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Volume 211, number 1,2 PHYSICS LETTERS B 25 August 1988

FIRST RESULTS FOR THE Air= ½ AMPLITUDE IN K DECAYS WITH QUENCHED LATTICE QCD AND WILSON FERMIONS

M.B. GAVELA l Departamento de Fisica Tebrica, Universidad Aut6noma de Madrid, E-28049 Madrid, Spain

L. MAIANI, S. PETRARCA 2 Dipartimento di Fisica "G. Marconi", Universit~ di Roma "La Sapienza", and lNFN, Sezione di Roma, 1-00185 Rome, Italy

G. MARTINELLI CERN, CH- 1211 Geneva 23, Switzerland

and

O. PlaNE LPTHE, [:-91405 Orsay Cedex, France

Received 2 June 1988

We present the results of a calculation of the AI= ½ K - n and K-nTt amplitudes in quenched lattice QCD with Wilson fermions. The results were obtained on a 20 × 102 X 40 lattice at fl= 6.0 for three values of the light quark masses and on a 163 X 48 lattice at /3=6.2 for two values of the light quark masses. Within large statitieal and systematic uncertainties our results support the observed enhancement. The eye diagrams, from which penguin operators are generated at low scales, are the sources of the enhancement. We also discuss the possible role played by octet-scalar particles in our quark mass range.

The enhancement of the AI= ½ non-leptonic amplitudes in strange particle decays is still a challenge for quantum chromodynamics.

From the basic theory, one can derive the effective non-leptonic hamiltonian down to a momentum scale /~ of the order of the charmed quark mass

Herr = ~ GF cos 0c sin 0c

x [c~-~(u)O- (u) +c~ +~(u)O + (u) ],

0 -+ (/A) = (fLPI, dL/TLp/~UL q- gLy/~/,/L/~L}'/'dL)

- ( u - , c ) . (1)

The coefficients c ( + ) (/~) give an enhancement of the octet ( ad= ½ ) over the 27-plet (A/= 3 ) part, of the

L On leave of absence from LPTHE-Orsay. 2 Partially supported by Ministero Pubblica lstruzione, Italy.

order of a factor 2-3, induced by hard gluon correc- tions [ 1 ]. This is still not enough to account for the experimental ratio

R = (~t+~-IHerf lKs) =21 .2 . (rt+~°lHen-I K+ )

(2)

Lowering the momentum scale below the charm threshold generates new operators (penguin diagrams), as first found in ref. [2] ~'. These new operators have been suggested to be the source of the further enhancement needed to reproduce the observed ratio, eq. (2). The answer, however, depends crucially upon the extrapolation of the renormaliza- tion group equations in a region where perturbation theory becomes questionable and upon the approximation scheme used to evaluate the hadronic matrix

~ For a recent review see ref. [ 3 ].

0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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Volume 21 l. number 1,2 PHYSICS LETTERS B 25 August 1988

elements. In refs. [2,3] the vacuum saturation method was used. Other approximation methods for the calculation of the operator matrix elements, in the framework of the 1 ~No expansion or QCD sum rules, have recently been proposed and discussed in ref. [ 4 ] and ref. [ 5 ], respectively•

Lattice QCD [ 6 ] offers in principle the possibility to compute the matrix elements of the weak hamiltonian in eq. ( 1 ), without further assumptions [7 ]. In practice, the calculations that can be done at present are limited in two respects: the suppression of internal quark loops (quenched approximat ion) and the use of rather large quark masses. The latter limi- tation implies an extrapolation to the physical region based on the hypothesis o f a smooth behaviour of the matrix elements as functions of the quark mass.

In this letter, we present a first at tempt to deter- mine the matrix elements of the A / = ½ effective hamiltonian in K decays ~2, in the quenched approximation with Wilson fermions. We have performed two independent calculations: a first one on a 20 X 102 X 40 lattice at f l= 6.0, with hopping param- eters for light quarks K=0.1515, 0.1530, 0.1545 (K- c,-~,c.~ = 0.1565) and K=Kch = 0.1350 for the charmed quark, and a second one on a 163×48 lattice at ,8=6.2, K=0.1500, 0.1510, 0.1520 (Kc,-it~c~ =0.1534) and K,.h=O.1350) ~3. In both cases, our results are based on 15 gauge configurations. The results have been obtained with different methods, based on chiral perturbation theory, that we briefly summarize in the following.

To lowest order in masses and momenta, chiral perturbation theory gives the expressions for the Ks--+n+n - amplitude [ 10,11 ]

A(Ks-+n+n -) m2-m~ =2GF sin 0c cos 0~

mK mKf~

× [ c ~ ) ( /z )7~-)+c~+)( /z )y ~+~], (3)

where the y ± are defined according to

i ( n + n - 1O-+ (/l) [K°) = f ( m 2 - m 2 ) y ¢-+~ (4)

In the same approximation, the y -+ appear in the following K - n and K - n n matrix elements:

~-~ Preliminary results have been presented in ref. [8 ]. "-~ Details about the calculation and the lattice calibration can be

found in ref. [ 9 ]; fl= 6~go, with go the lattice gauge coupling.

(n+ (O)lO+ (/~)lK+ (q) ) =r~+ m2+ 7+-E(q)m, (5)

(OlO+(Iz)lK+(O)n-(O))=r~+m2-7-+m2, (6)

• 2m2 7 -+ (7) (n+ ( 0 ) n - (0) I 0-+ (U) I K ° ( 0 ) ) = 1 ~ 7 - = ,

• 2m 2 + ( n + ( 0 ) 1 0 - + ( / ~ ) l n + ( 0 ) K ° ( 0 ) ) = - l - - f - y , (8)

where we have taken K and n degenerate in mass (mK=rn==2 rn2), q is the spatial momentum, E(q) =x/m2+q 2, and f=,-- 131 MeV.

The calculation on the lattice of the matrix elements on the LHS of eqs. ( 5 ) - ( 7 ) allows a determi- nation of y -+ i.e., of the physical amplitude, to lowest order in chiral perturbation theory.

As discussed in refs. [ 11,12 ], in the SU (3) sym- metric limit the renormalized operators 0 + (/~) are related to the lattice operators O + as follows:

g + + + , O+ ( / z )= ,a,(#a, go)(Oper,+cggd) (9)

where

O;ert = O-+ + 3 6 °+- " ~ 5 0 - + , (10)

is the naive lattice operator plus the appropriate operators of dimension six and five, whose finite coefficients can be computed safely in perturbation theory [ 13,14]. Zl+tt (/za, go) is introduced to normalize the lattice operators as the cont inuum ones, on quark states.

The coefficients c~ are quadratically divergent in the inverse lattice spacing and must be determined non-perturbatively. We have chosen the renormali- zation conditions

< n + ( 0 ) I0+- (~) I K + ( 0 ) ) = 0 , (11)

which correspond to 6 + = - 7 + ~4. Two remarks are important:

(i) The operators ~50 + are finite only if we have an explicit G IM cancellation on the lattice. For this reason the lattice must be such as to allow propaga- tion of a charmed quark, mca << 1.

(ii) In the SU (3) limit the counterterms to the bare operators are parity conserving [ 13,14 ]. Therefore

~4 As discussed in ref. [ 11 ] the ~-+ do not appear in the physical decay amplitude and in fact are arbitrary. Different values of a+ correspond to different and equivalent subtraction pre- scriptions needed to define 0 + (/t).

140


the matrix elements of the operators in eqs. (7) and (8) differ from the matrix elements of the bare ones only by the factor Zj±a~ and no subtractions are nec- essary to compute the K - ~ n matrix elements [ 15 ].

The smoothness assumption made in eqs. ( 5 ) - ( 8 ) may fail in the presence of octet scalar particles com- paratively light with respect to the pseudoscalar mass m. Exchange of scalar particles as shown in figs. 1 a - 1 c, gives non-smooth contributions to the matrix elements of eqs. ( 6 ) - ( 8 ), enhanced by the scalar propagator when M 2 _~4m 2. On the contrary, the matrix element in eq. (5) is relatively unaffected, because the momentum transfer is space-like. In the range of fl and quark masses where we are working a numeri- cal study of the scalar propagator indicates that the scalar octet mass Ms is close to 2m ~5

In this situation the only theoretically safe way to extract ? ± is from the matrix element ofeq. (5). Tak- ing into account the subtraction condition eq. ( 11 ), one has

~-+= (r~+ (0) I 0-+ (/z) IK+ ( q ) ) (12) m[E(q)-m]

The price to pay is that, because of the subtraction involved, the signal in the numerator of eq. (12) is small and affected by large statistical fluctuations. The direct calculation of the K - n n amplitude needs no subtraction, is subject to smaller statistical fluctuations, but may be contaminated by spurious octet scalar contributions unless M 2 >> 4m 2.

K - ~ and K-nrc matrix elements are computed from three- and four-point correlations of the operators and pseudoscalar sources. Multipoint correlation func-

,5 This is indicated by our data, with a statistics barely sufficient to see the exponential fall-offofthe scalar correlation function. A scalar mass even smaller than 2rn is found by high statistics preliminary results from lhe APE Collaboration. We thank the members of APE for communicating to us their preliminary results.

k (a) (b) (c) ~

Fig, 1. Feynman diagrams relative to the octet scalar contribution to K-n (a) and K-rift, (b) and (c), weak transitions. The symbol ® indicates the insertion of the weak hamiltonian.

k ~ , ~ " ~

(a) (b) (c)

Fig. 2. The "eight" (a), "eye" (b), and counterterm (c) diagrams for K-n three-point correlations.

tions are computed using the technique developed in ref. [ 16] for Susskind fermions. A closely related method is discussed in ref. [ 17 ]. The relevant diagrams are listed in figs. 2 and 3. The internal quark loop, where up and charm quarks propagate, charac- terizes the so-called eye diagrams. Only these diagrams require the non-perturbative subtraction of eq. ( 11 ). In the following we will denote the contribution to y-+ from the eye and from the other (eight- shaped) diagrams, by Y~ye and 7~gh, respectively. Matrix elements are extracted from the correlation functions according to the methods extensively de- scribed in ref. [ 9 ]. The errors quoted in the following are purely statistical. They are obtained by dividing the 15 configurations in three clusters and comput- ing, for each quantity, the dispersion over the clusters.

We discuss first K - n matrix elements. The correlation functions of figs. 2b and 2c give

the eye-contribution to the matrix elements o f O~e~t, and the matrix element of~d. The results are reported in table 1, which is the basis of the subsequent analysis, for the 20 × 102 × 40 lattice with f l= 6.0.

The momentum q is given along the direction with 20 lattice sizes, and A n is the min imum momentum in this direction. A number of consistency checks on the results in table 1 have been performed.

(a)

(c)

k

(b)

Fig. 3. The "eight" (a) and (b), and "eye' (c), diagrams for K- nn four-point correlation functions.

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Volume 211, number 1,2

Table 1

PHYSICS LETTERS B 25 August 1988

q Off~,~ X 103Xa 4 + Opcrt X 1 0 3 X a 4 ~ d x a E(q) X a

K-Tr O-Kn K-n O-Kn K-n O-Kn

0 7.4(0.8) 14.1(2.8) -6.3(0.4) 8.0(1.0) 12.2(2.5) -6.6(0.5) 8.0(0.8) 11.5(1.0) -6.4(0.5)

~n 4.0(0.5) -3.6(0.5) 3.4(0.3) - -3.2(0.4) 3.1(1.4) -2.9(1.3)

- 14.0(4.0) 1.52(18) 5.04(19) 0.523(6) -12.2(3.4) 1.49(20) 3.98(22) 0.436(9) -11.6(1.9) 1.37(18) 3.17(27) 0.327(14)

0.85(6) 0.620(6) - 0.71(8) - 0.548(12)

0.72(19) 0.465(20)

( 1 ) The e n e r g y - m o m e n t u m relat ion is well satis- fied for q ~ 0 (see also ref. [ 18 ] ).

(2) The K - n matr ix e lement of.qd at zero momentum transfer has to satisfy the Ward ident i ty

Om 2 < ~+ (0) I~qdl K+ (0 ) > I ch*al ~'m~t-- 0 ( I / K ) " (13)

We f ind **6

( n + (0) lYdl K + ( 0 ) ) ] chiral limil = 1 . 3 0 + 0 . 2 0 ,

Om 2 - - - - 1 . 3 0 + 0 . 0 3 . (14) 0 ( l / K )

(3) The logar i thmic der iva t ive with respect to _ ]q] 2 o f the K - n matr ix e lement o fgd is related, via s tandard current algebra arguments and the Ca l l an - Tre iman relat ion, tofK/f~:

m2K 0 l n ( ~+ (0 ) I ;d l K+ ( q ) ) 0 ( - q 2 ) q=O - - fK--f~f,~ (15)

Using a - ~ = 1.8 GeV and the exper imenta l value of -y

m ~, we f ind from table 1

m 2 0 1 n ( ~ + ( 0 ) l y d l K + ( 0 ) ) = 0 . 3 8 + 0 . 1 2 , (16) 0 ( _ q 2 )

which is quite consistent with our previous de termi- nat ion offK/f~ [9] . Detai ls o f this analysis are given in ref. [ 19 ].

(4) F rom table 1 one can der ive the coefficients c~ def ined by eq. ( 11 ), by taking the ratio of the K -

matr ix elements of Oo-~t at q = 0 with the corresponding matr ix e lements ofgd. The values o f c ~ ob- ta ined in this way agree in sign and magni tude with

~6 A similar result applies at fl= 6.2 where the LHS of eq. ( 13 ) is 0.68 + 0.14 and the RHS is 0.89_+ 0.01.

the leading divergent coefficients der ived in ref. [ 14 ] f rom a comple te calculat ion at two-loop per turba t ion theory.

One has

(cg)p~rt =goAp~rt ½ h

(17)

with A~ert = + 9 . 1 3 X 10 -3. F r o m table 1, using eq. (17) with g2 = 1 ( f l= 6.0),

we find

A-+ = (10.4, 10.1, 10.0))<10 -3

= ( - 12.1, - 12.2, - 12.4) X I0 -3 , (18)

at K = 0.1515, 0.1530 and 0.1545 respectively. It is seen from table 1 that the matr ix elements

( n l ~ d l K ) q = 0 and ( 0 l ~ d l K n ) , which should coin- cide in the chiral l imit , differ by a factor o f about 3. This is more than the l inear var ia t ion one can esti- mate from the slope in eq. (15) computed from the space-like region, and is suggestive o f the effect of the scalar octet propagator .

F rom the matr ix elements in table 1 one obtains **7 the values of ~e~e repor ted in table 2 and plot ted in figs. 4a and 4b versus ( m a ) 2. Different methods to extract the matr ix elements from the correlat ion funct ions lead to consistent results at K = 0.1515 and 0.1530, but to large var ia t ions at K=0 .1545 . This re- flects the large noise in the corresponding correlat ion functions. For this reason, to extrapolate to the chiral l imit we have either f i t ted 7~e to a constant taking the weighted average of the three points ( I ) or ex-

~7 For 7/-y~ we have estimated the statistical error also with the jacknife method [20 ], with practically identical results.

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Table 2


- 3 K-Wilson ma ~%~(q#0) X 10 Xa- y~(q#0) × 103×a a 7~gh, × 103Xa 2 Y~ght × 103×a z

0.1515 0.523 -3_+ 9 -2_+11 0.1530 0.436 -8_+ 10 0_+ 13 - 0.1545 0.327 -24_+ 17 + 11 _+ 16

extrapolation -8_+ 9 (I) +2_+ 12 (1) -2.9_+ 1.5 -20_+ 12 (II) +5_+ 19 (II)

3.9_+1.5

trapolated linearly discarding the last point (II) . The results are given in table 2.

The results for the eight-diagram contr ibut ions are given in the same table. They have been obtained by fitting the corresponding matrix elements as

( n + (0) 10 +- (# ) I K + (0))eight =a-+ "~ Y+ght m2 ,

( 0 l O ± ( / 0 IK+ (0)n- (0) )~ igh t = a ± -- ~ght+ m 2. (19)

0.01

0 .00

-0.01

-0.02

- 0 . 0 3 - -

- 0 . 0 4 - -

-0.05 L 0 0.2 0.25 0.$

(m,O z 0.05

[ 1 0 . I 0 .15

(-)

0 . 0 3

0 .02

0 .01

0.00

-0 .01

0 0 .05 0 . I 0 .15 0 .2 0 .2 5 0 .3

( ~ 1 z O)

Fig. 4. 7~ (a) and y~ (b) versus m~-a 2 from the K-n matrix elements at q# 0, see eq. (12) of the text. fl= 6, V= 20 × 102 × 40, average and errors from 15 configurations.

Details of the analysis are given in ref. [9] ,8 Using eq. ( 1 ) with the experimental values for inK,

m~ andf~ and using a - ' = 1.8 GeV we find the amplitude for Ks-- .n+n - reported in table 3, where we have used c ( - ) ( / t ) = - l . 7 and c ( + ) ( / t ) = 0 . 7 5 at /t--- 1.8 GeV ~9

The errors in the ratio R, eq. (2) , are amplified by a big fluctuation to a very small (posit ive) value of the M = 3 ampli tude for one of the clusters. The re- suits for A (Ks--, n + n - ) are still compatible with zero,

given that 7~ye is negative for two clusters and posi- tive for the other one. The signal for 7+e is definitely worse than the one for 7~ye. The trend of the data is however encouraging. In particular they favour a po- sitive sign for R, eq. (2), in agreement with the fact that B ( K s ~ n + n - ) > 2 B ( K s - , n ° n ° ) . More statistics are clearly needed.

If we use the matrix element in eq. (6) to extract ~'~ye we obtain results with smaller statistical errors, corresponding to a large signal, but R comes out to be negative. These results have been presented and discussed in length in ref. [ 8 ].

The anomalous sign may be due to the presence of a nearby scalar pole, which may affect the result in

two ways: ( i) A contr ibut ion of the scalar pole as in fig. 1

which dominates the ampli tude in this range of

masses. ( i i) It may be that the truly renormalized opera-

tors are not much coupled to the octet scalar particle,

~8 Non-vanishing a + represent a breaking of the chiral behaviour which we have interpreted in ref. [ 9 ] as a residual lattice arti- fact which should vanish as a-~0.

~9 In ref. [ 9 ] we have reported a smaller value for the Al= ~ amplitude obtained from 30 configurations at fl= 6 and from a quadratic f i t to Y4/f~ as a function of m 2. As already noticed in ref. [9 ], the present 15 configurations alone tend to give a larger value for this amplitude.

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Volume 21 1, number 1,2

Table 3


fl Source A (Ks--,rt+n - )/mK A(K+-~n+r~°)/mK R

6.0 K - n ( q # 0 ) (1.2_+ 1 .3 )×10 6 (I) (16.0_+6.0);<10 .8 8_+24 (I) (2.5_+1.8);<10 6 ( i i ) 16_+40(11)

6.2 K-~rt (2.1 _+ 1.3)?< l0 -6 (5.9_+ 1 .4)× 10 -8 35_+31

experiment 0.78;< 10 -6 3.8;< 10 -8 21.2

contrary to Yd. In this case an imprecision of the subtraction procedure could leave spurious terms, pro- portional to gd, which are enhanced by the scalar pole.

Only under the latter hypothesis is the K - x x calculation, where subtractions are not needed because of C P S symmetry [ 15 ], reliable even at not too low pseudoscalar masses.

At fl= 6.2, corresponding to a - ~ = 2.6 GeV, we have computed the K - ~ matrix elements given in eqs. (7) and (8) at K=0.1500, 0.1510 (m~0.84 , 0.78 GeV respectively). The results for the eye and eight diagrams are reported in table 4. The values of 2m 2 7 ~ / f ( m ) have been obtained from the average of the matrix elements eqs. (7) and (8) ~°. We give also 2 m 2 ) , 4 / f ( m ), which refers to the A J = 3 operator [2,3],

0 4 ( , / / ) = ~(LT/, dL/~L 7/~UL "Jr" fiLTh, ULaLyPdL

-- gL T,,dL dL y"dL . (20)

Assuming a constant behaviour in m 2 of all 7's, we obtain the physical amplitudes reported in table 3.

Within errors the K ~ results are quite compatible with those obtained from the q# 0 K ~ matrix elements, and do not show any non-smooth variation with the pseudoscalar meson mass m. Within the present data, we are unable to decide whether any contribution of the scalar particle is hidden in the errors, or is really small.

~"~ f ( m ) is the pseudoscalar meson axial coupling as function of the quark mass.

In conclusion, we have completed a first lattice calculation of the M = ½ amplitude in K decays with three different methods all based on the validity of chiral perturbation theory. The first one, based on the calculation of K-~ matrix elements with a non-vanishing space momentum, q, supports the observed enhancement of the A/= 1 amplitude, although within very large statistical fluctuations. The eye-diagrams are the source of the enhancement, an indirect con- firmation of the scheme of refs. [2,3 ] since penguin diagrams are indeed generated by eye-diagrams (see R.C. Brower et al., quoted in refs. [7,17] ). The vacuum saturation approximation gives - + ~2eight / ~eight = - ½, while we find a larger ratio (see table 2 ), which seems to be just the continuation of the short-dis- tance enhancement of ref. [ 1 ].

The K-xx method [ 15 ] ~ gives results in agreement with the previous one, but its validity is subject to the demonstration that the scalar pole has little influence, in the present quark mass range. Finally, the K-~ timelike region method [ 8 ] gives results incom- patible with the previous ones, which may be due to the effect of the scalar pole and/or imperfect subtraction.

An increase in statistics of a factor of about 10 should show the validity of the first method, and pro- vide clean evidence for the At/= ½ enhancement. A detailed monitoring of the influence of the scalar oc-

"~ Results at fl= 5.7 with the K-nn method have also been presented in ref. [21 ].

Table 4

K-Wilson ma [2m2/f(rn)]y~y~ [2m2/f(m)])'~e [2rn2/f(m)]Yc~ght [2m2/f(m) ] 7eight+ [2m2/f(m)]74 X 104Xa 3 X 104Xa 3 X 104Xa 3 X lOaXa 3 X 104Xa 3

0.1500 0.358 - -8 .0+5 .4 5 .4+2.3 --0.33_+0.28 0.51_+0.07 0.59_+0.13 0.1510 0.300 --5.5_+3.7 3.7_+1.7 --0.14_+0.24 0.38_+0.06 0.37_+0.11

144


tet will put the K-nn method , wh ich has the advan -

tage o f need ing no subt rac t ion , on m o r e sol id ground.

In perspec t ive , at real is t ical ly smal l qua rk masses,

all m e t h o d s should agree, which w o u l d p r o v i d e an

u n q u e s t i o n a b l e answer to the A / = ½ puzzle.

We thank all the m e m b e r s o f the C o m p u t e r Cen-

ters o f the C C V R o f the Ecole P o l y t e c h n i q u e and o f

Cineca for con t i nuous help and assis tance in devel -

op ing and runn ing our c o m p u t e r codes. The suppor t

f r o m C N R S and I N F N is grateful ly acknowledged .

We have bene f i t t ed dur ing this work f r o m con t inu-

ous and frui t ful d i scuss ions wi th G.C. Ross i and M.

Testa.

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