B-physics from non-perturbatively renormalized HQET in two-flavour lattice QCD
Fabio Bernardoni, Benoit Blossier, John Bulava, Michele Della Morte, Patrick Fritzsch, Nicolas Garron, Antoine Gerardin, Jochen Heitger, Georg M. von Hippel, Hubert Simma
MMS-TP-12-13LPT-Orsay / / / / CPP-12-74
B-physics from non-perturbatively renormalized HQETin two-flavour lattice QCD
LPHA A Collaboration
Fabio Bernardoni a , Benoˆıt Blossier b , John Bulava c , Michele Della Morte d , Patrick Fritzsch e , Nicolas Garron f ,Antoine G´erardin b , Jochen Heitger g, ∗ , Georg M. von Hippel d , Hubert Simma a a NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany b CNRS et Universit´e Paris-Sud XI, Laboratoire de Physique Th´eorique, Bˆatiment 210, F-91405 Orsay Cedex, France c CERN, Physics Department, TH Unit, CH-1211 Geneva 23, Switzerland d Johannes Gutenberg Universit¨at Mainz, Institut f¨ur Kernphysik, Becher Weg 45, D-55099 Mainz, Germany e Humboldt Universit¨at Berlin, Institut f¨ur Physik, Newtonstraße 15, D-14289 Berlin, Germany f Trinity College, School of Mathematics, Dublin 2, Ireland g Westf¨alische Wilhelms-Universit¨at M¨unster, Institut f¨ur Theoretische Physik, Wilhelm-Klemm-Straße 9, D-48149 M¨unster, Germany
Abstract
We report on the ALPHA Collaboration’s lattice B-physics programme based on N f = a ) improved Wilsonfermions and HQET, including all NLO e ff ects in the inverse heavy quark mass, as well as non-perturbative renormal-ization and matching, to fix the parameters of the e ff ective theory. Our simulations in large physical volume cover 3lattice spacings a ≈ (0 . − .
05) fm and pion masses down to 190 MeV to control continuum and chiral extrapolations.We present the status of results for the b-quark mass and the B (s) -meson decay constants, f B and f B s .
1. B-physics and lattice QCD
Lattice simulations of QCD have established as a soundtool to compute strong interaction e ff ects for accuratephenomenology in heavy flavour physics. For B-mesonweak decays, which constrain the CKM Unitarity Tri-angle, lattice QCD results for the involved low-energyhadronic matrix elements in conjunction with experi-mental studies decisively contribute to stringent tests ofthe self-consistency of the Standard Model and comple-ment direct searches for New Physics. Since the sig-nificance of such precision tests in the beauty sector ispredominantly limited by theoretical uncertainties, lat-tice computations with an overall accuracy of a few %are highly desirable. Let us highlight the ” V ub –puzzle”illustrated in Fig. 1, which has drawn the community’sattention in the recent past. | V ub | can be determined frominclusive semi-leptonic processes B → X u (cid:96)ν , from ex-clusive semi-leptonic B → π(cid:96)ν decays and from the lep-tonic one, B → τν . In the latter two, the hadronic form ∗ Speaker at
QCD 2012, Montpellier, France, 2 – 6 July 2012Email address: [email protected] (Jochen Heitger) B → π‘ν B → X u ‘ν B → τν | V ub | Figure 1:
Observed tension among di ff erent | V ub | –determinations [1]; ± σ bands are shown. factor f + ( q ) and the B-meson decay constant f B enter,respectively, so that lattice QCD input is required to ex-tract | V ub | . As currently there is a ∼ σ tension betweenits two exclusive (semi-leptonic and leptonic) determi-nations , as well as an inconsistency with the estimatefrom inclusive decays, precision lattice QCD calcula-tions can contribute to resolve this tension [4–6]. At the ICHEP 2012 Conference, the Belle Collaboration reporteda new result for B (B → τν ) [2, 3], obtained on basis of a new data setusing a more sophisticated tagging of the B. Taken alone, this wouldyield a value for | V ub | compatible with the exclusive semi-leptonic de-termination. However, this result has not yet been confirmed by othercollaborations, and more data and a careful study of all systematicsare still required before drawing final conclusions. Preprint submitted to Nucl. Phys. B Proc. Suppl. June 11, 2018 a r X i v : . [ h e p - l a t ] O c t . Non-perturbative HQET The particular challenge of B-physics on the lattice liesin the many disparate scales, ranging from its inverseextent (as IR cuto ff ) over the hierarchy of di ff erentlyflavoured hadron masses up to the inverse of the latticespacing a (as UV cuto ff ), to be treated simultaneouslyin the numerical simulations. Since lattice sizes that arecomputationally manageable today have am b >
1, dis-cretization e ff ects get most severe for heavy quark sys-tems with b-quarks and escape brute force simulations.Our approach to lattice B-physics is therefore based onthe Heavy Quark E ff ective Theory (HQET) for the b-quark [7, 8], which consists in a systematic expansionof its QCD action and heavy-light correlation functionsin Λ QCD / m b (cid:28) m b → ∞ ).The Lagrangian entering the heavy quark field’s lat-tice action S HQET = a (cid:80) x L HQET ( x ) in HQET at NLO,i.e., including O(1 / m b ) terms, reads: L HQET = ψ h D ψ h − ω kin O kin − ω spin O spin , (1)with ψ h satisfying P + ψ h = ψ h , P + = + γ . The pa-rameters ω kin and ω spin are formally O(1 / m b ) and mul-tiply the dimension–5 operators O kin = ψ h D ψ h and O spin = ψ h σ B ψ h , representing interaction terms due tothe motion and the spin of the heavy quark. Thus, S HQET has O( Λ / m ) truncation errors, and lattice artifactsonly scale as ( a Λ QCD ) n rather than ( am b ) n . Analogously,local composite fields are introduced in the e ff ective lat-tice theory. For instance, the NLO HQET expansion ofthe zero-momentum projected time component of theheavy-light axial vector current can be written as A HQET0 , R ( x ) = Z HQETA a (cid:80) x (cid:104) A stat0 ( x ) + c (1)A A (1)0 ( x ) (cid:105) , A stat0 ( x ) = ψ l ( x ) γ γ ψ h ( x ) , A (1)0 ( x ) = ψ l ( x ) γ γ i (cid:0) ∇ s i − ←−∇ s i (cid:1) ψ h ( x ) , (2) ∇ s i being the spatial covariant derivative. The relation f PS √ m PS = (cid:104) | A , R (0) | PS( p = (cid:105) to the pseudoscalardecay constant will be used to calculate f B (s) below.HQET treats the O(1 / m b ) interactions terms in (1) aslocal space-time insertions in static correlations func-tions. For correlators of some multi-local fields O andup to 1 / m b –corrections to the operator itself (irrelevantwhen spectral quantities are considered), this means (cid:104)O(cid:105) = (cid:104)O(cid:105) stat + a (cid:88) x (cid:110) ω kin (cid:104)OO kin ( x ) (cid:105) stat (3) + ω spin (cid:104)OO spin ( x ) (cid:105) stat (cid:111) , where (cid:104)O(cid:105) stat is the expectation value in the static theory.Still, for lattice HQET applications to lead to preciseand controlled results, two issues had to be solved. 1.) The exponential growth of the noise-to-signal ra-tio in static-light correlation functions with Euclideantime, caused by the linear divergence in the binding en-ergy E stat of the static-light system, which is particularlysevere for the Eichten-Hill action [9]. This is overcomeby so-called ”HYP-smeared” [10] discretizations of thestatic quark action, improving the statistical precision ofthe correlators substantially [11, 12].2.) Operator mixing in the e ff ective theory inducesUV power divergences in the lattice spacing that mustbe subtracted non-perturbatively: The formal definitionof lattice HQET and its composite fields in (1) and (2)involves the (a priori free) e ff ective couplings ω ≡ (cid:110) m bare , ln Z HQETA , c (1)A , ω kin , ω spin (cid:111) . (4)Here, the energy shift m bare is an additive mass renor-malization. It absorbs the 1 / a –divergence of the staticenergy, E stat , and a 1 / a –divergence at O(1 / m b ). Hence,a phenomenologically relevant predictive power of lat-tice HQET is only guaranteed, once these HQET pa-rameters ω = { ω i } have been fixed non-perturbatively such that no uncancelled power divergences in 1 / a ,which would remain in perturbation theory [13], canpreclude to take the continuum limit.A solution to 2.) was developed in [14] and reliesupon a non-perturbative matching of HQET and QCDin finite volume . The implementation of this strategy byour collaboration has led to NLO HQET computationsof the b-quark mass, B-meson spectroscopy and decayconstants in the quenched approximation ( N f =
0) [15–18], as well as in the more realistic two-flavour theory[19–22], of which we give an overview in the following.
The computational strategy of our approach [14, 16, 22],in which matching and renormalization are performedsimultaneously and non-perturbatively , is sketched inFig. 2. The matching part is performed in a smallvolume of L ≈ . am b (cid:28) ω i of the La-grangian and the time component of the heavy-light ax-ial current are fixed by imposing matching conditions Φ HQET ( z , a ) ! = Φ QCD ( z , = lim a → Φ QCD ( z , a ). The As soon as 1 / m b –corrections are included, matching must bedone non-perturbatively in order not to spoil the asymptotic conver-gence of the series. Otherwise, the perturbative truncation error fromthe matching coe ffi cient of the static term becomes much larger thanthe power corrections ∼ Λ QCD / m b of HQET, as m b → ∞ . igure 2: Idea of lattice HQET computations for B-physics phe-nomenology via a non-perturbative determination of HQET param-eters from small-volume QCD simulations. The step scaling methodmakes contact to physically large volumes L ∞ . The whole construc-tion is such that the continuum limit can be taken for all pieces. quark mass dependence of (non-perturbatively renor-malized) QCD is inherited by the HQET parameters ω i .It enters through the dimensionless variable z ≡ L M ,where M is the renormalization group invariant (RGI)mass [20]. Then a recursive finite-size scaling step L → L = L is used to reach larger volumes and lat-tice spacings a , by which connection with phenomenol-ogy in L ∞ (cid:38) max(2 fm , / m π ) can be made. As a resultof [22], the HQET parameters (4), ω = ω ( z , a ), absorb-ing the logarithmic and power divergences of HQET, arenow non-perturbatively known for renormalized QCDquark masses from the charm to beyond the beauty re-gion (parameterised by z ∈ { , , , , , , , , } )and for a ’s corresponding to the bare gauge couplingsof the available two-flavour configuration ensembles inlarge volume used to compute B-physics observables. Our large-volume gauge configuration ensembles arecharacterised by the plaquette gauge action and a seaof N f = a ) im-proved dynamical Wilson quarks. To be able to extrap-olate to the physical pion mass, several pseudoscalar(sea) masses in the range (190 (cid:46) m PS (cid:46) L m PS (cid:38) ff ects are expected to be negligible. Moreover, the con-figurations cover 3 β –values { . , . , . } with latticespacings a ∈ { .
08 fm , .
07 fm , .
05 fm } [23] to controlthe extrapolation to the continuum limit. For the numerical simulations to generate these two-flavour QCD configurations, we employ M. L¨uscher’simplementation of the Hybrid Monte Carlo (HMC) al-gorithm with domain decomposition [24] and an adap-tion [25], which combines the deflated SAP solver [26–28] with mass preconditioning [29], chronological in-version [30] and multiple time scale integration [31, 32].Large trajectory length [33, 34] and long runs are usedto ensure that our ensembles are not biased by the crit-ical slowing down of the QCD simulations. For a care-ful and conservative error estimation, a binned jackknifeprocedure is applied (which is being cross-checked bythe method advocated in [34]). All large-volume con-figuration ensembles have been produced and are sharedwithin the Coordinated Lattice Simulations (CLS) e ff ortby several lattice QCD teams in Europe [35].Our determination of the B-meson spectrum and de-cay constants in two-flavour QCD is based on theirHQET expansions in terms of the known HQET pa-rameters and associated HQET energies and matrix el-ements at the static and 1 / m b –order. The latter are ex-tracted from static-light correlation functions evaluatedon the available large-volume CLS ensembles by solv-ing the Generalised Eigenvalue Problem (GEVP) dis-cussed and applied in [36, 37, 17, 18], allowing for abetter control of excited state contaminations of the cor-relators. More specifically, the GEVP analysis amountsto compute a ( N × N )–matrix of correlators with the de-sired static and O(1 / m b ) insertions, where each entry ofthe matrix corresponds to a di ff erent Gaussian smearinglevel [38] of the light quark field in the B-meson in-terpolating quark bilinear. In these computations, vari-ance reduction in the light quark sector is achieved bystochastic all-to-all propagators (with 8 noise sourcesand full time-dilution) [39, 17], while for the staticquark propagators, we use two variants of the HYP-smeared static actions, HYP1 / ff ective energies and hadron-to-vacuum matrix elements, which converge faster withEuclidean time separation than standard ratios, since alarger gap governs the excited state corrections. I.e.,corrections to ground state energies and matrix ele-ments fall o ff in t and t as ∝ exp {− ( E N + − E ) t } and ∝ exp {− ( E N + − E ) t } × exp {− ( E − E ) ( t − t ) } , re-spectively, where t < t < t and N labels the N th excited state (and N = t min for fixed t max suchthat O (cid:0) e − ( E N + − E ) t (cid:1) ∼ σ sys (cid:46) σ stat , thereby minimis-ing our systematic errors. For more details we referto [37, 17, 18, 40, 41].3 . Results In this section we summarise the status of results of our N f = s -meson sector with the valence quark tuned tothe physical strange quark [23], corresponding to a par-tially quenched setup. .
05 0 .
10 0 . m m B ( z , m P S , a , HY P n ) m expB z b z m B ( z , m π , ) Figure 3:
Joint chiral and continuum extrapolation of the heavy-lightpseudoscalar meson mass (5) in NLO HQET to the ansatz (7) for fixed z . Recall that the z –dependence originates from the initial matchingstep to finite-volume QCD determining the HQET parameters. (Blue,red and green points refer to β = . , . and . , respectively, whilefilled / open symbols belong to the HYP1 / To begin with, we apply the non-perturbative results onthe HQET parameters from the matching step togetherwith the HQET energies from the large-volume GEVPanalysis of the CLS ensembles to calculate the b-quarkmass. To this end one writes down the NLO HQET ex-pansion (i.e., to first order in 1 / m b ) of the heavy-lightpseudoscalar meson mass as m B = m bare + E stat + ω kin E kin + ω spin E spin . (5)Beside the dependence on the light pseudoscalar (sea)mass m PS and the lattice spacing a of the CLS configu-rations, we also have to account for the apparent heavyquark mass ( z ) dependence of the HQET parameters. In fact, this z –dependence can now be exploited to fix theHQET parameters ω i = ω i ( z , a ) for once by imposingthe condition m B ( z , m π , a ) | z = z b ≡ m expB = . defines the physical value ofthe b-quark mass at NLO of HQET.For given z , the l.h.s. is obtained by evaluating (5)for each value of the lattice spacing and sea quark massof the CLS ensembles, followed by a global fit of m B (simultaneously for two variants of the aforementionedHYP-smeared actions, HYP n , n = ,
2) to the ansatz fora combined chiral ( m PS → m π ≡ m exp π =
135 MeV [1])and continuum ( a →
0) extrapolation, m B ( z , m PS , a , HYP n ) = B ( z ) + Cm − g π f π m + D HYP n a , (7)with f π ≡ f exp π = . g = . ∗ B π –coupling in the static approximation for theb-quark. These extrapolations to the physical point areshown in Fig. 3. .
05 0 .
10 0 . m m B ( z , m P S , a , HY P n ) m expB z b z m B ( z , m π , ) Figure 4: z –dependence of m B in the continuum limit and graphicalsolution of (6), which determines the physical b-quark mass z b . ( z = L M denotes the dimensionless RGI heavy quark mass.) The solution of (6) for a = z b ≡ L M b and is illustrated inFig. 4. Converting L to physical units, achieved viasetting the lattice scale through f K in [23], and translat-ing with 4– / / mass to4he conventional MS scheme, we obtain as our result forthe b-quark’s mass in HQET at O(1 / m b ) in the N f = z b = . z ⇔ m MSb ( m b ) = . z GeV . (8)The first error covers all statistical and systematic errors,including those from the GEVP analysis, the various ex-trapolations and the scale setting, while the second un-certainty of about 1% stems from the quark mass renor-malization in QCD, entering the finite-volume match-ing step [20, 22], and has to be added in quadrature.Since we find the di ff erence of the NLO HQET result(8) and the corresponding number in the static approxi-mation (LO HQET) to be very small, we conclude thatthe truncation error of O( Λ / m ) to (8) in the HQETexpansion is negligible compared to our overall error.Our result (8) also compares very well with other re-cent determinations and the value quoted by the ParticleData Group, see the (incomplete) compilation in Tab. 1.Note that some determinations claim very small errors,although they are based on perturbation theory or latticedata with heavy quark masses in lattice units close to 1. m MSb ( m b ) / GeV remarks, method ref.4.347(48) lattice, N f =
0, NLO HQET [15]4.22(11) lattice, N f =
2, NLO HQET eq. (8)4.29(14) lattice, N f =
2, extrapolation [44]4.164(23) lattice, N f =
3, extrapolation [45]4.163(16) perturbation theory & data [46]4.236(69) perturbation theory & QCD inputs [47]4.18(3) PDG average 2012 [1]
Table 1:
Compilation of some recent determinations of m b . As for thelattice results [44] and [45], the former uses extrapolations of relativis-tic data around the charm to known static limits, while the latter em-ploys moments of current-current correlators with Highly ImprovedStaggered Quarks (HISQ) [48] extrapolated to the b-scale. [46] and[47] rely on QCD sum rules. For more details, see the cited references. After the determination of the value of the physical b-quark mass (and thus z b ) from m B , we can fix the HQETparameters to ω i ( a ) = ω i ( z b , a ) and use those in anysuccessive HQET computation of B-physics obervables. To determine the B-meson decay constants f B and f B s ,we now combine the HQET parameters with the ma- trix elements resulting from the GEVP analysis. Dis-tinguishing the heavy-light B r –meson decay constants,where the light flavour can be either a valence ( = sea)quark flavour r = { u , d } ≡ l or a valence strange one,r = s, their NLO HQET expansions in terms of theHQET parameters ω i read f B r (cid:113) m Br = Z HQETA (cid:16) + b statA am q , r (cid:17) p statr (9) × (cid:16) + ω kin p kinr + ω spin p spinr + c (1)A p A (1) r (cid:17) . As explained before, the ω i are understood to be takenat the physical b-quark mass, z b , and the p X , X ∈{ stat , kin , spin , A (1) } , denote the previously extractedGEVP plateau values of the associated e ff ective matrixelements of Sect. 2.2. The improvement coe ffi cient b statA is known to 1–loop perturbation theory from [49].Due to non-perturbative O( a ) improvement employedin our computations, f B r (with r = l , s and f B l ≡ f B ),approaches the continuum limit quadratically in the lat-tice spacing. In order to estimate a systematic error inour combined chiral and continuum extrapolation, wechoose fits, where the sea quark dependence is mod-elled according to the prediction of Heavy Meson Chi-ral Perturbation Theory (HM χ PT) [50, 51], as well asonly linear in m , f B ( m PS , a , HYP n ) = (10) b (cid:48) (cid:34) −
34 1 + g (4 π f π ) m ln (cid:0) m (cid:1)(cid:35) + c (cid:48) m + d (cid:48) HYP n a , f B r ( m PS , a , HYP n ) = b r + c r m + d r , HYP n a ; (11)here, f π = f exp π and ˆ g = . f B s , not all CLSensembles were analysed yet.From the figures one can infer that whether we do ordo not include the chiral logarithm of HM χ PT in theextrapolation (10) of f B induces a very small change atthe physical point only. We thus take the HM χ PT ex-trapolation as the central value and the di ff erence to thelinear fit to account for a part of the systematic error ofour final result. For the B- and B s -meson decay con-stants from HQET at O(1 / m b ) in two-flavour QCD wepreliminarily give f B = χ MeV , (12) f B s = , (13) For the subtracted bare quark masses appearing here, the additiveimprovement term b statA am q , r is numerically very small in practice. π .
05 0 . . . . . . . . . . .
25 [ m PS / MeV] HM χ PT: f B ( z b , m PS , a ) / GeV
Figure 5:
Joint chiral and continuum extrapolation to the physi-cal point of the B-meson decay constant (9) in NLO HQET to theHM χ PT–motivated ansatz (10). The colour coding is the same as inFig. 3. (I.e., blue, red and green points refer to β = . , . and . ,while filled / open symbols belong to the HYP1 / where the quoted errors again cover all sources of sta-tistical and systematic uncertainties.Our results (12) and (13) are in line with computa-tions of other groups, see, e.g., summaries in [52, 47].
4. Outlook
The non-perturbative treatment of NLO HQET withcontrolled chiral and continuum extrapolations leads toresults for B-physics phenomenology with a few–% ac-curacy. They can contribute to resolving current ten-sions in precision CKM analyses of the B-meson sec-tor. As our computations are the only ones, which haveno perturbative uncertainties, including the renormal-ization of the axial current, this introduces a new quality,albeit for N f = ff erence between the vector B ∗ - and the pseu-doscalar B-meson is dominated by a pure O(1 / m b ) ef-fect from the contribution of O spin to the e ff ective HQETLagrangian (1) and, hence, is of particular interest.By extending our finite-volume matching strategy toall components of the axial and vector currents, we aimat a NLO HQET calculation of B → π semi-leptonic de-cay form factors as possible application. A status reportin the LO (static) approximation has been given in [53]. m π [ m PS /MeV ] f B ( z b , m PS , a ) /GeV m π N f =
0: 216 ( ) MeV f B s ( z b , m PS , a ) /GeV 0.170.180.190.20.210.220.230.240.25 Figure 6:
Joint chiral and continuum extrapolation to the physicalpoint of f B ( left ) and f B s ( right ) in NLO HQET, where only a lin-ear dependence on the squared light pseudoscalar (sea) mass m is assumed, cf. (11). In case of f B s , the NLO HQET result f B s = obtained in the quenched approximation ( N f = ) [18](where the scale was set through r = . ) is included for compar-ison. The colour coding is the same as in Fig. 5. Acknowledgments
This work is supported by the DFG in the SFB / TR 9,“Computational Particle Physics”, and was formerlyso through EU Contract No. MRTN-CT-2006-035482,“FLAVIAnet”. We thank for further funding by thegrants STFC ST / G000522 / / N f = References [1] Particle Data Group, J. Beringer et al., Phys. Rev. D86 (2012)010001, http: // pdg.lbl.gov.[2] BELLE, Y. Yook, talk at ICHEP 2012.[3] BELLE, I. Adachi et al., 1208.4678.[4] E. Lunghi and A. Soni, Phys. Lett. B697 (2011) 323, 1010.6069.[5] J. Laiho, E. Lunghi and R. Van de Water, PoS LATTICE2011(2011) 018, 1204.0791.[6] CKMfitter, A. Lenz et al., Phys. Rev. D86 (2012) 033008,1203.0238.[7] E. Eichten and B. Hill, Phys. Lett. B234 (1990) 511.[8] E. Eichten and B. Hill, Phys. Lett. B243 (1990) 427.[9] E. Eichten and B. Hill, Phys. Lett. B240 (1990) 193.[10] A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (2001) 034504,hep-lat / / / / /
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