Volume 48B, number 5 PHYSICS LETTERS 4 March 1974

IS T H E Q U A R K M A S S AS S M A L L AS 5 M e V ?

H. LEUTWYLER Institut fiir theoretisehe Physik, 3012 Bern, Switzerland

Received 27 December 1973

The constituent symmetry group U(6) is incorporated in the null plane quark model of mesons. We obtain the value m u = 5.4 MeV, and the estimate m s = 125 - 160 MeV for the masses of the non-strange and the strange quarks respectively. Some checks of the model in terms of meson mass formulas are given.

The purpose of this paper is to show that the null plane quark model described recently [1] can be extended to incorporate the constituent symmetry group U(6). The basic quantities of the model are the quark wave functions defined by the overlap of the state I M,p) which describes a meson of momentum p with the quark-antiquark sector of the quark Fock space

(Oiq~.O,~ q+(x)alM;p) =

i exp { - 5 p(x+y)) ~afl(~_y;p, M).

The points x and y are taken on the same null plane x 0 _ x 3 =yO y3 = 0 and the "good" components q+(x) of the quark field q(x) are obtained through the projection q+ = 21/4Gq with G = ½(1 +3'0")'3). The symbol ~ stands for the three coordinates within the null plane x = (x +, x 1, x 2) with x -+ = (x 0 + x 3 ) / ~ .

In order to discuss the symmetry properties of the wave functions under U(6) it is convenient to work in momentum space:

q+(x) =

(21r) -3/2 f d3k(Q(k)exp(-~x)+b+(k)exp(~x)} k ~ 0

where k = (k+, k l , k2), k x = k+x + + klxl + k2 x2. Our constituent symmetry scheme is a variant of the models proposed by Melosh, de Alwis and Stern and others [2]. We make the following two assumptions:

1) We postulate that there is a constitutent sym- metry operator Wp(S) which transforms the 36-plet of pseudoscalar affd vector mesons among themselves in the standard fashion [3]

W p(S)IM, E) = I SMS +, p>.

Here both S and M are 6 X 6 matrices; S is unitary, M is hermitean. The correspondence between the individual members of the multiplet and the associated hermitean matrix M is the following:

I M, p> = ¼ ~ tr [~,~M] I V, p, 0> V

+¼ ~ ~ tr[(?,ver(h)~/r)+MllV, pp, h) V h=+-i

_L ~ t r [(Xe~5)+M] le, p). 4 p ~

In this equation M is form~!y considered as a 12 × 12 matrix which vahishes except on the subspace selected by the projector G: GM = MG = M. The vector meson states IV, p, h) correspond to helicity h = -+ 1,0; it is important for the consistency of our scheme that these states are boosted from the rest frame by means of Lorentz transformations which belong to the stability group of the null plane - the Wigner rotation which connects these states with the conventional helivity states depends on the mass of the meson and therefore interferes with the sylmmetry group. The SU(3)-matrices for the pairs (r/, *7') and (w, ¢) involve two mixing angles, 0p and 0 V. We use the convention ?~n = diag(cosOp, cos0p, - x / ~ s i n 0 p ) , ?'n' = diag(sin0p, sin 0p, X/-2 cos0p) and analogously for (~o,~).

2) We assume that the constituent symmetry also transforms the "current" quarks among themselves:

We(S)* Q(k) ~ Wp(S ) = W°¢(S, k , p ) Q(k) t~ .

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Of course, if we would require the matrix W ~ to coincide with S we would effectively be identifying current and constituent quarks. We instead only use the fact that non-trivial 6 × 6-representations of U(6) are equivalent either to S or to S*. Since only the first case is consistent with the known transformation properties under charge or baryon number gauge transformations we must have

W(S, k,p) = T(_k, p ) ST(k, p ) t

The matrix T connects current and constituent quarks:

c(k, p)= T(k, p)tO(O.

Note that the constituent quark annihilation operator c(k, p) not only depends on the momentum k of the quark but also on the momentum p of the physical state in which this quark occurs.

Isospin invariance requires T to be diagonal in SU(3)-space; furthermore, the transformation matrices for up- and down-quarks must be the same. Lorentz invariance, time reversal and parity constrain the form of T further [4] :

Tq(k, p) = exp(-iKr~/r/3q) ; q = u ,d , s

where K r = kr-prk+/p+. The only freedom left in the transformation from current to constituent quarks is contained in the two spin-orbit mixing angles/3 u =/3 d and/3 s which depend only on the longitudinal momen- tum fraction [ = k+/p+ of the quark and on the square K 2 of its transverse momentum,/3q =/3q(~, ~2)$.

Invariance under charge conjugation fixes the transformation from current to constituent antiquarks

d(k, p) = b(k)T(k , p)? .

An immediate consequence of these assumptions is that the quark wave functions for pseudoscalar and vector involve only two invariant form factors f , g:

(01b(k')~ Q(k)~iM, p) = (27r) a/~ 6 3 ( p - k - k ' )

X {T(k, p) (Mf+¼Gg trM) T(k ' , p )}aa .

It is straightforward to evaluate the corresponding wave functions for the individual members of the multiplet. Lorentz invariance implies that in the frame

$ We see no justification in our framework for the additional assumptions put forward by Melosh, [,2], who abstracts the momentum dependence of the spin-orbit mixing angles from the free quark model.

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Pl = P2 = 0 the momentum space wave functions

~0(k, p, M) =

(2rr) -3/2 fd3z exp { i ( k - ½ p ) z } ~(z, p, M)

have the general form

~0(k, p) = iG {~0X-kr'yrtp2} ~'5

for the pseudoscalar mesons and

~ k , p, k = + 1) = G "[er'~r~o3+erkr~o4

+ (krk s - ~ 6 rsk2) er'gstp5 + ~ erk s [~,r, 7s] ~06 }

for the vector meson respectively. The invariant wave functions ~01 ..... ~06 depend only on ~ and ~2. U(6)- symmetry allows us to express these quantities in terms of fife invariant form fac to r s f and g and the spin- orbits mixing angles/3 u ,/3 s e.g.

" =/cOSOu+Yu) ~01

k¢~ = fsin03. + 13' u) U

2 ' with/3 u = 13u(~, k ),/3 u = 13u(1 - ~ , k2). From these explicit expressions one obtains a set of relations which are independent of the mixing angles 13 u and /3s, e.g.

- -4 , (1)

Analogous relations hold between the other members of the nonet.

3) Our third hypothesis concerns the structure of the terms in the Hamiltonian which break SU(3) × SU(3). We assume that this symmetry is broken only by the quark masses. More precisely, we assume that the bflocal field q(x)~0') satisfies the "equation of mot ion" [5]

{ - i . ' ~au+ M}"q(x) i~(Y)ix=y = ~'5O(x) (2)

at x --- y . Matrix kl is the quark mass matr ix/d --- diag(mu, mu, ms). This condition is satisfied in the soft gluon model [5] provided the bflocal field q(x)q-(y) includes a phase factor of the form

Volume 48B, number 5 PHYSICS LETTERS 4 March 1974

T expig f Bu dzU. In this model, the field ¢(x) is proportional to the square of the gluon field strength, GuvG~V. For our purpose, the specific properties of the field ~ are irrelevant as long as it is pseudoscalar and neutral, such that all vacuum to meson matri~ elements of it vanish except for (0l ¢(x)l r/) and (01 ¢(x)l ~7'). The relation (2) in particular implies that the divergences of vector and axial currents are proportional to quark mass terms - except for the SU(3)-singlet axial current which has an anomaly. We emphasize that the quark masses enter in our scheme not as inertial masses of free quarks, but as symmetry breaking parameters. In particular the mass m u = m d associated with the up- and down-quarks measures the breaking of SU(2) × SU(2), whereas the difference m s - m u is the symmetry brekking parameter of SU(3).

Note that these assumptions about breaking of SU(3) × SU(3) are perfectly consistent with an un- broken constituent symmetry U(6). The constituent symmetry could in principle be valid exactly despite the fact that the group SU(3) × SU(3) generated by the currents is broken by the nonvanishing meson masses. As was shown in ref. [1] the "equation o f motion" implies a set of sum rules for the meson wave functions, e.g.

/ ' d3k 2 2 ~r ~ {(mTr- 4mu) ~o 1 -- 6muk2tp~} = 0 d

(3) fd3k { mo~ l - 2mu~3- 2k2~6} = O .

The integrals over the wave functions ~o~, ~0~ are known - they determine the couplings of the weak and the electromagnetic currents to the pseudoscalar and vector mesons respectively (F . = f,~/x/-}):

2(2rt) -3'2 fd~d2k~:l = F = 0 . 6 8 m ,

2(2~) -~'2 f d ~ d 2 k ~ - - F -- (1.04 -+ 0.05)m .

Using the symmetry relations (1) in (3) we obtain the mass formula

2 2 3mu(mo-mu)/7o = ( m l r - m u ) F ~

which has a very small and a very large solution for m u. We reject the large solution m u ~ mp because we require that the SU(2) X SU(2) - symmetry limit m u ~ 0 entails mTr ~ 0, mp ~ 0. The small solution is, approximately

m 2 F 7r ~ - 5 . 4 M e V . ( 4 )

mu - 3m F p p

4) In order to simplify the analysis of the remaining sum rules we make use of the observation that the pseudoscalar coupling constants FTr , F K show very little SU(3)-breaking, much less than the meson masses. This indicates that the mixing anlges 3u and 3s are not very different. In the following we neglect the difference flu - 3 s and assume that the meson wave functions are SU(3)-symmetric to a good approxima- tion (10-20%). (A more detailed analysis which does not make use of this assumption is given in ref. [4] .) As an immediate consequence of ~07 = ~0 K we get the relation

m 2 - a m m 2 - ( m u + m s )2 = (5)

2m u (mu+m s)

which leads to m s = 125 MeV, ms/m u = 23 in agreement with the strong PCAC picture o f Gell-Mann et al . [6].

The ~um rules involving the wave functions of co and ~0 imply that the mixing angle 0 V is proportional to (mto-mp)/(m~-mp). Since this ratio is very small, we have 0 V = 0 to a good approximation, i.e. the w is made out of up- and down-quarks only, the ~o con- tains only strange quarks (as far as the exclusive pro- babilities for Finding nothing but two quarks are con- cerned). The same sum rules also show that the singlet form factor g is proportional to into - mp. In the same approximation we therefore get the well-known SU(3)-results based on 0 v = 0: Fp = 3Fro = - 3 F J x / 2 = FK*. These relations are satisfied by the data on Fro, F~o to within the same accuracy as F~r = FK: 10-15%. In the approximation 3u = 3s, g = 0v = 0 the sum rules for ~0 and K* may be written

m - m p = (ms -mu) (1 + F J F p ) , (6)

mK. -- m o = ~l, ms--mu)(1 +F~/Fo) •

These sum rules imply, in particular, equal spacing between m o, inK. and m~o - this rule is satisfied by the experimental values. Moreover, the spacing is determined by the quark masses. Inserting the values found above we get inK. = 864 MeV, m~ = 963 MeV to be compared with the experimental values inK. = 892 MeV, m~o = 1019 MeV. The values of the quark

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Volume 48B, number 5 PHYSICS LETTERS 4 March 1974

masses found above thus indeed give the correc t

spacing to within 20% $~:.

Final ly, the sum rules for the r / and ~ ' wave func-

tions lead to a de te rmina t ion o f the pseudoscalar

mixing angle 0p which comes out to be close to 45 ° :~:~.

At the same t ime these sum rules lead to a nonl inear

mass formula involving the pseudoscalar masses and

the parameters m u and m s. This formula can be

solved for the mass o f the B' wi th the result mr/, =

1270 MeV which is larger by 30% than the exper iment-

al value mr/, = 957 MeV. It should be not iced that in

contrast to the mass formulas for K* and ¢ the non-

linear mass formula for the rl' depends rather sensitively

on the approximat ions used here$:~:~; a more detailed

discussion which does not neglect the SU(3)-spli t t ing

in the coupling constants F~r, F K etc. m a y be found in

ref. [4] , where also some est imates o f the difference

in the shape o f the lr and K wave funct ions are given.

~t~ If one uses the mass formulea (6) instead of (5) to deter- mine the strange quark mass one finds m s = 160 MeV. This should give an idea of the uncertainty in the determination of the strange quark mass due to SU(3) breaking effects in the coupling constants. For a more detailed estimate of this uncertainty see ref. [4].

:~:~t If one assumes the r/to be the eigth member of the octet, the r/' to represent the singlet, i.e. tg0p = ~ then one obtains the mass formula 3rn~ + m~ - 4m K:= 4 (m s - rnu )2, which leads to mrl = 584 MeV. In this limit the mass formula for the r/' however requires the 71' to be infinitely heavy.

It is interest ing to observe that our mode l predicts

a rather sizeable anomaly in the divergence o f the

SU(3)-singlet axial current . F r o m the s.um rules for

the ~ wave funct ions one obtains

cos0 ( r n 2 - m 2 ) F . (010u~ 'y~75q-2 i~m75qi r l ) = 3 p . . . . .

The applicat ion o f this mode l to the ba ryon wave

func t ion is under s tudy.

References

[1] H. Leutwyler, Phys. Lett. B48 (1974) 45. [2] H.J. Melosh, Enrico Fermi Institute preprint EFI 73/26

(1973); S.P. de Alwis and J. Stern, CERN preprint TH 1679 (1973); See ref. [3] for a review of related work.

[ 3 ] J. Weyers, Intern. Summer School on Particle interactions at very high energies, Louvain, August 1973, CERN preprint TH 1743 (1973).

[4] H. Leutwyler, Mesons in terms of quarks on a null plane, preprint Univ. Bern, Nov. 1973, to be published; Light Cone Physics and PCAC, Lectures gives at the Adriatic Summer Meeting, Rovinj, September 1973, to be published.

[5] H. Fritzsch and M. Gell-Mann, Proc. Intern. Conf. on Duality and symmetry in hadron physics (Weizmann Science Press, 1971); H. Fritzsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. B47 (1973) 365.

[6] M. Gell-Mann, R.J. Oakes and B. Renner, Phys. Rev. 175 (1968) 224. S.L. Glashow and S. Weinberg, Phys. Rev. Lett. 20 (1968) 224.

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