Results from overlap valence quarks on a twisted mass sea
aa r X i v : . [ h e p - l a t ] O c t Results from overlap valence quarks on a twistedmass sea
N. Garron,
NIC, DESYPlatanenallee 615738 Zeuthen, GermanyE-mail:
L. Scorzato ∗ , ECT*Strada delle tabarelle, 28638100 Trento, ItalyE-mail: [email protected]
For the ETM Collaboration
We present results of lattice computations using overlap fermions on a twisted mass background. N f = a ≈ .
09 fm) and largevolumes ( V / a = × B K , for which we present first results. The XXV International Symposium on Lattice Field TheoryJuly 30-4 August 2007Regensburg, Germany ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ verlap valence on a Twisted Mass sea
L. Scorzato
1. Introduction
Dynamical overlap simulations are extremely expensive. One interesting possibility is to use adifferent regularization for valence and for sea quarks. In fact, valence quarks are much less criticalfrom the costs point of view (since they only appear in the final measurements, as in the quenchedcase), but much more critical from the point of view of the symmetries. In particular we can use the“twisted mass” (tm) regularization for the sea quarks, and the “overlap” (ov) regularization for thevalence quarks. This so called
Mixed Actions approach [1] is very promising, since it can stronglyreduce (or even completely eliminate) the operator mixing problem and has the potentiality ofdelivering the most precise and cost effective results in the near future.Violations of unitarity by lattice artifacts, which are expected, can be studied analyticallywithin ChPT. They may also take the form of ( O ( a ) suppressed) double poles, just like in Par-tially Quenched QCD, but a closer inspection suggests that these might be small in practice [2].Moreover, since the exact (twisted mass) sea quark matrix is available, they can also be studiednumerically. This is important in order to keep lattice artifacts under control. A first test is re-ported in [3]. Numerical simulations using a similar “mixed” approach has been reported by othercollaborations also in this conference [4, 5, 6, 7, 8, 9].In this proceedings we present our first physical results obtained with this approach. Thepresent analysis, which is done with limited statistics, is mainly meant to check the set-up that weare using in order to decide on possible improvements.The outline of this work is as follows. In the next section we describe the detailed set-up of ourcomputation. In section 3. we give physical results on the pion sector. In section 4. we discuss thecomputation of renormalization factors, which is done using the RI-MOM method and the WardIdentities. In section 5. we comment on our preliminary computation of B K .
2. Details of the computation
The gauge background that we use in the present work consists in the Twisted-Mass gaugeconfigurations which have been produced by the ETM Collaboration [10, 11]. We summarize herethe main features. We use twisted mass fermionic action at full twist with N f = b = . a ≈ .
09 fm. The volume is V / a = ×
48. In the present study we consider a single valueof the sea quark mass a m = .
004 (the lightest available), which corresponds to a pseudo-scalarmass m p ≈
300 MeV. As mentioned in the introduction, these first results are obtained with a lowstatistics of 54 independent gauge configurations. For more comments about the choice of thisbackground for sea quarks we refer to [10, 11].Valence quarks are described by the overlap operator [12]: D ( m ) = ( r − am ) D + m , D = a ( + A √ A † A ) , A = aD W − r , (2.1)where D W is the Wilson Dirac operator and r is a parameter that we set equal to one, in order tooptimize the locality properties of D [3]. Before applying the overlap operator we perform a single2 verlap valence on a Twisted Mass sea L. Scorzato
HYP-smearing transformation [13]. The computation of the propagators is done with point-likesources chosen randomly on the whole lattice. The inversions are performed by computing exactlythe lowest 40 eigenvalues and then using the SUMR algorithm [14, 15] with adaptive precision[16]. Thanks to a multiple mass procedure [17], which can be extended to the SUMR solver [16],we produced propagators for a wide range of bare masses down to am = .
006 and covering theStrange and Charm range. This brought a negligible loss of precision at high masses. The cost ofthe computation of one full propagator is equivalent to the cost of producing a few independentgauge configurations. In order to understand whether the continuum limit is convenient in thisapproach, it will be important to check how the above cost ratio will scales when a →
0, at fixedphysical volume.In a previous report [3] we discussed a wide range of tests performed on smaller lattices andwe will not repeat them here. We only mention that the comparison of the scalar correlator shownin [3] was not repeated in the larger lattice, since the low-mode averaging [18]– that is necessaryto have a clean scalar propagator – is rather expensive and we prefer to look at more physicalquantities first.
3. Results in the pion sector
The first quantity that we consider is the pion mass, since this is also what we use to matchthe valence quark mass with the sea quark mass. This is shown in Fig. 1. The horizontal line(with tiny error-bars) marks the pion mass obtained in the “unitary” (tm-valence, tm-sea) set-up.From this comparison, the matching point is estimated to be (in the overlap bare quark mass) at am = . ( ) . The matching of one quantity implies of course that other quantities are onlymatched up to lattice artifacts. The hope of this approach is that these are not too large in physicalquantities.The pion decay constant f p can be computed in a number of ways. The most interesting oneis the one which does not rely on any renormalization factor: f p = mm p |h | P | p i| . This can be compared directly with the tm-valence tm-sea result [10], which is also O ( a ) improved.In this approach f p turns out to be about 10-15 % larger than in [10], at the matching point, butalso the error-bars are of the same order of magnitude, and therefore still compatible. It is clear,from this analysis, that some kind of noise reduction techniques as those employed in [10] wouldbe important.It is also possible to compare our results for the pion masses and the pion decay constantswith Chiral Perturbation Theory. The necessary Partially Quenched formulae have been computedin [19] and the corresponding finite volume corrections in [20]. This comparison is shown inFig. 2. The dashed lines show the fit at finite volume, while the solid ones show the correspondingextrapolations at infinite volume. This gives a value of f which is larger than [10], as is clear fromthe considerations above. 3 verlap valence on a Twisted Mass sea L. Scorzato ( a m p ) am Figure 1:
Matching the valence-valence and the sea-sea pion masses. ( a m p ) / a m am a f p am Figure 2:
Fit of the data against Chiral Perturbation Theory at finite volume (dashed lines). The solid linesare the extrapolations of the curves at infinite volume. The pion mass is plotted in a way to make the presenceof non linear corrections more evident.
4. Renormalization constants
The renormalization factors have been computed with the RI-MOM method [21]. This ispossible since the gauge configurations had been (Landau) gauge fixed before the computation ofthe propagators.It is important to note that the tree level overlap operator is different from the Wilson oper-ator and for r = verlap valence on a Twisted Mass sea L. Scorzato Z V Z A 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.01600.0020.0040.0060.0080.010.0120.014 m m P C A C Figure 3:
On the left: Plateaux for the RI-MOM determination of the renormalization factors Z A and Z V .On the right: PCAC Ward Identity. renormalization constant Z y as: Z y ( m , g ) = − i
148 Tr [ g n p n S − ( p )] w ( p ) (cid:229) m sin ap m | p = m w ( p ) = h sin ( ap ) + ( sin ( ap ) − r ) i − The other definitions are unchanged with respect to [21]. We computed the renormalization factorsfor all bilinear fermionic operator and for some choices of four fermions operators. In general wefind that the chiral extrapolation is very stable, although the plateaux are not always completelyclear. As an example, we show in Fig. 3 (left) the plateaux for the renormalization factors ofthe Vector and Axial currents. In these cases we obtain, in the chiral limit, Z V = . ( ) and Z A = . ( ) , where the errors are only statistical. These can be compared with the renormalizationfactors obtained from the PCAC Word Identities. The relation between bare and PCAC quarkmasses is displayed on the right hand side of Fig. 3. From this Ward Identity one can derive Z A = . ( ) , where the errors are only statistical.The RI-MOM method can also be used to determine the renormalization factors of the fourfermions operators. In particular, in the next section, we are going to use the renormalization factorof the operator O D S = = [ ¯ s g n ( − g ) d ][ ¯ s g n ( − g ) d ] . The RI renormalization factor can then beconverted into the renormalization group invariant one using the anomalous dimension computedin [22]. This gives us Z RGI B K = . ( ) . The momentum dependence of Z RI B K ( m ) and Z RGI B K in thechiral limit are shown in Fig. 4.
5. To-wards the computation of B K . Comments and conclusions An obvious quantity which is particularly interesting in this approach is B K , the Kaon bagparameter, which is related to the mixing of ¯ K and K by the expression: h ¯ K | O D S = ( m ) | K i = M K F K B K ( m ) In fact a precise non perturbative determination of B K would have a strong impact on the deter-mination of the associated CKM matrix elements. Moreover, it is only with an exactly chirallysymmetric regularization that the operator O D S = cannot mix with other operators (without need of5 verlap valence on a Twisted Mass sea L. Scorzato Z RI ( m )Z RGI m p BK Figure 4:
Left: The momentum dependence of Z RI B K ( m ) (bottom) and Z RGI B K (top) at the chiral limit. Right:The bare factor B K as a function of the pseudo-scalar mass. The Kaon mass corresponds here to am p ≃ . relying on any tuning procedure). Finally, we have now the possibility to remove the quenchingerrors.We computed B K in a standard way employing the propagators described above. More pre-cisely we use the same procedure described in [23], although we use lighter quark masses. In par-ticular it was important to use the left hand current. Our results are shown in Fig. 4 and imply forthe bare B-parameter B lat K = . ( ) and for the renormalization group invariant one ˆ B K = . ( ) (errors are only statistical). Although the error-bars become very large at light masses, they are stillreasonable at the Kaon mass, which is relevant for B K . Nevertheless, some kind of noise reductiontechnique would be probably helpful and we are currently exploring those used in [10].Comparison with ChPT has been performed using the formulae in [19, 20], and the inclusionof appropriate lattice artifacts can be done following the procedure in [24]. Acknowledgments
We wish to thank the members of the ETM Collaboration who have contributed to the resultspresented here and in particular O. Bär, K. Jansen, S. Schaefer, A. Shindler. L.S. acknowledgesINFN for support and NIC/DESY for hospitality. Numerical work has been done in the SGI altixat HLRB (München), in the IBM p690 at ZIB (Berlin) and in the BEN cluster at ECT* (Trento).We are grateful to A. Vladikas and V. Lubicz for useful discussions about the RI-MOM method.
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