weak matrix elements from lattice qcd

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Weak Matrix Elements from Lattice QCD Damir BeCireviC”

“Laboratoire de Physique Theorique (Bbt.210), Universite Paris Sud, Centre d’orsay, 91405 Orsay-Cedex, France. This is a critical overview of the current status of the computation of the weak hadronic matrix elements involving kaons and pions on the lattice. A special attention has been devoted to the computation of the BK parameter, for which the world average remains BK = 0.87(6)(13). The second (dominant) error is the uncertainty due to the use of the quenched approximation. Most obstacles in computing the K -+ 7rrr matrix elements are also related to the use of quenched approximation. The main message is: to make a significant progress in computing the weak matrix elements on the lattice, the unquenching is necessary.

The theoretical description of very many experimentally measured weak interaction processes involving kaons and pions is usually plagued by the lack of a good control over the hadronic uncer- tainties. Even though we know quite a bit about the underlying dynamics governed by &CD, we are still lacking a thorough quantitative understanding of the non-perturbative nature of the confinement and its interplay with the sponta- neous chiral symmetry breaking effects. Lattice QCD is the only well defined framework that, in principle, allows the precision determination of the hadronic matrix elements. To do the calcula- tions, however, one has to introduce various approximation, each inducing some systematic uncertainty in the final results, the amount of which is difficult to assess. The whole beauty of the lattice approach lies in the fact that all approx- imations are removable [or, at least, reducible] by either sufficiently increasing the computing power or by some clever theoretical improvements of the lattice approach. Among the approxima- tions nowadays used in the lattice QCD simula- tions, the most infamous is the quenching. Soar- ing problem is also our inability to work with light quarks close enough to the physical r&/d, so that the matrix elements should be extrapolated from the directly accessed “heavy pions” to the physical ones. This is where the chiral perturbation theory (ChPT) becomes helpful. ChPT is an effective theory of QCD at the very low en- ergies, allowing one to compute the matrix elements in terms of Goldstone bosons. The pre-

cision determination of the matrix elements by means of ChPT alone is not possible: (i) The very matching of the expressions derived in ChPT with their high energy QCD counterparts is necessar- ily model dependent; (ii) Besides the non-analytic quark mass dependence, one also has to include the terms that are powers of quark mass, multi- plied by the low-energy constants, the values of which are a priori unknown. The status of the calculation of the hadronic matrix elements by means of ChPT has been nicely covered at the last year’s Lattice conference [l].

Before I embark on the specific topics, I need to say that this brief review is somewhat different from the ones presented in [2]. Instead of cov- ering all aspects of the kaon physics, I chose to discuss the topics that are most directly relevant to the CKM phenomenology in which the input from lattice QCD is often decisive [3]. A bulk of this talk is devoted to the BK-parameter, for which the broad HEP community expects the lattice QCD physicists to provide the accurate value. A detailed situation with the challenging problem of dealing with the non-leptonic K + TUT decay on the lattice has been excellently reviewed last year [4], to which I will only give a few comple- mentary information and comments.

1. Kaon decay constant

The simplest weak decay hadronic matrix element involving a kaon is

W?-Y~~~~IW = i.fK& 7 (1)

0920-5632/$ - see front matter 0 2004 Elsevier B.V All rights reserved. doi:l0.1016/j.nuclphysbps.2OO3.12.083

D. BeMrevid/Nuclear Physics B (Proc. Suppl.) 129&130 (2004) 34-45 3.5

parameterised by the decay constant f~, whose value is known from the experimentally measured I’(K-+~~~), namely f~ 21 160 MeV. So far, on the lattice, kaons were not computed directly. We computed a pseudoscalar meson (P) whose valence quarks (4’) were degenerate, with m4j tuned in such a way as t;h;Fduce the physical kaon mass, mp = mK . A matrix element for such a kaon is then computed and ‘<fK” extracted. If, instead, we fix the strange quark to its physical value and push the light quark mass as light as possible, a chiral extrapolation, m,+m,/d, will be necessary. The quenched ChPT (QChPT in the following) [5], indicates that the non-degeneracy of the valence quarks im- plies the appearance of the quenched chiral logs which diverge as the light quark is sent to the chiral limit. A clear numerical evidence for the presence of the quenched chiral logs in the case of f~ is still missing. A promising strategy for such a study would be the extension of the work done in ref. [6].

A thorough unquenched study of fK, f* and f~/f~ has been (for the first time) performed this year, and the results presented at this conference [7]. By working with the improved staggered quarks, they were able to reach the very light quarks, T E m,/m, E (0.12,0.7). The main diffi- culty of the method with staggered quarks is the problem of dealing with extra flavours (tastes): each of the dynamical quark in the simulation with nf =2$-l, has its three copies, whose effect is removed by simply taking the fourth root of the dynamical Dirac determinant. In ref. [8], the authors constructed the ChPT for the staggered fermions that contains the very same assumption, i.e. (Det)lj4, so that they could guide the re- maining extrapolation to the physical light quark mass, i.e. to r,/d = 0.04 (see fig. 1). From the fit of the numerical data to such a ChPT expressions, they were also able to estimate the size of the artifacts due to the contamination of the operators with LLwrong taste”. For fK such effects are at the -10% level, and are much smaller for fK/fX. The result they quote is

fK = 155.0 & 1.8 & 3.7 MeV, fK/f* = 1.201 Zk 0.009 f 0.015, (2)

where the systematic error in fK is mostly due to the scale setting uncertainty. Their fit to the ChPT expressions also allowed them to extract the low energy constants LQ, at p = m,. In units of 10p3, they obtain

L* = 0.3(3)(‘;), L5 = 1.9(3)(‘;). (3) Before promoting these as complete lattice QCD results, it is important to investigate the potential problems of the approach with dynamical staggered fermions induced by the non-locality of the “(Det)‘i4” practice.

0.22

0.18

dL = 0.85; 0.17 0 coarae 0 fine

extrap; &yfltematic err _ iii experiment

0.18 ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ( 0.10 0.15 0.20

(mx+myh

Figure 1. Preliminary unquenched (nf = 2+1) result by MILC [7]. Staggered quark action is used and the fit in the light quark mass is made by using the expressions derived in ref. [8].

To disentangle the presence of the chiral logs on the lattice, besides going to ever smaller quark masses, it is important to keep the finite volume effects under control. Those have been recently investigated in ref. [9] by confronting the chiral loop corrections in the finite and infinite volumes, namely

+m2, log m2,],

fK (L) “lOg’l 1 -- f

tree 8f2L3 (4)

36 D. Be&reviC/Nuclear Physics B (Proc. Suppl.) 129&130 (2004) 34-45

where w; = rn$+f2 (P = r, K,v), and <= Fri To solve the problem of eq. (5), besides the (n’ E 2”) and, for simplicity, I set p = 1 GeV. As ongoing experimental effort to reduce the errors it can be seen from fig. 2, the chiral log behav- and improve the consistency among various Kz3- ior gets modified by the finiteness of the volume. experiments, a theoretical improvement to com- That plot shows how hard it is [if at all possible!] pute the relevant form factors is needed. Recall to disentangle the physical chiral log (the one cor- that the hadronic matrix element that appear in responding to L -+ co) from the chiral behavior the Kes decay amplitude is parametrised by the that can be measured in the finite lattice box. two form factors as

MN$V-(PK)) = (PK +P.,,) F+(q2)

__ L=l.Ofm < ,., L=l.S fm

/

- - L=2.0 fm ^. p2.5 fm

- L-e -

‘e 0: 0 0.2 0.4 0.6 0.8

r

Figure 2. Finite volume effect modify the chiral behavior in fK in which the strange quark mass is kept to its physical value, whereas the light quark r = m4/rn~hys~ is varied between 1 and the physical r,/d = 0.04.

2. Semileptonic K-decay

The standard direct way to determine of the famous Cabbibo angle, or the (V&]-entry in the CKM matrix, is to divide the experimentally measured width, I’(K’ -+ &epe), by its theoretical estimate. On the other hand, the unitarity of the CKM matrix gives IVud/2 + ~Vus~2 + II&j2 = 1, so that IV,, 1 can be deduced indirectly, from the accurate value for lVudl = 0.9739(5), with lVz,b12 being totally negligible. After recent theoretical and experimental progress, it turned out that the two determinations differ by about 2a [lo], namely

Iks)direct = 0.2201(24), I&s/indirect = 0.2269(21) . (5)

To those, we could add the third estimate, by using the preliminary result for f~ from ref. [7] and the measured l?(K+ -+ p+vti + p+v,y), from which I get IV,,l = 0.2266(74) [ll]. This should finally convince a reader that the accurate full QCD estimate of f~ is highly important.

+4/1 m$-m2,

q2 (Fo(q2) - F+(q2)) >(6)

where Vzs = @y,s, q = PK -P,, q2 E [mz, q,$& with qkax = (mK - m.,,)2. In addition, F+(O) = PO(O) E F(0). The form factors are parameterised as

F+,o(q2) = JYO) (1 + 4” X+/o) > (7)

(I+,, is a parameter) or by a simple pole dom- inance formulae. The main concern is to determine the value of F(0). The ChPT result gives [12,13]

F(0) = 1 - 0.023 o(p4) + O.O1b(p6) + f > (8)

where the first term reflects the conservation of the vector current, the second and third terms are the chiral loop corrections, and the term that contains the analytic corrections to the Ademollo- Gatto theorem is denoted by f. Lattice QCD can be used to evaluate f, related to the slope (a) in

F(o,mK,mr) = 1 - o!(m& - mz)2. (9)

In their preliminary study [14], the SPQcdR adopted the strategy previously proposed by the Fermilab group in the study of the Isgur-Wise function [15]. By taking the mesons at rest, p’K = & = o’, the following double ratio

(10)

gives a very accurate value of the scalar form fac- tor Fo(q,&,). To get F(0) from from Fo(q,&,) one can consider the kinematics $K = 8, and p’ E jif 0, and from the ratios

(11)

D. Bedirevik/Nuclear Physics B (Proc. Suppl.) 129&130 (2004) 34-45 31

extract Fe($)/F+(q2). From the results at several kinematical configurations one can reconstruct F(O), for various values of m, and mK.

From the fit to a form (9), the preliminary result iS

a = 0.12(3) GeV4, (12)

where the error is statistical only. The SPQcdR results are obtained at the single value of the lattice spacing and in the quenched approximation. Thus these results can be improved in many ways.

3. K” - K” mixing amplitude

In this section I try to make a critical overview of the various strategies employed to compute the K” - ?@ mixing amplitude on the lattice. A critical outlook is important for at least two reasons: (1) a huge impact of BK on the CKM phenomenology, (2) this is the benchma,rk calculation for any 4-quark matrix element, and without clarifying the issues appearing here, there is little (or no) hope to handle more complicated matrix elements appearing in K -+ TT.

EK=d(KL -+ 7r7r)/d(Ks -+ 7rn), with (~7r)~,~, is the quantity that measures the indirect CP- violation in the system of neutral kaons. In Stan- dard Model, the only hadronic matrix element that appears in this calculation is

(~IOI(P)IK~) = $k.fiBK(p), (13)

where Ol=VV+AA = ~y,di?y,d+ .?y,y5d~y,y~d, and BK(/L) is the so called bag-parameter. Oi is logarithmically divergent and its scale dependence is described by the renormalisation group equation, with the leading order anomalous dimension coefficient being universal, yol - (‘I - 4. To specify the renormalisation scheme it is necessary to go to NLO. 7;; has already been computed in several renormalisation schemes [16-181. For an easier comparison of B~(p)‘s, computed by various groups in different renormalisation schemes, it is convenient to integrate out the scale dependence and define the renormalisation group in- variant bag-parameter, Bx. For example from a frequently quoted BgS(2 GeV), one gets &K = - [1.388(8) . BzS(2 GeV)], where I used nf = 0

and its corresponding A”f=’ MS = 250(25) MeV. *

By combining the fi)K with the perturbatively computed Wilson coefficient, Inami-Lim func- tions [20] and the accurately determined ]ez”.] = 2.280(13) x 10p3, one gets the hyperbola in the p-?j plane, shown in fig. 3. The width of the hyperbola almost completely reflects the error on g, and hence such a great interest in reducing it.

G=

I

P

Figure 3. The constraint onto the shape of the CKM unitarity triangle coming from &K hyperobola is shown together with the constraint on sin 2p coming from B -+ J/$Ks. The contoured region comes from the B-physics constraints.

The matrix element (13) has been computed by using various formulations of QCD on the lattice, each having its advantages and disadvan- tages. A summary of results is given in table 1. In the computation of the matrix element (13) the chirality is very important: From the UV point of view, the renormalisation procedure is very much simplified if the chirality is de facto preserved. Otherwise, before the standard multi- plicative renormalisation, the subtraction of the dangerous mixing with other AS = 2 operators should be made. On the other hand, the lack of chirality invalidates the use of “the natural strategy” to compute BK(,u), because in that case the chiral expansion goes like

(K”IO~(p)IKo) = bo + blrn& + bzrn& + . . . (14)

*With three-active flavors, and by usingd$F3 =

338(40) MeV [19], I get BK = [1.382(15) I?$‘(:! GeV)], i.e. the nf-dependence in the anomalous dimension is en- tirely balanced by the on in the running coupling.

38 D. BeCireviC/Nuclear Physics B (Proc. Suppl.) 129&130 (2004) 34-45

Reference Quark Action Renormalisation Lattspacing Ratio

CP-PACS [26] DWFIwasaki l-1oopm o* R; 0.80 f 0.03

P71 RII 0.97f 0.03

Garron et al. [28] Overlap ~p~R’/hto” 0.1 fm RI 0.87f 0.08

MILC [29] Overlap+HYP I-1oopm 0.13 fm RI 0.75zk0.04

0.09 fm 0.76 f 0.03

Table 1 Collection of the quenched lattice estimates of i?lx. New results are written in bold. ‘i,” means that the particular point is suspect and is further discussed in the text. RI refers to eq. (15), and RII to eq. (18)

where both kaons are kept at rest. Only the explicit chirality ensures that a0 = 0 [otherwise it is O(an)].

3.1. Extracting BK The above mentioned “natural strategy” means

simply

(~mw (A(O) %7>'it,))

RI zz -'-

;&i> hJAo@N~ AoWW ty>)

If the chirality is preserved

(15)

R,P -+ blm‘$ + bzrn& + . .

~.fi$m& = BK + . . . , (16)

but if it is not preserved then

mK stands for a generic pseudoscalar meson, so that for the degenerate valence quarks in the “kaon” (which is the common practice), the first term on the r.h.s. of eq. (17) diverges as l/m,.

A backup strategy to avoid the latter problem is to assume that BK does not vary with mass and extract it from the ratio [30]

The denominator now provides the needed mg- term to kill the divergence in the numerator, so that one has RyIp 4 O(an) + b’,mk + . . . To extract the BK, one then fits to

RNP_> 60 + blrn$ + . . . Wf7 ma WO) 8 IWOI~~)I~ I

if&m&

= 2 + BK + . . . (17) mK ((Op(Koj(2 = co + ““3 ((oppP)[2 + . ‘. (19)

D. BeCirevid/Nuclear Physics B (Proc. Suppl.) 129&130 (2004) 34-45 39

and the resulting CO -+ 0, as we approach the continuum limit, while cl -+ BK. Since such an extracted BK is assumed to be universal, nothing can be said about the difference between the physical BK and the one obtained by sending ?&-+O.

To summarise, the lattice actions explicitly pre- serving chirality (overlap and staggered quarks) can use the ratio RI to extract the BK. If the chirality is not manifest (such is the case with the Wilson quarks), the use of RI is ill when close to the chiral limit. t The studies with the domain wall fermions are problematic because the chirality is only in principle preserved. In practice, it is explicitly broken as the residual mass term mres # 0, most probably due to the finiteness of the 5th dimension. From the above discussion; it is clear that nothing conclusive can be said about about the behavior of BK as the (quark) meson mass is lowered. For the physical BK, both strategies RI and RII are necessary, at least for the systematic errors estimate. Since the error bars in the available domain wall fermions results are very small, the world average lattice estimate would be almost completely determined by those values. If RII is used to extract BK, a very different value for BK is obtained [27]. More research is needed to settle this problem before the domain wall results enter the world average for BK.

The new results for BK obtained for several pseudoscalar mesons around the physical kaon mass are plotted in fig. 4. All three groups obtain the results by using RI. Before promoting the de- viation of the small quark mass behavior as the physical chiral log behavior, I should stress that a plot very similar to the one shown in fig. 1 ap- plies also to BK: finite volume effects overwhelm the physical chiral-log behavior (see ref. [9] for details). For that reason, the full control over the chiral extrapolations (to obtain B$), is difficult even if the quark action preserves the chirality.

t “Close to the chiral limit” is loose enough so that even when working with kaons > 500 MeV, one should use also RII and include the difference of the respective results in the systematic error estimate.

0.7

I”““” “““““““‘i

0.5 1 1.5 2 2.5

Figure 4. BK parameter for several masses around the physical kaon mass, obtained by using RI. The corresponding references are: MILC [29], Garron et al. 1281 and Lee et al. (241.

3.2. Renormalisation and matching All the quark actions, except for the staggered

one, have a nice feature that the form of the operator that is computed on the lattice is the same a,s one computed in continuum. The general formula that ensures the matching of the operators computed in the lattice regularisation scheme (e.g. 01 (a)) to the one renormalised in the continuum is

01(p) = 21(w) [ 01(a) + $A,(u)O,(a) , (20)

k=2 1 where A, (a) E 0 for the chiraly symmetric quark actions, and Ak(u) # 0 otherwise (dAk/dp = 0). If the chirality is not preserved, the spurious mixing with 02-~(u) should be substracted away. To see the necessity for subtracting, let us choose the standard basis of parity even, AS = 2, operators, 02 =VV-AA, 03 =SS-PP, 04 =SS+PP, 05 =TT, in an obvious notation. After sandwich- ing eq. (20) by kaons and assuming the vacuum saturation approximation (VSA) for the matrix elements (Ok(u)) 2 (K”]O~]Ko), we have

(01(p)) = &(W)(Ol(U)) [l- 6A2-;3-A5 4A2-6A3i-5A.+-0As L 4 -

8 (w(u) + ms(a))2 1 (21)

40 D. Be&revid/Nuclear Physics B (Proc. Suppl.) 129&130 (2004) 34-45

We again see that the last term diverges when close to the chiral limit (and with degenerate quarks), again showing the illness of extraction of BK by using the ratio RI if the chirality is not fully preserved on the lattice. The above use of the VSA is only illustrative. It however shows that even if the mixing coefficients, A,(a), are small, the overall effect of subtractions cannot be ignored because (Os-s(a))/(Or(a)) > 1. The methods for computing Ah (a) non-perturbatively exist [31,32]. The groups using the domain wall fermions so far ignored the mixing indicated in eq. (20). The RBC-group indeed verify that Ak (u) are small, and argue that those coefficients scale as r$,, , but in practice they do not subtract the chirally divergent mixing. That is certainly a dangerous practice if one uses RI to compute the BK-parameter.

There are two ways to compute (01(p)) and avoid the troublesome mixing in spite of the lack of chirality [33,34]. First method relies on the use of the chiral Ward identity to connect the matrix element of the parity violating operator &=VA+AV, to the wanted Or. SPQcdR already implemented this method in the actual computations of BK [21]. The second method is to use the twisted mass QCD (see review in ref. [36]). The axial rotation equivalent to the one used in the first method is made by a specific choice of the twisting angle in the extra mass term in the tmQCD action (which contains the rs-matrix). Alpha group pursued this line and the first results presented at this conference are encouraging [22], although the nice property of tmQCD, that there are no exceptional configurations, has not been explored to get to ever smaller quark masses.

Matching of the lattice regularised staggered Optt (which is not of the same form as in the continuum) onto Oynt is not feasible beyond 6(a,) in perturbation theory because of the tremen- dous proliferation of the lattice operators with the same quantum numbers but with ‘(wrong ta,stes”, which mix with the “correctly tasting” ones. The problem is that the I?(a,)-corrections are large implying an a priori large systema.tic error in BK. This year it has been shown that for the staggered Optt defined in the 24-hypercube using the fat

(HYP)-links [37], the C?(a,)-corrections become much smaller [38]. The preliminary numerical results of such a study with HYP-links have also been presented at this conference [24].

3.3. Continuum limit Although the disretisation errors on staggered

BK are of 0(a2), the scaling to the continuum limit looks quite bad [23]. It is not clear if this feature is a consequence of the contamination by the ‘Lwrong tastes”, or of the large O(a,)-corrections in the renormalisation. It will be interesting to see what a study at various lattice spacings by using the HYP-links, will reveal in that respect.

On the Wilson action side, the values of BK are accompanied by 0(a) artifacts. By using the method without subtractions, and by computing at three fine lattice spacings, the SPQcdR extrapolated BK to continuum limit, but only linearly in a. At least one more point in a is needed to make that continuum extrapolation more trustworthy.

The CP-PACS group showed that their BK as obtained at two lattice spacings by using the domain wall fermions remain the same. Instead of the Wilson plaquette gauge action, they worked with the improved one (a la Iwasaki) which results in the smaller chirality breaking (smaller residual quark mass). t

It is fair to say that none of the continuum extrapolations is not fully satisfactory at present. The improvement in that respect is desirable.

3.4. Final result Before quoting the final lattice result, two more

comments are necessary. (1) The most troublesome is the use of quenched approximation. Only exploratory partially quenched studies with nf = 2 have been made so far [39-411, each indicating no difference between Bz=’ and Bz=‘. Before the results of a complete unquenched study with nf = 2 (and even better with nf=2+1) becomes available we should live with the Sharpe’s guesstimate of 15% of the systematic error due to the use of quenched approximation [42].

iFor the same reason, the RBC group nowadays use an improved gauge action (known as DBWB). For new results see ref. [25] (also displayed in tab. 1).

D. Bedirevib/Nuclear Physics B (Proc. Suppl.) 129&130 (2004) 34-45 41

(2) The effect of non-degeneracy of the valence quarks in kaon, compared to the commonly used kaons with degenerate quarks is expected to be small (2 - 3% from the chiral logarithms alone). Although the scaling violations are quite unpleas- ant with staggered fermions, the chirality, the extensive precision measurement over a wide range of lattice spacings, are good enough arguments to quote the JLQCD value as the current lattice estimate. The other estimates agree with JLQCD (within the error bars). So finally we have

l?K = 0.87 -+ 0.06 zt 0.13, (22)

where the last error is irreducible within the quenched approximation. Therefore, unquenching is the most important thing to do in the years to come.

4. K t TUT and E//E

The direct CP-violation in the K + m de- cays has been detected and recently accurately measured [43]: E’/E = (16.6 f. 1.6) x 10m4. This quantity is still a nightmare for the theory since various contributions tend to cancel one another. Moreover, the computation of many relevant 4- quark operators on the lattice is tremendously difficult. Let us briefly recall that

ieix14 ReAa ImA E’ = - fi ReAs ReAz --k-3 > (23)

w P2 PO

where the index in the amplitudes refer to the to- tal isospin of the resulting two pion state I = 0,2. The observed large enhancement of AI = l/2 amplitude w.r.t. AI = 3/2 (l/uexp. = 22.1), is a long standing puzzle. The standard application of OPE indicates that the short distance physics brings only l/w!$O w 2, while the rest is non- perturbative effect [44]. That is where the lattice QCD is expected to play a crucial role. I will only briefly report on the recent “progress” made in dealing with this complex issue. For a more complete discussion of the lattice strategies please see ref. [4].

There are three strategies to compute the weak matrix elements of the operators appea,ring in the A~,J amplitudes: (1) indirectly, by computing the

matrix element of K --+ 0, K -+ 7r transitions, and then relate them to K -+ 7~ be means of ChPT [47]; (2) directly, but in the unphysical kinematics, and (again) rely on the ChPT to get to the physical Ao,x amplitudes [48]; (3) directly, by using the Lellouch-Liischer (LL) formula [49].

4.1. K -+ m from K + 0, K + T This strategy has been recently actively pur-

sued by RBC and CP-PACS groups [50,51]. In measuring ReAs, although they do not fully agree with each other, they both reach a very small value for (St’,“), current-current operators. The calculation of these matrix element is equivalent to the computation of the BK parameter in the chiral limit, since (r+IQf$jKi) =

CWLf~~2,B& As we discussed in the previous section, the explicit breaking of the chiral symmetry (which generate the mixing with chirally divergent operators), as well as the modified chiral- log behavior due to the finite volume, are suffi- cient reasons to question the validity of the claim that (Qf’,“} are very small. Notice that the small- ness of these matrix elements renders their l/w close to l/wexp,. The ReAo amplitude contains the same operators but obviously with AI = l/2, which complicates the renormalisation procedure: the matrix elements of such operators contain the “eye’‘-shaped diagrams (“penguin” contractions) which induce mixing with the lower dimensional operators Qsub = (m, + md)sd + (m, - m&y&. The subtraction prescription to get rid of that power divergent mixing is needed. Such prescription exists [47,56] and consists in using the CPS symmetry to impose (01&i - aiQsubIK) = 0, and thus determining the subtraction coefficients oi. The calculation of ReAs is however easier if the charm quark is not integrated out, but kept dynamical, thus allowing the GIM mechanism to act efficiently. In such a picture, by a clever adjust- ment of the twisting angles in the tmQCD, the power divergent mixing can be eliminated alto- gether [52]. Notice also that the power divergent mixing for the current-current opera.tors in the lattice formulation of QCD in which the chirality is de facto preserved, such as overlap fermion action, is absent [53]. Those facts have not so far

42 D. BeCireviC/Nuclear Physics B (Proc. Suppl.) 129&130 (2004) 34-45

been explored in the numerical studies. 5 Back to the picture in which the charm is in-

tegrated out. This year RBC presented new results [25] which corroborate their previous (un- fortunate?) finding that the operator and the subtraction are of similar size. Moreover, ReAs also includes the famous QCD penguin operator Q6 = (gadb)v-~ Co=u.d,s(Qbqa)v+A, which is widely expected to provide the enhancement of ReAs, and thus the solution to the AI = l/2 puz-

zle. The results presented earlier [50,51], as well as the new one [25], indicate that this is not the case: the signal is smaller than expected and almost completely washed out by the subtraction. A much larger value has been presented this year by Lee et al. [24], who computed (Qs) by using the staggered+HYP quarks. In terms of the bag parameter Bg, defined as

((7r7r)1=OlQ6(~)IEo = - f7T)

( mk > 2

’ m&u) + m&CL) BB(PL)> (24)

they quote -

BUS = 0.73(g), (25)

which is much larger than BfS(m,) = 0.3, obtained in ref. [51], . There is probably no need to get too much upset about this disagreement, because the computation of the AI = l/2 operators in the quenched approximation is known to be plagued by several severe problems. In ref. [57] it has been observed that while in the full theory Qs E (8~, 1~) irreducible representation of the sum @ sum group, in the quenched theory that is not the case, because SU(3) -+ SU(3[3) when passing from the full to the quenched theory. Qs is not a singlet under sU(3]3)~. There- fore the quenched (and/or partially quenched)

Q6 =Q,“+Qt? (26)

or singlet (S)+ non-singlet (NS) piece, the latter being the unphysical operator. This year it has

IAnother interesting possibility, proposed in ref. [54], and known as “OPE without OPE” has not been attempted in &CD, although the extensive tests with non-linear sigma model brought optimism that the method might be feasible in QCD as well. [55].

been argued that (Qfs)/(Qf) >> 1 [58]. Thus, any quenched lattice evaluation of the (Qs) matrix element seems completely doubtful. A possible way out, proposed in ref. [57], is to simply omit the tjq contractions in the ‘(eye” and anni- hilation diagrams. This practice, however, not only kills the unwanted (unphysical) Q,““, but it also kills a part of the physical Qf matrix element . The recent study, by means of ChPT combined with the large N, expansion suggests that the contribution of the “eye’‘-contraction to the physical (Qs) is essential [59]. Therefore the Golterman-Pallante (GP) recipe can lead to a wrong answer for (Qs), too. With that remark in mind, I quote the new result by Lee et al. [24], in which they use the GP-recipe to get

- BFS(m,) = 0.98(7)GP. (27)

Once again, this like all the other presently available results- is obtained in the quenched approximation which is known to suffer from multiple problems, of which the computation of the QCD penguin operators seems to be the most affected.

As for the computation of the matrix elements of the electro-weak penguin operators, the main problem is the uncertainty induced by the use of ChPT in relating the actually computed K + rr matrix elements to the K + rrr ones: the use of the ChPT expressions derived at tree- level and l-loop leads to significantly different results [50,51]. As it can be seen from fig. 5, there is one more problem: the chiral logs can be included everywhere (dot-dashed line) or starting from the lowest point accessed by the lattice data for (rr+(Qi’2 IKf) (dashed line), which are otherwise nicely fit by a quadratic curve. As a result, the extrapolated value will move up-and-down resulting in the rather large systematic uncertainty of the final results.

The application of tmQCD and overlap fermion actions in getting to K + nrr amplitudes via the K -+ rr and K + 0 ones in the framework in which the charm is not integrated out, would be highly welcome, even though the ambiguity in the matrix element of the AI = l/2 penguin operators will remain with us. For the latter reason, unquenching (at least partially) is necessary. In ref. [60] it has been shown that from the K --+ TT,

D. BeCireviC/Nuclear Physics B (Proc. SuppI.) 129&130 (2004) 34-45 43

< TC+ Q3;‘l K+> fr. 5 5.

fi (GcV3)

which are obtained after including the physical (full) chiral logarithms in extra,polation, starting from (m,, mu) = (0.4,0.5) GeV, down to the physical (mghys, mghys ). The effect of variation of that point at which the chiral logs are included is

Qllaaralic (1, indicated in the la,st errors, whereas the other sys-

------ CWT.polynomd malching tematics mostly reflects the uncertainty in non- perturbatively determined renormalisation con- .-.-.-. Rlii c,,,ra, log 0, sta,nts.

I I 0.1 0.4 0.6 0.6

M; (M,,, in GcV) 2,

Figure 5. The resuks of the chiral extrapolation of the lattice the data of ref. (50] are highly sensitive to the choice of the point at which the chiral logarithms

1.5

I

08 - polynomial fit

are included in extrapolation. I thank David Lin for 11 I i I

this plot. 1 1

K -+ 0 and K + 7r,j7r,=j matrix elements computed in partially quenched &CD, with all mesons at rest and the weak operator carrying a momentum (to ensure the energy-momentum conservation), one can extract the complete set of low energy constants needed to reconstruct the general K -+ ~7r amplitudes at NLO in ChPT. The specific point with mK = m,, however, should be avoided as it would contaminate the results by the large finite volume effects [61]. Thus the partially quenched study is useful.

0.5 t

o M,(M,,M,,EJ/FIT p=O

1 0 M,(M,,M,,E,)/FIT p=2rc/L

ooi4 . I 0.8 1.2

M, (GeW

2L _ /

08 - quenched logarithy

1.5

1

0.5 0 M,(M,,M,,E,)/FIT p=O 0 M,(M,,M,,E,)/FIT p=PrclL

4.2. Direct K -+ m The SPQcdR group presented their final results

for the matrix elements of the electro-weak penguin operators MT,~ =: (mrjQf.$jK) [62]. The results are obtained by working in the specific kinematics with KG -+ 7~,yr~, where p’ E {a, F}. In the latter case, it is important to symmetrise, [(l~‘(p3~0(8)) + jh(@~~(p3)]/2, to remove the 1 = 1 component from the final 2-pion I = 2 sta,te. The l-loop ChPT formulae for this specific kinematics have been worked out in ref. [63]. As we can see in fig. 6, their data are much better fit if the (quenched) chiral logs are turned off, the phenomenon most likely due to the fact that their pions are not light enough (rn-,>l > 500 MeV). Their final results are

(mrlQ;“\K) = 0.664(57) (t;;) (‘ii) ,

(~nlQ;‘~lK) = O.lll(lO) (‘i) (5;) , (28)

0 0.4 1 . 018 i 1.2

M, (GeV)

Figure 6. The polynomial in m,, E, and mK fit better the lattice data for (x~lQ~‘21K), when the [quenched) chiral logs are left out. See ref. [62] for details.

The best way to avoid the Maiani-Testa theorem [64] and compute the K --+ (7r7r)1=~ amplitude directly on the lattice, is by means of the LL-formula [49]. To implement that proposal one needs a huge lattice L = 6.09 fm, which is not doable with the current computing power. The clever idea to get around that requirernent was proposed last year and further developed this year [65]. They propose to impose (what they call) the H-parity, i.e. H [u dT = [-u d] T , on the quark fields in the z-direction. In terms of pions, HIT*) = -ITT*), H~TT’) = ITO), i.e. it

44 D. BeCireui6/Nuclear Physics B (Proc. Suppl,) 129&130 (2004) 34-45

ensures the anti-periodic boundary condition on the pions. As a result, the lowest lying pion will have pr = n/L,, instead of zero. The important consequence is that in such a situation the LL formula, becomes valid for L M 3 fm, thus opening the possibility for the direct numerical study of the proposal of ref. [49].

The direct computation of the matrix elements of AI = l/2 operators on the lattice is very difficult and even if we were able to make them in the quenched (or partially quenched) approximation, the results would probably be useless. The recent QChPT study of ref. [46] shows that the princi- pal problem of (partially) quenched theory [lack of unitarity] creates insurmountable problems: (i) Watson theorem is not valid, i.e. the phase of the final state interactions is not universal (it de- pends on the operator used to create two pion state); (ii) there is no unambiguous way to form the time independent ratios of correlation func- tions to extract the desired amplitudes from the lattice data; (iii) amplitudes increase with the size of the lattice volume; (iv) Liischer’s quantisation condition [66] and LL formula [49] are not valid any more. In other words, the direct computation of the matrix elements of the AI = l/2 operators is sensible only if one is working in the full (unquenched) &CD. Personally, I would prefer to see some numerical work confirming (or contradict- ing) the pessimistic QChPT scenario of ref. [46].

5. Concluding remarks

Lattice QCD is our best tool to calculate the weak matrix elements. In dealing with the kaons and pions, the chiral properties of the lattice action and operators are often crucial. The computation of the decay constant f~, K~s decay form factors and numerous matrix elements of the 4-quark operators, are essential for our better understanding of the CP-violation in and beyond the Standard Model. The wide high-energy- physics community expects from us to reduce the errors on BK parameter as soon as possible. The way to do that is by unquenching. So my main message is: unquench, unquench, unquench... The necessity for unquenching is be- coming evident if we are to make a pertinent es-

timate of the K + 7r7r amplitudes and E//E, in particular.

Acknowledgements

I would like to thank the organisers for inviting me to this enjoyable conference, partial financial support by the organisers of “Lattice 2003”) as well as the EU network [HPRN-CT-2000-00145 Hadrons/LatticeQCD]. I also tha,nk all the para,l- lel speakers of the “Weak Matrix Elements” ses- sion, and to all my collaborators. Finally, I apol- ogise to everyone whose work has not been prop- erly cited.

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weak matrix elements from lattice qcd

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