Atomic Number Dependence of High Q² interactions in Nuclei

S. Frankel and W. Frati

Physics Department, University of Pennsylvania

Abstract

The time evolution of the internal structure of a composite proton colliding and passing through nuclear matter is contrasted with the different time evolution of scattered point-like constituents like the quarks and gluons involved in low Q² and high Q² interactions. This is examined in a review of Feynman x_Fdata on $J/\Psi$and Drell Yan production. We demonstrate analytically that S_p, the ratio of $J/\Psi$ production in nucleus(A)-nucleus(B) interactions relative to p-p production, cannot fall exactly exponentially with A^1/3 + B^1/3. Further, the shape of the falloff should be different for the ratios of the $J/\Psi$A-B cross section relative to the $J/\Psi$p-p cross section, S_p, compared with the ratio of the $J/\Psi$A-B cross section relative to the Drell-Yan dimuon background, S_dy. Using Woods-Saxon nuclear spatial distributions and a Monte Carlo simulation, to obtain distributions of the numbers of collisions prior and subsequent to the charmonium production in nuclei, we relate the mean number of nucleon-nucleon collisions, <n>, to A^1/3 + B^1/3, and give a new analytic functional form for the atomic number dependence of the $J/\Psi$ ``suppression''.

I. Time-Evolution

It is well known that, after a composite particle like a proton interacts with a nucleon in a nucleus, it is ``off-shell'' and that the final asymptotic ``on-shell'' state of the proton must take a time at least of the order of r_p/c to evolve, r_p being the proton size and c the velocity of light. Thus, immediately after such a collision the proton is not a ground state proton. However quarks and gluons are pointlike. Their states befare and after a collision differ only in that the momentum of the quark or gluon is changed in a collision, there being no internal quark or gluon structure that has to be reestablished. Because of this, soft low Q² collisions of protons in a nucleus, prior to the collision that results in a $J/\Psi$[#!def!#] or or Drell-Yan pair, can change the momentum of the constituents and affect the production of these high Q² reactions. This is a complex and interesting quantum-mechanical problem that is not easily understood quantitatively because perturbative QCD does not allow examination of the soft processes. However the data on these reactions, especially the Feynman x distributions, show that both cross sections are not independent of the soft initial state interactions.[#!qmatter!#] [#!upr!#]
In Fig. 1 we reproduce data on two representative cases of $J/\Psi$production in nuclei.[#!Katsenevas!#] [#!Alde1!#] The fits to the data are from ref. [#!absorption!#]. In Fig. 2 we reproduce some low energy data on the Feynman x distributions in pion-induced Drell-Yan pairs.[#!Bordalo!#] (It shows the experimental fall-off of the ratio of the Tungsten to Deuterium data with the invariant function, $\sqrt \tau = M/\sqrt s$, with M the mass of of the Drell-Yan pair and $\sqrt s$, the cm energy (in Gev), as well as our theoretical predictions of the fall-off as a function of $d\sqrt s/ dn$.)

In Fig. 3 we reproduce similar data in 800 GeV proton induced Drell-Yan pairs.[#!Alde!#] This case includes our calculations showing the expected limit at x_F= 1 which is model independent. That limit depends only on the Glauber coefficient for the probability that the dimuon is made in the first collision.

 

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Clearly the data show that the deviations from unity are much larger in the $J/\Psi$ as compared with the Drell-Yan distributions. However they are expected to be different for $J/\Psi$and Drell-Yan production for two reasons:

a) The Feynman x_F distributions are quite different in the two cases. The ratio of the p-A to p-p production in the $J/\Psi$case is known to fall rapidly as $\sim (1-x_F)^{5.2}$ while, in the Drell-Yan case, falls less rapidly as $\sim (1-x_F)^{2}$ and

b) There is no reason for the quarks and antiquarks producing Drell-Yan pairs to lose the same energy per collision as the gluons producing charmonium.

In our early work Ref. [#!dependence!#] studying the soft initial state energy loss of the incoming proton prior to $J/\Psi$or Drell-Yan production we observed empirically that the value of the loss per collision, $d\sqrt s/ dn$, was smaller, $\cong$ .2 -.4 GeV/collision, for Drell-Yan pairs. This latter estimate is very crude, but provided a rough fit to our analyses of various early experiments. In agreement with this, a theoretical estimate of the ratio of the quark and gluon soft energy losses was given, some time later,[#!Gavin!#] as about 4/9.

In fact, deviations from an A¹ dependence of the total Drell-Yan cross-section in p-A collisions are also shown in a review by C. Lourenço et al.[#!Lourenco!#], although study of the total cross section is very insensitive to the exponent of A.

Thus, there is no reason to neglect initial state interactions in the study of $J/\Psi$and Drell-Yan production in nuclei and it has been shown that Monte-Carlo calculations can reproduce the present data without need for any new physics.[#!jplettedsent!#]

The remainder of this work studies simultaneously the effects of initial and final state interactions. The formulas that are derived can be examined in the limit of no final state interactions (Drell-Yan) or no initial state interactions (photoproduction of $J/\Psi$).

II. Analytic Investigation of the Functional Form of the Atomic Number (A,B) dependence of $J/\Psi$ and Drell-Yan production

One can learn a great deal by first considering the simple case of the interactions of two lines of nucleons containing N and M nucleons, to examine how the production of high Q² particles varies with N and M. For completeness, both the effects of initial state energy loss and the disappearance of the $J/\Psi$into open-charmed particles in the final state are included. It is however trivial to examine the effects separately. Even photoproduction of the $J/\Psi$, where there are no initial state interactions of the photon, will be able to be read from the final formulas.

We shall see that the suppressions of the production in nuclei do not follow an exact exponential dependence on nuclear atomic numbers. Further we shall show analytically that the deviations from exponential behaviour are different when comparing the $J/\Psi$suppression relative to p-p data as opposed to comparing dimuon pairs from $J/\Psi$decay to the background of Drell-Yan dimuon pairs.

As shown in Fig. 4, N and M are the number of nucleons in each nucleus along a collision line and n and m are the nucleon positions in the line. In a nucleus-nucleus collision there will be a distribution of such line-on-line collisions and the final result will require averaging over the N + M distribution. We shall describe such Monte Carlo results later in this paper but the analytic calculation for the line-on-line expression for the $J/\Psi$suppression contains the basic information.

 

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We shall assume for this calculation that all the measurements are taken at the same projectile energy so that $e^{-\alpha}$ represents the reduction in yield per soft collision due to energy loss prior to the Drell-Yan or $J/\Psi$production and $e^{-\beta}$ represents the $J/\Psi$ absorption loss, i.e., disappearance into open charm, in a single $J/\Psi$-nucleon collision subsequent to the production. (No difference in any of the cross sections due to proton-neutron differences appears in this simple analysis, although the differences will affect yields at high A^1/3 + B^1/3.)

Consider the charmonium production in the scattering of nucleon n = 2 on nucleon m = 3 as shown in Fig. 4: n = 2 is slowed down by soft interactions with m = 1 and m = 2, while m = 3 has been slowed by interaction with n = 1. Our calculation sums over the n, m variables. Similarly, nucleons 3-6 in N and 4-7 in M interact with the produced charmonium, these collisions absorbing the charmonium by the breakup into free charmed particles, with an absorption cross-section, $\sigma_{oc}$. Because of the high Q² (low cross-section) nature of charmonium production, the number of $J/\Psi$'sproduced is proportional to NM, all nucleons being equally likely to interact to make a $J/\Psi$.

The energy loss effect comes from the strong §dependence of the $J/\Psi$cross section which is known to be reproduced by the single parameter expression:

1) $\sigma_{J/\Psi} = e^{-\gamma M/\sqrt s }$

with $\gamma$ = 14.5 and M = 3.1 GeV.

It is important to note that this expression is §dependent and that the measurements cover a wide variety of §.

Taking into account prior energy loss, this becomes:

2) $e^{-\gamma M/[\sqrt s_0 - n (d\sqrt {s}/dn) ]} = e^{-\frac{\gamma M}{\sqrt s_0} \times \frac{1}{ [1 -\frac{ d\sqrt {s}/dn}{\sqrt s_0} n ] }}$

s₀ is the initial energy, $d\sqrt s/ dn$is the energy loss per collision n is the number of soft, minimum bias type, collisions prior to the $J/\Psi$production. (Energy loss parameterizations of $d\sqrt s/ dn$, have been studied in low Q² nuclear interactions in Ref. [#!Predicting!#].)

Expanding the denominator, it can be approximated by $e^{-(\gamma M/\sqrt s) (1 + n d\sqrt s/dn //\sqrt{s_o})})$.

Thus we have the simple linear approximation:

2') $e^{-\alpha n} \cong e^{-\frac{\gamma M d\sqrt{s}/dn}{s_o}\times n }$

(Note that the energy loss effect depends on s₀ not $\sqrt s_0$ as in the cross section for $J/\Psi$production and that the lower the energy of the reaction, the larger the suppression, a point to consider when comparing data at different bombarding energies.

As we shall see, this is a fair approximation. We shall use it in our analytic studies. But we shall use the exact formulation of eq. 2 in our Monte Carlo calculations reported later in this paper.

For the initially interacting nucleons shown in Fig. 4, we have:

3) energy loss contribution: $e^{-\alpha[(n-1) + (m-1)] }$

Similarly we can parameterize the final state loss into open charm, where $\beta$ is proportional to the open charm cross section $\sigma_{oc}$. We assume, although it has never been experimentally determined, that $\sigma_{oc}$ is energy independent.

4) open charm contribution: $e^{-\beta[(N-n) +(M-m)]}$

This term can be separated as open charm: $e^{-\beta [(N-1) + (M-1)] } \times e^{+\beta [(n-1) + (m-1)]}$

5) Including both energy loss (eq. 3) and open charm (equ. 4) we obtain:

$S_p = e^{-\beta [(N-1) + (M-1)] } \times e^{-(\alpha -\beta) [(n-1) +(m-1)]}\times 1/NM$

S_p is defined as the ratio of the nucleus-nucleus to p-p cross-sections normalized by the factor NM.

Summing over the nucleons, this can be rewritten as:

6) $S_p = e^{-\beta (N+M)} \times \sum_1^N \sum_1^M e^{-(\alpha - \beta)(i-1) } e^{-(\alpha - \beta) (j-1) }$

Carrying out the sums and rearranging, we get the final result:

7) $S_p = [e^{- (\alpha + \beta) (N-1 + M-1)/ 2 }$] $\times$ $[ \frac {sinh (\alpha - \beta)N/2}{ N sinh (\alpha - \beta)/2}] [ \frac {sinh (\alpha - \beta)M/2}{ M sinh (\alpha - \beta)/2} ]$

By setting $\alpha$ or $\beta$ equal to zero one can get the expressions for both photoproduction of the $J/\Psi$and Drell-Yan production, relative to interactions with free protons.

(N-1 + M-1)/2 can be considered the mean number of prior and subsequent collisions in the row. (We shall see later how this is related to the mean number of collisions calculated from a Woods-Saxon distribution which we relate to A^1/3 + B^1/3).

The left-hand bracket in 7) represents exponential fall-off, but this dependence is modified by the right-hand bracketed term, denoted as C.

Since C increases with N and M, C is an enhancement factor as opposed to the exponential suppression factor it multiplies. It increases with N and M but inclusion of both initial and final state absorption enters as the difference between $\alpha$ and $\beta$ tending to make the deviation small for not too heavy nuclei. Note especially that, if initial state interactions were zero, so $\alpha = 0$, the enhancement effects would be larger. The tendency to cancel does not appear in photoproduction, since in that case $\alpha$ is zero.

A useful approximation for C is:

8) $C \cong [ 1 + (\alpha - \beta)^2 (N^2-1)/6 ] [ 1 + (\alpha - \beta)^2 (M^2-1)/6 ]$

Thus we have demonstrated that the cross section does indeed fall off exponentially with N + M except for the factor, C.

The enhancement effect of C is understood by recognizing that the interactions at the very front of the nuclei have only open charm absorption while the interactions at the very rear of the nuclei have only energy loss. However, at other positions the correlation effects appearing in the terms ($\alpha$ - $\beta$) are both present.

We now turn to a study of the ratio of $J/\Psi$production of muons to Drell-Yan production of muons. (Experimentally, the dimuons from both the Drell-Yan and the $J/\Psi$decays are detected simultaneously, so experimental errors are in common and the ratio should be more reliable than comparing nuclear to proton production, often in different experiments.) In this case there is no open charm contribution for the Drell-Yan production, only energy loss entering into the ratio, denoted as S_dy. The Drell-Yan production in nuclei can be obtained from equation 7) by setting $\beta$ equal to zero and replacing $\alpha$ by the Drell-Yan initial state parameter $\alpha^\prime$. Taking the ratio of ratios, S_dy, we have the final result:

9) $S_{dy} = e^{- (\alpha -\alpha^\prime + \beta )) (N-1 + M-1)/ 2 }$ $\times ~~ C_{dy}$, where

C_dy = $[ \frac {sinh (\alpha - \beta)N/2}{ N sinh (\alpha - \beta)/2}] [ \frac {sinh (\alpha - \beta)M/2}{ M sinh (\alpha - \beta)/2} ]$ $\left / [ \frac {sinh (\alpha^\prime )N/2} { N sinh (\alpha^\prime /2 )}] [ \frac {sinh (\alpha^\prime M/2} { M sinh (\alpha^\prime /2 )} ] \right.$

In this expression $\alpha^\prime$ is the Drell-Yan energy loss parameter which is smaller than that for $J/\Psi$production. [#!absorption!#] However, if $\alpha - \beta$ is smaller than $\alpha^\prime$ which appears in the denominator, one would expect that the exponential plot will turn downward at large values of N and M. In this case, C_dy would represent a depression at large N + M.

III. Relating the Line Formalism to Real Nuclei

The question we address here is how $\frac {(N-1) + (M-1)}{2}$ depends on the quantity K (A^1/3 + B^1/3 -2) since this is the variable that many workers use in plotting the suppression data. (We choose to keep the -2 in these expressions since, for p-p collisions, A^1/3 + B^1/3 = 2, so S is unity.)

It is clear that equation 7) would have to be integrated over the probability that one gets N and M nucleons in a line, averaged over the impact parameter in a nucleus-nucleus collision.

To answer these questions we do not need to go to a full scale Monte Carlo calculation of the $J/\Psi$suppression but first consider only the distribution of prior (nucleon-nucleon) and subsequent (charmonium-nucleon)collisions, n_prior = n_subsequent = n, before and after the $J/\Psi$production. It is important to emphasize that the struck nucleon cannot be counted in the initial state energy loss or in the final state absorption, as has been shown previously in the theory of color transparency in p-2p high p_tscatterings in nuclei. [#!physlett92!#] As we shall demonstrate, the determination of these basic parameters allows one to study another (linear and not exponential) way to plot the suppression data.

These collision parameters are needed to examine eq. 7. We obtained these distributions from a full Monte Carlo, using Woods-Saxon representation of the spatial distribution of nucleons within the nucleus. [#!privatecomm!#]

The averages for p-A and A-A interactions are shown in Fig. 5. In this figure n_A + n_B = <n>, is the sum of the prior scatters in both nuclei, A and B.. (For deuterium and helium we have used the best spatial distributions, as was done in our search for deconfinement in p, d, and $\alpha$ experiments at the ISR [#!akesson!#]. This shows up in the slight curvatures at very low A.)

 

Figure:

The calculations show that the slope of <n> vs (A^1/3 + B^1/3) is essentially the same for p-A and A-A interactions, and a little reflection will allow the reader to realize that this is what is to be expected. We can then ask what value of K is needed to calculate <n>. What we find from the slopes in Fig. 5 is that K $\cong .46$. The fits to a straight line, <n> = a +b( A^1/3 + B^1/3-2), are given in Table 1.

	b	a
p-A average (prior)	.46	.05
A-A average (prior)	.46	.00
p-A central (prior)	.63	.00
A-A central (prior)	.51	-.10

We also note from the curves for central, i. e., zero impact parameter, collisions, Fig 6, that for a central AB collision the values of <n> are somewhat different than those of an AA collision with the same (A^1/3 + B^1/3) value. So care must be taken in plotting data with A $\neq$ B.

However, as expected, A-B and A-A average collisions should fall on the A-A curve at the appropriate A^1/3 + B^1/3.

 

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One might guess that a good approximation for the averaging over all n might be to replace (N -1)/2 by <n> in eq. 7. Instead, we will carry out the averaging to test this hypothesis and then turn to an alternate functional form for the depression, S_p.

If one examines our Monte Carlo distribution of prior collisions, n, when the average number is <n>, one observes that the normalized distribution is given approximately [#!tabulation!#] by:

11) F(n) = e^-n/<n> / <n>

Therefore we can average S(n) over the F(n) distribution to obtain S(<n>)

Table 1.

12) $S_p(<n>) = \int e^{-\alpha n } e^{-n/<n>} dn / <n>$

This is just

13) S(<n>) = $\frac{1}{ [\alpha + (1/<n>)] <n>} = \frac{1}{1 + \alpha <n>}$

Thus we have arrived at a new extrapolation formula:

14) $1/S(<n>) = 1 + \alpha <n>$

 

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To verify this formula, we can plot 1/S_p(<n>) vs $\alpha$, which should then be a straight line whose slope is the mean number of collisions. To do this we have used our complete Monte Carlo calculation for $S_p(\alpha)$ for average Pb-Pb collisions to determine the validity of the approximate calculation of eq. 14. The complete Monte Carlo determines the suppression for each nucleon in each of the nuclei and finds the total suppression. Our approximate expression uses the mwan number of collisions from eq. 11 to calculate the suppression directly.

We have chosen $\alpha$'s corresponding with hypothetical absorption cross-sections, $\sigma_{oc}$, from 4.08 to 9.4 mb.

Fig 7 shows this plot. It is indeed linear with a slope of 0.44. which is, within error, the same as the value determined from the Monte Carlo calculation of <n> for a Pb-Pb collision. The use of an exponential shape of the n distribution is apparently a good approximation.

Further, we appear to have a simpler form than the distorted exponential form of equation 7) and it is actually based on real values of average collisions obtained from Woods-Saxon nuclear distributions.

As a second check, we can examine, for fixed $\alpha$, how the Monte Carlo calculation of <n>, the number of mean prior scatterings, and the approximate calculation of <n> from the approximate functional form of equation 12) that led to equation 14) each depend on A^1/3 + B^1/3 - 2 for p-B collisions. The exact value of S_p is used to calculate n_app. This plot is presented in Fig. 8 showing that both methods appear to agree for a range of B's from C to U.

It is very useful, therefore, to note that one can use the analytical expression of equation 14) and the parameter K = .46 to calculate effects of absorption whether they are due to disappearance of the $J/\Psi$or due to decrease in yield as the result of prior energy loss. Thus we note that an effective L for plotting the suppression should be L_eff = .46 (A^1/3 + B^1/3 - 2)

IV. Conclusions

We have shown, from simple analytic calculations, that the $J/\Psi$cross sections in nuclei, at fixed bombarding energy, cannot be exactly exponential in A^1/3 + B^1/3and that, therefore, the deviations in such a plot from a pure exponential shape cannot be attributed to formation of a quark-gluon plasma. Further, we show, from study of the Monte Carlo parameters obtained from Woods-Saxon density distributions, that an alternate plot, based on our Monte Carlo calculation of collision probabilities, would be the linear expression, 1/S_p = 1 + J [0.46 (A^1/3 + B^1/3 -2)], where J is a constant depending on the initial state and/or the final state parameters, $\alpha$ and $\beta$. Further, we remark that for central collisions, which might be selected with an E_t trigger, A-B collisions and A-A collisions of the same L_eff will not have the same suppressions.

Figures

Fig. 1 Measured Feynman x distributions, for the nucleus to proton target ratio, R, of $J/\Psi$production. (Also shown are fits to the data, including a crude estimate of the $J/\Psi$- nucleon inelastic cross sectiom in the final state taken from Reference [#!absorption!#].

Fig. 2 Feynman x distributions for Drell-Yan pion induced reactions.[#!Bordalo!#]

Fig. 3 Feynman x distributions for Drell-Yan proton induced reactions.[#!Alde!#]

Fig. 4 Schematic of interactions of nucleons in a line: representation of the collision of a line of N nucleons with a line of M nucleons.

Fig. 5 Results of Monte Carlo Woods-Saxon calculation of the mean number of collisions prior to (or succeeding) an interaction producing charmonium in p-A and A-A average collisions.

Fig. 6 Results of Monte Carlo Woods-Saxon calculation of the mean number of collisions prior to (or succeeding) an interaction producing charmonium in A-A central collisions of zero impact parameter as well as for average A-A collisions. Several A-B collisions show departures from the straight line slopes, but only for central collisions.

Fig. 7 Plot of $1/S_p = \alpha + <n>$ from the exact full Monte-Carlo calculation, using the values of <n> for Pb-Pb and varying $\sigma_{oc}$from 4.1 to 9.4 mb. This plot is used to demonstrate the validity of this new equation which has been derived using the n distribution of equation 11.

Fig. 8 Comparison of the A^1/3 + B^1/3dependence of the mean number of scatters <n>_mc, taken from the full exact Monte Carlo calculation of S_p with the value, <n>_approx, using the derived formula, $1/S_p = \alpha + <n>_{approx}$, for the case $\sigma_{oc}$= 7.4 mb. The x's are the exact Monte Carlo points and the dots are the points for the new functional form of eq. 14)