Semileptonic B c decays from full lattice QCD
Andrew Lytle, Brian Colquhoun, Christine Davies, Jonna Koponen, Craig McNeile
SSemileptonic B c decays from full lattice QCD Andrew Lytle ∗ , Brian Colquhoun, Christine Davies, Jonna Koponen SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UKE-mail: [email protected]
Craig McNeile
Centre for Mathematical Sciences, Plymouth University, Plymouth, PL4 8AA, UK
HPQCD Collaboration † We present first lattice QCD results for semileptonic form factors of the decays B c → η c l ν and B c → J / ψ l ν over the full q range, using both improved non-relativistic QCD (NRQCD) andfully relativistic (HISQ) formalisms. These can be viewed as prototype calculations for pseu-doscalar to pseudoscalar and pseudoscalar to vector decays involving a b → c transition. Inparticular we can use information from the relativistic computations to fix the NRQCD currentnormalisations, which can then be used in improved computations of decays such as B → D l ν and B → D ∗ l ν . ∗ Speaker. † c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] M a y emileptonic B c decays from full lattice QCD Andrew Lytle
1. Introduction
Semileptonic decays of B -mesons provide the main inputs for exclusive determinations of | V ub | and | V cb | (a status update of lattice QCD’s impact on these quantities was presented at this confer-ence [1]). The precise determination of | V cb | requires precision in both theoretical computationand measurement of b → c processes. Treatment of c and especially b quarks is a challenge forlattice simulations due to lattice artifacts which grow as ( am q ) n where a is the lattice spacing and m q is a quark mass. We have two complementary approaches to the treatment of b quarks: usinga highly improved relativistic action at small lattice spacings to simulate masses approaching m b ,and working directly at m b with an improved non-relativistic (NRQCD) effective theory formalism. Methodology
We use a highly improved staggered quark (HISQ) action [2] which systematically removeslattice artifacts, allowing for simulation of charm quarks with small discretisation effects. We caneven simulate quarks with mass significantly larger than m c , especially on the ensembles with veryfine lattice spacings of a ≈ .
045 fm [3]. This motivates one of our approaches for doing b -physics.By working in a regime am h (cid:46) .
8, say, but on finer and finer lattice spacings, we calculate thephysics of interest over a range in m h and then extrapolate that data to m b .We are also able to work directly at the b mass, without the need for extrapolation, using animproved non-relativistic (NRQCD) formalism [4]. This approach is complementary to the fullyrelativistic approach described above. The NRQCD Hamiltonian is expressed as an expansion inthe velocity of the heavy quark. In addition the current operators have a relativistic expansion, e.g.the temporal axial-vector current A nrqcd0 = ( + α s z ( ) ) (cid:104) A ( ) + ( + α s z ( ) ) A ( ) + α s z ( ) A ( ) (cid:105) + . . . , where A ( ) , A ( ) , . . . are higher order current corrections, with matrix elements proportional to1 / m b . The matching to the continuum current above is only known in QCD perturbation theory to O ( α s ) , and so has a systematic uncertainty from missing α s terms. One output of the present workwill be to improve the normalisation of the currents using the fully relativistic calculation wherethe normalisation is simpler and nonperturbative.We use gauge ensembles generated by the MILC collaboration, which include the effects of u / d , s , and c quarks in the sea, and with lattice spacings of a ≈ . , .
06, and 0 .
045 fm. Althoughwe have ensembles with physical u / d quark mass, all results presented here use m s / m u / d =
5, i.e.unphysically heavy pions.
Obtaining the form factors
In both formalisms we compute the matrix element of the V − A operator between the statesof interest. We work in the frame where the B c is at rest. The matrix elements are parametrisedin terms of form factors which are functions of q , where q = P − p is the difference in four-momentum between the B c and outgoing particle. The kinematic endpoint q is where the out-going hadron is at rest, whereas the energy of the outgoing hadron is a maximum at q = emileptonic B c decays from full lattice QCD Andrew Lytle
For the B c → η c decay there are two form factors to determine, f + and f . (cid:104) η c ( p ) | V µ | B c ( P ) (cid:105) = f + ( q ) (cid:20) P µ + p µ − M − m q q µ (cid:21) + f ( q ) M − m q q µ For the relativistic case we can determine f using a scalar current, which is absolutely normalised. (cid:104) η c ( p ) | S | B c ( P ) (cid:105) = M − m m b − m c f ( q ) There are five form factors to determine for the B c → J / ψ decay, one from the vector currentand four from the axial-vector current (three of which are independent). (cid:104) J / ψ ( p , ε ) | V µ − A µ | B c ( P ) (cid:105) = i ε µνρσ M + m ε ∗ ν p ρ P σ V ( q ) − ( M + m ) ε ∗ µ A ( q )+ ε ∗ · qM + m ( p + P ) µ A ( q ) + m ε ∗ · qq q µ A ( q ) − m ε ∗ · qq q µ A ( q ) Results
Figure 1 (left) shows our results for the B c → η c form factors f + ( q ) and f ( q ) computedusing improved NRQCD on the a ≈ .
09 fm ensemble. q [GeV ]0 . . . . . . . HPQCD Preliminary f + ( q ) f ( q ) M η h [GeV]11.21.41.61.822.22.42.62.8 f ( q ) / f H c [ G e V − ] HPQCD Preliminary q = 0 HISQ a ≈ .
09 fmHISQ a ≈ .
06 fmHISQ a ≈ .
045 fmNRQCD a ≈ .
09 fm q a ≈ .
09 fmHISQ a ≈ .
06 fmHISQ a ≈ .
045 fmNRQCD a ≈ .
09 fm
Figure 1: (Left) Results for B c → η c form factors f and f + from lattice NRQCD, determined over thefull q range. (Right) Extrapolation results in heavy quark mass m h for f ( q ) / f H c and f ( ) / f H c usingthe fully relativistic (HISQ) formalism. The rightmost points are the corresponding NRQCD results withphysical b mass. In Figure 1 (right) we show results for f ( q ) / f H c and f ( ) / f H c using the relativistic for-malism on ensembles with lattice spacings of a ≈ . , .
06, and 0 .
045 fm. For each ensemble weuse valence masses m h such that am h ≤ .
8, which correspond to heavier physical masses as we goto smaller lattice spacings. As m h approaches m b the results join smoothly with results computedusing NRQCD (rightmost points).Figure 2 is another extrapolation plot, this time showing the B c → J / ψ form factor A ( q ) .This is the only form factor that contributes to the decay rate at zero recoil. Furthermore Luke’stheorem tells us the 1 / m b current corrections vanish there so that the comparison between NRQCDand relativistic data is particularly simple. 2 emileptonic B c decays from full lattice QCD Andrew Lytle M η h [GeV]0.70.80.911.11.2 A ( q m a x ) HPQCD Preliminary
HISQ a ≈ .
06 fmHISQ a ≈ .
045 fmNRQCD a ≈ .
09 fm
Figure 2:
Extrapolation in heavy quark mass m h for the B c → J / ψ form factor A ( q ) using the relativisticHISQ formalism. The rightmost point is the NRQCD result at physical b mass. Conclusions
We have presented results for the B c semileptonic decays to η c and J / ψ using two comple-mentary approaches. • The B c → η c results provide proof-of-principle for our strategy; we are able to control thecalculation over the full q range and find good agreement between the NRQCD and fullyrelativistic approaches. • Our first results for the B c → J / ψ decay also appear promising. The full lattice calculationof this decay will allow the extraction of | V cb | if the decay is measured at LHCb. • The NRQCD b → c currents also mediate the decays B → D and B → D ∗ . Using informationfrom the relativistic calculation we will improve the normalisations of the currents, whichwill lead to improvements in theoretical precision for these decays. Acknowledgements
AL would like to thank the organisers for an enjoyable conference, and Gagan Mohanty, GregCiezarek, Barbara Sciascia, Lucio Anderlini, and Jorge Martin Camalich for useful discussions.This work was performed on the Darwin supercomputer, part of STFC’s DiRAC facility.
References [1] R. S. Van de Water, these proceedings.[2] E. Follana et al. , Phys. Rev. D , 054502 (2007) [hep-lat/0610092].[3] C. McNeile et al. , Phys. Rev. D , 074503 (2012) [arXiv:1207.0994 [hep-lat]].[4] G. P. Lepage et al. , Phys. Rev. D , 4052 (1992) [hep-lat/9205007]., 4052 (1992) [hep-lat/9205007].