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IL NUOV0 CIMENTO VOL. L A, N. 2 21 Luglio 1967
D o u b l e P h o t o n E m i s s i o n in e• - Col l is ions.
P. DI VECCHIA ~nd ~1. GRECO
Laboratori Naz iona l i del C N E N - Frascat i
(ricevuto il 16 Dicembre 1966)
Summary. - I n th i s work we e w d u a t e , for t he process e ' : ~ e - ~ - - > e • 2 4 7 2 4 7 the angu l a r d i s t r i b u t i o n of the two pho tons , t h e i r ene rgy s p e c t r u m a n d t he toted cross-sect ion for e m i t t e d p h o t o n s of energy ~ s , in t he ex t r e m e re la t iv i s t ic l imit .
1 . - I n t r o d u c t i o n .
I n the ,~ttempt to observe two-photon annihil,~tion of electron-posi tron p~irs wi th ADA (1) ~ n u m b e r of events were detec ted in which the two-
pho ton coincidences between two le~d-glass ( ' e renkov counters could be inter- p re ted ~s due to the process bf double bremsst r~hlung
(1) e + + e - - + e + + e- + 2 y .
I f the energy of the incident electrons ~md posi trons is sufficiently high,
process (1) would domina te over two-qu~nt~ ~nnihih~tion by ,~ f~etor of the
order ot"(E/m)" bec~mse under the conditions of the exper iments the four- m o m e n t u m tr~msfer betweelI the colliding p~rticles of process (1) is sm~ller
t lmn the corresponding qu~mtity for two-qu~mt~ ,,mnihil~tion. This p rope r ty
of the process (1) :~llowed one to pu t it on the list of possible moni tor ing
processes, defined ~s collision processes between electrons ~nd posi tron sin which
the m o m e n t u m t ransfer could be considered sufficiently sm~ll, so theft its
description by q m m t u m ele~'trodymmlics with unit form f~ctors ~nd unmodified
(1} (!. BEI~NARDINI, G. 1 ~. CORAZZA, G. DI GIU6NO. J. HAISSINSKI. P. MARIN, R. QUE~lZOLI ~md B. TOI'SCIIEK: N'ttOCO Cimot to , 34. 1473 (1964).
(2) p. DI VECCHIA: Nuovo CimeMo, 45. 249 (1966).
320 e. DI ViECCtIIA ~nd M. GRECO
propaga tors could be considered accurate . Besides it seemed tha t process (1) presented some par t icular advantao'es wi th regard to other moni tor ing pro-
cesses (2). The cross-section for process (1) has been calculated previously by
BANDER (3) and by BAYEI~ and GALITSKY (4), bu t thei r results did not aRTee.
We set out to unders tand the reason for this discrepancy. In view of the
lengthiness and complicat ion of the k inemat ic pa r t of the calculat ion this
inves t igat ion was per formed along two independent lines by the authors .
The results will be compared wi th those publ ished by other authors , so as to
obta in a complete pic ture of process (1). Our results agree wi th those obta ined by BAYER and GALITSKY.
2. - Results.
The Feynman diagra,ns cont r ibu t ing to (1) in lowest order are ob ta ined
by adding in all possible ways two ex te rna l pho ton lines to the two graphs
of the elastic scat ter ing and of the annihilation. I n the high-energy and small-angle app rox ima t ion only the leading gTaphs
in Fig. l a have been considered, the cont r ibu t ion of the graphs of Fig. lb and Fig. l c being considered negligible. This is cer ta inly correct for an exper iment which looks main ly at the photons in the fo rward -backward direction, but is not
k~ k k, k
P2 h P" P-~ P" P~ P" P~ &
k 2 k 2 k 2 k 2
Fig. la. - Leading graphs.
cf,Jk' f k= ~ "-~k2
k k 2
Fig. lb. - Some negligible scattering graphs.
(a) ]V[. BANDER: SLAC-TN-64-93 (1964). (4) V. •. BAYER ~tnd V. M. CxALITSKV: J E T P Lett.. 2, 165 (1965).
P3
&
k~ k 2
DOUBLE PIIOTON EMISSION IN tt~:~ ' - COLLISIONS 321
P~ P,, Pz P4, P~ P~ P~ P~
Fig. lc. - Some ncgli[tible annihih~tion graphs.
so obvious to hold for the to ta l cross-section. The reason tha t makes the
annihi la t ion graphs negligible can be inferred f rom an inspect ion of the expres-
sion of the pho ton propaga tor . Indeed the order of magn i tude of the pho ton
p ropaga to r in the annihi la t ion ~ a p h s is E(E--o)I.2) in c.m.s., while the prop-
~gator of the sca t te r ing gr~phs ~oes to .m: (m is the electron mass). The resul t ing cross-section is
,(2) cr =
[ ( p ~ p ~ ) 2 ,~q~ E~ E~ o,~ "
f d~.~. ,~(p~ + p~-- ~,~-- p,-- kl-- 1,~) y l M I ~ �9 (:0. ( P l - - 1)3 - k l ) 4 spin~ I~o] 4
where P~(P2) and p.~(p4) are the f o u r - m o m e n t a of the incoming ~nd outgoing
e lect ron (positron), k~, k2 those of the two photons and the m a x t r i x element M is
(3)
with
(4a)
"C4 M = u(pa)J)uu(pl)v(p~.)(, r(P4) -')(,')7~) ;'
~ , = V,Y'P~-- y ' k ~ + m y 'P3 + y ' k ~ + m 2(t, lkz) ~ ' "(~1 ~ ~] ' e l 7#,
- - 2(pf lq )
- - Y "P4-- y" k2 + m Y" e2 ~ Y" ('~ - - y . p,., + y . k,, + m (4b) C~, ~F, 2(pfl,:2) -- 2 (pd,~)
After el imination of the b-function the cross-section (2) becomes
(5) = [ ( p ~ p S - "~%~ ~'~ ~,~ "
. l ' d ~ [ d[lJP3] /)2 ]r 1)4 p~O1Spi n t%'1 ~Ej, J L dE, E~E,(~,-- ~ - - ~ l ~ I~ ' P ~ P f
where P~(E~) and Pf(E:) are the to ta l m o m e n t u m (energy) of the sys tem in the init ial and final states. The calculat ion of the integral (5) has been per- formed in the center-of-mass frame, where E l = E 2 = E and P ~ = - - p 2 .
21 - l l N~eovo Cimento A.
322 r. DI VECCIIIA and 5L GR, ECO
I n order to obtMn information about the high-energy beh,~viour of the process,
the special case in which one photon follows exact ly the direction of the
incident positron has been t reated in detail. The cross-section for this forw,~rd-
backward emission is (~)
(6) ro dXl dx,2 (1 - - x~) ( 1 do'j~..1~. ~ - ~ 3 :iel d ' 2
- x~) + (l - , r : )x~ +
(1 - .~,) ,r~ ~-x~,~t )
y~ ds dD._, ,
where x,,~ = r ~nd dD,.~ are the elements of solid angle of the two photons.
The result (6) is vMid only for d~Q~.2 << 7~[7~ ~nd this is too s trong ~ limita-
t ion on a detection device.
The total section, whi('h gives the energy distribution of the radiation, has
independent ly by the two authors, who found been calculated from (5)
respectively
(Ta,) d~ =
(7b) d~
where
o 2 8 ~ " r o d x 1 d3L 2
77; ;,t~ i W. 2
3'1 ;~'2
(M.G.) ,
(P.I).V.) ,
~(3) - - - . n = l ?~3
5/o , and both The result (7a) is given with an error about the order of o/
agree in the limit of low frequencies with the Bloeh-Nordsieck cross-section
daB.~, if):
(8) do-~.~, s ~ r o d~,~ d<. + ,~ ~( , ) (.) - - - .
The cross-sections (6) and (7) have been evaluated in this paper for the case
(') The numerical coefficient 2, which appears in a previous paper (2), is not correct and must be substituted with 5/4. The same results have been found by BAYER and GALITSKr (%
(~) V. N. BAYER and V. M. GALIXSK,': Phys. Lett., 13, 355 (1964).
I ) O I ' B L ) ; P H O T O N F M I S S 1 O N ] N ( ' s C O L L I S I O N S 323
Ea )> m where Ea could be for example the energy of the final e lectron; the results presented here therefore :~re wron~ only where E--(o~.2 is abou~ the order of m. In e,~ch other l)~rt of the spectrum the previous results are valid o(1/7 ) where 7:-=E/re. The result (7) at.n'ees with the w~lues of the cross-se('tion given by BAYER and GAr~ZTSKY, which is identical to (7b).
lit has been noted by SH>oR(>V th~tt the cross-section (7) can be pu t in the approximat ion form
(9) dcr ~_ Sa~r~ d.q_ (l:r= l"(.r~) I,",x._) , '1'I d',,
where F(x) is
(lo) F(,,,) :-- -2"(]I--X) [- t~ 2 .
This result is very impor tan t from the experimentM point of view. lit s:~ys theft the, energy spectrum of one photon is completely independen~ from the
energy spectrum of the other photon. This separabil i ty of the cross-section has been experimentMly confirmed with VEP-1 in hrovosibirsk (G).
The expressions of the forward-backward cross-section (6) and of the to ta l
cross-section (7) allow us to (,ah.ul,~te the mean angle (~ of the two-photon emission by the pro('ess (I)
(l 1) ~'4g~ ___ 8
, 5 ( 1 - - ,/;1)(l - - ae) [4 i ~ : ( : : ) ] : [ . t~(1-- .v. ,) :_ . r ~ ( l - - a : l ) J [ � 8 9 V' 7 2 ,8~(,~).,p, 2 e .2
. 4 ( ] _._ ,(,1)( | - - . 2 . 2 ) ! .*'~(1 -- '"el ! X~(1-- X~) 4-:r~x~
y404/8 is a slowly varyinff funct ion of x~ and x~, whose values lie l)etween 1.0 and t.8.
[n order to obtain information about the angular behaviour of the emi t ted photons, we evaluated the ('ross-section da(t, xl, x2) where t = y O = y R / l . R is the radius of both the ( 'erenkov ('ounters and 1 is the distance between a ( 'erenkov counter and the crossing point of the electron-positron beams. This resulting cross-section is
( l a g .,',~,~)- s~b'g d,q d,,,., { G ( t ) ( 1 - .q)(] --,.~'..,) + '['1 ;1!"
+ (;..,(t~ ~ x0 + J..;(l ~ ~ [*~(J --- ' -.,1)11 + G(;)a,;~,;~}.
(6) p . T. (~OL1;BNICIJ, A. P. 0NI;CIN, S. G. POPOV and V. A. SIDOROV: p resen ted �9 ~t the I~ternational Symposium o~. Electro~. and Positrons. Storage ]~,b~gs, Orsay Seplember 1966.
324 ~'. D[ u and ~. G~ECO
The functions G1, G2, Ga are indicated in Table I . This result agrees wi th an
analogous calculation pe r fo rmed b y BAYEIr et al. (7).
t
1 2 3 4 5 6 7 8 9
10 11 12 13 14
GI G2
0.081 0.065 0.41 0.31 0.74 0.55 1.02 0.74 1.23 0.88 1.39 0.99 1.52 1.07 1.62 1.14 1.70 1.19 1.77 1.23 1.82 1.27 1.87 1.30 1.91 1.32 1.94 1.34
TABL}~ I.
6 3
0.05 0.24 0.41 0.54 0.64 0.71 0.76 0.80 0.84 0.86 0.89 0.90 0.92 0.93
0.16 0.74 1.27 1.70 2.00 2.25 2.41
0.065 0.05 0.31 0.24 0.55 0.41 0.74 0.54 0.88 0.64 0.99 0.71 1.07 0.76
p q
m
_ _ I
~ is ob ta ined b y The cross-section for emi t t ed photons of energy wl.~/E~>e
in tegra t ing (12) over x~ and x2 and turns out to be
(13) cr(s~ t ) = 8v'%'~ {Gl(t)[ l ~ 1 Jr -~s] ~@
-~ G.(t)(l - - e 2) [log 1 ] ~_s]+G.~(t)(1 - - s2~2 t 2-/)"
A plot of a(s, t) is drawn in Fig. 2 for t ---> oo. An analogous calcula t ion per-
formed by BAN])E~ (4) has given the following result :
...., ]' [ . ] ' } (14) a(s,t)---- [~ l ( t ) log - - 1 . ~ s @M2(t) l o g - - - l ~ - s @ ~ Ma(t) .
The funct ions Ml(t), M2(t), M3(t) are indicated in Table I. We believe t h a t (14)
is va l id only in the l imit ~2<< 1, because the cross-section (14) does no t go
to zero when e goes to 1. I n this app rox ima t ion the formulas (13) and (14)
present the same analyt ical fo rm wi th regard to the dependence on t and s.
(~) V. N. BAYER, V. S. FADIN aIl(]. V. A. KItOZE: Emission of two photons into a given angle in electron collisions, presented at the International Symposium o.~ Electro.a and Positron Storage Rings, Orsay, September 1966.
D O U B L E P H O T ( ) N ~ E M I S S I O N I N C ~ C - C O L L I S I O N S 325
Besides, Table I shows an exact coincidenec between the functions G2(t), G3(t) and M2(t), Ma(t). The functions F~(t) and M~(t), which give the main con-
cross-section a t the t r ibut ion to the
low frequencies, present ins tead an
a lmost cons tan t discrepancy, whose
order is of ~bout 50 ~ / 0 "
Therefore our results agree com-
pletely witll those ob ta ined by BAYEg
and by BA~])ER apa r t f rom the dis-
crepancy wi th Bander ' s calculations, concerning the functions F~(t) and M~(t).
A rcm~rk~ble fea ture (ff the (:ross-
section (7) is the absence of a lo~arith-
nlic t e rm of the fo rm log2y, which
is so character is t ic of the single-brems-
s t rahlung to ta l cross-section (s). This
logar i thmic t.erm does not ~ppear in
the cross-section of process (1) because
the squar(~ modulus of the (.urrent
m a t r i x e lement is, because of the trans-
vers~dity of the emi t t ed photons, pro-
por t ional to the four th power of the
m o m e u t m n t ransfer and the probabi l i ty
of a m o m e n t u m t ransfer z is propor-
t ional to dz/z 3, so t ha t the differential cross-section of the process (1) ap- pro~wJms zero when there is no mo-
m e n t u m transfer . The single-brems- s t rahlnng differential cross-section be- haves as dz/z for h)w m o m e n t u m trans- fer and this creates the logar i thmie term. The lack of a logar i thmic te rm puts double bremsstrahluno- a t a. disad-
van tage opposi te the background of
created ei ther in collisions wi th the
genuine electron-posi t ron collisions (').
10 : ' 8
k r- 1
10 29~
b b
L
31 I
k
i
1 0 0 0.2 0.4 0.6 0.8 1.0
Fig. 2.
tile s inglc-bremsstrahlung processes
molecules of the residual gas or in
(8) G. ALTARELLI and F. BUCCELLA: .u Cimento, 34, 1337 (1964). (9) S. TAZZARI: unpublished paper presented ,at the Congressino Adone, z%'ascati
Febru~ry 1966.
326 P. DI VECCHIA and M. GRECO
We wish to express our gra t i tude to Profs. F. A~lVIAN and B. TOUSCI~IEK for having proposed the problem and for their kind assistance during the work.
We also thank Prof. C. PELLEGRINI and Dr. S. TAZZARI for many discussions and Dr. M. ]~ASSETTI for his essential contr ibut ion to the numerical calculations.
APPENDIX
In this Appendix we exhibi t some details of the calculation of da and dar.B., of which before we have given only the results. The total cross-section (5) can be wri t ten in the following form:
(A.~)
where
(27~) j d L /~'aE4(Pl - P3-- k~) 4 dE~ fE,=E,' Pi =Py
(A.2) X = ~ ~ Tr {D,A_(pl)D~A_(p~)} Tr {C"A+(p~)C"A+(p~)}. I)ol
Since we consider only the terms which in the limit of high energies give the ma x imum contr ibut ion to the cross-section (2), we obtain the following valuo for the expression (A.2) by making use of well-known trace theorems (~o):
9
(A.3) X = ZA~,
where
(A.~)
_/t - - !-- ~ ( pl01 P3 ~112 ( p4C2 -- P2 e'~) 2 1-- 2m 4 p/~ol~plkl p3]~l] \p4k 2 P2k2 "[(PlP~)(P2P3) + (PdJ.~)(P3P4)],
A:-- 2m 4 ~ \pikl pakl] \p4k.., p2k~!
[p~ k,, }q e~ ]
~- ~)4]i. ~ ((pl/~'2)(p2~)3) ~- (~'1P2)(P3]l'2)) ,
(10) j . M. JAUC]{ and F. ROIIRLICH: The Theory o] Photons alul Electrons (Reading, Mass., 1959), p. 439.
I)OUBLE PHOTON E_~IISSION IN (~:(~- COLLISIONS 327
(A.4)
P4 e., ] + p~k., ((l, zk~,)(p.,.kl) ~- (p~p.,)(klk~)) - -
_ p~ e~ [p . , e.,
i,,k~ Lplk l ((l'd~:~)(t'fl~''-') + (/q/,'~)(p~p~)) ~-
~- ])4~.2]'4 (~2 ((kl~'2)(~)2"3)+ (~2k,)(P3k2))]}
!p,~,,)(p~k 4 + (~,v,...) (p.~,~)] + p., k,., J §
_[_ ]13.01~)3 ~'1 i- r ! ~!.111:2)(/.)2 ~:1. !],4k2i- (plp2)(l~1k2) T' (],h P,) (k1,~:2) p,, k,,-F- (P 1 ]~',.,) (:P4 k,) ]] _L
+ ._,,,,~ [U', ,:1) (~,:, ~,.~)(~,.~k...) L plk~ (pj~.,)(p.,.k~) + p:,/,'~ (p._,k~)(p~k~)J) '
(pJq) (p3G) § (klk..,)(pd~4) 2 m 4 [ (pflq)(p4k~) ~ (lqk~)(p.,k~)
~_ !~'~f!'~,!(]/~]~'l) + (P1]t2)(klk2) + (['lP4)(klk2) -- (Plk2)(P4k,) ~_ (p.~k,)(p~k.,) (p~k,)(pJ,...,)
+, , ,~ [_ !#1#~!(~,:~/,..,) (p,#,;)(~.j,.~)
_..,._ (k,/,.~)(p,.k,) .4- (P'/~")(~: '#~) ] ' iP-flQ)(PflQ)(PflQ) ' (pltQ)(Pfl~'I)(Ps/;u)J -I
+ (p~k~)(pJ..)(p~k3(pJ._)~"
-iT, As, A~ ure obtuined from A~, A4, A5 by tile following substitutions:
328 P. DI VECCIIIA a.nd ~f. GRECO
Before performing the integrat ions in (A.1) over dg2, dg2x, df2: the follow- ing subst i tut ions must be made:
P a - - - - p 3 - - h i - - k=, , ~'4 : 2 E - - ~o l - - ('>.2-- Ea ,
(A.6) E3-- 2E'~-- 2E(oi-- 2E(o~ + coio~2-- kxk? 2E-- o,i-- ( ')~-- ~ a k l -~ ~ak2 '
where according to the previous approximat ions fl and fia are
'D't 2 ? ~,~ 2
(A.7) fi = ] -- 2E~, fl~ = 1 -- 2 (E - - ~Ol)""
Since the energy of the incident particles is ve ry high, the m ax in m m contribu- t ion to the integral (A.1) is obta ined when one pho ton follows the incident electron, the other the incident positron, and the scat ter ing angles of the electron and posi tron are ve ry small.
Therefore it is possible to develop the in tegrand in a series around the points 0 - - 0 1 = 0, 0~=.~ so tha t in the integral (A.I) are considered only the contributions around the cusp of the differential cross-section. Owing to this
Pl k,
/ ', p~
0 L I . / /
I i i I
,
/ / \ ~ I I
/ / N \ I
q
a)
Fig. 3.
"N
"N
b)
development tile angles 0, 01, 0, can be considered as vectors (Fig. 3) of the plane orthogonal to the colliding beams and the cr0ss-section (A.1) results
(A.8) 2r:(d lfd " .P(dx
D O U B L E P I I O T O N ].~MISSION IN O• - C O L L I S I O N S .'~29
where
(A.9)
x = : y O ~ , y = 7 0 = , z = y [ ( ] - - x O 0 + , , , ~ O ~ ] ,
,}?2
(1 -+ x~') ~ ' ( x - z) 2 2 x ( x - z)
[~ + ( x - z)~ - (~ • ~,,") i~ § (x - z)2] '
~2
F ( x , ~) = ~] t: :~,.o) i f !- (X-- z)'-] "
I f we eliminate the integrations over d.q, dx:, dx and dy, the forward-backw'~rd cross-section daF.~, can be obtained provided tha t we put x = y - - 0 and integrate over z. :By making use of polar co-ordinates i~ is easy to perform the integration over the two photon :~zimuths TlO. and then on .c and y from 0 to yO. The resulting ~ross-seetion is
(A.] 0) dG
where
co
2 ~ 2 r o &q d'r" I " - - ~ {1(1--a'JJ(,z, . ) '~ . . . . ~,} " - i - d { l ( I ( " ' "
3/,' 1 ,F 2 ,j o
�9 { ~ ( 1 .... . r . _ , ) g ( : , u ) --t--' '.~(;(-," ' ~ ~ ) ] - ,
(A.] ])
,) .~ - t - Z 2
J (Z, U) = ')~X/z" I i 4
(~ § +4) ~ �9 log .{:[: ~- v : ~ ~ 4 ]1 )4 ( : , . . , ) - ] ~ ,,,(~ t-:~)]-+ .~(:, ~)}-
3 1 (2 ~ z 2) u - - 1
Z (; (z, u)
�9 l og - (: + ~/~2 + 4) ' ._
A(z , u ) = [ 1 - 2z-'u + ~,:=2(z~- ~- 4)1~,
I 1
] i - t ~ ! ) '~ 'O'"
Because of the convergence of the integral (A.10) tile upper l imit of integration has been set equal to infinity and the calculation has been accouplished ~mMyt- ic~fl]y only if u z 0. Otherwise the integration over z has been performed numerically.
lns tead of developing the integrand of (A.1) in a series ~rround the forward- backward direction, we perform the angular integration ill another way,
330 r. DI VECCtIIA and M. GRECO
shown below. Let us introduce a slightly different definition of the w~rious angles, as in Fig. 4.
Pl k~
P~ . _~ -o / ,
~o ,., I
\ I \ I
\ I q
Fig. 4.
By making use of the (A.6) we find following expression for the pho ton propagator :
(A.12) (Pl-- P3-- kl) 4=-
4 (a - -6 cos0 +.'2 sin0) ~ ((2E-- ("~--(').2) + s cos 0i(ol cos 0~-- (o..eos-0~)§ sin 0~o; sin 0;cos q~ § o J: sin0~ cos (q~- Zi]~" '
where
(7 : ~ ( E - - ( o t ) ~ - (o..,) 2- [(Pl-- k l ) - m2] [ 2 E - - eo~-- ~o,,] ,
(~ = ( 2 ( E - - o)l)(W-- (,)2)(l)1-- (01 cos 01 ) - [ ( P l k l ) - 'm 2] [0) 1 cos 01 -- (02 cos 02]}/~3 ,
---- {2(E-- oh)(E-- (0~)(01 sin01 cos ~ +
-~- [(p~k~)- m 2] [0~ sin01 cos ~ + 0)2 sin0o cos (ff~-- Z)]} fi3 �9
By subst i tut ing (A.12) into (A.1), in order to perform the in tegrat ion over the tmomMy 0 of the outgoing electron, we must cMeul~te some integrals of this kind
(A.13) o
sin 0 d0P(eos 0, sin 0) 0 t- ~o sin0)2(~ - u cos0 ~ v s ] n 0 ? ( ~ ' - - u' co s0 - - v' sin0) 1~'
where P(cos O, sin 0) is a polynomial in cos 0, and h, k can take any value ~mong 0, 1,'2. The terms ( ~ - - u c o s 0 vs in0) , ( a ' - - u ' c o s O - - v ' s i n O ) result
I ) O U B L E r l l O T O N E M I S S I O N I N (~&(~- C O L I , I S I O N 5 331
f rom the expressions of (p4k=.) and (pale1). With the t r ans fo rmat ion z : : tg (0/2) we obtain some integrals of the form
(A.I~) L.~,~= ~:d.,(Z_z...).(z_z:,)~( z zj~(:_70,~(.:_ .),~, 0
where ~z is an inteo'er and
Z 1 ' 2 - - - O" , ~ r , $3 f l . . . . ~ -~'" U '
V' ~ /'. V/o~ ' ~ - - u 'a IV 2
The integrat ion can be per formed in the complex plane of the var iable z; the pa th is shown in Fig. 5.
We obtain
(A.15) 1,, h~,, = - - . eosl d [ i R e s ( z * ) ! " : r c ~ d~ i = l
For 0~ = 02 = 0 the funct ion result ing f rom (A.15), af ter in tegrat ion over 0, is indepen- den t of q,. This resul t therefore gives the fo rward-backward cross-section.
For tile general case, we havc to per form the integ'ration over the az imuth ,d angle q,.
/
/ Rez
/
F i g . 5.
This can be accomplished in a ra ther difficult manner , because the large number of t e rms which eome from the in tegrat ion over 0 m~kc the calculation very cumbersome. The result is expressed by the followino" formula :
(A.I(;) d~r 8 g a ro d(Ol d(o., d ( e o s 0~) d ( e o s 0.2) 1
~,,~ .~, ( 1 - 5 , , o s 0 y ( 1 /~:os0.)2:b, ~'
1
1
-~] lg] '-'r'(~ fl ' ~Y! \ E ! "
+
1 I
Tile t e rm I1 is not inter ' ra ted over (; because of its v e r y ( 'onq)lieated analy t ic expression, l [owever it does not contr ibute to the to ta l cross-section, which is
332 1,. DI VECCI[IA and ~. CREte
obt~ined pe r fo rming the in tegr~ t ion over 01 and 0._, in the first p~r t of the r ight - h a n d side of (A.16). The dependence of B on 01 and 0,, is of the f o r m R(cos 01, cos 0~,) log (cos 01, cos 00), R(eoS 01, cos 0~) and log (cos 01, cos 02) are a r'~tiomfl trod a logar i thmiea l f u n c t i o n of cos 01 and cos 0. and do .~ontribute to the ang~ular d is t r ibut ions .
R I A S S U N T O
In qucst(~ l~w)ro si ealcola, no[ limite rclativistico cstremo, p~r il processo e ~:+ c--~ ~ ' +~,- § 2~, lt~ distribuziom~ angolar~ dei due fotoni, il lore spettro di energia, e 1,~ sezion~ d'm'to totalc per l'emissione di fotoni di cmergi~ ~ s.
Pe3/oMe aBTOpOM He Ilpe~CTaB~eHo.
