We present an exploratory calculation of the standard model Delta S=2 matrix
element relevant for indirect CP violation in K -> pi pi decays. The
computation is performed with overlap fermions in the quenched approximation at
beta=6.0 on a 16^3x32 lattice. The resulting bare matrix element is
renormalized non-perturbatively. Our preliminary result is B_K^{NDR}(2
GeV)=0.61(7), where the error does not yet include an estimate of systematic
uncertainties.Comment: 3 pages, 2 figures, Lattice2002(matrixel). Parallel session talk by
  L. Lellouch at the 20th International Symposium on Lattice Field Theory
  (Lattice 2002), Boston, Massachusetts, 24-29 June 200

Ali Khan

Aoki

Blum

Buras

C. Hoelbling

C. Rebbi

Capitani

Ciuchini

Giusti

Hasenfratz

L. Giusti

L. Lellouch

Lellouch

Lu¨scher

Martinelli

N. Carron

Neuberger

Sharpe

English

arXiv

We present an exploratory calculation of the standard model Delta S=2 matrix element relevant for indirect CP violation in K -> pi pi decays. The computation is performed with overlap fermions in the quenched approximation at beta=6.0 on a 16^3x32 lattice. The resulting bare matrix element is renormalized non-perturbatively. Our preliminary result is B_K^{NDR}(2 GeV)=0.61(7), where the error does not yet include an estimate of systematic uncertainties

Garron, N

Giusti, Leonardo

Hölbling, C

Lellouch, L P

Rebbi, C

CERN Document Server

1CPT-2002/P.4416CERN-TH/2002-350BUHEP-02-41BK from quenched overlap QCDN. Garronacpt]Centre de Physique Theoriquez, Case 907, CNRS Luminy, F-13288 Marseille Cedex 9,France , L. Giustia,bcern]Theory Division, CERN, CH-1211 Geneva 23, Switzerland , C. Hoelblingc[cpt],L. Lellouchd[cpt] y and C. R bbiebu]Department of Physics, Boston University, 590 CommonwealthAvenue, Boston MA 02215, USAa[b[e[We present an exploratory calculation of the standard model ∆S = 2 matrix element relevant for indirect CPviolation in K → pipi decays. The computation is performed with overlap fermions in the quenched approximationat β = 6.0 on a 163 × 32 lattice. The resulting bare matrix element is renormalized non-perturbatively. Ourpreliminary result is BNDRK (2 GeV) = 0.61(7), where the error does not yet include an estimate of systematicuncertainties.1. IntroductionK0 − K0 mixing induces indirect CP violationin K !  decays and is governed, in the stan-dard model, by the matrix elementh K0jO∆S=2()jK0i = 163 M2K F2K BK() ; (1)where O∆S=2 is the four-quark operator given be-low (Eq. (2)) and where BK parametrizes devia-tion from the vacuum saturation approximation.The benet of using overlap fermions [1]to compute this matrix element is their exactSU(Nf)L  SU(Nf)R flavor-chiral symmetry atnite lattice spacing [2]. This implies that theS = 2 operator is simple to construct on thelattice (unlike with staggered fermions) and doesnot mix with chirally dominant operators [3] (asit does with Wilson fermions). Overlap fermionsfurther guarantee full O (a)-improvement. On adierent note, this exploratory computation teststhe feasibility of calculating matrix elements offour-quark operators with overlap fermions.2. Simulation detailsWe use Neuberger’s overlap action [1]∗Work supported in part by EU HPP contract HPRN-CT-2000-00145 and US DOE grant DE-FG02-91ER40676.We thank Boston University for use of its supercomputerfacilities and the Gauge Connection for its gauge configu-rations.†Presented by L. Lellouch at Lattice 2002, Boston, MA.‡ Unite´ Propre de Recherche 7061.L = a4Xxq1− 12amqD + mqq ;withD =a1 + X=pXyXand X = DW − =a ;where DW is the massless Wilson-Dirac operator.The renormalized S = 2 operator of Eq. (1)is related to the bare lattice operator throughO∆S=2() = Z∆S=2(a; g0)Obare∆S=2(g0) ; (2)where the lattice operator with correct chiralproperties isObare∆S=2 = [sγµ(1− γ5)d^][sγµ(1− γ5)d^] ;with d^ = (1− aD=2)d.The computation is performed with the Wil-son gauge action in the quenched approximationat  = 6:0 (a−1(r0) ’ 2:1 GeV) on a 163  32lattice with a statistics of 80 congurations andwith  = 1:4. The pseudoscalar mesons simulatedare composed of degenerate quarks with massesmq = 0:04; 0:055; 0:07; 0:085; 0:1 corresponding toa range from  ms=2 ! 1:3ms. Details regard-ing the implementation and the inversion of theoverlap operator can be found in [4].2The three-point function used to extract theS = 2 matrix element and its asymptotic timebehavior areXxyhJL0 (x)Obare∆S=2(0)JL0 (y)i Ty0>x01−! (3)jhKjJL0 j0ij2(2MK)2h KjObare∆S=2jKineMK(y0−T−x0) + RKe−MK(x0+y0)−δx0 + eMK(x0+y0−2T )+δ(y0−T )o;with JLµ = dγµ(1 − γ5)s^. We allow for a time-reversed contribution because our lattice is rathershort in the time direction and this contributioncan be as large as 3-4% for our lightest quarkmass at x0 = T − y0 = 8a, a point which isincluded in our ts. In Eq. (3)  measures thenite-volume shift in the energy of the two kaonstate. This shift varies from about 2% to 5% ingoing from our heaviest to our lightest mass andis consistent with zero within roughly three to onestandard deviations. RK is clearly intimately re-lated to h0jObare∆S=2jKKi, but nite volume eectscan be signicant and we choose not to pursuethe study of this time-reversed matrix element.The t that we retain for determining the for-ward matrix element is to times x0=a and y0=a inthe intervals 6 ! 8 and 24 ! 26 with y0 > x0. Toobtain jhKjJL0 j0ij and MK we compute two-pointfunctions of left-handed currents which we sym-metrize in time and t in the time range 5 ! 15.The motivation for using left-handed currentsis that they eliminate spurious zero-mode contri-butions which appear in nite volume and in thequenched approximation because the overlap op-erator satises an index theorem [5], as a goodDirac operator should. In our JLµ correlators,topological zero modes, 0(x), contribute asγµ(1 − γ5)0(x)y0(y)mqγµ(1 − γ5) ;which vanishes because of the chirality of thesemodes. The use of left-handed currents is pro-posed in [6] where it is suggested that the weakinteractions of Goldstone bosons be studied in the-regime of Gasser and Leutwyler.3. Chiral behavior of BKWe obtain BbareK fromBbareK =38h KjObare∆S=2jKih Kj JL0 j0ih0j JL0 jKi;where JL0 is the conjugate of JL0 . Quenched chiralperturbation theory (QPT) predicts [7]BK=B"1− 6M4F2lnMB+ bM4F4#;(4)where B, F and M are the leading order val-ues of the B-parameter, leptonic decay constantand pseudoscalar mass, respectively. B is a scalewhich corresponds to O (p4 contact terms in theeective theory and the term proportional to b isadded to parametrize likely higher order contribu-tions. We t this functional form to our results forBbareK , with M = MK , F = Fphyspi or FphysK andb  0 or not. The t parameters are Bbare, Band b when it is not xed to zero. 3 The resultsof these ts are shown in Fig. 1. We nd that themass-dependence of BK is consistent with expec-tations from PT , though results at lower massesand with better statistics would be necessary toconrm the coecient of the logarithm. Untilsuch results become available, we choose not toquote a number in the chiral limit. At the physi-cal point MK = MphysK , however, where the valueof the B-parameter obtained is insensitive to thechoice of functional form, these considerations arenot important. 44. Non-perturbative renormalizationWe perform all renormalizations non-perturbatively in the RI/MOM scheme a la [8].Thus we x gluon congurations to Landau gaugeand compute numerically approriate, amputatedforward quark Green’s functions with legs of mo-mentum p pp2. Then we use the fact that withoverlap fermions the renormalization constantsZJL = ZV and Z∆S=2 = ZV V +AA to deneZRIBK (ap; g0) =ΓV (pa)2ΓV V +AA(pa);3Note that the definition of b changes with F .4The result for BK given below is obtained from the chiralfit with F = F physpi and b 6= 0.3Figure 1. Chiral behavior of BK . Fits are to Eq.(4), as described in the text. The vertical dashedline indicates the physical point.where ΓO is the value of the non-perturbative,amputated Green’s function of operator O pro-jected onto the spin-color structure of O.We nd that in a range from approximatively1.5 to 2 GeV the p-dependence of ZRIBK matches2-loop running obtained by combining the resultsof [10] and [11], with Nf = 0 and s taken from[9]. This is shown in Fig. 2 where we plot, vs p,ZRGIBK which we dene as the ratio of ZRIBKto the2-loop running expression. Also shown in Fig. 2are the non-perturbative ZRIBK (p) and its 1-loopcounterpart ZRI,PTBK (p), obtained by expanding ins the ratio of ZRI,PT∆S=2 (p) to (ZRI,PTA )2. The lat-ter are taken from [12] and are properly matched.The value of ZRGIBK that we use is the result of t-ting our data to a constant, in the momentumrange shown in the plot. The renormalizationconstant in the MS − NDR scheme is then com-puted through multiplying this value by the 2-loop running expression of [10] with Nf and schosen as above.5. ConclusionThough costly, it is entirely feasible to com-pute weak matrix elements with overlap fermions,at least in the quenched approximation. The ad-vantages of this approach are the chiral symmetryand the O (a)-improvement.Our preliminary results are ZRGIBK = 1:29(1)for the renormalization constant and, for the B-parameter:BRGIK = 0:85(10) ; BNDRK (2 GeV) = 0:61(7);Figure 2. Scale dependence of the renormalizationconstants for BK defined in the text. The threecurves for ZRI,PTBK (p) are obtained by allowing thescale parameter in the coupling of [9] to take thevalues MSQCD=2, MSQCD and 2MSQCD.where the errors do not yet include an estimate ofsystematic uncertainties. For the B-parameter,agreement with continuum-limit world averages(see e.g. [13]) based on the staggered result of [14]is excellent. Agreement with the results of [15],obtained with a dierent formulation of overlapfermions, perturbative matching and larger quarkmasses is good. Finally, our result is respectivelya half and one standard deviations above the re-sults of [16] and [17], both obtained using domain-wall fermions with a nite fth dimension.REFERENCES1. H. Neuberger, Phys. Rev. D 57 (1998) 5417.2. M. Lu¨scher, Phys. Lett. B 428 (1998) 342.3. P. Hasenfratz, Nucl. Phys. B 525 (1998) 401.4. L. Giusti et al., Phys. Rev. D 64 (2001)114508; Nucl. Phys. 106 (PS) (2002) 739.5. P. Hasenfratz, V. Laliena and F. Nieder-mayer, Phys. Lett. B 427 (1998) 125.6. L. Giusti et al., arXiv:hep-lat/0212012.7. S. R. Sharpe, Phys. Rev. D 46 (1992) 3146.8. G. Martinelli et al., Nucl. Phys. B445 (1995)81.9. S. Capitani et al. [ALPHA Collaboration],Nucl. Phys. B 544 (1999) 669.10. A. J. Buras, M. Jamin and P. H. Weisz, Nucl.Phys. B 347 (1990) 491.11. M. Ciuchini et al., Z. Phys. C 68 (1995) 239.412. S. Capitani and L. Giusti, Phys. Rev. D 62(2000) 114506.13. L. Lellouch, Nucl. Phys. 94 (PS) (2001) 142.14. S. Aoki et al. [JLQCD Collaboration], Phys.Rev. Lett. 80 (1998) 5271.15. T. DeGrand, this volume.16. A. Ali Khan et al. [CP-PACS Collaboration],Phys. Rev. D 64 (2001) 114506.17. T. Blum et al. [RBC Collaboration],arXiv:hep-lat/0110075.

$B_{K}$ from quenched overlap QCD

Crossref

BK from quenched overlap QCD

B_K from quenched overlap QCD

B_K from quenched overlap QCD

Abstract

Similar works

Full text

Available Versions

CERN Document Server

Crossref