# Investigating the origin of the long range pseudo rapidity correlation in 2d
di-hadron measurements from STAR††thanks: Presented at Strange Quark Matter
conference, September, 2011, Krakow, Poland.

L. C. De Silva  The Department of Physics, University of Houston, 617 SR
Building 1, Houston, Texas 77204-5005

###### Abstract

Angular di-hadron correlations reveal novel structures in central Au+Au
collisions at $\sqrt{S_{NN}}$ = 200 GeV. One of them, known as the ridge, is
elongated in pseudo rapidity and peaks on the near side ($\Delta\phi$
$\approx$ 0). Investigating the origin of the ridge structure helps to
understand the hot dense matter that is created in ultra relativistic heavy
ion collisions. Results showing the $\langle$$p_{T}$$\rangle$ dependence of
the ridge structure are presented. Evidence for possible jet and non-jet
contributions to the ridge structure will be discussed.

##  1 Introduction

Recent triggered di-hadron correlation studies by STAR report a ridge
structure in two dimensions ($\Delta\eta$,$\Delta\phi$)[1]. Based on their
studies, the near side was assumed to consist of two independent structures, a
jet-like structure and the ridge. A complementary approach, presented here, is
carried out by using all possible charged particle pairs. The approach does
not require a trigger particle but with appropriate kinematic cuts reproduces
qualitatively similar correlation structure to that of triggered analysis. In
this approach, two empirical 2d fit functional models are used to extract
ridge properties. Possible model dependent quantitative evidence for jet and
non-jet phenomena contribution to the near side structure can be extracted
from the fits.

##  2 Data and analysis

The data used in this analysis were collected during run 4, year 2004, from
the STAR detector at Relativistic Heavy Ion Collider (RHIC) Brookhaven
National Lab (BNL), Long Island, New York. 32M Au+Au collisions at
$\sqrt{S_{NN}}$ = 200 GeV were analyzed. Charged tracks reconstructed using
the Time Projection Chamber (TPC) with 2$\pi$ azimuthal coverage and $|\eta|$
$\leq$ 1 in pseudo rapidity were selected. An earlier centrality evolution
analysis[2] was based on particles in the full transverse momentum range above
0.15 GeV/c. For the study presented here the lower threshold of the transverse
momentum was raised for both particles. The selected event vertexes are within
$\pm$ 25cm of the detector center and primary tracks are selected to come from
within a distance of closet approach of 3cm to the event vertex.

The correlation function for this analysis is a construct of Pearson s
correlation coefficient. Mathematically it can be derived as[2]:

$\displaystyle\frac{\Delta\rho}{\sqrt{\rho_{ref}}}$
$\displaystyle=\frac{\rho_{sib}\\--\rho_{mix}}{\sqrt{\rho_{ref}}}$ (1)

$\rho_{sib}$ is the normalized charged particle pair density within a single
event, which carries both correlated and uncorrelated pairs. The uncorrelated
background is subtracted by mixed event pairs. The resulting uncorrelated
density is denoted as $\rho_{mix}$. The difference is the number of correlated
pairs which is normalized by the square root of mixed pair density to yield
the _correlated particle pairs per final state charged particle_ as the
equivalent correlation measure to the Pearson s correlation coefficient. In
order to avoid possible artifacts due to pseudo rapidity acceptance, the
correlation measure is carried out in sub bins of the z-vertex. The event
mixing artifacts were eliminated by mixing events of a particular centrality
within a multiplicity window of 50 charged tracks. To correct for pair loss a
pair cut is implemented on both mixed and same event pairs. The effects due to
pileup events were removed by one of the standard pileup eliminating
procedures in STAR[7]. Finally the conversion electron positron pair
background is reduced by using a 1.5$\sigma$ dE/dX cut on electrons in
momentum ranges, 0.2 <$p_{T}$ <0.45 GeV/c and 0.7 <$p_{T}$ <0.8 GeV/c.

Figure 1: Evolution of the raw normalized correlation structure for selected
$\langle$$p_{T}$$\rangle$ bins at 0-1% centrality bin. The lower momentum
threshold of both particles have been increased in generating the spectrum of
correlation plots.

##  3 Fit functions

The initial empirical fit function carries three mathematical model components
that lead to seven free fit parameters.

F = $c_{0}$ \+ $c_{1}$cos($\Delta\phi$) +
$c_{2}$exp(-0.5*(($\Delta\phi$/$c_{3}$)2 \+ ($\Delta\eta$/$c_{4}$)2) +
$c_{5}$exp(-0.5*(($\Delta\phi$/$c_{6}$)2 \+ ($\Delta\eta$/$c_{6}$)2))

The first component of the fit is an offset parameter followed by a
cos($\Delta\phi$) term to extract the away side structure as seen in Fig. 1.
The assumption is that this structure corresponds to correlations due to a
recoiling jet[3]. The two remaining terms are introduced based on
$\langle$$p_{T}$$\rangle$ evolution of data (see Fig. 1). The asymmetric 2d
Gaussian attempts to address the long range correlation properties excluding
the peak structure around (0,0) that appears as $\langle$$p_{T}$$\rangle$
increases. This structure is described via the symmetric 2d Gaussian component
of the fit.

Modifications to the model were introduced after indications of higher order
harmonics were first predicted [4, 5] and then observed (see Fig. 2).

F = $c_{0}$ \+ $c_{1}$cos($\Delta\phi$) + $c_{2}$cos(2$\Delta\phi$) +
$c_{3}$cos(3$\Delta\phi$) + $c_{4}$cos(4$\Delta\phi$) +
$c_{5}$cos(5$\Delta\phi$) + $c_{6}$exp(-0.5*(($\Delta\phi$/$c_{7}$)2 \+
($\Delta\eta$/$c_{8}$)2))

The resulting fit carries higher order harmonics up to 5th order and an
asymmetric 2d Gaussian. It is important to note that in such a Fourier
expansion, the first order term serves as a momentum conserving term
(correlation due to recoil jet). In essence, the new model help in further
constraining the knowledge gathered from the initial model study.

Figure 2: Correlations at high $p_{T}$ very central data show evidence for the
existence of higher order harmonics.

##  4 Results

The focus of comparing the initial model study to the study which includes fit
components for higher harmonics based on initial energy density fluctuations
is to determine whether any information can be extracted from the asymmetric
2d Gaussian fit to the away side structure.

###  4.1 Initial model study

Resulting asymmetric 2d Gaussian parameters are reported in Fig. 3. The long
correlation strength approaches zero at higher $\langle$$p_{T}$$\rangle$ as
indicated by the asymmetric 2d Gaussian amplitude and volume parameters. The
$\Delta\eta$ and $\Delta\phi$ width evolution clearly indicate the asymmetry
shown in data. Also the $\Delta\eta$ width suggests that within the STAR
acceptance the ridge is flat whereas the $\phi$ width shows a very smooth
narrowing. An important observation is that the $\langle$$p_{T}$$\rangle$ of
the ridge ($\langle$$p_{T}$$\rangle$ = 0.72 GeV/c) is greater than that of the
inclusive spectrum ($\langle$$p_{T}$$\rangle$ $\approx$ 0.42 GeV/c [8]). The
hardness of the ridge spectrum is an indication of possible Jet contributions
to the ridge.

(a) Amplitude

(b) $\Delta\eta$ width

(c) $\Delta\phi$ width

(d) Symmetric width

Figure 3: Asymmetric and Symmetric 2d Gaussian parameter evolution as a
function of $\langle$$p_{T}$$\rangle$.

On the other hand, the symmetric 2d Gaussian component reveals important
information about an unmodified jet emerging at higher momenta (Fig. 3). The
width is comparable to the jet width in PP collisions. In order to further
constrain the above conclusion, amplitude and yield comparisons need to be
made.

(a) $\frac{v_{3}}{v_{2}}$ ratio

(b) $v_{n}\hskip 2.84544ptscaling\hskip 2.84544ptat\hskip 2.84544pt0\\--10\%$

Figure 4: Comparison to predicted higher harmonic ratios[5] and scaling
relations[4]

###  4.2 Higher harmonics study

Recent advancements in heavy ion collision models, predict significant
contributions from higher order harmonics to the observed correlation spectra
due to initial energy density fluctuations[4, 5]. STAR data also support the
predicted observation in high $p_{T}$, very central AuAu 200 GeV data(Fig. 2).
The $\frac{v_{3}}{v_{2}}$ ratio predictions were tested and reasonable
agreement to theory is evident (see Fig. 4a). However it is to be noted at low
$p_{T}$, deviations from the trends are observed for $N_{part}$ <150\. One
should note, thought, that the data is extracted fitting 2d di-hadron
correlations, whereas, theory uses single particle spectra and event plane
calculations.

A predicted hydro scaling relation[6] using extracted higher order Fourier
coefficients is studied, and reasonable agreement to theory has been observed
(Fig. 4b).In summary, given the different approaches used by theory
predictions tested in this proceedings and experiment, the data trends seem
support the hydro based theory predictions to a reasonable extent.

The remaining structure on the near side, after subtracting the Fourier
components still shows a $\Delta\eta-\Delta\phi$ asymmetry (see Fig. 3, red
data markers). This $\langle$$p_{T}$$\rangle$ evolution indicates that this
$\Delta\eta$ elongation is visible up to 2.7 GeV/c and is strongest at 0.9
GeV/c. The structure thus suggests possible jet modification at low
$\langle$$p_{T}$$\rangle$ which evolves to an unmodified jet at high $p_{T}$.
The amplitude and volume of the asymmetric structure scales lineary with
Glauber scaling as a function of centrality, which hints at a jet origin. The
rise of amplitude at high $\langle$$p_{T}$$\rangle$ is expected due to an
increase in per charge particle pair yield in jets. Further studies will be
carried out using the comparison to p+p collisions.

##  5 Summary and discussion

Un-triggered 2d di-hadron correlation studies reproduce qualitatively similar
results to that of a triggered analysis. The observed near side correlation is
modeled via an empirical fit function which extracts short and long range
structure properties. The $\langle$$p_{T}$$\rangle$ evolution of the extracted
parameters suggests a possible correlation between jets and the observed long
range correlation. Further constraint to data has been imposed modifying the
empirical model to incorporate higher order Fourier model components
cos(n$\Delta\phi$), n = 1– 5, based on theoretical predictions and evidence in
the data. The extracted higher order component strengths show reasonable
agreement to predicted hydrodynamical trends. The remainder on the near side
still reveals an asymmetric 2d Gaussian suggestive of possible modified jet
phenomena. A Glauber linear scaling was successfully applied to the un-
triggered centrality evolution (see Fig. 5) which indicates a jet origin for
the observed near side Gaussian.

Figure 5: Comparison of near side asymmetric 2D Gaussian amplitude to Glauber
linear scaling as a function of centrality.

## References

  * [1] B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C 80, 064912 (2009)
  * [2] M Daugherity et al. (STAR Collaboration), J. Phys. G 35, 104090 (2008)
  * [3] J. Porter and T. Trainor, J. Phys.: Conf. Ser. 27, 98 (2005)
  * [4] Xin Niang Wang et al., Phys. Rev. Lett. 106, 162301(2011)
  * [5] B. Alver et al., Phys. Rev. C 81, 054905 (2010)
  * [6] C. Gombeaud et al., Phys. Rev. C 81, 014901 (2010)
  * [7] Nuclear Phys. Lab. Annual Report, University of Washington (2009) p. 58
  * [8] J. Adams et al. STAR Collaboration


Fermilab Lattice and MILC Collaborations

# Refining new-physics searches in $B\to D\tau\nu$ decay with lattice QCD

Jon A. Bailey  Department of Physics and Astronomy, Seoul National University,
Seoul, South Korea  A. Bazavov  Physics Department, Brookhaven National
Laboratory, Upton, NY, USA  C. Bernard  Department of Physics, Washington
University, St. Louis, Missouri, USA  C.M. Bouchard  Department of Physics,
The Ohio State University, Columbus, Ohio, USA  C. DeTar  Physics Department,
University of Utah, Salt Lake City, Utah, USA  Daping Du<EMAIL_ADDRESS>Physics Department, University of Illinois, Urbana, Illinois, USA  A.X. El-
Khadra  Physics Department, University of Illinois, Urbana, Illinois, USA  J.
Foley  Physics Department, University of Utah, Salt Lake City, Utah, USA  E.D.
Freeland  Department of Physics, Benedictine University, Lisle, Illinois, USA
E. Gámiz  CAFPE and Departamento de Fìsica Teórica y del Cosmos, Universidad
de Granada, Granada, Spain  Steven Gottlieb  Department of Physics, Indiana
University, Bloomington, Indiana, USA  U.M. Heller  American Physical Society,
Ridge, New York, USA  Jongjeong Kim  Department of Physics, University of
Arizona, Tucson, Arizona, USA  A.S. Kronfeld  Fermi National Accelerator
Laboratory, Batavia, Illinois, USA  J. Laiho  SUPA, School of Physics and
Astronomy, University of Glasgow, Glasgow, UK  L. Levkova  Physics Department,
University of Utah, Salt Lake City, Utah, USA  P.B. Mackenzie  Fermi National
Accelerator Laboratory, Batavia, Illinois, USA  Y. Meurice  Department of
Physics and Astronomy, University of Iowa, Iowa City, Iowa, USA  E.T. Neil
Fermi National Accelerator Laboratory, Batavia, Illinois, USA  M.B. Oktay
Physics Department, University of Utah, Salt Lake City, Utah, USA  Si-Wei Qiu
Physics Department, University of Utah, Salt Lake City, Utah, USA  J.N. Simone
Fermi National Accelerator Laboratory, Batavia, Illinois, USA  R. Sugar
Department of Physics, University of California, Santa Barbara, California,
USA  D. Toussaint  Department of Physics, University of Arizona, Tucson,
Arizona, USA  R.S. Van de Water<EMAIL_ADDRESS>Physics Department, Brookhaven
National Laboratory, Upton, NY, USA  Ran Zhou  Department of Physics, Indiana
University, Bloomington, Indiana, USA

###### Abstract

The semileptonic decay channel $B\to D\tau\nu$ is sensitive to the presence of
a scalar current, such as that mediated by a charged-Higgs boson. Recently the
BaBar experiment reported the first observation of the exclusive semileptonic
decay $B\to D\tau^{-}\overline{\nu}$, finding an approximately 2$\sigma$
disagreement with the Standard-Model prediction for the ratio
$R(D)=\text{BR}(B\to D\tau\nu)/\text{BR}(B\to D\ell\nu)$, where $\ell=e,\mu$.
We compute this ratio of branching fractions using hadronic form factors
computed in unquenched lattice QCD and obtain $R(D)=0.316(12)(7)$, where the
errors are statistical and total systematic, respectively. This result is the
first Standard-Model calculation of $R(D)$ from ab initio full QCD. Its error
is smaller than that of previous estimates, primarily due to the reduced
uncertainty in the scalar form factor $f_{0}(q^{2})$. Our determination of
$R(D)$ is approximately 1$\sigma$ higher than previous estimates and, thus,
reduces the tension with experiment. We also compute $R(D)$ in models with
electrically charged scalar exchange, such as the type II two-Higgs doublet
model. Once again, our result is consistent with, but approximately 1$\sigma$
higher than, previous estimates for phenomenologically relevant values of the
scalar coupling in the type II model. As a byproduct of our calculation, we
also present the Standard-Model prediction for the longitudinal polarization
ratio $P_{L}(D)=0.325(4)(3)$.

###### pacs:

13.20.He, 12.38.Gc

††preprint: FERMILAB-PUB-12/333-T

Motivation. – The third generation of quarks and leptons may be particularly
sensitive to new physics associated with electroweak symmetry breaking due to
their larger masses. For example, in the minimal supersymmetric extension of
the Standard Model, charged-Higgs contributions to tauonic $B$ decays can be
enhanced if $\tan\beta$ is large. Thus the semileptonic decay $B\to D\tau\nu$
is a promising new-physics search channel Grzadkowski:1992qj ; Tanaka:1994ay ;
Garisto:1994vz ; Tsai:1996ps ; Wu:1997uaa ; Kiers:1997zt ; Miki:2002nz ;
Itoh:2004ye ; Chen:2006nua ; Nierste:2008qe ; Kamenik:2008tj ; Tanaka:2010se ;
Fajfer:2012vx .

The BaBar experiment recently measured the ratios $R(D^{(*)})=\text{BR}(B\to
D^{(*)}\tau\nu)/\text{BR}(B\to D^{(*)}\ell\nu)$, where $\ell=e,\mu$, and
reported excesses in both channels which, when combined, disagree with the
Standard Model by $3.4\sigma$ :2012xj . BaBar also interpreted these
measurements in terms of the type-II two-Higgs-doublet model (2HDM II) and
claimed to exclude the theory at 99.8% confidence-level. In this work, we
update the prediction for $R(D)$ in the Standard-Model and in new-physics
theories with a scalar current (such as the 2HDM II) using unquenched lattice-
QCD calculations of the $B\to D\ell\nu$ form factors $f_{0}(q^{2})$ and
$f_{+}(q^{2})$ by the Fermilab Lattice and MILC collaborations Bailey:2012rr .
This is the first determination of $R(D)$ from ab initio full QCD.

With lepton helicity defined in the rest frame of the virtual $W$ boson, the
general expressions for the differential rates for semileptonic $B\to
D\ell\nu$ decay are given by

$\displaystyle\frac{d\Gamma_{-}}{dq^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{24\pi^{3}}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}|\bm{p}_{D}|^{3}\left|{G}_{V}^{\ell
cb}f_{+}(q^{2})-\frac{m_{\ell}}{M_{B}}G_{T}^{\ell
cb}f_{2}(q^{2})\right|^{2}\,,$ (1) $\displaystyle\frac{d\Gamma_{+}}{dq^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}\frac{|\bm{p}_{D}|}{q^{2}}\left\\{\frac{1}{3}|\bm{p}_{D}|^{2}\left|m_{\ell}{G}_{V}^{\ell
cb}f_{+}(q^{2})-\frac{q^{2}}{M_{B}}G_{T}^{\ell
cb}f_{2}(q^{2})\right|^{2}\right.$ $\displaystyle+$
$\displaystyle\left.\frac{\left(M_{B}^{2}-M_{D}^{2}\right)^{2}}{4M_{B}^{2}}\left|\left(m_{\ell}{G}_{V}^{\ell
cb}-\frac{q^{2}}{m_{b}-m_{c}}G_{S}^{\ell
cb}\right)f_{0}(q^{2})\right|^{2}\right\\}\,,$

where the subscript denotes helicity, and $q=(p_{\ell}+p_{\nu})$ is the
momentum carried by the charged lepton-neutrino pair. The total semileptonic
width is the sum of the partial widths,
$\Gamma_{\text{tot}}=(\Gamma_{+}+\Gamma_{-})$. At tree-level of the Standard
Model electroweak interaction, the scalar- and tensor-exchange couplings are
$G_{S}=G_{T}=0$, while the vector coupling is ${G}_{V}^{\ell ij}=G_{F}V_{ij}$.
In the infinite heavy-quark-mass limit, the form factors $f_{+}(q^{2})$ and
$f_{2}(q^{2})$ are related via

$f_{2}(q^{2})=-f_{+}(q^{2})-\frac{M_{B}^{2}-M_{D}^{2}}{q^{2}}\left[f_{+}(q^{2})-f_{0}(q^{2})\right]\,.$
(3)

The form factor $f_{2}(q^{2})$ is only relevant for theories with tensor
currents, however, which we do not consider here.

Because the Standard-Model positive-helicity contribution to semileptonic
$B\to D$ decay is proportional to the lepton mass-squared, it can be neglected
for the light leptons $\ell=e,\mu$; thus experimental measurements of $B\to
D\ell\nu$ decays are sensitive only to the vector form factor $f_{+}(q^{2})$.
On the other hand, the differential rate for $B\to D\tau\nu$ is sensitive also
to the positive-helicity contribution and, hence, $f_{0}(q^{2})$. Existing
Standard-Model estimates of $d\Gamma_{-}/dq^{2}$ for $B\to D\tau\nu$ have
relied on the kinematic constraint $f_{0}(0)=f_{+}(0)$, dispersive bounds on
the shape Caprini:1997mu , relations from heavy-quark symmetry, and quenched
lattice QCD (neglecting $u$, $d$, and $s$ quark loops) deDivitiis:2007ui ;
deDivitiis:2007uk . See Refs. Kamenik:2008tj ; Nierste:2008qe ; Tanaka:2010se
; Fajfer:2012vx  for details. In this letter, we replace quenched QCD and
heavy-quark estimates with a full, 2+1-flavor QCD calculation. In particular,
we determine the following ratios within the Standard Model (where
$\ell=e,\mu$):

$\displaystyle R(D)$ $\displaystyle=$ $\displaystyle\text{BR}(B\to
D\tau\nu)/\text{BR}(B\to D\ell\nu)\,,$ (4) $\displaystyle P_{L}(D)$
$\displaystyle=$ $\displaystyle\left(\Gamma_{+}^{B\to
D\tau\nu}-\Gamma_{-}^{B\to D\tau\nu}\right)/\Gamma_{\rm tot}^{B\to
D\tau\nu}\,.$ (5)

These quantities enable particularly clean tests of the Standard Model and
probes of new physics because the CKM matrix elements and many of the hadronic
uncertainties cancel between the numerator and denominator.

Lattice-QCD calculation. – Here we briefly summarize the lattice-QCD
calculation of the $B\to D\ell\nu$ semileptonic form factors $f_{+}(q^{2})$
and $f_{0}(q^{2})$ Bailey:2012rr . Our calculation is based on a subset of the
(2+1)-flavor ensembles generated by the MILC Collaboration Bazavov:2009bb . We
use two lattice spacings $a\approx 0.12$ and 0.09 fm, and two light-quark
masses at each lattice spacing with (Goldstone) pion masses in the range
315–520 MeV; the specific numerical simulation parameters are given in Table I
of Ref. Bailey:2012rr .

This relatively small data set is sufficient for ratios such as those studied
here and in Ref. Bailey:2012rr , given the mild chiral and continuum
extrapolations. We use the Fermilab action ElKhadra:1996mp  for the heavy
quarks (bottom and charm) and use the asqtad-improved staggered action
Bazavov:2009bb  for the light valence and sea quarks ($u,d,s$). We minimize
the systematic error due to contamination from radial excitations in 2-point
and 3-point correlation functions by employing fits including their
contributions, as described in Sec. III of Ref. Bailey:2012rr . We renormalize
the lattice vector current $\bar{c}\gamma^{\mu}b$ (and other heavy-heavy
currents) using a mostly nonperturbative method ElKhadra:2001rv  in which we
determine the flavor-conserving normalizations nonperturbatively. The
remaining correction is close to unity and can be calculated in one-loop
tadpole-improved lattice perturbation theory Lepage:1992xa . When
extrapolating the lattice simulation results to the physical light-quark
masses and the continuum limit, we carefully account for the leading
nonanalytic dependence on the light-quark masses at nonzero but small momentum
transfer Chow:1993hr  including the effects of lattice artifacts (generic
discretization errors and taste-symmetry breaking introduced by the staggered
action) Aubin:2005aq ; Laiho:2005ue . The chiral-continuum extrapolation
results are plotted in the left panels of Figs. 6 and 7 of Ref. Bailey:2012rr
; for details see Eqs. (4.1) and (4.2) of the same work and the surrounding
text.

Figure 1: The form factors $f_{+}$ (top, red) and $f_{0}$ (bottom, blue) from
lattice QCD. The range of simulated recoil values is to the left of the
vertical line. The filled colored bands show the interpolation/extrapolation
of the numerical lattice data over the full kinematic range using the $z$
parameterization. For comparison, the experimental measurement from BaBar
Aubert:2009ac  is shown as solid filled circles (using $|V_{cb}|=41.4\times
10^{-3}$ deDivitiis:2007ui ; Aubert:2009ac ).

Figure 1 shows the results for $f_{+}(q^{2})$ and $f_{0}(q^{2})$ Bailey:2012rr
. The simulated data are in the range $w<1.17$ (to the left of the dashed
vertical line), where $w=(M_{B}^{2}+M_{D}^{2}-q^{2})/(2M_{B}M_{D})$. In this
region, we parameterize the $w$ dependence of the form factors by a quadratic
expansion about $w=1$, which works well for small recoil. To extend the form-
factor results beyond the simulated recoil values (to the right of the dashed
vertical line) we reparameterize the form factors in terms of the variable $z$
Boyd:1997kz , and then extrapolate to large recoil using a model-independent
fit function based on general quantum-theory-principles of analyticity and
crossing-symmetry. The functional forms used to extrapolate $f_{+}(q^{2})$ and
$f_{0}(q^{2})$ are defined in Eqs. (5.1)–(5.6) of Ref. Bailey:2012rr . The
fits are plotted in the left panel of Fig. 9, and the results are given in the
upper panel of Table V of the same work. As seen in Fig. 1, our result for
$f_{+}(q^{2})$ agrees very well with experimental measurements Aubert:2009ac
over the full kinematic range. This nontrivial check gives confidence in the
extrapolation of $f_{0}(q^{2})$, which cannot be obtained experimentally and
for which lattice-QCD input is crucial. In particular, lattice-QCD
uncertainties are smallest near $q^{2}=(M_{B}-M_{D})^{2}$, so the discussion
below hinges principally on our calculation of $f_{0}(q^{2})$ near this point,
the validated $f_{0}(0)=f_{+}(0)$, and a smooth connection between the two
limits.

We calculate the Standard-Model $B\to D\ell\nu$ partial decay rates into the
three generations of leptons using these form factors and Eqs. (1) and
(Refining new-physics searches in $B\to D\tau\nu$ decay with lattice QCD) with
$G_{S}=G_{T}=0$, $G_{V}=G_{F}V^{*}_{cb}$. The resulting distributions are
plotted in Fig. 2. To illustrate the role of the scalar form factor
$f_{0}(q^{2})$, we also show the rates with only the contributions from
$f_{+}(q^{2})$. Due to the significant helicity suppression, the differential
decay rates into light leptons are well-approximated by a single contribution
from the form factor $f_{+}(q^{2})$. For $B\to D\tau\nu$, however, the
contribution from the scalar form factor $f_{0}(q^{2})$ comprises half of the
Standard-Model rate.

Figure 2: Differential decay rates for $B\to De\nu$ (green), $B\to D\mu\nu$
(blue), and $B\to D\tau\nu$ (red) in the Standard Model. The black dot-dashed
curves show the rates calculated with $f_{0}(q^{2})=0$.

Given the lattice-QCD determinations of $f_{+}(q^{2})$ and $f_{0}(q^{2})$ we
can obtain the Standard-Model values for $R(D)$ and $P_{L}(D)$. These are the
primary results of this letter, and we now discuss the sources of systematic
uncertainty. In Ref. Bailey:2012rr , many statistical and several systematic
errors cancelled approximately or exactly in the ratio $f_{0}^{B_{s}\to
D_{s}\ell\nu}/f_{0}^{B\to D\ell\nu}$ studied there. Some of these do not
cancel (as well) in $R(D)$ and $P_{L}(D)$, however, because they affect
$f_{+}(q^{2})$ and $f_{0}(q^{2})$ differently.

Table 1 shows the error budgets for $R(D)$ and $P_{L}(D)$. The statistical
error in $R(D)$ is significant (3.7%) due to the different phase-space
integrations in the numerator and denominator, whereas for $P_{L}(D)$ the
correlated statistical fluctuations largely cancel. For the same reason, the
errors in $R(D)$ arising from the extrapolation to the physical light-quark
masses and the continuum limit ($1.4\%$) and to the full $q^{2}$ range
($1.5\%$), are much larger than for $P_{L}(D)$. We estimate the error from the
chiral-continuum extrapolation by comparing the results for fits with and
without next-to-next-to-leading order analytic terms in the chiral expansion.
We estimate the error from the $z$ extrapolation by varying the range of
synthetic data used in the $z$ fit, including an additional pole in the fit
function, and including higher powers of $z$. The specific chiral and $z$-fit
variations considered are enumerated in Table VI of Ref. Bailey:2012rr  and
discussed in detail in the surrounding text. The remaining sources of
uncertainty in Table 1 do not contribute significantly to the quantities
studied in Ref. Bailey:2012rr , so we describe them in greater detail below.

We determine the bare heavy-quark masses in our simulations by tuning the
parameters $\kappa_{b}$ and $\kappa_{c}$ in the heavy-quark action such that
the kinetic masses of the pseudoscalar $B_{s}$ and $D_{s}$ mesons match the
experimentally-measured values ElKhadra:1996mp . In practice, it is easier to
work with the form factors $h_{\pm}(w)$ on the lattice, which are linear
combinations of $f_{+,0}(q^{2})$ Bailey:2012rr . We study how the form factors
$h_{\pm}(w)$ depend on $\kappa_{b,c}$ by re-computing the form factors on some
ensembles at values of $\kappa_{b,c}$ slightly above and below the default
ones, and extracting the slopes with respect to $\kappa_{b,c}$. We use these
slopes to correct our results for $R(D)$ and $P_{L}(D)$ slightly from the
simulated $\kappa$ values to the physical ones, and conservatively take the
full size of the shift as the error due to $\kappa$-tuning.

We remove the leading taste-breaking light-quark discretization errors in the
form factors with the chiral-continuum extrapolation, and estimate the
remaining discretization errors from the heavy-quark action with power
counting Kronfeld:2000ck . We compute both the coefficient of the dimension 5
operator in the Fermilab action $c_{SW}$ and the rotation parameter $d_{1}$
for the heavy-quark fields at tree level in tadpole-improved lattice
perturbation theory ElKhadra:1996mp . Then the leading heavy-quark errors in
$h_{+}(1)$ are of $\mathrm{O}\left(\alpha_{s}(\Lambda/2m_{Q})^{2}\right)$ and
$\mathrm{O}\left((\Lambda/2m_{Q})^{3}\right)$, where $\Lambda$ is a typical
hadronic scale. Using the values $\alpha_{s}=0.3$, $\Lambda=500$ MeV, and
$m_{c}=1.2$ GeV, we estimate that heavy-quark discretization errors in
$h_{+}(1)$ are $\sim$1–2%. At nonzero recoil, $w>1$, there are corrections to
$h_{+}(w)$ of $\mathrm{O}\left(\alpha_{s}\Lambda/2m_{Q}\right)$, but these are
suppressed by $(1-w)$ because they vanish in the limit $w=1$ by Luke’s theorem
Luke:1990eg . We expect them to be largest at our highest recoil point
$w=1.2$, and estimate their size to be $\sim$ 1%. Thus we estimate the
uncertainty in $h_{+}(w)$ from heavy-quark discretization errors to be 2%,
which leads to negligible errors in $R(D)$ and $P_{L}(D)$. The leading heavy-
quark error in $h_{-}(w)$ is of
$\mathrm{O}\left(\alpha_{s}\Lambda/2m_{Q}\right)$, which we estimate with the
input parameters above to be $\sim$ 6%. To be conservative, we take the error
in the ratio $h_{-}(w)/h_{+}(w)$ to be 10%, which leads to small errors in
$R(D)$ and $P_{L}(D)$.

Our methods for computing $B\to D$ transitions incorporate the bulk of the
matching of the lattice vector current to continuum automatically, leaving a
factor $\rho_{V_{cb}^{\mu}}$ close to unity Harada:2001fj . For $R(D)$ and
$P_{L}(D)$, only the relative matching of the spatial and temporal components
of the current matters, $\rho_{V_{cb}^{i}}/\rho_{V_{cb}^{0}}$. Although we
have a one-loop calculation of $\rho_{V_{cb}^{0}}$ in hand, no nontrivial
result for $\rho_{V_{cb}^{i}}$ is available. We take
$\rho_{V_{cb}^{i}}/\rho_{V_{cb}^{0}}=1.0\pm 0.2$ to estimate the uncertainty
from this source. In similar calculations, we have never seen $\rho$ factors
that differ from unity by more than 5%, so this range is extremely
conservative. The uncertainty in the current renormalization factors leads to
a small error in $R(D)$, but is the second-largest source of error in
$P_{L}(D)$, after statistics.

We also consider the systematic uncertainties from tuning the light-quark
masses and determining the absolute lattice scale $r_{1}$, but these produce
negligible errors in both $R(D)$ and $P_{L}(D)$.

Table 1: Error budgets for the branching fraction and longitudinal polarization ratios discussed in the text. Errors are given as percentages. Source | $R(D)$ | $P_{L}(D)$
---|---|---
Monte-Carlo statistics | 3.7 | 1.2
Chiral-continuum extrapolation | 1.4 | 0.1
$z$-expansion | 1.5 | 0.1
Heavy-quark mass ($\kappa$) tuning | 0.7 | 0.1
Heavy-quark discretization | 0.2 | 0.3
Current $\rho_{V_{cb}^{i}}/\rho_{V_{cb}^{0}}$ | 0.4 | 0.7
total | 4.3% | 1.5%

Results and Conclusions. – We obtain the following determinations for the
branching-fraction and longitudinal-polarization ratios for $B\to D\ell\nu$
semileptonic decay:

$\displaystyle R(D)$ $\displaystyle=$ $\displaystyle 0.316(12)(7)\,,$ (6)
$\displaystyle P_{L}(D)$ $\displaystyle=$ $\displaystyle 0.325(4)(3)\,,$ (7)

where the errors are statistical and total systematic, respectively. The value
of $R(D)$ is approximately 1$\sigma$ larger than the recent Standard-Model
values obtained using estimates of $f_{0}(q^{2})$ from Refs. Kamenik:2008tj ;
Fajfer:2012vx , but it is still 1.7$\sigma$ lower than the recent BaBar
measurement, $R(D)=0.440\pm 0.058\pm 0.042$ :2012xj . The results for $R(D)$
from Belle Adachi:2009qg  agree with those of BaBar, but have larger
uncertainties. Current experimental measurements of $R(D)$ are statistics-
limited, so the luminosities available at Belle II and SuperB should enable
significant improvement on $R(D)$ and possibly a determination of $P_{L}(D)$.

We also re-examine the interpretation of the BaBar measurement of $R(D)$ as a
constraint on the 2HDM II; the result is plotted in Fig. 3.

Figure 3: Lattice-QCD calculation (red) and experimental measurement :2012xj
(blue) of $R(D)$ vs. $\tan\beta/M_{H^{\pm}}$ in the 2HDM II. For the BaBar
result, the dark- (light-) blue regions denote the 1$\sigma$ (2$\sigma$) error
bands. For comparison, the 2HDM II curve based on Refs. Kamenik:2008tj ;
Fajfer:2012vx  and used in Ref. :2012xj  is shown in green.

For this theory, the scalar-exchange coupling in Eq. (Refining new-physics
searches in $B\to D\tau\nu$ decay with lattice QCD) is given by

$G_{S}^{\ell
cb}=G_{F}V_{cb}\frac{m_{\ell}(m_{c}+m_{b}\tan^{2}\beta)}{M_{H^{\pm}}^{2}}.$
(8)

As soon as $\tan\beta\gtrsim 4$, $m_{c}$ in Eq. (8) can be neglected. On the
other hand, evaluating Eq. (Refining new-physics searches in $B\to D\tau\nu$
decay with lattice QCD) depends sensitively on $m_{c}/m_{b}$ via the prefactor
multiplying $f_{0}(q^{2})$. We take $m_{c}/m_{b}=0.22$, following Refs.
Fajfer:2012vx ; :2012xj . Our improved calculation of the scalar form factor
$f_{0}(q^{2})$ increases the prediction for $R(D)$ by about $1\sigma$ for all
$\tan\beta/M_{H^{+}}<0.6~{}\text{GeV}^{-1}$. This sensitivity of $R(D)$ to
differences in $f_{0}(q^{2})$ suggests that one should be cautious in using
indirect estimates of the form factors to constrain new-physics models in
other decay channels such as $B\to D^{*}\tau\nu$.

Inspection of the general formulas for the differential decay rates, Eqs. (1)
and (Refining new-physics searches in $B\to D\tau\nu$ decay with lattice QCD),
shows that many new-physics explanations for the $\sim 3\sigma$ tension
between measurements of $\\{R(D),R(D^{*})\\}$ and the Standard Model are
possible. Even in the context of 2HDM, variants other than type II can have
different phenomenology Crivellin:2012ye . Lattice-QCD calculations of
$f_{+}(q^{2})$ and $f_{0}(q^{2})$ can be used to provide reliable predictions
for $R(D)$ in any model given values for the couplings
$\\{G_{S},G_{V},G_{T}\\}$. We note, however, that the Dalitz distribution for
the lepton and $D$-meson energies in the $B$-meson rest frame may be more
sensitive to tensor interactions than $R(D)$ Kronfeld:2008gu .

The largest source of uncertainty in our determinations of $R(D)$ and
$P_{L}(D)$ is statistical errors, which we expect to reduce with an analysis
of our full data set in a future work Qiu:2011ur . The ratio $R(D)$ is
correlated with $P_{L}(D)$ as well as other observables such as $R(D^{*})$ or
$R(D_{s})$ in many new-physics models; thus the pattern of experimental
results for these quantities can help distinguish between new-physics
scenarios, such as those with and without a charged-Higgs boson Tanaka:2010se
. We will present lattice-QCD results for $R(D^{*})$ and $P_{L}(D^{*})$ in a
future paper on the $B\to D^{*}\ell\nu$ form factors, and also note that we
could easily obtain Standard-Model predictions for $R(D_{s})$ and
$P_{L}(D_{s})$ if measurements of these quantities were possible with an
$\Upsilon(5S)$ run at a $B$ factory.

Given the present tensions between experimental measurements and Standard-
Model predictions for both $\\{R(D),R(D^{*})\\}$ and the leptonic branching
fraction $\text{BR}(B\to\tau\nu)$ Bona:2009cj ; Lunghi:2010gv ; Charles:2011va
; Laiho:2012ss , lattice-QCD calculations of $B\to D\tau\nu$ form factors and
other hadronic weak matrix elements can play a key role in revealing whatever
theory beyond the Standard Model is realized in Nature.

Acknowledgements. – We thank Manuel Franco Sevilla, Jernej Kamenik, and Bob
Kowaleski for valuable discussions. We thank Yang Bai for emphasizing the
importance of the choice of scale for $m_{c}$ in Eq. (Refining new-physics
searches in $B\to D\tau\nu$ decay with lattice QCD). Computations for this
work were carried out with resources provided by the USQCD Collaboration, the
Argonne Leadership Computing Facility, the National Energy Research Scientific
Computing Center, and the Los Alamos National Laboratory, which are funded by
the Office of Science of the United States Department of Energy; and with
resources provided by the National Institute for Computational Science, the
Pittsburgh Supercomputer Center, the San Diego Supercomputer Center, and the
Texas Advanced Computing Center, which are funded through the National Science
Foundation’s Teragrid/XSEDE Program. This work was supported in part by the
U.S. Department of Energy under Grants No. DE-FG02-91ER40628 (C.B.), No. DOE
FG02-91ER40664 (Y.M.), No. DE-FC02-06ER41446 (C.D., J.F., L.L., M.B.O.), No.
DE-FG02-91ER40661 (S.G., R.Z.), No. DE-FG02-91ER40677 (D.D., A.X.K.), No. DE-
FG02-04ER-41298 (J.K., D.T.); by the National Science Foundation under Grants
No. PHY-1067881, No. PHY-0757333, No. PHY-0703296 (C.D., J.F., L.L., M.B.O.),
No. PHY-0757035 (R.S.); by the Science and Technology Facilities Council and
the Scottish Universities Physics Alliance (J.L.); by the MICINN (Spain) under
grant FPA2010-16696 and Ramón y Cajal program (E.G.); by the Junta de
Andalucía (Spain) under Grants No. FQM-101, No. FQM-330, and No. FQM-6552
(E.G.); by European Commission (EC) under Grant No. PCIG10-GA-2011-303781
(E.G.); and by the Creative Research Initiatives program (3348-20090015) of
the NRF grant funded by the Korean government (MEST) (J.A.B.). This manuscript
has been co-authored by employees of Brookhaven Science Associates, LLC, under
Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy. Fermilab is
operated by Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359
with the U.S. Department of Energy.

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