthor would like to thank M. Csan´ad, M. I. Nagy, S. L¨ok¨os and D. Kincses for the fruitfuldiscussions and useful suggestions along the way.9 eferences [1] T. Cs¨org˝o. “Particle interferometry from MeV to TeV”. In:
A. Phys. Hun. A , 1 (2002).[2] J. Bolz et al. “Resonance decays and partial coherence in Bose-Einstein correlations”.In: Phys. Rev. D , 3860 (1993).[3] L. Adamczyk et al. [STAR]. “Beam-energy-dependent two-pion interferometry and thefreeze-out eccentricity of pions measured in heavy ion collisions at the STAR detector”.In: Phys. Rev. C , no. 1, 014904 (2015).[4] A. Adare et al. [PHENIX]. “L´evy-stable two-pion Bose-Einstein correlations in √ s NN =200 GeV Au + Au collisions”. In:
Phys. Rev. C , no. 6, 064911 (2018).[5] S. S. Adler et al. [PHENIX]. “Evidence for a long-range component in the pion emissionsource in Au + Au collisions at √ s NN = 200 GeV”. In:
Phys. Rev. Lett. , 132301 (2007).[6] M. Csan´ad [PHENIX]. “Measurement and analysis of two- and three-particle correla-tions”. In: Nucl. Phys. A , 611-614 (2006).[7] B. Kurgyis [PHENIX]. “Three dimensional L´evy HBT results from PHENIX”. In:
ActaPhys. Pol. B Proc. Suppl. vol. 12 (2), pp. 477 - 482 (2019).[8] Y. Sinyukov et al. “Coulomb corrections for interferometry analysis of expanding hadronsystems”. In:
Phys. Lett. B , 248-257 (1998).[9] M. Bowler. “Coulomb corrections to Bose-Einstein correlations have been greatly exag-gerated”. In:
Phys. Lett. B , 69-74 (1991).[10] M. Csan´ad et al. “Expanded empirical formula for Coulomb final state interaction in thepresence of L´evy sources”. In:
Phys. Part. Nucl. , no.3, 238-242 (2020).[11] T. Cs¨org˝o et al. “Bose-Einstein correlations for Levy stable source distributions”. In: Eur.Phys. J. C , 67 (2004).[12] S. Pratt et al. “Detailed predictions for two pion correlations in ultrarelativistic heavy ioncollisions”. In: Phys. Rev. C , 2646-2652 (1990).[13] G. Bertsch et al. “Pion Interferometry in Ultrarelativistic Heavy Ion Collisions”. In: Phys.Rev. C , 1896-1900 (1988).[14] L.D. Landau, E.M. Lifshitz. “Quantum. Mechanics: Non-Relativistic Theory 3-rd Ed.”In: Pergamon Press, Oxford ().[15] N. Metropolis et al. “Equations of state calculations by fast computing machines”. In:
J.Chem. Phys., (6), 1087–1092. (1953).[16] W. Hastings. “Monte Carlo sampling methods using Markov chains and their applica-tion”. In: Biometrika, , 97–109, 97–109