Calculation of $\bar B \rightarrow D^\ast \ell \bar \nu$ form factor at zero recoil using the Oktay-Kronfeld action

We present the first preliminary results for the semileptonic form factor $h_{A_1}(w=1)/\rho_{A_j}$ at zero recoil for the $\bar B \rightarrow D^\ast \ell \bar \nu$ decay using lattice QCD with four flavors of sea quarks. We use the HISQ staggered action for the light valence and sea quarks (the MILC HISQ configurations), and the Oktay-Kronfeld (OK) action for the heavy valence quarks.


Introduction
The 4.1σ tension between recently updated values of the exclusive and inclusive |V cb | [1,2], and the 4.0σ tension between the standard model prediction of ε K using the exclusive |V cb | determined with lattice QCD inputs and the experimental value of ε K [2] motivates highter precision lattice calculations of |V cb |.
At present, the largest error in exclusive |V cb | comes from the about 1% heavy quark discretization error in the Fermilab action used in current calculations [3]. To reduce this error, we are simulating the Oktay-Kronfeld action [4], which is an O(λ 3 ) improved version of the Fermilab action 1 . In this talk, we present the first results on the semileptonic form factor h A 1 (w = 1)/ρ A j at zero recoil for thē B → D * ν decay. The calculations were done on the a12m310 2+1+1-flavor HISQ lattices generated by the MILC collaboration [5].
2 Nonperturbative determination of κ crit , κ c and κ b The Oktay-Kronfeld (OK) action [4] is a highly improved version of the Fermilab action. It contains counter-terms up to λ 3 , whereas the Fermilab action is improved only up to λ 1 order. In both actions, e-mail: wlee@snu.ac.kr Speaker 1 In the HQET power counting, λ ≈ Λ/(2m Q ) where Λ ≈ 300 MeV and m Q is the heavy quark mass. For the charm quark, λ ≈ 1/8 and for the bottom quark, λ ≈ 1/30. arXiv:1711.01786v2 [hep-lat] 7 Nov 2017 the bare quark mass is defined as where u 0 is the tadpole improvement factor determined from the 1 × 1 Wilson loops. In case of the Fermilab action, the coefficients of the dimension 5 counter terms added for improvement are independent of m 0 . It is therefore straight-forward to determine κ crit by fitting the pion spectra with respect to am 0 . In case of the OK action, some of the coefficients of the dimension 5 and dimension 6 counter terms, such as c E , c 1 , c 2 = c 3 , c EE defined in [4], depend on m 0 and κ crit . Hence, for the OK action we determine κ crit by iteration as described below.
2. Make an initial guess for κ crit . We choose κ crit = 0.96κ tree crit based on Ref. [6], where it was shown that the nonperturbative κ NP crit is 4% smaller than κ tree crit for the Fermilab action. 3. Determine the OK action coefficients using κ crit and measure 2-point pion correlators with point sources and sinks. We use 3 different κ values such that pions have masses in the range of 600 ≤ m π ≤ 950 MeV. Also, we investigate 9 different pion momenta p.

4.
We determine the pion kinetic mass, M 2 , by fitting the ground state energy, E(p), to the dispersion relation Here M 1 is the rest mass, M 4 is the quartic mass and W 4 is the Lorentz symmetry breaking term.
5. Determine the new κ crit by requiring A = 0 in the quadratic fit to the following function: where the kinetic quark mass m 2 is related to the m 0 by the tree-level relation: Since m 2 (κ crit , κ crit ) = 0, M 2 (κ, κ crit ) vanishes at κ = κ crit .
6. Update κ crit = κ crit and go back to step 3. Fig. 1 shows the convergence of κ crit as a function of the iteration number. After the two iterations, we declare convergence to κ NP crit = 0.051211(33) within statistical uncertainty. Further details will be reported in Ref. [7].
We measure the two point correlation functions for the B s and D s mesons (pseudo-scalar channel) using the OK action with the above nonperturbatively determined κ crit = κ NP crit . To determine κ c and κ b , 0.96κ crit tree (Initial guess) Figure 1. κ crit as a function of the iteration number. The black circle is the initial guess, the red square is κ crit after one iteration, and the blue square is the κ crit after two iterations.
we simulate at four values of κ about estimates for the charm and the bottom quarks. For each κ, we obtain data for 11 momenta and determine the kinetic masses by fitting to the dispersion relation in Eq. (3). The final values of κ c and κ b , obtained by requiring the meson masses match the experimental B s and D s masses, are given in Table 1, for two determinations of the lattice spacing a, a f π + and a r 1 .
Matching the pseudoscalar masses gives estimates with the smallest uncertainty. Table 1. Results for κ b and κ c . The lattice spacing a is set in two ways: using a f + π = 0.12520(22) fm from f + π in [8] and a r 1 = 0.1207(11) fm using r 1 from [5]. The first error is statistical, the second is from the experimental error in M X , and the third is a systematic error due to the fitting ambiguity.

Inconsistency
The inconsistency parameter I is defined as where δM X ≡ M 2X − M 1X , (X = Qq, QQ). In a relativistically invariant theory, the binding energy B 1 is equal to B 2 . The inconsistency I probes the binding energy difference δB = B 2 − B 1 . This difference comes from the discretization errors in the O((ap) 4 ) terms of the action or in the O(v 4 ) in NRQCD power counting. I vanishes at tree level for the OK action, but not for the Fermilab action [6,9]. In Fig. 2, we show I as a function of the pseudo-scalar heavy-light meson mass. Our previous results presented in Refs. [10][11][12] were obtained using the tree-level κ tree crit with asqtad strange quarks with point sources, while the results presented in Fig. 2 are obtained using the nonperturbatively determined κ NP crit with HISQ strange quarks with covariant Gaussian smearing applied to the heavy quarks. We find that the inconsistency parameter vanishes within statistical uncertainty near the B s region and it is smaller than that of Fermilab action by order of magnitude near the D s region. This improvement is due to the combination of using the OK action on HISQ ensembles and smeared sources. It is observed for both κ tree crit and κ NP crit .

Form factor h A 1 (1)/ρ A j at zero recoil
To extract the form factor h A 1 (1) at zero recoil, we calculate the double ratio R on the lattice [13][14][15]: Here, ρ A j is the matching factor at a 0. C X→Y J µ (t, t f ) is a 3-point correlation function: for example, if X = B, Y = D * and J µ = A j , We define the axial and vector currents as follows: where d 0 = 1, R 0 = 1, R 1 = aγ · D, and so on. Here, Ψ and ψ are improved and unimproved quark fields, respectively, and the coefficients d m are real-valued analytic function of the bare mass am 0 [16]. The currents are improved up to order λ 3 at the tree level in the HQET power counting. We obtain the improved fields Ψ using the field rotation defined in Eq. (10). In short, using the field rotation is sufficient for the current improvement at the tree level [16][17][18].
We can rewrite the currents as follows, A cb j,mn ≡ψ c R † m γ j γ 5 R n ψ b (12) and There are 12 different rotation operators for ψ b andψ c . In total sum, we have 144 = 12 2 terms. However, some of them are O(λ 4 ), which we exclude in this analysis. Hence, we end up with 30 terms up to O(λ 3 ). Other 3-point functions occuring in Eq. (7) can be rewritten in a similar way.
In the measurements, we use point sources for the HISQ light quarks and choose their coordinates randomly. For the c and b quarks, we apply the covariant Gaussian smearing with (σ, N GS ) = (1.5, 5) at the source and sink points [19]. We also use the coherent source method [19] which gives a statistical gain by factor of 3 in the measurements. In the next subsections, we describe the extraction of R 1/2 = h A 1 /ρ A j using two different analyses.

Direct analysis on 3-point functions C B→D
The fitting function used for analyzing C B→D * where B B→D * = D * |A cb 1 |B , B B→D * mn = D * |Ā cb 1,mn |B , andĉ B→D * andĉ B→D * mn represent the contamination from the excited states of B and D * mesons. Once we extract the B parameters from the data, we can obtain R using the following relation: The amplitude B of 3-point correlation function with O(λ p ) improved current is The merit of Eq. (18) is that the ratio of B mn /B 00 can be obtained by a simple constant fit to the following ratio: We find that the corrections to the leading term, contamination from the excited states, are small and under control. In fact, we can further reduce the contamination using the following linear combination [14,15]:F The coefficients of contamination terms from the parity partners are suppressed by factor of 1/2 ∼ 1/12 inc mn compared withĉ mn Hence, once we determine B 00 from the exponential fitting, the rest of analysis is just a simple fit ofF mn to a constant. In Fig. 3  √R as a function of time. Horizontal lines represent results of the constant fit. The red symbols represent those data points used for fitting. We use the O(λ 3 ) improved currents to obtainR. data to a constant as an example.

Analysis of R
By construction, the leading exponential dependence cancels out in the double ratio R defined in Eq. (7), so the correction due to the contamination from the excited states is further reduced. To further suppress this contamination, we take the same linear combination as forF mn in Eq. (20): and fit the results to a constant. In Fig. 3 (b), we show results of fitting data for √R , obtained using the currents improved up to O(λ 3 ), to a constant.  R] n as a function of n. Here, n represents the data points obtained using the currents improved up to O(λ n ). Data for R at t f = 12 and t f = 13 is used to reconstructR.
In Fig. 4 (a), we present results for √ R obtained in three different ways: (i) fit √R to a constant; (ii) obtain B 00 by fitting the data at t f = 12 and combine it with results ofF mn , and (iii) obtain B 00 by fitting the data at t f = 13 and combining it with results ofF mn . Results from the first method are given using red crosses, second method using blue squares, and the green circle for the third method. We find all three are consistent within statistical uncertainty.
To evaluate the corrections to √ R from the improvement terms in the currents consider the quantity where the subscript n in [R 1/2 ] n represents results obtained using the currents improved up to O(λ n ). In Fig. 4 (b), we present results for [∆R 1/2 ] n as a function of n with n = 1, 2, 3 using the same convention for the symbols as in Fig. 4 (a). Due to a complicated structure of cancellation in the double ratio, we do not expect to see a simple scaling behavior in λ n in [∆R 1/2 ] n . However, we observe a kind of scaling behavior in [∆B] n defined similarly as in Eq. (23). For a detailed analysis of these behaviors, we refer the reader to Ref. [7].