Quark-pion coupling constant in a chiral quark model
N B A R I K and B K D A S H *
Department of Physics, Utkal University, Bhubaneswar 751 004, India
*On leave from Department of Physics, S.C.S. College, Purl 753 001, India MS received 30 October 1984; revised 11 February 1985
Abstract. Incorporating chiral-symmetry to the potential model of quarks with confining potential U(r)= i(I +7°)ar z with m, = IOMeV and a = 2.273 fm -3 that gives a reason- able quark-core contribution to ~/,, ~r 2 )~/2 and gA, the quark-pion coupling constant for quarks in a nucleon is estimated. GqcJ4n obtained between 0-4 and 0-5 is consistent with those extracted from experimental vector meson dceay-width ratios by Suzuki and Bhaduri. The nucleon-pion coupling constant G~NJ4n comes out to be of the order of 13" 1 in reasonable agreement with the experimental value.
Keywords. Quark; confinement; vertex function; quark-pion coupling constant; chiral symmetry.
PACS No. 12-40; 13-75; 14.20; 14.80
I. I n t r o d u c t i o n
In a phenomenological model o f baryons, if one considers the quarks as point Dirac particles moving independently in an effective potential taken as an equal admixture o f scalar and vector parts, the static electromagnetic properties o f low lying baryons can be explained reasonably well (Ferreira 1977; Ferreira et ai 1980; Barik and Das 1983a, b;
Barik et al 1985). However unlike the electromagnetic and isospin currents, the axial vector current carried by the quarks is not conserved in this model. Such a situation is inherent with all the potential models confining quarks including the bag model. But in view o f the experimental success of ecAc and hence the fact that chiral SU(2) x SU(2) is one o f the best symmetries o f strong interaction, it is desirable to conserve the total axial vector current in any of these models describing h~idrons. This is usually done at a phenomenological level (Chodos and T h o r n 1975; Brown et al 1979a, b; Vento et al 1980; Theberge et al 1980, 1981; T h o m a s et al 1981; T h o m a s 1983) by introducing elementary pion field that also carries an axial current such that the four divergence o f the total axial vector current satisfies the PCAC condition. In spite o f m a n y successful applications o f chiral bag models, it is not totally free f r o m certain objections particularly for its insistence on excluding pions from the interior o f the static, spherical bag. Therefore we a t t e m p t a simpler alternative a p p r o a c h to formulate a chiral potential model with equally mixed scalar and vector harmonic potential used (Barik et al 1985) for studying the static properties of baryons. O u r main objective here is to determine the quark-pion coupling constant in this model to examine its consistency with the estimates made earlier by other workers (Suzuki and Bhaduri 1983; Faimen and Hendry 1983; Hendry 1982).
707
708 N Barik and B K Dash
2. Independent quark model with chiral s y m m e t r y
We consider that quarks in a hadronic core move independently in an effective central potential
U(r) = ~(l + : )
V(r),
O)obeying the Dirac equation and implying thereby a Lagrangian density
.LPq = ~ F 1 (x)yJ'gu q ( x ) -- mq ~ ( x ) q ( x ) -- ~ ( x ) U (r)¢l (x). (2) Then under a global infinitesimal chiral transformation
5/r.~\
the axial vector current o f the quarks
A~ (x) = ~ (x)T~'y s ~ q(x), T (4)
associated with such a transformation, is not conserved since its four divergence is
,:?# A u (x) = iG(r)Fl(x)7 s ~q(x), (5)
where G ( r ) = (V(r)/2+raq). This is due to the fact that just like the surface term - ½ ?:/q A~ in the bag model Lagrangian density, the term G (r)~lq in .L~q corresponding to the quark mass mq and the scalar potential a 2 V(r), is chirally odd. The vector part o f the potential poses no problem in this respect. Now to restore chiral symmetry in the usual manner, we can introduce a zero mass pion field with the interaction Lagrangian density,
- i
~ , = ~ [ G(r)~l(X ) 7 ~ (x" ~ )q(r), (6)
when f~ = 93 MeV is the pion decay constant. Then the total axial vector current due to quark and pion together, i.e. A ~ ( x ) = [~7~ysx/2q+f~d#r~] gets conserved with
~ A ~ ( x ) = 0. However if we give the pion field a mass m~, then
-f~m~dp, (7)
c~ A~(x) = 2
yielding the usual r,c^c relation in the current quark level.
First o f all, neglecting the pion coupling with the quarks, one can study the bare hadrons in terms of its individual quarks obeying the Dirac equation
[i7#t3 u -- mq -- U(r)] q ( x ) = 0. (8)
Taking U (r) in (1) with V(r) = a&, (a > 0) and m~ as the current quark mass, the spatial orbits of all the individual quarks in the low lying baryon ground states can be written in their 1S ½ configuration as,
1 (ig(r)/r
q (r) = (4n)'/2 \a" Pf(r)/r f (9)
when, with 2q = (Eq + mq) and ro = (a2q)- 1/a, the reduced radial parts o f the upper and lower components can be written as,
g(r) = N~ (r/r o) exp ( - r2/2r~),
f ( r ) = - ,~qro (r/r°)2 exp(--r2/2r*°)"
Nq
(10)Here Ea is the ground state (1Si) individual quark binding energy obtainable from the energy eigen value condition
().~/a)t/2 (Eq - m+) = 3, (I 1)
and N~ is the overall normalisation factor satisfying the relation,
N 2 x/nro/82q = I/(3E, + m,). (12)
These solutions resulting from (8) can be utilized to describe the bare nucleons represented by the quark-core alone. In fact, in an independent quark model approach, where the quarks in a nucleon are assumed to satisfy the Dirac equation as given in (8) with V(r) = ar 2, we obtained (Batik et ai 1985) a fairly reasonable description o f the bare nucleons with its static properties in terms o f magnetic moment/~p, charge radius (r 2 ) pt/2 and the axial constant OA for neutron//-decay being estimated after centre o f mass correction as
(tip,
( r 2 -p'~t/2, gA) =-- (2"6 jura 0"72fro, 1"02). (13) Here, the potential parameter a = 2.273 fm -3, the quark masses m. = mj = 10 M e V and as a consequence of (11), the quark binding energy in the 1S½ configuration E. = E~ = 540 MeV, have been used. Therefore in the present work, where our main objective is to build such a potential model for nucleons incorporating the chiral symmetry to study the pion coupling to quarks, we would adopt the same set o f parameters that describes the bare nucleon properties in a reasonable manner.3. Pion-quark coupling constant
We intend to study mainly the coupling of quarks in a nucleon to pions, in a chiral symmetric potential model. Therefore, in view of the fact that chirai SU(2) x SU(2) is experimentally found to be an excellent symmetry of strong interaction having its physical realization in pion with its small mass as the corresponding Goldstone boson, we concentrate our discussion mainly in the (u, d) flavour sector only. Then as a first step in this direction, let us assume that the interaction Lagrangian density in (6) can be written effectively as,
-ffl = - iGq+~ ~/(x)y s (¢" ~)q (x), (14)
with Gqq. as the effective quark-pion coupling strength. Then in a classical field approximation, taking the emitted pion field ~j in the process q --. q + n as a plane wave with momentum k, we can write the interaction Hamiltonian as,
Hin t ~- i Gqq~ f d 3 rF] (r)y s q (r) exp (ik" r) ¢j. (15) Now from (6) we can also similarly obtain
i Id3r~ (r) ~ 5 q ( r ) G ( r ) e x p ( i k , r)tj. (16) Hint "~-~
. l
710 N Barik and B K Dash
Then comparing (15) and (16), we can obtain a much simpler estimate o f G4q ~ as, 1 S d 3 rG(r)~/(r)~ 5 q (r) exp (ik" r)
G ~ = f ~
~ d3 r~(r)yS q(r) exp (ik. r) (17)N o w taking the IS~ spatial wave functions o f the quarks as given in (9) and (10), we obtain,
1 f o drrS/2G(r)Ja/z(kr)exp(-r2/r2°)
= -- foo (18)
G~q~ f~ I" drrS/2 J312(kr)exp(-rZ/r~) .I0
Using the standard integral result,
dXX~' exp ( - a X 2 l J , ( f l X )
=
~-,-T a - t 0 ' + ' + l ~0
e x p ( - f l 2 / 4 a )
F ~ , v + l, fl2/4a ,
(19) r ( v + 1)expression (18)can be simplified to give,
1[-5
r2 [
(20)Then with a soft pion approximation we can approximate
I [ 5arg
-I (5Eq + 7mq)J2f. (21)
so that, the quark-pion coupling constant comes out as,
G2qq~ = 0.49. (22)
4n
This is in good agreement with the estimate obtained by Suzuki and Bhaduri (1983) from the ratio
F(~ ~ n~)/F(p ° ~ e + e- ).
A better estimate of the quark-pion coupling constant can be made in a more reasonable way by looking at the NN~-vertex. For the interaction Lagrangian density (6), the NN~-vertex function, in a point pion approximation, can be written as,
i -tSd3rG(r)exp(ik.r)(N'/Y.~(r)~,Sq(t)¢~lN ).
(23)q
-~-- 2 1 2
H e r e j is the isospin index and ok (k 2 + m,) ; is the pion energy. Since for the N N , - vertex, the spatial orbits of all the quarks in the initial and final nucleon state are the same IS~, using (10) and (11) in (23), we can obtain,
i
x/ k -3 2
vN'N(k)=-~.f~.f,~ (2cOk)-tN~ -~qr-~- I(kj(N'lY(eCq'k)'rJ[N)'q
where,
fo G(r) Js/2
I(k)
= 2 dr r s/2 (kr) ¢xp ( - r2/ro2). (24)Using (19) for
l(k)
we can writeV~¢'# (k) = ~-~ (2 oJk)-I
gnu(k) <N'I ~:(¢,. k)xslN >
q
= (N'[Ev~"(k)]N).
(25)Hence we obtain the quark-pion vertex operator function
v~q(h) = ~-~(2co,)-t
aAu(k)
(o,'k)z s, (26)where, g~ is the axial vector coupling constant which can be obtained in this model as (Batik et
al
1985)u,,
= _5 (sEo
9\ 3Eq+m~ ) + 7mq
and
u(k)
is the form factor given by,[ (E'-m')kZr~]
u(k)= 1 2(5Eq+ 7m,)
exp(-kZr2°/4)' (27)which for k --. 0 reduces to one. Now comparing (26) with the corresponding expression in Chew-Low type model (Chew 1954; Chew and Low 1955; Wick 1955), which is written in terms of the pseudo-vector qqn-coupling
fqq,,
as,v~ q
(k) = i(2co~) -i ~/4n(f,q,,/m,,) u(k) (oq.
k)~j, we have,~/4-~ (J'q,j,,/m,,) = ~
(304/5).(28)
(29) This is the equivalent Goldberger-Treiman relation, which with the familiar equiva- lence of pseudo-scalar and pseudo-vector coupling constants yields,
= ~/~n (f,q,,/m,,)
= ~ (304/5), (30)(Gqq,,/2M,)
where, M ~ is the effective constituent quark mass taken as one-third of the N --, A spin --* isospin average mass
i.e.
390 MeV.Then we have,
= ,
~- 0-449. (31)
However if we consider the CM correction for g4, then using the corrected OA value from (13), we get,
G~,~/41t ~-
ff524. (32)P 4
712 N Barik and B K Dash
The pion coupling to the quarks has been considered so far to be a point particle. But one can introduce the finite size of the pion according to the prescriptions of (de Kam and Pirner 1982; Phatak 1983) by visualizing the pion absorption as a process in which a quark of the bare nucleon is replaced by a quark of the pion after it is annihilated by the antiquark of the pion. Then the NN~-vertex function can be written as,
fd
V~W(k) = - ~ - (2 tOk) -t 3rd3pG(r) exp (ik. r) P (p)
( N' I ~. ?/(r + p/2)75~sq (r -
p/2) I
N >. (33) Here 'p' is the qF/-separation distance and P ( p ) is the probability function for finding such a q~-pair in the pion. Introducing a size parameter R, for the pion, one can choose,3 l
P ( p ) = ~ - - ~ O ( R , - p). (34)
atlt l(~t
However with a reasonable approximation to replace 0. (r + pl2)llr +_
p/21
by o" FandIIr+p/21+lr-p/21)---2r,
(33)can be simplified to give the quark-pion vertex operator function[ 112 _ ~
v~,(k) ___ 2-f~ (2o, k)- u(k)F(R~)(a,, k)~s,
F (R.) = 4n f o dp p2 eLo) exp ( - p2/4r~), _ 12r~ 7(3/2, R~/4r~),
R~
where,
(35)
I ~o (R'/r°)'l
~- 1 + exp( -- R 2 J 4 r o ). 2
Then proceeding as before and taking the CM correction into account for gA in (35), we can obtain,
G~Zq~ = ( G ~ ~
2
F2(R~) ' (36)4-n \ 4n Jo
when (G~q~/4n)0 is the coupling constant obtained with point pion approximation (equation (32)). It is obvious that the effect of the finite size is to reduce the coupling constant depending upon the size-parameter R~" R~ is expected not to be the pion charge radius, but rather the radius of the qq-pair distribution within the pion, which is observed to be considerably smaller (Oset et a11984) than the charge radius of the pion.
According to the estimate of Brodsky and Lepage (Brodsky and Lepage 198 l; Brodsky 1982) R~ -~ 0-4 fin or smaller. Similar values are also obtained in a microscopic chiral model of the pion (Bernard et al 1984). Therefore taking a range of values for R~ as 0-4, 0.3 and 0.2 fm respectively we obtain,
G~q~/41z = [-0-463, 0-489, 0.508]. (37)
4. Conclusion
T h e c o u p l i n g s t r e n g t h G~q,/4n d e t e r m i n e d b y S u z u k i a n d B h a d u r i (1983) f r o m t h e v e c t o r m e s o n d e c a y r a t i o s w i t h a s t a t i c a p p r o x i m a t i o n c a n be c i t e d h e r e f o r a c o m p a r i s o n . T h e y o b t a i n it as a b o u t (i) 0-4 f r o m F ( ~ --, n ~ ) / F ( p --, e+e - ) (ii) 0-5 f r o m F(co - , ~°3,)/F(co --, e + e - ) a n d (iii) 0.88 f r o m F ( p - , n - ~ ° ) / F ( p --* ~ - ~ ) . W e find t h a t e x c e p t f o r case (iii), t h e v a l u e s o f t h e q u a r k - p i o n c o u p l i n g c o n s t a n t e x t r a c t e d f r o m t h e e x p e r i m e n t a l v e c t o r m e s o n d~,,ay w i d t h s a r e q u i t e c o m p a r a b l e w i t h o u r t h e o r e t i c a l e s t i m a t e s in this m o d e l given in (22), (31), ( 3 2 ) a n d (37). H o w e v e r f r o m t h e o b s e r v a t i o n s o f H e n d r y (1982) e x a m i n i n g the d e c a y o f e x c i t e d N a n d A, o n e o b t a i n s G2q,/4n ~- 1-1, which is m u c h l a r g e r t h a n o u r e s t i m a t e .
T h e n u c l e o n - p i o n c o u p l i n g c o n s t a n t (G2NN,/4~) 0 in this m o d e l c o m e s o u t t o be o f t h e o r d e r o f 13.1 w h i c h c o m p a r e s well w i t h t h e e x p e r i m e n t a l v a l u e 14-4. T h e finite size o f the p i o n , h o w e v e r , r e d u c e s the values o f (G2N./4n) t o 11.59, 12"23 a n d 12.71 for R~
e q u a l s 0-4, 0.3 a n d 0-2 fm respectively.
Acknowledgements
T h e a u t h o r s t h a n k P r o f . B B D e o for c o n s t a n t i n s p i r a t i o n s a n d v a l u a b l e s u g g e s t i o n s a n d M D a s for useful discussions. O n e o f the a u t h o r s ( B K D ) g r a t e f u l l y a c k n o w l e d g e s the s u p p o r t o f t h e G o v e r n m e n t o f O r i s s a , E d u c a t i o n D e p a r t m e n t for p r o v i d i n g s t u d y leave.
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