SNSN-323-63

$b\rightarrow s\gamma$ and $b\rightarrow d\gamma$ (B factories)

Wenfeng Wang

University of Notre Dame Du Lac, South Bend, IN 46556, USA

> The photon spectrum in $B\rightarrow X_{s,d}\gamma$ decay, where $X_{s}(d)$
> is any strange (non-strange) hadronic state, is studied using data samples
> of
> $e^{+}e^{-}\rightarrow\Upsilon(4S)\to{{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
> decays collected by the BABAR and Belle experiments. Here I present the
> latest measurements of the branching fraction and spectral moments from
> $B\to X_{s}\gamma$ decays by Belle and the direct $C\\!P$ asymmetry
> $A_{CP}(B\to X_{s+d}\gamma)$ measured at BABAR. The determination of
> $|V_{td}/V_{ts}|^{2}$ is also presented.

> PRESENTED AT
>
>
>
>
> 6th International Workshop on the CKM Unitarity Triangle(CKM2010),

##  1 Introduction

The electromagnetic radiative process $b\rightarrow q\gamma$ ($q=s,d$)
proceeds at leading order via the loop diagram in the Standard Model (SM).
Here the SM predication of the inclusive rate
$\Gamma({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})$
can be equated with the precisely calculable partonic rate
$\Gamma({{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})$
at the level of a few percent [1] (heavy quark duality). An extraordinary
theoretical effort has led to a precision SM prediction for the branching
fraction at the next-to-next-to-leading order (four-loop), $\mbox{\rm
BR}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})=(3.15\pm
0.23)\times 10^{-4}$ ($E_{\gamma}>1.6\mathrm{\,Ge\kern-1.00006ptV}$) [2],
where $E_{\gamma}$ is the photon energy measured in the rest frame of the
${{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$ meson. The possibility
for new heavy particles to enter into the loop at leading order could cause
significant deviations from the SM prediction. A recent review can be seen in
[3].

New physics can also significantly enhance the direct $C\\!P$ asymmetry for
${{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$
and
${{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{d}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$
decay [4] without changing the branching fraction. We define

$A_{CP}=\frac{\Gamma({{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}+{{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{d}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})-\Gamma({{}{}{{}{{}{{{{{{{}{}{{\overline{b}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}\to{{}{}{{}{{}{{{{{{{}{}{{\overline{s}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}+{{}{}{{}{{}{{{{{{{}{}{{\overline{b}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}\to{{}{}{{}{{}{{{{{{{}{}{{\overline{d}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})}{\Gamma({{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}+{{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{d}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})+\Gamma({{}{}{{}{{}{{{{{{{}{}{{\overline{b}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}\to{{}{}{{}{{}{{{{{{{}{}{{\overline{d}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}+{{}{}{{}{{}{{{{{{{}{}{{\overline{b}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}\to{{}{}{{}{{}{{{{{{{}{}{{\overline{d}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})}$
(1)

which is $\sim 10^{-6}$ in the SM, with nearly exact cancellation of opposite
asymmetries for
${{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$
and
${{}{{}{{{{{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to{{}{{}{{{{{{{}{}{{d}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$.
Thus any non-zero measurements of this joint asymmetry is an indication of new
physics.

The shape of the photon energy spectrum, which is insensitive to non-SM
physics [5], can be used to determine the Heavy Quark Expansion
(HQE)parameters, $m_{b}$ and $\mu_{\pi}^{2}$, related to the mass and momentum
of the ${{{{}{}{{b}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$ quark within the
${{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$ meson. These parameters
can be used to reduce the error in the extraction of the CKM matrix elements
$V_{cb}$ and $V_{ub}$ from the inclusive semi-leptonic
${{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$-meson decays,
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{c}\ell\nu$ and
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{u}\ell\nu$ ($\ell=e$ or $\mu$) [6] .

The inclusive rate for $b\rightarrow d\gamma$ is suppressed compared to
$b\rightarrow s\gamma$ by a factor $|V_{td}/V_{ts}|^{2}$ in the SM. This ratio
can also be obtained from the $B_{d}$ and $B_{s}$ mixing frequencies [7]. New
physics effects would enter in different ways in mixing and radiative decays.
Measurements of $|V_{td}/V_{ts}|$ using the exclusive modes
$B\rightarrow(\rho,\omega)\gamma$ and $B\to K^{*}\gamma$ [8, 9] are now well-
established, with theoretical uncertainties of 7% [10]. A measurement of
inclusive $b\rightarrow d\gamma$ relative to $b\rightarrow s\gamma$ would
determine $|V_{td}/V_{ts}|$ with reduced theoretical uncertainties. We
parametrize the inclusive ratio (following [11]) by:

${{\mbox{\rm BR}(b\rightarrow d\gamma)}\over{\mbox{\rm BR}(b\rightarrow
s\gamma)}}=\zeta^{2}\left|{{V_{td}}\over{V_{ts}}}\right|^{2}(1+\Delta R)$ (2)

where $\zeta$ accounts for any remaining SU(3) breaking and $\Delta R$
accounts for weak annihilation in $B^{+}$ decays.

B factories, BABAR [12] and Belle [13], already accumulated more than one
billion of
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
events, which allows BABAR and Belle collaborations to perform precision
measurements on $b\to s\gamma$ and $b\to d\gamma$ processes [14]. Here I
summarize the latest experimental achievements on the above inclusive
processes.

##  2 Direct CP asymmetry in $B\rightarrow X_{s,d}\gamma$

The result presented***The branching fraction of $B\to X_{s}\gamma$ and its
spectra shape from same analysis will be present in near future. is based on a
data sample of
${{}{{}{{{{{{{}{}{{e}_{\mspace{-2.0mu}{}}^{+}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{e}_{\mspace{-2.0mu}{}}^{-}}\mspace{-0.6mu}}}}}}}}}\to{{{}{{}{{{{{{{}{}{{\Upsilon}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}}{{}{{{{{{{}{}{{\left({4S}\right)}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}\to{{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
collisions collected with the BABAR detector at the PEP-II asymmetric-energy
${{}{{}{{{{{{{}{}{{e}_{\mspace{-2.0mu}{}}^{+}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{e}_{\mspace{-2.0mu}{}}^{-}}\mspace{-0.6mu}}}}}}}}}$
collider. The on-resonance integrated luminosity is 347 $fb^{-1}$ and 36
$fb^{-1}$ of off-resonance data, taken 40 $\mathrm{\,Me\kern-1.00006ptV}$
below the
${{{}{{}{{{{{{{}{}{{\Upsilon}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}}{{}{{{{{{{}{}{{\left({4S}\right)}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}$
resonance energy, are used to estimate the continuum background
(${}{{}{{{{{{{}{}{{q}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}{}{}{{}{{}{{{{{{{}{}{{\overline{q}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}:q={{}{{}{{{{{{{}{}{{u}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{d}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{s}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{c}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}},{{}{{}{{{{{{{}{}{{\tau}_{\mspace{-2.0mu}{}}^{+}}\mspace{-0.6mu}}}}}}}}}{{}{{}{{{{{{{}{}{{\tau}_{\mspace{-2.0mu}{}}^{-}}\mspace{-0.6mu}}}}}}}}}$).

Figure 1: Left: The photon spectrum in $347fb^{-1}$ of data after background
subtraction. The inner error bars are statistical only, while the outer
include both statistical and systematic errors in quadrature; Right:
Measurements of $A_{CP}$
(${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$),
with statistical and systematic errors. The three published results, top to
bottom, are from references [15].

The analysis begins by requiring a high-energy photon, characteristic of $B\to
X_{s}\gamma$ decays, while photons from $\pi^{0}$ and $\eta$ are vetoed. The
background from continuum events is significantly suppressed by charged lepton
tagging and by exploiting the more jet-like topology of the $q\overline{q}$ or
$\tau^{+}\tau^{-}$ events compared to the isotropic $B\overline{B}$ decays.
The remaining continuum backgrounds are estimated with off-resonance data. The
non signal
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
background arises predominantly from
${{{{}{}{{\pi}_{\mspace{-2.0mu}{}}^{0}}\mspace{-0.6mu}}}}$,${{{{}{}{{\eta}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$
decay but also from decays of other light mesons, mis-reconstructed electrons
and hadrons, which are estimated using Monte Carlo simulation and corrected
the data and MC difference using appropriate control samples. Figure 1 shows
the observed photo spectrum after subtracting off-resonance data and the
corrected
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
backgrounds. Two prior selected control regions,
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
control ($1.53<E^{*}_{\gamma}<1.8\mathrm{\,Ge\kern-1.00006ptV}$) and Continuum
control ($2.9<E^{*}_{\gamma}<3.5\mathrm{\,Ge\kern-1.00006ptV}$), are used to
validate the background estimation. In the
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
control region we find $1252\pm 272(stat.)\pm 841(syst.)$ events, dominated by
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
background with a small signal contribution component ( 200-400 events
depending on models); the continuum region yields s $-100\pm 138(stat.)$
events, consistent with zero which showing good estimation of off-resonance
subtraction.

The direct CP asymmetry, $A_{CP}$
(${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$)
is measured by dividing the signal sample into
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}$ and
${{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
decays according to the charge of the lepton tag to measure
$A^{\mathrm{meas}}_{CP}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})=\frac{N^{+}-N^{-}}{N^{+}+N^{-}}$,
where $N^{+(-)}$ are the positively (negatively) tagged signal yields. The
asymmetry must be corrected for the dilution due to the mistag fraction
$\omega$,
$A_{CP}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})=\frac{1}{1-2\omega}A^{\mathrm{meas}}_{CP}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}).$
the missing fraction $\omega$ is found to be $0.131\pm 0.007$, from
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}^{0}}\mspace{-0.6mu}}}}}}}}}-{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{0}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
oscillation, the fraction of events with wrong-sign leptons from the
${{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}$ decay chain and the
similar fraction due to misidentification of hadrons as leptons.

The theoretical SM predictions for a near-zero asymmetry do not require the
entire spectrum to be measured. To reduce the sensitivity to background, the
signal region is restricted to
$2.1<E^{*}_{\gamma}<2.8\mathrm{\,Ge\kern-1.00006ptV}$. In this selected energy
region, the tagged signal yields are $N^{+}=2623\pm 158(stat.)$ and
$N^{-}=2397\pm 151(stat.)$ giving an asymmetry of
$A^{\mathrm{meas}}_{CP}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})=0.045\pm
0.044\ .$ Finally the
$A^{\mathrm{meas}}_{CP}({{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}\to
X_{s+d}{{}{{}{{{{{{{}{}{{\gamma}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}})$
is corrected for mistagging and bias to give $A_{CP}=0.056\pm 0.060(stat.)\pm
0.018(syst.)$, where the systematic error is mainly from non signal
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
background and the lepton tagging efficiency. The result is consistent with no
observed asymmetry, consistent with SM expectation and previous measurements.
A comparison of the result to published measurements is shown in figure 1. The
current measurement is the most precise to date.

##  3 Branching fraction and moments of $b\rightarrow s\gamma$

Currently the most precise measurement of the inclusive $B\to X_{s}\gamma$
branching fractions has been done by Belle [16]. The data consists of a sample
of 605 $fb^{-1}$ taken on the $\Upsilon(4S)$ resonance. Another 68 $fb^{-1}$
sample was taken at an energy 60 MeV below the resonance.

The signal spectrum is extracted by collecting all high energy photons,
vetoing those originating from $\pi^{0}$ and $\eta$ decays to two photons. The
non $B\overline{B}$ background, mainly $e^{+}e^{-}\rightarrow q\overline{q}$
($q=u,d,s,c$) events, is subtracted using the off-resonance sample. The
remaining background from $B\overline{B}$ are subtracted using Monte-Carlo
simulated distributions normalized using data control samples.

The analysis proceeds in two different streams, with lepton tag (LT) and
without (MAIN). Two samples give similar sensitivity to the signal while being
largely statistically independent. After these selection criteria, $41.1\times
10^{5}$ ( $24.6\times 10^{4}$) and $3.5\times 10^{5}$ ($0.9\times 10^{4}$)
photon candidates survive in the MAIN (LT) stream of the on- and off-resonance
data samples, respectively.

The photon candidates from $B\overline{B}$ background is divided into six
categories:(i) $\pi^{0}$; (ii)$\eta$; (iii) other real photons from decays of
$\omega$, $\eta^{\prime}$ and $J/\psi$ mesons; (iv) mis-identified calorimeter
clusters from $K^{0}_{L}$ and $\overline{n}$; (v) electrons misidentified as
photons and; (vi) beam background. Each category is checked using appropriate
control samples as described in Ref.[17]. Each background yield, scaled by the
described procedures, is subtracted from the data spectrum. The photon energy
ranges 1.4-1.7 GeV and 2.8-4.0 GeV were chosen a prior as control regions to
test the integrity of the background subtraction since in the low energy
region the little signal expected is negligible with respect to the
uncertainty on the background, and no signal is possible in the high energy
region above the kinematic limit. The yield in the high energy region are
$1245\pm 4349$ and $292\pm 410$ candidates in the MAIN and LT stream,
respectively, while corresponding yields in the low energy region are
$-1629\pm 3071$ and $-745\pm 623$, respectively.

To obtain the true spectrum, a three-step unfolding procedure is used to
correct the raw spectrum. The procedure does not distinguish between $B\to
X_{s}\gamma$ and $B\to X_{d}\gamma$. Assuming the shape of the corresponding
photon energy spectra are equivalent, the contribution of $B\to X_{d}\gamma$
is subtracted using the ratio $R_{d/s}=(4.5\pm 0.3)\%$. Boost corrections,
obtained from MC simulation, are used to derive the measurements in the rest
frame of the $B$ meson. The two streams, MAIN and LT, are combined taking the
correlation into count.

The measured branching fraction in the $B$-meson rest frame is $BF(B\to
X_{s}\gamma)=(3.45\pm 0.15\pm 0.40)\times 10^{-4}$ for the photon energy range
from 1.7 GeV to 2.8 GeV. The most accurate measurement is given in the photon
energy range 2.0 GeV to 2.8 GeV, $BF(B\to X_{s}\gamma)=(3.02\pm 0.10\pm
0.11)\times 10^{-4}$. Here the errors are statistical and systematic,
respectively. The measured branching fractions are in agreement with the
latest theoretical calculation. The measured spectral moments can be used to
reduce the uncertainty on $|V_{ub}|$ [18, 19].

##  4 $b\rightarrow d\gamma$

Here we present the first significant observation of the $b\to d\gamma$
transition in the hadronic mass range
$M(X_{d})>1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, used in the
determination of $|V_{td}/V_{ts}|$ via the ratio of inclusive widths.Inclusive
$B\to X_{s}\gamma$ and $B\to X_{d}\gamma$ rates are extrapolated from the
measurements of the partial decay rates of seven exclusive final states in the
hadronic mass ranges
$0.5<M(X_{d})<1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$1.0<M(X_{d})<2.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Here the $X_{d}$
includes $\pi^{+}\pi^{-}$, $\pi^{+}\pi^{-}$ $\pi^{0}$, $\pi^{+}\pi^{-}$
$\pi^{+}$, $\pi^{+}\pi^{-}$ $\pi^{+}\pi^{-}$, $\pi^{+}\pi^{-}$ $\pi^{+}$
$\pi^{0}$ and $\pi^{+}\eta$; while the $X_{s}$ includes $K^{+}$ $\pi^{-}$,
$K^{+}$ $\pi^{0}$, $K^{+}$ $\pi^{+}\pi^{-}$, $K^{+}$ $\pi^{-}$ $\pi^{0}$
$K^{+}$ $\pi^{-}$ $\pi^{+}\pi^{-}$, $K^{+}$ $\pi^{-}$ $\pi^{+}$ $\pi^{0}$ and
$K^{+}\eta$. We combine these measurements and make a model-dependent
extrapolation to higher hadronic mass to obtain an inclusive branching
fraction (${\cal B})$ for $b\to(s,d)\gamma$. These measurements use a sample
of $471\times 10^{6}$
${{}{{}{{{{{{{}{}{{B}_{\mspace{-2.0mu}{}}}\mspace{-0.6mu}}}}}}}}}{}{{}{}{{}{{}{{{{{{{}{}{{\overline{B}}_{\mspace{-2.0mu}{\mspace{1.0mu}{}}}^{\mspace{1.0mu}{{\raisebox{-1.65764pt}{{${{{\scriptstyle{{{{{{}}}}}}}}}$}}}}}}\mspace{-0.6mu}}}}}}}}}}$
pairs collected by the BABAR experiment.

The signal yields in the data for the combination of all seven decay modes are
determined from two-dimensional extended maximum likelihood fits to the
$\Delta E^{*}$ and $m_{\rm ES}$ distributions after all event selections,
where $\Delta E^{*}=E^{*}_{B}-E^{*}_{\rm beam}$, $E^{*}_{B}$ is the energy of
the $B$ meson candidate and $E^{*}_{\rm beam}$ is the beam energy, and
$\mbox{$m_{\rm ES}$}=\sqrt{E^{*2}_{\rm beam}-{\vec{p}}_{B}^{\;*2}}$,
${\vec{p}}_{B}^{\;*}$ is the momentum of the $B$ candidate. Table 1 gives the
signal yields, efficiencies and partial branching fractions.

Table 1:  Signal yields ($N_{S}$), efficiencies ($\epsilon$). partial branching fractions ($BF$) and inclusive branching fractions ($\cal{B})$ for the measured decay modes. The first error is statistical the the second systematic (including error from extrapolation to missing decay modes, for the inclusive $\cal{B}$). | $M(X_{s})0.4-1.0$ | $M(X_{d})0.4-1.0$ | $M(X_{s})1.0-2.0$ |  $M(X_{d})1.0-2.0$ (GeV/$c^{2}$)
---|---|---|---|---
$N_{S}$ | $804\pm 33$ | $35\pm 9$ | $990\pm 42$ | $56\pm 14$
$\epsilon$ | 4.5% | 3.1% | 1.6% | 1.9%
$BF(\times 10^{-6})$ | $18.9\pm 0.8\pm 0.8$ | $1.2\pm 0.3\pm 0.1$ | $65.7\pm 2.8\pm 5.9$ | $3.2\pm 0.8\pm 0.5$
${\cal{B}}(\times 10^{-6})$ | $38.3\pm 1.6\pm 1.5$ | $1.3\pm 0.3\pm 0.1$ | $192\pm 80\pm 45$ | $7.9\pm 2.0\pm 3.3$
$\frac{{\cal{B}}(b\to d\gamma)}{{\cal{B}}(b\to s\gamma)}$ | $0.0033\pm 0.009\pm 0.003$ | -

To obtain inclusive ${\cal B}(b\to s\gamma)$ and ${\cal B}(b\to d\gamma)$ we
need to correct the partial $\cal B$ values in Table 1 for the fractions of
missing final states. After correcting for the 50% of missing decay modes with
neutral kaons, the low mass $B\to X_{s}\gamma$ measurement is found to be
consistent with previous measurements of the rate for $B\to K^{*}\gamma$ [14].
For the low mass $B\to X_{d}\gamma$ region, we correct for the small amount of
non-reconstructed $\omega$ final states ($\omega\to\pi^{0}\gamma$ and others),
and find a partial branching fraction consistent with previous measurements of
$\mbox{\rm BR}(B\to(\rho,\omega)\gamma)$ [14]. We assume that non-resonant
decays do not contribute in this region.

In the high mass region, the missing fractions depend on the fragmentation of
the hadronic system and are expected to be different for $X_{d}$ and $X_{s}$.
We explore the uncertainty in the correction for missing modes by considering
several alternative models. The resulting missing fractions vary by up to 50%
relative to the nominal model. We therefore independently vary final states
with $\geq 5$ stable hadrons, or with $\geq 2\pi^{0}$ or $\eta$ mesons, by
$\pm$50%.

Combining the two mass regions, taking into account a partial cancellation of
the missing fraction errors in the ratio of $b\to d\gamma$ to $b\to s\gamma$ ,
we find ${\cal B}(b\to d\gamma)/{\cal B}(b\to s\gamma)=0.040\pm
0.009(stat.)\pm 0.010(syst.)$ in the mass range
$M(X)<2.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. For the unmeasured region
$M(X)>2.0$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ the differences between
$b\to s\gamma$ and $b\to d\gamma$ are small and almost completely cancel in
the ratio.

Conversion of the ratio of inclusive branching fractions to the ratio
$|V_{td}/V_{ts}|$ is done according to [11], which requires $\overline{\rho}$
and $\overline{\eta}$ as input. However, since these are partially determined
from previous measurements of $|V_{td}/V_{ts}|$, we instead re-express
$\overline{\rho}$ and $\overline{\eta}$ in terms of the independent CKM angle
$\beta$. This procedure yields a value of $|V_{td}/V_{ts}|=0.199\pm
0.022(stat.)\pm 0.024(syst.)\pm 0.002(th.)$ competitive with more model-
dependent determinations from the measurement of the exclusive modes
$B\to(\rho,\omega)\gamma$ and $B\to K^{*}\gamma$ [8, 9].

##  5 Summary

Here I summarized the experiment progresses on the inclusive $b\to s\gamma$
and $b\to d\gamma$ after the last CKM workshop. Belle measured the inclusive
branching fraction in the $B$-meson rest frame, $BF(B\to X_{s}\gamma)=(3.45\pm
0.15\pm 0.40)\times 10^{-4}$, for the photon energy range from 1.7 GeV to 2.8
GeV. BABAR presents the most precise direct CP asymmetry measurement to date,
its preliminary result is consistent with SM prediction. BABARalso measured
the ratio of $b\rightarrow d\gamma$ over $b\rightarrow s\gamma$ using seven
exclusive modes, providing the independent determination of $|V_{td}/V_{ts}|$.

ACKNOWLEDGEMENTS

I am grateful to the wonderful works from BABAR and Belle collaborations.
Thanks also to the organizers of the CKM2010 for all efforts in making this
venue successful.

## References

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# First global QCD analysis of the TMD $\bm{g_{1T}}$ from semi-inclusive DIS
data

Shohini Bhattacharya  Physics Department, Brookhaven National Laboratory,
Bldg. 510A, Upton, NY 11973, USA  Department of Physics, Temple University,
Philadelphia, PA 19122, USA  Zhong-Bo Kang  Department of Physics and
Astronomy, University of California, Los Angeles, CA 90095, USA  Mani L.
Bhaumik Institute for Theoretical Physics, University of California, Los
Angeles, CA 90095, USA  Center for Frontiers in Nuclear Science, Stony Brook
University, Stony Brook, NY 11794, USA  Andreas Metz  Department of Physics,
Temple University, Philadelphia, PA 19122, USA  Gregory Penn  Department of
Physics, Yale University, New Haven, CT 06511, USA  Department of Physics,
Temple University, Philadelphia, PA 19122, USA  Daniel Pitonyak  Department of
Physics, Lebanon Valley College, Annville, PA 17003, USA

###### Abstract

The worm-gear transverse momentum dependent (TMD) function $g_{1T}$ is one of
the eight leading-twist TMDs and has the probabilistic interpretation of
finding a longitudinally polarized quark inside a transversely polarized
hadron. In this work, we present the first simultaneous extraction of $g_{1T}$
from all the available experimental measurements. The study involves the
analysis of COMPASS, HERMES, and Jefferson Lab data on semi-inclusive deep-
inelastic scattering using Monte Carlo techniques. We also compare $g_{1T}$
obtained from this experimental data with different theoretical approaches,
such as the large-$N_{c}$ approximation, the Wandzura-Wilczek-type
approximation, and lattice QCD.

##  I Introduction

An important class of functions required to understand the structure of
hadrons in terms of their underlying partons are transverse momentum dependent
parton distribution and fragmentation functions (TMD PDFs and TMD FFs,
collectively called TMDs). We use TMDs in the description of hard scattering
processes such as semi-inclusive deep-inelastic scattering (SIDIS), Drell-Yan,
and $e^{+}\,e^{-}$ annihilation into hadron pairs. The presence of both hard
and soft scales in these reactions allows one to probe the intrinsic motion of
partons. For example, in SIDIS one has the photon virtuality
$q^{2}\equiv-Q^{2}$ and transverse momentum $q_{T}$ such that if $q_{T}\ll Q$,
one is sensitive to TMD physics. While the usual collinear PDFs describe the
probability to find a parton in a fast-moving hadron with a particular
fraction $x$ of the hadron’s longitudinal momentum, TMD PDFs in addition
encode the probability that the parton has a specific transverse momentum
$\vec{k}_{\perp}$. The TMD PDFs can therefore be regarded as the natural
extension of collinear PDFs from one to three dimensions in momentum space.

When the spin of the hadron and that of the quarks are taken into account,
certain constraints of QCD, such as Hermiticity, time reversal, and parity,
allow us to define eight TMDs at leading twist (twist-2) as functions of
$(x,\vec{k}^{2}_{\perp})$: $f_{1}$ (unpolarized function), $g_{1L}$ (helicity
function), $h_{1}$ (transversity function), $f^{\perp}_{1T}$ (Sivers
function), $g_{1T}$ and $h^{\perp}_{1L}$ (“worm-gear” functions),
$h^{\perp}_{1}$ (Boer-Mulders function) and $h^{\perp}_{1T}$ (pretzelosity
function). Only three of them, $f_{1}$, $g_{1L}$, and $h_{1}$, survive after
carrying out the $\vec{k}_{\perp}$ integral, while the remaining five vanish.
Those five TMDs provide novel information about spin-orbit correlations. Just
like the collinear PDFs, extracting TMDs calls for global fits of experimental
data. There has been tremendous progress in our understanding of TMDs and
their extraction from data in the last few years, thanks to new
theoretical/phenomenological ideas and experimental measurements at, e.g,
Belle, COMPASS, HERMES, Jefferson Lab (JLab), and the Relativistic Heavy Ion
Collider (RHIC) at Brookhaven National Lab (BNL). From this perspective, one
of the least known TMDs is $g_{1T}$, which is the focus of this work. This
function has the probabilistic interpretation of finding a longitudinally
polarized quark inside a transversely polarized hadron Tangerman and Mulders
(1995); Kotzinian (1995); Mulders and Tangerman (1996); Boer and Mulders
(1998); Goeke et al. (2005). Since the quark “spins” in one direction while
the hadron “spins” perpendicular to that, it is also known as a “worm-gear”
function.

There does exist some prior information on $g_{1T}$ from various sources.
Quite a few model calculations for the function exist, e.g., in the light-cone
constituent quark model Bacchetta et al. (2000); Ji et al. (2003); Pasquini et
al. (2008); Boffi et al. (2009); Bacchetta et al. (2010), spectator diquark
model Jakob et al. (1997); Gamberg et al. (2008); Bacchetta et al. (2008,
2010), MIT bag model Avakian et al. (2010), and the covariant parton model
Efremov et al. (2009). All of these calculations suggest that the up quark
function $g^{u}_{1T}$ is positive, down quark function $g^{d}_{1T}$ is
negative, and the peak amplitude for $g^{u}_{1T}$ is larger in magnitude than
that of $g^{d}_{1T}$. In addition, the first lattice QCD calculations of TMDs
were presented in Refs. Hagler et al. (2009); Musch et al. (2011); Yoon et al.
(2017), which demonstrated the $k_{\perp}$-moment of
$g_{1T}(x,\vec{k}^{2}_{\perp})$, denoted $g^{(1)}_{1T}(x)$, gives rise to
distortions in the quark densities in the transverse momentum plane. The size
of the distortion was characterized by a quantity called the “transverse
momentum shift” $\langle\rho_{x}\rangle^{q}_{TL}$ for a quark flavor $q$:
$\langle\rho_{x}\rangle^{q}_{TL}\sim\int^{1}_{0}dx\,\big{(}g^{(1)q}_{1T}(x)-g^{(1)\bar{q}}_{1T}(x)\big{)},$
where $g^{(1)\bar{q}}_{1T}(x)$ is the antiquark distribution. This shift was
found to be positive for up quarks and negative for down quarks. Furthermore,
this study determined that the amount of shift was larger for the up quarks.
These lattice results certainly indicate that $g^{u}_{1T}$ and $g^{d}_{1T}$
differ in sign and possibly also have relatively different magnitudes.
Finally, there are two existing theoretical predictions for $g_{1T}$ based on
certain approximations. The large-$N_{c}$ approximation Pobylitsa (2003)
states that $g^{u}_{1T}=-g^{d}_{1T}$ up to $1/N_{c}$ corrections. The
Wandzura-Wilczek (WW)-type approximation Avakian et al. (2008); Accardi et al.
(2009); Kanazawa et al. (2016) neglects quark-gluon-quark correlators to
arrive at $g_{1T}^{(1)}(x)=x\int_{x}^{1}dy\,g_{1}(y)/y$, where $g_{1}$ is the
helicity PDF. The WW-type approximation has been used to make predictions for
the relevant asymmetries in SIDIS Kotzinian et al. (2006); Avakian et al.
(2008); Bastami et al. (2019), single-inclusive (collinear twist-3) reactions
in lepton-nucleon collisions Kang et al. (2011); Kanazawa et al. (2015), and
vector boson production from proton-proton collisions Huang et al. (2016).
Still, to date, there has been no proper extraction of $g_{1T}$ from
experimental data.

In this work, we perform for the first time, using Monte Carlo (MC)
techniques, a global QCD extraction of the TMD $g_{1T}$ from all available
data, namely, SIDIS measurements from HERMES Airapetian et al. (2020), COMPASS
Adolph et al. (2017); Parsamyan (2018); Avakian et al. (2019), and JLab Huang
et al. (2012). We also provide a quantitative comparison of $g_{1T}$ obtained
from experimental data to the large-$N_{c}$ approximation, the WW-type
approximation, and lattice QCD calculations. The work presented in this paper
is an important step towards fully understanding the transverse (partonic)
structure of the nucleon. We organize the manuscript as follows. In Sec. II,
we provide the definition of $g_{1T}$ and discuss the large-$N_{c}$ and the
WW-type approximations. We also provide a brief sketch of how SIDIS acts as a
probe for $g_{1T}$. In Sec. III, we discuss the parameterization for $g_{1T}$
and give an overview of the MC techniques that we use in performing the fit.
In Sec. IV, we present our results and also make the aforementioned
comparisons to other information we have on $g_{1T}$. In Sec. V, we summarize
our work and provide a brief outlook.

##  II Theoretical Background

The TMD $g_{1T}(x,\vec{k}_{\perp}^{2})$, for a quark with momentum $k$, is
defined as the following projection of the quark-quark correlator for a
transversely polarized nucleon with momentum $P$ and spin
$S_{\perp}=(0^{+},0^{-},\vec{S}_{\perp})$ Tangerman and Mulders (1995);
Kotzinian (1995); Mulders and Tangerman (1996); Boer and Mulders (1998); Goeke
et al. (2005),

$\displaystyle\Phi^{[\gamma^{+}\gamma_{5}]}(x,\vec{k}_{\perp},\vec{S}_{\perp})$
$\displaystyle=\frac{1}{2}\int\frac{dz^{-}d^{2}\vec{z}_{\perp}}{(2\pi)^{3}}e^{ik\cdot
z}\langle P,S_{\perp}|\bar{\psi}(0)\gamma^{+}\gamma_{5}\,{\cal
W}(0,z)\psi(z)|P,S_{\perp}\rangle\bigg{|}_{z^{+}=0}$
$\displaystyle=\dfrac{\vec{k}_{\perp}\cdot\vec{S}_{\perp}}{M}g_{1T}(x,\vec{k}^{2}_{\perp})\,,$
(1)

where $x\equiv k^{+}/P^{+}$, $M$ is the nucleon mass, and ${\cal W}(0,z)$ is a
Wilson line connecting the quark fields that ensures color gauge invariance.
The main outcome of this paper is the first global fit of $g_{1T}$ from
experimental data. There is already some information about $g_{1T}$ from
certain approximations based on theoretical analyses:

  * •

The large-$N_{c}$ approximation states that, in the limit of a large number of
colors $N_{c}$, $g_{1T}$ for up and down quarks are related as Pobylitsa
(2003),

$\displaystyle g^{u}_{1T}(x,\vec{k}^{2}_{\perp})$
$\displaystyle\stackrel{{\scriptstyle\text{Large-$N_{c}$}}}{{\approx}}-g^{d}_{1T}(x,\vec{k}^{2}_{\perp})\,,$
(2)

where terms of relative $\mathcal{O}(1/N_{c})$ have been neglected.

  * •

The first $\vec{k}_{\perp}$-moment $g_{1T}^{(1)}(x)$ can be related to an
integral of the helicity PDF $g_{1}(x)$ and a term involving quark-gluon-quark
correlators Avakian et al. (2008); Accardi et al. (2009); Kanazawa et al.
(2016). The Wandzura-Wilczek (WW)-type approximation neglects the latter to
arrive at

$\displaystyle g^{(1)}_{1T}(x)$ $\displaystyle\equiv\int
d^{2}\vec{k}_{\perp}\bigg{(}\dfrac{k^{2}_{\perp}}{2M^{2}}\bigg{)}g_{1T}(x,\vec{k}^{2}_{\perp})\stackrel{{\scriptstyle\text{WW-
type}}}{{\approx}}x\int^{1}_{x}\dfrac{dy}{y}g_{1}(y)\,.$ (3)

In Sec. IV, we will use our global fitting results from experimental data to
test for the first time these theoretical relationships as well the
calculations from lattice QCD Hagler et al. (2009); Musch et al. (2011); Yoon
et al. (2017).

The SIDIS process is an excellent channel to probe the transverse structure of
the nucleon. We can write the reaction as

$\displaystyle\ell(l,\lambda_{l})\,+\,N(P,S)\rightarrow\ell^{\prime}(l^{\prime},\lambda^{\prime}_{l})\,+\,h(P_{h})\,+\,X\,,$
(4)

where, $\ell(l,\lambda_{l})$
($\ell^{\prime}(l^{\prime},\lambda^{\prime}_{l})$) denotes an incoming
(outgoing) lepton, $N(P,S)$ denotes a nucleon, and $h(P_{h})$ denotes a
measured hadron in the final state, with other unobserved particles denoted by
the symbol $X$. The momenta and polarizations of the particles involved are
indicated in parentheses. There are six independent kinematical variables for
this process,

$\displaystyle x_{B}=\dfrac{Q^{2}}{2P\cdot q}\,,\quad
Q^{2}=-(l-l^{\prime})^{2}\,,\quad\phi_{S}\,,\quad z_{h}=\dfrac{P\cdot
P_{h}}{P\cdot q}\,,\quad P_{hT}\,,\quad\phi_{h}\,.$ (5)

In the one-photon exchange approximation, $Q^{2}$ is the virtuality of the
photon. We work at leading order, where $x_{B}$ ($z_{h}$) is equivalent to the
fraction $x$ ($z$) of the incoming nucleon’s (fragmenting quark’s) momentum
carried by the struck quark (produced hadron), so we drop the subscripts for
brevity. In the photon-nucleon center-of-mass frame, the azimuthal angle
$\phi_{S}$ gives the orientation of the (transverse) spin vector of the
nucleon $\vec{S}_{\perp}$ relative to the lepton plane (plane formed by the
incoming and outgoing lepton), $P_{hT}=(0^{+},0^{-},\vec{P}_{hT})$ is the
transverse momentum of the produced hadron, and $\phi_{h}$ gives the
orientation of $\vec{P}_{hT}$ relative to the lepton plane. One can also work
in a reference frame where the hadron has no transverse momentum. In that
case, the virtual photon has transverse momentum $q_{T}$, and one has
$q_{T}=P_{hT}/z$ up to $\mathcal{O}(1/Q^{2})$ corrections. The factorization
of this process in terms of TMDs requires that $q_{T}\ll Q$. One can provide a
model-independent decomposition of the (one-photon exchange) SIDIS cross
section in terms of a certain number of structure functions $F_{XY}$, with $X$
and $Y$ denoting the polarization (unpolarized ($U$), longitudinal ($L$), or
transverse ($T$)) of the incoming lepton and nucleon, respectively. The (six-
fold) differential cross section reads Bacchetta et al. (2007); Bastami et al.
(2019),

$\displaystyle\dfrac{d\sigma}{dx\,dy\,d\phi_{S}\,dz_{h}\,d\phi_{h}\,dP^{2}_{hT}}$
$\displaystyle=\dfrac{\alpha^{2}_{\rm
em}}{x\,y\,Q^{2}}\bigg{\\{}\bigg{(}1-y+\dfrac{1}{2}y^{2}\bigg{)}F_{UU}$
$\displaystyle+\lambda_{l}\,|\vec{S}_{\perp}|\,y\bigg{(}1-\dfrac{1}{2}y\bigg{)}\,{\rm
cos}(\phi_{h}-\phi_{S})F^{{\rm
cos}(\phi_{h}-\phi_{S})}_{LT}+\dots\bigg{\\}}\,,$ (6)

where $\alpha_{\rm em}$ is the fine structure constant, $y=P\cdot q/P\cdot l$,
and $\dots$ denotes terms that are irrelevant for our study. The structure
functions $F_{XY}$ depend on $(x,z,P^{2}_{hT},Q^{2})$, which we did not write
explicitly in the above expressions. One can separately isolate each structure
function since each has its own unique angular modulation associated with it.

In the limit $q_{T}\ll Q$, we can use TMD factorization Collins and Soper
(1981); Collins et al. (1985); Meng et al. (1996); Ji et al. (2004); Collins
(2011) to write the cross section in terms of perturbatively calculable hard
interactions and (non-perturbative) TMDs. The resulting expressions for the
structure functions then involve convolutions of the TMD PDFs and FFs
(generically denoted $f(x,\vec{k}_{\perp}^{2})$ and
$D(z,\vec{P}_{\perp}^{2})$, respectively) Bacchetta et al. (2007); Bastami et
al. (2019),

$\displaystyle C\Big{[}w\,f\,D\Big{]}=x\sum_{q}e^{2}_{q}\int
d^{2}\vec{k}_{\perp}\int
d^{2}\vec{P}_{\perp}\,\delta^{(2)}\big{(}z\vec{k}_{\perp}+\vec{P}_{\perp}-\vec{P}_{hT}\big{)}\,w(\vec{k}_{\perp},\vec{P}_{\perp})\,f^{q}(x,\vec{k}^{2}_{\perp})\,D^{q}(z,\vec{P}^{2}_{\perp})\,,$
(7)

where $w$ is a weight factor depending on the transverse momenta
$\vec{k}_{\perp},\vec{P}_{\perp}$. The structure functions in Eq. (6) read
Bacchetta et al. (2007); Bastami et al. (2019),

$\displaystyle F_{UU}$
$\displaystyle=C\Big{[}w^{\\{0\\}}\,f_{1}(x,\vec{k}^{2}_{\perp})\,D_{1}(z,\vec{P}^{2}_{\perp})\Big{]}\,,$
(8) $\displaystyle F^{{\rm cos}(\phi_{h}-\phi_{S})}_{LT}$
$\displaystyle=C\Big{[}w^{\\{1\\}}\,g_{1T}(x,\vec{k}^{2}_{\perp})\,D_{1}(z,\vec{P}^{2}_{\perp})\Big{]}\,,$
(9)

where $f_{1},D_{1}$ are the unpolarized TMDs, and
$w^{\\{0\\}}=1\,,w^{\\{1\\}}=\vec{P}_{hT}\cdot\vec{k}_{\perp}/(|\vec{P}_{hT}|M)$.
We will work with the leading order (parton model) expression for the cross
section and a Gaussian ansatz for the TMDs,

$\displaystyle g^{q}_{1T}(x,\vec{k}^{2}_{\perp})$
$\displaystyle=g^{(1)q}_{1T}(x)\dfrac{2M^{2}e^{-\tfrac{\vec{k}^{2}_{\perp}}{\langle
k^{2}_{\perp}\rangle|_{g_{1T}^{q}}}}}{\pi\big{(}\langle
k^{2}_{\perp}\rangle|_{g_{1T}^{q}}\big{)}^{2}}\,,\quad
f^{q}_{1}(x,\vec{k}^{2}_{\perp})=f^{q}_{1}(x)\dfrac{e^{-\tfrac{\vec{k}^{2}_{\perp}}{\langle
k^{2}_{\perp}\rangle|_{f_{1}^{q}}}}}{\pi\langle
k^{2}_{\perp}\rangle|_{f_{1}^{q}}}\,,\quad
D^{q}_{1}(z,\vec{P}^{2}_{\perp})=D^{q}_{1}(z)\dfrac{e^{-\tfrac{\vec{P}^{2}_{\perp}}{\langle
P^{2}_{\perp}\rangle_{q}}}}{\pi\langle P^{2}_{\perp}\rangle_{q}}\,,$ (10)

which has been quite successful in describing a wide variety of reactions
Anselmino et al. (2005a); Anselmino et al. (2001); Anselmino et al. (2005b);
Vogelsang and Yuan (2005); Collins et al. (2006, 2005); Anselmino et al.
(2007, 2009); Schweitzer et al. (2010); Qiu et al. (2011); Anselmino et al.
(2013); Signori et al. (2013); Anselmino et al. (2014); Boer and den Dunnen
(2014); D’Alesio et al. (2020); Callos et al. (2020); Cammarota et al. (2020)
and is sufficient to gain the first information on $g_{1T}$ from experimental
data. In Eq. (10), $\langle k^{2}_{\perp}\rangle$ and $\langle
P^{2}_{\perp}\rangle$ are the TMD PDF and the FF widths, which can in
principle be flavor dependent. Also, lattice QCD data are compatible with the
Gaussian shape for the TMDs at small transverse momenta Hagler et al. (2009);
Musch et al. (2011). Using these, we obtain the following expressions for the
structure functions,

$\displaystyle F_{UU}$
$\displaystyle=x\,\sum_{q}\,e^{2}_{q}\,f^{q}_{1}(x)\,D^{q}_{1}(z)\,g^{q}_{f_{1}}\,,$
(11) $\displaystyle F_{LT}$
$\displaystyle=x\,\sum_{q}\,e^{2}_{q}\,g^{(1)q}_{1T}(x)D^{q}_{1}(z)\,(2M)\,(z\,P_{hT})\,\dfrac{g^{q}_{g_{1T}}}{\lambda^{q}_{g_{1T}}}\,,$
(12)

with

$\displaystyle\lambda^{q}_{f}=z^{2}\langle
k^{2}_{\perp}\rangle\big{|}_{f^{q}}\,+\,\langle
P^{2}_{\perp}\rangle\big{|}_{D^{q}_{1}}\,,\qquad
g^{q}_{f}=\dfrac{e^{-P^{2}_{hT}/\lambda^{q}_{f}}}{\pi\lambda^{q}_{f}}\,,$ (13)

where $f=g_{1T}\;{\rm or}\;f_{1}$. These results agree with Refs. Mulders and
Tangerman (1996); Bastami et al. (2019). We use them to fit $g^{(1)}_{1T}(x)$
and $\langle k^{2}_{\perp}\rangle\big{|}_{g_{1T}^{q}}$ to the experimental
data for the asymmetry $A^{{\rm cos}(\phi_{h}-\phi_{S})}_{LT}$, which is
defined as

$\displaystyle A^{{\rm cos}(\phi_{h}-\phi_{S})}_{LT}$
$\displaystyle=\dfrac{F^{{\rm cos}(\phi_{h}-\phi_{S})}_{LT}}{F_{UU}}\,.$ (14)

##  III Overview of the Fitting Methodology

As put forth in Eq. (10), we use a Gaussian function for the
$k_{\perp}$-dependence of $g_{1T}(x,\vec{k}_{\perp}^{2})$, with $\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1T}^{q}}$ a free parameter, while the
$x$-dependence is encoded in its first $k_{\perp}$-moment $g_{1T}^{(1)}(x)$.
We parameterize this latter function, which also depends on $Q^{2}$, as

$\displaystyle g_{1T}^{(1)}(x,Q^{2})$
$\displaystyle=\dfrac{N}{\widetilde{N}}x^{\alpha}(1-x)^{\beta}f_{1}(x,Q^{2})\,,$
(15)

where $N$ is a normalization parameter, $\alpha$ and $\beta$ are parameters
that determine the behavior of $g^{(1)}_{1T}(x)$ as $x\rightarrow 0$ and
$x\rightarrow 1$, respectively, and $f_{1}(x,Q^{2})$ is the unpolarized PDF.
We normalize the function so that, at the initial scale $Q_{0}\equiv 2\,{\rm
GeV}$, $\int^{1}_{0}dx\,x\,g^{(1)}_{1T}(x,Q^{2}_{0})=N.$ That is,
$\widetilde{N}\equiv\int_{0}^{1}dx\,x^{\alpha+1}(1-x)^{\beta}f_{1}(x,Q^{2}_{0}).$
We take the evolution of $g_{1T}^{(1)}(x)$ to be the same as the DGLAP
evolution of $f_{1}(x)$ and use CT10 PDFs Lai et al. (2010) at next-to-leading
order for the latter. Our ansatz for the functional form and evolution of
$g_{1T}(x,\vec{k}_{\perp}^{2})$ is sufficient to work with at the present
stage since the available data is not precise enough, nor spans a large enough
range in $Q^{2}$, to be sensitive to the finer details of TMD evolution or a
more flexible functional form. If data on $A_{LT}$ from vector boson
production in proton-proton collisions becomes available in the future, such
issues will need to be revisited. Note that for simplicity of notation, we
have not shown the explicit flavor dependence of the parameters in Eq. (15).
For the actual fit, we do assign flavor dependence to most of the parameters,
as we discuss below. We also assume
$g^{(1)\bar{u}}_{1T}(x)=g^{(1)\bar{d}}_{1T}(x)=g^{(1)s}_{1T}(x)=g^{(1)\bar{s}}_{1T}(x)=0$
since most of the data included in our fit is in the moderate-$x$ region (see
Fig. 1).

We start with seven free parameters: $\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1T}^{u}}=\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1T}^{d}}\equiv\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1T}}$ and $N^{q}$, $\alpha^{q}$, $\beta^{q}$
for $q=u,d$. However, we find that the presently available data is
insufficient to constrain $\langle k^{2}_{\perp}\rangle\big{|}_{g_{1T}}$,
$\alpha^{d}$, $\beta^{u},\beta^{d}$. Therefore, for the final fit we work with
three free parameters: $N^{u},N^{d}$, and $\alpha^{u}=\alpha^{d}\equiv\alpha$.
As for $\langle k^{2}_{\perp}\rangle\big{|}_{g_{1T}}$, we fix it according to
what is supported by the lattice QCD calculation of Ref. Hagler et al. (2009)
involving the widths of the unpolarized and helicity PDFs ($\langle
k^{2}_{\perp}\rangle\big{|}_{f_{1}}$ and $\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1}}$, respectively),

$\displaystyle\dfrac{\langle k^{2}_{\perp}\rangle\big{|}_{g_{1}}}{\langle
k^{2}_{\perp}\rangle\big{|}_{f_{1}}}\approx\dfrac{{\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1T}}}}{\langle
k^{2}_{\perp}\rangle\big{|}_{f_{1}}}\approx 0.76\,,$ (16)

where we have assumed $\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1}}\approx\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1T}}$. We note the approximation (16) was
first used in Ref. Bastami et al. (2019). In our case, since we take $\langle
k^{2}_{\perp}\rangle\big{|}_{f_{1}}=0.53$ $\rm{GeV}^{2}$ from Ref. Cammarota
et al. (2020), this leads to $\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1T}}=0.40\,\rm{GeV}^{2}$. As for
$\beta^{u},\beta^{d}$, HERMES and JLab do not provide measurements at very
large $x$. On the other hand, COMPASS does, but the errors increase
significantly in that region. Therefore, because of the lack of precise data
at large $x$, we are unable to constrain the $\beta$ parameter and instead fix
it according to the WW-type approximation (3): since
$g_{1}(x)\sim(1-x)^{\beta_{g_{1}}}$ at large $x$, the WW-type approximation
makes it reasonable to assume that $g^{(1)}_{1T}(x)$ falls off as
$(1-x)^{\beta_{g_{1}}+1}$, and because $\beta_{g_{1}}\approx\beta_{f_{1}}$, we
set $\beta^{u}=\beta^{d}\equiv\beta=1$. We have also explored different values
of $\langle k^{2}_{\perp}\rangle\big{|}_{g_{1T}}$ and $\beta$ within the
ranges $0.1\,\textrm{GeV}^{2}<\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1T}}<\langle
k^{2}_{\perp}\rangle\big{|}_{f_{1}}$ and $0<\beta<6$. (Note that for $\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1T}}$, it is important to stay within the
mentioned range in order to be consistent with the positivity bound Bacchetta
et al. (2000),
${k_{\perp}}/{M}\,\big{|}g_{1T}(x,\vec{k}^{2}_{\perp})\big{|}<f_{1}(x,\vec{k}^{2}_{\perp})\,.)$
We checked explicitly that none of the qualitative conclusions of our work
change by choosing different values for $\langle
k^{2}_{\perp}\rangle\big{|}_{g_{1T}}$ and $\beta$. Certainly more precise data
will be required to better constrain the values of these parameters in the
future.

We fit the experimental data and 200 additional replicas of the data, where
each “pseudo-data” point $P_{j}$ in the replica is given by
$P_{j}=D_{j}+R_{j}\cdot e_{j}$, where $D_{j}$ is the actual experimental data
point, $R_{j}$ is a random number generated from a Gaussian distribution
centered at $0$ with standard deviation of $1$, and $e_{j}$ is the quadrature
sum of statistical and systematic experimental errors. The starting parameters
for the fit are chosen from a flat sampling of the parameter space and then
Monte Carlo techniques are used to calculate the functions and observables
from the posterior distributions. Namely, the average value $E[\mathcal{O}]$,
used to create a central curve for a function or observable $\mathcal{O}$, is
determined by $E[\mathcal{O}]=\frac{1}{n}\sum_{i}\mathcal{O}(\vec{p}_{i})$,
where $\vec{p}_{i}=\\{N^{u},N^{d},\alpha\\}_{i}$ is the set of parameters that
minimizes the $\chi^{2}$ for replica $i$, with $n$ being the total number of
replicas. The 1-$\sigma$ confidence level (CL) error band is found from the
standard deviation of $\mathcal{O}$:
$S[\mathcal{O}]=\sqrt{\frac{1}{n}\sum_{i}\left(\mathcal{O}(\vec{p}_{i})-E[\mathcal{O}]\right)^{2}}$.

##  IV Phenomenological Results

Figure 1: The $x$-$Q^{2}$ coverage for the HERMES Airapetian et al. (2020)
(squares), COMPASS Adolph et al. (2017) (triangles), and JLab Huang et al.
(2012) (circles) data sets. Colored (open) points are for data that do (do
not) satisfy the cut of $q_{T}/Q<0.50$.

Before presenting our final fit results, we provide an overview of the
experimental data. A set of 152 data points were reported by the HERMES
Collaboration using a proton target with a pion ($\pi^{\pm}$ or $\pi^{0}$)
detected in the final state Airapetian et al. (2020). While the asymmetries
for the neutral pion were extracted in one-dimensional kinematical bins, the
ones for the charged pions were extracted (for the first time) in three-
dimensional kinematical bins of $(x,z,P_{hT})$. A set of 396 data points were
reported by the COMPASS Collaboration for a proton target with an unidentified
charged hadron ($h^{\pm}$) in the final state Adolph et al. (2017); Parsamyan
(2018); Avakian et al. (2019). These data were extracted in two-dimensional
kinematical bins of $(x,Q^{2})$, $(z,Q^{2})$ and $(P_{hT},Q^{2})$.
Furthermore, these data were sub-divided into three $z$ intervals: $z>0.1$,
$0.1<z<0.2$, and $z>0.2$. Here we choose to work with data from the $z$
intervals $0.1<z<0.2$ and $z>0.2$ since we found that they allow us to better
constrain $g_{1T}$ than the $z>0.1$ data set alone. A set of eight data points
were reported by Jefferson Lab Hall A for a neutron target, and the measured
hadrons were $\pi^{\pm}$ Huang et al. (2012). For the neutron, we apply
isospin symmetry on $g_{1T}$ and $f_{1}$ for the up and down quark flavors
($u\leftrightarrow d,\bar{u}\leftrightarrow\bar{d}$), while leaving strange
quarks ($s,\bar{s}$) unchanged.

In order to ensure the applicability of TMD factorization, one needs $q_{T}\ll
Q$. In the literature, cuts $q_{T}/Q\lesssim 0.25$ have been used recently
Scimemi and Vladimirov (2018, 2020); Bacchetta et al. (2020); Bury et al.
(2021). In our case, with such a cut we lose a significant number of data
points and are unable to sufficiently constrain all the parameters in our fit.
Therefore, we apply a cut of $q_{T}/Q<0.50$ to balance the condition for TMD
factorization with the need to have enough data points to meaningfully extract
$g_{1T}$. (This is similar to the approach in Ref. Echevarria et al. (2021),
where a cut of $q_{T}/Q<0.75$ was used for a fit of the Sivers function.)
After applying this cut, we are left with 60 data points from HERMES, 64 data
points from COMPASS, and 4 data points from JLab. In Fig. 1, we show the
distribution in $x$ and $Q^{2}$ of the data points along with which ones
survive our cut.

As mentioned above, for COMPASS the measured hadrons are unidentified charged
hadrons, and we make the approximation
$D_{1}^{h^{\pm}}=D_{1}^{\pi^{\pm}}+D_{1}^{K^{\pm}}$. (All FFs are taken from
DSS de Florian et al. (2007) at leading order.) We assign favored widths
$\langle P^{2}_{\perp}\rangle\big{|}_{\rm{fav}}$ (unfavored widths $\langle
P^{2}_{\perp}\rangle\big{|}_{\rm{unfav}}$) to $u$ and $\bar{d}$
($\bar{u},d,s$, and $\bar{s}$) for asymmetries associated with $\pi^{+}$
production and employ charge conjugation for the $\pi^{-}$ FF; for $\pi^{0}$,
we use $D_{1}^{\pi^{0}}=(D_{1}^{\pi^{+}}+D_{1}^{\pi^{-}})/2$. The explicit
values for the widths of the FFs, which we take from Ref. Cammarota et al.
(2020), are $\langle P^{2}_{\perp}\rangle\big{|}_{\rm{fav}}=0.124\,{\rm
GeV}^{2}$ and $\langle P^{2}_{\perp}\rangle\big{|}_{\rm{unfav}}=0.145\,{\rm
GeV}^{2}$. We mention that HERMES does have data for
$A_{LT}^{\cos(\phi_{h}-\phi_{S})}$ for $K^{\pm}$. However, since we are
focused on extracting $g_{1T}$ for up and down quarks, and have set antiquarks
and strange quarks to zero, we do not consider this data in our analysis.

###  IV.1 Main results: weighted $\chi^{2}$ method

Figure 2: HERMES data for $\pi^{0}$ compared with our theory curves at
$1$-$\sigma$ CL. The dark (open) points are data that were (were not) included
in our fit after the $q_{T}/Q<0.5$ cut.

In this section, we present our final results for $g^{(1)}_{1T}(x)$ extracted
simultaneously from HERMES, COMPASS, and JLab data in the so-called “weighted
$\chi^{2}$ method”. The reason we utilize this approach, which we describe
below in more detail, is that JLab has very few data points compared to HERMES
and COMPASS. Therefore, in the process of fitting with the usual definition of
$\chi^{2}$,

$\chi^{2}=\sum_{i}\left(\frac{T_{i}-D_{i}}{e_{i}}\right)^{\\!\\!2}\,,$ (17)

where $T_{i}$ is the theory value for an experimental data point $D_{i}$ with
error $e_{i}$, our description of the JLab data, specifically for $\pi^{-}$
(see Table 2 in Sec. IV.4), was not as good as the charged pion data sets from
HERMES and COMPASS. (Observing such a feature probably is not surprising. Due
to the larger number of data points from HERMES and COMPASS, the fit tries to
find solutions to accommodate these data sets better and compromises on the
quality of the fit to the JLab data where the number of data points is much
less, and, consequently, so is their contribution to the overall $\chi^{2}$.)
In an attempt to achieve an equally good description of all the experimental
data sets, we weight the $\chi^{2}$ so as to emphasize the JLab data as much
as the data from HERMES and COMPASS. This approach was recently used in Ref.
Echevarria et al. (2021) for a fit of the Sivers function. Following Ref.
Echevarria et al. (2021), we define a weighted $\chi^{2}$ ($\chi^{2}_{w}$) as

$\displaystyle\chi^{2}_{w}$
$\displaystyle=\chi^{2}\big{|}_{\rm{H+C}}+w\,\chi^{2}\big{|}_{\rm J}\,,$ (18)

where $\chi^{2}|_{\rm{H+C}}$ ($\chi^{2}|_{\rm J}$) is the $\chi^{2}$ for the
HERMES and COMPASS (JLab) data points using the definition (17). In Eq. (18),
we choose the weight factor $w$ as $w\equiv N_{\rm H+C}/N_{\rm J}=124/4$,
where $N_{\rm H+C}$ ($N_{\rm J}$) is the total number of points from HERMES
and COMPASS (JLab). We must also take

$\displaystyle N_{\rm pts.}$ $\displaystyle=N_{\rm H+C}+w\,N_{\rm J}\,,$ (19)

where $N_{\rm pts.}$ is the effective number of data points in the weighted
$\chi^{2}$ method. In Sec. IV.4, we provide a comparison of our final fit
results in the weighted $\chi^{2}$ method to the fit results obtained with the
usual definition of $\chi^{2}$ (17). (Such a scenario corresponds to simply
choosing $w=1$.)

In Figs. 2–9 we plot our theoretical curves against the experimental data,
with Figs. 2, 3, and 4 displaying the HERMES data for $\pi^{0}$, $\pi^{+}$,
and $\pi^{-}$, Figs. 5, 6 (7, 8) giving the COMPASS data for the $z$ interval
$0.1<z<0.2$ ($z>0.2$) for $h^{+}$ and $h^{-}$, and Fig. 9 showing the JLab
data for $\pi^{+}$ and $\pi^{-}$. In Table 1, we summarize
$\chi_{w}^{2}/N_{\rm pts.}$ for each data set along with the global
$\chi^{2}_{w}/N_{\rm pts.}$. For HERMES, the $\chi^{2}_{w}/N_{\rm pts.}=1.20$
for $\pi^{+}$ and $\chi^{2}_{w}/N_{\rm pts.}=0.88$ for $\pi^{-}$,

Figure 3: HERMES data for $\pi^{+}$ compared with our theory curves at
$1$-$\sigma$ CL. The dark (open) points are data that were (were not) included
in our fit after the $q_{T}/Q<0.5$ cut. Figure 4: HERMES data for $\pi^{-}$
compared with our theory curves at $1$-$\sigma$ CL. The dark (open) points are
data that were (were not) included in our fit after the $q_{T}/Q<0.5$ cut.
Figure 5: COMPASS data for $h^{+}$ for the bin $0.1<z<0.2$ compared with our
theory curves at $1$-$\sigma$ CL. The dark (open) points are data that were
(were not) included in our fit after the $q_{T}/Q<0.5$ cut. Figure 6: COMPASS
data for $h^{-}$ for the bin $0.1<z<0.2$ compared with our theory curves at
$1$-$\sigma$ CL. The dark (open) points are data that were (were not) included
in our fit after the $q_{T}/Q<0.5$ cut. Figure 7: COMPASS data for $h^{+}$ for
the bin $z>0.2$ compared with our theory curves at $1$-$\sigma$ CL. The dark
(open) points are data that were (were not) included in our fit after the
$q_{T}/Q<0.5$ cut. Figure 8: COMPASS data for $h^{-}$ for the bin $z>0.2$
compared with our theory curves at $1$-$\sigma$ CL. The dark (open) points are
data that were (were not) included in our fit after the $q_{T}/Q<0.5$ cut.

Figure 9: JLab data for $\pi^{+}$ (left) and $\pi^{-}$ (right) compared with
our theory curves at $1$-$\sigma$ CL. The dark (open) points are data that
were (were not) included in our fit after the $q_{T}/Q<0.5$ cut.

thus

indicating good agreement between our theory and the data. On the other hand,
$\chi^{2}_{w}/N_{\rm pts.}=1.94$ for $\pi^{0}$, thus implying that the
agreement with our theory is just fair for this data set. The reason for the
larger $\chi^{2}_{w}/N_{\rm pts.}$ most likely is the few points that deviate
from the overall trend of the data. For COMPASS, $\chi^{2}_{w}/N_{\rm
pts.}=0.97$ for $h^{+}$ and $\chi^{2}_{w}/N_{\rm pts.}=0.71$ for $h^{-}$ data.
For JLab, $\chi^{2}_{w}/N_{\rm pts.}=0.31$ for $\pi^{+}$ data and
$\chi^{2}_{w}/N_{\rm pts.}=1.13$ for $\pi^{-}$. These values suggest strong
compatibility between our theory and the data.

Figure 10: Main global fit results for $xg_{1T}^{(1)}(x)$ at $Q^{2}$ = 4 ${\rm
GeV}^{2}$ for up quarks (left) and down quarks (right) obtained in the
weighted $\chi^{2}$ method.

In Fig. 10 we show our final results for $x\,g^{(1)}_{1T}(x)$ for the up
quarks (left panel) and the down quarks (right panel) at $Q^{2}=4\,\rm
GeV^{2}$. We emphasize that this is the very first information on
$g_{1T}(x,\vec{k}_{\perp}^{2})$ from experimental data, which we have obtained
from an analysis of the world SIDIS data on
$A_{LT}^{\cos(\phi_{h}-\phi_{S})}$. Figure 10 indicates that $g_{1T}$ for the
up quark is positive and for the down quark is negative, although with large
error bands for the latter. This is most likely because, even though JLab does
have neutron data, the $\pi^{+}$ errors are larger compared to those for
$\pi^{-}$ (see Fig. 9), so one cannot achieve as precise an extraction for the
down quark $g_{1T}$. Additional data from JLab on a neutron target, or COMPASS
on a deuteron target, would be needed to obtain a better flavor separation.
Nonetheless, the first prominent qualitative feature that we observe here is
that our results are compatible (to some extent) with the large-$N_{c}$
approximation (2), which implies that $g_{1T}^{u}$ and $g_{1T}^{d}$ have
relative signs. Recall that such was the conclusion also from lattice QCD as
well as calculations in constituent quark models. Note that we mention “to
some extent” because of the relative sizes of the two distributions. The
values of the parameters from our fit are $N^{u}=0.026\pm 0.007$,
$N^{d}=-0.012\pm 0.010$, and $\alpha=1.9\pm 0.6$, so the magnitude of
$g_{1T}^{u}$ does slightly overlap with that for $g_{1T}^{d}$. We will present
results for a fit that imposes the large-$N_{c}$ constraint
$g_{1T}^{u}=-g_{1T}^{d}$ in the next subsection.

###  IV.2 Large-$N_{c}$ and WW-type approximation results

Figure 11: Comparison of our main extraction of $xg_{1T}^{(1)}(x)$ at $Q^{2}$
= 4 ${\rm GeV}^{2}$ for up quarks (left) and down quarks (right) with the
results obtained by imposing the large-$N_{c}$ approximation on the fit. Both
functions were obtained using the weighted $\chi^{2}$ method.

In this subsection, we study how well the experimental data support the
large-$N_{c}$ and WW-type approximations for $g_{1T}$ (see Sec. II) by
separately imposing these constraints and analyzing the $\chi^{2}_{w}/N_{\rm
pts.}$ for each case.

To perform the fit in the large-$N_{c}$ approximation, we set
$g^{(1)d}_{1T}(x)=-g^{(1)u}_{1T}(x)$. Therefore, this analysis involves
fitting two free parameters, $N^{u}$ and $\alpha$, and we also carry this out
using the weighted $\chi^{2}$ method. In Fig. 11, we provide a comparison of
our final fit from the previous subsection with the large-$N_{c}$ results. We
find that the large-$N_{c}$ curves and our final results overlap rather well
within error bands. In Table 1, we also compare the $\chi^{2}_{w}/N_{\rm
pts.}$ for the two cases. We observe that our main fit gives a better global
$\chi^{2}_{w}/N_{\rm pts.}$ than the large-$N_{c}$ fit. However, one must keep
in mind that these $\chi^{2}_{w}/N_{\rm pts.}$ values correspond to the
central curves of the theory. Since every replica has its own
$\chi^{2}_{w}/N_{\rm pts.}$ value, it is actually more informative to make a
comparison between the $\chi^{2}_{w}/N_{\rm pts.}$-distribution for our main
fit and the large-$N_{c}$ fit to determine which of the two approaches
produces better results (if at all). In Fig. 13, we show histograms of
$\chi^{2}_{w}/N_{\rm pts.}$ for each data set for the two cases. We observe a
significant overlap of the $\chi^{2}_{w}/N_{\rm pts.}$-distributions,
especially for (but not limited to) the COMPASS data sets for $h^{\pm}$ and
for the JLab data set for $\pi^{+}$. This implies that, although there is a
slight preference for $g_{1T}$ to violate the large-$N_{c}$ approximation,
there is actually no statistically significant difference between the two fits
(main and large-$N_{c}$). That is, the large-$N_{c}$ approximation is
consistent with the experimental data.

In Fig. 12, we provide a comparison of our main fit with a calculation of
$g_{1T}^{(1)}(x)$ using the WW-type approximation (3) with $g_{1}(x)$ taken
from NNPDF Nocera et al. (2014), JAM Sato et al. (2016), and DSSV De Florian
et al. (2019). Overall, we observe a qualitative agreement with the WW-type
approximation in terms of the general behavior of $g_{1T}$. However, there are
slight differences that may hint at a violation of the WW-type approximation
for the up quark. As for the down quark, there seems to be an agreement
between our main fit and the WW-type approximation, admittedly because of the
presence of large uncertainties in our extraction. More precise data will
certainly be needed to affirm the degree of violation (if any) of the WW-type
approximation. This is especially important to resolve since a clear breaking
of the WW-type approximation would be a direct signal of quark-gluon-quark
correlations in the nucleon Avakian et al. (2008); Accardi et al. (2009);
Kanazawa et al. (2016). In Table 1, we compare the $\chi^{2}_{w}/N_{\rm pts.}$
for our main fit with that from calculating $A_{LT}^{\cos(\phi_{h}-\phi_{S})}$
in the WW-type approximation. (We note that the WW-type approximation results
are not from a fit since $g_{1T}^{(1)}(x)$ is fixed by using Eq. (3) with
$g_{1}(x)$ taken from NNPDF, JAM, or DSSV, and the width $\langle
k_{\perp}^{2}\rangle|_{g_{1T}}$ is fixed to be the same as the one used in our
main fit.) We not only obtain the same or somewhat better $\chi^{2}$ for each
of the data sets as the WW-type approximation, but the global
$\chi^{2}_{w}/N_{\rm pts.}$ for our main fit is better than that for the WW-
type approximation. However, as before, this does not imply that our fit is
favored over the WW-type approximation. As shown in Fig. 13, the statistical
spread of the $\chi^{2}_{w}/N_{\rm pts.}$-distribution for our fit results are
rather large and hence overlaps significantly with the $\chi^{2}_{w}/N_{\rm
pts.}$-distribution from the WW-type approximation. This confirms that at
present the WW-type approximation is compatible with the experimental data.

Figure 12: Comparison of our main extraction of $xg_{1T}^{(1)}(x)$ at $Q^{2}$
= 4 ${\rm GeV}^{2}$ for up quarks (left) and down quarks (right) with the
results obtained from the calculation of $xg^{\rm WW}_{1T}(x)$ using Eq. (3)
with $g_{1}(x)$ taken from NNPDF Nocera et al. (2014), JAM Sato et al. (2016),
and DSSV De Florian et al. (2019). Summary of $\bm{\chi^{2}_{w}/N_{\rm pts.}}$
---
Data set | $\chi^{2}_{w}/N_{\rm pts.}\big{|}_{\rm Main}$ | $\chi^{2}_{w}/N_{\rm pts.}\big{|}_{{\rm Large}{\text{-}}N_{c}}$ |  $\chi^{2}_{w}/N_{\rm pts.}\big{|}_{\rm NNPDF}$ |  $\chi^{2}_{w}/N_{\rm pts.}\big{|}_{\rm JAM}$ |  $\chi^{2}_{w}/N_{\rm pts.}\big{|}_{\rm DSSV}$
$\rm{HERMES}\;\pi^{+}$ | 1.20 | 1.23 | 1.19 | 1.19 | 1.19
$\rm{HERMES}\;\pi^{-}$ | 0.88 | 0.88 | 0.85 | 0.85 | 0.85
$\rm{HERMES}\;\pi^{0}$ | 1.94 | 2.01 | 1.98 | 1.95 | 1.96
${\rm COMPASS}\;h^{+}$ | 0.97 | 0.51 | 0.71 | 1.02 | 0.89
${\rm COMPASS}\;h^{-}$ | 0.71 | 0.53 | 0.71 | 0.81 | 0.80
$\rm{JLab}\;\pi^{+}$ | 0.31 | 0.06 | 0.81 | 0.78 | 0.96
$\rm{JLab}\;\pi^{-}$ | 1.13 | 2.23 | 1.15 | 0.93 | 0.93
Global | 0.86 | 0.99 | 0.95 | 0.94 | 0.97
Table 1: Summary of the $\chi^{2}_{w}/N_{\rm pts.}$ for our main fit and for
the large-$N_{c}$ and the WW-type approximation results in the weighted
$\chi^{2}$ method. For the latter, we separately give results using $g_{1}(x)$
from NNPDF, JAM, and DSSV. Figure 13: Distribution of $\chi^{2}_{w}/N_{\rm
pts.}$ for our main fit results with those obtained in the large-$N_{c}$ and
the WW-type approximations. For the latter, we have shown as the
representative case results using NNPDF. The results do not qualitatively
change with using JAM or DSSV.

###  IV.3 Comparison with lattice QCD

In this section, we explore the so-called worm-gear shift $\langle
k_{x}\rangle_{TL}$ for $g_{1T}$,

$\displaystyle\big{[}\langle k_{x}\rangle_{TL}\big{]}\\!(Q^{2})$
$\displaystyle\equiv
M\,\dfrac{\int^{1}_{0}dx\,\Big{[}g^{(1)u}_{1T}(x,Q^{2})-g^{(1)d}_{1T}(x,Q^{2})\Big{]}}{\int^{1}_{0}dx\,\Big{[}f^{u}_{1}(x,Q^{2})-f^{d}_{1}(x,Q^{2})\Big{]}}\,.$
(20)

The motivation for calculating this quantity is that it has also been computed
using lattice QCD Hagler et al. (2009); Musch et al. (2011); Yoon et al.
(2017). The results are shown in Fig. 14 for our main fit, the WW-type
approximation, and a lattice point from Ref. Yoon et al. (2017) (namely, the
rightmost data point from Fig. 13 of Ref. Yoon et al. (2017) in the domain-
wall fermion (DWF) scheme). We find consistency between lattice and our main
fit as well as the WW-type approximation and our main fit, but a slight
discrepancy between lattice and the WW-type approximation, again perhaps
hinting at a breaking of the latter. We also note that from the large-$N_{c}$
fit (Fig. 11), $\langle k_{x}\rangle_{TL}=0.14\pm 0.05$, similar to that of
our main fit. We mention some caveats in making a direct comparison between
phenomenology and lattice for the worm-gear shift. The scale for the main fit
and WW-type approximation points is $Q^{2}=4\,{\rm GeV^{2}}$. For lattice,
there is not a definitive scale, but it can be approximated by the inverse
lattice spacing, from which one finds $Q^{2}\approx 5.52\,{\rm GeV^{2}}$ from
the spacing $a=0.084\,{\rm fm}$ used in Ref. Yoon et al. (2017) for the DWF
scheme. In addition, the full correspondence between the theoretical and
lattice calculations of $\langle k_{x}\rangle_{TL}$ occurs in the
$\hat{\zeta}\to\infty,b_{T}\to 0$ limit of the lattice result, where
$\hat{\zeta}$ is a Collins-Soper type parameter and $b_{T}$ is the separation
between the quark fields Yoon et al. (2017). The specific lattice point we
compare with in our Fig. 14 has $\hat{\zeta}\approx 0.42$ and
$b_{T}=0.34\,{\rm fm}$, which were the closest values used in Ref. Yoon et al.
(2017) to the aforementioned limit. Nevertheless, the dependence of $\langle
k_{x}\rangle_{TL}$ on these lattice parameters seems very mild Yoon et al.
(2017). It is encouraging at this stage that experimental data and lattice are
in reasonable agreement.

Figure 14: Comparison of the worm-gear shift $\langle k_{x}\rangle_{TL}$
calculated using our main fit (blue circle) with those obtained in the WW-type
approximation (red squares) as well as lattice QCD (green triangle). The
latter is the rightmost data point from Fig. 13 of Ref. Yoon et al. (2017) in
the domain-wall fermion (DWF) scheme. The phenomenological results are at
$Q^{2}$ = 4 ${\rm GeV}^{2}$.

###  IV.4 Results in the unweighted $\chi^{2}$ method

Figure 15: Global fit results for $xg_{1T}^{(1)}(x)$ at $Q^{2}$ = 4 ${\rm
GeV}^{2}$ for up quarks (left) and down quarks (right) obtained in the
unweighted $\chi^{2}$ method (light brown) compared with our main fit (light
blue).

In this section, we discuss what happens to $g_{1T}$ if we do not weight the
JLab data at all, which corresponds to $w=1$ in Eq. (18). In Fig. 15, we
compare our main fit results to the ones with $w=1$. We see that the impact is
significantly larger on the up quark, while for the down quark the two cases
overlap rather well within error bands. We also notice that the JLab data for
$\pi^{-}$ is not described particularly well if we do not weight the
$\chi^{2}$ (cf. the relevant entry on the second column of Table 2 with that
of Table 1). Since JLab data are for neutrons, the $\pi^{-}$ channel, which
has somewhat precise data, has the most impact on the proton $g_{1T}$ for an
up quark (which is the function plotted in the left panel of Fig. 15). We
therefore conclude that what we currently have in Fig. 10 is the best fit that
one can provide at this stage, certainly because of the strong compatibility
that we obtain simultaneously with all the data sets. (In the Appendix we
compare the fit results obtained in the unweighted $\chi^{2}$ method with the
results obtained in the (unweighted) large-$N_{c}$ fit and the WW-type
approximation.)

Summary of $\bm{\chi^{2}/N_{\rm pts.}}$
---
Data set | $\chi^{2}/N_{\rm pts.}\big{|}_{\rm Main}$ | $\chi^{2}_{/}N_{\rm pts.}\big{|}_{{\rm Large}{\text{-}}N_{c}}$ |  $\chi^{2}/N_{\rm pts.}\big{|}_{\rm NNPDF}$ |  $\chi^{2}/N_{\rm pts.}\big{|}_{\rm JAM}$ |  $\chi^{2}/N_{\rm pts.}\big{|}_{\rm DSSV}$
$\rm{HERMES}\;\pi^{+}$ | 1.24 | 1.23 | 1.19 | 1.19 | 1.19
$\rm{HERMES}\;\pi^{-}$ | 0.89 | 0.88 | 0.85 | 0.85 | 0.85
$\rm{HERMES}\;\pi^{0}$ | 2.03 | 2.03 | 1.98 | 1.95 | 1.96
${\rm COMPASS}\;h^{+}$ | 0.39 | 0.40 | 0.71 | 1.02 | 0.89
${\rm COMPASS}\;h^{-}$ | 0.54 | 0.53 | 0.71 | 0.81 | 0.80
$\rm{JLab}\;\pi^{+}$ | 0.42 | 0.15 | 0.81 | 0.78 | 0.96
$\rm{JLab}\;\pi^{-}$ | 1.88 | 2.23 | 1.15 | 0.93 | 0.93
Global | 0.83 | 0.83 | 0.93 | 1.02 | 0.99
Table 2: Same as Table 1 but for the unweighted $\chi^{2}$ method.

##  V Summary and outlook

In this work, we have performed the first global extraction of the TMD
$g_{1T}(x,\vec{k}^{2}_{\perp})$ using all experimental measurements available,
namely data from HERMES, COMPASS, and JLab on the SIDIS asymmetry
$A_{LT}^{\cos(\phi_{h}-\phi_{S})}$. We have used a so-called weighted
$\chi^{2}$ method in order to allow the JLab data, which has few data points,
to also contribute on the same footing as the HERMES and COMPASS data sets.
Our main fit indicates that the up quark $g_{1T}$ is positive and the down
quark is negative, with the up quark $g_{1T}$ being somewhat larger in
magnitude than the down quark. Such a feature is qualitatively compatible with
the large-$N_{c}$ approximation and several model calculations. Actually, the
current experimental data cannot rule out the strict large-$N_{c}$
approximation, namely that $g_{1T}^{u}=-g_{1T}^{d}$, as we confirmed by
conducting a fit imposing this constraint. Furthermore, we have provided for
the first time a quantitative comparison of $g_{1T}$ from the experimental
data with the WW-type approximation and a lattice QCD calculation. Our final
fit yields a value for the so-called worm-gear shift that is compatible with
lattice QCD. The agreement between experiment and lattice is encouraging and
motivates continued comparisons in the future. Moreoever, our results give a
slight indication of a breaking of the WW-type approximation, even though the
approximation is still compatible with experimental data. More precise data
will be needed to reliably determine how much (if at all) $g_{1T}$ violates
the WW-type approximation. Since such a breaking is directly connected to
quark-gluon-quark correlations in the nucleon, it is of great interest to
rigorously address this question. This includes not only further theoretical
improvements but also additional measurements of $A_{LT}$ from JLab and
COMPASS in SIDIS as well as complimentary measurements at RHIC of vector boson
production, which will allow one to study evolution effects in more detail.

###### Acknowledgements.

We thank B. Parsamyan and G. Schnell for providing us with the COMPASS and
HERMES data that was used in our fit. We thank D. Callos for useful
discussions. We are also grateful to M. Engelhardt for providing us with the
results for the worm-gear shift from Ref. Yoon et al. (2017) and for fruitful
discussions about the lattice results for $g_{1T}$. D.P. thanks R. Sassot for
providing the tables from Ref. De Florian et al. (2019) and for help with the
authors’ Fortran code. S.B. was supported by the U.S. Department of Energy
under Contract No. DE-SC0012704. This work has been supported by the National
Science Foundation under grants No. PHY-2110472 (S.B., A.M. and G.P.), No.
PHY-1945471 (Z.K.), and No. PHY-2011763 (D.P.). This work has also been
supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, within the framework of the TMD Topical Collaboration.

*

##  Appendix A Additional plots from fits in the unweighted $\chi^{2}$ method

In this Appendix, we compare the fit results obtained in the unweighted
$\chi^{2}$ method (Fig. 15) with the large-$N_{c}$ and the WW-type
approximations. We notice that the large-$N_{c}$ results overlap rather well
with these fit results. (Here the large-$N_{c}$ fit has been carried out using
the unweighted $\chi^{2}$.) Furthermore, the results for the WW-type
approximation are very interesting. There seems to be a consistent hint for a
slight violation of the WW-type approximation for the up quark in the
moderate-to-small-$x$ region, both in the weighted and unweighted $\chi^{2}$
approaches. However, the differences in the large-$x$ region seems to be
relatively stronger in the weighted $\chi^{2}$ set up. One can speculate that
such a feature is driven by the JLab data. The qualitative conclusion for the
down quark remains unchanged from the main fit.

Figure 16: Comparison of our main extraction of $xg_{1T}^{(1)}(x)$ at $Q^{2}$
= 4 ${\rm GeV}^{2}$ for up quarks (left) and down quarks (right) with the
results obtained by imposing the large-$N_{c}$ approximation on the fit. Both
functions were obtained using the unweighted $\chi^{2}$ method. Figure 17:
Comparison of our unweighted $\chi^{2}$ extraction of $xg_{1T}^{(1)}(x)$ at
$Q^{2}$ = 4 ${\rm GeV}^{2}$ for up quarks (left) and down quarks (right) with
the results obtained from the calculation of $xg^{\rm WW}_{1T}(x)$ using Eq.
(3) with $g_{1}(x)$ taken from NNPDF Nocera et al. (2014), JAM Sato et al.
(2016), and DSSV De Florian et al. (2019).

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