tW-i^'-fVf, h^ | L ^ F

Hotice 1691

INVARIANT AMPLITUDES FOR COMPTON SCATTERING Or OFF-SHELL PHOTONS ON POLARIZED NUCLEONS

&

M. PERROTTET

ABSTRACT : We descr ibe a se t of 18 invar ian t amplitudes f ree from kinematical s i n g u l a r i t i e s in p and t ( V i s proport ional t o the incoming photon energy in the laboratory frame : t i s the squared momentum t r ans fe r ) for Compton s c a t t e r i n g of o f f - she l l photons on polar ised nucléons. Special, cases r e l a t e d to the forward and the non-forward s c a t t e r i n g l i m i t s are discussed.

JUNE 1972

72/P.465

••Attaché de Recherches C.N.R.S.

, POSTAL_ADDRESS : Centre de Physique Théorique - C.N.R.S. 31 , Chemin J . Aiguier 13 MARSEILLE Cedex 2 (France)

TABLE OF CONTENTS

I - INTRODUCTION

I I - DECOMPOSITION OF THE GENERAL COMPTON AMPLITUDE INTO

INVARIANT AMPLITUDES FREE FROM KINEMATICAL SINGULARITIES

1 . R e s t r i c t i o n s due t o . t i m e r e v e r s a l i n v a r i a n c e

2 . C r o s s i n g symmetry p r o p e r t i e s

3 . R e s t r i c t i o n s due t o t h e h e r m i t i c i t y o f t h e Compton a m p l i t u d e

I I I - THE FORWARD SCATTERING LIMIT ( t = 0 )

IV - STUDY OF PARTICULAR NON-FORWARD SCATTERING LIMITS

1 . One r e a l p h o t o n , one o f f - s h e l l pho ton

2 . O f f - s h e l l p h o t o n s w i t h e q u a l masses (q = q ' )

3 . Rea l Comptoil s c a t t e r i n g

ACKNOWLEDGMENTS

APPENDIX : Summary of r e s u l t s

FOOTNOTES

REFERENCES

FIGURE CAPTIONS

T>fa AfK

1)

I - MTHODUCTION

We sha l l be concerned with Compton sca t t e r ing of o f f - she l l photons

from polarized nucléons. The corresponding amplitude i s defined by the

following expression

where J [%) i s the electromagnetic hadronic current and | fc>, A > ,

| b ( i V > are one-nucleon s t a t e s with energy momenta p and p ' and

polar iza t ions A and A ' The T product i s understood to be the

covariant time-ordered product.. A diagrammatic representa t ion of the ampli-AD,'

tude (£ (P .q ,q ' ) i s given in Fig. 1 .

These general non-forward Compton amplitudes appear in many physical

processes . As examples we quote the two-photon exchange in te r ference with the

Born amplitude in e l a s t i c electron-proton sca t t e r ing (see p ig . 2) ; the s o -

ca l led Compton diagrams in the Bethe-Heitler processes (see Fig. 3b and 4b) .

I t i s conceivable that su f f i c i en t ly precise experiments wi l l be performed

which are sens i t ive to these general nucléon Compton amplitudes. In pa r t i cu ­

l a r , with the use of polarized hydrogen t a r g e t s , i t i s poss ible to obtain

precious information on observables governed by o f f - she l l nucléon Compton

amplitudes. An example i s the e l a s t i c s ca t t e r ing of e lect rons from polar ized

protons where there already exis t experimental r e s u l t s [ l j . In t h i s case ,

the up-down asymmetry i s sens i t ive to the absorptive par t of the spin inde­

pendent as well as the spin dependent nucléon Compton amplitudes f 2 j . The

absorptive Compton amplitude s a t i s f i e s p o s i t i v i t y condit ions, which have been

discussed recent ly in d e t a i l s by De EÛjula and de Rafael f 3 j . These pos i ­

t i v i t y restr ict ions are most eas i ly expressed using a set of invar iant ampli­

tudes , the so-cal led t r ansve r s i t y amplitudes,. which however are not free from

kinematical s i n g u l a r i t i e s .

2)

lu \£ew of possible applications it would be nier: to have a set of

invariant amplitudes for Compton scattering of off-shell photons from polarized

nucléons which are free from kinematical singularities. This is already an

interesting problem per se, since a full exploitation of dispersion relations

in Compton scattering requires, as a preliminary ÎJexercise", the construction

of such a set of amplitudes. The purpose of this paper is precisely to give

the details of the construction of a set of invariant amplitudes free from

kinematical singularities in the variables V and t

Such invariant amplitudes have already been partially constructed by

various authors f4,5 f6 J. Their results, however, are given in spinorial

form which requires further algebraic elaboration to use them in applications.

Moreover we have not seen sxi explicit construction which takes into account

the gauge invariance of the Compton amplitude and avoids kinematical singula­

rities in y and C . The choice of a set of tensor covariants for the

construction of the nucléon Compton amplitudes is rather delicate ; indeed,

due to an unsuitable choice of the set of tensor covariants, many authors [7,

8,9 J haVe obtained, either invariant amplitudes with kinematical .singula­

rities, or an incorrect number of independent invariant amplitudes.

The organization of the paper is as follows. In section IT, we

define the general Compton amplitude (LMV ( ^,c\ic^1) and the corresponding

kinematics. The absorptive part *'uv ( 'Pjli^') °£ th e general Compton

amplitude plays an imoortant role in the applications ; we give the definition

u/44' u/M' /r W

of VVii* and indicate the relation between W^ and u-„ y

Then we describe the method used for the construction of invariant amplitudes

free from kinematical singularities. On some crucial points a brief review of ;

the literature is given. The selection rules due to time reversal invariance,

to crossing symmetry properties and to hermiticity are discussed in sections

II.1 to II.3 .

72/P.465

3)

In section I I I , we study the forward sca t t e r ing l imi t ( t = 0 ) ,

which i s r e l a t ed to deep i n e l a s t i c electron-proton s c a t t e r i n g . The r e l a t i o n

between our invariant amplitudes and the usual s t ruc tu re functions ' Wi i

W 2 , G, , and Gl2 i s given.

In section IV, we consider some p a r t i c u l a r non-forward s c a t t e r i n g 2 2

l i m i t s . The cases q = 0 or q' = 0 which are respec t ive ly r e l a t e d t o

photoproduction of e lectron (or muon) pa i r s on nucléons and bremsstrahlung

of e lec t ron (or muon) on nucléons are discussed in sect ion IV.1 . In 2 2

sect ion IV.2 we study the case q '= q' and in sect ion IV.3 the r e a l 2 2

Compton sca t t e r ing case (q = q' = 0) .

For the sake of c l a r i t y the r e s u l t s of sect ions I I to IV are summa-

r i t e d in an appendix where t he reader can immediately f ind t he t ensors c o r r e s ­

ponding t o each case which has been s tudied.

' • )

I I - DECOMPOSITION OF THE GENERAL COMPTON AMPLITUDE INTO INVARIANT

AMPLITUDES FREE FROM KINEMATICAL SINGULARITIES.

The a m p l i t u d e (Ln v I i . , q , q ' / f o r Compton s c a t t e r i n g of o f f -

s h e l l pho tons on p o l a r i z e d n u c l é o n s has a l r e a d y been d e f i n e d by Eq . ( l . l ) .

The s t a t e s j b , A > and | h', A'> a r e o n e - n u c l e o n s t a t e s w i t h momenta p

and p ' and s p i n p r o j e c t i o n e i g e n v a l u e s Ot, A c o r r e s p o n d i n g t o d i r e c -

2) t i o n s n ( p ) and n ' ( p ' ) such t h a t '

n ( p ) . p = 0 , n ( p ) = -1

n ' ( p ' ) . J > ' = 0 , n ' C p ' ) 2 = -1 ( 2 . 1 )

and

n ' ( P ' ) = A ^ . it(f>) . ( 2 . 2 )

where A.p + pi i s t h e p u r e L o r e n t z t r a n s f o r m a t i o n ( b o o s t ) which b r i n g s p

i n t o p ' . J\. h-+pi c a n b e w r i t t e n e x p l i c i t e l y ' i n t e r m s of t h e momenta ;"

and p 1 and t h e nudeon mass M „ The c o n s e r v a t i o n o f energy-momentum i s

c o n t a i n e d i n t h e d e f i n i t i o n ( 1 . 1 ) o f ; we h a v e

P + q = P ' + q ' ( 2 . 3 )

For c o n v e n i e n c e , we i n t r o d u c e t h e f o l l o w i n g 4 - v e c t o r s

p = i (f» + f>0

Q = 2 (l+l') (2-4)

A = i h-^'J

The i n v a r i a n t s V and t h a v e been d e f i n e d i n Eq . ( 1 . 2 ) . I t i s e a s y t o

2 2 s e e t h a t q , q ' , V and t a r e a s e t of i n d e p e n d e n t i n v a r i a n t s f o r

t h e problem,

H E absorptive part Wi, K (7,^,«j') of the nudeon Cômpton amplitude is defined by

jT l ' ' ' >T/ r< " • ~r*-> -^ '

V>/=y.65

5)

The summation 2-t i s extended to a l l possible on mass-shell s t rongly inter— V

act ing s t a t e s "P with phase space in tegra t ion understood.

Using t r ans l a t i on invariance for the electromagnetic hadronic cur^

rent J y (x) and the corresponding transformation of s t a t e s | l o , ^ > and

lf>,A'> i t can be shown tha t '

By means o£ the def in i t ion of the T -product and using agpin r sns l a t ion i n -

variance, we have for the absorptive par t of (L^ ( i.,c\,q) the expected

resul t

/\ts ^!r?,*<}') = v ^ r r ? j (2.7) We sha l l separate (Lj „ and VK ^ in to nucleor. spin inde­

pendent and dependent parts» For t h i s purpose we wri te

as a 2x2 matrix in the spin space of t he nucléons

,44 ' t

The matrices 6 are the usual Pauli matr ices , n ' . ( p ' ) are th ree 4-vectors

such tha t ( i = 1,2,3)

«.'. (fo'J • f,' = o , H(' f^'J. H.J ffi'J - - <T- (2.10)

The terms C f (f,<j,q'y and l/V , (^q,^ 1 ! are spin independent tensors ;

they are second rank tensors in Minkowski space. The terms ^Ljuvf [*-lqtc\')

and <<urf ( ^ ' , ( ) , q 7 are t h i r d rank spin pseudotensors. The quantizat ion

axis n ' ( p ' ) (see Eqs» (2.1) and (2„2)) of the outgoing nucléon i s chosen

along the t h i r d d i rec t ion n ' ( p ' ) . V/e emphasiee tha t the choice of the

4-vectors n ' . ( p ' ) Ci = 1,2,3) i s a rb i t r a ry , but the tensors Qytv l ^ i ^ , ^ ' / ,

^ ( * , « ! . < ? ' J . 'Cpi, CT.1,1-) «id \ j (P.I.I') are intrinsic obj ects. •

72/P.465

6)

Let us assume the following decomposition for the spin independent

tensors C^, {'Slcjlq') and W„„ ($, q,q<)

W^ (Z, <,,<,') - Z ^ W. (V,^, .>, fj (2.12)

I ,,„ are tensor, covariants built up with the three independent 4-vectors

P, q, and q' , and Sl^ [cj*, Cj'Z, V, t) are invariant amplitudes. Then

with the definitions (2.8) , (2.9) and Eq. (2.?) we have

W i Wl", ',t) « } !«• ^ 4 fotyî V, t) (2.13)

The same relation holds for invariant amplitudes associated wivh the spin

tensors.

The general structure of Compton amplitudes for scattering of off-

shell photons on polarized nucléons in terms of tensor covariants and inva­

riant amplitudes can be specified under the restrictions of Lorentz invariance

gauge invariance and discrete symmetry requirements (parity and time reversal)

As mentioned in the left hand side of Eq.,, (1.1) we shall use a basic varia­

bles the three 4-vectors P , q , q' which allow to express eas'ly the

gauge invariance and the crossing properties of (C . These va. labiés will

also be very useful to deal with the problem of kinematical singular, ties and

to discuss directly the particular cases studied in sections III and I,' „

In the Dirac formalism, we can write

C ^ l t w ) - 2 M fïpïfr ÙftfA'J A r [V,M') Hff>,A) (2.14)

where n , ( is the most general second rank tensor constructed with the

Dirac matrices, the three 4-vectors P , q , q' and invariant amplitudes

a i (q a , q , 2 , v , t )

V (f>W) = S ^ ai llW, fi t) (2.15) 1 .

The following l is t contains all the possible tensor covariants t »</ that

we can consider.at first sight

72/P.465

1)

I -VI' « ; " ^

'% -£* , « ' • i *

t 7 = i , ^

t 8 = ^ ^

-V^

*" - VK# *? -^r» ^ - i ^

t ^ - f ^ - ^ ; ) j i t2^ - < j r ) > I

' i

In wr i t ing t h i s l i s t , we have already taken in to account pe.rity conservation

which -allows to el iminate a l l the tensor covariants containing Ys . More­

over Dirac equation has been used to keep only $

Let us remark tha t 0 and A , which wi l l be used in the following, have :> I

be understood as a shortening for _' ( cj + cj 'I and I ( cf - a 'J , s ince P , \

q and q' are the bas ic va r i ab le s , t and t , have been chosen i n j /*» /*" !

t h i i pa r t i cu l a r way ( ins tead of 7uC\v(h . - a n d Tv ClL $ ) i " order to -j J '' '!

preserve simple crossing proper t ies in fur ther e l iminat ions . . -•

Nctice tha t a simple h e l i c i t y counting y i e l i s t o 32 independent '•]

amplitudes. Unfortunately we have 34 tensor- covariants in (2.16) ; thus

thereuust be two independent r e l a t i ons among the 34 tensor covariants t„„ . r

Following a method described in Eef. [ 6 J , and based on the fac t t h a t the

ex te r io r product of f ive a rb i t r a ry 4-vectors vanishes i d e n t i c a l l y , we can

obtain two independent r e l a t i ons by computing successively 72/P.465

8)

JT*

= o

These are

(2.19)

These r e l a t i ons are understood to be taken between Dirac spinors W [tp',fy') and

The f i r s t r e l a t i o n i s the same as one of the r e l a t i o n s

given by Gerstein [ 4 ] ; and de Calan and Stora [ë] . The r e l a t i o n (2.19)

d i f f e r s from tha t given by de Calan and Stora by numerical coe f f i c i en t s . We

have checked the second r e l a t i on given by Gerstein, which he has chosen i t

d i f f e r e n t l y to our Eq. (2.19) . We disagree with the two r e l a t i o n s giver, by

Conway [5 J i n some of t he numerical coe f f i c i en t s . Equations (2.18) and (2.19)

have been checked by means of a Bouchiat-Michel type formula fio] . These two

independent r e l a t i ons (2.18) and (2,19) allow to. el iminate two tensor cova­

r i a n t s from the l i s t given i n (2.16) without introducing any kinematical s i n ­

g u l a r i t i e s . We have chosen to el iminate t ' and t I l u , Then we are l e f t

with a se t of tensor covariants which under crossing (see sec t ion I I . 2),. are

e i the r invar iant or transform one in to another. Some authors have not used.

r e l a t i o n s (2.18) and (2.19) (or analogous r e l a t i o n s ) and . theirs '. .• choice 7)

of amplitudes suffers from various diseases .

72/P.465' ' '• ; .

9)

Next we express gauge invariance with respect to each photon, i . e . ,

(J/* Ar ( T, o/j£?'j = o (2.20)-;

h r l%ci,q') l" = O (2.21)

8) We obtain 16 equation^ and 14 of them are independent . Thus we are l e f t

with 34-2-14 = 1b independent invar iant amplitudes for Compton sca t t e r i ng

of o f f - she l l photons on polar ized nucléons.

The elimination of 14 amplitudes from Eqs. (2.20) and (2.21) must

be done in such a way as t o preserve the crossing proper t ies of the remaining

amplitudes. We sha l l only give an example of the method used. The following

equations come from Bq. (2.20) (coeff icient of ay ) and Eq. (2.21) (coef f i ­

c ien t of a' )

a, + cj*a3 + ( Œ4J as + (q-y) a, 0 - z (T-cfj a3l =0 (2.22)

a r +q | ! l a* . + f*q'jas+ f^')a, 0 + 1 (f-qj a 3 2 =0 (2.23)

It is possible to solve these equations without introducing kinematical sin­

gularities in y and t . Moreover we can preserve crossing properties by a

symmetrical elimination, i.e.,

a.j = - -t f a, + ( <2-q) a 5 + (w) ala - 1 (f-y) a3,J (2.24)

a„ = _ ±_ [ a, +• (f.cjj ae + fc^'j a t o + 2 fî^ar J (2.25) T

„ , „ . . , - . . ^ . . .. , „ • .3 , . 4 nie J.ISL {ïtioj AS ci yiuac xor LIULU a io i ce , s in^e u arid L Lidrisxunu /» ' /*»

in to each other by cross ing. The same method can be applied t o the remaining

12 equations» We have checked tha t two equations ( the coeff ic ient of q'v <Â

in Eq. (2.20) and tha t of CU^L in Eq. (2.21)) are i d e n t i c a l l y s a t i s f i e d

byoi r so lu t ion . From t h i s solut ion of Eqs„(2.20) and (2.21) we can wr i t e the

general tensor A w l^ify'j ') with 18 independent invar iant amplitudes

f ree from Icinematical s i n g u l a r i t i e s in \) and t

in the following way '

72/P.465

10)

+ K^r, az + fyS, as + Sjtf; a 6 ^ S ^ c L , , (2.26)

The 4-vectors E , E' , S and S' are defined by

E = ?_<*!>g S = c j - f î i ^ ' <T 7 1 f ^ (2.27)

I t i s easy t o verify tha t each tensor coeff ic ient in the decomposition (2.26) -

i s gauge invar iant ; l e t us remarie tha t

E.q = S ' -q = K ' - q ' = S-Cj' = o . (2-28)

^ 44', _ ,\ In order to obtain the general Compton amplitude tt,«r [ Jr, c j , q7

under the form (2.8) we can use a Bouchiat-Michel type formula [lOJ . We

sha l l choose i t in a way best adapted to our purposes :

mt>\v) V ( r < ' H ' ) W ( M ) = ^I«o^J® s(fW) V (*.«?.?')J C2.29)

Eq,(2.26) requires the evaluation of the t races

where I T = 4L , fy , fy, , fyY fj + fa fy* •

The computation of these four t races can be done eas i ly ; rox.-:h more tedious i s

the use of these t r aces by means of Eq. (2.29) in order t o put C„,, {x,Cj,Q'/

in the form (2.8) . We find tha t there are five invar iant amplitudes

72/P.465

11)

Hi ( p 2 , a1', V , tj for the spin independent tensor C„y ( T ^ e ^ a ' J

and th i r t een invariant amplitudes CJ. I CJ , a'1, V t t ) for the spin

dependent .pseudotensor ^uve U i t ) / ! i | ' / (see Appendix, Eqs.(A.l)

and (A.4)) . The r e l a t i on between invar iant amplitudes -T2,- , tTU • and y •

Cl j i s given by (2.31)

Jl, '(f.tfvrt) = I [(zMz. t)a, + E ^ a , j

-0-z (^cj'^v^) = cM[(ZMl-L)at+ zMFa { + î M ( a u + a t t J

-a 3(c} 2

)c ?

, 2,y, tJ = cM[(2M J _ i Ja s + 2Mi/a„ H M a „ + 2va S f - ( y - 7 " J a i ( - z a 3 j J

- 2 v a 2 f -Cc? J -<7' J ja 3 l -za 3 3 ]

- r t s ( ^ y > , t , U t M [ ( E M 2 - | J a f o - 2 y ( a 3 , - a 3 j j

and

S, W / ^ = 2 * « ' < * *

S 3 ^ , 9 " , ^ t ; - ^ M ' a , 5

S k ( ^ , 9 * v,t) = zL M 2 a„

3 s C^, 9", v, tj = î i N' ÛÎJ

S 6 ( < f , 7 ' 1 , ^ / = i i M ' a J S C2.32)

S , f eyS«y**, y,t) = * i M ! a J S

S, (f.tfv.t) = - H M 3 a 8

S 9 (<f,<?V,^ - - K M s a i t

- 1 O W l , 9V, U = - 'HM 3CX 3,

S„ f ^ ' W , ej = -St M'a»

no fo

12)

The invar iant amplitudes il; and ^j ; are free from kinematical s ingula-v

r i t i e s in t> and t . They allow to wri te dispersion r e l a t i ons in the var-

r i a b l e y at fixed t . Powers of the nucléon mass M have been introduced

in Eqs.(2.31) and (2.32) in order to have invar iant amplitudes Jh( and Ci ; J

with same dimension'.

The exp l i c i t r e su l t for the general Compton sca t t e r i ng of o f f - she l l

photons on polarized nucléons in terms of the invar iant amplitudes

-Q-t i^1, q'1/ V / t) and S • ( Cj}, qu, V, t) i s given in the Appendix (see Eqsv (A.1) to (A.6)) . Each tensor covariant in Eqs. (A.2) and (A.5) i s

gauge inva r i an t .

1. Restr ict ionsdue to Time Reversal Invariance.

From time reversa l invariance, the general Compton sca t t e r i ng s a t i s ­

f i e s the following condition

Cfd.W) - (-/""' ( <T_£, (?,?, ?}f (2.33)

With fc> = ((0° la ) f the « operation on the 4-momenta i s simply : fc> = fh° ~~h).

T.'.ie symbol * means complex conjugate, Eq. (2.33) implies r e s t r i c t i o n s on the

i n t r i n s i c tensors Q-„r (T,o,a'J and ^are l 2 3 ; c J i c ) 7 • We have

C (*,«,,«,') = - f C (T,S[,$f (2.34)

and

dL ( T . ^ U -(V/"\ /? * S'il*

The restrictions due to Eqs. (2.34) and (2.35) impose that

-ÏÎ £ I «J 1 ^' < / "j and CJ . (of* a'* v1, b) are pure imaginary invariant

J amplitudes.

£. Crossing Symmetry Properties.

With our conventions the crossing symmetry proper t ies - read

€r {e,w') = C y / 4 (<P,- ?;-<jj (2.36)

72/P.465

13)

The corresponding r e s t r i c t i o n s on the invar iant amplitudes are as follows (r=e

Appendix, Eqs.(A.2) , (A.5) and (A.7))

(2.37)

and

3. Restrictions due to the Hermiticity of the Compton Amplitude.

From the definition of the general Compton amplitude in Eq,(1.l) , and the hermiticity of the electromagnetic current, one can show that "-«if \ * - i c [ < c i ' ) obeys the following hermiticity property

The corresponding restrictions on the invariant amplitudes are the following :

Ai ( f , ^ ^ = *i*(<l"i<i\*>lk) t~,,2,s

A.iWvtf-JLKlW'.t) (2-40)

and

&AW>,t)--s? if,?,»,*-')

14)

Using time reversa l invariance and hermi t ic i ty , the invar iant ampli­

t u d e s IL ; , C J • have the following proper t ies under the exchange

q - — • - q '

and

£« ( " P ^ / ^ = 2< («j'^*, V, ^ 1 = 1,2,11,13

2 2

In the case of equal "phcton masses" q = q' , these r e l a t i ons play an

important ro le to decrease the number of invar iant amplitudes (see Section

IV.2) .

I l l - THE FORWARD SCATTERING LIMIT ( t = o ) .

We wi l l be concerned in t h i s sect ion with the forward sca t t e r ing :

l i m i t ( t = o) of the absorptive Compton amplitude. I t i s well-faiown that

t h i s s i tua t ion i s c losely re la t ed to deep i n e l a s t i c s ca t t e r ing of e lectrons

(or muons) on nucléons (see Pig. 5 ) . Talcing in to account he rmi t i c i ty and

time reversa l invariance, there are two invar ian- amplitudes for the spin

independent tensor and two invar iant amplitudes for the spin dependent t ensor .

This r e su l t i s obtained s e t t i n g

q = q ' , P = P' (3.1)

in Eqs. (A.2) and (A„5). Then we have

S = S' = 0

(3.2) R = R' ' = P " (p.q) 2

q

q

A = 0

E '

H'

= *-ctg-Pi/ ~*

F 0 T "

- c«r/y3 ? ' 1z

and we are left with the following tensor covariants

and £ • £ *r*>

q l (3.3)

^ • ^ p Y ^ f - r ^ .. (3-4)

S 1 « " M K ^ J ^V" <f ^ + <V^r)

1)

« M3

The comparison between r e l a t i o n s (3.3) and the usual de f in i t ion 1 ' ' of V„v

in terms of s t ruc tu re functions W, fflf'yj and Wj. (ql

lv) y ie lds

the r e s u l t 9 ^ ' 1 0 )

72/P.4S5

16)

(3.5)

The se t of tensors in Eq„ (3,4) seems to ind ica te that there are four inva­

r i an t amplitudes for the spin dependent par t whereas we expect th ree s t r u c ­

tu re functions or two with time reversa l invariance plus he rmi t i c i ty . In

f a c t , using the i den t i t y

1 1 } where V is any 4-vector, it can be shown (with V = p) that

IK, "^ - S'^i - ^ i * / W - o (3.7) r j « M* /"'/ /"/

Thus we have actually three independent amplitudes for the spin dependent

part, which is the correct result. In order to avoid kinematical singulari-8 9

S and S , 8 9

t i e s in V and to preserve crossing symmetry between. S and S , we choose 13 to el iminate S • Then

B „ ,.t A (3-8)

(3.9)

where, by def in i t ion

gi(«l» = Ta ty») - T«tyv)

The comparison between (3.8) and the usual def in i t ion ' of Wurf in

terms of s t r uc tu r e functions <?, (tj*v) , G2(^z

lv) and bz(a)v)

yie lds the r e s u l t

<j3(«f,v) = G,fij«,vJ + G.Ojf'.v]

17)

Using now time reversa l invariance plus hermi t ic i ty for the invar iant ampli­

tudes J: (c j^vj (see Eq. (2.43)) we have

% (c{)v) = - ^ ( Y ; V j i . e . , in terms of s t r uc tu r e functions G. , G (q ,v) = 0 , as i s . we l l -

2 2 Jcnown. Final ly , we choose g (q ,v) and g,(q ,v) as bas ic invar iant

. . . A 4 ' , i amplitudes for the spin dependent part of yy v ( l 0 !^ ) •

The r e l a t i on between g ,g and G ,G i s

• frW-^W'*) (3.12)

In the Appendix we write explicitely the tensor covariants correspon­

ding to the situation we have just described. Of course each tensor covariant

in Eqs.(A.1l) and (A.13) is gauge invariant.

18)

;

IV - STUDY OF PARTICULAR NON-FORWARD SCATTERING LIMITS.

We want t o c o n s i d e r i n t h i s s e c t i o n some p a r t i c u l a r n o n - f o r w a r d s c a t -

t e r i n g l i m i t s , f o r t h e g e n e r a l Compton a m p l i t u d e (£ ( (Y) Q^a'j d e f i n e d

i n Eq. ( 1 . 1 ) .

1 . One Rea l Pho ton , One O f f - S h e l l ' P h o t o n .

Two p a r t i c u l a r c a s e s d i r e c t l y r e l a t e d t o p h y s i c a l p r o c e s s e s can be

c o n s i d e r e d w i t h one r e a l photon and one o f f - s h e l l p h o t o n .

A") Rea l_ incoming pho ton (q = 0 ) , o f f - s h e l l o u t g o i n g p h o t o n .

A p h y s i c a l p r o c e s s r e l a t e d t o t h i s s i t u a t i o n i s t h e p h o t o p r o d u c t i o n o f e l e c t r o n

( o r muon) p a i r s on nuc léons . . To lowes t o r d e r i n t h e f i n e s t r u c t u r e c o n s t a n t

o(- = , the diagrams which contribute to this process are given

in Fig. 4 . More precisely we have in Fig. 4a one of the Bethe Heitler dia­

grams and in Fig. 4b the so-called Compton diagram. For the Compton ampli-

2

tude which appears in this diagram, we have with q =0 three invariant ampli­

tudes for the spin independent tensor C. (T,cf,a') and nine invariant

amplitudes for the spin tensor "'M-VP ('^•C[IC\') • These results, and

the corresponding tensor covariants are obtained from Eqs. (A.1) and (A.4)

1 setting the coefficient of — equal to zero. For example, one obtains

q

(CP-cjJji, ( o ^ ' J ^ t ; + ( T q ' J J l ^ ((>,<#»>, t) = o (4.1)

We solve this equation setting

JLt {o, c,>) v, t) » £L2? -n*. (oyc,') v, t) ( 4 . a )

SLlto,<f,Y,t) - -'^SL^lcf, h)

For the spin independent tensor C (x,q,cj'/ s , we choose the

and -fl-* (°, 1'\ »,tr) • . -Is (° • <?'* v > b)

-fttK. \0/ <f'. i ^ i t~J as basic invariant ampli-thides. The corresponding

tensor covariants are written in the Appendix (see Eqs. (A.14) and (A.15)) ;

i t can be verified that they are gauge invariant.

72/P-465

19)

<f=o

In the same way as in Eq. (4 .2 ) , we can introduce a new invar iant

amplitude S ^ (o, £f{i, V,t) for the spin dependent tensor Cm.» (î.cj.oj'

The nine tensor covariants for (Tuw> C 3?, cj, cj 'J I ; are also wri t ten in

the Appendix (see Eqs. (A.16) and (A.17)) ; each of them i s gauge invar ian t .

B°) Off-shell incoming_photon, r ea l putgoing_ photon (q^ _= 0 ) .

A physical process r e l a t ed to t h i s s i t ua t i on i s the electron bremsstrahlung

in the react ion

electron + nucléon — • electron + nucléon + photon .

To lowest order in cc , the diagrams which contr ibute t o t h i s process

are given in Fig. 3 . We have in Fig. 3a one of the 3ethe-Hei t ler diagrams

and in Fig. 3b the Compton diagram. For the Compion amplitude which appear

2 in t h i s case, we have with q' = 0 three invar iant amplitudes for the

spin dependent tensor \Lu.v [T, q, 9'J , E and n ine j invar iant amplitudes

for the spin tensor ^uve K^iclicll) , j _ • These r e s u l t s , and the cor res­

ponding tensor covariants are obtained from Eqs. (A.l) and (A.4) s e t t i n g

the coeff icient"of —— equal to zero . As previously, we introduce new q'

invar iant amplitudes -&„„£ (cj)o,\>,t) and £„„£. [ y, o, i/, tj .

The t e n s o r s . £^v ( î . t ^ q ' j I ( j _ and C (T^.cj') I ^ are

e x p l i c i t e l y wri t ten in the Appendix, Eqs. (A.18) to (A.21) .

Cases A") and B") tha t we have j u s t discussed are r e l a t ed to each

other by crossing symmetry p r o p e r t i e s . In fac t under cross ing ,

-O-Ui"'^ >>h) - ^ouf Li') °, -*, h) (4.3)

and furthermore, (see Eqs,(A.15), (A.17) , (A.19) and (A.21))

n3 , ^

'r r U (4.4)

72/P.465

s 20)

T 1 — - UL

Fi un T 5 _ U\ > ' J vfs

T 4 _ _ ufu F! u n T 6 — .

- " T v ri /"/

T 7 — - I I e

n Yï T 9 — • - I I s

V'f 7"-f T 1 0 — * 'C Ft FS T

i n —* ~vu

FS n

2,3

(4.5)

2 . Off-Shell Photons with Equal Masses (g = g' ) .

In t h i s case, using time reversa l invariance plus he rmi t i c i ty

(see Eqs. (2.42) and (2.43)) . we see tha t there are four invariant amplitudes

-^-£ ( °| / 9 i v i t) for the spin independent tensor C ^ \'Slcjlc^1]

and eight invar iant amplitudes 3 . ( d\ q z V, f") for the spin tensor

^Mvt '-f'< *3f# *ï j . The corresponding tensor covariants are wr i t t en in

t he Appendix, Eqs. (A.22) t o (A.25) .

2 2 3 . Real Compton Scat ter ing (g = q' = o ) .

As i s well-known [ l1 j , r e a l Compton sca t t e r i ng i s described

by s ix invar iant amplitudes, two for the spin independent p a r t , four for

the spin dependent p a r t . These amplitudes can be obtained from the case

2 2 ' 1 q = q' , s e t t i n g the coeff icient of — equal to zero in Eqs. (A.22)

• q

and (A.24) . As previously, we introduce new invar ian t amplitudes XI (v,t)

and . The tensor covariants for r e a l Compton s c a t ­

t e r i n g are wr i t ten i n t he Appendix, Eqs. (A.26) t o (A.29) . Each of them

i s gauge inva r i an t .

72/P.465

H)

A C K N O W L E D G M E N T S

I t i s a pleasure to thank E. de Rafael for suggesting t h i s

problem and for h i s constant and f r iendly col laborat ion along t h i s

work. The author wishes to thank E. Stora , C P . Korthals Altes

for f r u i t f u l discussions and Professor A. Visconti for h i s i n t e r e s t

on t h i s work and h i s h o s p i t a l i t y at the "Centre de Physique Théorique"

at Marse i l l es .

A P P E H D I

SUMMARY OF RESULTS

We recall the decomposition of the general Compton scattering of

off-shell photons on polarized nucléons into spin independent and dependent

parts

£ «£"(*,,,,.) - «xWia, +1 «>7 K<M'KVJ ^ We write for the spin independent tensor

5 £„, (?,<,,,') - p l ' fo^jH; ( < ^ », t) (A.1)

where

I L S r (A.2)

(A. 3)

i l = -i 5' < r M« /" T 5 - ' -S ' Ç

The 4-vectors R , R" , S and S' are defined by

E - Î - l™c, , 8 - « , - f - ^ a ' q* " " <jit "

E( =T- ( I±la ' , S' = a'- <lAa

and we r e c a l l t ha t

P = K p + p«)

Q = K i + V )

4 = | ( q - q ' ) .

We write for the spin dependent tensor

24)'

E

r —* K

s , —* - * ; E '

*tfV — E<trf^

H V —* "ttrfj*

G*W —* ** «iff

(A.7)

The absorptive Comptontamplitude i s decomposed in to spin independent

and spin dependent par t s in the following way

We have

Wr (T.^j - É 1^ CP1<H'J «O, (ft*,*, t) (A.8)

J

where the tensor covariants I „ y and sjj • are defined in Eqs. (A.2) and

(A.5) . Invariant amplitudes Wj , T". are r e l a t e d to SL( , Û=J .• by

W, ( ? < ? ", w, tj = 1 IK. Ji £ fy£, < , v, i - ,,,,..,, s

f ; («,«, 9", •», t j - i r * s f ? « v ; t ; y - i,t,.-, >3

The forward scattering limit of the absorptive Compton amplitude

(q = q') is described by two invariant amplitudes W, (q!, v"j and

w 2 . ( cf i V) for the spin independent part and two invariant amplitudes

gr(l ,v) and g„(q ,v) for the spin dependent part

W/V {M) = £j I J ' ( < l m < i ' ) U' <•?>*) CA.10) where

^ M - (s^(ri'i-^v(rfJifcf?îy) CA.12)

•t. . v . .

72/P.465

25)

where

•S r h'ï) -S/y (1"ï) =

Eqs. (A.12) and (A.13) have been obtained by talcing in to account he rmi t i c i ty

plus time reversa l invariance of the Compton amplitude.

2 The non-forward sca t t e r ing l imi t g = 0 of the general Compton

amplitude i s described by th ree invar iant amplitudes for the spin indepen­

dent par t and nine invar iant amplitudes for the spin dependent part

c

r (*.i.i')U. -._£. V (*.w')*-il°.1)"'6J ( A- 1 4 )

where .' J

j " 1,3,^, 5,6,^,3, to, in.

where (A.17)

- ^ (J-diK^fî.ffiJ-CÇ^M2- |)J - fs Z. (?SA«)<Slr (7-0!) ]

T 4

n -~ [~z *** **Mf2s$ + ^ W + c r j - fifïrr+ ^Jf?-fijj

7S/P.465

26)

~ ts«r ( îK(T-(Bj - CC" (M2- I )) - Ts €„r ( ?"+ ZTJ lS r J

i [-2 * w ^ + ? ; r r + es-; - f v i t ^ + ç r )M

T 6

2 ^

The non-forward sca t t e r ing l imi t q ' = 0 of the general Compton

amplitude i s described by three invar iant amplitudes for the spin independent

par t and nine invar iant amplitudes for the spin dependent par t

c... c r l * w ) \ r . o = S I 5 I U

n r ^'?<?'J^ W°M) (A-l8> where

î-ùtîr+M-'W+s^) ( A, 9 )

3 = 2 , 3 , 4 , 5 , 6 , 7 , 8 , 1 1 , out

where

72/P.4S5".

27) (A.21)

- d^ fTq-)fr(T-«) - csff MJ - {)) -fj^Ar^m^')]

2 2

The non-forward scattering limit g = g' of the general Comptor_

amplitude is described by four invariant amplitudes for the spin independent,

part and eight invariant amplitudes for the spin dependent part €r [T'^\.. „ °? Lr tew)-*1' '?* "' ( A , 2 2 )

1 , 1 i - 1,2,3,5 where

72/P.465

28)

L 1

m a _ Ml Irti , WL a ,

v^i.rj , -^ O ^ v j s. ii;i',',tj j = 1,2,4,6,8,10,12,13 {A. 24)

where (A.25)

v 6 - ' <P*Ar[(a ^ V l E + s'(F -F Qff")!

8 ^ ^«- /" f t ( C + <f d ^ ' l - (2 ~(f,<s]Q')H 1

v'Q _ i vw[(n (1-rLi ) /•/. - s ' f f + f y ? M l via = _, £^f £^_ ? ^ {e^ _ ^ ^ + V J

+ % £ ^ ( £ " + ^ t l T j 2 2

The non-forward sca t t e r ing l imi t q = q ' = 0 ( rea l Compton s c a t ­

t e r ing) of the general Compton amplitude i s described by two invar iant am­

pl i tudes for the npin independent par t and four invar iant amplitudes for the 12) spin dependent part '

% (*'1"»'JL ,. - 2 M'(?,<,,<,<) Jlc (y,t) (A.26)

where

<^J~^^') + &*] .5

75/P.465

29)

V;(*"<«') \^o - f xj* {<î'^ ~J ("'t} (A" where

30;

F O O T N O T E S

1) See e .g . ref . [ 3 j .-.'

2) The non-zero components of the metric tensor are : g = 1 , g... = g , ,

= g 3 3 = -1 . We take f = J" 'f'f- £ 3 ' , £ o l l 3 = 1 .

3) A - 1 - J££2*f£l£! + 2 ±!î£ . that is, in

components ( A ^ ) ' = 4 ' / - W l ' V » ^ + 2 J ! ^

4) Eq. (2.6) i s often used as a def in i t ion of <C„y ( £ , a , q ' J

However we sha l l prefer Eq. (1.1) whih exhibi t s c l e a r l y the crossing

symmetry proper t ies of the Compton amplitude.

5) As usua l , / i s defined by (/ = b„ If/1 • W e have a lso

«WHO'LL • 6) Throughout t h i s paper, any Dirac matrix i s understood to be taken between

Dirac spinors S(|»' ,A') and n(fc>,A)

7) For example, Gourdin £ 7 ! el iminates t and t and thus obtains

invar iant amplitudes with kinematical s i n g u l a r i t i e s in V , as can be

seen from Eqs. (2.18) and (2.19) . Meyer f 8 ] el iminates

(Tr (<J V + <,',) - % [ V + ^ ] ) $ and TA f v - % rr

but the f i r s t does not appear at a l l in Eqs'. (2.18) and (2.19) and so i s

ac tua l ly independent while the second one appears in Eq. (2.18) with coef-

2 2 2 2

f i c i e n t q - q' which i s zero when q = q ' . Amati, Jengo and

Remiddi £ 9 ] use a l l 34 tensor covariants and therefore have a

redundant s e t .

72/P.4S5

31)

8) Eight of the 16 equations come from Eq. (2.20) setting equal to zero the i

coeff ic ients of the following quan t i t i e s

The other eight equations come from Eq. (2.21) in the same way.

Only seven equations in each of those se t s are independent.

9) " i i<?i<\";Vik) and fj ( ?"> f", "/ t ) are r e a l

invar iant amplitudes defined as (see Eq. (2.13))

, y. ( f, <,«,», t), J. Tm Sj (f,q* v,t) j - f,z„..,ra

10) Throughout t h i s sec t ion , we wr i te Co (q 2, e) , J: (tj'/Vj a s

a short-hand nota t ion for tot- ( cjr', oz, V, o) , T- (a^q*, \>', o\ .

11) I would l i k e t o thank Prof. E. Stora for a discussion on t h i s po in t .

12) We wri te Slt (v,t) , 3 • f )>11) as a short-hand nota t ion for

•0-t (°,°,V, t) , S . f o , o , V, t j .

FIGURE CAPTIONS

33)

Figure 1 : A diagrammatic representa t ion for Compton sca t t e r i ng of o f f - she l l photons on polarized nucléons.

Figure 2 : Two-photon exchange contr ibut ions to e l a s t i c electron-proton s c a t t e r i n g .

Figure 3 '• a) One of the e lectron bremsstrahlung diagrams in the process : e lectron + nucléon - • electron + nucléon + photon.

b) Compton diagram in the same process .

Figure 4 : Diagrams for photoproduction of electron pa i r s on nucléons. a) One of t he Bethe-Heitler diagrams. b) Compton diagram.

Figure 5 : I n e l a s t i c electron-proton s ca t t e r i ng to lowest order in the electromagnetic coupling.

I,/» q'* 34)

FIGUEE 2

FIGUEE

72/P.465

35)

FIGURE 3

FIGURE 5

b)

7 2 / P . 4 6 5