Efficiency corrections for factorial moments and cumulants of overlapping sets of particles
EEfficiency corrections for factorial moments and cumulants of overlapping sets ofparticles
Volodymyr Vovchenko ∗ and Volker Koch † Nuclear Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720 (Dated: January 7, 2021)In this note we discuss subtleties associated with the efficiency corrections for measurementsof off-diagonal cumulants and factorial moments for a situation when one deals with overlappingsets of particles, such as correlations between numbers of protons and positively charged particles.We present the efficiency correction formulas for the case when the detection efficiencies follow abinomial distribution.
I. INTRODUCTION
Fluctuations of conserved charges as a probe of the phase structure of strongly interacting matter have recentlyreceived considerable interest theoretically as well as experimentally (for a recent review, see [1]). For example, higherorder cumulants of the net baryon density are sensitive to the existence of a critical point [2], and may also provideinsights about the chiral criticality governing the cross-over transition at vanishing baryochemical potential [3]. Theso-called off-diagonal cumulants, i.e. correlations between two different conserved charges, such as baryon numberand strangeness, on the other hand, provide insight about the effective degrees of freedom in the medium [4, 5].Experimentally, these cumulants are measured by analyzing event-by-event distributions of particles produced inheavy-ion collisions. For these measurements to reveal the true fluctuations of the system created in these collisions,one needs to take into account and remove fluctuations induced by the detector measurement process itself. Thesedetector induced fluctuations, often referred to as efficiency fluctuations [6–8], arise from the finite detection probability W D ( n, N ) of an actual detector, where W D ( n, N ) is the probability to observe n particles given N ≥ n particles in anevent. The probability distribution of observed particles, p ( n ), is related to the distribution of true particles, P ( N ),by p ( n ) = (cid:88) N W D ( n, N ) P ( N ) . (1)Consequently, the cumulants of the observed distribution p ( n ) differ from those of the true distribution, P ( N ).Therefore, an unfolding procedure is needed, which mathematically corresponds to finding the inverse of W D ( n, N ).This is not an easy task in general (see discussion e.g. in [9]). However, if W D ( n, N ) can be approximated by abinomial distribution – a reasonably good approximation in a number of cases (see [10, 11]) – the relevant formulasfor the efficiency corrections of cumulants can and have been derived [6, 7, 12, 13].However, certain subtleties arise when efficiency corrections are performed for off-diagonal cumulants, such asthe correlation of net-proton or net-kaon number with the net-charge number. These subtleties have not yet beenaddressed in the literature. It is the purpose of this note to discuss and provide the necessary efficiency correctionformulas. These may be useful for the ongoing and future heavy-ion experiments, in particular as an effort to measuresuch off-diagonal cumulants is underway (see e.g. [14]). II. A REMINDER ON EFFICIENCY CORRECTIONS FOR NONOVERLAPPING SETS OFPARTICLES
In the following we shall denote all true quantities with upper case letters and all measured quantities, which areaffected by detector efficiencies, with lower case symbols. Following [6] it is convenient to express the cumulants interms of factorial moments, as efficiency corrections for those are simpler. For example the co-variance or off-diagonalcumulant between two distinct particle species, A and B ( A ∩ B = ∅ ), defined asΣ , A,B = (cid:104) N A N B (cid:105) − (cid:104) N A (cid:105) (cid:104) N B (cid:105) , (2) ∗ E-Mail:[email protected] † E-Mail:[email protected] a r X i v : . [ nu c l - t h ] J a n may be repressed in terms of the factorial moments F i,jA,B = (cid:28) N A !( N A − i )! N B !( N B − j )! (cid:29) (3)so that Σ , A,B = F , A,B − F , A,B F , A,B . (4)The second moments are given by (cid:10) N A (cid:11) = F , A,B + F , A,B (5)and also the higher moments can be expressed as combinations of the factorial moments.Given the multiplicity distribution function P ( N A , N B ) for particles of type A and B , the corresponding factorialcumulants are conveniently obtained through the generating function G ( Z A , Z B ) = (cid:88) N A ,N B Z N A A Z N B B P ( N A , N B ) (6)via F i,jA,B = d i + j dZ iA dZ jB G ( Z A , Z B ) | Z A =1 ,Z B =1 . (7)Consider now a case when the probability to detect a particle is governed by a binomial distribution, W ( n, N ) = B ( n, N ; (cid:15) ) = N ! n !( N − n )! (cid:15) n (1 − (cid:15) ) ( N − n ) (8)with (cid:15) being the so-called efficiency. The distribution of measured particles is then given by p ( n A , n B ) = (cid:88) N A ,N B B ( n A , N A ; (cid:15) A ) B ( n B , N B ; (cid:15) B ) P ( N A , N B ) . (9)The resulting factorial moment generation function which provides the measured factorial moments is then g ( z A , z B ) = (cid:88) n A ,n B z n A A z n B B p ( n A , n B )= (cid:88) N A ,N B (1 − (cid:15) A + z A (cid:15) A ) N A (1 − (cid:15) B + z B (cid:15) B ) N B P ( N A , N B ) , (10)where we used the fact that (cid:80) Nn =1 z n B ( n, N, (cid:15) ) = (1 − (cid:15) + z(cid:15) ) N . The measured factorial moments reduce simply to f i,jA,B = (cid:15) iA (cid:15) jB F i,jA,B , (11)thus, the true factorial moments, F A,Bi,j , can be recovered by dividing the measured ones by the appropriate powersof the efficiencies F i,jA,B = f i,jA,B (cid:15) iA (cid:15) jB , A ∩ B = ∅ . (12)As result the true co-variance or off-diagonal cumulant is given byΣ , A,B = σ , A,B (cid:15) A (cid:15) B , A ∩ B = ∅ . (13)Therefore, as long as we have different particle species or even distinct groups of particles, such as protons and pions,correcting cumulants for efficiency simply entails expressing the cumulants in terms of factorial moments and thenmake use of Eq. (12), as discussed in detail in [6].However, one needs to be more careful if one is dealing with overlapping sets of particles, such as for example inthe case of proton-charge correlations, Σ , p,Q . The protons do carry charge, thus producing a self-correlation term inΣ , p,Q . This requires a separate efficiency correction treatment. III. EFFICIENCY CORRECTIONS FOR OVERLAPPING SETS OF PARTICLES
There are many off-diagonal cumulants of interest which involve overlapping sets of particles. For example, inaddition to the aforementioned proton-charge correlations, which may serve as a proxy for baryon number-chargecorrelations, kaon-charge correlations as proxy for strangeness-charge are also being studied [14]. In the following wewill derive the efficiency corrections for the case of proton-charge correlations noting that the resulting formulas doalso apply in other, similar situations, such as kaon-charge correlations etc.Typically one studies correlations of net numbers, for instance the correlation Σ , N net − p ,Q of the net-proton numberwith the net charge, Q = Q + − Q − . In order to apply efficiency corrections it is better to consider the individualterms contributing to Σ , N net − p ,Q Σ , N net − p ,Q = Σ , N p ,Q + + Σ , N ¯ p ,Q − − Σ , N p ,Q − − Σ , N ¯ p ,Q + (14)The last two terms involve only non-overlapping sets of particles, thus the efficiency corrections for those follow Eq. (12)with the appropriate efficiencies. The first two terms, on the other hand, involve overlapping sets of particles, namelythe correlation between numbers of protons and positive charges in the first term, and between anti-protons andnegative charges in the second term.Let us focus on the case of protons and positive charges to derive the necessary efficiency correction formulas.These will then straightforwardly apply to all other cases of overlapping sets of particles. To be more specific, wewill discuss what we believe is a common scenario in the experiment, where first the sign of the charge is determinede.g. by the bending of tracks in a magnetic field and then the protons are selected from these tracks by an additionalparticle identification, for instance via a time-of-flight detector. Let us assume that the efficiencies for charge andproton identification both follow a binomial distribution. We denote the efficiency for charge identification by (cid:15) q andthe efficiency associated with proton identification by ˜ (cid:15) p so that the total proton detection efficiency is (cid:15) p = (cid:15) q ˜ (cid:15) p .It is best to separate all positive charges in into protons, N p and all other positive charges, which we denote byˆ Q + , so that the total positive charge is given by Q + = N p + ˆ Q + . Next, let us denote the true probability to have N p protons and ˆ Q + charges other than protons by P ( N p , ˆ Q + ). The true probability to have N p protons and Q + positivecharges is then P ( N p , Q + ) = (cid:80) ˆ Q + P (cid:16) N p , ˆ Q + (cid:17) δ Q + ,N p + ˆ Q + . The efficiency for charge identification affects both theprotons and the other charges, so that the distribution p (cid:15) q (˜ n p , ˆ q + ) to have ˜ n p protons and q + positive charges after charge identification but before proton identification is p (cid:15) q (cid:0) ˜ n p , q + (cid:1) = (cid:88) ˆ q + , ˆ Q + ,N p δ q + , ˆ q + +˜ n p B (˜ n p , N p ; (cid:15) q ) B (ˆ q + , ˆ Q + ; (cid:15) q ) P ( N p , ˆ Q + )= (cid:88) ˆ Q + ,N p B (˜ n p , N p ; (cid:15) q ) B ( q + − ˜ n p , ˆ Q + ; (cid:15) q ) P ( N p , ˆ Q + ) . (15)The inclusive probability to measure q + charges is then p (cid:15) q ( q + ) = (cid:88) ˜ n p p (cid:15) q (˜ n p , q + ) = (cid:88) ˆ Q + ,N p B ( q + , N p + ˆ Q + ; (cid:15) q ) P ( N p , ˆ Q + ) (16)as it should. Of course we still need to account for the efficiency associated with identifying proton, ˜ (cid:15) p , by foldingwith the (binomial) probability distribution B ( n p , ˜ n p , ˜ (cid:15) p ) for proton identification. Thus the probability p data ( n p , q + )to measure n p identified protons and q + charges is given by p data (cid:0) n p , q + (cid:1) = (cid:88) ˜ n p B ( n p , ˜ n p ; ˜ (cid:15) p ) p (cid:15) q (˜ n p , q + )= (cid:88) ˜ n p , ˆ Q + ,N p B ( n p , ˜ n p ; ˜ (cid:15) p ) B (˜ n p , N p , (cid:15) q ) B ( q + − ˜ n p , ˆ Q + , (cid:15) q ) P ( N p , ˆ Q + ) (17)Given p data ( n p , q + ) the factorial moment generating function is g ( z p , z q ) = (cid:88) n p , ˆ q + z n p p z q + q p data (cid:0) n p , ˆ q + (cid:1) = (cid:88) n p , ˆ q + , ˜ n p , ˆ Q + ,N p z nq z q + q B ( n p , ˜ n p ; ˜ (cid:15) p ) B (˜ n p , N p , (cid:15) q ) B ( q + − ˜ n p , ˆ Q + , (cid:15) q ) P ( N p , ˆ Q + )= (cid:88) N p , ˆ Q + [1 − (cid:15) q + (cid:15) q z q (1 − ˜ (cid:15) p + ˜ (cid:15) p z p )] N p (1 − (cid:15) q + z q (cid:15) q ) ˆ Q + P ( N p , ˆ Q + ) (18)where we used again that (cid:80) Nn =0 z n B ( n, N ; (cid:15) ) = (1 − (cid:15) + z(cid:15) ) N . The factorial moments of the measured distribution, f i,j ( p, q + ) are then easily obtained . For example the first factorial moments of the measured proton and positivecharge distributions are given by f , n p ,q + = (cid:104) n p (cid:105) = ddz p g ( z p , z q ) | z p =1 ,z q =1 = (cid:88) N p , ˆ Q + (cid:15) q ˜ (cid:15) p N p P ( N p , ˆ Q + ) = (cid:15) p (cid:104) N p (cid:105) = (cid:15) p F , N p ,Q f , n p ,q + = (cid:10) q + (cid:11) = ddz q g ( z p , z q ) | z p =1 ,z q =1 = (cid:88) N p , ˆ Q + (cid:15) q ( N p + ˆ Q + ) P ( N p , ˆ Q + ) = (cid:15) q (cid:10) Q + (cid:11) = (cid:15) q F , N p ,Q (19)Here we used (cid:15) p = ˜ (cid:15) p (cid:15) q . Similarly, for the diagonal second order factorial moments one finds, after some algebra, f , n p ,q + = (cid:104) n p ( n p − (cid:105) = d dz p g ( z p , z q ) | z p =1 ,z q =1 = (cid:88) N p , ˆ Q + (cid:15) q ˜ (cid:15) p N p ( N p − P ( N p , ˆ Q + )= (cid:15) p (cid:104) N p ( N p − (cid:105) = (cid:15) p F , N p ,Q f , n p ,q + = (cid:10) q + ( q + − (cid:11) = d dz q g ( z p , z q ) | z p =1 ,z q =1 = (cid:88) N p , ˆ Q + (cid:15) q ( N p + ˆ Q + ) ( N p + ˆ Q + − P ( N p , ˆ Q + )= (cid:15) q (cid:10) Q + ( Q + − (cid:11) = (cid:15) q F , N p ,Q . (20)Thus we have for the second proton number moment (cid:10) n p (cid:11) = f , n p ,q + − f , n p ,q + = (cid:15) p F , N p ,Q + + (cid:15) p F , N p ,Q + = (cid:15) p (cid:0)(cid:10) N p (cid:11) − (cid:104) N p (cid:105) (cid:1) + (cid:15) p (cid:104) N p (cid:105) . (21)The expression for (cid:68) q +2 (cid:69) is analogous.One can see that the measured and true factorial moments involving only protons or only charges do follow thestandard relation for a binomial efficiency distribution, Eq. (12). However, this is no longer the case for the mixedfactorial moments involving both charges and protons. The measured mixed factorial moment, f , n p ,q + , is given by f , n p ,q + = d dz p dz q g ( z p , z q ) | z p =1 ,z q =1 = (cid:88) N p , ˆ Q + (cid:15) q ˜ (cid:15) p N p (cid:104) (cid:15) q (cid:16) N p + ˆ Q + − (cid:17)(cid:105) P ( N p , ˆ Q + )= (cid:15) p (cid:104) N p (cid:105) + (cid:15) p (cid:15) q (cid:10) N p Q + − N p (cid:11) = (cid:15) p (cid:15) q (cid:20) F , N p ,Q + + 1 − (cid:15) q (cid:15) q (cid:104) N p (cid:105) (cid:21) . (22)We see that the “standard” relation, Eq. (12), does not hold anymore. Instead, expressing the true factorial moment, F , ( N p , Q + ), in terms of measured quantities we get, using (cid:104) N p (cid:105) = (cid:104) n p (cid:105) (cid:15) p [Eq. (19)], F , N p ,Q + = f , n p ,q + (cid:15) p (cid:15) q − (cid:104) n p (cid:105) − (cid:15) q (cid:15) p (cid:15) q , P ⊆ Q + . (23) See Appendix for a more elegant and efficient method to calculate the measured factorial moments.
The “standard” correction (12), on the other hand, would have only given rise to the first term in this expression.Since (cid:15) q <
1, the second term is negative and, therefore, applying the “standard” correction would overestimate thetrue off-diagonal factorial cumulant. For the co-variance between protons and positive charges we then getΣ , N p ,Q + = (cid:10) δN p δQ + (cid:11) = F , N p ,Q + − F , N p ,Q + F , N p ,Q + = 1 (cid:15) p (cid:15) q (cid:16) f , n p ,q + − f , n p ,q + f , n p ,q + (cid:17) − (cid:104) n p (cid:105) − (cid:15) q (cid:15) p (cid:15) q = 1 (cid:15) p (cid:15) q σ , n p ,q + − (cid:104) n p (cid:105) − (cid:15) q (cid:15) p (cid:15) q , P ⊆ Q + . (24)where we have the same extra correction term as compared to the “standard” method. This term vanishes in thelimit (cid:15) q → (cid:104) δN ¯ p δQ − (cid:105) , the efficiencycorrection being given by Eq. (24). On the other hand, as the co-variances between protons and negative charges, (cid:104) δN p δQ − (cid:105) , and anti-protons and positive charges, (cid:104) δN ¯ p δQ + (cid:105) do not involve overlapping set of particles, the standardformulas for the efficiency corrections apply, for example,Σ , N p ,Q − = (cid:10) δN p δQ − (cid:11) = 1 (cid:15) p (cid:15) ¯ q (cid:16) f , n p ,q − − f , n p ,q − f , n p ,q − (cid:17) = 1 (cid:15) p (cid:15) ¯ q (cid:10) δn p δq − (cid:11) (25)and analogous for the (cid:104) δN ¯ p δQ + (cid:105) . Here (cid:15) ¯ q denotes the efficiency for detecting negative charges. Therefore, theco-variance between net-protons and net-charges, Σ , N net − p ,Q [Eq. (14)] is given byΣ , N net − p ,Q = 1 (cid:15) p (cid:15) q σ , n p ,q + − (cid:104) n p (cid:105) − (cid:15) q (cid:15) p (cid:15) q − (cid:15) p (cid:15) ¯ q σ , n p ,q − + 1 (cid:15) ¯ p (cid:15) ¯ q σ , n ¯ p ,q − − (cid:104) n ¯ p (cid:105) − (cid:15) ¯ q (cid:15) ¯ p (cid:15) ¯ q − (cid:15) ¯ p (cid:15) q σ , n ¯ p ,q + = (cid:15) p = (cid:15) ¯ p ,(cid:15) q = (cid:15) ¯ q (cid:15) p (cid:15) q σ , n net − p ,q − (cid:104) n p + n ¯ p (cid:105) − (cid:15) q (cid:15) p (cid:15) q (26)where in the last line we assumed identical efficiencies for particles and anti-particles. IV. LOCAL EFFICIENCY CORRECTIONS
Our considerations in the previous section are based on the assumption of constant efficiencies. In a real experiment,however, the efficiency is usually not constant but depends on the momentum of a particle. For this reason the phasespace is typically partitioned into momentum bins, each having its own value of the efficiency parameter. The localefficiency corrections for the case of a binomial detector response in each bin have been worked out in Refs. [7, 15]for the case of non-overlapping sets of particles. Here we extend these considerations to cover the case of overlappingsets of particles. As in Sec. III, we shall consider the off-diagonal cumulants of protons and charged particles as aconcrete example.Let us assume an arbitrary partition of the phase space into bins. Following Ref. [7], we introduce a variable x to enumerate the bins. The numbers of (anti)protons and positively (negatively) charged particles are obtained bysumming over all the bins: N p (¯ p ) = (cid:88) x N p (¯ p ) ( x ) ,Q ± = (cid:88) x Q ± ( x ) ,n p (¯ p ) = (cid:88) x n p (¯ p ) ( x ) ,q ± = (cid:88) x q ± ( x ) . (27)Here N p (¯ p ) ( x ) and Q ± ( x ) correspond to the numbers of (anti)protons and positive (negative) charges in a phase-spacebin x . As before, the lowercase n p (¯ p ) ( x ) and q ± ( x ) correspond to the numbers of measured particles.Consider now the factorial moments involving protons and positive charges. The first-order proton number momentsread F , N p ,Q + = (cid:104) N p (cid:105) = (cid:88) x (cid:104) N p ( x ) (cid:105) ,f , n p ,q + = (cid:104) n p (cid:105) = (cid:88) x (cid:104) n p ( x ) (cid:105) . (28)As the binomial efficiency corrections in different phase space-bins are independent, one has (cid:104) n p ( x ) (cid:105) = (cid:15) p ( x ) (cid:104) N p ( x ) (cid:105) and, thus, F , N p ,Q + = (cid:104) N p (cid:105) = (cid:88) x (cid:104) n p ( x ) (cid:105) (cid:15) p ( x ) , (29)and, analogously for positive charges, F , N p ,Q + = (cid:10) Q + (cid:11) = (cid:88) x (cid:104) q + ( x ) (cid:105) (cid:15) q ( x ) . (30)The second-order diagonal proton number factorial moments are F , N p ,Q + = (cid:104) N p ( N p − (cid:105) = (cid:88) x ,x (cid:104) N p ( x )[ N p ( x ) − δ x ,x ] (cid:105) ,f , n p ,q + = (cid:104) n p ( n p − (cid:105) = (cid:88) x ,x (cid:104) n p ( x )[ n p ( x ) − δ x ,x ] (cid:105) . (31)The binomial efficiency correction is applied independently for all pairs of bins in (31), as per Eq. (20), therefore, F , N p ,Q + = (cid:104) N p ( N p − (cid:105) = (cid:88) x ,x (cid:104) n p ( x )[ n p ( x ) − δ x ,x ] (cid:105) (cid:15) p ( x ) (cid:15) p ( x ) , (32)and, analogously, F , N p ,Q + = (cid:10) Q + ( Q + − (cid:11) = (cid:88) x ,x (cid:104) q + ( x )[ q + ( x ) − δ x ,x ] (cid:105) (cid:15) q ( x ) (cid:15) q ( x ) . (33)These results for the diagonal factorial moments are the same as obtained in Ref. [7]. Consider now the mixedfactorial moments F , N p ,Q + = (cid:10) N p Q + (cid:11) = (cid:88) x ,x (cid:10) N p ( x ) Q + ( x ) (cid:11) ,f , n p ,q + = (cid:10) n p q + (cid:11) = (cid:88) x ,x (cid:10) n p ( x ) q + ( x ) (cid:11) . (34)The efficiency correction proceeds independently for all pairs ( x , x ) of bins, as advocated above. The terms with x (cid:54) = x correspond to protons and positive charges from different phase-space bins, which, therefore, correspond tonon-overlapping particles. For these terms the “standard” correction [Eq. (12)] applies. However, the terms with x = x in Eq. (34) correspond to mixed factorial moments involving overlapping sets of particles: protons andpositive charges in the same phase-space bin. This means that the generalized correction (23) should be used in thesecases. Combining the corrections for the x (cid:54) = x and x = x terms together, we arrive at F , N p ,Q + = (cid:10) N p Q + (cid:11) = (cid:88) x ,x (cid:104) n p ( x ) q + ( x ) (cid:105) (cid:15) p ( x ) (cid:15) q ( x ) − (cid:88) x (cid:104) n p ( x ) (cid:105) − (cid:15) q ( x ) (cid:15) p ( x ) (cid:15) q ( x ) . (35)The relation between the true co-variance between protons and positive charges and the measured moments readsΣ , N p ,Q + = (cid:10) δN p δQ + (cid:11) = F , N p ,Q + − F , N p ,Q + F , N p ,Q + = (cid:88) x ,x σ , n p ( x ) q + ( x ) (cid:15) p ( x ) (cid:15) q ( x ) − (cid:88) x (cid:104) n p ( x ) (cid:105) − (cid:15) q ( x ) (cid:15) p ( x ) (cid:15) q ( x ) . (36)The expression for the co-variance between antiprotons and negative charges is analogous to (36). For the co-variances between protons and negative charges, as well as between antiprotons and positive charges, one does notencounter overlapping sets of particles, thus the second term in the r.h.s. of Eq. (36) does not appear. For completeness,we list here the results for all the remaining proton-charge covariances:Σ , N ¯ p ,Q − = (cid:10) δN ¯ p δQ − (cid:11) = (cid:88) x ,x σ , n ¯ p ( x ) q − ( x ) (cid:15) ¯ p ( x ) (cid:15) ¯ q ( x ) − (cid:88) x (cid:104) n ¯ p ( x ) (cid:105) − (cid:15) ¯ q ( x ) (cid:15) ¯ p ( x ) (cid:15) ¯ q ( x ) , (37)Σ , N p ,Q − = (cid:10) δN p δQ − (cid:11) = (cid:88) x ,x σ , n p ( x ) q − ( x ) (cid:15) p ( x ) (cid:15) ¯ q ( x ) , (38)Σ , N ¯ p ,Q + = (cid:10) δN ¯ p δQ − (cid:11) = (cid:88) x ,x σ , n ¯ p ( x ) q + ( x ) (cid:15) ¯ p ( x ) (cid:15) q ( x ) . (39)The co-variance between net protons and net charges is then given by Eq. (14). Equations (36)-(39) reduce to theresults of Sec. III for the case of uniform efficiencies. V. EXAMPLESA. Poisson distributed particles
To illustrate the above findings let us consider the case where all particle multiplicities are independent and followPoisson distributions. Let us consider again the case of protons and positive charges, and uniform binomial efficiencies,i.e. a single phase-space bin. With Q + = N p + ˆ Q + the true co-variance between protons and positively charged particlesis given by Σ , N p ,Q + = (cid:10) δN p δQ + (cid:11) = (cid:68) δN p δ ˆ Q + (cid:69) + (cid:68) ( δN p ) (cid:69) = (cid:68) ( δN p ) (cid:69) = Poisson (cid:104) N p (cid:105) (40)Here we used the fact that Σ , N p , ˆ Q + = (cid:68) δN p δ ˆ Q + (cid:69) = 0 for independently distributed N p and ˆ Q + . The only source ofcorrelations between numbers of protons and positive charges is the proton self-correlation.Next let us calculate the same co-variance for the measured particles, σ , n p ,q + = (cid:104) ( δn p δq + ) (cid:105) = f , ( n p , q + ) −(cid:104) n p (cid:105) (cid:104) q + (cid:105) .Since the particles are distributed independently, the probability distribution P ( N p , Q + ) factorizes into that forprotons and that for all other positively charged particles, P ( N p , ˆ Q + ) = P p ( N p , (cid:104) N p (cid:105) ) P p ( ˆ Q + , (cid:68) ˆ Q + (cid:69) ) (41)with P p ( N, Λ) = exp( − Λ)Λ N /N ! denoting a Poisson distribution with mean (cid:104) N (cid:105) = Λ. With (cid:80) ∞ N =0 z N P p ( N, Λ) =exp(Λ( z − g ( z p , z q ) = exp [ (cid:104) N p (cid:105) (cid:15) p z q ( z p − (cid:2)(cid:10) Q + (cid:11) (cid:15) q ( z q − (cid:3) (42)where we have used (cid:104) Q + (cid:105) = (cid:68) ˆ Q + (cid:69) + (cid:104) N p (cid:105) . Given the generating function, the factorial moment f , ( n p , q + ) is f , n p ,q + = ∂ ∂z p ∂z q g ( z p , z q ) | z p =1 ,z q =1 = (cid:15) p (cid:15) q (cid:104) N p (cid:105) (cid:10) Q + (cid:11) + (cid:15) p (cid:104) N p (cid:105) . (43)With (cid:104) n p (cid:105) = (cid:15) p (cid:104) N p (cid:105) and (cid:104) q + (cid:105) = (cid:15) q (cid:104) Q + (cid:105) we get for the measured co-variance σ , N p ,q + = f , n p ,q + − (cid:104) n p (cid:105) (cid:10) q + (cid:11) = (cid:15) p (cid:104) N p (cid:105) (44)which is the expected result for Poisson distributed particles since the binomial efficiency correction result again inPoisson distributed measured particles. And, applying the efficiency correction, Eq. (24), we recover the result forthe true distribution, Eq. (40)Σ , N p ,Q + = σ , n p ,Q − (cid:15) p (cid:15) q − (cid:104) n p (cid:105) − (cid:15) q (cid:15) p (cid:15) q = (cid:104) N p (cid:105) (cid:15) q − (cid:104) n p (cid:105) − (cid:15) q (cid:15) p (cid:15) q = (cid:104) n p (cid:105) (cid:15) q = (cid:104) N p (cid:105) . (45)However, if we were to apply the “standard” efficiency correction, Eq. (12), which applies only to non-overlappingsets of particles, we would get σ , n p ,Q − (cid:15) p (cid:15) q = (cid:104) N p (cid:105) (cid:15) q = Σ , N p ,Q + (cid:15) q (46)which for (cid:15) q < B. Monte Carlo simulation of a case with a non-trivial proton-charge correlation
Finally, we shall test the developed efficiency correction for a more general case when non-trivial proton-chargecorrelations are present. Here we perform a Monte Carlo simulation, which resembles more closely the actual proceduredone in experiment. As in the previous example, we only look here at protons and positive charges.We assume the following setup. Each event is characterized by three non-negative integer numbers, N , N ,and N . These three numbers are all independent and distributed in accordance with a Poisson distribution, i.e. P ( N i ) = P p ( N i , (cid:104) N i (cid:105) ) for i = 1 , ,
3. The values of N , N , N define the numbers of protons N p and other positivecharges ˆ Q + in a given event as follows: N p = N + N , (47)ˆ Q + = N + N . (48)The fact that N contributes to both the number of protons and that of other positive charges generates a non-trivialproton-charge correlation. Evaluating Σ , N p ,Q + explicitly yieldsΣ , N p ,Q + = (cid:10) δN p δQ + (cid:11) = (cid:10) δN p (cid:11) + (cid:68) δN p δ ˆ Q + (cid:69) = (cid:10) δN (cid:11) + 2 (cid:10) δN (cid:11) + 3 (cid:104) δN δN (cid:105) + (cid:104) δN δN (cid:105) + (cid:104) δN δN (cid:105) = (cid:104) N (cid:105) + 2 (cid:104) N (cid:105) = (cid:104) N p (cid:105) + (cid:104) N (cid:105) . (49)Normalizing Σ , N p ,Q + by the mean number of protons, one can explicitly see the additional proton-charge correlationin excess of the proton self-correlation: Σ , N p ,Q + (cid:104) N p (cid:105) = 1 + (cid:104) N (cid:105)(cid:104) N (cid:105) + (cid:104) N (cid:105) . (50)The Monte Carlo simulation procedure is the following:1. For each event first the numbers N , N , and N are sampled from the three independent Poisson distributions.The numbers of protons and other positive charges are then evaluated as N p = N + N and ˆ Q + = N + N .2. The charge identification efficiency is simulated by applying a binomial filter with efficiency (cid:15) q to both thenumber of protons N p and other positive charges ˆ Q + . This gives the number of charge-identified protons ˜ n p and other positive charges ˆ q + . The total measured charge in the given event is registered as q + = ˜ n p + ˆ q + .3. The proton identification efficiency is simulated by applying a additional binomial filter with efficiency ˜ (cid:15) p ≡ (cid:15) p /(cid:15) q to the number of the charge-identified protons ˜ n p . This gives the number of identified protons n p in the givenevent.4. The factorial moments f i,kn p ,q + of identified protons and positive charges are evaluated as statistical averages overall the simulated events. One example for such a correlation would be the decay of the ∆ ++ into a proton and a positively charged pion, thus contributing toboth N p and ˆ Q + . (cid:2)(cid:1)(cid:6) (cid:2)(cid:1)(cid:7) (cid:2)(cid:1)(cid:8) (cid:2)(cid:1)(cid:9) (cid:2)(cid:1)(cid:10) (cid:3)(cid:1)(cid:2)(cid:2)(cid:1)(cid:7)(cid:2)(cid:1)(cid:9)(cid:3)(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:3)(cid:1)(cid:5)(cid:3)(cid:1)(cid:7)(cid:3)(cid:1)(cid:9)(cid:4)(cid:1)(cid:2) (cid:1) (cid:1) (cid:5) (cid:14) (cid:13) (cid:18) (cid:9) (cid:1) (cid:4) (cid:6) (cid:16) (cid:12)(cid:14) (cid:3) (cid:1) (cid:15) (cid:16) (cid:14) (cid:15) (cid:9) (cid:16) (cid:1) (cid:9) (cid:10) (cid:10) (cid:11)(cid:7) (cid:11)(cid:9) (cid:13) (cid:7) (cid:22) (cid:1) (cid:7) (cid:14) (cid:16) (cid:16) (cid:9) (cid:7) (cid:18) (cid:11)(cid:14) (cid:13)(cid:1) (cid:1) (cid:5) (cid:14) (cid:13) (cid:18) (cid:9) (cid:1) (cid:4) (cid:6) (cid:16) (cid:12)(cid:14) (cid:3) (cid:1) (cid:2) (cid:17) (cid:18) (cid:6) (cid:13) (cid:8) (cid:6) (cid:16) (cid:8) (cid:2) (cid:1) (cid:9) (cid:10) (cid:10) (cid:11)(cid:7) (cid:11)(cid:9) (cid:13) (cid:7) (cid:22) (cid:1) (cid:7) (cid:14) (cid:16) (cid:16) (cid:9) (cid:7) (cid:18) (cid:11)(cid:14) (cid:13)(cid:1) (cid:1) (cid:9) (cid:21) (cid:15) (cid:9) (cid:7) (cid:18) (cid:9) (cid:8) (cid:1) (cid:20) (cid:6) (cid:12)(cid:19) (cid:9) S (cid:2) (cid:1) (cid:2) (cid:3) (cid:2) (cid:1) (cid:4) (cid:1)(cid:2)(cid:1) Æ (cid:3) (cid:5) æ e (cid:1) FIG. 1. The reconstructed values of the scaled proton-charge correlator Σ , N p ,Q + / (cid:104) N p (cid:105) for various values of the chargeidentification efficiency (cid:15) q from Monte Carlo simulations of a toy model described in Sec. V B. For each value of (cid:15) q one millionevents was generated. The black symbols depict the results obtained by applying the proper efficiency correction via Eq. (24).The red symbols correspond to the “standard” efficiency correction (13), which is applicable only for the case of non-overlappingparticles. The horizontal dashed line corresponds to the true value of Σ , N p ,Q + / (cid:104) N p (cid:105) = 1 . Here we present results of the simulations for parameter values (cid:104) N (cid:105) = 90, (cid:104) N (cid:105) = 10, and (cid:104) N (cid:105) = 200. We fixthe proton identification efficiency at ˜ (cid:15) p = 0 . . < (cid:15) q < (cid:15) q we generate one million events in accordance with the procedure described above.The true proton-charge co-variance Σ , N p ,Q + is reconstructed using the factorial moments f i,kn p ,q + of measured particlenumbers, calculated as averages over the sampled events, and the efficiency correction via Eq. (24). The results arecompared with the “standard” efficiency correction [Eq. (13)].Figure 1 depicts the reconstructed values of the scaled proton-charge correlator Σ , N p ,Q + / (cid:104) N p (cid:105) for various values of (cid:15) q . The black symbols correspond to the results obtained using the proper efficiency corrections given by Eq. (24). Forall the (cid:15) q values the efficiency corrected Monte Carlo results are consistent with the true value of Σ , N p ,Q + / (cid:104) N p (cid:105) = 1 . , N p ,Q + / (cid:104) N p (cid:105) is shown in Fig. 1 by the red symbols,systematically overestimates the true value of proton-charge correlations. The error is larger for smaller values of (cid:15) q and only disappears in the limit of perfect charge identification, (cid:15) q →
1. The realistic range for (cid:15) q in heavy-ioncollisions, however, is of order (cid:15) q = 0 . − . (cid:15) q leads to an overestimation of Σ , N p ,Q + by as much as 20-50%. This underscores the importance of taking into accountthe subtleties associated with the efficiency corrections for overlapping sets of particles discussed in the present note. VI. DISCUSSION AND SUMMARY • We note that the difference between the true correlation F , N p ,Q + = (cid:104) N p Q + (cid:105) and that obtained from the measuredquantities via the “standard” efficiency correction, Eq. (22), F , N p ,Q + − f , n p ,q + (cid:15) p (cid:15) q = − (cid:104) n p (cid:105) − (cid:15) q (cid:15) p (cid:15) q (51)vanishes in the limit of perfect charge detection, (cid:15) q →
1. This is not surprising, since in this case one only needsto correct for the proton detection efficiency and the corrections for protons only do agree with the standardprocedure, as shown above.0 • It may be instructive to consider the measured correlation between protons and all other positive chargedparticles, as this corresponds to a correlation of non-overlapping measured particles. Using Eqs. (19),(20), and(23) one gets (cid:10) n p ˆ q + (cid:11) = (cid:10) n p q + (cid:11) − (cid:10) n p (cid:11) = f , n p ,q + − (cid:10) n p (cid:11) = (cid:15) p (cid:15) q (cid:68) N p ˆ Q + (cid:69) + (cid:15) p ( (cid:15) q − (cid:15) p ) (cid:0)(cid:10) N p (cid:11) − (cid:104) N p (cid:105) (cid:1) (52)Even in this case, of seemingly non-overlapping particles, the “standard” correction does not work in general,since (cid:68) N p ˆ Q + (cid:69) − (cid:104) n p ˆ q + (cid:105) (cid:15) p (cid:15) q = ( (cid:15) p − (cid:15) q ) (cid:15) q (cid:0)(cid:10) N p (cid:11) − (cid:104) N p (cid:105) (cid:1) (cid:54) = 0 . (53)Only if we have perfect proton identification, i.e. ˜ (cid:15) p = (cid:15) p /(cid:15) q = 1 does the “standard” correction work. This iseasy to understand. With perfect proton identification we remove all protons from the measured charge whenwe calculate ˆ q + = q + − n p . Otherwise, q + always contains protons, which are identified as charges but not asprotons. This implies that the finite detection efficiency induces artificial correlations between the measuredprotons, n p , and the measured other positive charge, ˆ q + . Indeed, calculating the co-variance between protonsand other charges, σ , n p , ˆ q + in the case where the true distribution is uncorrelated, Σ , N p , ˆ Q + = 0, one finds σ , n p , ˆ q + = (cid:15) p ( (cid:15) q − (cid:15) p ) (cid:2)(cid:10) ( δN p ) (cid:11) − (cid:104) N p (cid:105) (cid:3) (54)which vanishes only in the special case of Poisson distributed protons. • The above procedure can be extended to higher order factorial moments and cumulants. As detailed in theAppendix, given the factorial cumulant generating function, Eq. (18), one can calculate the measured factorialcumulants in terms of the true ones and then simply needs to invert these relations in order to obtain expressionfor the true factorial moments in terms of measured quantities. • The local efficiency corrections proceed by correcting all the relevant local factorial moments, as originallydevised in Ref. [7]. In contrast to [7], however, the generalized efficiency correction must be used for those localfactorial moments that involve overlapping particles from the same phase-space bin. • While we have mostly concentrated on the specific case of protons and positive charges, the above results do applyto all cases of overlapping particles, such as kaon-charge correlators (cid:104) δN K + Q + (cid:105) and (cid:104) δN K − Q − (cid:105) , and others,as long as detection of the identified particle, say the proton or kaon, involves the same charge identificationprocess as all other charges. If, on the other hand, one had two distinct detectors, one to measure all charges,one to measure the protons or kaons without making use of the charge measurement of the other detector, thenthe standard procedure works. As in this case the identification of a proton does not influence in any way theidentification of all the charged particles, including the same proton, and vice versa.In summary, we have derived the formulas for the efficiency corrections of co-variances involving overlapping setsof particles. These formulas apply when the same detector is used for the initial identification of all the particles, asis the case e.g. for the reconstruction of charged tracks in heavy-ion collision experiments, and then a subset of theseparticles, such as protons among all the charged particles, is identified with an additional detector. Our main resulthere is Eq. (24), which shows that an extra term arises compared to the case of distinct particles, which would resultin apparent larger correlation if not properly taken into account. The result has also been generalized for the case oflocal efficiency corrections [Eq. (36)], which permit variations in the efficiencies between different phase-space bins. ACKNOWLEDGMENTS
We acknowledge fruitful discussions with Arghya Chatterjee and Dmytro Oliinychenko. This material is basedupon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under con-tract number DE-AC02-05CH11231. The work of V.V. received support through the Feodor Lynen program of theAlexander von Humboldt foundation.1
Appendix A
Here we present a more elegant and efficient way of relating the measured factorial moments, f i,jn p ,q + with those ofthe true distribution, F i,jN p ,Q + . We again restrict ourselves to the specific case of protons and positive charges notingthat the results can be directly translated to other equivalent cases, such as K − and negative charges etc. Let us startwith the factorial moment generating function for the true distribution, G ( Z P , Z Q ). Given the probability function P ( N p , Q + ) = (cid:80) ˆ Q + P ( N p , ˆ Q + ) δ Q + , ˆ Q + + N p , the generating function is given by G ( Z P , Z Q ) = (cid:88) N p ,Q + Z N p P Z Q + Q P ( N p , Q + ) = (cid:88) N p , ˆ Q + Z N p P Z ˆ Q + + N p Q P ( N p , ˆ Q + ) (A1)The true factorial moments are F i,jN p ,Q + = ∂ i + j ∂Z iP ∂Z jQ G ( Z P , Z Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z P = Z Q =1 (A2)Comparing with the expression of generating function for the factorial cumulants of the measured distribution, g ( z p , z q ) [Eq. (18)], we find that g ( z p , z q ) can be expressed in terms of G ( Z P , Z Q ) g ( z p , zq ) = G [ Z P ( z p , z q ) , Z Q ( z q )] (A3)with Z P ( z p , z q ) = 1 − (cid:15) q + (cid:15) q z q (1 − ˜ (cid:15) p + ˜ (cid:15) p z p )1 − (cid:15) q + z q (cid:15) q Z Q ( z q ) = 1 − (cid:15) q + z q (cid:15) q (A4)Therefore, as we have Z P ( z p = 1 , z q = 1) = Z Q ( z q = 1) = 1, the measured factorial moments f i,jn p ,q + = ∂ i + j ∂z ip ∂z jq g ( z p , z q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z p = z q =1 = ∂ i + j ∂z ip ∂z jq G ( Z P ( z p , z q ) , Z P ( zq )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z p = z q =1 (A5)can be easily related to those of the true distribution by applying the chain rule. For example f , n p ,q + = ∂∂z p G ( Z P ( z p , z q ) , Z P ( z q )) (cid:12)(cid:12)(cid:12)(cid:12) z p = z q =1 = ∂∂Z P G ( Z P , Z Q ) (cid:12)(cid:12)(cid:12)(cid:12) Z P = Z Q =1 × ∂∂z p Z P ( z p , z q ) (cid:12)(cid:12)(cid:12)(cid:12) z p = z q =1 = (cid:15) p F , N p ,Q + (A6)where in the last step we used the expression for the full detection efficiency for the protons, (cid:15) p = (cid:15) q ˜ (cid:15) p . Many termsin these expressions will vanish as only a few of the derivatives of Z P ( z p , z q ) and Z Q ( z q ) are nonzero: ∂∂z q Z Q ( z q ) (cid:12)(cid:12)(cid:12)(cid:12) z p = z q =1 = (cid:15) q ,∂ n ∂z nq Z Q ( z q ) (cid:12)(cid:12)(cid:12)(cid:12) z p = z q =1 = 0 , n > ∂∂z p Z P ( z p , z q ) (cid:12)(cid:12)(cid:12)(cid:12) z p = z q =1 = (cid:15) p ,∂ n ∂z np Z P ( z p , z q ) (cid:12)(cid:12)(cid:12)(cid:12) z p = z q =1 = 0 , n > ∂ n +1 ∂z p ∂z nq Z P ( z p , z q ) (cid:12)(cid:12)(cid:12)(cid:12) z p = z q =1 = ( − n +1 n ! (cid:15) p (1 − (cid:15) q ) (cid:15) n − q . (A7)2This procedure, which can be easily automatized for high-order factorial moments using tools such as Mathematica,provides the measured factorial moments expressed in terms of the true factorial moments. These can then be invertedto obtain the relations that express the true factorial moments in terms of the measured ones, which, in turn, providethe necessary relations required for efficiency corrections. For example, the third-order mixed factorial moments aregiven by F , N p ,Q + = f , n p ,q + (cid:15) p (cid:15) q − (cid:15) q − f , n p ,q + (cid:15) p (cid:15) q F , N p ,Q + = f , n p ,q + (cid:15) p (cid:15) q − (cid:15) q − (cid:16) f , n p ,q + − f , n p ,q + (cid:17) (cid:15) p (cid:15) q (A8) [1] A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov, and N. Xu, Phys. Rept. , 1 (2020), arXiv:1906.00936 [nucl-th].[2] M. Stephanov, Phys.Rev.Lett. , 032301 (2009), arXiv:0809.3450 [hep-ph].[3] K. Morita, B. Friman, K. Redlich, and V. Skokov, Phys. Rev. C88 , 034903 (2013), arXiv:1301.2873 [hep-ph].[4] V. Koch, A. Majumder, and J. Randrup, Phys. Rev. Lett. , 182301 (2005), arXiv:nucl-th/0505052 [nucl-th].[5] A. Bazavov, H. T. Ding, P. Hegde, O. Kaczmarek, F. Karsch, et al. , Phys. Rev. Lett. , 082301 (2013), arXiv:1304.7220[hep-lat].[6] A. Bzdak and V. Koch, Phys. Rev. C86 , 044904 (2012), arXiv:1206.4286 [nucl-th].[7] A. Bzdak and V. Koch, Phys. Rev.
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