Gauge fields, hidden local symmetries, and meson exchange currents

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 624 (1997) 655-686

Gauge fields, hidden local symmetries, -ha n d meson exchange currents J. Smejkal "~, E. Truhl~ a, H.

G611er b,1

a Institute of Nuclear Physics, Czech Academy of Sciences, CZ-25068 l~ei, Czech Republic b Institute fiir Kernphysik, Johannes Gutenberg-Universita't, D-55099 Mainz, Germany

Received 16 December 1996; revised 18 July 1997

Abstract The model dependence of the electromagnetic isovector meson exchange current (MEC) operator at higher momentums transfer is studied. For this purpose, the MEC operator constructed from a Lagrangian of the 7rpal system obtained within the framework of the hidden local symmetries (HLS) approach is compared with the analogous operator derived earlier from a minimal Yang-Mills (YM) Lagrangian. Numerical results for the deuteron disintegration near threshold show that the model dependent part of the exchange current modifies the tail of the cross section sensibly and that the presence of the al meson both in the MEC and in the nuclear interaction provides a consistent description of the cross sections up to the momentums transfer t ~ 1.5 GeV 2. It is also found that the standard ~r + p MEC does not possess the correct chiral properties. (~) 1997 Elsevier Science B.V. PACS: 25.10.+s; 25.30.Dh; 27.10.+h

1. I n t r o d u c t i o n The idea of gauge fields introduced by Yang and Mills [ 1 ] turned out to be extremely useful in describing physical phenomena. Actually, the whole modern subnuclear physics is based on the concept of gauge fields [2]. These fields do play an important role also in the hadron physics at intermediate energies from the very beginning. Namely, effective chiral Lagrangians of hadron systems, formulated already in sixties in terms of mesons * Supported by grants Nos. 202/94/0370 (GA CR) and 148410 (GA AV CR) and partially by the grant No. 202/97/0447 (GA CR) and by the Deutsche Forschungsgemeinschaft(SFB 201 ). I Present address: repas ProzeB-Automation,VoltastraBe 8, D-63303 Dreieich, Germany. 0375-9474/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PIIS0375-9474(97)00369-7

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J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

and baryons as elementary local fields, were based on the concept of the Yang-Mills (YM) gauge fields [3,4]. This phenomenological description of hadron physics at intermediate energies was quite successful in spite of the fact that the underlying theory was internally inconsistent. The problem was in introducing the mass terms of the gauge fields, interpreted as physical vector mesons, into the effective Lagrangians, which was in contradiction with the idea of the massless YM gauge fields. On the other side, the presence of these mass terms allowed one to incorporate the Vector Meson Dominance (VMD) idea into the scheme. We shall label here any Lagrangian constructed within the concept of the massless YM compensating fields with the superscript YM. The above mentioned concept defect was removed in eighties in the approach of HLS [5,6]. Its idea lies in an extension of a given global symmetry group Gg of a system Lagrangian to a larger one, Gg × Hi. The extension of the symmetry group by a local group Ht allows one to introduce both the gauge and (non-physical) compensator fields. The role of the compensators is to supply the mass via the Higgs mechanism to the physical particles represented by the gauge fields. Actually, the compensators disappear from the Lagrangian which can be written in a form independent explicitly also of the transformations from the group Ht (the local symmetry becomes hidden not only because being spontaneously broken, but it is hidden literally). Moreover, the new mass terms do not violate the local symmetry in this case. For the case of groups Q, = [SU(2)c × SU(2)R]g and Ht =~ [SU(2)c × S U ( 2 ) n ] t the gauge particles are standardly identified [5,6] with the p and al mesons. Another prominent feature of the HLS approach is the direct way of including external gauge fields by gauging the global symmetry group Gg with the elementary gauge fields which are directly related with the bosons y, W~ and Z ° of the Standard Model. These bosons couple to the particles of the system independently of the gauge bosons of the HLS. In contrast to this, the approach using the concept of the massless YM fields allows an external excitation Ag to interact with the particles of the system only via an interaction Lagrangian of the type £int = - J u A u ,

(1.1)

where the hadron current Ju is obtained in Ref. [7] from the Lagrangian of the system by the Gell-Mann-L6vy method [ 8 ]. YM of the ~rpal system constructed by Ogievetsky and A minimal chiral Lagrangian £,w,~ Zupnik (OZ) in Ref. [7], based on the chiral group Gt = [SU(2)L x S U ( 2 ) n ] t of local transformations, broken only by the heavy meson mass terms, was used in Refs. [9-12] as a starting point for constructing the electroweak nuclear meson exchange currents (MECs). This model has been extensively used [11,13] to describe the cross section of the backward deuteron electrodisintegration e + d

~e t +np,

(1.2)

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J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

measured in Refs. [ 14-16]. Its specific feature is reflected in the existence of a transverse vM MEC Ar~,,u, the spatial component of which is in the non-relativistic approximation of the form, YM

JTr.~ = - i

g gA I~V [ 3 a 2 ---t ~k)(71 × r2) d ~ ( q l ) k 2 M 2 f~r

+ ( 1 ~--*2).

x (o'1 × ( k + q 2 ) ) A ~ ( q ~ ) ( O >

-

" q~) -

(1,3)

We refer for details to Section 2.5 of Ref. [11]. This current contains a part ~ k 2 (k;, is the momentum transfer), the origin of which can be traced to the corresponding part of the photoproduction amplitude, which makes it compatible with the soft pion theorem ] 1 1,17,18]. This piece of the current should be present in any correct theory of MECs. A part of the current (1.3) which is ~q2 is not fixed by any low energy theorem. Then the question arises, how this part depends on the model Lagrangian of the system. Actually, the answer to this question is the ultimate goal of the paper. For that purpose, ¥r,4 [7] with those we compare the predictions following from the YM Lagrangian £~p,,~ obtained from the Lagrangian constructed within the framework of the HLS approach as presented by Bando, Kugo and Yamawaki (BKY) in Ref. [5]. However, in contrast to Ref. [7], the Lagrangian presented in Section 7 of Ref. [5] and later generalized in Ref. [ 19] to the ~ p a l w f l system, is not in a form suitable for our purposes. Indeed, after eliminating the non-physical a l - T r coupling the use of equation of motion for the p and al mesons has been made. This step restricts the application of such a Lagrangian only to the processes with these particles on-shell. We construct the Lagrangian £nLs - - "rrp a l exempt from this restriction in Sections 2 and 3. Besides, we reanalyze the choice of higher order Lagrangians needed to correct the behaviour of amplitudes at higher energies. Our construction is ideologically close to that of Ref. [5 ]. In Section 2.1 we write down the basic Lagrangian. Essentially, this is nothing new in comparison with Ref. [5] and it serves us to fix our notations. We start from the non-linear o- model possessing the global chiral symmetry G~ ~ [SU(2)L × SU(2)R]~. We extend it by adding a local symmetry Ht =- ]SU(2)L × SU(2)R];. Doing this, we introduce both local gauge fields and additional variables corresponding to nonphysical fields-compensators. Admitting at the quantum level the dynamical generation of the kinetic terms corresponding to gauge fields [5], we have the chance to interpret these fields as corresponding to real particles. In the next step, the compensators are absorbed via the Higgs mechanism by the gauge fields, which then represent real massive vector mesons. In this stage, the Lagrangian contains only the fields invariant under the considered local symmetry, which becomes actually hidden. The remnant of its presence is reflected in the form of the Lagrangian and in a set of relations between entering constants. The same result can be obtained by fixing the gauge [5,6]. YM is written in terms of the vector meson fields belonging to The Lagrangian £~p,,~ the linear realization of the local chiral G; symmetry, while the fields, entering the Lagrangian ['HLS _~;~,, so far transform according to the non-linear realization of the HLS H;. In order to simplify the comparison, we accomplish in Section 2.2 the Sttickelberg trans-

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J. Smejkal et aL /Nuclear Physics A 624 (1997) 655-686

formation [6,20,21] upon these fields. Simultaneously, this step allows us to introduce external fields. The transformed fields belong to the linear realization of the HLS and we shall label the corresponding Lagrangian with the superscript I. This Lagrangian is equivalent to the starting --TT,Oal F.HLs but it differs from the YM Lagrangian by the presence of the heavy meson mass terms which do not violate the HLS and the interaction terms emerge in it naturally. In other words, the YM Lagrangians form a subclass of the HLS Lagrangians [5,6]. The new Lagrangian £HLS contains a serious defect in the form of the non-physical "n"p~:t 1 aj-7"r coupling. This shortage can be standardly removed by a redefinition of the al meson field. However, after removing this non-physical coupling, new problems arise. In Section 3 we introduce in the spirit of Ref. [5] higher order Lagrangians serving to remove defects of the constructed Lagrangian, which it possesses so far. We could also choose the method of Ref. [7] and we would get a minimal Lagrangian of the HLS scheme. Generally, our construction given in Section 2 and partially also in Section 3 differs from that of Ref. [7] as follows: (i) In Ref. [7], the Lagrangian is supposed to contain no more than two field derivatives. We demand, in accord with Ref. [5] that too a strong momentum dependence of the coupling gp~r~ would be eliminated and the p meson dominance of the decay al ~ pTr would be maintained. (ii) Consequently, different higher order correction Lagrangians in Section 3 and Ref. [7] are chosen so that previous point were satisfied. (iii) Different redefinition of the al meson field is chosen, in order to suppress the non-physical a j - T r coupling. Actually, only the different higher order correction terms influence the physical content of the Lagrangian. Our construction of Sections 2 and 3 differs from that of Section 7.4 of Ref. [5] in that: (i) We do not use the equations of motions for the gauge fields, which is justified only for the on-shell particles. (ii) Our choice of the correction terms corresponds to the value of the anomalous magnetic moment of the al meson •/za = - 1 , while the analogous choice made in Ref. [5] yields ~/z,t = +3. We also show explicitly that the form of the Lagrangian does not depend on the redefinition of the al meson field. In Section 4 we discuss the elimination of the aj meson degrees of freedom from the Lagrangian. This step was discussed earlier in Refs. [6,22] and serves to get rid of a particle with somewhat fuzzy known mass and decay width. Intuitively, this procedure can be acceptable for considering the processes at low energies, where the direct influence of a heavy particle might be expected to be small. However, it may be not true generally. As discussed in Ref. [6], the full Lagrangian describes better even the static baryon properties. Besides, as we show here, the cross section of reaction (1.2) is influenced by explicit absence of the al meson degrees of freedom in the Lagrangian,

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

659

from rather low momentums transfer. In Refs. [6,22] the elimination of the as meson field is achieved by imposing an additional chiral invariant constraint upon the fields. The basis of our method consists in freezing out the as meson degrees of freedom by sending the a] meson mass to infinity. As a consequence, the kinetic term of the as meson disappears from the Lagrangian, the al meson field becomes redundant and it can be eliminated by solving the corresponding equation of motion. The procedure of eliminating the redundant fields from Lagrangians is widely used in Ref. [5] and a short remark about it can be found also in Ref. [6]. In Section 5 we extend the construction scheme to the NTrpal system. We give the corresponding Lagrangians and we derive the one- and two-body current operators corresponding to various Lagrangians discussed so far. In Section 6 we apply the derived model exchange currents in calculations of the cross sections for reaction (1.2) at threshold and we make the comparison with the data [ 1416]. In Section 7 we present the results and make conclusions. We notice that the present data [ 14-16] is reasonably described by the studied models with the at meson included, up to the momentums transfer t ~ 1.5 GeV z (cf. Figs. 2-4). The formally correct current model derived from the Lagrangian with the a~ meson eliminated fails to describe the data from rather low momentums transfer (cf. Fig. 5). Preliminary results were reported in Refs. [23,24].

2. Generalities

The standard derivation of the soft pion theorems [4,25] is based on the fundamental commutation relations between charges and currents and on the PCAC hypothesis. These theorems can be effectively obtained from any Lagrangian reflecting the global chiral symmetry G~. The most popular model used is the non-linear tr model formulated using non-linearly transforming pion fields H a ( x ) . In Section 2.1 we discuss shortly the generalization of this model within the HLS approach as outlined in Ref. [5], fixing simultaneously our notations. In this way, the vector mesons fS~,, flu are introduced. They belong to the non-linear realization of the local chiral algebra ~¢ related to the chiral group Gt. In Section 2.2 we perform the Stiickelberg transformation of the vector meson fields. The Sttickelberg construction served in Refs. [6,21] to study generally the similarity between the YM and HLS approaches. By doing this transformation, we transform the fields /Su,~ l, so that the new fields p~,,a u transform according to prescriptions of the linear realization of the chiral symmetry. Such fields were used in Ref. [7] when constructing the YM-type Lagrangian of the 7rpal system, an HLS analogue of which is discussed in Section 2.1. This step allows us to compare directly both approaches to constructing the needed concrete Lagrangians and also to introduce external fields in a simple way.

J. Smejkalet al./Nuclear PhysicsA 624 (1997)655-686

660

2.1. Construction of the rrpal Lagrangian within the framework of HLS As it is discussed at length in Ref. [5], the non-linear o- model, possessing the Gg symmetry, is gauge equivalent on the classical level to a model having G~ x/41 symmetry, where Ht is an arbitrary group of local transformations. The corresponding Lagrangian is written in terms of invariants which are formed, besides pion field, by gauge fields and compensators. At the classical level, the kinetic terms of the gauge fields are absent. Then the gauge fields can be eliminated in terms of pions and compensators applying the equations of motion. The compensators disappear after fixing the gauge and the equivalence of the o- and extended models follows. However, the new situation arises at the quantal level when gauge bosons of the HLS are generated dynamically. According to Ref. [ 5], the dynamical generation of the gauge bosons is a common phenomenon not restricted to a particular model. It actually means that the Lagrangian includes also the kinetic terms of the gauge bosons, which are now independent fields and cannot be eliminated from the theory simply by employing the equations of motion. In this way, the p and al mesons are introduced, after extending the o- model by the local chiral group Ht = GI. Then the full symmetry group of the model is

Gu x GI ~ [SU(2)L x SU(2)R]g x [SU(2)L x SU(2)R]I.

(2.1)

We now fix our notations and shortly discuss the derivation of the basic Lagrangian of the model. The field variable ( ( x ) of the model is parametrized as

( ( x ) = ( + ( p ) ( ( s ) ( ( H ) ~ ~+(p)((s, H)

(2.2)

= exp(ip(x)/f~) exp(-is(x)/f~) exp(-iH(x)/f~), p(x)=Zpa(x)X",

H(x)=ZHa(x)Xa,

a

a

s(x)=Zs"(x)S".

(2.3) (2.4)

Here the fields H'~(x) correspond to pions, pa(x) and o-~'(x) are compensators and S~'(X~) is the set of unbroken (broken) generators of the group G u, which is spontaneously broken to its isospin subgroup [SU(2)V]g. We demand that

((p)--,((p')=h(p,~(x))((p)~+(x), ( ( H ) ---+( ( H ' ) = h ( H , g ) ( ( H ) g +,

~(x) cGt,

h(p,~) ~ [SU(2)v]t, (2.5)

g E G~,, h(lI, g) E [SU(2)v]g.

(2.6)

If the function sO(s) transforms as

( ( s ) --~ ( ( s ' ) = h(p, ~(x) ) ( ( s ) h+(H, g),

(2.7)

then the field variable ~:(x) transforms under the transformations of the group (2.1) according to

( ( x ) ~ ( ' ( x ) = ~ ( x ) ~ ( x ) g +,

~,(x) E Gt,

We also need to introduce the Maurer-Cartan l-forms,

g c G~.

(2.8)

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

66 I

fl/~( x) = - i ( cgu~( s, I1) )sC+( s, 11) = flu(s) + ( ( s)ceu( I I ) ( + ( s), flu(l) = -i(a/,~(1))~+(1),

l = s,p,

(2.10)

oeu(H) = - i ( c g u ~ ( I 1 ) ) ( + ( r l ) = ceu±(//) + aul I ( / / ) , e e u i ( H ) = ( 2 t r X a c e u ( I I ) ) X a,

(2.9)

(2.11)

aull ( H ) = ( 2 t r S " c e u ( H ) ) S '*.

(2.12)

and the covariant derivative of the field sO(p) (2.13)

D u E ( p ) = Ou((p) - i ~ ( p ) W ~ ( x ) .

In what follows, the perpendicular and parallel components of any quantity are defined in analogy with Eq. (2.12), The vector field W u ( x ) is an element of the local algebra H/ -- [ S U (2) L • S H (2) R] I transforming under the local gauge transformations ot' the group G/ as

W~,(x) -~ W~(x) = ~,(x)Wu(x)~,+(x) + i(c~u~C(x))~+(x),

(2.14)

but it does not change under the transformations from the global group G e. The last objects we need for writing down the Lagrangian are the covariantized tbrms /3u(x) a n d / 3 u ( P ) which are defined as

~u(x) =flu(x),

(2.15)

~ u ( P ) = -i(79/~((p) )sc+ (p) = flu(P) - ~C(p) W u ( x ) ( + ( p ) .

(2.16)

The analysis of the properties of the covariantized forms under the transtbrmations of the group (2.1) based on Eqs. ( 2 . 5 ) - ( 2 . 7 ) , (2.14) allows us to write the most general Lagrangian of the considered system as the linear combination

£j = a£v, + b£a, + C£p + d £ ~ + £kin(Wu),

(2.17)

/2~ = -- f~tr(/3u± (x) )2,

(2.18)

El , = _ fZtr(/~u ± ( p ) ) 2 ,

(2.19)

£a, = --fZtr(/3u± (P) - - / 3 u ± ( x ) )2,

(2.20)

£ v ' = --f~tr(/3ul[ (P) --/3ull ( x ) ) 2 ,

(2.21)

where the kinetic term is defined as

/~kin(Wu)=

1

_ _2.)/2 _ t r ( , F(W) uv )2,

(w) = Wu ~ + i[ W u, W~ ], F~zv

Wu v = Ou W~ _c~,Wu. (2.22)

The Lagrangian £ , is written in terms of the fields H ( x ) and W u ( x ) and of the nonphysical compensators. Let us now identify the parallel and perpendicular components of the field W u ( x ) with the physical vector and axial vector fields Vu(x) and A u ( x ) , respectively

Vu(x ) = Wz,ll(X),

A n ( x ) = Wu± (x ).

(2.23)

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

662

The vector field Vu(x) belongs to the representation of the algebra 7-/11 - ( S H ( 2 ) v ) t , while the axial vector field A u ( x ) belongs to the orthogonal space 7-/.L -= [ ( S H ( 2 ) v ) t ] .L for which it holds,

[/-/.L,~±] C 7-/11,

[7-/11,7-/i] C 7-[.L,

(2.24)

and 7-/1 = 7"/11® 7-/.L. In order that ~1 describes fully the physics, the non-physical compensator fields s~(x) and pa(x) should be removed. This can be done by a series of redefinitions

W~(x) =~(p)Wjz(x)(+(p) - flu(P) = -/~u(P) -= W,~I[ + W,~.L; V~

S - W~I I,

A uS

~

S W~±,

(Vu( x) =?+ ( s) WS~(x)?( s) - i(+ ( s)au(( s) =_ l,~'ulI + 1,1,'u.L,

(2.25) (2.26)

(2.27)

Vu = gzull,

'4u = Wu.L,

(2.28)

1Pu = -~Vu,

lgtu = -~Au"

(2.29)

Now the Lagrangian £I of Eq. (2.17) reads in terms of the physical fields

£1 = a £ v + bgA + c£p + d£.~. +/~ldn(/Su, ~U),

(2.30)

£V = -- f~tr(~ull )2,

(2.31 )

£ a = --f2~tr( ~u-L ) 2,

(2.32)

/~p = --f2tr(y~u) 2,

(2.33)

£,~ = --f~tr(au.L ( H ) ) 2 ,

(2.34)

where we defined the Maurer-Cartan 1-forms gauged by the gauged fields related to the group of the HLS, &~ll = cecil( H ) + Y/~u,

&u-L = ceu-J-( H ) + ya~,

(2.35)

and the field tensors .~'(~)'(~) u~ read

Y(u~) = Pu~ + iy[Pu, P,' ] + iy[ fiu, fi~],

(2.36)

F(u~) =fiu~ + iy[~u, gt,,] + i y [ f i u , ~ l .

(2.37)

Actually, the quantities entering Eq. (2.30) change only under the transformation from the global chiral group G~ and they are not affected by the local chiral group Gt at all. In other words, the local symmetry realized by this group becomes hidden. It can be shown that at the classical level, the considered model Lagrangian of Eq. (2.30), taken without the kinetic terms of the gauge fields, is equivalent to the non-linear o" model Lagrangian if d +

bc - 1. b+c

Eq. (2.30) tells us that the p and a~ mesons acquire the mass

(2.38)

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

m 2a = ( b + c ) y

mp2 ---a 3/ 2,.2 y~,

2 f~r, 2

663

(2.39)

by the Higgs mechanism: the fields/5 n and fin absorbed the compensators and obtained the longitudinal component. If we demand that the constant y is the universal coupling of the vector mesons, y = gp~rrr = . . . = g p then we should choose a = 2 and the first of Eqs. (2.39) reproduces the KSFR relation [26]. 2.2. The Stiickelberg transformation

We did not discuss the inclusion of external fields until now. They can be introduced by gauging the global chiral group Gg present in the model before the additional HLS were introduced. In our model describing the 7rpal system, the external gauge vector ~)u and axial ,4 n fields correspond to the bosons of the Standard Model. The presence of the external gauge fields induces modification of terms £ v , £ a and £~, of Eq. (2.30), because the entering variables change as follows, 0

0

-

a u ( H ) --~ &u = - i ( D~,(( H ) )sc+ ( H ) ,

where the covariant derivative D u 6 ( H )

(2.40) (2.41)

is here defined as

D ~ ( ( I1) = cgn~c(n ) - ie6( I I ) (])u + ,An).

(2,42)

and e is the coupling constant of the external gauge fields• Instead of Eqs. (2.30)-(2•32) and Eq. (2.34) we have, £2 = a12°v + b £ ° + CCp + d E ° + £kin(/5,, h u) + £kin (]2,, ,A,),

(2.43)

12o = - f ~ t r ( aul 0 I ) 2,

(2.44)

120

2 = -,f~tr( a n0 ± ) 2 ,

( 2.45 )

12° = - f~tr(c~u± ) 2.

(2.46)

Eq. (2.43) is an analogue of Eq. (7•78) from Ref. [5]. The Stiickelberg transformation with the external fields included is e

_eA

=Y

The new fields Pu, an belong to the linear realization of the global chiral group G~ wu -- Pu + au --~ wuI = g w u g +,

(2.48)

while for the transformations from the gauged chiral group G u it holds I

w u ---+ w u = g ( x ) w n g + ( x )

+ t(3ng(X))g+(x). Y

(2•49)

After some algebra, we obtain for the Lagrangian Z22 in terms of the new fields Pu and a/z

664

J. Smejkal et aL /Nuclear Physics A 624 (1997) 655-686 1 £,2 = - ~ ( a + b) f ~ [ t r ( y p u - eV~,) 2 + tr(ya~ - e.A~) 2 ]

_ l (a - b) f Z t r { ( 2 ( f I ) [ (yp~ - elJu ) + (ya u - e.A~,) ] 2 x ( ( + ( / / ) ) 2 [ (yp~ _ e]ju ) _ (yau _ eAu) ] } l

z

- - ~ c f ; t r [ D~sC2(H ) O u ( ( + ( H ) )21 - Idf~tr[ b u ( a ( II) bu(~+ ( f l ) )21 +/~kin(Pu, a~) + £kin(])u, A~z),

(2.50)

where Ot~(2(//) = 0 ~ ( 2 ( H ) + ie(l)t, - .Au)(2(fI) - i e ( Z ( l I ) ( V u + A u ) ,

(2.51)

lOu(~+(H) )z = c~ ( ( + ( H ) ) z _ i e ( ( + ( i 1 ) )2(1j~ _ .Au) + ie(1)~ + A~,) (so+(//))2. (2.52) The analysis of Eq. (2.50) shows that the choice a = b makes the second Weinberg sum rule g~ = g] valid. In order to maintain the strict p meson dominance, we should accept the condition d = 0. Then in view of Eq. (2.38) c = b = 2 and the second of Eqs. (2.39) implies the Weinberg relation m,,z = 2m~. Summing up we have,

a=b=c=2,

d = 0.

(2.53)

Ifwemakeachoice

m~ = a f 2 y 2,

a=b,

d=0,

c = d 2,

(2.54)

then

~2 _/~HLS(I) 2e -~pal + mo y (Pt' " V~, + a u • .A u) ( V 2 -~, `4"2) + £'kin(V/z, ,.'At,u.),

-lm2

(2.55)

£Hes(l) is identical in the form as the starting Lagrangian E of Ref. [7]. where the term -~pa~ It is seen from Eq. (2.55) that the essential difference between the YM-type Lagrangian EHLS(~) and the HLS-type one lies in the presence of the interaction term. Equally as the q'/'pO 1 heavy meson mass terms, the interaction term should be supplied in the YM-approach by hands as

£",

=

•v + a',,a

.a

(2.56)

with the hadron currents j~A derived in the presence of the gauge fields by the GlashowGell-Mann method [27] and not by the Gell-Mann-L4vy method [8] as supposed in Ref. [7]. This point will be discussed in Section 7. However, neither the Lagrangian equation (2.43) nor that of Eq. (2.50) are in the final form. Actually, two defects are present:

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

665

(i) The kinetic pion term is not correctly normalized, because generally, the factor 62

1 + b+c

4= 1.

(2.57)

(ii) Non-physical r r - a l coupling is present via the interaction term £;,~,,, = - 2by f~tr( gt~,auz ( H ) ).

(2.58)

In the next section, we discuss how these defects can be removed.

3. Correction Lagrangians of higher order The removing of the defects present in the Lagrangians equation (2.43) or Eq. (2.50) can be done by adding correction terms of higher order to them. The correction terms used in Refs. [5,7] differ and the resulting Lagrangians are also different. Here we analyse the approach applied in the HLS [5,28] and discuss shortly which correction terms were employed in Ref. [7]. First, we consider the biderivative terms. 3.1. Correction terms in the non-linear realization of chiral symmetry It is possible to construct the following six independent invariants [5] which are in our notations, i (,~) [a,ll(H),a~L o o I( H ) ] } , £ ] = ~tr{F;~

3.1)

£~ =/ytr{ F~(~) [ a#, fi~ ] },

3.2)

- ( H ) , &,± ( H ) ] }, /~3c -- ~t t r { F ;(t~) , [ au±

3.3)

C4c = t__tr{F~)[aOz(a) ' a ~0± ( / / ) ] y

3.4)

},

£c=itr{F#(a)[ofl 5 ull ( H ) ' ?t~]},

3.5)

C~ = ~tr{ F~(~)[ ±°It ( H ) , &.± (H) ] }.

3.6)

We correct the Lagrangian of Eq. (2.43) as 6

£3 = £2 -t- ~

Cnl2',i,

( 3.7 )

n=l

with the coefficients e, to be determined in what follows• In order to remove the non-physical r r - a l coupling (2.58), the field au is redefined as

a . = a~ -o + ~ fr ~ c9~H,

(3.8)

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666

where we denoted r -

b

(3.9)

b+c"

It is straightforward to prove that the coupling of the type (2.58) between the new field 0 and the pion field ~ does not exist any more and the kinetic term of the pion field is correctly normalized. However, the redefinition (3.8) brings new problems: too a strong momentum dependence o f the vertex pTrTr appears and the decay amplitude f o r the process al --+ y + 7r as well as the vertices p p V and a l a I V are not dominated by the p meson.

The analysis shows that in order to remove these defects, the coefficients ci should satisfy a set of r-dependent equations. Moreover, the solution is also r-dependent. If we accept the validity of the KSFR relations and of the Weinberg sum rules then 1

r = 2'

(3.10)

and the corresponding equations reduce to - 1 + c 2 + 4 c 3 --~-c4 = 0 , -1

-

(3.11)

c 2 --~ c 4 = 0 ,

(3.12)

Cl = c5 --- c6 =0.

(3.13)

The simplest choice would be, C4

C2 = C3 ----"0 ,

(3.14)

= 1.

Nonetheless, in Ref. [5] the preferred values are, c2 = - 2 ,

c3

=

--C4

=

1.

(3.15)

The choice (3.14) leads to the generally accepted moment of the at meson 6~,, = - 1 , while the set side, our results for F ( a l --~ pTr) = 360 MeV and with those of Ref. [5]. Let us extract the vertex 7rpal. We find that it of coefficients c~, . r

~0

-

2i

_

[7] value of the anomalous magnetic (3.15) yields 8~° = +3. On the other for F(a~ -+ yTr) = 300 keV coincide does not depend on the particular set

0

-

E~.toa~ = - 2 t ~ t r { a u , , [pu, 0,,H] } - ~-~tr{/Su~ [ flu' a~H] jr.

(3.16)

In the isovector formalism and for r = ½ this equation reads after redefining f o __~ flu, ~'oa, = f-'--~ Pv.~ " at~ x O~,II + 7 p " • O~II x ?tu, .

(3.17)

The gauge fields/51,, f u entering the Lagrangian (3.7) are given in the parametrization which corresponds to the non-linear realization of the chiral symmetry. We now pass to

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

667

investigate the case, when the Sttickelberg transformation (2.47) is applied to the fields /5~,, ~u- The new fields Pz,, a~ correspond to the linear realization of the chiral symmetry. Actually, we perform in the next section the same program for the Lagrangian (3.7) with £2 from Eq. (2.50).

3.2. Correction terms in linear realization of chiral symmet~' Our starting Lagrangian is given in Eq. (2.50). The analogue of Eq. (3.8) is l-r o _ --O~Tr.

a~ = a/~

(3.18)

Yf~

As a consequence, the term of the type a~O~Tr o will be absent, but the problems encountered in the previous section persist. Again, these defects can be removed by applying the correction Lagrangians equations (3.1)-(3.6), written in terms of the fields Pt,, au, 7r. Generally, it is rather complicated matter to do it, because the Stiickelberg transformation mixes the parallel and perpendicular operators. However, we need only the linear parts of the transformed fields, which simplifies the matter essentially. It is then enough to substitute these parts to the invariants £ ]C - £ 6C and to extract the needed redefined three-particle terms. Actually, these linear parts are [/Su (pv, a,, 7"r)] 1 =P~z,

(3.19)

-0 0 t =au. 0 [au(p,,,a~,~r)]

(3.20)

Eqs. (3.19) and (3.20) simplify the study of the problem. Indeed, the same threeparticle terms, expressed via the fields p u , a #0, 7"r, correspond formally to the threeparticle terms of the invariants £~-£~. The correction terms we use are again of the form given in Eqs. (3.1)-(3.6) and the corrected Lagrangian (2.50) is 6

c4 = c2 +

(3.21) n=l

As in the previous section, we can derive for the coefficients dn equations, analogous to those tbr ci. However, the difference is essential in a sense that now the set of equations has a solution independent of r and given by equation analogous to Eq. (3.14), de = d3 = 0,

d4 = 1.

(3.22)

On the other side, only the value r = 1 is of the physical interest. In this case, both sets of equations coincide and therefore, the linear combinations of the correction Lagrangians entering Eqs. (3.7) and (3.21) are the same. Nevertheless, the resulting Lagrangians L:3 and £4 differ in that they are expressed in terms of the fields of various parametrizations. If we now extract the vertex 7rpal, we obtain, in contrast to Eq. (3.16), 1-r -0 £t~,~LS = 2 i - - ~ t r { a ~ , ~ [Pu, O~Tr]}.

(3.23)

668

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

I Eq. (3.23) holds for the arbitrary value of r. If we choose r = 3, then the Lagrangian (3.23) is of the same form as the first term in Eq. (3.16) but the sign is opposite. In the isovector formalism,

/~/],HLS

~p~, -

r--

~

1 o at*~ .pt* × 0/n'.

(3.24)

This Lagrangian could be compared with the analogous vertex £YM ~rpal obtained from the Lagrangian of the 7rpal system in Ref. [7] within the framework of the YM fields, YM

'

£~rpaj =

f~

'

(

pt*p . a u X OpTr + -~Ptzp "~r x amp

).

(3.25)

However, the redefinition of the axial gauge field applied in Ref. [7] differs from the redefinition (3.18) used in this section. If we write down it in the needed approximation we have, r-1 at* = a~ztq_ ~

[01z,Tr_+_y.TT× ,Ot*] _k (..9([~.[3).

(3.26)

Going back to the formalism used in Ref. [5], we write Eq. (3.26) as r-1

a u = at* + - - { 0 t * T r + i y [ p t * , r r ] } + O(Izrl3).

rf,~

(3.27)

We now perform the same procedure as we did after Eq. (3.18), but applying the redefinition (3.27). The linear part of the field a m ' follows from this redefinition, t

1

l-r

[at*(pt*, at*, rr) ] = al, + - - 3 t * r r ,

rf.

(3.28)

and from the equation, t

~

~0

[at*(pt*,at*,II)

]1 _ ~ 0

- at*,

(3.29)

which can be inverted as -0

t

[at*(pt*,a m,rr)

]1

/

= a~,.

(3.30)

Comparing Eqs. (3.29) and (3.30) we can see that the linear parts of the fields at,~°and a~, correspond each other. It follows that substituting Eq. (3.27) for Eq. (3.18) does not bring into the analysis any change. The subsequent analysis shows that the equations OZ for the coefficients d, are the same as before. However, the new Lagrangian £,~p~, for the vertex ¢rpal differ from that given in Eq. (3.23), £oz .r- 1 _ rrpal = t - - ~ t r { r r [ Pl*p, at*p] }. In the isovector formalism,

(3.31)

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

£OZ I -- r a~]. ~rpa, = 2f~- ['n'. p,~ x

669 (3.32)

We considered up to now only the biderivative terms in Lagrangians for the 7"rpal vertex. But non-derivative terms can also appear in these interaction Lagrangians. They did not play any role in the previous analysis, but they are needed now, in order to compare the full results following from the redefinitions (3.18) and (3.26). As to the redefinition (3.18), non-derivative term for the vertex 7"rpal appears from the Lagrangian £ p o f Eq. (2.33) after applying the Sttickelberg transformation, /2~p,,t/"HLS= , y 2 f r ~ C [ ~ " RU X a0u] .

(3.33 I

For the redefinition (3.26), the non-derivative terms also appear but they cancel exactly oz ~ gets unchanged. each other and the Lagrangian 12~roa Summing up Eqs. (3.24) and (3.33) we have o + r-

I.HLS = r - 1 2

£"rrp,,,

.f~ man" P , × a/~

f,~

lao ~,,

" P u x O,,¢r.

(3.34)

In deriving Eq. (3.34), we used equation 9

b 9 2 c = r l-r

roT, = - Y ' f ;

2 ~

(3.35)

Y f~r,

following from Eqs. (2.39), (3.9). The second term on the r.h.s, of Eq. (3.34) can be modified by adding a fourdivergence l-r

0 3~[a,~ . p , x or],

(3.36)

which yields, £t,HLS

1 -- r

~P"' = 2 f ~

{¢r'Pu~

0

x a~

2 0

0

-- 2[maa~z + ( G a ~ , ) l

• ~r × p , } .

(3.37)

For the on-mass-shell al meson, the equation of motion is 2 o + c)~,a°,, = O, maa~

(3.38)

the second term on the r.h.s, o f Eq. (3.37) disappears and in view of Eqs. (3.29) OZ and (3.30) the Lagrangians £~oa, and /~/,HLS -~r,a~ coincide. In other words, the different redefinitions of the aj meson field do not have any physical consequence. The same is true also when the al meson is off-mass-shell. Putting in Eq. (3.37) r = ½ and omitting the upper label HLS we obtain after redefining auo __+ a u ' L;t

1

,rp,,, = 4 f ~ ~ r . Pu~ x a ~ .

(3.39)

In Eqs. (3.17), (3.25) and (3.39), different Lagrangians of the 7rpal system are presented. The tbllowing point does influence essentially the form of these vertices:

670

J. Smejkal et aL /Nuclear Physics A 624 (1997) 655-686

The Lagrangian (3.25) is obtained after demanding that the starting Lagrangian £YM(~) of Ref. [7] would contain no more than two field derivatives in each term, after 'tl"p¢l 1 redefining the al meson field in the most general form, I au = am + F ( ~ 2 ) (0uzr + "y(zr x p u ) ) + G(Tr2)~'(~ . a u ~ ) .

(3.40)

The correction Lagrangian is constructed in terms of the field tensors F(p) ~,, , Fu<~) and of the pion field. The unwanted trilinear and quadrilinear terms are compensated by the proper choice of the functions F and G. In order to get the Lagrangians (3.17) and/or (3.39), the correction terms are chosen in the form (3.1)-(3.6). Combining properly these term, it can be achieved that too a strong momentum dependence of the vertex p~Tr is eliminated and the decay amplitude for the process al --* y + 7"r as well as the vertices pp)2 and alal~) are dominated by the p meson. However, in this case, the multiple particle terms stay present. On the other side, either the parametrization of the gauge fields or different redefinitions of the al meson field do not influence essentially the physical content of the Lagrangian. In the Section 4 we eliminate the al meson in a chiral invariant way.

4. Eliminating al meson

In this section, we remove the al meson field from our model Lagrangian in such a way that the chiral invariance would not be violated and that the p meson dominance, universality and the KSFR relation would continue to hold. A similar task was considered in Ref. [6] and earlier in Ref. [22]. However, the starting Lagrangian of the 7rpal system of Refs. [6,22] is actually a non-physical one. Only the resulting Lagrangian, with the al meson degrees of freedom eliminated, is the physical HLS Langrangian of the 7"rp system [29]. Moreover, this method of eliminating the al meson is not easy to extend to the NTrpal system. As discussed in Ref. [22], pions have no coupling to quarks after eliminating the al meson, which is a major defeat of this model. Here we consider a different task. We eliminate the aj meson from our physical Lagrangian £3 of Eq. (3.7). Actually, we start from our Lagrangian £2 of Eq. (2.43), because the redefinition (3.8) plays no role here and the procedure made in what follows is applicable also to the correction Lagrangians /:~ entering Eq. (3.7). As a result, we obtain the physical HLS Lagrangian of the ~ p system. We also show how to extend this procedure in the presence of the N N a l coupling. Strictly speaking, the method of freezing out particle degrees of freedom is not defined uniquely in the case of the al meson when the nucleon field is present in the model Lagrangian. Actually, there exist different ways of moving parameters of the Lagrangian in order to send the a~ mass to infinity. Generally, different ways will provide different results and some of them will cause undesirable effects. We have found that the best way for our purpose is to use the method of solving equation of motion of Ref. [5]. It means

J. Smejlcalet al,/Nuclear Physics A 624 (1997) 655-686

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that we eliminate the kinetic term of the a~ meson field from the model Lagrangian and subsequently, we solve the equation of motion for that field, expressing it as a function of the other fields. Actually, it is not enough to omit simply the al meson kinetic term, because the presence of the al meson field in the p meson kinetic term would make the equation of motion practically unsolvable. Instead of it, we introduce into the Lagrangian 122 of Eq. (2.43) a parameter dependence,

121 (A) = 122 - 12kin (P/~, a# ) + 12kin (/9/.~, a/~; A),

( 4. l )

where we define 12ki~(/Su, aF,; A) =

l f t r G F ( ~ ) ~ A ~ 2 + t r ( F~. (~,) (A)) 2].

(4.2)

Here in contrast to Eqs. (2.36) and (2.37) we have

1

F
(4.3)

F ~ ) ( A ) = 1 { 5 . . + iy[~u, fi.] + i7[fi.,,6.] },

(4.4)

and the t e n s o r ,~(¢) u~ is defined as

5~,(f') = ~g. + iT [ ~g, ~ ], p

~g~ =_ c~,~. - O. Pu.

(4.5)

It is seen that £~(A = 1) =/22. Different situation appears for l i m A _ s . Indeed,

ltr(~'(~) 2- "" ~" )2,

lira 12kin(/)/~,a/~;A) = /[~OO

(4.6)

and it follows from Eqs. (4.6), (4.1) and (2.43) that lim 12~(A) ~ 12~ = - a f ~ .2t r ( a ~0l I ( H ) )2 _ f ~ t r ( ~ u ± ) 2 A~oo -(b+

c)f~tr

y~ + ~-~c~u±(H)

-ltrt'~'(P))22_,_ ~ . + 12kin(],f#,,Ap~).

(4.7)

The Lagrangian g~~ does not contain the kinetic term of the axial field a~,, which can be now eliminated by solving the corresponding equation of motion and we obtain for the redundant field,

~ -

1

b

y b + c&~x ( H ) .

Substituting tor the field ~

(4.8)

its value (4.8) back into the Lagrangian 12~t we have

122 2 0 I ( H ) )2 _ ltr(~-(~))2 + 12kin(Vu,.A~,) ' - " = _f~tr(6tu_L(H))2 - af~.tr(aul 2 "- ~"

(4.9)

J. Smejkalet aL/Nuclear PhysicsA 624 (1997) 655-686

672

From the other side, this Lagrangian can be derived from the HLS Lagrangian of the rrp system possessing the symmetry G~, × [SU(2)v]t by gauging the global group Gg and using the standard HLS methods. It is easy to verify that the Lagrangian £~2(A) of Eq. (4.1) is invariant under the local chiral transformations for any value of the parameter A. In the next step, we add to the Lagrangian E~t of Eq. (4.7) an interaction Lagrangian of the form,

£i = - t r ( y a / z ) u ) ,

(4.10)

where )/z is a perpendicular current containing some other fields not considered until now. Again, we eliminate the field fi~ from the model by solving the corresponding equation of motion. After substituting this solution back we obtain,

/:2 2i-/:i----->•2 - ~ i , I!

-II

-

(4.11)

where £~' is given in Eq. (4.9) and 1

b

/~i = 4 f 2 ( b + c) tr(S'u) 2 + b--~cctr()uf~/z± ( / 7 ) ) .

(4.12)

In the isospin formalism and taking into account Eq. (2.53) we have, 1

2

1

~i = 3--~ J/~ -'}-4(J/z" alz±(//))"

(4.13)

As an example of the interaction Lagrangian of the type (4.10) we consider the Lagrangian describing the a l N N vertex, ~-~i -~

(4.14)

-igAT( /VT#T57"N) • a~z.

The related Lagrangian (4.13) obtained after eliminating the axial meson is, g2 i ~i = -- ~ ( AIY/~Y5r N ) 2 + ~ga ( NTuy5 ~'N) • &u± (/7)"

(4.15)

It means that the four-fermion interaction is present and the vertex rrNN is renormalized. Moreover, the vertices NN,A and N N ~ V appear, because the Maurer-Cartan 1-form &u± is gauged by the external fields ,Au and Vu [cf. Eq. (2.4l)]. In the next section, we extend the scheme to the Ncrpal system. Besides the Lagrangians, we derive also the currents.

5. Lagrangians and currents o f the

N1rpal system

In Section 5.1 we extend the developments of Sections 2-4 to constructing the Lagrangians of the NTrpal system, using the procedure which has been worked out earlier [9,10] within the framework of the YM fields approach. As to the HLS approach,

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

673

we restrict ourselves to the non-linear parametrization of the gauge fields, because the choice of the parametrization cannot change the physical content of the model Lagrangian. In Section 5.2 we derive the one-body hadron currents from this Lagrangian. In contrast to Ref. [7], we apply the Glashow-Gell-Mann method [27], which allows one to take correctly into account the presence of gauge fields in the Lagrangian. The 7r-MECs are derived in Section 5.3. 5.1. Lagrangians

We write down the total Lagrangian of the NTrpal system in the following form, £× = £ × x Nlrpal -~ V~ pal '

X = YM , nl.

( 5.1 )

where the Lagrangian £ xNrrpal contains always the interaction between the nucleon N and the mesons, while the Lagrangian ~ a m describes only the interaction between mesons• The superscript 'YM' indicates that the gauge fields are introduced within the framework of the YM fields, while 'nl' tells one that the gauge fields belong to the non-linear realization of the HLS. As to the Lagrangians ~a,~,' only the proper vertex rrpal is of the interest here and YMm and £nt~p,,, respectively. we refer to Eqs. (3.25) and (3.17) for the Lagrangians /2~p x Next we consider the Lagrangian £m,p,,,. We start from the case × = YM. In the nonlinear parametrization of the nucleon and pion fields and in the linear parametrization of YM YM the heavy meson fields, the Lagrangian £N~rpa, related to the vertex £~p,, can be found in Refs, [9-11,30]. Using the StiJckelberg transformation, we can write it in terms of non-linearly parametrized gauge fields, YM

ff-N~rpm ( n l ) - - ~ l y t ~ (cg~ + igpp~ " 2 ) N - M A l N -

igagp~lT~T57N " ¢t#

gp t
(5.2)

which is in the operator formalism of the form, £YM

" I" gp KV KI~ ~'(P) hi 2 2 M ' ' ~ / ~ #" ""

(5.3)

nl If we construct the analogous Lagrangian £N,rpa, within the concept of the HLS, we obtain the wider class of Lagrangians, which will contain the Lagrangian of Eq. (5.3) as a special case. This is due to the fact that we have now richer choice of local chiral invariants. The most general Lagrangian is of the form,

£,,I = -~IT~O~N NTrpa~

-

MNN - igp#lyu~uN

g# KV A[o" Ff#) N 2 2M u~ uu

+iK~ N T u f i u N + iK2No'u~, [ flu' fi~ ] N + iK3AITu&uj N • o I, a~l o I] N + i K 6 1 ~ o . u ~ [ 6 ~ u ± , & ~ l [ N +iK4l~yz, ee°ll N + tKsNo'u~ [ ce#l

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

674

-

0

-

0

~

+ i KT Al O'u~, [ 6Lu ±, fi~ ] N + K 8 N o'uv [ atilt, &p± ] N + K9 N o'u~, [ a ~ lI , a~, ] N, (5.4) with the real parameters Kl . . . . . Kg. All the terms --,K/ with i/> 3 have no analogues in the concept of the YM fields. Actually, the present data does not support the need for them. It can be shown that the term "~K2 has an analogue in the concept of the YM fields if 2 K2 - gp Kv 2 2M'

(5.5)

Besides, only the terms ~K1, K3 survive. These terms should satisfy the constraint imposed by the weak axial form factor gA ( k ) , 1

2gpKl + K3 = gA(O) ~ ga.

(5.6)

If we demand that the axial form factor would be al meson dominated, then K3 = 0 and

K~ = --2gAgp,

(5.7)

which makes the Lagrangians (5.3) and (5.4) identical, ~nl YM Nlrpal = £N~'pal

(nl),

(5.8)

In order to compare the case × = nl with the case × = YM considered in Refs. [9-11 ], we pass in Eq. (5.2) to the pseudoscalar (ps) ~ N coupling, using the Foldy-Dyson transformation, . ga N=exp(-t~-~,5("r.lI))N

t.

(5.9)

From Eqs. (5.8), (5.2) and the redefinition (3.8) we have in the leading order, £~t

NTrpal =

£nt(o)

i gog2A Fl

N'n'p -~- ~

" ~/Iz ~ 7 . 11 x ~ u ) N -

igAgpNy~zys(7" ~ u ) N + O([//12),

(5.10) where •

~U~-prm(°)

gA

-

= - A l y u a u N - M1VN + igNy5( H . 7 ) N - tgp~-f~Ny,~y5('r. ~ _i?19

× H)N

(y~,~iz _ i KV o . "x'(~) ~ . T N 4 M ~z~,~~l, j

q_igpga 2MNySO'u~, KV (11. --~"T(P)~N , . . + O(11112) 4f~.

(5.11)

In completing this section, we consider the Lagrangian with the al meson eliminated. We perform its elimination in the Lagrangian £ ,Nn'pa l 1 of Eqs. (5.8), (5.2). Using the results of Section 4 we obtain,

675

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686 ~,,I NTrp

( Ojz + igp~z • -~ ~') N - MI~IN

= --~[~Iz

2 gA

+i

gp T ( ~, 4 2KV M ~o. zvT.N. --u~ -

9

/9T~Ts~'N • ¢i#±(H) - ~ 5 - ( N y u y s r N ) - .

(5.12)

8f, r

The application of the Foldy-Dyson transformation (5.9) in £ ' ~ p yields in the leading approximation, £,,/ . f r N-y # y 5 ( ' r . H × V a ) N NT"rp = r"t~°) ~ N T r p - ie-~/qyz, ys(~'. , , 4 u ) N - t"e ~ga 6Ag2

+ie

NT,(~-. H

×

.,4~) N

2.f~

e2

-

8~= (/~/y/zys"rN) 2 + .

O(I//[ z)

(5.13) '

r"t(°) defined in Eq. (5.11). The pion production amplitudes derived from this with ~N=R Lagrangian reproduce fully the results provided by the low energy theorems, due to the presence of the contact terms N N I I V and N N I I A . 5.2. One-body currents

The one-body currents were derived from the Lagrangians of the previous sections by using the Glashow-Gell-Mann method [27]. Let us note that the Gell-Mann-Ldvy method [8] used in Ref. [7] leads to the different currents. However, both e.m. currents yield the same conserved charge. Therefore, their difference is unimportant. The one-body vector and axial-vector currents with the superscript YM and in the nonlinear parametrization of the nucleon and pion fields and in the linear parametrization of the heavy meson fields can be found in Refs. [9-11,30]. In obvious notations, for the new currents we have m2 v,t •"V,,u

=

---~P- 2f~gp?t~ × 1 I + 0 ( 1 1 I ] 2 ) , g p Pt~

(5.14)

m 2

v':

- ___e_aft + f~c)~lI - 2 f ~ g p # u x H

•.~A,tz --

go

#

+--

gp

O~H-gp~t~+e.A~ × #~,,+O(I//12),

(5.15)

The existence of a contact e.m. vertex can be deduced in the HLS approach from the form of the current j~,t in Eq. (5.14), /2~y,, = 2ef~rgpA~(~t~ × //)~3).

(5.16)

nl The currents generated from the Lagrangians/21v~r p of Eq. (5.13) and Z~' of Eq. (4.9) are m 2

INTrP ---"~-P ~ -~'V,,iz = gp P/x

i f--~--a~Fly~,ys( Fl Zf~r

x 7 ) N + e f ~ ( l l x .A,u) + (9(1H12),

(5.17)

676

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

p'~

t

P] 7r

y"

Q

I

1

¢

F~

7¢

I¢"

j3

t

, . .,¢

k."

P2

pl

b

a

c

d

rL

e

Fig. I. Two-nucleon e.m. Feynman one-pion exchange amplitudes constructed from the Lagrangian (5.1); B (a),(b) nucleon Born terms J~(k), (c) contact term J~(k); (d) mesonic term J~(k); (e) non-potential amplitude J~q .~ ( k ) , x = Y M , nl, I. jNrrp_ F c9 ! 1 --a~r ~ . l -- 2 f ~ g p ~ u X / / + A,Iz --ef~r(//×

.gA t-~NyuysrN

V u) + O(I//12).

. g2 _ + t~f Nyu(//

x 7)N

(5.18)

5.3. Pion exchange currents

Let us consider the pion exchange amplitude constructed from the chiral Lagrangians /:YM and Z:'t of Eq. (5.1). The exchange amplitude is graphically represented in Fig. 1. In the approach of the YM fields, the full amplitude is JuvM(ps) = JuB(ps) + J~,' + J~(ps) + J~V~l,u. The amplitudes with the superscripts B and doscalar (ps) 7rN coupling] and mesonic demanded by the local chiral invariance and VM of Eq. (3.25). It is known Lagrangian E~pat

(5.19)

m are the standard Born [with the pseuterms. The contact amplitude J~(ps) is the last term J~V~,,u 2 is derived from the from Refs. [ 10,11 ] that

2 We consider here the a] -~" current as a pion contact current with a form factor at the vertex NNyrr.

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

677

kuJVM(ps) = k/~(J~(ps) + J~) = kuJ"~(ps) = k/,JVM,,,• = 0.

(5.20)

Let us now discuss the structure of the amplitude J~t(ps). Inspecting the structure of the Lagrangian (5.1) for the case when × = nl and of the current J~,~ of Eq. (5.14) we can see that there are two more terms for the amplitude j~t (ps) in comparison with Eq. (5.19), J~/(ps) = J~(ps) + J~ + J~(ps) - J~u(p~') + J~t,,,u + J~tpa,. u.

(5.21)

The term jnl7 r y a l , # stems from the part of the one-body current of Eq. (5.14) where the photon interacts directly with the ~'al system, giving the interaction Lagrangian (5.16), in contrast to the current J~t~l,U, where the photon interacts with this system via the p meson. In deriving the last current, we employed the Lagrangian £,,t 7rpo I of Eq. (3.17). The term J ~ ( p v ) is the contact current well known from the model with the pseudovector (pv) ~ N N coupling. Here, it enters with the negative sign [it corresponds to the fourth term on the r.h.s, of Eq. (5.11)] and its presence is crucial for the current conservation. Indeed, it can be verified that k/~( j,,t + jnl77"pal ~'yU I ,~L

,~

- JCu(p~') ) = 0,

(5.22)

which together with Eq. (5.20) yields k/~J~t(ps) = 0.

(5.23)

Let us note that the comparison of the currents JVU(ps) and J~t(ps) provides a good example of how the particular components of the current depend on the formalism used in the derivation. The next step is to study the difference between the amplitudes JuTM (ps) and j~t (ps), which will reveal the model dependence of the exchange current. Actually, this task vu reduces to comparison of the amplitude JTrpal,l~ with the amplitude jnl'rJ'd I ,/,t defined as jnl nl nl c ) 7rm ,tz ~- J~yal,tz d- J~rpal,l~ - Ju (Pt )"

(5.24)

The amplitude J~p,,,/~ ¥~ can be split naturally into two parts, YM

J~rpal.tz = Au + BI~,

(5.25)

with A~, and B~, derived from the first and the second term of the Lagrangian gYM 77*p t l l of Eq. (3.25), respectively. On the other side, the analysis shows that j~t .,/~ = B/~.

(5.26)

The standard non-relativistic reduction of the amplitude (5.26) yields for the spatial nl part of the MEC j~,,~,/~, J =,,t, , , = - - -i ~ 2g f +(1 ~ 2).

F.(k)(~'l

×

~'2)3A"F(q~)k × (o'~ × ( k - q 2 ) ) A ~ ( q ~ ) ( t r 2

.

q2) (5.27)

J. Smejkal et aL /Nuclear Physics A 624 (1997) 655-686

678

Let us remind that the similar procedure applied to the amplitude J~V~,u yields the YM of Eq. (1.3). Comparing the current of Eq. (5.27) with that of Eq. (1.3) current j~.,,~ shows that the model dependent parts of these currents differ in the sign. The impact of this difference on the cross section of reaction (1.2) will be studied in the next section. Let us note that the vertex £t~pal of Eq. (3.39) coincides with the second term of the YM of Eq. (3.25). Therefore, the current jt7rpal ,It constructed from this vertex vertex £~p,,~ will coincide with the current J~rl,,~.~ of Eq. (5.26) as it should do, because the physical content of a model is independent of the parametrization of the particle fields. At last, we consider the two-nucleon amplitude Ju(ps) coming from the Lagrangian nl £N~w of Eq. (5.13). Graphically, it is given by the same set of graphs as in Fig. 1, but the graph le, Ju(ps) -- JuB(ps) + J~,' + J~(ps) + J~'z,p,u'

(5.28)

where

J~ye,u = 3~,~ - jc (pv).

(5.29)

The contact current J~,u corresponds to the third term from the end on the r.h.s, in Eq. (5.13). The calculation yields the result,

gag F V ( k ) (kukv - k28~v)e3ab~(p~)yvT"ay5u(pl ) A ~ ( q 2) J~z'p,u - 2 f ~ m~ ×R(p~)ysrbu(p2) + (1 ~ 2), k~JCyp,~ = 0.

(5.30) (5.31 )

In the non-relativistic limit, the space part of the MEC coming from the amplitude of Eq. (5.30) is • g

jcyp

=

ga F ~ ( k )

_t 2 M 2 f ~

-

-

m 2 (~'z × ~'2)3k × (o'1 × k)A~(q2)(~r2 "q2)

+(1 ,--,2).

(5.32)

In this case, the c u r r e n t jCyp is completely fixed by the low energy theorem and no model dependent part is present. We compare the currents of Eqs. (5.27) and (5.32) by calculating the ratio,

FV(k)a~'(q~) 2 o 2 1F FV(k)/m~ = mp/tF(ql) = ~ a ( ~ ) .

(5.33)

This result indicates that the two models can differ considerably at large momentums transfer.

679

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

10 5 ~-~

10 4

.... , ~~q- ~

i0 a

T

'

I

'

I-- '

©

0 3

:-"a O

10 2

a. ~ D

0

%:~

C

b

Enp=

0.2

0.4

l.SMeV

w ~-~ .... "E3-

0.6 t

0.8

1.0

1 .~

[c~v~]

Fig. 2. The cross sections for reaction e + d ~ e' +np; solid curve: the calculation without the al MEC included, long-dashed curve: HLS model of the MEC with the current f ~'a 1 of Eq. (5.27), short-dashed curve: YM model of the MEC with the current JYaM of Eq. (1.3). The data is from Ref. 1141. For the details s e e text.

6. Application We used the obtained currents J~-~ of Eq. (5.27), j~rp o f Eq. (5.32) and the current YM j~,,, of Eq. ( 1.3) in the calculations of the cross sections for reaction (1.2). The impact o f the change of the sign of the model dependent part in Eq. (5.27) on this observable can be deduced from Figs. 2-4. The deuteron and the n - p final state wavefunctions correspond to the Bonn OBEPQB potential modified by including the a~ meson exchange in Ref. [31]. The waves with L <~ 3 were taken into account. The cross sections were calculated without the a l - ~ " MEC (solid curve), with the a l - T r MEC derived within the HLS scheme (long-dashed curve) and with this MEC obtained earlier within the Y M approach (short-dashed curve). All three curves correspond to the H7.1 parametrization o f the e.m. nucleon form factors of Ref. [32]. Besides the a l - T r exchange current, 3 the M E C operator contains the contribution from the rr and p exchanges. The 7r M E C includes the full set of leading relativistic corrections of the order O ( I / M 3 ) , including boost and retardation [33,34], the p MEC is the standard one [ 3 5 - 3 7 ] so far. We also calculated the cross sections with the current operator of the HLS scheme and using many other available parametrizations of the e.m. nucleon form factors, including the dipole fit, dipole fit with G~ = 0 and those o f Refs. [ 3 2 , 3 8 - 4 3 ] . With one exception, all the cross sections are between the long-dashed curve and dotted curve, The al mesic current plays no role.

680

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

which corresponds to the dipole fit with G~: = 0. The exception is the long-dashed curve labelled by the squares, resulting from the e.m. nucleon form factors of the model 1 of Ref. [39]. We disregard this parametrization from now. Then it follows from Fig. 2 that the variation of the cross section due to the model dependence of the al MEC is about 30% of the dependence on the parametrization of the e.m. nucleon form factors, up to t ~ 1 GeV 2. On the other side, it is at the level of the present experimental errors [ 14]. In Fig. 2 the dotted curve labelled by the circles corresponds to the model of the MEC developed recently in Ref. [44]. All other ingredients of the model coincide with those discussed above, the e.m. nucleon form factors are from Ref. [32]. In Ref. [44], the operator of the nuclear 7"r MEC, J/zmec,S is constructed, starting from the one-nucleon current in the Sachs parametrization and the two-nucleon 7"r exchange Born amplitude in an arbitrary mixing of the p s - p v 7rN couplings. The 7r MEC has the correct chiral properties and up to the considered order O ( 1 / M 3 ) , it satisfies the same nuclear continuity equation as the analogous standard ¢r MEC Jtzmec'D'°,derived directly from the Lagrangians [33,34] (or by using the one-nucleon Dirac-Pauli current in the construction of Ref. [44]). These two currents differ by a relativistic four-current Aju =- ( A j , Ap) which satisfies the continuity equation in the considered order in I / M , q. Aj (3) - q o A p (2) = O.

(6.1)

As it is seen from Fig. 2, this model does not conform with the data. The reason should be related to different off-shell behaviour of the one-nucleon currents in different parametrizations. This result illustrates also the importance of the relativistic corrections due to the 7r MEC in the considered kinematical region. In order to make fully consistent calculations, we should add higher order corrections into the p MEC and ~/, 6, o" and o~ MECs. These heavy meson exchanges are present in OBEPQB, but not yet in our MEC operator. As it is known from the earlier calculations [ 13], this part of MEC tends to rise up the cross section in the considered kinematical region, thus worsening the agreement of our theory with the data [ 14] at the large momentums transfer. On the other side, the results of the Bates measurement [ 15] of the cross section (the circles in Fig. 3) are for t ~> 1.2 GeV 2 larger and our calculation describes them well up to t ~ 1.5 GeV 2, showing no dip in this region. The last data point od Ref. [ 15] at t = 1.65 GeV 2 is underestimated by our calculation. It might indicate the onset of new degrees of freedom, not present in our theory or shortcomings of the used approximations. On the other side, our calculations agree much better with the last data point of Ref. [ 15 ] averaged over the interval (0 - 10)Me V of the relatives energies of the np system. In Fig. 4 we present the analogous results for the region of even higher momentums transfer investigated experimentally in Ref. [ 16]. Again, for the values t /> 1.5 GeV 2, our calculations evidently underestimate the data. In Fig. 5 we present the cross sections for reaction (1.2) calculated using the model currents with the al meson eliminated. As it has been found in Section 5, the correct elimination of the al meson degrees of freedom from the Lagrangian yields a pion-like

J, Smejkal et at./Nuclear Physics A 624 (1997) 655-686

] 05

F

~

~-

[

7 ----~

~----~

681

-

104 q9 i0 a ~D 10 2

v~ I

101 e~ b 10 0

0 e, = 10 -1

I

160

L

o

I

0.4

,

I

0.8

,

1.2 t

[

1.6

,

20

[ G e V ~]

Fig. 3. The cross sections for reaction e + d ---, e' + n p ; the meaning of the various curves is similar as in Fig. 2, the data is from Ref. [ 14] (triangles) and from Ref. [ 151 (circles).

•

T

F

~

T

101

lo 0

"<::-:~

'z3 b ~,~

10 -1

np 0 e. = ,

10 -2 1.0

180 ° I

1.5

,

I 2.0

L~ _ _ 2.5

3.0

t [ ~ v ~] Fig, 4. The cross sections for reaction e + d - ,

e ' + rip; the meaning of the various curves is similar as in

Fig. 2, the data is from Ref. 1 1 6 ] .

contact c u r r e n t jCp of Eq. ( 5 . 3 2 ) , which should be added tO the standard rr+p MEC. The solid curve corresponds to the rr+p MEC, with all the relativistic corrections of the order 69( 1 / M 3) in the one-nucleon current and in the rr MEC included but without the contact current ( 5 . 3 2 ) . It describes the data [ 14] well up to t ~ 1.5 GeV 2. The dotted curve was obtained by adding the contact c u r r e n t jerryp to the rr+p MEC taken as for

J. Smejkal et a l . / N u c l e a r Physics A 624 (1997) 6 5 5 - 6 8 6

682

10 5

I-

--I~

l

- ~ - -

\ 10 4

-~

103

10 2 "u 101 b rtp 10 °

0 e, = 1 6 0 °

i0 -1

, 0.2

I 0.6

,

I 1.0

,

t 1.4

, .8

t [~eV z] F i g . 5. T h e c r o s s s e c t i o n s f o r r e a c t i o n e + d --~ e t + np, c a l c u l a t e d u s i n g t h e c u r r e n t s w i t h t h e a l

meson

eliminated; solid curve: ~r+p MEC, with the ~r MEC containing all the relativistic corrections of the order O(I/M3), dotted curve: the ~r + p MEC as before + jCrpof Eq. (5.32), short-dashed curve: the current operator as before, but with the ~- MEC from Ref. 144], long-dashed curve: fully non-relativisticone- and two-nucleoncurrents. For other details see text.

the solid curve. The short-dashed curve represents the same calculation but with the ~r MEC operator from Ref. [44]. The long-dashed curve is due to the fully non-relativistic one- and two-nucleon current operators and without the contact current (5.32). It does not conform with the data, in contrast to the similar result of Ref. [45]. In Ref. [45], the agreement with the experiment was achieved by varying the cutoffs A~ and Ap. Here we have no free parameters, All the curves of Fig. 5 were calculated with the standard OBEPQB nuclear wave functions [46] and with the e.m.nucleon form factors of Ref. [32]. As it is seen from Fig. 5, the effect of the current j~rp is not negligible. Comparison of the solid and dotted curves leads to a seeming paradox: The standard r r + p MEC describes the data better than the correct chiral model ¢r + p MEC + j~p. Actually, this agreement with the data masks the essence of the problem. We remind that the Lagrangian (5.13) with the aj meson eliminated was derived for m, --* oo. Because m,, ,-~ 1.3 GeV, the model with the al meson eliminated works well near threshold. But its validity becomes questionable if one would like to apply it at higher energies. The Fig. 5 indicates that this happens already at rather low momentums transfer. We conclude that the presence of the al exchange both in the MEC and in the potential provides a reasonably consistent description of the data up to the momentums transfer t ,~ 1.5 GeV 2 and the corresponding Lagrangian model reproduces correctly the current algebra and PCAC predictions at threshold.

Z Smejkal et al./Nuclear Physics A 624 (1997) 655-686

683

7. Results and conclusions 7.1. Results

In this paper we studied the model dependence of the e.m. isovector MEC. This study concerns its transverse part, the longitudinal part being fixed by the continuity equation. The transverse component of the MEC was constructed for the first time in Ref. [ 10] within the framework of the hard pion method based on the minimal YM Lagrangian [ 7] of the 7rpal system. Its space component is given in Eq. (1.3). A piece of this current Nq2 is not fixed at threshold by a low energy theorem and its variation with the change of the model Lagrangian can induce the model dependence in the MEC. As it is known [5,6], Lagrangians constructed within the framework of the HLS contain the YM Lagrangians as a subclass and we choose the HLS Lagrangians as a tool for studying the above discussed model dependence. In Sections 2 and 3 we constructed the HLS Lagrangian of the 7rpal system, invariant on the group Gg × GI (2.1). We followed the method of Ref. [5], which consists in adding to the basic Lagrangian a set of higher order Lagrangians so that the final Lagrangian possesses the needed properties: the p dominance and the universality are respected, the KSFR relation and the Weinberg sum rules are satisfied. Our construction differs from the one of Ref. [ 51 in that we do not use the equations of motion for the gauge fields and that it provides lbr the anomalous magnetic moment of the al meson the commonly accepted value 6#,, = - 1 , in contrast to 6/za = + 3 obtained in Ref. [5]. The resulting rrpaj interaction of Eq. (3.17)4 differs from the analogous one of Eq. (3.25), obtained within the YM fields approach. In Section 4 we have shown how the al meson degrees of IYeedom can be eliminated in our scheme of construction so that the p dominance, universality and the KSFR relation continue to hold and the chiral invariance is preserved. In Section 5 we extended the construction to the NTrpal system. The obtained Lagrangian r,,/N'rrptt I of Eq. (5.4) is more general than the YM Lagrangian £YM N T r p ~1j of Eq. 5.3), but the present data does not support the need for more rich structure. Further we generated the one-body and two-nucleon currents. Instead of the Gell-Mann-L6vy method [81, we used the Glashow-Gell-Mann method [27], which takes correctly into account the presence of the gauge fields when generating the currents. Further we have lound that the HLS Lagrangian yields a different model dependent part of the transverse MEC, which differs by the sign from the one constructed within the YM fields approach [cf. Eqs. (1.3) and (5.27)]. Besides, we have shown that our exchange current of the reduced NTrp system differs from the commonly used 77-+ p MEC by a transverse contact 7ryp current [cf. Eq. (5.32)]. We applied the generated currents to the description of the cross sections of reaction ( I . 2 ) at threshold in Section 6. The studied model dependence of the MEC "~The equivalent interaction written in terms of Ihe gauge fields belonging to the linear realization of the chiral symmetry can be tbund in Eq. (3.39).

684

J. Smejkal et aL /Nuclear Physics A 624 (1997) 655-686

manifests itself at large momentums transfer (cf. Figs. 2-5). In Fig. 2 the calculated cross sections are compared with the data of Ref. [ 14]. For the values of t ~ 0.8 GeV 2, this model dependence is about 30% of the change in the cross sections coming from the model dependence of the e.m. nucleon form factors. For t ~> 0.8 GeV 2, our theory slightly overestimates the data. On the other side, the new data [ 15] obtained in the similar kinematics, is well described (cf. Fig. 3) up to t ~, 1.5 GeV 2. Our calculation does not show any dip at these values of the momentum transfer. In Fig. 4 we compare our predictions with the data of Ref. [ 16]. In view of the made approximations, the agreement of our calculation with the data is acceptable for t ~< 1.5 GeV 2. The application of the current obtained for the reduced Nrrp system (see Fig. 5) shows that such a model loses its applicability already at rather low momentums transfer. On the contrary, the standard ~- + p MEC is also able to describe the data reasonably, even though it does not possess the correct chiral properties. This case can serve as a good example of the incorrect current which can describe the data, thus masking the essence of the studied effect.

7.2. Conclusions

Our main conclusions are (i) In principle, the HLS scheme offers much richer structure of the Ncrpal Lagrangian than that of the massless YM compensating fields. However, the physics at intermediate energies is not enough rich to profit from it. On the other side, the incorporation of the heavy mesons within the HLS approach is being made naturally: the masses are generated by the Higgs mechanism and the heavy meson mass terms do not violate the underlying local symmetry. Of course, the schemes with the heavy meson fields belonging either to the linear (YM like fields) or to the non-linear realizations of the HLS are equivalent. (ii) The model dependence of the e.m. isovector MEC induced by the non-uniqueness of the studied Lagrangians manifests itself at higher momentums transfer. However, it is less pronounced than the model dependence coming from the uncertainty in the e.m. nucleon form factors. (iii) The presence of the al meson exchange both in the MEC and in the nuclear interaction provides a consistent description of the cross sections [14-16] for the backward deuteron disintegration (1.2) up to the momentums transfer t 1.5 GeV 2. Simultaneously, the underlying model Lagrangian respects the current algebra and PCAC predictions at threshold and the p dominance, the universality, the KSFR relation and the Weinberg sum rules. (iv) The standard ~" + p MEC does not possess the correct chiral properties.

J. Smejkal et al./Nuclear Physics A 624 (1997) 655-686

685

Acknowledgements We thank Prof. Dr. H. Arenhrvel for the interest and discussions. One of the authors (E.T.) would like to thank him for the hospitality during his stays at the University of Mainz. We also thank Dr. W.M. Schmitt for sending us the results of the measurement [15] prior to publication and for the critical remarks on Section 6 and Dr. S. Platchkov for providing us with the parameters a and b of the two-parameter Galster formula for the form factor G~, for our improved deuteron wavefunction.

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