PHYSICAL REVIEW D VOLUME 33, NUMBER 1 1 JANUARY 1986

Coherence and Coulomb effects on pion interferometry

Scott Pratt School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455

(Received 24 July 1985)

The effects of coherence and final-state Coulomb interactions on the Bose-Einstein correlations of identical particles are studied. It is shown that these corrections can be important, reducing the ap- parent source size by 20%. Coherence effects are shown to significantly reduce the coherence pa- rameters for small sources. Other causes of correlation are also discussed.

I. INTRODUCTION

Identical particles produced in a finite volume and dur- ing a finite lifetime from a random source are correlated in their momenta distribution due to interference.'z2 This effect is used to measure the size of the region from which the particles were emitted. Experiments have measured the correlation function

where P(p, . . . , q) is the probability of observing particles of momentum p . . . q all in the same event. We consider noninteracting particles which are emitted with the proba- bility per unit of space-time:

The correlation function due to this Gaussian emission function is3

for bosons; for fermions it is

C ( ~ , q ) = l - e x p [ - ( p - q ) ~ ~ ~ / 2 - ( ~ ~ - ~ ~ ) ~ ~ ~ / 2 ] ,

where up is the energy of a pion with momentum p. Thus the width of the peak in the correlation function in- dicates the size and lifetime of the source. Smaller sources yield broader peaks. Section I1 gives a general derivation of the correlation function in terms of an emis- sion function. The extent of the spatial and temporal in- formation contained in the correlation function is dis- cussed there.

Experiments have usually seen the height of the peak to be less than one. Data have been fitted to the function

The coherence parameter h should be unity for indepen- dently emitted particles when final-state interactions have been neglected. Charged particles will be less correlated since two identical charged particles will repel each other, especially if they have small relative momentum. This ef-

feet has been taken into account previously for charged pions, but only in the approximation that the source size is small compared to the Bohr radius. Typical source sizes are few fm while the Bohr radius for pions is 400 fm. The first-order correction for a finite source size is derived in Sec. 111. Including this correlation reduces the coherence parameter further and yields a significantly smaller source size.

Particles emitted from a coherent source such as a laser do not interfere, which could be an explanation for the small coherence parameter measured in experiments. Par- ticles emitted from the same quantum state are coherent. The uncertainty principle requires that for small sources a fraction of the particle pairs do come from the same state. This effect on the coherence parameter and apparent source size is evaluated in Sec. IV. Section V contains a brief discussion of other correlations such as varying im- pact parameters and various kinds of decays.

The most common particles studied are charged pions. Since they interact with each other rather weakly, the correlation function is not dominated by final-state in- teractions. The pion correlation function has been used to measure source sizes produced in e +e - (Ref. 4), pp (Refs. 5 and 61, pp (Refs. 6 and 7), meson-proton (Ref. 7), aa (Ref. 5 ) and nucleus-nucleus (Refs. 8-10) collisions. Correlations between kaons6 and have also been measured in pp, pp, and aa collisions, and correlations between baryons and light nuclei have been measured in nucleus- nucleus" and proton-nucleus12 collisions. The accuracy of these experiments improves as the number of emitted particles is increased. Ultrarelativistic heavy-ion col- lisions should produce such a large number of pions that perhaps the correlation could be measured two within a few percent. If all the other sources of correlation are well understood the spatial and temporal structure of the source might also be estimated to within a few percent. The promise of increased precision motivates this study of other sorts of correlation on the interferometry. The remainder of this paper is concerned with pion emission, where the greatest precision lies.

11. GENERAL DERIVATION OF THE CORRELATION FUNCTION

Hydrodynamic and cascade models are examples of theoretical tools that can predict emission functions for

72 0 1 9 8 5 The American Physical Society

33 - COHERENCE AND COULOMB EFFECTS ON PION INTERFEROMETRY 73

various particles. These are semiclassical treatments which can yield the classical emission probability g ( p l , x I , . . . , p, ,x , ) where g d 3 p l d 4 x is the probability of emitting a particle along a straight-line trajectory from the space-time points x with corresponding momenta p , . We will derive-an ex~ression for the correlation function in terms of the emission function g. Since interference is a quantum effect one must always make some assumption about the phases of particles emitted at different points or from different sources.

Here we give a general derivation of the correlation function in terms of the quantum-mechanical emission functions. A state 1 q ) could be produced by some set of sources i = 1, . . . , N. We are interested in the probability of observing particles of momenta p and q and we neglect all other particles.

where a and a t are creation and destruction operators for particles of a given momentum.

The structure of the wave function that describes / q ) cannot be predicted without solving the Schrodinger equa- tion or the full field theory. However, it is likely that wave packets which are orthogonal to each other have random phases with respect to each other. A general description of a multiparticle state is

where It, J', and K t are creation operators for the wave packets denoted by i, j, and k. It will be assumed that each term has a random phase. Then for any operator x ,

+ a ? * a f ( 0 1 I~J'XJI 1 0 ) + . . . . (2 .3 ) P ( p , q ) = ( q / a t ( p ) a t ( q ) a ( p ) a ( p ) I q ) ,

(2.1) With this assumption the expressions for P ( p , q ) and P ( p ) P ( p ) = (7 1 a t ( p ) a ( p ) 1 q ) , simplify:

where q i ( p ) is the wave function of the packet and P ( i , j ) is the probability of emitting particles from the wave packets i and j in the same event.

Now one can define the emission functions g in terms of the wave packets. This is in analogy to Wigner functions,

g ! i , ~ , x ; j , ~ , y ) = ~ ( i , j ) I d ~ ~ d 3 q q f (P+p/2)q;(Q+q/2)qj(Q-q/2)7)i(P-p/2)exp[ipx + q . y ) ] , (2.6)

g ( i , P , x ) = P ( i ) I d 3 p q ; ( ~ + p / 2 ) q ~ ( ~ - p / 2 ) e x p l i ~ . x ) . Using this definition one can calculate P ( p , q ) which shows the physical interpretation of g ( i , P , x ; j , Q , y ) as being the probability of observing one particle emitted from source i at point x and momentum P and another from source j at point y with momentum Q:

P ( p , q ) = x IS d 3 x d 3 y (g(i,p,x;j,q,y)+g(i,~,x;j,~,y)exp[ik(x - y ) ] / - // d 3 x d3yg(i,~,x,i,K,y)exp[ik(x y ) ] , (2 .7 )

P ( p ) = X I d 3 x g ( i , p , x ) , where K is the average momentum of the pair and k is the relative momentum. K= ( p + q ) / 2 and k = p - q . If the probability of two pions having been emitted from the same source is small compared to the probability of having come from different sources, then the last term can be neglected. This is the case when the system is large.

One can define a continuous function g ( p , x ; q , y ) in terms of the sums over the sources. Semiclassical models such as cascade models or hydrodynamic models usually predict continuous emission functions. For any limits on x o or y o :

This yields the two-particle distribution P ( p , q ) :

74 SCOTT PRATT - 3 3

This is the general expression for the correlation func- tion. The function g, is the correction for the last term in Eq. (2.7). Its interpretation is that g, is the probability of emitting two pions from the same source or from the same wave packet. This can become important for certain collisions. It will be discussed in Sec. IV.

If different sources are independent in their emission then g(p,x;q,y) can be factored into g(p,x)g(q,y). This is the case for hydrodynamic models where all events have the same impact parameter. If g, can be neglected and g(p,x;q,y) can be factored, then C(K,k)-+2 as k ap- proaches zero for any K. In Sec. V the effect of averag- ing over various impact parameters is discussed. This raises the intercept above 2. In Sec. V the effects of de- cays and conservation laws are also discussed. These ef- fects would tend to lower the intercept, except they are usually negligible.

Equation (2.9) does not yet include corrections for final-state interactions. That will be discussed in Sec. 111. Short-range interactions between the pions and other par- ticles are accounted for since emission is defined as the point where the particle left upon its straight-line trajecto- ry. Long-range interactions due to the Coulomb force of the other particles do not affect the relative phase between the pions; therefore they do not affect the interference in an appreciable way. Long-range interactions between the pions do have a significant effect on the correlation, and this is what is derived in Sec. 111.

Neglecting these interactions one can see that the in- terference term in Eq. (2.9) is the Fourier transform of the two-particle emission function for two pions with exactly the same momentum. If one boosts to the frame where K=O, P(p ,q) becomes (neglecting g,)

P(p,q)= d4x'd4y'[g(k'/2,x'; -k'/2,y1)+g(0,x';0,y')

where the primed quantities are measured in the rest frame of the pion pair. Here kb is zero so one can see that one measures the spatial structure of the source in the frame where the total momentum is zero. Also, since the argument of the exponential is a Lorentz invariant and kb is zero, the interference term can be written in terms of the relative position of all the particles emitted from x and y moving with the same velocity vK, [(x- y)-vK(xo -yo I]. This relative position does not change since the particles have the same velocity; thus, any point in the final trajectory can be chosen as the emis- sion function:

x e x p { i k * [ ( ~ - y ) - v K ( x g - ~ 0 ) ] ] ) ,

(2.1 1) where

Perpendicular and parallel refer to the direction of K, and V K is the velocity of the center of the pion pair. Thus the interference term can be considered the Fourier transform of the spatial structure function of particles of the same momentum after they have both been emitted. Models with different emission functions can yield the same correlation functions if the particles of the same momen- tum leave the same distance apart in the final state. This is illustrated in Fig. 1. This might be expected since the correlation function for a given total momentum is a function of three variables k l , k2, and k 3 , while the emis- sion function for a given total momentum is a function of four variables XO, x x2, and x3.

111. FINAL-STATE INTERACTIONS

Even if two charged pions were not identical there would be a correlation due to the Coulomb repulsion. The hadronic interaction is very weak between pions. It would be most convenient if one could detect aO's and measure their correlation function. Unfortunately, they decay much too quickly to be detected directly. The Coulomb repulsion between charged pions does affect the correla- tion function, but it does not dominate it. In the limit of a very small source (as compared to the Bohr radius) the Coulomb interaction can be accounted for with a simple multiplicative correction, the Gamov factor. A general treatment for final-state interactions is given in this sec- tion. It is not feasible to solve the problem exactly for any complicated emission functions.

However, the Coulomb distortion can be well approxi- mated in the case where the emission functions are con- sidered to be functions only of x, and g, in Eq. (2.9) is neglected. The correlation function is derived to first or- der in an expansion of the ratio of the source size to the Bohr radius (which is 400 fm for pions). This is one order better than the Gamov factor. For larger sources this is most important. Although this correction is an expansion in terms of a very small parameter for most collisions, the effective source sizes are shown to be reduced by around 20% even for sources of only a few fm across.

If the momenta of the particles could be measured at the time of emission final-state interactions would have no effect. However, the particles are measured far from the source at time T = m . Consider an instantaneous source that emits at t =0:

LONG LIVED SOURCE a- ,- ------C

PIONS WITH IDENTICAL MOMENTUM

1

SHORT LIVED SOURCE m- FIG. 1. A short-lived source can emit pions of a given veloci-

ty with the same spatial distribution as a longer-lasting small source. These sources yield the same correlation function.

33 - COHERENCE AND COULOMB EFFECTS ON PION INTERFEROMETRY 7 5

This problem could either be formulated in perturbation theory in terms of the exact solution to the Schrodinger equation, where a two-body force is assumed to account for the interaction. This nonrelativistic treatment should be appropriate when the relative momenta of interest to interferometry is small compared to the effective pion mass. The effective pion mass is 70 MeV, the pion mass. This is not much smaller than the relative momen- ta of interest in collisions with small source size such as nucleon-nucleon or e +e -. However, the Coulomb force mainly affects pairs with low relative momenta of about an inverse Bohr radius, which is around 1 MeV. So the nonrelativistic treatment for the Coulomb force between pions is valid.

The creation operators at T can be related to creation

operators at t = O by the solution to the Schrodinger equa- tion

where Tt ( r ) is a creation operator at t=O, and @(p,r;q,rl) is the two-particle wave function found by solving the Schrodinger equation. The creation operators T t can be expressed in terms of creation operators for wave packets denoted by I t :

Inserting these definitions into the expression for P (p ,q ) yields

P ( ~ , ~ ) = J' d3x d 3 x ' d 3 y ' @ * ( p , y ; q , y ' ) @ ( q , x ' ; p , x 12 [ ~ ; ( y ) ? 7 ; ( y ' ) l ? i ( x ) ) 7 1 i ( x )+q:(ylvf ( y ' ) ~ i i ~ ' i ~ ~ ( x ) ] P ~ . (3.4)

The 7's can be replaced using the definition of the emission function g in Sec. 11:

P(p ,q)= Jd3x d3x'd3y d 3 y ' d 3 p ' d 3 q ' @ * ( p , y ; q , y ' ) @ ( q , x ' ; p , x i

x (g[pl,(x +y i / 2 ; q ' , ( x ' + y ' ) / 2 ] e x p [ i p ' ( x -y )-iq1(x'--y')]

+g[pl,(x + y ' i / 2 ; q 1 , ( x ' + y ) / 2 ] e x p [ i p ' ( x -y')+iq'(xf-y)]] . (3.5)

This expression is not very transparent. However, if g is independent of p' and q' the integrals over p' and q' can be simplified. Replacing g(p, x;q,y ) with g(x ,xl), the usual answer is obtained:

The two-particle wave function can be factored into the product of the center-of-mass wave function and the rela- tive coordinate wave function:

Here 4 is the wave function of the relative coordinates. It is found by solving the Scrhijdinger equation:

The primes refer to the quantities measured in the center- of-mass frame of the pion pair. Since we have neglected the momentum dependence in g we need only consider the case where p = -q, and up =wq. In this frame the solu- tion to the Schrijdinger equation is13

where y =me2/q, r=x- y, and @ is the confluent hyper- geometric function:

We insert the expression for 4 into Eqs. (3.6) and (3.7) to obtain an answer for the two-particle probability:

P(q , - q i = ~ ( q ) 2 [ ~ : ( q ) ~ l ( q ) + ~ ~ (q)Z,(q)] , (3.1 1)

where G(q) is the Gamov correction factor,

G ( q ) = [ 2 ~ y / ( e ~ v - l ) ] ,

and the functions I are for the integrals

where x - i (qr-q-r) and g ( r ) is the single-particle emis- sion function. The function @ can be expanded in powers of y . To first order in y,

Here ci and si are the cosine and sine integral functions. If g ( r ) is spherically symmetric the integral over the an- gular coordinates can be done explicitly:

SCOTT PRATT - 3 3

The integral over r was done by a Monte Carlo method for a Gaussian source to within 1% accuracy. Then the expressions for II and I2 were inserted into Eq. (3.1 1 ) to obtain an expression for the correlation function. This is shown for three different source sizes in Fig. 2. There are two plots for each source size. One is the correlation function if there were no final-state interactions. The second is the correlation function where I I and I 2 have been calculated to first order in y so that the distortion is

C l k l for Neutrol Plans

Gomov Corrected C l k l

2 0 0 fo r Charged Pions

1 7 5 C(k1

1 50

I

correct to first order in the source size. This correlation function is then Gamov corrected [divided by ~ ( q ) ~ ] . The second correlation function therefore shows only the effects of this first-order expansion in y since the Gamov factor is divided out.

Since one is interested in the correlation function when qR is of order unity, the expansion in y can be considered as an expansion in the size of the system divided by the Bohr radius which is I / yq . For sufficiently small sources these corrections would go to zero and the Coulomb in- teractions would be taken into account by the Gamov fac- tor. In this zeroth-order approximation there is no depen- dence on the size of the system. One would guess that the first-order correction would be insignificant for sources of the order of a few fm, but Fig. 2 shows otherwise. The combination of raising the intercept and lowering the outer parts of the correlation functions in Fig. 2 leads to half-widths shorter by 25% even though the ratio of the system size to the Bohr radius is about 1 %.

IV. COHERENCE EFFECTS

2 2 5 --- C l x l fo r Neutral P o n s

- Gornov Corrected C l x )

for Charged Pions

5 0 -

2 5 -

0 0 I I / ----. 4 0 8 0 120 160

k ( M e V )

- - - --- C l k l f o r Neu l ro , P ~ o n s

2 . - for G o m w Charged Corrected Pions C l k l

7 5 --

FIG. 2. (a) The Gamov-corrected correlation function is shown for charged pions emitted from an instantaneous Gauss- ian source of radius 2 fm. Compared with the correlation func- tion for neutral pions, this shows the additional Coulomb distor- tion to first order in the ratio of the source size to the Bohr ra- dius. The half-width falls appreciably even for a small source. Taking this correction into account would yield smaller source sizes from experiment. (b) The correlation functions are shown for charged pions emitted from a Gaussian source of radius 4 fm. The intercept is higher for larger sources. (c) The addition- al Coulomb correction is demonstrated for a very large source of radius 8 fm. Although the intercept rises more for larger sources the half-widths are decreased by nearly the same factor.

Pions emitted from the same coherent state or wave packet do not exhibit interference effects.14 This is shown in Eq. (2.9) in the term g,. An example of this kind of coherent emission is a laser where all the pions are in phase. Pions emitted from the same wave packet should be in phase. The spatial width of a wave packet must be greater than the inverse of the spread of the single-particle momentum distribution due to the uncertainty principle. A small source with a small spread in momentum can not be considered as a collection of unrelated incoherent sources. In this section the effect of coherence is studied for an instantaneous Gaussian shaped source where all other types of correlation are neglected:

R is the size of the source and Q is the spread of the mo- menta distribution. The uncertainty principle demands Q.R > 1 . The probability of emitting two pions from the same source will also be chosen to be Gaussian:

This can also be written in terms of the total and relative momentum:

COHERENCE AND COULOMB EFFECTS ON PION INTERFEROMETRY 77

r=x-y . Since the operators for k and r do not commute ( [k , , r l ] =2) there is an uncertainty relation between Qo and Ro, QoRo > 1. R o is always less than R and Qo is al- ways less than Q. In the limit that the entire distribution becomes a single packet, L and Q * both go to infinity. The two-particle probability can be calculated from Eq. (2.9):

(4.4) When Q = cc , the correlation function becomes

The first two terms are the usual expression for the corre- lation function for a Gaussian source. The last term is of the form of the interference term due to a source of size R o multiplied by the ratio of the source volumes. As L approaches infinity Ro goes to R and C ( p , q ) goes to 1. This would be the case for a fully coherent source. As L goes to zero, R o also approaches zero and the last term can be neglected.

R o is the spatial width of a single wave packet. It must be larger than l / Q due to the uncertainty principle, but there is no definite upper bound. However, if the col- lisions are considered to be random, one would expect R o to be not much greater than the distance between the pions' last interactions. This distance could be 1 or 2 fm. R o could be large if there were long-lived collective states that emitted pions simultaneously. The states must live long enough so that the pions could traverse the entire state. The pions must also be emitted without further col- lisions. This seems unlikely. This effect is shown in Fig. 3 for R = 2 fm and for several different ratios of L to R. This effect depends only on the ratio of L to R if the rela- tive momentum is plotted in units of 1/R.

k (MeV I

FIG. 3. The corrections to the correlation function due to coherence effects are shown for different coherence lengths. This effect is expected to be significant for small sources. Con- sidering the mean free path and the uncertainty principle, a coherence length of 1 or perhaps 2 fm is not unreasonable.

The coherence parameter h is a measure of the height of the interference term. Experiments have fit data to the forms

c ( p , q ) ~ l+hexp(-k2R2/2-ko22/2) , (4.6)

c ( p , q ) o r 1 +h4~1~(k, ,R ) / [ ( l + k 0 ~ 2 ) k ~ e r p ~ ~ ~ ] .

The first parametrization is for a Gaussian source, and the second is Kopylov's formula for a radiating spherical shell where J 1 is the first-order Bessel function and kperp is the projection of k perpendicular to the total momen- tum. Several experiments have measured h to be less than one. Hadron-hadron collisions at CERN yield a h less than one. Heavy-ion collisions also have yielded low values of A. It is much more difficult to imagine coher- ence over a large proportion of the source in a heavy-ion collision.

V. OTHER CORRELATIONS

Averaging over different impact parameters adds a pos- itive correlation. Decays usually induce a negative corre- lation. There are three different kinds of decays that af- fect an experiment in different ways. A decay inside the hot region will only affect the correlation if it emits more than one identical particle, and then if the produced pions retain this correlation by suffering no further collisions. This correlation would be due to conservation of momen- tum. It is hard to imagine a scenario where this would be a large effect. If a particle decays into pions outside the source region but not far enough for the products to have their straight-line trajectories distinguishable from those from the ;ource, then the effect is handled simply by in- cluding this in the source function. If there is a probabili- ty that a pion was formed out of the source region (perhaps by a millimeter which is common for elec- tromagnetic decays), then the interference term in the correlation function is reduced since the decay products contribute to the single-particle spectra but not to the in- terference term (they are well separated from the rest of the particles). If the pions are formed by a decay far from the source region where the pions cannot be tracked then this affects the correlation function's numerator and denominator in the same fashion; both are reduced by the factor ( 1 -q )2 where q would be the probability such a de- cay occurred. This does not affect the correlation func- tion at all. This is also true if the pion is unmeasurable for some other reason such as the pion itself decays. Of course a given experiment can have spurious effects due to such mundane problems as particle identification. For in- stance, a muon from a pion's decay could be mistaken for a pion.

Decays just outside the source region are probably the most important in their effect on the correlation function. Two pions emitted a millimeter apart will not contribute to the interference term in Eq. (2.9) unless the correlation function could be measured for relative momenta of less than one inverse millimeter. which is not feasible. If ei- ther of the particles were produced from such a decay they would not interfere. Thus if the probability of such an electromagnetic decay is PEM then the interference term would be reduced by the factor ( 1 -PEM while the denominator in the correlation function would not be af-

78 SCOTT PRATT - 33

fected at all. Heavy-ion experiments that have been per- formed thus far are not energetic enough to produce any particles that decay into pions electromagnetically. Ha- dronic decays should all occur within a few fm of the source and weak decays should mostly occur after the pions have been measured. If the pion itself decays into a muon (the dominant mode) and the muon is created early enough that it appears to have come from the source and is misidentified as a pion then it will have this same effect on the correlation.

Averaging over impact parameters means that the prob- abilities for emission P ( q ) are measured by averaging over events with different multiplicities. If a pion is observed with momentum q there is an enhanced probability that a pion with momentum p will be seen in the same event. Observing the first pion biases the measurement towards those events with higher multiplicities. A streamer chamber experiment has the advantage of measuring the multiplicity so that only similar events are averaged to- gether. A narrow-range magnetic spectrometer does not have this capability. The value of the correlation function at k=O will be increased due to this effect. C(K,k=O) can be written as an average over impact parameters'5 [here we neglect g, from Eq. (2.911:

(5.1) x p i = l . The sum over pi is the average over different impact pa- rameters denoted by the subscript i. pi is the probability of the specific impact parameter. This can be considered as twice the average of the square of the probability of ob- serving a pion of momentum K divided by the square of the average of the probability of observing the pion:

This last identity shows that the numerator in the expres- sion for the correlation function is always larger than the denominator; therefore, this always makes the intercept rise. However it would be expected that this enhancement to the correlation function could be the same for large values of k. This would be true if the probabilities are enhanced only by a multiplicative power for smaller im- pact parameters. The entire correlation function could be then normalized and C( K,k ) would go to 2 as k - 0 and C(K,k)wouldgoto 1 as k-a , .

In general the average over impact parameters can be incorporated into the two-particle emission function. These effects can be taken care of by choosing the ap- propriate g(p,x; q,y ), where g(p,x;q,y)#g(p,x )g(q,y) when events of different multiplicities or different distri- butions are averaged together. In principle the two- particle classical emission probability can be derived from cascade codes and hydrodynamic calculations.

VI. CONCLUSIONS

Pion interferometry could be a very important tool if the accuracy could be improved to within 5% or 10%.

Other experiments can measure the temperature (if an equation of state is assumed) and total energy of the source when it breaks up, but detailed dynamical informa- tion can only be acquired through knowledge of the spa- tial structure of the source. A previous paper'6 shows how collective expansion manifests itself when the corre- lation function is measured for different values of the to- tal momentum. If the correlation function can be mea- sured for different directions of the relative momentum then information about the lifetime can be obtained. In- terferometry is performed for experiments with a high multiplicity where statistical models are used and one is usually trying to infer an equation of state. Without a measurement of the lifetime one cannot know whether or not equilibrium is obtained. Collective expansion should also be a sign of a source that remained in contact over several collision times. If the equation of state is to be measured the collision volume must be known. Present experiments measure the radius to within about 20%. Since the volume goes as the radius cubed it is necessary to measure the radius to a greater accuracy.

The corrections shown in this paper are of the order of 10%. They will be more useful if the statistics and accu- racy of correlation measurements improves. This is ex- pected for experiments with higher multiplicities such as the oxygen beam planned for CERN. The correction in Sec. I11 for final-state interactions is most important for larger sources. It should yield smaller values for the ra- dius and smaller values for the coherence parameter. The correction for coherence in Sec. IV is dependent on the ra- tio of the wave-packet length to the source size. This would be more important for small sources. The correc- tions for decays and varying impact parameters in Sec. V need further study. They could be estimated with statisti- cal codes that simulate processes like decays of reso- nances.

Many of the experiments performed thus far show that the coherence factor which measures the height of the in- terference term in the correlation function is less than one. Only the correction in Sec. IV for multiple from a single wave packet and the possibilities for decays altering the correlation function show effects in the right direction to account for a lower coherence parameter. It does not seem that these effects should be strong enough to ac- count for coherence parameters as low as 0.6 in heavy-ion collisions which some experiments have measured. Large sources should not be coherent over a large fraction of " their size unless there is some peculiar reason that the pions should be emitted from the same quantum state (e.g., pion condensation). However, these experiments do have systematic and statistical errors which may be elim- inated some time and provide coherence parameters around 1. These must be understood if the interferometry is to be trusted.

ACKNOWLEDGMENTS

This work was supported by the U.S. Department of Energy under Contract No. DOE/DE-AC02-79er- 10364. The author wishes to thank Miklos Gyulassy, Tom Hu- manic, and Joseph Kapusta for many helpful discussions.

33 - COHERENCE AND COULOMB EFFECTS ON PION INTERFEROMETRY 79

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