IL NUOVO CI~ENTO VOL. XXVII , N. 6 16 5Iarzo 1963

C a l c u l a t i o n of the Y u k a w a Coupl ing C o n s t a n t .

R. DELBOURGO (*)

Department o] Physics, Imperial College - .Londo,t

(ricevuto il 17 Ottobre 1962)

S u m m a r y By expressing Feynman digrams as dispersion integrals with a finite cut-off, we calculate to lowest order the renormalization constants Z1, Z2, Z3, for a Yukawa coupling. By setting each Z equal to zero we obtain t~he approximate solutions g2/47:=0 (30) and tt/m :~ 0.2 with a cut-off of about 3 nucleon masses.

1 . - I n t r o d u c t i o n .

On the basis t h a t each par t ic le in n a t u r e is to be r ega rded as a b o u n d

s ta te of all o ther exis t ing part icles, SALAN (1) has p roposed t h a t t he th ree

cons tan t s 8m, 8/~ ~ and Z1 all vanish. (Here m is the fe rmion mass and # is

t he boson mass.) F o r ins tance if all the par t ic les in n a t u r e were e i ther fer-

mions or pseudosca la r bosons i n t e r ac t i ng via the Y u k a w a coupl ing gv~ysyJ~0 t h e n

(1) Z1 (g, m, re) ---- 0

(2) ~m (g, m, #) = o

(3) ~ ( g , m, ~) = 0

SALASI (1) has f u r t he r po in ted ou t t h a t in order to sa t isfy eqs. (2) and (3) i t

is sufficient to p u t Z2----0 and Z3 = 0.

(') The research reported in this document has been sponsored in part by Air Force Office of Sodntific Research, OAR, through the European Office, Aerospace Research, United States Air Force.

(1) A. SALAd: Nuovo Cimento, 25, 224 (1962).

1432 R. D : E L B O U R G O

I t is well known, of course, t h a t all these constants diverge, bu t if we car ry out a pe r tu rba t ion calculation with the inclusion of a finite cut-off A, we m a y

well be able to solve Z~(g, m, #, A ) = O, etc.

The question remains as to how one mus t introduce this cut-off in a more or less unique fashion. Thus, we cannot proceed in the s t andard m a n n e r (~) b y separa t ing out the divergent f rom the finite par t , because such methods

rely ent irely on the fact t ha t A is ve ry large, so t ha t var ious theorems on shift ing the origin of integration, can be established. Ins t ead we shall express the F e y n m a n diagrams through a dispersion representa t ion by the CUTKOSKY

prescr ipt ion (3), and in this way, the cut-off appears only in the final inte-

gra t ion over masses. I n fact the dispersion integrals will jus t be equivalent

to the LEHMANN spectral representa t ion (4) in lowest order. The advan tage

of this procedure is t h a t a calculation of even the four th order diagrams is not too difficult, and in par t icular the self energy diagrams which contain

over lapping divergences. H a v i n g calculated the Z to lowest order in pe r tu rba t ion theory, we find

t h a t (1 - -Za)~>2(1- -Z~) so t ha t i t is impossible for all three Z ' s to vanish simultaneously. However , we do solve the pairs of equations: (i) Z1 = Z2 = 0 and (ii) Z 3 = Z 2 = 0 , for g2/4~ and AU/m 2 in te rms of ~t2/m 2. B y compar ing magni tudes (see Section 5 for discussion of this point) we arr ive at. the es t imates

g2/4~ = 0 (30), #/m <0.2, Aim = 0 (3).

Final ly we take our fermion to be an isospinor, and our boson to be an

isovector ; in this case we have Z~ > 1 and it is only possible to solve (ii). Moreover the a rguments used in obtaining the previous solutions are no longer applicable, so one can no longer reach any conclusion. However , dis- regarding the difficulty due to the nonvanishing of Zr, ~nd choosing #/m ~0.1,

we still have g2/4~ ~ 25 and Aim .~ 3.

2. - B o s o n s e l f m a s s .

Figure l a gives the loop contr ibut ion as

ig 2 f l 4 4(p 2 - p ' k - - m 2)

(4) H ( k 2 ) - - (~)~)4J ~ P [ p 2 _ m 2 ] [ ( p _ 1,~)2- m '~] "

We relate this to the constants in question b y the expansion

(5) //(1, '2) = - - ~t~ 2 + (Z~ 1 - - 1)(k ~ - ~2) + / 6 ( k ~) (k ~ - - ~ )~ ,

(2) j . M. JAU~K and R. ROHRLICH: The Theory o] Photons and Electrons (Cam- bridge, 5~ass, 1955).

(a) R. E. CUTKOSKY: Journ. ]lath. Phys., 1, 429 (1960). (4) H. LEHMANN: NUOVO Cimento, 11, 342 (1954).

C A L C U L A T I O N O F T H E Y U K A W A C O U P L I N G C O N S T A N T 1433

SO

(6) s ~ = - H ( ~ )

and

~H(k ~ ) (7) Z ~ I = 1 + --~k ~ ~,=~,

or, w h a t we use ins tead ,

(79 Z~ = 1 - - / / ' ( ~ ) .

-

+q a) b) c)

Fig. 1. - a) Self mass of boson; b) self mass of fermion; c) vertex part.

The in t eg ra l (4) has a n o r m a l t h re sho ld s ingu la r i ty (5) a t k 2 - - 4 m s and

across t h e n o r m a l b r a n c h cut, the C u t k o s k y p re sc r ip t ion (3) gives

(8) _ l I m H ( k s ) = . ( )3 p . k S ( p s - ms)O(p)5(k 2 - 2 p . k ) O ( k - p ) d 4 p = 7g

g 2

= 8~r- ~ O(k s - 4m ~) {k ~ - 4m2k ~} .

W r i t i n g down the usua l d i spers ion re la t ion w i th ~ cut-off A -~,

A 2

(9) i i(k2) ~_ _1 f I m / / ( K '2) d K 2 K s -- k s -- is '

and us ing (6) and (7)

(10) ~/A2 - -

A ~

g2 f K ( K s _ 4mS)�89 8r~. o K s _/~2 dK2 '

4m 2

(11) Z a

A 2

g~ f K ( K 2 - - 4m2)�89 = 1 - s ~ ~ _ - ~s)~ dKs.

4m a

(5) See for instance, R. J. E])E~-: Lectures on the use o] Perturbation Mettwds i~t Dispersion Theory, technical report no. 211 (U~iversity of lVlaryland, 1961).

1 4 3 4 R, DELBOURGO

3. - Fermion self mass .

I n Fig. lb, t he loop c o n t r i b u t i o n is

(12) ig ~ f d 4 7 . k - - m ~ mA(pO Jr y'pB(p2) . Z (p) - (2~), k [k~-- m~][(k-- pl ~ - ~ ]

B u t , accord ing to the usual ident i f ica t ion ,

(13) (p) -~ ~m q- (Z~-~-- 1 ) ( 7 . p - -m) -k ~ (P)(~'P - -m) 2, ]

80

(34)

a n d

(15)

or eq t f iva len t ly

(15') Z2 = 1

(3m ---- ~ (p)1~.~=,,---- m[A(m2) + B(m2)]

z ; ~ 3 + p" ~ (p) m ~p~ ~,.~m

P~' ~ (P) ---- 1 - - 2m 2 [ A ' ( m 2) ~- B'(m2)] - - B(m2).

W e can the re fo re d e t e r m i n e ~m and Z2 f r o m the in tegra l r e p r e s e n t a t i o n s

of A a n d B :

A(p~ ) _ T r [ ~ ( p ) ] _ ig 2 f 1 d4 k 4m (2~)4 J (k s - - m s] [(k - - p)2_/~2] '

B(P 2) = T r [ 7 . p ~ ( p ) ] - ig ~ f p 'k d4 k 4p~ p~(.~)~j [ks_ m q [ ( k - p ) ' - ~ ] '

(16) l i m A ( p s ) = - gS f - 5(k 2 --m 2) O(k) (~(p2 --2p. k q-m 2-#2)0(p --k) d4k ---- (2n)~J

= - eS{(ps_ m S - ~2)s_ 4m~r (m + t~)~]/36~p ~ ,

(17) 1 -- I m B(p s) =

_ g2 f (2:~)ap2" k.pd(k 2 - ms)O(k)d(p 2 - 2p.k + m 2 - # 2 ) O ( p - k)d4k=

= g~(p~- + m ~ - ~ ) {(p~- m s _ ~ ) ~ _ 4m~}~O[p~- (m + ~)~]/32~p'.

C A L C U L A T I O N O F T H E Y U K A W A C O U P L I N G C O N S T A N T 1435

t t e n c e ,

Os)

,(19)

A 2

8m = m:~ f Im A(K2) + ImB(K: ) dK 2 - m ~ , _-

on§ ~

A ~

( r e+ i t ) ~

K e_ m ~ dK2 ,

A 2

z~ -- 1 1 i 2ms I m A ( K 2) + (K ~ + m 2) h n B ( K 2) d K 2 j ( K 2 - - m 2)

( m + # ) z

A~

g2 f ((K2 m2_ /t2)2_ 4m2 u2}�89 ] _ ~2(K2 _~ m z) = 1 - - 327~ ~ K4 1 ( K 2 _ m2 )

( r e + p ) z

d K 2 .

4 . - V e r t e x p a r t .

T h e m o s t s y m m e t r i c a l w a y of ob t a in ing a d ispers ion r e l a t ion for the v e r t e x f u n c t i o n is to i n v e s t i g a t e t he ana ly t i c b e h a v i o u r in the m o m e n t u m square of t he boson , w i t h b o t h nucleons p u t on the i r mass shells. A l t e r n a t i v e l y we could m a k e one b o s o n a n d one f e rmion real, b u t besides be ing u n s y m m e t r i c a l

we expec t the s a m e resu l t for Z~ a t la rge cut-offs, as in the s y m m e t r i c a l t r e a t -

m e n t . W e m u s t m e n t i o n t h a t our A m a y no t t u r n ou t large, so the resu l t for Z1 is c e r t a i n l y a m b i g u o u s . This a m b i g u i t y is t r ue in a n y t r e a t m e n t of t he v e r t e x pa r t .

The loop c o n t r i b u t i o n of Fig. l c is

ig 3 fd4k [V- (p ' - - k) - - m] V5 IV" (P - - k) - - m] gFs(q~) _-- _ (~7~)~ ( p ' - - k) ~ - - m 2] [k 2 - #'~] [ ( p - k )~ - -m2] '

S O

(20) ~t(p')Fs(q")u(p) (2~) 4 j [ ( p , k)2_ m2][k2 _#~][ (p_ k) 2 - m2 ] .

The s t a n d a r d def in i t ion is ( Z [ 1 - - 1 ) 7 5 = / " 5 (#2), or i n s t ead

(21) z l - - i - r~ (~2)y;1 .

T h e in tegra l (20) has a n o r m a l b r a n c h cut s t a r t i n g a t q 2 _ 4m ~, and the discon-

1436 R. DELBOURGO

t i n u i t y across i t is given by

1 (22) - I m (~Fsu) =

7~

_ g~,uY_su f d(k2 - 2p .k ) (~(k 2 -- (2~) ~ 3

g2~ysuO(q2-- 4m 2)

1.6~2 {q4__ 4m 2 q2}�89 q2 __ 4m2 _

2p" k)O(p'-- k)O(k-- p) -k.2. -

lo ( k 2

#2

A 2

g2 f K 2 _ 4m 2 _ #2 l o g [ ( K 2 4m 2 + #2)/#2] (23) .'. Z1---- 1 - 16--- ~ K ( K 2 _ # 2 ) { K 2 _ 4m2}�89 dK2"

4m 2

d4k _--

5. - Calculation of the costants.

W e summar ize the results conta ined in eqs. (10), (11), (18), (19) and (23)

by in t roduc ing the abbrev ia t ions ,

x = K~/m 2, y = #2/m2, Z = A~ /m ~ and a = g2/8~2

(10') 8#2 f <x=-- = ~ d x x y

4

(11') Za = 1 -

z

~ / ( x ~ - 4x)+ ( x - y))i dx ,

4

(18') 8m

( i + f F ) ~

1 + dx X 2 X �9

(19') Z2- - 1 - f{(x- (1+]I~-) 2

i- y)~ X~ ( x - 1

/ x - - 4-- y log [(x-- 4 § y)/y] dx (23') Z, = 1 - -~ .~ ( x - - y ) (x 2 - ix)�89 .......... "

4

As expected, the in tegrands in (10') and (18') are a lways posi t ive wi th in

the ranges of in tegra t ion and we cannot have 3m--3#2___ 0. However , it

C A L C U L A T I O N OF T H E Y U K A W A C O U P L I N G CONSTAI~T 1437

is entirely feasible for the Z to vanish. Examina t ion of the integrands in (11')

and (23') shows tha t

(2~) (1 - - z~) > 2 (1 - - z j ,

the equality sign holding in the limit y -+ 0. Equat ion (24) means tha t it is

impossible for Z~ and Za to vanish simultaneously, bu t we can still hope to

solve the pairs of equations (i) Z1----Z~ = 0 and (ii) Z3 = Z2 = 0, numerically,

for a and z terms of y. These solutions are plot ted in Fig. 2 and 3 for the range

0 . 0 0 1 < y 4 l . When y 4 0 . 0 1 , Za ~ - - 1 in (i) and Z1 ~ �89 in (ii).

~, 15 q - g~ ~ , '

2 0 0 4 T r ~ 10

I 0 0 ~ ,/ 5 / ,y : y

0 L s ~ - ':" 0 10 ~ 10 ~ 10 " 1

Y , L ) .

10 ~ 10 ~ 10 -~ 1

Fig. 2. ge/4= as a function of y-- Fig. 3. - z=A'~/m ~ as a function of = ff2/m~: (i) for Z 1 - Z . = 0; (ii) for y=ff2/m2: (i) for Z1-Z2--0; (ii) for

Z~-- Z 2 - 0. Z 3 - Z 2=0 .

The fact tha t it is impossible to obtain exact solutions to Z1 = Z~ ~-- Za = 0 is perhaps not surprising, in the sense tha t a fixed cut-off for all three con-

stunts is probably too stringent. The existence of a cut-off corresponds to

some nonlocality, and its value is always quoted to an order of magnitude.

This suggests tha t we only compare the solutions of (i) with those of (ii),

and in doing so, it is most impor tant tha t the g~ solutions (rather than z,

which as we have stated, is a loose quanti ty) be close to one another. As

another justification of this approach, we must add tha t the calculation of

the Z, being done to lowest order only, must be subject to higher order cor-

rections; therefore the solutions from the lowest order diagrams are only

expected to be correct up to an order of magnitude.

As an artificial criterion, which is open to criticism, we have required the

ratio of the g~ solutions to be within 1 and 2. Figure 2 then shows tha t

y ~<0.05, and in this range g2/47~= 0 (30) and z----0 (10), i.e. #/m ~<0.2 and

A / m = 0 (3). Surprisingly these values are very close to the meson-nucleon

physical constants, al though our model is unrealistic so far, as no account

has been taken of isospin.

1438 n. D~LBOURGO

I f we include the no rma l isospin proper t ies in to a new coupl ing gTVsT" ~bT~

the ne t effect is to leave ~ u n c h a n g e d in Z3 and to replace i t b y { a in Z2

and - - 1 ~ in Z~. I m m e d i a t e l y we fall in to difficulties because Z ~ > I . The new

solut ions to Z~ = Za = 0 are g r a phe d in Fig. 4 and 5 and are m u c h the same

g 2

~oo 2 ~

5 0

0 - - 10 -~

I

l O t z / ,

( i i ) !

' 'o-' " 1 0 -2 1 -

Fig. 4. - g2/4r~ as a function of y = =f f~ /m 2 for Z a : Z 2 = 0 , isospin beig

included.

Y

Fig. 5. - z = A ~ / m 2 as a function of y=f f 2 / m2 for Z a = Ze= 0, isospin being

included

as before. However , the fac t t h a t Z1 > 1 m i g h t lead us to conclude t h a t a t

least ano the r b o u n d - s t a t e par t ic le exists in na tu re , if we are to bel ieve t he

a p p r o x i m a t e va l id i ty of lowest order p e r t u r b a t i o n t h e o r y and the t r u t h of t he t h e o r y (i).

I a m indeb ted to Professor A. SALA~ for p ropos ing this p rob lem, a n d

n u m e r o u s suggest ions in connec t ion wi th i t ; I also wish to t h a n k Mr. J .

STRATHDEE for m a n y helpful comments .

R I A S S U N T 0 (*)

Esprimendo i diagrammi di Feynman come integrali di dispersione con un ~aglio finito, si calcolano, nell'ordine pifl basso, le costanti di rinormalizzazione Z 1, Z 2, Z 8, per un accoppiamento di Yukawa. Ponendo ciascuna Z uguale a zero, si ottengono le sohzioni approssimate g~/4n = 0 (30) ed f f / m ~ 0 . 2 con un ~aglio di circa 3 m a s s e

nucleoniche.

(*) Traduzione a cura della Redazione.