Pion photoproduction in the skyrme model and low-energy theorems

Nuclear Physics A526 (1991) 733-750

North-Holland

PION PHOTOPRODUCTION IN THE SKYRME MODEL

AND LOW-ENERGY THEOREMS*

Stefan SCHERER’ and Dieter DRECHSEL

Institut fiir Kernphysik, Johannes Gutenberg-Universitiit, D-6500 Maim, Germany

Received 24 October 1990

Abstract: Threshold pion photoproduction has been calculated for the nucleon, within the framework

of the Skyrme model. The calculation involves two different types of coupling between the pion

and the nucleon, either as a small amplitude fluctuation or as a chiral perturbation about the nucleon soliton solution. It is shown that a correct determination of the axial coupling constant

in the chiral limit is a compulsory prerequisite in order to reproduce the leading term of the charged

production (Kroll-Ruderman theorem). Both methods fail to predict the higher order terms of

low-energy theorems, which are of particular importance for the neutral pion production. This

shortcoming is not due to the adiabatic treatment of the rotation matrices, but is a consequence

of the approximations for the pion-nucleon coupling mechanism and of the neglect of the Dirac

sea in the theory.

1. Introduction

Pion photoproduction at threshold has regained a lot of attention over the past

few years. This renewed interest is mainly triggered by recent experiments 1*2) which

indicate a serious discrepancy between experiment and generally accepted theoreti-

cal predictions for the threshold amplitude. Since these low-energy theorems (LET)

have been derived on the grounds of Lorentz invariance, gauge invariance and the

partial conservation of the axial current (PCAC), they should be valid in any model

obeying these basic symmetries. The lagrangian of the Skyrme model incorporates

all the symmetries required to derive LET. However, the standard procedure to

obtain nucleons and pions in the framework of this model involves serious approxi-

mations which eventually violate LET. Therefore it is the aim of this contribution

to evaluate the model prediction of the threshold amplitude and to compare it with

LET.

The content of LET may be summarized as follows. The threshold production

amplitude may be expanded in a power series in the momentum of the photon, k,. Due to the conservation of the electromagnetic current the coefficients of the leading

terms of order k;’ and k”, are completely determined. This result is known as the

Kroll-Ruderman theorem ‘) and was derived within a field-theoretical framework.

l Supported in part by Deutsche Forschungsgemeinschaft (SFB 201).

’ Present address: National Institute for Nuclear Physics and High Energy Physics, Section K

(NIKHEF-K), P.O. Box 41882, 1009 DB Amsterdam, The Netherlands.

0375-9474/91/$03.50 @ 1991 - Elsevier Science Publishers B.V. (North-Holland)

134 S. Scherer, D. Drechsel / Pion photoproduction

Kroll and Ruderman proved that the photoproduction of charged pions at threshold

computed to lowest order in the pion-nucleon mass ratio, p = m,/m,, but to

arbitrary order in the pion-nucleon coupling constant is equivalent to a calculation

in second order perturbation theory with pseudoscalar coupling, provided that the

pion-nucleon coupling constant and the nucleon mass are replaced by their renor-

malized values. It was also shown that in the static limit, p + 0, the -ir’-production

amplitude vanishes. The predictions of the Kroll-Ruderman theorem may be ex-

tended by using the PCAC hypothesis. Such soft-pion techniques require an extrapo-

lation from the unphysical soft pion limit, pion four-momentum qi = 0, to qt = mf,.

As a first application to pion photoproduction, Fubini, Furlan and Rossetti “) derived

dispersion relations within the framework of current algebra, connecting the isoscalar

and isovector anomalous magnetic moments of the nucleon with the forward produc-

tion amplitude for soft pions. A different approach was used by de Baenst ‘) by

splitting the production amplitude into a Born part and a residual amplitude. By

imposing the PCAC relation (which is not automatically incorporated in pseudo-

scalar coupling) he derived a relation between the Born amplitude and the residual

part. This additional constraint allows a prediction of the threshold amplitude up

to terms of order p(~*) for the charged (neutral) pion production. The coefficients

of the power series expansion depend on macroscopic properties of the nucleon,

such as charge, mass and magnetic moment. A more ambitious approach including

the most general off-shell formfactors recovers the old predictions of de Baenst 6).

Finally, one may also use Weinberg’s effective lagrangian ‘) to calculate the Born

diagrams which now include a seagull term. An expansion of the amplitude up to

terms of second order in p agrees with the predictions of LET, since the Weinberg

lagrangian is constructed in such a way as to reproduce the current algebra results.

While the predictions of LET for the production of charged pions are in good

agreement with experimental data, recent experiments for the reaction y + p + p + r”

[refs. ‘,‘>I show a strong discrepancy between theory and experiment. This observa-

tion has triggered a series of new theoretical calculations of the yrr process. Typical

investigations are concerned with rescattering, higher order contributions, unitarity

and the role of chiral and isospin symmetry breaking *-12).

Despite the discrepancy between experiment and theory, LET provides a powerful

tool for testing models of the nucleon (and the pion), since most of the QCD inspired

models include the basic ingredients necessary to derive LET (CVC, PCAC) and

for that reason should reproduce the LET predictions. In the case of quark models

it can be shown that a correct treatment of the CM motion is crucial for all the

terms beyond the Kroll-Ruderman theorem 13,14).

The Skyrme model 15,i6) is an effective description of hadronic phenomena at low

energies in terms of mesonic degrees of freedom. Its revival was motivated by ‘t

Hooft’s observation that QCD in the large N, limit reduces to a theory of weakly

interacting pions 17) and that baryons may be regarded as solitons in such a theory *8).

Static properties were calculated in the pioneering work of Adkins, Nappi and

S. Scherer, D. Drechsel / Pion photoproduction 735

Witten 19) and were found to agree with experimental values typically at the 30%

level. Pion photoproduction in the Skyrme model has been investigated by several

authors. The photoproduction of nucleon resonances up to one GeV has been

calculated in semiquantitative agreement with experiment *‘,‘i). Since the model

does not contain any parameters specifically adjusted to this process, this result is

quite encouraging. It has been stressed by Hoodbhoy that the topological current

of the Skyrme model has special consequences for yr” production **).

In sect. 2 we discuss the formalism of pion photoproduction and the predictions

of LET. The Skyrme model is briefly described in sect. 3. The axial coupling constant

and the mechanism for pion photoproduction in the Skyrme model are discussed

in sects. 4 and 5, respectively. Finally, we present our results on the threshold

production amplitude in sect. 6.

2. Low-energy theorems for pion photoproduction

Threshold kinematics corresponds to a c.m. photon momentum l&l =

m,(l+&)/(l+~), where p=mm,/m, ~0.15. The differential cross section for

unpolarized particles may be parametrized in the c.m. frame as

(1)

where 9 is an operator, acting on the Pauli-spinors Ii) and If). Its multipole

decomposition in terms of electric and magnetic multipole amplitudes E,, and M,, ,

leading to final TN states of orbital angular momentum 1 and total angular momen-

tum j = I*;, was given first by Chew et al. *‘). At threshold the pion is produced

in an s-state and, due to the conservation of angular momentum and parity, the

only contributing multipole is E,, ,

9= io * EE,,+.

Consequently the differential cross section at threshold is given by

(2)

$=fi (Eo+12. Y

Assuming isospin symmetry of the pion-nucleon interaction, E,,, has the following

structure in isotopic spin space,

where xi and xi denote the isospinors of the initial and final nucleon, respectively.

Defining the operators 7, = fi( 2 r, i iT2), the four physical amplitudes are linear

combinations of the three amplitudes Eh$‘(P = 0, *)>,

E,,+(T+) =d(E;y+ EL;‘), (5)


E,,+(T-)=a(EbO,)-EL-,‘), (6)

E,+(p.ir’) = Ebi,‘+ Eri, (7)

E,,+(nr”) = Eb+,‘-- Ehy. (8)

Due to the conservation of the electromagnetic current and the partial conservation

of the axial current, the threshold amplitudes for E&’ and Eby+’ are determined

up to terms of order /1 and p2, respectively,

(9) E(;,)_ eO &NN l+tF

4~7 2m, (l+j.~)~‘~ (I + [P21) ,

E(o/+) -3 g,,, 1+$/L o+ -

47r 2m, (l+j~)~” (-5/J +:P2(l + KS/V )f[P31). (10)

In our numerical calculations we have used the coupling constants ei/4v = & and

giNN/4rr = 14.3. The anomalous magnetic moments of the nucleon are K, = Kp - K, =

3.70 and K, = Kp+ K, = -0.12.

Table 1 compares the experimental and theoretical values for the electric dipole

photoproduction amplitudes. If isospin symmetry breaking is negligible, relations

(5)-(8) may be inverted,

E:;;,’ = &E,+( r+) - Eo+( n-)) , (11)

E~O,‘=&E,+(T+)+E,+(T-)), (12)

EC:,’ = Eo+( pn”) - Ef2 , (13)

and one obtains as experimental values for the amplitudes

E&_‘=(21.3*0.4)~ 1O-3/m,+, (14)

EL? = (-1 3*0 4) x 10e3/m,+ . . 7 (15)

TABLE 1

Experimental and theoretical values of the electric dipole

photoproduction amplitude E,, in units 10W3/ mrr+

Channel Experiment LET

(16)

yp-f nv+

yn + p6 YP-fP?rO

yn+ n7f”

28.3 kO.5 [ref. *“)I

-31.9*0.5 [ref. *“)I -0.5 *0.3 [ref. ‘)I

-0.35 f 0.1 [ref. ‘)I unknown

21.5

-32.0

-2.4

0.4


which have to be compared with the theoretical predictions,

EL-,’ = 20.9 x 10m3/m + ?r 9 (17)

E(O)= -1 6x 10e3/m o+ . + n > (18)

EdI’ = -0.8 x 10-“/m,+ . (19)

While there is a good agreement between theory and experiment for EL;’ and EgJ,

the prediction for E,+ (+I differs substantially from the experimental result, and even

the overall sign is not described properly.

3. The Skyrme model and pion-nucleon coupling

The lagrangian of the original Skyrme model 15,16) is given by

LZ=$cTr(a,Lia*U’)+& Tr ([ Utd,U, U’a,U][ U*d@U, U’a”U]) , (20)

where U is an arbitrary SU(2) matrix, fr the pion decay constant (experimental

value f, = 93 MeV) and e a parameter determining the size of the soliton. Eq. (20)

is invariant under global SU(2) x SU(2) transformations. The corresponding Noether

currents

sz s2? j”=Tr -sLJs-

6l3,iJ 6d,lY 6Uf

> (21)

may be constructed from

&Au= -${U,T}, (22)

s, u = ii[ u, T] . (23)

If we add a symmetry breaking term to the lagrangian of eq. (20), e.g.

L?~=~rn’,f~Tr(UU-l), (24)

we obtain, to lowest order in the pion field, U = 1 + iT. m/f, + [r2], the PCAC

relation

a w A”+ = rnz fwrr" . (25)

The construction of a nucleon soliton solution, using the hedgehog ansatz U,(x) =

exp (in * i%(r)), and its subsequent quantization is discussed in detail in refs. ‘9*25).

The momentum and kinetic energy of the soliton may be included by replacing x

by s(t) = x -R(t) and treating R as a collective coordinate 2”). The corresponding

hamiltonian reads

(26)


where the expressions for the static mass M and the moment of inertia 0 are as in

ref. 19). In our convention an active rotation of the isovector, i.e. i + D( a (t), p(t),

y( t))i, is chosen for quantization. Using Euler angles and the position of the center

of the soliton as the collective coordinates of the baryon, its wave function is given

by

(cu,p,y;R(T= J,M,,M,;P)=(2rr))3’2exp(iP*R)(-)”+M~

X Di$-~,b, P, Y) , (27)

where the convention of ref. *‘) for the Wigner functions is used.

Before attacking the process of pion photoproduction, one has to solve the question

of how to couple a pion fluctuation to the nucleon soliton solution. There are two

different approaches to this problem. The first,

U(x, t) = exp D(a, P,

regards the pion as a small amplitude fluctuation and is discussed in detail in

refs. 28*29). The ansatz of eq. (28) is inserted into the Skyrme lagrangian eq. (20)

and expanded up to terms of second order in m. In the adiabatic approximation,

couplings between the fluctuation n and the rotational and translational motion of

the nucleon are neglected, i.e. all terms containing a time derivative of the rotation

matrix D or the c.m. coordinate R are dropped. Since the hedgehog ansatz minimizes

the action in the B = 1 sector, there is no linear pion-nucleon coupling in the

adiabatic approximation. Pion wave functions of the pion-nucleon system are

obtained by expanding the action into normal modes and solving the (coupled)

second-order differential equations.

In the second approach 30,3’) the pion is introduced as a chiral perturbation about

the nucleon soliton solution,

U(x, t) = u’,u,u’, , (29)

where

Lr=(x, t)=exp(i** yj: ‘)). (30)

Inserting eq. (29) into the Skyrme lagrangian leads to 30)

_Y = ZN + _YM + 5&, + corrections , (31)

where ZN, XM, ~i,f stand for free nucleons, mesons and their interaction, respec-

tively. The keyword “corrections” contains hard pion terms (see ref. *“)). The

interaction term is given by

1 ~i,,,=la,rr.A~[Li,l-fif,mz,~.Tr(l~~)+-_a,nxn

.Lr 2f2, . VN[&l

+$ritm2Tr (1- V,)+[G-~]. (32)

S. Scherer, D. Drechsel / Pion photoproduction

After comparing eqs. (31) and (32) with the effective Weinberg lagrangian ‘)

739

(33)

Schnitzer concludes that the soft-pion soliton lagrangian satisfies all soft-pion

theorems in a model independent way if the Goldberger-Treiman relation

gA g,NN __=-

fw mN (34)

is satisfied. We will show that this argument will not automatically hold for the pion

photoproduction amplitude.

4. Axial coupling constant in the chiral limit

The matrix element of the axial current is defined as

(N’(Pz)(A”“(O)(N(P,)) = I ; ( ?““GA(c?~)+ q+@&(p~), (35)

where ql* = p$ -py . For simplicity we have omitted the Pauli spinors and the

isospinors of the nucleon. The momenta p, and p2 obey the on-shell condition

pf = pi = M:. If the axial current is conserved,

(P~-PJJ~‘(P~)~AYO)IN(~~))=O,

the form factors GA( q2) and GP( q2) may be related by applying the Dirac equation,

Wq2) = -4d&&!2) cl2 .

Assuming GA( q2) to be constant and nonvanishing for q2 + 0, GP( q2) must have a

pole at q2 = 0, corresponding to the pion exchange of fig. 1. In the Breit frame

A a.K

q

f

I n”

,i.

N(P,) N(P,)

Fig. 1. Pion contribution to the axial current matrix element between nucleon states.

740 S. Scherer, D. Drechsel J Pion photoproduction

(qb = (0, q)), a nonrelativistic reduction of eq. (35) leads to

(AQq)(A”(O)lN( +))‘+ra(GA(q2)a -9 q u * q) . N

(37)

Inserting eq. (36) into eq. (37) results in

lim (N’($q)IA”(O)IN( -~q))=$Pg,(a-a . ii), C&+0

(38)

where gA = GA(O). This reduction is not unique, since it still contains the direction

of 4. For that reason it is suggested in ref. i9) to perform an averaging procedure

with respect to all directions of 4,

=pu$gA.

In the Skyrme model the static axial current matrix element is given by 19)

I d3x A”(x) = +T’

I d’x (A(r)u+B(r)u * $2))

where

A(r)= -; f’,

(39)

(40)

(41)

and

Taking account of the asymptotic behaviour of F(r) for large r, F(r) - rp2, the

integral of eq. (40) diverges when the r-integration is performed first. On the other

hand, interchanging the order of spatial and angular integration leads to a finite

result,

I

00 d3x A”(x) = $ra

I J dr r2 d0 (A(r)u+B(r)u . ii). (43) 0

In ref. 19) this is interpreted as the analogue of taking the symmetric limit in eq.

(39). For that reason the axial coupling constant is determined by setting the right

sides of eq. (39) and eq. (43) equal,

gA= -$r jomdrr2[/2,(Ft+s)

2F’sin2 F+sin 2F sin’ F

r2 F12+-

r r2 (44)


There are several arguments why such a reasoning cannot be correct. In the first

place, including a pion mass term leads to a more rapid convergence of the function

F(r) for large r. In that case the order of integration does not matter and the

procedure of eq. (43) may then be regarded as the continuous limit for m, +O.

There is no convincing argument why the axial coupling constant g, should be

discontinuous for m, + 0, that is multiplied by $, as suggested by eq. (44). Second,

the integrals also converge for any finite momentum transfer q, if we take account

of the recoil of the Skyrmion. Finally, in order to apply the argument of current

conservation, one has to make sure that the axial current is conserved for any

momentum transfer q, and not just in the static limit. As we will show next the

nucleonic contribution is in fact not conserved when going beyond the static

approximation.

Taking account of recoil by introducing the collective coordinate R and omitting

terms involving two time derivatives, the operator for the axial current is

A”(x, R) =$“[A(s)a+ B(s)a . s^$] , (45)

where s = x -R and the functions A and B are given by eqs. (41) and (42). Evaluating

this operator between nucleon states at x = 0 and leaving out the normalization

factor (27r) ‘I2 in the nucleon soliton wave function, we obtain

(N’(iq)lA”(O)[N( -iq))= d’R eiWR A”(0, R)

-ha @(q*) -2 G+,(q2)a-a. qq- , (46)

N

with

GA( q2) = 47r J m dr ~*(4rhdqb) +SB(r)(j,(lqlr)+j2(lqlr))) , (47) 0

G:(q2) = “,;;’ J 0adrr2R(r)j,(lqlr) .

The index N of GF(q2) in eq. (48) refers to the fact that it only contains the nucleonic

contribution to the form factor Gp(q2). Taking account of the behaviour of the

spherical Bessel function j,(x) for small x,j2(x)+&x2, it is obvious that Gp(q2)

does not show the pole behaviour expected for small q2. Consequently the nucleonic

contribution to the axial current matrix element is not conserved. Indeed, this is

not a surprise since it is well-known that a massive fermion does not conserve its

helicity. In fact, axial current conservation is restored by coupling a massless pion

to the nucleon in a chirally invariant way. At first sight there seems to be a

contradiction between the nonconservation of the nucleonic axial current and the

fact that the equation of motion for the static soliton can be cast into the form

diAa3’[ U,] = 0, (49)

742 S. Scherer, D. Drechsell Pion photoproduction

but eq. (49) is valid only for zero momentum transfer, and therefore equivalent to

the trivial statement 4 * A” = 0 for q = 0.

The pionic contribution to the axial current matrix element may be calculated by

inserting the ansatz of eq. (29) into the expression for the axial current,

Aq’“[ U; UN U;] = A”,@[ U,] + A”+[ U;,] +. . . , (50)

where U 2,, = U,Uw. To lowest order in the pion field the pionic part reads

A”+[ U:,] =~+YT~. (51)

The pion field in eq. (51) may be eliminated with the help of the equation of motion

O&‘(x) = --f d,Aa+[ Up,]. (52) XT

Taking the Fourier transform of eq. (52) yields

P(q)= I

d4x ei4.“,rr”(X)= -+exp (-iq. R)$ A”‘[ U,(q)]. (53) Tr

Transforming eq. (53) back to coordinate space gives the pionic contribution

Aqp[ I!.&] = - J d4P (2rr)4 e

-ip.x e-ip.R Pp -3 PJY WP)l . P

(54)

The operator has to be evaluated between nucleon states for x = 0, giving

(N’(+q)(A”[U:,(x=O)](N(-fq))=+vq^ GA(q2)+qzT)$ (

(55) N

The sum of the nucleonic and pionic contributions, eqs. (46) and (55), is

(~‘(;q)~(A”[U,(X=O)]+A”[U:,(X=O)])\~( -$q))=&“G,(q’)(a-u * ii>,

which is now exactly of the

constant in the chiral limit

g,= -&i-

(56)

form of eq. (38). Hence we find for the axial coupling

Jam dr ri[f:( .I+?) 2F’sin2 F+sin 2F

r2 -( Ft2+y)] )

r (57)

which is 3 of the result quoted in ref. 19).

The Goldberger-Treiman relation is established by comparing the pseudoscalar

pion-nucleon interaction

L!? ,rNN = -igvNN!&g. nq, (58)

with the pion-nucleon interaction of eq. (32)

S. Scherer, D. Drechsel / Pion phoioproduction 743

A nonrelativistic reduction of the pseudoscalar pion-nucleon vertex, omitting the

usual normalization factors, yields

g,NN iJZmNN --, --

2mN - * 47,) (60)

where q =pf-pi. Comparing this expression with the pion-nucleon vertex in the

Skyrme model, evaluated in the Breit frame and omitting terms of order (q12,

i2?TNN+-gAcr.qra 2fT

gives the Goldberger-Treiman relation.

(61)

5. The pion photoproduction mechanism

The coupling of the electromagnetic potential to the electromagnetic current of

the Skyrmion,

_Y?s[ U] = -e,A,($W[ U] + V33p[ U]) , (62)

contains the third component of the isovector current,

V,,[U]=aif2,Tr(aPUr[U,~])+~ Tr ([ lJ+a,U, U’[ U, T]][ lJ’a”U, U+a'*U]) ,

(63)

as well as the anomalous topological current

Tr ( U+avUU+aPUU+a”U) , +,,23 = 1 .

When using the amplitude fluctuation as the pion-nucleon coupling mechanism,

the ansatz of eq. (28) is inserted into eq. (62) and expanded to first order in the

pion field. The S-matrix is calculated in lowest-order perturbation theory,

Sfi = i(f) d4x LL’,&x)li) , (65)

where Ii) consists of a (static) nucleon and a real photon with momentum k, and

polarization E and (f] is a correlated TN state in an s-wave (see refs. 29320)). This

pion wave function is given by

(n-“(k), L= M =0/n-cp”(x)~O)

(66)

744 S. Scherer, D. Drechsel / F’ion photoproduction

(a) (b) Cc) (dl

Fig. 2. Born diagrams for pion photoproduction at threshold in the chiral perturbation ansatz. (a):

isovector seagull term as well as the anomalous contact interaction, (b) and (c): s- and u-channel diagrams

with a nonrelativistic nucleon in the intermediate state, (d): pion pole term.

where ~‘p”(x) is the spherical component of the pion field. Note that eq. (66) contains

a spherical harmonic Yzm(.?) which results from the tensor type of interaction

between the fluctuation and the hedgehog in the intrinsic system. Due to the isovector

and isoscalar part of the interaction, there are two contact interactions leading to

multipoles with a ( - ) and (0) but no ( + ) isospin structure. Consequently this ansatz

automatically fails to reproduce the T’ production amplitude.

Introducing the pion as a chiral perturbation leads to a linear pion-nucleon vertex.

The interaction lagrangian reads

zint = g~N + zyN + zy7r + zyNm 7 (67)

with ZnN given by eq. (32) and

TyN = -e,(fW‘[ U,] + V3+[ U,])A, , (68)

Lfym = -eo(m x 8~r)~A~, (69)

2 yN?T= -0; (= x A*“[ UN])~& - eo%,d @r UN ut,lA, . (70) ?i

A calculation in second-order perturbation theory contains the usual Born diagrams

shown in fig. 2. Once again, as a peculiarity of the Skyrme model, there is a contact

interaction resulting from the anomalous current, which leads to a (0) isospin

amplitude.

6. Results and discussion

Following the notation of ref. 20), the isovector electromagnetic current e, V3,‘“[ U] yields as contribution to the S-matrix for the ( -) isospin amplitude,

$;‘=-a * &$[T_,, T0]27rCqEy-m,) l 1 I’-’

&qzg&j5z ’ (71)

S. Scherer, D. Drechsel / Pion photoproduction

where I’-’ is given by

I’-’ = $reo 2 2 I,:dx{sin'F[~(-~~)*,+$R:'*)

745

The functions I?‘_’ and Rc,-’ describe s-wave fluctuations, parallel and perpendicular

to f in the intrinsic system 29),

RI-)(&, x) = R$;‘(&, x)+@-)(&, x) ) (73)

R’?(&, x) = R:b-‘(&, x) -JzR$‘(&,, x) . (74)

For further details on the notation see ref. 20). The index (-) in eqs. (73) and (74)

refers to the asymptotic boundary condition of the final scattering state, i.e., incoming

spherical waves in all open channels and a purely outgoing wave in an s-state (see

ref. “)). Eqs. (71) and (72) have been evaluated in the Coulomb gauge. All terms

containing a time derivative of rotation matrices have been neglected. Furthermore,

the nucleon has been assumed to be infinitely heavy, i.e., recoil has not been

considered. Finally, only the leading term in the plane wave of the photon has been

kept, exp (ik, . r) = I+ ik, * r+ . . . . Fig. 3 shows the EAT’ amplitude, which is

obtained by multiplying eq. (72) with (49~( 1 + p)))l, as a function of m,. For m, = 0

the Kroll-Ruderman theorem is numerically reproduced. It is important to notice

200

% 150

5

% 100

c .?- 050

T -+

Gp

C

I

‘i i 20 40 60 ti0 IdO 12’0

pion mass [MeV]

Fig. 3. E&’ amplitude as a function of the pion mass WI,. A factor (I+ p)-’ = 1 -CL + [p’] has been

omitted. The prediction of the Kroll-Ruderman theorem for m, = 0 is E&) = 1.629 10m4 MeV-‘, where

the Goldberger-Treiman relation has been used.


that the axial coupling constant for m, = 0 is evaluated as in sect. 4 and not as

suggested in ref. 19). The parameters fn and e were adjusted for each value of mrr

such as to reproduce the experimental values of frr (93 MeV) and g, (1.26). In fig.

4 the resulting values of e are plotted against msr.

The anomalous current contributes to the isospin (0) component,

S~)=-a.~~_,2rr6(E,-m,) 1 1 I(O)

me ) (75)

where

(76)

The S-matrix element of eq. (75) has been obtained with the same assumptions as

above. The proportionality to mrr results from a time derivative of the pion wave

function. The expression of eq. (76) is identical with that of ref. 22), provided the

functions Rr'* and R'_'* are replaced by x,, of ref. 22). When comparing with eqs.

(66), (73) and (74) one finds that the part proportional to Y2, in the wave function

has been omitted by ref. 22).

Our result has the same sign as the LET prediction and is consistent with ref. 20),

i.e., it increases the cross section for yr” production, if one looks at the anomalous

seagull interaction as an additional contribution beyond LET, whereas the cross

section is reduced in ref. 22). It should be noted that the phase of our result is fixed

- ----- -- - --r--yl -YT---,

1 20 40 60 00 100 120 140

pion nlass [MeV]

Fig. 4. Skyrme parameter e as a function of the pion mass m,. While the pion decay constant is fixed at its experimental value, f,, = 93 MeV, the parameter e is determined such as to reproduce the experi-

mental value of the axial coupling constant g, = 1.26.

S. Scherer, D. Drechsei / Pion photoproduction 747

by the Kroll-Ruderman theorem. Hoodbhoy’s additional contribution reads 0.52 x

10p3/m,+ in contrast to our result of -0.98 x 10m3/m,+. When comparing the numbers

one has to take into consideration that Hoodbhoy’s result was derived with the

parameters of ref. 19), g, = 0.41 (in our convention) and fm = 64.5 MeV, whereas we

used g, = 1.26 and fr = 93 MeV. In our opinion, when considering pion photopro-

duction at threshold, the parameters should be fixed in such a way as to reproduce

the Kroll-Ruderman result, which is proportional to gA/fv. Finally, it is stated in

ref. 2”) that the Born amplitude for r” production was calculated with the pion-

nucleon interaction

2 ,rNN = -6 Tr (T . VA-‘7 * =A) + e U.V?.?T (77) N

of ref. r9) and that almost the same values as with a chiral lagrangian calculation 32)

were obtained. This is hard to believe, since the leading terms of LET in a calculation

in PV coupling result from the backward propagation of negative-energy states,

whereas a calculation with the nonrelatistic reduction of eq. (77) and the nonrelativis-

tic wavefunctions of the Skyrme model only reproduces the less important terms of

order p2 in LET.

As one can see from fig. 5, the integral of eq. (76) depends very little on the mass

of the pion, i.e. the anomalous contribution is almost linear in m,. The predictions

of LET, calculated with the values of the Skyrme model for the mass and the

magnetic moment of the nucleon, are shown in comparison with the Skyrme model

calculations.

w -9

10 7--

0 20 40 7s --- --7 ---.

80 1OT 120 120

pion rnass [MeV]

Fig. 5. Anomalous contribution E&‘2 using amplitude fluctuations in comparison with the LET prediction. LET has been calculated with the values of the Skyrme model for the mass of the nucleon mN and the

isoscalar magnetic moment p,. Factors (1 +p)-’ and (1+~/2)( 1 + P)-~” have been omitted from the

Skyrme model calculation and LET, respectively.


When using the chiral perturbations of eq. (29) as the pion-nucleon coupling

mechanism, the interaction term

2? YNT = -7 (m x A’“[ U&A+ (78) 77

immediately leads to the Kroll-Ruderman theorem in lowest order perturbation

theory. The resulting S-matrix at threshold is

S~;)=-(2,rr)464(pf+q~-Pi-ky) ’ 1 1

s m (27r)3

xc ;[~-a, T&J * G,(-Ikyl*). 77

In contrast to the above calculation recoil is taken into account, leading to the momentum conserving delta function and the additional normalization factor

1/(27r)3 of the nucleon wave function in eq. (79). The expansion of the formfactor,

G,(-lk,12) = gA+[ki21, results in the EL:’ threshold amplitude

(79)

As in the former case there is an anomalous seagull graph with a (0) isospin

structure. However, its contribution is of order mi,

Ep!““” = !?? 1 m3 OD -2L

f, 47r(l+/_~) 18~ I dr r2 sin2 F. 0 (81)

The higher-order contributions are calculated in second order perturbation theory.

The vertex for the absorption of a real photon on a nucleon is given in Coulomb

gauge by

where pi+ k =pf, and the formfactors B(lkl), ps((kl) and p,(lk\) are defined as

B(\k() = -a lrn drj,(lklr) F’ sin2 F, 0

Ik(u,(lkl) = -& Ia dr rj,(lk)r)F’sin2 F, 0

tkb-dkl) =; j-m

0

drrj,()k)r) sin’F(ft+-$ [ Ff2+F]) .

(84)

(85)


Since we have omitted terms in V3+ containing two time derivatives, there is no

isovector convection current. Expressions containing the velocity fi of the nucleon

soliton, such as f(s)k, were quantized after symmetrization, i.e.

f m -$pb)vR+vRf(S))~ This is the reason why the static mass M of the soliton appears (see eq. (82)) instead

of the mass of the nucleon mN. For vanishing photon momentum the above

expressions reduce to

B(0) = 1 , (87)

/-G(O) = Pu, 7 (88)

P”(O) = PL, . (89)

The pion-nucleon vertex for the emission of a pion of charge aeo may be derived

from the interaction lagrangian of eq. (32) and it reduces at threshold to

From eq. (90) it is easily seen that the nucleon pole in the s-channel does not

contribute at threshold, because the momenta of the nucleon before and after

emission vanish.

The contribution from the nucleon pole diagrams reads

(91)

(92)

However, this ansatz does not produce any terms of order p, which are responsible

for the reaction y + p + no+ p at threshold. A comparison with a calculation of the

nucleon pole Born diagrams in pseudovector coupling shows that the terms of order

p are completely due to the backward propagation (negative-energy states). Further-

more, the terms proportional to the magnetic moments result from the forward

propagation of a nucleon in the u-channel. This is the complete analogue to the

Skyrme model results of eqs. (91) and (92) in the chiral perturbation ansatz. From

that point of view it is clear that the Skyrme model does not reproduce LET with

respect to terms of order p. In so far the analogy between the chiral perturbation

ansatz of ref. 30) and the effective Weinberg lagrangian ‘) only holds for processes

where the relativistic nature of the nucleon can be neglected. Due to the intrinsic

negative parity of the pion, there occurs a strong coupling to the negative-energy

part of the propagator even at threshold. This effect is even more dramatic in

pseudoscalar coupling where even the Kroll-Ruderman term is completely due to

backward propagation.


In conclusion the present treatment of pion photoproduction in the Skyrme model

does not give a correct description of the threshold amplitude beyond the Kroll-

Ruderman term. In particular the classical nonrelativistic quantization does not

respect Lorentz invariance which is one of the basic symmetries necessary to derive

LET.

One of the authors (S.S.) is grateful to Prof. G. Holzwarth and Dr. B. Schwesinger

for fruitful discussions and for making available a computer code for calculating

pion wave functions of the Skyrme model.

References

1) E. Mazzucato et al., Phys. Rev. Lett. 57 (1986) 3144

2) R. Beck, PhD thesis, Mainz 1989

3) N.M. Kroll and M.A. Ruderman, Phys. Rev. 93 (1954) 233 4) S. Fubini, G. Furlan and C. Rossetti, Nuovo Cim. 40 (1965) 1171 5) P. de Baenst, Nucl. Phys. B24 (1970) 633

6) H.W.L. Naus, J.H. Koch and J.L. Friar, NIKHEF-preprint (1989)

7) S. Weinberg, Phys. Rev. Lett. 18 (1967) 188

8) R. Dahm and S. Scherer, in Proc. of the Int. School on nuclear physics XII, Erice, 1989, ed. A.

Faessler, Prog. in Part. and Nucl. Phys. 24 (1989)

9) L.M. Nath and S.K. Singh, Phys. Rev. C39 (1989) 1207

10) S. Nozawa, T.-S.H. Lee and B. Blankleider, Phys. Rev. C41 (1990) 213

11) D. Drechsel, in Yamada Conf. XXIII, nuclear weak processes and nuclear structure, Osaka, Japan (1989), ed. M. Morita et al. (World Scientific, Singapore, 1989)

12) L. Tiator and D. Drechsel, Nucl. Phys. A508 (1990) 541~

13) D. Drechsel and L. Tiator, Phys. Lett. B148 (1984) 413

14) S. Scherer, D. Drechsel and L. Tiator, Phys. Lett. B193 (1987) 1

15) T.H.R. Skyrme, Proc. R. Sot. A260 (1961) 127

16) T.H.R. Skyrme, Nucl. Phys. 31 (1962) 556

17) G. t’Hooft, Nucl. Phys. B72 (1974) 461

18) E. Witten, Nucl. Phys. B160 (1979) 57

19) G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552 20) G. Eckart and B. Schwesinger, Nucl. Phys. A458 (1986) 620

21) B. Schwesinger, H. Weigel, G. Holzwarth and A. Hayashi, Phys. Reports 173 (1989) 173

22) P. Hoodbhoy, Phys. Lett. B173 (1986) 111

23) G.D. Chew, M.L. Goldberger, F.E. Low and Y. Nambu, Phys. Rev. 106 (1957) 1345

24) M.I. Adamovitch, Proc. P.N. Lebedev Physics Institute 71 (1976) 119

25) G. Holzwarth and B. Schwesinger, Rep. Prog. Phys. 49 (1986) 825

26) E. Braaten, S. Tse and C. Willcox, Phys. Rev. D34 (1986) 1482

27) A. Bohr and B. Mottelson, Nuclear Structure, vol. I (Benjamin, New York 1969) ch. 1

28) H. Walliser and G. Eckart, Nucl. Phys. A429 (1984) 514 29) G. Eckart, A. Hayashi and G. Holzwarth, Nucl. Phys. A448 (1986) 732

30) H.J. Schnitzer, Phys. Lett. 8139 (1984) 217

31) H.J. Schnitzer, Nucl. Phys. B261 (1985) 546

32) R.D. Peccei, Phys. Rev. 181 (1969) 1902

Pion photoproduction in the skyrme model and low-energy theorems

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