Cumulants from short-range correlations and baryon number conservation - next-to-leading order Micha l Barej1and Adam Bzdak1y

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Cumulants from short-range correlations and baryon number conservation -

next-to-leading order

Micha l Barej1, ∗and Adam Bzdak1, †

1AGH University of Science and Technology,

Faculty of Physics and Applied Computer Science, 30-059 Krak´ow, Poland

We calculate the baryon number cumulants within acceptance with short-range correla-

tions and global baryon number conservation in terms of cumulants in the whole system

without baryon conservation. We extract leading and next-to-leading order terms of the

large baryon number limit approximation. Our results extend the ﬁndings of Refs. [1, 2].

These approximations are checked to be very close to the exact results.

I. INTRODUCTION

The phase diagram of the strongly interacting matter is not yet well explored. Especially, the

search for the ﬁrst-order phase transition between the hadronic matter and quark-gluon plasma,

and the corresponding critical endpoint, which are predicted by the eﬀective models, remains a big

challenge in high-energy physics [3–6]. It is known that the ﬂuctuations of conserved charges, e.g.,

baryon number, electric charge, or strangeness are sensitive to the relevant critical phenomena.

Therefore, many theoretical projects, as well as experiments in relativistic heavy-ion collisions,

have been established to study them [3, 7–26].

Cumulants are commonly used to quantify these ﬂuctuations because they naturally appear in

statistical mechanics [12, 27–34]. On the other hand, the factorial cumulants might be easier to

interpret since they represent integrated multiparticle correlation functions [6, 35–43]. However,

both the cumulants and factorial cumulants are aﬀected also by ﬂuctuations unrelated to the

phase transition, for instance, the impact parameter ﬂuctuations and the conservation laws, e.g.,

the baryon number conservation [28, 32, 43–52].

In our previous paper [2], we derived analytically the baryon number factorial cumulant gener-

ating function in a ﬁnite acceptance, assuming short-range correlations and global baryon number

conservation. Among other results, we calculated the factorial cumulants and cumulants within

the limit of small short-range correlation strengths, αk, and large baryon number, B. We also

reproduced the relations between cumulants in a subsystem with all correlations and cumulants in

the whole system without baryon conservation, initially obtained in Ref. [1].

In this paper, we extend this study and present a method of obtaining the ﬁrst correction to

the cumulants in the large baryon number limit. We also note that the short-range correlations

strengths cannot assume arbitrary values. Finally, we compare our approximate analytic results

with the brute-force computations.

In the next Section, we show our method of extracting the baryon number cumulants assuming

short-range correlations and global baryon number conservation. Then, we present the leading-

order and next-to-leading order terms of cumulants in the subsystem with all correlations expanded

in the large Blimit with respect to cumulants in the whole system without baryon conservation.

This is our main result. In the fourth section, we show how our approximate analytic formulas

work by comparison with the exact results. The alternative approach and the discussion on the

limitations of αk’s originating from the probability theory can be found in the Appendixes.

∗michal.barej@ﬁs.agh.edu.pl

†adam.bzdak@ﬁs.agh.edu.pl

arXiv:2210.15394v1 [hep-ph] 27 Oct 2022

II. METHOD

A. Previous study

In our previous paper [2], we considered a system of ﬁxed volume and some number of particles

of one kind, say baryons. We divided it into two subsystems (inside and outside the acceptance,

see Fig. 1) which can exchange particles. Let P1(n1) and P2(n2) be the probabilities that there are

n1particles in the ﬁrst subsystem and n2particles in the second one, respectively. The probability

that there are n1particles in the ﬁrst subsystem and n2particles in the second one is P(n1, n2) =

P1(n1)P2(n2) if there are no correlations between the two subsystems or (approximately) if there are

only short-range correlations. Assuming the global baryon number conservation, this probability

becomes

PB(n1, n2) = A P1(n1)P2(n2)δn1+n2,B ,(1)

where Ais the normalization constant and Bis the total baryon number. In this case, the proba-

bility that there are n1particles in the ﬁrst subsystem (within acceptance) reads

PB(n1) =

∞

n2=0

PB(n1, n2).(2)

inside the

acceptance

outside the

acceptance

FIG. 1: The system is divided into two subsystems with n1particles in the ﬁrst subsystem (inside

the acceptance) and n2particles in the second one (outside the acceptance).

Then, we calculated the factorial cumulant generating function for the ﬁrst subsystem with

baryon number conservation:

G(1,B)(z) = ln "A

dxBexp ∞

k=1

(xz −1)kˆ

C(1)

k+ (x−1)kˆ

C(2)

k!!x=0#,(3)

where

C(1)

k=hn1iαk=fhNiαk,

C(2)

k=hn2iαk= (1 −f)hNiαk

(4)

are the short-range factorial cumulants in the ﬁrst and the second subsystem, respectively (see

[2, 6]), for the multiplicity distribution without global baryon conservation. Here hNi=hn1i+hn2i

is the mean total number of particles in the system, f=hn1i/hNiis a fraction of particles in the

ﬁrst subsystem, and αkdescribes the strength of k-particle short-range correlations (α1= 1). We

assumed that the total average number of particles hNi=hn1i+hn2i=B. Introducing the global

baryon number conservation further requires that the total number of particles N=n1+n2equals

Bin every event.

Using the factorial cumulant generating function (3), one can obtain the factorial cumulants in

the ﬁrst subsystem (within acceptance) with baryon number conservation:

C(1,B)

k=dk

dzkG(1,B)(z)z=1

.(5)

We obtained [2] the analytic as well as approximate formulas for the factorial cumulants in the

simple case of only two-particle short-range correlations. Then, we also derived the approximate

factorial cumulants and cumulants in the limit of B→ ∞ assuming multiparticle short-range

correlations. These results reproduced the ﬁndings of Ref. [1] obtained originally using a diﬀerent

approach.

B. Next-to-leading order correction

We extend the previous study and obtain the next-to-leading order terms of the expansion of

the cumulants in the limit of B→ ∞. We derive the relevant expressions in two diﬀerent ways.

One method, presented in Appendix A, is based on analyzing the ﬁrst few terms of the expansions,

deducing the following terms, and then summing the inﬁnite series. Here we present another

method that is simpler.

In order to make the computations, we approximate Eq. (3) with Eq. (4) by

G(1,B)(z)≈ln "A

dxB"exp (xz −1)fB + (x−1) ¯

fBM

m=0

m!#x=0#,(6)

where ¯

f= 1 −fand

k=2 (xz −1)k

k!fBαk+(x−1)k

k!¯

fBαk,(7)

with Mand Kbeing the upper limits. Note that in Eq. (7) we allow for up to K-particle

short-range correlations.

To calculate the Bth derivative with respect to xin Eq. (6), we use the general Leibnitz formula

as described in Ref. [2]. In this way we evaluate the factorial cumulant generating function (6)

and then the factorial cumulants (5). Having the factorial cumulants, we calculate the cumulants

according to

κn=

k=1

S(n, k)ˆ

Ck,(8)

where S(n, k) is the Stirling number of the second kind [53].1For instance, the second cumulant

reads

κ(1,B)

2=hn1i+ˆ

C(1,B)

2,(9)

where ˆ

C(1,B)

1=κ(1,B)

1=hn1i=fB.

1See also Appendix A of Ref. [6] for explicit formulas for the ﬁrst six cumulants.

The global (both subsystems combined) short-range factorial cumulants, without the baryon

number conservation, are deﬁned as (compare with Eqs. (4)):

C(G)

n=Bαn.(10)

The global cumulants without the baryon number conservation, κ(G)

n, are obtained by Eq. (8).

As shown in Ref. [2], the cumulants, κ(1,B)

n, can be expressed as a power series in terms of B

where the highest-order term is linear in B. Namely,

κ(1,B)

n≈κ(1,B,LO)

| {z }

un,1B1

+κ(1,B,NLO)

| {z }

un,0B0

+κ(1,B,NNLO)

| {z }

un,−1B−1

+. . .

|{z}

O(B−2)

,(11)

where κ(1,B,LO)

n,κ(1,B,NLO)

n, and κ(1,B,NNLO)

ndenote the leading-order, next-to-leading-order, and

next-to-next-to-leading-order terms of the power series in B, respectively.

Let us focus on the second cumulant. Using the method presented in Appendix A, we deduced

that in order to extract LO and NLO terms, it is convenient to multiply κ(1,B)

2by (κ(G)

2)2=

[B(1 + α2)]2. We deﬁne

eκ(1,B)

2=κ(1,B)

2(κ(G)

2)2=κ(1,B)

2[B(1 + α2)]2.(12)

Then, we expand eκ(1,B)

2into the power series in αkup to the order of M, obtaining eκ(1,B,ser)

eκ(1,B)

2≈eκ(1,B,ser)

2=u2,1(1 + α2)2B3+u2,0(1 + α2)2B2+. . .

|{z}

O(B)

.(13)

The coeﬃcients of the expansion are calculated as follows,

u2,1=1

(1 + α2)2lim

B→∞ eκ(1,B,ser)

B3,

u2,0=1

(1 + α2)2lim

B→∞ eκ(1,B,ser)

2−u2,1(1 + α2)2B3

B2.

(14)

Clearly, it is possible to extract even higher terms in an analogous way. Using Eqs. (8) and (10),

we express these coeﬃcients in terms of the global short-range cumulants (without the baryon

conservation), κ(G)

The same technique is applied to obtain the leading and next-to-leading order terms of κ(1,B)

Namely, we multiply κ(1,B)

3by (κ(G)

2)2.

It turns out that κ(1,B)

4needs to be multiplied by (κ(G)

2)4. Namely,

eκ(1,B)

4=κ(1,B)

4(κ(G)

2)4=κ(1,B)

4[B(1 + α2)]4.(15)

In this case, equations corresponding to Eqs. (13) and (14) read:

eκ(1,B)

4≈eκ(1,B,ser)

4=u4,1(1 + α2)4B5+u4,0(1 + α2)4B4+. . .

|{z}

O(B3)

,(16)

and

u4,1=1

(1 + α2)4lim

B→∞ eκ(1,B,ser)

B5,

u4,0=1

(1 + α2)4lim

B→∞ eκ(1,B,ser)

4−u4,1(1 + α2)4B5

B4.

(17)

The results are computed using Mathematica software [54].

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Cumulantsfromshort-rangecorrelationsandbaryonnumberconservation-next-to-leadingorderMichalBarej1,andAdamBzdak1,y1AGHUniversityofScienceandTechnology,FacultyofPhysicsandAppliedComputerScience,30-059Krakow,PolandWecalculatethebaryonnumbercumulantswithinacceptancewithshort-rangecorrela-tionsandglobal...

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