Chapter 10 - Electron-positron Pair Production

Chapter 10

E l e c t r o n - P o s i t r o n Pa ir P r o d u c t i o n

10.1 I n t r o d u c t i o n

The production of electron-positron pairs in collisions of charged cosmic ray particles with nuclei has been a subject of theoretical interest soon after the discovery of the positron in the early 1930s [LaL34, Bha35, Rac37, NIT35]. These calculations were mainly based on the equivalent-photon or Weizs/icker-Williams method [Wei34, Wi134, Wi135], which describes pair production as the decay into an e +e- pair of one of the virtual photons composing the transient electromagnetic field of the projectile in the presence of the static Coulomb field of the nucleus (Bethe-Heitler process). A simple estimate [LaL34] (see also [Hei54]) for the total pair production cross section at large values of the Lorentz factor ~ is given by

28 (O~ZT)2(c~Zp)2 ~ ln-~ (10.1) o-~+ ~ 27--~

where ~ is the electron Compton wavelength and c~ is the fine-structure constant. The cross section increases with ~ and becomes large for high-Z collision partners.

At the present time, there is no need to rely entirely on cosmic rays for investigating pair production. In addition to the accelerators for relativistic heavy ions already in operation, there is the promise for the near future of the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven and the Large Hadron Collider (LHC) at CERN that will allow the study of heavy-ion reactions up to 20 and 3.5 TeV/u, respectively, of equivalent laboratory

281

282 CHAPTER 10. ELECTRON-POSITRON PAIR PRODUCTION

mc 2

ISII 2

0

2 -mc

a} Ib}

Figure 10.1. Schematic representation of (a) bound-electron-free-positron pair production and (b) free pair production.

energy. For a given collider Lorentz factor %o11, the equivalent projectile Lorentz

factor ~FT for a fixed target is given according to Eq. (2.30) by

7FT = 2~c2oll- 1. (10.2)

In this way, extreme relativistic collision energies can be reached for which pair production is one of the dominating processes. When considering pair production we usually have in mind very large values of 7.

Wi th the advent of accelerators for relativistic heavy ions, the interest in this process has been revived [Sof80, NIP82]. The process of producing a single electron-positron pair is best visualized by referring to Fig. 4.2 for the free Dirac equation. If the Coulomb field of the nucleus is included, the Dirac equation also supports bound states. Consequently, we can distinguish pair production in which the electron is created in a bound state and pair production in which the electron is created in a continuum state. This is illustrated in Fig. 10.1. We denote these two reaction channels as "bound- electron-free-positron pair production" (or "bound-free pair production") and "free pair production," respectively. 1

In Sec. 10.2, we discuss the production of free electron-positron pairs, in Sec. 10.3 of bound-electron-free positron pairs and turn to multiple pair production in Sec. 10.4. We do not discuss the creation of heavier leptons (p

1We avoid the notation "pair production with (or followed by) capture" which is sometimes found in the literature because it would suggest a two-step process thus dis- regarding the complete analogy of the processes (a) and (b) in Fig. 10.1.

10.2. PRODUCTION OF FREE ELECTRON-POSITRON PAIRS 283

and 7- pairs), let alone exotic particles, like magnetic monopoles, W-pairs, b-quarks, and the Higgs particle, see, e.g., [GrV93]. Already for muons, the length scale connected with their mass is strongly contracted compared to the electronic length scale. This will render it difficult to clearly separate electromagnetic pair production from hadronic processes, which are beyond the scope of this book.

Following conventional usage, we adopt relativistic ("natural") units in this chapter with h = c = rne = 1, (see Appendix). However, where clarity demands it, the electron mass me, the velocity of light c, or the Compton wavelength ~ are displayed explicitly. Similarly, we usually keep/3 = v/c instead of replacing it by v. Electron and positron quantities are labelled by "e" and "p", e.g., as in Ee and Ep. Only if for free electrons or positrons the momenta serve as quantum numbers, we refer to positive and negative energy states (corresponding to electrons and positrons, respectively) by the labels p+ and p_, e.g., for the energies, as Ep+ and Ep_.

10.2 Produc t ion of free pairs

e lectron-positron

Figure 10.1 illustrates two competing processes for single-pair production in heavy-ion collisions. Conceptually, these processes are very similar; however, there are two distinctions which suggest a separate treatment. (1) In contrast to a free electron, an electron created in a bound state has to accommodate its momentum within the available momentum distribution of the target (if it is bound in the target) or of the projectile (if it is bound in the projectile). This entails the typical charge number dependence Z5T known from electron capture, see Chap. 8, and also an energy dependence different from that for the production of free electrons. (2) Since an electron bound in a specific state has no further degrees of freedom, bound-electron- free-positron pair production is more accessible to calculations.

The creation of free electron-positron pairs is by far the dominant pair production process for extreme relativistic energies. It had been observed already in the 1930s, while bound-free pair production has been detected only recently [BEG93, BEG94]. Both processes occur with a relatively small probability so that it is a reasonable first step to employ perturbation theory. In this section, we confine ourselves to free pair production, deferring bound-free pair production to See. 10.3. We start with perturbative calculations, first in lowest-order quantum electrodynamics (QED), in which electrons and positrons are represented by plane waves, and then in a first- order distorted-wave approximation. In See. 10.2.3, we discuss the applica-


tion of the equivalent-photon or Weizss method for extremely high projectile energies.

10.2.1 Lowest-order QED calculations

The theoretically most clear-cut way to describe the production of free electron-positron pairs is a formulation in terms of lowest-order QED with external electromagnetic fields. In the center-of-mass system, the target and projectile nuclei merely serve as sources of time-dependent electromagnetic fields. Since energy-momentum conservation rules out pair creation by a single freely moving charge, the lowest-order QED process involves two moving nuclei, each of them interacting with the electron-positron field via the exchange of a single photon. A representation in terms of Feynman diagrams is given in Fig. 10.2 to be discussed below.

The lowest-order nonvanishing term in the expansion (5.66) of the S- operator is therefore of second order and reads

F f 1 ~(2) _ (_i)2 dtl dr2 Hi( t1 )Hi ( t2 ) , (10.3)

where for an external electromagnetic field (5.68)

H(t) = / ~(r,t)V(t)~(r,t) d3r (10.4)

with

V(t) = -e~~ = -e((I) - c~. A). (10.5)

Here, for brevity, we display in the potential V(t) = V(r, t) only the time dependence explicitly. Within QED, the space-time-dependent field operators r are expanded in terms of plane waves according to Eq. (5.8).

Alternatively, we could choose target or projectile eigenstates (5.53) or (5.54) instead of plane waves. In the language of QED, such states are produced by exchanging infinitely many photons between the electron (or the positron) and the target or projectile nucleus. Since these states do not possess a definite momentum, energy-momentum conservation does not impose a constraint, so that a first-order distorted wave process as described in Eq. (5.72) and implied by Fig. 10.1 is possible. We defer a discussion to Sec. 10.2.2.

In terms of the creation and annihilation operators introduced in Sec. 4.2.2, the S matrix element describing the creation from the vacuum of a single free pair with electron momentum and energy p+, Ep+ and positron


b

c d

Figure 10.2. Lowest-order Feynman diagrams for the production of a free electron-positron pair in a collision between two heavy ions. The time arrow points upwards. The heavy vertical lines denote the nuclei 1 and 2, while photons are depicted as wavy lines and electrons and positrons are described by thin solid lines with arrows directed upwards and downwards, respectively.

momentum and energy p_ , Ep_, respectively, is obtained from Eq. (10.3) in the form

= (_i)2 dtl dt2 O(3 (X)

• /~)tg(t2)~)d37"2 10>. (10.6)

Here, as a first step in the process, a pair is created at the earlier t ime t2 by the interaction V(t2), mediated by a photon emit ted either from the target or from the projectile. Both alternatives contribute to the matr ix element. In order to decompose the vacuum expectat ion value in Eq. (10.6) into a first and a second reaction step, we insert a complete set

E ~k-b~k+ I0) (O'bk+dk_ -- 1 (10.7) k+k_

of plane-wave electron-positron states (with momenta and energies k+, Ek+


and k_, Ek_) between the two space integrals. The matrix element (10.6) h is now reformulated by introducing the expansion (5.8) of r in terms of

a complete set of single-particle states. From ~ t~ , we obtain four types of operator products, namely ~t~, d ~ , ~td ~ , and bd, corresponding to electron scattering, positron scattering, pair production, and pair annihilation, respectively. In the matrix element at the earlier time t2, only the pair production term contributes, while in the matrix element at the later time t l, only the electron or positron scattering terms yield a nonzero result. In a first step, the matrix element at t2 can be written explicitly if, according to Eq. (4.39), we write the wave function in the usual form ~k+ (r, t) = ~k• ( r )exp (--iEk+ t). Then

= (-i) Z k+ k_ cc

X <Olb~p+dp_ / ~*V(t l )~d3rl ~k_btk+ 10>

x (wk+ Iv(t2)I~k_ > e-*(E~- -E~+)t~, (10.8)

where the last bracket no longer involves field operators and simply denotes a space integral. In the single-particle matrix element at time tl, only one state can change so that we have two nonvanishing contributions: the first one with k_ = p_ describing the scattering of the electron k+ --, p+ and the second one with k+ = p+ describing the scattering of the positron k_ --, p_. These possibilities are illustrated in the Feynman diagrams of Fig. 10.2. Diagrams (a) and (b) show the creation of a pair by nucleus 1 and the subsequent scattering of the electron and positron, respectively, by nucleus 2. In the "crossed" diagrams (c) and (d), the role of the two colliding nuclei is interchanged. All four diagrams contribute to the transition amplitude.

In order to distinguish the four diagrams, let us introduce the following notation. The label s = q- denotes positive- or negative-energy intermediate scattering states, while the label n = 1 or n = 2 denotes the nuclear potential that creates the electron-positron pair. Then the diagrams (a), (b), (c), and (d) are assigned to the combinations (+,1), (-,1), (+,2), and (-,2). The S matrix element corresponding to the diagram (a), in which the pair is created by nucleus 1 and the intermediate scattering state is an electron state, is written as

/ ? /___tlcK ) ~(+,l)p+p_ __ (_i)2 E dti dt2(~p+lV2(ti)l~k+>e -i(sk+-Ep+)tl k+ oo

X <(ilk+ Igl (t2)I~p_ > e-i(Ep- -Ek+ )t2. (10.9)

10.2. P R O D U C T I O N OF FREE E L E C T R O N - P O S I T R O N PAIRS 287

For the diagram (c), the potentials V1 and V2 are interchanged. It is now convenient to express the t ime-dependent potentials V(t) by

their Fourier transforms V(w), thus performing a frequency analysis, in analogy to Eq. (3.41). Then the t ime dependence resides entirely in the resulting exponential factors. Wi th the aid of a convergence-enforcing damp- ing factor, the t ime integration can be performed as in ordinary perturbation theory, see, e.g., [Sch61], with the result

i?i? S (+,~)p+p_ = - i ~ d~ d~'<~p+lV2(~)l~k+><~k+lVl(~')l~p_> k+ ~

+ Ep Ep+) x - (10.10)

w ~ + E p _ - E k + + i r l '

where the limit r/ + +0 has to be taken. After integrating over w', this form can be rewrit ten as

/ 2 d <Pp+lV2(w + Ep+)l~k+)<Pk+ lV l ( -Ep - w)lq@_} S (+'1) = i Z w - .

P+P- Ek+ + w - it/ k+

(10.11)

In complete analogy , the contributions from the remaining Feynman did- grams in Fig. 10.2 can be calculated.

Bottcher and Strayer [BOS89] derived the S matr ix in a different way, taking advantage of the particular symmetr ic si tuation in a collider experiment. In this case, Zp = ZT = Z, and it is convenient to use the center- of-mass frame in which the two nuclei 1 and 2 have the impact parameters +b/2 and velocities +v. Because of symmetry, the matr ix elements are simpler and the contributions with s = + denoting positive- and negative- energy scattering states can be combined. The S matr ix is a sum of all four

r terms ~p+p_. The "direct" diagrams (a) and (b) yield the matr ix elements

- _

(s,1) Sp+p_ - i ~ - ~ d (~p+lV2(w Ep+)l~k~)<~k~lYl(Ep a~)l~p_>

ks ~ Eks -- w -- irl '

(10.12)

and, by interchanging V1 and 1/2, one obtains the corresponding expressions S (~,2) for the diagrams (c) and (d). The space matr ix elements occurring in Eq. (10.12) are evaluated using the frequency representation V(w) of the potentials derived in analogy to Eq. (5.106), with q0 = a;/v, so that

f d eiqb'rb• _ Ze2 2 V1,2(w) - 7rvx /~ qbq2 ~ + (1 -- ~2)~2/v2 e+~q~'b/2(1 T Zaz), (10.13)


where we used the decomposition r = (rb, z) into a transverse and a longitudinal part (parallel to v) and, similarly, q = (qb, oz/v). When inserting the plane-wave states (4.62)and (4.63) (whose time dependence has already been taken into account) into Eq. (10.12), the spatial integrations yield delta functions in the momenta for each of the matrix elements. While the transverse parts simply eliminate the qb integrations, the longitudinal parts fix the frequency w and the longitudinal momentum kz of the intermediate states. As a result, for given momenta p+, p_ of the final electron- positron states, the S-matrix elements (10.12) contain only a summation (or a two-dimensional integration) over the transverse momenta kb,• of the intermediate states. In order to obtain the total cross section, the squares of the sums over all contributions have to be integrated over all electron and positron momenta. It has been shown and confirmed by numerical integration that the sum of all contributions in Fig. 10.2 is gauge invariant. The results for the total cross sections obtained by the Monte-Carlo technique of integration are discussed in Sec. 10.2.2 and are included in Fig. 10.8.

1 0 . 2 . 2 F i r s t - o r d e r d i s t o r t e d - w a v e theory As illustrated in Fig. 10.1 (b), the pair production process can be viewed as the ionization of a negative-energy electron by the projectile, a process that can occur even in collisions between bare nuclei. In contrast to Sec. 10.2.1, here we adopt Coulombic target wave functions for the electron and positron states and consider the projectile-electron interaction as a perturbation. This means that the projectile-electron interaction is taken into account in first order, while the target-electron interaction is included to all orders of aZ . In the description of the continuum states, we have two options. We may either use a partial-wave expansion of the exact Dirac wave function, similarly as for ionization in Sec. 6.3, or we may employ a Sommerfeld-Maue wave function which is approximate, but can be given in a closed form, see Sec. 4.4.3.

Partial-wave expansion of the continuum wave functions

Becker et al. [BeG86a] use the analogy to the ionization process, see Sec. 6.3.2, to write the double-differential production probability in the form

d2p(b) _ - E lAp+p_ (b)l 2, (10.14)

dEp+ dEp_ t~p+ ,~p+ ~t~p_ ~p_

where Ep+ and Ep_ are the electron and positron energies, and the summation extends over the corresponding Dirac quantum numbers n and the


angular momentum projections p. The transition amplitude (5.96) is (in natural units)

Ap+p_ - i',/Zpe 2 f dt ei(Ep+ +Ep_)t I~p+ 1 (1 - ~C~z) D--

r p ~p_ } . (10.15)

Here, the brackets denote a spatial integration. Similarly, as in Eq. (5.105), one may introduce

qo = Ep+ + Ep_ (10.16) V

to obtain, by multipole decomposition in momentum space (cf. Sec. 6.3.2), an equation for the transition amplitude that is formally identical to Eq. (6.44) and need not be repeated here. By integrating Eq. (10.15) over the impact parameter plane, one derives the double-differential cross section for pair creation (in units of )~) in the form

(Zpe2; _ v (q2 _ 32q2)2 IMp+p_ (q)12,

/'i;p+ ~/./,p+ ~N;p_ ~ p _ 0

(10.17) where the matrix element Mp+p_ is given by Eq. (6.47) with the labels f,i replaced by p+,p_. If one is interested in angular correlations, one has to use wave functions (4.123) or (4.124) with well-defined asymptotic momenta, which implies a coherent summation over partial waves.

Compared to the case of ionization [VaB84], the evaluation of the matrix elements Mp+p_ occurring in Eq. (10.17) is complicated by the presence of two continuum Coulomb-Dirac wave functions. In actual calculations, the summation over the partial waves has to be truncated at a maximum value of the Dirac quantum number [gmaxl for the electron and positron. This implies that at very large projectile energies, the method of partial wave expansion is bound to fail because too large values of I~maxl would be required for a proper description of the resulting high-energy electron and positron wave functions.

Similarly, as in See. 6.1.2, it should be stated here that the Coulomb boundary condition expressed by Eq. (6.8) are automatically taken into account in (10.15). If one introduces the exact solutions of the asymptotic Dirac equation, one obtains a relation in analogy to Eq. (6.9) containing an additional term with 1/R'. Again, within first-order perturbation theory, the contribution of this term drops out after time integration [TOE91].

Becker et al. [BeG86a] show that in the extreme relativistic limit 7 ~ oc the double-differential probability d2P(b)/(dEp+ dEp_ ) approaches a constant value and that the double-differential cross section d2o/(dEp+ dEp_ )


x10-'1

. ~o

>

1.o

0.1 1 2 ~ 6 8 10 12 14.

Ep (mr z)

Figure 10.3. Single,differential probability for the creation of free electron- positron pairs in p + U 92+ collisions as a function of the positron energy Ep = Ep_ at an impact parameter b = 100 fm for various projectile energies El~b = 1, 10, and 100 GeV/u. From [BeG86a].

increases as In 7- This behavior is caused solely by the space-time dependence of the projectile field and not by the particular choice of the wave functions.

Figure 10.3 shows the single-differential probabilities for pair creation as a function of the positron energy. The steep decrease of the probabilities for small positron energies is caused by the repulsion between positron and target nucleus at the given impact parameter b. The decrease at large Ep_ is caused by the unavailability of high-momentum components in the virtual- photon spectrum. As a result, the probabilities peak at Ep_ ~ ( 2 - 3)mec 2.

Total cross sections for free pair production are shown in Fig. 10.8 and are compared there with those derived with the QED method of See. 10.2.1 and with Sommerfeld-Maue wave functions discussed below.

Sommerfeld-Maue wave functions for electrons and positrons

Since exact Dirac continuum wave functions for a Coulomb potential can be represented only by partial-wave expansions, it is meaningful to introduce approximate Sommerfeld-Maue or Furry wave functions, see Sec. 4.4.3. Their advantage is twofold: the wave functions can be given in a closed ana-


lytical form and, furthermore, are characterized by well-defined asymptotic momenta. According to Bethe and Maximon [BeM54], who first applied them to the production of electron-positron pairs by real photons in the field of a nucleus, these wave functions are very good approximations to the exact continuum states for electron energies large compared to the rest energy mec 2.

The sixfold differential probability for emitting an electron-positron pair with momenta p + , p _ and (undetected) spin projections s+,s_, respectively, is given by [Dec91, IOE93]

d6p(b) p+p_Ep+ Ep_ = Z IAp+p - (b)12' (10.18) dEp+ dEp_dt2p+ df~p_ (27r) 6

8A_~s_

where, in analogy to Eqs. (6.4) and (6.6) or directly from Eq. (10.15),

_ [ Ap+p_ (b) ~v J qb 2 + (%/,7) 2 e-iqb'bMp+p_(qb, qo) (10.19)

with Mp+p_(qb, qo ) = (qpp+ [(1 -- flaz)eiq'rlczp_ ). (10.20)

The longitudinal momentum transfer q0 resulting from the time integration is again given by Eq. (10.16).

Before evaluating the expression (10.20) with Sommerfeld-Maue wave functions, it is necessary to adopt the reformulation of the matrix element according to Eq. (5.109) in order to avoid spurious contributions, index- Matrix elements!of the alpha@of the a-operator

The calculations [IOE93] show that even for high collision energies, the term proportional, to 1/V 2 in Eq. (5.109) cannot be neglected compared to the other terms. However, when the electron wave function (4.132) with incoming spherical waves and the positron wave function (4.133) with outgoing spherical waves is inserted into Eq. (10.20), we ignore the cross term, in which the derivatives of the 1F1 function is taken both for the electron and for the positron. This term is of higher order in Za, that is, of the same order as terms already omitted in the Sommerfeld-Maue wave function.

By integrating the probability (10.18) over the impact parameter plane, one obtains the sixfold differential cross section

d6a(p+, p_) = 4 P+p-Ep+Ep - (Zpe2"~ 2 dEp+ dEp_ df~p+ dt2p_ (27r) 6 \ ] v

f ~_,~+ IMp+p_ (qb, %)12 ,s_ 62 x + (lo.21)


Calculations for the production of free electron-positron pairs in collisions with relativistic heavy ions using Sommerfeld-Maue wave functions have been carried out by introducing simplifying approximations [NIP82, BeB88], as well as by using fully numerical methods [Dec91]. Some inac- curacies in the latter work are corrected and the analytical reduction is pushed farther in [IOE93] by use of the trace technique for performing the calculations in spinor space. All integrals can be evaluated numerically using adaptive Monte-Carlo techniques [Lep87]. For the total cross section, seven-dimensional integrations are required.

A combination of existing perturbative results with the equivalent- photon method (Sec. 10.2.3) is used by Eby [Eby89, Eby91], who also points out a connection between the two methods and analyzes it in detailed calculations.

Results with Sommerfeld-Maue wave functions

In the following, we present some results obtained from first-order distorted- wave calculations with Sommerfeld-Maue wave functions [IOE93]. In some cases, a comparison with other theoretical or experimental data is possible.

In a first step, the calculation of the angular correlation of pair production as a function of the relative azimuthal angle between the electron and the positron direction [DeG90, IOE93] shows that the electron and positron are preferably emitted with opposite transverse momenta yielding a distribution that is symmetric about 180 ~ .

Next, by averaging over the azimuthal correlation and fixing the electron and positron energy, one may derive angular distributions in the laboratory system as a function of the electron and positron polar angles 0e and 0p. In Figs. 10.4 and 10.5, the polar angular distributions for the collision system U 92+ -~- A u 79+ at 960 MeV/u and for S 16+ -Jr- A u 79+ at 200 GeV/u are presented. In the latter case, the increase of the Lorentz factor 7 causes a striking localization of the emitted pairs in the forward direction. This behavior is also observed in the process of pair creation by real photons [BeM54, DAB54].

Having calculated the angular distribution, one is in a position to discuss energy distributions of electrons and positrons. In Figs. 10.6 and 10.7, we consider the single-differential cross sections da/dEp and da/dEe (with Ep = Ep_, E~ = Ep+) for the collision system p + U 92+ for three different Lorentz factors "~. The cross sections for other projectiles may be easily obtained in perturbation theory by scaling these results with the square Z 2 of the projectile charge. With increasing values of 7, the curves have a decreasing slope, showing that more and more electrons and positrons with high energies are created. While for positrons, the cross sections vanish at

10.2. P R O D U C T I O N OF F R E E E L E C T R O N - P O S I T R O N PAIRS 293

1 0 -~

I L_ if)

?' 10-2 >

..(3 v

Q. C:: -o 1 0-3

" o

" o

" o ~ 1 0 -4

b

"0

�9 i i , i i , i

U 92+ -t- A u 79+ 0 0

Eu, b=960 MeV/u

E,=2.01 mc 2, Ep=2 mc z

1 2 0

�9 1 8 0

0 50 60 90 120 150 t 8 0 Op (deg)

F i g u r e 10.4. Angular distribution in the laboratory system of e+e - pairs (av-

eraged over the azimuthal distribution) as a function of the positron polar angle 0p for different electron polar angles 0e. The collision system is U 9e+ + Au ~9+ at

960 MeV/u. From [IOE93].

1 02

01 , 1

2 1 0 ~

]~ I0-I

10-2

" 0 ~ 10_3

L~ 10-4

~ 10-5

"~ lO-e

i , i , I , i i ,

0 $16+ + Au79+

~~o EL,,,,= 200 geV/u E,=8.01 mc 2, Ep=8 mc 2

60

90

120

I 0 - 7 i , i i I i I ,

3'0 6'0 910 120 150 180 Op (deg)

F i g u r e 10.5. Same as Fig. 10.4 but for the collision system S 1 6 + § Au 79+ at

200 GeV/u. From [IOE93].


10-I

> (D

..Q

,., 10-2 !,1 'O

Io "O

10-3

!

' I ' I ' I '

7 ' - 1 0 ~ P + U92+ -

.

- !

1 5 10 15 20 Ep ( m c 2)

Figure 10.6. Single-differential cross section in p + U 92+ collisions as a function

of the posi t ron total energy Ep for several values of the Lorentz factor "7. For

other projectiles, the cross section scales with Z~. From [IOE93].

10-1

> (13

--Q 0 - 2

'O

13 "O

10-3

' 1 ' I ' I I

7=1 03 p "!- U 92+

3,,=1 02

"7=10 i

,

0 t 5 10 15 20 Ee (mc 2)

F i g u r e 10.7. Single-differential cross section in p + U 92+ collisions as a function

of the electron total energy Ee for several values of the Lorentz factor "7. For

other projectiles, the cross section scales with Zp 2. From [IOE93].


1 0 5 !~ . . . . . . . ' . . . . . . . . ' . . . . . . . . ' . . . . . . . . ' !

F 104 ~

, ,2" . , - -- I~ / . , f . " o .-

102

o ,"/ (D

101

10 o 1 01 1 02 1 0 3 1 04

Lorentz foctor y

Figure 10.8. Total cross section for pair production as a function of the Lorentz factor 3' in a U 92+ + U 92+ collision. Dashed-dotted curve, first-order perturbation theory with partial-wave expansion [BeG86a]. Dashed curve, QED calculation [BOS89]. Circles, combination of perturbative QED calculation with the equivalent-photon method [Eby91]. Solid curve, first-order perturbation theory with Sommerfeld-Maue wave functions. From [IOE93].

the threshold energy Ep -- me c2 owing to Coulomb repulsion, the electron cross sections tend to a finite value owing to the Coulomb attraction. The dependence on the positron energy Ep is similar as that for the pair production probability at a fixed impact parameter in Fig. 10.3. However, for very large values of 7, the partial-wave expansion with I~l _< 10 is no longer valid, see Fig. 10.8.

In Fig. 10.8, we show the dependence of the total cross section on the Lorentz factor 7 in a U 92+ + U 92+ collision. The 7 values considered cover the energy range 1 GeV/u _< Elab _< 20 TeV/u. Per turbat ion theory with Sommerfeld-Maue wave functions is represented by the solid curve. The dash-dotted curve corresponds to first-order per turbat ion theory using the partial-wave expansion, see Eq. (10.17); the dashed curve represents


16

~12 > Q)

c~

Q.

I,I -0

b 4 -(3

I ' I ' I ' I ' I '

"'t', $16+ + AU79+ "l, ELob= 200GeV/u

! |

, I , I ~ I , I i I i

5 lO 15 20 25 50

Ep(mc 2)

Figure 10.9. Single-differential cross sections in S 16+ ~- Au 79+ collisions at E]~b = 200 GeV/u as a function of the total positron energy Ep. Dashed curve, QED calculation [BOS89]. Solid curve, first-order perturbation theory with Sommerfeld-Maue wave functions [IOE93]. Experimental data are from [VaD92].

results of QED calculations, see Sec. 10.2.1; and the circles represent the equivalent-photon method for low energy transfer combined with perturbative QED calculation for high energy transfer. The last results agree with corresponding calculations by Racah [Rac37]. The Sommerfeld-Maue results are in good accord with those of the partial-wave expansion for

_< 10. For high relativistic energies, the partial-wave expansion, which was confined to 1 <_ 10, begins to fail. At the highest energies, the results of [BOS89] and those of [Eby91] slightly exceed the Sommerfeld-Maue cross sections (20% at - t ' - 100 and 10% at higher values).

Figure 10.9 gives a comparison of theoretical and experimental single- differential cross sections da/dEp for the collision system S 16+ + Au 79+ at Elab = 200 GeV/u as a function of the total positron energy Ep. The different symbols represent measured cross sections at different magnetic- field settings. The solid curve shows results of perturbation theory with Sommerfeld-Maue wave functions, while the dashed curve is derived from QED calculations. The discrepancy between the former theory and the data may be due to a failure of the Sommerfeld-Maue wave functions at


low positron energies.

1 0 . 2 . 3 T h e e q u i v a l e n t - p h o t o n m e t h o d

It is shown in Sec. 3.4 that the electromagnetic field produced by an extremely fast moving charge can be replaced with a short but intense pulse of electromagnetic radiation composed of (almost) real photons. A real (on-shell) photon is characterized by the relation k 2 = a~ 2 - k 2 = 0 and by transverse polarization.

Soon after the discovery of the positron, the equivalent-photon method was applied to the production of electron-positron pairs in the collision between very energetic charged particles. For early references, see Heitler [Hei54]. In these calculations, one convolutes the cross section for the production of an electron-positron pair by a real photon in the field of a static charge, with the photon spectrum N(a~) representing the transient field of the projectile. In a more symmetric formulation, both nuclear charges in their center of mass are replaced with equivalent photon pulses. This is the two-photon particle production mechanism, analyzed in detail in the review article of Budnev et al. [BuG75]; a recent summary is given in [BOS92]. In this description, the two-photon pair production cross section ~ is convoluted with the photon spectra N(w) and N(w') arising from both collision partners. Such a description should be most appropriate for colliders in which extremely relativistic beams of counter-propagating identical charges Ze intersect, so that the center-of-mass system coincides with the laboratory frame.

In the current subsection, we discuss this situation, that is, we reduce the process

Z + Z --~ Z + Z + e +e- (10.22)

to the subprocess "~ + "7 ---, e + e- . (10.23)

For the maximum frequency occurring in Eq. (10.23), a rough estimate is given in Eq. (3.47)

j_~c. (10.24) C0max ~ bmin

For example, for the proposed Large Hadron Collider at CERN, one has ~ 3500, and if bmin is estimated by the nuclear radius of Pb, one may

achieve ~COma x ~ 100 G e V .

Differential cross section for pair production in distant collisions

If one is interested in distant collisions and the impact-parameter dependence is not relevant, then the pair production cross section may be


obtained from the impact-parameter-integrated photon flux N(02). One may use Eq. (3.51) in conjunction with Eq. (3.48) for a point charge or Eq. (3.66) for an extended charge distribution. As a cut-off parameter entering in Eq. (3.48), it is reasonable to choose the Compton wavelength Ac = 1~me = 386 fro. Adopting this choice, Baron and Baur [BaB92] calculate the differential pair production cross section in the form

-- / d021/d022 N(021)N(022)do-,), 3, (10.25) d o - p a i r

for a collider geometry, i.e., in the center-of-mass frame of the heavy ions. However, since the energies 021 and 022 of the colliding photons are usually different, the center-of-mass (or center-of momentum) frame of the 77 sub- system is different for each combination of 021 and 022 and does not coincide with the laboratory frame. In such a situation, it is convenient to express the cross section o-~ in terms of the invariant Mandelstam variables s, t, and u defined in ]Eqs. (2.20) to (2.22). In the case of colliding photons, we have

s = W 2 = 4021022, (10.26)

and, if Ep_, p_, and 0 are energy, momentum, and emission angle of the positron with respect to the direction of propagation of the photon 022 in the laboratory system,

2 2021 (F_,p_ p_ COS 0 ) t - - m e - -

2 2022(Ep_ + p_ cos0), (10.27) U - - / r t e - -

where the transverse momenta of the equivalent photons have been neglected, since they are of the order 1/7 compared to the longitudinal ones. This can be seen from the Poynting vector (3.39) and its relative contributions to the frequency spectrum (3.46) or from Fig. 3.6. The relation (2.23) connecting s, t, and u then translates into the kinematical relation

2 0 2 1 0 2 2 - - 0 2 1 (gp_ - - p_ cos 0) + 022 (Ep_ + p_ cos 0). (10.28)

The covariant formulation of the on-shell photon-photon cross section o-~ within perturbation theory can be found in [BUG75] and [BeC82]. In the laboratory system, one obtains [BaB92]

do-.)..), z

2 2 a 2 1 2 W l W 2 - - m e + (021022 - - m e ) s i n 2 0

2wl w2 Ep+ Ep_ m 2 --1- (021022 - - m 2) sin2 0

2)2 sin 4 0 _ 2 ( ~ 1 ~ 2 - rn~ ~(col + x 2 - E p + -

[me 2 + (021022 - me 2) sin 2 0] 2 ; Ep_ ) d3p_.

(10.29)


10*4

"C" 10 *= > .

10 §

~'~ 10 §

10 ~ "o -'~ 10-' "o

10-=0

10"

L-IO

10-" ..Q

10-"

C 10-'

"~ 10_ = "0

~ ' 'LHC': =~;b :0.p; '

\\ -,..4

l ~ ~ ,'o 1'2 ,'~ ,'6 ,'8 ~o (.,v)

.~ % ~ ' 'LHC':'~;b-:~ '

b) : - 1~ 2'o ,'o ~'o go ,~o 1~o ,~o ,~o ,~0 20o

(MeV)

F i g u r e 10.10. Double-differential cross section for Pb s2+ + Pb s2+ collisions at ~/ = 3400 and fixed angles at 0 = 30 ~ 60 ~ 900 as a function of the energy of the emitted positron (electron). Full lines correspond to Eq. (10.25) with (10.29), while the dashed line is calculated from Eq. (10.30) for 0 = 90 ~ (a) Energy range between threshold and 20 MeV, and (b) energy range between 10 and 200 MeV. From [BaB92].

The same angular d i s t r ibu t ion also holds for the emi t t ed electron. Baron and Baur [BaB92] also derive a compac t analyt ical formula valid if (OJlaJ2-

2 which reads rne 2) sin 2 0 >> me,

d2~ = ct2Z4 8 P l [ em] 2 2 - sin20 de dft 37r2 ~ e3 In --~ sin 6 0 ' (10.30)

where e ,p are electron or pos i t ron energy and m o m e n t u m , e m = 0.68 ~rne, and it is a ssumed tha t the logar i thm is a slowly varying funct ion of the energies. If e << era, one is not very sensit ive to the cut-off p a r a m e t e r bmin enter ing in Eq. (3.48).

F igure 10.10 shows calcula ted energy spec t ra of posi t rons (or equiva- lently electrons) from Pb 82+ + Pb s2+ collisions for LHC condit ions, i.e.,


b ~ b2

b

Figure 10.11. Geometry in the impact parameter plane. Within the equivalent- photon picture, the two photons collide at the point P with distances bl from nucleus 1 and b2 from nucleus 2. The impact parameter b is directed from nucleus 1 (with radius R1) to nucleus 2 (with radius R2). From [BaB93, Bau90a].

for ~ = 3400. The spectra for all values of 0 show a rather sharp peak near threshold. The analytical formula (10.30) is seen to give a rather good approximation to the numerically evaluated equation (10.25).

The treatment described here is intended for peripheral collisions, in which the distance of closest approach in the collision stays much larger than the sum of the nuclear radii. Only then is it permissible to use photon fluxes N(w), in which the integration over the impact parameters b has already been performed. We now include the impact parameter explicitly.

Impact-parameter-dependent cross sections

Since according to Eq. (10.24), the maximum frequency of virtual photons and hence the probability for pair production strongly increases with decreasing impact parameter, it is imperative to investigate the behavior of the cross sections as a function of the impact parameter b by using the photon flux N(wl b) defined in Eq. (3.70) for a point charge and in Eq. (3.69) for an extended nucleus with the form factor f(k2). In analogy to Eq. (10.25), the b-dependent probability for pair production in the frequency intervals (~1, ~dl -~-dwl) and (w2, ~2 + dw2) is given by [BaB93]

d2gPair(b)dwl dw2 = / d2bl / d252 (~(b -~- b2- bl)N(wl,bl)N(w2, b2)

• (s) sin 2 ~a ], (10.31) • [ cos: +

where s = W u = 4 Wlwu. The collision geometry projected into the impact-


parameter plane is illustrated in Fig. 10.11 for colliding nuclei with radii R1 and R2. The angle between bl and b2 is denoted by ~, and a ~ and cr:y~ are the photon-photon cross sections for parallel and perpendicular relative polarization of the photons.

The cross section for pair production in the equivalent-photon approximation is obtained by integrating over the photon frequencies and over the impact-parameter plane as

O'pair - /d2b f dcul f d(M2/d2bl/d2b2c~(b--[-b2-bl)W(a21,bl) • (s) sin 2 ~] (10.32) • c o s +

• (s) -- a ~ ( s ) , the integration over bl and b2 can be carried If a ~ (s) - a~/ out, and one arrives at Eq. (10.25). The result, valid for extreme relativistic collisions, has also been derived in a perturbative treatment in [BaB93, ViG93].

Figure 10.12 shows the one-photon distributions entering in Eq. (10.32) in their dependence on the impact parameter b for various choices of the form factor and a fixed value of the Lorentz factor 7 = 3500 corresponding to the maximum energy of the Large Hadron Collider [ViG93]. The figure confirms that all form factors lead to the same result outside the extended charge distribution. For small impact parameters, the results have to be regarded with some caution since the validity of the equivalent-photon method breaks down if the assumption of a constant field across the transverse linear dimension of the system is no longer satisfied, see See. 3.4.1.

The cross sections for the electromagnetic production of a fermion pair in lowest order of a are given in the covariant form [BUG75, ViG93] as

c@y (s) -

and

47ra 2 ( 4me 2 12m~) ( 6m~) ] 4m~) 1 + s s2 L - 1 + s M O ( s -

(10.33)

0",7, ~

where

4 ~ [( s 8

s ~ L - 1 + s M O ( s -

(10.34)

M

) - 21n ~m--~m + - 1

_ i l 4m~.s (10.35)


10 2 ~_",. ; ; ; ; _ ' . . . . . . _

I

>= lo 0

IE _ 10 -2

3 "~ 10-~

N \ 10-6

10-o

I

10 Meu

m

i

. i

m

0 5 10 15 20

b (fm) F i g u r e 10.12. Photon distribution N(w,b) - n(~,b) for a Pb nucleus and "7 = 3500 as a function of the impact parameter b for different photon energies. The dash-dotted curve corresponds to a point-like nucleus, Eq. (3.70), the other curves to photon distributions (3.69) of extended charges: the dashed line refers to the form factor (3.57) of a homogeneously charged sphere with radius R0 -- 7.107 fm and the solid line to the Gaussian form factor (3.58) with Q0 = 60 MeV. From [ViG93].

Here, s - 4WlW2 _~ 4m~ is ensured by the Heaviside s tep funct ion O. Equa t ions (10.33) and (10.34) are equally valid for the e lec t romagnet ic produc t ion of o ther fermion pairs if the electron mass me is replaced by the cor responding fermion mass. The cross sections refer to the p roduc t ion by real (i.e., on-shell) photons . Off-shell correct ions can also be appl ied [BUG75, BaB93] and t u rn out to be i m p o r t a n t for e+e - p roduc t ions wi th invariant masses W - ~ _<1 GeV.

As is cus tomary, we have used here the invariant mass W - v ~ - V/4CdlCd 2 since the final d i lepton s ta te depends on a number of pa rame te r s in the collider ( labora tory) system. Ano the r convenient p a r a m e t e r is the rapid i ty X defined in Eq. (2.15). Transverse m o m e n t a being negligible in the collider frame, one can define the to ta l energy as Cd 1 -~-(2 2 and the to ta l m o m e n t u m in the b e a m direct ion as pz - Cdl - - ~ M 2 . From Eq. (2.18) it follows tha t

~ = P z D J1 - - ~22

m~c v/4 ~1~2

10.2. P R O D U C T I O N OF FREE E L E C T R O N - P O S I T R O N PAIRS 303

1 Wl w2 - s i n h ~ l n (10.36) 2 a~2

so that from Eq. (2.15) the rapidity is

1 cO 1 X - ~ i n - - , (10.37)

O32

w eX and w2 w e-X. or Wl - y - -5- Rewriting Eq. (10.31) in terms of the variables W and X and adopting an

off-shell corrected photon cross section ~ averaged over the polarizations, one obtains [BaB93]

/0- /0- /0 d2a - 27r bl dbl b2 db2 d~ O ( b - 2Ro) W dW dx

• N(Tw N(T - (10.38)

Here, it is explicitly assumed that the nuclei with radii R0 do not penetrate each other so that no nuclear debris and other hadronic background arises. Figure 10.13 shows the differential cross section for e+e - pair production as a function of the invariant mass M - W for Lorentz factors 7 - 10 corresponding to the SPS, 7 - 100 corresponding to RHIC, and 7 - 3400 corresponding to LHC. The rapidity is taken as X - 0, that is, Wl - w2. It is seen that with increasing "7, more collision energy becomes available for producing the invariant mass W. The cross sections for b _> 2R0 exceed the cross sections for electromagnetic pair production at b < 2R0 by almost one order of magnitude [BaB93]. If the sharp cut-off function O ( b - 2R0) is replaced with a soft cut-off Fermi function {1 + exp[ (b - 2Ro)/a]} -1, where a is a thickness parameter, the cross sections are not affected significantly.

Cautioning remarks

In concluding this section, it is appropriate to recall the limitations of the equivalent-photon method. When replacing the transient electromagnetic field of a fast-moving charge, two decisive approximations are made: (a) the virtual (off-shell) photons produced by the moving charge are replaced with real (on-shell) photons and (b) the impact-parameter-dependent electromagnetic fields produced by the moving charge are replaced with plane waves whose amplitudes do not vary in the transverse direction. That is, even if the photon spectrum is taken to be b dependent, there is no complete equivalence between the original field and its substi tute for per turbed systems that occupy a transverse space region comparable or larger than the range Ab, over which the original field changes appreciably.


10 s

0 4 > I -

r~ 05 1 - .Q E 10 2-

=o 101-

~- , 0 o >.- 1 - "1o ~Z lO - t "1o

~ " 10 -z_ N

-u o-3_ -4

IO

I I I I I I I I I I I I I I

LHC

e*e- production ~ 2OSpb + zOSpb ~ '

o.1 1.0 M (GeV)

Figu re 10.13. Double differential cross section for e + e- production as a function of the invariant mass M - W for Y - X = 0 in Pb + Pb collisions. The impact parameter b > 2R0 is chosen so that the nuclei do not penetrate each other during the collision. The Lorentz factors are 7 -- 10 for the CERN Super Proton Synchrotron (SPS), 3' - 100 for the Relativistic Heavy-Ion Collider (RHIC), and 7 = 3400 for the Large Hadron Collider (LHC). From [BaB93].

This limits the applicability of the method to impact parameters b >> Ab. For pair production, one may take Ab _~ )~, since the localization of a Dirac electron to be t te r than the Compton wavelength ~c in itself requires the use of negative-energy states [NEW49]. In the absence of rigorous tests of the viability of the equivalent-photon method for impact parameters b << )~, results for b comparable to the nuclear radii should be viewed with some caution.

In a quant i ta t ive way, the applicability of the equivalent-photon method for electron-positron pair product ion is examined by Hencken et al. [HeT94] in a comparison with exactly calculated second-order cross sections for im-

pact parameters b - 0. In Eq. (10.32), then r - 0 and only a~.~(s) con- t r ibutes to the cross section. It turns out tha t the differential probabilities calculated with the equivalent-photon method are by more than an order of magni tude too large for small emission angles (< 0.1 ~ of the posi tron and only begin to agree with the exactly evaluated second-order probabilities at angles of more than a few degrees. As the total probabili ty is dominated by forward emission, the equivalent-photon method is not suitable for a description of electron-positron creation at small impact parameters .

10.3. B O U N D - E L E C T R O N - P O S I T R O N PAIR P R O D U C T I O N 305

10.3 B o u n d - e l e c t r o n - positron pair production

When an electron-positron pair is created in a relativistic heavy-ion collision, the electron may find itself in a bound state of the target or of the projectile, provided its momentum can be accommodated in the bound-state momentum distribution. Since K-shell wave functions offer the broadest momentum spread, production in the K-shell is dominant.

In this subsection, we consider electron-positron pair formation by the transient electromagnetic field of a moving charge for the case where the electron ends up in a bound state. This production mechanism is not a capture or transfer process which is characterized by a transition between two inertial frames moving with respect to each other. However, in a full two-center description, in principle it could happen that the electron is asymptotically bound to one of the nuclei while the positron is in a continuum state of the other nucleus. We may denote this transfer-like reaction, which has not been theoretically investigated so far, as "charge transfer from the negative-energy continuum". Presumably, this mechanism would not have the charge dependence Z 2 characteristic for a perturbing nuclear charge Ze and would be less important at high energies than the ionization- like bound-free pair production discussed here.

Bound-e lec t ron- free-positron (or bound-free) pair production changes the charge state of one of the colliding ions, therefore, this process is of great interest for the design and operation of relativistic heavy-ion colliders. The reason is that heavy ions whose charge state is decreased by one unit get lost from the beam in the storage ring, thus limiting its luminosity and lifetime. The potential harm is even greater because, in contrast to all electron capture processes, the cross section for bound-free pair production increases rather than decreases with the collision energy.

10.3.1 First-order perturbation theory

Let us consider the process in which the electron is created in a K-shell orbit of the target. Since creation in a bound projectile state may be analogously described in the projectile system, this choice can be made without any loss of generality.

In first-order t ime-dependent perturbation theory, the differential probability for the production of an electron in a bound state of the target and a positron with an energy Ep can be written as

dP(b) _ E IAep(b) 12 (10.39)

dEp t~p ~ p ~te

306 C H A P T E R 10. E L E C T R O N - P O S I T R O N P A I R P R O D U C T I O N

10 -9

/ ~ " ~ - . ~

2 u I0-I0 'E

e,~O.

N 10-11

._. 1 0 - 1 2

10-13 . . . . I . . . . 2.5 5.0 7.5 10.0

Ep(mC z) 12.5 15.0

Figure 10.14. Differential cross section for bound-free pair production into the K-shell in Zp + S 16+ collisions as a function of the positron energy Ep for the laboratory projectile energies E = 1 (bottom), 15 (middle), and 100 (top) GeV/u. Solid curves: partial-wave analysis; dashed curves: Darwin and Sommer- feld-Maue wave functions. All quantities are given in relativistic units such that h 2 / m 3 c 4 - 2.9 barn/keV and the kinetic energy of the positron is (Ep - 1)meC 2. From [Bec87].

with Aep(b) being defined by Eq. (10.15), subject to the only change that qPp+ now is a bound-state Coulomb-Dirac wave function ~e, see Eqs. (4.100) and (4.101), while the exact positron continuum wave function is given by the partial wave (4.113) and (4.115). The evaluation of Eq. (10.39) proceeds along the same lines as for ionization, see See. 6.3.

As an alternative to exact Dirac wave functions [BeG87a, Bec87], one may also use approximate electron and positron states [Bec87] and thus avoid the partial-wave analysis. As an approximation to the K-shell wave function, one may choose the Darwin form (4.109) and for the positron state the Sommerfeld-Maue wave function (4.133). These "quasirelativistic" wave functions are valid if c~Z << 1 and, consequently, only the leading terms of c~ZT are retained in the derivation. Similarly, as in Sec. 10.2.2, care has to be taken at which stage the approximation is introduced in evaluating Eq. (10.15). Rewriting Eq. (5.109) for bound-e lec t ron- positron

10.3. BO U N D - E L E C T R O N - P O S I T R O N P A I R P R O D U C T I O N 307

pair production, we obtain from Eq. (10.15)

(~el(1 -- ~Ctz)eiq'rl~p} =

+

1

~2 ~ ( ~ e ] ( q x C t x + qyC~y)ezqrl~p). (10.40) Ep n t- Ee

The first term on the right-hand side of Eq. (10.40) proportional to 1/? 2 = 1 - /32 comes about as the difference between two matrix elements, which almost cancel for large ~/ only if they are evaluated between exact eigenstates. With the reformulation (10.40), the correct asymptotic behavior of the cross section as (ln ?) for y ~ oc is ensured even when approximate wave functions are used in evaluating the right-hand side.

Using Darwin and Sommerfeld-Maue wave functions (Sees. 4.3.3 and 4.4.3) for an electron and a positron in the target potential and keeping the lowest-order terms in c~Zw, the angle-differential cross section d2cr/dEp df~p and the angle-integrated cross section dcr/dEp can be expressed as one- dimensional integrals over the momentum q. For the resu l t ing- somewhat lengthy f o r m u l a s - the reader is referred to the original paper by Becker [Bec87].

Figure 10.14 shows the angle-integrated cross section for bound-free pair production in Zp + S 16+ collisions at 1, 15, and 100 GeV/u as a function of the positron total energy. The solid curves are calculated with approximate wave functions while the dashed curves represent results obtained from a partial-wave analysis. The results scale with Z~. For high projectile energies, the curves differ because the partial-wave expansion is cut off for ]np] > 10. The positron spectra are seen to follow a similar behavior as in Fig. 10.3 for a fixed impact parameter b and in Fig. 10.6 for the b-integrated cross section. With higher target charges, the discrepancy between the methods of calculation increases, so that for ZT -- 92 the t reatment using an c~Z expansion underestimates the cross section by about a factor of 4 compared to the t runcated partial-wave treatment.

Combining the projectile and the target charge dependence, the cross sections scale approximately as

dcr

dEp o( (aZp)2 (aZT) 5 (10.41)

for c~Zp,w << 1. Here, the Z~ dependence reflects the availability of high- momentum components in a bound-state wave function and also occurs for charge transfer. In order to exhibit this dependence in Fig. 10.15, the total cross sections obtained by numerically integrating over the positron spectra are plotted in such a way that Eq. (10.41) gives a horizontal line


10-2

v e~a . 1',4

10 i--

N

N ...,,

c)

10 -~,

! ! ! ! ! ! i 1 [ i ! ! ! ! ! l ! i

\ .

B

" \ ~ \ !

\ \ .

\ .

i i i i ! , i l l l I I I I I i i

5 10 50 100 ZT

Figure 10.15. Cross section for bound-electron (K-orbit) - positron pair production as a function of the target charge for laboratory projectile energies E = 1 (bottom) and E = 15 (top) GeV/u. Solid curves: partial-wave expansion; dashed curves: Darwin and Sommerfeld-Maue approximations; and dash-dot curves: Z 5 scaling. From [Bec87].

[Bec87]. The results derived from the a Z expansion (solid line) and from the partial wave expansion (dashed line) are in rather good agreement with one another for not too high target charges. The simple scaling behavior of Eq. (10.41) does not hold except for the lowest charges Zp. A charge dependence different from Eq. (10.41) has been obtained by a convolution prescription [RhBSg]; however, it has been criticized in [Eic90, Bau911.

Perturbative estimates of the cross sections for bound-free pair production have been obtained in a closed analytical form by Bertulani and Baur [BeB88] for a Z << 1 (in order to be able to use quasirelativistic wave functions). For the dominant K-shell capture they get

1 [ l n ( ~ ) 5 ] .bound ,-,o 3371" (o~Zp)2(OzZT)6 j~ e 27r~Zw 1 a~p ~ 10 _ - g (10.42)

where d = 0.681 is a number related to Euler's constant and ~ = h/mec is the Compton wavelength. Equation (10.42) is valid [BeBS8] for 7 > 50 and has the same dependence on target and projectile charges for aZT << 1 as expressed by Eq. (10.41). The ratio of the total cross section for bound-free

10.3. B O U N D - E L E C T R O N - P O S I T R O N P A I R P R O D U C T I O N 309

pair production as compared to free pair production is obtained as [BeB88]

. b o u n d 337r [ ( ~ ) ] - 2 O e P ~ (OLZT) 3 In . (10.43)

r r f r e e 2 0 ~ e p

This means that bound-free pair production becomes relatively more important for high-Z target atoms, as is to be expected since high-Z atoms provide the high-momentum components in their wave functions needed to accommodate energetic electrons.

Some other earlier calculations at tempting to include into the formulation the distortions of the electron and positron wave functions by the projectile are discussed in [Eic90]. Some of the remarks made in Sec. 8.6 apply here as well. So far, there seems to be no fully satisfactory treatment of these distortion effects, which remain an important problem for the future.

For very high values of the Lorentz factor 3' typical for colliders, a sim- plified treatment is possible. Baltz et al. [BAR91] show that a considerable simplification is achieved by expanding the multipole matrix elements defined in analogy to (6.47) in terms of the small parameter 1/3, and retaining only the first-order term. Moreover, this expansion allows one to separate the interaction into 3,-independent and 7-dependent terms. In analogy to the technique of removing the troublesome 1/R long-range terms in the formulation of excitation, ionization, and charge transfer by a suitable gauge transformation [ToE90b] (see Secs. 6.1.2, 8.3, and 8.7), the 7 dependence for small impact parameters b can also be removed up to higher-order terms in 1/7. This means that the contributions to the cross section from the small b range, which requires a nonperturbative calculation because of the strong Coulomb interaction (see Sec. 10.3.2), are independent of "7. On the other hand, in the perturbative regime for b larger than about 5)~, the 7 dependence is given by In 7, in accordance with %% (10.42). As a result, Baltz et al. [BaR91] predict the total cross section for bound-e lec t ron- free-positron pair production to be of the form

b o u n d _ A In 7 + B (10.44) e p ~

where 7 = 7FT is the equivalent fixed-target Lorentz parameter (10.2) and A as well as B are independent of 7 (to within higher orders of 1/~). The quantity A has to be estimated from perturbation theory, while B receives contributions from the perturbative as well as the nonperturbative impact parameter range. In a detailed estimate, Baltz et al. [BaR93a] arrive at a relation for Au 79+ -t- Au 79+ collisions as

o . b o u n d ep ~ (11.2 In 7 - 14.0) barn. (10.45)

310 CHAPTER 10. ELECTRON-POSITRON PAIR PROD UCTION

For a (fixed-target) value ")/FT - - 2.33 x 104, one has _bound ~ 99 barn Oep ,~

For further results on Pb + Pb collisions using very large basis sets, see also [BaR94]. It is worthwhile mentioning that the simple estimate of Eq. (10.42), applicable to a Z << 1, has the same structure as Eq. (10.44).

10.3.2 Nonperturbat ive calculations

One expects that perturbation theory breaks down for the description of pair production in collisions of very heavy ions (Au + Au, U + U) at very small impact parameters b since the interaction then becomes very strong. Carrying the treatment beyond lowest-order perturbation theory will lead to two consequences: first, there will be a modification of the perturbative results for single pair production; and second, one will obtain a finite probability for the production of multiple electron-positron pairs, to be discussed in Sec. 10.4 together with the bounds imposed by the unitarity requirement.

While nonperturbative calculations for free electron-positron pairs have to cope with the difficulty of handling two continuum states, bound-free pair production is more accessible to a theoretical treatment since it involves one bound and one continuum state, similarly as ionization, see Secs. 6.4 - 6.6. We start our discussion with numerical solutions of the time-dependent Dirac equation on a grid and then proceed to coupled-channel calculations.

Finite-difference methods

The most direct way to solve the time-dependent Dirac equation (5.39)

[ ZT e2 Zpe 2 i ~ ~(rT, t) -- [ - - i a - V T ~(1 --/3a~) ~ + me7 ~ t~(rT, t)

r T r p

(10.46) is by purely numerical methods on a grid. Together with their treatment of ionization, see Sec 6.4, Thiel et al. [ThB92] solve Eq. (10.46) for U 92+ + U91+(ls) collisions at 10 GeV/u assuming an impact parameter b = 0. In this way, they enforce cylindrical symmetry and keep the problem at two dimensions in space. The impact parameter b = 0 carries zero weight for the cross section, but it should be most suitable to study nonperturbative effects. The numerical methods are outlined in Sec. 6.4. In the initial state, at a time when the projectile is about 5.7 uranium K-shell radii away, the hydrogenic target atom is assumed to be in an unperturbed lSl/2 ground state. The time evolution of the electron density is depicted in Fig. 6.11. The final wave function is projected onto excited target states, positive-energy continuum states for ionization, and negative-energy continuum states for bound-free pair production. The numerical results show

10.3. B O U N D - E L E C T R O N - POSITRON PAIR PRODUCTION 311

the strongly nonperturbative character of pair production in collisions between the heaviest ions at zero impact parameters. The probability for pair production is about two orders of magnitude larger than the value obtained in perturbation theory. This is in qualitative agreement with earlier results of Strayer et al. [StB90] obtained by solving the Dirac equation on a lattice using a so-called B-spline collocation method.

As has been pointed out in [Eic90] and [BaR93b], the observed nonperturbative effects may be strongly dependent on the gauge chosen for solving Eq. (10.46). To put it simply, the long-range electromagnetic interaction in Eq. (10.46) cannot be expected to leave the target atom still unperturbed at a distance of a few K-shell radii. For a more realistic solution, it is necessary to remove the long-range part of the interaction by a suitable gauge transformation (e.g., by imposing Coulomb boundary conditions), see Sec. 6.1.2.

Wells et al. [WeO92] use a computationally efficient basis-spline collocation method [BOS87, UmW91] to calculate bound-free muon pair production (for computational ease) in Au 79+ + Au 79+ collisions at 100 GeV/u collider energy. In particular, they discuss the influence of the gauge chosen for the calculation. In agreement with the observations made above, they find the Lorentz gauge embodied in Eq. (10.46) too demanding computationally because of the long transverse range and the compression in the z-direction of the interaction which requires very small time steps. Instead, they propose an "axial gauge," which is similar to the gauge discussed in See. 6.5.4 and is regarded to be more suitable than the Lorentz gauge for the lattice implementation in extreme relativistic collisions.

Single-center coupled-channel calculations

Another method to investigate the nonperturbative behavior at small impact parameters is furnished by single-center coupled-channel calculations [RUM91, RuS93]. These calculations are discussed in the context of ionization in Sec. 6.6.2. The continuum states of the positron are described by stationary wave packets (6.74) and the interaction is decomposed into multipoles in coordinate space, see See. 6.3.1.

Figure 10.16 shows the dependence of bound-free pair creation probability on the impact parameter b for 1.2-GeV/u Pb + Pb collisions. At small impact parameters, we notice the large deviation of perturbation theory from the results of coupled-channel calculations. With increasing impact parameters, the ratio of the probabilities quickly decreases, and for b > 600 fin ~ aK, perturbation theory yields about the same results as coupled- channel calculations. The rate of pair production then decreases almost exponentially with the impact parameter.


10-3

A

. 0

10

lO-S 0

" ' I ' I ' I ' I ' I ' o,

%%%%%% %%

.. ...

.. %%%%%%%%%%%%%%%%%%%%% ..

, I | I | I i I i I - ~ i

I00 200 300 tOO 500

b If m) Figure 10.16. Probability for bound-electron (K-orbit) - positron pair production in 1.2-GeV/u Pb + Pb collisions as a function of the impact parameter b. Dashed line: coupled-channel calculations; solid line: perturbation theory. From [RuS93].

While coupled-channel calculations with an infinitely large basis set are equivalent to solving the Dirac equation exactly and hence are gauge- independent, calculations with a small basis size depend strongly on the choice of the gauge [BaR93b]. Therefore, the comparison between nonperturbative and perturbative calculations is affected by the gauge and the size of the basis set. Baltz et al. [BAR94] have investigated the 1.2-GeV/u Pb + Pb system in detail, using the standard Lorentz gauge of [RuS93]. They find that the enhancement of the nonperturbative over the perturbation theory results is decreased from almost two orders of magnitude to a factor of 9 at b ~ 0, if the basis size is increased t o Inmax[ = 6 and to the maximum positron energy IEmaxl = 14.4m~c 2. Quite generally, the nonperturbative enhancement decreases at every impact parameter with increasing size of the basis set. For the total cross section, there is almost no enhancement over the perturbative value.

Two-center coupled-channel calculations

It would be desirable to perform two-center coupled-channel calculations, which treat target and projectile on equal footing. They allow the pair to be created by the projectile in the target frame and by the target in the

10.3. BOUND-ELECTRON- POSITRON PAIR PRODUCTION 313

projectile frame. As has been discussed in Sec. 6.5.1, calculations in which the basis states are centered around the target as well as around the projectile nucleus, provide more flexibility of the expansion in the interaction region and allow for transfer channels in the intermediate states. In particular, charge transfer from the negative-energy continuum of the target to a bound state of the projectile (and vice versa) would be included in this description as a possible reaction mechanism. Unfortunately, calculations are not available at present owing to the difficulties of including continuum channels at both centers.

Magnus expansion of the time evolution operator

A different nonperturbative approach starts from the time development A

operator U(t2,tl), which describes the time evolution of the system from the earlier time tl to the later time t2 and is defined by I~(t2)) =

u(t2,t l)[~(t l)} . In the interaction representation, for tl --+ - o c and t2 ~ +ec, it is given explicitly by the expression [GoW64]

g ( + o c , - e c ) - T exp - g /-/i(t) dt , (10.47) ( X 3

A A

where T is the time-ordering operator and Hi(t ) is the interaction Hamil- tonian in the interaction picture [Sch61]. Instead of expanding the exponential into the usual perturbation series (5.66), one may use a commutator expansion of the exponent known as the Magnus expansion [Mag54, PeL66, BrM92]. It has the merit of manifestly ensuring unitarity and gauge invari- ance in each order of the approximation. Such a treatment was applied to nonrelativistic ionization of hydrogen by high-Z projectiles [Eic77] in order to avoid the overestimates of ionization probabilities at small impact parameters obtained from first-order perturbation theory. In the extension of this approach to pair production lion94], the interaction of the electron and positron with the projectile is included to all orders in the projectile charge Zpe by adopting two basic assumptions: (a) the correlations between the interactions at different times [effected by the time-ordering operator

A

T in Eq. (5.66)] are neglected. This is a reasonable approximation if the effective collision time At _~ b/~/v, see Eq. (3.37), is short compared to the relevant electronic transition time ~- _~ h /AE , where A E is the energy difference between initial and final states. (b) Intermediate multipair states are disregarded.

Let ~ep(b) be the partial first-order probability for creating a pair in the specific state (e,p), where e = {p+, s+} and p = {p_, s_} denote a set of momentum and spin quantum numbers of electron and positron, respectively, see also See. 10.4.2. If now P(1)(b) - }-~e,p 7)ep(b) is the total


perturbative probability for creating a single pair, then the partial probability of pair production within the Magnus approximation is derived as [Ion94]

P~Mag(b) -- Pep(b) sin2 v/P(1)(b) (10.48) p(1) (b) "

An analogous expression was derived in [Wi183] for nonrelativistic ionization and in [RyW78] for nonrelativistic electron transfer. By summing over all electron-positron states e, p, one verifies that the total probability cannot exceed unity, as required by unitarity.

One may regard Eq. (10.48) as a result of unitarizing perturbation theory [MoG93]. In fact, the total pair production probability within the Magnus approximation is always less than the total first-order probability. On the other hand, a comparison of the results of coupled-channel calculations with those of perturbation theory for bound-free pair production, see Fig. 10.16, shows that, according to [MoG93], the nonperturbative probabilities exceed the perturbative results at small impact parameters and moderate relativistic energies.

1 0 . 3 . 3 E x p e r i m e n t a l r e s u l t s

The first observation and measurement of bound-electron- positron pair production has been reported by Belkacem et al. [BeG93]. The experiment was performed at Lawrence Berkeley Laboratory's BEVALAC accelerator using 0.956 GeV/u U 92+ ions incident on thin, fixed targets of At, Ag, Cu, and mylar.

The positrons were detected in coincidence with the charge state of the emerging projectile using an advanced positron spectrometer, see Fig. 13.7. The spectrometer has a very high acceptance of electrons and positrons emitted both forward and backward with respect to the beam direction. By measuring the time of flight of the electrons and positrons for a given energy in the strong field of the solenoid, it is possible to determine the angle of emission with respect to the beam direction.

A comparison of the positron spectra at forward and backward angles with respect to the beam direction indicates that positrons emitted at higher energies are also emitted at smaller (more forward) angles. The measured total cross section for bound-free pair production by 0.956 GeV/u U 92+ on a Au target is 2.19 + 0.25 b. The cross section for free pair production in the same collision is 3.30 + 0.65 b. This is to be compared with a theoretical perturbative value of 1.01 b for bound-free pair production [Bec87] and for free pair production with the values of 5.1 b [BeG86a] and 1.25 b of [Dec91]. The cross section for bound-free production varies ap-

10.4. M U L T I P L E PAIR P R O D U C T I O N 315

proximately as Z~ s+~ as compared to the Z~ dependence expected from perturbation theory.

Experiments [BEG94] including La aT+ projectiles yield a target charge dependence as Z~ 95+~176 for 0.956 GeV/u and Z~ "65+~ at 1.3 GeV/u, while the projectile charge dependence is Z~ 54+~ This is roughly in line with Eq. (10.41) if ZT and Zp are interchanged, because the bound electron is created in the projectile, not in the target.

10.4 Multiple pair production In the discussion of bound-free pair production, Sec.10.3, indications emerge that for collisions between high-Z ions at small impact parameters b, perturbation theory breaks down and higher-order effects have to be taken into account. These calculations, while still burdened with the problem of gauge dependence, give a hint that similar effects must also be expected for the production of free electron-positron pairs.

10.4.1 Limits of perturbative probabilities for single- pair production

Limitations become already evident in perturbative calculations following the early work of Bethe, Maximon, and Davies [BeM54, DAB54]. Disregard- ing Coulomb corrections proportional to the logarithm in 7 of the Lorentz factor and assuming Er Ep >> m~, one obtains an approximate analytical expression [Bau90b] for the first-order impact-parameter-dependent pair production probability in the form

14 (C~ZT) 2 2/~ (7dAc) (10.49) g ( x ) ( b ) - ~ 2 ( o / Z p ) ~-~ In 2 2b '

where d - 0.681 [compare Eq. (10.42) for bound-free pair production]. This expression is valid for )% _< b _< 7dAr For Pb + Pb collisions at 7FT = 3 • 107, see Eq. (10.2), and b - )%, one obtains a probability p(1)(~) _ 5.3, which clearly shows the need for a nonperturbative treatment.

Another violation of unitarity is observed in the total cross section at asymptotically large values of 7. Froissard has derived an upper bound on the total cross section of two strongly interacting particles colliding with the invariant energy W - v G, see Eq. (2.20), by using the Mandelstam representation [Fro61] (for a review see [Val91]) as

~tot _< const. • ln2 ( 8 ) - - as s -~ oc, (10.50) 80


where So is a constant. It is usually assumed that this relation is also valid for electromagnetic processes. If this is so, and since Ins o( In 7 for large values of 7, we have O'to t ~ const, x ln2(7/70), in contradiction to the asymptotic 7 dependence of the partial cross section (10.1), proportional to ln3(7/70), for single-pair production. Another brief discussion of the asymptotic behavior of pair creation can be found in [Bau90c], where also further references are given. While such discussions of unitarity may be only of academic interest as long as it is not specified what asymptotically large values of 7 mean, they point in the same direction as Eq. (10.49), namely that higher-order effects become important.

1 0 . 4 . 2 M u l t i p l e p r o d u c t i o n o f i n d e p e n d e n t e l e c t r o n -

p o s i t r o n p a i r s

Violation of unitarity in perturbative results for extreme relativistic collisions between high-Z ions indicates that some of the probability flux is diverted into other reaction channels, in particular into multiple pair production.

We give here a very simple discussion of multiple production of free electron positron pairs, 2 which, in fact, is the common root of various more sophisticated treatments that have been published so far. The basic assumptions, without which a theory would become exceedingly complicated, are the following: (a) The energy of the colliding high-Z ions is so large that even the production of several pairs does not change their energies or trajectories. (b) The pairs are created completely independently of each other. This means that electrons and positrons do not interact with one another (this is a 1/Z correction) and are not subject to the Pauli principle. Both assumptions should be well justified for extreme relativistic collisions of, say, Au on Au or Pb on Pb.

Let us now suppose that we have calculated an impact-parameter- dependent transition amplitude Ap+p_ (b) = A~p(b), for example from Eqs. (10.15) and (10.19), where e = {p+ , s+} and p = { p _ , s _ } denote a set of momentum and spin quantum numbers of electron and positron, respectively. Then the first-order probability of finding an electron in a state e after the collision is

7)~(b) - E 7)~p (b) - ~ IA~p ]2. (10.51) P P

As long as P~(b) << 1, there is no violation of unitarity and we may take [1 - P~(b)] as the probability of not having created an electron in the state

2Multiple bound-free pair production is not expected to compete with free pair creation because of the restricted phase space.

10.4. MULTIPLE PAIR PROD UCTION 317

e. In general, however, it is necessary to obtain the single-electron probabilities from more exact methods like coupled-channel calculations, which rigorously satisfy the unitari ty requirement. The probability that only one electronic state e and no other positive-energy state is occupied is then

P~(1, b) - T'~(b) H [1 - P~,(b)], (10.52)

where the number N - 1 in P~(1, b) indicates that only one electron has been created. In writing down the product (10.52), the assumption (b), listed above, of independent probabilities has been used. From here we get the probability that exactly one electron (i.e., because of summation over positron states, one pair) is created in any state as

P(1, b) - E T'~(b) H [1 - 7)~,(b)]. (10.53) e Ctr

Rewriting the product term in Eq. (10.53), we find

P(1, b) - EP~(b) exp ( E ln[l - Pc,(b)] ~'r

,~ Z T'~(b) exp ( - Z T)~,(b)) ~, (10.54)

1 Z 2 where we have used the first term in the expansion l n ( l + x) - x - ~ + - . . of the logarithm and no longer exclude the term e ~ - e from the sum, owing to the smallness of P~,(b) for any given state d.

Proceeding to two electrons 11,12, irrespective of their permutation, we have

1 Pele2(2 ' b) -- -~.~ p e l ( b ) p e 2 ( b ) H [1 - 7)~, (b)], (10.55)

e' :J=e 1, ct-~= 62

and, discarding higher-order terms in the small quantities P~,(b),

P(2, b) ~ ~ T'r exp - P~,(b) . (10.56)

In general, for N pairs, with

P(b) - E Pc(b)- E Ar 12' (10.57) e e , p


we obtain the Poisson distribution

P(N, b) - [T'(b)lN N! e-~'(b)' (10.58)

which one would expect for independent events. One verifies that the total probability for creating 0, 1, 2,- . . pairs in a single collision is in agreement with the unitarity requirement. The average number of pairs

(x)

E N P(N,b) - P(b) (10.59) N = I

provides an interesting interpretation of the perturbative (or nonperturbative) single-pair probability P(b). Since the derivation of Eqs. (10.58) and (10.59) is based on the assumption that each individual 7)r << 1 but does not require P(b) = }-~r T'r << 1, Eq. (10.59) offers a possibility to give meaning to unitarity-violating perturbative total pair creation probabilities. Under all the assumptions outlined above, creation probabilities larger than one express, in an approximate fashion, that - on the average

- more than one electron-positron pair is produced. In this sense, perturbation theory can still be used when it yields probabilities exceeding unity.

The Poisson distribution (10.58) has been derived in several ways. Baur [Bau91] used a "sudden approximation" to the S-matrix and a quasi-Boson approximation for describing electron-positron pairs, thus discarding the Pauli principle. Rhoades-Brown and Weneser [RhW91] perform a summation of a perturbational subseries of selected Feynman diagrams including the interaction between the colliding ions and the electron-positron pairs to all orders but dropping the interaction between the leptons. They also discuss coupled-channel calculations. Best et al. [BEG92] adopt a similar nonperturbative approach neglecting quantum mechanical interference terms. While these works differ in the theoretical methods used, they are based essentially on the same assumptions, namely the complete indepen- dence of the individual pair production processes. In each case, it is the goal to derive the structure of Eq. (10.58), not the input probability 7:'(5).

The single-pair probabilities for extreme relativistic collisions may be estimated in several ways, as discussed in Sec. 10.2. One may use lowest- order QED calculations (Sec. 10.2.1), first-order perturbation theory with distorted waves (Sec. 10.2.2), or the equivalent-photon method (Sec. 10.2.3). Coupled-channel calculations for free electron-positron pairs have not been performed so far, owing to computational difficulties.

As a simple application, we show in Figs. 10.17 and 10.18 the N- pair probability distributions [RhW91] calculated from Eq. (10.58) and

10.4. MULTIPLE PAIR PRODUCTION 319

101

10 0

10 -1

~, 10 .2

a~ 10. 3

10 .4

10-5

I I I I I I I I I

%= 38.2 kbarn

~ N1 = a 3 PAIR = 437 barn

a 4~PAIR = 114 barn ~~. ~ . ~ = 28 barn

f',,.=\ N=3 ~ \ N=4 !L" \ N-5

1-PAIR = 32.3 kbarn a 2-PAIR = 1.96 kbarn

~. " P1(b) l ~ . ' I~ ~ x,q

10-6 I ' \ I I I I I I I 0 8 16 24 32 40 xl 03

IF"

! i ' ' . " \

\ .

I

b (fro)

F i g u r e 10.17. Probabil i ty distr ibution for N-pair production in Au + Au colli-

sions at the highest RHIC energy with ~FT -- 2 • 104 as a function of the impact parameter b (in units of 103). Here, P1 (b) is the per turbat ive result corresponding

to Eq. (10.49). It deviates from P(1, b) of Eq. (10.58) only at the smallest impact parameters shown. From [RhW91].

101 I I I I I I I I I

|

10 0 ~ % = 240.7 kbarn a 1-PAIR = 192.4 kbarn _ ...L N 1 a 2-PAIR = 9.9 kbarn

10.1 ~-'~ = a 3-PAIR = 3.3 kbarn _ ~\ ~ ~4-PAIR = 1.5 kbarn ~ i \ ~ a5PAIR =820barn

._. 10 .2 -

" " 10 .3 ~i" .W---N=2 ~ - I

I ~, "- "x '~-" 10 .4 --I~ ".. -',,

I IL.," Pl(b) I ~- ".. "-. ,N=4.

10 .5 I I "... o-6 l ' , 1

0 8

~'.,,. N=5 ~o~

I 1~... 16

! I I I !

b (fm) 24 32 4o x o3

F i g u r e 10.18. Same as Figure 10.17, but for Pb + Pb collisions at the highest

LHC energy with Y E T - - 3 x 107. From [RhW91].


from Eq. (10.49) for the approximate single-pair production probability 7)(b) - P(1)(b). Also shown in these figures are the original lowest-order perturbative results (10.49). The curves are plotted for the highest envi- sioned beam energies and charge numbers of the respective facilities. It can be seen that single-pair creation dominates the picture at RHIC as well as at LHC. The N-pair production probabilities with N _> 2 are seen to decrease veryrapidly for the depicted impact-parameter range b _> ~ .

The method outlined in this section will certainly have limitations when the impact parameters become very small, that is when they get close to the sum of the nuclear radii. Then the probability for single-pair production has to be calculated nonperturbatively and may become very large. It is not certain whether the basic assumptions for deriving the Poisson distributions (10.58) are justified any longer. However, for the total pair multiplicity, the regime of very small impact parameters carries little weight.

Documents

Chapter 10 - Electron-positron Pair Production