Pair-production cross-sections for photons bombarding protons at high energies

Volume 2, number 2

PHYSICS

PAIR-PRODUCTION BOMBARDING

LETTERS

CROSS-SECTIONS PROTONS AT HIGH

15 August 1962

FOR PHOTONS ENERGIES

W. RUIIL Institut fur Theoretisehe Physik d e r Universit~t K~ln Received 20 July 1962

In two e a r l y p a p e r s 1 , 2 ) a f o r m u l a f o r t h e t o t a l c r o s s - s e c t i o n of t h e p r o d u c t i o n of e l e c t r o n - p o s i t r o n pairs was derived under the assumption that the p r o t o n can b e d e s c r i b e d a s a s o u r c e of a s t a t i c Coulomb field. This (Bethe-Heitler) formula is

~ 3 1 (28 2k = - ~ m 2 -~-in m

218~ 27 / '

(1)

m b e i n g t h e e l e c t r o n m a s s , ~o t h e f i n e s t r u c t u r e c o n s t a n t , a n d k t h e e n e r g y of t h e i n c i d e n t photon. In a m o r e r e c e n t p a p e r 3) c o r r e c t i o n s to t h i s f o r m u l a , w h i c h c o n s i s t of two p a r t s , w e r e c a l c u lated. First, there is a purely kinematical correct i o n , w h i c h t a k e s into a c c o u n t t h e f i n i t e m a s s of the proton and its recoil energy. Second, there are two n e w d i a g r a m s w h i c h c o n t r i b u t e to t h e p r o c e s s when t h e p r o t o n i s not k e p t f i x e d . T h e s e d i a g r a m s c o n t a i n a v i r t u a l C o m p t o n p r o c e s s in t h e p r o t o n c u r r e n t , and a r e t h e r e f o r e c a l l e d V C - d i a g r a m s . T h e a u t h o r s a l s o t r y to i n c l u d e t h e e f f e c t of t h e p r o t o n s t r u c t u r e a s known f r o m H o f s t a d t e r ' s e x p e r i m e n t s . The two d i a g r a m s t h a t c o n t r i b u t e to t h e B e t h e - H e i t l e r f o r m u l a can now be f u l l y d e s c r i b e d in t e r m s of t h e H o f s t a d t e r f o r m f a c t o r s . On t h e o t h e r h a n d , t h i s c a n n o t be done e x a c t l y in t h e V C d i a g r a m s , but o n l y in a f i r s t a p p r o x i m a t i o n t h a t m a y b e v a l i d up to a photon e n e r g y of a b o u t a h u n d r e d MeV. T h e r e f o r e , t h e V C - m a t r i x e l e m e n t and t h e i n t e r f e r e n c e t e r m s a r e c a l c u l a t e d o n l y in t h e l o w e s t o r d e r of t h e m a g n e t i c m o m e n t , a n d a r a t h e r complicated formula for the differential crosssection is obtained after a first integration. It i s of i n t e r e s t now to c o m p l e t e t h e s e c a l c u l a t i o n s in two d i r e c t i o n s . We w a n t to g e t a t o t a l c r o s s - s e c t i o n f o r m u l a and to c o n s i d e r a l s o t h e e n e r g y r a n g e a b o v e t h e p r o t o n r e s t e n e r g y . In t h i s e n e r g y r a n g e t h e a p p r o x i m a t i o n of t h e V C - d i a g r a m s m e n t i o n e d a b o v e can no l o n g e r be u s e d . Since t h e p o l a r i s a t i o n p r o p e r t i e s of t h e p r o t o n a r e unknown, we d i s r e g a r d t h e s e d i a g r a m s . B e c a u s e of an a n t i s y m m e t r y in t h e m o m e n t s of t h e e l e c t r o n a n d of the p o s i t r o n , t h e i n t e r f e r e n c e t e r m s e x a c t l y v a n i s h a f t e r i n t e g r a t i o n . T h e r e f o r e , we t a k e into a c c o u n t t h e B e t h e - H e i t l e r d i a g r a m s o n l y , and a p p l y a f o r m f a c t o r t h a t c o r r e s p o n d s to an e x p o n e n t i a l c h a r g e

and m o m e n t distribution in the non-relativistic limit. A s the main resuR of our calculation w e get again the Bethe-Heitler formula (1) independent of the form-factor used. Corrections to this formula that contain the contributions of the anomalous magnetic m o m e n t and the form-factors as well as the kinematic corrections, are of the order m 2 / M 2, m2/kM or m2/k 2, where M is the proton mass. Certainly these corrections are negligible in the energy range of interest. This striking result is a consequence of the high singularity in the forward direction of the pair-production amplitude. This resuR is related to one stated in an earlier paper 4) where the distribution of the recoil m o m e n t of the proton was discussed, assuming that the proton h a s an i n f i n i t e m a s s , a n d i t w a s shown t h a t with i n c r e a s i n g e n e r g y t h e m a x i m u m of t h i s d i s t r i b u t i o n t e n d s to z e r o . W h e n w e c o n s i d e r t h e e n e r g y r a n g e k >> M , t h e r e r e m a i n s o n l y a c o r r e c t i o n of t h e o r d e r m 2 / M 2 w h i c h r e p r e s e n t s j u s t t h e e f f e c t of t h e a n o m a l o u s moment. We get Crmagn. = ~ g 2

I Rtv(k) M-2

(2)

(gA i s the Land~ f a c t o r of t h e a n o m a l o u s m o m e n t ) with 1 in2 2 k M R°(k) = 3 rn 2

19 2kM 301 18 in - ~ + 216

~r2 0

(a = 0) ,

(a) Rdk)

1 m2 M2

=5

~ m2

8

9In

M2

11

czm2

18

+-~

(~ ~ 0)

(4~ where ~ is a distribution parameter defined by p(r) = A e - r / ) t ,

c~ = M 2 k 2 .

To ), = 0.23 f t h e r e c o r r e s p o n d s c~ = 1.20 a n d R a = 60.8. The m o s t i n t e r e s t i n g f e a t u r e of t h i s f o r m u l a i s t h e v a n i s h i n g of t h e l o g a r i t h m i c d i v e r g e n c e , when we c o n s i d e r a m o m e n t d i s t r i b u t i o n i n s t e a d of a point moment. The calculation shows that this p r o p e r t y i s r a t h e r l i k e l y i n d e p e n d e n t of t h e s p e c i a l 69

Volume 2, number 2

PHYSICS

f o r m - f a c t o r . F u r t h e r m o r e , t h e d e p e n d e n c e on the parameter a is very weak. T h e a p p e a r a n c e of t e r m s w i t h a s q u a r e d l o g a r i t h l n b e c o m e s p l a u s i b l e when we c o m p a r e f o r m u l a (2) w i t h t h e t o t a l c r o s s - s e c t i o n f o r p a i r p r o d u c t i o n in a s t a t i c f i e l d of a m a g n e t i c d i p o l e . We do n o t r e g a r d a d i p o l e d i s t r i b u t i o n but o n l y a p o i n t d i p o l e , and get 1 4 2k a m a g n . , s t a t i c = ~o3 g 2 M-2 ( ~ ln2--m 41n2k 3'I -~ m -5-~-

~)

,

(5)

w h e r e g A a n d t h e h y p o t h e t i c m a s s M of t h e p a r t i c l e are introduced by e

~A = ~ g A

•

LETTERS

15 August 1962

c o n t a i n s t h e f a c t o r m2/M 2, w h i c h i s of o r d e r 10-6. T h i s i s n o t t r u e f o r a n e u t r o n , t h e t o t a l c h a r g e of w h i c h v a n i s h e s . Then we w i l l h a v e an e l e c t r i c e f f e c t of t h e s a m e o r d e r a s t h e m a g n e t i c o n e , if there is an electric effect at all. Until now n o t h i n g h a s b e e n s a i d a b o u t t h e V C p a r t of t h e t o t a l c r o s s - s e c t i o n . We e x p e c t i t to b e a l s o of an o r d e r m 2/M2 c o m p a r e d w i t h t h e B e t h e H e i t l e r f o r m u l a in a m e a n e n e r g y r a n g e , but w e cannot say anything about its energy dependence. The c a l c u l a t i o n of t h e a b o v e given f o r m u l a s i s g e n e r a l l y v e r y t e d i o u s . One of t h e m a i n p o i n t s i s to f i n d a p p r o p r i a t e L o r e n t z s y s t e m s f o r t h e i n t e g r a t i o n s . The d i s c u s s i o n of t h i s and of f u r t h e r c a l c u l a t i o n m e t h o d s a n d of s o m e f o r m u l a s f o r d i f f e r e n t i a l c r o s s - s e c t i o n s w i l l b e p u b l i s h e d in t h e Annalen der Physik.

References

H e r e , ~A i s t h e m a g n e t i c m o m e n t of t h e f i x e d p a r ticle. It i s c l e a r t h a t t h e t o t a l c r o s s - s e c t i o n f o r p a i r p r o d u c t i o n in t h e f i e l d of a p r o t o n i s w h o l l y d e scribed by the old Bethe-Heitler formula apart from the VC-parts, since the magnetic contribution

1) W. Heitlerand F.Sauter, Nature 132 (1933) 892. 2) H.A.Bethe and W.Heitler, Proc. Roy. Soc. A 146 (1934) 83. 3) J . D . Bjorken, S.D. Drell and S. C. Frautschi, Phys. Rev. 112 (1958) 1409. 4) R. Jost, J.M. Luttinger and M. Slotmick, Phys. Rev. 80 (1950) 189.

**~**

THE

A FURTHER NOTE ON POLES OF GREEN FUNCTIONS D. T E R H A A R a n d W. E. P A R R Y The Clarendon Laboratory, Oxford Received 9 July 1962

W e s h o u l d l i k e to m a k e a c o r r e c t i o n and a d d a n o t e of c l a r i f i c a t i o n t o o u r r e c e n t l e t t e r 1). Eq. (6) should read: <(tx+a;mx+>) = 0 ;

2 ((a~tx;a+tx)~ + ((air+; atx+>~ = 21rE "

T h i s m e a n s t h a t e q s . (9) and (12) now a g r e e , a s of c o u r s e t h e y m u s t , s i n c e both a r e e x a c t . S e c o n d l y , t h e r e i s a n e g a t i v e s i g n m i s s i n g in t h e s e c o n d t e r m of ('/), a n d c o n s e q u e n t c h a n g e s in ( l l a , b , c) a r e n e e d e d *.

L a s t l y , t h e p o l e w h i c h we o b s e r v e d a t - 25e i s in f a c t t h e p o l e a t t h e o r i g i n d i s p l a c e d b y t h i s a m o u n t . Since we a r e w o r k i n g to f i r s t o r d e r in 5 ( o r r a t h e r , to s e c o n d o r d e r in 5~), t h i s i l l u s t r a t e s t h a t to g e t t h e e x c i t a t i o n e n e r g i e s to a given o r d e r i t i s not enough to w o r k to t h a t o r d e r w i t h t h e G r e e n f u n c t i o n , a n d u n d e r l i n e s t h a t we m u s t p r o c e e d w i t h c a r e when e q u a t i n g t h e p o l e s of G r e e n functions with excitation energies.

RGference

* We a r e grateful to Dr. A. K. Rajagopal for pointing this out. *****

70

1) D . t e r Haar and W. E . P a r r y , Physics L e t ~ r s 1 (1962) 145.