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On the anomalous magnetic moment of the muon
Volume 20. n u m b e r 6
PHYSICS
LETTERS
1 A p r i l 1966
w h e r e m A i s t h e m e s o n m a s s . W e c a n t r y to e s t i m a t e i t b y l o o k i n g a t t h e f o r m f a c t o r s s l o p e a t ±2 = 0. i n e q s . (75, (10) t h e t e r m s w i t h t h e p i o n p o l e a r e e n h a n c e d i n t a k i n g d e r i v a t i v e s , m a s k i n g t h e s m o o t h e r b e h a v i o u r of t h e o t h e r t e r m s . F r o m e q s . (8), (9) a s s u m i n g F V2 () ~. = F ~ ( ~ 2 ) / 3 . 7 1 = ~rn2/(~ 2 + r a p , we o b t a i n r e s p e c t i v e l y m A = 1.07 GeV a n d m A = 1.18 GeV. In a p r e v i o u s w o r k [4], f r o m t h e c o m m u t a t o r s
[7 (a), 3)] = 0,
[7(-),
:
a(-) P
a n d in t h e s a m e a p p r o x i m a t i o n a s a b o v e , w e d e r i v e d
4rn ~ F r o m t h e s l o p e a t A2 = 0, w e o b t a i n e d m A = 1,24 GeV (in a m o r e r e f i n e d c a l c u l a t i o n of G(A 2) t h e r e s u l t w a s 1.1 GeV). T h e a g r e e m e n t a m o n g t h e v a r i o u s e s t i m a t e s s e e m s to b e r a t h e r e n c o u r a g i n g .
References 1. 2. 3. 4. 5.
M.Gell-Marm, Phys. Rev, 125 (1962) 1067. S.Fubini, G. F u r l a n and C.Rosetti, Nuovo Cimento A40 (1965} 1171. S. Fubini, G. F u r l a n and C. Rosetti, Nuovo Cimento, to be published. G. F u r l a n , R. Jengo and E. Remiddi. Istituto di F i s i c a di T r i e s t e , p r e p r i n t (submitted to Nuovo Cimento). A.H. Rosenfeld, A. B a r b a r o - G a l t i e r i , W.H. B a r k a s , P . L . Bastien. J. Kirz and M. Roos, UCRL 8 0 3 0 - P a r t I, August 1965. 6. D . A m a t i and S.Fubini, Ann. R e v . N u c l e a r Science 12 (1962) 359, eq. (10.13}. 7. S.M. B e r m a n and M. Veltman, Nuovo Cimento 38 (1965) 993.
ON
THE
ANOMALOUS
MAGNETIC
MOMENT
OF
THE
MUON
H. H. E L E N D
Ins titut p~r theoretische Physik der Universit~it, Freiburg i.Br. Received 15 F e b r u a r y 1966
An exact calculation of the anomalous magnetic m o m e n t of the muon to second o r d e r in the fine s t r u c t u r e c o n s t a n t is given. The u n c e r t a i n t i e s which l i m i t the possible p r e c i s i o n in testing quantum e l e c t r o d y n a m i c s with this effect a r e also discussed.
In r e c e n t y e a r s m u c h i n t e r e s t h a s b e e n d e v o t e d to t e s t i n g of q u a n t u m e l e c t r o d y n a m i c s . Up to n o w the most severe limit is given by the anomalous m a g n e t i c m o m e n t of t h e m u o n [1]. T h e s e c o n d o r d e r c o r r e c t i o n s of t h i s q u a n t i t y i n t h e f i n e s t r u c ture constant a have already been calculated by s e v e r a l a u t h o r s [2], w h e r e , h o w e v e r , t e r m s of the order m / M and higher terms are neglected, rn a n d M b e i n g t h e e l e c t r o n a n d t h e m u o n m a s s e s r e s p e c t i v e l y . T h u s t h e r e i s a n u n c e r t a i n t y of a b o u t 5 x 10 - 7 i n t h e t h e o r e t i c a l v a l u e of t h e g f a c t o r of t h e m u o n . On t h e o t h e r h a n d a n e w m e a s u r e m e n t of (g= 2) of t h e m u o n i s t o b e m a d e at CERN in the near future, the experimental er= 682
r o r of w h i c h i s h o p e d to b e of t h e o r d e r of 2 X 10-7 or smaller. These facts stimulated an e x a c t c a l c u l a t i o n of t h e s e c o n d o r d e r c o r r e c t i o n s , w h i c h w i l l b e r e p o r t e d o n h e r e . M o r e o v e r , we shall discuss the other uncertainties which limit the possible precision in testing quantum electrodynamics with this effect. Up to s e c o n d o r d e r i n ~ t h e a n o m a l o u s m a g n e t i c m o m e n t of t h e m u o n i s r e p r e s e n t e d b y t h e same Feynman diagrams as for the electron exc e p t f o r v a c u u m p o l a r i z a t i o n of a v i r t u a l e l e c t r o n p a i r b y t h e m u o n [2]. T h e c o n t r i b u t i o n of t h i s diagram is given in the standard way
Volume 20; number 6
PHYSICS LETTERS
1
T h e r e a r e s e v e r a l u n c e r t a i n t i e s involved in the r e s u l t (4). F i r s t of a l l we have to c o n s i d e r the influence of the e r r o r s in a and X. F r o m (2) we obtain for the a b s o l u t e e r r o r of g p
1
o
v2) + X (1 - u) ' _-
(1)
2
1 April 1966
Ag~ = 0.32 Aa + 0.019 /x~. A f t e r a s o m e w h a t tedious c a l c u l a t i o n we obtain the following e x a c t e x p r e s s i o n 62=(_~)2 / --~ 2s + ½ In ~
+
-__(~'X-1~hS-~\F~ 2 ) L~ - +
-f(-~-2)
rx + ' .~rtt~, _ 3 T in2.+~lnX] + v--~ ~
{_ _~IX_ ~ 2 +4 +f\2_ jX / +
½ln~-ln~:~
F~'2 - 2 f(½,,~) + 2 f ( - ½JX)~ . + - ~X2 L-6- - I n 2 2 +
+ ~ln2~+ ½in41n(4-X)-½f(~X)l) =
+
4 M -~ + olnm+
=
(2)
y2~m
[i
k~-~-) ~ +
+ 4 ( 2 - In M~ m / \ (- ~~ '/2 + O ( ~ ) }
2
~ 1.09586 (a),~,
H e r e f(x) d e n o t e s the Spence function [3]. The two l e a d i n g t e r m s of (2) a r e j u s t those which have been given p r e v i o u s l y [2]. The c o n t r i b u t i o n of the o t h e r s e c o n d o r d e r d i a g r a m s can be taken f r o m the c a l c u l a t i o n of the a n o m a l o u s m a g n e t i c m o m e n t of the e l e c t r o n [4] 61 = ( ~ ) 2 ~ '97[ lr2 , + 4 ~(3)- ~ 2 In 2} |-~+-i-~ (3) -0.32848 ( ~ ) 2 . F r o m (2) and (3) the following n u m e r i c a l value for the total r a d i a t i v e c o r r e c t i o n s of the g - f a c t o r of the muon up to the s e c o n d o r d e r in o~ i s obtained (g#-2)=2[~+61+52+O(~3)1
(6)
A c c o r d i n g to (5) the e r r o r of a c a u s e s an u n c e r tainty of 1 x 10 - 8 f o r g ~ , w h e r e a s the influence of the e r r o r of X i s l e s s i m p o r t a n t : 9 × 10 -11. M o r e o v e r , the t h i r d o r d e r c o r r e c t i o n s in a a r e of c o u r s e not included in (4), s i n c e they a r e unknown at p r e s e n t . But they can r e a s o n a b l y be a s s u m e d to be at m o s t of the o r d e r (ol/Tr) 3 = 1.25 x 10 -8. Thus, the value of g ~ given in (4) is the p r e d i c t i o n of p u r e quantum e l e c t r o d y n a m i c s within about one unit of 10-8. Deviations f r o m p u r e quantum e l e c t r o d y n a m i c s have to be e x p e c t e d f i r s t of a l l by weak and s t r o n g i n t e r a c t i o n s . The influence of weak i n t e r a c t i o n s has been s t u d i e d by s e v e r a l a u t h o r s . T h e i r r e s u l t s differ c o n s i d e r a b l y , but in g e n e r a l they a r e of the o r d e r of 10 -9 o r s m a l l e r [7]. The s i t u a t i a n i s different with s t r o n g i n t e r a c t i o n . H e r e it has been shown [8] that the e x i s t e n c e of the p - and wr e s o n a n c e s r e s u l t s in a c o n s i d e r a b l e pionic cont r i b u t i o n to the photon p r o p a g a t o r a s s o c i a t e d with the r a d i a t i v e c o r r e c t i o n s to the g - f a c t o r of the muon. If we use the d a t a of R o s e n f e l d et al. [6], this contribution is e s t i m a t e d to about 5 x 10--'8. The influence of the s t r o n g l y i n t e r a c t i n g p a r t i c l e s thus e x c e e d s the c o n t r i b u t i o n of the t h i r d o r d e r t e r m s a s well a s the i n a c c u r a c y of g # c a u s e d by the e r r o r s of ot and X. T h e r e f o r e , t h i r d o r d e r t e r m s can p r o b a b l y not be t e s t e d without a m o r e d e t a i l e d knowledge of the influence of s t r o n g int e r a c t i o n s . N e v e r t h e l e s s to t e s t quantum e l e c t r o d y n a m i c s a s f a r a s p o s s i b l e with this effect it would be v e r y worthwhile to i m p r o v e the a c c u r a c y of the planned e x p e r i m e n t at l e a s t half an o r d e r of magnitude f u r t h e r .
=
= 2 [ ~ ' + 0 . 7 6 7 4 ( ~ ) 2 + O(a3)] =
(4)
= 2 x 0.001 165 5 2 6 . In r e f s . 2-4 the following n u m e r i c a l d a t a have been u s e d [5, 6], w h e r e the e r r o r of ~ is s u b j e c t to s o m e d i s c u s s i o n [5]
The author is v e r y indebted to P r o f e s s o r H. S a l e c k e r , who s u g g e s t e d the p r o b l e m and s t i m u l a t e d the work in s e v e r a l m o s t helpful d i s c u s s i o n s . He i s a l s o v e r y g r a t e f u l to Dr. T. A n d e r s , who p e r f o r m e d an independent c a l c u l a t i o n of the integ-ration of (1) and s i m p l i f i e d the r e s u l t in (2). F i n a n c i a l s u p p o r t f r o m the B u n d e s m i n i s t e r i u m fttr w i s s e n s c h a f t l i c h e F o r s c h u n g i s g r a t e f u l l y a c knowledged.
ot -1 = 137.0388 ± 0.0006
M= (206.767
(5) ± 0.005)rn.
T h i s r e s u l t i n c l u d e s a l s o the d i a g r a m with v a c u u m p o l a r i z a t i o n of a ~ + - p a i r .
References 1. G.Charpak, F . J . M . F a r l e y , R.L.Garw,:n, T.Muller, J.C.Sens and A. Ziehichi, Physic8 Letters 1 (1962) 16; H. Salecker, Nuovo Cimento 29 (1963) 92. 683
Volume 20, n u m b e r 6
P HYSI CS L E T T E RS
2. H. Suura and E. H. Wicbmann, Phys. Rev. 105 (1957) 1930; A. Petermarm, Phys. Rev. 105 (1957) 1931. 3. K. Mitchell, Phil. M a g . 4 0 / I (1949)351. 4. R. Karplus and N. M. Kroll, Phys. Rev. 77 (1950) 536; C. M. Sommerfield, Phys. Rev. 107 (1957) 328; A . P e t e r m a r m , Helv. P h y s . A c t a 30 (1957) 407.
THE
SATURATION QUARK MODEL
1 A p r i l 1966
5. E . R . C o h e n and J . W . M . DuMond, Rev.Mod. Phys. 37 (1965) 537. 6. A. H. Roeenfeld, A. B a r b a r o - G a l t i e r i , W.H. B a r k a s , P. L. Bastien, J. Kirz and M. Roos, Rev. Mod. Phys. 37 (1965) 633. 7. See e . g . R . A . Shaffer, Phys. Rev. 135 (1964) B 187 and the l i t e r a t u r e quoted t h e r e . 8. L . D u r a n d , Phys. Rev. 128 (1962) 441.
PROBLEM IN THE OF ''ELEMENTARY''
NON-RELATIVISTIC PARTICLES
G. M O R P U R G O Istiluto di Fisca dell Universit~-Genova lstituto Nazionale di Fisica Nucleate - Sezione di Genova
Received 26 F e b r u a r y 1966
It is shown how a situation in which a t h r e e q u a r k s y s t e m is strongly bound to f o r m a proton, while a four or m o r e q u a r k s y s t e m is unbound or has a binding energy much s m a l l e r than expected at f i r s t sight may produce itself in the q u a r k model of " e l e m e n t a r y " p a r t i c l e s .
Among the questions which we have listed as needing clarification in the non-relativistic quark m o d e l of e l e m e n t a r y p a r t i c l e s [1], t h a t of " s a t u r a t i o n " i s a p r o m i n e n t o n e ; in v i e w of t h e m a n y s u c c e s f u l l t e s t s of t h e m o d e l [ 2 - 4 ] i t n o w appears appropriate to propose a possible qualitative solution to this question. This is the p u r p o s e of t h i s n o t e . In w h a t f o l l o w s we c o n c e n t r a t e o u r a t t e n t i o n on t h e b a r y o n s a l t h o u g h w h a t we a r e g o i n g to s a y m a y c l e a r l y e x t e n d t o t h e m e s o n s ; we m a y t h i n k of q u a r k s h a v i n g a m a s s M -~ 10 G e V * , b u t o u r r e m a r k s w i l l b e i n d e p e n d e n t on t h e p r e c i s e v a l u e of t h e m a s s of t h e q u a r k . W e n e e d only assume: 1) t h a t t h e m a s s M of t h e q u a r k i s m u c h l a r g e r t h e n t h e m a s s M B of t h e b a r y o n s : M >> M B 2) t h a t
I V ! ~ ~ / - ~ - 5'/p M>>
~t
w h e r e p - 1 i s t h e r a n g e of t h e i n t e r a c t i o n b e t w e e n two quarks. * Compare ref. 5, where a crude e s t i m a t e of 10 Ge¥ is given for the m a s s M of the 5 q u a r k b a s e d on the diff e r e n c e between the a v e r a g e m a s s of the odd p a r i t y and even p a r i t y m e s o n s . 684
If t h e f o r c e b e t w e e n q u a r k s i s d o m i n a n t l y t r a n s m i t t e d b y v e c t o r m e s o n s /~-1 ~ 5rnTr. In t h e f o l l o w i n g we m a y t h i n k , f o r c o n c r e t e n e s s , of t h e p o t e n t i a l b e t w e e n t w o q u a r k s a s a p o t e n t i a l w e l l w i t h a r a d i u s h a v i n g t h e a b o v e o r d e r of m a g n i t u d e . N o t e t h a t 1) a n d 2) a r e t h e b a s i c a s s u m p t i o n s of t h e n o n - r e l a t i v i s t i c q u a r k m o d e l . T h e q u e s t i o n of s a t u r a t i o n c a n in i t s e s s e n tials be formulated as follows: why systems with 4 or 5 quarks are not much more bound (or, i n o t h e r w o r d s , why t h e y a r e n o t m u c h l i g h t e r ) t h a n a s y s t e m of t h r e e q u a r k s , s a y , t o b e d e f i n i t e , a p r o t o n ? A n d a l s o : why a s y s t e m of 6 q u a r k s i s a d e u t e r o n a n d n o t a c o l l a p s e d a g g r e g a t e ** To clarify this question we begin by recalling that, assuming we deal only with two body f o r c e s , t h e a v e r a g e p o t e n t i a l e n e r g y of a p a i r of q u a r k s in t h e p r o t o n i s : (1)
In f a c t , in t h e n o n r e l a t i v i s t i c q u a r k m o d e l t h e k i n e t i c e n e r g y i s n e g l i g i b l e , w i t h r e s p e c t to ** Note that the " s a t u r a t i o n " question is masked, but n e v e r t h e l e s s exists also in all the m o r e a b s t r a c t approaches; it can be stated in this way: why a r e the r e p r e s e n t a t i o n s _8 for the m e s o n s and 8, 10 for the b a r y o n s the lowest ones and in fact the only ones so far o b s e r v e d
PHYSICS
LETTERS
1 A p r i l 1966
w h e r e m A i s t h e m e s o n m a s s . W e c a n t r y to e s t i m a t e i t b y l o o k i n g a t t h e f o r m f a c t o r s s l o p e a t ±2 = 0. i n e q s . (75, (10) t h e t e r m s w i t h t h e p i o n p o l e a r e e n h a n c e d i n t a k i n g d e r i v a t i v e s , m a s k i n g t h e s m o o t h e r b e h a v i o u r of t h e o t h e r t e r m s . F r o m e q s . (8), (9) a s s u m i n g F V2 () ~. = F ~ ( ~ 2 ) / 3 . 7 1 = ~rn2/(~ 2 + r a p , we o b t a i n r e s p e c t i v e l y m A = 1.07 GeV a n d m A = 1.18 GeV. In a p r e v i o u s w o r k [4], f r o m t h e c o m m u t a t o r s
[7 (a), 3)] = 0,
[7(-),
:
a(-) P
a n d in t h e s a m e a p p r o x i m a t i o n a s a b o v e , w e d e r i v e d
4rn ~ F r o m t h e s l o p e a t A2 = 0, w e o b t a i n e d m A = 1,24 GeV (in a m o r e r e f i n e d c a l c u l a t i o n of G(A 2) t h e r e s u l t w a s 1.1 GeV). T h e a g r e e m e n t a m o n g t h e v a r i o u s e s t i m a t e s s e e m s to b e r a t h e r e n c o u r a g i n g .
References 1. 2. 3. 4. 5.
M.Gell-Marm, Phys. Rev, 125 (1962) 1067. S.Fubini, G. F u r l a n and C.Rosetti, Nuovo Cimento A40 (1965} 1171. S. Fubini, G. F u r l a n and C. Rosetti, Nuovo Cimento, to be published. G. F u r l a n , R. Jengo and E. Remiddi. Istituto di F i s i c a di T r i e s t e , p r e p r i n t (submitted to Nuovo Cimento). A.H. Rosenfeld, A. B a r b a r o - G a l t i e r i , W.H. B a r k a s , P . L . Bastien. J. Kirz and M. Roos, UCRL 8 0 3 0 - P a r t I, August 1965. 6. D . A m a t i and S.Fubini, Ann. R e v . N u c l e a r Science 12 (1962) 359, eq. (10.13}. 7. S.M. B e r m a n and M. Veltman, Nuovo Cimento 38 (1965) 993.
ON
THE
ANOMALOUS
MAGNETIC
MOMENT
OF
THE
MUON
H. H. E L E N D
Ins titut p~r theoretische Physik der Universit~it, Freiburg i.Br. Received 15 F e b r u a r y 1966
An exact calculation of the anomalous magnetic m o m e n t of the muon to second o r d e r in the fine s t r u c t u r e c o n s t a n t is given. The u n c e r t a i n t i e s which l i m i t the possible p r e c i s i o n in testing quantum e l e c t r o d y n a m i c s with this effect a r e also discussed.
In r e c e n t y e a r s m u c h i n t e r e s t h a s b e e n d e v o t e d to t e s t i n g of q u a n t u m e l e c t r o d y n a m i c s . Up to n o w the most severe limit is given by the anomalous m a g n e t i c m o m e n t of t h e m u o n [1]. T h e s e c o n d o r d e r c o r r e c t i o n s of t h i s q u a n t i t y i n t h e f i n e s t r u c ture constant a have already been calculated by s e v e r a l a u t h o r s [2], w h e r e , h o w e v e r , t e r m s of the order m / M and higher terms are neglected, rn a n d M b e i n g t h e e l e c t r o n a n d t h e m u o n m a s s e s r e s p e c t i v e l y . T h u s t h e r e i s a n u n c e r t a i n t y of a b o u t 5 x 10 - 7 i n t h e t h e o r e t i c a l v a l u e of t h e g f a c t o r of t h e m u o n . On t h e o t h e r h a n d a n e w m e a s u r e m e n t of (g= 2) of t h e m u o n i s t o b e m a d e at CERN in the near future, the experimental er= 682
r o r of w h i c h i s h o p e d to b e of t h e o r d e r of 2 X 10-7 or smaller. These facts stimulated an e x a c t c a l c u l a t i o n of t h e s e c o n d o r d e r c o r r e c t i o n s , w h i c h w i l l b e r e p o r t e d o n h e r e . M o r e o v e r , we shall discuss the other uncertainties which limit the possible precision in testing quantum electrodynamics with this effect. Up to s e c o n d o r d e r i n ~ t h e a n o m a l o u s m a g n e t i c m o m e n t of t h e m u o n i s r e p r e s e n t e d b y t h e same Feynman diagrams as for the electron exc e p t f o r v a c u u m p o l a r i z a t i o n of a v i r t u a l e l e c t r o n p a i r b y t h e m u o n [2]. T h e c o n t r i b u t i o n of t h i s diagram is given in the standard way
Volume 20; number 6
PHYSICS LETTERS
1
T h e r e a r e s e v e r a l u n c e r t a i n t i e s involved in the r e s u l t (4). F i r s t of a l l we have to c o n s i d e r the influence of the e r r o r s in a and X. F r o m (2) we obtain for the a b s o l u t e e r r o r of g p
1
o
v2) + X (1 - u) ' _-
(1)
2
1 April 1966
Ag~ = 0.32 Aa + 0.019 /x~. A f t e r a s o m e w h a t tedious c a l c u l a t i o n we obtain the following e x a c t e x p r e s s i o n 62=(_~)2 / --~ 2s + ½ In ~
+
-__(~'X-1~hS-~\F~ 2 ) L~ - +
-f(-~-2)
rx + ' .~rtt~, _ 3 T in2.+~lnX] + v--~ ~
{_ _~IX_ ~ 2 +4 +f\2_ jX / +
½ln~-ln~:~
F~'2 - 2 f(½,,~) + 2 f ( - ½JX)~ . + - ~X2 L-6- - I n 2 2 +
+ ~ln2~+ ½in41n(4-X)-½f(~X)l) =
+
4 M -~ + olnm+
=
(2)
y2~m
[i
k~-~-) ~ +
+ 4 ( 2 - In M~ m / \ (- ~~ '/2 + O ( ~ ) }
2
~ 1.09586 (a),~,
H e r e f(x) d e n o t e s the Spence function [3]. The two l e a d i n g t e r m s of (2) a r e j u s t those which have been given p r e v i o u s l y [2]. The c o n t r i b u t i o n of the o t h e r s e c o n d o r d e r d i a g r a m s can be taken f r o m the c a l c u l a t i o n of the a n o m a l o u s m a g n e t i c m o m e n t of the e l e c t r o n [4] 61 = ( ~ ) 2 ~ '97[ lr2 , + 4 ~(3)- ~ 2 In 2} |-~+-i-~ (3) -0.32848 ( ~ ) 2 . F r o m (2) and (3) the following n u m e r i c a l value for the total r a d i a t i v e c o r r e c t i o n s of the g - f a c t o r of the muon up to the s e c o n d o r d e r in o~ i s obtained (g#-2)=2[~+61+52+O(~3)1
(6)
A c c o r d i n g to (5) the e r r o r of a c a u s e s an u n c e r tainty of 1 x 10 - 8 f o r g ~ , w h e r e a s the influence of the e r r o r of X i s l e s s i m p o r t a n t : 9 × 10 -11. M o r e o v e r , the t h i r d o r d e r c o r r e c t i o n s in a a r e of c o u r s e not included in (4), s i n c e they a r e unknown at p r e s e n t . But they can r e a s o n a b l y be a s s u m e d to be at m o s t of the o r d e r (ol/Tr) 3 = 1.25 x 10 -8. Thus, the value of g ~ given in (4) is the p r e d i c t i o n of p u r e quantum e l e c t r o d y n a m i c s within about one unit of 10-8. Deviations f r o m p u r e quantum e l e c t r o d y n a m i c s have to be e x p e c t e d f i r s t of a l l by weak and s t r o n g i n t e r a c t i o n s . The influence of weak i n t e r a c t i o n s has been s t u d i e d by s e v e r a l a u t h o r s . T h e i r r e s u l t s differ c o n s i d e r a b l y , but in g e n e r a l they a r e of the o r d e r of 10 -9 o r s m a l l e r [7]. The s i t u a t i a n i s different with s t r o n g i n t e r a c t i o n . H e r e it has been shown [8] that the e x i s t e n c e of the p - and wr e s o n a n c e s r e s u l t s in a c o n s i d e r a b l e pionic cont r i b u t i o n to the photon p r o p a g a t o r a s s o c i a t e d with the r a d i a t i v e c o r r e c t i o n s to the g - f a c t o r of the muon. If we use the d a t a of R o s e n f e l d et al. [6], this contribution is e s t i m a t e d to about 5 x 10--'8. The influence of the s t r o n g l y i n t e r a c t i n g p a r t i c l e s thus e x c e e d s the c o n t r i b u t i o n of the t h i r d o r d e r t e r m s a s well a s the i n a c c u r a c y of g # c a u s e d by the e r r o r s of ot and X. T h e r e f o r e , t h i r d o r d e r t e r m s can p r o b a b l y not be t e s t e d without a m o r e d e t a i l e d knowledge of the influence of s t r o n g int e r a c t i o n s . N e v e r t h e l e s s to t e s t quantum e l e c t r o d y n a m i c s a s f a r a s p o s s i b l e with this effect it would be v e r y worthwhile to i m p r o v e the a c c u r a c y of the planned e x p e r i m e n t at l e a s t half an o r d e r of magnitude f u r t h e r .
=
= 2 [ ~ ' + 0 . 7 6 7 4 ( ~ ) 2 + O(a3)] =
(4)
= 2 x 0.001 165 5 2 6 . In r e f s . 2-4 the following n u m e r i c a l d a t a have been u s e d [5, 6], w h e r e the e r r o r of ~ is s u b j e c t to s o m e d i s c u s s i o n [5]
The author is v e r y indebted to P r o f e s s o r H. S a l e c k e r , who s u g g e s t e d the p r o b l e m and s t i m u l a t e d the work in s e v e r a l m o s t helpful d i s c u s s i o n s . He i s a l s o v e r y g r a t e f u l to Dr. T. A n d e r s , who p e r f o r m e d an independent c a l c u l a t i o n of the integ-ration of (1) and s i m p l i f i e d the r e s u l t in (2). F i n a n c i a l s u p p o r t f r o m the B u n d e s m i n i s t e r i u m fttr w i s s e n s c h a f t l i c h e F o r s c h u n g i s g r a t e f u l l y a c knowledged.
ot -1 = 137.0388 ± 0.0006
M= (206.767
(5) ± 0.005)rn.
T h i s r e s u l t i n c l u d e s a l s o the d i a g r a m with v a c u u m p o l a r i z a t i o n of a ~ + - p a i r .
References 1. G.Charpak, F . J . M . F a r l e y , R.L.Garw,:n, T.Muller, J.C.Sens and A. Ziehichi, Physic8 Letters 1 (1962) 16; H. Salecker, Nuovo Cimento 29 (1963) 92. 683
Volume 20, n u m b e r 6
P HYSI CS L E T T E RS
2. H. Suura and E. H. Wicbmann, Phys. Rev. 105 (1957) 1930; A. Petermarm, Phys. Rev. 105 (1957) 1931. 3. K. Mitchell, Phil. M a g . 4 0 / I (1949)351. 4. R. Karplus and N. M. Kroll, Phys. Rev. 77 (1950) 536; C. M. Sommerfield, Phys. Rev. 107 (1957) 328; A . P e t e r m a r m , Helv. P h y s . A c t a 30 (1957) 407.
THE
SATURATION QUARK MODEL
1 A p r i l 1966
5. E . R . C o h e n and J . W . M . DuMond, Rev.Mod. Phys. 37 (1965) 537. 6. A. H. Roeenfeld, A. B a r b a r o - G a l t i e r i , W.H. B a r k a s , P. L. Bastien, J. Kirz and M. Roos, Rev. Mod. Phys. 37 (1965) 633. 7. See e . g . R . A . Shaffer, Phys. Rev. 135 (1964) B 187 and the l i t e r a t u r e quoted t h e r e . 8. L . D u r a n d , Phys. Rev. 128 (1962) 441.
PROBLEM IN THE OF ''ELEMENTARY''
NON-RELATIVISTIC PARTICLES
G. M O R P U R G O Istiluto di Fisca dell Universit~-Genova lstituto Nazionale di Fisica Nucleate - Sezione di Genova
Received 26 F e b r u a r y 1966
It is shown how a situation in which a t h r e e q u a r k s y s t e m is strongly bound to f o r m a proton, while a four or m o r e q u a r k s y s t e m is unbound or has a binding energy much s m a l l e r than expected at f i r s t sight may produce itself in the q u a r k model of " e l e m e n t a r y " p a r t i c l e s .
Among the questions which we have listed as needing clarification in the non-relativistic quark m o d e l of e l e m e n t a r y p a r t i c l e s [1], t h a t of " s a t u r a t i o n " i s a p r o m i n e n t o n e ; in v i e w of t h e m a n y s u c c e s f u l l t e s t s of t h e m o d e l [ 2 - 4 ] i t n o w appears appropriate to propose a possible qualitative solution to this question. This is the p u r p o s e of t h i s n o t e . In w h a t f o l l o w s we c o n c e n t r a t e o u r a t t e n t i o n on t h e b a r y o n s a l t h o u g h w h a t we a r e g o i n g to s a y m a y c l e a r l y e x t e n d t o t h e m e s o n s ; we m a y t h i n k of q u a r k s h a v i n g a m a s s M -~ 10 G e V * , b u t o u r r e m a r k s w i l l b e i n d e p e n d e n t on t h e p r e c i s e v a l u e of t h e m a s s of t h e q u a r k . W e n e e d only assume: 1) t h a t t h e m a s s M of t h e q u a r k i s m u c h l a r g e r t h e n t h e m a s s M B of t h e b a r y o n s : M >> M B 2) t h a t
I V ! ~ ~ / - ~ - 5'/p M>>
~t
w h e r e p - 1 i s t h e r a n g e of t h e i n t e r a c t i o n b e t w e e n two quarks. * Compare ref. 5, where a crude e s t i m a t e of 10 Ge¥ is given for the m a s s M of the 5 q u a r k b a s e d on the diff e r e n c e between the a v e r a g e m a s s of the odd p a r i t y and even p a r i t y m e s o n s . 684
If t h e f o r c e b e t w e e n q u a r k s i s d o m i n a n t l y t r a n s m i t t e d b y v e c t o r m e s o n s /~-1 ~ 5rnTr. In t h e f o l l o w i n g we m a y t h i n k , f o r c o n c r e t e n e s s , of t h e p o t e n t i a l b e t w e e n t w o q u a r k s a s a p o t e n t i a l w e l l w i t h a r a d i u s h a v i n g t h e a b o v e o r d e r of m a g n i t u d e . N o t e t h a t 1) a n d 2) a r e t h e b a s i c a s s u m p t i o n s of t h e n o n - r e l a t i v i s t i c q u a r k m o d e l . T h e q u e s t i o n of s a t u r a t i o n c a n in i t s e s s e n tials be formulated as follows: why systems with 4 or 5 quarks are not much more bound (or, i n o t h e r w o r d s , why t h e y a r e n o t m u c h l i g h t e r ) t h a n a s y s t e m of t h r e e q u a r k s , s a y , t o b e d e f i n i t e , a p r o t o n ? A n d a l s o : why a s y s t e m of 6 q u a r k s i s a d e u t e r o n a n d n o t a c o l l a p s e d a g g r e g a t e ** To clarify this question we begin by recalling that, assuming we deal only with two body f o r c e s , t h e a v e r a g e p o t e n t i a l e n e r g y of a p a i r of q u a r k s in t h e p r o t o n i s : (1)
In f a c t , in t h e n o n r e l a t i v i s t i c q u a r k m o d e l t h e k i n e t i c e n e r g y i s n e g l i g i b l e , w i t h r e s p e c t to ** Note that the " s a t u r a t i o n " question is masked, but n e v e r t h e l e s s exists also in all the m o r e a b s t r a c t approaches; it can be stated in this way: why a r e the r e p r e s e n t a t i o n s _8 for the m e s o n s and 8, 10 for the b a r y o n s the lowest ones and in fact the only ones so far o b s e r v e d