Vittorio Lubicz Dip. di Matematica e Fisica, Università Roma Tre & INFN, Sezione di Roma Tre, Rome, Italy

Transcription

1 mesons in Lattice QCD with N f = twisted-mass fermions Vittorio Lubicz Dip. di Matematica e Fisica, Università Roma Tre & INFN, Sezione di Roma Tre, Rome, Itay E-mai: ubicz@fis.uniroma3.it Aurora Meis Dipartimento di Matematica e Fisica, Università Roma Tre, Rome, Itay E-mai: me.aurora.16@gmai.com Sivano Simua Istituto Nazionae di Fisica Nuceare, Sezione di Roma Tre, Rome, Itay E-mai: simua@roma3.infn.it for the ETM Coaboration We present a attice cacuation of the decay constants and masses of D (s) and B (s) mesons using the gauge configurations produced by the European Twisted Mass Coaboration (ETMC) with N f = dynamica quarks and at three vaues of the attice spacing a fm. Pion masses are simuated in the range m π MeV, whie the strange and charm quark masses are cose to their physica vaues. We computed the ratios of vector to pseoscaar decay constants or masses for various vaues of the heavy-quark mass m h in the range 0.7mc phys m h 3mc phys. In order to reach the physica b-quark mass, we expoited the HQET prediction that, in the static imit of infinite heavy-quark mass, a the considered ratios are equa to one. We obtain: f D / f D = 1.078(36), m D /m D = (79), f D s / f Ds = 1.087(20), m D s m Ds = (56), f B / f B = 0.958(22), m B /m B = (15), f B s / f Bs = 0.974(10) and m B s /m Bs = (10). Combining them with the corresponding experimenta masses from the PDG and the pseoscaar decay constants cacuated by ETMC, we get: f D = 223.5(8.4) MeV, m D = 2013(14) MeV, f D s = 268.8(6.6) MeV, m D s = 2116(11) MeV, f B = 185.9(7.2) MeV, m B = (7.6) MeV, f B s = 223.1(5.4) MeV and m B s = (5.3) MeV. 34th annua Internationa Symposium on Lattice Fied Theory Juy 2016 University of Southampton, UK Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercia-NoDerivatives 4.0 Internationa License (CC BY-NC-ND 4.0).

2 Aurora Meis 1. Introduction The decay constants of D (s) and B (s) mesons are important ingredients in the phenomenoogica description of various processes, ike semieptonic and non-eptonic decays of heavy hadrons. It is we known that in the imit of infinite heavy-quark mass the Heavy Quark Effective Theory (HQET) predicts that the ratios of vector (V) to pseoscaar (PS) decay constants or masses are equa to one, i.e. im mh (M H /M H ) = 1 and im mh ( f H / f H ) = 1. When the heavy quark is either the charm or the beauty, the spin-favor symmetry is broken and the above ratios deviate from one because of power corrections in 1/m h. Ti now there are ony few attice cacuations of the D (s) and B (s) decay constants using gauge configurations with N f = 2 [1, 2] and N f = 2+1(+1) [3, 4] dynamica quarks. These resuts exhibit a surprisingy non-negigibe dependence on N f. In this contribution we present the resuts obtained for the V to PS ratios of decay constants and masses in the charm and beauty sectors using the gauge ensembes generated by the European Twisted Mass Coaboration (ETMC) with N f = dynamica quarks [5, 6, 7]. In the ETMC set-up the guon interactions are described by the Iwasaki action, whie the fermions are reguarised in the maximay twisted-mass (Mtm) Wison attice formuation. We considered three vaues of the attice spacing, namey a = (36), (30) and (18) fm, with the owest simuated pion mass being equa to 210 MeV. The vaence quark masses are chosen to be in the ranges: 3m phys m 12m phys, 0.7mphys s m s 1.2ms phys and 0.7mc phys m c 1.1mc phys. To extrapoate up to the b-quark sector we have aso considered higher vaues of the vaence heavyquark mass in the range 1.1mc phys m h 3mc phys 0.7m phys b. The attice scae was determined using the experimenta vaue of f π + [8], whie the physica up/down, strange, charm and bottom quark masses were obtained [8, 9] by using the experimenta vaues for m π, m K, m D and m B, respectivey. In Ref. [8] eight branches of the anaysis were adopted. They differ in: i) the continuum extrapoation adopting for the scae parameter either the Sommer parameter r 0 or the mass of a fictitious PS meson made up of strange(charm)-ike quarks; ii) the chira extrapoation performed with fitting functions chosen to be either a poynomia expansion or a Chira Perturbation Theory (ChPT) ansatz in the ight-quark mass; iii) the choice between the methods M1 and M2, which differ by O(a 2 ) effects, used to determine in the RI -MOM scheme the mass renormaization constant (RC) Z m = 1/Z P. In the present anaysis we made use of the input parameters corresponding to each of the eight branches of Ref. [8]. 2. Extraction of masses and decay constants The decay constants of V and PS mesons are defined in terms of the matrix eements 0 hγ µ H ( p,λ) = f H m H ε λ µ, (2.1) 0 hγ 5 H ( p) = 0 µ ( hγµ γ 5 ) H ( p /(m h + m ) = f H m 2 H /(m h + m ), (2.2) where m h and m are the heavy- and ight-quark masses with h = {c,b} and = {,s}, and ε λ µ is the vector meson poarization. Ground-state masses and decay constants can be determined by stying two-point correation functions at arge time distances, viz. C V (t) = 1 3 i, x V i ( x,t)v i (0,0) t t min i 0 V i (0) H (λ) 2 cosh[m H (T /2 t)] e mh T /2, (2.3) 3m H 1

3 Aurora Meis C P (t) = x P( x,t)p (0,0) 0 P(0) H 2 cosh[m H (T /2 t)] e m H T /2, (2.4) t t min m H where t min stands for the minimum time at which the ground state can be considered we isoated. In Eq. (2.3) V i Z A hγ i is the oca vector current, which in our Mtm setup renormaizes mutipicativey with the RC Z A, whie in Eq. (2.4) P (m h + m )Z P hγ 5 = (µ h + µ )hγ 5, where µ h and µ are bare quark masses, is the pseoscaar interpoating fied, which in our Mtm setup is renormaization group invariant and does not require any RC. The meson masses m H and m H are extracted from the pateaux of the effective mass at arge t t min, whie the correation functions (2.3) and (2.4) for t t min contain the required matrix eements. In Eqs. ( ) we considered oca source and sink operators, but we anayzed aso the whoe set of four correation functions given by the combinations of oca interpoating operators with those obtained from a Gaussian smearing procedure in both the sink and the source, namey and CSS P,V, where L and S denote oca and smeared operators, respectivey. It is straightforward to check that the required oca matrix eements in Eqs. ( ) can be extracted from the LL correation functions as we as from an appropriate combination of the SL and SS ones, that is CP,V SL (t)/ C SS C LL P,V,CLS P,V,CSL P,V P,V (t). For the reasons expained in the Introduction we have considered the foowing ratios R m H = m H /m H and R f H = f H / f H. (2.5) In Fig. 1 we show an exampe of the above quantities by comparing the extraction from Gaussiansmeared and/or oca correation functions. The smearing techniques aows pateaux to start at earier time distances, and the SL correation functions exhibit the best signa to noise ratio. The vaue of t min corresponds to the smaest time distance where the effective masses obtained from SL and SS correation functions intercept each other. Figure 1: The ratios R m H and R f H (see Eq. (2.5)) using ony oca (red points) or aso Gaussian smeared operators (bue points). The vaue of t min is shown as the vertica dot-dashed ine. The grey bands are the mass and the decay constant, obtained as constant fits in the pateaux regions. 2.1 The D (s)-meson masses and decay constants We perform a smooth interpoation of the attice data for the ratios R m H and R f H to the vaues of the physica strange and charm quark masses ms phys (MS, 2 GeV) = 99.6(4.3) MeV and 2

4 Aurora Meis mc phys (MS, 2 GeV) = 1.176(39) GeV [8]. The dependence of R m D (s) and R f D (s) on the renormaized up/down quark mass m = aµ /(az P ) and the attice spacing a is investigated by performing a combined chira and continuum extrapoation, based on a poynomia expansion of the form R f it D (s) (m,a) = P 0 + P 1 m + P 2 a 2 + P 3 m 2 + P 4 a 4, (2.6) where we have taken into account that for our Mtm setup the automatic O(a)-improvement impies that discretization effects invove ony even powers of the attice spacing. The resuts obtained with quadratic m 2 and quartic a4 terms have not been inced in the fina average (which therefore corresponds to P 3 = P 4 = 0), but they have been considered to estimate the uncertainty reated to the chira and continuum extrapoation, respectivey. The atter ones are shown in Fig. 2, where the physica point corresponds to m phys (MS, 2 GeV) = 3.70(17) MeV [8]. Figure 2: Chira and continuum extrapoations of R m D and R f D based on the poynomia fit (2.6) with P 3 = P 4 = 0. The green points represent the vaues at the physica point m phys (MS, 2 GeV) = 3.70(17) MeV [8]. Simiar resuts hod as we in the case of the ratios R m D s and R f D s. In this way at the physica point we get m D /m D = (71) stat (30) input (13) tmin (8) disc (5) chir [79], (2.7) m D s /m Ds = (49) stat (27) input (8) disc (4) tmin (2) chir [56], (2.8) f D / f D = 1.078(31) stat (9) chir (8) disc (6) tmin (5) input [36], (2.9) f D s / f Ds = 1.087(16) stat (7) disc (6) input (6) tmin (5) chir [20], (2.10) where tota uncertainty (in the square brakets) is the sum in quadrature of the statistica and a the systematic uncertainties, which have been written in order of reevance for each of the ratios. The various systematic uncertainties are estimated in the foowing way: i) the uncertainty abeed tmin is computed by repeating the anaysis with a vaue of t min shifted by two units and taking the haf difference with the fina estimates; ii) the chira and discretization uncertainties, abeed respectivey as chir and disc, are obtained by considering either P 3 0 or P 4 0 in Eq. (2.6) and taking again haf of the difference with the fina resuts; iii) the uncertainty abeed input is given by the spread of the resuts over the eight branches of the input parameters of Ref. [8]. By combining R m D (s) with the experimenta vaues of the D (s) -meson masses [10] we obtain m D = 2013 (14) MeV and m D s = 2116 (11) MeV, (2.11) 3

5 Aurora Meis that compare we with the experimenta meson masses m exp D (4) MeV [10]. = (5) MeV and mexp D = s As for the decay constants, existing attice cacuations for R f D (s) have been carried out ony with N f = and N f = 2 dynamica quarks. The N f = estimate f D s / f Ds = 1.10(2) [3] is in good agreement with our resut, whie the N f = 2 resuts f D / f D = 1.208(27) [2] and f D s / f Ds = 1.26(3)[1] are 10% arger than our predictions. Using the pseoscaar decay constants cacuated by ETMC in Ref. [11] we get f D = (8.7) MeV and f D s = (6.5) MeV. (2.12) 2.2 The B (s)-meson masses and decay constants We have computed the ratios R m(k) H and R f (k) H for a series of masses {m (k) h } m c with k = 1,...,8. The resuts are extrapoated to the chira and continuum imits, as shown in Fig. 3. Figure 3: Chira and continuum extrapoations of R m(k) H and R f (k) H (2.6) with P 3 = P 4 = 0. Simiar resuts hod as we in the case of R m(k) H s for k = 3, based on the poynomia fit and R f (k) H s. The HQET predicts that the ratios R m H and R f H R f H /C W (m h ), where C W (m h ) is the perturbative matching correction between fu QCD and HQET (computed in Ref. [12] up to next-to-nexteading order), are equa to one in the static heavy-quark imit, viz. im m h Rm H = 1 and im m h R f H = im m h R f H /C W (m h ) = 1. (2.13) Thus, we perform correated poynomia fits in 1/m h imposing the static imit constraint, namey R m H f it phys = 1 + D 2/m 2 h + D 3 /m 3 h + D 4/m 4 h, R f H f it phys = 1 + D 1/m h + D 2 /m 2 h + D 3 /m 3 h, (2.14) where we have taken into account that, according to HQET, the inear term is absent in the case of the mass ratio (i.e., D 1 = 0). In Fig. 4 the interpoations of the various ratios in the inverse heavy-quark mass are shown together with the resuts at the b-quark physica point obtained using the vaue m phys b from Ref. [9]. 4

6 Aurora Meis Figure 4: The dependence of R m H (s) phys and R f H (s) phys, extrapoated to the chira and continuum imits, on the inverse heavy-quark mass 1/m h (MS, 2 GeV). The fits are based on Eqs. (2.14) and the correation matrix among the data points is accounted for. The vertica bands correspond to 1/m phys b from Ref. [9]. Our fina resuts for B (s) mesons are m B /m B = (8) stat (8) chir (7) tmin (5) disc (2) input [14], (2.15) m B s /m Bs = (6) stat (7) chir (6) disc (3) tmin (2) input [11], (2.16) f B / f B = 0.958(18) stat (10) disc (6) chir (5) tmin (2) input [22], (2.17) f B s / f Bs = 0.974(7) stat (6) disc (3) tmin (2) input (1) chir [10], (2.18) where the error bget accounts for the same sources of uncertainties aready considered for the charm sector in Sec Mass ratios can be combined with the experimenta vaues of B (s) -meson masses [10] to obtain m B = (7.6) MeV and m B s = (6.2) MeV, (2.19) that compare nicey with the experimenta vaues m exp B = (32) MeV and mexp B = (1.6) s MeV [10]. As for the decay constant ratios, we can compare our resuts with a recent computation [4] obtained from N f = simuations (ike the ones considered in this work), f B / f B = 0.941(26) and f B s / f Bs = 0.953(23), as we as with a recent determination based on the QCD sum rue approach [13], f B / f B = 0.944(23) and f B s / f Bs = 0.947(30). A these estimates are nicey consistent with our resuts. On the contrary we find again a 10% difference with the N f = 2 determination f B / f B = 1.051(17) from Ref. [2]. 5

7 Aurora Meis Eventuay, combining our resuts for the ratios with the pseoscaar decay constants cacuated by ETMC in Ref. [9] yieds 3. Concusions f B = (7.1) MeV and f B s = (5.6) MeV. (2.20) We have computed the masses and the decay constants of vector heavy-ight mesons using ETMC gauge configurations with N f = dynamica quarks. Our resuts reproduce very we the experimenta vaues of both D (s) - and B (s)-meson masses. We have found that f D (s) / f D(s) > 1 and f B (s) / f B(s) < 1 with a spin-favor symmetry breaking effect of +8% in the charm sector and 4% in the beauty sector. Our resuts for the decay constant ratio exhibit a tension with the corresponding attice determinations obtained by ETMC at N f = 2 [1, 2], whie they are consistent with the findings of Refs. [3, 4] obtained by HPQCD with N f = 2 + 1(+1) dynamica quarks. Since our present anaysis foow amost the same steps of the previous ETMC anayses at N f = 2, the observed 10% tension may be due to a dependence on the number of sea quarks, and in particuar to the incusion of the strange quark. The possibiity that the observed difference can be attributed to a quenching effect of the strange quark is a quite interesting issue, because its size wo be arger than what typicay expected. Further investigations at different N f vaues are therefore required. Acknowedgments We gratefuy acknowedge the CPU time provided by PRACE under the project PRA067 on the BG/Q system Juqueen at JSC (Germany) and by CINECA under the specific initiative INFN-LQCD123 on the BG/Q system Fermi at CINECA (Itay). References [1] D. Becirevic et a., JHEP 1202 (2012) 042 [arxiv: [hep-at]]. [2] D. Becirevic et a., arxiv: [hep-ph]. [3] G.C. Donad et a. [HPQCD], Phys. Rev. Lett. 112 (2014) [arxiv: [hep-at]]. [4] B. Coquhoun et a. [HPQCD], Phys. Rev. D 91 (2015) 11, [arxiv: [hep-at]]. [5] R. Baron et a. [ETMC], JHEP 1006 (2010) 111 [arxiv: [hep-at]]. [6] R. Baron et a. [ETMC], Comput. Phys. Commun. 182 (2011) 299 [arxiv: [hep-at]]. [7] R. Baron et a. [ETMC], PoS LATTICE 2010 (2010) 123 [arxiv: [hep-at]]. [8] N. Carrasco et a. [ETMC], Nuc. Phys. B 887 (2014) 19 [arxiv: [hep-at]]. [9] A. Bussone et a. [ETMC], Phys. Rev. D 93 (2016) no.11, [arxiv: [hep-at]]. [10] K. A. Oive et a., Chin. Phys. C 40 (2016) no.10, [11] N. Carrasco et a. [ETMC], Phys. Rev. D 91 (2015) 5, [arxiv: [hep-at]]. [12] D. J. Broadhurst and A. G. Grozin, Phys. Rev. D 52 (1995) 4082 [hep-ph/ ]. [13] W. Lucha et a., Phys. Rev. D 91 (2015) no.11, [arxiv: [hep-ph]]. 6