Form factors for B s →Kℓν decays in Lattice QCD
Felix Bahr, Fabio Bernardoni, John Bulava, Anosh Joseph, Alberto Ramos, Hubert Simma, Rainer Sommer
DDESY 14-217SFB/CPP-14-90October 19, 2018
Form factors for B s → K (cid:96)ν decays in Lattice QCD Felix Bahr , Fabio Bernardoni, John Bulava, Anosh Joseph,Alberto Ramos, Hubert Simma, Rainer Sommer John von Neumann Institute for Computing (NIC), DESY, Platanenallee 6,D-15738 Zeuthen, Germany
We present the current status of the computation of the form factor f + ( q ) for the semi-leptonic decay B s → K (cid:96)ν by the ALPHA collabora-tion. We use gauge configurations which were generated as part of theCoordinated Lattice Simulations (CLS) effort. They have N f = 2 non-perturbatively O ( a ) improved Wilson fermions, and pion masses downto ≈
250 MeV with m π L ≥
4. The heavy quark is treated in non-perturbative Heavy Quark Effective Theory (HQET).We discuss how to extract the form factors from the correlation func-tions and present first results for the form factor at q = 21 .
23 GeV extrapolated to the continuum. Next-to-leading order terms in HQETand the chiral extrapolation still need to be included in the analysis.PRESENTED AT Speaker a r X i v : . [ h e p - l a t ] N ov Introduction
Determinations of the CKM matrix element | V ub | from different exclusive (and in-clusive) decays tend to disagree at the ∼ − σ level [1]. Both theoretical andexperimental improvements are needed to clarify the situation.In this work, we report on our ongoing effort to non-perturbatively determine theform factors for B s → K (cid:96)ν decays from Lattice Quantum Chromodynamics (LQCD)with N f = 2 sea quarks. Although no experimental data is available yet for thisdecay, the heavier spectator s -quark renders the LQCD computations technicallysimpler than for B → π(cid:96)ν , and thus provides a good starting point to gain solidcontrol on all systematic errors (and to make an LQCD prediction).The decay amplitude for B s → K (cid:96)ν is proportional to | V ub | times the hadronicmatrix element (cid:10) K( p K ) (cid:12)(cid:12) V µ (cid:12)(cid:12) B s ( p B s ) (cid:11) of the vector current V µ ( x ) = ψ u ( x ) γ µ ψ b ( x ). Thematrix element is parametrised by two form factors, f ( q ) and f + ( q ), which dependon q = ( p B s − p K ) , the invariant mass of the lepton pair. In the limit of vanishinglepton masses only f + ( q ) contributes to the decay rate. | V ub | can then be determined by combining the differential decay rate from ex-periment with f + ( q ) from theory. In principle, it is sufficient to do this at a singlevalue of q . In practice, experimental data is provided over a range (of bins) of q ,and one can use the BCL paramterisation [2] to express the form factor f + ( q ) as acontinuous function of q . Then, a theoretical prediction of f + ( q ), e.g. from LQCD,for at least a single q allows to extract | V ub | .Here we report on preliminary work to study the feasibility of a precise determi-nation of the form factor in the continuum limit and at a fixed q . On the lattice (with spatial extent L and lattice spacing a ) the large mass of the b quark gives rise to a hierarchy of scales L − (cid:28) m π ≈
140 MeV (cid:28) m B ≈ (cid:28) a − , (1)which cannot be directly simulated with present computing resources. Instead, wefollow the strategy devised by the ALPHA collaboration [3] to treat the heavy quarkwithin the framework of non-perturbative Lattice HQET. It is an expansion in inversepowers of the heavy quark mass m h and valid for kaon momenta p K (cid:28) m h . In practice,we require p K (cid:46) /m h , and thus the computationshave a well-defined continuum limit. The expectation value of a product of local fields, O , up to and including O (1 /m h ) in HQET on the lattice is (cid:104)O(cid:105) = (cid:104)O(cid:105) stat + ω kin a (cid:88) x (cid:104)OO kin ( x ) (cid:105) stat + ω spin a (cid:88) x (cid:104)OO spin ( x ) (cid:105) stat , (2)1here (cid:104) . . . (cid:105) stat is the expectation value in the static approximation. On the right handside, also O needs to be expanded in 1 /m h , for instance,V HQET k ( x ) = Z HQET V k (cid:20) V stat k ( x ) + (cid:88) j =1 c V k,j V k, j ( x ) (cid:21) , (3)for the spatial components of the vector current, and analogous for V . The HQETparameters ω k in , ω s pin , and c V µ,j are of order 1 /m h , while Z HQET V and Z HQET V k are oforder 1. They can be determined fully non-perturbatively by matching HQET andQCD [4]. Thus, perturbation theory can be avoided at any stage of the computation.Since the non-perturbative matching is still in progress, we present in this ex-ploratory work only results in the static approximation, i.e. setting ω kin = ω spin = c V = 0. For the renormalisation constants we follow the lines of [5, 6] and write Z HQET as a product of matching factors, C PS or C V , which are known at three loops in per-turbation theory [7], and Z statA , RGI which is known non-perturbatively [8]. Truncationof the lattice theory at static order is expected to be a 10-20% effect.To perform the continuum extrapolation of the form factors at a fixed value of q , we employ flavour twisted boundary conditions [9] for the s quark, ψ ( x + L ˆ k ) =e i θ k ψ ( x ). In this way, the quark momentum is altered from (cid:126)p = 2 π(cid:126)n/L to (cid:126)p θ =(2 π(cid:126)n + (cid:126)θ ) /L , with (cid:126)n ∈ N . Choosing the twist angle θ k , one can freely tune themomentum of the s quark, and thus of the kaon. The heavy quark is twisted by thesame angle to remain in the rest frame of the B s meson.Our computations are performed on gauge field ensembles generated with N f = 2dynamical sea quarks within the CLS effort. They use non-perturbatively O ( a )-improved Wilson fermions and the scale is set via f K [10]. All ensembles have m π L (cid:38)
4. In this work, we present results from measurements on the three CLS ensemblesA5, F6 and N6. Their properties are listed in ref. [10]. Error estimates take intoaccount correlations and autocorrelations [11].
On the gauge configurations we measure the two- and three-point correlation functions C K ( x − y ; (cid:126)p ) = (cid:88) (cid:126)x,(cid:126)y e − i (cid:126)p · ( (cid:126)x − (cid:126)y ) (cid:104) P su ( x ) P us ( y ) (cid:105) , C B ij ( x − y ; (cid:126)
0) = (cid:88) (cid:126)x,(cid:126)y (cid:104) P sb i ( x ) P bs j ( y ) (cid:105) , C µ, j ( t K , t B ; (cid:126)p ) = (cid:88) (cid:126)x K ,(cid:126)x V ,(cid:126)x B e − i (cid:126)p · ( (cid:126)x K − (cid:126)x V ) (cid:104) P su ( x K )V µ ( x V ) P bs j ( x B ) (cid:105) , (4)where P q q i ( x ) are interpolating fields, like ψ q ( x ) γ ψ q ( x ), for the mesons. Thesubscripts i or j indicate different levels of Gaussian smearing [12] of the s quark2n the heavy-light meson, i.e. different trial wave functions. In the limit of largeEuclidean times, t B ≡ x − x V and t K ≡ x V − x , the ratio f ratio µ, i ( t B , t K ; q ) ≡ C µ, i ( t K , t B ) (cid:112) C K ( t K ) C B ii ( t B ) e E K t K / e E B t B / (5)will then give the desired matrix element (for any suitable smearing i ) (cid:104) K( p θ K ) | V µ | B s (0) (cid:105) = lim T,t B ,t K →∞ f ratio µ, i ( t B , t K ; q ) (6)Alternatively, we can parameterise the correlation functions as C K ( t K ) = (cid:88) m ( κ ( m ) ) e − E ( m )K t K , C B ij ( t B ) = (cid:88) n β ( n ) i β ( n ) j e − E ( n )B t B , (7) C µ, i ( t B , t K ) = (cid:88) m,n κ ( m ) ϕ ( m,n ) µ β ( n ) i e − E ( n )B t B e − E ( m )K t K , (8)and determine { κ ( m ) , E ( m )K } from a fit to C K , and { β ( n ) i , ϕ ( n,m ) µ , E ( n )B } from a combinedfit to C µi and C B ij . Then, ϕ (1 , µ is equal to the matrix element of eq. (6). We take m = 1 and n = 1 ,
2, i.e. we include the first excited B s state, but only the kaonground state.In fig. 1 we show the ratio f ratio µ of eq. (5) at fixed t K = 20 as a function of t B . Forcomparison, we also indicate the value of ϕ (1 , µ resulting from the fit. . . . . . . . . . . . . . t B / fm f r a t i o0 Figure 1: The ratio f ratio µ (blue points)and the fit result ϕ (1 , µ (red band) for lat-tice N6, µ = 0 and fixed t K = 20. In the rest frame of the B s meson, thematrix elements have the form (cid:104) K | V | B s (cid:105) = (cid:112) m B s f (cid:107) ( q ) , (cid:104) K | V i | B s (cid:105) = (cid:112) m B s p i K f ⊥ ( q ) , where the form factors ( f (cid:107) , f ⊥ ) are relatedto ( f + , f ). In particular, we have f + = 1 √ m B s f (cid:107) + 1 √ m B s ( m B s − E K ) f ⊥ . (9)Fig. 2 shows f + , as extracted from thefitted ϕ (1 , µ , for different lattice spacings.Working in the static approximation ofHQET, we are free to keep or drop termsof order 1 /m h in eq. (9) for computing f + .To illustrate this O (1 /m h ) ambiguity, we3how in fig. 2 (and 3) two sets of data points: the upper one corresponds to usingall terms in eq. (9), the lower one to dropping the term proportional to f (cid:107) . Once weinclude all O (1 /m h ) terms of HQET, this ambiguity will disappear. For both setswe show a constant continuum extrapolation and one linear in a . The latter has byfar the larger error and within this error is consistent with the result of the constantextrapolation.In fig. 3, we compare our results from the linear continuum extrapolation of f + ( q )to recent results of HPQCD [13] (at their smallest a = 0 .
09 fm and m π = 320 MeV). − . . . . . . . · − . . . . . a / fm f + DataExtrapolations
Figure 2: Continuum extrapolation of ourdata for f + at q = 21 .
23 GeV .
17 17 . . . . . . . . . . q / GeV f + HPQCD, ref. [13]This work, preliminary
Figure 3: Comparison of LQCD resultsat various values of q . We presented the current status of our computation of the form factor f + ( q ) for thesemi-leptonic decay B s → K (cid:96)ν at a fixed value of q = 21 .
23 GeV using HQET onthe lattice. We compare two different methods to extract the form factors, eitherfrom the plateau value of a suitable ratio of correlators, or from a simultaneous fit tothe functional form of the correlators.We also have performed a continuum extrapolation of our lattice data and findsmall O ( a ) effects. The preliminary results reported here are still computed in thestatic approximation and an extrapolation to the physical pion mass has yet to beperformed. Our preliminary value of f + at this stage is in rough agreement with theresults from other collaborations.All O (1 /m h ) effects of HQET will be included in the analysis once the HQETparameters are known non-perturbatively. We also plan to extend the computationto B → π(cid:96)ν decays, several values of q , and N f = 2 + 1 flavours of sea quarks.4 CKNOWLEDGEMENTS
We thank the Leibniz Supercomputing Centre for providing computing time on Su-perMUC. The gauge configurations were produced by the CLS effort and for the usedcomputing resources we refer to the acknowledgement of ref. [10].
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