Radiative corrections to pion-nucleon coupling constants

B Wb)

The single-particle term is

G ,gn, 1 d3n

(n I Jr-(O) vlUp = GGW9 gDT-(&h

(55)

where G(0) is defined by
I I+(O) I n> = a36 - 7;) &ps$W).

(56)

This is precisely the matrix element which enters in the weak, neutron decay. Hence G(0) is the electromagnetic renormalization of the weak, vector, ,&decay coupling constant.

PION-NUCLEON

COUPLING

2.5

CONSTANTS

It is given by G(0) = Gw(0)/G;(“), where Gl”’

is the unrenormalized,

vector coupling constant for p decay (44)

Gv(~) = GE&$ - (em corrections) 0 According

(57)

= (1.431 & .002)

x

10-4gerg cm3.

(58)

to Cabibbo (45), Gvca’ = G”e;;&os

d,

(59)

where G~$,)~is the experimental, vector coupling constant for neutron decay, and 0 N .23 is the “Cabibbo angle.” Using (46) Gl”dt = (1.420 f .003) x IO-“. erg. cm3, we find G(0) =

(I + 0.0129)

with

0 = 0.23,

(60)

I (1 - 0.0077)

with

8 = 0.

(61)

The value of G(0) - 1 is fairly sensitive to 19. The form factors for p + w t+ n in Eq. (55) is evaluated at q2 = nz$,. It is related to the renormalized nN coupling constant by

+ (m$- mf->&&$-> + -a-,

(62)

g’(x) = dgldx, g,,-Cm”> = g,,- . In terms of G(O), the single-particle term, Eq. (55), becomes

(64) In order to determine /3, we must estimate g’. There are several ways of doing this. For example we could write a dispersion relation for process(seeFig. 7) n-(q2# mT2)---f N + N

(65)

26

MORRISON

or we might consider an approximate solution to the Omnes equation (47). In any event we might expect the low q2 part to be dominated by the lowest intermediate state of Eq. (65) that has the same quantum numbers as the pion. The most likely candidates might be the NN state of the Al meson (48) (since Al is an axial vector, derivative coupling to the pion would be required). We will now consider several models which predict the functional form of g(q2). We will see that the ratio, g’/g, cannot be predicted with certainty. (A) Let us first consider the case that the dominant intermediate state in Fig. 7 is given by the NN [see Fig. 8(a)]. If this is the case, we would expect the following behavior: g(q2) x l/(q2 - 4m2). (66)

FIG.

7.

nNN

vertex.

Thus the ratio, g’/g, is given by

g’h”) _ dmm2)

1 4m2-mT2

1110.25 BeV-2.

(67)

(B) For the case of the Al intermediate state [Fig. S(b)], we use the axial vector coupling determined from the weak interaction (49) with the result (8"'

cq2

dq2)

Oc [2mFA(q2)

+

q2~~(q2)1f,Al(~2)~

_cr"q"/mfd m2 ) Al

qvf~A1(q2),

(68)

(69)

where FA , Fp , and frrAl are the axial vectors, induced pseudoscalar, and Al meson decay form factor, respectively.

PION-NUCLEON

COUPLING

CONSTANTS

27

( b)

FIG.

8.

Models for g(q2).

In this model the derivative of g(q2) is related to the derivatives of other form factors, Fi , Fi , and unfortunately, nothing is known about these derivatives, so no estimate will be given here. (C) The form of g(q2) given by Eq. (69) is approximately the same that is predicted by the “partially conserved axial current” hypothesis (PCAC) which states that the divergence of the axial vector current can be used as an interpolating field for the pion. Using PCAC, Gasiorowicz finds (49)

dq’) CC(9’ - n~w2PmFAq2)+ q2Fdq2)1.

(70)

If Eqs. (69) and (70) are both valid, then we must have,fmA1(qz)cc (q” - WZ,,~). From Eq. (70) it follows that either F,(q2) or FD(q2) has a pole at q2 = rn,2.

28

MORRISON

According to Gasiorowicz (49) the pole is in F,(q2). In any case, however, it would be difficult to obtain the derivatives of g(q2) from Eq. (70) at the point, q2 = m,3, unless one could say more about the exact nature of the singularity in F, . One can make further simplifing assumptions that (a) g’(m,“) 5 g’(O), and (b) ~z,~FL(O)< FA(0) ‘v 1.2. We expect these assumptions to be approximately true since the massof the pion is “small”. [we mean F~z,~< I+, rnn2< m”A1, etc. In the limit m, + 0, assumptions (a) and (b) become exact]. Thus, taking the derivative of Eq. (70) at q2 = 0, we get

g’ -N-mi2+ &$jj * gA

(71)

Using [see (SO)] F,(O)/FA(O) = 8/m,, where m, ‘v 106MeV is the muon mass,we get g’/g N - 13 BeV2. (72) From Eqs. (67) and (72) we get for NN intermediate state, from PCAC [Eq. (72)].

(m$ -

VW (73b)

It should be emphasized that entirely different estimatesof g’/g may be obtained by choosing different models for g(q2). For this reason a reliable estimate for g’/g, and hence for j3 [Eq. (64)], cannot be given. Combining Eqs. (60), (64), (73), and (74) we get for 0 ‘v 0.23,

P-j

0.0188 0.0352

for NN intermediate state, from PCAC.

(74) (75)

Returning now to Eq. (54b) we denote the continuum terms on the left side of this equation by CTP”: Ny contribution = CT$ , N*y contribution = CT::,, , etc. As previously mentioned, we are neglecting the nonelectromagnetic contributions such as NT which give a feedback term relating Sg to Snr. We will neglect the resonant NT (ie, the N*) intermediate state as well. This contribution would require knowledge of the matrix elements (N 1I* j N*) which could in principle, be estimated by means of the divergence condition (51) in the following manner (N I I$,, j N*) oc e(N I J,*(O,G) A@(O,G)/ N*).

(76)

To estimate this, another sum over intermediate states would be introduced. The sum would be truncated after a certain number of low-lying states.

PION-NUCLEON

29

COUPLING CONSTANTS

However, the contribution from the NY intermediate state can be found in the following way: CT;;

=

c C, j. d3nd3k(p I I+(O) j w, k)(n, k / Jm-(0) vlU, . SPiTIS

(77)

The matrix element of I+ between a nucleon state and a nucleon-photon state may be reduced by meansof the divergence condition (51) to give

(P I Z+(O)I n, k> =

2/z GkGGv’(k) pO- .O - k0

-

u,r,vu,(277)3

P(E; - T;- i;),

(78)

where C,,, = [(27r)32k0]-l/“, EAu(k)is the photon polarization, and rNv is the isovector, elect_romagnetic vertex function whose argument is (p - u)~ = (p” - no)z - k2. We note that energy is not conserved at this vertex and that the matrix element is of order e. The matrix element, 8, k j J,-vlUDC, is related to the (off-sehll) invariant, photopion production amplitude. From the reduction formula (52) (n, k / pr)

= iC?i(27r)4 lj4(p + q - n - k)(n, k 1Jn-(0) 1p)

(79)

= iC&‘,,k.CDCn(2r)4a4(p + q - n - k) e:*(k) unT,,*UD , where oPTuUn is the photoproduction amplitude, and C, = [(2~-)~2q”]-1/2. Generalizing to the off-shell case, we have (n, k I J,-(O) vlU,C,

= C,,,CnCD~z*(k)DnTu*vlUP.

(80)

Using Eqs. (77), (78), and (80), we find a simple expression for the continuum

x (~Per,vUn)(~,Tu*v,UD).

031)

Or, from Eqs. (54b), (63), and (81), the / nr) contribution to A- is

If we compare this with Eq. (43), found by the method of dispersion relations, we can immediately see the similarities. We should not be surprised however, since we determined the absorptive part of the amplitude in dispersion relations by a sum over intermediate states. We note in particular the following points.

30

MORRISON

(A) Equation (82) depends on ); while Eq. (43) does not. This difference is easily alleviated by choosing the i; = 0 frame. In this frame,

P(G +fc- ;;) P(i+Z) no + kO - po +

no $ k” - m ’

which makes the integrals in Eqs. (43) and (82) much more similar in form. Also we have C, = C, = 1 and a: = /3 in this frame. (B)

Equation (43) contains an extra factor,f,

f=[ (C)

no+ k0 no + k” + m

1

, l

where

and

&f


L o* + k+x

(83)

A more subtle difference is the fact that the matrix element (P I I+ I n, k) 0~ ~J’uv~,+@)

contained in Eq. (75) does not conserve energy whereas the corresponding one in dispersion relations does, although the final nucleon is not on its mass shell. Writing the general vNN form factors in the form, F(p2, p”, k2), then for+Eq. (75), we have F(m2, m2, (p - n)“) where (p - n)” = 2m2 - 2m(m2 + k2)l12 < 0 for s = 0; while for Eq. (43) we have F(@ + q)2, m2, 0). The off-shell form factors of Section II thus provide a natural cutoff for the sum rule but not for dispersions relations. This is because of our ignorance of the form factors when any nucleon is off-shell. (D) Only the isovector part of the vNN vertex, ruv, enters in Eq. (82) while both the isoscalar, rMs, and isovector parts contribute to Eq. (43). If we explicitely decompose the pion photoproduction amplitude for different isospin state, we get

@W T;y+9no = 0f l/2 S” + ; (21/,“,, -

v;,~>,

@4b)

where P and Vu are the photoproduction amplitudes for isoscalar and isovector photons, respectively. The subscript on V refers to the isospin of the NOTfinal state. Using Eqs. (84a) and (84b) we find for dispersion relations [Eq. (43)]

PION-NUCLEON

31

COUPLING CONSTANTS

while from sum rules [Eq. (82)], we get

WI We see that quite different combinations and vertices enter in the two cases.

of isoscalar and isovector amplitudes

(E) Finally we note the obvious difference of the term -/3 (for j = 0) depending on the electromagnetic renormalization of the p-decay constant. This term probably contains information about rUs (which was absent in Eq. (82)). It may also explain the discrepancies described in (D) above. We can derive an expression for ATy = (g,,+/z/Z + g,,,o)/g by considering Eq. (47b) taken between neutron states. Proceeding as above, we find

Any +=

(86)

In order to evaluate Eqs. (82) and (86), we will need the pion photoproduction amplitude, Tu, when the incident nucleon is off its mass shell! Since we assume the integral is dominated by the low-energy region, we may again appeal to the TABLE

IV

SUMMARY OF SUM RULE RESULTS (a) Ny contribution ANY = -0.0028 A$"' = -0.0023 Al

= $0.0088

(b) Total TN coupling constant shift (0 = .23, excluding N*y contribution)

with from with from

g’/g Nfi model: g’/g PCAC: Atots 0 ZAP Atotal~A~-

*

-0.0216

-0.0211

-0.0380

-0.0375

= +0.0088 ANY-$‘%’

-

= -0.0083

32

MORRISON

result of Kroll and Ruderman (4) by taking only the Born terms (see Fig. 6) which uniquely define the off-shell extrapolation. The anomalous magnetic moment coupling was neglected since the form factor from the matrix element of rWv provided a cutoff slightly above threshold. The infrared divergent was removed as in Section II and III. The results are given in Table IV. Eq. (48) gives no new results, but it gives a consistency check of our assumptions. Taking the appropriate matrix element of Eq. (48) between a physical, one-nucleon state, / N >, and an off-shell state vlU,, , we see that the single-particle term cancels with the right hand side. This means the continuum terms must vanish. This is indeed the case since the physical one-nucleon states are exact eigenfunctions of P (this follows from charge conservation for example). The difference, A,, = (g,+ + g,,,O)/g, can be obtained from Eqs. (47a) and (82) by (a) taking the matrix element of Eq. (47a) between one-neutron states with final neutron off-shell, and (b) subtracting the resulting equation from Eq. (82). The coupling constant, gDT- , will vanish leaving an expression for A,, in terms of a difference of continuum terms (which is nonzero). It should be emphasized NSC0 Y

proton

v

\

a

\

t (a)

‘1\ \\

(b)

N*+

proton

neutron (off-shell) (cl

FIG. 9. Born terms for N $ ?r ---f N* f y,

\\ \

PION-NUCLEON

COUPLING

33

CONSTANTS

that d, , obtained in this manner, is independent of G(0) and g’/g. The result is given in Table IV(a). We also give the result ford+ which can be obtained from d, , A- , and A,, , and which is also independent of G(0) andg’/g. An estimate of all higher order terms in the sum over intermediate states in Eq. (54b) would be impossible. However we can make an estimate of the N*(1238 MeV) plus photon state. The analysis of the N*y state follows along the same lines as the NY state, so the details will not be given. The Born terms in Fig. 9 were used. We used the notation of Gourdin and Salin (53) for the N* t--t NT and N* t-) Ny couplings. The interaction Hamiltonian for these processes are given by H N*

Na

H N*Ny

= G

$@all$ ?r

+ Hermitian

= eC1$y5@A,

+ $

conjugate,

$y“y5#Y$AU

+ Hermitian

conjugate,

c

where I,!J” is the spin-$ field of the N* and A, is the electromagnetic coupling constants are given by (53) h = 2.07, The free-particle @ - w

wavefunction,

C, = 5.6,

UU, for the N* has the following

properties (54):

UU(P, A) = 0,

(8% Wb)

Q,(P, A) VU@, A’) = a,,< , U,(P, A) B”(P, A) = [L” - ; $$

field. The

c, = 0.37.

YUU(P, A) = P”U,(P, A) = 0,

jl

(88)

(89~) + (P”Y, - PJJl(3M)

- Y,Y”/3] 2*

*

The diagram in Fig. 9(a) has a singularity due to the proton propagator. This presumably occurs because we have treated the N* as a stable particle with a real mass, M > m + m, . A correct treatment would take into account the imaginary part of the mass, namely the width. The diagrams which contribute to A:*, [i.e., those in Figs. 9(c) and 9(d)] do not have this problem, so we will confine our attention to these diagrams. Unfortunately the derivative coupling given by Eqs. (87) and (88) and the polarization sum given by Eq. (89d) introduce enough extra momentum factors into the Born terms (Fig. 9) that the resulting integral over the 3-momentum of the N* diverges. This is, of course, because we have neglected the form factors which should accompany the coupling given by Eqs. (87) and (88).

34

MORRISON

If we again assume that the largest contribution arises from the low-energy region, we can provide a cutoff, k,, , corresponding to some maximum photon energy. In terms of k,, , we find that the contribution of the N*y state to A+ = (gnm+/d2 + gn,d/g is given by

“Y *y =

0.0007 0.0015 i 0.0020

for for for

km,, = m, k,,, = 2m, km, = 3m

(90)

This contribution is very sensitive to the cutoff, but we seethat for a reasonable value of the cutoff, namely k,,, E M, the contribution from the N*y is a factor of three smaller than that of the Ny (seeTable IV).

V. CURRENT

COMMUTATOR

APPROACH

Becauseof recent concern (X5)-(58) with the finiteness of radiative calculations, it is of interest to inquire about the validity of our present work. In this section we will show that the electromagnetic correction to the TN coupling constant contains a model dependent, logarithmic divergence. We will also relate the coupling constant shifts to other electromagnetic corrections. We again assumethat the strong renormalization has been carried out in the absence of electromagnetic interactions. The strong Hamiltonian density, HS , is to be given in terms of ms , m, and g, the strong renormalized massesand rrNN coupling constant. We then ask, what is the effect on g,,, of “turning on” a small electromagnetic interaction, which has the form: HI = e:JzmAu: - Sm:$,h:

-

Sm,2:c$$t:

(91)

where : : means normal ordering. In Eq. (91) Jz” is the total electromagnetic current, and Sm and Sm,2 are the electromagnetic mass shifts with neglect of feedback effects from the strong interactions. In terms of Feynman diagrams to order e2, 8m and Smv2 are obtained from Figs. l(c) and l(i), respectively. A convenient expression for Sm to this order is given by (9) Sm=$S$+

Wk2)

TAP,

k),

where T,,(p, k) = j d4xeik’X

VJ:%)

Jye”@>> I P>.

(93)

PION-NUCLEON

COUPLING

35

CONSTANTS

The IITNN form factor, g(q2), is defined by (N’ I &t((O) I N) = C,C,,a,~7n5U~g=lN(q3[iS:i(q2)]

(no sum on i),

(94)

where i is the isospin, g,&z$) = grriN is the exact coupling constant, S$(q2) is the electromagnetic renormalized, pion propagator which we will assume is well-behaved for q -+ 0, and & is the electromagnetic renormalized pion field. The TNN coupling constant defined by Eq. (94) is correct to all orders in e. To extract the order e2 part of this, we use the T function and its electromagnetic perturbation expansion. If we define g(q’) = g(0)(q2) + e2gc2)(q2)+ --a ,

(95)

yp(q”) = S$)(q2) + e?sy(q2) + *** ,

(96)

then we have e2(AG)i = e2~N’7iY5UN[g!2’(q2)(iS(q!(q2))

+ gf)‘(q2)(i,S~(q2))]

and we find to order e2, e2CNC,,(AG)i

= q

i -$$-

Duy(k2) / d*yd*z

(times) e-i(a.J+k*z)(N’ 1 T(&i(y)

J&(z) J&(O)) I N)

+ iC,C,,8,,[T,i(N,

6mN + Gm,+SF(lV’))

q)(i&(N))

F,,i(N, q)

+ rr,(N, a>(iS~i(q>> Sm$l ~N> + Kz;~/2 - 1) + (z;;!2 - 1) + (Z;l” - l)] C,C,~U,*r,,~(N,

q) U&!Q(q)), (98)

where S&V) is the full, strong renormalized, fermion propagator, Z, and Z,, are the electromagnetic wavefunction renormalization constants for the fermion and pion respectively, and r, , the proper pion vertex, is defined by

= s d4xd4yexp{W + d - x - q * YIW I T(h&d MY) &dW I 0). (9% We may represent the various terms in Eq. (98) by the diagrams in Fig. 10. The terms proportional to the mass shifts are singular but will just be cancelled by the pole contributions of the first term.

MORRISON

36

e2 C, C,,(

A G Ii =

FIG. 10. Pictorial

representation

of (AC), .

There are two routes open at this stage: (a) one may keep the 9 field (or better, replace it by the source current, J,,). In this case, one needsa model of the strong interactions such as that used in the Feynman graph approach. (b) Or one may use PCAC: that is, use a,JbA (JS1\is an axial-vector current) as an interpolating field for rj and assumea smooth behavior in the limit qh ---f 0. We will choose the second route and take

&i(x)

= $ a&(x). 77

WV

Using the PCAC assumption given by Eq. (IOO), we will now try to simplify the expression for the coupling constant shift, Eq. (98).

PION-NUCLEON

COUPLING

37

CONSTANTS

We start by defining the following: Iy(N,

q, k) = j d4yd4z exp[--i(q

* y + k * z)]

(times) W’ I T(~RYY) J&&9 JkdW

I W.

(101)

By means of Eq. (100) this becomes I!‘”I = d4yd4z exp[-i(q s

’ ’ + k ’ “I (N’ 1 T(a hJ?25 (y) J”em(z)J em ” (0)) 1N)

x7 = d4yd4zexp[--i(q - Y + k ' z)l P,,W i .L - %Y’ - z”W

- W’W’

I T(Ji6(y) J,“&) J&n(O)) I W

I WJi’5(zo, h J&&)1 J&(O)) I N)

I ~(JL~(z)[J~‘~@, ;I>, Jhn(O)l) I W

= f-77j d4yd4zev--i(q

- Y + k * z>q&V’ I T(J&)

J,“,(z) J&,(O)) I N)

- !!.@??/ d4ze-ik.z[e-i~*z(N’ 1T(J&(z) J&(O)) I N)

.Ll

+ W I ~(J&&)

J&(O)) I N)l,

W)

where we have integrated by parts, dropping the surface terms (24), and we used the well-known SU(3) x N(3) current commutation relations of Gell-Mann (59).

[Ji”(x), Jj”(y)l &x0 -

Y”)

= $ikJk“(x)

a4(x - y),

(103a)

LJi5(x), J,“(y)1 %x0 -

Y”)

= &JQx)

a4(x - i),

(103b)

LJ$x), JY6(y)l %x0 - y”) = iJ;ljkJ~‘W a4(x -

Y).

(103c)

Also we have J&,, = J$ + J8u, where J$ and J8* are the isovector and isoscalar currents respectively. Thus only the isovector current, JsaLc, contributes since i, the isospin of the pion, is limited to 1, 2, and 3. Furthermore, we have used the fact fis,,, = cizrnfor i = 1,2, and 3. We notice that for i = 3 (pprr” or nnrO), the last two terms of Eq. (89) vanish since eaSrn = 0 ! We will say more about this later.

MORRISON

(M? +

Crik) F (c)

(Mpv1 9

(is;)

(b)

--iCi3m

-I-

-GL

11. Representation

FIG.

Cd 1 (e) for the terms of Zy’ with the one-nucleon

terms separated out.

Separating our the one-nucleon terms, we get (see Fig. 11) 1~ = $k CNCN&,[T;6(N, where

;M“‘(N’,

(WN’))

r?&N, q)GWW)

= W&N

s

d4xd4y

M“(N, =

expW’

q)(iS;i(N)) M““(N) k)

WMN’)) F;&N, Q)+ j%““(N, q, WI UN,

* x -

q * y)l
I U&b)

J~&Y) $I@)) I O>,

(104)

(105)

W%(N))

I d4xd4yd4z exp{i[N’

* x + k * (Y -

z)lW I ~(#R(x)J&(Y) JL&)

&dO)) I 0). UW

PION-NUCLEON

COUPLING

39

CONSTANTS

M@” is the generalization of TuY[Eq. (93)] off the mass shell, and q,&f~ contains all other diagrams. In particular, fi AuVwill contain a pion pole which is proportional to qa[S~(q)]z. In the limit q -+ 0, this term vanishes if we assume the pion propagator is well behaved. Referring to Fig. 1l(a) (i.e., tiAtiY) we see that in this limit,

(107) is the electromagnetic correction to the axial-vector form factor. To evaluate Zr, we will need a Ward identity for I’f6 . From Eq. (105) we find iq,FfJN,

q) = f,(iS~i(q))

FJN,

q) - i9L1(N’)

y5 % + ‘ys T (igil(N

(108)

where we have used Eq. (100) and we have assumed that the zero component of the axial vector current obeys the free-field commutation relations

[J:Ax), $(Y)I Xx0 -

Y”)

(109a)

= -J&Y) y5 % 64(x - y>,

[J:.+>, RY)I Go - Y”> =

~5

7 #(x>S4(x-

(109b)

Y).

That is, we assume that Jd behaves like 1/&(7~/2) #, where we have

i1Cri+(c 3, ~i(~,;>>= &J3G- h, {~i+o,3, +i+o,31= {$w,3, ~j(~,3>> = 0.

(109c) (109d)

Using Eq. (108), we may write

+ f

?I

(-(SF’(N))

+ M‘V’, +

(~5

+

y5 $! (i&(N))

k) [(isg(NY %

+

G%(W)

r,,4N y5

2

- y5 +)]

APV,

k)

qW;&)) (j&-‘(N)))]

?r

+ 9

@F(N,

q, k)/ U, .

(110)

Now we go back to our expression for (dG)i , Eq. (98) which we recall is related to the TN coupling constant shift to order e2 by Eq. (97). We see that we will

40

MORRISON

need to multiply If” by D,,(k2) and integrate over d4k. We may use the expression in Eq. (1 IO) for Ir (where of course we must let TV-+ 7i , etc.) but we must add to it the extra terms in Eq. (102) which were not present for i = 3. Furthermore we notice that the variable, k, enters into Eq. (1 IO) only in the terms MuY(N, k), M”“(N’, k), and L%@“(N, q, k). The integral of D,,Muv over d4k is just the mass operator (or self mass) of the fermion (9), (36) to order e2: ztN)

= $ j $$

D,,(k2)

(111)

M@“(N, k).

Using this expression for Z, we find that (AG)i can be written e2CNCN,(AG)i

= C,C,t

!

- i-Q(N, qW;(N)MN)

- i[Z(N’) + ;

- Sm,*](iS&(N’))

[(iS;l(N’))

- $ Z(N’)

y5 $-

[y” +

- ~%vlW:iW

TJN,

(z&(N))

Q)(iS;i(q)) + y5 +]

+ (i&(N’))

y5 +

Z(N)

(jr,-‘(N))

t i&N,

q)(iS;i(q))2 Sm:il UN

+ [(z;;‘”

- 1) + (&I;!‘,/” - 1) + (2,-1’2 - l)]

X u,(js%l))

+ $-

j” &

as follows:

D,,(k2)

C,C,,O,jr,,(N

q)

/C&J,,

(x ) q&$@“(N, q, k) U, - l isrn d4ze-ik.z[e-iq.z s

(x) W’ I T(JE&) J,‘dON I W + W’ I ~(JX4 JA,(W I WI/ (112) We know how Sk(N) and Z(N) behave as w -+ m (60): Z

&(N) PO+M~w -z”m, = H : mhr11+ W2)1, Z(N) - 6mN-CM - md@Z;: - 1). D4+mN

(113) (114)

Since Sk is multiplied by terms of order e2, namely 2 and am, we may replace Sk by the Feynman propagator [i.e., we may drop the term of order e2in Eq. (113)].

PION-NUCLEON

COUPLING

41

CONSTANTS

Thus Eq. (112) becomes e2C&,,(AG)i

= C,C,,U,, x

i

&(N,

q>(iS;,(q))[(Z,“,“2

-

1) + (Z;;

- 1) + (Z;1’2 -

SmN,) + iT,,,(N, q)(iS;i(q))2

- %nsd4ze-ik.z[e-iq.z(N’

/ T(J&(z)

J&(O))

1)l

Sm,,2/ UN

1N)

+ W’ I ~(Jk&> JL,@>>I WI 1,

(115)

where, to order e2, (zp2

-

1) + (1 - Z,‘)

z z,1’2 -

1

(116)

since Z, = 1 + O(e2). The contributions from the factors of Z, agree with the Feynman calculation, Eq. (20). The contribution from the pion wave function renormalization constant, Z, , is not the same however. Eq. (115) is really quite interesting in that it relates the r - NN coupling constant shift, Sg = e2gt2), for q --f 0 to several other electromagnetic shifts in addition to the Z factors: (A) II - p and + - no mass differences am, - am,* = 0 and presumably 8rn$ = 0), (B)

e2 correction

to the axial-vector

for g,+nr (for

g,,ohi we have

form factor, and

(C) for n*, to a term involving a time ordered product of an axial-vector current with the electromagnetic current. (It can easily be shown that the last two terms are equal.) We recognize some of the terms which sections.

occur in Eq. (115) from the previous

(A) The dependence on the n - p mass difference was found in all approaches, but was systematically neglected. We found a dependence on the pion mass difference only in Section IV. (B) Section IV also yielded an electromagnetic correction to the P-decay coupling constant which is related to the electromagnetic correction of the axial vector form factor by means of the V-A theory of weak interactions. CC) The terms in Eq. (115) involving to be unique to this approach.

the time-ordered

products

appear

42

MORRISON

This relationship appears to be somewhat different from that which is obtained from the Goldberger-Treiman formula (61) if we assume the latter to be valid to order e2: g=

Fs”,

)T n

I;Rm ,

(117)

where

Thus we have (119)

where again we have defined g

=

g’o’

+

,2,(2)

+

. . ..

w9

Comparing Eqs. (115) and (119) we see that, in both equations, gc2) depends on the e2 correction to the axial vector form factor, FA“). At this point the similarity stops; Eq. (115) does not depend on F, (2), but it has several more terms which are absent in Eq. (119). Now we will show that g$,,, for i f 3, contains a logarithmic divergence which is model-dependent. In particular, we will consider the last two terms in Eq. (115), which are (essentially) equal to a quantity Pm5 , defined in the following way : Pm, = j d4kD,,(k2) j d4zeik.Z(N’ I T(J&,(z) J:,(O)) I N) (121a) (121b) where P”*P = &yp~6 3 (122) Pr&(N, q, k) E

s

d4zeik.Z(N’

1 T(J&(z)

J&,(O)) 1 N).

To determine the degree of divergence with respect to k of Eq. (121b), it is sufficient to look at the asymptotic behavior of the integrand when one compenent of the momentum, say k”, becomes large. This problem has been considered by Bjorken (55) who finds what the large k” dependence of Pg’ , given by Eq. (122), is given by the equal-time commutator of the currents. His result is PK5 -

k%co

I

d3ze-s.;(N’ 1[J&,(0,

t), J&,(0, $1 I N).

(123)

PION-NUCLEON

COUPLING

43

CONSTANTS

In Bjorken’s calculation, say of the mass shifts, he needed (N ) [J& , JL’,] 1N) which vanished when you take the quark model and contract on p and v. If we use quark-model commutators, we get the following:

[J&(0, 21,J&(O)] = +? [; EuOvoJo*(0) f (g%:, - g““J$ + g”‘J;,)]

a”@).

(124)

But

so we have to lowest order in e:

In order to do the integral in Eq. (121b), it is mof;e convenient to put P&, in a covariant form. This is easily done, since for finite k, we have k2 -

kblm

k”:

k.q----+ kO-tim k”qo, etc.

The result is pL

Unfortunately,

z

Tx,C,~ ka

-

uN’[tiFzdq”)+ k ’ qF,(q2)1y57dN

,

(126~)

this form is not unique, since as k, -+ ice, we have

In fact either form is sufficient to show that there is no linear divergence. This will be demonstrated for the form given in Eq. (126~). From invariance we may write

p&l = C,vCrdN’[k *qG,(q2, k2,~1, ~2 , ~3)+ ltG(q2, .*-> + plG(q2, a*-> + lfQIG(q2, **.)Iwi-uN,

(127)

44

MORRISON

where v1 E N * q, v2 = N * k,

(128)

v3 = k - q, N’ = N + q. Comparing

with the asymptotic

form, we see that

G(q2,k2,VI

3 v2 > v3

Wi2,

I----+ kbim

‘-> s

F-‘2Fdq2) k2 F 2Fb-i”) k2

’ 7

(129a) (129b)

G,(q2, -*-I x

Wlk2),

(129c)

G(q2, -*-I G

Wk3).

(129d)

Thus the most divergent part of P is given by

or P$” = 0 since it is odd in k. However this is the part which would have been linearly divergent! We are unable to say anything about the next order terms (logarithmic divergence) unless we take some model for the strong Hamiltonian and find the (l/k0)2 term which, according to Bjorken (55), is proportional to

(j-f’ I [[JhdO, $7 HI, J&V01I W. Furthermore, this may also We notice the Feynman The model investigated.

(131)

we must know the correct way to make Eq. (126b) covariant since introduce a logarithmic divergence. that the cancellation of the linearly divergent term also occurs in calculation of Z, [i.e., z(N)] so this is not an unexpected result. dependence of the logarithmic divergence is presently being

VI. SUMMARY

AND CONCLUSIONS

We have presented four methods for obtaining the electromagnetic corrections to the nNN coupling constant. Numerical estimates of these corrections were given for the first three methods with a summary of the results given in Table V.

PION-NUCLEON

COUPLING CONSTANTS TABLE

45

V

SUMMARY OF RESULTS" Sum Rules Feynman Graphs

Ai

= km+

4

= km0

- g,,-l/x’2

DWith only Ny intermediate

NN model PCAC model for W) for gW) and 0 = .23 and 0 = .23

-0.0070

+0.0009

-0.0211

-0.0375

-0.0046

-0.0049

-0.0216

-0.0380

0

+0.0009

-0.0083

+0.0049

+0.0088

g

+ g,,d/g

Dispersion Relations

-0.0023

(independent of model) (independent of model)

state.

Several different techniques were used because none of them are completely trustworthy. The hope was that they would give roughly the same numerical results. However, this was not found to be the case presumably because of the importance of high energy contributions in one form or another (cutoffs, offenergy-shell form factors, high-spin intermediate states, etc.). The effects are small, as expected, but exact magnitudes and even signsare not easily determined. This is brought out by Table V and by comparison with Riazudding (17). Nevertheless there are some definite trends which should be emphasized: (a) d- 1~ -0.5 % (the Ny contribution from the sum rule approach gives O?’ ‘v -0.3 %); (b) d * is generally small compared to d, . (From Feynmann graphs, d* N 0, and from dispersion relations, d* N d,/5. However, the sum rule result for d, is large, / Ll* ( ‘v 0,). The contributions which we have considered in Sections II and III lead to m2, > m, . Since experimentally m, > m, , it is unlikely that all significant contributions to dg have been included. It is possible that a major source of error comes from the neglected radiative corrections to the strong renormalizations which would have an unknown cutoff dependence. The contribution from the anomalous magnetic moment coupling of the yN vertex wascalculated for the Feynman approach and estimated in the dispersion approach. In both casesthis contribution was small (about 10%) and we expect it to be small for the other casealso in the approximation of low energy dominance. As we have pointed out this approximation is probably invalid, so one might expect the magnetic contribution to be significant. Furthermore, the dispersion technique is dependent on an artificially introduced, ultraviolet cutoff. Since

46

MORRISON

this cutoff, h, enters logarithmically, value:

the results are relatively insensitive to its

I

for 2 d A2 < 5, we have and

.0005 < d* < .0014, .oo1 1 G d, B .oo78.

All methods have an infrared divergence which is cancelled by the soft-photon emission terms. There will always be left a term which depends on energy resolution and the logarithm of some cutoff, (1. We have already mentioned in the Feynman calculation that the results are insensitive to this term. In the dispersion relation approach we assumed that this cutoff was the same as the cutoff which occurred in the dispersion integrals, namely (h2 - 1) m2. This need not be the case; however, the results are rather insensitive to this parameter: for m2 < A2 < 4m2, we have and I

.0033 < A, < .0065, .oOO09 < d ( oo17 \ f-l. -

An accurate estimate of d, or d- cannot be given in the sum rule approach because of the large uncertainty in g’/g. It is possible that the large contributions from G(0) and g’/g approximately cancel. We see that for this to occur, we must have g’/g > 0. The NiJ model for g(q2) satisfied this condition, but the derivative was too small in magnitude. However, we should point out that d+ and d,, are independent of both g’/g and G(0). Thus d+ and d, , as calculated by the sum rule, should be as reliable as the Feynman graph or dispersion relation calculations. In all the approaches, we have assumed that the intermediate state of lowest mass dominates (nearest pole approximation). Since use of the current commutators has shown that a model-dependent logarithmic divergence is present, this approximation may be suspect. The contribution of an N* intermediate state for d, was estimated in the sum rule approach. The result is badly divergent and so is very sensitive to the cutoff. For this reason, the calculation of the N* contribution is untrustworthy. A large uncertainty in all approaches was the lack of knowledge of the electromagnetic form factors for one or both nucleons off the mass shell. However, when the intermediate state is a nucleon (i.e., not an N*), the dependence on the cut-off parameter, from a form factor or otherwise, is generally small. Thus we expect the use of elastic, on-shell form factors to be true at least in some approximate sense for the N-N-y vertex. However, for higher-spin intermediate states such as the N*, we have seen that there is a stronger cutoff dependence. Because these higher mass states are likely to be important, one will need the offshell form factors. Referring to the Feynman-diagram calculation in Table II we see that the breaking of charge symmetry is less than that of charge independence. (d* N 0,

PION-NUCLEON

COUPLING

CONSTANTS

47

d, N L/2, and d,, _N 4+/3). Furthermore we see that the n* coupling constants are less than the TPJcoupling constants by 0.5 to 0.7 % which is in rough agreement with Reference (2) [referring to Table II of Reference (2), we see that for

mp - mpo = 2 MeV,

and a value of the f-n coupling constant, r12,of 1.6681. The results of Sections IV and V are unique in that they relate the electromagnetic shifts of the TN coupling constants to electromagnetic shifts of other quantities which are measureable in principle. In Section V, we have demonstrated that the linear divergence vanishes as expected. Although the next-order divergence was not determined, we expect a nonvanishing result in view of the current persistence of divergences in radiative calculations. Finally we point out a possible method for the experimental determination of the TN coupling constant shifts. It has been known for some time that in the limit, m,/m --f 0, the pion photoproduction amplitude at threshold is exactly given in terms of the Born amplitude. Recently, however, Gaffney (62) has determined the terms of order m,/m for z-TT+*O production. He finds satisfactory agreement with experiment for r* photoproduction. In particular, he finds R_

L,, dQcm &(Y

+ n-p

+n->

+ p + n + 7rf)

N -&?C2x13 ( 1 gm+ . at threshold,

whereas the experimental value is [see (62), (6.91 R = 1.265 4 0.075. Equating the theoretical prediction of R to the experimental value, we find N 0.986 i 0.058, N 0.014 i 0.058. Referring to Table V, we see that the “experimentally determined” d* has the same sign as the dispersion relation result for d*, while the sum rule result for d* has the opposite sign. However, no conclusion can be drawn from this since the experimental error in dyPt exceeds its magnitude. Gaffney’s prediction for no photoproduction does not agree with experiment, presumably because of the importance of the N* [see (62)]. For this reason, it is doubtful that the rrN coupling constant differences, do, d, , and d- can be experimentally determined from the photoproduction data. In conclusion, the present techniques are not sufficiently reliable to adequately

MORRISON

48

determine the radiative corrections to the pion-nucleon coupling constants. Since this problem is of considerable importance in nuclear physics (determination of scattering lengths, energy levels, etc.), there exists a need for new techniques that give trustworthy results.

ACKNOWLEDGMENT

I wish to express my gratitude to Professor Ernest M. Henley, my research advisor, for his continued encouragement and stimulation in this research. I am also very thankful to Professor Marshall Baker who contributed greatly to the development of Section V. In addition, my thanks go to Professor D. Boulware, Professor I. Muzinich, and to Dr. M. Gundzik for many valuable and stimulating discussions. RECEIVED: February 28, 1968

REFERENCES 1. 2. 3.

4. 5. 6.

7. 8. 9.

A review of charge dependent effects was given by E. M. HENLEY, Bull. Am. Phys. Sot. Ser. 2, 11, 623 (1966). E. M. HENLEY AND L. K. MORRISON, Phys. Rev. 148, 1489 (1966). H. FESHBACH,E. LOMON, AND A. TUBIS, Phys. Rev. Letters 6, 635 (1961); H. FESHBACH AND E. LOMON, “Congres International de Physique Nucleaire,” Vol. II, p. 189. Centre National de la Recherche Scientific, Paris, 1964, Suppl. Progr. Theoretical Phys. (Kyoto) Suppl. No. 3 (1956); see also G. CHEW, Phys. Rev. 112,138O (1958). N. KROLL AND M. RUDERMAN, Phys. Rev. 93, 233 (1954). G. CHEW AND F. Low, Phys. Rev. 101, 1579 (1956). G. CHEW AND F. Low, Phys. Rev. 101, 1570 (1956); S. BARNES et al., ibid. 117, 226 (1960). Another point of view is that there are no elementary, strongly interacting particles. See, for example, G. CHEW, “s-Matrix Theory of Strong Interactions.” Benjamin, New York, 1962. R. FEYNMAN AND G. SPEISMAN,Phys. Rev. 94,500 (1954). M. CINI, E. FERRARI, AND R. GATTO, Phys. Rev. Letters 2, 7 (1959).

IO. RMZLIDDIN, Phys. Rev. 114, 1184 (1959). Il. R. SOCOLOW, Phys. Rev. 137, B1221 (1965).

12. T. DAS, G. S. GURALNIK, V. S. MATHUR, F. E. Low, AND J. E. YOUNG, Phys. Rev. Letters 18, 759 (1967). They use the soft pion technique of S. WEINBERG, ibid. 16, 879 (1966). 13. H. PAGELS, Phys. Rev. 144, 1261 (1966); see also Reference (31). 14. H. M. FRIED AND T. N. TRUONG, Phys. Rev. Letters 16, 559 (1966). 15. M. SUZUKI AND F. ZACHARUSEN, Phys. Rev. Letters 17, 1033 (1966). 16. H. HARARI, Phys. Rev. Letters 17, 1303 (1966). 17. RIAZUDDIN, Nucl. Phys. 7, 223 (1958). 18. M. GELL-MANN AND Y. NE’EMAN, “The Eightfold Way.” Benjamin, New York, 1964. 19. R. DASHEN, Y. DOTHAN, S. FRAUTSCHI, AND D. SHARP, Phys. Rev. 151, 1127 (1966). 20. S. BOSE AND Y. HARA, Phys. Rev. Letters 17, 409 (1966). 21. M. MURASKIN AND S. GLASHOW, Phys. Rev. 132,482 (1963). 22. S. FUBINI AND G. FURLAN, Physics 1, 229 (1965).

PION-NUCLEON 23. 24. 25. 26. 27. 28.

29. 30.

31. 32. 33. 34. 35. 36.

49

COUPLING CONSTANTS

FURLAN, F. LANNOY, C. ROSSETTI, AND G. SEGRE, Nuovo Cimento 40, 597 (1965). FUBINI, G. FURLAN, AND C. ROSSETTI, Nuovo Cimento 40, 1171 (1965). FUBINI, Nuovo Cimento 43, 41.5 (1966). GATTO, L. MAIANI, AND G. PREPARATA, Physics 3, 1 (1967). WARD, Phys. Rev. 78, 1824 (1950). TAKAHASHI, Nuovo Cimento 6, 370 (1957). M. GELL-MANN, Phys. Rev. 125, 1067 (1962). A. BINCER, Phys. Rev. 118, 855 (1959). S. DRELL AND H. PAGEM, Phys. Rev. 140, B397 (1965). E. KAZES, Nuovo Cimento 13, 1226 (1959). L. CHAN, Phys. Rev. 141, 1298 (1966). D. YENNIE, M. Lfivu, AND D. RAVENHALL, Rev. Mod. Phys. 29,144 (1957). D. BOULWARE (private communication). J. BJORKEN AND S. DRELL, “Relativistic Quantum Mechanics,” p. 166. McGraw-Hill,

G. S. S. R. J. Y.

New

York, 1964. 37. Reference (36), p. 169. 38. J. JAUCH AND F. ROHRLICH, The Theory of Photons and Electrons. Addison-Wesley, 1955. 39. D. YENNIE, S. FRAUTSCHI, AND H. SUURA, Ann. Phys. (N.Y.) 13, 379 (1961). 40. L. LANDAU, A. ABRIKOSOV, AND 1. HALATNIKOV, Nuovo Cimento Suppl. 3, 80 (1956).

Mass.,

41. There is an error in Pagels’ calculation of the coupling constant shifts (ref. 13). Referring to his Appendix III, Eq. (III-l), we see that his most general T-N vertex is of the form Hence the v-Ncoupling

constant should be, g = K(nP) + 2nX(mz)

but he used g = K(&) in his work. This error affects his numerical calculations. 42. See Reference (31) or S. MANDELSTAM, Phys. Rev. 115, 1741 (1959). 43. K. MITCHELL, Phil. Mag. 40, 351 (1949). 44. S. M. BERMAN, Phys. Rev. 112,267 (1958), T. KINOSHITAAND A. SIRLIN, ibid. 113,1652 (1959). 45. N. CABIBBO, Phys. Rev. Letters 10, 531 (1963). 46. D. HENDRIE AND J. GERHART, Phys. Rev. 121, 846 (1961). 47. G. BARTON, “Dispersion Techniques in Field Theory,” p. 150. Benjamin, New York, 1965. 48. S. WEINBERG, Phys. Rev. Letters 18, 507 (1967). 49. S. GASIORO~ICZ, “Elementary Particle Physics,” p. 581. Wiley, New York, 1966. 50. J. SAKURAI, “Invariance Principles and Elementary Particles,” p. 241. Princeton University Press, Princeton, New Jersey, 1964. 51. M. VELTMAN, Phys. Rev. Letters 17,553 (1966); M. NAUENBERG, Phys. Rev. 154,1455 (1967); D. BOULWARE AND L. BROWN, ibid. 156, 1724 (1967). 52. J. BJORKEN AND S. DRELL, “Relativistic Quantum Fields.” McGraw-Hill, New York, 1965; or see Ref. (47). 53. M. GOURDIN AND P. SALIN, Naovo Cimento 27, 193 (1963); see also S. JACKSON AND H. PILKUHN, ibid. 33, 906 (1964). 54. W. RARITA AND J. SCHWINGER, Phys. Rev. 60, 61 (1941); H. UMEZAWA, “Quantum Field Theory.” North Holland, Amsterdam, 1956; V. GLASSER AND B. JAKSIC, Nuovo Cimento 5, 1197 (1957). 55. J. BJORKEN, Phys Rev. 148, 1467 (1966). 56. E. ABERS, R. NORTON, AND D. DICUS, Phys. Rev. Letters 18, 676 (1967). 595/50/7

-4

50

MORRISON

57. G. KALLEN, Nucl. Phys. Bl, 225 (1967). 58. K. JOHNSON, F. Low, AND H. SWRA, Phys. Rev. Letters 18, 1224 (1967). 59. See Reference (IQ, p. 224. 60. See Reference (52), p. 304. Remember we are dealing with strong renormalized so Z, means Za . 61. M. GOLDBERGER AND S. TREIMAN, Phys. Rev. 110, 1178 (1958). 62. G. W. GAFFNEY, Phys. Rev. 161, 1599 (1967). 63. J. BURG, Ann. Phys. 10, 363 (1965).

operators,