Volume 224, number 1,2 PHYSICS LETTERS B 22 June 1989

HIGHER ORDER PERTURBATION THEORY FOR THE SU(3) SKYRME MODEL

N.W. PARK, J. S C H E C H T E R and H. W E I G E L Physics Department, Syracuse University, Syracuse, NY 13244-1130, USA

Received 22 March 1989

We show that the results obtained by Yabu and Ando treating the symmetry breaking exactly can easily be reproduced by perturbation theory, carried out to next to leading order. An explanation is provided for the success of the Gell-Mann-Okubo

I

mass formula in this model even though first order perturbation theory is inadequate. The important role of SU(3) 10 and 27 "impurities" in the octet baryon wavefunctions is pointed out. A number of observables are discussed in the framework of per- turbation theory.

There has recently been a revival of interest in the SU (3) Skyrme model as a " toy" model for investigating such quest ions as the "strangeness content" of the nucleon [1 ] and the "spin of the p ro ton" [2 ]. As poin ted out, however, by many authors [3 ] the most s t raightforward extension of the min imal SU (2) Skyrme model to SU (3) leaves quite a lot to be desired. For example, a t tempt ing to fit the nucleon and delta masses requires choosing an extremely small value for the pion decay constant , F~. Fur the rmore the pat tern of octet mass split- tings is incorrect. It seems to be accepted that the origin of the difficulty is related to the relatively large SU (3) symmetry breaking in nature and also to the quest ion of how reasonable it is to treat the K meson as a Golds tone boson. In the "bound state" approach [ 4 ] to strangeness, which has been pursued with some success, the strange baryons are t reated in an SU (2) f ramework and the ord inary nucleons do not (wi thout further embel l ishments of the mode l ) acquire any strange content.

Consider ing the great phenomenologica l successes of the SU (3) analysis of low lying part icle multiplets, it does seem worthwhile to see if the SU (3) Skyrme model can somehow be t reated to improve its predict ions. Encouraging progress along these lines has been made in recent work by Yabu and Ando [ 5,6 ]. They were able to get much improved fits by essentially taking two new features into account: ( i ) a k ind of zero-point subtrac- t ion to the energy, ( i i ) t reat ing the SU (3) mass spli t t ing exactly ra ther than by first order per turba t ion theory. Feature ( i ) seems to us plausible but far from well established. Feature ( i i ) on the other hand is undoubtedly correct and o f great qual i ta t ive interest. All the previous works on the SU (3) Skyrme model considered it suf- ficient to treat the mass spli t t ing to first order in symmetry breaking. There is a strong mot iva t ion for doing so since the famous G e l l - M a n n - O k u b o mass formulas [7] are der ived with just this assumption. Yabu and Ando have in effect demons t ra ted that this is wrong for the SU (3) Skyrme model. How then can one unders tand the success of the G e l l - M a n n - O k u b o formula i f first order per turba t ion theory is not valid? Fur the rmore many a t tempts to extract mat r ix elements of physical ly relevant operators are similarly based on neglecting SU (3) symmetry breaking corrections. Can these results be trusted?

In order to investigate these quest ions we now study the SU ( 3 ) Skyrme model with higher order per turba t ion theory. We will see that second order is actually sufficient to recover the Y a b u - A n d o mass results. The pertur- bat ion method is, of course, easier than that of Yabu and Ando which requires one to solve a set of coupled differential equat ions for each isospin mult iplet . The per turba t ion method also enables us to develop some physical in tui t ion about what is going on and to easily make est imates for a large number o f physical quantit ies. Another advantage of the present approach is that it can be readily extended to more compl ica ted models.

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Following essentially the notation of ref. [ 5 ], the quantized hamiltonian for the collective modes of the min- imal SU (3) Skyrme model is

( 1 ~ 2 ) j ( j + l ) + 1 3 H=Mc,+ 2~x2 2 ~ C 2 [ S U ( 3 ) ] - ~ [ B ( U o ) ] 2 + ½ 7 [ 1 - D g s ( A ) ] . (1)

To briefly explain [3,5] ( 1 ), we first remind the reader that the starting chiral lagrangian is built on the 3 × 3 unitary matrix U describing the pseudoscalar mesons. The static soliton solution Uo has baryon number, B(Uo)= 1. The dynamical variables, A( t ) describing the baryon degrees of freedom are defined by U(r, t) =A (t)Uo (r)A* (t). Generators of the spin and flavor SU (3) groups can be constructed in terms of the mo- menta conjugate to A (t). These generators appear implicitly in H via the spin Casimir operator eigenvalue J(J+ 1 ) and the SU (3) Casimir operator eigenvalue

C2[SU(3) ] = ~ (p2+pq+q2) + ( p + q ) , (2)

for a tensor with p upper symmetric indices and q lower symmetric tensor indices. Mc~ in ( 1 ) is the energy of the static soliton solution Uo while c~ 2 and f12 are "moments of inertia" depending on Uo. The last term in ( 1 ) represents the SU(3) symmetry breaking induced by the kaon-pion mass difference and Dij(A) is the octet representation matrix defined by

Do(A ) =D,~(A) = 1Tr (2,A2jA*). (3)

The 2~ are the Gell-Mann SU (3) matrices. Numerical evaluation of the coefficients in (1) using the soliton solution Uo (r) and including the effects of a finite pion mass term yields [ 5 ]

37.21 12.95 1 6 . 1 6 ( m 2 - m 2 ) 57.37F~ , f12= M d - (4) e~ - F~e3, F~e 3 ' Y= F~e 3 ' e

In (4) F~= 132 MeV is the pion decay constant (often considered to be a free parameter) while e, the dimen- sionless Skyrme constant, is a free parameter (typically taken in the range 4-5 ). To the extent (about 15%) that the pion mass is negligible the quantities in (4) will scale as shown with respect to F~ and e.

To start we take as our perturbation

V= - lyD88 (A) . (5)

This transforms as the eighth component of a flavor octet: if the SU (3) flavor generator is denoted by G, we have [3,5] [G, A] = -½2~A so (3) yields [Gi, Djk(A)]=ifj iDlk(A) showing that only the left index of Djk is affected. Thus the Gell-Mann-Okubo mass formula will hold automatically at first order. The wavefunctions of the free hamiltonian with baryon number 1, spin J, hypercharge Y, isospin I and belonging to the flavor SU (3) representation It may be written [ 3,5 ] in terms of the SU (3) representation matrices:

_ D ( ~ ) * T(/2, Y, I, 13; J, J3; A) = ( 1 )J-J3~/dim// y,I,13;1,J_j3(A) . (6)

Only those representations # are possible which contain a state with Y= 1 and I=J. In this note we will confine ourselves to the J = 1 baryons so the lowest lying states will be 8, 10, 27, .... For perturbation theory we must ask which intermediate states are accessible from the 8 with the perturbation (5). Using the decomposition 8 × 8 = 1 + 8 + 8 ' + 10+ 10+27, it is easy to see that only the 10 and 27 states are relevant up to third order in mass splitting. The needed matrix elements in (5) are obtained [noting that D88 (A) ~8) =Dooo.ooo(A) ] from the formula

(# ' , Y ' , I ' , I '3;J ' ,J '31Dss(A)II t , Y,I, I3;J, J3)

= ( _ l ) j + j _ j 3 j~ ~ ( 8 /z /t;, ' ] ( 8 /z /~'Y "] (7) N/dim/z ' ~ 000 VII3 V'I 'I '3J\O00 1J,-J3 1J', - J ~ J '

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Table 1 Matrix elements in eq. (9).

PHYSICS LETTERS B 22 June 1989

-2Vs,8/y -2Vs,~/y -2Vs,27/Y -2V~,~/7 -2V~,27/y --2V27,27/Y

p 3/10 xf5/lO x/-6/lO 1/8 x /~/80 0.2446 A 1/10 0 3/10 0 0 13/70 Z + - 1 / 1 0 xfl5/10 1/5 0 xf5/10 13/280 E ° -2 /10 0 ,/6/10 0 0 0.0571

in terms of the SU (3) C l e b s c h - G o r d o n coefficients [ 8 ]. We also need the zeroth order mass differences from (1) and (2) :

3 5 E?o-E8 = 2fl 2, E 2 7 - E 8 - 2fl"" (8)

The s tandard expression [ 9 ] for energy shift up to third order then yields

BE= V8,8-2f12( 1 2-- I 2 Vs, ,o+sVs,zv)+4f14[~ Vs,lo(V?6T6-Vs,s)2-- +~3V8,27(Vzy,27_Vs,s)+ 2 2 (9)

where the matr ix elements for members o f the baryon octet as obta ined from (7) are l isted in table 1. Notice that V,j = V~ = Vj<. Numer ica l ly we have for the dimensionless quant i ty 2flZSE (which is eSB--3--~fl2 in ref. [5]):

2flZSE(p ) = - 0.37fl 2 - 0.0287 (7fl 2 ) 2 + 0.0006 (Tfl 2 ) 3 + ....

2flZSE(A) = - 0 . 1 7 f l 2 - 0.0180 (7fl 2) 2 - 0.0003 (Tfl 2) 3 + . . . ,

2f128E(Z ) = 0.17fl 2 - 0.0247 (Tfl 2 ) 2 + 0.0002 (7fl 2 ) 3 + ....

2f128E(E) = 0.27fl2_ 0.0120 (7fl2) 2_ 0.0006 (7fl2) 3+ .... (10)

A compar ison of the nucleon results with the exact ones obta ined by numerical ly solving the eigenvalue eq. (2.17 ) of ref. [ 5 ] is given in table 2. For the relevant values of ~,f12 (a round 3 ) the agreement is quite good. This gives us confidence that per turba t ion theory is fairly reliable for this model i f the next to leading order contri- but ion is included. Now we can make several observat ions concerning the results in (10) . First, for the typical fits in ref. [5] , yfl2~ 3.33, the th i rd order terms are pret ty much negligible (3% at most ) . Even though the expansion coefficient is large, the product o f the C lebsch -Gordon coefficients, each of which is substantial ly less than unity, is very small. On the other hand the second order terms are very impor tan t (as much as 80% of the first order term in magni tude for the case o f the Z) . While they do not satisfy the G e l l - M a n n - O k u b o formula [ 7 ] m ( Z ) + 3 m (A) = 2m ( p ) + 2m ( E ), they do not devia te by much from it either. This can be part ial ly under- s tood by noting that all second order correct ions must be negative in per turba t ion theory so that i f they are all of the same order o f magni tude (only roughly t rue) they will not affect the splittings. It can be seen that the (7fl2)2 terms do differ somewhat but the pat tern of differences is such as to min imize the violat ion of the G M O

Table 2 Comparison of the perturbation result e~) = 3 + 7ff + et + e2 + e3 with the exact result 6S(~3 xacl ) for the nucleon.

y~ < ~ ~ ~ E~ ~°~''

1.0 - 0.3 - 0.0287 0.0006 3.6719 3.6720 3.0 - 0.9 - 0.2583 0.0162 4.8579 4. 8638 5.0 - 1.5 -0.7175 0.0750 5.8575 5.8907 7.0 -2.1 - 1.4063 0.2058 6.6995 6.7939

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formula. Specifically, the fact that the rn (p) piece is above average is compensated by m (E) being below average on the RHS. To the extent that the Skyrme model is a good representation of QCD these remarks may explain the success of the GMO formula. The mass differences are predicted to stand in the ratios m(A) - m ( N ) : m ( Z ) - m ( A ) : m ( E ) - m ( Z ) = 1:0.81:0.59. These are in significantly better agreement with the experimental numbers 1:0.43:0.69 than are the first order predictions 1 : 1 : ½.

The preceding discussion has highlighted the important role played by the 10 and 27 multiplets in determining the properties of the octet baryons in the SU (3) Skyrme model. We can make this more quantitative by using (for simplicity) first order perturbation theory to find the approximate 10 and 27 "contamination" of the spin 1/2 baryons:

ibaryon) ~ [ 8) _ zozv. - - 2 2 - s f l 127) 3~' 8.10110 ) V8,27 •

Specialized to the proton, using table 1, this becomes

[ p ) ~ [ 8 ) + 0.07452:/? 21]O) + 0.04907/? 2 [ 27 ) , ( 11 )

which shows that for a typical value 7/72~ 3.33, the proton contains about a 25% amplitude admixture of the 10 and 15% of the 27. This will be seen to have a significant effect for matrix elements. What about the properties of these states themselves? They must have spin ½ and positive parity (or else their transition matrix elements to the 8 would vanish). Because of their SU (3) representations they cannot be qqq states but should be of the form qqqq(t. So physically they seem to correspond to states with an extra q(t pair excited. With the typical fits of ref. [5] /?~ 3 GeV -~, so (8) shows that the 10 proton-like state lies about 500 MeV above the proton while the 27 state lies about 800 MeV above the proton. One might then try to identify the 10 with the Roper state N (1440) and the 27 with the N ( 1710). The latter two states are, however, generally considered "breathing modes". Perhaps a significant mixture between breathing mode states and excited pair states is called for. Or perhaps the proton simply breathes by creating a pair. Further pursuit of this point is beyond the scope of the present note.

As an application of formula ( 11 ) let us consider the corrections to the pure SU (3) results for the u, d and s "content" of the proton. This was done recently by Yabu [ 6 ] using a different method. The content fractions for the proton are defined [ 1 ] as

(p ] ~l~q~ [p) - (0 D ~l~q~ ]0) Xa= ( p l f i u + d d + g s l p ) - ( 0 [ f i u + a d + ~ s l 0 ) ' (12)

for a = 1, 2, 3 (ql =u, etc. ). Evidently Xu+Xd+Xs= 1. In the sigma model the subtracted matrix elements of the quark bilinears are interpreted as

<pl ( a u - a d ) I p > ~c ( p I T r [ Z 3 ( U + U * - 2 ) ] Ip) ocxfl3 ( o l D 3 8 ( / ) [ P ) ,

(Pl ( O u + d d ) I P ) oc ( p l T r [ T ( U + U t - 2 ) ] I p ) oc ( p I [ 2 + D a s ( A ) ] I p ) ,

(p lgs lp )oc ( p I T r [ S ( U + U t - 2 ) ] I p ) o c (Pl [ I - D 8 8 ( A ) ] I p ) , (13)

where 23=diag( 1, - 1, 0), T=diag( 1, 1, 0), S=diag(0 , 0, 1 ) and the vacuum subtraction has not been explic- itly indicated on the LHS. To find Xs we need the matrix elements of D88(A) evaluated between I P) given approximately by ( 1 1 ). This yields

Xs ~,.~, ~ 0 43 R2-v (14) -- 2-5T6 ~" . . . . .

It is interesting to see that the corrections reduce the predicted strange content of the proton. For a typical choice 7/? 2 = 3.33, Xs is reduced from 0.23 to roughly 0.17. The reduction of the isovector density in (13) is even more dramatic. To evaluate this we first need to know that ( 1-01D38 (A) 1 8 > = - ~ o x / ~ and (271338 (A) 18 ) = ~o,f2. Then, using (13) and ( 11 ) we find

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V o l u m e 2 2 4 , n u m b e r 1,2 P H Y S I C S L E T T E R S B 2 2 J u n e 1 9 8 9

Xu - X o = ~ - ( ~ - ~75x/2)yf12 + .... (15)

For a typical choice 7fl2=3.33, X u - X a is reduced from ~o to slightly less than half that value. This has an important bearing on the non-electromagnetic part of the neut ron-pro ton mass difference which would be pro- portional to X u - X o in the present model. Taking the SU (3) breaking corrections into account shows that the predicted mass difference in the SU (3) Skyrme model o f pseudoscalar mesons is much too small. Elsewhere [ 10 ] it has been shown how this problem can be solved by including "short distance" effects like vector mesons or explicit quarks in the picture.

As another application of formula ( 1 1 ) let us consider the symmetry breaking corrections to the neutron beta- decay constant g~ ( = 1.25 experimentally). This does not seem to have been previously discussed in the frame- work of the SU (3) Skyrme model. At the SU (2) level, the usual choice [ 1 1 ] of F~ and e leads to a poor predic- tion g~ ~ 0.6. What changes when we go to SU (3)? First note that the third component of the isovector, axial- vector current operator is proportional to D33 (A). The matrix element of D33 (A) between proton states is - with SU (2) Skyrme wavefunctions but only - ~o with pure octet SU (3) wavefunctions. Thus, if symmetry breaking corrections are not included and if the same values of F~ and e are used one has

g j ( S U ( 3 ) ) = ~ g 4 ( S U ( 2 ) ) . (16)

This makes the prediction even worse. It is interesting that the effect of symmetry breaking correction is to partially restore the damage. Taking matrix elements of D33 (A) between the corrected proton states in ( 11 gives instead of ( 16 )

gA(SU(3 ) ) = ~0 [ 1 + 7 f l 2 ( ~ + 1o ~-~)]gA(SV(2)) (17)

For 7fl 2 = 3.33 this represents an increase to gA (SU ( 3 ) ) = 0.82gA (SU (2) ). This result may be pictured physi- cally as being due to a reduction in the "strangeness content" of the nucleon due to the symmetry breaking corrections, the "strangeness content" being the culprit for (16). As the symmetry breaking coefficient 7 in- creases gA (SU (3) ) gets closer to ga (SU (2) ). Of course we cannot trust the result ( 17 ) for values of 7fl 2 >/10. One might expect that g/, (SU (3) ) ~gA (SU (2) ) as 7--* ~ . One sees this by writing the original chiral lagrangian in terms of the rt, K and I1 fields. When 7 - * ~ the K and rl fields get infinitely heavy and cannot be excited; the model then reduces to the SU (2) model containing just the pions. We have checked this point by evaluating the matrix element of D33 (A) in the exact proton states gotten using the differential equation of Yabu and Ando [5]. For example, gA(SU (3) )/gA (SU (2) ) =0 .87 and 0.97 for 7fl2= 10 and 200, respectively.

In light of the above it is reasonable to expect that all physical quantities should reduce to the SU (2) Skyrme model values as 7--,oo. Yabu and Ando [5] observe that this does not hold for the energy eigenvalues due essentially to the linearity in 7 of the last term in the hamiltonian ( 1 ). They find that a proper SU(2 ) reduction in the limit is obtained if one subtracts from the energy for each state the energy eigenvalue Eo of the state corresponding to the SU ( 3 ) singlet eigenstate of the free hamiltonian, Do(~*ooo (A). It would be interesting if one could find an a priori reason for this particular subtraction. In any event it noticeably improves the fit to exper- iment. We remark that the subtraction energy can be evaluated in perturbation theory as

2f12Eo = 7fl 2 [ 1 - ~ , f 1 2 1 - - 3 ~ ( ~ ] ~ 2 ) 2 ] "~ - . . . .

Yabu and Ando [ 5 ] have produced several overall fits. These still suffer from the drawback that F= and mK are not quite fixed at their experimental values. I f one wishes to do this it requires taking e roughly in the neighborhood of 3 to explain the mass splittings. But this would [ see (4) ] increase Mc~ significantly. One might consider [ 12 ] an arbitrary energy subtraction to overcome this. An amusing feature of such a fit would be that since g, scales as 1/e 2 it would come out in the correct range ~ 1.25. Furthermore the electromagnetic charge radii would remain reasonable. Actually it may be premature to insist on a perfect fit for this model since other effects like the existence of vector and scalar mesons, cranking, etc. have not been included.

The main conclusion of this paper is that perturbation theory carried out to next to leading order can repro-

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Volume 224, number 1,2 PHYSICS LETTERS B 22 June 1989

duce the exact results o f Yabu and A n d o for the SU (3) Skyrme mode l o f pseudoscalars. The per turba t ion m e t h o d

enables one to deve lop some physical in tu i t ion about the model . In par t icular , the i m p o r t a n t role o f SU (3) 10

and 27 con ten t for the " o c t e t " baryons has been discussed. An amus ing po in t which emerges is that , in spite o f

the fact that first o rde r pe r tu rba t ion theory is inadequa te , the G e l l - M a n n - O k u b o mass fo rmula never the less is app rox ima te ly satisfied.

We would like to thank A.P. Ba lachandran , R. J o h n s o n and U. MeiBner for helpful discussions. Th is work

was suppor ted by the U S D e p a r t m e n t o f Energy unde r cont rac t No. D E - F G 0 2 - 8 5 E R 4 0 2 3 1 and in part by the Deu t sche Forschungsgemeinschaf t .

Note added. Afte r this le t ter was submi t ted , we lea rned that Prasza lowicz [ 13 ] has m e n t i o n e d the results o f

calcula t ing second order cor rec t ions to the ba ryon masses and magne t i c m o m e n t s for a cer ta in choice o f parameters .

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