K* vector and tensor couplings from Nf = 2 tmQCD
aa r X i v : . [ h e p - l a t ] A p r ROM2F/2011/04,ICCUB-11-129, UB-ECM-PF-11-48December 20, 2017 K ∗ vector and tensor couplings from N f = 2 tmQCD ETMC
P. Dimopoulos a , F. Mescia c , and A. Vladikas da Dipartimento di Fisica, Universit`a di Roma “Tor Vergata”Via della Ricerca Scientifica 1, I-00133 Rome, Italy c Departament dEstructura i Constituents de la Mat´eria and Institut de Ci´encies del Cosmos,Universitat de Barcelona, Diagonal 647,E-08028 Barcelona, Spain d INFN, Sezione di “Tor Vergata”c/o Dipartimento di Fisica, Universit`a di Roma “Tor Vergata”Via della Ricerca Scientifica 1, I-00133 Rome, Italy
Abstract
The mass m K ∗ and vector coupling f K ∗ of the K ∗ -meson, as well as the ratio of the tensorto vector couplings f T f V (cid:12)(cid:12)(cid:12) K ∗ , are computed in lattice QCD. Our simulations are performedin a partially quenched setup, with two dynamical (sea) Wilson quark flavours, having amaximally twisted mass term. Valence quarks are either of the standard or the Osterwalder-Seiler maximally twisted variety. Results obtained at three values of the lattice spacing areextrapolated to the continuum, giving m K ∗ = 981(33)MeV , f K ∗ = 240(18)MeV and f T (2 GeV) f V (cid:12)(cid:12)(cid:12) K ∗ = 0 . . Basics
The aim of the present letter is to present novel lattice results for the mass of the K ∗ -meson, as well as its vector and tensor couplings ( f V and f T respectively), definedin Euclidean space-time as follows: h | V j | K ∗ ; λ i = − if V ǫ λj m K ∗ , (1.1) h | T j | K ∗ ; λ i = − i f T ǫ λj m K ∗ . (1.2)In the above expressions, V j = ¯ sγ j d is the vector current (spatial components only; j = 1 , , T j = i ¯ sσ j d is the tensor bilinear operator (temporal component), and ǫ λj denotes the polarization vector.Our results are based on simulations of the ETM Collaboration (ETMC) [1],with N f = 2 dynamical flavours (sea quarks) and “lightish” pseudoscalar mesonmasses in the range 280 MeV < m PS <
550 MeV. With three lattice spacings ( a =0 .
065 fm, 0 .
085 fm and 0.1 fm) we are able to extrapolate our results to the continuumlimit. Our simulations are performed with the tree-level Symanzik improved gaugeaction. For the quark fields we adopt a somewhat different regularization for seaand valence quarks. The sea quark lattice action is the so-called maximally twistedstandard tmQCD (referred to as “standard tmQCD case”) [2]. The N f = 2 light seaquark flavours form a flavour doublet ¯ χ = (¯ u , ¯ d ) and the fermion lattice Lagrangianin the so-called “twisted basis” is given by L tm = ¯ χ h D W + iµ q γ τ i χ , (1.3)where τ is the isospin Pauli matrix and D W denotes the critical Wilson-Diracoperator. By “critical” we mean that, besides the standard kinetic and Wilsonterms, the operator also includes a standard, non-twisted mass term, tuned at thecritical value of the quark mass ( κ cr in the language of the hopping parameter), soas to ensure maximal twist. With only two light dynamical flavours, strangenessclearly enters the game in a partially quenched context. For the valence quarks weuse the so-called Osterwalder-Seiler variant of tmQCD, which consists in maximallytwisted flavours which, unlike the standard tmQCD case, are not combined intoisospin doublets: L OS = X f = d,s ¯ q f h D W + iµ f γ i q f , (1.4)with sign( µ f ) = ± K ∗ -related quantities) we only need down- and strange-quark flavours in thevalence sector. Note that the choice of maximally twisted sea and valence quarksimplies O ( a )-improvement of the physical quantities (i.e. the so-called automatic1mprovement of masses, correlation functions and matrix elements) [6]. Thus unitar-ity violation, which plagues any partially quenched theory at finite lattice spacing,is an O ( a ) effect.The sign of µ s may be that of µ d or its opposite. We conventionally refer to thesetup in which sign( µ d ) = − sign( µ s ) as the “standard twisted mass regularization”(denoted by tm) and the setup with sign( µ d ) = sign( µ s ) as the “Osterwalder-Seilerregularization” (denoted as OS). Quenched pseudoscalar masses and decay constantsin tm- and OS-setups have already been studied [7, 8].The continuum operators of interest are expressed, in terms of their latticecounterparts, as follows: V cont µ = Z A A tm µ + O ( a ) = Z V V OS µ + O ( a ) , (1.5) T cont µν = Z T T tm µν + O ( a ) = Z T ˜ T OS µν + O ( a ) , (1.6)where ˜ T µν = ǫ µνρσ T ρσ . The vector and axial currents are normalized by the scaleindependent factors Z V and Z A , while Z T ≡ Z T ( µ ) runs with a renormalizationscale µ (i.e. it is defined in a given renormalization scheme).The vector boson mass, m V , as well as f V and f T , are obtained form two-pointcorrelation functions at zero spatial momenta and large time separations. These aredefined in the continuum (Euclidean space-time) as C cont V ( x ) ≡ X j Z d x h V j ( x ) V † j (0) i cont → f V m V − m V T /
2] cosh (cid:2) m V ( T / − x ) (cid:3) , (1.7) C cont T ( x ) ≡ X j Z d x h T j ( x ) T † j (0) i cont → f T m V − m V T /
2] cosh (cid:2) m V ( T / − x ) (cid:3) . (1.8)The asymptotic expressions of the above equations correspond to the large timelimit of the correlation functions (symmetrized in time), with periodic boundaryconditions for the gauge fields and (anti)periodic ones for the fermion fields in the(time)space directions (i.e. 0 ≪ x ≪ T / C cont V ( x ) = Z A C tm A ( x ) + O ( a ) = Z V C OS V ( x ) + O ( a ) , (1.9) C cont T ( x ) = Z T C tm T ( x ) + O ( a ) = Z T C OS˜ T ( x ) + O ( a ) . (1.10)The meaning of the notation C tm A , C OS˜ T , etc. should be transparent to the reader.The ratio f T /f V is computed from the square root of the ratio of correlations func-tions C cont T /C cont V , in which many systematic effects cancel. We compute the vector2eson mass and decay constant from C cont V and the ratio f T /f V from the ratioof correlation functions C cont T /C cont V . The tensor coupling f T is then obtained bymultiplying f T /f V by f V .Note that f V is a scale independent quantity, while f T ( µ ) depends on the renor-malization scale µ , as well as the renormalization scheme. The scale and schemedependence of the latter quantity is carried by the renormalization factor Z T ( µ ); weopt for the MS-scheme and for µ = 2 GeV. ETMC has generated N f = 2 configuration ensembles at four values of the inversegauge coupling; in this work we make use of only three of them. Light mesons consistof a valence quark doublet, with twisted mass aµ ℓ equal to that of the sea quarks; aµ ℓ = aµ sea . Heavy-light mesons consist of a valence quark pair ( aµ ℓ = aµ sea , aµ h ).As already stated, these bare quark mass parameters are chosen so as to have lightpseudoscalar mesons (“pions”) in the range of 280 ≤ m PS ≤
550 MeV and heavy-light pseudoscalar mesons (“Kaons”) in the range 450 ≤ m PS ≤
650 MeV. Thesimulation parameters are gathered in Table 1. β a − ( L × T ) aµ ℓ = aµ sea aµ h N meas ×
48 0.0080, 0.0110 0.0165, 0.0200 180( a ∼ . ×
48 0.0040 0.0150, 0.0220 4000.027024 ×
48 0.0064, 0.0085, 0.0150, 0.0220 2000.0100 0.02703.90 32 ×
64 0.0030, 0.0040 0.0150, 0.0220 270/170( a ∼ .
085 fm) 0.02704.05 32 ×
64 0.0030, 0.0060, 0.0120, 0.0150 200( a ∼ .
065 fm) 0.0080 0.0180Table 1: Simulation detailsOur calibrations are based on earlier collaboration results. The ratio r /a ,known at each value of the gauge coupling β from ref. [9], allows to express our rawdimensionless data (quark masses, meson masses and decay constants) in units of r . Knowledge of the renormalization constant Z P in the MS scheme at 2 GeV (seeref. [10]) enables us to pass from bare quark masses to renormalized ones (again in r units). Using only data with light valence quarks in the tm-setup, we have appliedthe procedure described in refs. [1,9] for the determination of the physical continuum3ight quark mass µ MS u/d . From the data concerning light and heavy valence quarkmasses in the tm-setup [9], we determine the physical continuum strange quark mass µ MS s (2 GeV). These quark mass values are listed in Table 3. The Sommer scale weuse, based on an analysis with three values of the lattice spacing, is r = 0 . r computation, derived with two β ’s, cf. ref. [1].We see from eqs. (1.9) and (1.10) that we need to know the renormalizationparameters Z V , Z A , and Z T . These quantities, as well as Z P , have been computedin ref. [10], in the RI/MOM scheme; Z P and Z T are perturbatively converted to MS.In the same work a Z V estimate, obtained from a Ward identity, is also provided.In Table 2 we gather the most reliable estimates of ref. [10], which we have used inthe present analysis, as well as our estimates of the r /a ratio. β Z V Z A Z MS T (2 GeV) Z MS P (2 GeV) r /a r /a values ateach gauge coupling. Z V is obtained from a lattice vector Ward identity, while theother renormalization constants are obtained from the RI/MOM scheme; for detailssee ref. [10]. µ MS u/d (2 GeV) µ MS s (2 GeV)3.6(2) MeV 95(6) MeVTable 3: The quark mass values (in the MS scheme), used in our analysis; see ref. [9].As can be seen in Table 1, at β = 3 .
90 we have performed more extensivesimulations, which enable us to check in some detail the quality and stability of themeasured physical quantities. We wish to highlight straightaway the two problemswe have encountered in these tests, performed for the tm-setup: (i) For all sea quark4asses, when the valence quark attains its lightest value aµ ℓ = 0 . aµ ℓ = 0 . − ( m V − m PS ) x ]) the ρ -meson mass and decayconstant can still be extracted (see results presented in ref. [11]). (ii) A poor qualityvector meson effective mass is also seen when µ ℓ < µ sea . This problem is absent inthe pseudoscalar channel.The above problems are easily avoided in the present work, since the quantitiesof interest are related to the K ∗ -meson, consisting of a down and a strange valencequark mass ( µ u/d < µ s ). We thus proceed as follows: at each β value, we computethe necessary observables (vector meson mass m V , vector decay constant f V , andthe ratio f T /f V ), for all combinations of aµ ℓ = aµ sea and aµ h (with µ ℓ < µ h ). Inthis way unitarity holds in the light quark sector, while the heavy valence quarkmass, in a partially quenched rationale, spans a range around the physical value µ s .Examples of the quality of our signal are given in Figs. 1 and 2; the lightest massis aµ min and the heavy mass, corresponding to the physical strange value aµ s , isobtained by interpolation, as will be explained below. = 4 . , = 0 . . , = 0 . β = 3 . aµ h = 0 . x /T r m t m V ( ℓ , h ) (a) = 4 . , = 0 . . , = 0 . β = 3 . aµ h = 0 . x /T r m O S V ( ℓ , h ) (b) Figure 1: Effective vector meson mass r m V at three values of the lattice spacing.The light quark mass is aµ min (see Table 1) and the heavy quark mass aµ h is closeto that of the physical strange quark. (a) tm-setup ; (b) OS-setup. Plateau intervalsare indicated by straight lines.Statistical errors are estimated with the bootstrap method, employing 1000bootstrap samples. A reliable direct determination of the ratio f T /f V in the OS-setup is not possible, because the ratio of correlation functions C OS˜ T /C OS V do notdisplay satisfactory plateaux, due to big statistical fluctuations of the tensor cor-relator C OS˜ T . We only present f T /f V results in the tm-setup, obtained from thebetter-behaved correlation function C tm T . In Fig. 3 we show results for this ratio at5 µ min and also at a heavier light quark mass. = 4 . , = 0 . . , = 0 . β = 3 . aµ h = 0 . x /T r f t m V ( ℓ , h ) (a) = 4 . , = 0 . . , = 0 . β = 3 . aµ h = 0 . x /T r f O S V ( ℓ , h ) (b) Figure 2: Vector decay constant r f V at three values of the lattice spacing. Thelight quark mass is aµ min (see Table 1) and the heavy quark mass aµ h is close tothat of the physical strange quark. (a) tm-setup ; (b) OS-setup. Plateaux intervalsare indicated by straight lines. = 4 . , = 0 . . , = 0 . β = 3 . aµ h = 0 . x /T [ f T / f V ] t m ( ℓ , h ) (a) = 4 . , = 0 . . , = 0 . β = 3 . aµ h = 0 . x /T [ f T / f V ] t m ( ℓ , h ) (b) Figure 3: The ratio f T /f V in the tm-setup, at three values of the lattice spacingand heavy quark mass aµ h , close to that of the physical strange quark. (a) Forthe lightest quark mass aµ min ; (b) for the next-to-lightest quark mass. Plateauxintervals are indicated by straight lines.Regarding vector meson masses m V and couplings f V , both tm- and OS-resultsdisplay similar plateau quality and statistical accuracy. At finite lattice spacing andfor equal bare quark masses, tm- and OS-estimates of m V are compatible withinerrors. Agreement is also very good for f V , with occasional discrepancies, interpreted6s cutoff effects, showing up at the coarsest lattice . Contrary to the well knownlarge O ( a ) isospin breaking effects in the neutral to charged pion splitting mass,no numerically large differences are observed between tm and OS results for f V and m V . This fact is in agreement with theoretical expectations, see ref. [12]. β aµ l r m tm V ( ℓ, s ) r m OS V ( ℓ, s ) r f tm V ( ℓ, s ) r f OS V ( ℓ, s ) [ f T /f V ] tm ( ℓ, s )3.80 0.0080 2.443(41) 2.471(30) 0.642(18) 0.700(13) 0.764(38)0.0110 2.508(32) 2.500(23) 0.651(14) 0.706(15) 0.792(35)3.90 0.0040 2.410(41) 2.381(38) 0.610(21) 0.643(17) 0.755(19)0.0064 2.441(32) 2.427(35) 0.626(22) 0.659(12) 0.726(20)0.0085 2.484(48) 2.441(33) 0.628(16) 0.652(16) 0.776(27)0.0100 2.468(54) 2.481(32) 0.619(20) 0.657(16) 0.774(31)0.0030(L=32) 2.259(75) 2.335(45) 0.577(20) 0.639(16) 0.714(20)0.0040(L=32) 2.364(32) 2.371(50) 0.599(22) 0.640(21) 0.722(19)4.05 0.0030 2.305(86) 2.263(80) 0.568(49) 0.588(40) 0.742(27)0.0060 2.439(67) 2.295(76) 0.618(41) 0.578(46) 0.768(30)0.0080 2.512(65) 2.427(48) 0.649(31) 0.648(27) 0.741(31)CL µ u/d Table 4: Results for three values of lattice spacing and several light quark masses aµ ℓ , interpolated to the physical strange mass µ s . Vector mass and vector decayconstant results are presented for both tm- and OS-setups . The ratio f T /f V resultsare given only in the tm-setup. Our extrapolations at the µ u/d physical point andin the continuum limit are also shown. In the last row the experimental results forthe vector mass and the vector decay constant, in units of r , have been added.The extrapolation to the physical quark masses is carried out in two steps.First, for fixed values of the gauge coupling β and light quark mass aµ ℓ = aµ sea ,we perform linear interpolations of r m V , r f V and f T /f V to the physical strangequark mass µ s . The second step consists in using these interpolated results for acombined fit of our data at three lattice spacings and all available light quark masses,in order to determine the continuum value of the quantity of interest ( r m V , r f V and f T /f V ). The fitting function we use is m V r = C ( µ s r ) + C ( µ s r ) µ ℓ r + D ( µ s r ) a r , (2.1) Given the large fluctuations of f T /f V in the OS-setup at the finer lattice spacing, we only quoteresults for this ratio in the tm-setup. f V r and f T /f V . The results of the interpolations in the heavyquark mass µ h to the physical value µ s , at each β and aµ ℓ , are gathered in Table 4.In the same Table we also display the results of the combined chiral and continuumextrapolations. Note that for the three quantities of interest, m V , f V and f T /f V ,the value of χ / d . o . f . is less than unity. The linear dependence of our data on thelight quark mass agrees with the predictions of chiral perturbation theory for theratio f T /f V in the K ∗ mass range; see refs. [13, 14].Our final results, extracted in the tm-setup, are m K ∗ = 981(31)(10)[33]MeV , (2.2) f K ∗ = 240(18)(02)[18]MeV . (2.3)The first error includes the statistical uncertainty and the systematic effects re-lated to the simultaneous chiral and continuum fits, mass interpolations and ex-trapolations, and uncertainties in the renormalization parameters. The second errorarises from that of r . These two errors, combined in quadrature, give the to-tal error in the square brackets. It is encouraging that these results agree withthe ones obtained in the OS-setup (which is a different regularization), namely m K ∗ = 969(27)(10)[29]MeV and f K ∗ = 231(13)(02)[13]MeV. Compared to the ex-perimentally known values, m K ∗ = 892MeV and f K ∗ = 217MeV, the vector mesonmass is 2-3 standard deviations off, while the decay constant is compatible withinabout one standard deviation.Our final estimate (tm-setup) for the ratio of vector meson couplings is f T (2 GeV) f V (cid:12)(cid:12)(cid:12) K ∗ = 0 . . (2.4)This is compatible with the continuum limit quenched result [ f T (2 GeV) /f V ] K ∗ =0 . N f = 2 + 1 dynamical fermions at a single lattice spacing,they quote [ f T (2 GeV) /f V ] K ∗ = 0 . f T (2 GeV) /f V ] K ∗ = 0 . Acknowledgements
We thank G.C. Rossi and C. Tarantino for having carefully read the manuscriptand for their useful comments and suggestions. We acknowledge fruitful collabo-ration with all ETMC members. We have greatly benefited from discussions withO. Cata, C. Michael, C. McNeile, S. Simula and N. Tantalo. F.M. acknowledges thefinancial support from projects FPA2007-66665, 2009SGR502, Consolider CPAN,and CSD2007-00042. 8 xpt. point β = 4 . β = 3 . β = 3 . r ˆ µ ℓ r m t m V ( ℓ , s ) (a) expt. point β = 4 . β = 3 . β = 3 . r ˆ µ ℓ r m O S V ( ℓ , s ) (b) Figure 4: r m V plotted against the renormalized light quark mass r ˆ µ ℓ ; (a) tm-setup; (b) OS-setup. The continuous lines are combined chiral and continuum ex-trapolations to the physical point. The bottom (black) line corresponds to eq. (2.1)at a = 0. The separation among the four lines in (a) is invisible to the naked eye(i.e. small scaling violations). expt. point β = 4 . β = 3 . β = 3 . r ˆ µ ℓ r f t m V ( ℓ , s ) (a) expt. point β = 4 . β = 3 . β = 3 . r ˆ µ ℓ r f O S V ( ℓ , s ) (b) Figure 5: r f V plotted against the renormalized light quark mass r ˆ µ ℓ ; (a) tm-setup;(b) OS-setup. The continuous lines are combined chiral and continuum extrapola-tions to the physical point. The bottom (black) line corresponds to eq. (2.1) at a = 0. References [1]
ETM
Collaboration, R. Baron et al. , “Light Meson Physics from MaximallyTwisted Mass Lattice QCD”,
JHEP (2010) 097, [ ].9 = 4 . β = 3 . β = 3 . r ˆ µ ℓ [ f T / f V ] t m ( ℓ , s ) Figure 6: f T /f V plotted against the renormalized light quark mass r ˆ µ ℓ in the tm-setup. The continuous lines are combined chiral and continuum extrapolation to thephysical point. The bottom (black) line corresponds to eq. (2.1) at a = 0. The fourlines are almost indistinguishable (i.e. small scaling violations).[2] ALPHA
Collaboration, R. Frezzotti et al. , “Lattice QCD with a chirallytwisted mass term”,
JHEP (2001) 058, [ hep-lat/0101001 ].[3] K. Osterwalder and E. Seiler, “Gauge Field Theories on the Lattice”, Ann.Phys. (1978) 440.[4]
ALPHA
Collaboration, C. Pena, S. Sint, and A. Vladikas, “Twisted massQCD and lattice approaches to the Delta I = 1/2 rule”,
JHEP (2004) 069,[ hep-lat/0405028 ].[5] R. Frezzotti and G. C. Rossi, “Chirally improving Wilson fermions. II: Four-quark operators”, JHEP (2004) 070, [ hep-lat/0407002 ].[6] R. Frezzotti and G. C. Rossi, “Chirally improving Wilson fermions. I: O(a)improvement”, JHEP (2004) 007, [ hep-lat/0306014 ].[7] ALPHA
Collaboration, P. Dimopoulos et al. , “Flavour symmetry restorationand kaon weak matrix elements in quenched twisted mass QCD”,
Nucl. Phys.
B776 (2007) 258–285, [ hep-lat/0702017 ].[8]
ALPHA
Collaboration, P. Dimopoulos, H. Simma, and A. Vladikas,“Quenched B K -parameter from Osterwalder-Seiler tmQCD quarks and mass-splitting discretization effects”, JHEP (2009) 007, [ ].109] ETM
Collaboration, B. Blossier et al. , “Average up/down, strange and charmquark masses with Nf=2 twisted mass lattice QCD”,
Phys. Rev.
D82 (2010)114513, [ ].[10]
ETM
Collaboration, M. Constantinou et al. , “Non-perturbative renormaliza-tion of quark bilinear operators with Nf=2 (tmQCD) Wilson fermions and thetree- level improved gauge action”,
JHEP (2010) 068, [ ].[11] ETM
Collaboration, K. Jansen et al. , “Meson masses and decay constantsfrom unquenched lattice QCD”,
Phys. Rev.
D80 (2009) 054510, [ ].[12]
ETM
Collaboration, P. Dimopoulos, R. Frezzotti, C. Michael, G. Rossi, andC. Urbach, “O(a**2) cutoff effects in lattice Wilson fermion simulations”,
Phys.Rev.
D81 (2010) 034509, [ ].[13] O. Cata and V. Mateu, “Chiral perturbation theory with tensor sources”,
JHEP (2007) 078, [ ].[14] O. Cata and V. Mateu, “Chiral corrections to the f(V)-perpendicular /f(V)ratio for vector mesons”,
Nucl.Phys.
B831 (2010) 204–216, [ ].[15] D. Becirevic et al. , “Coupling of the light vector meson to the vector and tothe tensor current”,
JHEP (2003) 007, [ hep-lat/0301020 ].[16] RBC-UKQCD
Collaboration, C. Allton et al. , “Physical Results from 2+1Flavor Domain Wall QCD and SU(2) Chiral Perturbation Theory”,
Phys.Rev.
D78 (2008) 114509, [ ].[17] P. Ball, G. W. Jones, and R. Zwicky, “B → V gamma beyond QCD factorisa-tion”,
Phys.Rev.
D75 (2007) 054004, [ hep-ph/0612081hep-ph/0612081