Nuclear Physics A515 (1990) 686-700

North-Holland

STRANGENESS VIBRATIONS AND THE STRANGENESS CONTENT OF

THE NUCLEON

Chun Wa WONG, Duy VUONG and Keh-cheng CHU

Department of Physics, University of California, Los Angeles, CA 90024, USA

Received 22 January 1990

Abstract: A schematic model of strangeness vibrations is used to show the importance of continuum

contributions to strangeness correlations in both S = -1 and S = 1 baryon channels. The strangeness

content of the nucleon due to vibrational correlations of the skyrmion is found to be convergent

in the sum over vibrational eigenmodes, but the associated correlation energy and the number of

strange mesons in the nucleon both diverge. The divergence comes from an angular momentum

transmutation potential caused by the “cowlick” of the hedgehog pion field located at the origin.

The model dependence and other problems of the calculated vibrational contributions to the

strangeness content of the nucleon are discussed.

1. Introduction

The description of baryons as rotating skyrmions appears to work well for

nonstrange baryons I), but not for strange baryons ‘). Callan, Hornbostel and

Klebanov ‘) (CHK) have observed that the rotational treatment fails in the strange-

ness directions because of the breaking of flavor symmetries. They have proposed

a much more successful approach in which strange baryons are described instead

as quantized strangeness vibrations of the skyrmion. The physical picture is that

the increase of meson masses in the strangeness directions gives rise to an effective

restoring force which destroys the rotational symmetry on which the rotational

model is based.

It has been shown by Blaizot Rho and Scoccola “) (BRS) that in the usual quadratic

approximation, these strangeness vibrations can be described in terms of the familiar

random-phase approximation (RPA) with the skyrmion as the classical, or uncorre-

lated, vacuum. Since the RPA vibrations introduce correlations into the physical

vacuum ‘), these authors are able to extract important information concerning the

strangeness content of the proton by studying these strangeness vibrations. They

find that the fractional “strangeness content”

C,=(NIS~~N)/(NIS~+~~+;I~~N) (1)

of the nucleon’s quark sea is only about 3% at the physical kaon mass when the

correlation caused by the single strangeness S = - 1 bound-state vibration is included.

A similar single-mode result of 1.8% has been obtained by McGovern and Birse “)

in their study of strangeness vibrations in the related chiral quark-meson model of

baryons.

037%9474/90/$03.50 @ Elsevier Science Publishers B.V. (North-Holland)

C. W. Wong et al. / Strangeness ~ibraij~ns 687

These vibrational results of 2-3% are in gross disagreement with a previous result

of up to 23% obtained from the model of strange baryons as rotating skyrmions ‘**). Of course, vibrations and rotations might involve significantly different degrees of freedom, so that this discrepancy is hard to interpret unless there is a unified

treatment of both. In addition, each of these models has its own problems. The rotational skyrmion

model is open to suspicion because it gives a poorer description of strange baryon masses. On the other hand, the vibrational result contains a sum of contributions from an infinity of vibrations of strangeness -1 and 1. Only one term (with S= -1) in this sum has been evaluated in the results quoted above.

The purpose of this paper is to obtain a more complete understanding of the vibrational contribution to the proton’s strangeness content.

We shall start in sect. 2 by studying a soluble model of strangeness vibrations based on a “degenerate schematic” model of RPA vibrations well known in nuclear physics. We shall show that the strangeness -1 and 1 vibrations in this model contribute in equal amounts to the strangeness content. This seems to suggest that the complete sum over all vibrations is required.

The contributions left out by BRS are those from vibrations in the continuum in both strangeness channels. These continuum contributions can be calculated by putting the system in a box [as already proposed but not executed in ref. 6)], and taking the limit as the size of the box goes to infinity. The formalism is reviewed in sect. 3, where we point out that in the case of the CHK model, two distinct ways of defining the conjugate momentum for vibrations (the G-form and the K-form) give the same vibrational energies but different results for strangeness correlations.

The calculated results are given in sect. 4. We find that the BRS model gives a strangeness content of 1.4% for the single S = -1 bound-state vibration, i.e. only about half of the result reported by BRS. However, the S = -1 vibrational contribu- tions from the continuum are important; they give a roughly equal contribution. The S = 1 vibrations, all in the continuum, are found to give roughly the same contribution as all the S = -1 vibrations, in agreement with the schematic model. The total vibrational contribution to C, turns out to be 6% for the BRS model at the physical kaon mass. We show explicitly how these results change with the kaon mass.

For the CHK model, onIy the G-form of the vibrational momentum leads to the expected positive values for the strangeness content. The calculated strangeness content turns out to be about three times larger than that of the BRS model. This large model dependence shows that it is necessary to study more realistic models of baryons in order to pin down the value of the nucleon’s strangeness content from strangeness vibrations. The correlation energy AE caused by strangeness correlations in the nucleon is found to be divergent in the ultraviolet in both models.

In sect. 5, we find that the primary source of this ultraviolet divergence is an angular momentum transmutation potential associated with the “cowlick” of the

688 C. W. Wang et al. / Strangeness vibrations

hedgehog at the origin. The divergence in AE is shown to be of the type K In (K/m),

where K is the upper limit of the momentum integration. The strangeness correlation

itself is shown to be convergent if calculated from a formula obtained in ref. “).

However, the number of virtual strange mesons diverges.

The implications of our results are briefly discussed in sect. 6.

2. Schematic model of strangeness vibrations

The equation of motion for strangeness vibrations can be written in the standard

RPA form 9*10)

(2)

where A, B, and D are differential operators diagonal in coordinate space “). The

matrices X and Y contain the vibrational wave functions columnwise. The structure

of this RPA equation is of course independent of the representation. The same

equation holds in momentum space. In a discrete representation, such as that

appropriate to a system in a box, the operators A, B and D are simply (infinite

dimensional) matrices. This discrete form is familiar in nuclear physics where nuclear

vibrations are often represented in discrete shell-model states.

Unlike most nuclear RPA equations, D # A in eq. (2). Rather

A=A,+SL, D=A,-SL, (3)

where L is a potential which comes from the Wess-Zumino (WZ) term in the

lagrangian. This WZ potential is known ‘) to lower the energies of the strangeness

S = -1 vibrations, and to raise those of the S = 1 vibrations.

The operator A,, in eq. (3) can be separated into two parts h + B, where B is the

off-diagonal matrix shown in eq. (2). By a schematic model 9), we mean the approxi-

mation in which

h = E& , Bij = gV[Vj , (4)

in a certain discrete R-dimensional model space which does not have to be specified

explicitly. This model is said to be degenerate if the “unperturbed energies” &i have

the same value E for all basis states i. In our schematic approximation to eq. (2),

we shall take

L,=/M,, (5)

where A (20) describes a constant shift of the energy which is in opposite directions

for opposite strangeness.

This degenerate schematic model is easily solved ‘). The vibration is concentrated

entirely in one collective mode of energy ws (S = + or -), where

ws=sh+wo, (6)

C. W. Wong et al. / Strangeness vibrations 689

with

w;= E2+2.sgA, A=Cuj. (7)

The remaining fi - 1 states appear at the same unperturbed energy SA + E. They are

noncollective in that they contain no vibrational component at all; they will not

appear further in the discussion below.

The collective wave function is

xi Vi a-- yi &FWg’

withC(Xf-Yf)=l. I

Thus the collective wave functions are actually independent of S, even though the

vibrational energies ws have been shifted in different directions.

We shall see later that the strangeness content is proportional to a quantity AK

which takes the form

(9)

in the present model. This is independent of S, leading to the expectation that in

general S = 1 vibrations are likely to give as much contribution to the strangeness

content as S = -1 vibrations.

3. Strangeness vibrations in skyrmions

Of course, a vibrating skyrmion is much more complicated than this model system.

An explicit calculation of the continuum contributions is therefore needed.

Let us define the quantities to be calculated while reviewing how strangeness

vibrations introduce strangeness correlations into nonstrange baryons “). The hamil-

tonian density for the pseudoscalar kaon field K(r) around a nonstrange skyrmion

in the quadratic approximation is known to have the general form

%‘(r)=~+~+K+(--V’+rnZ,+ V+h2)K+iA(Kn-r+K+). (10)

Here rr is the momentum conjugate to K, and V = V(r) and A = A(r) describe the

kaon-skyrmion interaction. The resulting hamiltonian is

H= j d”rX(r)=(C:C-)(; ;)(%), (11)

where the r-integration has not been made explicit in the last step. The annihilation

operators C, for strangeness S

C+=f[d%K+imn], C_=b[&K-i-r], (12)

690 C. W. Wong et al. / Strangeness vibrations

can be expressed in terms of the physical strangeness vibrational boson (i.e. interact-

ing kaon) operators bs(n), b:(n):

(13)

where the subscripts r and n have been inserted to make explicit the intermediate

sum over the physical vibrational states n. The orthogonality of the RPA transforma-

tion matrix

(14)

can be used to obtain the vibrational forms of the hamiltonian

H =c (w+b:b++w_b_br),) (19 n

The “physical” nucleon is the vacuum 16) of bs, while strange baryons are states

with one or more strangeness vibrations.

In the absence of the potentials V(r) and A(r), eq. (10) describes free kaons. The

vacuum IO) of the free kaon operator as(p) of momentum p represents a “bare”

nucleon free of any strangeness correlation. It is useful to relate the interacting kaon

operators to the free kaon operators by the expression

(16)

where the momentum wave functions xs( n, p) and ys( n, p) can be calculated from

the X’s and Y’s.

We are interested in the following vibrational correlations, i.e. the differences of

operator expectation values in the physical vacuum from their values in the bare

vacuum:

AE = (61HI@-(OlHlO) = - C 0,s I

dp IY,s(P)~’ , (17) n,S

where

AS=C dp[l~,-(~)l*-Iy,+(~)l*l=CASs, I n S

(18)

S= dp [C&Z+-a:~_] I

is the strangeness operator, and

AK =(6~K+K~6)-(O~K+K~O)

(19)

= 5 j- dp G+-’ Yns(P)[Yns(P)-xns(P)l* * (20)

C. W. Wong et al. / Strangeness vibrations 691

In these expressions, we have used a slightly different notation for the momentum

wave functions.

Only the term n = 1, S = -1 in eqs. (17) and (20) have been calculated in refs. 4*6).

To calculate the remaining terms, which come from unbound interacting kaon states,

we enclose the system in a box of radius R, and take the limit as R +CO. The

momentum integration is now replaced by the summation over free-kaon eigen-

momenta pi = q/R in the box:

(21)

The operator KtK in eq. (20) is related to the strange sea for the following

reason “): Donoghue and Nappi ‘) have proposed that the correlation in the quark

scalar density for flavor i in the nucleon can be extracted from skyrmions by using

the relation

(6]~iqi16)-(O]~jq,]0)ocTr(QiU+Q:Ut-Qi-Q~). (22)

With Qs= diag (O,O, 1) and the Callan-Klebanov ansatz U = U~‘U,U~‘, the

surviving term in the quadratic approximation is proportional to KtK.

A more direct way of measuring the strangeness content of 16) in the RPA formalism

is to use the kaon number operator of either strangeness

dp [a:a++a’a_]. (23)

This gives a result

which is different from that implied by eq. (20). We shall show their numerical

difference in sect. 4.

The vibrational equation of motion is conveniently solved not in the RPA form

of eq. (2), but as a second-order wave equation ‘):

(0+fw;-2Swsh)ks =O, (25)

where

k,=X,+Y,, (26)

(J= r-2; hr2$_;- V,,+fm2,(1 -cos F), (27)

ve,= 1(1+1)/?+ v,+1* LV,, (28)

and F is the skyrmion profile function (chiral angle). Explicit forms of J; h, V, and

V, , and other details of the calculation can be found in refs. 3*4).

692 C. W. Wong et nl. / Strangeness vibratj~ns

We should note that eq. (25) has an eigenvalue w- = -w+ in the S = -1 channel for each eigenvalue u+ in the S = fl channel. In particular, the zero mode o = 0 appears simultaneously in both channels. Its appearance signifies the flavor SU(3) symmet~ of the lagnngian.

The WZ potential A(r) is known ‘) to have a repulsive effect in the S = 1 channel, making all solutions unbound. The summation over n in eqs. (17), (18) and (20) is over the o B ?nK states for S = 1, and over the w a -mK states for S = -1.

In sect. 4, we shall report results not only for the BRS model where the skyrmion is stabilized by coupling to a vector meson I’), but also for a model 12) with a qua&c stabilizing term and a massive pion. (The latter model shall be called the massive CHK model below.) We should note that the quartic term introduced scaling functions f(r) and h(r) (both Zl) into the lagrangian

=4lr I

r2 dr {G;‘G+ - . +}. (29)

As a result, there are two nonequivalent ways of introducing the conjugate momen- tum - momentum conjugate to the pseudoscalar kaon field K (the K-form), or momentum conjugate to the scaled field C =x@ K (the G-form). The G-form is unique in that it leads to the standard form of the hamiltonian shown in eq. (lo), while the K-form gives a hamiltonian with a kinetic energy of the form r’(l/f)n: The vibrational frequencies turn out to be the same in both forms, but not the vibrational correlations. This is because the G-form integrals (17)-(20) are obtained from their I( form analogs by replacing the wave function k by sf k, wns by wJf and h by A/J: The numerical differences will be given in the next section, where we shall see that only the G-form gives reasonable results.

4. Numerical results

As the kaon mass decreases from the physical value, we expect the lowest vibrational frequency w = w,=~,~=_, to decrease. At the point where w is zero, the restoring force vanishes, and rotational symmetry in strangeness directions is restored. As a check on our computer program, we have confirmed that this zero mode appears as expected when mK - - rn, in the massive CHK model. The situation

is more complicated in the BRS model because the coupling to the vector meson turns out to be flavor asymmetric. We shall come back to this unusual feature later.

The result of our calculation are shown in table 1. Here “n term” denotes the number of terms included in the partial sum over n. The numerically estimated sum for the infinite series is denoted “est. sum”. The quantities calculated are AK (in units of the inverse mass scale), AS and AE (in MeV). The mass scale is M, = 782.4 MeV in the BRS model, and eF,, = 4.84 x 108 MeV in the massive CHK model.

C. W. Wong et al. / Strangeness vibrations 693

The calculated results are found to converge rapidly as the (dimensionless) radius

R of the box increases. In table 1, we show the results for relatively small values

of R since the energy range covered for the same number of terms increases as R

decreases. We use R = 20 in the BRS model, but R = 10 in the massive CHK model

to partially compensate for the difference in mass scales.

Table 1 shows that AK is convergent in the BRS model, and probably converges

also in the massive CHK model. In both models the S = 1 states give roughly the

same contribution as the S = -1 states, in agreement with the soluble model studied

in sect. 2. The continuum contribution in the S = -1 channel is a little more than

the contribution from the single bound state. Thus the total contribution is a little

more than four times that of the single bound state obtained by previous authors.

The two models differ in absolute values, the massive CHK results being about 3

times larger.

TABLE 1

Strangeness correlations AK (in units of the mass scale), AS and AE (in MeV) for the BRS model and

for two versions of the massive CHK model

s=-1 S=l

n term

AK AS AE AK AS AE

(a) BRS model:

1 0.0664 0.0229 -4.4 0.0019 -0.0004 -0.2

10 0.1077 0.0528 -29 0.0638 -0.0156 -15

20 0.1219 0.0794 -78 0.1068 -0.0508 -90

30 0.1284 0.0986 -136 0.1201 -0.0785 -180

100 0.1399 0.1652 -619 0.1379 -0.1587 -766

170 0.1423 0.1982 -1154 0.1404 -0.1883 -1231

est. sum 0.1461 --co 0.1448 --co

(b) Massive CHK model (G-form):

1 0.1585 0.1779

10 0.2553 0.2977

20 0.2908 0.3744

30 0.3107 0.4300

100 0.3659 0.6777

170 0.3891 0.8622

est. sum 0.46

-26 0.0404 -0.0260 -14

-135 0.2294 -0.2374 -235

-312 0.2789 -0.3402 -489

-527 0.3023 -0.4051 -753

-2919 0.3613 -0.6664 -3304

-6772 0.3851 -0.8542 -7266

-‘xJ 0.44 --CD

(c) Massive CHK model (K-form):

1 -0.0227 0.0494

10 -0.1163 0.3654

20 -0.1826 0.9367

30 -0.2269 1.584

100 -0.3714 6.632

170 -0.4385 11.9 est. sum

-7 -0.0014 -0.0012 -1

-344 -0.0924 -0.1790 -243

-1709 -0.1674 -0.6855 -1558

-4240 -0.2160 -1.305 -4100

-55 800 -0.3679 -6.308 -56 300

-167 000 -0.4362 -11.6 -168 000 --co -cc

694 C. W. Wong ef al. / Strangeness vibrations

To convert AK to the fractional strangeness content C, of eq. (l), we need the BRS results

(NISsjN) = aAK, (Nfuu+;id{N)=a(2a/m2,)+.‘., (30) C,=AK/[(2a/mZ,)+AK+..,]. (31)

The result for Cr, is independent of a = mi/( m,i- md), but depends on the sigma term (+ from the skyrmion. With u = 54.2 MeV in the BRS model (using m, =

782.4 MeV), we find C, = 5.9%, to which the single S = -1 bound state contributes 1.4%. This leads us to suspect that the result of 3% reported by BRS contains a spurious factor of 2. BRS might also have left out the pion mass term in eq. (27), but this is not too important, except in the study of the zero mode.

Eq. (31) is reliable only when quadratic vibrations dominate in the strangeness directions. This is likely to occur with large tlavor symmetry breaking. It is neverthe- less interesting to see how C, and the bound-state energy w depend on mK. The results are shown in fig. 1 for the BRS model. We see that w passes through 0 when m, = 200 MeV, not at the mass symmetry point mK = m,. This unexpected result is a consequence of the neglect of strangeness vibrations of the stabilizing vector meson field in the BRS model. This point will be discussed further in sect. 5 below.

Fig. 1 shows that when w = 0, C, is about 13%, and that it decreases monotonically as mK increases. The vibrational quenching of C, by mk is qualitatively similar to, but considerably stronger than, that found in rotating skyrmions “).

For the massive CHK model, table 1 shows that the K-form of this model (as defined in sect. 3) gives negative values for the strange sea correlation AK. Since this is not a physically reasonable result, we shall henceforth consider only the G-form of this model which gives the expected positive values for AK. Table 1

Fig. 1. The vibrational energy w (in MeV, broken curve) of the lowest strangeness -1 state and the

strangeness content C, (in %, solid curve) of the nucleon for the BRS model“) as functions of the pseudoscalar kaon mass mK.

C. W. Wang et ai. j Strn~geness uibraf~ons 695

shows that the convergence of the summations for AK is much poorer than that for the BRS model, but it does appear that AK is about 3 times larger than that for the BRS model. Both the mass scale (eF, = 523 MeV) and the sigma term (o. = 37.7 MeV) are smaller than those for the BRS model, leading to a large C, of about 27% at mK = 495 MeV and about 40% at mk = 138 MeV (where o goes to zero), as calculated from eq. (31) with the 170-term results. We should point out that the vibrational contributions to (pllu+ddlp) is also large in this model, and this has not been included in our calculations, thus leading to an overestimate of C,. To partially compensate for this overestimate, we have not included terms beyond 170 in C, for the massive CHK model.

To properly account for the vibrational contributions to (pltiu + ad/p), we need to include not only strangeness vibrations but also nonstrange vibrations such as virtual pions. This would require a much more elaborate calculation than those attempted in this paper. As a result, the values for C, estimated from eq. (31) are not very precise. It is clear however that the massive CHK model gives rise to much stronger vibrational correlations than the BRS model. This is not surprising because the bound kaon energy w (146 MeV) is lower than the value (194 MeV) for the BRS model.

Another indication of the unusual strength of the kaon-skyrmion interaction in the massive CHK model is the rapid increase in the number of bound vibrations with increasing pseudoscatar meson mass. For example, at the D-meson mass of 1868 MeV, there are four S = - 1 and two S = 1 bound states.

It is also clear that as mk is decreased towards m,, quadratic vibrations will become less and less dominant, and we will have to worry more and more about rotational “) and other degrees of freedom. It would be interesting to treat these effects together in a simple and consistent way.

On the other hand, if AK is as small as that for the BRS model, eq. (31) might not be too bad at the physical kaon mass. For this reason, it would be interesting to repeat our calculation for other models of kaon-nucleon interaction to see if they give rise to strong or weak strangeness correlations. The contribution of 1.8% to C, found for the single S = -1 bound vibration in the quark-gluon model of ref. “) indicates that this model is quite similar to, though a little stronger than, the BRS model.

Table 1 shows that in both the BRS and the massive CHK models, AS probably diverges separately for S = -1 and I, but with opposite signs. The sum over both channels appears to be small (about 0.01, and consistent with the expected value of 0). On the other hand, the number nk of strange kaons in Ifi), which is the sum of their absolute values, probably diverges. This suggests that AK is related to only a part of nk.

Table 1 also shows that AE certainly diverges in both models, the divergence being worse than a linear one. (We know this because the contribution to AE from each vibrational eigenmode in the box is increasing slowly as n increases.) Thus

696 C. W. Wong et al. / Strangeness vibrations

this simplest of quantum fluctuations beyond the classical skyrmion solution is not

under control.

5. Sources of the divergences

The divergence in AE, AS, and nK is obviously an ultraviolet one. We therefore

look for its source near the origin. There V,, is dominated by certain terms ‘)

Ve,= (h/r2)[Z(1+ 1)+2c2+4cl. L-j, (32)

where

cssin’+F=l (33)

near the origin. As a result, the vibration near the origin has an effective orbital

angular momentum lefi = If 1, when the “grand” spin13) t = I + L has the value of

t=l*i. The t=i, S=-1 bound state has I=1 and therefore Z,,=O.

We shall now show that this 1 transmutation potential, which has such an amusing

effect in skyrmion vibrations, is a source also of our divergence.

Let us begin with the simpler BRS model, for which h = 1 because of the absence

of the quartic term. Keeping only the transmutation term (defined by c = 1 in eq.

(32)) and excluding also the Wess-Zumino term h (which is not an important factor

in the following discussion) in the kaon-skyrmion interaction, we find that eq. (25)

has solutions j,( k,r), where k, = rm/ R. These solutions differ in both r dependence

and in eigenvalues (features which contribute to AK) from thej,( p,r) eigenfunctions

of the free kaon in the box. The RPA vibrational correlation energy AE can now

be calculated from eq. (17) using the result

1 k

=-[z-Ro~,,w(p)]“~ P+~;.,* (34)

Thus

MU = I a dp lx&)12= (~RwJ’ ln (WmK) (35) 0

keeping only the leading term for large k. Finally

dkw,(k)l,(k)--5 K hl (K/m,), (36)

to the leading term in K. We see that AE does indeed diverge more strongly than

K as K+m.

C. W. Wong et al. / Strangeness vibrations 697

We are also in a position to calculate the vibrational change in strangeness. The

leading term

ASAW = -2 [In (KlmK)12 (37)

is divergent, but AS = AS+ + AS- = 0 since there is no S-dependence in Is when the term A is neglected.

On the other hand, the strangeness sea correlation (20)

AK-& In (K/m,)+finiteterms (38)

is finite as K -+ CO.

Thus the transmutation potential alone is sufficient to cause divergences. This potential comes from a term involving (Vf.Jo)*, where fJO is the SU(2) chiral field. It is singular at the origin because the profile function F(0) = 7r, but the pion hedgehog field points in every direction. This is an essential feature of the skyrmion, because without this transmutation potential the zero mode (w = 0) of the CHK model does not appear correctly at the mass symmetry point mK = m,. However, this essential feature turns pathological when quantum fluctuations are included.

There is an interesting alternative way of looking at the situation. The skyrmion profile function must have the value F( r = CO) = 0 (mod r) at infinity so that the pion field will vanish there. The required baryon number (=l) of the solution is obtained if F(0) - F(a) = F. There are thus two possible solutions: (1) F(O) = 7r {mod Zm), and (2) F(O) = 0 (mod 2~). Our discussion so far is based on the first of these solutions. If we take the second solution, we find that

c=sin’iF=O (39)

near the origin. This means that the transmutation potential is now absent, while the zero mode does not appear correctly any more at the mass symmetry point for the massive CHK model. In fact, the lowest S = -1 state becomes unbound when mK falls below ==190 MeV.

Nevertheless, we can ask if the absence of the transmutation potential will ensure convergence. Table 2 shows this is the case for the BRS model, but the divergences persist in the massive CHK model. Since the massive CHK model differs from the BRS model primarily in the presence of the scale functions f(r) f 1 and h(r) # 1 introduced by the stabilizing quartic terms, we see that these scale functions are a second source of ultraviolet instability.

This is an interesting situation because these scale functions also appear in models in which the stabilizing quartic term is replaced by the coupling to the complete nonet of vector mesons 14). They arise from terms involving the vector kaon fields when these are eliminated in favor of the pseudoscalar kaon field through the approximation that these two fields are proportional to each other. It would be interesting to determine if this second source of divergences in this model exists in

698 C. W. Wong ef al. / Strangeness vibrations

TABLE 2

Strangeness correlations AK (in units of the mass scale), AS and AE (in MeV) for the BRS model and

for the massive CHK model (G-form) using the second solution of the profile function

n term

s=-1 S=l

AK AS AE AK AS AE

(a) BRS model:

1 0.0086 0.0047 -2.0 0.0099 -0.0003 0.2 10 0.0267 0.0073 -4.1 0.0445 -0.0025 -1.9 20 0.0260 0.0078 -4.9 0.0285 -0.0069 -11.2 30 0.0255 0.0079 -5.2 0.0255 -0.0075 -13.1

100 0.0250 0.0080 -5.5 0.0240 -0.0077 -13.9 est. sum 0.0250 0.0080 -5.5 0.0239 -0.0077 -14.0

(b) Massive CHK model (G-form):

1 0.0776 0.0381 10 0.1506 0.0694 20 0.1783 0.0921

30 0.1947 0.1122

100 0.2440 0.2440

est. sum

-13 0.0248 -0.0065 -3 -44 0.1338 -0.0606 -60 -98 0.1714 -0.0854 -123

-178 0.1894 -0.1060 -209 -1520 0.2409 -0.2382 -1580

--co --co

the original coupled-equation form of strangeness vibrations or if it arises from the elimination of the vector kaon field.

The situation with the BRS model is different in several respects. In this model, the kaon is coupled to both the skyrmion and a vector meson whose strangeness component is controlled by a parameter q (chosen to be 0.09). The vector meson is an SU(3) singlet when q = 0, and a nonstrange meson (i.e. with magic mixing) when q = G 3_ At the mass symmetry point, w = -35 MeV at the BRS vector meson coupling parameter /3 = 15.3. The zero mode appears instead at q = 0.000 if all the other parameters are unchanged. This might appear satisfactory, but it is not because where the zero mode appears at the mass symmetry point in this model actually depends on model parameters such as /3 and m,. For example, the zero mode appears at q =0.485 when /3 is halved to 7.65.

This strange result appears to be the artificial consequence of an oversight in the BRS model, namely that the vibration in the strangeness directions has accidentally been neglected for the stabilising vector meson of the static solution. If the vector kaon is properly included, the model might actually be quite similar to the massive CHK model 14).

Since the BRS model is not formulated in a flavor symmetric way when mK = m,,

it is actually possible to obtain a zero mode for the second solution of the profile function by adjusting q to 0.6935, although this parameter will not give good baryon masses. We confirm numerically that AE is still finite for this model We do not believe that this model is physically interesting because of the diffculty in the

C. W. Wang et al. / Strangeness uibrutions 699

formalism mentioned earlier. It seems to us likely that a skyrmion model which

reproduces the expected zero mode correctly for the first solution of the profile

function will not do so for the second solution.

While the first solution of the profile function might well be the only permissible

one for skyrmions, it is quite possible that for other models of baryons in which a

hedgehog pion field appears, a solution of the second type with F(0) = 0 is the only

correct one. In fact, the second solution appears rather naturally in the chiral-meson

model of baryons of ref. “). In any case, a more detailed study of the relation between

the zero mode and the value of F(0) appears to be desirable.

6. Discussion

We have found that Skyrme models do not have the correct high-energy behavior

in harmonic vibrations. This is not surprising since Skyrme models are effective,

low-energy models. What is perhaps unpleasant is that the problem is connected

with an angular momentum transmutation potential which appears to be such an

essential feature of vibrating skyrmions.

Strangeness vibrations of skyrmions cannot be a quantitative tool until the unsatis-

factory high-energy behavior is brought under control. One can immediately think

of simple procedures such as a momentum cutoff analogous to the usual form factor

(1 +pz/A2)-* and equivalent to smearing the ultraviolet divergence over a distance

of the order A -‘. Assuming that this can be done without ruining the appearance

of the desirable zero mode at the llavor symmetry point, the next problem is the

choice of the cutoff parameter A. To do this sensibly even from a model-making

point of view, we need to find additional divergences in physical observables so

that the usefulness and consistency of a cutoff procedure can be tested against

known hadronic properties. Information on the energy alone is insufficient for model

making.

It is likely that the divergence found here is ultimately a reflection of the unsatisfac-

tory ultraviolet behavior of the skyrmion picture, such as the absence of asymptotic

freedom. It would be much more useful if the cutoff parameter could be related

directly to these missing physical features.

In this paper, the fractional strangeness content has been estimated with the help

of a formula [eq. (31)] dependent on a quantity AK which is convergent at least

under favorable conditions. In view of the divergence in nK, the validity of this

formula should be critically examined.

The harmonic vibrational effects calculated here are expected to dominate for

strong flavor-symmetry breaking. Since all quadratic terms have been included in

the present formalism, rotational effects have automatically been included in the

same quadratic approximation. However, near the flavor-symmetry point, large

excursions into strangeness directions beyond these quadratic terms are expected.

These must be handled differently, for example by perturbation from the rotational

700 C. W. Wang et al. / Strangeness vibrations

picture “). It would be helpful if the rotational and the vibrational descriptions can

be combined in a consistent way.

Of the two models studied here, the massive CHK model shows the expected

behavior of the zero mode. It gives a rather large strangeness content as the result

of the strong quartic term used to stabilize the skyrmion. The skyrmion is stabilized

in the BRS model by coupling to a vector meson instead, a procedure which is

“milder” and perhaps more realistic. The resulting strangeness content is a factor

of 3 smaller. But the model appears to require additional work to bring out the

expected zero mode correctly. In spite of this uncertainty, it is clear that there is

significant model dependence in the calculated vibrational effects.

We are inclined to believe that the kaon-nucleon coupling in the massive CHK

model might be too strong, and that the strangeness content of the nucleon’s quark

sea at the physical kaon mass might be closer to the 6% obtained here for the BRS

model. It would be interesting to see if this is in fact the case for the more realistic

models of refs. 6V14).

This work is supported in part by NSF grant PHY88-19084.

References

1) G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552 2) A. Monohar, Nucl. Phys. B248 (1984) 19;

M. Praszalowicz, Phys. Lett. B158 (1985) 264;

M. Chemtob, Nucl. Phys. B256 (1985) 600

3) C.G. Callan, K. Hornbostel and 1. Klebanov, Phys. Lett. B202 (1988) 269;

C.G. Callan and I. Klebanov, Nucl. Phys. B262 (1985) 365

4) J.-P. Blaizot, M. Rho and N.N. Scoccola, Phys. Lett. B209 (1988) 27;

N.N. Scoccola, H. Nadeau, M.A. Nowak and M. Rho, Phys. Lett. B201 (1988) 425

5) GE. Brown and C.W. Wong, Nucl. Phys. Al00 (1967) 241

6) J.A. McGovern and M.C. Birse, Nucl. Phys. A506 392 (1990) 7) J.F. Donoghue and C.R. Nappi, Phys. Lett. B168 (1986) 105

8) H. Yabu and K. Ando, Nucl. Phys. B301 (1988) 601;

H. Yabu, Phys. Lett. B218 (1989) 124

9) G.E. Brown, Unified theory of nuclear models and forces (North-Holland, Amsterdam, 1967) ch. V

10) W. Broniowski and T.D. Cohen, Nucl. Phys. A458 (1986) 652

11) G.S. Adkins and C.R. Nappi, Phys. Lett. B137 (1984) 251

12) G.S. Adkins and C.R. Nappi, Nucl. Phys. B233 (1984) 109

13) T.D. Cohen and W. Broniowski, Phys. Rev. D34 (1986) 3472

14) N.N. Scoccola, D.P. Min, H. Nadeau and M. Rho, Nucl. Phys. A505 (1989) 497