Dynamical and Bose-Einstein correlations in hadronization

Nuclear Physics A555 ( 1993) 793-808 North-Holland

Dynamical

and Bose-Einstein 0. Scholten’

Kernfysisch

Versneller Instituut,

correlations and H.C. Wu2

NL-9747

Received

in hadronization

AA Groningen,

The Netherlands

23 May 1992

Abstract: Pion correlations in the hadronization process are studied. A distinction is made between “dynamical”, due to the mechanism of the fragmentation scheme, and Bose-Einstein correlations, due to the statistics. It is found that in a string hadronization model not based on the usage of fragmentation functions, the dynamical correlations are at least as important as statistical correlation for identical charged pions. Other correlation functions are dominated by resonance decay. The importance of dynamical correlations imply that a pure chaotic assumption for the hadronization process is not applicable and thus that observed correlations should not be interpreted as measuring the spatial and temporal extent of sources. Comparisons are made with data from hadronic (e+, e-) annihilation.

1. Introduction One of the major outstanding problems in high-energy (multi GeV per constituent quark) reactions is the description of the hadronization process. Even though the reaction itself may involve a high momentum transfer and thus could be described using perturbative QCD, the hadronization process is dominated by non-perturbative processes. Since this precludes an ab initio QCD approach, one is obliged to introduce some phenomenological modelling. In the present investigation we have studied

this hadronization

electron-positron

process

annihilation

in detail. To do so we have selected the hadronic

process

as a test case

since

the initially

formed

quark-gluon state has a simple structure. The initial state is a (q, 4) state in which, due to asymptotic freedom, the full energy is carried as kinetic energy by the two partons. In time, as the partons move apart, the kinetic energy is converted into the gluon-color field. This picture lies at the basis of the color-string model in which the gluon field is modelled as a classical relativistic string ‘-3). The strings that will be considered presently are of the simplest variety, two dimensional with massless quarks at their ends, the so-called yo-yo strings. The hadronization process is described in the string model as the fragmentation of this high-energy string in Correspondence to: Dr. 0. Scholten, Kernfysisch Groningen, The Netherlands. ’ e-mail:[email protected]. ’ Permanent address: Suzhou University, Suzhou, Academia Sinica, Beijing, China. 0375-9474/93/$06.00

@ 1993 - Elsevier

Science

Versneller

China

Publishers

Instituut,

Zernikelaan

and The Institute

25, 9747 AA

of Theoretical

B.V. All rights reserved

Physics,

0. Scholten, H.C. Wu / Bose-Einstein

794

correlations

smaller bits and pieces corresponding to stable particles. String breaking is seen as the spontaneous creation of a (q, q) pair in the strong gluon field described by the string which forms two color singlet The earlier hadronization A fragmentation the leading

function

quarks

this description

states.

models “) emphasized was introduced

to emit a hadron

is very successful

the description

which specifies

with a certain in explaining

fraction

of inclusive

the probability

of its momentum.

cross sections,

data.

of one of

this approach

While will

fail to describe correlation phenomena as shown in ref. ‘). In the Lund model “) for example, in which also a fragmentation-function based approach is used, correlations are not due to the fragmentation process itself but are rather due to the decay of resonances and Bose-Einstein interferometry 7,8). In a dynamical string fragmentation model one rather formulates a string-breaking mechanism instead of a fragmentation function which only parametrizes the distribution of produced particles. One of the earliest dynamical string fragmentation models is due to Artru and Mennessier (AM) ‘) and throughout this paper we refer to this model unless explicitly mentioned otherwise. Imposing the conditions of covariance, causality and locality on the string-breaking mechanism allows essentially only the simplest possible mechanism: the string can break with equal probability at any point in space-time. In sect. 3 this formalism will be discussed in more detail. Recently it was shown ‘) that the dynamics of the fragmentation process could introduce strong pion correlations. In sect. 4 we distinguish two types of correlations: one is dynamical, non-statistical correlations due to the mechanism of the fragmentation process, and the other is statistical, due to Bose-Einstein symmetrization. We find that the dynamical correlations dominate, even in correlations between equal charged pions. This important result implies that, in the framework of a dynamical processes and probably fragmentation model, in (e+, e-) annihilation scattering experiments, pion correlations cannot be used to determine

also in other source sizes

and decay times.

2. Strings and string-breaking

formalisms

process interact The (q, 4) quark pair created in the hadronic (e+, e-) annihilation through the QCD gluon field. In analogy with magnetic fields in a superconductor and supported by lattice QCD calculations, this gluon field is modelled as a string. The quarks are the color charges and are located at the end points. One of the reasons for the success of this very simple model is the fact that it satisfies the two limits of QCD, asymptotic freedom at high momenta or short distances, and confinement. The latter is guaranteed since the energy of the string is proportional to its length, K = 1 GeV/fm. A detailed discussion of the dynamics of relativistic strings is given in refs. le3). To introduce the notation a short review is presented focussed on the simplest of all relativistic strings, the two-dimensional “yo-yo” strings.

0. Scholten, H.C. Wu / Bose- Einstein correlations

A classical full

relativistic

string is described

(3 + l)-dimensional

possible

Minkowski

form for the action S=

describing

Z’dzdt,

space.

by a (1 + I)-dimensional Using

gauge

the motion

invariance,

795

surface

in the

the simplest

of the string x”(z, t) is

_Y= [(ii * x’)‘-

(i)‘(x’)‘]‘,

I

(2.1)

where the usual conventions P = aP/at, xfcl = a.?/az have been used. In eq (2.1) a particular gauge has been selected such that the string is oriented along the z-direction. It is also assumed that the quarks at the endpoints of the string are massless and have no transverse momentum. Since the endpoints are massless they move with the light velocity (c = 1) and their trajectories are straight 45” lines in the (t, z) plane. The motion of the endpoints for half an oscillation is thus described by a rectangle (ADBC) in fig. 1. Instead of working in the space-time coordinate system it is often easier to work in the energy-momentum frame. The light-cone momenta of the string can be deduced immediately from fig. (l), p_=E-p,=t&AC,

p+=E+j&=~fiAD.

From eq. (2.2) it will be obvious that the invariant mass of a string is proportional to its space-time area, W* = p+p- = ~K*AC . AD. The rapidity calculated as n = i In p+/p_ = In AD/AC. It follows that in the center of mass q = 0 one has AC = AD and the string is represented by a square. These rules facilitate thinking in terms of a string picture.

(2.2)

simply can be where simple

In a string model the hadronization process is described by the breaking of the string, as schematically depicted in fig. lb. At the breakpoint x a (q, q) pair at rest is created in the non-perturbative color field of the string. The color quantum

a

b

Fig. 1. (a) A yo-yo modelling a stable particle; (b) fragmentation in the yo-yo string model. At the point x the string breaks into two substrings through the spontaneous creation of a (q. 4) pair in the strong gluon field. The letters refer to the notation used in the text.

796

numbers

0. Scholten,

H.C. Wu / Bose-Einstein

are such that two color

singlets

correlations

can be formed

with the quarks

of the

original string. In between the created q and q the color field therefore vanishes and the string is broken into two. From the string equations of motion it follows that (see fig. la) CIA, = A,x and DIAl = AIC and similarly for the other substring. It is a simple exercise to check that total energy and momentum have been conserved in the process. determination

Much

of the physics

of the position

of this string-breaking

process

enters

in the

of the breakpoints.

This dynamical fragmentation procedure we want to contrast with one based on the usage of fragmentation functions 4), as is for example the Lund model l”,ll). A “salami” tactics is followed to cut the original heavy string into pions. Starting at one end of the string, using a Monte Carlo procedure, the production of a pion is assumed with a momentum fraction given by the fragmentation function. This procedure is applied until all energy of the original string has been converted into pions. As shown in for example refs. ‘“,ll) this model is extremely successful in explaining inclusive cross sections. This procedure, however, does not take into account any of the dynamics of the hadronization process since each pion is peeled off from the original string independent from all others, except for the condition of total energy and momentum conservation. The fragmentation process itself does not introduce any correlations, as shown in refs. 5,7512).Only through the symmetrization of the pions to the maximal extent, in particular ignoring the formation of intermediate

resonances

‘), agreement

is obtained

3. The AM string-breaking

with the data.

formalism

As mentioned before, the string-breaking process is inherently making perturbative QCD arguments inapplicable. Instead one arguments of gauge invariance and causality. On this basis, Artru originally proposed a formalism in which a string break can probability

in any infinitesimal

area of the string equation

non-perturbative, should be led by and Mennessier ‘) occur with equal

(2.1), provided,

of course,

that the string has not been broken anywhere in its backward lightcone (defined as the absolute past). This procedure is one of the few that is covariant, causal and local. An additional point in its favor is that it is simple. It can easily be seen that this breaking law implies that the chance for breaking at a particular point is only a function of the string area in its absolute past. In the numerical evaluation of the breaking formalism the method derived in ref. “) has been used. In this breaking formalism there is only a single free parameter, the breaking probability per unit space-time area, (Y. In analogy to pair creation in an electromagnetic field 13), the quarks will carry transverse momentum when created. Part of this transverse momentum could be due to a Schwinger-like mechanism and reflect the transverse size of the string 14), and part could also be though to mimic quantum oscillations of the string due to zero-point motion or gluon emission. The structure of the transverse-momentum

0. Scholten, H.C. Wu / Bose-Einslein

distribution

is however

not known.

In our calculations

correlations

797

we have assumed

an exponen-

tial distribution, P (p,) = pt empJ w. In addition momentum

to this exponential distribution distribution, we have included

effects of hard-gluon

emission,

without

the transverse-momentum spectrum, The orientation of the transverse momentum of the created quark and total momentum conservation. As substring has gone through a yo-yo

(3.1)

which simulates the small transversea more flat distribution to mimic the

which the observed

see fig. 3, momentum anti-quark a technical motion the

energy

dependence

in

could not be reproduced. is taken random, but such that the are equal and opposite. This ensures detail, it is assumed that after a transverse momentum is distributed

homogeneously over the string. In the breaking prescriptions no quantum-mechanical effects have been incorporated, in the sense that the mass spectrum of the produced particles is continuous (in a later section this point will be readdressed). To take into account that the lightest particle is the pion, the ad hoc condition has been imposed that no strings with an invariant mass less than the pion mass can be created. If, following the numerical procedure, the selected breakpoint would be such that the mass of one of the substrings does not obey this limit the breakpoint is dropped as such. Any breakpoint for which the string masses are above the cut-off is accepted. It should be noticed that any such condition breaks covariance, the results become dependent on the procedure used ‘). In the present calculation this mass condition for the substrings is imposed, checking all breakpoints in sequence according to their proper time. All (sub)strings with invariant masses larger than 2m, will fragment again. The mass spectrum of the final “stable” strings will thus be continuous ranging from m, to 2m,. All of these should correspond of course to pions with a fixed mass. The additional mass is considered as “intrinsic” transverse momentum and has been added to the transverse momentum created in the fragmentation process. This procedure will induce some violation of momentum conservation which is however minor on the scale of energies considered. For the sake of simplicity, only “up” and “down” quark flavors have been considered. This allows for a non-statistical determination of the pion charges. In the calculation of correlation phenomena this is important. Two equal charged pions, for example, cannot be created at neighboring sites in the string fragmentation due to flavor conservation. Some care should be taken in assigning quark flavor at the breaking points. When the substrings still are of an appreciable mass, isospin puts no constraint and both flavors are created with equal probability. When one or both of the substrings represents a stable pion, isospin puts constraints on the relative probability for creating a (u, U) versus a (d, d) pair, depending on the flavor of the endpoints. Strangeness and also baryon production, string breaking due to

798

the creation

0. Scholten,

of (qq, 44) pairs,

H.C. Wu / Bose-Einstein

is not considered

correlations

in the present

calculations.

This

simplification can be justified by the TPC data 16), which shows that more than 80% of all particles with momenta below 1.5 GeV/ c produced in (e+, e-) annihilation are pions. The parameters in the AM breaking formalism are adjusted inclusive data, in particular the pion rapidity, the transverse-momentum multiplicity

as measured

in hadronic

(e+, e-) annihilation

to reproduce spectra and

I’). As mentioned

in the

above, the produced particles are mainly pions with only a small percentage of photons, kaons and baryons i.e. there is only a minor violation of our pions only assumption. In fig. 2 the experimental results 15) for the rapidity spectra at 14, 22 and 34 GeV are compared with calculations. The width of the rapidity plateau determines the breaking probability, (Y= 0.75 GeV-‘. Increasing the breaking probability will cause the initial

string

to break

Fig. 2. The normalized

rapidity

into fragments

spectrum

with higher

(l/a,) da/dy calculation.

relative

momenta

versus rapidity y, panel a:

and thus

data “1, panel b:

0. 10'

Scholten,

H.C.

I

Wu

/ Bose-Einstein

I

I

correlations

1

I

199 1

I

-4

l 14 GeV

l lb GeV .22 GeV n 31 GeV

2

8

n

9 t

I

10-z 0

I 1

I

I

2

3

0

2

1

3

4

Pi (GeV/O’ Fig. 3.

Same as fig. 2 for the normalized

transverse-momentum

pt spectrum,

(l/a,)

do/dp:

versus

pf.

cause a widening of the plateau with an even stronger peaking of the cross section near the rim. The transverse-momentum spectrum fig. 3 is reproduced by using W = 0.4 GeV/c in eq. (3.1). Due to constraints in the string fragmentation procedure there is no

10' _

I

I

I

I

l 14GeV l 22 GeV . 34 GeV

Ial

-- (b)

I”“I”“I”“I’ +

.

14 GeV 22 GeV

/r, “=

10' -

10“ 0

'r

0:

I OS

I 1.0

nCH’<%tl>

I 1.5

2.0

0

0.5

1.0

n/

1.5

2.0

I 2.5

Fig. 4. The calculated normalized multiplicity spectra (n)9’( n) is the probability for observing an event with multiplicity n, are compared with data I’). The average multiplicities (n) are given in table 1.

800

0. Scholten, H.C. Wu / Bose-Einstein correlations TABLE

1

Measured “) and calculated charged-particle multiplicities (n,,)

E [GeV]

Exp.

AM

14 22 34

9.1 11.2 13.5

9.2 10.8 12.4

direct one-to-one correspondence between the transverse momentum of the created pair and the observed pion spectrum. The energy-independent normalized multiplicity spectra are compared with the data i5) in fig. 4. The data show a considerable tail extending to lower multiplicities. This may be explained by the fact that in the present simplified calculations strangeness and baryon production have been ignored. Also bremsstrahlung of the annihilating (e+, e-) pair has not been considered. The difference in the width can be expressed by the quantity D’ = (n)/m which in the data “) is approximately D’ = 2.8 while the calculation gives a much larger value, D’ = 3.7. The calculation, see table 1, gives a slight underestimate of the energy dependence in the absolute charged-particle multiplicities.

4. Correlations In general

a correlation

function

is defined d6@)

WI,

where

P2)

=

d3p,

d3p2

I

as d3&‘)

d3#)

--1, d3p,

a(‘) and a(*) are the one- and two-pion

(4.1)

d3p2

cross sections,

respectively.

When

the pions are emitted independently, the correlation function is identically equal to zero and any deviation is thus indicative of correlations. Instead of analyzing a six-dimensional quantity it is more convenient to work with a one-dimensional correlation

function,

defined

as (4.2)

where

Q, the Lorentz

momenta

invariant

four-momentum

difference,

is a function

(4.3)

Q(P~,P~)=J(EI-E~)~-((PI-P~)~, The

of the

of the two particles,

uncorrelated

pair cross-section

appearing

in eq. (4.2) is defined

as (4.4)

801

0. Scholten, H.C. Wu / Bose-Einstein correlations

In analogy

to eq. (4.2) one can define a two-dimensional

arguments

18*12)

correlation

function

with

AE = IE, - E21

(4.5)

AP, = I(P, -PZ)~),

(4.6)

and

the component of the three-momentum difference which is perpendicular to the center-of-mass momentum of the two pions. These variables have been selected since they supposedly allow for the investigation of the transverse size of the emitting source and the time dependence. In the calculation of the correlation function, two sources for pion correlations are distinguished namely dynamical and Bose-Einstein correlations. In the following first the correlation function is discussed, not considering any particular symmetry for the pion wave function. This calculation shows the extent of the dynamical correlations which are due to the particularities of the fragmentation procedure. As a next step the string will be quantized to allow for a more proper treatment of the pion symmetrization. 4.1. DYNAMICAL

CORRELATIONS

In figs. 5-7 the calculated correlation functions are compared with the data for identical charged pions. Please note that in the calculation there are no free parameters. The curves drawn represent a best fit to the data as determined in ref. 12) using

1‘.O ?

.‘:-

0 .5Y

E

l

CY

0

-0. 5 0.0

I

I

I

0.5

1.0

1.5

0.0

Q

Fig. 5. of the where shown

(GeV/cl

I

I

I

0.5

1.0

1.5

The Q correlation functions for identical charged pions. The panels on the left show the results calculations (the error bars are reflecting the statistical errors in the Monte Carlo calculation) the diamonds display the results including the effect of Bose-Einstein correlations. The curve is a best fit to the data ‘*) for 29 GeV (e+, e-) annihilation as displayed on the right-hand side.

0. Scholten, H.C. Wu / Bose-Einstein correlations

802

A---_l_.15 0.0

0.5

1.0

0.0

AP,. (GeV/r)

0.5

1.0

Fig. 6. One slice of the two-dimensional (Ap,, AE) correlation functions for identical charged pions. In calculating the correlations in Ap,, AE ~0.2 GeV has been selected. The panel on the left show the results of the calculations where the diamonds display the results including the effect of Bose-Einstein correlations. The curve shown is a best fit to the data I*) as shown on the right-hand side.

5 Fig. 7. One slice of the two-dimensional (Ap,, AE) correlation In calculating the correlations in AE, Ap, < 0.2 GeV/c has been the left show the results of the calculations where the diamonds of Bose-Einstein correlations. The curve shown is a best fit to the

functions for identical charged pions. selected, and vice versa. The panel on display the results including the effect data “) as shown on the right-hand side.

0. Scholten, H.C. Wu / Bose-Einstein correlations

803

the parametrizations R(Q)=N[l+A

e-Q*‘*](l+rQ)

(4.7)

with h = 0.61, r = 0.65 fm [ref. “)I and WP,,

5)/4~51*/[l+(~~W*1

AE) = l+h[2J,(&,,

with A = 0.62, 5 = 1.27 fm and CT= 0.62 fm [ref. ‘*)I. To facilitate

(4.8)

comparison,

the

same curves have also been drawn through the calculation. Understanding the observed dynamical correlations quantitatively is complicated. One quite remarkable observation is that the width of the correlation function is essentially independent of the energy of the string. Only the absolute amount of correlation varies with energy. The correlations between unequal charged pions are dominated by last-generation substring (1.g.s.) or equivalently resonance decay. Simple SU(2) counting shows that for the decay into two pions the charged pair ratios are NW+,-/ N,o,Q = 5

(4.9)

and N,+,+ = 0, i.e. two equal charged pions cannot arise from a single binary decay. This is reflected in the fact that, see fig. 8, the correlation function for (v+, 7~~) is considerably larger than that for (TO, 7~‘) which in turn is considerably larger than that for (r+, P+) which was considered in fig. 5. The correlations induced by the decay of these last-generation substrings can be related directly to the mass of the l.g.s., using only energy-momentum conservation, (4.10)

AQ=hf:,.,.-4tnZ,.

If the 1.g.s. decay is the main source for the correlations, one expects that the width of the 1.g.s. mass distribution as given in fig. 9 to be reflected directly in the width of the correlation function, fig. 8. A detailed comparison shows that the measured LO-

I

’

r

I

I

I

I

nono

n+Jl-

30-

0.0

I

I

I

0.5

1.0

1.5

0.0

1 0.5

I 1.0

I 1.5

2.0

CL(GeV/c) Fig. 8. The calculated

(m+, T-) and the (T?‘, TO) correlation

functions

in 0.

0. Scholten,

804

Fig. 9. Mass distribution

H.C. Wu / Bose-Einstein

of first-generation

strings

correlations

for the fragmentation

of a 29 GeV string.

correlation function is narrower than would follow from the width of the 1.g.s. mass distribution, a clear indication that there must be also other mechanisms for generating the observed correlations. In addition the 1.g.s. decay does not explain the dynamical correlations between identical particles since these cannot originate from a binary decay of a resonance. In order to obtain a better insight in the origin of the dynamical correlations we have investigated its dependence on the only two parameters in the model, the breaking probability (Y and the width of the transverse-momentum distribution W. At this point it should be emphasized that these parameters have not been fine tuned to improve the agreement of the correlation function with experiment, instead these

parameters

have

been

obtained

from

a calculation

of inclusive

data,

as

discussed in sect. 3. The calculated correlation functions appear to be hardly affected by large changes in the breaking probability. Diminishing the width of the transversemomentum distribution by an order of magnitude hardly affects the rapidity spectrum, but has a dramatic effect on the 1.g.s. mass distribution and the correlation functions. The dynamical correlations do not allow for an interpretation in terms of source sizes. This is shown most convincingly by comparing the decay time obtained from a fit to the data by eq. (4.8) with the string decay time in the calculation. As a measure for the life-time of the original string we used the average proper time of the breakpoints. We find approximately 2.5 fm almost independent of the energy

0. Scholten, H.C. Wu / Bose-Einstein

correlations

805

of the string, using the usual string tension of 1 GeV/fm. These times are considerably longer than the 0.62 fm which follows from a source time interpretation of the energy correlation function ‘*). The extent of the dynamical

correlations

As shown in an earlier investigation depend

on the particulars

fragmentation

depends

9), the amount

of the dynamical

on the fragmentation of correlations

fragmentation

procedure.

(or intermittency)

scheme.

In the Lund

procedure, as an extreme case, one does not obtain any dynamical I*,‘) since, as explained earlier, each pion is peeled off from the original

correlations string independently ‘). Without the explicit unequal charge-correlation functions would 4.2. STATISTICAL

inclusion vanish.

of resonance

decay even the

CORRELATIONS

Up to this point the symmetry properties of the As pioneered by Goldhaber et al. 19) there has been tion of statistical or Bose-Einstein correlations as a on the spatial and temporal extention of the emitting

pions have not been considered. much emphasis on the interpretameasure of providing information source, see ref. *‘) and references

therein. The original (q, 4) pair produced in the annihilation has isospin T = 0 or 1. Since the hadronization process is governed by the strong interaction, isospin is conserved. For each event the pions thus should couple to a total T = 0 or 1. This puts a strong restriction on the symmetry of the pion wave function. Since the pion carries T = 1 the symmetry group is SU(3) 1 SO(3). For example if in an event 15 pions are produced which have to couple to T = 0 or 1, the possible SU(3) irreps are, assuming five pions of each kind, (15,0), (13, l), . . . , (1, l), (0,O) totaling 21 irreps. Using the usual conventions, (15,0) stands for full symmetrization between all 15 pions, while (O,O), the other extreme, implies full anti-symmetrization between nonidentical pions and symmetrization for identical pions. On the average there is thus no particular symmetry between non-identical pions. The wave function of identical pions

should

be fully symmetrized.

The symmetrization and quantization effects should be considered since the fragmentation process occurs on scales comparable to the rest mass and spatial extention of the pion. As argued in refs. ‘,‘) quantization of the string and interference phenomena are important to understand observed correlations on a quantitative level. The string model as discussed so far is a classical model in which event probabilities are calculated, not amplitudes. To quantize the model and to allow for the calculation of amplitudes we will follow the procedure originally proposed by Bowler “) and by Artru and Bowler *I). In the following only a short review is given of the essentials. The string model

can be quantized

by assigning

an amplitude

“)

JU=eiS, to any particular

diagram

where S is the classical

(4.11) string action,

S=

-K

~~~f:~~

d& =

806 -K&z,

0. Scholten, H. C. Wu / Bose- Einstein correlations

used to derive the string equations

one particular

classical

the imaginary

part of the complex

probability,

Z(K)

= $a.

string

Feynman’s

one can show *l) that only when

eq. (2.1). The string area for

from (x1, tl) to (x2, t2) is denoted

path leading Using

of motion tension

measures

K

sum-over-history

the mass obeys

by &i2 and

the string

quantization

rn? = 2rKn (n integer)

breaking condition there

is

constructive interference for the propagation from 1 to 2 and thus one obtains a discrete mass spectrum. The imaginary part of m,, related to the breaking probability of the string, represents the decay width of the resonance. The natural extention of the above amplitude interpretation of a single particle to that for an event, producing N particles is

=ew[-~4,,..,,NJ

J%,...,W,

for the propagation ordered { 1, . . . , IV},,

ii Qi,

(4.12)

i=l

where Qi is the phase for propagating the ith particle from the point of creation. The space-time area covered by the string depends on the ordering { 1, . . . , IV}, of the produced particles, while the phases @i are independent. As argued the full amplitude for an event must be symmetric under the interchange of like particles, Je Sym=CJ&l,...,N),,

(4.13)

x

where the sum runs over all interchanges of identical 1 and 2 are identical, interchanging them introduces ‘#‘12=-id&,2,....rvj

In general

one should

particles. Assuming a (complex) phase

(4.14)

-~~~,,,...,NI)=-~KA~~~.

thus for every diagram

particles

consider

also the diagram

in which

two identical particles, i and j, have been interchanged, introducing a phase & = -iddv. This effect should be taken into account through the explicit inclusion, for each diagram obtained in the Monte Carlo procedure, of all additional diagrams where two, or more, identical particles are interchanged with the phase given in eq. (4.14). Effectively

this implies

to the decay probability

that the contribution

is multiplied

by a weighting

of each Monte

Carlo diagram

factor (4.15)

In principle

one ought

to include

not only the diagrams

where

a single

pair has

been interchanged, but also those in which more then just two particles are interchanged. Due to the large numbers involved this is technically impossible. However, the weighting factor will only differ appreciably from unity when two pions have almost identical momenta. Since the frequency for this to happen for two pions in a single event is very low, it can be argued *l) that the net effect of multiple interchanges is negligible. In the actual calculations for technical reasons the above scheme is applied in a somewhat simplified fashion. The most important simplification is that in calculating

0. Scholten, H.C. Wu / Bose-Einstein

the phase factor eq. (4.14) the formation

of unstable

correlations

substrings

807

that go through

one

or more yo-yos is not taken into account. In ref. 22) this effect has been estimated to reduce the calculated BE effect by roughly a factor two. A minor reduction of the effect could symmetrization pair separately

come from taking

transverse

momentum

into consideration

in the

‘). Also the weighting factor eq. (4.15) has been applied to each instead of to a full single event. If the weighting factor is applied

to a whole event, also the inclusive

cross section

would be affected,

probably

by an

overall normalization constant which will again reduce the BE correlation. Since there is only a minor percentage of low relative-momentum pairs, the net result would be only a minor change in the correlation function. The present estimate of the BE interference effect should thus be regarded as an upper estimate, with the real effect being roughly half mainly due to the formation of intermediate resonances. In figs. 5-7 the calculated correlation functions are given for identical charged pions with and without including the Bose-Einstein effect. As argued the present estimate of the effect of BE correlations forms an upper limit of the effect. The value of the correlation function at small relative momenta suggests that two-thirds of the observed identical charged-pion correlations are due to the dynamics of the fragmentation process and only about one-third due to statistical correlations. This result is in strong disagreement with the results of a Lund model fragmentation calculation ‘) where all correlations are due to the BE effect. One implication of our result is that the & and r-parameters are extracted from a fit to the correlation function thus should not be interpreted as a source size and decay time. As discussed in refs. 23224)there are in principle also exchange correlations between contribution to (TO, rr’) which does not exist (r+, Y) and there is an additional for identical charged pions. These additional correlations arise from the exchange of a neutral pion pair 24). However, it can be argued that these effects are minor and can safely be neglected.

5. Summary and conclusions Pion correlations in a string fragmentation model have been calculated, guishing dynamical and statistical correlations. In hadronization calculations the Artru-Mennessier dynamical string fragmentation scheme we find dynamical

correlations

in hadronic

(e+, e-) annihilation.

Correlations

distinusing strong

introduced

by properly symmetrizing the pions, the Bose-Einstein correlations, appear to account for roughly one-third of the observed correlations between identical charged particles while the dynamical correlations give the dominant contribution. This result is in strong contrast with the results of hadronization calculations based on the use of fragmentation functions ‘). The correlation function for neutral pions is dominated even more strongly by the dynamical correlations. The presence of strong dynamical correlations has important consequences for the interpretation of pion correlations in more complicated reactions as reflecting the size of the source emitting

0. Scholten, H.C. Wu / Bose-Einstein

808

correlations

the pions. This interpretation in terms of sizes is only valid when the emitting source is purely chaotic which has been invalidated with the present investigation for processes

where

mentation. Our results is supported dominantly

only

a minor

on the relative by experiment.

amount

importance

of rescattering of dynamical

occurs

after

and statistical

string

frag-

correlations

The strong

(rTT+,C) and (TO, r”) correlations prereflect the decay of small strings. Only (TO, 7~‘) correlations may have

an additional contribution due to Bose-Einstein correlations. These small substrings go through one or more yo-yos before breaking and can thus be regarded as some kind of averaged resonances. Since resonance decay is also included in most non-dynamical hadronization models, there are only minor differences predicted in the calculated correlation functions for other than identical charged pions. For identical charged pions large differences between different hadronization models can be expected since these cannot be created pair-wise from the decay of a substring. We gratefully acknowledge illuminating discussions with K. Werner. This work is supported by the Foundation for Fundamental Research (FOM) and the Netherlands Organization for the Advancement of Science Research (NWO). References 1) X. Artru and G. Mennessier, Nucl. Phys. B70 (1974) 93 2) A. Patrascioiu, Nucl. Phys. B81 (1974) 525 3) I. Bars and A.J. Hanson, Phys. Rev. D13 (1976) 1744; W.A. Bardeen, I. Bars, A.J. Hanson and R.D. Peccei, Phys. Rev. D13 (1976) 2364 4) R.D. Field and R.P. Feynman, Nucl. Phys. B136 (1978) 1 5) J.F. Amundson, Phys. Rev. C41 (1990) 1292 6) B. Andersson, G. Gustafson and B. Nielsson-Almqvist, Nucl. Phys. B281 (1987) 389 7) B. Andersson and W. Hofmann, Phys. Lett. B169 (1986) 364 8) M.G. Bowler, 2. Phys. C29 (1985) 617 9) 0. Scholten, Z. Phys. A343 (1992) 235 10) B. Andersson, G. Gustafson, G. Ingelman and T. Sjiistrand, Phys. Reports 97 (1983) 31 11) B. Andersson, J. of Phys. G17 (1991) 1507 12) H. Aihara et al., Phys. Rev. D31 (1985) 996 13) F. Sauter, Z. Phys. 69 (1931) 742; W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714; J. Schwinger, Phys. Rev. 82 (1951) 664 14) K. Sailer, Th. Schonfeld, A. Schafer, B. Miiller and W. Greiner, Phys. Lett. B240 (1990) 381 15) TASS0 ~011. R. Brandelik et al., Phys. Lett. Bs9 (1980) 418; TASS0 ~011. M. Althoff et al., Z. Phys. Cl7 (1983) 5 16) H. Aihara et al., Phys. Rev. Lett. 52 (1984) 577 17) San Lau Wu, Phys. Reports 107 (1984) 59 18) G.I. Kopylov, Phys. Lett. BSO (1974) 472 19) G. Goldhaber, S. Goldhaber, W. Lee and A. Pais, Phys. Rev. 120 (1960) 300 20) D.H. Boal, C.K. Gelbke and B.K. Jennings, Rev. Mod. Phys. 62 (1990) 553 21) X. Artru and M.G. Bowler, Z. Phys. 37 (1988) 293 22) M.G. Bowler, Particle World 2 (1991) 1; Phys. Lett. B180 (1986) 299 23) I.V. Andreev, M. Pliimer and R.M. Weiner, Phys. Rev. Lett. 25 (1991) 3475 24) M.G. Bowler, Phys. Lett. B276 (1992) 237