skip to main content

Title: Semileptonic form factors for $$B\rightarrow D^*\ell \nu $$ at nonzero recoil from $$2+1$$-flavor lattice QCD: Fermilab Lattice and MILC Collaborations

We present the first unquenched lattice-QCD calculation of the form factors for the decay$$B\rightarrow D^*\ell \nu $$BDνat nonzero recoil. Our analysis includes 15 MILC ensembles with$$N_f=2+1$$Nf=2+1flavors of asqtad sea quarks, with a strange quark mass close to its physical mass. The lattice spacings range from$$a\approx 0.15$$a0.15fm down to 0.045 fm, while the ratio between the light- and the strange-quark masses ranges from 0.05 to 0.4. The valencebandcquarks are treated using the Wilson-clover action with the Fermilab interpretation, whereas the light sector employs asqtad staggered fermions. We extrapolate our results to the physical point in the continuum limit using rooted staggered heavy-light meson chiral perturbation theory. Then we apply a model-independent parametrization to extend the form factors to the full kinematic range. With this parametrization we perform a joint lattice-QCD/experiment fit using several experimental datasets to determine the CKM matrix element$$|V_{cb}|$$|Vcb|. We obtain$$\left| V_{cb}\right| = (38.40 \pm 0.68_{\text {th}} \pm 0.34_{\text {exp}} \pm 0.18_{\text {EM}})\times 10^{-3}$$Vcb=(38.40±0.68th±0.34exp±0.18EM)×10-3. The first error is theoretical, the second comes from experiment and the last one includes electromagnetic and electroweak uncertainties, with an overall$$\chi ^2\text {/dof} = 126/84$$χ2/dof=126/84, which illustrates the tensions between the experimental data sets, and between theory and experiment. This result is in agreement with previous exclusive determinations, but the tension with the inclusive determination remains. Finally, we integrate the differential decay rate obtained solely from lattice data to predict$$R(D^*) = 0.265 \pm 0.013$$R(D)=0.265±0.013, which confirms the current tension between theory and experiment.

more » « less
Award ID(s):
Author(s) / Creator(s):
; ; ; ; ; ; ; ; ; ; ; ; ; ; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
The European Physical Journal C
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    We study the distribution over measurement outcomes of noisy random quantum circuits in the regime of low fidelity, which corresponds to the setting where the computation experiences at least one gate-level error with probability close to one. We model noise by adding a pair of weak, unital, single-qubit noise channels after each two-qubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution$$p_{\text {noisy}}$$pnoisyand the corresponding noiseless output distribution$$p_{\text {ideal}}$$pidealshrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmarkFthat measures this correlation behaves as$$F=\text {exp}(-2s\epsilon \pm O(s\epsilon ^2))$$F=exp(-2sϵ±O(sϵ2)), where$$\epsilon $$ϵis the probability of error per circuit location andsis the number of two-qubit gates. Furthermore, if the noise is incoherent—for example, depolarizing or dephasing noise—the total variation distance between the noisy output distribution$$p_{\text {noisy}}$$pnoisyand the uniform distribution$$p_{\text {unif}}$$punifdecays at precisely the same rate. Consequently, the noisy output distribution can be approximated as$$p_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1-F)p_{\text {unif}}$$pnoisyFpideal+(1-F)punif. In other words, although at least one local error occurs with probability$$1-F$$1-F, the errors are scrambled by the random quantum circuit and can be treated as global white noise, contributing completely uniform output. Importantly, we upper bound the average total variation error in this approximation by$$O(F\epsilon \sqrt{s})$$O(Fϵs). Thus, the “white-noise approximation” is meaningful when$$\epsilon \sqrt{s} \ll 1$$ϵs1, a quadratically weaker condition than the$$\epsilon s\ll 1$$ϵs1requirement to maintain high fidelity. The bound applies if the circuit size satisfies$$s \ge \Omega (n\log (n))$$sΩ(nlog(n)), which corresponds to onlylogarithmic depthcircuits, and if, additionally, the inverse error rate satisfies$$\epsilon ^{-1} \ge {\tilde{\Omega }}(n)$$ϵ-1Ω~(n), which is needed to ensure errors are scrambled faster thanFdecays. The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; for example, it was an underlying assumption in complexity-theoretic arguments that noisy random quantum circuits cannot be efficiently sampled classically, even when the fidelity is low. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds.

    more » « less
  2. Abstract

    It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$Lβ,γ=-divDd+1+γ-nassociated to a domain$$\Omega \subset {\mathbb {R}}^n$$ΩRnwith a uniformly rectifiable boundary$$\Gamma $$Γof dimension$$d < n-1$$d<n-1, the now usual distance to the boundary$$D = D_\beta $$D=Dβgiven by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$Dβ(X)-β=Γ|X-y|-d-βdσ(y)for$$X \in \Omega $$XΩ, where$$\beta >0$$β>0and$$\gamma \in (-1,1)$$γ(-1,1). In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$Lβ,γ, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$D1-γ, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$|D(ln(GD1-γ))|2satisfies a Carleson measure estimate on$$\Omega $$Ω. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).

    more » « less
  3. Abstract

    We prove that$${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$poly(t)·n1/D-depth local random quantum circuits with two qudit nearest-neighbor gates on aD-dimensional lattice withnqudits are approximatet-designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was$${{\,\textrm{poly}\,}}(t)\cdot n$$poly(t)·ndue to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for$$D=1$$D=1. We also improve the “scrambling” and “decoupling” bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy ($${{\,\mathrm{\textsf{PH}}\,}}$$PH) is infinite and that certain counting problems are$$\#{\textsf{P}}$$#P-hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called “anti-concentration”, meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent experiment by the Google Quantum AI group to perform such a sampling task with 53 qubits on a two-dimensional lattice (Arute in Nature 574(7779):505–510, 2019; Boixo et al. in Nate Phys 14(6):595–600, 2018) (and related experiments by USTC), and confirms their conjecture that$$O(\sqrt{n})$$O(n)depth suffices for anti-concentration. The proof is based on a previous construction oft-designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anti-concentration and to establish equivalence between these various norms for low-depth circuits. For random circuits with long-range gates, we use different methods to show that anti-concentration happens at circuit size$$O(n\ln ^2 n)$$O(nln2n)corresponding to depth$$O(\ln ^3 n)$$O(ln3n). We also show a lower bound of$$\Omega (n \ln n)$$Ω(nlnn)for the size of such circuit in this case. We also prove that anti-concentration is possible in depth$$O(\ln n \ln \ln n)$$O(lnnlnlnn)(size$$O(n \ln n \ln \ln n)$$O(nlnnlnlnn)) using a different model.

    more » « less
  4. Abstract

    We introduce a family of Finsler metrics, called the$$L^p$$Lp-Fisher–Rao metrics$$F_p$$Fp, for$$p\in (1,\infty )$$p(1,), which generalizes the classical Fisher–Rao metric$$F_2$$F2, both on the space of densities$${\text {Dens}}_+(M)$$Dens+(M)and probability densities$${\text {Prob}}(M)$$Prob(M). We then study their relations to the Amari–C̆encov$$\alpha $$α-connections$$\nabla ^{(\alpha )}$$(α)from information geometry: on$${\text {Dens}}_+(M)$$Dens+(M), the geodesic equations of$$F_p$$Fpand$$\nabla ^{(\alpha )}$$(α)coincide, for$$p = 2/(1-\alpha )$$p=2/(1-α). Both are pullbacks of canonical constructions on$$L^p(M)$$Lp(M), in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of$$\alpha $$α-geodesics as being energy minimizing curves. On$${\text {Prob}}(M)$$Prob(M), the$$F_p$$Fpand$$\nabla ^{(\alpha )}$$(α)geodesics can still be thought as pullbacks of natural operations on the unit sphere in$$L^p(M)$$Lp(M), but in this case they no longer coincide unless$$p=2$$p=2. Using this transformation, we solve the geodesic equation of the$$\alpha $$α-connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman–Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of$$F_p$$Fp, and study their relation to$$\nabla ^{(\alpha )}$$(α).

    more » « less
  5. Abstract

    We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West’s stack-sorting map s. Associated to each permutation$$\pi $$πis a particular set$$\mathcal V(\pi )$$V(π)of integer compositions that appears in a formula for the fertility of $$\pi $$π, which is defined to be$$|s^{-1}(\pi )|$$|s-1(π)|. These compositions also feature prominently in more general formulas involving families of colored binary plane trees calledtroupesand in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that$$\mathcal V(\pi )$$V(π)is a transversal discrete polymatroid when it is nonempty. We define thefertilitopeof$$\pi $$πto be the convex hull of$$\mathcal V(\pi )$$V(π), and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of$$\pi $$πdirectly from$$\pi $$πusing Bousquet-Mélou’s notion of the canonical tree of $$\pi $$π. As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we callquasicanonical trees. We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density 1, and we determine all infertility numbers of size at most 126. Finally, we reformulate the conjecture that$$\sum _{\sigma \in s^{-1}(\pi )}x^{\textrm{des}(\sigma )+1}$$σs-1(π)xdes(σ)+1is always real-rooted in terms of nestohedra, and we propose natural ways in which this new version of the conjecture could be extended.

    more » « less