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Sign and magnitude of the nucleon-nucleon parity-violating π-exchange potential in the Weinberg-Salam model of weak interactions
SIGN AND MAGNITUDE n -EXCHANGE
20June1977
PHYSICS LETTERS
Volume 68B, number 4
OF THE NUCLEON-NUCLEON
PARITY-VIOLATING
POTENTIAL IN THE WEINBERG-SALAM OF WEAK INTERACTIONS
MODEL
Bertrand DESPLANQUES Institut de Physique Nuclbire,
Division de Physique Thiorique
*, 91406 Orsay Cedex, France
and Jacques MICHELI Laboratoire de Physique Thdorique et Hautes Energies *, Universitt? de Paris&d,
91405 Orsay, France
Received 13 April 1977 We show that, inside a quark model of baryons, the sign of the nucleon-nucleon potential can be determined.
parity-violating n-exchange
The discovery of neutrino events related to a weak neutral current [l] , and the possibility of constructing renormalizable unified theories of weak and electromagnetic interactions [2] have encouraged the development of several models of weak interactions. Consequently, there appeared a new interest for calculating, in these models, the nNN weak coupling constant [3-61 which determines the weak n-exchange NN potential. Because of its long range, this potential is expected to give the dominant contribution to parity-violating (p.v.) AZ = 1 transitions in the proton neutron system. Depending upon its magnitude, but also upon its sign to which little attention has been given in previous studies, it may explain p.v. effects observed in such nuclei as lgF, 41K, 175Lu and 181Ta where the AI= 1 as well as the fY = 0 parts of the weak interaction (WI.) contribute [7] . Because currents in recent models are no longer of the V-A type, nor members of a SU(3) octet, the experimental information on hyperon decays does not suffice to determine all the quantities which appear in the calculation of this coupling constant. To make a prediction, models suppressing all the unknown quantities [3, 51 or determining them in a simple valence colored quark model of baryons [6] have been proposed. In this letter, we want to show that this last model allows us to determine the relative sign of the strong and weak KNN coupling constants and therefore the sign of the weak n-exchange potential which contains their product. Afterwards, we will show that further contributions to this potential, due to including qq pairs in the quark model of baryons, have also a definite sign. Results will be presented for the SU(4) version of the Weinberg-&lam model [8] which has the advantage of illustrating most of the difficulties raised by the new W.I. models. Using the P.C.A.C. hypothesis, the L.S.Z. formula and current algebra, we can write the amplitude, f,, for n -+ pn- as:
where “HA1=l” differs from the AZ = 1 component metric in V and A currents. It can be written as:
of the p.c. part of the weak hamiltonian
because it is not sym-
* Laboratoire associe au C.N.R.S.
339
Volume 68B, number 4
uH~=l,>
-_ DG
sin20, -3
PHYSICS LETTERS
sin20,(%%
- &a
20 June 1977
- iicCu t &Ed) - (1 - 2 sin2ew) (Uu - ad) (ss - cc)
_ (uu - ;id) (;u + ;id + ;s
sin28 VV+AA - +
(uu - ;Td)(iTu + ;id + Es +;c)~_/,~
,
(2)
where u and d quarks form an isodoublet (I= f) of strangeness S = 0 and s and c quarks carry strangeness and charm respectively. In this expression of “EfAzZ1”, we have separated terms symmetric and antisymmetric in V and A currents because they will play a different role later on. Taking care of the different signs which enter in the derivation off,, the weak rr-exchange potential can be written as:
v, =
&NN 16nM2g,
i(Tl X 72)2(cl+
02)
--+
eCmd
p1 - p2,
7
) 1(Pzlw~~=~pz)
(3)
where gA has its usual value of t1.25. This expression for V,, shows that its sign is independent of the convention chosen to define the strong nNN coupling since gzNN is a positive quantity. Hence the sign depends only upon the matrix element (-1 “HAzZ1 “lm). We will now concentrate on this matrix element. In a first approximation, we assume that --quarks in weak currents act on the constituent quarks of the baryons --in such a way that only the terms, (uuuu - dddd)VV+AA, and, (uuuu - dddd)vv_AA, contribute to (n-1“HAz=l”l*). Since the Pati-Woo theorem [9] applies to the first of these two terms, the non zero contribution will come only from the second term, which, because of its antisymmetric character in V and A currents, has been neglected in most past studies. Assuming a non-relativistic quark model, and recalling that the weak coupling constant, G/a, has a well defined sign in any WI. model mediated by a vector boson, as is the case here, its contribution can be expressed as: (*I ‘rH+$~AAy’l~)
= 5 sin2Bw x ,
with d3(r1 - r2)l ti(rl, r2, r3)12 %l-ri)
.
Although the spatial wave function of the quarks, Jl(rl, r2, r3), is unknown, this expression, which is metric independent, shows that x is a positive quantity. Therefore, the corresponding contribution to the p.v. potential, V,,has a well defined sign. To obtain the magnitude of x, one can use one of the relations which, in the present quark model, allows us to express this quantity in terms of the S-wave amplitudes for hyperon decays:
kNN sin~,c0s8,/Mg~Ix=IAO_I/J?;=15~1/2~=1~~1/3~.
(5)
Since the different experimental amplitudes do not conform very well to relation (5) *, the uncertainty in the magnitude of x may be large. We now consider corrections to the quark model of baryons used here by allowing for the possibility of
340
[IO]. It corresponds to the relation between
20 June 1977
PHYSICS LETTERS
Volume 68B, number 4
We will assume that the contribution of Cc pairs is negligeable, which breaks the SU(4) symmetry, but consider that the SU(3) symmetry still holds. Furthermore, in order to estimate the contribution of the remaining, Al= 1, VV + AA terms: --iis% - &;d , [email protected] - ad) Ss and uuuu - dddd , (6) we will assume that terms symmetric ($,VPtir$,V,tiS
in SU(3) indices (or color indices):
+ $,A,tiY$#,tiS
+ terms (-Ycf 6)))
(7)
give negligeable contributions. This hypothesis is suggested by the small S-wave, Z+ + nrr+ amplitude which depends upon the 27-plet part of such terms and,implies relations between contributions of W t AA terms which belong to different octects. It might reflect the existence of a dynamical mechanism which, such as the gluon exchange, enhances and suppresses contributions of W t AA terms respectively antisymmetric and symmetric in quark indices (or color indices) [ 121. As a consequence of our model, the contributions of the two first terms in (6) are equal, whereas the contribution of the 3rd term is negligeable. The total contribution to @l“H A1=l”l*) of W + AA terms, due therefore to the presence of Ss pairs in nucleons, can be written in the form:
Using the SU(3) symmetry and the absence of contributions matrix element can be expressed as:
due to the 27-plet part of W
t AA terms, this last
(9) The magnitude of the matrix elements in the right-hand side are usually determined from the known experimental S-wave amplitudes for A -, pn- and E- --, hrr- decays. We assume they have respectively the positive and negative sign of the dominant contribution calculated in the simplest quark model (see eqs. (4) and (5)). The sign of the contribution of W + AA terms to (rzl “H ApI=l”l~) is thus related to the sign of the difference, 21At I - IZI 1,by the relation:
The same relative sign would be obtained if we had assumed that the small, AI= 3/2, S-wave amplitudes in A --, NT and Z -, An decays are due to those calculated in the factorization approximation. Gathering the different results, we now write the weak potential V, as:
observed
(11) where h,, which has the magnitude sin2B, h, =
3lsin ec cos e,i
off,
but is free of any phase convention,
has the following expression:
(sin2eC t 1 - $.in20vv) I sin 8, cos eci
&2IA!I
- IKI).
(12)
The first term represents the contribution of terms of “HAzzl” antisymmetric in V and A currents in a valence quark model, whereas the second term represents the contribution of terms symmetric in V and A currents which 341
Volume 68B, number 4
PHYSICS LETTERS
20 June 1977
are due to 44 pairs. Unfortunately, we cannot calculate exactly the first contribution. Existin ex erimental data determine only the differences between the corrections due to ?jq pairs to the amplitudes 2 &lA!l, 1and $IZZ I, but not their absolute magnitude. Because we do not know which of these amplitudes best represents the contribution of W - AA terms, there is a resulting uncertainty. Except for some details, this contribution is quite comparable to what Donoghue [6] calculated, as far as the magnitude is concerned. Our contribution for the second term of eq. (12) differs from the results of Gari and Reid [3] . Taking the experimental S-wave amplitudes for hyperon decays and the value 0.36 for sin2ew,, we get
filz1
t4.8 X 1O-7
< t6.1 X 1O-7 .
(13)
Up until now, we have assumed that SU(3) was a good symmetry. To obtain a better fit with experimental data, some SU(3) breaking corrections have been suggested. Taking them into account we get: +3.5 X IOF
< t4.0 X 10P7 (ref. [13])
t3.6 X 10e7
< t4.0 X low7 (ref. [14])
t4.0X10-7
342
20June1977
PHYSICS LETTERS
Volume 68B, number 4
OF THE NUCLEON-NUCLEON
PARITY-VIOLATING
POTENTIAL IN THE WEINBERG-SALAM OF WEAK INTERACTIONS
MODEL
Bertrand DESPLANQUES Institut de Physique Nuclbire,
Division de Physique Thiorique
*, 91406 Orsay Cedex, France
and Jacques MICHELI Laboratoire de Physique Thdorique et Hautes Energies *, Universitt? de Paris&d,
91405 Orsay, France
Received 13 April 1977 We show that, inside a quark model of baryons, the sign of the nucleon-nucleon potential can be determined.
parity-violating n-exchange
The discovery of neutrino events related to a weak neutral current [l] , and the possibility of constructing renormalizable unified theories of weak and electromagnetic interactions [2] have encouraged the development of several models of weak interactions. Consequently, there appeared a new interest for calculating, in these models, the nNN weak coupling constant [3-61 which determines the weak n-exchange NN potential. Because of its long range, this potential is expected to give the dominant contribution to parity-violating (p.v.) AZ = 1 transitions in the proton neutron system. Depending upon its magnitude, but also upon its sign to which little attention has been given in previous studies, it may explain p.v. effects observed in such nuclei as lgF, 41K, 175Lu and 181Ta where the AI= 1 as well as the fY = 0 parts of the weak interaction (WI.) contribute [7] . Because currents in recent models are no longer of the V-A type, nor members of a SU(3) octet, the experimental information on hyperon decays does not suffice to determine all the quantities which appear in the calculation of this coupling constant. To make a prediction, models suppressing all the unknown quantities [3, 51 or determining them in a simple valence colored quark model of baryons [6] have been proposed. In this letter, we want to show that this last model allows us to determine the relative sign of the strong and weak KNN coupling constants and therefore the sign of the weak n-exchange potential which contains their product. Afterwards, we will show that further contributions to this potential, due to including qq pairs in the quark model of baryons, have also a definite sign. Results will be presented for the SU(4) version of the Weinberg-&lam model [8] which has the advantage of illustrating most of the difficulties raised by the new W.I. models. Using the P.C.A.C. hypothesis, the L.S.Z. formula and current algebra, we can write the amplitude, f,, for n -+ pn- as:
where “HA1=l” differs from the AZ = 1 component metric in V and A currents. It can be written as:
of the p.c. part of the weak hamiltonian
because it is not sym-
* Laboratoire associe au C.N.R.S.
339
Volume 68B, number 4
uH~=l,>
-_ DG
sin20, -3
PHYSICS LETTERS
sin20,(%%
- &a
20 June 1977
- iicCu t &Ed) - (1 - 2 sin2ew) (Uu - ad) (ss - cc)
_ (uu - ;id) (;u + ;id + ;s
sin28 VV+AA - +
(uu - ;Td)(iTu + ;id + Es +;c)~_/,~
,
(2)
where u and d quarks form an isodoublet (I= f) of strangeness S = 0 and s and c quarks carry strangeness and charm respectively. In this expression of “EfAzZ1”, we have separated terms symmetric and antisymmetric in V and A currents because they will play a different role later on. Taking care of the different signs which enter in the derivation off,, the weak rr-exchange potential can be written as:
v, =
&NN 16nM2g,
i(Tl X 72)2(cl+
02)
--+
eCmd
p1 - p2,
7
) 1(Pzlw~~=~pz)
(3)
where gA has its usual value of t1.25. This expression for V,, shows that its sign is independent of the convention chosen to define the strong nNN coupling since gzNN is a positive quantity. Hence the sign depends only upon the matrix element (-1 “HAzZ1 “lm). We will now concentrate on this matrix element. In a first approximation, we assume that --quarks in weak currents act on the constituent quarks of the baryons --in such a way that only the terms, (uuuu - dddd)VV+AA, and, (uuuu - dddd)vv_AA, contribute to (n-1“HAz=l”l*). Since the Pati-Woo theorem [9] applies to the first of these two terms, the non zero contribution will come only from the second term, which, because of its antisymmetric character in V and A currents, has been neglected in most past studies. Assuming a non-relativistic quark model, and recalling that the weak coupling constant, G/a, has a well defined sign in any WI. model mediated by a vector boson, as is the case here, its contribution can be expressed as: (*I ‘rH+$~AAy’l~)
= 5 sin2Bw x ,
with d3(r1 - r2)l ti(rl, r2, r3)12 %l-ri)
.
Although the spatial wave function of the quarks, Jl(rl, r2, r3), is unknown, this expression, which is metric independent, shows that x is a positive quantity. Therefore, the corresponding contribution to the p.v. potential, V,,has a well defined sign. To obtain the magnitude of x, one can use one of the relations which, in the present quark model, allows us to express this quantity in terms of the S-wave amplitudes for hyperon decays:
kNN sin~,c0s8,/Mg~Ix=IAO_I/J?;=15~1/2~=1~~1/3~.
(5)
Since the different experimental amplitudes do not conform very well to relation (5) *, the uncertainty in the magnitude of x may be large. We now consider corrections to the quark model of baryons used here by allowing for the possibility of
340
[IO]. It corresponds to the relation between
20 June 1977
PHYSICS LETTERS
Volume 68B, number 4
We will assume that the contribution of Cc pairs is negligeable, which breaks the SU(4) symmetry, but consider that the SU(3) symmetry still holds. Furthermore, in order to estimate the contribution of the remaining, Al= 1, VV + AA terms: --iis% - &;d , [email protected] - ad) Ss and uuuu - dddd , (6) we will assume that terms symmetric ($,VPtir$,V,tiS
in SU(3) indices (or color indices):
+ $,A,tiY$#,tiS
+ terms (-Ycf 6)))
(7)
give negligeable contributions. This hypothesis is suggested by the small S-wave, Z+ + nrr+ amplitude which depends upon the 27-plet part of such terms and,implies relations between contributions of W t AA terms which belong to different octects. It might reflect the existence of a dynamical mechanism which, such as the gluon exchange, enhances and suppresses contributions of W t AA terms respectively antisymmetric and symmetric in quark indices (or color indices) [ 121. As a consequence of our model, the contributions of the two first terms in (6) are equal, whereas the contribution of the 3rd term is negligeable. The total contribution to @l“H A1=l”l*) of W + AA terms, due therefore to the presence of Ss pairs in nucleons, can be written in the form:
Using the SU(3) symmetry and the absence of contributions matrix element can be expressed as:
due to the 27-plet part of W
t AA terms, this last
(9) The magnitude of the matrix elements in the right-hand side are usually determined from the known experimental S-wave amplitudes for A -, pn- and E- --, hrr- decays. We assume they have respectively the positive and negative sign of the dominant contribution calculated in the simplest quark model (see eqs. (4) and (5)). The sign of the contribution of W + AA terms to (rzl “H ApI=l”l~) is thus related to the sign of the difference, 21At I - IZI 1,by the relation:
The same relative sign would be obtained if we had assumed that the small, AI= 3/2, S-wave amplitudes in A --, NT and Z -, An decays are due to those calculated in the factorization approximation. Gathering the different results, we now write the weak potential V, as:
observed
(11) where h,, which has the magnitude sin2B, h, =
3lsin ec cos e,i
off,
but is free of any phase convention,
has the following expression:
(sin2eC t 1 - $.in20vv) I sin 8, cos eci
&2IA!I
- IKI).
(12)
The first term represents the contribution of terms of “HAzzl” antisymmetric in V and A currents in a valence quark model, whereas the second term represents the contribution of terms symmetric in V and A currents which 341
Volume 68B, number 4
PHYSICS LETTERS
20 June 1977
are due to 44 pairs. Unfortunately, we cannot calculate exactly the first contribution. Existin ex erimental data determine only the differences between the corrections due to ?jq pairs to the amplitudes 2 &lA!l, 1and $IZZ I, but not their absolute magnitude. Because we do not know which of these amplitudes best represents the contribution of W - AA terms, there is a resulting uncertainty. Except for some details, this contribution is quite comparable to what Donoghue [6] calculated, as far as the magnitude is concerned. Our contribution for the second term of eq. (12) differs from the results of Gari and Reid [3] . Taking the experimental S-wave amplitudes for hyperon decays and the value 0.36 for sin2ew,, we get
filz1
t4.8 X 1O-7
< t6.1 X 1O-7 .
(13)
Up until now, we have assumed that SU(3) was a good symmetry. To obtain a better fit with experimental data, some SU(3) breaking corrections have been suggested. Taking them into account we get: +3.5 X IOF
< t4.0 X 10P7 (ref. [13])
t3.6 X 10e7
< t4.0 X low7 (ref. [14])
t4.0X10-7
342