B -> D* l nu at zero recoil: an update
Jon A. Bailey, A. Bazavov, C. Bernard, C.M. Bouchard, C. DeTar, A.X. El-Khadra, E.D. Freeland, E. Gámiz, Steven Gottlieb, U.M. Heller, J.E. Hetrick, A.S. Kronfeld, J. Laiho, L. Levkova, P.B. Mackenzie, M.B. Oktay, J.N. Simone, R. Sugar, D. Toussaint, R.S. Van de Water
aa r X i v : . [ h e p - l a t ] N ov BBB →→→
DDD ∗ lll nnn at zero recoil: an update Jon A. Bailey, a A. Bazavov, b C. Bernard, c C.M. Bouchard, a , d C. DeTar, e A.X. El-Khadra, d E.D. Freeland, c , d E. Gámiz, a Steven Gottlieb, f , g U.M. Heller, h J.E. Hetrick, i Andreas S. Kronfeld ∗ , a J. Laiho, j L. Levkova, e P.B. Mackenzie, a M.B. Oktay, e J.N. Simone, a R. Sugar, k D. Toussaint, b and R.S. Van de Water l a Fermi National Accelerator Laboratory, † Batavia, IL, USA b Department of Physics, University of Arizona, Tucson, AZ, USA c Department of Physics, Washington University, St. Louis, MO, USA d Physics Department, University of Illinois, Urbana, IL, USA e Physics Department, University of Utah, Salt Lake City, UT, USA f Department of Physics, Indiana University, Bloomington, IN, USA g National Center for Supercomputing Applications, University of Illinois, Urbana, IL, USA h American Physical Society, One Research Road, Ridge, NY, USA i Physics Department, University of the Pacific, Stockton, CA, USA j SUPA, Department of Physics and Astronomy, University of Glasgow, Glasgow, UK k Department of Physics, University of California, Santa Barbara, CA, USA l Department of Physics, Brookhaven National Laboratory, ‡ Upton, NY, USAE-mail: [email protected]
Fermilab Lattice and MILC Collaborations
We present an update of our calculation of the form factor for ¯ B → D ∗ l ¯ n at zero recoil, with higherstatistics and finer lattices. As before, we use the Fermilab action for b and c quarks, the asqtadstaggered action for light valence quarks, and the MILC ensembles for gluons and light quarks(Lüscher-Weisz married to 2+1 rooted staggered sea quarks). In this update, we have reduced thetotal uncertainty on F ( ) from 2.6% to 1.7%.At Lattice2010 we presented a still-blinded result, but this writeup includes the unblinded resultfrom the September 2010 CKM workshop.
The XXVIII International Symposium on Lattice Field Theory, Lattice2010June 14–19, 2010Villasimius, Italy ∗ Speaker. † Operated by Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the United StatesDepartment of Energy. ‡ Operated by Brookhaven Science Associates, LLC, under Contract No. DE-AC02-98CH10886 with the UnitedStates Department of Energy. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ → D ∗ l n Andreas S. Kronfeld
1. Introduction
The
W bc vertex is proportional to the coupling V cb , which is an element of the Cabibbo [1]Kobayashi-Maskawa [2] (CKM) matrix. Along with the quark masses, it represents the observ-able part of the quarks’ coupling to the Higgs sector and is, thus, a fundamental part of particlephysics. The CKM matrix has four free parameters, and it is convenient to choose one of them tobe (essentially) | V cb | . Consequently, | V cb | appears throughout flavor physics [3]. | V cb | is determined from semileptonic decays ¯ B → X c l ¯ n , where X c denotes a charmed finalstate. In exclusive decays, X c is a D or D ∗ meson, and the decay amplitudes can be written h D ( v D ) | V m | ¯ B ( v B ) i = √ M B M D [( v B + v D ) m h + ( w ) + ( v B − v D ) m h − ( w )] , (1.1) h D ∗ ( v D , a ) | V m | ¯ B ( v B ) i = √ M B M D ∗ e mn rs ¯ e ( a ) n v r B v s D ∗ h V ( w ) , (1.2) h D ∗ ( v D , a ) | A m | ¯ B ( v B ) i = i √ M B M D ∗ ¯ e ( a ) n (cid:8) g nm ( + w ) h A ( w ) − v n B [ v m B h A ( w ) + v m D ∗ h A ( w )] (cid:9) , (1.3)where e ( a ) is the D ∗ polarization vector, v B and v D ( ∗ ) denote the mesons’ 4-velocities, and w = v B · v D ( ∗ ) is related to the invariant mass of the l n pair, q = M B + M D ( ∗ ) − wM B M D ( ∗ ) . The formfactors h ± , h V , and h A i ( i = , ,
3) enjoy simple heavy-quark limits and are linear combinations ofthe form factors f ± , V , and A i used in other semileptonic decays.The differential decay distributions are d G ( ¯ B → Dl ¯ n ) dw = G F p m D ( M B + M D ) ( w − ) / | V cb | | G ( w ) | , (1.4) d G ( ¯ B → D ∗ l ¯ n ) dw = G F p m D ∗ ( M B − M D ∗ ) ( w − ) / | V cb | c ( w ) | F ( w ) | , (1.5)neglecting the charged lepton and neutrino masses. The physical combinations of form factors are G ( w ) = h + ( w ) − M B − M D M B + M D h − ( w ) = √ M B M D M B + M D f + ( q ) , (1.6) F ( w ) = h A ( w ) + w s H ( w ) + H + ( w ) + H − ( w ) c ( w ) → h A ( ) , (1.7)where the zero-recoil ( w →
1) limit of F is shown. The function c ( w ) is chosen so that the squareroot in Eq. (1.7) collapses to 1 if h V = h A = h A and h A =
0, as in the heavy-quark limit withoutradiative corrections. Expressions for H ± ( w ) , H ( w ) , and c ( w ) can be found in Ref. [3].The messy formula for F ( w ) indicates the advantage of the zero-recoil limit for ¯ B → D ∗ l ¯ n :one must compute only h A ( ) , not four functions. In addition, the heavy-quark flavor symmetry islarger when v D ∗ = v B , and Luke’s theorem applies. For determining | V cb | , the key aspect of Luke’stheorem is that it helps control systematic errors. In particular, in lattice gauge theories that respectheavy-quark symmetry, one can compute h A ( ) with heavy-quark discretization errors that areformally ¯ L / m Q times smaller than those of h A ( w ) , w =
1, or those of G ( w ) even at w = B → D ∗ l ¯ n at zero recoil, describing our calculations of F ( ) = h A ( ) .Starting in 2001, experimental determinations of | V cb | used a quenched calculation [4] F ( ) = . + . − . ± . + . − . + . − . + . − . , (1.8)where the errors stem, respectively, from statistics, matching lattice gauge theory to QCD, lattice-spacing dependence, chiral extrapolation, and the quenched approximation. A notable feature of2 → D ∗ l n Andreas S. Kronfeld
Table 1:
Parameters of the MILC ensembles used for heavy-quark physics. Here C denotes the number ofconfigurations in each ensemble; ( m ′ l , m ′ s ) the asqtad sea-quark masses; m q the asqtad valence masses; k and c SW the hopping parameter and clover coupling of the heavy quark. Standard nicknames for the latticespacings are noted ( a ≈ .
045 fm is “ultrafine”). Data are being generated on all ensembles for all m q insidethe {· · ·} , but the present analysis uses at most two, namely m q = m ′ l and m q = . m ′ s . a (fm) Lattice C ( am ′ l , am ′ s ) m q k b k c c SW ≈ .
15 16 ×
48 596 (0.0290,0.0484) {0.0484, 0.0453, medium ×
48 640 (0.0194,0.0484) 0.0421, 0.0290, coarse ×
48 631 (0.0097,0.0484) 0.0194, 0.0097, 0.0781 0.1218 1.57020 ×
48 603 (0.0048,0.0484) 0.0068, 0.0048} ≈ .
12 20 ×
64 2052 (0.02,0.05) {0.05, 0.03, 0.0918 0.1259 1.525 coarse ×
64 2259 (0.01,0.05) 0.0415, 0.0349, 0.0901 0.1254 1.53120 ×
64 2110 (0.007,0.05) 0.02, 0.01, 0.0901 0.1254 1.53024 ×
64 2099 (0.005,0.05) 0.007, 0.005} 0.0901 0.1254 1.530 ≈ .
09 28 ×
96 1996 (0.0124,0.031) {0.031, 0.0261, 0.0982 0.1277 1.473 fine ×
96 1946 (0.0062,0.031) 0.0124, 0.0979 0.1276 1.47632 ×
96 983 (0.00465,0.031) 0.0093, 0.0062, 0.0977 0.1275 1.47640 ×
96 1015 (0.0031,0.031) 0.0047, 0.0031} 0.0976 0.1275 1.478 ≈ .
06 48 ×
144 668 (0.0072,0.018) {0.0188, 0.0160, 0.1052 0.1296 1.4276 superfine ×
144 668 (0.0036,0.018) 0.0072, 0.1052 0.1296 1.428756 ×
144 800 (0.0025,0.018) 0.0054, 0.0036,64 ×
144 826 (0.0018,0.018) 0.0025, 0.0018} ≈ .
045 64 ×
192 860 (0.0028,0.014) {0.014, 0.0056, 0.0028}Eq. (1.8) is that an estimate of the error associated with quenching has been made. Nevertheless, itis necessary to incorporate the light- and strange-quark sea. The first calculation with 2+1 flavorsof sea quarks obtained [5] F ( ) = . ± . ± . ± . ± . ± . ± . ± . , (1.9)where, now, the errors stem from statistics, the g D ∗ D p coupling, chiral extrapolation, discretizationerrors, matching, and two tuning errors. (The catch-phrases for the errors do not have exactlythe same meaning in Refs. [4, 5]; for example, the g D ∗ D p error in Eq. (1.8) is incorporated into thechiral-extrapolation error.) This paper presents an update of the 2+1-flavor calculation, with mostlythe same ingredients, but with higher statistics and without the second of the tuning errors.The new data set is shown in Table 1, based as before on the MILC ensembles [6] with theLüscher-Weisz gauge action [7], with the g N c [8] but not g N f corrections [9], and the asqtad-improved [10] rooted staggered determinant for the sea quarks. For the valence quarks, we usethe asqtad action for the light quark and the Fermilab interpretation [11] of the clover action [12]for the heavy quark. In this report, we use all ensembles in Table 1 with entries for the heavy-quark couplings ( k b , k c , and c SW ), except the fine 32 ×
96 lattice. These data are being generatedas part of a broad program of heavy-quark physics, including other semileptonic decays [13] andneutral-meson mixing and decay constants [14]. 3 → D ∗ l n Andreas S. Kronfeld
Improvements to F ( ) are timely [3], because the values of | V cb | that follow from inclusivedecays are in a 2 . s tension with those that follow from Eq. (1.9) and also from ¯ B → Dl ¯ n and G ( ) [15]. The result described below is but one aspect of a resolution of the discrepancy. Others includea re-examination of the extrapolation to zero recoil, unquenched lattice-QCD calculations at w = F ( ) has been studied so much in the past, any new analysis could be influenced in subtle human ways.To circumvent any such bias, we hide the numerical value of F ( ) via an offset in the matchingfactor r A cb , explained in Sec. 3. We present our preliminary results, with all sources of uncertaintyestimated, in Sec. 4. We include the unblinded value here, which was revealed after Lattice 2010 but before these proceedings.
2. Data analysis
As in Ref. [5], we aim for the direct double-ratio R A = h D ∗ | ¯ c g j g b | ¯ B ih ¯ B | ¯ b g j g c | D ∗ ih D ∗ | ¯ c g c | D ∗ ih ¯ B | ¯ b g b | ¯ B i = | h A ( ) | , (2.1)where the expressions here are all in (continuum) QCD. To this end, we use lattice gauge theory tocompute the three-point correlation functions C B → D ∗ ( t i , t s , t f ) = (cid:229) xxx , yyy h | O D ∗ ( xxx , t f ) Y c g j g Y b ( yyy , t s ) O † B ( , t i ) | i , (2.2) C B → B ( t i , t s , t f ) = (cid:229) xxx , yyy h | O B ( xxx , t f ) Y b g Y b ( yyy , t s ) O † B ( , t i ) | i , (2.3) C D ∗ → D ∗ ( t i , t s , t f ) = (cid:229) xxx , yyy h | O D ∗ ( xxx , t f ) Y c g Y c ( yyy , t s ) O † D ∗ ( , t i ) | i , (2.4)where O B and O D ∗ are interpolating operators coupling to the B and D ∗ mesons, and Y c g m Y b and Y c g m g Y b are improved currents [11, 4, 17, 5]. Then the lattice ratio R A ( t ) = C B → D ∗ ( , t , T ) C D ∗ → B ( , t , T ) C D ∗ → D ∗ ( , t , T ) C B → B ( , t , T ) (2.5)should reach a plateau for a range of t , T ≫ t ≫
1. The relationship between the plateau value of R / A and h A ( ) is discussed in Sec. 3.With staggered fermions, O B and O D ∗ couple to both parities, and three-point correlation func-tions have four distinct contributions: C X → Y ( , t , T ) = (cid:229) k = (cid:229) ℓ = ( − ) kt ( − ) ℓ ( T − t ) A ℓ k e − M ( k ) X t e − M ( ℓ ) Y ( T − t ) (2.6) = A X → Y e − M X t − M Y ( T − t ) + ( − ) T − t A X → Y e − M X t − M ′ Y ( T − t ) +( − ) t A X → Y e − M ′ X t − M Y ( T − t ) + ( − ) T A X → Y e − M ′ X t − M ′ Y ( T − t ) + · · · (2.7)with time-dependent factors of − → D ∗ l n Andreas S. Kronfeld ¯ R A ( , t , T ) = R A ( , t , T ) + R A ( , t , T + ) + R A ( , t + , T + ) , (2.8)which should tend more quickly to a plateau. The key here is to have t f = T and T + m q = m ′ l and m q = . m ′ s (or the single m q when m ′ l = . m ′ s ). The choice of afixed m q , here 0 . m ′ s , for all m ′ l matches, by design, our plans for the ultrafine lattice ( a ≈ .
045 fm),to anchor future analyses even closer to the continuum limit. Typical plateaus are shown in Fig. 1for a coarse, a fine, and a superfine ensemble. As one can see, the plateau in ¯ R A emerges readily,and the statistical errors are 1% or smaller.
3. Matching, blinding, and discretization effects
The ratio combination ¯ R A tends to a ratio of matrix elements like R A in Eq. (2.1) but withlattice currents. Each current must be multipled by a matching factor Z A or Z V , defined nonpertur-batively in Ref. [17]. The lattice ratio R A must, therefore, be multiplied by a matching ratio r A cb = Z A cb / Z V cc Z V bb . (3.1)A subset of the collaboration has computed r A cb in the one-loop approximation. The result is veryclose to unity, but the deviation is, or could be, comparable to h A ( ) −
1. Our numerical analysisreplaces r A cb with F blind r A cb , where the blinding factor F blind is again close to unity, but known onlyto those engaged in the one-loop calculation. In this way, choices of fitting ranges, etc., cannot beinfluenced by a human desire to (dis)agree with results for F ( ) already in the literature.The HQET-Symanzik formalism used to define the Z J can also be used to control and suppresscutoff dependence [18, 17]. In the general case, several operators—both corrections to the currentand insertions of the effective Lagrangian—generate cutoff effects. For details, see, e.g. , the dis-cussion of Eq. (2.40) in Ref. [17]. For zero recoil, v D ∗ = v B , and the heavy-quark flavor symmetryenlarges from U ( ) × U ( ) to SU ( ) . The leading discretization errors drop out, and the remaindercan be found by applying the formulas of Ref. [18] to ¯ R / A and h A ( ) . One finds r A cb ¯ R / A = h A ( ) + O ( a s a ¯ L / m c ) + O ( a s a ¯ L ) + O ( a s ) , (3.2)where the last error acknowledges the one-loop calculation of r A cb . A study of the asymptotic t R A t R A t R A Figure 1:
Ratio combination ¯ R / A ( , t , T ) vs. t with m q = m ′ l = . m ′ s . From left to right: the coarse ensemblewith T =
12 and ( am ′ l , am ′ s ) = ( . , . ) ; the fine ensemble with T =
17 and ( am ′ l , am ′ s ) = ( . , . ) ;the superfine ensemble with T =
24 and ( am ′ l , am ′ s ) = ( . , . ) . → D ∗ l n Andreas S. Kronfeld behavior of Fermilab actions provides a reasonable guide to the dependence on m Q a of the correc-tions. We see in our data little dependence on the lattice spacing, in accord with Eq. (3.2).
4. Preliminary result
Figure 2 provides a glimpse into our systematic error analysis, which closely follows Ref. [5].We use our previous study of heavy-quark-mass dependence to fine-tune a posteriori the hoppingparameters and to assess the tuning errors. We fit the light-quark mass dependence to one-loopchiral perturbation theory, suitably modified for staggered quarks [19]. The cusp is a necessary,physical effect that appears because the D p threshold sinks below the D ∗ mass.With the blinding factor in place, we find F blind F ( ) = . ± . ± . ± . ± . ± . ± . , (4.1)where the errors again stem from statistics, the g D ∗ D p coupling, chiral extrapolation, discretizationerrors, matching, and tuning k c and k b . To show how the errors have been reduced, it helps to scalethis result to the old central value ( F F is the needed ad hoc factor): F ( ) = . ( )( )( )( )( )( )( ) [5] , (4.2) F F F ( ) = . ( )( )( )( )( )( ) [this work] . (4.3)The higher statistics and wider scope of this dataset has reduced the statistical error with C − / . Thequoted heavy-quark discretization error is smaller, because with the superfine data we can movebeyond pure power counting and combine the (lack of) trend in the data with the detailed theoryof cutoff effects [18]. After Lattice 2010, we continued to examine the heavy-quark discretizationand k -tunings errors, reducing them somewhat, and the chiral-extrapolation error, increasing itsomewhat. For the 2010 Workshop on the CKM Unitarity Triangle , we removed the blindingfactor, finding [20]: F ( ) = . ( )( )( )( )( )( ) . (4.4)This result reduces the tension with | V cb | from inclusive decays to 1 . s . k b,c h A (1) k b k c r m x h A (1) medium coarse (0.15 fm)coarse (0.12 fm)fine (0.09 fm)superfine (0.06 fm) c /dof = 8.9/12, CL = 0.72 Figure 2:
Left: dependence of h A ( ) on the heavy-quark hopping parameters (with data of Ref. [5]). Right:chiral extrapolation showing only points with m q = m l and a fit to all data. → D ∗ l n Andreas S. Kronfeld
Computations for this work were carried out in part on facilities of the USQCD Collaboration,which are funded by the Office of Science of the U.S. Department of Energy. This work was sup-ported in part by the U.S. Department of Energy under Grants No. DE-FC02-06ER41446 (C.D.,L.L., M.B.O), No. DE-FG02-91ER40661 (S.G.), No. DE-FG02-91ER40677 (C.M.B., A.X.K.,E.D.F.), No. DE-FG02-91ER40628 (C.B, E.D.F.), No. DE-FG02-04ER-41298 (D.T.); the Na-tional Science Foundation under Grants No. PHY-0555243, No. PHY-0757333, No. PHY-0703296(C.D., L.L., M.B.O), No. PHY-0757035 (R.S.), No. PHY-0704171 (J.E.H.) and No. PHY-0555235(E.D.F.). C.M.B. was supported in part by a Fermilab Fellowship in Theoretical Physics and by theVisiting Scholars Program of Universities Research Association, Inc. R.S.V. acknowledges supportfrom BNL via the Goldhaber Distinguished Fellowship.
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