NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 604 (1996) 455-465
Heavy-meson production cross sections from proton-proton collisions * A.A. Sibirtsev Joint Institute of Nuclear Research, 141980 Dubna, Russia
Received 26 June 1995; revised 23 January 1996
Abstract
The p-, to- and ~b-meson production cross sections in pp-interactions are calculated within the one-pion exchange model. We use the rrN amplitudes from the experimental data on pion-induced reactions, which are parametrized by simple functions. The model reasonably reproduces the available experimental data from proton-proton collisions. The results obtained from the one pion exchange model are fitted with scaling functions. New parametrizations of the elementary cross sections of p-, to- and qb-mesons in pp interactions are proposed which may be used for proton-nucleus and heavy ion studies.
1. I n t r o d u c t i o n
Heavy-meson production in p r o t o n - p r o t o n collisions is one of the most unstudied subjects in nuclear physics. The average age of the experimental results is around 20 years and all o f them were obtained by hydrogen bubble chambers and at high energies. The first precise experimental data on the reaction p p - ~ p p ' q were measured at Saturne at beam energies close to the threshold and were published in 1994 [1]. It was found that the experimental results strongly contradict available theoretical predictions. The calculations with the meson exchange model [ 2 - 4 ] substantially underestimate the experimental data. The most serious problems are concerned with the N * ( 1 5 3 5 ) resonance, for which mass and partial widths are not well defined [5,6], and the strong
* Supported in part by FZ Jiilich, Germany and RFFR, Russia.
0375-9474/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0375-9474(96)00102-9
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A.A. Sibirtsev / Nuclear Physics A 604 (1996) 455-465
"tIN final-state interaction [7]. Also the splitting between the contributions from the pion and the p meson exchange graphs were discussed controversially. The production of p-, to- and d~-mesons in proton-proton collisions so far have not been studied theoretically. Here we calculate the cross sections of vector-meson production at kinetic energies lower than 10 GeV. A further implication of our study is related to the properties of heavy mesons in hot and dense matter, which are presently the subject of intense investigations [8,9]. The modification of the masses and the widths of the vector mesons in nuclear matter may serve as a probe for the nuclear equation of state. Thus the heavy-meson production cross sections are needed in transport models especially for the simulation of heavy-ion collisions. In the following we account for the one-pion exchange graph only. Our idea is to compare the model results with available experimental data and to look for the room left for the contribution from heavier meson exchange graphs.
2. The model The pion exchange diagram for the reaction NN--> NNM is shown in Fig. 1, where M stands for the p-, to- or @meson. The relevant cross section for M-meson production in nucleon-nucleon interaction is given as [10-13]
fw kw
3m~ (r( UU ---~NNM; V~') = 21.r2p2is Wmin +
2dW
2
× f t f~NN F 2 ( t ) D 2 ( t ) t r ( T r N . . , M N ; ~tp2
W, t)t dt,
(1)
where ~ - and W are the invariant masses of the colliding nucleons and the produced meson-nucleon system, respectively. Obviously Wmin - - m M + m N and
Wmax = V~s - m N
(2)
with m s and m M denoting the masses of the nucleon and M-meson. In Eq. (1) t is the squared four-momentum transfered from the initial to the final nucleon and
t += 2m 2 - 2EiEf +_-2pipf,
(3)
where E i, pi are the energy and momentum of the initial nucleons in the center-of-mass frame. Ey, py are those for the final nucleons, while i~ and k denote the mass and momentum of the exchange pion. With the K~illen function
(x--y--z)2--4yz k ( x , y, z ) = one can define =
4x
'
(4)
A.A. Sibirtsev / Nuclear Physics A 604 (1996)455-465
457
M
Fig. 1. One-pion exchange diagram for the NN ~ NNM reaction.
# = X(s, W2, k2= X(W2, 4 ,
(5)
The pion propagator in free space is taken in the form 1
D(t) = t-
Ix2"
(6)
Furthermore the coupling constant was taken as f~UN = 1.0 [2--4]. TO account for the off-shell modification of the vertex we use the pion form factor as [14] A 2 _ i~2 F(t)
(7)
Az - t
with a cut-off parameter A = 1 GeV. In Eq. (1) cr(~rN--* MN; W, t) is the M-meson production cross sections in the "rrN interaction averaged over all possible members of the isospin multiplets of the initial and final channels. We neglect the t-dependence of the ~rN ~ MN cross section in calculating the total production cross section. For the neutral meson production in the pp-interaction we have to account for the ~r°p reaction channel. The isospin symmetry leads to the following relations between ~ 0p and the experimentally observed cross sections of to- and ~b-mesons: or(,n-Op --,, top) = 2or ' ( "n"- p
ton),
or(,rrOp _,, + p ) = 2¢r('rr-p l ~ ~bn)
(8)
Defining the amplitudes between the isospin decomposed components we get for p-meson production
ff(TrOp __, pOp) = ½[ tr( 7r-p ~ p - p ) + o'( rr +p ~ P + P) - or(,rr-p --* p ° n ) ] .
(9)
A.A. Sibirtsev / Nuclear Physics A 604 (1996) 455-465
458
c)
o)
_o
E b I
I
]-I
t0-"
\ 10-
i,
]-2
4
6
2
4
6
~
2
~-~
4
-~-
6
S ~/2 [ C e V ] Fig. 2. The cross sections for the reactions l r - p ~ p ° n (a), Ir-p--*p-p (b) and - r r + p ~ p + p Experimental points are from Ref. [15]. The solid lines show the parametrization (10).
(c)
Fig. 2a shows the cross section of the reaction r r - p ~ p°n [15] and our fit to the data by cr(~r-p ~ p p ) = 9 . 2 7 ( ~ - - s ~ 0 ) [ m b ] at ~ < 2 GeV, = 6 4 . 1 s - 2 l l [ m b ] at ~s ~> 2 GeV
(10)
with v~- being the invariant mass of the 7r-p system and s~0 =mp + m e, where mp and the masses of p-meson and proton, respectively. In Fig. 2b, 2c, we show the cross sections of the reactions ~r-p ~ p - p and 7r+p ~ p+p and the approximation (10) plotted with the same parameters. In Fig. 3 the dots show the experimental cross section of the reaction -rr-p ~ ton and the squares present those for the reaction "rr+n --* top [15]. The solid line shows our fit mp are
~(~r-p--*top) = 40.43(fs-s- s~0 ) [mb] at fs- < 1.8 GeV, =61.57s -2'55 [mb] at ~ >/ 1.8 GeV
(11)
A.A. Sibirtsev / Nuclear Physics A 604 ( 1996 ) 455-465
459
53
8 tD
lO
,\
'x,
10-
1 0 -3!
~
......
2
~
'
'
3
z.
5
6
'
~
7
8
s ~/2 [ C e V 21 Fig. 3. T h e c r o s s s e c t i o n s f o r the r e a c t i o n s ~ - p ~ e n ( d o t s ) a n d ~+ n --* o J p ( s q u a r e s ) . E x p e r i m e n t a l d a t a are f r o m Ref. [15]. T h e solid line is the p a r a m e t r i z a t i o n (11), w h i l e the d a s h e d line s h o w s the fit f r o m R e f . [16].
with v/s- being the invariant mass of the ¢r-p system and ~ o = m,,, + mp, where mo, stands for the o-meson mass. The dashed line is the parametrization of Cugnon et al. [16] cr('rr-p ~ t o n ) = 0 at p~ < 1.095 G e V / c , p~= 13.76
_3.33
p~
1.095 - 1.07
[mb] at p~ >/ 1.095 G e V / c ,
(12)
where p~ is the pion momentum in the laboratory system. The cross section of the reaction av-p---* ~bn is taken from the compilation of Flaminio et al. [15] and shown in Fig. 4. The solid line is our fit cr(~-p ~bp)
= 0.47(fs's - S~-o) [mb] at vCs-s< 2.05 GeV, = 23.7S -4"4 [mb] at ¢]-s >/2.05 GeV,
(13)
A.A. Sibirtsev /Nuclear Physics A 604 (1996) 455-465
46O
---} b
10
lO
-1
2
.3
4
s ~/2 [ G e V ] Fig. 4. The cross section for the reaction ~ - p ~ t k n the dashed line (14).
[15]. The solid line shows the parametrization (13) and
where fs- is the "rr-p invariant mass and s~0 = moo + rap, while moo stands for the qb-meson mass. The dashed line shows the parametrization from [17] 18~/s - - s o c r ( ~ - p ~ qb.) = 0.1285 + ( s - s0) 2 [Ixb]
(14)
with notations similar to (13) and s expressed in GeV 2.
3. Results
We calculate the cross sections of p-, to- and qb-meson production in proton-proton collisions with (1) corrected by isospin factors and compare them with available experimental data. In the present study we did not account for the contribution from the heavy-meson (-q, p, to, etc.) exchange graphs. As found by Laget et al. [3] and Moalem et al. [18] the contribution from p-exchange to xl-meson production in pp-collisions is
A.A. Sibirtsev / Nuclear Physics A 604 (1996) 455-465
S b
P+P--> P+P+P
461
0
10 2
JAA
10
"'"
J
1
•
Z"
•
*
•
•
jJ
10-
~-
•
•
-2
10 0.2
10 -1
1
10
10 2
S--So [ O e V 2] Fig. 5. The cross section for the reaction pp --* pppO. The dots are the experimental data from [15,24]. The squares are the results from the one-pion exchange model, while the solid line shows the function (15) with the parameters given in Table 1. The triangles show the results from the string model [21]. The dashed line is the parametrization (16).
about twice that from -tr-meson exchange. The contribution from "q and to-exchange is negligible because of the small coupling constants fnN~ and f~NN" In our calculations we account for the one-pion exchange only and look to find room for the contribution from other mesons. In Fig. 5 we show the cross section of the reaction pp ~ pppO as a function of s - s o with s being the squared invariant mass o f the initial protons and s o = (2rap + mp) °. The dots are the experimental data from [15] and the squares are the results o f the calculations with the pion exchange model. The experimental points available so far are reproduced rather well. The solid line shows a fit to the model results in the form
or( pp --* ppM) = a(1 - x) bxc,
(15)
where x = So/S and the parameters a, b and c as well as s o are given in Table 1.
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A.A. Sibirtsev / Nuclear Physics A 604 (1996) 4 5 5 - 4 6 5
Table 1 Parameters of the approximation (15) Reaction
s o [GeV 2]
a [mb]
b
c
p p ~ pppO p p --* ppto p p ~ ppqb
7.01 7.06 8.38
1.01 5.7 0.06
1.8 2.3 2.24
1.6 2.4 2.7
The dotted line shows the parametrization
B 2 + ( (~__~_s~_ , s/~_] 2 ,
~(pp~ppp)=
~v\,
(16)
Iv!
which quite reasonably describes the pseudoscalar meson (Tr and r I) production from pp-collisions [19]. Here the coefficients A = 240 i~b and B 2 = 1.4 GeV 2 were taken from [20].
10 3
S b
p+p
> p+p+co
10 2
/," •m 10
,.
•
1
-1 10
10-2
10 . 2
•
•
10 -~
1
10
10 2
s - s o [CeV 2] Fig. 6. The cross section for the reaction p p ~ pp~o. The dots show the experimental data [15,23], while the squares are the results calculated with the one-pion exchange model. The solid line shows the function (15) with parameters given in Table 1. The dashed line is the parametrization from [19], almost identical forms are used in Refs. [20,21].
A.A. Sibirtsev / Nuclear Physics A 604 (1996)455-465
463
10 .,.0
p+P
> p+P+¢ e
0
E i-
~~
A A
10-~ [._
~o• AA~
10
-3
, ~i 10 -2
t
10-1
: i i ,t,,J
i
1
, , i i,itl
10
i
, , , ,,,,I
10 2
s-So [OeV 2] Fig. 7. The cross section for the reaction pp -~ ppqb. Experimental dots are from Refs. [22,23]. The squares are the results from the one-pion exchange model calculated with (13) and triangles with (14). The solid line shows the function (15) with parameters given in Table 1.
The triangles are the results of the calculations with the string model performed by Cassing [21], which are in good agreement with the one-pion exchange model. The model results together with the experimental data on the reaction p p ~ p p t o are shown in Fig. 6. The solid line is the function (15) with parameters given in Table 1. The dotted line shows the parametrization (16) with A = 330 ixb and B E = 1.05 GeV 2 from Cassing [ 19], which have been used with tiny modifications also by Wolf [20] and Golubeva et al. [22] in proton- and heavy-ion induced reactions. At s - sth < 1 GeV 2 this parametrization drastically deviates from the calculations with the pion exchange model. The cross section of the reaction p p - ~ ppd? is shown in in Fig. 7. There are only two experimental points available at incident proton energies of 10 and 24 GeV [23,24]. The squares in Fig. 7 show the model calculations with the ¢rN ~ qbN cross section from (13); the solid line is the parametrization (15) with coefficients given in Table 1. The triangles show the results of the one-pion exchange model calculated with (14).
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A.A. Sibirtsev / Nuclear Physics A 604 (1996) 455-465
The difference between the model results obtained with (15) and (14) at s - Sth < 2 GeV 2, which corresponds to the incident proton kinetic energy of 2.59 < T, < 3.65 GeV, demonstrates a strong influence of the "rrN ~ ~bN amplitude introduced to the calculations. We hope that a more reliable rrN amplitude might be obtained from calculations within the resonance model by Tsushima et al. [25,26], which are in progress.
4. Summary Within the pion exchange model we have calculated the cross section for the reaction pp ~ p p M (for M = p0, to and ~b) at energies from threshold to s = 100 GeV 2. We used the experimental ~ N amplitudes parametrized by simple functions. The available experimental data are quite reasonably reproduced by the calculations by accounting for the one-pion exchange only. From the comparison with the experimental data we found almost no room for the contribution from heavy-meson (xI, p, to, etc.) exchange graphs. However, the very limited experimental data available so far in a rather small range of the beam energy do not constrain the model very much. To make more definite conclusions one needs further experimental data in a wider kinematical range. We notice that the scaling behaviour of the vector-meson production cross section drastically deviates from those for the pseudoscalar meson at energies close to threshold. The cross sections calculated within the one-pion exchange model have been parametrized by simple functions that can easily be implemented in transport simulations of proton-nucleus and heavy-ion collisions.
Acknowledgements The author gratefully acknowledges many helpful discussions with Professors W. Cassing and M.G. Sapozhnikov. I want to thank Professor O. Schult for the warm hospitality extended during various visits to the Forschungszentrum Jiilich. This work was partially supported by Forschungszentrum Jiilich and Russian Foundation for Fundamental Research under contract 93-02-3745.
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A.A. Sibirtsev / Nuclear Physics A 604 (1996) 455-465
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