Novel calculation of weak meson-nucleon couplings

Nuclear Physics A499 (1989) 699-726 North-Holland. Amsterdam

NOVEL

CALCULATION

OF WEAK

MESON-NUCLEON

COUPLINGS*

N. KAISER Niels Bohr Institute,

Blegdamsvej

Ulf-G.

Centerfor

Theoretical

Physics, Laboratory

Massachusetts

Institute

qf

17, Dk-2100

0,

Denmark

MEISSNER** .for Nuclear

Technology,

Received

Copenhagen

Science

Cambridge,

25 January

and Department MA 02139,

of’ Physics,

USA

1989

Abstract:

We investigate parity-violating (pv) meson-nucleon vertices within the framework of a nonlinear chiral effective lagrangian which includes pions and the vector mesons p and w. Nucleons emerge as topological solitons. We find a value for the weak pv fl N coupling (-2 x lo-‘) which is considerably smaller than the standard quark-model values. The w-meson-nucleon couplings ht and 11; are comparable to the quark-model results. The values for the p-meson-nucleon couplings hz, hb and ha are somewhat smaller than the best quark model predictions. Furthermore, we find a non-zero and non-negligible value for the coupling constant h;’ (hz = -2 x lo-‘). We compare our results with available data by using existing nuclear-structure calculations. The strong meson-nucleon vertices entering into these calculations are also predicted by the soliton scenario, with values close to what is derived from phenomenological boson-exchange models of the nucleon-nucleon interaction. The agreement of the soliton-model predictions with the data is satisfactory. We also discuss possible improvements and limitations of our approach.

1. Introduction Nuclear parity violation is an important tool to study the standard mode1 of strong and electroweak interactions ‘). In the nucleon-nucleon system there are in genera1 five independent parity-violating (pv) amplitudes of isoscalar, isovector and isotensor type. Equivalently, one can work in the framework of meson-exchange and parametrize strong and weak interaction through parity-conserving and parityviolating meson-nucleon vertices, respectively, as shown in fig. 1. Due to the CP invariance, there exists no parity-violating coupling of neutral scalar or pseudoscalar mesons to nucleons ‘). Therefore, standard parametrizations of the pv NN potentials generally involve the exchange of charged pions (.x*) as well as the vector mesons p and w. These mesons are also playing the dominant role in the semi-phenomenological models of the one-boson exchange potentials ‘) with the addition of a scalar-isoscalar particle (u) which accounts for most of the intermediate-range attraction. To test the standard model, one needs to calculate the weak meson-nucleon vertices. Only a few calculations based on the quark model exist 4,5). These calculations start from the observation that there are essentially three types of diagrams, l This work is supported in part by funds provided by the U.S. Department under contract # DE-AC02-76ER03069, and by Deutsche Forschungsgemeinschaft. l * Heisenberg-Fellow.

0375-9474/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V

of Energy

(D.O.E.)

700

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

couplings

him

N

N

Fig. 1. Weak (/I,,,) and strong (g,,,) meson-nucleon coupling constants in the one-boson-exchange model. Due to Barton’s theorem 2), only charged pions can contribute to this pv Feynman diagram.

which can be categorized as factorization, quark-model and sum-rule contributions. Renormalization group techniques and baryon wave functions based on models are needed to evaluate them. This introduces a variety of uncertainties (e.g. through the determination of the quark masses, strong interaction enhancements, . . . ) which lead DDH in ref. “) to introduce a “reasonable range” for the values of the weak meson-nucleon couplings. In particular, the weak nN coupling (G,) is very sensitive with respect to these uncertainties, e.g. the values of G, in refs. 4,5) differ by a factor of 3, whereas the results for the vector meson-nucleon couplings are more stable. In ref. 6), we proposed an alternative scheme to get a handle on the weak pion-nucleon vertex strength. It is based on the soliton picture of the nucleon which takes into account important non-perturbative effects of QCD at low energies. The starting point is an effective non-linear chiral lagrangian, which incorporates the pions and the low-lying vector mesons, p and w. The effective theory therefore encodes the two main features of low-lying strong interaction physics, namely the spontaneous breakdown of chiral symmetry with the appearance of light pseudoscalar Goldstone bosons and the fact that the lowest lying states are indeed the pseudoscalars and the vectors (as they are in the quark model). Nucleons arise as quantized topological solitons, with all their properties determined from the meson sector. The beauty of this approach is that once one has determined the mesonic input parameters from experiment, all nucleon properties can be calculated parameter-free. In the context of the problem at hand, one simultaneously gets a prediction for the strong and weak meson-nucleon vertices together with many other observables like electromagnetic and axial form factors. The calculation of the weak MNN vertices can be summarized by the following step-by-step procedure: First, one calculates the classical soliton of the effective action, and then projects onto states of good spin and isospin by virtue of the collective quantization method first proposed by Adkins et al. ‘). The nucleon and the A(1232) arise as rotational excitations of the ground-state hedgehog configuration. At this point, we should mention that this quantization procedure is not without its own problems. In particular, if one chooses to take the meson parameters on their empirical values (in particular the pion decay constant fW), the nucleon mass generally comes out

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

couplings

701

too large by several hundred MeV. Once one has constructed the nucleon, it is straightforward to extract information about the weak MNN vertices. For that, one brings the lagrangian current

x current

of the standard

interaction

electroweak

model into the form of an effective

of the Fermi type and rewrites it in terms of the isoscalar,

isovector and axial currents of the soliton. These currents are now functionals of the meson fields rr, p and w. One can then read off the pertinent vertices by expanding these meson fields as ~uctuations around the soliton background and pick out the terms linear in the fluctuations. Similarly, the soliton acts as a source for the strong meson-nucleon vertices, which can then be evaluated by Fourier transformation of the respective source functions “). Therefore, weak and strong MNN vertices follow uniquely from the soliton model once one has fixed the meson parameters and constructed the soliton. We should point out here that these calculations are fairly straightforward, quite in contrast to the more involved quark-model calculations 45s). The paper is organized as follows. In sect. 2 we briefly review the effective lagrangian of pions, p and o mesons. We will make use of the recently developed “realistic” pseudoscalar-vector lagrangian (RPV lagrangian) of ref. “). We outline the procedure to construct the classical soliton solution and the nucleon; however, we omit any calculational details and refer to the vast literature “). Since we will need the nucleonic currents as constructed from the soliton model, we comprehensively summarize them at the end of sect. 2, and also in appendix A. Sect. 3 contains the main part of the paper. We recall how to rewrite the Weinberg-Salam-Glashow model in the form of an effective current x current interaction (for details, see ref. “)). We then proceed to caiculate the pv ?rN coupling in a more elegant method than in ref. “). There, we had used PCAC and the current algebra to evaluate G, from the isospin-breaking, parity-conserving piece of the effective weak interaction. Here, we use the aforementioned method of fluctuating meson fields on top of the soliton background, which can easily be generalized to the case of the vector mesons. Using this method, we systematicalIy evaluate the weak wN and pN vertices for the “minimal” model of ref. lo). This is done to make the calculational procedure more transparent, the more involved resuhs for the RPV lagrangian are given in appendix B. Sect. 4 is devoted to the discussion of our results. First, we compare them to recent quark-model calculations. Then, we use existing nuclear structure calculations to calculate the pertinent pv observables. We essentially follow Adelberger and Haxton ‘) and concentrate on what they argue is the best-known set of pv observables, i.e. the longitudinal asymmetry in pp, pd and p 4He scattering, as well the paritymixing matrix-elements in “F, “F and “Ne. Of course, we correct these calculations in that we use the strong meson-nucleon couplings as predicted by our model rather than the “standard values” generally taken in these calculations. A thorough discussion of the results together with comparison to experimental data and quarkmodel predictions is presented. The summary and outlook is surveyed in sect. 5, together with a critical discussion of our approach and its limitations.

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

702

2. Chiral

soliton

couplings

model

In this section, we will briefly review the effective lagrangian of pions, p and w mesons. Nucleons arise as solitons, with all their properties determined through the mesonic parameters, which are the weak pion decay constant f= = 93 MeV, the “gauge” coupling g = g,,, = 6, the pion mass m, = 138 MeV, and three pseudoprocesses. In scalar-vector couplings ( &v4, L, K) related to the unnatural-parity the case of electromagnetic anomalous interactions (like p”-+ rr’y) another parameter (d,) comes in. We also will write down the currents which arise in this model since they will be used in sect. 3 to calculate

2.1. EFFECTIVE

LAGRANGIAN

the weak meson-nucleon

OF PSEUDOSCALARS

couplings.

AND VECTORS

We will be short here and only write down the effective lagrangian we will use and add some pertinent remarks. For more details, the reader is referred to the extensive review in ref. “) as well as to ref. “). The effective action which encodes the two main features of low-energy hadron physics, spontaneous chiral-symmetry breaking and the fact that the pseudoscalars and vectors are the lightest mesons, has the form (we restrict ourselves to the two-flavor sector): (2.1)

r&-(=, P, w I= r”a + ra” 7

r,, is the so-called non-anomalous action, whereas the anomalous part ra:,, embodies all unnatural-parity processes proportional to the totally antisymmetric Levi-Civita r,, is uniquely given by tensor E_~.

r,,, = %, =

d4x Tn,,

171: Tr b,p”l

-$Tr [~,JP)F'""(P)I+

+2ig,,,f2,Tr[p~((a,55++a,5’5)]+If~m’xTr(U-l) +$‘,(a with F,,(p) with g,,,

+ 1) Tr [a,&P“5’]

= ;)w~y -avpP - ig[p,,

=6. The constant

-;f’,(a

- 1) Tr [5’a,&Y‘5’1,

Pa,]. We have identified

the “gauge”

(2.2) coupling

g

a is given by

a =

%&,f 2 mi

(2.3)

and follows to be a ~2.2 for the empirical values off=, g,,, and mp. We will, however, use the KFSR value a = 2 for simplification. Since pP = fw, + $7 . pp, this leads to m, = m, and g = 5.8545. 5 in (2.2) embodies the pseudoscalar field 4, in our case the pion fields, 5 = ~07 = exp {icP/2f,),

u=5*5,

with 4 = n 1 + T - n. The effects of the isoscalar-pseudoscalar

(2.4) v-meson

are very small

N. Kaiser,

as shown

in ref. “) and

symmetric

anomalous

U.-G.

/ Weak meson-nucleon

we will set n = 0 throughout.

action

r,,=T,,(U)+

Meissner

has been derived

Tr{ic,fi[A

couplings

The most general

703

chirally

in ref. ‘I) and reads

&]+2cz[dALcuAL-A,a

dAL+ALcxALa]

I ALaALa]},

+2&,[-2iA:(u+(l/g) where we used the notation

of differential

(2.5)

one-forms,

a=(a,Cr)U’dx’*, A, = (5p,5++ (M&5’)

dx’*.

(2.6)

r,,,,, is the standard Witten-Wess-Zumino action of pseudoscalars. “) In ref. ‘), two from purely strongof three unknown constants c, , c2 and cj could be determined interaction processes, i.e.,

6= -8i[c,

-&fc,/g-

c,/g’]

= 10.4.

(2.7)

The third parameter, c3, could not be determined, however for a best fit to the nucleon properties it was found that K = O.l2c,/ c? = 0 . . . 2. cj is, of course, calculable from meson reactions like K*+ Kmr, but these involve complicated SU(3) symmetry-breaking terms and are also not well measured. In what follows, we will call the lagrangian based on the anomalous action (2.5) the “realistic” pseudoscalarvector lagrangian (RPV lagrangian). To make our calculations more transparent, we will also use the so-called “minimal” model of ref. I”), which follows from (2.5) by setting c? = cj = 0 and choosing i= -g/(fir*), i.e. I$“=~,,(U)+;g B,(x) is the topological baryon current, not use the so-called “complete” model symmetry breaking Bardeen-subtracted comment on some results obtained using

2.2.

NUCLEONS

w,(x)BF(x)d4x.

(2.8)

with 5 d3r B,(r) = 1 for a baryon. We will of ref. “) since it makes use of the chiralWess-Zumino action (however, we will this model.)

AS SOLITONS

Nucleons arise as solitons from the effective action discussed above. To construct them, one goes through a two-step procedure. First, one finds the classical soliton which has neither good spin nor good isospin. Then, an adiabatic rotation of the soliton is performed and it is quantized collectively. For details we again refer the reader to ref. “), here, we will only give the pertinent formula to define the meson profiles.

704

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

The classical

soliton

follows

from (2.1) by virtue

couplings

of the hedgehog

U(r) = exp { i7 * tF( r)} t(r) p”“(r)

(2.9a)

= exp ($7 - iF( r)}

(2.9b)

= E ikaikG(r)/gr

(2.9~)

cop(r) = o(r)Sp” In what follows,

ansiitze

we will call F(r),

(2.9d)

G( r ) and w(r) the pion-,

p- and w-profile,

respectively. The pertinent boundary conditions to ensure baryon number one and finite energy are F(0) = V, F(W) = 0, G(0) = -2, G(oo) = w’(0) = W(W) = 0. To project out baryonic states of good spin and isospin, we perform a time-dependent SU(2) rotation U(r, t)=A(t)U(r)Ai(t), 5(r, r) = A(r)S(r)A’(r) ~-p’(r, T -

with 2K the angular

(2.1Oc)

- pi(r)

pi(r, t) = AT

(2.10d)

t)=wKxt, r

frequency lagrange

L(t)=

(2.10b)

,

t)=iA(f)~-(Kl,(r)+GK*rlZ(r))At(f),

w(r,

the time-dependent

(2.10a)

(2.10e)

of the spinning

soliton,

iT - K = A’A. This leads to

function

d3r~=--MH[F,G,w]+O[F,G,w;5,,52,~]Tr(AAt)

(2.11)

I Minimizing the classical mass M,[F, G, w] leads to the coupled differential equations for F, G and w subject to the aforementioned boundary conditions. In the spirit of the large NC expansion, one then extremizes the moment of inertia O[ . . .] which gives the coupled differential equations for 5,) & and 4 in the presence of the background profiles F, G and w. The pertinent boundary conditions are L!(W) = l;(O) = l;(O) = 12(co) = 4(O) = C#J(CO) = 0. The nucleon mass is then given by MN= M,+3/(80)

2.3. CURRENTS

OF THE

(2.12)

NUCLEON

In what follows, we will need the electromagnetic (2.1). For that, let us introduce an U(2),x U(2), fields, B, and BR, which transform as SB L,R

=

-[BL,R,

and axial currents of the action multiplet of non-strong gauge

&,A - id&,R,

(2.13)

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

gauge transformations, with E,,, the left- and right-handed the anomalous action (2.5), one gets

couplings

705

respectively.

Gauging

where F(B) = dB - iB2, aI = Q:- igA,+ iUBRUt, a2 = igA,-- iB,, @, = U’a, U and /!_ZZ = UfcrZ iJ.* There appears a new constant d,, which is related to processes like pO-* rrOy or w + ~‘y. d, can be easily determined from the width of w + nOy, one finds “) (2.15) The sign of R together with the sign of gvv+ and i could be dete~ined in ref. *) from a best fit to the n&eons’ properties, one finds l? ~0, .&v,+ > 0 and &> 0. Specializing to electromagnetism, one sets B L,R+ eW, with J& the photon choice is

field and

Q = diag($, -$).

(2.16) For axial currents,

the appropriate

(2.17) with B an external axial field. We will now derive the electroweak currents of the nucleon. For the sake of simplicity, however, we wilt only give the results for the “minimal” model (the equivalent formula for the RPV lagrangian can be found in appendix A). Denoting by I”(X), V+‘(x) and A’“(x) the isoscalar, isovector and axial currents, respectively, one fmdss.9’13) (2.18)

* For more details,

the interested

reader

should

consult

refs. 8.9).

N. Kaiser,

706

U.-G. Meissner

/ Weak meson-nucleon

couplings

Apu = f ', Tr[ra( i~~tP&-- i~~#‘&-~itf‘UU -$U’WlJ

+$g.$RT.

pP&-$gt[~

. p”&)]

•t 3gi EI-‘“~~w, Tr[ra(a,UU’a,UUi+ 32rr2 Here, we have U = [I.$,. In the unitary details, see ref. ‘). Inserting the hedgehog

IO(r) =

-2

gauge

U'a,UUta,U)]

.

(2.20)

one sets .$L= .& = .$ (for further

anstiitze (2.9) and (2.10), this leads to:

w(r) = Z,(r),

(2.21)

Z’(r)=-$$qKxq’=z,(r)(Kxr^)l, (2.22) v”.“(r)

= - Uo( r)Tr (AT - KA+r”) (2.23)

+[Uo(r)-2f~(~,+~,)]Tr(Ar~iK~rAtr”)

(2.23a)

Uo(r)=f~[4sin4~F+2j,cosF]+~~F’sin2F, Vi.a(r) = Film V,(r);’

(2.24)

Tr(Ar*A’-r“),

2

(2.24a)

V,(r)=f-‘(2sin4fF)-GcosF)+~~F’sin2F, r

A’+‘(r)

= aa

Tr (AT

-(K x ;)A+.T~),

(2.25) 3g 9 F’sin 16~~ r2

a,(r)=ftsinF(2-25,-cosF)+-

2F 2

(2.25a) (2.26)

A’~“(r)=~A,(r)Tr(A~‘A’~“)+~A2(r)iiTr(Ar~~Atr~), 2

A,(r)=f”sinF(2G+2-cosF)+~~F’sin2F,

(2.26a)

A,(r)=-A,(r)+f:F’+s;sin’F.

(2.26b)

r

The equivalent radial functions Zo(r), Z,(r), U,(r), . . . , Az( I) for the RPV lagrangian can be found in appendix A. At this point we have now assembled all the tools to calculate the weak meson-nucleon vertices. This calculation will be presented in the following section. 3. Calculation 3.1. EFFECTIVE

WEAK

CURRENTx

of the weak meson-nucleon CURRENT

vertices

LAGRANGIAN

In this section, let us briefly recall how we can bring the standard electroweak interaction lagrangian into the form of an effective current xcurrent lagrangian of

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

707

couplings

the Fermi type. For that, one starts with the standard coupling of the electroweak gauge bosom to (point-like) nucleons and then goes to the limit of small momentum (q2<< I’U~~,~).

transfer

The details

of this procedure

just want to give the final results current

x

current

lagrangian

are outlined

and add some pertinent

in ref. ‘), here we

remarks.

The effective

reads

cos’ Bc ; [ V;(x)-A;(x)]’ o=,

+[cos20wV~(x)-A:(x)-2sin2

&Z,(x)]’

I

,

(3.1)

with G, = 1.16 x lo-’ GeV-’ for the Fermi constant, 0< (cos’ 0, = 0.946) the Cabbibo angle and &, (sin’ 0w = 0.23) the Weinberg angle. I,, V, and A, are the isoscalar, isovector and axial currents of the chiral soliton model discussed previously. The first term in (3.1) comprises the Cabbibo model (charged currents), where as the neutral currents of the standard model are given by the second term. It should be mentioned here that one could derive an Z~r,- like (3.1) directly from the quark flavor currents like tjyw+‘q, . . . , which would give a similar result up to normalization factors. In the form (3.1), the effective lagrangian is directly amenable to the soliton (pv) as well as parity-conserving currents. Clearly, L!Ye, contains parity-violating (PC) pieces. Here, we are only interested in the pv pieces, which are given by LP~$=fiGF 1

cos’ Bc i V~(X)A~““(X)+A~(X) a=,

[cos 2O,V’~(x~-2

,

sin2 fIwZ’(x)] I

(3.2) Here, and in the following, all products symmetrized to guarantee hermiticity.

of currents

are understood

to be properly

Note that the pv effective lagrangian (3.2) includes isoscalar, isovector and isotensor pieces. This property will, of course, be used when we extract the pertinent weak meson nucleon couplings, as we will demonstrate in the next sections. 3.2. WEAK

PION-NUCLEON

The relativistic

COUPLING

form ofthe parity-violating

pion-nucleon

vertex, 6pyN, has the form

x .rr)3N.

(3.3)

L5?rN= -J;G,N(r N denotes

a nucleon

Dirac-spinor,

and

G,

is the weak

TN

coupling*.

For our

purposes, it is most convenient to write down (3.3) in the Breit frame and perform the non-relativistic reduction (the limit 4 + 0), Z’~N = -J~G,x:(T

X nTT)3Xi

(3.4)

Notice that we use the symbol .j;, exclusively for the weak pion decay constant to avoid confusion. The normalization of G, used here is the conventional one. We have been informed by W. Haxton, that the normalization of G, in eq. (3) of ref. ‘) contains a misprint and should read as here. l

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

708

couplings

with x: and xi the conventional two-spinors. To generate the parity-violating nucleon vertex from the effective lagrangian in (3.2), we write &=&,+&,,r with GM the pertinent

meson-field

(M==,P,~), of the soliton

meson-

(3.5)

configuration

and c$~,~ are small

fluctuations. One then simply reads off the terms linear in d,,,,f to get the wanted weak meson-nucleon vertices. Let us illustrate this method for the case of the pion (M = 7~). Our procedure amounts to an axial rotation of the fields U and ,$ defined in (2.9a) and (2.9b), respectively, as has been introduced by Schnitzer 14):

incorporating the pionic fluctuation srP Now the parity-violating TN coupling is obviously of isovector type so that only the term proportional to Ai(x)l’“(x) in (3.2) will contribute. Therefore, we need A, as a functional of ?j (since Z“(x) does not depend on the pion-field). Straightforward algebra leads to A’“[ ij] =fiTr

+

for the “minimal”

{T * ( ij’i[Ldl”&jj

- iji[~dp&ijt

- ij$d’“LJUtfjt

3gi

-&~“Y”P~yTr{7.(77iJuUUti)pUUtl7i+71tU’a,UUta,U77)}, 32~’ model. A similar analysis

leads to the following

relations

for the vector current

(as long as chiral symmetry

A“[nf] = A“ + f

V’” x nf+.

..

V’“[~i]=V~+IA~~~f+.... L These relations are clearly leads to the Kroll-Ruderman pv TN vertex reads

(3.9b)

exhibiting the chiral current algebra theorem of pion photo-production.

-feF& = -2v’?Gr

( V” x mf)Jp d3r,

sin’ 8,;

V&(x) immediately

is maintained)

and e.g. eq. (3.9b) Using (3.9a), the

(3.10)

n I so that ~GF TG, = sin’ Bw Ip(x)Vw(x) I fVT

d3r,

(3.11)

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

Inserting ation

the soliton

rules*,

currents

exhibited

in eqs. (2.21)-(2.24)

we get with V”(r) = fir U,(r) +f?J5, &r~+“F

+

709

couplings

and using the quantiz-

&)I

@Imr2[lo(r)VO(~)-$1,(r)V,(r)1 dr

=k,

(3.12)

0

so that (3.13) This equation agrees with eq. (4.5) of ref. “) up to normalization factor fi cos 20w. In ref. ‘), we used PCAC and current algebra to evaluate G, using the isospin breaking part of the parity-conserving piece of the effective lagrangian. However, one has to be careful with normalization factors, when a current, which is not of pure V-A type, such as the neutral weak current is involved. The proper normalization is obtained by investigating the following commutators for the neutral current contributions,

[Qs, Hk’~~,;-.,~l =2fiGsin2 b[Q5, A~I@l = 2fiGF sin’ O,[ Q, Vi] Ip 1 = -~ cos 20,

[Qv H%,iwI

(3.14)

where we have used the chiral charge algebra. The results in ref. “) for G,, should therefore be multiplied with (a cos 20w)-’ - 1.3. Equation (3.13) obviously holds for any chirally symmetric lagrangian like the “minimal” model as well as the “realistic” pseudoscalar-vector lagrangian of ref. “). 3.3. w-MESON-NUCLEON

The relativistic

COUPLINGS

form of the parity-violating

wN couplings

read

~~~=-N(hO,+h~~~)w~‘y,y,N which is the Breit-frame

and q+O

(3.15)

limit is

Ze”N =&/IO,

* oxi.

+ r’h&r

(3.16)

We will now calculate the two parity-odd coupling constants hz and hk. To demonstrate the calculational procedure, we will only use the “minimal” model since it is less involved algebraically but exhibits all relevant features. It is most instructive to first calculate h:, which goes with the third component of an isovector and therefore stems from the term proportional to AiIp in eq. (3.2). We have

=e&1, = -2fiGF * These thereof).

are

sin’ Ow

(r = -2iO Tr [A’AT],

For further

aI (x) + A+(x)+

I,(x)

9

w’(x) d3r.

(3.17)

I 7 = 2iO Tr [AA+71

details, see e.g. ref. ‘).

and

r’,~,, = -5 Tr [AT,A’T,]

(and

products

710

hf. Kaiser,

couplings

U.-G. Meissner / Weak meson-nucleon

From (3.17) we can read off h: once we know the derivatives For the minimal model

aA’“” --_dw’ -

3gi a’“0 ijk E Tr{ra(i)iUUti)kCJU’+ 32rr=

and integration,

UdiUtakUUt)} .

_2sinzF-F’sin2F

riF’sin2F+iiT.i r After quantization

~~~/~~i and aA”“/?k~‘.

r

r2

we end up with sm 2F+sin’

F)w dr

1.

(3.18)

(3.19)

The first term stems from f d3rA”(x) =igA x,:Ur”xi and the fact that in the “minimal” model we have exact VMD in the isoscalar channel. In general, this will not hold. Also, the second term has its particular form because of VMD. As pointed out by Shmatikov “), who derived h: in the framework of the w-stabilized of Adkins and Nappi I’), the isoscalar current decomposes as l~=f(I+hg”)B~+$?r~w,.

skyrmion

(3.20)

Onty for the VMD choice, h = -l/g,, g, =zg, we then end up with the result of (3.19). The formula for h: using the RPV lagrangian (2.5) can be found in appendix B. It is important to note the second term in eq. (3.19). It clearly shows that our approach goes beyond the factorization approximation, which would give only the term proportional to g,. Let us now turn to the calculation

of h:. For that, we have to pick out the isoscalar

components of (3.2). These are obviously in the terms proportional to VzAFa. Clearly, there are two isovectors which can couple to an isoscalar, isovector or isotensor. However, to ensure hermicity, we have to work with symmetrized products, i.e. V;A’” so that only the symmetric isoscafar part,

+ $( V;Apa + A; V”“) ,

components

(isoscalar

(3.21)

and isotensor)

survive.

For the

then gives ZrN:“,(,, = $‘? Gr(2 cos2 %,+ cos 2%w) X

aA,(x) aw,

F

V (x)+A,W~

Wl*(x)

Iw’(x)d3r.

(3.22)

N. Kaiser,

For the “minimal” Evaluating

U.-G. Meissner / Weak meson-nucleon

model,

and using the collective hO,= -G

o

of At and

only the time-components dA~/dw’

the derivatives

and rl V~/dw’, putting

quantization

711

couplings

Vz depend

pieces together,

on w.

integrating

rules we are left with

(2 cos’ O,+COS 20,)

X’

X

I

[rF’U,,(r)sin2F+2f~sin2F(~,+~,)-2rF’ao(r)sin2F]dr

(3.23)

0

where the radial

U”(r) and a,,(r)

functions

3.4. WEAK p-MESON-NUCLEON

For the p-meson,

are defined in eqs. (2.23a) and (2.25a), for the RPV lagrangian are summarized in

for h:’

respectively. The formulae appendix B.

COUPLINGS

one can write down four independent

parity-violating

couplings

+h;N[7xpC]3~ysN,

(3.24) N

with MN the nucleon mass. In quark-model calculations4’5”7) one generally neglects hj,‘, but we will also calculate this vertex. After the non-relativistic reduction, (3.24) order in q:

reads to leading

-p

ih” 2Miv

(3.25)

XFa * q(T x p”).JXi.

Notice that the conventional weak pN couplings are proportional to the space components of the p-field, whereas h L’ is proportional the time component pO. First, let us calculate the constants hj and hz. They stem from the terms VzA@, which can couple to form an isoscalar or an isotensor as already discussed in the context of ht. The pertinent lagrangian reads

37,u,,2,

= v-h=

j

[cos20~f,

dV’“‘(x) + cos 20w ~ apjb

(

(yA,(xj+

A;(x)

+ VP”(x) y

VV)$f)

>Ip”‘(x) dir,

(3.26)

712

N. Kaiser, U.-G. Meissner / Weak meson-nucleon couplings

with

aVP” -= itp’b

-igf”,Silj

l3AC””

-

apjb

=

Tr [A(T~ cos F+ I’m - ;(l -cos

s*j;f “,gc cdeje sin

F Tr [AT~A+P]

For b # 3, after integration and quantization whereas b = 3 leads to h: + hE,/&. Finally,

~+-!g

gG,f2,(2cos*

r2[4V,(r)

X

F))A’T”]

Tr[Ar”A’rb]

Tr [ArCAtrb]

(3.26a)

(3.26b)

one gets the combination we obtain

hi - hE/(2&),

@,+cos28,+,)

sin F+A,(r)(l+2

cos F)+A,(r)]

dr,

(3.27)

r2[A,(r)(l+2cosF)+~Az(r)(2+3cosF)-2V,(r)sinF]dr,

(3.28)

with the radial functions V,(r), A,(r) and A,(r) given in sect. 2.3. For the RPV lagrangian the equivalent formula can be found in appendix B. Next, consider the coupling constant hi. It is of isovector nature and stems from the IPAP term in (3.2). We have (3.29) For the “minimal” model, only Ai depends on pJb, but Z”(x)(dA~(x)/apJb) vanishes if hermiticity is respected after collective quantization. Therefore, we find hk = 0 (“minimal”

model)

.

(3.30)

For the more realistic RPV lagrangian, which also includes wp~ correlations in the anomalous action, the coupling hi is, however, non-vanishing (as can be read off from eq. (B.8) in appendix B) because I” in that case depends on p’“. Finally, we have to evaluate h:. Here, a small complication arises due to the fact that it is a coupling of the “magnetic” type (-aioq’Ys)+ To pick up the first non-trivial term when one takes the limes q +O, we have to re-instate the factor exp {iq - r}* in the integral of Lz?$&,,, i.e.

qkYw = -2fiG, * In the previous

calculations,

sin’ Bw we could identify

aIw -A;+I“ apOb this factor

with unity.

3A3 + aP

pob d3r I

(3.31)

N. Kaiser,

Picking

up the leading

U.-G.

/ Weak meson-nucleon

term proportional

k,’ =32an 27 P

Again, the formula

Meissner

.

sin

2

to 4, after collective

BwGrmZ,M,f

‘,

quantization,

we find

sin F dr .

r3w

for hr using the RPV lagrangian

713

couplings

is written

(3.32)

down in appendix

B.

4. Nuclear parity violation In this section, we will first present the results for the parity-violating mesonnucleon coupling constants and compare them to recent quark-model calculations. We will then proceed to calculate some relevant PNC observables following Adelberger and Haxton ‘) as well as Desplanques. ‘*) We will not perform any new nuclearstructure calculations but rather use existing calculations to estimate these observables.

4.1. RESULTS

FOR THE

WEAK

MESON-NUCLEON

COUPLINGS

In table 1, we exhibit the weak rrN coupling constant G,, the weak pN couplings h F’32, hc as well as the weak wN couplings h:’ for the “minimal” model (&= -0.4194, &v, = y3 = O)* and the RPV lagrangian with 6 = 0.4, &vs = 1.9, R = -0.04 and K = y3/ y2 = 0, 1 or 2 (the best description of all nucleon properties was found inref.‘)forK=+l.. + 2). For comparison, we also give the results of Desplanques TABLE

1

Weak meson-nucleon couplings as calctdated from the chirai soliton model for various _ _ input parameters (h, g,,,, K) in units of IO-‘. (I;, &v#, K) = (-0.419, 0,) gives the result of the “minimal” model, whereas the more realistic pseudoscalar-vector Lagrangian leads to the results in the column &vrh = 1.9, h = +0.4, and K =O, +I and 1-2. For comparison the quark-model calculations of Desplanques et al. (,DDH) [ref.“)] and Dubovik and Zenkin (DZ) [ref. ‘)] are aiso given. The vatues in brackets in the DDH column give the “reasonable range” estimate defined by these authors.

x gvv4 K

-0.419 0 0

0.4 I .9 0.

0.4 1.9 1.0

0.4 1.9 2.0

0.20 -0.8 0 -4.7

0.28 -0.5 +0.08 -3.8

0.19 -1.9 -0.02 -3.8

0.19 -3.7 -0.1 -3.3

-1.3 -2.1 -0.85

-0.6 -2.1 -0.9

-1.1 -2.2 -1.0

-1.4 -2.2 -1.0

* For the definitions

of the coefficients

DDH(range)J) 4.54 -11.4 -0.19 -9.5

(0 + 11.4) (-30.8-rl1.4) (-0.4-+0.1) (-11.0-t-7.6)

-,.9O -1.1

(-10.3-+5.7) (-1.9++0.76)

y, (i = 1, 2, 3) see appendix

DZ’) 1.3 -8.3 +0.39 -6.7 0 -3.9 -2.2

A, eq. (A.12) and also Eq. (2.7).

N. Kaiser,

714 et ai. “)

(DDH)

(including

U.-G. Meissner / Weak meson-nucleon

their “reasonable

ranges”)

couplings

and of Dubovik

and Zenkin

‘).

The results

for the “minimal” model are generally on the small side, in particular wprr for /z:‘,~. This is, however, expected since this model facks the important

correlations in the anomalous action. For the RPV lagrangian, we find results which are generally comparable to the quark-model results, with some exceptions. The weak rrN coupling constant is a factor -18 smaller than the DDH value, and a factor 5 below the value of Dubovik and Zenkin “). We should point out, however, that a best fit to most reliable data as discussed by Adelberger and Haxton ‘) gives also a smaller value for G, than the one of DDH. Furthermore, the P-decay rate upper limit for G, 19), “Ne(O+, 1) + *8F(O-, 0) sets a largely model-independent G, < 1.43 x 1o-7

(4.1)

13.45 for the strong pion-nucleon coupling. It is comforting when One USeS &NN= to realize that the soliton model predicts G, which lies below the empirical upper bound. Another difference to the quark-model is the weak pN coupling h:‘. Holstein has argued 17) that the assumption of taking hL’= 0 will have no significant effect upon the calculation of nuclear parity-violating observables*. While this might be true in general, our model predicts a non-zero and non-negligible value for h:’ and we will therefore always include this term. Furthermore, the coupling hz comes out relatively small as compared to the quark-model values. We note, however, that it is most strongly dependent on the empirically unknown parameter K. For K = +2, which gives an overali satisfactory description of other nucleon properties, the value of hi is within a factor 3 to 4 of the quark-model results 4,5). We should point out that the best tit of Adelberger and Haxton ‘) gives a smaller value of hz (=5.7 x lo-’ for g, =2.79) than the one of DDH “). For the “complete” model of ref. r3) which employs the Bardeen-subtracted Wess-Zumino action, the results for h:’ and h:“.* are similar to the ones in the ‘“minimal” model. For hz, one finds a positive value (-1.4~ lo-‘). This opposite sign compared to the “minimal” or RPV model can be traced back to the chiralsymmetry breaking terms of the anomalous action and we therefore do not consider further the results obtained in the “complete” model. We should point out that in general the “complete” model gives results very similar to the best values of RPV model, with the exception of the pion-nucleon II term, the violation of the Goldberger-Treiman relation and the sign of the weak coupling hi. As we already mentioned in the Introduction, the soliton model also predicts the strong meson-nucleon vertices. In table 2 we summarize the strong meson-nucleon couplings calculated in refs. 8*2o)together with the commonly used products of weak and strong coupling constant I)** (4.2a) * In the framework of the MIT bag model, Holstein estimates ** Note that in ref. ‘) we called the first coefficient 6,.

!I:’ = -7

x

lo-‘.

N, Kaiser,

U.-G. Meissner / Weak meson-nucleon TABLE

couplings

71s

2

Strong and weak coupling constants. The first three columns give-the predictions of the chiral so&on model for various sets of the parameters (h, &+,. K). The st~~n~cou~li~~ strengths for the “minimal” model (first column) have been calculated in ref. ‘O), whereas the ones for the RPV lsgrangian are taken from ref. ‘). The column denoted “DDH” gives the best values of the quark-model calculation of ref. “), “AH’” the best fit of the weak-coupling constants to the “best set” of available PNC data as explained in ref. ‘), and “DZ” the recent calculation of ref. ‘).

h” &Wch K

g,,, & &” Kg1 KU C 6, 6 & CC, c, 6

-0.419 0 0

0.4 1.9 0

0.4 1.9 1.0

0.4 1.9 2.0

DDH”)

AH’)

14.34 2.67 8.78 5.38 - 0.21

12.51 3.11 8.76 5.02 - 0.19

15.24 3.14 8.29 5.52 - 0.29

16.60 3.15 7.95 5.82 - 0.33

13.45 2.79 8.37 3.70 - 0.12

13.45 2.79 8.37 3.70 - 0.12

0.5

0.6

1.o

0.8

0.6 3.0 0.04 5.9 4.6 4.2 1.7

0.6 5.9 + 0.17 5.3 5.7 4.1 1.7

10.X I 5.9

0.0 6.2 5.6 3.7 1.8

- 0.14 6.0 2.5 4.0 1.6

-;g,,hl,

5.0

8.0 4.8 0.0

X.0 0.3 9.8 2 7.0 i 0.0 0.0

Fz = -$g,h;

,

0.27

13.3

F<,= -+g,,h’:

F, =

G, = -$g,h:

G, = -$g,h$ ,

DZ’)

t 3.45 2.6 7.8 3.7 - 0.12 3.1 1 1.5 - 0.54 9.3 1 6.3 9.2 0.0

(4.2b) (4.2Cf

H, = -$g,h;,‘. Again, it is parameters not far from now calculate

interesting to note that the chiral solieon model for the best set of I h = 0.4, &v+ = 1.9, R = -0.04 and K = 1.0-2.0 gives results which are the quack-model predictions, with the exception of F,, and F,,. Let us some physical

4.2. CALCULA’flON

Here,

(4.2d)

OF PNC

observables

with these coupling

constants.

OBSERYABLES

we will use existing

nucfear

structure

calculations

to estimate

the PNC

observables AL ~~o~g~t~di~al asymmetry~ in ij Cp, e+ d, ijt4He and the paritymixing matrix elements in ‘*F, 19F, and 2’Ne. These have been discussed by Adelberger and Haxton ‘) as the most reliable set of data to give insight into parity-violation in the NN interaction. We will also give the results for some other observables like P, (circular polarization) in 4’K, “‘Lu and “‘Ta as well as c in “0. Of course, these results should only be considered as indicative for reasons discussed in ref. ‘1 and below.

716

N. Kaiser,

il.-G.

Meissner

/ Weak meson-nucleon

Our strategy to include the weak and strong is to write all PNC observables 6 as

coupling

couplings

constants

from our model

B = c,F, + i d+xF;:+ i ei#3Gi + d4aH, , i=O i=o

(4.3)

with (Y= 1 or (Y= (1 + ~y~“‘)/4.7,

(4.3a)

p = 1 or p = (1 + ~~~~~‘)/0.88,

(4.3b)

depending upon if we have to correct for the different vector-to-tensor coupling ratios predicted by our model to the conventional choice K~ = fcv = 3.7, K, = K, = -0.12. To clarify this point, let us write down the PNC NN potential in the one-boson-exchange approximation

+&

For(l)

.7(2)+4F,[7(1)+7(2)].

N [i

+iH,[7(1)x~(2)lZE~fl)+~(2)1~~,(r) , I

(4.4)

with r = r, - r,, p =p, -p2, u, = [p, e-“‘J/4m], v, = {p, e-m~~r/4m) and K, = G, -F, . It is obvious that all p-meson terms (a~ F,, F, and F2) are proportional to (1 +K~), whereas the w-exchange terms go with GO( 1-t K,) and G,(l+ K,). The nuclear structure coefficients c,, di, ei and d4 will be taken from the existing literature. If not stated otherwise, we use the structure calculations by Desplanques and Missimer”) including the terms propo~ional to hg. For the calculation of the parity-mixing matrix elements, we foltow Adelberger et al. 19) and Adelberger and Haxton ‘). To be specific, for the ‘*F matrix element we use the 2Aw shell mode1 calculation of ref. 22), for 19F the P-decay scaled configuration ‘) and for “Ne the Kuo and Brown23 ) 0-lhw G-matrix wavefunctions scaled by one-third for consistency with the ‘*F and r9F effective charges*. Finally, for A, in p+4He we use the calculation of Jacquemart et al. 24)*t. Again, we would like to stress that the results we are going to present should only be considered as indicative since some l l

For a thorough discussion on this point, we refer the reader to sects. 6 and 7 of ref. ‘). * Using a calculation of Lang ef aLZS) leads to somewhat smaller results (-15% difference).

N. Kaiser,

of these observables related

to nuclear

U.-G. Meissner / Weak meson-nucleon

are clearly sensitive structure

couplings

to short-range

correlations

In table

3 we summarize

calculations.

111

and other issues the pertinent

results for the “best set” of observables following ref. ‘). The overall agreement with the data is satisfactory. Let us stress two particular features. First, the parity-mixing matrix

element

in ‘*F is considerably

smaller

than

the quark-model

calculations

predict, for the simple reason that our weak G, is considerably smaller than theirs. The vector-meson contribution is enhanced as compared to the quark-model result, but still the sum is decreased. The opposite effect happens in the (VP,,) of 2’Ne - the dominating contributions proportional to F, and F. do not cancel each other as it happens in the quark-model calculations. Overall, the soliton picture works remarkably well if one keeps in mind the simplicity of our approach and the fact that we simultaneously predict the weak and strong meson-nucleon couplings. In table 4, we give some PNC observables in medium and heavy nuclei based on the nuclear-structure calculations of refs. ‘8,2’). Again, the overall agreement of our model (for the best set of parameters) with the recent quark-model calculation of

TABLE

3

Calculated PNC observables. We give the results of our best set of parameters 6 = 0.4, gvvd = I .9 and K = 2.0 together with R = -0.04 in comparison with the quark model calculations of refs. ‘.‘), the best fit of ref. ‘) and the experimental values (mostly taken from ref. ‘) Observable p+p. A;“‘(15 MeV) [x10’] p+p, A:“‘(45 MeV)[xlO’] p+d, A, (15 MeV) [x10’] p+‘He, A, (46 MeV) [x10’] lXF, -i (VraNe.) [eV] ‘“F, -i (V,,e) [eV] “Ne, -i (VP,,.) [eV]

(0.4, 1.9, 2.0)

DDH 4,

AH r)

DZ “)

Exp.

-0.88 -1.04 -0.30 -1.4 0.03 0.15 -0.17

-0.77 -1.35 -1.4 -3.3 0.28 0.46 -0.018

-0.79 -1.3 -0.77 -2.6 0.20 0.29 -0.036

-1.3 -2.3 -0.65 -3.5 0.10 0.28 -

-1.7*0.8 -1.5i0.22 “) -0.35 10.85 -3.3 *0.9 co.09 0.38 i 0.10 co.029

TABLET

PNC observables in medium and heavy nuclei. We consider the reaction ‘h0(2m) + “C+ 1y and the circular polarizations of y-quanta in “K(s- +$‘), “‘Lu(p-+;+) and ‘X’Ta($t + $‘) for our best set of parameters (i = 0.4, g,,, = 1.9, K = 2.0 and R = -0.04). For comparison, the quark-model results of ref. s) and the empirical values taken from that reference are also given RPV ‘60, c “K, P, x 1’5Lu, P, ‘*ITa, P,

x 10S[&v] 10s x lo5 x 10’

0.7 0.7 2.2 -1.9

DZ’

Exp.

1.1 1.3 4.9 -3.5

1.0*0.1 2.0 * 0.4 5.5 * 0.5 -5.210.5

Iv. Kaiser, U.-G. Meissner / Weak meson-nucleon

718

couplings

ref. ‘) and experiment is satisfactory, although the predictions tend to come out on the small side. However, the issues surrounding nuclear structure and possible medium modifications of the fundamental meson parameters* make these numbers much less reliable on both the theoretical and experimental sides. Nevertheless, it is comforting to see that the sofiton model of the nucleon produces numbers which are in the right ball park. It is obvious that further high-precision experiments are needed to give more restrictions on the pv nuclear amplitudes, FinaIliy, let us briefly discuss the five etementary S-Pamplitudes which characterize the two-nucleon matrix elements of the PNC interaction VPN’ = VI?:;+ Vf;“;‘zl+ Vf;‘;2z at low energies. These amplitudes are the ‘Su++3Pu (AI =0, 1,2), ‘Sr++‘Pr (A1 =O) and 3S, e3P, (A$= 1) transition. Near threshold, these can be related to the meson-exchange potential via ‘f m2Vp,= m’V,,

(F,+F;+~~F,)(~+K-,)+(G,+G,)(~~K,,,)=~.~x~O-~,

(4Sa)

=(F,-F,+&F,)(~+K,)+(G,-G,)(~+K,,)=~.~~~O-~,

(4Sb)

m2Vpn=(Fo-J’~F2)(2+~p)+Go(2+~,)=2.2~10-B,

(4Sc)

2

m2Wpn=2F,,

m +2H,+GG,+F,=4.3x10-6, ( m71>

m2Upn = G”K~

-3Fo~p

=

(4Sd)

-10.5 x 10e6,

(4Se)

with m = m, = m, and the numbers are for our best values of the RPV mode& i = 0.4, iv& = 1.9, K = 2.0, and R = -0.04. The values for these amplitudes are again simitar to typical quark-model calculations. 5. Summary and twtiook We have investigated parity-violating pion-nucleon and vector-meson-nucleon vertex functions within the framework of a non-linear chiral meson theory in which nucleons emerge as solitonic excitations. Such an approach is motivated by the idea that the underlying non-linear effective lagrangian (2.1 f approximates QCD at low energy and momentum transfers up to about 1 GeV, The physics of baryons, including their various vertex functions for electromagnetic, weak and strong couplings is thus described in terms of a few mesonic parameters: the pion decay constant ,f, = 93 MeV (which essentially sets the scale of the spontaneous breakdown of chiral symmetry), the pion mass m, = 138 MeV, and four universal vector-meson pseudoscalar coupling constants (gv,, = g,_, gvvm -g,,,, 6- g,,,, and K). The first three could be determined from meson reactions like p + 27r, 4 + JIT, 4 + vp and w -+ 3~. [ref* s)] The constant K could be fixed from a best fit to the nucleon properties. In the presence of electromagnetism, another parameter (R) related to processes like p”+ ray or w + t-rOy appears and can be fixed by these decays, with R = -0.04. * For a recent ref. y.

approach

to this problem

in the framework

of non-linear

chit4

lagrangians,

see e.g.

N. Kaiser, U.-G. Meissner / Weak meson-nucleon couplings

Our main interest

here is the following

question:

719

to what extent can such a unified

theory of mesons and baryons account for the parity-violating meson-nucleon couplings derived by either the quark-model or from a best fit to all reliable PNC observables? Our findings

can be summarized

as follows:

(i) The weak TN coupling constant G, comes out below the empiricat limit c zp< 1.4x lo-‘. The absolute value (-0.2 x LO-‘) is relatively small and rather stable against variations in the third parameter of the “realistic” pseudoscalar vector lagrangian “). We should point out, however, that a best fit to the most reliable PNC observables as discussed in ref. ‘) under the exclusion of the “Ne data leads to c&=0. (ii) The two pv w-meson-nucleon coupling constants h’l, and II: come out of the not far from the quark-model prediction of DDH “). ht is somewhat order -1 x lo-‘, sensitive to changes in K, whereas h: is very stable. (iii) The pattern of the relative strength in the three axial weak p-meson nucleon couplings is the same as in the quark model, we find lh:‘,Ib lh;,l> lh:,I for our best set of parameters. The absolute value of h: is, however, smaller than in the quark-model calculation 4,5). The fourth “magnetic” pv pN coupling, hc, is not small (==- 2 x lo-‘) and rather stable against variations in K. (iv) One of the main virtues of our approach is that it simultaneously predicts the weak as well as the strong meson-nucleon couplings. The latter have already been calculated in refs. 8,0) and good agreement with OBE phenomenology and dispersion theoretical approaches was found. In particular, the soliton model predicts a strong tensor-to-vector coupling K~ = 5-6 for the p-meson, and a dominating vector coupling for the w-meson (K, - -0.2). The form factors related to the strong meson-nucIeon vertices are of monopote type and have cut-offs of order of 1 GeV. (v) For our best set of parameters of the RPV lagrangian (h”=O.4, &,v+ = 1.9, K = +2, R = -0.04) we have used existing nuclear structure calculations to perform a systematic analysis of the best set of PNC observables (A, in fit-p, p+ d, p+‘He, { VpNc) in ‘*F, ‘“F, “Ne) as defined by Adelberger and Haxton ‘). The overall agreement with the data and comparable quark-model caiculations’.4.5) is satisfactory, although our results generally tend to be on the small end. This can be traced back to the relatively small values of G, and hz. Similar trends hold for PNC observables in heavier nuclei. These results, however, should only be considered as indicative due to experimental and theoretical uncertainties. In summary, we have given a unified treatment of weak and strong meson-nucleon vertices, based on the soliton model of the nucleon which reflects the symmetries of the fundamental strong interaction theory. We argue that most of the important non-perturbative effects of the quark-gluon interactions are included in our model; e.g. as noted in sect. 3.3 our approach goes beyond the factorization approximation. Of course, it is not without its own problems. As we already pointed out in the Introduction, the simplified quantization procedure to project onto states of good spin and isospin leads to too high masses for the nucleon on 4(1232). This might,

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

720

however, thermore, present

eventually be overcome by more elaborate quantization procedures. Furwe have restricted ourselves here to the two-flavor sector. In light of the discussion

it appears sector.

couplings

of possible

of utmost

While

strangeness

importance

our formalism

admixtures

to generalize

is general

to the proton

our formalism

wavefunction

27),

to the three-flavor

in the sense that includes

any

U(N,),x

flavor symmetry, the more involved symmetry breaking (i.e. m,/AocD1) in the three flavor sector will complicate an analysis similar to the one presented here. Of course, more precise experiments are also needed so that the large uncertainty in the value of the weak pion-nucleon coupling constant could be reduced. This would also give a more stringent test on the soliton scenario to describe the low-lying baryons. Note added: After finishing this work, we received a preprint by Grach and Shmatikov (ITEP 100-88, Moscow, 1988) in which the weak 77,p and w-meson nucleon couplings are calculated for the “minimal” model. They find G, = 0 and hi = 0, in agreement with our results. Their results for h:’ and h>’ are, however, considerably larger than ours. Unfortunately, they do not give any final formula so that we could not pin down the discrepancies. We believe that their results rest on assumptions of the NC behavior of certain terms which we did not neglect. U( N’)R

Appendix A CURRENTS

OF THE

“REALISTIC”

PSEUDOSCALAR-VECTOR

LAGRANGIAN

In this appendix we will give the explicit expressions for the nucleonic current densities of isoscalar, isovector and axial currents derived from the effective meson lagrangian eqs. (2.2), (2.5). Using the ansdze for the adiabatically after a somewhat lengthy - Isoscalar current:

rotated

hedgehog

eq. (2.10), we obtain

algebra*: Ip(x)

= (Mr),

K x rl,(r)),

(A.1)

with

+$(F’(l+sin’F-(G+l)cosF)-G’sinF) +3F’(G+1-cos 3gr2

F)2

-$(F’((G+l)cosF-cos2F)+G’sinF), * In what follows, are negligibly small.

we will neglect

all contributions

from the q-meson

(A.2) as discussed

in ref. “) since they

U.-G. Meissner / Weak meson-nucleon

N. Kaiser,

-2

(F’(l,(l-

cos F)+52+(G+2)cos

couplings

721

F)-2cos2F)

+ (G’ - 5:) sin F) . - Isovector

(A.4)

current:

V““(X)=(-U~(Y)T~(AT~KA~A~)+(U~(~)+U,(~)) Tr (AT. iK.

;A’?)

9

Tr (A;x?A++‘))

V,(r)

,

with cos F. {,)+T

UO(r)=ffJ4sin4$F+2

+-$(+‘sin

F-cos

-$&F’cos

+g

d,g

F))

F(G+l))-G’sin

F(G+l-cos

4

$F’sin’F

F)

F(G+l-2cos

+4(F’(1+2sin2

Yl 2r

F)

.

-sm2F-2@F’cos2F-+‘sin2F ( r

U1(r)=-2fZ,(b,+{2)+-$+F’f%+sin2F

>

(A-5) (A.61

r

2

F)+EwF’sin’F

V,(r)=L(2sin4$F-Gcos r

+E(w’sin

F(G+l-2cosF)

+w(F’(1+2sin2 -~wF’cos

F-cos

F(G+l-cos

F(G+l))-sin

FG’))

F)+y(2wF’sin’F-fw’sin2F)

(A.7)

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

722 _ Axial

couplings

current: A@(X) iA,

= (a,(r)

Tr (AT * (AK x ~)A+T~), (AT - ?A’T~))

Tr (AIAtT”)+$Az(r);Tr

(A.81

with c#JF’ sin F cos F

a,(r) = f ‘, sin F (2 - 25, - cos F) +$

+-$(4(F’sinF(G+lt2cosF)-G’cosF) +~‘((G+l-cos~)cos~+sin~F)) Y3

+~+F’sinF(G+l--cosF)

+~(+‘sit?F+Z~F’sinFcosF),

(A.9)

2

F(2G+2-cos

A,(r)=&sin

r

Fcos

F)+?/‘wF’sin r

F

+~(o(F’sinF(G+1+2cosF)-G’cosF) +w’fG+l-cosF)cos

F+sin*F))

+BwF’sinF(G41-cosF) r

+2d,g r(

+y,W(G+l-cos r*

w’sin2 F+wF’sin2F)

(A.lO)

F)‘+4d,gWsin2F r2

(A.1 1)

The coefficients y,, y2 and -yS are related to the parameters term ~??w, in eq. (2.5) c,, c2 and c, in the following way:

y1 = 6iJ3(c,

The corresponding

currents

-&&/g

- c,/g’)

of the Wess-Zumino

,

(A.12a)

y2 = 4ic21g,

(A.12b)

y3 = 2iJzc31g2.

(A.12~)

for the “minimal” y2 = 0

model follow immediately yPJ=o.

by setting (A.13)

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

couplings

723

Appendix B WEAK

VECTOR-MESON-NUCLEON

COUPLING

CONSTANTS

FOR THE

RPV LAGRANGIAN

In this appendix

we give the formula

for calculating

the weak parity-violating

p-

and w-meson-nucleon couplings h:‘, h>‘,2, and hi,’ as defined in sect. 3.1 for the “realistic” pseudoscalar-vector lagrangian, eqs. (2.2), (2.5). As already mentioned in sect. 3.2, the formula for the weak rrN coupling G, (eq. (3.13)) can also be used for the RPV model (with V,(r) =+[2U,(r) - U,(r)]). Eq. (3.13) is valid for any effective meson lagrangian with the feature that the only term which breaks chiral symmetry is the pion-mass term, i.e. PCAC is exact. In the RPV model, the isoscalar current Z@(x) depends explicitly on the pion field. It is however invariant under the axial transformations (eq. (3.6)) in accordance with the usual current algebra commutation relations. Therefore, a term proportional to A~(~)(;tZ~(x)/?m’) cannot contribute to G,. The calculation of the weak vector meson nucleon coupling constants for the RPV lagrangian turns out to be much more involved as in the “minimal” model discussed in sect. 3. The obvious reason is far more complicated dependence of the (isoscalar, isovector and axial) currents on the p- and u-meson fields as can be seen from the expressions for current densities in appendix A. Nevertheless, the calculation of h:l and h:132” 1s . straightforward and analogous to the one sketched in sect. 3. One has to calculate the pertinent derivatives of the various currents with respect to the vector meson fields and multiply with the corresponding current as it appears in the parity violating part of the weak current x current lagrangian. These expressions are then to be evaluated using the ans&ze of the adiabatically rotating hedgehog and then are spatially integrated. The remaining matrix elements depending on the rotational collective coordinates (A, K) have to be quantized in terms of nucleon spin and isospin operators. In this procedure, one has to be careful in order to guarantee hermiticity of the final results. After a quite lengthy algebra we obtain the following results for the two weak o NN coupling constants: h’ w

=~~sin’ew

G

9

with lo(r), A,(r),

F

A2(r)

d~r2[Z~tr)(3s,(r)+s2(r))+~(3A,(r)+A2(r))l given in appendix

03.1)

A and

aA, y1 S,(r) = -=-FF’cosFsinF+E(F’sinF(G+1+2cosF)-G’cosF) aw r

+~F’sinF(G+1-cosF)+4dlgF’sinFcosF, r

r

03.2)

724

N. Kaiser, U.-G. Meissner / Weak meson-nucleon

Sz(r)

a&(r) =r=

couplings

-S,(r)+~sinzF+$(1+sin2F-(G+l)cosF) 4d,g sin2 F. r2

++G+l-cosF)‘+r2 h;=$fi7r(2cos2

t3,+cos20w)G,/0

X

+ Ur))

dr r2[(S,(r)

(B.3)

-

K(r)

(B.4)

with aVI(r) y1 K(r)= -=-FF’sin2F+z(F’(l+2sin2 aw

F-(G+l)cos

2r

-E

~‘cos

The four weak pNN hb’=-

F(G+~-cos coupling

SJzrr sin2 BwGFM, 27

2&g F’ sin2 F.

F)+-

constants

F)-G’sin

F)

(W

r

read:

00 dr[12gftr3sin I0

xA,(r)+(y,+4d,g)(2r2F’(1-cos

FZO(r)+2yjr2F’(G+1-cos

F)

F)A,(r)

+ sin F( rA2( r) - r’A:( r)))].

(B.6)

For numerical purposes it is convenient to perform a partial integration on the last term in (B.6) proportional to A’,(r), -r2sin FA;(r)+A,(r)(r’F’cos F+2r sin F). In hc, which enters in a “magnetic” type coupling -i(a - q)(Tx p&, one has to include also total derivative terms of the form ai (Z,,(~JA~/~IY~~~~)),etc. For all other weak meson-nucleon couplings, which are of “charge” type such total derivative terms do not contribute. We have checked this carefully.

%&%-

hi=--

27

J

(2

cos2 OC+ cos 2Ow)gG,

cc

X

dr{r2V,(r)[4ftsin

F+y,w’cos

F+(y2+2y3)

0

+(wF’sin

+r2(A,(r)

F+f(l-cos

f2,(1+2cos

;F’cos

F))-2y,fGcos

F)-:w’sin

F+-

F

sin F r

F]

>

+ y3 _wG sin F r

1

N. Kaiser,

U.-G.

Meissner

/ Weak meson-nucleon

1 -$ Py,(cb’+ 4)cos

+r2A2(r)

[

F

.ft+(y2+2yJ)Esin

F+l

F+(y2+2y,)d(2rF’sin

- 2 y&G

725

couplings

-cos

F)

cos F]

+ U,(r) ,~~z[Yz(~+~~‘)+(Yz+~Y~)~(~-cos~)+~Y,~GI

aO(r) -s[-y2(~+rq5’)sinF+(y2+2y,)@(rF’cosF+fsinF)

sinFII

+ r&G dansin 27

h;=--

,

(B.7)

-GF

e

wo

a,

X

I

dr{a,(r)rF’[(y2+4d,g)(l-~osF)+2y,(G+1-~~~F)]

0

+A,(r)[(y2+4d,g)r2F’(cos

F-l)

+ 2 y3( r2F’( 5, + 1 + cos F) + r sin F(25, + 12+ G)] +A,(r)[2y,r

sin F(G+[,)]-3gr21,(r)[y2~‘($+cos

+3gMr)[yz(r4’+d)(i+cos h:, = -*(cos 27 X

+ r*A,( r)

F)+(y2+2yd@F’sin

20w-~0s’

cos

F+'

F+ y2w‘cos F+(y2+2y,)

r

sin F

- 2 y3 w G sin F r I

$fJ2+3cosF)-(y2+2y3)WsinF-$y,U)GsinF r

b(r) -s[y2(4+@‘)(3+7

cos F)+(Yz+~Yx)+

r

F] 03.8)

1-t 2 cos F) + y2w’ sin F

F’

-(Y2+2Y,h

WI,

O,)gG,

4r2V,(r)[-2ftsin

4ft(

F)+(~~+ty,)wF’sin

I

IV. Kaiser, U.-G. Meissner / Weak meson-nucleon

726

x

(7rF’sin

F-y{1

-cos

F))+y,4G(8

couplings

cos F-3)]

+~~y~(~+~~~)(2+3cosF)+(y,+2y,)~(3rF’sinF+f(l-cosF)) - y&G(2

-- adr)

402 [

-J

2

+ 3 cos F)] yz((t)+r@‘)

y3~G sin F

sin F-(y,+2y,)qS(rF’cos

11 .

F+ikin

F)

(B.9)

References 1) 2) 3) 4) 5) 6) 7) 8) 9)

10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27)

28)

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