Light quark masses and pseudoscalar decay constants from Nf=2 twisted mass QCD
aa r X i v : . [ h e p - l a t ] O c t Light quark masses and pseudoscalar decayconstants from N f = twisted mass QCD Vittorio Lubicz
Dip. di Fisica, Università di Roma Tre and INFN, Sez. di Roma Tre,Via della Vasca Navale 84, I-00146 Roma, ItalyE-mail: [email protected]
Silvano Simula
INFN, Sez. di Roma Tre,Via della Vasca Navale 84, I-00146 Roma, ItalyE-mail: [email protected]
Cecilia Tarantino ∗ † Dip. di Fisica, Università di Roma Tre and INFN, Sez. di Roma Tre,Via della Vasca Navale 84, I-00146 Roma, ItalyE-mail: [email protected] for the European Twisted Mass Collaboration (ETMC)
We present the results of the lattice QCD calculation of the average up-down and strange quarkmasses and of the light meson pseudoscalar decay constants, recently performed with N f = O ( a ) -improvement of the physical quantities. Quark masses are renormalizedby implementing the non perturbative RI-MOM renormalization procedure. Our results for thelight quark masses are m MS ud ( ) = . ± . ± .
40 MeV, m MS s ( ) = ± ± m s / m ud = . ± . ± .
2. We also obtain f K = . ± . ± . f K / f p = . ± . ± . G ( K → m ¯ n m ( g )) / G ( p → m ¯ n m ( g )) and the average value of | V ud | from nuclear beta decays, we ob-tain | V us | = . ( )( ) , in agreement with the determination from K l decays and the unitarityconstraint. The XXV International Symposium on Lattice Field TheoryJuly 30-4 August 2007Regensburg, Germany ∗ Speaker. † It is a pleasure to thank the organizers of “Lattice 2007” for the very interesting conference realized in Regensburg.We thank the other authors of the work presented here: B. Blossier, Ph. Boucaud, P. Dimopoulos, F. Farchioni, R. Frez-zotti, V. Gimenez, G. Herdoiza, K. Jansen, C. Michael, D. Palao, M. Papinutto, A. Shindler, C. Urbach, and U. Wenger. We are also grateful to D. Becirevic, G. Martinelli and G.C. Rossi for useful comments and discussions. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ight quark masses and pseudoscalar decay constants from N f = tmQCD Cecilia Tarantino t/a a m =0.0040a m =0.0064a m =0.0085a m =0.0100a m =0.0150 aM PS effective Figure 1:
Effectivemassesofpseudoscalarmesonswith m S = m = m ,asafunctionofthetime.
1. Introduction
We present our recent determination [1] of the light quark masses (strange quark mass m s andaverage up-down quark mass m ud ), of the kaon pseudoscalar decay constant f K , and of the ratio f K / f p . In order to investigate the properties of the K meson, we have simulated the theory with N f = m and m different between each other and from the sea quark mass m S .The calculation is based on a set of (ETMC) gauge field configurations generated with the tree-level improved Symanzik gauge action at b = .
9, corresponding to a = . ( ) fm ( a − ≃ . a m S = { . , . , . , . , . } and 8 values, a m , = { . , . , . , . , . , . , . , . } , for the valence quarkmass. The first five masses, equal to the sea masses, lie in the range 1 / m s < ∼ m , < ∼ / m s , being m s the physical strange quark mass, while the heaviest three are around the strange quark mass.At each value of the sea quark mass we have computed the two-point correlation functions ofcharged pseudoscalar mesons, on a set of 240 independent gauge field configurations, separated by20 HMC trajectories one from the other. To improve the statistical accuracy, we have evaluated themeson correlators using a stochastic method with a Z ( ) -noise to include all spatial sources [3, 4].Statistical errors on meson masses and decay constants are evaluated using the jackknife procedure,while those on the fit results, based on data obtained at different sea quark masses, are evaluatedusing a bootstrap procedure. Further details on the numerical simulation can be found in refs. [1, 5].The use of twisted mass fermions presents several advantages [6]: i) the pseudoscalar mesonmasses and decay constants are automatically improved at O ( a ) ; ii) at maximal twist, the physicalquark mass is directly related to the twisted mass parameter of the action, and it is subject only tomultiplicative renormalization; iii) the determination of the pseudoscalar decay constant does notrequire the introduction of any renormalization constant, and it is based on the relation f PS = ( m + m ) |h | P ( ) | P i| M PS . (1.1)The meson mass M PS and the matrix element |h | P ( ) | P i| have been extracted from a fit of thetwo-point pseudoscalar correlation function in the time interval t / a ∈ [ , ] . In order to illustratethe quality of the data, we show in fig. 1 the effective masses of pseudoscalar mesons, as a functionof the time, in the degenerate cases m S = m = m .2 ight quark masses and pseudoscalar decay constants from N f = tmQCD Cecilia Tarantino
2. Quark mass dependence of pseudoscalar meson masses and decay constants
The determination of the physical properties of K mesons requires to study the correspondingobservables over a large range of masses, from the physical strange quark down to the light up-down quark. In ref. [1], we have studied the quark mass dependence of pseudoscalar meson massesand decay constants by considering two different functional forms: i) the dependence predicted bycontinuum partially quenched chiral perturbation theory (PQChPT), ii) a polynomial dependence. PQChPT fits:
Within PQChPT we have considered the full next-to-leading order (NLO) ex-pressions with the addition of the local NNLO contributions, i.e. terms quadratic in the quarkmasses, which turn out to be needed for a good description of the data up to the region of thestrange quark. The PQChPT predictions [7] can be written as M PS ( m S , m , m ) = B ( m + m ) · (cid:20) + x ( x S − x ) ln 2 x ( x − x ) − x ( x S − x ) ln 2 x ( x − x ) ++ a V x + a S x S + a VV x + a SS x S + a VS x x S + a VD x D (cid:3) , (2.1) f PS ( m S , m , m ) = f · (cid:20) − x S ln 2 x S − x S ln 2 x S + x x − x S x ( x − x ) ln (cid:18) x x (cid:19) ++( b V + / ) x + ( b S − / ) x S + b VV x + b SS x S + b V S x x S + b V D x D (cid:3) , where x i = B m i / ( p f ) , x i j = B ( m i + m j ) / ( p f ) and x Di j = B ( m i − m j ) / ( p f ) . The param-eters B and f are the LO low energy constants (LECs) , whereas a V , a S , b V and b S are related tothe NLO LECs by a V = a − a , a S = a − a , b V = a , b S = a . The quadratic mass termsin eq. (2.1) represent the local NNLO contributions. The chiral logarithms, also known at two loopsin the partially quenched theory [8], involve a larger number of NLO LECs whose values cannotbe fixed from phenomenology in the N f = L = a ≃ . M PS L ≥ .
2. Since we have notperformed yet a systematic study on different lattice volumes, we have estimated the finite sizeeffects by including in the fits the corrections predicted by one-loop PQChPT [9] (for their explicitexpressions see ref. [1]).
Polynomial fits:
The inclusion of the local NNLO contributions in the PQChPT predictionsof eq. (2.1) is required by the observation that the pure NLO predictions are not accurate enough todescribe the quark mass dependence of pseudoscalar meson masses and decay constants up to thestrange quark region. Not having considered the full NNLO chiral predictions, we have evaluatedthe associated systematic uncertainty, considering as an alternative description a simple polynomialdependence on the quark masses, for both pseudoscalar meson masses and decay constants: M PS ( m S , m , m ) = B ( m + m ) · (cid:2) + a V x + a S x S + a VV x + a SS x S + a VS x x S + a VD x D (cid:3) , f PS ( m S , m , m ) = f · (cid:2) + ( b V + / ) x + ( b S − / ) x S + b VV x + b SS x S + b VS x x S + b VD x D (cid:3) . (2.2)The differences between the results obtained by performing either chiral or polynomial fits havebeen included in the final estimates of the systematic errors. The pseudoscalar decay constant f is normalised such that f p = . ight quark masses and pseudoscalar decay constants from N f = tmQCD Cecilia Tarantino
3. Chiral extrapolations
The input data of our analysis [1] are the lattice results for the pseudoscalar meson masses anddecay constants obtained at each value of the sea quark mass, with both degenerate and non degen-erate valence quarks. We have excluded from the fits the heaviest mesons having both the valencequark masses in the strange mass region, i.e. with a m , = { . , . , . } , consideringtherefore 150 combinations of quark masses. The number of free parameters in the combined fit of M PS and f PS is 14, but a first analysis shows that some of them (from 1 to 5 depending on the fit)are compatible with zero within one standard deviation, and are kept fixed to zero.In order to extrapolate the pseudoscalar meson masses and decay constants to the points cor-responding to the physical pion and kaon, we have considered three different fits: • Polynomial fit: a polynomial dependence on the quark masses is assumed for pseudoscalarmeson masses and decay constants, according to eq. (2.2). • PQChPT fit: pseudoscalar meson masses and decay constants are fitted according to thePQChPT predictions of eq. (2.1) including the finite volume corrections derived in ref. [9]. • Constrained PQChPT fit: this fit, denoted as C-PQChPT in the following, deserves a moredetailed explanation. The main uncertainty in using eqs. (2.1) and (2.2) to describe thequark mass dependence of M PS and f PS is related to the extrapolation toward the physicalup-down quark mass. On the other hand, we have shown in ref. [2] that pure NLO ChPT,with the inclusion of finite volume corrections, is sufficiently accurate in describing the latticepseudoscalar meson masses and decay constants when the analysis is restricted to our lightestfour quark masses in the unitary setup (i.e. m = m = m S ). In order to take advantage of thisinformation, when performing the C-PQChPT fit we first determine the LO parameters B and f and the NLO combinations a V + a S and b V + b S from a fit based on pure NLO ChPTperformed on the lightest four unitary points. By using these constraints, the other parametersentering the chiral expansions of M PS and f PS are then obtained from a fit to eq. (2.1) overthe non unitary points. For consistency with the previous unitary fit, we exclude also in thiscase from the analysis the data at the highest value of sea quark mass, a m S = . m ≥ m = m S , are not affected by dangerous chiral logarithms. The comparison betweenthe results obtained by considering the two different sets of quark masses is reassuring, as it showsthat the effects of potentially divergent chiral logarithms are well under control in our analysis.The mass dependence of the pseudoscalar meson masses and decay constants is illustratedin fig. 2, where lattice data are compared with the results of the polynomial, PQChPT and C-4 ight quark masses and pseudoscalar decay constants from N f = tmQCD Cecilia Tarantino a m a m S =a m =0.0040a m S =a m =0.0064a m S =a m =0.0085a m S =a m =0.0100a m S =a m =0.0150Polynomial FitPQChPT FitC-PQChPT Fit (aM PS ) (a m +a m )/2 a m a m S =a m =0.0040a m S =a m =0.0064a m S =a m =0.0085a m S =a m =0.0100a m S =a m =0.0150Polynomial FitPQChPT FitC-PQChPT Fit af PS Figure 2:
Latticeresultsfor a M PS / ( a m + a m ) (left) and a f PS (right)as afunctionofthevalencequarkmass a m ,with a m = a m S . Thesolid,dashedanddottedcurvesrepresenttheresultsofthethreefits. PQChPT fits. We have shown in the plots the cases in which one of the valence quark mass ( m )is equal to the sea quark mass, and the results are presented as a function of the second valencequark mass ( m ). The points corresponding to the physical pion and kaon are thus obtained byextrapolating/interpolating the results shown in fig. 2 to the limits m → m ud and m → m s .To investigate the impact of finite volume corrections we have compared, for the pseudoscalarmeson masses and decay constants, the PQChPT fits obtained with or without including thesecorrections. The differences turn out to be small [1]; however, to better quantify the systematicerror due to finite size effects, we plan to extend our analysis on lattices with different spatial sizes.By having determined the fit parameters, we have then extrapolated eqs. (2.1) and (2.2) to thephysical pion and kaon, as follows. We have first used the experimental values of the ratios M p / f p and M K / M p to determine the average up-down and the strange quark mass respectively. Once thesemasses have been determined, we have used again eqs. (2.1) and (2.2) to compute the values of thepion and kaon decay constants as well as their ratio f K / f p .
4. Physical results
In order to convert into physical units the results obtained for the strange quark mass andthe kaon decay constants we have fixed the scale within each analysis (polynomial, PQChPT andC-PQChPT fits) by using f p as physical input. In the case of the polynomial and PQChPT fitswe conservatively introduce for the dimensionful quantities a 6% and 3% of systematic error totake into account the different scale estimate derived in the analysis over the lightest four unitarypoints [1, 2].The determination of the physical strange and up-down quark masses also requires implement-ing a renormalization procedure. The relation between the bare twisted mass at maximal twist, m q ,and the renormalized quark mass, m q , is given by m q ( m R ) = Z m ( g , a m R ) m q ( a ) , where m R is therenormalization scale, conventionally fixed to 2 GeV for the light quarks. Z m is the inverse ofthe flavour non-singlet pseudoscalar density renormalization constant, Z m = Z − P . We have usedthe non-perturbative RI-MOM determination of Z P , which gives Z RI − MOM P ( / a ) = . ( )( ) at b = . m R = LO [11].5 ight quark masses and pseudoscalar decay constants from N f = tmQCD Cecilia Tarantino
Fit m MS ud (MeV) m MS s (MeV) m s / m ud f K (MeV) f K / f p Polynomial 4.07(9)(33) 109(2)(9) 26.7(2)(0) 158.7(11)(89) 1.214(8)(0)PQChPT 3.82(15)(25) 107(3)(7) 27.9(2)(0) 160.2(15)(54) 1.225(11)(0)C-PQChPT 3.74(13)(21) 102(3)(6) 27.4(3)(0) 161.8(10)(0) 1.238(7)(0)
Table 1:
Results for the light quark masses and pseudoscalar decay constants, in physical units, fromthe polynomial, PQChPT and C-PQChPT fits, analysing only the combinations of quark masses satisfy-ing m ≥ m = m S . The quoted errors are statistical (first) and systematic (second), the latter coming fromtheuncertaintiesinthedeterminationofthelatticescaleandofthequarkmassrenormalizationconstant. In table 1 we collect the results for the light quark masses and pseudoscalar decay constants, inphysical units, and for the ratios m s / m ud and f K / f p , as obtained from the polynomial, PQChPT andC-PQChPT fits. To be conservative, we consider the results obtained from the analysis of the quarkmass combinations satisfying the constraint m ≥ m = m S which, though being affected by largerstatistical errors, are safe from the effects of the potentially divergent chiral logarithms. In table 1we quote as a systematic error within each fit the uncertainty associated with the determination ofthe lattice spacing and of the quark mass renormalization constant.In order to derive our final estimates for the quark masses and decay constants, we performa weighted average of the results of the three analyses presented in table 1 and conservativelyinclude the whole spread among them in the systematic uncertainty. In this way, we obtain as ourfinal estimates of the light quark masses the results m MS ud ( ) = . ± . ± .
40 MeV , m MS s ( ) = ± ± , (4.1)and the ratio m s / m ud = . ± . ± . , (4.2)where the first error is statistical and the second systematic. For the kaon decay constant and theratio f K / f p we obtain the accurate determinations f K = . ± . ± . , f K / f p = . ± . ± . . (4.3)An interesting comparison of our results for the strange quark mass and the ratio f K / f p withother lattice QCD determinations is illustrated in fig. 3 (see ref. [1] for the full list of references).An important finding of our analysis [1] is that the use of non-perturbative renormaliza-tion turns out to play a crucial role in the determination of the quark masses. The estimate Z RI − MOM P ( / a ) = . ( )( ) obtained with the RI-MOM method is in fact significantly smallerthan the prediction Z BPT P ( / a ) ≃ . ( ) given by one-loop boosted perturbation theory (in thesame RI-MOM renormalization scheme) [10]. Had we used the perturbative estimate of Z P wewould have obtained m MS ud ( ) = . ± . ± .
36 MeV and m MS s ( ) = ± ± ight quark masses and pseudoscalar decay constants from N f = tmQCD Cecilia Tarantino
40 60 80 100 120 140 160 180 200 m s (2 GeV) [MeV] CP-PACS 01JLQCD 02ALPHA 05SPQcdR 05QCDSF-UKQCD 04-06ETMC 07HPQCD-MILC-UKQCD 04-06CP-PACS-JLQCD 07
PDG 06 Average (W-Clov, a-->0, PT)(W-Clov, a=0.09 fm, PT)(W-Clov, a=0.07 fm, SF)(Wilson, a=0.06 fm, RI-MOM)(W-Clov, a-->0, RI-MOM)(TM, a=0.09 fm, RI-MOM)(KS, a-->0, PT)(W-Clov, a-->0, PT) (Lattice only)
RBC 07(DWF, a=0.12 fm, RI-MOM) N f =2N f =2+1 f K /f p CP-PACS 01JLQCD 02RBC 04NPLQCD 06(DWF/KS, a=0.125 fm)ETMC 07MILC 04 - Lattice06CP-PACS-JLQCD Lattice06
PDG 06 Average (W-Clov, a-->0)(W-Clov, a=0.09 fm)(DWF, a=0.12 fm)RBC-UKQCD 07(DWF, a=0.12 fm)(TM, a=0.09 fm)(KS, a-->0)(W-Clov, a-->0)(KS, a-->0)HPQCD-UKQCD 07
Average value of Vus from Kl3 decayswith the updated N f =2N f =2+1 Figure 3:
Lattice QCD determinations of the strange quark mass (left) and of the ratio f K / f p (right)obtainedfromsimulationswith N f = N f = + f K / f p ,withtheaveragefromthe K ℓ determinationof V us [13]. Our result for the ratio f K / f p can be combined with the experimental measurement of G ( K → m ¯ n m ( g )) / G ( p → m ¯ n m ( g )) [12] to get a determination of the ratio | V us | / | V ud | [14]. We obtain | V us | / | V ud | = . ( )( ) , where the first error is the experimental one and the second is thetheory error coming from the uncertainty on f K / f p . It yields, combined with the determination | V ud | = . ( ) [15] from nuclear beta decays, the estimate | V us | = . ( )( ) , in agree-ment with the value extracted from K ℓ decays, | V us | = . ( ) [13], and leads to the constraintdue to the unitarity of the CKM matrix | V ud | + | V us | + | V ub | − = ( − . ± . ) · − . References [1] B. Blossier et al. [ETM Coll.], 0709.4574[hep-lat].[2] Ph. Boucaud et al. [ETM Coll.], Phys. Lett. B (2007) 304 [hep-lat/0701012].[3] M. Foster and C. Michael [UKQCD Coll.], Phys. Rev. D (1999) 074503 [hep-lat/9810021].[4] C. McNeile and C. Michael [UKQCD Coll.], Phys. Rev. D (2006) 074506 [hep-lat/0603007].[5] C. Urbach [ETM Coll.], PoS(LAT2007)022.[6] R. Frezzotti and G. C. Rossi, JHEP (2004) 007 [hep-lat/0306014].[7] S. R. Sharpe, Phys. Rev. D , 7052 (1997) [Erratum-ibid. D , 099901 (2000)] [hep-lat/9707018].[8] J. Bijnens and T. A. Lahde, Phys. Rev. D (2005) 074502 [hep-lat/0506004].[9] D. Becirevic and G. Villadoro, Phys. Rev. D (2004) 054010 [hep-lat/0311028].[10] P. Dimopoulos et al. [ETM Coll.], PoS(LAT2007)241.[11] K. G. Chetyrkin and A. Retey, Nucl. Phys. B (2000) 3 [hep-ph/9910332].[12] W. M. Yao et al. [Particle Data Group], J. Phys. G (2006) 1.[13] G. Isidori, conference summary talk at KAON’07, 0709.2438 [hep-ph], .[14] W. J. Marciano, Phys. Rev. Lett. (2004) 231803 [hep-ph/0402299].[15] W. J. Marciano and A. Sirlin, Phys. Rev. Lett. (2006) 032002 [hep-ph/0510099].(2006) 032002 [hep-ph/0510099].