NN interaction from Lagrangian field theory

NN interaction from Lagrangian field theory

Volume 39B, n u m b e r 5 N-N PHYSICS INTERACTION FROM LETTERS LAGRANGIAN 29 May 1972 FIELD THEORY D. BESSIS, G. T U R C H E T T I * a n d W...

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Volume 39B, n u m b e r 5

N-N

PHYSICS

INTERACTION

FROM

LETTERS

LAGRANGIAN

29 May 1972

FIELD

THEORY

D. BESSIS, G. T U R C H E T T I * a n d W. R. W O R T M A N ** Service de Physique Th~orique, Centre D'Etudes Nucleaires de Saclay, B.P-2, 91, Gif-sur-Yvette, France Received 26 March 1972

Nucieon-nucteon phase shifts are calculated by using the r e s u l t s of the p e r t u r b a t i o n expansion of two Lagrangian field theories, the Yukawa model and the non linear sigma m o d e l F o r this purpose the [1, 1] Padg approximant to a s u b m a t r i x of the 16 × 16 spin m a t r i x is taken which e l i m i n a t e s the problem of anomolous thr,eshoid behaviour. A d r a m a t i c i m p r o v e m e n t relative to the s c a l a r method is found and this e l i m i n a t e s the 1D2 and 1G4 r e s o n a n c e s while providing reasonable low energy behaviour for the S waves and i n v e r s i o n of the 3P o phase shift.

In the f r a m e w o r k of L a g r a n g i a n f i e l d t h e o r y the p r o b l e m of NN i n t e r a c t i o n h a s b e e n a t t a c k e d in two d i f f e r e n t w a y s . In o n e c a s e , s t a r t i n g f r o m the B o r n t e r m s of a p h e n o m e n o l o g i c a l L a g r a n g i a n , in w h i c h the n u c l e o n i s c o u p l e d to a l l the n o n - s t r a n g e m e s o n s , e i t h e r a r e l a t i v i s t i c N / D i s s o l v e d [1] o r a p o t e n t i a l i s d e r i v e d a n d t h e T m a t r i x i s t h e s o l u t i o n of a B l a n k e n b e c l e r - S u g a r l i k e e q u a t i o n [2]. Due to t h e l a r g e d e g r e e of f r e e d o m c o n c e r n i n g t h e s c a l a r m e s o n s , coupling constants and form factors (or cutoffs), good f i t s to a l l t h e p h a s e s h i f t s c a n be o b t a i n e d . T h e s e c o n d a p p r o a c h , f i r s t a p p l i e d to ~ s c a t t e r i n g [3], h a s a m o r e f u n d a m e n t a l c h a r a c t e r b e c a u s e i t a i m s a t e x t r a c t i n g the a c t u a l a n s w e r c o n t a i n e d in a g i v e n r e n o r m a l i z a b l e l a g r a n g i a n . T h e f o r m of t h e L a g r a n g i a n i s a l m o s t f i x e d o n c e the i n v a r i a n c e g r o u p i s s p e c i f i e d : t h e SU(2) s y m m e t r y l e a d s to the Y u k a w a L a g r a n g i a n w h i l e t h e SU(2) × SU(2) c h i r a l s y m m e t r y l e a d s to the l i n e a r cr m o d e l [4]. F o l l o w i n g the r i g o u r o u s e x a m p l e s by p o t e n t i a l s c a t t e r i n g [5, 6] o n e a s s u m e s t h a t the T m a t r i x c a n b e o b t a i n e d by r e p l a c i n g t h e n o r m a l i z e d p e r t u r b a t i o n s e r i e s w i t h a s e q u e n c e of P a d d a p p r o x i m a n t s . T h e p r o p e r t i e s of the P a d b a p p r o x i m a n t s a r e q u o t e d in d e t a i l in v a r i o u s p a p e r s [7, 8]. We w i l l o n l y n o t e h e r e t h a t the [M~N] P a d d a p p r o x i m a n t s to p a r t i a l w a v e amplitudes satisfies the elastic unitarity condit i o n f o r N >/ M a n d p r o d u c e s s m a l l c r o s s i n g * I. N. F.N. Sezione di Bologna and Department of Mathematics. ** P r e s e n t a d d r e s s : Department of Physics, U n i v e r s i ty of W e s t e r n Ontario, London 72, Ontario, Canada.

v i o l a t i o n s [9]; m o r e o v e r t h e i r s t a b l e p o l e s very likely represent physical singularities. The [1,1] Padd approximant for the Yukawa L a g r a n g i a n , in w h i c h no f r e e p a r a m e t e r s a r e a v a i l a b l e , h a s g i v e n a s e t of p h a s e s h i f t s [10, 11] in q u a l i t a t i v e a n d o f t e n q u a n t i t a t i v e a g r e e m e n t w i t h the e x p e r i m e n t a l d a t a . C o m p a r e d to t h e unitarized Born approximation the largest i m p r o v e m e n t s a r e f o u n d in the L = J - 1 f a m i l y of the c o u p l e d t r i p l e t w h e r e t h e [ 1 , 1 ] Padd approximant restores the correct threshold behaviour. However serious difficulties still e x i s t f o r the e v e n s i n g l e t w a v e s (the 1S o h a s a D w a v e b e h a v i o u r w h e r e a s t h e 1D~ a n d 1G 4 a r e r e s o n a n t a t low e n e r g i e s ) . T h e S1 e x h i b i t s a bound state with large binding energy and its p a r t n e r s 3D1, ¢ 1 a s w e l l a s the 3 P o a r e f a r from being satisfactory. These low-waved e f i c i e n c i e s c a n be e x p l a i n e d e i t h e r by the p o o r n e s s of the f i r s t a p p r o x i m a t i o n o r the i n a d e q u a c y of t h e Y u k a w a L a g r a n g i a n to e x p l a i n the s h o r t r a n g e p a r t of the NN i n t e r a c t i o n . We w i l l p r o p o s e h e r e a new s c h e m e of a p p r o x i m a t i o n a n d a p p l y it to t h e Y u k a w a L a g r a n g i a n a n d the non l i n e a r s i g m a m o d e l . T h e b a s i c idea is that much more information is contained in t h e s p i n m a t r i x t h a n in i t s p h y s i c a l e l e m e n t and therefore an approximation such as the [ 1 , 1 ] P a d d a p p r o x i m a n t to the s p i n m a t r i x i t self, which avoids all kinematic difficulties conn e c t e d w i t h the B o r n t e r m , i s m o r e r e l i a b l e . F r o m the 16 × 16 m a t r i x in s p i n s p a c e we s e l e c t the f o l l o w i n g 2 x 2 m a t r i x ~ d e f i n e d by:

gJ~ I g 1 = 601

PHYSICS LETTERS

Volume 39B, number 5

g O l ' ) ~(P2')(Fe 1® Fe 1 ) T( r e 2® r e

2)U(Pl)U (P2) (1)

w h e r e P l , P 2 ( P l ' , P 2 ' ) a r e the m o m e n t a of the i n c o m i n g (out-going) nucleons with F 1 = 1 and F2 = 75" This may be r e a d

~h

=~

'ele 2

E

e I el

Tu

u

el el

w h e r e u 1 = u and u2 = v a r e the p o s i t i v e and negative energy spinors. With this choice the k i n e m a t i c s and u n i t a r i t y condition a r e s t i l l m a n a g e a b l e . @ele2 d e c o m p o s e s on a b a s i s given by the F e r m i i n v a r i a n t s plus a 6th i n v a r i a n t defined by: O 6 = U(Pl ') 75 u (Pl) u(P2 ') 75 ~ 1 u (P2) +

+ E(Pl') r 5 & U(pl) ~(P2') ~'5 u(P2) •

(2)

Note that 0 6 is not t i m e r e v e r s a l i n v a r i a n t so it cannot a p p e a r in the p h y s i c a l e l e m e n t of the m a t r i x ~ . The u n i t a r i t y condition r e a d s :

-¢,+

tP~i

- f~P~+

d~22

(3)

w h e r e the i n t e g r a t i o n runs o v e r the 2 body p h a s e spa c e and P = (~ 0) is a p r o j e c t o r s e l e c t i n g the p h y s i c a l i n t e r m e d i a t e s t a t e s . F o r the p a r t i a l waves FE=

/11/12 j j

,

(4)

f21-f22 (in the case of coupled triplet thef]Ele2 are themselves 2 × 2 matrices) the unitarity condition becomes:

FJ - FJ+ =iFJPFJ+

29May 1972

It can be v e r i f i e d that 2~[],]] has the following properties: A) It s a t i s f i e s the u n i t ar i t y condition (5). B) The a n a l y t i c i t y p r o p e r t i e s a r e c o r r e c t (the d i f f i c u l t i e s involved by the p r e s e n c e of the lefthand cut in the s e a r c h of bound s t a t e s can be r e m o v e d by takin~ the,2nucleons ,2 p a r2t i a l l y off s h e l l , n a m e l y p 2 =p22 =p] =P2 =P C m 2, as s u g g e s t e d by p o t e n t i a l s c a t t e r i n g [12]. C) In the z e r o e n e r g y l i m i t the S wave a m p l i tudes c o i n c i d e with the r e s u l t obtained when taking the [1, 1] Pad~ a p p r o x i m a n t to the c o m p l et e 16 x 16 spin m a t r i x [13]. D) The t h r e s h o l d b e h a v i o u r is always c o r r e c t and f o r the J = 1 coupled t r i p l e t the z e r o e n e r g y l i m i t is s m o o t h (namely it is the s a m e as the [1, 1] Padd a p p r o x i m a n t to the z e r o e n e r g y amplitude). B e f o r e d i s c u s s i n g the r e s u l t s obtained f o r the Yukawa L a g r a n g i a n and non l i n e a r s i g m a m o d el we will account f o r the r e n o r m a l i z a t i o n s c h e m e s we have used. F o r the Yukawa L a g r a n g i a n the p r o c e d u r e is s t a n d a r d b e c a u s e we have an e x a c t s y m m e t r y so that no i n t e r m e d i a t e n o r m a l i z a t i o n is n e c e s s a r y . When looking f o r an i n v a r i a n c e group l a r g e r than SU(2), the c h i r a l SU(2) × SU(2) group, broken in o r d e r to s a t i s f y PCA C, is the n a t u r a l p a r t n e r ; a non l i n e a r r e a l i z a t i o n like the non l i n e a r s i g m a m o d el i n s u r e s at the lowest o r d e r the c o r r e c t nn and nN z e r o e n e r g y a m p l i tudes. The non l i n e a r s i g m a model f o r m a l l y d e r i v e d f r o m a Lagrangian which is not s t r i c t l y r e n o r m a l i z a b l e , and can be r e g u l a r i z e d by the r e n o r m a l i z e d G r een functions of the l i n e a r a m o d el and then r e n o r m a l i z e d by sending the a m a s s to infinity. The p r o c e d u r e , s y s t e m a t i c a l l y w o r k e d out at the one loop a p p r o x i m a t i o n fixed the subt r a c t i o n constants of the n o n - l i n e a r s i g m a model. The p a r a m e t e r s of the non l i n e a r s i g m a m o d el coincide t h e r e f o r e with the ones r e q u i r e d by the l i n e a r a m o d e l ; beyond the p h y s i c a l ~ and nucleon m a s s e s we have t h r e e p a r a m e t e r s :

(5)

c = (m//~)(1 + ~ and for the p h y s i c a l e l e m e n t FJit r e d u c e s to the usual condition I m ( F J ) -1 Given the p e r t u r b a t i o n expansion of FJ up to 4th o r d e r in the coupling constant G and u s i n g a=G2/4n, FJ=aFJ+a 2 FJ+ . . . the [1,1] Padd a p p r o x i m a n t is computed:

J - aFJ]-IFJ1 F[IJ,1]= a FIJ [F1 602

(6)

(0)/m 2 ) ~ m//~

,

(7)

the (~ m a s s M, connected to the r e n o r m a l i z e d NN v e r t e x by: GnN N = G(1

+go(M)G2+...)

,

and a 2 defined as the s u b s t r a c t i o n point f o r the nn amplitude. The pion l i f e t i m e f n ~ 94 MeV f i x es G = 10.

(8)

V o l u m e 39B, n u m b e r 5

PHYSICS

LETTERS

29 Ma y 1972

Table 1 NN phase shifts at several energies for the Yukawa lagrangian (Y), Y only irreducible graphs (Y'), the non linear sigma model (N), the s c a l a r Yukawa (YS), and the phase shift analysis. The values for the pion and nucleon m a s s e s we used both in the Yukawa model and the non linear sigma model are my = 138 MeV, m N = 938 MeV. E l ab

YS

Y

Y'

N

25.1 53.7 142. 320.

179. 178. 175. 169.

67.7 84.3 111. 127.

1So 123. 122. 125. 127.

85.5 95.3 108. 114.

48.2 54.7 20.4 -10.5

• ± ± ±

1.8 9.0 4.1 1.3

25.1 53.7 142. 320.

2.87 5.84 10.2 12.0

1.25 3.24 8.65 15.6

3P 2 4.42 11.7 31.4 50.5

0.68 1.83 5.25 10.8

2.5 • 0.1 5.8 + 0 . i 13.7 ± 0.1 16.3 ± 0.6

25.1 53.7 142. 320.

-4.86 -8.30 -15.4 -29.2

-4.27 -7.40 -13.1 -19.7

3P 1 -4.44 -7.45 -14.4 -24.3

-5.18 -9.28 -17.6 -28.3

-5.0 -8.4 -17.0 -28.8

± 0.2 ± 0.3 ± 0.2 ± 1.2

25.1 53.7 142. 320.

-0.25 -0.79 -2.09 -3.41

-0.19 -0.61 -1.75 -3.07

3F 3 -0.24 -0.79 -2.00 -2.93

-0.20 -0.67 -2.01 -3.79

---0.4 ± 0.4 -2.1 + 0.2 -3.6 + 0.6

25.1 53.7 142. 320.

-7.26 -12.2 -16.6 -16.1

1.85 0.81 -2.21 -5.64

-0.3 ± 0.7 3.5 ± 3.2 4.3 ± 1.0 2 0 . 7 ±5.5

25.1 53.7 142. 320.

0.61 1..92 4.60 3.05

0.45 1.44 3.94 6.77

£3 0.61 1.93 5.03 7.26

0.48 1.53 4.19 7.27

----4.4 ±0.4 3.9 + 1.3

25.1 53.7 142. 320.

0.12 0.44 1.49 2.77

0.09 0.32 1.08 2.11

3F 2 0.12 0.43 1.49 3.56

0.09 0.32 0.98 1.64

--0.0 ± 0,3 0.7 ±.0.2 0.5 ± 0.6

25.1 53.7 142. 320.

-5.69 -7.79 -10.2 -42.2

-4.80 -6.81 -8.75 -12.0

1P 1 -5.23 -6.39 -6.29 -12.2

-5.76 -8.93 -14.1 -22.9

-4.0 -1.3 -18.2 -26.1

± 0.7 ± 1.7 ±1.3 ± 8.8

25.1 53.7 142. 320.

-0.45 -1.31 -2.74 -3.20

-0.34 -0.99 -2.24 -2.92

1F 3 -0.46 -1.30 -2.66 -2.55

-0.37 -1.08 -2.54 -3.70

-----2.0 ± 0.8 -6.4 :e 4.0

25.1 53.7 142. 320.

110. 90.1 62.9 42.6

84.5 57.1 29.3 -11.2

± 2.7 ± 2.7 ~- 0.9 ± 4.4

25.1 53.7 142. 320.

0.09 0.67 5.55 20.3

-0.09 -0.28 -0.54 -0.33

3D3 0.08 0.52 3.45 9.81

-0.13 -0.44 -1.23 -2.03

--1.5 ± 0.9 2.5 ± 0~6 3.0 ± 1.2

25.1 53.7 142. 320.

25.1 53.7 142, 320.

17.3 29.3 36.7 34.0

11.8 22.8 36.3 12.0

3P o 16.6 24.4 -16.8 -58.8

11.3 20.3 28.8 20.9

8.5 10.6 6.3 -12.5

± 0.3 ± 0.7 ± 0.5 :e 1.6

25.1 53.7 142. 320.

-0.61 -0.47 2.00 4.57

-2.28 -5.54 -13.3 -24.1

3D 1 -2.15 -5.41 -13.~ -25.5

-2.33 -5.69 -13.9 -25.5

-3.2 -5.3 -14.4 -21.3

25.1 53.7 142. 320.

0.87 3.48 177. 179.

0.47 1.14 2.63 4.97

1D 2 0.80 2.14 6.93 19.4

0.49 1.05 1.99 2.93

0.7 1.7 5.1 9.3

E1 2.93 -2.95 2.39 -6.43 0.24 -13.3 -2.17 -20.2

EXP

E[ab

YS

Y

Y'

N

EXP

78.1 72.0 63.3 54.1

3S 1 104. 92.7 79.2 6B.7

85.4 77.7 68.0 58.8

4.22 11.4 29.5 48.2

2.87 7.30 17.3 27.5

3D2 4.20 11.2 27.6 42.2

3.04 7.66 17.7 27.5

--9.4 ~- 2.2 22.5 * 0.8 23.6 ± 3.0

25.1 53.7 142. 320.

-0.91 -2.11 -4.43 -6.18

-0.66 -1.54 -3.25 -4.65

E2 -0.91 -2.04 -2.98 -0.14

-0.70 -1.62 -3.39 -4.70

---1.6 ± 0.2 -2.9 + 0.1 -2.6 ± 0.4

± 0.2 ± 1.8 ±0.8 ±1.7

25.1 53.7 142. 320.

-0.06 -0.33 -1.79 -3.89

-0.04 -0.24 -1.23 -3.47

3G -~.06 -0.33 -1.84 -5.34

-0.05 -0.25 -1.36 -3.94

------- 6 . 1 ± 1.9

± ± ± ±

25.1 53.7 142. 320.

0.04 0.19 0.80 11.9

0.03 0.13 0.46 0.95

1G4 0.04 0.19 0.74 1.95

0.03 0.14 0.46 0.82

----0.6 + 0.1 1.2 :e 0 . 2

0.1 0.1 0.2 0.5

603

Volume 39B, n u m b e r 5

PHYSICS

T h e r e s i d u e a t t h e p i o n p o l e in the [1, 1] P a d e a p p r o x i m a n t i s to b e e q u a l to the p h y s i c a l c o u p l i n g a n d f i x e s g o (M) by

G2/{47r(1-2goG2)} = 15 .

(9)

O n l y a 2 i s a r b i t r a r y ; it c o n t r i b u t e s o n l y to J = 0, 1 a n d t h e b e s t f i t t i n g v a l u e i s a 2 = - 1 0 0 0 p 2. T h e e x p l i c i t f o r m of the non l i n e a r s i g m a m o d e l L a g r a n g i a n a n d the g r a p h s c o n t r i b u t i n g the o n e loop NN a m p l i t u d e c a n b e found in r e f . [15]. T h e r e s u l t s a r e d i s p l a y e d in t a b l e 1. We q u o t e the p h a s e s h i f t s f o r the Y u k a w a L a g r a n g i a n along with that for the Yukawa Lagrangian with o n l y i r r e d u c i b l e g r a p h s a n d t h e non l i n e a r cr m o d e l a s f o u n d u s i n g the [ 1 , 1 ] P a d d a p p r o x i m a n t on t h e 2 × 2 m a t r i x . T h e r e s u l t s f o r the s c a l a r P a d e a p p r o x i m a n t on t h e Y u k a w a L a g r a n g i a n a r e a l s o p r e s e n t e d a l o n g w i t h the r e s u l t s of the m o s t r e c e n t p h a s e s h i f t a n a l y s i s f r o m NN s c a t t e r i n g d a t a [17]. F o r the Y u k a w a L a g r a n g i a n the 1S o e x h i b i t s a r e s o n a n t b e h a v i o u r f o r a l a r g e r a n g e in t h e c o u p l i n g c o n s t a n t (10 < a < 20). T h e 3 P o i s r e markably improved since after a rapid rise it starts decreasing and becomes negative for T ~ 340 M e V ( w h e n ce = 15~; t h e P w a v e s f o r J = 1, on t h e c o n t r a r y , a r e q u i t e c l o s e to t h e r e s u l t s of the s c a l a r [ 1 , 1 ] P a d d a p p r o x i m a n t . T h e 3S1 s t i l l h a s a b o u n d s t a t e b e h a v i o u r v e r y s i m i l a r to the s c a l a r c a s e w h e r e a s i t s p a r t n e r 3D1 u n d e r g o e s a d r a s t i c c h a n g e a n d now f i t s t h e d a t a e x t r e m e l y w e l l ; a l s o the ~1, now v e r y small, is greatly improved. The m o s t r e l e v a n t p h e n o m e n a f o r the h i g h e r w a v e s a r e t h e d i s a p p e a r a n c e s of t h e r e s o n a n c e s f o r 1D2 a n d 1G4 w h i c h now b e h a v e q u i t e n i c e l y . T h e 3D 3 i s now e x t r e m e l y s m a l l a n d n e g a t i v e a n d a s w i t h t h e 3 P 2 a n d 3D2, f o r w h i c h m i n o r c h a n g e s a r e o b s e r v e d , m u c h c l o s e r to t h e experimental results. W h e n we d r o p the c o n t r i b u t i o n s of r e d u c i b l e g r a p h s t h e p i c t u r e i s e s s e n t i a l l y the s a m e ; t h e g e n e r a l e f f e c t i s to l o w e r the c o u p l i n g c o n s t a n t (in the s e n s e t h a t a v e r y s t r i c t c o m p a r i s o n c a n b e m a d e w h e n t a k i n g in the s e c o n d c a s e a c o u p l i n g 20 ~ 30~o l o w e r t h a n in the p r e v i o u s o n e ) . M o r e o v e r t h e 1So i s s t r o n g l y b o u n d a n d the i n v e r s i o n of s i g n in 3Po i s c l o s e r to t h r e s hold ( T - 100 M e V f o r a = 15). T h e m o s t r e l e v a n t e f f e c t s i n t r o d u c e d by the n o n l i n e a r m o d e l a r e t h a t the 1S o c a n b e o b t a i n e d w i t h a b o u n d s t a t e v e r y c l o s e to t h r e s h o l d a l t h o u g h it i s r e s o n a n t f o r s m a l l e r c o u p l i n g c o n s t a n t s w h i l e a t the s a m e t i m e the 3S 1 i s b o u n d w i t h b i n d i n g e n e r g y B = 0.5 M e V f o r 604

LETTERS

2 9 M a y 1972

G = 10. A t t h e s a m e t i m e the 1P 1 a n d 3 P 1 a r e e x t r e m e l y good a n d t h e 3 P o k e e p s a b e h a v i o u r q u i t e s i m i l a r to the o n e d e s c r i b e d f o r t h e Y u k a w a L a g r a n g i a n ; the 3S 1 p a r t n e r s a r e a l s o v e r y s a t i s f a c t o r y w h i l e f o r the h i g h e r w a v e s m i n o r d i f f e r e n c e s w i t h r e s p e c t to Y u k a w a e x i s t . ( n o t e t h a t f o r J > 1 the n o n l i n e a r s i g m a m o d e l h a s no a d j u s t a b l e p a r a m e t e r s j u s t a s the Y u k a w a Lagrangian. We c a n c o n c l u d e t h a t t h e p i c t u r e o b t a i n e d i s v e r y s a t i s f a c t o r y in b o t h m o d e l s . T h e P w a v e s a r e n i c e in the Y u k a w a L a g r a n g i a n a n d e x c e l l e n t in the non l i n e a r s i g m a m o d e l . F o r the h i g h e r w a v e s no m o r e s e r i o u s p r o b l e m s e x i s t . The S waves probably require a further approxim a t i o n ~or a m o r e c o m p l e x L a g r a n g i a n ) t h o u g h f o r t h e oS1, a t low e n e r g y , we a r e c l o s e to the experimental results both with the Yukawa L a g r a n g i a n a n d the n o n l i n e a r s i g m a m o d e l . F o r t h e l a t t e r we m a y s a y t h a t a t v e r y low e n e r g i e s the 1S o b e h a v i o u r i s r e a s o n a b l e , a n d t h e f a c t t h a t correct binding energy for the deuteron can be f o u n d a t the s a m e t i m e ( f o r the r i g h t v a l u e s of fT; a n d Gph) i s c e r t a i n l y n e i t h e r t r i v i a l n o r fortuitous.

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