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

FIRST RESULTS FOR THE Air= 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= 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 ob- served enhancement. The eye diagrams, from which penguin operators are generated at low scales, are the sources of the enhance- ment. We also discuss the possible role played by octet-scalar particles in our quark mass range.

The enhancement of the AI= non-leptonic am- plitudes 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= ) 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 dia- grams), as first found in ref. [2] ~'. These new oper- ators have been suggested to be the source of the further enhancement needed to reproduce the ob- served 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 approxi- mation 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 )

139

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 hamil- tonian in eq. ( 1 ), without further assumptions [7 ]. In practice, the calculations that can be done at pres- ent are limited in two respects: the suppression of in- ternal quark loops (quenched approximation) and the use of rather large quark masses. The latter limi- tation implies an extrapolation to the physical region based on the hypothesis of a smooth behaviour of the matrix elements as functions of the quark mass.

In this letter, we present a first attempt to deter- mine the matrix elements of the A /= effective ham- iltonian in K decays ~2, in the quenched approxi- mation 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 16348 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 (m2-m2)y -+~ (4)

In the same approximation, the y -+ appear in the fol- lowing K -n and K-nn 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)10-+( /~) ln+(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 ele- ments 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-+ +36 +- "~50-+ , (10)

is the naive lattice operator plus the appropriate op- erators of dimension six and five, whose finite coef- ficients can be computed safely in perturbation theory [ 13,14]. Zl+tt (/za, go) is introduced to normalize the lattice operators as the continuum 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

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

the matrix elements of the operators in eqs. (7) and (8) differ from the matrix elements of the bare ones only by the factor Zja~ 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 ele- ments of eqs. (6 ) - ( 8 ), enhanced by the scalar prop- agator 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-nn amplitude needs no subtraction, is subject to smaller statistical fluctua- tions, 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 contribu- tion 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) dia- grams 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 dia- grams 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 dia- grams require the non-perturbative subtraction of eq. ( 11 ). In the following we will denote the contribu- tion 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 of O~e~t, and the matrix element of~d. The results are reported in table 1, which is the basis of the subsequent anal- ysis, for the 20 102 40 lattice with fl= 6.0.

The momentum q is given along the direction with 20 lattice sizes, and An is the minimum 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.

141

Volume 211, number 1,2

Table 1

PHYSICS LETTERS B 25 August 1988

q Off~,~ X 103Xa 4 + Opcrt X 103Xa 4 ~dxa E(q) Xa

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 energy-momentum relation is well satis- fied for q ~ 0 (see also ref. [ 18 ] ).

(2) The K -n matrix element of.qd at zero momen- tum transfer has to satisfy the Ward identity

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

We find **6

(n + (0) lYdl K + (0 ) ) ] chiral limil = 1 .30+0.20,

Om 2 - - - -1 .30+0.03 . (14) 0 ( l /K )

(3) The logarithmic derivative with respect to _ ]q] 2 of the K -n matrix element ofgd is related, via standard current algebra arguments and the Cal lan- Treiman relation, tofK/f~:

m2K 0 ln( ~+ (0) I ;dl K+ (q ) )0 ( -q 2 ) q=O - - fK--f~f,~ (15)

Using a - ~ = 1.8 GeV and the experimental value of -y

m ~, we find from table 1

m 2 01n(~+(0) lyd lK+(0) ) =0.38+0.12 , (16) 0 (_q2)

which is quite consistent with our previous determi- nation offK/f~ [9]. Details of this analysis are given in ref. [ 19 ].

(4) From table 1 one can derive the coefficients c~ defined by eq. ( 11 ), by taking the ratio of the K -

matrix elements of Oo-~t at q=0 with the corre- sponding matrix elements ofgd. The values ofc~ ob- tained in this way agree in sign and magnitude 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 derived in ref. [ 14 ] from a complete calculation at two-loop perturbation theory.

One has

(cg)p~rt =goAp~rt h

(17)

with A~ert = +9.13X 10 -3. From table 1, using eq. (17) with g2 = 1 ( f l= 6.0),

we find

A-+ = (10.4, 10.1, 10.0))

Volume 211, number 1,2

Table 2

PHYSICS LETTERS B 25 August 1988

- 3 K-Wilson ma ~%~(q#0) X 10 Xa- y~(q#0) 103a a 7~gh, 103Xa 2 Y~ght 103a 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 (I I). The results are given in table 2.

The results for the eight-diagram contributions 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 lO ( /0 IK+ (0)n-(0))~ight =a -- ~ght+ m 2. (19)

0.01

0.00

-0.01

-0.02

-0 .03 - -

-0 .04 - -

-0.05 L 0 0.2 0.25 0.$

(m,O z 0.05

[ 1 0. I 0.15

(-)

0.03

0.02

0.01

0.00

-0.01

0 0.05 0. I 0.15 0.2 0.25 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 am- plitude for Ks--.n+n - reported in table 3, where we have used c ( - ) ( / t )=- l .7 and c (+)( / t )=0.75 at /t--- 1.8 GeV ~9

The errors in the ratio R, eq. (2), are amplified by a big fluctuation to a very small (positive) value of the M= 3 amplitude 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(Ks~n+n - ) > 2B(Ks - ,nn ) . 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 dis- cussed 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 contribution of the scalar pole as in fig. 1 which dominates the amplitude in this range of masses.

(ii) 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= ~ am- plitude obtained from 30 configurations at fl= 6 and from a quadratic fit 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.

143

Volume 21 1, number 1,2

Table 3

PHYSICS LETTERS B 25 August 1988

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);

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

tet will put the K-nn...

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