Large symmetry-breaking meson-baryon interactions

I

8.B

I

Nuclear Physics 60 (1964) 521--528; (~) North-Holland Publishing Co., Amsterdam

! Not to be reproduced by photoprint or microfilm without written permission from the publisher

LARGE SYMMETRY-BREAKING

MESON-BARYON

INTERACTIONS

ROBERT L. ANDERSON Research Laboratories, General Motors Corporation, Warren, Michigan and SURAJ N. GUPTA Department of Physics, Wayne State University, Detroit, Michigan ? Received 21 April 1964 Abstract: The general structure of the symmetry-breaking meson-baryon interactions within the framework of the three-dimensional unitary-unimodular group is investigated. It is suggested that the symmetry-breaking terms are not necessarily small compared with the symmetrypreserving terms, and that the mass formulae for mesons and baryons follow as a result of certain cancellations in the self-energies of these particles. An estimate based on the second-order self-energy calculations leads to an intrinsic mass of about 870 MeV for mesons as well as baryons, and accounts for all the observed mass splittings between the meson and baryon multiplets in terms of the symmetry-breaking interactions.

I. Introduction

The u n i t a r y u n i m o d u l a r g r o u p in three d i m e n s i o n s or SU3 has been applied to the t h e o r y o f mesons a n d b a r y o n s b y N e ' e m a n 1), G e l l - M a n n 2) a n d O k u b o 3). This s y m m e t r y t h e o r y implies t h a t all m e s o n s have the same intrinsic m a s s raM, a n d all b a r y o n s have the same intrinsic m a s s m B. I t is, therefore, expected t h a t the observed m a s s splittings between the v a r i o u s m e s o n s a n d the v a r i o u s b a r y o n s are d u e to some simple s y m m e t r y - b r e a k i n g i n t e r a c t i o n l e a d i n g to the m a s s f o r m u l a e ½(ran + m.-) = ¼(3m A + mz),

(1)

m~ = ¼(3m 2 + m2),

(2)

which are a p p r o x i m a t e l y satisfied b y the e x p e r i m e n t a l masses. Ok-ubo 3) has given a d e r i v a t i o n o f the a b o v e m a s s f o r m u l a e b y a s s u m i n g the s y m m e t r y - b r e a k i n g i n t e r a c t i o n terms as small c o m p a r e d with the s y m m e t r y , p r e serving terms. H o w e v e r , this a p p r o a c h leads to the difficulty o f a negative intrinsic m a s s for b a r y o n s at least in the s e c o n d - o r d e r a p p r o x i m a t i o n 4), a n d it is d o u b t f u l t h a t this difficulty will d i s a p p e a r on including the h i g h e r - o r d e r c o n t r i b u t i o n s . T h e a i m o f this p a p e r is to explore the general structure o f the s y m m e t r y - b r e a k i n g i n t e r a c t i o n t e r m s a n d to p r o v i d e a m o r e satisfactory d e r i v a t i o n o f the m a s s f o r m u l a e free f r o m the difficulty o f negative intrinsic masses. C o n t r a r y to the v i e w p o i n t o f t Work supported in part by the U.S. National Science Foundation 521

522

R.L.

ANDERSON AND S. N. GUPTA

the earlier authors, we do not assume that the symmetry-breaking interaction terms are small compared with the symmetry preserving interaction. On the other hand, we suggest that the mass formulae (I) and (2) follow as a result of certain cancellations in the self-energy contributions of the particles, and we shall demonstrate the feasibility of this approach by carrying out the necessary cancellations in the secondorder self-energies. We shall also show that it is reasonable to choose the same intrinsic mass for mesons as well as baryons, and then account for the mass splittings between all the meson and baryon multiplets as due to the self-energy effects of the symmetry-breaking interactions.

2. Interaction of Mesons and Baryons The symmetry-preserving Lagrangian density for a system of baryons and mesons can be expressed in terms o f ordered products as (taking c = h -- 1) L

(3)

-- L B +L M +Lint,

with j

(4)

2 j i ~ :, (oS; __IJ +,,,MZ/; !

(5)

\

L,, =

i

)

.

Ls = -- : w - i ~', ~ -

+ m s M i Nj

ox~

.,

\Oxa ~xx •

j

i

k.

•

.

j

i

k

Lint = -- ig 1 . M i 7 5 N k f j .-- ~ g 2 . M i T s f ~ N j . ,"

(6)

where the traceless tensors N~, M{ and f{ represent the field operators for the eight baryons and eight mesons as 3) Z'+ -- N~,

2:0 = x / ~ ( N I - N ~ ) ,

2:- = N~,

p -----N1a, ~,- = M ~ ,

~,+ = M ~ ,

n = N~,

= Na1,

K+ = f a,

K ° = f 2 a,

Z ° = Na2;

(7)

,~ = _ x / ~ M ] ,

~ o = M23,

x° = x / ' ~ ( f l - f ~ ) ,

~r- = f l ,

•-

~o = x / ~ ( M I _ M 2 ) ,

~ - = M 3, 7r+ = f ~ ,

A = -~/~N],

p = M~,

h = M];

(8)

tt = - , / l f s s, K + * = f a ~,

K°*=f].

(9)

I f we now assume that the symmetry-breaking interaction terms behave as the T33 component o f a tensor T~, we can write the general meson-baryon interaction as ,

Eint

=

.

j

--igt.MiysNkf

i

k.

j .-ig2:MiTsf:N~:-ig,(:MlTsN~af?:

_ , g• , , ( : M , j T s f ~lN ~ 3: + : M i a T s f 3 N ~ : ) _ i g 5

+ :M~7, N ~ f I : )

• . M sj ~5 N ff~ s i.. :M~3~5 f j , N 3j . - zg6

- i g T : M l r• 5 N ijf : s3". - i g s ( ..M i rs s f j Nis . +3.: M ] r s f / N } : ) ,

(10)

MESON-BARYON INTERACTIONS

523

where all the coupling constants are real to ensure that (10) is Hermitian. T h e above interaction can be written in a m o r e familiar f o r m as

+ igaz~(~175T'i+ ~ir5 A)ni + ignAK(Nrs KA + AK*y s N) + igNZK(NT5 "fiK2~t + £i K*zi ~'5N) + iganK(~'s K'A + AK'*7 s ~=) + igszz(27 s z, K'E, + ~, K ' * z, y s ~) + igNN.NT 5Nrl + igzz. ~7 s Zq + igaa, A?s An + igzz,~,Y5 ~7,t/] :,

(11)

where the n coupling constants are given by

g~N. = (½)~(g, + g & g-.--~. = -(½)~0~ + gs),

02)

g:~. = (½)'~(gl--g2),

ga~. = (i})*(g, + 0 2 - - 2 g s ) , the K coupling constants are given g n a t = -- ('~)½(2g I -- 92 + 2g3 -- g . + 2g6 + 2gs),

g~

= (½)~(g2 +g~),

03)

g.-al¢ = ({)~(g 1 - 202 + 03 - 2 g 4 - 2g 5 - 298),

g~

= _(½)~(g~ + g~),

the q coupling constants are given by ..qNNn = (~)~(g 1 -- 2g2 -- 494 + 96

-

-

2g7),

gzzn = - (~t)~(291 --if2 + 4ga --95 + 2 g 7 ) , g..Ian = --(~)~("* + r e +§g~ + } g ~ +~"~ + } g ~ + 2 g . + 4,.).

(14)

gzztl = ({O~(g, + g 2 - 2 g 7 ) , with N=

(:)

(7)

~= '

,

K=

K'= K o

'

\ K +* ] '

z l = (½)~(z + +,~-),

-r2 = i(½)~(z + - ~ , - ) ,

~,a = r, °,

7171 ~--- (½)*(7~+"[-~-),

~2 ~--~i ( ½ ) ' ~ ( ~ + - - ~ - ) ,

~3 = ~ 0

It is interesting to note that the coupling constants identically vanish, if gl

---~ g 2 =

--g3

---- - - g 4 .

---- - - g 5

=

--g6

(12)-(14), =

g7

=

(15)

and hence (10), gs,

(16)

which shows that (10) is invariant under the transformation g,, ~ g,~,

(17)

524

R. L. ANDERSON AND S. N. GUPTA

where gl = g x + L gl = g s - L

gl = g2+,t, g~ = g6-,~,

gl = g 3 - , t , g~ = g7+~,

g;, = g 4 - 2 , gl = gs+,t,

08)

and Z is an arbitrary constant. It also follows that the symmetry-preserving and symmetry-breaking terms cannot be separated from each other in a unique way.

3. Meson and Baryon Serf-Energies The meson and baryon self-energies can be obtained from the scattering operator in the interaction representation by using the contractions

N[,'~(x)Mk,'~(x ') = i(5~5[--½5{6~)SF,,t(x-- x'), f/'(x)ff'(X')

=

--

• k ~- '(5,5,

~ 5 , 6j , ) Ake ( x -

(19)

x ,) ,

(20)

where

Sr(x-x')

= lim 1 (dke,k(x_x.) i k y - m . .--, + o (2zO'*.J k 2+ m2 - i e '

(21)

Ar(x-x')

= lim 1-- (dke'k¢x-~') 1 ,-. + o (2n) 4,/ k 2 + m ~ - ie"

(22)

We find that the effect of the second-order self-energy of free baryons due to the interaction (10) is equivalent to adding to the Lagrangian density a term of the form

1_~ = --aIB:M[N}:--blB:M~N~:--clB:M~N~3:--dlB ".MaaNaa'.,

(23)

where a =

7

2

~gl--34-gl,y2

,,~ . ~ _ 7 ~ 2

~y2-l-2glga--~glg7--~glga +

2g2 g4 - - ~Jg2g7 -- Jg2 gs + g~ + g~ + ~g~ + ~g~,

b = - ~gl g3 - 2gz g4 + ~ g l --

g6 + 2gl gs - ~g2 ga + 4g2 g4

4g2 g6 + 2g2 g7 -- ~g~ -- ~g3 g,l. + ~ga g6 + ~g3 gs

+1o 2 2 "~-g4 - -

c

(24)

s

g4 g6 + ~-g4g7 - -

~7_2

2_

(25)

4

~g4gs ~ ~-,.q6---SY6 g7 -.I-3g6 gs,

4 i gs + 2gl g7 -- 2g2 g3 -- sZ-g2g4 = 4gt g3 - - ~gl g4 -- ~-g

+14 -~-g2 g5 + 2 g2 ga +1o -3-g32 -- ]g3 g4 -- 2g3 g5 + s3-gag7

(26)

- - ~ g s g s - - ~ ' g ~ + '~

,, + 7 2 ~ 4 ~g4g5 + ~g4gs 3g5 --~-g5 g7 +~-g5 gs,

2 2+s +40 +20 d=--993 ~gag4 W-Z~gags -~-gag6 -3-gags 2

2

40

16

20

1 2

S

(27)

-- ~g4 + ~-g4 g5 +-9-g4 g6 + ~-g4 gs + ~-g5 + vgs g6 16 +2gsg7+-~-gsgs

..~1

- ¢ g26 + 2 g 6 g v + ~ - g 6 g s + 4 g T g s +

lOg~,

while IB is a divergent constant, which on evaluation in an unambiguous manner by the method of auxiliary fields 5), is found to be positive.

MESON-BARYON INTERACTIONS

525

Similarly, the effect of the second-order self-energy of free mesons is equivalent to adding to the Lagrangian density a term of the form L'M = -- ½0dM'.f/f) : - - ½firM :f~ f 3 : _ ½tiM 'faaf3 :,

(28)

where a

7

2

ygi--~gtg2

--[.--[.-7_2

4

4

2

~g2+2gtg6--~glgs+2g2gs--~g2gs+gs+g6+-yg8,

2

4

2

fl = ~ - g I g 3 - - -s~ g t g , - - ~,*g l g s - - ~ - g l g 6 + 4 g t g s - - S g 2 g 3 + -2s 3 - g 2 g4 - - ~* g 2 g s 4 . 1, 2 8 4 --[.-8 4 --~g2g6+ g2gs+Tg3---~g3ga---~g3g5 -~g3g6+-~g3g8+~g~ + ~8g , g s - - ~ g , * g 6 4 2 s 2 8

+ ~g4 gs - ~'gs + ~-g5 gs -

4,6

2..[ 5 6 a

a

8a

a

8a

a

7 = -~-g3 9-Y3V4--9-V3YS--9-Y3b'6

~g6

--[.-32

8g4 g6 + ~-g4 g7 + 4g, gs + 8g 2 + 4g7 gs + 2g 2 ,

+

4-6

~'g6 gs, 2

(29)

(30)

S

~-g3g7+4g3gs+-v-g4--~g4g5 .4_ 1 _2.31_ {g5

~Y5

2 1 2 2 g6 -- 3"g5 g7 + vg6 -- 3"g6 g7

(3 I)

and the divergent constant IM, on evaluation by the method of auxiliary fields, is found to be negative. Thus the effect of the second-order self-energies is to replace the mass terms in the baryon and meson Lagrangian densities (4) and (5) as -mB:MIN~:--*

-- (roB +

alB):Msj N j, : - b i n :M3, N i3 :

- - ½ m 2 " . f i q j : .--', - - { ( m 2

--

c l n :M~ N~ : - d l B : M ~ N 3 : ,

+alM)'.fffj:--½fllM'.f~fi3:--½ylM'.f2f2:.

(32) (33)

We m a y expect that ultimately the divergent constants In and I u will acquire finite values, presumably due to some cut-off for large values of energy and momentum, and therefore we shall treat Ia and Ira as finite but unknown constants. However, it is reasonable to assume that the signs of In and Ira can be unambiguously determined by the present calculations, and hence, as already pointed out, we have

IB > 0,

Ira < 0.

(34)

4. Structure of Symmetry-Breaking Interaction Expressing the effective baryon mass term (32) as _ ( m , + a I B ) : M ! N ~ . : _ b I e : M , N3: - c I e .. M ia N ,3.. - d l B . .M 33 N 33.. =

-- m N

:NN: - m z :~E: - m z :~q E F - m a : f l A :,

(35)

we obtain for the experimental masses of baryons mz = m, + al B , m N = m a -st-(a + b ) I B , m e = m B + (a

+ C)IB,

m a = rn B + ( a + 2 b + { c + 2 d ) I n .

(36)

526

R . L . A N D E R S O N A N D S. N . G U P T A

Similarly, expressing the effective meson mass term (33) as -- ½(m 2 + m/M) ".fffj : -- ½film .f3+f+3 : _ ½7IM . f ~ f33 : 2. * . 2. . 2. . = --rnK.K K.-½m~.7~dr~.--½m~.qq.,

(37)

we find that the experimental masses of mesons are given by m n2 - ~

m 2 + ~XlM '

m~c = mR + (~ + ½fl)IM, 2

2

2

(38)

2

m~ = m M + (0~ + ~ f l + ~-7)IM •

The mass formulae (1) and (2) follow from (36) and (38), provided that d=0,

? =0.

(39)

We shall assume that the above conditions are exactly satisfied by the second-order self-energies, and that the small deviations from the mass formulae are to be attributed to the higher-order self-energy effects. Since the expressions (27)and (31) for d and y remain unchanged under an interchange of g3 and gs with g+ and g6, respectively, it is convenient to express them in terms of f = g3 + g4,

f ' = ga - g4,

g = g5 + g6,

g ' = g5 - g6,

(40)

and then the conditions (39) can be written as 28. * r , ~~, + 2 o ~ - - f g s + 1-~g 2 --~;g,2 -I- 2 ggT-1--3-ggs 16 + lOg2-1-4gTgs = O, -lf2 -- ~ f ' 2 . 1 _~-fg---~J

(41) 37

2

-~-f + f

t2

8

32

$

--~fg+-~fgT+4fgs+-f~g

2

p2

2

2

--~;g ---sggT+8gT+ 2g2 +4gTgs

= O.

(42)

Further ,it follows from the invariance of the interaction (10) under the transformation given by (17) and (18), that by taking 2 = ½(g3 + g , ) we may put without loss o f generality f=

0.

(43)

Thus, the conditions (41) and (42) reduce to

--•f

,2

,,, , , s 2 16 ---~f g +T~g --~sg,2 + 2 ggT+-3-ggs+lOg2+4gTgs = O, ff2 + -f~Y s - 2 - *1Y- ' 2 --sYY7 2 -.~ 8-2 yv+ 2g2 +4gTgs = 0.

(44) (45)

The reality of coupling constants severely limits the possible relationships between them such that the above conditions are satisfied. Indeed, assuming that the coupling constants appearing in the symmetry-breaking interaction either vanish or their ratios are given by small integers or ratios of small integers, we find that the only possible relationships satisfying (44) and (45) are f'

= g = O,

g7 =

- gs,

g' =

-I- 6 g 7 ,

(46)

MESON-BARYON INTERACTIONS

527

so that (43) and (46) give us g 3 ---- g 4 =

0,

g5 =

g7 = --gs,

--g6,

g5 = --3g7"

(47)

Hence, the symmetry-breaking interaction terms can be expressed in terms of a single coupling constant, which can be taken as g5.

5. Estimate of Coupling Constants We have confined ourselves to calculations of the second-order self-energies, but it is possible that the appropriate cancellations also occur at least approximately in the higher-order self-energies to satisfy the mass formulae. Another possibility is that for some reason, which is not yet fully understood, the second-order results represent the major contributions to the self-energies of particles, and therefore the higher-order self-energies are of secondary importance. These questions can be fully settled only after the problem of divergencies has been finally resolved. In order to obtain a rough estimate of the ratios of the coupling constants gx, g2 and gs, we shall compare our theoretical expressions for the baryon and meson masses, obtained by including the effect of the second-order self-energies, with the experimental values. We shall also make the reasonable assumption that the intrinsic mass of the meson cannot exceed the intrinsic mass of the baryon, so that mB ~ mM _>- 0,

(48)

which surprisingly puts a severe restriction on the ratios of the coupling constants. We now make use of the relations (47), which lead to two cases corresponding to the two values of g7 in terms of g5. I f g7 = +½gs, we obtain 7

2

4

b = - T1~ gl

.at. 2

2

2

+~gs,S 2

c = -~glgs+4g2gs 7

4

g2)+~-gs, g5 + 2 g2 g5 +3g~,

a = ~(gl--~gtg2

(.49)

4. . . . .

2x+58_2 14_ _ .1.22_ = ~(gl--'YYlY2"I-Y2) ~-ffY5 - - - - 9 - Y l Y5 9-Y2YS,

fl = --'~as(gl-1-i-g2 .1. g5), while, if g7 = - (~')gs, 7-

2

4-

a ---- W t g l - - v g t g 2

+

2~+

g2)

b = -4glgs+Zg2gs c =

-- 2 g t g5 7

2

4

4-

2

Tq-gs,

+~gS 2s,

+ ~-6-g2 g5 .1. 3 g 2 ,

o: = ~(gz--~-gtg2

"1" 2 x + 5 8

g2)

(50) 2

22

~-vgs---o-gzgs+~g2g5,

fl = ~ g a ( g l + g 2 - - g s ) . Using (49) or (50) together with (39), we get the theoretical expressions for the baryon and meson masses from (36) and (38), and equate them to the experimental values to obtain the ratios of the coupling constants and the values of the intrinsic masses.

528

R. L. ANDEIL$ON AND S. N. GUPTA

We find that when g7 = + ~'gs,

(51)

the condition (48) can be satisfied by positive as well as negative values of gl/g2, but for each sign of gz/g2 the magnitudes of gl and g2 come closest to each other when m M = ms. It is, therefore, reasonable to take m M ---- m a =

m 0,

(52)

and we then obtain either

gl/g2

=

gl/g5 = 1.77,

3.07,

mo = 868 MeV,

(53)

or

gl/g2 = - 2 . 4 5 ,

gl/gs = 0.59,

m o = 671 MeV.

(54)

Out of the above two sets of values, (53) seems preferable, because according to it gs < g t - I t is also reasonable to expect that in a more precise treatment, involving the effect of the higher-order self-energies, the ratios gt/g2 and gt/g5 will have simpler values. Therefore, presumably,

gl/g2 ~ 3,

gl/g5

~

2,

(55)

which give us 2 2 9NNJgNZX ~ 2.25.

(56)

On the other hand, when g7 = - ] # 5 , the condition (48) cannot be satisfied for any value of gx/g2, and therefore this case should be ignored. The present theory provides us with a reasonably consistent picture of the meson and baryon mass splittings, when the relationship between the coupling constants is given by (47), (51) and (55). It can also be extended to include the vector mesons and other resonances. In fact, most of the results of this paper remain unchanged if we replace the pseudoscalar mesons by vector mesons. However, according to our approach the mass formulae are exactly satisfied in the lowest order, while the departures from the mass formulae are due to the higher-order effects. It is conceivable that the higher-order effects are more important for vector mesons than pseudoscalar mesons, and this m a y explain why the mass formulae are found to be less accurate for the vector mesons. References 1) 2) 3) 4) 5)

Y. N e ' e m a n , Nuclear Physics 26 (1961) 222 M. Gvll-Mann, Phys. Rev. 125 (1962) 1067 S. Okubo, Prog. Theor. Phys. 27 (1962) 949 R. L. A n d e r s o n and S. N. Gupta, Nuclear Physics 60 (1964) 518 S. N. Gupta, Proc. Phys. Soc. A66 (1953) 129