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Anomalous electron-pair production in π−p collisions
Volume 237, number 2
PHYSICS LETTERS B
15 March 1990
A N O M A L O U S E L E C T R O N - P A I R P R O D U C T I O N IN n - p C O L L I S I O N S C. B O T T C H E R , M.R. STRAYER Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA C.J. ALBERT and D.J. E R N S T Center for Theoretical Physics and Physics Department, Texas A&M University, College Station, TX 77843, USA Received 6 November 1989; revised manuscript received 2 January. 1990
Anomalous e+e - pairs with invariant mass 0.2 ~
The emission o f e l e c t r o n - p o s i t r o n pairs has been observed [ 1,2] in ~ - p collisions with corresponding pion m o m e n t a o f 16 and 17 G e V / c . The experiments detected pairs over a limited range o f invariant pair mass, M - [ ( E I + E 2 ) 2 - ( R ~ + p 2 ) 2 ] I/2, given by 0 . 2 < M < 1 . 2 G e V / c 2. A large peak near M = 0 . 8 G e V / c 2 was observed. This peak is well understood as resulting from the direct electromagnetic decay of the p and to mesons, p--,e+e - , and to--,e+e - . A continuum of pairs is also seen at lower invariant masses. These pairs were d e e m e d a n o m a l o u s as conventional hadronic mechanisms failed to explain the data. A m o n g these possible mechanisms are the Dalitz decay o f the q and to, q--,),e+ e - and to--,~°e+e-. This process can be reliably calculated from the measured meson production cross section and the known branching ratio for Dalitz decay. However, it only produces a small fraction o f the observed cross section. The Dalitz decays together with the direct decay into leptons o f the p, to and rl are subtracted from the measured data, and the remaining events are termed a n o m a l o u s pairs. A n o t h e r mechanism which produces a continuous spectrum o f pairs is the D r e l i Yan process. For values o f the pair mass much larger than 0.8 G e V / c 2, this process usually agrees with experimental data [3,4]. However, for the low mass
pairs [ 1,2,4] observed in the experiments o f refs. [ 1,2 ], the Drcll-Yan results are more than an order o f magnitude below the data. Several exotic mechanisms have been proposed [ 5 - 8 ] , but none o f these have proven satisfactory. We d e m o n s t r a t e here that an ab initio calculation o f the two-photon mechanism produces results that are in quantitative agreement with the data. The two-photon mechanism for electromagnetic lepton-pair production was first investigated [9,10] in the 1930's. In ref. [ 10], the equivalent photon, or Weizs~icker-Williams, a p p r o x i m a t i o n was developed. This a p p r o x i m a t i o n replaces the electromagnetic fields by a spectrum of photons, which then directly create the pairs. Two-photon physics, within this a p p r o x i m a t i o n , has received considerable attention [ 11 ] of late. In particular, it has been suggested that this mechanism could produce large yields of electrons, muons, and tauons in relativistic heavy-ion collisions [12], and that the electrons and muons could mask signals originating from hadronic collisions. The closely related mechanism o f electron-pair production and capture [13] represents a serious beam loss mechanism in relativistic collider physics with heavy ions. The experimental results o f refs. [ 1,2 ] are shown
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
17 5
Volume 237. number 2
PHYSICS LETTERS B
in figs. 1,2, and 3. The qualitative features ofthc cross sections are clear. The cross section decreases with increasing pair mass, decreases with increasing Feynman x (defined as the total longitudinal momentum of the pair divided by the maximum value allowcd kinematically), and decreases with increasing transverse momentum. We calculate the two-photon process in the limit where the relative motion of the pion and proton is treated classically [ 14 ]. Using this approximation, the total cross section for electron-positron-pair production is given by
a=fd:bZ
-
d3k ' (2~t5~ ~ f
d3k I ( k ' ° " ISnlk¢~) I ~
•
(~)
15 March 1990
=-i
(T~).Z
~(k')
X [flA6 + ~(b: k' - p ) - a t ' A C+ ~(b: k' - p ) ] X(
~1
~ z '°' (,,,)z~,,,,(p)
+ - - - -1 - ~~" q/~." ~(p)g/,~ ..~(p) ) Po + Ep '. X [flA~-)(b:k+p)+at'.4~-'(b:k+p)]~°~(k).
(4) The most general form of the electromagnetic current of the proton is
dU(p'a', po) = e a ' " ~(p' ) [ 7"F, (p' - p ) ]u t") (p) e
a(" ~(p') [,~"~(p~,+p.)x&(p' -p) ]
where b is the impact parameter for the classical trajectory of the relative pion-proton motion. For a 17 GeV/c pion. we may use a straight-line trajectory,. The scattering amplitude has two terms corresponding to crossed and uncrossed photon lines: further details are given in ref. [ 15 ]. The contribution to the amplitude from the uncrossed diagrams is given by
By inserting explicit expressions for the proton spinors in cq. (5) and multiplying by the photon propagator, the vector potential in eq. (4) becomes
( k a ' ISb I k o ) =
A~,+ '(b: q) --a,,~ ~ )(b:q)+ Sa~,+ )(b:q) ,
In
- i ,,/E ~
f
d4p
~-~_~.~t~
ca)
× u~°~(p).
(2)
at/+ ~(b:q) =2rwd(qo-flq:)'/
x where the notation and normalization are those of ref. [ 16 ]. Here .4 ~,+ ~(q) and A J,- ) (q). which appear as prescribed external fields, are given by the photon propagator times the matrix element of the proton or pion current. Eq. (2) can be translated into the notation of ref. [ 15] by utilizing ,~+m
1
m
p e - m ' - - Po -15o F.p ~ u ' " ( p ) a ' " ' ( p ) 1
prl
+ --~-k'p/-:~ Z z " " (p)v~°'(p) , Po
"7
176
exp(-iq.b/2)
- q~ +,/q2_
a~,- ~ ' ( b : q ) = 0 ,
Gr~(q 2 )
,
al.+~(b:q)=O,
a~ + )(b:q)=fla~ + '(b:q) .
(6)
The electric form factor of the proton, GE (q2), is here taken to have a dipole form ( l + q - / A p ) - - . with Ap=0.71 GeV 2 [17]. The second term in A~,+~(q), which involves both k'~ (q) and l.~(q), corresponds to events in which the proton spin flips. This term yields
(3)
and. in addition, by replacing the invariantly normed spinors by Z ~ ( p ) = (m/Ep)~/:u<~'~(p) and ~,{,~(p) : (IFI/Ep)I/2l'~") (p),
(5)
where the term which corresponds to non spin-flip events is
(k')a'+'(bk'-p)
× p-YLm~4~-~(b:p+k)r~O~(k) •
2 :~Ip
6a{+)(b:q)=
2n
e
2 :t//p 6(qo-flq:)
X e x p ( - i q . b / 2 ) q: G M(q-) ~ , q~+ },-q± , 2
(7)
Volume 237, number 2
PHYSICS LETTERS B e
aal. + ' ( b : q ) = 2rt ~M-~o~(qo -flq:) x exp(-iq.b/2) ~ + 72q2_
iq:GM(q2),
6a~ + )(b:q) = - 2 n ~M~ 6(qo -flq:)7 X
eXpql-iq'b/2) + 72q2L
(qx+iqy)GM(q2) ,
5a~ + )(b:q) =fl3a~ + )(b:q) ,
( 7 cont'd)
where the magnetic form factor of the proton, GM(q2), is normalized so that G M ( 0 ) = I + K = 2Mo14,/e, where/t o is the magnetic moment of the proton. We use the same dipole form for the magnetic form factor, GM (q2) = ( 1 + X) ( 1 + q2/A~) - 2, as for GE (q2). For the spin-zero pion the electromagnetic current is given simply by
jU(p, p)=e(p,,+pU)GE( (p'_p)2) ,
(8)
which produces similar results forA ~- ) (q) as is given in eq. (6); in the center-of-velocity frame, A J,-)(q) is simply given by eq. (6) with f l - , - f l and b-, - b . We utilize a monopole form for the electric form factor of the pion, G~.(q2)=(l+q2/A~) -I, where A~ = 0 . 5 9 GeV 2 [ 17 ]. The expressions derived here for the total cross section is Lorentz invariant and is most casily evaluated in the center-of-velocity frame. Finally, the crossed photon term, which contributes to eq. (2), is added coherently to the direct term. In summary, we take the classical limit on the relative motion of the proton and pion of standard QED perturbation theory for the two-photon process. The arguments of ref. [ 14] indicate that the wavelength of the relative pion-nucleon motion is sufficiently large and the electromagnetic field sufficiently smooth that we are well into the classical region. In this region, the matrix element of the current which couples the proton or the pion to the electromagnetic field multiplied by the photon propagator takes the form of prescribed electromagnetic fields that have becn boosted from the rest frame to a frame with velocity - f t . as given in cq. (6), non spin-flip, and eq. (7), spin-flip. A different derivation of the non spin-flip amplitude can be found in ref. [15]. In ref. [ 15], identical numerical results were obtained both in the
15 March 1990
Lorentz gauge (as is used here) and when the calculation was repeated in the temporal gauge. The eight-dimensional integration required to calculate the total cross section in eq. ( 1 ) is done utilizing the Monte Carlo tcchnique described in ref. [ 15 ]. We impose on the Monte Carlo integration the experimental cut on the pair mass, 0.2~
....
,
....
,
....
/. . . . . . .
!01
:::L
, ....
, ....
Spin-Flip Non Spin-Hip
/
,0 o
I
.
EL_
c~______2~ ~°f
X "(3
"'"''"'-.......
1021 10 .3
13 .4/ 30
02
04
06
08
X
Fig. 1. The differential cross section for electron-positron-pair production versus Feynman x. The dashed curve is the result omitting the spin-flip terms. The data labeled a) are from ref. [1 ], those labeled b) are set "'PAIR A" from ref. [2], and those labeled c), are set "'PAIR B" from ref. [2]. 177
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Volume 237, number 2
lower cut on the invariant mass at exactly 0.2 G e V / c 2. If we decrease this value slightly, the cross sections at low x and low p~ are enhanced. The difficulty might also lie in our neglect of the final-state distortions o f the electron and positron. The cross sections for the slower-moving leptons might well be somewhat higher if distorted waves were used. The total cross section for electromagnetic pair production is about 55 ~tb, o f which only 0.5 ~tb lies within
10 2
S p m . t q l F' " - - "m""
Non £pm.Hip
10'
Ref 1
> (.9
10 0
.,0 ::L
10
1D
10:
10 3
....
oo
, . . . . . . . . . . . . . .
o~
02
,
03 M
....
i ....
04
os
, ....
06
07
(GeV/c21
Fig. 2. The differential cross section for electron-positron-pair production versus the pair mass. The dashed curve is the result omitting the spin-flip terms. The data are from ret: [ 1]. . . . .
10
,
....
, ....
,
....
,
. . . .
, . .
.
Spin-Hip •
--'~.-'"
Nk;fnlSpin-Flip
10 ~
10 0
10.1
13..
10-2
"~
10. 3
10 4 0.0
, , . . , .... 0.1
m .... 0.2
I .... 0.3
t .... 0.4
, .... 0.5
0.6
[GeV/c] 2
Fig. 3. The differential cross section for electron-positron-pair production versus the transverse momentum of the pair. The dashed curve is the result omitting the spin-flip terms. The data are from ref. [ I 1. The results for electron-positron-pair production including the magnetic spin-flip term, eq. (7), are also shown in figs. 1-3 as the solid curve. We see that the results are in reasonable agreement with the data, The calculation is m a d e ab initio with no adjustable parameters. Although the overall agreement is good, the cross sections in fig. 1 lie a little below the data, for low F e y n m a n x. This may be due to our use o f the 178
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the cut. T h e spin-flip term, eq. ( 7 ) , is of order q/Mp, as c o m p a r e d to the leading term, eq. (6). The reduction due to this factor is more than c o m p e n s a t e d for by two features of the spin-flip current: ( 1 ) the current has transverse components, and ( 2 ) the 0 and z c o m p o n e n t s are proportional to the transverse mom e n t u m o f the photon. Thus the spin-flip current produces pairs at the large values o f transverse mom e n t u m o f the photon. Thus the spin-flip current produces pairs at the large values o f transverse mom e n t u m emphasized by the experimental cut on the invariant mass. We conclude from the general quality o f the agreemerit of the calculation with the data that the lepton pairs observed in refs. [ 1,2 ] are produced, to a large extent, via the two-photon mechanism, specifically by the magnetic part o f the proton current. The calculation itself allows some internal consistency checks on this view o f the physics. The presence o f the proton and pion form factors reduces the predicted cross sections by less than a percent. This indicates that the cross section arises from a region where the scale for the m o m e n t u m or energy o f the virtual photon is given by several h u n d r e d MeV or less. At these mom e n t u m transfers, the d o m i n a n t response o f the proton or the pion is the coherent elastic response, although the excitation of the nucleon to a delta, or the pion to a rho might also contribute a non negligible a m o u n t in the kinematic region. We note that the coincidence data given in ref. [ 2 ] is consistent with this low multiplicity interpretation o f the data. This relatively low m o m e n t u m transfer from the hadrons does not justify a quark level calculation o f the two-photon process as was done in ref. [ 2 ] and we point out that the independent quark level processes are suppressed by a factor o f Z~Z~, with Z, the charge o f the quark, and thus can only become the d o m i n a n t term in kinematic regions where the elastic form factor
Volume 237, number 2
PHYSICS LETTERS B
suppresses the coherent elastic term. In our calculation we integrate over impact parameters starting at zero. For impact parameters less than one fermi the pion and proton will undergo strong interactions which would invalidate the pure electromagnetic calculation done here. However, we find that non negligible n u m b e r s of lepton pairs are emitted out to an impact parameter > 100 fm. Thus even the extreme measure of neglecting the contribution which arises from impact parameters below 1 fm would have little effect on our results. Lepton-pair production in particle collisions is important for several reasons. The two-photon process can produce large numbers [ 12,15 ] of soft pairs. Here we have found that it can also produce pairs with significant transverse m o m e n t u m . Such pairs would provide a background for the lepton pairs emitted in heavy-ion collisions. If the lepton pairs are to serve as a probe of the q u a r k - g l u o n plasma, this background must first be understood. Also, the Lorentz contracted fields represented by the vector potentials o f e q . (6) and eq. (7) become very large for the ultrarelativistic heavy ions that will be produced at R H I C and perhaps at the SSC. In this case, the perturbative approach utilized here will certainly be inadequate. Lepton production would then offer the opportunity to study QED in a region where it is not perturbative. The present results are needed as a baseline for this study. There exists two possibly related experiments [ 18,19] in which electron-positron pairs have been observed in collisions of a proton with 9Be. Because ~Bc is a composite nucleus, the dynamics are more complicated than those of the rt-p collisions considered here. For the 9Be data, we are examining the role played by the two-photon mechanism and will present results shortly. The work of D.J.E. was supported, in part, by Oak Ridge Associated Universities. The work of D.J.E. and C.J.A. was supported, in part, by the National Science Foundation. One of us, D.J.E., would like to thank the Physics Division of Oak Ridge National Laboratory for its hospitality during part of this work. This research was sponsored by the Division of Nuclear Physics of the US D e p a r t m e n t of Energy under contract No. DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc., and by the Division
15 March 1990
of Chemical Sciences, Office of Basic Energy Sciences.
References [I]R. Slroynowski, D. Blockus, W. Dunwoodie, D.W.G.S. Leith, M. Marshall, C.L. Woody, B. Barnett, C.Y. Chien, T. Fiefuth, M. Gilchriese, D. Hutchinson, W.B. Johnson, P. Kunz, T. Lasinski, L. Madansky. W.T. Meyer, A. Pevsner, B. Ratcliff, P. Schacht, J. Scheid, S. Shapiro and S. Williams, Phys. Lett. B 97 (1980) 315; D. Blockus, W. Dunwoodie, D.W.G.S. Leith, M. Marshall, R. Stroynowski, C.L. Woody, B. Barnett. C.Y. Chien, T. Fieguth, M. Gilchriese, D. Hutchinson, W.B. Johnson, P. Kunz, T. Lasinski,L. Mandansky,W.T. Meyer, A. Pevsner, B. Ratcliff, P. Schacht, J. Sheid, S. Shapiro and S. Williams, Nucl. Phys. B 201 (1982) 205. [2] J. Stekas, G. Abshire, M.R. Adams, C. Brown, L. Cormell, E. Crandall, G.J. Donaldson, J. Goldberger, H.A. Gordon, P.D. Grannis, B.T. Meadows, G.R. Morris and P. Rehak, Phys. Rev. Lett. 47 (1981) 1686" M.R. Adams, G. Abshire, ('. Brown, E.S. Crandall, J. Goldberger, P.D. Grannis, B.T. Meadows, J. Stekas. G.J. Donaldson, H.A. Gordon. G.R. Morris, P. Rehak and L. Cormell, Phys. Rev. D 27 (1983) 1977. [3] C.B. Newman, K.J. Anderson, R.N. Coleman, G.E. Hogan, K.P. Karhi, K.T. McDonald, J.E. Pilcher, E.I. Rosenberg, G.H. Sanders, A.J.S. Smith and J.J. Thaler, Phys. Rev. Lett. 42 (1979) 951. [4] J.D. Bjorkenand H. Weisberg,Phys. Rev. D 13 ( 1976) 1405. [ 5 ] V. (~erny,P. Lichard and J. Pi,~fit,Phys. Lett. B 70 ( 1977 ) 61;Acta Phys. Pol. B9 (1978) 901;B 10 (1979) 537. [6 ] G.R. Farrar and S.C. Fraulschi, Phys. Rev. Lett. 36 ( 1976 ) 1017; R. Ruckl, Phys. Len. B 64 ( 1976 ) 39; N.S. Craigieand H.N. Thompson, Nucl. Phys. B 141 (1978) 121. [ 7 ] E.L. Feinberg, Nuovo Cimento 34A ( 1976 ) 391 ; E.V. Shuryak, Phys. Len. B 78 (1978) 150. [8] T. Goldman, M. Duong-Vanand R. Blankenbecler.Phys. Rev. D20 (1979) 619. [9 ] W.H. Furry and J.F. Carlson, Phys. Rev. 44 ( 1933 ) 237: W. Heitlerand L. Nordheim,J. Phys. (Paris) 5 (1934) 449; Y. Nishina, S. Tomonaga and M. Kobayasi, Sci. Pap. Inst. Poincare, C. R. 27 (1935) 137; E.J. Williams,Nature 135 (1935) 66; H.J. Bhabha, Proc. R. Soc. A 152 (1935) 559. [ 10] C.F. von Wcizsacker,Z. Phys. 88 (1934) 612; E.J. Williams, Phys. Rev. 45 (1934) 729; J.R. Oppenheimer, Phys. Rev. 47 (1935) 146; L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 6 (1934) 244. [ 11 ] F.E. Low, Phys. Rev. 120 (1960) 582; R.H. Dalitz and D.R. Yennie,Phys. Rev. 105 (1957) 1598; S.J. Brodsky, T. K.inoshitaand H. Terazawa, Phys. Rev. D 4(1971) 1532; 179
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D.L. Olsen, B.L. Berman, D.E. Greiner, H.H. Heckman, P.J. Linstrom, G.D. Westfall and H.J. Crawford, Phys. Rev. C 24(1981) 1529; A. Goldberg, Nucl. Phys. A 240 (1984) 636; R. Anhoh and H. Gould, in: Advances in atomic and molecular physics, ed. B. Bederson (Academic Press, New York, 1987); C.A. Berlulani and G. Baur, Phys. Rep. 163 (1988) 300. [ 12 ] C. Bottcher and M.R. Strayer, Nucl. Instrum. Methods B 31 ( 1988 ) 122; in: Physics of strong fields, ed. W. Greiner, Vol. 153 (Plenum, New York, 1987) p. 629; Proc. Second Workshop on Experiments and detectors for the relativistic heavy ion collider (Berkeley, CA, May 1987), Lawrence Berkeley Laboratory Report LBL-24604; in: Frontiers of heavy-ion physics, eds. N. Cindro, W. Greiner and R. ~.aplar ( World Scientific, Singapore, 1987 ) p. 471 ; M. Grabiak. B. MiJller, W. Griener, G. Soft and P. Koch. J. Phys. G 15 (1989) L25. [13] M.J. Rhoades-Brown, C. Bottcher and M.R. Strayer, Phys. Rev. A 40 (1989) 2831.
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[ 14 ] A.S. Umar, M.R. Strayer, D.J. Ernst and K.R. Sandhya Devi, Phys. Rev. C 30 (1984) 1934. [ 15 ] C. Bottcher and M.R. Strayer, Phys. Rev. D 39 ( 1989 ) 1330; Ann. Phys. (NY) 175 ( 1987 ) 64. [16IJ.D. Bjorken and S.D. Drell, Relativistic quantum mechanics ( McGraw-Hill, New York, 1964 ). [17]F.E. Close, An introduction to quarks and parlons (Academic Press, New York, 1979 ). [ 18] S. Mikamo, A. Maki, M. Ono, M. Takasaki, A. Yamamoto, H. Hirabayashi, S. Kurokawa, A. Kusumegi, S. Naito, Y. Sakai, S. Higashi, F. Kajino, Y. Kawashima, M. Nakashita, S. Ozaki, T. Takahashi, T. Ishizuka. J. Kishiro and M. Mishina, Phys. Left. B 106 ( 1981 ) 428. [ 19] G. Roche, G. Claesson, D. Hendrie, G.F. Krebs, E. Lallier, A. Letessier-Selvon, H.S. Matis, T. Mulera, C. Naudet, L. Schroeder, P.A. Seidl. A. Yegneswaran, Z.F. Wang, J. Bystricky, J. Carroll, J. Gordon, G. Igo, S. Trentalange, T. Hallman, L. Madansky, J.F. Gilot, P. Kirk. D. Miller and G. Landaud, Phys. Rev. Lett. 61 (1988) 1069.
PHYSICS LETTERS B
15 March 1990
A N O M A L O U S E L E C T R O N - P A I R P R O D U C T I O N IN n - p C O L L I S I O N S C. B O T T C H E R , M.R. STRAYER Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA C.J. ALBERT and D.J. E R N S T Center for Theoretical Physics and Physics Department, Texas A&M University, College Station, TX 77843, USA Received 6 November 1989; revised manuscript received 2 January. 1990
Anomalous e+e - pairs with invariant mass 0.2 ~
The emission o f e l e c t r o n - p o s i t r o n pairs has been observed [ 1,2] in ~ - p collisions with corresponding pion m o m e n t a o f 16 and 17 G e V / c . The experiments detected pairs over a limited range o f invariant pair mass, M - [ ( E I + E 2 ) 2 - ( R ~ + p 2 ) 2 ] I/2, given by 0 . 2 < M < 1 . 2 G e V / c 2. A large peak near M = 0 . 8 G e V / c 2 was observed. This peak is well understood as resulting from the direct electromagnetic decay of the p and to mesons, p--,e+e - , and to--,e+e - . A continuum of pairs is also seen at lower invariant masses. These pairs were d e e m e d a n o m a l o u s as conventional hadronic mechanisms failed to explain the data. A m o n g these possible mechanisms are the Dalitz decay o f the q and to, q--,),e+ e - and to--,~°e+e-. This process can be reliably calculated from the measured meson production cross section and the known branching ratio for Dalitz decay. However, it only produces a small fraction o f the observed cross section. The Dalitz decays together with the direct decay into leptons o f the p, to and rl are subtracted from the measured data, and the remaining events are termed a n o m a l o u s pairs. A n o t h e r mechanism which produces a continuous spectrum o f pairs is the D r e l i Yan process. For values o f the pair mass much larger than 0.8 G e V / c 2, this process usually agrees with experimental data [3,4]. However, for the low mass
pairs [ 1,2,4] observed in the experiments o f refs. [ 1,2 ], the Drcll-Yan results are more than an order o f magnitude below the data. Several exotic mechanisms have been proposed [ 5 - 8 ] , but none o f these have proven satisfactory. We d e m o n s t r a t e here that an ab initio calculation o f the two-photon mechanism produces results that are in quantitative agreement with the data. The two-photon mechanism for electromagnetic lepton-pair production was first investigated [9,10] in the 1930's. In ref. [ 10], the equivalent photon, or Weizs~icker-Williams, a p p r o x i m a t i o n was developed. This a p p r o x i m a t i o n replaces the electromagnetic fields by a spectrum of photons, which then directly create the pairs. Two-photon physics, within this a p p r o x i m a t i o n , has received considerable attention [ 11 ] of late. In particular, it has been suggested that this mechanism could produce large yields of electrons, muons, and tauons in relativistic heavy-ion collisions [12], and that the electrons and muons could mask signals originating from hadronic collisions. The closely related mechanism o f electron-pair production and capture [13] represents a serious beam loss mechanism in relativistic collider physics with heavy ions. The experimental results o f refs. [ 1,2 ] are shown
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
17 5
Volume 237. number 2
PHYSICS LETTERS B
in figs. 1,2, and 3. The qualitative features ofthc cross sections are clear. The cross section decreases with increasing pair mass, decreases with increasing Feynman x (defined as the total longitudinal momentum of the pair divided by the maximum value allowcd kinematically), and decreases with increasing transverse momentum. We calculate the two-photon process in the limit where the relative motion of the pion and proton is treated classically [ 14 ]. Using this approximation, the total cross section for electron-positron-pair production is given by
a=fd:bZ
-
d3k ' (2~t5~ ~ f
d3k I ( k ' ° " ISnlk¢~) I ~
•
(~)
15 March 1990
(T~).Z
~(k')
X [flA6 + ~(b: k' - p ) - a t ' A C+ ~(b: k' - p ) ] X(
~1
~ z '°' (,,,)z~,,,,(p)
+ - - - -1 - ~~" q/~." ~(p)g/,~ ..~(p) ) Po + Ep '. X [flA~-)(b:k+p)+at'.4~-'(b:k+p)]~°~(k).
(4) The most general form of the electromagnetic current of the proton is
dU(p'a', po) = e a ' " ~(p' ) [ 7"F, (p' - p ) ]u t") (p) e
a(" ~(p') [,~"~(p~,+p.)x&(p' -p) ]
where b is the impact parameter for the classical trajectory of the relative pion-proton motion. For a 17 GeV/c pion. we may use a straight-line trajectory,. The scattering amplitude has two terms corresponding to crossed and uncrossed photon lines: further details are given in ref. [ 15 ]. The contribution to the amplitude from the uncrossed diagrams is given by
By inserting explicit expressions for the proton spinors in cq. (5) and multiplying by the photon propagator, the vector potential in eq. (4) becomes
( k a ' ISb I k o ) =
A~,+ '(b: q) --a,,~ ~ )(b:q)+ Sa~,+ )(b:q) ,
In
- i ,,/E ~
f
d4p
~-~_~.~t~
ca)
× u~°~(p).
(2)
at/+ ~(b:q) =2rwd(qo-flq:)'/
x where the notation and normalization are those of ref. [ 16 ]. Here .4 ~,+ ~(q) and A J,- ) (q). which appear as prescribed external fields, are given by the photon propagator times the matrix element of the proton or pion current. Eq. (2) can be translated into the notation of ref. [ 15] by utilizing ,~+m
1
m
p e - m ' - - Po -15o F.p ~ u ' " ( p ) a ' " ' ( p ) 1
prl
+ --~-k'p/-:~ Z z " " (p)v~°'(p) , Po
"7
176
exp(-iq.b/2)
- q~ +,/q2_
a~,- ~ ' ( b : q ) = 0 ,
Gr~(q 2 )
,
al.+~(b:q)=O,
a~ + )(b:q)=fla~ + '(b:q) .
(6)
The electric form factor of the proton, GE (q2), is here taken to have a dipole form ( l + q - / A p ) - - . with Ap=0.71 GeV 2 [17]. The second term in A~,+~(q), which involves both k'~ (q) and l.~(q), corresponds to events in which the proton spin flips. This term yields
(3)
and. in addition, by replacing the invariantly normed spinors by Z ~ ( p ) = (m/Ep)~/:u<~'~(p) and ~,{,~(p) : (IFI/Ep)I/2l'~") (p),
(5)
where the term which corresponds to non spin-flip events is
(k')a'+'(bk'-p)
× p-YLm~4~-~(b:p+k)r~O~(k) •
2 :~Ip
6a{+)(b:q)=
2n
e
2 :t//p 6(qo-flq:)
X e x p ( - i q . b / 2 ) q: G M(q-) ~ , q~+ },-q± , 2
(7)
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PHYSICS LETTERS B e
aal. + ' ( b : q ) = 2rt ~M-~o~(qo -flq:) x exp(-iq.b/2) ~ + 72q2_
iq:GM(q2),
6a~ + )(b:q) = - 2 n ~M~ 6(qo -flq:)7 X
eXpql-iq'b/2) + 72q2L
(qx+iqy)GM(q2) ,
5a~ + )(b:q) =fl3a~ + )(b:q) ,
( 7 cont'd)
where the magnetic form factor of the proton, GM(q2), is normalized so that G M ( 0 ) = I + K = 2Mo14,/e, where/t o is the magnetic moment of the proton. We use the same dipole form for the magnetic form factor, GM (q2) = ( 1 + X) ( 1 + q2/A~) - 2, as for GE (q2). For the spin-zero pion the electromagnetic current is given simply by
jU(p, p)=e(p,,+pU)GE( (p'_p)2) ,
(8)
which produces similar results forA ~- ) (q) as is given in eq. (6); in the center-of-velocity frame, A J,-)(q) is simply given by eq. (6) with f l - , - f l and b-, - b . We utilize a monopole form for the electric form factor of the pion, G~.(q2)=(l+q2/A~) -I, where A~ = 0 . 5 9 GeV 2 [ 17 ]. The expressions derived here for the total cross section is Lorentz invariant and is most casily evaluated in the center-of-velocity frame. Finally, the crossed photon term, which contributes to eq. (2), is added coherently to the direct term. In summary, we take the classical limit on the relative motion of the proton and pion of standard QED perturbation theory for the two-photon process. The arguments of ref. [ 14] indicate that the wavelength of the relative pion-nucleon motion is sufficiently large and the electromagnetic field sufficiently smooth that we are well into the classical region. In this region, the matrix element of the current which couples the proton or the pion to the electromagnetic field multiplied by the photon propagator takes the form of prescribed electromagnetic fields that have becn boosted from the rest frame to a frame with velocity - f t . as given in cq. (6), non spin-flip, and eq. (7), spin-flip. A different derivation of the non spin-flip amplitude can be found in ref. [15]. In ref. [ 15], identical numerical results were obtained both in the
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Lorentz gauge (as is used here) and when the calculation was repeated in the temporal gauge. The eight-dimensional integration required to calculate the total cross section in eq. ( 1 ) is done utilizing the Monte Carlo tcchnique described in ref. [ 15 ]. We impose on the Monte Carlo integration the experimental cut on the pair mass, 0.2~
....
,
....
,
....
/. . . . . . .
!01
:::L
, ....
, ....
Spin-Flip Non Spin-Hip
/
,0 o
I
.
EL_
c~______2~ ~°f
X "(3
"'"''"'-.......
1021 10 .3
13 .4/ 30
02
04
06
08
X
Fig. 1. The differential cross section for electron-positron-pair production versus Feynman x. The dashed curve is the result omitting the spin-flip terms. The data labeled a) are from ref. [1 ], those labeled b) are set "'PAIR A" from ref. [2], and those labeled c), are set "'PAIR B" from ref. [2]. 177
PHYSICS LETTERS B
Volume 237, number 2
lower cut on the invariant mass at exactly 0.2 G e V / c 2. If we decrease this value slightly, the cross sections at low x and low p~ are enhanced. The difficulty might also lie in our neglect of the final-state distortions o f the electron and positron. The cross sections for the slower-moving leptons might well be somewhat higher if distorted waves were used. The total cross section for electromagnetic pair production is about 55 ~tb, o f which only 0.5 ~tb lies within
10 2
S p m . t q l F' " - - "m""
Non £pm.Hip
10'
Ref 1
> (.9
10 0
.,0 ::L
10
1D
10:
10 3
....
oo
, . . . . . . . . . . . . . .
o~
02
,
03 M
....
i ....
04
os
, ....
06
07
(GeV/c21
Fig. 2. The differential cross section for electron-positron-pair production versus the pair mass. The dashed curve is the result omitting the spin-flip terms. The data are from ret: [ 1]. . . . .
10
,
....
, ....
,
....
,
. . . .
, . .
.
Spin-Hip •
--'~.-'"
Nk;fnlSpin-Flip
10 ~
10 0
10.1
13..
10-2
"~
10. 3
10 4 0.0
, , . . , .... 0.1
m .... 0.2
I .... 0.3
t .... 0.4
, .... 0.5
0.6
[GeV/c] 2
Fig. 3. The differential cross section for electron-positron-pair production versus the transverse momentum of the pair. The dashed curve is the result omitting the spin-flip terms. The data are from ref. [ I 1. The results for electron-positron-pair production including the magnetic spin-flip term, eq. (7), are also shown in figs. 1-3 as the solid curve. We see that the results are in reasonable agreement with the data, The calculation is m a d e ab initio with no adjustable parameters. Although the overall agreement is good, the cross sections in fig. 1 lie a little below the data, for low F e y n m a n x. This may be due to our use o f the 178
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the cut. T h e spin-flip term, eq. ( 7 ) , is of order q/Mp, as c o m p a r e d to the leading term, eq. (6). The reduction due to this factor is more than c o m p e n s a t e d for by two features of the spin-flip current: ( 1 ) the current has transverse components, and ( 2 ) the 0 and z c o m p o n e n t s are proportional to the transverse mom e n t u m o f the photon. Thus the spin-flip current produces pairs at the large values o f transverse mom e n t u m o f the photon. Thus the spin-flip current produces pairs at the large values o f transverse mom e n t u m emphasized by the experimental cut on the invariant mass. We conclude from the general quality o f the agreemerit of the calculation with the data that the lepton pairs observed in refs. [ 1,2 ] are produced, to a large extent, via the two-photon mechanism, specifically by the magnetic part o f the proton current. The calculation itself allows some internal consistency checks on this view o f the physics. The presence o f the proton and pion form factors reduces the predicted cross sections by less than a percent. This indicates that the cross section arises from a region where the scale for the m o m e n t u m or energy o f the virtual photon is given by several h u n d r e d MeV or less. At these mom e n t u m transfers, the d o m i n a n t response o f the proton or the pion is the coherent elastic response, although the excitation of the nucleon to a delta, or the pion to a rho might also contribute a non negligible a m o u n t in the kinematic region. We note that the coincidence data given in ref. [ 2 ] is consistent with this low multiplicity interpretation o f the data. This relatively low m o m e n t u m transfer from the hadrons does not justify a quark level calculation o f the two-photon process as was done in ref. [ 2 ] and we point out that the independent quark level processes are suppressed by a factor o f Z~Z~, with Z, the charge o f the quark, and thus can only become the d o m i n a n t term in kinematic regions where the elastic form factor
Volume 237, number 2
PHYSICS LETTERS B
suppresses the coherent elastic term. In our calculation we integrate over impact parameters starting at zero. For impact parameters less than one fermi the pion and proton will undergo strong interactions which would invalidate the pure electromagnetic calculation done here. However, we find that non negligible n u m b e r s of lepton pairs are emitted out to an impact parameter > 100 fm. Thus even the extreme measure of neglecting the contribution which arises from impact parameters below 1 fm would have little effect on our results. Lepton-pair production in particle collisions is important for several reasons. The two-photon process can produce large numbers [ 12,15 ] of soft pairs. Here we have found that it can also produce pairs with significant transverse m o m e n t u m . Such pairs would provide a background for the lepton pairs emitted in heavy-ion collisions. If the lepton pairs are to serve as a probe of the q u a r k - g l u o n plasma, this background must first be understood. Also, the Lorentz contracted fields represented by the vector potentials o f e q . (6) and eq. (7) become very large for the ultrarelativistic heavy ions that will be produced at R H I C and perhaps at the SSC. In this case, the perturbative approach utilized here will certainly be inadequate. Lepton production would then offer the opportunity to study QED in a region where it is not perturbative. The present results are needed as a baseline for this study. There exists two possibly related experiments [ 18,19] in which electron-positron pairs have been observed in collisions of a proton with 9Be. Because ~Bc is a composite nucleus, the dynamics are more complicated than those of the rt-p collisions considered here. For the 9Be data, we are examining the role played by the two-photon mechanism and will present results shortly. The work of D.J.E. was supported, in part, by Oak Ridge Associated Universities. The work of D.J.E. and C.J.A. was supported, in part, by the National Science Foundation. One of us, D.J.E., would like to thank the Physics Division of Oak Ridge National Laboratory for its hospitality during part of this work. This research was sponsored by the Division of Nuclear Physics of the US D e p a r t m e n t of Energy under contract No. DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc., and by the Division
15 March 1990
of Chemical Sciences, Office of Basic Energy Sciences.
References [I]R. Slroynowski, D. Blockus, W. Dunwoodie, D.W.G.S. Leith, M. Marshall, C.L. Woody, B. Barnett, C.Y. Chien, T. Fiefuth, M. Gilchriese, D. Hutchinson, W.B. Johnson, P. Kunz, T. Lasinski, L. Madansky. W.T. Meyer, A. Pevsner, B. Ratcliff, P. Schacht, J. Scheid, S. Shapiro and S. Williams, Phys. Lett. B 97 (1980) 315; D. Blockus, W. Dunwoodie, D.W.G.S. Leith, M. Marshall, R. Stroynowski, C.L. Woody, B. Barnett. C.Y. Chien, T. Fieguth, M. Gilchriese, D. Hutchinson, W.B. Johnson, P. Kunz, T. Lasinski,L. Mandansky,W.T. Meyer, A. Pevsner, B. Ratcliff, P. Schacht, J. Sheid, S. Shapiro and S. Williams, Nucl. Phys. B 201 (1982) 205. [2] J. Stekas, G. Abshire, M.R. Adams, C. Brown, L. Cormell, E. Crandall, G.J. Donaldson, J. Goldberger, H.A. Gordon, P.D. Grannis, B.T. Meadows, G.R. Morris and P. Rehak, Phys. Rev. Lett. 47 (1981) 1686" M.R. Adams, G. Abshire, ('. Brown, E.S. Crandall, J. Goldberger, P.D. Grannis, B.T. Meadows, J. Stekas. G.J. Donaldson, H.A. Gordon. G.R. Morris, P. Rehak and L. Cormell, Phys. Rev. D 27 (1983) 1977. [3] C.B. Newman, K.J. Anderson, R.N. Coleman, G.E. Hogan, K.P. Karhi, K.T. McDonald, J.E. Pilcher, E.I. Rosenberg, G.H. Sanders, A.J.S. Smith and J.J. Thaler, Phys. Rev. Lett. 42 (1979) 951. [4] J.D. Bjorkenand H. Weisberg,Phys. Rev. D 13 ( 1976) 1405. [ 5 ] V. (~erny,P. Lichard and J. Pi,~fit,Phys. Lett. B 70 ( 1977 ) 61;Acta Phys. Pol. B9 (1978) 901;B 10 (1979) 537. [6 ] G.R. Farrar and S.C. Fraulschi, Phys. Rev. Lett. 36 ( 1976 ) 1017; R. Ruckl, Phys. Len. B 64 ( 1976 ) 39; N.S. Craigieand H.N. Thompson, Nucl. Phys. B 141 (1978) 121. [ 7 ] E.L. Feinberg, Nuovo Cimento 34A ( 1976 ) 391 ; E.V. Shuryak, Phys. Len. B 78 (1978) 150. [8] T. Goldman, M. Duong-Vanand R. Blankenbecler.Phys. Rev. D20 (1979) 619. [9 ] W.H. Furry and J.F. Carlson, Phys. Rev. 44 ( 1933 ) 237: W. Heitlerand L. Nordheim,J. Phys. (Paris) 5 (1934) 449; Y. Nishina, S. Tomonaga and M. Kobayasi, Sci. Pap. Inst. Poincare, C. R. 27 (1935) 137; E.J. Williams,Nature 135 (1935) 66; H.J. Bhabha, Proc. R. Soc. A 152 (1935) 559. [ 10] C.F. von Wcizsacker,Z. Phys. 88 (1934) 612; E.J. Williams, Phys. Rev. 45 (1934) 729; J.R. Oppenheimer, Phys. Rev. 47 (1935) 146; L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 6 (1934) 244. [ 11 ] F.E. Low, Phys. Rev. 120 (1960) 582; R.H. Dalitz and D.R. Yennie,Phys. Rev. 105 (1957) 1598; S.J. Brodsky, T. K.inoshitaand H. Terazawa, Phys. Rev. D 4(1971) 1532; 179
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D.L. Olsen, B.L. Berman, D.E. Greiner, H.H. Heckman, P.J. Linstrom, G.D. Westfall and H.J. Crawford, Phys. Rev. C 24(1981) 1529; A. Goldberg, Nucl. Phys. A 240 (1984) 636; R. Anhoh and H. Gould, in: Advances in atomic and molecular physics, ed. B. Bederson (Academic Press, New York, 1987); C.A. Berlulani and G. Baur, Phys. Rep. 163 (1988) 300. [ 12 ] C. Bottcher and M.R. Strayer, Nucl. Instrum. Methods B 31 ( 1988 ) 122; in: Physics of strong fields, ed. W. Greiner, Vol. 153 (Plenum, New York, 1987) p. 629; Proc. Second Workshop on Experiments and detectors for the relativistic heavy ion collider (Berkeley, CA, May 1987), Lawrence Berkeley Laboratory Report LBL-24604; in: Frontiers of heavy-ion physics, eds. N. Cindro, W. Greiner and R. ~.aplar ( World Scientific, Singapore, 1987 ) p. 471 ; M. Grabiak. B. MiJller, W. Griener, G. Soft and P. Koch. J. Phys. G 15 (1989) L25. [13] M.J. Rhoades-Brown, C. Bottcher and M.R. Strayer, Phys. Rev. A 40 (1989) 2831.
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