Unravelling γγ dileptons in central relativistic heavy ion collisions

Z. Phys. C 60, 95-100 (1993) ZEITSCHRIFT FOR PHYSIK C �9 Springer-Verlag 1993

Unravelling dUeptons in central relativistic heavy N. Baron, G. Baur

Institut ffir Kernphysik (Theorie), Forschungszentrum Jfilich, D-52425 Jfilich, Germany

Received 12 March 1993; in revised form 5 May 1993

ion collisions

Abstract. Using an impact parameter formulation, differential production probabilities for 77 dileptons in central relativistic heavy ion collisions at RHIC and LHC energies are calculated and compared to Drell-Yan and thermal ones. The angular distributions of the leptons give a handle for the discrimination. Nuclear stopping leads, apart from Bremsstrahlung pair production, to a modification of the pure 77 dilepton production. These modifications are studied in a simple model and are found to be of minor influence.

1 Introduction

Dileptons are an important diagnostic tool for the study of highly excited matter which is formed in central relativistic heavy ion collisions [1, 2]. In the early stage of the collision dileptons are due to the Drell-Yan mechanism. Subsequently, there will be pre-equilibrium dileptons and finally, thermal dileptons from the quark-gluon-plasma. Dilepton production probabilities for these mechanisms were calculated in [3] and [4]. In addition to these mechanisms, there are dileptons from the collision of the two intense electromagnetic fields which accompany the relativistic heavy ions. So far, cross sections for the production of these ?:;~ dileptons have been calculated where attention has primarily been paid to dilepton center of mass variables. The order of magnitude of this effect in comparison to Drell-Yan or thermal pairs render it necessary to inves- tigate this mechanisms in more detail. We concentrate on differential cross sections and on possible effects due to the strong interactions in central collisions. Dilepton spectra have been calculated by many groups [5-10] where the emphasis has been put on cross sections integrated over the impact parameter. In a central collision we select b = 0. In any case b is smaller than the sum of the nuclear radii; therefore, due to the insensitivity of the cross section on b in this region (see Fig. 7 of [11]), this choice is appropriate. Thus it is convenient to use a semiclassical method to specify explicitly the impact parameter. We use the for-

malism of [11] and [12] where a factorization property was found. This greatly simplifies the numerical analysis. This formulation is readily extended to angular differential cross sections and evaluated in Sect. 2. Within the same method strong interaction effects can be included in a simple nuclear stopping model. The modifications of the dilepton production due to nuclear stopping are studied in Sect. 3. Our conclusions are given in Sect. 4.

2 Angular distributions of dilepton production probabilities

In [11] it was shown that for 77 produced dileptons in relativistic heavy ion collisions there are probabilities of a comparable magnitude to Drell-Yan or thermal ones over a wide range of invariant masses. From this point of view, 77 dileptons seem to be a hazard for the detection of dileptons originating from the different stages of the collision. On the other hand, the different creation mechanisms have different angular characteristics. Therefore it is appropriate to compare angular distributions which pro- vide information on the importance of different creation mechanisms in certain parts of the solid angle.

The differential creation probability is calculated fol- lowing the formalism of [11]. We restrict ourselves to central A-A collisions. At rapidity y = 0 of the pair the differential creation probability in the c.m.s, is given by:

dMdyd~2 J , = o - M dg2 S pdpN ~ , p , (2.1)

where da~7/df2 is the on-shell elementary 77 cross section in the c.m.s (see for example [13]). In principle, due to the off-shell character of the photons art has to be used, though, for large invariant masses a~y is a good approximation (see Fig. 4 of 1-11]). For smaller values of the invariant mass we use the off-shell correction function of [11]. The generalization to y#:0 could be obtained straightforwardly.

To deduce similar distributions for Drell-Yan and thermal pairs we use the results of [-4]. For fixed values of

96

M and y we have

d2p dap dMdy- ~ dO d M d y d O ' (2.2)

where the left-hand side is given in [-4]. For thermal pairs in relativistic heavy ion collisions the angular distribution is isotropic [21, while for Drell-Yan pairs it is given by [2, 14]:

dd~r~ (1 + O). (213) C O S 2

Using (2.2) we find

. d 3 p T u 1 rd2pTH 1 dM dydO ]M,y= Ld yJM,,

[ l 3 dMdydO M,r = 20re [_dMdYAM, y (1 + cosaO). (2.4)

In Figs. 1-3 we plot the differential creation probability d3p/dM dydO for dielectron production as a function of the angle O for y = 0 and fixed values of the invariant mass M in central 2~176 collisions for the different creation mechanisms. For reasons of symmetry we only plotted the range from 0 to 90 deg. For the form factor we use a gaussian with a width corresponding to Q0=60 MeV [15].

In Fig. 1 we set M=0 .5 GeV and 7 = 3400, appropriate for LHC. The full line corresponds to the y7 dileptons, while the dashed one shows the thermal production probability (for the Drell-Yan dileptons there were no calculations at this invariant mass). The extreme forward peak of the ?7 dielectrons is transparently seen from the sharp peak of about 4 orders of magnitude in the distribution at 0 deg. Already below 20 deg this contribution gets smaller than the thermal one. At around 90 deg the thermal contribution dominates the 77 one by two orders of magnitude.

In Fig. 2 the same quantity as in Fig. 1 is shown for an invariant mass of M=3 .5 GeV. The long dashed line shows the Drell-Yan contribution, the short dashed one

~ 102

10' I

"2- 1~176 I ._.

")' = 3400

M = 0 . 5 G e V

10 -~

C 10 -2

I0 -3 -o

10 -4.

~-~ 10-s 110 210 310 410 510 610 710 / 810

(~) (deg) 9 0

Fig. 1. Differential probabi l i ty for dielectron p roduc t ion as a func- t ion of the angle 0 for y = 0 and M = 0 . 5 GeV in central 2~176 collisions for L H C energies (7 = 3400). For the form factor we used a gaussian with a width cor responding to Qo = 60 MeV [15]. The 77 con t r ibu t ion (full line) is compared to the the rmal one (dashed line)

I 0 ~ ~ , , , , , , , ""C" O3 > 10 -~

(D 10 -2

I0 -3

C "~ I0-"

~ 10 -s

10 -6

"Y = 3400

M = 3.5 GeV

1 0-70 ll0 213 310 410 510 610 710 810 90

(~ (deg)

Fig. 2. Same quantity as in Fig. 1 but for M=3.5 GeV. The ~7 contribution (full line) is now compared to the Drell-Yan (long dashed line) and the thermal (short dashed line) contribution to the dielectron production

10-~ "Z"

> 10 -2

10-"

~10- '

C �9 "u I0 -s

I0 -6

~ - 10-7

" ~ 10-%

i i i i i i i i

")1 = 100

M = 3.5 GeV

. . . . . F . . . . . F . . . . . + . . . . . 4 . . . . . 4 . . . . . 4 . . . . . 4 . . . . . 4 . . . . .

t0 20 30 40 50 60 70 80 ~0

0 (deg)

Fig. 3. Same quant i ty as in Fig. 2 but for R H I C energies (7 = 3400). The labell ing of the lines is the same as in Fig. 2

the thermal one, while the full line again shows the 77 contribution to the dielectron production. Except for a sharper peaking resulting in smaller probabilities of one order of magnitude for larger angles, the behaviour of the 77 contribution is the same as in Fig. 1. The weak angular dependence of the Drell-Yan contribution leads to very similar values with the thermal contribution at angles around 90 deg. In a small range around O = 90 deg these dominate over the 77 one only by about a factor of two.

In Fig. 3 we again set M = 3.5 GeV, but now calculated for RHIC conditions, i.e. 7 = 100; the labelling of the lines is the same as in Fig. 2. The relation between the three contributions to the dielectron production has changed in such a way, that the Drell-Yan contribution now clearly dominates over the other ones at large angles. Between the Drell-Yan and the 77 contribution there is a factor of ten, while the thermal one is suppressed by another order of magnitude.

From Figs. 1-3 it is obvious, that a large part of the 77 dileptons is produced in forward (backward) direction in an angular range below 10 deg to the beam direction. The characteristic of the angular distributions should allow

the discrimination of Drell-Yan and thermal dileptons by simply measuring at 90 deg where the 77 probability has fallen off. For the thermal dileptons this is, in the low invariant mass region, a very good method since the 77 contribution is only about 1% of the thermal one. For the Drell-Yan pairs this is more difficult. For LHC conditions even at high invariant masses (up to 4 GeV) the 77 contribution is still in the order of 50% of the Drell-Yan one which may be a serious problem for unravelling. A discrimination may be possible at much higher energies, though, the overall magnitude of the probabilities is very small. Due to the suppression of the 77 production probability under RHIC conditions, such a discrimination seems to be possible in this case.

3 Modification of 77 dilepton production due to strong interactions

The electromagnetic fields accompanying the colliding nuclei are generated by their charge distributions. Due to the strong interaction there will be modifications of the charge and current distributions. These effects can become very important at e.g. SPS energies, where nuclear stopping prevails. In the present paper, we are mainly con- cerned with RHIC and LHC energies. After the collision, almost all the baryon numbers (and thus also the net electric charges) are in the forward and backward regions. Of course, due to the strong interaction, these charges are no longer contained in a volume as small as an atomic nucleus (see Fig. 4). However, we can assume that coherence still holds. This condition can be formulated as

1 Iql• Iq2•

7 (3.1) (D1,2 ~ , where R denotes the size of the system.

We find that coherence is maintained even for high equivalent photon energies; for ? = 3400 and R ~-7 fm we obtain from (3.1) a maximum energy (Dma x of equivalent photons of COmax ~ 100 GeV where coherence holds.

Without referring to the various elementary processes, after the collision, due to the strong interaction the velocity of the net charge distributions is smaller than the initial one. The simple effect of velocity reduction leads to a modification of the 77 dilepton production, but also to Bremsstrahlungs dilepton production in the stopping phase of the collision. This Bremsstrahlungs dilepton production was extensively studied by the Giel3en group 1-16]. We restrict therefore ourselves to the study of the modification of the 77 dilepton production due to the stopping in the collision of the two heavy ions (the nuclection of interference terms between the Bremsstrahlungs and the 77 dilepton production is surely a good approximation).

The modification of the Y? dilepton production in the framework of the equivalent photon method is due to the modification of the net nuclear charge current which determines completely the equivalent photon spectra.

In the semiclassical approach the equivalent photon spectrum of a charge distribution is generated by

before

97

z

" V X A Z V ' W

i i

after Fig. 4. Schematic picture of the relativistic heavy ion collision. Before the collision the two nuclei move with 71 along the z-axis acc6mpanied by equivalent photons wi th 4-momentum qi=(coj, q~• j= l ,2 , where [q~zl=~oj/vi. The net charge, in- dicated by the shade, is located in the small volume of the nuclei with a longitudinal width of Ri/7~. After the collision the baryon numbers (and thus the net charge distribution) move with yy in the same direction and therefore are still accompanied by equivalent photons; the longitudinal width is given by RH7 I. The central region repres- ents the excited vacuum

a Fourier transform with respect to time of the corresponding Lienard-Wiechard potential. We assume for sim- plicity that the size Rr of the net charge distribution after the collision is still more or less the same as before, i.e. R = Ri~-R I. Assuming a collision at t =0 the velocity of the heavy ions is shown in Fig. 5. For t ~ - oc the heavy ions move with constant velocity vi, after the collision this velocity is reduced to v I. During the collision time A t a deceleration takes place. This deceleration time is of the order of 10-23/7(s ) [16]. In contrast to the Bremsstrah- lungs dilepton production, where due to the sensitivity of the probabilities a deceleration parameter has to be introduced [16], for the equivalent photon spectrum it is suffi- cient to use a sudden deceleration at t = 0 because an integral over the very small deceleration time A t does not contribute significantly. The velocity of the heavy ions may therefore be written as

v(t) = v, 0 ( - t) + v I 0 ( t), (3.2)

where O(t) is the usual step function. The equivalent photon spectrum in the semiclassical approach is then generated by the field

E" , Z e p I ei~ t c~ ~.Ti-~i at[pZ+?iv~t2]3/2 eic~ }

+ (3.3)

Introducing variables

7 i / f l~i/f Z i / j - - t

P

cop xi/y = (3.4)

7 i l l 1Jill

the integration can be carried out leading to an equivalent

98

Vi

%

V

,, Vf

:< >- A t t

Fig. 5. Velocity profile of the colliding relativistic heavy ions. Ini- tially the ions move with velocity vl, during the collision phase A t the ions are decelerated and afterwards move with velocity v/. For the study of the modification of the Y7 dilepton production for small At the smooth deceleration may be replaced by a sudden one

photon spectrum of:

_ 1 Z2c~ 1 R(x~(p), x;(p), p ) - ~ ~ (P(xi, x/).

For convenience we defined

(3.5)

1 (P(x,, x~)=~ {[f,x,Iq(x,)+f~x~K:(xs)] ~

+~ [f,x,I:(xO-flx;I~ (xj.)

+ fy x/ L -1 (Xf ) - ~ x i L - : (xi)] 2 }, (3.6)

whereJ~/l = 1/fl~//and K1/I 1 are modified Bessel functions while L_ ~ is a modified Struve function.

We stress at this point that (3.5) is only valid for p > R. In the limit of no stopping x ~ x ~ we have (p(xi, x / )=r of [17]. Therefore we have

N(x,, x/) x l~x~ N(x = xi), (3.7)

as it should be. To get a measure of the strength of the stopping the

rapidity change 6 Y= Yi - Yy of the heavy ions is conveni- ently introduced. The rapidity of the heavy ions is given by

Y~//= In [7~/y (1 + v~//)]. (3.8)

Introducing the rapidity change 5Y of the heavy ions, we can replace x I by 6Y and write (p(x~, x l ) = (par(X~). The function (PaY characterizing the equivalent photon spectrum (3.5) is plotted in Fig. 6 and Fig. 7 for fixed cop as a function of 6Y for LHC and RHIC conditions, i.e. 7i=3400 and 7i= 100, respectively. We set cop= 10 and cop = 50 in Fig. 6 and Fig. 7, respectively. The possible values, of 5Y are restricted from 0 to 5Y= Yi.

In Fig. 6 for cop = 10 it is obvious that for SPS energies the stopping affects the equivalent photon spectrum already for very small rapidity losses. Losses of 1 to 2 units even reduce the characteristic function (Pay by more than 50%. The increase for very large rapidity changes is due to interference from the imaginary part of the integral (3.3). For RHIC energies the reduction of (Por sets in slowly; the strong reduction sets in above 5Y= 2. In the case of LHC the characteristic function stays nearly constant up to

1.20

.00

0.80

g 0,60

0.40

0.20

O.OC

i i i i i i i i

~ ")'i = 5400

oo

6Y

Fig. 6. Characteristic function <b~r of the equivalent photon spectrum as a function of the rapidity loss for cop = 10. The calculations are performed for SPS, RHIC and LHC energies. The possible values of c~Y are restricted from 0 to 5Y= Yi

1,20

1.00

0.80

0.60

0.40

0.20

0.00

i i i i i :

~ i= 3400

6Y

cop = 50

~ ~ ~o

Fig. 7. Same quantity as in Fig. 6 for cop = 50. Calculations are only shown for RHIC and LHC energies, since for SPS energies qS~y is supressed by 3 orders of magnitude for this value of cob

5Y= 5. For even larger rapidity changes, reduction also in this case sets in.

For cop = 50 the characteristic function for SPS energies is reduced by more than two orders of magnitude over the whole range of 5Y. Therefore Fig. 7 contains only curves for RHIC and LHC energies. The reduction for both sets in for smaller values of 5Y, though, for LHC there is still a large region of constant (bay.

Figure 6 and Fig. 7 clearly reflect that the modification due to the stopping in the collision phase measured by the rapidity loss of the heavy ions originates only from the second integral in (3.3). The equivalent photon spectrum as a function of the rapidity loss shows a sharp decrease around

cop

1 7 2 = X f ~ - 1. (3.9)

Qualitatively, we have: For x i< 1 the function (PaY is nearly constant below this critical value, while for xi > 1 the critical value is shifted in such a way that there is no constant region any more (see SPS case in Fig. 6), i.e. the exponential decrease of (PaY already sets in at 6 Y= 0. Also above the critical value of xy the function (Par stays nearly constant for x~_< 1. In this region the second integral in

(3 3) does not contribute. Therefore in the limit of &Y~ Y; which corresponds to x1~oo we have q~0r-~ 1/4 (which is due to the first integral in (3.3)).

This general property of the equivalent photon spectrum can also be used to discuss the question of coherence. Assuming a nearly constant velocity of the ions, i.e. vK -~ vl, the same value ofxz may be obtained by a spreading of the net charge distributions after the collision, i.e. R/>_R~. This can be interpreted as a reduction of coherence after the collision. From (3.3) we find that even if coherence after the collision is completely destroyed there remains the contribution originating from times before the collision, where coherence holds. In general an inclusion of the effect of spreading may be possible by introducing a time- dependent size of the system in addition to the time- dependent velocity.

Since for the calculation of the production probability the spectra have to be integrated up, the different reduction profiles for different cob interfere and produce a net effect on the probabilities. Since for our purpose form factor effects are expected to be small, we use (3.5) and introduce a sharp lower cut-off. We define the integral

_ ~ dp �9 or(co)- R P (co, p)' (3.1o)

where we explicitly denoted the dependence of co and p instead of x~ in {%y. For the lower cut-off R we use for the 2~ system R = 7 fro. From (2.1) we get

d2P(M, b =0)1 4n a~(M)Cb~y(M). (3.11)

For a constant value of the invariant mass the modification of the cross section due to the stopping may again be studied by simply looking at the function (boy which is dependent on the colliding nuclei via the lower cut-off radius R.

In [t8] rapidity losses for A-A collisions at RHIC and LHC energies are taken from values of p-A collisions:

[6 Y] RmC = [ 6 Y ] LBC --~ 2. (3.12)

Due to the still remaining uncertainty in the determina- tion of 6Yfor the calculation of the production probabilities we take into account also higher values of 6Y. In Fig. 8 {/'or for central 2~176 collisions at LHC energies is plotted as a function of the rapidity loss 6Y for different values of the invariant mass M. For invariant masses up to 3 GeV the reduction of the production probability is not significant for rapidity losses up to &Y= 2. To get sufficiently large effects due to the stopping for LHC energies we have to go to invariant masses well above 5 GeV. In Fig. 9 the same quantity is shown for RHIC energies at the same invariant masses. For invariant masses below 1 GeV the reductions are small while for higher invariant masses they become essential. These characteristics are due to the fact that the stopping reduc- es the maximum equivalent photon energy available in a collision at given initial beam energy, i.e. stopping affects mainly the high equivalent photon energy part of the spectra. For LHC invariant masses of several GeV are generated by equivalent photons with energies well below the maximum possible equivalent photon energy. Due to

99

0"20 l

T 0"15 I 0.10

>_.

).e.~ o.o~

o .oo

i l i

LHC: "Y~ = 3400

M = 0.5 GeV

.50 L 2.50 3 . 0 0 1.00 1.50 2.00 ~ ' 3.50 4 .00

6Y Fig. 8. ( ~ O y in (fn1-2) a s a function of the rapidity loss &Y for LHC energies. Calculations are performed for different values of the invariant mass M, For the cut-off radius we used for the 2~176 system R=7 fm

0.20

0.15

i i i / i

E ~ 0.10

~ 0 . 0 5

o.O%o

RHIC: ")/r = I00

M = 0,5 GeV

- M : I GeV

300

dY

Fig. 9. Same quantity as in Fig. 8 but for RHIC energies

the essential lower initial beam energy of RHIC for the generation of the same invariant mass equivalent photons of the high energy part of the spectrum are required.

4 Conclusions

We calculate angular distributions for 77 dilepton production in the impact parameter dependent equivalenl photon method for central relativistic heavy ion collisions. These are compared to angular distributions of lepton pairs due to the Drell-Yan mechanism and of thermal pairs to study the possibility of discrimination in such collisions. It is found that for small invariant masses a discrimination between thermal and 77 dileptons may simply be achieved by measuring around 90 deg to the beam direction where the 77 contribution is only about 1% of the thermal one. For the Drell-Yan dileptons the situation is more complicated; for LHC energies the differ- ence to the 77 contribution is only of the order of 50% at angles around 90 deg. Even at RHIC energies where the 77 production is strongly reduced it is still a 10% contribution in comparison to the Drell-Yan production. The question of coherence in the presence of strong interactions is discussed and found to hold up to high equivalent

100

pho ton energies. The effect of s trong interactions in central relativistic heavy ion collisions on the 77 dilepton rate is studied in a simple nuclear s topping model. This model characterizes the s topping by the rapidity loss 6Y of the beam. The modifications are manifest in the equivalent pho ton spectrum which may be calculated explicitly. It may be conceivable that the a rgument could be turned around: The modifications of the equivalent pho ton spec- t rum could be a handle on the t ime-development of the central collision process. Since the s topping mainly modifies the high equivalent pho ton energy part of the spectrum, modifications can only be expected for invariant masses of the order of the max imum equivalent pho ton energy available. For L H C energies where we take the rapidity loss to be of the order 6 Y = 2 the yy production probabil i ty is not affected significantly for invariant masses up to several GeV. For R H I C energies where the same s topping is taken we find essential reductions for invariant masses above 1 GeV.

Acknowledgement. One of the authors (N.B.) would like to thank the Studienstiftung des deutschen Volkes for the support of his studies.

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Unravelling γγ dileptons in central relativistic heavy ion collisions