- Email: [email protected]
Relativistic spin corrections in proton-proton bremsstrahlung
Volume 32B, number 2
PHYSICS
RELATIVISTIC
SPIN
LETTERS
CORRECTIONS BREMSSTRAHLUNG
IN
8 June 1970
PROTON-PROTON *$
J. H. M c G U I R E
Department of Physics, Texas A and M University, College Station, Texas 77843, USA Received 3 April 1970 Relativistic corrections to the ppy cross sections from the electromagnetic vertex are much s m a l l e r than experimental e r r o r if these corrections are calculated in the center of mass. In the lab frame they may be larger than experimental error•
In t h e i r t h e o r y of p r o t o n - p r o t o n b r e m s s t r a h lung, C r o m e r and Sobel [1] h a v e e x p r e s s e d the s c a t t e r i n g a m p l i t u d e in t e r m s of a n o n - r e l a t i v i s t i c e l e c t r o m a g n e t i c v e r t e x f o r the photon e m i s s i o n and a h a l f - o f f - s h e l l n u c l e a r s c a t t e r i n g a m plitude calculated from non-relativistic scattering p a r a m e t e r s . In t h i s t h e o r y , w h i c h has b e e n a d o p t e d by v a r i o u s a u t h o r s [2], the e l e c t r o m a g n e t i c v e r t e x is c a l c u l a t e d in the l a b o r a t o r y f r a m e , w h i l e the n u c l e a r a m p l i t u d e is c a l c u l a t e d in the c e n t e r of m a s s . It is the p u r p o s e of t h i s l e t t e r to note that (i) s u c h a c a l c u l a t i o n g i v e s p o o r a g r e e m e n t w i t h e x p e r i m e n t , (ii) r e l a t i v i s t i c c o r r e c t i o n s to the e l e c t r o m a g n e t i c v e r t e x m a y be added, (iii) good a g r e e m e n t w i t h e x p e r i m e n t can be o b t a i n e d u s i n g t h e s e c o r r e c t i o n s in the C r o m e r Sobel t h e o r y , and (iv) t h e s e r e l a t i v i s t i c e l e c t r o m a g n e t i c c o r r e c t i o n s a r e q u i t e s m a l l if the e l e c t r o m a g n e t i c v e r t e x is c o r r e c t l y c a l c u l a t e d in the c e n t e r of m a s s f r a m e . In the C r o m e r - S o b e l t h e o r y , the e l e c t r o m a g n e t i c i n t e r a c t i o n is g i v e n by
a(p) =p. ~ + l i
Upa" (~xK)
(1)
H e r e , p is a n u c l e o n m o m e n t a , 6 is the p o l a r i z a t i o n v e c t o r of the photon, K its m o m e n t u m , and the p r o t o n ' s m a g n e t i c m o m e n t . It is e a s y to d ~ r i v e t h i s e x p r e s s i o n , a s w e l l a s c o r r e c t i o n s to t h i s e x p r e s s i o n by c o n s i d e r i n g m a t r i x e l e m e n t s of the c o v a r i a n t o p e r a t o r a o p = eU(~/x - 1• Kp ~ - a/~ u g r )
(2)
* Supported in part by a grant from The National Science Foundation. ;~ Supported in part by the U.S. Air Force Office of Scientific Research Office of Aerospace Research under Grant No. 69-1817.
5':El Z
\
,,
2 .D
T
x.,\
/
I c%._~
//¢l:--±---.~-~S~
Jo. F~"
I 30-
_
I
- - -
60*
Fig.
|
i
~
8r
Izo °
l
~---=.J
150 °
i.
H e r e ~_ is the a n o m a l o u s m o m e n t of the p r o t o n , and e o ~ 0. If m a t r i x e l e m e n t s a r e e v a l u a t e d b e t w e e n a p p r o p r i a t e p o s i t i v e e n e r g y s p i n o r s (we s u p p r e s s n e g a t i v e e n e r g y s t a t e [3]), t h e n t e r m s of the f o r m r e s u l t .
a(p) = e ~ p ~ + 1 i (Kp+ 1 ) o " {e × [K - p k / m [~+
+ ~ ( p ( p / m ) 3)
(3)
H e n c e , we h a v e s i m p l e (v/c) c o r r e c t i o n s to the m a g n e t i c t e r m s . We h a v e u s e d eq. (3) to r e c a l c u l a t e the c o e f f i c i e n t s in t a b l e 2 of C r o m e r and S o b e l ' s p a p e r . O u r r e s u l t s a r e g i v e n in t a b l e 1. In fig. 1 we p r e s e n t t h e r e s u l t s of v a r i o u s c a l c u l a t i o n s at 158 MeV 30 ° - 30 ° and c o m p a r e the r e s u l t s to the e x p e r i m e n t a l d a t a of G o t t s c h a l k et al. [4]. A l l of t h e s e c a l c u l a t i o n s i n c l u d e i n t e r n a l s c a t t e r i n g t e r m s to o r d e r K °. T h e c a l c u l a t i o n 73
Volume 32B, number 2 i
PHYSICS
LETTERS
8 J u n e 1970
Oi
1
1
-
(c~ -~cb%~c%.,Cd,d) (Aaaa -+A b a b -÷Bc a c -+Bdad)
2
al'h
-
3
(~2" ~
- (Ba
*Bba b ~Aca e +Adad)
4
al.na2.~
- (c~
-+C b a ~ %%.~ Ca~d)
5
( q q' %" 4
6
"1"~%'i'
7
'~1"~o2"p
Xi ax= 1 - p ' p x / r n
cx = ~. Px/m
i(Eaa a - E b a b - F e ac t Fdad) -
i(Faa a - F b a b - E c a c +Eda d) ' i ( G a a a - Gbab +Gca c - Gda d)
- i ( G a a a - G b a b ~Gcac-Gdad )
Yi
9
~l'q
(Aaaa+Abab +Eeae ~ Edad) - (CaCa - CbCb) + G c
10
(/2'
(Eaa a+ Eba b +A c a c! Adad) + Ga a a + Gb c b - i ( C c c c - C dcd)
II ~rl"~ 12
0"2"/~
la
%.q%.
i(Caa c - C b a b ) ~ G ca c ~ Gdad+ Ab Cb ~-AbCb + Fc Cc ~ FaC d Gaac ~ Gba n ' i ( C c a c - Cdad) ~ FaCa +FbCb+AcCc ~ AdC d
h
14 al.i~al. ~ 15 al.ha2. ~ 16
al.pa2.n
c~% + % % - i(G¢%- Gdad - i(B~ ca - BdCd - E ¢% -Ed~d ) - i(Gaac - Gba b )+ Cc ac +Cdad + i(EaC a - E b c b - Bc c c + Bd cd) -
i(Faaa - V b a b - B c a c + B d a d) + i(aaC a - abc b) + CcCc + CdCd i(Ba aa -Bb ab - F c a c ~ Fd ad) ~ C Ca* CbCb + i(Gc Cc - Gacd)
u s i n g eq. (1) e v a l u a t e d in the lab f r a m e , d e n o t e d by L o, g i v e s poor a g r e e m e n t w i t h e x p e r i m e n t a l data. The c u r v e l a b e l e d L c u s e s eq. (3) (i. e., inc l u d e s r e l a t i v i s t i c s p i n c o r r e c t i o n s ) and is again e v a l u a t e d in the lab f r a m e . T h i s c u r v e g i v e s quite good a g r e e m e n t w i t h the data. C u r v e s C o and C c are c a l c u l a t i o n s done in the c e n t e r of m a s s without and w i t h spin c o r r e c t i o n s r e s p e c t i v e l y . We s e e that in the c e n t e r of m a s s o v e r r e l a t i v i s t i c s p i n c o r r e c t i o n s are m u c h s m a l l e r than the p r e s e n t e x p e r i m e n t a l e r r o r . C u r v e ONB s h o w s the e f f e c t of l e a v i n g out the s p i n t e r m s . In the n e u t r o n - p r o t o n b r e m s s t r a h l u n g c r o s s s e c t i o n , the spin t e r m s are m u c h l e s s i m portant [2, 3], so that w e e x p e c t our s p i n c o r r e c tions to be l e s s i m p o r t a n t in npv than in ppv.
74
+ GdCd
The author w i s h e s to thank A. H. C r o m e r , V . R . B r o w n and W. A. P e a r c e for p r o f i t a b l e d i s c u s s i o n s this w o r k . R e f e~'c~ces [1] A. H. C r o m e r and H. I. SobeL P h y s . Rev. 152 (1966) 1 3 5 1 : 1 6 2 (1967) 1174 (E). [2] M. I. Sobel and A. H. C r o m e r , P h y s . Rev. 132 (1963) 2698; P. Signel and D. M a r k e r . P h y s , L e t t e r s 26B (1968) 559; D. D r e c h s e l , L . C . M a x i m o n and R. E. W a r n e r . P h y s . Rev. 181 (1969) 1720; J. H. M c G u i r e , P h y s . Rev.. to be published. [3] J. H. M c G u i r e , T h e s i s , Northeastern University, Boston. Mass, [4] B. G o t t s c h a l k . W . J . S h a l e r and K. H. Wang, Nuct. P h y s i c s 94 (1967) 491.
PHYSICS
RELATIVISTIC
SPIN
LETTERS
CORRECTIONS BREMSSTRAHLUNG
IN
8 June 1970
PROTON-PROTON *$
J. H. M c G U I R E
Department of Physics, Texas A and M University, College Station, Texas 77843, USA Received 3 April 1970 Relativistic corrections to the ppy cross sections from the electromagnetic vertex are much s m a l l e r than experimental e r r o r if these corrections are calculated in the center of mass. In the lab frame they may be larger than experimental error•
In t h e i r t h e o r y of p r o t o n - p r o t o n b r e m s s t r a h lung, C r o m e r and Sobel [1] h a v e e x p r e s s e d the s c a t t e r i n g a m p l i t u d e in t e r m s of a n o n - r e l a t i v i s t i c e l e c t r o m a g n e t i c v e r t e x f o r the photon e m i s s i o n and a h a l f - o f f - s h e l l n u c l e a r s c a t t e r i n g a m plitude calculated from non-relativistic scattering p a r a m e t e r s . In t h i s t h e o r y , w h i c h has b e e n a d o p t e d by v a r i o u s a u t h o r s [2], the e l e c t r o m a g n e t i c v e r t e x is c a l c u l a t e d in the l a b o r a t o r y f r a m e , w h i l e the n u c l e a r a m p l i t u d e is c a l c u l a t e d in the c e n t e r of m a s s . It is the p u r p o s e of t h i s l e t t e r to note that (i) s u c h a c a l c u l a t i o n g i v e s p o o r a g r e e m e n t w i t h e x p e r i m e n t , (ii) r e l a t i v i s t i c c o r r e c t i o n s to the e l e c t r o m a g n e t i c v e r t e x m a y be added, (iii) good a g r e e m e n t w i t h e x p e r i m e n t can be o b t a i n e d u s i n g t h e s e c o r r e c t i o n s in the C r o m e r Sobel t h e o r y , and (iv) t h e s e r e l a t i v i s t i c e l e c t r o m a g n e t i c c o r r e c t i o n s a r e q u i t e s m a l l if the e l e c t r o m a g n e t i c v e r t e x is c o r r e c t l y c a l c u l a t e d in the c e n t e r of m a s s f r a m e . In the C r o m e r - S o b e l t h e o r y , the e l e c t r o m a g n e t i c i n t e r a c t i o n is g i v e n by
a(p) =p. ~ + l i
Upa" (~xK)
(1)
H e r e , p is a n u c l e o n m o m e n t a , 6 is the p o l a r i z a t i o n v e c t o r of the photon, K its m o m e n t u m , and the p r o t o n ' s m a g n e t i c m o m e n t . It is e a s y to d ~ r i v e t h i s e x p r e s s i o n , a s w e l l a s c o r r e c t i o n s to t h i s e x p r e s s i o n by c o n s i d e r i n g m a t r i x e l e m e n t s of the c o v a r i a n t o p e r a t o r a o p = eU(~/x - 1• Kp ~ - a/~ u g r )
(2)
* Supported in part by a grant from The National Science Foundation. ;~ Supported in part by the U.S. Air Force Office of Scientific Research Office of Aerospace Research under Grant No. 69-1817.
5':El Z
\
,,
2 .D
T
x.,\
/
I c%._~
//¢l:--±---.~-~S~
Jo. F~"
I 30-
_
I
- - -
60*
Fig.
|
i
~
8r
Izo °
l
~---=.J
150 °
i.
H e r e ~_ is the a n o m a l o u s m o m e n t of the p r o t o n , and e o ~ 0. If m a t r i x e l e m e n t s a r e e v a l u a t e d b e t w e e n a p p r o p r i a t e p o s i t i v e e n e r g y s p i n o r s (we s u p p r e s s n e g a t i v e e n e r g y s t a t e [3]), t h e n t e r m s of the f o r m r e s u l t .
a(p) = e ~ p ~ + 1 i (Kp+ 1 ) o " {e × [K - p k / m [~+
+ ~ ( p ( p / m ) 3)
(3)
H e n c e , we h a v e s i m p l e (v/c) c o r r e c t i o n s to the m a g n e t i c t e r m s . We h a v e u s e d eq. (3) to r e c a l c u l a t e the c o e f f i c i e n t s in t a b l e 2 of C r o m e r and S o b e l ' s p a p e r . O u r r e s u l t s a r e g i v e n in t a b l e 1. In fig. 1 we p r e s e n t t h e r e s u l t s of v a r i o u s c a l c u l a t i o n s at 158 MeV 30 ° - 30 ° and c o m p a r e the r e s u l t s to the e x p e r i m e n t a l d a t a of G o t t s c h a l k et al. [4]. A l l of t h e s e c a l c u l a t i o n s i n c l u d e i n t e r n a l s c a t t e r i n g t e r m s to o r d e r K °. T h e c a l c u l a t i o n 73
Volume 32B, number 2 i
PHYSICS
LETTERS
8 J u n e 1970
Oi
1
1
-
(c~ -~cb%~c%.,Cd,d) (Aaaa -+A b a b -÷Bc a c -+Bdad)
2
al'h
-
3
(~2" ~
- (Ba
*Bba b ~Aca e +Adad)
4
al.na2.~
- (c~
-+C b a ~ %%.~ Ca~d)
5
( q q' %" 4
6
"1"~%'i'
7
'~1"~o2"p
Xi ax= 1 - p ' p x / r n
cx = ~. Px/m
i(Eaa a - E b a b - F e ac t Fdad) -
i(Faa a - F b a b - E c a c +Eda d) ' i ( G a a a - Gbab +Gca c - Gda d)
- i ( G a a a - G b a b ~Gcac-Gdad )
Yi
9
~l'q
(Aaaa+Abab +Eeae ~ Edad) - (CaCa - CbCb) + G c
10
(/2'
(Eaa a+ Eba b +A c a c! Adad) + Ga a a + Gb c b - i ( C c c c - C dcd)
II ~rl"~ 12
0"2"/~
la
%.q%.
i(Caa c - C b a b ) ~ G ca c ~ Gdad+ Ab Cb ~-AbCb + Fc Cc ~ FaC d Gaac ~ Gba n ' i ( C c a c - Cdad) ~ FaCa +FbCb+AcCc ~ AdC d
h
14 al.i~al. ~ 15 al.ha2. ~ 16
al.pa2.n
c~% + % % - i(G¢%- Gdad - i(B~ ca - BdCd - E ¢% -Ed~d ) - i(Gaac - Gba b )+ Cc ac +Cdad + i(EaC a - E b c b - Bc c c + Bd cd) -
i(Faaa - V b a b - B c a c + B d a d) + i(aaC a - abc b) + CcCc + CdCd i(Ba aa -Bb ab - F c a c ~ Fd ad) ~ C Ca* CbCb + i(Gc Cc - Gacd)
u s i n g eq. (1) e v a l u a t e d in the lab f r a m e , d e n o t e d by L o, g i v e s poor a g r e e m e n t w i t h e x p e r i m e n t a l data. The c u r v e l a b e l e d L c u s e s eq. (3) (i. e., inc l u d e s r e l a t i v i s t i c s p i n c o r r e c t i o n s ) and is again e v a l u a t e d in the lab f r a m e . T h i s c u r v e g i v e s quite good a g r e e m e n t w i t h the data. C u r v e s C o and C c are c a l c u l a t i o n s done in the c e n t e r of m a s s without and w i t h spin c o r r e c t i o n s r e s p e c t i v e l y . We s e e that in the c e n t e r of m a s s o v e r r e l a t i v i s t i c s p i n c o r r e c t i o n s are m u c h s m a l l e r than the p r e s e n t e x p e r i m e n t a l e r r o r . C u r v e ONB s h o w s the e f f e c t of l e a v i n g out the s p i n t e r m s . In the n e u t r o n - p r o t o n b r e m s s t r a h l u n g c r o s s s e c t i o n , the spin t e r m s are m u c h l e s s i m portant [2, 3], so that w e e x p e c t our s p i n c o r r e c tions to be l e s s i m p o r t a n t in npv than in ppv.
74
+ GdCd
The author w i s h e s to thank A. H. C r o m e r , V . R . B r o w n and W. A. P e a r c e for p r o f i t a b l e d i s c u s s i o n s this w o r k . R e f e~'c~ces [1] A. H. C r o m e r and H. I. SobeL P h y s . Rev. 152 (1966) 1 3 5 1 : 1 6 2 (1967) 1174 (E). [2] M. I. Sobel and A. H. C r o m e r , P h y s . Rev. 132 (1963) 2698; P. Signel and D. M a r k e r . P h y s , L e t t e r s 26B (1968) 559; D. D r e c h s e l , L . C . M a x i m o n and R. E. W a r n e r . P h y s . Rev. 181 (1969) 1720; J. H. M c G u i r e , P h y s . Rev.. to be published. [3] J. H. M c G u i r e , T h e s i s , Northeastern University, Boston. Mass, [4] B. G o t t s c h a l k . W . J . S h a l e r and K. H. Wang, Nuct. P h y s i c s 94 (1967) 491.