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Charge-dependent radiative corrections to the fermi matrix element for pure fermi β-decays
(4.c
Nuclear Physics A162 (1971) 97-l Not to be reproduced
CHARGE-DEPENDENT MATRIX
by photoprint
10; @ North-Holland Publishing Co., Amsterdam or microfilm without written permission from the publisher
RADIATIVE
ELEMENT
CORRECTIONS
TO THE FERMI
FOR PURE FERMI B-DECAYS
W. JAUS Institut fiir Theoretische Physik der Universitiit Zurich, Switzerland Received 28 September 1970 A&&act: The charge-dependent radiative corrections of order Z’c? to the Fermi matrix element for pure Fermi p-decays is calculated neglecting the effect of excited nuclear states. The calculation treats the nucleus as an elementary particle and makes use partly of the technique of dispersion relations and partly of perturbation theory. The charge-dependent corrections are free from ultraviolet divergences and the electromagnetic form factor reduces the relevant dispersion and perturbation integrals to their low-energy limit. This fact enables the application of a low-energy theorem to determine the structure-dependent terms and an approximation method to estimate the two-photon contribution. The result is a correction to the Fermi matrix element of a few hundredths of a percent which is entirely negligible.
1. Introduction
Precise ft values for Fermi p-decays whose endpoint energies and half-lives are experimentally well determined, are of interest to calculate the nuclear vector p-decay coupling constant Gv, and thus test the Cabibbo theory of weak interactions. Though there still exist uncertainties in the calculation of radiative corrections of order a [ref. ‘)I to semileptonic decays (where a is the fine structure constant), theft values of Fermi decays can be compared among themselves (because the unknown modeldependent part of the radiative correction is the same for all decays) and another, more restricted aspect of the theory can be tested, namely the hypothesis of the conserved vector current (CVC). The near equality of the measuredft values in seven 0+-O+ superallowed Fermi transitions ‘) has always been regarded as a strong evidence in favor of the CVC hypothesis. Theft value for pure Fermi transitions is defined by 2n3 In 2 ft = ~ G;IWZ ’ where MF is the Fermi matrix element, which for a 0+-O+ superallowed decay, whose initial and final state are members of the same isospin T multiplet, is given by MF(T, T,&l
-+ T,
T,) = d(TfT,)(T+T,+l).
For T = 1 multiplets we find MF = J2. This result is independent of the details of the wave function of the nuclear states involved in the decay, but follows from the 97
98
W. JAUS
assumption that these states are pure isospin states. This statement is not true in reality because electromagnetism destroys the conservation of isospin, and therefore causes deviations from the exact CVC hypothesis. In the presence of the eiectromagnetin interaction the divergences of the isospin current VS* (x) are no longer zero but are proportional to terms of order e, the electromagnetic coupling constant. Furthermore the initial and final states are not eigenstates of isospin any more. The existence of a nonvanishing mass difference of nuclei within the same isospin multiple& i.e. the endpoint energy, is one obvious manifestation of this influence of the electromagnetic field. Consequently the Fermi matrix element is not strictly J2 and the deviation of A4r from 42 is usually expressed by “)
where 6, represents the effect of the charge-dependent corrections. We do not consider the radiative corrections of order ct, i.e. corrections which are independent of the nuclear charge, because these have already been discussed extensively, see for example ref. ‘). A general theorem has been proved “) which states that 6, is of second order in the difference of the mass of the decaying nucleus of charge Z-t 1 and the mass of the daughter nucleus of charge 2, i.e. of order Z2a2. This means that 6, could be of order 0.35 % for the decay of I40 and 3.9 % for 54Co, which would lead to a rather large correction to theft values. On the other side, we have the above mentioned near equality of theft values which fits the theory so nicely and which makes one expect that 6, is small. It is the purpose of this paper to calculate 6,. This has been attempted in the past within the frame of specific nuclear models by various authors (see ref. ‘) for a review), who study the wave functions of the initial and final nuclear states, where the physical state is a superposition of the ground state and excited states of different isospin. Of course, this approach does not consider radiative corrections coming from the interaction of the decaying and the daughter nucleus with the electromagnetic field without exciting the nucleus. We shall discuss just this aspect of the problem, i.e. we shall determine S, neglecting the effect of excited nuclear states. For this purpose the nucleus is treated as an elementary particle which has in our case spin 0 and positive parity, and the complex nuclear structure manifests itself only in the electromagnetic form factor. Electromagnetic corrections to the Fermi matrix element can then be described as usual as emission and absorption of virtual photons by the nucleus as a whole. In the actual calculation of 6, we partly use sidewise dispersion relations. These have been developed first by Bincer “) in connection with the electronlagnetic form factor of the nucleons, then used to determine electron and nucleon anomalous moments by Drell and Pagels “). This technique allows us, for example, to calculate quite simply in terms of the Fermi matrix element, including corrections of order ti, that contribution of order Z2a2 to a,, coming from the exchange of two virtual photons between the nuclei. The application of dispersion relations is cful only when no subtractions are required or when
FERMI /T-DECAYS
these are known. is evaluated
That part of 6, which depends
via perturbation
theory,
where
99
upon unknown
subtraction
the structure-dependent
constants
terms
of the
relevant amplitudes are determined by an exact low-energy theorem. This is possible because we are interested only in the charge-dependent corrections to Mr and these are convergent by virtue of the electromagnetic form factor, which practically reduces the amplitudes in the dispersion and perturbation integrals to their low-energy limit. As mentioned above, we did not consider the effect of excited nuclear states in this calculation, but believe that we have found at least the right order of magnitude of the charge-dependent corrections to the Fermi matrix element.
2. Method of calculation The general form of the matrix element of the charge lowering component of the isovector current VA-(O) between two nuclei of charge Z+ 1 momentum p; mass M and charge 2, momentump, mass M, both nuclei having spin 0, is determined by two form factors
= (2n)-3(2p,2p~)-iJZr,(M2,
M’2, I”) = (P+~‘)~F,(M~,
M’2, l’),
M”, Z2)+ Z,F2(M2, M”, Z’),
(la) (lb)
1 = p’-p. The factor J2 in (la) interactions, isospin F2(M2, M2, I’) = 0. factor of a nucleus of We shall investigate magnetic interactions.
is a Clebsch-Gordan coefficient. If we neglect electromagnetic is conserved and we have FI(M2, M2, I’) = G(Z2) and Here G(12), with G(0) = 1, is the usual electromagnetic form charge 2. the behaviour of P,,,(M2, Mf2, 12) in the presence of electroFor nuclei at rest we can cast (1) into a more familiar form
(M, p = OlV,-IM’, p’ = 0) = (27~)-~$(1-@~) which is up to a factor electromagnetic
equal
correction -+S,
to the Fermi
which expressed =
matrix
element.
(2a) The quantity
by the form factors
6, is the
reads
F,(M2, M”, 6M2) - 1 + $j F,(M’).
Here 6M = M’ -M is the mass difference of the two nuclei. General theorems have been proved 3), which assure that the deviation of the Fermi matrix element MF from ,/2 is of second order in the mass splitting. The mass difference can therefore be ignored in F2 and we have put F2(M2, M2, 0) E F,(M’). Following the work of Bincer “) we shall examine the analytic properties of the vertex function m(M2, M’2, I”) of eq. (1) as a function of the mass of one of the nuclei at fixed momentum transfer I2 = 6M2. This then leads to a dispersion relation for the form factor F, .
W. JAUS
100
We shall see below that second-order terms in rA lead to contributions to 6,) which are of third order in the mass splitting, and therefore we can put Z2 = 0 in r, . Using the standard reduction formalism to contract one of the nuclei in the matrix element of eq. (1) we obtain
ra(W2,M’2,
0) =
(2n)q2p,)fiSd4xB(x)eip”“
where p12 = W2, p2 = M”, Ti(M2,
1’ = 0
V;(O)]lp),
(34
and
W2, 0) = (2x)i(2po)ti~d4xO(--)e-ip”“
qz+l(x)]lO),
P-4
where now p2 = M2, p” = W2, I2 = 0 and with the source operator y’(x) = (cl + M2)4Z(x), where 4”(x) is the field of a nucleus of charge 2. In writing down eq. (3) we have omitted equal-time commutator terms which can affect only the subtractions in the final dispersion relations. Using the projection operator Pa(M2) =
with I = p’-p
we
w&
la,
(4a)
find from eq. (3) F,( W2, M’2, 0) = P,(M2)T,(W2,
M’=, O),
(4b)
F,(M’,
W=, 0).
(44
W=, 0) = I$(M’=)T,(M=,
Dispersion relations for the form factor as function of the complex variable W2 can be proved along the same lines of reasoning as those given by Bincer “) for the electromagnetic form factors. Nothing can be said generally about possible subtractions, so we used perturbation theory as a guide. In a first step we assume a once subtracted dispersion relation for F1 2
F,(M’,
M’2, 6M2) = F,(M=,
M2, 6M2)+
M’2-M n
a3
Im F,(M’,
W2,
0)
s iv’2dW2 (W2_M2)(W2-M’2)’
Similarly we have m
F,(M2,
M’=, 6M=) = F,(M”,
M’=, 6M=)+
M2-
n
M’2
Im F,( W2, M’=, 0) dW2 (W2-M2)(W2-M”) s M2
101
FERMI ~-DECAYS and the subtractions
are given by another
F,(M~,
F,(M’2,
Mu, 6~‘)
set of disperion
= A+ 1 O”dW2 Xs M2
Mt2, 6M2) = B+
1 71s
relations
Im F,(M2,
W2, 0)
W2-M2
’
m d w2 Im F,( W2, M’2, 0) W2-Mf2
M’Z
(6b)
’
where A, B are unknown subtraction constants. The function Fl( W2, W2, 6M2) is equal to the form factor Fl for both external particles having the same mass W, while the intermediate states which saturate the dispersion integrals in eq. (6) have physical masses M and M’ (we do not consider excited nuclear states). The two subtractions (6a) and (6b) transform into each other by simply interchanging the masses M and M’. Consequently the sum F,(M2, M2, &IL?~)-~F,(M’~, hT2, 6M2) is free of terms linear in the mass splitting and this leads at once to two important results: (i) The two dispersion integrals entering this sum can be evaluated for all nuclei (external and intermediate) having the same mass. (ii) The sum of the two subtractions +(A+B)
= F,(a)
depends only quadratically upon the mass splitting. Therefore Fl (a) does not contain charge-dependent corrections (which are of order Za 6M) and is given by the uncorrected form factor F&o)
Taking
= G(6M2).
(7)
the sum of eqs. (5a) and (5b) we have F,(M’, AF,,
AF,,
M”,
6M2)
= F,(co)+AF,,
* dW2 Im F,( W2, M2, O)+Im = 1 2TT W2-M2 s M2 = 2M6M mdW2 271 s M2
Im F,(M’,
+AF,,, F,(M’,
W2, 0)-Im (W2-M2)2
F,(W’,
(84 W2, 0)
,
(t(b)
M2, 0) ’
(8C)
In writing the expression for AF,, use has been made of the fact that the dispersion integral is multiplied by 6M and therefore the mass splitting can be neglected in the integrand. Eq. (8) for F,(M2, Mf2, 6M2) is the basic relation for the following discussion. It has the great advantage that the absorptive part of F, can be evaluated for equal masses. The correction due to the non-zero mass difference is given by AF,, . Furthermore we assume low-energy dominance, namely that it is the threshold contribution to Im Fl that gives the dominant part of the electromagnetic correction. In the work of Drell and Pagels “) where the anomalous magnetic moment of the electron is computed, the above assumption is made ad hoc and then justified by the success of the calculation. We are in a slightly better position because the perturbation theory for the charge-dependent correction is finite due to the electromagnetic
W. JAUS
102
form factor, and it is this effect which we approximate by a suitable cut-off in the dispersion integrals. Let us therefore keep only the contributions between M2 5 W2 s A2M2, M”
5 W2 5 it2M,
in eqs. (9) and (10) with 1’ - 1 << 1. It is not possible to calculate F2(M2) in dispersion theory and we shall present alternative perturbation theoretic approach in the appendix. 3. Determination
an
of the cut-off 12’
The two nuclear states of charge 2 and Z+ 1 with which the matrix element of eq. (1) was formed, belong to an approximate isotriplet. It is the electromagnetic interaction that breaks isospin conservation and causes a mass shift between the two states. We shall now calculate this mass shift using the method discussed in the previous section and by comparison with the experimental value for the mass difference determine the value of A’. The electromagnetic self-mass
of a nucleus
of charge Z and spin 0 is given by
c(I-“>= - i ($/d4, For the Compton
amplitude
T&J,
k).
(104
we use the Born approximation
TB@ flv
$
kj ’
=
(2p-k),(‘b-kj, (P_kjz_Mz
-gcv’
PW
In writing down eq. (10) we have neglected the contribution of all excited states. The self-mass c( W”) as a function of the mass p2 = W2 of the nucleus is analytic in the cut Wz plane with a branch cut from M2 to + co. Using the hypothesis of low-energy dominance we have the dispersion relation
We derive the above expression from perturbation theory and by comparison one finds that one subtraction is needed. We shall show below that c(cc) is small and does not contribute. The absorptive part of I( W2) is obtained by the replacement of the propagators of the internal lines according to ‘) (k2+i.s)-1(q2-M2+iE)-1 The absorptive
amplitude ImC(W’)
+ 2~26(q2-M2)B(q,)6(k2)~(k,)).
is then given by = Z2f-12
zd’(p-q-k)(2q+k)‘. 0
103
In the c.m. system with p = (W, 0) we have Im~(W2)=Z2~(W2-MZ)~$, 7c2 C(W”)
Perturbation
= C(co)+Z”
~2M2~(A2-l+ln12). x
theory gives for the subtraction
i.e. the subtraction
constant
is two orders of magnitude
from the dispersion integral. For the mass difference 6M of a nucleus 6;
constant
= (22+
smaller than the contribution
of charge Z-t- 1 and one of charge Z one has
(11)
1) 3 {&(A”- 1) + O[(A’ - l)“]}.
Comparison with the experimental gives the result
end-point
140.
AZ-1
energies for the decays of 14O and 5 4Co
-
3.2
Mr ’ 54co:
p-1
=:
(12)
!I!!
MT-’ with the rms radius Y = 1.03 A?‘ fm. We note that (MY)-~ which we are interested is of order 10e3.
4. The charge-dependent
for the range of nuclei in
corrections to the Fermi matrix element
We turn now to the absorptive parts of the form factor near threshold, follow from eq. (3) by replacing S(X) and i0( -x) by + [ref. ‘)I Im Ti(W2,
which
M’2, 0) = (27~)~(2p~)%c
x ~~,c
~,-(o)lPP4(P’-
4-
WI ~,-(o)ln’>
a* (13)
The sum over states ~1’cannot contribute since Z2 = 0. The only admissible state II’ is the vacuum and (Ol~z(0)lp) = 0. The lightest state 12that contributes to the sum in eq. (13) is the state n = iVy; i.e. the state with one nucleus and one photon, and generally any multiparticle state with the same quantum numbers as the one-nucleus state.
W. JAUS
104
In the previous section nuclear states. Therefore, the form factors we must then enable us to estimate The same considerations
we have already emphasized that we shall neglect excited to calculate the electromagnetic corrections of order a to only include the state n = NY in the sum. This result will the contribution of order c?. are true for lm rA(M2, W2, 0) and we have
Im r,(W2, M2, 0) = (2n)f(2p,))i,/d3kd3q64(p+1-q-k)
Im TA(M2, W2, 0) = (2n)f(2p,)tn/d3kd3q
a4(p+ I-q - k)
Here q is the momentum of the intermediate nucleus, and k and p the momentum and spin of the intermediate photon. For the transition matrix element for a virtual nucleus to go into a spin 0 nucleus, one-photon state with q2 = M2, k2 = 0 we have (4; kplr/Z+‘IO) = -!L-~ 1 (Z + l)e& (2743 (%o 2ko)+
+ k),h),
1 1 ‘“‘rlz14’ kPu) = (.&$3 (.&,$k,)+ Z4% + k)y 44
(15)
where E&) is the polarisation vector of the photon. Let us put
k,lI/,-IP)
= N,v+).
(161
In order to calculate In MnV and InNAYwe use the principle of minimal electromagnetic coupling to write for the divergence of the vector current ‘)
a,V;(x)= - ied,(x)V;(x),
(17)
where AA(x) is the electromagnetic vector potential. Using eq. (17) gives the following divergence conditions 1 1 __ (244 (2k,)+ e
1, M,, = __
1 441 Kelp). (24* (2k,)’
1, NAY = - -
1
(18)
Neglecting electromagnetic corrections we have
=
= 1 w3
’ ,/%~+dvG((q-~)~). c&o 2PoY
(19)
105
FERMI ,&DECAYS
We are now in a position to discuss the absorptive part of the form factor F,; using (4) (14), (15) and (18) we find 1 Im F,(W2, M2, 0) = z ~ 2x W2-M2
d3k d3q d4(p+Z-q-k)
s 2k,
2q,
x [ -CQ + kMW3(2qo ‘&d%d
(204
V;b>19
d3k d3q 4 Im F,(M2, W2, 0) = JL ___-L--8 (p+Z-q-k) 27~ W2-MM2 s 2ko 2q,
x (2q+ k>v(W3(2qo %d”(Z+l)
G(-2Z.k)
we write F,(M’,
M’2, 6M2) = F; +F:‘.
(214
The dispersion relation (8) is valid for the structure dependent part F:’ F, = G(6M2) - 1 + AF:‘, + AF:‘, .
(2lb)
The term which does not depend upon G must be treated with greater care, for the dispersion integral AF\2 is singular at threshold, i.e. has no meaning. For 6M = 0 we have F: = l+AF:,.
(214
But F: is a number independent of the mass and therefore independent of the mass splitting. The relation (21~) for F: is consequently valid also for 6M # 0. Regarding this point the dispersion relation (8) takes the form F,(M’,
M”, 6M2) = G(6M2) + AF, 1 + AF:‘, .
(22)
In calculating the absorptive part of Fy we expand the electromagnetic form factor in powers of -2 I * k. This expansion converges rapidly and we consider only the first two terms f Im Fy(W’,
M’, 0) = - -!L. z+1
= aM2G’(0)(W2-M2)2
8W4
F:‘(M2, W2, 0) 2 W2+M2 M2 -aM4G”(0)
W2-M2
I
o~~w2_M2~2~]
M2 (W2-M2)4 8W8
+ 0((W2-M2)‘)
1 .
(23)
106
W. JAUS
The corresponding contribution dF,,
to dF,,
is
= atIF’:, = - 6M2G’(0)[ 1 - ;(A’ - 1) + O((A” - I)“)]
+6M2M2G”(0)[&(AZ - l)‘+ O((A” - i)“)].
(24)
In writing down this equation use has been made of the expression (11) for 6M. Until now we have discussed the charge-dependent corrections to Fl(M2, Ml2 6M2) coming from including only the lightest state in the sum of eq. (13), namely the state with one nucleus and one photon. Of course there is also the state with one nucleus and two photons which gives rise to corrections of the same order. We shall estimate this contribution, applying the approximation scheme developed by Drell and Pagels “). For this purpose, we replace the matrix element in eq. (20) by the matrix element, which to order CIis exact at t~eshold: (2n)3(2~,2q,)~<~l~~Iq>
+ (~iq),F,(M’)f(q-p),F,(M’),
(25)
where we have put Fi(M2, M2, 0) = Fi(M2), i = 1,2. It has already been shown above that F,(M2) is independent of the nuclear charge Z to order CIand consequently only those terms proportional to F2(M2) need to be considered (for as we shall see below F2(M2) is of order Za). The corresponding contribution to the absorptive part of Fl is given by i Im @(W2, M2, 0) = Z%
Im Ffy(M2, W2, 0) = -~F~(~z)~~2-~2’2 16W4
(26)
and this result has been obtained neglecting the mass splitting, i.e. for ~7~= p2 = M2. Inserting eq. (26) into eq. (8b) and cutting off the integral at IV2 = A2M2 gives for the correction coming from the intermediate two-photon state AF;;
= -&(A2
- 1) ‘+
P2(M2).
(27)
This is the only charge-dependent contribution to AF,, . Collecting the results of eqs. (22), (24) and (27) the form factor F,(M’, M”, 6M2) including only chargedependent corrections of order Z2ct2 is now given by F,(Mz,
I%‘~, &%f2) = G(~~2)-~~~2G’(0)~l
-+(A” - l)]
+~~2~zG”(0)~(~z
- 1)‘~&(A’ - 1) ‘$ F&V’).
(28)
We have calculated the second form factor F,(M’) in the appendix via perturbation theory. The structure-dependent part which gives the leading contribution, was determined by means of a low-energy theorem. The result is
107
FERMI P-DECAYS
5. Conclusion and discussion The charge-dependent correction a,, eq. (2b), to the Fermi matrix element in the first non-vanishing order Z2cr2 is now given, using (28) and (29), by $8, = -G(GMZ)+ 1 +6M2G’(0)[1 -$(,I” - l)] -6M2M2G”(0)~(12 Expanding the electromagnetic order Z2a2 finally gives
-1)2.
(30a)
form factor G(6M2) and retaining only the term of
6, = - 6M2G’(0)$(A2 - 1) -6M2kf2G”(0)~(12
- l)‘,
(30b)
where we have written down the leading terms only. We see that the large term 6M2 G’(0) has been cancelled against the contribution from the electromagnetic form factor. Furthermore, we note that the contribution (27) from the two-photon state is one order of magnitude smaller than the leading terms of eq. (30b). To estimate 6, we use a generalized shell model for the electromagnetic form factor 11) G(k2) =
1 + &&)]
exp ($)
3
(314
where a=-,
z-2 3
which gives for 7 s Z 5 26 G’(0) = 4 r2, G”(0)
=
0.024 r4.
@lb)
To test this form factor we have calculated the electromagnetic self-masses of the nuclei under consideration in perturbation theory, using eq. (10) and replacing Ii$,(p, k) by ?$,(p, k) G2(k2). Neglecting again the effect of excited nuclear states, one finds that (31a) explains more than 90 ‘A of the mass difference between 140 and 14N but only 55 ‘Aof the mass difference between 54Co and s4Fe. This result suggests that the numerical value of G”(0) is smaller, especially for the heavy nuclei, than the shell-model value given in (31b). Using (31b) and eqs. (11) and (12) one finds that the term proportional to G”(0) in (30b) gives the leading contribution to 6, which is 140:
6, = -0.013 %,
54Co: 6, = -0.08 %,
(32a) (32b)
where the absolute value of 6, for 54Co should be considered as an upper limit.
W. JAUS
108
The result of our calculation of 6,, as expressed in (32), was obtained neglecting the influence of excited nuclear states, but we expect that these do not change the order of magnitude of 6,. The effect of excited nuclear states has been estimated in the work that has been summarized by Bhn-Stoyle in ref. “). The contribution to 6, calculated in the frame of the shell model, for example for I40 is positive and less than 0.1 76 (other nuclear models lead to higher values). Relying upon this estimate of the magnitude of the effect of excited nuclear states, we can therefore conclude that the charge-dependent corrections to the Fermi matrix element are onIy a few hundredths of a percent and entirely buried in the experimental errors. We also notice that it is the other class of charge-dependent corrections to theft values of Fermi beta decays, arising from the electromagnetic interaction of the emitted positon with the nucleus which alone seems to be important. The bulk of this effect is considered in the Fermi function. Corrections of order Za2 have been found to be quite large 12), and for heavy nuclei even contributions of order Z2a3 cannot be neglected. The author wishes to thank Prof. G. Rasche for his interest and many stimulating discussions. Appendix
We shall now discuss a perturbation theoretic approach which is equivalent to the dispersion theoretic formulation, but which allows the calculation of the second form factor F2. We shall use the technique developed for the c~culation of radiative corrections to order a [ref. ‘)I, but allowing for a small momentum transfer E = p’--p, where p’ and p are the momentum of the incoming and outgoing nucleus respectively. The correction of order a to the uncorrected matrix element (l), (plYAelp’), is given by M, =
_ ez
s
d4k/j -z(@Fj-4
MP,,A(p’,p, k, k’) = i2
s
gpv3Mp,,(p’, P, k, - k),
d4x d4y eik*“eik’“<~IT*(je;m.(X)j~m.(y)~~(0))I~‘),
(A-1) (A-2)
with 1 = p’-p-k-k’. At the end, we shall put k’ = -k. We have written above a T* product of two electromagnetic currents and the weak current VL to denote the fact that the r.h.s. of eq. (A-2) is a sum of the usual Tproduct and equal-time commutator terms. To first order in e, the following divergence condition is valid ~~~~~~(P’, P, k, k’) = eM~“(p’, P, k)+e~~~(~‘,
P, k’),
(A-3)
109
FERMI B-DECAYS
where
M,,(p’, p, k) = i d4xeik’X(plT*(j~m(x)VV-(0))lp’). s
(A.4
In addition, we shall make use of the Ward identities which express the conservation of the electromagnetic current k,, M,,&
‘,P, k) = - eM,&
ki fifPvrl(p’, P, k) =
- eM,,(p’,
t p, k’),
(A-5)
P, k).
(A4
The three divergence conditions (A.3) (A.5) and (A.6) are sufficient to determine the tensor MflyAfor small photon momentum k, k’ and small momentum transfer I, by extending the procedure first given by Low lo). In the following, we again neglect excited intermediate states. For the tensor M,,” we make the ansatz &,(P’,
P, k) = #,,(P’,
P, k)G( +21.
k’)+M;v(p’,
P, k),
(‘4.7)
with M;,(P’, P, k)(2pb 2~0)+(27+”
= (z + l)eJ2 (2~‘- k),(p + p’ - k)v + zeJZ (2~‘+ k),(p + p’ + k)y k=-2p’.
k
k2+2p
*k
-(2Z+l)eJZg,,.
(A.8)
Here Miy consists of(i) terms in which the photon is radiated from an external nucleus line and (ii) the contact term. We are interested in the low-energy part only, so we have neglected the dependence upon k2 of the electromagnetic form factor G(k’), i.e. we put G(k2) = 1, G((Z+k’)=) = G(+2 I - k’). The term M:t in eq. (A.7) describes the emission of the photon at the weak vertex and is independent of the nuclear charge Z; therefore it does not contribute to the charge-dependent corrections. Similarly we split MPyl ~$,L(P’,
P, k k’)
= @wr(p’,
P, k k’) + @fvl(p’,
P, k k’),
(A-9)
where again Mivn stands for those terms in which two photons are radiated from an external nucleus line plus corresponding contact terms. The M$ term depends upon the structure of the weak vertex and from (A.3), (A.5), (A.6) and (A.7), we find 11M:‘,,(p’, p, k, k’) = e2Z * k’G’(O)Mi,(p’, p, k) + e22 * kG’(O)M&(p’, p, k’), k,M:‘,,(p’,
P, k k’)
= e[ -22
. kG’(O)Mt,(p’,
k: MFJp’,
P, k, k’)
= e[ -2E
* k’G’(O)Mf,,(p’,
p, k’)],
p, k)],
where the expansion G(2 I * k) = 1+2 I - k G’(0) has been used.
(A.lO) (A.ll) (A.12)
W. JAUS
110
From eq. (18) and conservation of electromagnetic current, we find (I + k’)~~~=~P’, P, k) = O(ZO)Y (A.13)
kjl M,,(P’, P, k) = O@*),
i.e. the divergence of MflBis independent of the nuclear charge 2. From eqs. (A.lO)(A.13), the charge-dependent part of the amplitude i@,, can be determined up to terms linear in I, k and k’ M$(P’,
P, k k’) = e[2k;g,,-2~,g,,+(~-k’),g,,lGf(0)M:,(p’, + 42%
P, k)
ga. - 24 gna + (I - kh g~~l~(O)M~~(p’,
i- O(E * k, 1. k’,
kk’, k2, k”,
P, k’)
2’)-l- O(Z’),
(A.14)
where O(I * k, k2, 1’) denotes terms of order I - k and of second order and higher degree in k and 1. We have derived above the exact low-energy part of the amplitude MNvn(p’,p, k, k’). If we compute the integral in eq. (A.l) with this low-energy part, we get a divergent result. To be consistent with the dispersion theoretic approach, we approximate the influence of the electromagnetic form factor by cutting off the integral at k2 = Kg with K, +J(Kii-M2) = IM and A was given in eq. (12). The calculation of F2 in this frame is now straightforward and we only give the result
F,(M’) = -+(P? - l)M~~G’(O)+
0
~~ i
a
(A.15)
1
References 1) R. E. Marshak, Riazuddin and C. P. Ryan, Theory of weak interactions in particle physics (Wiley-Interscience, New York, 1969) p. 366 2) R. Blin-Stoyle, Phys. Lett. 29B (1969) 12 3) R. E. Behrends and A. Sirlin, Phys. Rev. Lett. 4 (1960) 186; M. Ademollo and R. Gatto, Phys. Lett. 13 (1964) 264 4) R. Blin-Stoyle, Isospin in nuclear B-decay, in Isospin in nuclear physics, ed. D. H. Wilkinson (North-Holland, Amsterdam, 1969) p. 115 5) A. M. Bincer, Phys. Rev. 118 (1960) 855 6) S. D. Drell and H. R. Pagels, Phys. Rev. 140 (1965) B397; H. R. Pagels, Phys. Rev. 144 (1966) 1250, 1261 7) S. Mandelstam, Phys. Rev. 115 (1959) 1741; R. Cutkosky, J. Math. Phys. 1 (1960) 429 8) R. Oehme, Phys. Rev, 100 (1955) 1503 9) M. Veltman, Phys. Rev. Lett. 17 (1966) 553 10) F. E. Low, Phys. Rev. 110 (1958) 974; S. L. Adler and Y. Dothan, Phys. Rev. 151 (1966) 1267 11) R. Hofstadter, Nuclear and nucleon structure (Benjamin, New York, 1963) 12) W. Jaus and G. Rasche, Nucl. Phys. Al43 (1970) 202
Nuclear Physics A162 (1971) 97-l Not to be reproduced
CHARGE-DEPENDENT MATRIX
by photoprint
10; @ North-Holland Publishing Co., Amsterdam or microfilm without written permission from the publisher
RADIATIVE
ELEMENT
CORRECTIONS
TO THE FERMI
FOR PURE FERMI B-DECAYS
W. JAUS Institut fiir Theoretische Physik der Universitiit Zurich, Switzerland Received 28 September 1970 A&&act: The charge-dependent radiative corrections of order Z’c? to the Fermi matrix element for pure Fermi p-decays is calculated neglecting the effect of excited nuclear states. The calculation treats the nucleus as an elementary particle and makes use partly of the technique of dispersion relations and partly of perturbation theory. The charge-dependent corrections are free from ultraviolet divergences and the electromagnetic form factor reduces the relevant dispersion and perturbation integrals to their low-energy limit. This fact enables the application of a low-energy theorem to determine the structure-dependent terms and an approximation method to estimate the two-photon contribution. The result is a correction to the Fermi matrix element of a few hundredths of a percent which is entirely negligible.
1. Introduction
Precise ft values for Fermi p-decays whose endpoint energies and half-lives are experimentally well determined, are of interest to calculate the nuclear vector p-decay coupling constant Gv, and thus test the Cabibbo theory of weak interactions. Though there still exist uncertainties in the calculation of radiative corrections of order a [ref. ‘)I to semileptonic decays (where a is the fine structure constant), theft values of Fermi decays can be compared among themselves (because the unknown modeldependent part of the radiative correction is the same for all decays) and another, more restricted aspect of the theory can be tested, namely the hypothesis of the conserved vector current (CVC). The near equality of the measuredft values in seven 0+-O+ superallowed Fermi transitions ‘) has always been regarded as a strong evidence in favor of the CVC hypothesis. Theft value for pure Fermi transitions is defined by 2n3 In 2 ft = ~ G;IWZ ’ where MF is the Fermi matrix element, which for a 0+-O+ superallowed decay, whose initial and final state are members of the same isospin T multiplet, is given by MF(T, T,&l
-+ T,
T,) = d(TfT,)(T+T,+l).
For T = 1 multiplets we find MF = J2. This result is independent of the details of the wave function of the nuclear states involved in the decay, but follows from the 97
98
W. JAUS
assumption that these states are pure isospin states. This statement is not true in reality because electromagnetism destroys the conservation of isospin, and therefore causes deviations from the exact CVC hypothesis. In the presence of the eiectromagnetin interaction the divergences of the isospin current VS* (x) are no longer zero but are proportional to terms of order e, the electromagnetic coupling constant. Furthermore the initial and final states are not eigenstates of isospin any more. The existence of a nonvanishing mass difference of nuclei within the same isospin multiple& i.e. the endpoint energy, is one obvious manifestation of this influence of the electromagnetic field. Consequently the Fermi matrix element is not strictly J2 and the deviation of A4r from 42 is usually expressed by “)
where 6, represents the effect of the charge-dependent corrections. We do not consider the radiative corrections of order ct, i.e. corrections which are independent of the nuclear charge, because these have already been discussed extensively, see for example ref. ‘). A general theorem has been proved “) which states that 6, is of second order in the difference of the mass of the decaying nucleus of charge Z-t 1 and the mass of the daughter nucleus of charge 2, i.e. of order Z2a2. This means that 6, could be of order 0.35 % for the decay of I40 and 3.9 % for 54Co, which would lead to a rather large correction to theft values. On the other side, we have the above mentioned near equality of theft values which fits the theory so nicely and which makes one expect that 6, is small. It is the purpose of this paper to calculate 6,. This has been attempted in the past within the frame of specific nuclear models by various authors (see ref. ‘) for a review), who study the wave functions of the initial and final nuclear states, where the physical state is a superposition of the ground state and excited states of different isospin. Of course, this approach does not consider radiative corrections coming from the interaction of the decaying and the daughter nucleus with the electromagnetic field without exciting the nucleus. We shall discuss just this aspect of the problem, i.e. we shall determine S, neglecting the effect of excited nuclear states. For this purpose the nucleus is treated as an elementary particle which has in our case spin 0 and positive parity, and the complex nuclear structure manifests itself only in the electromagnetic form factor. Electromagnetic corrections to the Fermi matrix element can then be described as usual as emission and absorption of virtual photons by the nucleus as a whole. In the actual calculation of 6, we partly use sidewise dispersion relations. These have been developed first by Bincer “) in connection with the electronlagnetic form factor of the nucleons, then used to determine electron and nucleon anomalous moments by Drell and Pagels “). This technique allows us, for example, to calculate quite simply in terms of the Fermi matrix element, including corrections of order ti, that contribution of order Z2a2 to a,, coming from the exchange of two virtual photons between the nuclei. The application of dispersion relations is cful only when no subtractions are required or when
FERMI /T-DECAYS
these are known. is evaluated
That part of 6, which depends
via perturbation
theory,
where
99
upon unknown
subtraction
the structure-dependent
constants
terms
of the
relevant amplitudes are determined by an exact low-energy theorem. This is possible because we are interested only in the charge-dependent corrections to Mr and these are convergent by virtue of the electromagnetic form factor, which practically reduces the amplitudes in the dispersion and perturbation integrals to their low-energy limit. As mentioned above, we did not consider the effect of excited nuclear states in this calculation, but believe that we have found at least the right order of magnitude of the charge-dependent corrections to the Fermi matrix element.
2. Method of calculation The general form of the matrix element of the charge lowering component of the isovector current VA-(O) between two nuclei of charge Z+ 1 momentum p; mass M and charge 2, momentump, mass M, both nuclei having spin 0, is determined by two form factors
= (2n)-3(2p,2p~)-iJZr,(M2,
M’2, I”) = (P+~‘)~F,(M~,
M’2, l’),
M”, Z2)+ Z,F2(M2, M”, Z’),
(la) (lb)
1 = p’-p. The factor J2 in (la) interactions, isospin F2(M2, M2, I’) = 0. factor of a nucleus of We shall investigate magnetic interactions.
is a Clebsch-Gordan coefficient. If we neglect electromagnetic is conserved and we have FI(M2, M2, I’) = G(Z2) and Here G(12), with G(0) = 1, is the usual electromagnetic form charge 2. the behaviour of P,,,(M2, Mf2, 12) in the presence of electroFor nuclei at rest we can cast (1) into a more familiar form
(M, p = OlV,-IM’, p’ = 0) = (27~)-~$(1-@~) which is up to a factor electromagnetic
equal
correction -+S,
to the Fermi
which expressed =
matrix
element.
(2a) The quantity
by the form factors
6, is the
reads
F,(M2, M”, 6M2) - 1 + $j F,(M’).
Here 6M = M’ -M is the mass difference of the two nuclei. General theorems have been proved 3), which assure that the deviation of the Fermi matrix element MF from ,/2 is of second order in the mass splitting. The mass difference can therefore be ignored in F2 and we have put F2(M2, M2, 0) E F,(M’). Following the work of Bincer “) we shall examine the analytic properties of the vertex function m(M2, M’2, I”) of eq. (1) as a function of the mass of one of the nuclei at fixed momentum transfer I2 = 6M2. This then leads to a dispersion relation for the form factor F, .
W. JAUS
100
We shall see below that second-order terms in rA lead to contributions to 6,) which are of third order in the mass splitting, and therefore we can put Z2 = 0 in r, . Using the standard reduction formalism to contract one of the nuclei in the matrix element of eq. (1) we obtain
ra(W2,M’2,
0) =
(2n)q2p,)fiSd4xB(x)eip”“
where p12 = W2, p2 = M”, Ti(M2,
1’ = 0
V;(O)]lp),
(34
and
W2, 0) = (2x)i(2po)ti~d4xO(--)e-ip”“
qz+l(x)]lO),
P-4
where now p2 = M2, p” = W2, I2 = 0 and with the source operator y’(x) = (cl + M2)4Z(x), where 4”(x) is the field of a nucleus of charge 2. In writing down eq. (3) we have omitted equal-time commutator terms which can affect only the subtractions in the final dispersion relations. Using the projection operator Pa(M2) =
with I = p’-p
we
w&
la,
(4a)
find from eq. (3) F,( W2, M’2, 0) = P,(M2)T,(W2,
M’=, O),
(4b)
F,(M’,
W=, 0).
(44
W=, 0) = I$(M’=)T,(M=,
Dispersion relations for the form factor as function of the complex variable W2 can be proved along the same lines of reasoning as those given by Bincer “) for the electromagnetic form factors. Nothing can be said generally about possible subtractions, so we used perturbation theory as a guide. In a first step we assume a once subtracted dispersion relation for F1 2
F,(M’,
M’2, 6M2) = F,(M=,
M2, 6M2)+
M’2-M n
a3
Im F,(M’,
W2,
0)
s iv’2dW2 (W2_M2)(W2-M’2)’
Similarly we have m
F,(M2,
M’=, 6M=) = F,(M”,
M’=, 6M=)+
M2-
n
M’2
Im F,( W2, M’=, 0) dW2 (W2-M2)(W2-M”) s M2
101
FERMI ~-DECAYS and the subtractions
are given by another
F,(M~,
F,(M’2,
Mu, 6~‘)
set of disperion
= A+ 1 O”dW2 Xs M2
Mt2, 6M2) = B+
1 71s
relations
Im F,(M2,
W2, 0)
W2-M2
’
m d w2 Im F,( W2, M’2, 0) W2-Mf2
M’Z
(6b)
’
where A, B are unknown subtraction constants. The function Fl( W2, W2, 6M2) is equal to the form factor Fl for both external particles having the same mass W, while the intermediate states which saturate the dispersion integrals in eq. (6) have physical masses M and M’ (we do not consider excited nuclear states). The two subtractions (6a) and (6b) transform into each other by simply interchanging the masses M and M’. Consequently the sum F,(M2, M2, &IL?~)-~F,(M’~, hT2, 6M2) is free of terms linear in the mass splitting and this leads at once to two important results: (i) The two dispersion integrals entering this sum can be evaluated for all nuclei (external and intermediate) having the same mass. (ii) The sum of the two subtractions +(A+B)
= F,(a)
depends only quadratically upon the mass splitting. Therefore Fl (a) does not contain charge-dependent corrections (which are of order Za 6M) and is given by the uncorrected form factor F&o)
Taking
= G(6M2).
(7)
the sum of eqs. (5a) and (5b) we have F,(M’, AF,,
AF,,
M”,
6M2)
= F,(co)+AF,,
* dW2 Im F,( W2, M2, O)+Im = 1 2TT W2-M2 s M2 = 2M6M mdW2 271 s M2
Im F,(M’,
+AF,,, F,(M’,
W2, 0)-Im (W2-M2)2
F,(W’,
(84 W2, 0)
,
(t(b)
M2, 0) ’
(8C)
In writing the expression for AF,, use has been made of the fact that the dispersion integral is multiplied by 6M and therefore the mass splitting can be neglected in the integrand. Eq. (8) for F,(M2, Mf2, 6M2) is the basic relation for the following discussion. It has the great advantage that the absorptive part of F, can be evaluated for equal masses. The correction due to the non-zero mass difference is given by AF,, . Furthermore we assume low-energy dominance, namely that it is the threshold contribution to Im Fl that gives the dominant part of the electromagnetic correction. In the work of Drell and Pagels “) where the anomalous magnetic moment of the electron is computed, the above assumption is made ad hoc and then justified by the success of the calculation. We are in a slightly better position because the perturbation theory for the charge-dependent correction is finite due to the electromagnetic
W. JAUS
102
form factor, and it is this effect which we approximate by a suitable cut-off in the dispersion integrals. Let us therefore keep only the contributions between M2 5 W2 s A2M2, M”
5 W2 5 it2M,
in eqs. (9) and (10) with 1’ - 1 << 1. It is not possible to calculate F2(M2) in dispersion theory and we shall present alternative perturbation theoretic approach in the appendix. 3. Determination
an
of the cut-off 12’
The two nuclear states of charge 2 and Z+ 1 with which the matrix element of eq. (1) was formed, belong to an approximate isotriplet. It is the electromagnetic interaction that breaks isospin conservation and causes a mass shift between the two states. We shall now calculate this mass shift using the method discussed in the previous section and by comparison with the experimental value for the mass difference determine the value of A’. The electromagnetic self-mass
of a nucleus
of charge Z and spin 0 is given by
c(I-“>= - i ($/d4, For the Compton
amplitude
T&J,
k).
(104
we use the Born approximation
TB@ flv
$
kj ’
=
(2p-k),(‘b-kj, (P_kjz_Mz
-gcv’
PW
In writing down eq. (10) we have neglected the contribution of all excited states. The self-mass c( W”) as a function of the mass p2 = W2 of the nucleus is analytic in the cut Wz plane with a branch cut from M2 to + co. Using the hypothesis of low-energy dominance we have the dispersion relation
We derive the above expression from perturbation theory and by comparison one finds that one subtraction is needed. We shall show below that c(cc) is small and does not contribute. The absorptive part of I( W2) is obtained by the replacement of the propagators of the internal lines according to ‘) (k2+i.s)-1(q2-M2+iE)-1 The absorptive
amplitude ImC(W’)
+ 2~26(q2-M2)B(q,)6(k2)~(k,)).
is then given by = Z2f-12
zd’(p-q-k)(2q+k)‘. 0
103
In the c.m. system with p = (W, 0) we have Im~(W2)=Z2~(W2-MZ)~$, 7c2 C(W”)
Perturbation
= C(co)+Z”
~2M2~(A2-l+ln12). x
theory gives for the subtraction
i.e. the subtraction
constant
is two orders of magnitude
from the dispersion integral. For the mass difference 6M of a nucleus 6;
constant
= (22+
smaller than the contribution
of charge Z-t- 1 and one of charge Z one has
(11)
1) 3 {&(A”- 1) + O[(A’ - l)“]}.
Comparison with the experimental gives the result
end-point
140.
AZ-1
energies for the decays of 14O and 5 4Co
-
3.2
Mr ’ 54co:
p-1
=:
(12)
!I!!
MT-’ with the rms radius Y = 1.03 A?‘ fm. We note that (MY)-~ which we are interested is of order 10e3.
4. The charge-dependent
for the range of nuclei in
corrections to the Fermi matrix element
We turn now to the absorptive parts of the form factor near threshold, follow from eq. (3) by replacing S(X) and i0( -x) by + [ref. ‘)I Im Ti(W2,
which
M’2, 0) = (27~)~(2p~)%c
x ~~,c
~,-(o)lPP4(P’-
4-
WI ~,-(o)ln’>
a* (13)
The sum over states ~1’cannot contribute since Z2 = 0. The only admissible state II’ is the vacuum and (Ol~z(0)lp) = 0. The lightest state 12that contributes to the sum in eq. (13) is the state n = iVy; i.e. the state with one nucleus and one photon, and generally any multiparticle state with the same quantum numbers as the one-nucleus state.
W. JAUS
104
In the previous section nuclear states. Therefore, the form factors we must then enable us to estimate The same considerations
we have already emphasized that we shall neglect excited to calculate the electromagnetic corrections of order a to only include the state n = NY in the sum. This result will the contribution of order c?. are true for lm rA(M2, W2, 0) and we have
Im r,(W2, M2, 0) = (2n)f(2p,))i,/d3kd3q64(p+1-q-k)
Im TA(M2, W2, 0) = (2n)f(2p,)tn/d3kd3q
a4(p+ I-q - k)
Here q is the momentum of the intermediate nucleus, and k and p the momentum and spin of the intermediate photon. For the transition matrix element for a virtual nucleus to go into a spin 0 nucleus, one-photon state with q2 = M2, k2 = 0 we have (4; kplr/Z+‘IO) = -!L-~ 1 (Z + l)e& (2743 (%o 2ko)+
+ k),h),
1 1 ‘“‘rlz14’ kPu) = (.&$3 (.&,$k,)+ Z4% + k)y 44
(15)
where E&) is the polarisation vector of the photon. Let us put
k,lI/,-IP)
= N,v+).
(161
In order to calculate In MnV and InNAYwe use the principle of minimal electromagnetic coupling to write for the divergence of the vector current ‘)
a,V;(x)= - ied,(x)V;(x),
(17)
where AA(x) is the electromagnetic vector potential. Using eq. (17) gives the following divergence conditions 1 1 __ (244 (2k,)+ e
1, M,, = __
1 441 Kelp). (24* (2k,)’
1, NAY = - -
1
(18)
Neglecting electromagnetic corrections we have
=
= 1 w3
’ ,/%~+dvG((q-~)~). c&o 2PoY
(19)
105
FERMI ,&DECAYS
We are now in a position to discuss the absorptive part of the form factor F,; using (4) (14), (15) and (18) we find 1 Im F,(W2, M2, 0) = z ~ 2x W2-M2
d3k d3q d4(p+Z-q-k)
s 2k,
2q,
x [ -CQ + kMW3(2qo ‘&d%d
(204
V;b>19
d3k d3q 4 Im F,(M2, W2, 0) = JL ___-L--8 (p+Z-q-k) 27~ W2-MM2 s 2ko 2q,
x (2q+ k>v(W3(2qo %d”(Z+l)
G(-2Z.k)
we write F,(M’,
M’2, 6M2) = F; +F:‘.
(214
The dispersion relation (8) is valid for the structure dependent part F:’ F, = G(6M2) - 1 + AF:‘, + AF:‘, .
(2lb)
The term which does not depend upon G must be treated with greater care, for the dispersion integral AF\2 is singular at threshold, i.e. has no meaning. For 6M = 0 we have F: = l+AF:,.
(214
But F: is a number independent of the mass and therefore independent of the mass splitting. The relation (21~) for F: is consequently valid also for 6M # 0. Regarding this point the dispersion relation (8) takes the form F,(M’,
M”, 6M2) = G(6M2) + AF, 1 + AF:‘, .
(22)
In calculating the absorptive part of Fy we expand the electromagnetic form factor in powers of -2 I * k. This expansion converges rapidly and we consider only the first two terms f Im Fy(W’,
M’, 0) = - -!L. z+1
= aM2G’(0)(W2-M2)2
8W4
F:‘(M2, W2, 0) 2 W2+M2 M2 -aM4G”(0)
W2-M2
I
o~~w2_M2~2~]
M2 (W2-M2)4 8W8
+ 0((W2-M2)‘)
1 .
(23)
106
W. JAUS
The corresponding contribution dF,,
to dF,,
is
= atIF’:, = - 6M2G’(0)[ 1 - ;(A’ - 1) + O((A” - I)“)]
+6M2M2G”(0)[&(AZ - l)‘+ O((A” - i)“)].
(24)
In writing down this equation use has been made of the expression (11) for 6M. Until now we have discussed the charge-dependent corrections to Fl(M2, Ml2 6M2) coming from including only the lightest state in the sum of eq. (13), namely the state with one nucleus and one photon. Of course there is also the state with one nucleus and two photons which gives rise to corrections of the same order. We shall estimate this contribution, applying the approximation scheme developed by Drell and Pagels “). For this purpose, we replace the matrix element in eq. (20) by the matrix element, which to order CIis exact at t~eshold: (2n)3(2~,2q,)~<~l~~Iq>
+ (~iq),F,(M’)f(q-p),F,(M’),
(25)
where we have put Fi(M2, M2, 0) = Fi(M2), i = 1,2. It has already been shown above that F,(M2) is independent of the nuclear charge Z to order CIand consequently only those terms proportional to F2(M2) need to be considered (for as we shall see below F2(M2) is of order Za). The corresponding contribution to the absorptive part of Fl is given by i Im @(W2, M2, 0) = Z%
Im Ffy(M2, W2, 0) = -~F~(~z)~~2-~2’2 16W4
(26)
and this result has been obtained neglecting the mass splitting, i.e. for ~7~= p2 = M2. Inserting eq. (26) into eq. (8b) and cutting off the integral at IV2 = A2M2 gives for the correction coming from the intermediate two-photon state AF;;
= -&(A2
- 1) ‘+
P2(M2).
(27)
This is the only charge-dependent contribution to AF,, . Collecting the results of eqs. (22), (24) and (27) the form factor F,(M’, M”, 6M2) including only chargedependent corrections of order Z2ct2 is now given by F,(Mz,
I%‘~, &%f2) = G(~~2)-~~~2G’(0)~l
-+(A” - l)]
+~~2~zG”(0)~(~z
- 1)‘~&(A’ - 1) ‘$ F&V’).
(28)
We have calculated the second form factor F,(M’) in the appendix via perturbation theory. The structure-dependent part which gives the leading contribution, was determined by means of a low-energy theorem. The result is
107
FERMI P-DECAYS
5. Conclusion and discussion The charge-dependent correction a,, eq. (2b), to the Fermi matrix element in the first non-vanishing order Z2cr2 is now given, using (28) and (29), by $8, = -G(GMZ)+ 1 +6M2G’(0)[1 -$(,I” - l)] -6M2M2G”(0)~(12 Expanding the electromagnetic order Z2a2 finally gives
-1)2.
(30a)
form factor G(6M2) and retaining only the term of
6, = - 6M2G’(0)$(A2 - 1) -6M2kf2G”(0)~(12
- l)‘,
(30b)
where we have written down the leading terms only. We see that the large term 6M2 G’(0) has been cancelled against the contribution from the electromagnetic form factor. Furthermore, we note that the contribution (27) from the two-photon state is one order of magnitude smaller than the leading terms of eq. (30b). To estimate 6, we use a generalized shell model for the electromagnetic form factor 11) G(k2) =
1 + &&)]
exp ($)
3
(314
where a=-,
z-2 3
which gives for 7 s Z 5 26 G’(0) = 4 r2, G”(0)
=
0.024 r4.
@lb)
To test this form factor we have calculated the electromagnetic self-masses of the nuclei under consideration in perturbation theory, using eq. (10) and replacing Ii$,(p, k) by ?$,(p, k) G2(k2). Neglecting again the effect of excited nuclear states, one finds that (31a) explains more than 90 ‘A of the mass difference between 140 and 14N but only 55 ‘Aof the mass difference between 54Co and s4Fe. This result suggests that the numerical value of G”(0) is smaller, especially for the heavy nuclei, than the shell-model value given in (31b). Using (31b) and eqs. (11) and (12) one finds that the term proportional to G”(0) in (30b) gives the leading contribution to 6, which is 140:
6, = -0.013 %,
54Co: 6, = -0.08 %,
(32a) (32b)
where the absolute value of 6, for 54Co should be considered as an upper limit.
W. JAUS
108
The result of our calculation of 6,, as expressed in (32), was obtained neglecting the influence of excited nuclear states, but we expect that these do not change the order of magnitude of 6,. The effect of excited nuclear states has been estimated in the work that has been summarized by Bhn-Stoyle in ref. “). The contribution to 6, calculated in the frame of the shell model, for example for I40 is positive and less than 0.1 76 (other nuclear models lead to higher values). Relying upon this estimate of the magnitude of the effect of excited nuclear states, we can therefore conclude that the charge-dependent corrections to the Fermi matrix element are onIy a few hundredths of a percent and entirely buried in the experimental errors. We also notice that it is the other class of charge-dependent corrections to theft values of Fermi beta decays, arising from the electromagnetic interaction of the emitted positon with the nucleus which alone seems to be important. The bulk of this effect is considered in the Fermi function. Corrections of order Za2 have been found to be quite large 12), and for heavy nuclei even contributions of order Z2a3 cannot be neglected. The author wishes to thank Prof. G. Rasche for his interest and many stimulating discussions. Appendix
We shall now discuss a perturbation theoretic approach which is equivalent to the dispersion theoretic formulation, but which allows the calculation of the second form factor F2. We shall use the technique developed for the c~culation of radiative corrections to order a [ref. ‘)I, but allowing for a small momentum transfer E = p’--p, where p’ and p are the momentum of the incoming and outgoing nucleus respectively. The correction of order a to the uncorrected matrix element (l), (plYAelp’), is given by M, =
_ ez
s
d4k/j -z(@Fj-4
MP,,A(p’,p, k, k’) = i2
s
gpv3Mp,,(p’, P, k, - k),
d4x d4y eik*“eik’“<~IT*(je;m.(X)j~m.(y)~~(0))I~‘),
(A-1) (A-2)
with 1 = p’-p-k-k’. At the end, we shall put k’ = -k. We have written above a T* product of two electromagnetic currents and the weak current VL to denote the fact that the r.h.s. of eq. (A-2) is a sum of the usual Tproduct and equal-time commutator terms. To first order in e, the following divergence condition is valid ~~~~~~(P’, P, k, k’) = eM~“(p’, P, k)+e~~~(~‘,
P, k’),
(A-3)
109
FERMI B-DECAYS
where
M,,(p’, p, k) = i d4xeik’X(plT*(j~m(x)VV-(0))lp’). s
(A.4
In addition, we shall make use of the Ward identities which express the conservation of the electromagnetic current k,, M,,&
‘,P, k) = - eM,&
ki fifPvrl(p’, P, k) =
- eM,,(p’,
t p, k’),
(A-5)
P, k).
(A4
The three divergence conditions (A.3) (A.5) and (A.6) are sufficient to determine the tensor MflyAfor small photon momentum k, k’ and small momentum transfer I, by extending the procedure first given by Low lo). In the following, we again neglect excited intermediate states. For the tensor M,,” we make the ansatz &,(P’,
P, k) = #,,(P’,
P, k)G( +21.
k’)+M;v(p’,
P, k),
(‘4.7)
with M;,(P’, P, k)(2pb 2~0)+(27+”
= (z + l)eJ2 (2~‘- k),(p + p’ - k)v + zeJZ (2~‘+ k),(p + p’ + k)y k=-2p’.
k
k2+2p
*k
-(2Z+l)eJZg,,.
(A.8)
Here Miy consists of(i) terms in which the photon is radiated from an external nucleus line and (ii) the contact term. We are interested in the low-energy part only, so we have neglected the dependence upon k2 of the electromagnetic form factor G(k’), i.e. we put G(k2) = 1, G((Z+k’)=) = G(+2 I - k’). The term M:t in eq. (A.7) describes the emission of the photon at the weak vertex and is independent of the nuclear charge Z; therefore it does not contribute to the charge-dependent corrections. Similarly we split MPyl ~$,L(P’,
P, k k’)
= @wr(p’,
P, k k’) + @fvl(p’,
P, k k’),
(A-9)
where again Mivn stands for those terms in which two photons are radiated from an external nucleus line plus corresponding contact terms. The M$ term depends upon the structure of the weak vertex and from (A.3), (A.5), (A.6) and (A.7), we find 11M:‘,,(p’, p, k, k’) = e2Z * k’G’(O)Mi,(p’, p, k) + e22 * kG’(O)M&(p’, p, k’), k,M:‘,,(p’,
P, k k’)
= e[ -22
. kG’(O)Mt,(p’,
k: MFJp’,
P, k, k’)
= e[ -2E
* k’G’(O)Mf,,(p’,
p, k’)],
p, k)],
where the expansion G(2 I * k) = 1+2 I - k G’(0) has been used.
(A.lO) (A.ll) (A.12)
W. JAUS
110
From eq. (18) and conservation of electromagnetic current, we find (I + k’)~~~=~P’, P, k) = O(ZO)Y (A.13)
kjl M,,(P’, P, k) = O@*),
i.e. the divergence of MflBis independent of the nuclear charge 2. From eqs. (A.lO)(A.13), the charge-dependent part of the amplitude i@,, can be determined up to terms linear in I, k and k’ M$(P’,
P, k k’) = e[2k;g,,-2~,g,,+(~-k’),g,,lGf(0)M:,(p’, + 42%
P, k)
ga. - 24 gna + (I - kh g~~l~(O)M~~(p’,
i- O(E * k, 1. k’,
kk’, k2, k”,
P, k’)
2’)-l- O(Z’),
(A.14)
where O(I * k, k2, 1’) denotes terms of order I - k and of second order and higher degree in k and 1. We have derived above the exact low-energy part of the amplitude MNvn(p’,p, k, k’). If we compute the integral in eq. (A.l) with this low-energy part, we get a divergent result. To be consistent with the dispersion theoretic approach, we approximate the influence of the electromagnetic form factor by cutting off the integral at k2 = Kg with K, +J(Kii-M2) = IM and A was given in eq. (12). The calculation of F2 in this frame is now straightforward and we only give the result
F,(M’) = -+(P? - l)M~~G’(O)+
0
~~ i
a
(A.15)
1
References 1) R. E. Marshak, Riazuddin and C. P. Ryan, Theory of weak interactions in particle physics (Wiley-Interscience, New York, 1969) p. 366 2) R. Blin-Stoyle, Phys. Lett. 29B (1969) 12 3) R. E. Behrends and A. Sirlin, Phys. Rev. Lett. 4 (1960) 186; M. Ademollo and R. Gatto, Phys. Lett. 13 (1964) 264 4) R. Blin-Stoyle, Isospin in nuclear B-decay, in Isospin in nuclear physics, ed. D. H. Wilkinson (North-Holland, Amsterdam, 1969) p. 115 5) A. M. Bincer, Phys. Rev. 118 (1960) 855 6) S. D. Drell and H. R. Pagels, Phys. Rev. 140 (1965) B397; H. R. Pagels, Phys. Rev. 144 (1966) 1250, 1261 7) S. Mandelstam, Phys. Rev. 115 (1959) 1741; R. Cutkosky, J. Math. Phys. 1 (1960) 429 8) R. Oehme, Phys. Rev, 100 (1955) 1503 9) M. Veltman, Phys. Rev. Lett. 17 (1966) 553 10) F. E. Low, Phys. Rev. 110 (1958) 974; S. L. Adler and Y. Dothan, Phys. Rev. 151 (1966) 1267 11) R. Hofstadter, Nuclear and nucleon structure (Benjamin, New York, 1963) 12) W. Jaus and G. Rasche, Nucl. Phys. Al43 (1970) 202