Strange things in the proton?

Volume 209, number 2,3 PHYSICS LETTERS B 4 August 1988

S T R A N G E T H I N G S IN T H E P R O T O N ?

Judi th A. M c G O V E R N and Michael C. BIRSE Department of Theoretical Physics, University of Manchester, Manchester M I 3 9PL, UK

Received 28 March 1988; revised manuscript received 26 May 1988

We use the RPA to calculate the energy of strange baryons in a three-flavour version of the chiral quark-meson model. For realistic symmetry breaking this energy is found to depend non-linearly on the strength of the SU (3)-breaking term in the hamiltonian. This supports Jaffe's contention that a large nN sigma commutator need not imply a large strange-quark content in the proton.

The pr incipal measure o f chi ra l -symmetry breaking in baryons is X~N, the p ion -nuc l eon sigma commuta to r "~. It can be def ined as the double commuta to r o f the hami l ton ian with the axial isospin charges:

S~N=-~ ~ (P l [Q~, [ Q ~ , H ] ] I p ) • (1) i

In QCD, where chiral symmet ry breaking is broken by quark mass terms, this reduces to

s~N=m(pl~u+adlp), (2)

where r~ is the average o f the up and down quark masses. While not directly observable, a value for --Y~N can be found by extrapolat ing nN scattering to the Cheng-Dashen poin t [2] . The commonly quoted value is --Y~N ~ 55 MeV [ 1,3 ]. An al ternat ive es t imate can be obta ined by relating it to the matr ix e lement of the SU ( 3 ) -breaking term in the hami l ton ian [ 4 ]:

1 + x f 2 2 " ~ n N = - - 2 2 ( p l H s B ( S U ( 3 ) ) I p ) •

2 ( m ~ . f K / m j ~ - - l )

(3)

The quant i ty 2 is the rat io of f lavour-singlet to octet quark condensates in the proton, x / 2 ( f i u + d d + gs) / ( •u + dd - 2gs ) . The matr ix e lement can be de- t e rmined f rom the mass split t ings of the baryon octet using first-order per turbat ion theory and the Wigner - Eckart theorem, giving

~ See ref. [ 1 ] for a review.

0370-2693 /88 /$ 03.50 © Elsevier Science Publishers B.V. ( Nor th -Hol land Physics Publishing Divis ion )

( p l H s B ( S U ( 3 ) ) [P) = M A - M _ - . (4)

I f one assumes that the strange-quark content o f the pro ton is small, so that 2-~ x/2, then this suggests that X~N should be a round 25 MeV. Recently it has been suggested [ 5 ] that the measured value is a signal for a substant ial ~s content in the proton, in viola t ion of the OZI rule. This has lead to speculat ion about the possibili ty of kaon condensat ion at densities only two or three t imes those found in ord inary nuclei [6,7 ].

On the basis o f chiral-bag-model results [ 8 ], Jaffe has challenged the analysis leading to these surpris- ing results, point ing out that mass splitt ings of the octet baryons may not be reliably es t imated from first- order per turba t ion theory [9] . This was suppor ted by a calculat ion with Bernard and Meissner [10] o f quark mixing in a version o f the Nambu- Jona -Las i - nio model. Here we use the chiral quark-meson model [ 11 ] to examine the range of val idi ty o f f i rs t-order per turba t ion theory for the octet baryons ~2.

To study strangeness we use a three-f lavour ver-

~2 It is worth noting that the "experimental" value of 55 MeV is unlikely to be correct. The extrapolation to the Cheng-Dashen point has a long "lever arm" which makes the result very sen- sitive to any systematic errors in the data [ 12]. Banerjee and Cammarata [ 12 ] have been able to fit the s-wave nN scattering data directly using a Chew-Low theory with X~N -~ 25 MeV. The disagreement between "'experiment" and "theory" may, in the end, prove to be illusory. However, here we are inter- ested in the question of whether the assumptions leading from a large sigma commutator to a large strange-quark content in the proton are valid.

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sion of the model [13]. This is based on an SU (3) L × SU (3) R linear a model [ 14 ]. It involves quarks coupled to eighteen meson fields. At a fundamental level, the mesons should be described as clq bound states, and so the phenomenological meson fields play the role of scalar and pseudoscalar quark bi-linear operators. We denote the singlet and octet scalar fields by a and ~., and the corresponding pseudoscalar fields by r /and ~.. The model hamiltonian has the form

H = f d3r(gt*{

- g f l [ x / ~ ( a + it/y, ) + 2 " ( ~ + i ~ ,y s ) ] }~u

+ ½ ( ~ ) : + ½ (~,,)2 + ½ (~ . )2+ ½ (n~.)~ + ½ (v~)~

+ ½ (v~)~+ ½ (v~ )2+ ½ (vo. Y - u) , (5)

where n~ is the conjugate momentum to a and so on, and 2 ~ are the usual Gell-Mann matrices. The meson interaction potential can be written in the form

U~.-.m- ~/~2( ~Tr [MtM] - u~') 2

+ ¼x2{_~Tr[ (M'M) 2 ] - ( -~Tr [M*M] )2}

+7[de t M + d e t M*] +coa+c8~8, (6)

where for convenience we define

M = a + i r / + x / ~ ( ~ + i 0 , ) 2 ~ .

The term involving det M ensures that there is no axial U( 1 )A symmetry, and gives the 1]' meson a large mass. The parameter fitting is slightly more compli- cated in the presence o f this term but the results are very similar to those obtained previously [ 11,13 ].

For our present purposes, the important terms in H are those which break chiral symmetry [ 1,15 ]. I f we choose to fit the pion and kaon masses and decay constants, then by PCAC the coefficients of the symmetry-breaking terms are

Co = ( 1/. ,f6) (2m~FK +m2F~),

Cs = (2 /xf~) ( 2 2 m~F~--mKFK) . (7)

It is possible to choose the other parameters to give "ideal" mixing between the a and ~8 fields. In this case the mass eigenstates of the scalar fields are the combination which is coupled only to non-strange quarks, a(2)=~/~[a+ (1 /x /2)~s] , and the one cou-

pled only to strange quarks, (=x~-~ ( a -x /2~8 ) . The mean-field equations then have solutions which are just embeddings o f solutions to the SU (2) model. As usual we write the soliton fields in the form [ 11 ]

e x p ( - i ~ t ) f G(r) qo= x / ~ ~itx.iF(r)JZh,

1 ~h ~--- ~ (XU$--Xdt) ,

0"(2) = O ' t2 ) ( ? ' ) , q i = / h ( r ) . (8)

In particular there are hedgehog solitons in which the ( f ie ld remains at its vacuum value. Since the mean at2~ and ( fields stand in for the non-strange and strange quark condensates which would be present in a more fundamental description, the model shows no indication of a large ~s content in non-strange baryons. The exact vanishing of the scalar field corresponding to gs holds only if the scalar mesons are ideally mixed. For other sets of parameters which give reasonable masses, we find that ( g s ) / ( au + dd + ~s) is at most _+ 0.05, as opposed to about 0.2 obtained by Donoghue and Nappi [ 5 ]. (Note that both signs are possible for the strange condensate in our model. ) These parameter sets are ones for which the two scalars are quite strongly mixed and have masses in the range 700-1500 MeV. A large gs content is obtained only for parameter sets where one scalar field (mainly singlet) has a low mass ( < 500 MeV).

The sigma commutator in this model is propor- tional to the volume integral of a~2),

X,~N=F~m 2 f d3r(a~2) +F~) , (9)

and is found to be 92 MeV [ 11 ]. This is rather larger than any experimental value, although the inclusion of vector mesons does improve matters [ 16].

We have calculated the energy splitting between baryons of different strangeness using the random- phase approximation (RPA) [17]. This involves solving the linearised equations of motion for small- amplitude oscillations o f the meson fields coupled to particle-hole excitations o f the quarks. We look for bound states with S = 1, grand spin ~3 G = ½ and even parity, built on the hedgehog ground state. This gives an average of the energy differences between Z, A and N (as well as Z* and A). From this we can get only

~3 Grand spin denotes the vector sum of spin and isospin [ 18 ].

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one of the octet reduced matrix elements for the split- tings of the baryon octet. The other octet matrix element, which splits the Z and A, as well as a possible 27 could be obtained by spin and isospin cranking [ 18 ] of the RPA state. Such calculations have been done in the Skyrme model [ 19 ]. They find good agreement with the observed baryon spectrum: the octet matrix element given by the RPA does give the largest contribution to the SU (3) breaking, and the 27 is very small. We expect that our model would give similar results.

Due to strangeness conservation, the RPA equations decouple for particle-hole and hole-particle amplitudes [ 17 ], leaving a simpler equation. In terms of the fields and their conjugate momenta this equation can be symbolically written:

Hc,,q H~¢,, Ho.o, 0 II~o.q H~,, H~o , 0

0 0 6~ ~Tt,,,] 0 0 0 6ab] \nab /

=09 7to. , (10)

where He.o,, etc. are the second derivatives of H with respect to the fields. Details of the equations will be given elsewhere [ 20 ]. The isospin doublet of anti- kaon fields (which have S= - 1 ) has the form

( _ _ ~ 0 ) _ ( _ _ 06 +i~7 (04 +iq~5)/

1 ( - z ) ( l l a ) = 0 ( r ) e x p ( - i 0 9 t ) - ( x+ iy ) '

which has G= ½ and MG= + ½. The eigenstates of charge for the ~ mesons are defined similarly, and they have the form

• 1 ~ ° ) = ~ ( r , e x p ( - 1 0 9 t ) ( 0 ) ( l i b ) (_~_

The corresponding strange-quark spinor is

£ ( A ( r ) ) qs= e x p [ - i ( e + 0 9 ) t ] gsr • ( l l c )

\ i~ . /B( r )

To solve these equations we expand the fields in a Bessel-function basis and diagonalise the RPA matrix. For realistic symmetry breaking, we find a bound excitation with 09 = 315 MeV (compared to an experimental value of ~ 18 5 MeV).

In this model we take 2 in eq. (3) to be the ratio of f d 3 x ( g - a v ) to f d3x(~8-~v), which is exactly xf2 for ideal mixing. Using this and "~N = 92 MeV we get (P IHsn(SU(3) ) [ p ) = 8 4 0 MeV. This is far larger than the RPA energy, indicating that the first-order perturbation theory, eq. (4), is not valid and sub- stantial non-linearities are present.

To see where these non-linearities become important we vary c8, the coefficient of the SU (3)-breaking term in H. In the symmetric case (Ca=0) we choose Co such that the pion mass and decay constant have their experimental values; the kaon and pion are de- generate. At the same time as we vary c8 we adjust Co by ~Co = - x / ~ c 8 , so that the pion mass and decay constant are unchanged while the kaon mass and decay constant increase. Because of the embedding property of the solitons mentioned above, this pro- cedure changes only the vacuum expectation values of the scalar fields, from av = - _3x~F~, ~v = 0 to O'v=-- I ~ ( F ~ + 2 F K ) , ev=2/,f/3(FK-F=). The deviations of the fields from their vacuum expectation values are unchanged.

For unbroken SU (3) the degeneracy of the soliton means that the RPA equations have a zero mode; the radial forms of the fields in this mode are given by [13]

(A,B,~, ¢) = ( (1 /v /2 )G, (1/~/2)F, - x / 3 Cs, h ) , (12)

in terms of the soliton fields, (8); the conjugate momenta are equal to zero. As the symmetry breaking is turned on, this mode evolves into the lowest-energy RPA mode for strange excitations.

If we write (10) in the short-hand form H"x=0911x, then for unbroken symmetry we have Hgxo = 0. The change in the mode as the symmetry breaking is turned on, 6x, satisfies the equation

GH"xo + H~ax= 0911(Xo + ax) , (13)

where 6H" is the change in the RPA matrix when the symmetry breaking is turned on. To first order in 6H" the energy 09 is given by

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x; aH" xo =COX; nXo . (14)

The leading term of the LHS of (14) is propor- tional to the sigma commutator. To see this, note that the zero mode can be produced by acting on the hedgehog with A - =A 6 - iA 7. Hence Xo can be written

/ [A- , qo] k IRA-, 8-¢vl /

xo= t ). (15,

Once the symmetry is broken, ¢8 can have a non-zero vacuum expectation value, but the form of the zero mode is still given by rotating only the localised part of the field, {8- {v. The quantity x~ H ")co is therefore a sum of terms like [ q~, A + ] Hq,q [A - , qo ], giving

x ; H " x o = ( [ [ A - , H ] , A + ] ) , (16)

where the A +- are to be understood as acting only on the localised fields.

The commutators pick out only the parts of H which change as the SU(3) symmetry breaking is turned on. There are two of these: one comprises the additional terms c8~8 + 8Cog, and the other comes from the changes in the vacuum expectation values. Since the hedgehog is a solution of the mean-field equations, the first order changes in H due to these van- ish. Thus the change in H is, to second order,

1 ( ~2H 792H 8H=cs~s + 8coa+ ~ k-O~a2 (Say)2+ ~ (8~v) 2

O2H ) + ~ (SavS~v) • (17)

Evaluating (16) in the mean-field approximation, we get

< [ [ A - , H I , A + ] )

f 18).2js(m) = -3c8 (~s -~v ) d3r+ 4 C2' (18) mo

where m 2 =322f 2 +m~ is the mass of the 0 at zero symmetry breaking. The quantity o ~ m) is the mesonic contribution to the moment of inertia for strange rotations [ 1 3 ]

~m =n I- r2dr( h2+ 3~28) • (19) d

The RHS of eq. (14) is

oJ[q~, A + ] [A- , qo] = 3o). (20)

Hence, to first order in c8, the RPA energy is

& N ¢o= ( p ] H s B ( S U ( 3 ) ) ]p ) = - , /~ f~m 2 Cs, (21)

which agrees exactly with the perturbative results (3) and (4).

For non-linearities, we must work to second order in c8. As well as keeping both terms in eq. ( 18 ), we need to use second-order perturbation theory. The energy can be found from

x~ 8H"xo + 8x* 8H"xo +x~ 8H" 8x+ 8x*H {~ 8x

= w(x~ ilXo + 8X*ilXo +x~ ilSX), (22)

where, from (13) and (14), the first-order changes in the fields satisfy

H~ax= - aH"xo + (x~ aH"xo) 11Xo. (23)

The form of this is very similar to the cranking equation [13], but, since the right-hand side of (23) is orthogonal to Xo, the solution must be finite.

The equations for the conjugate momenta decouple from those for the changes in the quark and meson fields. From (10) the former are just

6x ,<=(Ss~ ']=w(~) (24) - k a = ~ )

From the similarity to the cranking equations [ 13 ], the changes in the radial forms of the fields might be expected to be small, as they are in cranking. Even for realistic symmetry breaking we find numerically that this is indeed the case. The dominant second- order contribution to the energy comes from the conjugate momenta; this is easily found, using (23), to be

t ,, t 8x=H o ~)x==SXrtllx o = 4 0 ) 2 J ~ m) (25)

(since 8x~ 8H'xo = 0). Neglecting the contribution from the changes in the

radial fields, as well as higher-order terms, we write eq. (22) as

S~N 122aJ~ ") (J)'~-8j~m)o)2-~ -- ~ C 8 " J I - C 2 (26) x/ 3 f~rn ~ m 4o •

Since it contains a factor of,~-2, the second term on

166


the RHS is also small. Even for realistic symmetry 8~(m),,~2 is the dominan t second-order breaking ~Js w

term. Thus first-order per turbat ion theory breaks 8 ( ~ ( m ) )l -- 1 ,.~ 2 0 0 MeV. This down when 09 is of order 3 ~--s ,

is reasonable, since in the symmetric l imit ~ deter- mines the mass splitting between the octet and the

multiplets which are coupled with it by the SU (3)

breaking interaction, for example the spin - ½10" and 27, the next lowest multiplets with members of the

same spin and flavour quan tum numbers as the oc-

tet. When the splitting between S = 0 and S = - 1 states is comparable to the splitting between the mul-

tiplets, significant mixing will occur, invalidating first-

order perturbat ion theory. In summary: we have found that, although it gives

a large value for X~N, the chiral qua rk -meson does not predict any unusual modificat ion of the strange-

quark condensate in the proton. These features are

consistent because, for realistic SU(3 ) symmetry

breaking, first-order perturbation theory breaks down for the strange baryon masses. Jaffe has already found

similar results in a chiral bag model [ 8,9 ]. We expect

such effects to occur in any model where the nucleon

has a significant mesonic component . In particular

we expect them in the Skyrme model, where the RPA-

type approach gives a good description of strange baryons [ 19 ]. Since the moment of inertia for strange

rotations [21 ] is very similar to that in our model,

we do not expect first-order perturbat ion theory to be valid for realistic symmetry breaking. This would

contradict the large condensate found in ref. [ 5 ], us-

ing a perturbative approach to the Skyrme model.

Since completing this work we have learned that

such a calculation has been done [22 ]. However, the approach is complementary to ours in terms of the

quanti t ies studied; Blaizot et al. [ 22 ] have looked at

the ground-state correlations implied by the RPA, and have calculated their contr ibut ions to the nucleon en-

ergy and (gs ) . At least in a one-mode approximation, they find that these contributions are very small.

Since the forms of the fields in our model are very similar to those of the skyrmion, we do not expect ground-state correlations to alter our results significantly.

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We are grateful to M.K. Banerjee, W. Broniowski, G.E. Brown and R.L. Jaffe for useful discussions. J.A. McG. acknowledges support from the SERC.

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Strange things in the proton?