Rigid quantization of the Skyrmion

Nuclear Physics A500 (1989) 573-588 North-Holland, Amsterdam

RIGID QUANTIZATION

OF THE SKYRMION

H. VERSCHELDE’,’ Physics Department,

State University

of New York at Stony Brook, Stony Brook, NY

11794, USA

H. VERBEKE Seminarie

uoor Theoretische Vaste Stof en Lage Energie Kernfysica, Krijgslaan 281.S9, B-9000 Gent, Belgium Received

21 December

Rijksuniversiteit-Gent,

1988

Abstract: We present a detailed discussion of the rigid quantization scheme for rotating skyrmions. We pay special attention to ordering problems and their repercussion on the pion-baryon coupling. We derive the pion-baryon vertex in the rigid gauge and discuss the baryon form factor. We show that the “radiation problem” for rotating skyrmions is due to neglect of operator ordering and restriction to K = 0 waves.

1. Introduction In recent years there has been a revival of Skyrme’s old idea that baryons can be described as solitons of an effective meson theory I,‘). The fundamental theory of strong interactions, QCD, was shown to reduce to a theory of weakly interacting mesons (an infinity of them) when the number of colors N, is taken to infinity. At low energy and in the chiral limit (m, = 0), the Skyrme lagrangian:

(1.1) is supposed to be a good approximation of this meson theory, and is known to have solitons. This lagrangian can be rewritten as a generalized non linear o-model ‘). Defining

the pion field 4; as: U=exp

one can rewrite

i-

d?.r

f,

(1.1) as ~==~,~,iG,i(~,V~)a,~j-~(4,V~).

The isospin

(1.2)

metric

can be written

(1.3)

as (1.4)

On leave from Seminarie voor Theoretische 281-S9, Gent, Belgium. 2 Research associate N.F.W.O.

Vaste Stof en Lage Energie Kernfysica,

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

B.V.

R.U.G., Krijgslaan

H. Verschelde, H. Verbeke / Rigid qunntizntion

574

where gti is the well known

non linear

v-model

metric and G, is the contribution

from the fourth-order Skyrme term. The equations time-independent hedgehog solution (the skyrmion) 4,9(r)

=fwies(r)

of motion

have

a classical

.

(1.5)

The chiral angle 0,(r) goes to zero as l/r2 for r + 00 and approaches For the hedgehog, the metric can be decomposed as:

rr at the origin.

>

G,(~,)=f(8,)6,+~i~jh(B,)

(1.6) This will be the only explicit information we need about G. The hedgehog (1.5) is a baryon number one configuration but does not represent a physical baryon because it breaks both spin and isospin. It is only invariant under combined rotations and isorotations and has K = 0, where K-spin is defined as: K=I+J.

Therefore, coordinate

(1.7)

one has to restore rotational invariance and this can be done by collectivequantization 475). Th e collective coordinates cu are defined as: 4i

=

u(a),(4~+

77j)

9

(1.8)

where 4 is the pion field in the lab frame, C& the Skyrme intrinsic pion field which obeys the gauge condition

d3x@(4,) T&s = 0, This gauge

restricts

the pion

fluctuations

hedgehog

and

a=l,2,3.

to be orthogonal

r] the

(1.9)

to the rotational

zero

mode T&, . We call it a rigid gauge because the skyrmion field must adjust instantly to an infinitesimal rotation just as a rigid rotor does. Static baryon properties were calculated using this rigid gauge and agreed qualitatively well with experiment.

However,

for non-static

properties

(where

retardation

comes into play) it was shown that the rigid gauge presents problems 6*7,9).The A-decay width comes out a factor of 4 too large and the pion-baryon vertex causes infrared divergences in higher order. We showed that the problem of the pion-baryon coupling in the Skyrme model can be solved by working in a non-rigid gauge 7P9) where retardation can be described properly. Several other attempts have appeared in the literature 12-15) which try to find a solution to the pion-baryon coupling problem within the rigid quantization scheme. They all arrive at essentially the same coupling as the one we found in the rigid gauge “) but there are small but important differences dependent on whether one symmetrizes and at what point in the course of the calculation one symmetrizes or takes the plane wave limit. Therefore, in this paper we will give a detailed discussion

H. Verschelde, H. Verbeke / Rigid quantization

of rigid associated

quantization ordering

of the rotational problems

degrees

of freedom

of the skyrmion,

and how they affect the pion-baryon

points, we have chosen to give the calculations with existing results possible. In sect. 2 we discuss

575

the quantization

the

vertex. At some

in full detail to make comparison

of the skyrmion

in the rigid gauge using

the Dirac formalism for constrained systems. We pay careful attention to the ordering problem and derive a properly symmetrized hamiltonian. In sect. 3 we calculate the mass shift due to symmetrization and find a result which disagrees with a previous treatment lo). In sect. 4, we expand around the hedgehog and derive the pion-baryon coupling. We discuss in detail the symmetrization corrections to the pion-baryon coupling and conclude that only part of these corrections have been included in existing treatments 13*14).In sect. 5, we calculate the pion-baryon vertex in the soft-pion limit. We make a detailed comparison with other treatments and show that, in the rigid gauge, a satisfactory value for the A-decay width can only be found by smuggling in a coupling of K = 1 waves to the rotational degrees of freedom of the skyrmion (which is identically zero in the rigid gauge) via an incorrect plane-wave limit. Finally, in sect. 6 we discuss the baryon form factor in the rigid gauge. We show that the diagonal form factor (I, = I,.) has a classical static pole at q2 = 0. This means that the behavior at infinity of the pion field is still given by the classical skyrmion field and that the Goldberger-Treiman relation is obeyed. This result depends crucially on the symmetrization procedure. We show how neglect of symmetrization and restriction to the hedgehog ansatz (only K = 0 emission) is responsible for the radiation problem for rotating skyrmions as found in ref. “). The dynamical pole structure for the off-diagonal form factor (I, # I,,) is pathological in the rigid gauge and reflects the inability of the rigid gauge to describe retardation properly (at least in lowest order). We end with some conclusions in sect. 7.

2. The Dirac formalism

and rigid quantization

When collective coordinates are introduced, one creates redundancy in the degrees of freedom, which can be eliminated by using the Dirac formalism for constrained systems. We will briefly recapitulate the results of ref. “). There we showed there are three first-class constraints: 4,+i

a =

d3xZZ”T,(&+n)=0, I

where 4;, is the body fixed isospin and ZZ’ the conjugate to the conjugate momentum ZZ’ of 4 through

1,2,3, momentum

(2.1) of n related

ZP = ZJ(of)ZZ~. Using the gauge condition

(1.9), the constraints

can be solved by decomposing

(2.2) ZZV

H. Verschelde, H. Verbeke / Rigid quantization

576

into longitudinal

and transverse

components: IP=n”+II~,

(2.3)

with

I

d3x Il=T,&

= 0.

(2.4)

We find

(2.5) where &,

is the moment-of-inertia

tensor:

d3x(4s+ ~)TaG(4,)TbA.

Aa6=

(2.6)

Note that due to the gauge condition (1.9), Aab = &. The collective coordinates CY,,4, commute with the intrinsic coordinates n and IIT. For the commutation relations of the intrinsic coordinates, we find:

[Ti(x, t), n:<~, t)l=

i~qSz(x-Y)+~ (T,4,(x))j(G(4,(y))T,4,(y))j

(2.7)

s

with A, the moment

of inertia

of the skyrmion:

d3x4,LG(4,) To ensure hermiticity, (2.2) and (2.5):

we will have to symmetrize

IP

(2.8)

G4s = As&b. appropriately

= ;{ U(cu), IP}+ + u(a)n’

n’=-iG(~,)T=~~~{A~~,,,+iI

)

the expressions

(2.9)

d3xIITTbr)}+.

(2.10)

Since [ u(a),

$a,]= u(a) To,

(2.11)

we have: Ii” = U(CY)(~T~+IT~)+~~U(~)T~G(~,)T,~~,A~~ =(~“+~‘)U-‘(~)-~~~,T,G(~,)T~U-‘(~Y)A~~.

The hamiltonian

as a function H=;

of the original

d3xIIdG-‘(4)17++ I

variables

(2.12)

reads: (2.13)

S-II

H. Verschelde, H. Verbeke / Rigid quantizaiion

Since A(W)=&(4)

>

G( U+) = UC(+) and after carefully

manipulating

so that they compensate H=f

the U matrices

each other,

(2.14)

UP’, to the correct position

(using (2.12))

one finds:

d3xIITGP1(&+n)L7’+

d3x.A(+,+n) I

I

where 4” is the body-fixed

isospin

of the skyrmion

4”,=9,+i

defined

d3x IIT T,v

.

through (2.16)

The dots stand for commutator terms of G-‘(c$,+ 7) which, as we discuss in the next section, do not contribute in the soft-pion limit. The first two terms in (2.15) constitute the intrinsic hamiltonian; it provides the classical mass of the skyrmion and describes the interactions of the pion fluctuations with the skyrmion background and with themselves. The third term is the rotational energy of the skyrmion which appears now in an appropriately symmetrised form. The fourth term is a skyrmion-pion coupling which because of (2.4) vanishes in lowest order*. The last two terms in (2.15) are generated by the symmetrisation procedure. Besides providing a correction term to the pion-skyrmion vertex, they result in an overall mass shift which we will discuss in the next section. 3. Mass shift due to symmetrization It is easy to prove the following

identity: (3.1)

l This is a direct consequence of the rigid gauge condition. gauge described in refs. ‘,‘)) h ave a non-vanishing O(N;“‘) of freedom.

Other gauge choices (such as the non-rigid coupling of IIT to the skyrmion degrees

578

H. Verschelde, H. Verbeke / Rigid quantization

Furthermore,

using (1.6), one finds

=

~~s[&bf&d

+

kzc~bdl

-rs[f&b&d

+

hc8bd

+

&zd8bcl

,

(3.2)

where cc

I

r, '&Tf2,

0

r2

dr

**

f(w4&)

s(f(R)+h(e))

.

(3.3)

In this section and also in the rest of the paper, we will replace G-‘( c#J~+7) by G-l(&). For calculating mass shifts this is of course natural because we only keep the zeroth-order terms in the expansion around &. When we will calculate the pion-skyrmion vertex, the rationale behind this simplification is that it is correct in the soft-pion limit (expanding G around 4, will introduce a form factor which goes as I$- l/r4 at infinity and therefore does not contribute in the soft-pion limit) and this is the regime we are most interested in. Using (3.2), we find for the fifth term in (2.15) (3.4)

[UT, Therefore

using

n,l=-i~G(~,)Tj~,-t&~ijG(~s)Tf~, *

(3.5)

(3.1), (3.2) and (3.5), we have:

(3.6)

(3.7) so that T6= -&A,[3A;;A,-,‘+A;;A;;]+;&[7A;;A;;+A;:A;;]+O(q2) Therefore,

(3.8)

the fifth and sixth term in (2.15) add up to: T5+ T6= -~A,A,-,‘AIr,‘+~~,[3A,-,‘A,-,‘-A;;,’A~~]+O(~*).

(3.9)

It is easy to see that the terms proportional to r, will not contribute to the mass shift and to the pion-skyrmion vertex (linear term in 7). Indeed, for A,,b = AsSnb( v= O), these terms cancel exactly. Therefore, they will only start to contribute in second order in 7. Hence, we finally find: T,+ TV= -~A,A,-,‘A;;,‘+o($).

(3.10)

H. Verschelde, H. Verbeke / Rigid quantization

Putting

519

&, = AsSob in (3.10), we find for the mass shift: AM=+

(3.11) c

This mass shift disagrees be traced

with the one found

by Fujii et al. lo). The discrepancy

can

back to two points:

(i) The classical mass is purely potential energy. The mass shift on the contrary comes from kinetic energy. So if one goes from lagrangian to hamiltonian formalism, the former will change sign while the latter will not. Therefore because the mass shift was identified in ref. lo) in the lagrangian formalism, there is a sign error. Modulo a minus sign, the result of Fujii et al. corresponds with our fifth term which gives 3/4A, - 15r,/2A%. (ii) The treatment in ref. lo) neglected sixth term. This sixth term is important fifth term and turns a into -2.

the intrinsic

coordinates,

because

it cancels

4. The pion-skyrmion

coupling

so it missed the

the r, dependence

in the

The pion-skyrmion coupling can now be found by expanding (2.15) around the hedgehog and keeping terms linear in q or IZT. Because of the equation of motion and the gauge condition (1.9) respectively, the second and fourth term do not contribute. Therefore, the only way we can find linear terms in 77 (at least in the soft-pion limit where we do not expand G) is by expanding the moment-of-inertia tensor. Using the commutation relations (2.7) and the gauge condition (1.9), one can prove:

[ Expanding

A and moving

the contribution

Kl,

I

1

d3x nT,G(A)T,+,

the isospin

operators

= 0.

(4.1)

to the right using (4.1) we find for

of the third term in (2.15)

Using (3.4) and (3.8) we find for the contribution

of the fifth and sixth term in (2.15):

d3xn&f(e5). -3s I d3m#dV,)+& d’xv$,f(e,)=& s sI When added,

* The procedure term, when added,

the pion-skyrmion

coupling

(4.3)

becomes:

of Fujii et al. works for translational degrees just changes the sign of the fifth term.

of freedom

“) because

there the sixth

H. Verschelde, H. Verbeke / Rigid quantization

580

It is trivial the skyrmion moved

to check that our interaction

hamiltonian

(4.4) is hermitian.

body-fixed

operators

are properly

symmetrised

the isospin

operators

isospin

back and forth through

K = 0 and the skyrmion

body-fixed

isospin

operators

Indeed,

and n can be

using

(4.1). Because

& has

appear

in symmetrised

form,

it is clear that (4.4) will couple the skyrmion degrees of freedom only to K = 0 and K = 2 pion fluctuations. The coupling to K = 1 waves in ref. ‘*) is due to neglect of operator ordering. Pion-skyrmion couplings which appeared in the literature miss the second term in (4.4) [Holzwarth et al. ‘“)I or have a different coefficient for this term [Saito 13), Uehara ‘“)I. To compare our pion-skyrmion coupling with the one of Saito or Uehara, let us reintroduce the lab pion fluctuation 4 = Us. Using (1.4) and (1.5) we can rewrite our pion-skyrmion coupling as:

where

Note that U(a) and 9, do not appear in symmetrized form. This is not in contradiction with the fact that we have a symmetrized hamiitonian (2.15). In refs. ‘3,14), a symmetrization procedure was followed by working from the start with lab pion fluctuations

4 and at the end symmetrising

Oab. Using

$:=-.I:,

1{

9:}+ = IS,

Uaj,

where

1’ and

J” are respectively

pion-skyrmion

(4.7)

9

the skyrmion

isospin

and

spin,

the following

vertex was obtained:

where OL* =4{ GA=), and the dots stand

.C]++f{f”,,

Jsb1t

for terms which do not contribute

(4.9)

in the soft-pion

limit. Using

(4.7) and [V(o),

KJ = U(a) T,

(4.10)

it is easy to prove that Ohb = Oab -$( Wo)T,),&+ The second

term in (4.11) does not contribute. f,

I d3x gbU(
B,).Pp:=

u(a),,

.

(4.11)

Indeed d3x ‘~fG(#d T+P,E = 0

(4.12)

581

H. Verschelde, H. Verbeke / Rigid quantization

where

we used the gauge

condition

(1.9). Therefore

going

back to intrinsic

pion

fields, one finds:

(4.13) So, comparing the pion-skyrmion coupling of Saito 13) or Uehara 14) (4.13) with our expression (4.4), we notice a difference in the coefficient of the second term (symmetrization correction). The origin of this difference can be cleared up by noticing that the symmetrization correction in (4.13) is exactly the one coming from the fifth term in (2.15) (see (4.3)). Therefore, just as in Fujii et al. lo), Saito and Uehara missed the sixth term in (2.15).

5. The pion-baryon To calculate matrix elements of our interaction intrinsic pions into isovector spherical harmonics numbers:

vertex hamiltonian (4.4), we expand the r6,17)which have good I( quantum

with nzZ”‘(;)=CC(L’, I*

K~-~,l,~;E(,icz)Y~‘.“:-~(;)~,.

(5.2)

R?F( r) is the radial wave function with energy E, and outgoing angular momentum L. For K = L, one has L = L’ and the scattering is elastic. For K = I,* 1, the diagonal component L = L’ represents elastic scattering while the off -diagonal one represents inelastic scattering with AL = 2. I( = 0 is a special case with only elastic scattering. For soft pions, there is only elastic scattering and only the L= L’ part will be important. A one-pion-one-sky~ion momentum L for the pion, pion skyrmion is given by ‘6*3): I& M,JY ml/$, n, “‘=Z

(-l)K(-l)

basis with good total I and J, outgoing angular energy E,, and I: = Jt = $f = $( ,J$+ 1) for the

L+‘-+‘+J(-l)‘--J+“((2K

+1)(2$+1))“2

(5.3) where II,M,J,mIK,n,L)=CC(I,m-K,,K,K,;J,m)(a,K:,K:)+II,M,m-K,), KZ

(5.4)

H. Versrhefde, H. Verbeke / Rigid qua~~iza~ion

582

and [I, M, m> is the pure one-skyrmion basis with I = J, 1, = M, J, = m. The pionskyrmion interaction hamiltonian (4.4) can be written as the sum of three terms:

sJ +$$ rl&&.f(~,)~~“, ,.6,)+ -+y Jd3x J

H ~Blb = -2

d3x &j(

e,).F:

d3x #?J(

Carrying

out the angular

J

t9,) .

s

S

integration,

one can check that:

do $] I, M, m) = -v%

T -&= Ry;f, (a’;‘:0,)+/ I, M, m) n R?!, 14 M I, ml% n, 1) .

Using the Wigner-Eckart

theorem

and Racah

Q&{-E, 61,/1,M, m> Jdf2 =2I(I+

(5.5)

algebra,

(5.6)

we find:

I)

x (-1)*‘((2K

+ 1)(21+1))

II,MLmlKn,I)

~2I(I+l~~[~(-~)~R~~~lI,~I,m,O,~,l~ 1 (41(1+1)-3)“* ‘&z Consequently, we find: H,+a211,

1 m

(I(I”F1>)‘/2

performing

n

the radial integration

@$I 4 M, I, m 12, n, Q and adding

M, m)=.L~~~&A:::‘II,

I

.

(5.7)

the three terms in (5.5),

M, I, m/O, n, 1)

+~~~[~~-~)~A’:iirM,~,jo,,,ii 1 (41(1+1)-3)“2 +;z

(I(I-tl))“” -

where

1 mA%/I, n

M, I, mj2, n, I)

A?f, iI, M I, m IO, n, I>,

1 (5.8)

H. Versckelde,

For soft pions there is elastic

H. Verbeke / Rigid

scattering

quantization

and only the L = 1 term will contribute

I’*

(-lP-Jj

1 ( -2$-tl 21+1 >

1 ( 2dp+1 21+ 1 )

(5.10)

,

1/2 (4i(r-t1)-3)-‘~‘{~(r+l))-‘/’

x[~(dp(&p+1)-z(I+1))2-~(~(~+1)-I(I+1))-41(1+l~f3l. Adding

the various

contributions,

(I’, M’,J’,

in

in 1I, A4, I, m 118,n, 1) are respectively

(5.8). The K = 0 and K = 2 coefficients WP’~

583

m’ll$,

(5.11)

one finds:

@,NLB,E&

M m>

x~~I(i+l)A~~+~$(~(~+l1)-I(~+l)~*-~~~(~+l) - I(1 -t 1)) -$I(1 In the plane-wave

approximation,

-t- 1) ++]A:;: -+A:+]

(5.12)

one has A:: = A;$ = A,,, where

A,= In the soft-pion

.

limit, A, goes to a constant

(5.13) “) (5.14)

The pion-baryon

vertex finally becomes:

(5.15) Two things should be noted about the pion-baryon vertex (5.15). Firstly, the vertex is zero for B = B’. This will be important in the next section when we discuss the pole structure of the baryon form factor. As one can see from (5.12), this is due to a cancellation which involves all three terms in (5.5). In particular, this cancellation will not occur when the third term, due to symmetrization, is omitted or has a wrong

584

H. Verschelde, H. Verbeke / Rigid qunntization

coefficient. ges as l/G

Secondly, because A,,+ constant for q + 0,the pion-baryon vertex diverfor q -+ 0. The nature of this infrared divergence has been discussed at

length in ref. “). If one calculates the A-decay width using (5.15) one finds it is too large by a factor of 4 [ref. “)I (see also the next section). Depending

on what one chooses

to neglect

and at what point

in the course

of

the calculation, one can get a pion-baryon vertex which differs from our expression (5.15). If one neglects the coupling to K =2 waves and the correction term due to symmetrization (last term in (5.12)), one finds

This is the pion-baryon vertex used in ref. “). As we show in the next section, it leads to the well known radiation problem for rotating skyrmions. One can arrive at a pion-baryon vertex which gives good results for A-decay 13,r4) by taking an incorrect low-energy limit (plane-wave approximation) at the end of the calculation. Let us start from the coupling (4.8) (which misses part of the symmetrization correction) where we use Ohb as given by (4.11). After some angular-momentum algebra,

one finds:

x (-1)2’((2K

+ 1)(21+

1))“’

= -(21(1+1))1’2 The K = 1 coefficient

]I,MI,m]K,n,Q

K:,f,lI,M,I,m]1,n,Q.

in 1I,M, I,m 1) 8, n,1) is l/2

(1(1+1))-“2[&3+1)-1(I+1)-2]. Adding (elastic

the contributions scattering):

from K = 0, 1,2 waves,

one finds in the soft-pion

(5.18) limit

H. Verschelde, H. Verbeke / Rigid quantization

If one takes for K = 0, K = 1 and K = 2 plane waves, one has A:: and the pion-baryon

vertex reduces

585

= At:: = A:$ = A,

to:

x~(1,~(1,~+1)-1,(1,+1))2

(5.20)

which coincides with the expression of Saito 13) and Uehara ‘4). For A-decay, the matrix element (5.20) is exactly a factor of 2 smaller than the corresponding one using (5.15). This reduces r, by a factor of 4 and hence one finds good agreement with experiment 13). However the way in which the low energy limit (and hence the plane-wave limit) was taken in (5.20) is incorrect. Indeed, in the rigid gauge one has A’,’,,n = 0 for all energies because the K = 1 scattering waves are orthogonal to the rotational zero mode (we already noted in the previous section that the second term in (4.11) does not contribute due to the gauge condition). At the origin, the K = 1 scattering waves differ appreciably from plane waves (they look more like zero modes) and it is precisely this behavior at the origin which makes Ai;: = 0. A pion-baryon vertex such as (5.20), although phenomenologically successful, cannot be derived in the rigid gauge using some low-energy approximation “). A gauge which allows that the K = 1 as well as the K = 0 and K = 2 scattering waves can be traded for plane waves in the low energy limit, should subtract the zero mode-like behavior for K = 1 at the origin and has been discussed in 7,9). In this non-rigid gauge one finds a proper description of retardation already in lowest order and a matrix element for A-decay which in the plane-wave limit is exactly equal to the one given by (5.20). 6. The baryon form factor in the rigid gauge As we stressed

in refs. 6*9), the rigid gauge poses problems when one tries to calculate processes where soft pions are emitted. The reason is that the adiabatic approximation which assumes that the skyrmion rotates as a rigid object breaks down when retardation is important. This problem is reflected in the pole structure of the baryon form factor. In lowest order (semi-classical approximation) given by (see appendix in ref. ‘)): (I,., M’, ~‘1

d3r epi4.‘U(cx)&lIB,

XX

M-M’,

M-M’dr(_i)

form factor

is

M, m)

C(l, X c(1,

the static baryon

m -m’,

I*,, m’; I*, m)

IB’, M’; I,, M) y*'."-m'(G)

J

_3

g ,rNN _4 4?T ~MN q2’

(6.1)

586

H. Verschelde, H. Verbeke / Rigid quantization

Because

(b= ut~Ms+ higher-order

corrections

to the baryon

via the lab pion fluctuation

(i

(6.2)

form factor should be given by pion emission

field 4. The matrix element

(5.15) describes

pion emission

B * B’+ r for Is = iW * 1. By interchanging 1, and IB’, (5.15) gives the corresponding matrix element for pion absorption B + rr + B’. Both processes will contribute to the matrix element of 4 in first-order perturbation theory. One has (fB’,M’,,‘I~lI,,M,mllf,,,n,l) = C(1, M-M’, XX

M-M'

Y

IBr, M’; IB, M)C(l,

m - m',IBc, m';IB, m)

l,m-m’

(6.3)

and

(6.4) Using (5.15), (6.3) and (6.4) and adding contributions from both processes, one finds for the next order correction* to the baryon form factor (soft-pion limit): (-lfQ-IS.

xC(1,

21,.+ 1 r/= 21+1 C(1, m-m',I,,,m';Is, m) ( R > M-M’,

3 gnNN --47~ 2M,

Iw, M’; Ig,M)~M-M’4~(-i)Y*1,m-m’(~)

1

q=

&-w,+q)+&4&3+q)

Ii *

1 En-(E,,+q)_E,,-(E*+q)

Adding

(6.5) to (6.1), the static pole gets converted

4

1 -i+

1 q2-(&-I?&

to:

(&[ 1+3_ 211,

(6.5)

J%,)2 q3

1.

(6.6)

For B= B’, we recover the static pole at q*=O of the classical skyrmion field. This means that the Goldberger-Treiman relation is obeyed. This result depends crucially on the fact that the pion-baryon vertex (5.15) vanishes for B= B’ and ’ Because we are only interested in the pole structure, we neglect the one loop correction which is o(N;‘/*).

H. Verschelde, H. Verbeke / Rigid quantization

hence couples

on the operator

ordering.

Using

a pion-baryon

only the K = 0 pions to the skyrmion 1

1

4 ‘-*sz

For B = B’, the static pole becomes

94

>+A 1

[ If 21(1+1) 3n,’

1

3&

1 s’

vertex such as (5.16) which

and neglects symmetrization,

21(1+1) [ *+

587

q2-(Fe-Es*)”

1 Y=+

one finds:

1. 1

(6.7)

(6.8)

q’-21(1+1)/3/l:

In the large NC limit (A, + co), the pole is shifted from q2 = 0 to q* = 21(1+ 1)/3At. This means that the skyrmion field will oscillate at infinity with frequency This is the famous radiation problem *) which is not due to (21(1+ 1)/3n:>“‘. breakdown of the semi-classical expansion in l/N, but a direct consequence of neglecting the coupling to K = 2 pions and symmetrization. If one uses the pionbaryon coupling (4.13) refs. 13,14),one finds that the static pole at q2 = 0 gets shifted to q2=3/2Af. For B # B’ (6.6) has a pathological pole structure which reflects the inability of the rigid gauge (at least in lowest order) to describe retardation properly. For d-decay, the pole at q = Ea -EN =3/2A, has residue 2 so that the decay matrix element is too large by a factor of 2 and hence r, is too large by a factor of 4 [ref. “)I.

7. Conclusions We have discussed in detail the quantization of the skyrmion in the rigid gauge. We stressed

of the rotational degrees of freedom the importance of proper symmetriz-

ation to arrive at the correct result for the pion-baryon vertex. In the soft-pion limit, the baryon form factor has the correct static pole structure for diagonal elements (Z, = Iu,) and hence is in agreement with the Goldberger-Treiman relation. For off-diagonal elements, the baryon form factor has a pathological pole structure in the rigid gauge.

This reflects

the problems

one has in rigid-rotor

quantization,

to

describe retardation effects. Higher order corrections to this rigid behavior should restore the correct dynamical pole structure. However, due to the infrared behavior of the rigid pion-baryon vertex (-I/& for q --, 0), higher order corrections will be beset with infrared divergences. A way out of this is to choose a non-rigid gauge which does not have this infrared problem and which already in lowest order in the pion-baryon coupling yields the correct dynamical pole structure for the baryon form factor 7,9). One of the authors (H.V.) would like to thank Prof. G.E. Brown and the nuclear theory group at Stony Brook for their hospitality.

588

H. Verschelde, H. Verbeke / Rigid quantization

References 1) T.H.R. Skyrme, Proc. R. Sot. London A260 (1961) 127; I. Zahed and G.E. Brown, Phys. Reports 14X (1986) 3 2) E. Witten, Nucl. Phys. B223 (1983) 422 3) H. Verschelde, Habilitation thesis, R.U.G. (1987) 4) G.S. Adkins, C. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552 5) H. Verschelde, Phys. Lett. B181(1986) 203; Rotating skyrmions in a hamiltonian formalism, solitons, ed. K.F. Liu (World Scientific, Singapore, 1987) 6) H. Verschelde, Phys. Lett. B209 (1988) 34 7) H. Verschelde, Phys. Lett. B215 (1988) 444 8) M. Bander and F. Hayot, Phys. Rev. D30 (1984) 1837; E. Braaten and J.P. Ralston, Phys. Rev. D31 (1985) 598 9) H. Verschelde and H. Verbeke, Nucl. Phys. A495 (1989) 523 10) K. Fujii, K.I. Sato, N. Toyota and A.P. Kobushkin, Phys. Rev. Lett. 58 (1987) 651; K. Fujii, A. Kobushkin, K.I. Sato and N. Toyota, Phys. Rev. D35 (1987) 1896 11) E. Tomboulis, Phys. Rev. D12 (1975) 1678 12) G. Holzarth, A. Hayashi and B. Schwesinger, Phys. Lett. B191 (1987) 27 13) S. Saito, Nucl. Phys. A463 (1987) 169~; Prog. Theor. Phys. 78 (1987) 746 14) M. Uehara, Prog. Theor. Phys. 78 (1987) 984 15) C. Adami and I. Zahed, Stony Brook preprint (1987) 16) A. Hayashi, G. Eckart, G. Holzwarth and H. Walliser, Phys. Lett. B147 (1984) 5; H. Walliser and G. Eckart, Nucl. Phys. A429 (1984) 514 17) M.P. Mattis and M. Karliner, Phys. Rev. D31 (1985) 2833

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