Volume 145B, number 1,2 PHYSICS LETrERS 13 September 1984

FLAVOUR-SINGLET MESONS IN SU(2) LATI'ICE GAUGE THEORY:

VIOLATION OF THE OZI RULE AND MIXING WITH GLUEBALL S T A T E S

M. FUKUGITA a, T. KANEKO b and A. UKAWA e, 1 a Research Institute for Fundamental Physics, Kyoto University, Kyoto, 606 Japan b National Laboratory for High Energy Physics, Tsukuba, Ibaraki, 305 Japan c Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo, 188 Japan

Received 14 June 1984

The OZI rule violation and the mixing with glueballs of the flavour-singlet mesons are studied in SU(2) lattice gauge theory. For the scalar the OZI rule violating component rapidly becomes comparable to the non-violating piece towards the light quark mass, while for the vector the former is strongly suppressed. A rapid increase of the correlation function be- tween the scalar meson and glueball are also observed for the light quark mass.

Study of flavour-singlet mesons provides us with a variety of interesting problems which are not met in the case of the non-singlet mesons: The propagator of flavour-singlet mesons receives a contribution from a disconnected quark loop which is associated with the OZI rule violating phenomenon [ 1 ]. Another interest- ing aspect is that the flavour singlet ~q state generally mixes with glueballs. These two aspects are related in that the mixing with glueballs might be the source of a contribution to the disconnected quark loop. Con- sideration of this mixing phenomenon is important not only to understand the light meson spectrum [2,3] but also to find realistic estimates of the glue- ball masses [4].

The two-loop (disconnected loop) contribution leads to the mixing between quark states with differ- ent flavours. Empirically we know that the magnitude of this mixing depends not only on the mass of quarks but also on the quantum numbers of the states. The mixing is small in the vector and tensor channel, while it contributes largely to the pseudoscalar and scalar channel. Understanding this phenomenon has been an important problem in hadron dynamics.

This problem should be attacked using the method of lattice gauge theories. There are, however, a num-

1 Present address: Institute of Physics, University of Tsukuba, Ibaraki, 305 Japan.

ber of technical difficulties which restrict the feasibili- ty of the study. One of the limitations stems from the large amount of computer time needed to evaluate the disconnected contribution in the ~q Green's function. This requires the quark propagator G(n, n') for a large number of pairs of sites n and n', as compared with the case of flavour nonsinglet mesons where one can fix n' at a particular value. Also the magnitude of two loop contribution is generally much smaller than the one- loop contribution and much better statistics is neces- sary for its evaluation.

The mixing between quark and glueball states may be studied through estimate of the Green's function (~q, Og) with Og the glueball operator. A technical limitation we meet here is that for the study of low- lying meson states with JP = O- and 1- , one needs for Og a complicated operator, numerical estimate of which is not easy because of its very small magnitude. For the 2 + state the glueball operator takes a simple form, but for the ~q sector one needs an extended operator such as qn Unuqn+~. This consideration indi- cates that the only channel feasible for study at pres- ent is 0 +

At this point, one should also note that the scaling behaviour of the glueball mass has been observed only for a limited range of ft. With the single plaquette ac- tion which we shall use, this range for SU(2) is/3 = 2 . 0 - 2.2, beyond which the weak-coupling contribu-

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Volume 145B, number 1,2 PHYSICS LETTERS 13 September 1984

tion dominates that completely masks the scaling be- haviour [4]. On the other hand, the string tension [5] and the p-meson mass [6] (the latter with the Wilson's quark action) start to scale at/3 ~- 2.2 [for the 0 + me- son, the onset of scaling might be somewhat earlier (/3 ~ 2.0) [6]]. Thus if one wants to study mixing within the scaling region, the value 13 = 2.2 seems to be the unique choice and examining the scaling behav- iour with larger values of 3 is practically not possible.

To make our computation within a reasonable amount of computer time we are forced to work on the lattice with a size as small as possible. For our val- ue 3 = 2.2 the effect of fake quark loops [7] disap- pears for the spatial lattice size L ~ 2.2~a ~ 3.7 [8]. We then fix our lattice size at 43 × 8. We employ the standard single plaquette action for the SU(2) gauge field and the Wilson action for quarks with the hop- ping parameter K. Periodic boundary conditions are imposed on both gauge and quark fields.

In this paper we study both the one-loop and two- loop Green's function for the pseudoscalar, vector and scalar mesons. Examination of the mixing be- tween the meson and glueball states is limited to the 0 + channel because of the reason noted above.

We denote the ~q operator by O~iq(n ) = V:lnPaqn (oL = PS, V, S) and the scalar glueball by the rotation- ally invariant combination OS(n) = Tr(Unxy + Uny z + Unzx)/~f3 with Uni ] the plaquette variable in the (i]) plane at site n. The Green's functions with zero spatial momenta, relevant for our study, are then

M/~(t) = L- ~ n~,n ' (O~(t + t', n) O~(t', n'))e ,

i = ~q, g , (1)

where the subscript c means exclusion of the vacuum from the intermediate states. In a full treatment of mixing, one of course needs to include the effect of vacuum quark loops. Nevertheless we adopt the quenched approximation [2], because of lack of prac- tical numerical techniques for computing the quark determinant. In this approximation one can integrate out the quark field in terms of the quark propagator G(n, n') in the background gauge field. Assuming the exact flavour symmetry, one can rewrite the expecta- tion value in (1 )more explicitly as

O g - ( . ') =

( 2 )

94

" ' " ' Og(n ) )c -- - ( T r P G(nn ) O~(n'))

+ (Tr F'~G(nn))(O~(n')), (3)

(O~q(n) O-~(n') )c = - (Tr V c, G (nn') P aG (n'n))

+ (Tr PaG(nn) Tr PC'G(n'n'))

- (Tr FC'G(nn))(Tr FC'G(n'n')), (4)

where (...) denotes the average over gauge configura- tions. In (4) the second line represents the contribu- tion of the diagram containing two disconnected quark loops. We shall denote this part as M~q,~q and the con-

. . a I , nected piece (the first hne) as M~l,~q. For n = n a term

1 . (Tr F~G(n,n))Z~nn , 4NfNe

should be added in (4). This term is the gauge invari- ant piece of the disconnected counterpart of the first term given by

( {paGia,jb(n, n'))uv)( { p°~GJb, ia(n'n)}vu)

where (i, j), (a, b) and (/a, v) are the flavour, colour and Dirac indices. This addition is necessary because the conventional definition of the hadron Green's function (4) does not respect the Grassmann feature of the quark wave function at n = n'. Without addition of this term the Green's function for the scalar ~q meson will be negative at n = n' [6]. At the strong coupling limit/3 = 0 in particular, the scalar meson Green's func- tion evaluated from (4) has the value -4NfNc6nn, in contradiction to its exact value zero. The additional term exactly cancels this Kronecker delta contribution• This term vanishes for the pseudoscalar and vector states at/3 = 0. We confirmed that this addition makes the scalar meson Green's function develope an expo- nential fall-off at a sufficiently large 3, 13 = 2.5 say. At 3 = 2.2 that concerns our study, however, the scalar meson Green's function exhibits a concave form for t = 0, 1, 2, which may be a remnant of the behaviour at strong coupling.

We now describe briefly our numerical procedure for estimating (2)-(4) . The heat bath algorithm was used to generate 10 independent sets of gauge configu- rations each containing 1400 sweeps• For the plaquette correlation (2) we used the entire set of configurations

Volume 145B, number 1,2 PHYSICS LETTERS 13 September 1984

except for the initial 50 sweeps in each set to avoid the relaxation effects. To estimate (3) and (4), we

selected 600-, 800-, 1000-, and 1200-th sweeps from each set. On these 40 configurations, we evaluated the quark propagator employing the incomplete Gauss conjugate-residual algorithm developed in ref. [9]. Compared with the relaxation or Gauss-Seidel proce- dure [2,6], this method turned out to be about 3 times faster in achieving the same accuracy. It has an additional advantage of converging for large values of K where the relaxation or Gauss-Seidel iteration di- verges. We have calculated G ( n , n ' ) for arbitrary site n

2 ~ n x >~ ny >f n z and for the 12 spatial sites satisfying >/ ' ' ' /> 0 at each time slice n~ = 0, 1, 2. With the periodicity and discrete rotational invariance on the spatial lattice of size 43, this allows us to estimate M~qq(t) and

M~q,~q(t) in (1) for the temporal separation t = 0, 1, 2. It is well known [4], that the plaquette correlation

M~g(t) becomes very small and it is difficult to have statistically credible data beyond t = 2. This is also the

• ~ ~ , ILl case with Mg,-qq and M~q,~q. For the scalar state a large subtraction of order uni ty is necessary to obtain the connected expectation value. With the statistics presently available this practically prohibits us from exploring the correlation function for t > 2. For this reason we did not estimate the ~q Green's function

beyond t = 2. We start our discussion with a consideration on

the magnitude of the two quark loop t e r m lV/~q,~q • . OL~ I

• relatave to the one loop p]eceM~q, qq. In fig. 1 we show them for the scalar, vector, and in fig. 2 the pseu- doscalar as a function of the hopping parameter at

1.0 0.8 0.6 0.4 0.2 10 I I I I I I I I I I

i0-1

i0-~.

10 I 0.12

(d) M s - q q,qq

o

o

o

1T

mq (GeV) 0.0

I I I I 0.16 K c K

0.4 0.2 1 1 I I

1.0 0.8 0.6 I 0 I I I I I

(b) M v

I Or 0

i0-~

iO-Z

1 0 - 3 I 0.12

0 0

0 0 0 0

T

t i i

I

I 0.14 0.14 0.16

m q (GeV) OD

I

I I K K

Fig. 1. (a) Scalar meson propagator at t = 1 as a function of K. The open and filled circles respectively represent one- and two- quark loop contributions. The vertical line marked K c represents the critical value of the hopping parameter at which the pseudo- scalar meson mass vanishes The quark mass at the top is estimated from the relation mq = (1/K - 1/Kc)/2a with a -1 = 0.68 GeV. The error bars shown are obtained by dividing the gauge configurations into two clusters in a variety of ways and comparing the averages obtained in the clusters The number of flavor N F = 2. (b) Vector meson propagator at t = 1. The meanings of symbols are the same as in (a).

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Volume 145B, number 1,2 PHYSICS LETTERS 13 September 1984

mq (GeV) 1.0 0.8 0.6 0.4 0.2 0.0

lO l l , l , l l , l l

O O O

O O

O

I o o I I

10-2

lI 1 io-~ t l

,o-, I -.t

4

iO I I I I I I I I ] QI2 O.14 0.16 K K

Fig. 2. PseudoscaIar meson propagator at t = 1. The meaning of K c and the quark mass estimate at the top is the same as in fig. 1.

the temporal separation t = 1. (NF = 2 is assumed.) In order to provide an idea of the physical mass scale in- volved, we also show in figs. 1 and 2 the quark mass defined by mq = a - l ( 1 / K - l IKe) /2 [2,3]. Here K c

0.177 is the critical value at/3 = 2.2 [6] of the hop- ping parameter, at which the pion mass vanishes, and the inverse lattice spacing a -1 = 0.68 GeV is taken from the Monte Carlo estimate AL/~/-o = 0.013 [5] with ~ = 0.40 GeV. Let us discuss first the case for the vector and scalar. A trend common to both is the fact that the two-loop contribution increases in mag- nitude relative to the one-loop piece as quark becomes lighter. However, there exists a significant difference between the scalar and the vector: For the scalar the two-loop contribution becomes comparable to the one-loop piece near Kc, while for the vector the two- loop piece is still quite small near K "" Kc (mq ~ 0). These features nicely reproduce our empirical know- ledge established in the phenomenology of hadrons.

(A similar trend is also seen for t = 0 and t = 2. The data for the latter suffers from large statistical errors, however.)

In order to extract the magnitude of OZI rule vio- lating amplitude, Fourier transformation of the Green's function in time is necessary, but the statistics of our data does not allow such an analysis. Let us use ~ = M-~lu,~q(t)/M-~t~q(t) at t = 1 as an estimate of the

~ T

OZI rule violation since it depends on t only weakly. We observe that ~V increases from +3.6 × 10 -3 at K~- O.12(rnq~ 1 GeV) to + ( 6 - 1 0 ) X 10-2 a t K ~ Kc(mq ~ 0). For the strange quark (mq ~ 0 .1-0 .15 GeV) we expect ~v ~ ( 3 - 6 ) × 10 -2. The analysis of q~-~o mixing phenomena has given the magnitude of mixing to be +0.07 [10]. We may also estimate the mixing for the charmed quark from the suppression of the decay ff-+hadrons to be of the order of ( 2 -3 )X 10 -3 (the sign cannot be determined). The agreement of our numbers and their signs with these empirical values seems impressive, though we should bear in mind the uncertainty in the extraction of numerical values in our analysis. For the scalar state ~s is significantly larger and it is ~0.1 a tKe ~ 0.12 (or mq ~ 1 GeV), implying a loose suppression of the OZI rule even for heavy quarks.

We see from fig. 1 that the propagator MS~q, gq(t) will be substantially modified by the two loop term f o r K 2 0 .15-0 .16 or mq ~ 0 .3 -0 .2 GeV. Calculating the ratio-ln[MSgq,~q(t = 2)/3~qq, qq(t = 1)] with and without the two-loop term, we find that the two-loop term increases the meson mass. The numerical frac- tion of increase is again a function of K; negligibly small for K ~ 0.12 and increasing to about 10% at K = 0 .15-0.16.

On the other hand the behaviour of the two-loop contribution for the pseudoscalar (fig. 2) seems 9uzzl- ing in that it is still small for K ~ O. 165, rather close to Kc, where one expects a large contribution of ~O(1) . The inclusion of two-loop term increases the pseudoscalar meson mass, but only by a small amount (~1% at most). One also observes that the increase of ~PS as a function of K is much more rapid in compari- son with the case of the scalar and vector. The pseu- doscalar meson is special in that it is the would-be Nambu-Golds tone boson of the U(1) chiral symme- merry which, however, is made massive by the topo- logically non-trivial gauge configurations. Of course this is not the case with the vector and scalar channel

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where we have obtained a good result. For the mo- ment we do not know whether the anomalous behav- ior in the pseudoscalar state be ascribed to the explicit chiral symmetry breaking due to the Wilson mass term as studied in some detail in ref. [11] or to an in- sufficient excitation of the topological charge on the small lattice [ 12]. (A sufficient excitation could give the correct n--r/mass splitting [13].)

Let us now turn our attention to the question of mixing between ~lq and glueball states. Fig. 3 shows our data for the transition propagatorM~g, q q ( t ) _ _ at t 0, 1, 2 with Nf = 2. The magnitude of mixing propa- gator increases exponentially with the hopping param- eter, and it becomes a substantial amount for light quark mass. An interesting question concerning the OZ1 rule violation is whether the intermediate state in the two-loop term of M~q,~q is dominated by the glue- balls or the multi-quark contributions [10]. In the

1.0 0.8 I 0 I I I

M S q

I

I° ' l t

iO-Z b

iO-3 I O.12

0.6 0.4 0.2 I I I I I I

I I I I 014 0.16

T : O

T : I

T=2

mq (GeV) 0.0

' 1 q

1

l

1

1

]

1

1 I I

K c K

Fig. 3. Mixing propagator between the scalar ~q and the scalar glueball operator O~a t t = 0, 1 and 2 as a funct ion of K. The meaning o f K c and the quark mass estimate at the top is the same as in fig. 1. The number o f flavor N F = 2.

PHYSICS LETTERS 13 September 1984

case the two-loop term h'fi~l,~q should be rough- former ly equal to (M~g,~q) 2 divided by the mass difference of

- - • I I the glueball and qq meson. Since both ] l ' f i ~ q and M~g, qq increase exponentially in K, this implies a rela-

• , I I t l o n b e t w e e n t h e slope Ot~q ~q ~ 2Otg,qq with/l ' f i~q,~q

exp(o~qq,~qK) and~g ,~q ~ exp(ag,~qK). Fitting the data at t = 1 by an exponential gives a-qq,~q/ag,~q 1.2, indicating that there is perhaps a substantial con- tribution to the intermediate states other than glue- balls [10] at least for a quark lighter than 1 GeV.

We see from figs. la and 3 that the mixing element M~g,~q between the glueball and the T:l~state becomes comparable to the diagonal element ~ q , ~ q for K 0.15. A sizable mixing naturally shifts the eigenvalues of the hamiltonian. These eigenvalues_may be ob- tained by diagonalizing the matrix MiT(t), i, f f f ~q, g, with t #= 0 and fLxing the normalization ofM~/(t = 0). I f Xi(t) (i = 1, 2) denotes the eigenvalue of Mij thus obtained, the masses mi are given by

mi(t ) = - t -1 In Xi(t), i = 1, 2 . (5)

This procedure may be applied to the Green's function at t = 0 and t = 1. The Green's function at t = 0, however, may still receive a strong coupling ef- fect at fl = 2.2 as discussed above. This effect (which reduces the value at t = 0 perhaps by a factor of two) puts a bias towards a small mass. To reduce effects caused by this, we use the data at t = 0 and t = 2, al- though our data at t = 2 suffers from large statistical uncertainties. We should emphasize therefore that the numerical values given below are indicative only. The result of our analysis is as follows. (i) The unperturbed glueball mass mg 0 is equal to 1.75. The unperturbed ~q meson mass decreases with increasing K, crossing the glueball value mg 0 --- 1.75 at K ~ 0 .14-0 .15 where mq ~ 0.4 GeV. (ii) Once the mixing is taken into ac- count, these unperturbed levels split into two branches at the crossing point as is familiar from the quantum mechanics of two-level systems. (iii) Above the cross- ing point, the lower state is made mainly of~q, with about 30% of the glueball component at K = 0 . 1 6 - 0.17, and the mass of the upper state which is made mainly of the glueball is pushed up from the unper- turbed value mg 0 ~- 1.75.

A feature relevant for a realistic estimate of the glueball mass is that the mixing o f ~ q component pushes up the glueball mass for large values of K. In the (u, d) quark sector, using the physical pion mass as

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Volume 145B, number 1,2 PHYSICS LETTERS 13 September 1984

input Fixes the value ofKu (=Kd) at 0.176 [6]. Thus to the extent that one trusts the value Kcr -~ 0 .14-0 .15 as the crossing point of the unperturbed levels, the true glueball mass should be larger than the Monte

Carlo estimates in pure gauge theory [4]. This is a general result as long as Ku, d > Kcr holds. The actual amount of increase, on the other hand, depends on

the precise values of parameters such as Ku, d, Kcr and unperturbed masses. The statistical quality of our data is not good enough for such a quantitative estimate.

We would like to thank Y. Oyanagi for informative discussions on his matrix inversion algorithm. M.F. and A.U. express their gratitude to the Theory Division of the National Laboratory for High Energy Physics for the hospitality extended to them while this work was

being carried out.

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