We report on form factors for the B->K l^+ l^- semi-leptonic decay process.
We use several lattice spacings from a=0.12 fm down to 0.06 fm and a variety of
dynamical quark masses with 2+1 flavors of asqtad quarks provided by the MILC
Collaboration. These ensembles allow good control of the chiral and continuum
extrapolations. The b-quark is treated as a clover quark with the Fermilab
interpretation. We update our results for f_\parallel and f_\perp, or,
equivalently, f_+ and f_0. In addition, we present new results for the tensor
form factor f_T. Model independent results are obtained based upon the
z-expansion.Comment: 7 pages, 4 figures, presented at The XXXth International Symposium on
  Lattice Field Theory - Lattice 2012, June 24-29, 2012 Cairns, Australia, to
  appear as PoS(Lattice 2012)12

/Fermilab

/Illinois U., Urbana

Bailey, Jon A.

Du, Daping

El-Khadra, Aida X.

Gottlieb, Steven

Jain, R.D.

Kronfeld, Andreas S.

Liu, Yuzhi

Meurice, Yannick

U., /Indiana

U., /Iowa

U., /Seoul

Van de Water, Ruth S.

Zhou, Ran

English

arXiv

Jain, R. D.

UNT (University of North Texas) Digital Library

arXiv:1211.1390v1  [hep-lat]  6 Nov 2012Form factors for semi-leptonic B decaysRan Zhou, Steven Gottlieb∗Department of Physics, Indiana University, Bloomington, IN 47405, USAE-mail: sg@indiana.eduJon A. BaileyDepartment of Physics and Astronomy, Seoul National University, Seoul 151-747, South KoreaDaping Du, Aida X. El-Khadra, R.D. JainPhysics Department, University of Illinois, Urbana, Illinois 61801, USAAndreas S. Kronfeld, Ruth S. Van de WaterFermi National Accelerator Laboratory, Batavia IL 60510, USAYuzhi Liu, Yannick MeuriceDepartment of Physics and Astronomy, University of Iowa, Iowa City, IA 52240, USA(Fermilab Lattice and MILC Collaborations)We report on form factors for theB → Kl+l− semi-leptonic decay process. We use several lat-tice spacings froma= 0.12 fm down to 0.06 fm and a variety of dynamical quark masses with2+1 flavors of asqtad quarks provided by the MILC Collaboration. These ensembles allow goodcontrol of the chiral and continuum extrapolations. Theb-quark is treated as a clover quark withthe Fermilab interpretation. We update our results forf‖ and f⊥, or, equivalently,f+ and f0. Inaddition, we present new results for the tensor form factorfT . Model independent results areobtained based upon thez-expansion.The 30th International Symposium on Lattice Field TheoryJune 24–29, 2012Cairns, Australia∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/FERMILAB-CONF-12-603-TForm factors for semi-leptonic B decays Steven Gottlieb1. Motivation and theoretical backgroundThe rareBmeson decayB→Kll is mediated by a flavor changing neutral current (FCNC). TheStandard Model (SM) contribution occurs through a penguin diagram for theb→ sll process. SinceFCNCs are small in the SM, this decay presents an opportunityto detect beyond-the-Standard-Model (BSM) physics. This decay has been studied by several experiments including BaBar [1],Belle [2], CDF [3], and LHCb [4]. We expect that LHCb will conti ue to improve its precision,and newB factories such as SuperB and SuperKEKB will greatly add to our understanding of thisdecay.Recently, the CDF Collaboration published results for theB+ →K+µ+µ− differential branch-ing ratio [3]. They compared their measurement with a decade-ol prediction of the decay formfactors based on light cone sum rule (LCSR) calculations [4]. There is considerable uncertainty inthe LCSR form factors at small momentum transferq2. Unless the uncertainty in the form factorsis reduced, our ability to search for BSM physics will be sever ly limited by lack of knowledge ofthe SM decay rate.Lattice QCD enables first-principles calculations of the form factors with controlled errors thatare systematically improvable. In fact, the first study of this decay, using the quenched approxima-tion, was done over 15 years ago [5]. Since then, there have been three additional studies using thequenched approximation [6, 7, 8]. Recently, there have beentwo groups studying this decay using2+1 flavor improved staggered quark configurations from the MILC collaboration [9, 10].2. EnsemblesTable 1 shows the ensembles that were used in the current calculation. The MILC ensem-bles [11, 12, 13] were generated using three flavors of dynamical quarks. The quarks are asqtad-improved staggered quarks and a Symanzik improved gauge action is used. The common mass (inlattice units) of the two light sea quarks is denotedaml and that of the strange sea quark isams.We use four coarse (a ≈ 0.12 fm), five fine ((a ≈ 0.09 fm) and two super-fine (a ≈ 0.06 fm) en-sembles, in order to control the chiral and continuum extrapolations. The ratio of light-to-strangesea-quark masses varies from 0.4 to 0.05. On each configuration, we use four source times toincrease statistics.3. Form factorsSemileptonic decays of a heavy-light pseudoscalar meson toa pseudoscalar or vector mesonare characterized by form factors that describe matrix elemnts of a hadronic current between initialand final states. We concentrate here on the decayB→ Kll for which we need two matrix elements〈K(pK)|is̄γµb|B(pB)〉 and〈K(pK)|is̄σ µνb|B(pB)〉, where theB meson momentum ispB and that ofthe kaon ispK . For the vector current matrix element, we define two form factors f+ and f0, whilefor the tensor operator, we only need a single form factorfT :〈K|is̄γµb|B〉 = f+(q2)(pµB + pµK −M2B−M2Kq2qµ)+ f0(q2)M2B−M2Kq2qµ , (3.1)2Form factors for semi-leptonic B decays Steven Gottlieb≈ a(fm) size aml/ams Nmeas0.12 203×64 0.02/0.05 20520.12 203×64 0.01/0.05 22590.12 203×64 0.007/0.05 21100.12 203×64 0.005/0.05 20990.09 283×96 0.0124/0.031 19960.09 283×96 0.0062/0.031 19310.09 323×96 0.00465/0.031 9840.09 403×96 0.0031/0.031 10150.09 643×96 0.00155/0.031 7910.06 483×144 0.0036/0.018 6730.06 643×144 0.0018/0.018 827Table 1: Ensembles of asqtadNf = 2+1 configurations analyzed.〈K|is̄σ µνb|B〉 = 2 fT(q2)MB+MK(pµB pνK − pνBpµK), (3.2)In these equations,qµ = pµB − pµK is the momentum transfer, andq2 is the outgoing dilepton invari-ant mass squared. We can alternatively describe the vector current form factors byf‖ and f⊥. Wedefine〈K|is̄γµb|B〉 =√2MB[vµ f‖(EK)+ pµ⊥ f⊥(EK)], (3.3)wherevµ = pµB/MB is the four-velocity of theB meson andpµ⊥ = pµK − (pK ·v)vµ .In lattice QCD, it is convenient to work in the rest frame of theB meson. The new form factorsare considered to be functions of the kaon energyEK :f‖(EK) =〈K|is̄γ0b|B〉√2MB, (3.4)f⊥(EK) =〈K|is̄γ ib|B〉√2MBpiK, (3.5)fT(EK) =MB+MK√2MB〈K(k)|ibσ0is|B(p)〉√2MBpiK, (3.6)where there is no sum on the indexi in the equations forf⊥ and fT . In theB-meson rest frame,q2 = (pB− pK)2 = M2B+M2K −2MBEK, so smallEK corresponds to largeq2.4. Chiral-continuum extrapolationBecause we work with light quark masses larger than in Naturend at non-zero lattice spacing,we must take the chiral and continuum limits of our results. We use heavy-light meson staggeredchiral perturbation theory (HMSχPT) [14] to perform extrapolations. We work to zeroth order in1/mb and next-to-leading order in the light-quark masses, kaon recoil energy, and lattice spacing.3Form factors for semi-leptonic B decays Steven Gottlieb1.21.41.61.822.22.40.6 0.8 1 1.2 1.4 1.6 1.8 2r1/21f ‖(r1EK)p-value=0.35continuum0.40.60.811.20.6 0.8 1 1.2 1.4 1.6 1.8 2r−1/21f ⊥(r1EK)p-value=0.35continuumFigure 1: The form factorsf‖ (left) and f⊥ (right) as a function ofEK in lattice units. The legend in Fig. 2explains the meaning of all the symbols.The form factorsf‖ and f⊥ are fit to the following forms:f‖ =C(0)‖fπ[1+ logs+C(1)‖ ml +C(2)‖ (2ml +ms)+C(3)‖ EK +C(4)‖ E2K +C(5)‖ a2], (4.1)f⊥ =C(0)⊥fπ[1EK +∆∗Bs+ logs+logsEK +∆∗Bs]+C(0)⊥fπ(EK +∆∗Bs)[C(1)⊥ ml +C(2)⊥ (2ml +ms)+C(3)⊥ EK +C(4)⊥ E2K +C(5)⊥ a2], (4.2)where the notation in Ref. [15] is used, and∆∗Bs = mB∗s −MB. The 1-loopSU(2) chiral logarithms(denoted as “logs” above) are non-analytic functions of theps udoscalar meson masses. We usethe same form forfT as for f⊥, because the Isgur-Wise relation [16] requires thatfT = f+ in thelargeMB limit, and the shape of+ is dominated byf⊥ at low recoil. Figures 1 and 2 show ourresults forf‖, f⊥ and fT including the continuum, physical quark-mass curve plotted as a black linewith a cyan error band.5. z-expansionBecause only relatively small kaon momenta are accessible in the lattice calculation, the rangeof q2 directly calculable is close to the maximum value. However,the experimental results are bestat low q2. We use thez-expansion to extend the kinematic range [17]. This expansion i based onthe field theoretic principles analyticity and crossing symmetry. It is systematically improvable byadding more orders to the expansion. Theq2 region is mapped tozbyz(q2, t0) =√t+−q2−√t+− t0√t+−q2+√t+− t0, (5.1)4Form factors for semi-leptonic B decays Steven Gottlieb0.811.21.41.61.822.22.42.60.8 1 1.2 1.4 1.6 1.8 2f T(r1EK)p-value=0.990.02/0.050.01/0.050.007/0.050.005/0.050.0124/0.0310.0062/0.0310.0047/0.0310.0031/0.0310.00155/0.0310.0036/0.01880.0018/0.0188contFigure 2: The form factorfT as a function ofEK in lattice units.00.050.10.150.2-0.15 -0.1 -0.05 0 0.05 0.1 0.15f +,0(q2)B(q2)φ(q2,t0)zp-value=0.95f+Bφf0Bφ00.010.020.030.04-0.15 -0.1 -0.05 0 0.05 0.1 0.15f T(q2)B(q2)φ(q2,t0)zp-value=0.85fTBφFigure 3: Fit of form factor multiplied by Blaschke factor and outer function to a polynomial expansion inz. Only statistical errors are shown.wheret± = (MB±MK)2. We are allowed to chooset0 as we wish to minimize the range of|z|. Itis convenient to chooset0 = t+(1−√1− t−t+). This results in the full range ofq2 for this decaybeing mapped to|z|< 0.16. The form factors are then expressed as a function ofz:f (q2) =1B(z)φ(z)∞∑k=0akzk, (5.2)where the Blaschke factorB(z) = z(q2,m2R) is used to account for the pole structure of the formfactor, and the outer functionφ(z) is selected such that∑∞k=0 a2k ≤ 1 [18].In Fig. 3, we show our polynomial fits to the form factors multiplied by the Blaschke factor andouter function,i.e., f (q2)B(z)φ(z). To generate these plots, we calculate synthetic data at selectedvalues ofz in the range accessible to our simulations based on the chiral- ontinuum extrapolation5Form factors for semi-leptonic B decays Steven Gottlieb00.511.522.530 5 10 15 20f +andf 0q2(GeV2)f+f0Cambridge Group00.511.522.530 5 10 15 20f Tq2(GeV2)fTCambridge GroupFigure 4: Preliminary results for form factors as a function ofq2, including comparison with Ref. [9].of the lattice data in Figs. 1–2. We see that the negativez region corresponds to maximumq2,and that is the region in which the lattice calculation is done. We impose the kinematic constraintf+(q2 = 0) = f0(q2 = 0) in the z-fit. We use terms up toz3 for f+ and f0, andz2 for fT . Theform factors as a function ofq2 are shown in Fig. 4. Both statistical and systematic errors areshown, but these results are preliminary and do not include the final renormalization factors or allthe ensembles that we will analyze.At large q2, where we can directly compute the form factors, the error isabout 5%. Asq2decreases, the errors from statistics, chiral fit andz-expansion grow to more than 30% forf+ andf0 and to about 100% forfT . One reason that the relative errors grow at smallq2 is that the formfactors themselves are getting much smaller.6. ConclusionsWe have presented some preliminary results for the rare decay B→ Kll form factors. Thereare additional ensembles not yet included in the analysis. We still need to include the perturbativerenormalization factors, which are expected to be close to unity and will not change our results sig-nificantly. Our results forf‖, f⊥ and fT can be used to calculate the SMB→ Kll branching fractionor to make predictions for this process in BSM theories. Other semileptonic decays currently beinganalyzed includeB→ π lν andBs → Klν .AcknowledgmentsComputations for this work were carried out with resources provided by the USQCD Collab-oration, the Argonne Leadership Computing Facility, and the National Energy Research ScientificComputing Center, which are funded by the Office of Science ofthe U.S. Department of Energy;and with resources provided by the National Institute for Computational Science, and the Texas6Form factors for semi-leptonic B decays Steven GottliebAdvanced Computing Center, which are funded through the Nation l Science Foundation’s Tera-grid/XSEDE Program. This work was supported in part by the U.S. Department of Energy underGrants No. DE-FG02-91ER40661 (S.G., R.Z.), No. DE-FG02-91ER40677 (D.D., R.D.J., A.X.K.),No. DE-FG02-91ER40664 (Y.L., Y.M.). Fermilab is operated by Fermi Research Alliance, LLC,under Contract No. DE-AC02-07CH11359 with the United States D partment of Energy. J.A.B. issupported by the Creative Research Initiatives program (2012-0000241) of the NRF grant fundedby the Korean government (MEST).References[1] B. Aubertet al. [BABAR Collaboration], Phys. Rev. Lett.91, 221802 (2003) [arXiv:hep-ex/0308042].[2] K. Abe et al. [BELLE Collaboration], Phys. Rev. Lett.88, 021801 (2002) [arXiv:hep-ex/0109026].[3] T. Aaltonenet al. [CDF Collaboration], Phys. Rev. Lett.106, 161801 (2011) [arXiv:1101.1028[hep-ex]].[4] R. Aaij et al. [LHCb Collaboration], JHEP1207, 133 (2012) [arXiv:1205.3422 [hep-ex]].[5] A. Abadaet al. [APE Collaboration], Phys. Lett. B365, 275 (1996) [arXiv:hep-lat/9503020].[6] L. Del Debbioet al. [UKQCD Collaboration], Phys. Lett. B416, 392 (1998) [arXiv:hep-lat/9708008].[7] D. Becirevic, V. Lubicz and F. Mescia, Nucl. Phys. B769, 31 (2007) [arXiv:hep-ph/0611295].[8] A. Al-Haydari et al. [QCDSF Collaboration], Eur. Phys. J. A43, 107 (2010) [arXiv:0903.1664[hep-lat]].[9] Z. Liu, S. Meinel, A. Hart, R. R. Horgan, E. H. Müller and M.Wingate, arXiv:1101.2726 [hep-ph].[10] R. Zhouet al. [Fermilab Lattice and MILC Collaborations], PoS LATTICE2011, 298 (2011)[arXiv:1111.0981 [hep-lat]].[11] C. W. Bernardet al. [MILC Collaboration], Phys. Rev. D64, 054506 (2001) [arXiv:hep-lat/0104002].[12] C. Aubinet al. [MILC Collaboration], Phys. Rev. D70, 094505 (2004) [arXiv:hep-lat/0402030].[13] A. Bazavovet al., Rev. Mod. Phys.82, 1349 (2010) [arXiv:0903.3598 [hep-lat]].[14] C. Aubin and C. Bernard, Phys. Rev. D76, 014002 (2007) [arXiv:0704.0795 [hep-lat]].[15] J. A. Baileyet al. [Fermilab Lattice and MILC Collaborations], Phys. Rev. D79, 054507 (2009)[arXiv:0811.3640 [hep-lat]].[16] N. Isgur and M. B. Wise, Phys. Rev. D42, 2388 (1990).[17] C. G. Boyd, B. Grinstein and R. F. Lebed, Phys. Rev. Lett.74, 4603 (1995) [arXiv:hep-ph/9412324].[18] M. C. Arnesen, B. Grinstein, I. Z. Rothstein and I. W. Stewart, Phys. Rev. Lett.95, 071802 (2005)[arXiv:hep-ph/0504209].7

Form factors for semi-leptonic B decays

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