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Parity nonconservation in atoms and elastic electron scattering caused by nuclear parity violation
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
PARITY NONCONSERVATION IN ATOMS AND ELASTIC ELECTRON SCATTERING CAUSED BY NUCLEAR PARITY VIOLATION E.M. HENLEY and W.-Y.P. HWANG University of Washington, Seattle, WA 98195, USA
and G.N. EPSTEIN Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 9 August 1979
We demonstrate, on the basis of gauge invadance, hermiticity, and time-reversal invariance, that there can be no long range transmission of nuclear parity violation to electrons in an atom or in elastic electron scattering. It follows that the ratio of the nuclear induced parity violation in atoms or in elastic electron scattering to that due to weak neutral currents (Z-boson exchange) remains of the order of the f'me structure constant, even in the limit of zero momentum transfer.
During recent years, studies of parity nonconservation (PNC) in atoms have been carried out in attempts to elucidate the structure of the weak neutral currents [1,2]. In this connection, it is important to ascertain that the atomic PNC effects cannot be induced by nuclear parity violation; such parity violation is caused by charged as well as by neutral weak currents. Indeed, from time to time arguments are presented that nuclear effects may even be more important than effects due to Z-boson exchange. It is the main purpose of this comment to demonstrate, on the basis of gauge invariance, hermiticity, and time-reversal invariance, that, both for elastic electron scattering from nuclei and for atoms, the ratio of the PNC asymmetry due to parity mixing in the nuclear ground state to that due to Z-boson exchange is roughly of order ~, the fine structure constant. This conclusion follows from the fact that there can be no long range (e.g., 1/r or 1/r 2) transmission of the nuclear PNC to the electron. Our comment should not be taken to imply that nuclear parity violating effects, whatever their origin, can be neglected in electron scattering experiments. Indeed, in low energy inelastic electron scattering, the asymmetry due to nuclear PNC must be considered and may even be larger than the asymmetry caused by Z-boson exchange [3]. Nor should nuclear effects be neglected in the study of PNC in internal conversion [4]. The possible importance of nuclear PNC effects in atomic or electron scattering studies can be noted from a comparison of figs. 1a and 1b. The contribution of fig. 1a is roughly (we neglect atomic coherence or consider a light atom) (I)
cFIZ ~ 4 n a / 8 M 2 -~ G/47r ,
(a)
(b)
Fig. 1. Diagrams which interfere with the usual one-photon exchange and give rise to parity-violating signals. (a) represents Z-boson exchange (neutral weak current) while (b) describes abnormal-parity mixing, indicated by x, in the initial and final nuclear wavefunctions. 349
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
where M z is the mass of the Z boson and G is the weak coupling constant. On the other hand, the contribution of figs. 1b could be
C~N ~ (a/q2) ~r~ (a/q2) GM2 /4zr,
(2)
where qh is the four-momentum transfer (q0 = 0 in the CM frame), and we assume that the parity mixing in the nuclear ground state is roughly of order 5r-~ GM2/47r ~ 10 - 6 , where M is the nucleon mass. The ratio of the two matrix elements is thus of order
R
= C~Z/C~ N ~
c~M2/q 2 ,
(3)
and diverges as q2 approaches zero. In atomic studies, or in forward elastic electron scattering, the ratio R is much larger than unity. The divergence and large ratio occur because it is assumed in eq. (2) that a long range force (the photon propagator is proportional to Iq1-2 and corresponds to a 1/r Coulomb potential) transmits the nuclear PNC to the electron. We shall show that gauge invariance of the electromagnetic current, hermiticity, and timereversal invariance require the matrix c ~ N to be multiplied by a factor which is roughly q2/M2, so that the long range potential proportional to 1/q2 is transformed to a constant which corresponds to a contact potential in configuration space. Our comment could be stated as a theorem which is that contributions to ~r from figs. 2a and 2b, which correspond to the right hand parts of figs. lb, as well as that from fig. 2c, must approach zero proportionately to q2 as q2 _+ 0. Not only can there be no constant term, but there can also be no linear term in I q [ in an expansion of ~r about q = 0. In order to prove our statement, we combine figs. 2 a - 2 c into a single figure, which can be represented effectively by fig. 2c with a form factor that includes the parity admixtures of figs. 2 a and 2b. To see that this can be carried out formally, we note that, for a nucleus of four-momentum p, angular momentum j, and magnetic quantum number m, the ground state in the presence of weak forces can be written as
[ p , j , m ) = l p , j , m , + )+ i e l p , / , m , - ) ,
(4)
where e ~ 10 . 6 - 1 0 . 8 characterizes the abnormal parity admixture (which is real for bound states) and we have assumed that the normal parity of the state of our concern is positive. We then recall that the matrix element for the study of PNC in atoms or in elastic electron scattering can be written as
cl?~= 47ra q2 _ 2X,/~G
f d3r~O(r)
~,x ~(O(r)e_iq.r(pf, j ' m,iJx(O)lPi,j, m)
(5)
fd3r~(ef)(r) 7X(v e +acTs) ~(i)(r)e_iq.r(pf,
j ' m,lNx(O)lPi,f, m)
where re, a e are constants and N x (x) is the hadronic neutral weak current. In view of eq. (4), we can write the matrix element of the hadronic electromagnetic current in eq. (5) as
(o)
350
(b)
(c)
Fig. 2. Contributions to the abnormal-parity electromagnetic vertex (pf, j, m'lJh (0)I Pi, J, m)(-). (a), (b), and (c) descr~e, respectively, abnormal-parity mixing in the initial and final wavefunctions and the abnormal-parity vertex renormalizatlon.
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
(pf, j, m' IJx(0 ) IPi,/, m) =-(pf, j, m' I [J(+)(0) + J(-) (0)] IPi,1, m) (pf, f, m',+l J(+) (O)[Pi,], m, +) + (ie(pf,/, m', + ]J(+) (O)lPi,f , m , - )
(6)
- i e(pf,/, m', - I J(+)(0)]pi, f, m, +) + (pf,/', m', + IJ(-)(O)lPi,f, m, +)) --(pf, f, m' JJx(0)1Pi,l, m)(+) + (Pf,/, m' IJx (0)JPi,], m)(-), where the three terms in (pf, f, m' IJx(0)I pi,/, m)(-) correspond, respectively, to the contributions to ~r from figs. 2 a - 2c, the last one describing exclusively the abnormal-parity renormalization of the electromagnetic vertex by the presence of weak forces. Our theorem refers to the sum of these three terms and requires that
(pf, j, m'lJx(O) lPi,/, m)(-) (x q2.
(7)
We proceed to prove eq. (7) on the basis of gauge invariance (GI) and the combined operation of hermiticity and time-reversal invariance (HT). GI requires
qk(pf, f, m' IJx(O) lPi,], m) (-) = 0,
(8)
while HT yields [5]
( - ) m + m ' ( E i , - p i ; / , - m [ TJx(O) T -1 [Ef,-pf;j,-m')(-)=(pf;/,m'lJx(O)lPi;/,m)(-),
(9)
with
TJ(O) T -1 =-J(O),
TJo(O) T -1 = +J0(0).
(10)
Rather than considering the general case from the very beginning, we first examine the special case of spin 1 (e.g., 2H). The spin 1/2 case has been treated elsewhere [6]. For such a nucleus, we introduce the polarization four-vector ~x and write ,1 (pf, ~(f) Ijx(0) i pi, ~ (i))(-)
~)* ~---/PK g'~
^
qx
Gv(qZ)- ~
GR(q2)) + ePr/r~"
~(f'*
~(~'~.,..,,, 2M eg (-~1 eh (q2)) ,,..,,,, Gs(q2) - ~ GT
!
(1 la)
+ expn• ~(f). 2M qn 2M P~ ~(i). q 2M GK(q2) + expnK ~0) 2M qn 2M P~ ~(f)*.q 2M GL(q2)' with Px - (Pi +Pf)x and where expn~ is the fully antisymmetric Levi-Civita tensor. The GI constraint, i.e. eq. (8), yields Gv(q2 ) + (q2/4M 2) Gs(q2 ) = 0.
(1 lb)
HT requires that eq. (I 2a) be invariant under the following substitutions:
Qo -+ Qo ,
Q -->- Q;
qo -->- qo,
_, _
q -+ q;
(11c)
¢o* _,_ ¢i)
More explicitly, we obtain from HT GR(q 2) = 0,
GT(q 2) = 0,
GK(q 2) = -- GL(q2).
(1 l d)
~:1 This expression is used in a more complete paper in preparation. 351
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
Since all the form factors GV,S,K,L (q2) must be regular at q2 = 0 (if no zero-mass strongly interacting bosons exist), we expect, from eq. (1 lb), GV (q2) ~ (t q l/2M)2 •
(1 l e)
Hence, eq. (7) follows. For a nucleus of arbitrary spin/, we can use Holstein's parametrization [7] :
rc '.'~ r''2"~P~ (Pf,/,m'lJx(O)lPi,J,m)(-)=(/m;l kl/m')ei/k {ei/xn~. .' 4M-d(q 2) q-~-~] qx q~ h ( q 2 ) / + other form factors at least quadratic in qx, + e6~n 2M 43,/ 2M
(12a)
where all form factors are regular at q2 = 0 [7]. Now, the GI constraint, i.e. eq. (8), yields c (q2) + (q2/4M2) h (q2) = 0,
(12b)
while HT implies d(q 2) = O.
(12c)
Eq. (7) follows eqs. (13a)-(13c) since h (q2) is regular at q2 = 0. Our conclusion is not restricted to one-photon exchanges, but can be extended to two or more photons; see figs• 3. By a reduction akin to that which leads to the last equality of eq. (6), we can represent two-photon PNC effects by figs• 3d and 3e. In the evaluation of two-photon exchanges, one must take care not to include Coulomb effects which are already included in the bound state (or Coulomb-distorted) wavefunctions. The proof of our conclusion for two- or more-photon exchanges relies on a series of papers by Feinberg and Sucher [8]. They use dispersion relations to show that the longest-range part of the two-photon-exchange potential is related to on-mass-shell propagation (i.e. the pole of the propagator) and to the Compton scattering matrix element• In our case, this is a PNC Compton scattering, which has been treated by Kim and Dass [9]. They show, on general grounds, that this PNC Compton amplitude goes like q2 for small momentum transfers• Together with the Feinberg-Sucher work, it follows readily that there is no long-range force which transmits the nuclear PNC to the electron; that is, there is no 1/1 q l 2 or 1/I q I effect (I q I is the net three-momentum transfer). In many nuclear physics applications, the impulse approximation (IA) is used• This approximation preserves HT but often violates GI [7] ,1. For instance, if we use the usual IA to calculate the various deuteron form factors as defined by eq. (12a), we find that the form factor Gv (q2) behaves like a constant rather than Iq I2/M2. However, all form factors calculated in the same fashion are regular at q2 = 0. Such violation of GI (eq. (1 lb)) by the
(o)
(b)
(a)
352
(c)
(e)
Fig. 3. Two-photon exchange diagrams with abnormal-parity mixing in the initial, intermediate, and final nuclear wavefunctions ((a), (b), and (c)) and with abnormal-parity electromagnetic vertices ((d) and (e)).
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
usual IA thus can give rise to the incorrect result that the ratio R of the PNC effect due to parity mixing in the deuteron wavefunction to that due to Z-boson exchange is proportional to I q 1-2 as I q I -~ 0. As we have just shown, this conclusion would be incorrect. This illustrates the importance of including meson exchange effects in order to preserve gauge invariance and so avoid spurious results. In summary, we are led to conclude that, both for atomic physics tests and for elastic electron scattering from nuclei, PNC effects due to nuclear parity violation will be of the order of radiative corrections (fine structure constant) as compared to those due to Z-boson exchange, even if the contributions due to multiple photon exchanges are included. Of course, exceptions may occur if the parity mixing in the nuclear ground state is enhanced and the atomic (electron scattering) effects due to Z-boson exchange are inhibited. This work is supported in part by the U.S. Department of Energy.
References [1] For a review in theory, see, e.g., L. Wilets, in: Neutrinos-78, ed. E.C. Fowler (Purdue Univ., 1978) p. 437. [2] For a review in experiment, see, e.g., N. Fortson, in: Neutrinos-78, ed. E.C. Fowler (Purdue Univ., 1978) p. 417; for a recent Berkeley experiment, see R. Conti et al., Phys. Rev. Lett. 42 (1979) 343. [3] W.-Y.P. Hwang, Phys. Rev. C20 (1979) 331. [4] A. Barroso and D. Tadid, Phys. Rev. C17 (1978) 832; S.E. Koonin, Phys. Rev. C19 (1979) 2314. [5] See, e.g., H. Frauenfelder and E.M. Henley, Nuclear and particle physics (Benjamin, Reading, MA, 1975). [6] E.M. Henley, A.H. Huffman and D.U.L. Yu, Phys. Rev. D7 (1973) 943. [7] B.R. Holstein, Rev. Mod. Phys. 46 (1974) 789. [8] J. Sucher, in: Carg~se lectures in physics, ed. M. Levy (Gordon and Breach Science, New York, London, Paris, 1977) Vol. 7, p. 43. [9] K.J. Kim and N.D.H. Dass, Nucl. Phys. Bl13 (1976) 336.
353
PHYSICS LETTERS
17 December 1979
PARITY NONCONSERVATION IN ATOMS AND ELASTIC ELECTRON SCATTERING CAUSED BY NUCLEAR PARITY VIOLATION E.M. HENLEY and W.-Y.P. HWANG University of Washington, Seattle, WA 98195, USA
and G.N. EPSTEIN Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 9 August 1979
We demonstrate, on the basis of gauge invadance, hermiticity, and time-reversal invariance, that there can be no long range transmission of nuclear parity violation to electrons in an atom or in elastic electron scattering. It follows that the ratio of the nuclear induced parity violation in atoms or in elastic electron scattering to that due to weak neutral currents (Z-boson exchange) remains of the order of the f'me structure constant, even in the limit of zero momentum transfer.
During recent years, studies of parity nonconservation (PNC) in atoms have been carried out in attempts to elucidate the structure of the weak neutral currents [1,2]. In this connection, it is important to ascertain that the atomic PNC effects cannot be induced by nuclear parity violation; such parity violation is caused by charged as well as by neutral weak currents. Indeed, from time to time arguments are presented that nuclear effects may even be more important than effects due to Z-boson exchange. It is the main purpose of this comment to demonstrate, on the basis of gauge invariance, hermiticity, and time-reversal invariance, that, both for elastic electron scattering from nuclei and for atoms, the ratio of the PNC asymmetry due to parity mixing in the nuclear ground state to that due to Z-boson exchange is roughly of order ~, the fine structure constant. This conclusion follows from the fact that there can be no long range (e.g., 1/r or 1/r 2) transmission of the nuclear PNC to the electron. Our comment should not be taken to imply that nuclear parity violating effects, whatever their origin, can be neglected in electron scattering experiments. Indeed, in low energy inelastic electron scattering, the asymmetry due to nuclear PNC must be considered and may even be larger than the asymmetry caused by Z-boson exchange [3]. Nor should nuclear effects be neglected in the study of PNC in internal conversion [4]. The possible importance of nuclear PNC effects in atomic or electron scattering studies can be noted from a comparison of figs. 1a and 1b. The contribution of fig. 1a is roughly (we neglect atomic coherence or consider a light atom) (I)
cFIZ ~ 4 n a / 8 M 2 -~ G/47r ,
(a)
(b)
Fig. 1. Diagrams which interfere with the usual one-photon exchange and give rise to parity-violating signals. (a) represents Z-boson exchange (neutral weak current) while (b) describes abnormal-parity mixing, indicated by x, in the initial and final nuclear wavefunctions. 349
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
where M z is the mass of the Z boson and G is the weak coupling constant. On the other hand, the contribution of figs. 1b could be
C~N ~ (a/q2) ~r~ (a/q2) GM2 /4zr,
(2)
where qh is the four-momentum transfer (q0 = 0 in the CM frame), and we assume that the parity mixing in the nuclear ground state is roughly of order 5r-~ GM2/47r ~ 10 - 6 , where M is the nucleon mass. The ratio of the two matrix elements is thus of order
R
= C~Z/C~ N ~
c~M2/q 2 ,
(3)
and diverges as q2 approaches zero. In atomic studies, or in forward elastic electron scattering, the ratio R is much larger than unity. The divergence and large ratio occur because it is assumed in eq. (2) that a long range force (the photon propagator is proportional to Iq1-2 and corresponds to a 1/r Coulomb potential) transmits the nuclear PNC to the electron. We shall show that gauge invariance of the electromagnetic current, hermiticity, and timereversal invariance require the matrix c ~ N to be multiplied by a factor which is roughly q2/M2, so that the long range potential proportional to 1/q2 is transformed to a constant which corresponds to a contact potential in configuration space. Our comment could be stated as a theorem which is that contributions to ~r from figs. 2a and 2b, which correspond to the right hand parts of figs. lb, as well as that from fig. 2c, must approach zero proportionately to q2 as q2 _+ 0. Not only can there be no constant term, but there can also be no linear term in I q [ in an expansion of ~r about q = 0. In order to prove our statement, we combine figs. 2 a - 2 c into a single figure, which can be represented effectively by fig. 2c with a form factor that includes the parity admixtures of figs. 2 a and 2b. To see that this can be carried out formally, we note that, for a nucleus of four-momentum p, angular momentum j, and magnetic quantum number m, the ground state in the presence of weak forces can be written as
[ p , j , m ) = l p , j , m , + )+ i e l p , / , m , - ) ,
(4)
where e ~ 10 . 6 - 1 0 . 8 characterizes the abnormal parity admixture (which is real for bound states) and we have assumed that the normal parity of the state of our concern is positive. We then recall that the matrix element for the study of PNC in atoms or in elastic electron scattering can be written as
cl?~= 47ra q2 _ 2X,/~G
f d3r~O(r)
~,x ~(O(r)e_iq.r(pf, j ' m,iJx(O)lPi,j, m)
(5)
fd3r~(ef)(r) 7X(v e +acTs) ~(i)(r)e_iq.r(pf,
j ' m,lNx(O)lPi,f, m)
where re, a e are constants and N x (x) is the hadronic neutral weak current. In view of eq. (4), we can write the matrix element of the hadronic electromagnetic current in eq. (5) as
(o)
350
(b)
(c)
Fig. 2. Contributions to the abnormal-parity electromagnetic vertex (pf, j, m'lJh (0)I Pi, J, m)(-). (a), (b), and (c) descr~e, respectively, abnormal-parity mixing in the initial and final wavefunctions and the abnormal-parity vertex renormalizatlon.
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
(pf, j, m' IJx(0 ) IPi,/, m) =-(pf, j, m' I [J(+)(0) + J(-) (0)] IPi,1, m) (pf, f, m',+l J(+) (O)[Pi,], m, +) + (ie(pf,/, m', + ]J(+) (O)lPi,f , m , - )
(6)
- i e(pf,/, m', - I J(+)(0)]pi, f, m, +) + (pf,/', m', + IJ(-)(O)lPi,f, m, +)) --(pf, f, m' JJx(0)1Pi,l, m)(+) + (Pf,/, m' IJx (0)JPi,], m)(-), where the three terms in (pf, f, m' IJx(0)I pi,/, m)(-) correspond, respectively, to the contributions to ~r from figs. 2 a - 2c, the last one describing exclusively the abnormal-parity renormalization of the electromagnetic vertex by the presence of weak forces. Our theorem refers to the sum of these three terms and requires that
(pf, j, m'lJx(O) lPi,/, m)(-) (x q2.
(7)
We proceed to prove eq. (7) on the basis of gauge invariance (GI) and the combined operation of hermiticity and time-reversal invariance (HT). GI requires
qk(pf, f, m' IJx(O) lPi,], m) (-) = 0,
(8)
while HT yields [5]
( - ) m + m ' ( E i , - p i ; / , - m [ TJx(O) T -1 [Ef,-pf;j,-m')(-)=(pf;/,m'lJx(O)lPi;/,m)(-),
(9)
with
TJ(O) T -1 =-J(O),
TJo(O) T -1 = +J0(0).
(10)
Rather than considering the general case from the very beginning, we first examine the special case of spin 1 (e.g., 2H). The spin 1/2 case has been treated elsewhere [6]. For such a nucleus, we introduce the polarization four-vector ~x and write ,1 (pf, ~(f) Ijx(0) i pi, ~ (i))(-)
~)* ~---/PK g'~
^
qx
Gv(qZ)- ~
GR(q2)) + ePr/r~"
~(f'*
~(~'~.,..,,, 2M eg (-~1 eh (q2)) ,,..,,,, Gs(q2) - ~ GT
!
(1 la)
+ expn• ~(f). 2M qn 2M P~ ~(i). q 2M GK(q2) + expnK ~0) 2M qn 2M P~ ~(f)*.q 2M GL(q2)' with Px - (Pi +Pf)x and where expn~ is the fully antisymmetric Levi-Civita tensor. The GI constraint, i.e. eq. (8), yields Gv(q2 ) + (q2/4M 2) Gs(q2 ) = 0.
(1 lb)
HT requires that eq. (I 2a) be invariant under the following substitutions:
Qo -+ Qo ,
Q -->- Q;
qo -->- qo,
_, _
q -+ q;
(11c)
¢o* _,_ ¢i)
More explicitly, we obtain from HT GR(q 2) = 0,
GT(q 2) = 0,
GK(q 2) = -- GL(q2).
(1 l d)
~:1 This expression is used in a more complete paper in preparation. 351
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
Since all the form factors GV,S,K,L (q2) must be regular at q2 = 0 (if no zero-mass strongly interacting bosons exist), we expect, from eq. (1 lb), GV (q2) ~ (t q l/2M)2 •
(1 l e)
Hence, eq. (7) follows. For a nucleus of arbitrary spin/, we can use Holstein's parametrization [7] :
rc '.'~ r''2"~P~ (Pf,/,m'lJx(O)lPi,J,m)(-)=(/m;l kl/m')ei/k {ei/xn~. .' 4M-d(q 2) q-~-~] qx q~ h ( q 2 ) / + other form factors at least quadratic in qx, + e6~n 2M 43,/ 2M
(12a)
where all form factors are regular at q2 = 0 [7]. Now, the GI constraint, i.e. eq. (8), yields c (q2) + (q2/4M2) h (q2) = 0,
(12b)
while HT implies d(q 2) = O.
(12c)
Eq. (7) follows eqs. (13a)-(13c) since h (q2) is regular at q2 = 0. Our conclusion is not restricted to one-photon exchanges, but can be extended to two or more photons; see figs• 3. By a reduction akin to that which leads to the last equality of eq. (6), we can represent two-photon PNC effects by figs• 3d and 3e. In the evaluation of two-photon exchanges, one must take care not to include Coulomb effects which are already included in the bound state (or Coulomb-distorted) wavefunctions. The proof of our conclusion for two- or more-photon exchanges relies on a series of papers by Feinberg and Sucher [8]. They use dispersion relations to show that the longest-range part of the two-photon-exchange potential is related to on-mass-shell propagation (i.e. the pole of the propagator) and to the Compton scattering matrix element• In our case, this is a PNC Compton scattering, which has been treated by Kim and Dass [9]. They show, on general grounds, that this PNC Compton amplitude goes like q2 for small momentum transfers• Together with the Feinberg-Sucher work, it follows readily that there is no long-range force which transmits the nuclear PNC to the electron; that is, there is no 1/1 q l 2 or 1/I q I effect (I q I is the net three-momentum transfer). In many nuclear physics applications, the impulse approximation (IA) is used• This approximation preserves HT but often violates GI [7] ,1. For instance, if we use the usual IA to calculate the various deuteron form factors as defined by eq. (12a), we find that the form factor Gv (q2) behaves like a constant rather than Iq I2/M2. However, all form factors calculated in the same fashion are regular at q2 = 0. Such violation of GI (eq. (1 lb)) by the
(o)
(b)
(a)
352
(c)
(e)
Fig. 3. Two-photon exchange diagrams with abnormal-parity mixing in the initial, intermediate, and final nuclear wavefunctions ((a), (b), and (c)) and with abnormal-parity electromagnetic vertices ((d) and (e)).
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
usual IA thus can give rise to the incorrect result that the ratio R of the PNC effect due to parity mixing in the deuteron wavefunction to that due to Z-boson exchange is proportional to I q 1-2 as I q I -~ 0. As we have just shown, this conclusion would be incorrect. This illustrates the importance of including meson exchange effects in order to preserve gauge invariance and so avoid spurious results. In summary, we are led to conclude that, both for atomic physics tests and for elastic electron scattering from nuclei, PNC effects due to nuclear parity violation will be of the order of radiative corrections (fine structure constant) as compared to those due to Z-boson exchange, even if the contributions due to multiple photon exchanges are included. Of course, exceptions may occur if the parity mixing in the nuclear ground state is enhanced and the atomic (electron scattering) effects due to Z-boson exchange are inhibited. This work is supported in part by the U.S. Department of Energy.
References [1] For a review in theory, see, e.g., L. Wilets, in: Neutrinos-78, ed. E.C. Fowler (Purdue Univ., 1978) p. 437. [2] For a review in experiment, see, e.g., N. Fortson, in: Neutrinos-78, ed. E.C. Fowler (Purdue Univ., 1978) p. 417; for a recent Berkeley experiment, see R. Conti et al., Phys. Rev. Lett. 42 (1979) 343. [3] W.-Y.P. Hwang, Phys. Rev. C20 (1979) 331. [4] A. Barroso and D. Tadid, Phys. Rev. C17 (1978) 832; S.E. Koonin, Phys. Rev. C19 (1979) 2314. [5] See, e.g., H. Frauenfelder and E.M. Henley, Nuclear and particle physics (Benjamin, Reading, MA, 1975). [6] E.M. Henley, A.H. Huffman and D.U.L. Yu, Phys. Rev. D7 (1973) 943. [7] B.R. Holstein, Rev. Mod. Phys. 46 (1974) 789. [8] J. Sucher, in: Carg~se lectures in physics, ed. M. Levy (Gordon and Breach Science, New York, London, Paris, 1977) Vol. 7, p. 43. [9] K.J. Kim and N.D.H. Dass, Nucl. Phys. Bl13 (1976) 336.
353