Classical hadrodynamics: application to soft nucleon-nucleon collisions

NUCLEAR PHYSICS A

Nuclear Physics A560 (1993) 586-602 North-Holland

Classical hadrodynamics: application to soft nucleon-nucleon collisions * Brian W. Bush and J. Rayford

Nix

Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received

1 December

1992

Abstract:We present results for soft nucleon-nucleon

collisions at plab = 14.6, 30, 60, 100 and 200 GeV/c calculated on the basis of classical hadrodynamics for extended nucleons. This theory, which corresponds to nucleons of finite size interacting with massive neutral scalar and vector meson fields, is the classical analogue of the quantum hadrodynamics of Serot and Walecka but without the assumptions of the mean-field approximation and of point nucleons. The theory is manifestly Lorentz-covariant and automatically includes space-time nonlocality and retardation, nonequilibrium phenomena, interactions among all nucleons and particle production when used for applications such as relativistic heavy-ion collisions. We briefly review the history of classical meson-field theory and present our classical relativistic equations of motion, which are solved to yield such physically observable quantities as scattering angle,

transverse momentum, radiated energy and rapidity. We find that the theory provides a physically reasonable description of gross features associated with the soft reactions that dominate nucleon-nucleon collisions. The equations of motion are practical to solve numerically for relativistic heavy-ion collisions.

1. Introduction When Hans Weidenmiiller visited Los Alamos in September 199 1, we had the pleasure of discussing with him a new microscopic approach to relativistic heavy-ion collisions that we had undertaken. The goal of this work is to provide accurate predictions for relativistic heavy-ion hadronic

collisions

on the basis of conventional

degrees of freedom only. Significant

experimental

nuclear physics, taking into account

deviations

between these predictions

data would then signal the onset of new phenomena

and

such as a quark-

gluon plasma ‘,*). We were encouraged by Hans’ comments during the early stages of the development of this theory and are delighted to be able to report in this Festschrift our first results calculated for soft nucleon-nucleon collisions. Prior theoretical treatments of relativistic heavy-ion collisions on the basis of con-. ventional nuclear physics have utilized a variety of approximation methods and models, including relativistic nuclear fluid dynamics, nonrelativistic many-body equations of motion, geometrical models, intranuclear cascades, the Boltzmann-Uehling-Uhlenbeck Correspondence to: Dr. J.R. Nix, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. l

Dedicated

to Hans A. Weidenmiiller

0375-9474/93/$06.00

@ 1993-Elsevier

on the occasion of his 60th birthday. Science Publishers B.V. All rights reserved

587

B. W. Bush, J.R. Nix / Classical hadrodynamics

equation,

relativistic

quantum

molecular

dynamics

proaches have been useful for many purposes, validity are poorly satisfied for heavy-ion At these energies, the interaction

and string models * 1. These ap-

but unfortunately

the criteria for their

collisions at AGS, CERN and RHIC energies.

time is extremely short, the Lorentz factor y can be huge

and the nucleon mean free path, force range and internucleon separation are all comparable in size. In order for relativistic nuclear fluid dynamics to be valid, the interaction time would need to be sufficiently long to permit the establishment of local equilibrium, the mean free path would need to be short relative to the size of the system and many degrees of freedom would need to be present. The nonrelativistic approximation and absence of particle production in many-body equations of motion are inappropriate at AGS, CERN and RHIC energies. Finally, the other approaches would require for their validity a long mean free path relative to the force range, as well as the absence of correlations among nucleons. To satisfy a priori the basic conditions that are present at AGS, CERN and RHIC energies, a fully microscopic many-body treatment that is manifestly Lorentz-covariant and that allows for nonequilibrium phenomena, interactions among all nucleons and particle production appears necessary. On the other hand, at bombarding energies of many GeV per nucleon, the de Broglie wavelength of projectile nucleons is extremely small compared to all other length scales in the problem. Furthermore - in contrast to the situation in electrodynamics - the Compton wavelength of the nucleon is small compared to its radius, so that effects due to the intrinsic size of the nucleon dominate those due to quanta1 uncertainty. The classical approximation for nucleon trajectories should therefore be valid, provided that the effect of the finite nucleon size on the equations of motion is taken into account. Given these expectations, classical hadrodynamics for extended nucleons, corresponding to nucleons of finite size interacting with massive meson fields, is a natural approach to pursue. This theory represents the classical analogue of the quantum hadrodynamics of Serot and Walecka (QHD), which has been widely used for describing nuclear matter and nuclear structure 3), but without the assumptions of the mean-field approximation and of point nucleons. The physical input underlying

this approach

consists of Lorentz

invariance

(which

includes energy and momentum conservation), nucleons of finite size interacting with massive meson fields and the classical approximation applied in domains where it should be reasonably valid. This starting point builds upon the traditional hadronic description of nuclear processes that has in the past been so successful in a wide variety of situations. By treating the underlying quarks and gluons implicitly in terms of nucleons and mesons, we are able to solve the resulting classical relativistic many-body equations of motion numerically without further approximation. Of particular importance, neither a meanfield approximation nor a perturbative expansion in coupling strength is necessary within this approach. Our present version of the theory includes only neutral scalar and vector meson fields, which we identify with o and o mesons, respectively. This permits a qualitative discussion of several important physical points, including an inherent space-time nonlocality that

may be responsible

for significant

massive bremsstrahlung.

collective effects, as well as particle production

through

The CTand w mesons that are produced through this mechanism

will subsequently decay primarily into pions with some photons also. A more realistic treatment wilf require the inclusion of additional fields to account for at least charged 7~ and p mesons 4j, which would permit the study

ofisospineffects. In addition,

lhe present

version of the theory neglects excitations in the nucleons themselves; the d excited state may ultimately need to be included. Massive meson fields were of course introduced in the 193Os, but their classical treatment has suffered from previous m~su~d~rs~and~ngs ~o~c~~~ing the nucleon effective mass and how the finite nucleon size affects the fundame~~aI character oftbe dynamical motion We therefore begin our presentation in sect. 2 with a brief historical overview, which is followed in sect. 3 by a description of our classical relativistic equations of motion. Sect. 4 contains our results calculated for soft nucleon-nucleon collisions at 14.6, 30.60, 100 and 200 GeV/c, and sect. 5 our summary and conclusions. PIaL = Some ~~elimina~ results of our ~n~~sti~ations have been presented previously s-z0], ahhough not all of the equations appearing in these early publications are in their final form. A comprehensive discussion of the foundations of the theory will be submitted soon ‘t 1. In a future series of papers, we plan to systematically solve our newly derived equations of motion and compare with experimental data for proton-nucleus coliisions and nucleus-nucleus collisions. This wark will &on~entrate on such phys~~all~ observaMe quantities as transverse momentum distributions, energy loss, nuclear stopping, radiated energy and particle production.

Classical meson-field theory has a history dating back to the 1930s. Yukawa first introduced mesons in 1934, when he postulated the existence of a new “heavy particle” *2). The associated field of force had a potential function satisfying the Klein-Gordon equation and an interaction energy between two particles given by what is now known as the Yukawa potential. In 1936 Proca 13) elaborated on the theory by generafizing the Maxwefi equations to Gelds with nonzero mass {the Proca e~~a~~~~~~ ad proved torrent and energy-momentum conservation. This was the first rigorous discussion of a classical massive vector field. In the next year, Yukawa and Sakata “1 constructed the theory of the charged scalar field. Pais ts 1 provides a clear discussion of further developments and outlines the early theoretical and experimental thinking about mesons. Classical ~~~~~~~~ra~~~~~ particle motion was Erst discussed by Thomson in I Ei I f when he recognized that the electromagnetic se&interaction within a particle contributes to its effective mass 16J71. Lorentz, using his newly constructed microscopic theory of particle electrodynamics ‘s,‘9), first found the expression for the nonrelativistic electromagnetic self-force on a charged particle. This result motivated Abraham fo undertake a detailed investigation of the ele~~rom~gne~i~ self-mass ‘n 1, where he discovered that although the s&f-interaction contributes a term fA&ttf~*~to the kinetic energy, it contributes $M,tia

B. W. Bush, J.R. Nix / Classical hadrodynamics

to the momentum. Abraham

589

It was not until the discovery of the Lorentz transformation

realized the inconsistency

of the + factor with special relativity

2’ ) that

22). Electrody-

namics seemed to indicate pself = 54MSeir~ while the Lorentz transformation required pself = Mselfu [ref. “) 1, an apparent paradox that has generated a tremendous amount of speculation.

Fermi was the first to resolve it, in 1922, when he found that the : factor

resulted from an incorrect nonrelativistic limit of the relativistic electromagnetic momentum 23,24). Despite the repeated rediscovery “p2 ) of this important result that there is no $ paradox, it is still unknown to many. In 1938 Dirac provided the first rigorous derivation of the classical theory of radiating electrons 26) and wrote the equation of motion now known as the “Lorentz-Dirac” equation. He dealt with the singularity of the electromagnetic field of a point particle by separating the field symmetrically into retarded and advanced parts and then demonstrating that the radiation field, which is the difference of the two, has no singularity along the world line of a particle. The experimental mass in Dirac’s theory was identified by considering the energy-momentum flux out of a vanishingly small tube surrounding the particle’s world line. Because the Lorentz-Dirac equation needs three boundary conditions, Dirac required that the final acceleration of a particle be specified in addition to its initial position and velocity. Dirac also recognized that such boundary conditions require a particle to start accelerating before it feels a force which will act at a future time; this is the infamous “preacceleration” phenomenon. Thus in a single paper Dirac laid a rigorous foundation for classical point-electron theory and identified the conceptual and acausal problems with the theory. Bhabha did for massive vector mesons in 1939 [ref. 27) ] what Dirac had done the year before for photons 26) by constructing the meson-field analog of the Lorentz-Dirac equation using Dirac’s methods. Bhabha used the renormalization procedure of Pryce 28) to relate what he calls the “real” neutron mass to the “effective” neutron mass: his terminology here is unfortunately ambiguous. Assuming that his real mass is the experimental mass, his physical interpretation

of the equation

of motion is incorrect.

This same mis-

interpretation appears in the work of Harish-Chandra 2g), Havas 30) and others. Much of the later work on point particles interacting through classical meson fields was performed by Havas and his collaborators 30-33). The only applications of classical mesonfield theory have been simple calculations nucleons 2g,33). As noted above in our discussion

of the plane-wave

of the Lorentz-Dirac

scattering of mesons off of

equation,

the equations

of mo-

tion for a self-interacting point particle pose several conceptual problems. These problems are not limited to the case of electrodynamics, but also occur for meson-field selfinteractions. In general, the solutions to the equations of motion contain exponentially growing components called “runaway solutions”, which diverge for large times. To eliminate the component of the solution that grows uncontrollably, one must require that the acceleration of the particle vanish for a time long after the interaction. This requirement, however, introduces acausal behaviour in that it causes the particle to start accelerating before it reaches the interaction region - “preacceleration”. Preacceleration occurs on a

-

-0.0

(Fcurier-Sssal analysii) Expcmential (Rm = 0.862 fm)

0.5 1.0 Radial Distance r (fm)

1.5

Fig. 1x Comparison of the experimental proton charge density 4o) with an exponential form.

time scale of the classical radius divided by the speed of light. Many “solutions” to the problems of prea~~e~eratio~ and runaway motion have been proposed. Erber 34f reviews the different approaches to the subject, some of them involving a reinterpretation of the equations of motion and others involving the introduction of finite-size particles. The definitive explanation of the origin of runaway solutions appears in the work of Moniz and Sharp 3s-39), who analyzed a particle’s self-interaction in nonrelativistic quanfurn electrodynamics. They discovered that taking the cIassicaI limit of the quantum operator equation introduces runaway solutions which do not correspond to any of the solutions to the original quantum equation. Only the nonrunaway solution corresponds to the classical limit. This resolution of the runaway-solution problem seems to result from the quanta1 uncertainty in particle position, giving the particle an effective size large enough to overcome the point-particle pathology. The probIems of the point-particte theories can be avoided altogether by considering particies to have finite size: when the extent of the particle is iarger than its classical radius, no runaway solutions exist. In addition to eliminating preacceleration and runaway solutions, an extended-particle theory provides a much better model of the nucleon than does a point-particle theory. In fig. 1 we show the experimentally determined proton charge density 40), which has a root-mean-square radius of ~~.~~~~~~~~~) fm, Note that it is well approximated by the exponential density which we adopt for our calculations. Although the extended-particle theory does not possess runaway solutions or preacceleration, it still has problems with causality when special relativity is taken into account. It is a well-known fact that rigid bodies cannot be accelerated without violating special relativity ” )_ It is less well-known, however, that it is possible to formulate covariant equations of motion for rigid bodies r7) in spite of this causality difficulty.

B. W. Bush, J.R. Nix / Classical hadrodynamics

591

3. Equations of motion Our action for N extended,

unexcited

nucleons

interacting

with massive scalar and

vector meson fields is

--

’

/d4x

(jG2-m:V2)--Jd4x

811..

(i4+K.I/),

‘M interaction

vector field

where MO is the bare nucleon mass and qi is the four-position of the ith nucleon, whose trajectory is given by qi = q; (Ti). A dot represents the derivative with respect to 7i. In the action the four-velocities are not constrained so that 4i2 = 1 and ‘Siis not yet identified as the proper time; it is only in the equations of motion, which are derived as a result of the variation of I, that this is true. We use the metric g”” = diag( 1, - 1, - 1, - 1) and write four-vectors as q” = (q’, q) = (q’, qx, qy, 4’). We use units in which h = c = 1; the introduction of h does not imply that there is anything quanta1 in the approach, but simply enables us to use familiar units and facilitates comparison with other work. The scalar potential is denoted by 4, the four-vector mS,“. The vector field strength tensor is

potential

G”” = ,IpYYl s a”v”

by V and the meson masses by

_ $‘V”,

(2)

the scalar source density is

and the vector source density is (4) where p is the four-dimensional mass density of the nucleon, the spatial part of which we assume to be exponential in the nucleon’s rest frame. Table 1 lists the values 3S42743) of the six physical constants appearing in the theory that we have used here. In ref. ” ) we have derived exact equations of motion for the above action in two limits: (i) relativistic point particles and (ii) nonrelativistic extended particles. The nonlinearities present in the special theory of relativity prohibit the analytic evaluation of the seven integrals present in the calculation of the interaction. Instead of providing a rigorous derivation of the extended-particle relativistic equations of motion, we have found the simplest covariant generalization of the equations of motion for the two limits I’). These limits put such strong constraints on the possible generalization that there seems to be little ambiguity in the form of the generalization at which we arrive.

592

B. IV. Bush, J.R. Nix / Classical hydrodynamics TABLE 1 Values used for the constants appearing in the CHD model Quantity

Our complete

Symbol

Value

nucleon mass

M

scalar (a ) meson mass vector (u) meson mass scalar interaction strength vector interaction strength nucleon r.m.s. radius

ms

938.91897 MeV 550 MeV 781.95 MeV 7.29 10.81 0.862 fm

set of relativistic

equations

I?lv 2 &

g$ R rms

of motion

for extended

particles

can be

written as

MW = .Ll:+ fv:+CL., + f&t.i

’

(5)

with the effective mass given by

(8)

(9)

The hadrostatic self-energies of the scalar and vector fields are denoted by MS,” and their logarithmic derivatives with respect to meson mass by Ml,“. The self-forces are given by

B. W. Bush, J.R. Nix / Classical hadrodynamics

593

and the external forces by

&ttxt,i

=

$P:yc/pdo j#i

O

(13)

In the above PC” = g gv - r,!uV Vj = ij(?j), V; = ij(Tj-a), 1’ s!J = qj(rj)-qj(Tj-O), aj = & (Tj), ki = s; .L$ ani the retarded proper time rj is determined implicitly from the condition kj = 0. These equations are written in terms of the nucleon structure functions h and w and quantities derived from them. We define the interaction energy function to be d3X, d3xz ~(-d$d+2), (14) J I where R z /xi - x21 and r is the distance between the centres of the two particles. Here p(r) is the nucleon mass density normalized so that 471sow r’dr p(r) = 1. A Laplace transform relates the structure function w and its derivative ~‘(a, r) E r-‘0w/dr to the interaction energy W: W(m,r)

E

w(cT,r)

:: 2f.-“[W(m,r);2a]

.

(15)

The self-interaction structure function is h(a) E 32n2 imdo’

(a’Z-02)~(~+~‘)~(Ia-~11),

(16)

with h’(a) G dh(o)/da, h(c) s J,“da’h(a’) and h(o) z h’(a) - 6mzh((1.). The first-order Bessel function of the first kind is denoted by Jr. Inspection of the above equations of motion reveals that they are second-order, nonlocal, nonlinear, ordinary differential equations with four dimensions per particle. To solve them we use a fourth-order Adams-Moulton predictor-corrector algorithm with adaptive step sizes. The integrations over proper time are done with a special error-minimizing application of Lagrange’s four-point (cubic) interpolation formulae. 4. Results for soft nucleon-nucleon collisions We now present the first results obtained by solving our classical hadrodynamics equations of motion for the soft collision of two nucleons at laboratory momentum &+b= 14.6,

594

B. W. Bush, J.R. Nix / Ciassical ha~l~yn~rnic~

0.5

1.0

1.5

2.0

Impact Parameter b (fm)

Fig. 2. Calculated dependence of the centre-of-mass scattering angle upon impact parameter for soft nucleon-nucleon collisions at five incident laboratory momenta.

30, 60, 100 and 200 GeV/c. At three of these momenta substantial experimental data exist for heavy-ion collisions 2,44) at the AGS and CERN, and at the remaining two mo-

menta experimental data exist for proton-proton collisions 45). We will concentrate our discussion here on such physically observable quantities as scattering angle, transverse momentum, radiated energy and rapidity in the centre-of-mass system, in which frame the computations are performed. The qualitative behaviour of the results to be presented can be understood in terms of the nature of the external forces ( 12) and (13). The repulsive vector force scales as the Lorentz factor y in both the longitudinal and transverse directions, whereas the attractive scalar force scales as y2 in the longitudinal direction and as unity in the transverse direction. This implies that the vector force will dominate the transverse acceleration and the scalar force will dominate

the longitudinal

acceleration.

For a given impact parameter

the

scattering angle and transverse momentum will be essentially propo~ional to the vector interaction strength g,‘, and the radiated energy will be essentially proportional to y times the scalar interaction strength g,“. Figs. 2 through 5 illustrate how the centre-of-mass scattering angle km, transverse momentum PT, radiated energy per nucleon E rad,cm and rapidity y,, depend upon impact parameter b in these collisions. As shown in fig. 2, for a given incident momentum, the centre-of-mass scattering angle for the dominating soft reactions described by our theory has a maximum value at a certain impact parameter and decreases to zero for both head-on and distant collisions. With increasing incident momentum in this range both the maximum angle and the impact parameter at which this maximum occurs decrease. For u~trareiativistic collisions this impact parameter is approximately the distance at which the transversely dominating static vector force for extended nucleons ‘*J’ f has its

B. W. Bush, J.R. Nix / Classical hadrodynamics

n

595

1.0 1.5 Impact Parameter b (fm)

0.5

Fig. 3. Calculated dependence of the transverse momentum upon impact parameter for soft nucleon-nucleon collisions at five incident laboratory momenta.

maximum. At the other extreme of low incident momentum, the opposing scalar and vector forces are of similar magnitude and give rise for small impact parameter to the more complicated behaviour of the double-dot-dashed curve in fig. 2. The transverse momentum has a related behaviour, as shown in fig. 3. For a given incident momentum, the transverse momentum for soft reactions also has a maximum value at a certain impact parameter and decreases to zero for both head-on and distant collisions. The maximum transverse momentum increases slowly with increasing incident momentum in this range, and the impact parameter at which this maximum occurs decreases. The centre-of-mass has a maximum

radiated energy per nucleon for soft reactions shown in fig. 4 also

value at a certain impact parameter.

However, this quantity

decreases

to a finite value for head-on collisions and to zero for distant collisions. The maximum centre-of-mass radiated energy per nucleon increases strongly with increasing incident momentum. In the present version of the theory, this radiated energy will be in the form of u and w mesons, which will subsequently decay primarily into pions with some photons also. The classical approximation is expected to be valid only when the amount of radiated energy in the centre-of-mass system exceeds the mass of the lightest meson, which is 550 MeV. As seen in fig. 4, this condition is well satisfied for impact parameters of physical interest at the three highest incident momenta, but not at the two lowest incident momenta. The centre-of-mass rapidity for soft reactions has a related behaviour, as shown in fig. 5. For a given incident momentum, this quantity has a minimum value at a certain impact parameter and increases to a finite value for head-on collisions and,to the .initial rapidity for distant collisions. The impact parameter at which this minimum occurs decreases

594

Impact Parameter Lt (fm) Fig. 4. Caiculated dependence of the centre-of-mass radiated energy per nucleon upon impact parameter for soft nucleon-nucleon collisions at five incident laboratory momenta.

with increasing incident momentum From the calculated dependence

in this range. upon impact parameter

cally observable quantities discussed above, we determine differential cross section from the relationship

of each of the four physithe corresponding

classical

(17)

0.5

1.5 1.0 Impact Parameter b (fm)

Fig_ 5. Cakuiated dependence of the centre-of-mass rapidity upon impact parameter for soft nu&eon-nucleon collisions at five incident laboratory momenta.

597

B. W. Bush, J.R. Nix / Classical hadbdynamics 150

0

0

2 4 6 Scattering Angie 8, (deg)

6

Fig. 6. Calculated differential cross section as a function of centre-of-mass scattering angle for soft nucleon-nucleon collisions at !ive incident laboratory momenta. The classical cross section diverges when d&,,/db = 0.

where Q refers to either the centre-of-mass scattering,angle I%,, transverse momentum pr, radiated energy per nucleon Ed,, or rapidity y,. These classical differential cross sections for soft collisions are shown in figs. 6 through 9. The presence of dQ/db in the denominator of eq. ( 17) means that the classical cross section diverges when dQ/db = 0. The divergences at the origins in figs. 6 through 8 and at the values of the initial ra-

0.0

’

0

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

100 200 300 Transverse Momentum & (Me%)

’

’

’

400

Fig 7. Calculated differential cross section as a function of transverse momentum for soft nucleon-nucleon collisions at five incident laboratory momenta. The classical cross section diverges when dpr/db = 0.

598

3. W. Bush, J.R. Nix / Ckssical h~~r~yn~rn~cs

01 0

’ ’ ’ ’ ’ 1 2 3 Radiated Energy per Nucleon ’

’

’

’

’

4 Ersdsm (GeV)

5

Fig. 8. Calculated differential cross section as a function of centre-of-mass radiated energy per nucleon for soft nucleon-nucleon collisions at five incident laboratory momenta. The classical cross section diverges when dE,,,&db = 0.

pidity in fig. 9 arise from this occurring for distant collisions, as a consequence of the scalar and vector potentials in our theory remaining nonzero at large distances. The other divergences arise when dQ/db = 0 for a finite value of the impact parameter. The discontinuities in the curves in figs. 8 and 9 arise because E ra~,cmand ycm depend quadratically upon b for nearly head-on collisions.

0.5

1.0

2.0 1.5 Rapidity y,

2.5

3.0

Fig. 9. Calculated differential cross section as a function of centre-of-mass rapidity for soft nucleon-nucleon collisions at five incident laboratory momenta. The classical cross section diverges when dy&db = 0.

B, W. Bush, J.R. Nix / Classical hadrodynamics

+

g 100 -

g E I-

0

..‘.....

599

Experimental (ptataau in rapidity) Berot-Wale&aconstants(cakulated) Bryan-Scottconstants(estimated) I

“‘1”

I

I

I,,,,

1 Cen&f-Mass

Energy s’” (GeV)

lo*

Fig. 10. Comparison of experimental and theoretical values of the mean transverse momentum (pr) as a function of total centre-of-mass energy &. The experimental results 46,47) are peak values in the central region, where there is a short plateau in rapidity, and include high-m contributions arising from hard collisions. The dotted curve is estimated by multiplying the solid curve by the ratio of the vector interaction strength g: obtained by Bryan and Scott 48) to that obtained by Serot and Walecka 3 ).

As an initial test of the theory to describe the gross features of soft nucleon-nucleon collisions, we compare in fig. 10 experimental and theoretical values of the mean transverse momentum

@r) as a function

of total centre-of-mass

energy &. Because the classical

total cross section diverges, it is necessary when calculating such average quantities to select a particular cutoff in impact parameter, or equivalently, a value of the total cross section, for which purpose we have used 42) atot = 40 mb. Our results calculated with the values of the constants listed in table 1 are given by the solid curve in fig. 10 and are seen to be both lower in magnitude and to increase more rapidly with \/s than the experimental

values 46*47 ). However, the theoretical

curve is calculated

for all values of

rapidity corresponding to a 40 mb total cross section, whereas the experimental results are peak values in the central region, where there is a short plateau in rapidity. Furthermore, the experimental values include high-h contributions arising from hard collisions. Because of the rough proportionality of (PT) to the vector interaction strength gy’, we show with the dotted curve in fig. 10 a simple estimate obtained by multiplying the solid curve by the ratio of g: = 17.26 obtained by Bryan and Scott 48) from an analysis of nucleon-nucleon scattering at laboratory kinetic energies between 0 and 350 MeV to the value appearing in table 1.

3. W. Bush, J.R. Nix I Classical hadr~d~n~mics

6ocl

5. Summary and conclusions In this Festschrift at five incident nudeons.

article we have presented

momenta

This theory, which corresponds

sive neutral

results for soft nucleon-nucleon

calculated on the basis of classical hadrodynamics to nucleons

of finite size interacting

scalar and vector meson fields, is the classical analogue

hadrodynamics approximation

collisions for extended with mas-

of the quantum

of Serot and Walecka but without the assumptions of the mean-field and of point nucleons. The theory is manifestly Lorentz-covariant and

automatic~ly includes space-time nonlocality and retardation, nonequilibrium phenomena, interactions among all nucieons and partide production when used for applications such as relativistic heavy-ion collisions. Unlike previous models, this approach satisfies a priori the basic conditions that are present at AGS, CERN and RHIC energies, namely an interaction time that is extremely short and a nucleon mean free path, force range and internucleon separation that are all comparable in size. From the limited comparison with experimental results that we have made in tig. IO, we saw that the theory is capable - for a suitable choice of vector interaction strength - of describing the magnitude of the mean transverse momentum (pi}, but that its dependence upon the total centre-of-mass energy ,/X is too rapid in this region. Because the coupling constants that we have used were determined for point nucleons, their use for extended nucleons should be regarded as for illustrative purposes only. The next step in the systematic development of the theory will be to determine a consistent set of coupling constants for extended nucleons. Then, detailed comparisons should be made between solutions of our newly derived equations of motion and a wide range of experimental data for proton-proton collisions, proton-nucleus collisions and nucleus-nucleus collisions. For nucleus-nucleus collisions up to about “j0 + I60 f we expect to be able to perform the calculations on desktop workstations; for collisions involving

heavier targets and projectiles

supercomputers

will probably be required.

The theoretical development of classical hadrodynamics should be continued in several other ways as well. Isospin effects should be incorporated by including fields for the isovector n and p mesons, which would add other physically important degrees of freedom. It may also be possible to include some quantum-mechanical

effects 38) by in-

corporating the effect of quanta1 fluctuations in the nucleon’s position on the nucleon structure function h (a). Excited nucleon states such as the d resonance should also be included in the formalism, although it is not yet clear how to do this in a covariant manner. Finally, quantum mechanics should be taken into account when the radiation is quantized to determine the number of mesons (primarily pions) created in such collisions. In conclusion, we would stress that classical hadrodynamics for extended nucleons avoids further assumptions such as the mean-field approximation or perturbative expansion in coupling strength. It therefore represents an exact solution to the classical many-body problem of extended particles interacting via meson fields. Because it treats the conventional hadronic degrees of freedom covariantly and exactly, classical hadrodynamics for extended nucleons promises to be extremely helpful in understanding the importance of these degrees of freedom at AGS, CERN and RHIC energies.

601

B. W. Bush, J.R. Nix / Classical hadrodynamics

We are grateful to R.J. Hughes and A.J. Sierk for their participation

during early phases

of this work. We would also like to thank M. Bolsterli, AS. Goldhaber, J.I. Kapusta, Steadman,

SE. Koonin,

P.W. Milonni,

W.J. Swiatecki and H.A. Weidenmiiller

was supported

by the US Department

T.J. Goldman,

M.M. Nieto, E.A. Remler, D.H. Sharp, S.G. for stimulating

discussions.

This work

of Energy.

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