Heavy flavour precision physics from N f = lattice simulations

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1 Available online at Nuclear and Particle Pysics Proceedings (2016) Heavy flavour precision pysics from N f = lattice simulations A. Bussone a, N. Carrasco b, P. Dimopoulos d,a,, R. Frezzotti a, P. Lami c, V. Lubicz c, F. Nazzaro a, E. Picca c, L. Riggio c, G.C. Rossi a, F. Sanfilippo e, S. Simula b,c, C. Tarantino c a Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00173, Rome, Italy b INFN, Sezione di Roma Tre, Via della Vasca Navale 84, I Rome, Italy c Dipartimento di Fisica, Università Roma Tre, Via della Vasca Navale 84, I Rome, Italy d Museo Storico della Fisica e Centro Studi e Ricerce Enrico Fermi, Compendio del Viminale, Piazza del Viminale 1, I Rome, Italy e Scool of Pysics and Astronomy, University of Soutampton, SO17 1BJ Soutampton, U.K. Abstract We present precision lattice calculations of te pseudoscalar decay constants of te carmed sector as well as determinations of te bottom quark mass and its ratio to te carm quark mass. We employ N f = dynamical quark gauge configurations generated by te European Twisted Mass Collaboration, using data at tree values of te lattice spacing and pion masses as low as 210 MeV. Strange and carm sea quark masses are close to teir pysical values. Keywords: D (s) -decay constants, b-quark mass, Lattice QCD, ETMC 1. Introduction Pysical processes in te eavy quark sector offer te possibility to get some of te more stringest tests of te Standard Model and to searc for possible footprints of New Pysics dynamics, by directly callenging te unitarity constraints of te CKM matrix. Lattice QCD as already entered te precision era as te accuracy of numerical computations is becoming comparable to tat of experiments. For some of te relevant adronic quantities in Flavour Pysics te goal of per cent precision as been acieved. State-of-teart lattice calculations involve O(a)-improved fermionic actions wit N f = 2, 2+1 and dynamical flavours wit te smallest simulated pion masses being today at te pysical point or sligtly iger and employing tree or more values of te lattice spacing. For a review wit a critical evaluation of lattice results and averages, see [1]. First computations wit four non-degenerate quark flavours including electromagnetic effects ave also been presented recently [2]. Corresponding autor, dimopoulos@roma2.infn.it ttp://dx.doi.org/ /j.nuclpysbps / 2015 Elsevier B.V. All rigts reserved. Direct computations by many lattice collaborations ave sown tat te cutoff effects in te D-sector are small and under control. Moreover, considerable progress as been recently made in flavour pysics at te b mass, wit te elp of bot effective teories approaces and tanks to te implementation of some innovative metods. All tese progresses ave allowed te determination of a number of B-pysics parameters (e.g. m b, f B and f Bs ) wit controlled systematic uncertainties. Lattice metods are an invaluable tool to obtain direct determinations of adronic quantities relevant for te computation of many of te so called golden plated processes suc as decay constants, form factors and bag parameters. For instance, te widt of te D and D s leptonic decays is given, to lowest order, by Γ(D (s) lν) = G2 F f D 2 (s) m 2 l M D(s) 8π 1 m2 2 l MD(s) 2 V cd(s) 2. (1) Tus lattice computations of te quantities f D and f Ds gives access to te determination of te CKM matrix elements, V cd and V cs, respectively, as in Eq. (1) all te

2 A. Bussone et al. / Nuclear and Particle Pysics Proceedings (2016) rest is known experimentally. On te experimental side also te accuracy of te measurements of te D [3, 4, 5] and D s [6, 7] leptonic widt as progressively improved during te years. Lattice QCD provides a first principles way to compute quark masses. Tis is possible since quark masses enter as parameters in te QCD Lagrangian and teir values can be extracted by matcing adron masses calculated on te lattice wit teir experimental values. Te accuracy of quark mass estimates depends on te conversion from te lattice regularisation to continuum renormalisation scemes. Quark mass ratios instead can be computed in a fully non-perturbative way and are free of renormalisation sceme ambiguities. We notice, ere, tat te knowledge of te b-quark mass value and to less extent tat of te c-quark mass plays an important rôle in te study of te Higgs decay to b b and c c [8]. Te European Twisted Mass Collaboration (ETMC) as undertaken an extensive program of eavy quark pysics calculations on te lattice using two and four dynamical flavours. Here we present te results of te computation of te D (s) pseudoscalar meson decay constants (in te isospin symmetric limit) and te b to c quark mass ratio obtained using gauge configurations wit N f = dynamical quarks. Te main (preliminary) results in tese proceedings are f D = 208.7(5.2) MeV, f Ds = 247.5(4.1) MeV, (2) ( ) f Ds fds / ( ) fk = 1.186(21), = 0.998(14), (3) f D f D f π m b (MS, m b ) = 4.26(16) GeV, (4) m b /m c = 4.40(8) (5) For completeness we remind te recent ETMC determinations of te c-quark mass and te carm to strange quark mass ratio publised in [9]: m c (MS, m c ) = 1.348(42) GeV, m c /m s = 11.62(16) (6) For a preliminary ETMC computation of te B-meson decay constants, giving f B = 196(9) MeV, f Bs = 235(9) MeV and f Bs / f B = 1.201(25), we refer to [10]. 2. Lattice setup ETMC as produced gauge configurations wit N f = dynamical quarks [11] employing te Iwasaki gluon action [12] and te Wilson Twisted Mass fermionic action for te sea quarks [13]. Automatic O(a)-improvement is guaranteed bot for te ligt and eavier quarks by tuning at maximal twist wilst te drawback of te mixing between te strange and carm sectors [14] is avoided in te valence by using te Osterwalder-Seiler fermions [15]. We ave data ensembles at tree values of te lattice spacing in te range [0.06, 0.09] fm. Simulated pion masses lie in te interval [210, 440] MeV. Tanks to te properties of te twisted mass action ligt quarks in te sea and all types of quarks in te valence enjoy multiplicative mass renormalisation, Z m = 1/Z P, wic is computed nonperturbatively using te RI -MOM sceme [9]. Moreover owing to PCAC, at maximal twisted angle no normalisation constant is needed in te computation of te decay constants. In Ref. [9] we ave presented our computation for te quark masses of te (degenerate) ligt m u/d (MS, 2 GeV) = 3.70(17) MeV, strange m s (MS, 2 GeV) = 99.6(4.1) MeV and carm m c (MS, m c ) = 1.348(42) GeV, wic are determined by using te experimental values of te pion, kaon and D (or D s ) masses, respectively. Te penomenological value of f π as been used for setting te scale. In tis work te computation of te decay constants in te carmed region as well as te determination of te b-quark mass are performed using (Gaussian) smearing meson operators [16, 17] combined wit APE smeared links [18] in order to reduce bot te coupling of te interpolating field wit te excited states and te gauge noise of te links involved in te smeared fields. (For an alternative preliminary analysis of te carmed decay constants tat use local point correlators see Ref.[19].) A summary of te most important details of our simulations is given in Table Carmed decay constants We use two point correlation functions wit pseudoscalar interpolating operators, P(x) = q 1 (x)γ 5 q 2 (x), tat in periodic lattice ave te typical form: C PP (t) = (1/L 3 ) P( x, t)p ( 0, 0) t 0, (T t) 0 x ξ PP 2M ps ( e M ps t + e M ps(t t) ) (7) We take te Wilson parameters of te two valence quarks of te pseudoscalar meson to be opposite in order to guarantee tat te cutoff effects on te pseudoscalar mass are O(a 2 μ) [21, 22, 23]. We ten consider two cases, using smeared source only and source and sink bot smeared. As for ξ PP, tis is given by ξ PP = 0 P L ps ps P S 0 in te first case and ξ PP = 0 P S ps ps P S 0 in te second one, were L and S indicate local and smeared operators. By combining te

3 1640 A. Bussone et al. / Nuclear and Particle Pysics Proceedings (2016) β V/a 4 aμ sea = aμ l N cfg aμ s aμ c aμ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (fcs/mcs) (MDs) expt (GeV) β =1.90 β =1.95 β =2.10 CL-pys. point μ l (GeV) Table 1: Summary of simulation details. Gauge couplings β = 1.90, 1.95 and 2.10 correspond to lattice spacings a = 0.089, and 0.062, respectively; see Ref. [9]. We denote wit aμ l, aμ s and aμ c aμ, te ligt, strange-like, carm-like and somewat eavier bare quark masses, respectively, entering in te valence sector computations. Figure 1: Combined ciral and continuum fit (χ 2 /(dof) = 0.8) of ( f cs /M cs ) M expt Ds against te renormalised ligt quark mass expressed in MS-sceme at te scale of 2 GeV, μ l = μ sea. Te fit ansatz is linear bot in μ l and in a 2. Te vertical black tin line marks te position of u/d quark mass point. Te empty black circle is our result at te pysical u/d quark mass point in te continuum limit. two kinds of correlators it is easy to obtain te matrix element of te local operator g ps = 0 P L ps wic serves for computing te pseudoscalar decay constant (via PCAC) given by: g ps f ps = (μ 1 + μ 2 ), (8) M ps sin M ps were μ 1,2 are te masses of te valence quarks tat form te pseudoscalar meson wit mass M ps. Te use of sin M ps (rater tan M ps ) in Eq. (8) is beneficial for reducing te discretisation errors. For te computation of f Ds we tune, via well controlled interpolations, one of te valence quark masses to te value of te strange mass and te oter to te value of te carm mass, bot taken from Ref. [9]. In tis way, for eac value of te sea ligt quark mass and of te tree lattice spacings, we get estimates for te decay constant f cs. Ten a simultaneous extrapolation to te pysical value of te u/d quark mass and to te continuum limit can be performed in order to obtain f Ds. In te present analysis we consider te quantity ( f cs /M cs ) M expt Ds, were M cs is a pseudoscalar meson mass made of c and s quarks and is computed at eac value of te sea ligt quark mass, wile M expt Ds = (1.4) MeV is te experimental value of te D s mass. Te above coice of observable is advantageous because, first, in te determination of f Ds any scale setting uncertainty is avoided and, second, tis quantity presents very small discretisation effects. Te fit ansatz of te combined ciral and continuum extrapolation reads: [( f cs /M cs ) M expt Ds ] = C 0+C 1 μ l +Da 2, see Fig. 1. We control ciral fit uncertainties by adding in te above fit ansatz eiter a quadratic quark mass term or fitting data corresponding to ligt pseudoscalar masses wit M ll < 350 MeV. Finite volume systematics are estimated by fitting data corresponding to L > 2.6 fm. Discretisation systematic errors ave been estimated by fitting data eiter from te two finest lattice spacings or from te two coarsest ones, and also by estimating te difference of our results from te finest lattice to te continuum limit. Moreover, we ave also included te propagated error due to te m s,c uncertainties as well as te systematic effect of te quark mass renormalisation constant (RC) computed in two ways tat differ by O(a 2 )effects. Our central value is te weigted average over te results from all te analyses described above. Our (preliminary) result for f Ds reads f Ds = 247.5(3.0) stat+ fit (2.7) syst [4.1] MeV, (9) were we report in square brackets te total error ( 1.6%) tat is te sum in quadrature of te statistical and systematic uncertainties. For te full error budget see Table 2. In Fig. 2 we compare our result wit tose computed in oter lattice studies and wit te PDG estimate based on experimental results and unitarity assumptions. Some tension between te PDG estimate and te most precise lattice results is still present. In order to determine te SU(3) symmetry breaking ratio f Ds / f D we measure on our data sets te double ratio R f = ( f cs / f cl )( f ll / f sl ). Tis coice enjoys te

4 A. Bussone et al. / Nuclear and Particle Pysics Proceedings (2016) uncertainty (in %) f Ds f Ds / f D f D stat. + fit syst. from ciral fits syst. from discr. effects syst. from FSE syst. from f K / f π Total Rf β =1.90 β =1.95 β =2.10 (HM)CPT Fit Lin. Fit Table 2: Full error budget for f Ds, f Ds / f D and f D. Te different sources of uncertainty are self explanatory PDG (expt + pen) (tis work) FNAL/MILC 14 χqcd 14 N f =2 HPQCD 12 FNAL/MILC 11 f Ds [MeV] Figure 2: We compare te available continuum limit determinations for f Ds (in MeV) from lattice studies tat use N f = 2, 2+1 and dynamical flavours. result refers to te present work. For te results of oter lattice studies we refer to (from top to bottom) [24, 25, 26, 27, 28, 29, 30]). For te PDG result see [31]. property of very mild ligt quark mass dependence as expected from te large cancellation between te SU(2) ciral logaritms [32, 33]. Te quantity R f in te continuum limit and at te pysical pion mass point multiplied wit ( f K / f π ) (taken from Ref. [20]) will provide te result for f Ds / f D. We try te following fit ansätze: R f = c (1) 0 + c (1) 1 μ l + D (1) a 2, (10) [ R f = c (2) c (2) 1 μ l + ( 9ĝ ) ] ξ l log ξ l + D (2) a 2, (11) 2 were ξ l = (2B 0 μ l )/(4π f 0 ) 2 wit B 0 and f 0 determined in Ref. [9]. We ave applied finite size corrections using Ref.[34]. Among te available estimates for te (D Dπ) coupling we ave used ĝ = 0.61(7) tat in our case leads to te most conservative estimate for te ciral fit systematic uncertainty. Te ciral and continuum limit extrapolation is sown in Fig. 3. Moreover we ave performed an analysis similar to te one for f Ds in order μ l (GeV) Figure 3: Combined ciral and continuum fit for te quantity R f against te renormalised ligt quark mass expressed in MS-sceme at te scale of 2 GeV, μ l = μ sea. Te vertical black tin line marks te position of u/d quark mass point. Te empty black circle and empty triangle represent te results for te ratio f Ds / f D, using te fit ansatz of Eq. (10) (χ 2 /(dof) = 0.7) and Eq. (11) (χ 2 /(dof) = 1.1), respectively, at te pysical u/d quark mass point and in te continuum limit. to estimate our systematic uncertainties. Te full error budget is given in Table 2. Te central value corresponds to te weigted average over results from all te different analyses. Our (preliminary) results read ( f Ds / f D )/( f K / f π ) = (8) stat+ fit (11) syst [14], (12) f Ds / f D = (9) stat+ fit (19) syst [21], (13) and eac one of te total errors (in square brackets) is te sum in quadrature of te statistical error and te systematic one. We combine te results from Eqs. (9) and (13) to get our (preliminary) result for te decay constant of te D- meson, namely f D = f Ds /( f Ds / f D ), wic reads: f D = 208.7(3.3) stat+ fit (4.0) syst [5.2] MeV, (14) were also in tis case te total error written in square brackets ( 2.5%) is te sum in quadrature of te statistical and systematic uncertainties. For te complete error budget see Table 2. In Figs. 4 and 5 we present a world comparison between lattice results for f Ds / f D and f D, respectively. In bot figures te PDG estimate is also included. For some recent non-lattice estimates of te carmed decay constants, see Refs. [35, 36, 37, 38]. 4. Computation of m b and m b /m c We perform te determination of te b-quark mass employing te ratio metod described in detail in

5 1642 A. Bussone et al. / Nuclear and Particle Pysics Proceedings (2016) PDG (expt + pen) (tis work) FNAL/MILC 14 HPQCD 12 FNAL/MILC f Ds/f D Figure 4: We compare te available continuum limit determinations for f Ds / f D from lattice studies tat use N f = 2, 2+1 and dynamical flavours. result refers to te present work. For te results of te oter lattice studies we refer to (from top to bottom) [24, 26, 28, 29, 30]). For te PDG result see [31]. PDG (expt + pen) (tis work) FNAL/MILC 14 HPQCD 12 FNAL/MILC 11 N f =2 f D [MeV] Figure 5: We compare te available continuum limit determinations for f D (in MeV) from lattice studies tat employ N f = 2, 2+1 and dynamical flavours. result refers to te present work. For te results of te oter lattice studies we refer to (from top to bottom) [24, 25, 26, 27, 28, 29, 30]). For te PDG result see [31]. Refs. [30, 39, 40]. We present ere a variant of tis metod and we build te quantity Q = M s /(M l ) γ, were M s and M l are te eavy-strange and eavyligt pseudoscalar masses, respectively. Te parameter γ is a free one and may take values at will in te interval [0, 1). By HQET arguments we know tat for te asymptotic beaviour we get: lim M s/(m l ) γ = const., (15) μ pole (μ pole ) (1 γ) were μ pole is te eavy quark pole mass. We ten consider a sequence of eavy quark masses expressed in te MS-sceme at te scale of 2 GeV suc tat any two successive masses ave a common and fixed ratio i.e. μ (n) = λμ (n 1), n = 2, 3,... Te next step is to construct at eac value of te sea quark mass and te lattice spacing te following ratios: y Q (μ (n),λ; μ l, μ s, a) Q (μ (n) ; μ l, μ s, a) Q (μ (n 1) ; μ l, μ s, a) = λ (γ 1) Q (μ (n) ; μ l, μ s, a) Q (μ (n) /λ; μ l, μ s, a) μ (n) ρ(μ(n),μ ) μ (n 1) ρ(μ (n 1) ρ(μ (n) (γ 1),μ ) ρ(μ (n),μ (γ 1) ) /λ, (16) μ ) wit n = 2, 3,... and we ave used te relation μ pole = ρ(μ,μ ) μ (μ ) between te MS renormalised quark mass (at te scale of 2 GeV) and te pole quark mass. Te ρ s are known perturbatively up to N 3 LO. For eac pair of eavy quark masses we ten carry out a simultaneous ciral and continuum fit of te quantity defined in Eq. (16) to obtain y Q (μ ) y Q (μ,λ; μ u/d, μ s, a = 0). By construction tis quantity involves (double) ratios of pseudoscalar meson masses at successive values of te eavy quark mass, so we expect tat discretisation errors will be under control. In fact tis is te case even for te largest values of te eavy quark mass used in tis work, see Fig. 6. Since we ave taken into account te matcing of QCD onto HQET concerning te evaluation of a eavy-ligt pseudoscalar mass, M s/l, our ratio y Q (μ ) as been defined in suc a way tat te following ansatz is sufficient to describe te μ -dependence 1 y Q (μ ) = 1 + η 1 + η 2, (17) μ in wic te constraint lim μ y Q (μ ) = 1 as already been incorporated. Tis fit is reported in Fig. 7. Finally, we compute te b-quark mass via te cain equation y Q (μ (2) ) y Q(μ (3) )... y Q(μ (K+1) ) = = λ K(γ 1) Q (μ (K+1) ) (ρ(μ (K+1),μ )) γ 1 Q (μ (1) ) (18) ρ(μ (1),μ ) in wic te values of te factors in te (ls) are evaluated using te result of te fit function (Eq. 17) and λ, K (integer) and μ (1) are suc tat te quantity Q (μ (K+1) ) matces M Bs /(M B ) γ, were M Bs = (4) MeV and M B = (3) MeV are te experimental values of te B s - and B-meson masses [31], respectively. Notice tat te quantity Q (μ (1) ) for any value of μ(1) around 1 For more details on tis point see te Appendix of Ref. [40] and [30], section 4. μ 2

6 A. Bussone et al. / Nuclear and Particle Pysics Proceedings (2016) te carm quark mass is safely computed in te continuum limit and at te pysical pion mass. For instance, using quark mass RC of te M2-type (see [9]) and setting as input μ (1) = GeV and γ = 0.75 we find (λ, K) = (1.1588, 10). Tus, te b-quark mass in te MS-sceme at te scale of 2 GeV is given by μ b = λ K μ (1). Our preliminary result for te b- quark mass is given by te average over two estimates obtained using eiter M1 or M2-type quark mass RCs wile teir alf difference is taken as an additional systematic error. Tis reads m b (MS, m b ) = 4.26(7) stat+ fit (14) syst [16] GeV, (19) y (8) Q β =1.90 β =1.95 β =2.10 CL - pys. point μ l (GeV) were te total error (in brackets) is te sum in quadrature of te statistical and te systematic ones. For a complete error budget we refer to Table 3. We ave uncertainty (in %) m b m b /m c stat+fit syst. from lat. scale syst. from discr. effects syst. from ratios fits syst. from ciral fits syst. from RC Total Table 3: Full error budget for m b and m b /m c. Te different sources of uncertainty are self explanatory. verified tat for a large range of values of γ [0, 1) we get fully compatible final results 2 for m b. Te freedom of coosing γ allows for better control of systematic uncertainties stemming from discretisation effects and te fit ansatz Eq. (17). Finally, te ratio metod offers te advantage of determining te ratio m b /m c in a simple and fully nonperturbative way. By setting μ (1) = μ c we repeat te above ratio metod analysis and we find m b /m c = 4.40(6) stat+ fit (5) syst [8] (20) A complete error budget is also reported in Table 3. In Figs. 8 and 9 we present a comparison between lattice results for m b and m b /m c, respectively. For non-lattice estimate of m b see [41]. Acknowledgements We are grateful to all members of ETMC for fruitful discussions. We acknowledge te CPU time provided by 2 Tis systematic uncertainty as been included in te estimate called syst. from ratios fits of Table 3. Figure 6: Combined ciral and continuum fit of te ratio defined in Eq. (16) and corresponding to te case of te two largest values of eavy quark mass against te renormalised ligt quark mass μ l = μ sea. Te fit ansatz is linear bot in μ l and in a 2 wit χ/(dof) = 1.1. Te empty black circle is our result at te pysical u/d quark mass point in te continuum limit. In tis example we ave considered γ = yq(μ ) μ 1 b /μ (GeV 1 ) Figure 7: y Q (μ ) against 1/μ using te fit ansatz of Eq. (17) wit χ 2 /(dof) = 0.1. We ave used γ = 0.75, λ = and Λ N f =4 QCD = 296(15) MeV for te running coupling entering in te ρ(μ,μ) function. Te vertical black tin line marks te position of 1/μ b. Quark mass values, μ, μ b are expressed in te MS-sceme at te scale of 2 GeV. te PRACE Researc Infrastructure under te project PRA067 at te Jülic and CINECA SuperComputing Centers, and by te agreement between INFN and CINECA under te specific initiative INFN-lqcd123. References [1] FLAG, Aoki, S., et al., Eur.Pys.J. C74 (9) (2014) arxiv: , doi: /epjc/s [2] S. Borsanyi, et al. arxiv: [3] Belle, Zupanc, A., et al., JHEP 1309 (2013) 139. arxiv: , doi: /jhep09(2013)

7 1644 A. Bussone et al. / Nuclear and Particle Pysics Proceedings (2016) PDG (tis work) HPQCD m b (MS,m b ) [GeV] Figure 8: We compare te available continuum limit determinations for m b (m b ) (in GeV) from lattice studies wit N f = 2, 2+1 and dynamical flavours. result refers to te present work. For te results of oter lattice studies we refer to (from top to bottom) [42, 43, 44, 30]. For te PDG value see [31]. (tis work) m b /m c Figure 9: Comparison between te two available continuum limit determinations for m b /m c obtained from fully non-perturbative studies. For te HPQCD result see [43]. [4] BaBar, del Amo Sancez, P., et al., Pys.Rev. D82 (2010) arxiv: , doi: /pysrevd [5] CLEO-C, Naik, P., et al., Pys.Rev. D80 (2009) arxiv: , doi: /pysrevd [6] H.-B. Li, EPJ Web Conf. 72 (2014) doi: /epjconf/ [7] Y. Zeng, ICHEP 2014, tese proceedings. [8] A. Djouadi, Pys.Rept. 457 (2008) arxiv:epp/ , doi: /j.pysrep [9] ETMC, Carrasco, N., et al., Nucl.Pys. B887 (2014) arxiv: , doi: /j.nuclpysb [10] ETMC, Carrasco, N., et al., PoS (LATTICE 2013) (2013) 313. arxiv: [11] ETMC, Baron, R., et al., JHEP 1006 (2010) 111. arxiv: , doi: /jhep06(2010)111. [12] Y. Iwasaki, Nucl.Pys. B258 (1985) doi: / (85) [13] R. Frezzotti, G. Rossi, Nucl.Pys.Proc.Suppl. 128 (2004) arxiv:ep-lat/ , doi: /s (03) [14] ETMC, Baron, R., et al., Comput.Pys.Commun. 182 (2011) arxiv: , doi: /j.cpc [15] K. Osterwalder, E. Seiler, Annals Pys. 110 (1978) 440. doi: / (78) [16] S. Gusken, Nucl.Pys.Proc.Suppl. 17 (1990) doi: / (90)90273-w. [17] K. Jansen, et al., JHEP 0812 (2008) 058. arxiv: , doi: / /2008/12/058. [18] M. Albanese, et al., Pys.Lett. B192 (1987) doi: / (87) [19] ETMC, Dimopoulos, P., et al. PoS(LATTICE 2013) (2013) 314. arxiv: [20] ETMC, Carrasco, N., et al. in preparation. [21] R. Frezzotti, G. C. Rossi, JHEP 08 (2004) 007. arxiv:eplat/ [22] R. Frezzotti, et al., JHEP 0604 (2006) 038. arxiv:eplat/ , doi: / /2006/04/038. [23] P. Dimopoulos, et al., Pys.Rev. D81 (2010) arxiv: , doi: /pysrevd [24] FNAL-MILC, Bazavov, A., et al. arxiv: [25] χqcd, Yang, Yi-Bo, et al. arxiv: [26] HPQCD, Na, Heecang, et al., Pys.Rev. D86 (2012) arxiv: , doi: /pysrevd [27] HPQCD, Davies, C.T.H., et al., Pys.Rev. D82 (2010) arxiv: , doi: /pysrevd [28] FNAL, Bazavov, A., et al., Pys.Rev. D85 (2012) arxiv: , doi: /pysrevd [29] ALPHA, Heitger, J., PoS LATTICE2013 (2013) 475. arxiv: [30] ETMC, Carrasco, N., et al., JHEP 1403 (2014) 016. arxiv: , doi: /jhep03(2014)016. [31] PDG, Olive, K.A., et al., Review of Particle Pysics, Cin.Pys. C38 (2014) doi: / /38/9/ [32] D. Becirevic, S. Fajfer, S. Prelovsek, J. Zupan, Pys.Lett. B563 (2003) arxiv:ep-p/ , doi: /s (03) [33] ETMC, Blossier, B., et al., JHEP 0907 (2009) 043. arxiv: , doi: / /2009/07/043. [34] G. Colangelo, et al., Nucl.Pys. B721 (2005) arxiv:ep-lat/ , doi: /j.nuclpysb [35] S. Narison, Pys.Lett. B718 (2013) arxiv: , doi: /j.pysletb [36] W. Luca, D. Melikov, S. Simula, Pys.Lett. B701 (2011) arxiv: , doi: /j.pysletb [37] Z.-G. Wang, JHEP 1310 (2013) 208. arxiv: , doi: /jhep10(2013)208. [38] P. Gelausen, et al., Pys.Rev. D88 (1) (2013) arxiv: , doi: /pysrevd [39] ETMC, Blossier, B., et al., JHEP 1004 (2010) 049. arxiv: , doi: /jhep04(2010)049. [40] ETMC, Dimopoulos, P., et al., JHEP 1201 (2012) 046. arxiv: , doi: /jhep01(2012)046. [41] K. Cetyrkin, et al., Pys.Rev. D80 (2009) arxiv: , doi: /pysrevd [42] HPQCD, Lee, A.J., et al., Pys.Rev. D87 (7) (2013) arxiv: , doi: /pysrevd [43] HPQCD, McNeile, C., et al., Pys.Rev. D82 (2010) arxiv: , doi: /pysrevd [44] ALPHA, Bernardoni, F., et al., Pys.Lett. B730 (2014) arxiv: , doi: /j.pysletb