# Wavelength-Dependent Extinction and Grain Sizes in “Dippers”

Michael L. Sitko Center for Exoplanetary Systems, Space Science Institute,
4750 Walnut Street, Suite 205, Boulder, CO 80301, USA  Ray W. Russell The
Aerospace Corporation, Los Angeles, CA, USA  Zachary C. Long, Korash Assani,
Monika Pikhartova  Department of Physics, University of Cincinnati,
Cincinnati, OH 45221-0011, USA  Ammar Bayyari  Department of Physics,
University of Cincinnati, Cincinnati, OH 45221-0011, USA  Department of
Physics and Astronomy, University of Hawaii, 2505 Correa Road Watanabe 416.
Honolulu, Hawaii 96822 USA  Carol A. Grady  Eureka Scientific, Inc., 2452
Delmer Street, Suite 100, Oakland, CA 496002, USA  Carey M. Lisse JHU-APL,
11100 Johns Hopkins Road, Laurel, MD 20723, USA  Massimo Marengo Department of
Physics and Astronomy, Iowa State University, Ames, IA 50010, USA  John P.
Wisniewski Department of Physics and Astronomy, George Mason University, 4400
University Drive, Fairfax, VA 22030-4444, USA  William Danchi NASA Goddard
Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771-2400, USA

###### Abstract

We have examined inter-night variability of K2-discovered “Dippers” that are
not close to being viewed edge-on, as determined from previously-reported ALMA
images, using the SpeX spectrograph on NASA’s Infrared Telescope facility
(IRTF). The three objects observed were EPIC 203850058, EPIC 205151387, and
EPIC 204638512 ( = 2MASS J16042165-2130284). Using the ratio of the fluxes
between two successive nights, we find that for EPIC 204638512 and EPIC
205151387, we find that the properties of the dust differ from that seen in
the diffuse interstellar medium and denser molecular clouds. However, the
grain properties needed to explain the extinction does resemble those used to
model the disks of many young stellar objects. The wavelength-dependent
extinction models of both EPIC 204638512 and EPIC 205151387 includes grains at
least 500 microns in size, but lacks grains smaller than 0.25 microns. The
change in extinction during the dips, and the timescale for these variations
to occur, imply obscuration by the surface layers of the inner disks. The
recent discovery of a highly mis-inclined inner disk in EPIC 204638512 is
suggests that the variations in this disk system may point to due to rapid
changes in obscuration by the surface layers of its inner disk, and that other
“face-on” Dippers might have similar geometries. The He I line at 1.083
microns in EPIC 205151387 and EPIC 20463851 were seen to change from night to
night, suggesting that we are seeing He I gas mixed in with the surface dust.

Protoplanetary disks (1300); Circumstellar disks (235); Circumstellar grains
(239); Circumstellar gas (238)

††journal: AAS Journals††software: HOCHUNK3D (Whitney et al. 2013, ApJS, 207,
30), Spextool(Vacca et al. 2003, PASP, 115, 389; Cushing et al. 2004, PASP,
116, 362)

##  1 Introduction

The formation of planets in young protoplanetary disk systems by means of core
accretion requires that the dust within these systems undergo growth and
likely settling of the largest grains toward the disk midplane. The degree of
such grain growth, however, is often difficult to assess observationally. At
visible- to mid-infrared wavelengths, the disks may be optically thick, and
observations only give clues about the surface layers. Both the scattered
light and thermal emission are subject to the effects of radiative transfer,
and their interpretation will be model-dependent. At millimeter/submillimeter
wavelengths, the spectral slope provide clues as to grain sizes, but optical
thickness and grain size distributions will not be constant throughout the
planet-forming regions of the disk. In particular, gas/dust mass ratios
derived from such grains seem to unusually low (by factors of up to 30 times
compared to expected gas/dust ratios) (Birnstiel et al., 2010; Williams &
Best, 2014; Kama et al., 2016; Long, et al., 2017). Recent models of the
depletion of CO in young disk systems suggests that gaseous CO will condense
into ices, and coat more refractory grains, and the extent of the depletion
depends on radial distance and age of the system (McClure et al., 2016; Powell
et al., 2022). Mid-IR interferometry near 10 $\mu$m indicates that grains are
often larger near the star than further out (van Boekel et al., 2004; Lopez,
et al., 2022).

Occultations of the stellar light by dust along the line of sight, however,
can provide a more direct measure of the wavelength-dependent extinction of
the dust in these systems at visible and near-IR wavelengths. Many young stars
undergo variations in brightness that are most easily explained by dust
extinction in the upper layers of disks that are viewed at high inclinations.
Such “UX Orionis” stars, or “UXors” have been known for some time (Grinin et
al., 1991, 1996; Eaton & Herbst, 1995). In the case of disks with puffed-up
inner rims (Natta et al., 2001; Dullemond & Dominik, 2004) the occulting dust
would likely be located at the inner disk rim of highly self-shadowed disks
(Dullemond et al., 2003). The color changes in these systems have been studied
at visible wavelengths, and as expected for submicron-sized grains, the
extinction is accompanied by reddening, until the dust becomes optically
thick, after which ”bluing” may actually occur due to the remaining stellar
light reaching the observer relatively unaffected. This occurs when the disk
is very optically thick, but scattering at the poles reaches the observer
(Grinin et al., 1991).

The Spitzer Infrared Array Camera (IRAC) and the Convection, Rotation, and
Planetary Transit (COROT) satellite provided long time baseline photometric
observations of large numbers of Classical T Tauri stars (CTTs) in young
stellar clusters. Among these were stars exhibiting large irregular changes in
brightness over time scales of days or even hours. These “Dippers” became
important targets for the K2 mission, and Ansdell et al. (2016a) reported
finding many in the Upper Sco and $\rho$ Oph stellar associations. Similar
“dipper” activity was observed in stars in the young star-formation region NGC
2264 (Stauffer et al., 2016). While disks with near edge-on orientations would
would be most likely capable of occulting the light of the star, Ansdell et
al. (2016b) presented ALMA continuum observations of three Dippers, EPIC
203850058, EPIC 205151387, and EPIC 204638512, whose geometries were far from
edge-on. In the case of EPIC 204638512, the disk is nearly face-on ($\sim 6\pm
1.5^{\circ}$.), with a large dust cavity, yet it exhibits brightness
fluctuations of nearly 1 magnitude over a time span of a day. This suggests
that if the brightness changes were due to dust, a UXor-type geometry was
unlikely.

However, recent imaging data for EPIC 204638512 ALMA (Mayama et al., 2018) and
VLT/SPHERE (Pinilla et al., 2018) provide evidence for an inner disk that is
misaligned with the outer disk. With ALMA observations, a twisted “butterfly”
radial velocity map in CO (3-2) clearly shows a highly inclined inner disk,
despite the outer disk being seen nearly face-on. Shadows of the inner disk
cast on the outer disk are also visible in both dust continuum thermal
emission and gas emission maps. These shadows are also visible in the
scattered light images from SPHERE, where the depth and width of the shadows
change with time. In the case of this Dipper, at least, the light variations
are likely coming from irregularities in the orbiting dust in the inner disk.
In the other two objects discussed in this paper, we need more evidence to
conclude a similar geometry, because there are other ways to generate similar
variability in brightness.

Other possible mechanisms may also produce somewhat similar changes in the
observed brightness of a star. Using high spatial resolution imaging obtained
with the Goddard Fabry-Pérot imager on the Apache Point Observatory 3.5 m
telescope, Wassell et al. (2006) reported a string of luminous clouds along
two opposing outflows in the Herbig Ae star HD 163296 (MWC 275). Ellerbroek et
al. (2014), using photometry obtained by the All Sky Automated Survey
(Pojmanski, 1997), reported that a Dipper-like extinction event occurred in
2001 in the star, and this extinction event was traced, using the measured
proper motion of these clouds, to an ejection event at that time. it was
suggested that dust entrained in a disk wind (see their Figure 9), similar to
that suggested by Bans & Königl (2012) for the source of the near-IR emission
in protostellar systems, was the cause. Unfortunately, the ASAS photometric
observations provided no color information. However, in 2012 multi-filter
observations obtained by the American Association of Variable Star Observers
(AAVSO) captured a 1-day extinction event that could be subjected to modeling
using various types of grains. Grains typical of both the diffuse interstellar
medium and dark molecular clouds with star-formation regions provided poor
fits to the wavelength dependence of the extinction. However grains that had
undergone further growth, and were similar to those included in some radiative
transfer models of disks, such as those used in the HOCHUNK3D code package by
Whitney et al. (2013), were more successful at reproducing the extinction
properties of that one-day event Pikhartova et al. (2021)

Here we report near-infrared observations of the three Dippers in Ansdell et
al. (2016b) obtained with the SpeX spectrograph on NASA’s Infrared Telescope
facility (IRTF) taken on two consecutive nights. From these we obtain the
extinction curves for the two brighter “Dippers”, and subject them to the same
extinction analysis that Pikhartova et al. (2021) employed for HD 163296.

##  2 Observations

We observed the three “face-on” Dippers on two successive nights, 10 and 11
August, 2017, at an effective spectral resolution R = 750 (0.8” slit) from 0.7
- 2.4 $\mu$m using the SXD grating of the SpeX spectrograph (Rayner et al.,
2003) on NASA’s Infrared Telescope Facility (IRTF). In order to determine the
absolute fluxes of these slit spectra, which are subject to time-variable
throughput due to the effects of astronomical seeing and telescope guiding, we
also observed them in the K-band using the SpeX guide camera using a 9-point
dither pattern. Both spectra and images were flux-calibrated using the A0V
star HD 145127 at nearly the same airmass. The spectral data reduction was
carried out using the Spextool software, running under IDL. (Vacca et al.,
2003; Cushing et al., 2004)

Figure 1: 0.7-2.4 $\mu$m spectra of three Dippers obtained with the SpeX
spectrograph between 0.7-2.4 $\mu$m on 10 and 11 Aug. 2017 (UT). Due to poor
signal/noise, the spectrum of EPIC 203850058 was truncated at wavelengths
smaller than 0.8 $\mu$m. The spectra were calibrated in absolute flux using
K-band images obtained with the guide camera (filled circles). For EPIC
204628512 and EPIC 205151387, the fainter spectrum was also slightly redder,
and were used to examine the wavelength-dependence of the extinction
(reddening). The difference in the flux levels for EPIC 2035850058 were so
small that we did not attempt to derive a flux ratio for examining potential
reddening by dust.

In Figure 1 we show the SpeX spectra of all three Dippers, each normalized by
the K-band fluxes. The two brightest Dippers, EPIC 204638512 (K2; Köhler, et
al. (2000)) and EPIC 205151387 (M0.1V; Ansdell et al. (2016a)), underwent
marked changes in brightness between the two nights, while the third, much
fainter one, EPIC 203850058 (M5; Manora et al. (2015)), showed smaller
changes, being less than the “noise” at the shortest wavelengths.

Table 1: Dipper Fundamental Parameters Star Name | EPIC 203850058 | EPIC 204638512 | EPIC 205151387
---|---|---|---
_Dippers_ |  |  |
2MASS | 16253849-2613540 | 016042165-2130284 | 16090075-1908526
Gaia DR3 | 6049072316675781120 | 6243393817024157184 | 6245777283349430912
_Star Properties_ |  |  |
Gaia Distance (pc) | 141.577 | 145.309 | 137.397
Spectral Type | M5 | K2 | M0.5
Prot (d) | 2.88 | 5.0 | 9.55
Region |  $\rho$ Oph | Upper Sco | Upper Sco
K Band Magnitude | 10.766 | 8.508 | 9.475
_Disk Properties_ |  |  |
ALMA Disk Diam. (”) | 0.2 x 0.3 | 1 | 0.6 x 0.4
ALMA Disk Diam. (au) | 28.3 x 42.5 | 145 | 82.2 x 54.8
AlLMA Disk Incl. (∘) | 73$\pm$23 | 6$\pm$1.5 | 53$\pm$2
Dusk Type aa(Ansdell et al., 2016a) | Full | Transitional | Full
Age | 1 Myr | 10 Myr | 10 Myr
Table 2: Observed Dipper K-Band Fluxesaa K-band flux is $\lambda$Fλin units of 10-13 W m-2. Uncertainties are given in parentheses. Date (UT) | MJD | EPIC 203850058 | MJD | EPIC 204638512 | MJD | EPIC 205151387
---|---|---|---|---|---|---
170810 | 57975.23912 | 0.65(0.11) | 57975.25421 | 3.95(0.12) | 57975.27806 | 2.34(0.11)
170811 | 57976.24319 | 0.60(0.05) | 57976.24319 | 4.80(0.04) | 57976.27624 | 2.01(0.04)

##  3 Wavelength-Dependent Extinction

To characterize the dust we first selected the brightest of these objects,
EPIC 204638512 ( = 2MASS J16042165-2130284), an object that has been the
target of imaging programs using extreme adaptive optics (Dong et al., 2017;
Uyama et al., 2016; Canovas et al., 2017). The difference in extinction
between the two successive nights (we cannot assume that the brighter flux
state has no extinction) was determined by taking the ratio of the two flux-
calibrated spectra, and comparing the result with a number of published dust
extinction laws, as shown in Figure 2.

Figure 2: Extinction curve of EPIC 204638512. The model curves labeled “CCM”.
are those of Cardelli, Clayton, & Mathis (1989), using different values of RV,
and representative of ISM extinction. Also shown are DMC extinction curves
from McClure (2009) and Weingartner & Draine (2001) The curves labeled “ww04”,
“www003”, and “ww002” are grain files included with the HOCHUNK3D code package
of Whitney et al. (2013). For example, “www003”’ is Model 1 of Wood et al.
(2002), which has a power-law grain size distribution, with a maximum grain
size of 1000 $\mu$m and an exponential cutoff at starting an 50 $\mu$m and
going to the large grain limit. “Modified KMH” uses the grain size
distribution of Kim, Martin & Hendrey (1994) but increasing the minimum and
maximum grain sizes to 0.25 $\mu$m and 500 $\mu$m, respectively. These are
described more fully in the text.  Figure 3: Extinction curve of EPIC
205151387. The grain models are explained in the caption to Figure 2, except
we did not include grain models used in the HOCHUNK3D package, however it is
clear that the grain sizes involved are similar to those in EPIC 204638512.
The change in the He I feature is more pronounced in EPIC 205151387.

We began with the extinction laws of Cardelli, Clayton, & Mathis (1989) often
used in the literature for reproducing extinction by dust in the diffuse
interstellar medium. Despite changing the ratio of total-to-selective
extinction (RV) where larger values represent larger grains), no good fit was
obtained. Other recipes for denser cloud regions (Weingartner & Draine, 2001;
McClure, 2009) fared little better.

Next we tried grain prescriptions (e.g. from Cotera et al. (2001) and Wood et
al. (2002)) developed for circumstellar disks and included in the HOCHUNK3D
Monte Carlo radiative transfer code of Whitney et al. (2013). These use the
optical constants of silicates and carbonaceous grains from Rouleau & Martin
(1991) and Weingartner & Draine (2001). “www03” refers to the grain created to
the disk midplane providing the bet fit for the SED of HH30 in Wood et al.
(2002). Specifically, their Model 1, which uses a mixture of amorphous carbon
and astronomical silicates that have a power-law size distribution $n(a)\sim
a^{-p}$ with $p=3.0$, plus exponential cutoff starting at 50$\mu$m and going
to longer wavelengths, to a grain maximum size size of 1000 $\mu$m. “ww04” is
the grain model of Cotera et al. (2001), used to fit the scattered light in
HH30. It a power law index of 3.5 for amorphous carbon and 3.0 for silicates,
a turnover at 0.55 $\mu$m and grain maximum of 20 $\mu$m. “ww002” is the same
grain model as “w003” except having the exponential cutoff at 100 $\mu$m
instead of 50 $\mu$m.

Finally we used a similar grain file that used the ISM dust extinction of
(Kim, Martin & Hendrey, 1994) but adjusted the minimum and maximum size
present. This extinction (“Modified KMH”) has grains extending to at least 500
$\mu$m and lacks grains smaller than 0.25 $\mu$m. While larger grains were not
excluded, they produced little further change in the resulting extinction at
the observed wavelengths.

We also subjected the extinction curve of EPIC 205151387 to a similar
analysis, as illustrated in Figure 3. This uses the same modified KMH model as
in the case of EPIC 204638512 (with lower total extinction), grains more
consistent with those used in disk modeling provided a much better agreement
with the observed extinction than did either ISM-like or dark cloud-like
extinction models.

A similar analysis was applied to a short-duration extinction event in HD
163296 by Pikhartova et al. (2021). In the case of HD 163296, an extinction
event lasting only about 1 day was modeled, with the www003 grain model
providing the best fit, over all. The large extinction event of 2001
(Ellerbroek et al., 2014), likely due to the ejection of HH-like blobs in a
bipolar jet (with a disk wind) was mostly monitored at a single visible
wavelength, and could not be subjected to the same wavelength-dependent
extinction analysis.

##  4 The Gas

In both derived extinction curves, there is an added feature at a wavelength
consistent with the He I line at 1.083 $\mu$m often seen in other young
stellar objects. In EPIC 204638512, this line seems to exhibit exhibits a “P
Cygni” profile (blueshifted absorption component flanked by a redshifted
emission component) indicative of outflowing gas (with the blue absorption
wing being produced by gas between the observer and the stellar photosphere),
but the evidence is only suggestive. Silicia-Aguilar et al. (2020) reported
line profile changes in H$\alpha$ and H$\beta$. The former line exhibited
complex behavior, with epochs of inflowing and outflowing gas, while the
latter had one epoch with a very distindegreect inflowing gas component In
EPIC 205251387, the He I line is more indicative of infalling gas. In PMS
stars, the He I profile can exhibit strong time-dependent changes in intensity
and shape (Sitko et al., 2012; Fernandes et al., 2018). A significant fraction
of the time, HD 163296 seems to exhibit a combination of both outflowing and
inflowing gas (Pikhartova et al., 2021) at the same time. The data on EPIC
203850058 was deemed to ne of insufficient quality to determine the nature of
the gas flow characteristics.

Figure 4: The region near He I line for EPIC 204638512 on two consecutive
nights, along with the spectrum of HD 3765, a star of similar spectral type,
taken from the SpeX Atlas of Cool Stars (Rayner et al., 2009). The gap in the
data on EPIC 204628512 is the result of trimming out the spectrum heavily
affected by telluric absorption. Also shown are the locations of two nearby
hydrogen lines, often seen in emission in stars actively accreting material
from their disks, and the He I line. The Pa $\beta$ line is weaker than in HD
3765, suggesting it might be weakly in emission, partially filling the
photospheric absorption line (in K2 V stars this region also contains a blend
of Ti I, Ni I, Fe I, and Ca I lines - see Fig. 30 of Rayner et al. (2009)). In
the brighter continuum state, the He I line exhibits a weak emission component
in addition to a strong absorption core at shorter wavelengths. This resembles
a P Cygni-like profile, but with a relatively stronger short wavelength
absorption component. In the higher dust occultation data, the red wing of the
He I line is not evident.  Figure 5: The region near He I line for EPIC
205151387 on two consecutive nights, along with the spectrum of HD 209290, a
star of similar spectral type, taken from the SpeX Atlas of Cool Stars (Rayner
et al., 2009). During the dust extinction event (lower net flux) , the He I
absorption line became even stronger than in EPIC 204628512, The Pa $\beta$
line developed a distinct absorption component, and the emission on Pa
$\gamma$ was weakened. The absorption in He I and Pa $\beta$ are both shifted
toward longer wavelengths than the emission, inverse P Cygni-like profiles,
indicating infalling gas.  Figure 6: The region near He I line for EPIC
203850058 on two consecutive nights, along with the spectrum of Wolf 47, a
star of similar spectral type, taken from the SpeX Atlas of Cool Stars (Rayner
et al., 2009). The Pa $\beta$ and Pa $\gamma$ lines are in emission,
suggesting some accreting gas. The He I line exhibits absorption, but unlike
EPIC 204638512 and EPIN 501551387, does not show a detectable He I emission
component, although the quality of the spectra are inferior to those of the
other two dippers. Note that Wolf 47 is a dMe star exhibiting flares
(including at radio wavelengths White et al. (1989)) and so some line emission
(Pa $\gamma$?) is not unexpected.

For completeness, we also show the two spectra of EPIC 203850058 (Fig. 5), but
the data were of insufficient quality for an analysis of the extinction.

##  5 Discussion

Using ALMA dust continuum and CO line emission data of EPIS 204638512, Mayama
et al. (2018) have provided convincing evidence of an inner disk that is
misaligned from the outer disk, with an inclination of -45∘. Pinilla et al.
(2018) have also found that the shadows cast by the inner disk onto the outer
disk are variable in time. These results clearly indicate that the “dipper”
phenomenon EPIC 2064638512 are due to dust in the inclined inner disk
producing variable extinction by the inclined inner disk. While less-well
studied, it suggests that the extinction events seen in EPIC 205251387 have a
similar origin.

In this paper we have derived the extinction properties of the dust in these
twi sources that imply grains that are unlike those found in the general ISM
from which they originally formed. Rather, the occulting grains are those that
at some earlier epoch have undergone significant growth in size. Such grain
growth is expected in disks (D’Alessio, Calvet, & Hartmann, 2001; D’Alessio et
al., 2006; Furlan et al., 2005), and is generally thought to be accompanied by
settling of the larger grains to the disk mid-plane, as they begin to decouple
from the gas. In the case of EPIC 205151387 and EPIC 204638512, the types of
grains present are consistent with the sorts of grains expected in disk
systems. Indeed, the extinction properties of the grains used were derived
from the scattered light and thermal emission of the disk of HH30 (Cotera et
al., 2001; Wood et al., 2002). However, if the dust particles causing the
Dipper events are this large, then the “settled dust” dominates these
extinction events. In this case we may be seeing one possible near-final
evolutionary stage of a disk system. The structure of the CO maps in EPIC
204638512 might also hint at infalling material (Mayama et al., 2018), but the
evidence is inconclusive. The evidence seems stronger that this system
possesses a mis-inclined inner disk that produces variable amounts of
shadowing nearly 140-180∘. apart in azimuth in the outer disk. “Dipper”
occultation events would be more likely than if the inner regions had the low
inclinations originally reported for the outer disk (Ansdell et al., 2016b).

While the “Dipper” events are almost inevitable for highly inclined inner
disks, is there evidence for similar structure in disks not fortuitously
observed at high inclination? A number of disks exhibit outer disk shadows
that imply mis-aligned inner disks: HD 142527 (Avenhaus et al., 2014), HD
100453 (Benisty, et al., 2017; Long, et al., 2017), TW Hya (Debes et al.,
2017; Poteet et al., 2018), and HD 100453 (Garufi et al., 2016). A related
type of phenomenon might also be occurring in the non-Dipper star SAO 206462
(HD 135344B). Recent SPHERE imaging of its disk by Stolker et al. (2017)
showed moving shadows in the outer disk. These traveled many degrees in the
course of months, suggesting varying shadows by irregular dust clouds in the
inner disk. This apparent orbital motion was accompanied by changes in color
of the disk in general, suggesting that the outer disk was experiencing
changing illumination due to clouds that effectively reddened the light
reaching it. Were its inner disk highly inclined with respect to the viewer,
it might exhibit “Dipper”-like events. Careful monitoring of near-IR spectra
of systems with evidence for mis-aligned inner disks should be pursued.

##  6 Conclusions

We have found that the “dipper” events observed in EPIC 20435638512 are due to
changes in the extinction by dust grains whose sizes indicate significant
grain growth compared to their ISM sizes, and these are located in the surface
layers of an inner disk that is highly inclined to the outer disk, seen in
ALMA continuum images. Our results also suggest that other Dippers having
outer disks with modest inclinations (such as EPIC 205151387, with $i=53\pm
2^{\circ}$ might contain misaligned inner disks. The detection of shadows,
especially those with wide position angle separations and/or variable shadows
in the outer disk would help confirm such an hypothesis, as would
interferometric observations of the inner regions.

As to the gas velocity structures in EPIC 204638512 and EPIC 205151387, the
situation is complex. Both outflowing (wind) and inflowing (accretion) gas is
present in differing degrees at different epochs. Further work that includes a
number of emission/absorption lines simultaneous with thermal dust extinction
measurements are needed.

This paper was based on observations taken with the IRTF/SpeX 0.8-5.5 Micron
Medium-Resolution Spectrograph and Imager, funded by the National Science
Foundation and NASA and operated by he NASA Infrared Telescope Facility.The
authors would like to thank the tireless support of of the staff of NASA’s
Infrared Telescope Facility. We wish to emphasize the pivotal cultural role
and reverence that the summit of Maunakea has always had within the indigenous
Hawaiian community. We are most fortunate to have the privilege to conduct
scientific observations from this mountain. This work was supported in part by
NASA XRP grants NNX17AF88G and NNX16AJ75G (MLS) and 80NSSC20KO254 (CAG)..

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∎

11institutetext: K. Ocalan 22institutetext: Necmettin Erbakan University,
Faculty of Aviation and Space Sciences, Konya, Turkey
ORCID ID: http://orcid.org/0000-0002-8419-1400
22email<EMAIL_ADDRESS>
# Impact of lepton $p_{T}$ threshold on the charge asymmetry predictions for
the inclusive $W$ boson production in $pp$ collisions at 13 TeV

Kadir Ocalan

(Received: date / Accepted: date)

###### Abstract

This paper presents the impact of lepton transverse momentum $p^{l}_{T}$
threshold on the $W$ boson charge asymmetry predictions in perturbative QCD
for the inclusive $W^{\pm}+X\rightarrow l^{\pm}\nu+X$ production in proton-
proton ($pp$) collisions. The predictions are obtained with various
low-$p^{l}_{T}$ thresholds $p^{l}_{T}>$ 20, 25, 30, and 40 GeV in a fiducial
region encompassing both central and forward detector acceptances in terms of
the lepton pseudorapidty 0 $\leq\eta_{l}\leq$ 4.5. The predicted distributions
for the lepton charge asymmetry, which is defined by means of the $\eta_{l}$
($A_{\eta_{l}}$), at next-to-next-to-leading order (NNLO) accuracy are
compared with the CMS and LHCb data at 8 TeV center-of-mass collision energy.
The 8 TeV predictions are observed to reproduce the data fairly well within
the quoted uncertainties. The predictions from the CT14 parton distribution
function (PDF) model provide slightly better agreement with the data over the
other PDF sets that are being tested. The 13 TeV predictions by using various
$p^{l}_{T}$ thresholds are reported for the $A_{\eta_{l}}$ and also the charge
asymmetries that are defined in terms of the differential cross sections in
bins of the $W$ boson rapidity $y_{W}$ ($A_{y_{W}}$) and transverse momentum
$p^{W}_{T}$ ($A_{p^{W}_{T}}$). The NNLO predictions for the $A_{\eta_{l}}$,
$A_{y_{W}}$, and $A_{p^{W}_{T}}$ distributions are assessed to be in close
correlation with the $p^{l}_{T}$ value. The $A_{\eta_{l}}$ and $A_{y_{W}}$
distributions are particularly shown to be more correlated at a higher
$p^{l}_{T}$ threshold. The $A_{p^{W}_{T}}$ distributions are also reported
from the merged predictions with improved accuracy by the inclusion of next-
to-next-to-next-to-leading logarithm (N3LL) corrections, i.e., at NNLO+N3LL.
The predicted distributions from various $p^{l}_{T}$ thresholds represent also
a finer probe in terms of the capability to provide more constraints on the
ratio of $u$ and $d$ quark distribution functions in the parton momentum
fraction range $10^{-4}<x<1$.

###### Keywords:

High Energy Physics phenomenology Perturbative QCD calculations $W$ bosons $W$
boson charge asymmetries Lepton transverse momentum threshold

###### pacs:

12.38.-t 12.38.Bx 14.70.Fm 13.85.Qk

##  1 Introduction

Weak vector boson ($W$ and $Z$ boson) production plays a crucial role at
hadron colliders including the present proton–proton ($pp$) collider at CERN,
the Large Hadron Collider (LHC). Production of $W$ and $Z$ bosons in $pp$
collisions at the LHC enables several precision tests of the quantum
chromodynamic (QCD) and electroweak (EW) sectors of the Standard Model (SM).
Their precise measurements provide substantial inputs for constraining parton
distribution functions (PDFs) in the proton and improved background modeling
for several rarer SM processes such as top quark and Higgs boson productions
and beyond the SM searches such as supersymmetry and dark matter. Their
measurements in leptonic decay modes are very advantageous as they are
produced in abundance with clean experimental signatures and constitute a
major experimental benchmark to calibrate detector response for lepton, jet,
and missing transverse energy reconstructions. Their productions through
leptonic decays are not only important for experimental aspects, but they are
also essential to test Monte Carlo based event generators and fixed-order
calculations for the advancement of the field of theoretical predictions.

In particular, $W$ boson production is experimentally characterized by one
isolated lepton with high transverse momentum $p_{T}$ and large missing
transverse energy due to neutrino in its leptonic decay mode $pp\rightarrow
W^{\pm}\rightarrow l^{\pm}\nu$, where $l$ is either a muon $\mu$ or an
electron $e$. Dominant mechanism for $W$ boson production at the LHC proceeds
via annihilation of a valence quark from one of colliding protons with a sea
antiquark from other proton as $u\bar{d}\rightarrow W^{+}$ and
$d\bar{u}\rightarrow W^{-}$. The excess of two valence $u$ quarks over one
valence $d$ quark in the proton requires $W^{+}$ bosons to be produced more
often than $W^{-}$ bosons. This production asymmetry between $W^{+}$ and
$W^{-}$ bosons is referred to as $W$ boson charge asymmetry and is usually
defined with cross sections $\sigma(W^{+})$ and $\sigma(W^{-})$ differential
in $W$ boson rapidity $y_{W}$ as

$A_{y_{W}}=\frac{d\sigma(W^{+}\rightarrow
l^{+}\nu)/dy_{W}-d\sigma(W^{-}\rightarrow
l^{-}\bar{\nu})/dy_{W}}{d\sigma(W^{+}\rightarrow
l^{+}\nu)/dy_{W}+d\sigma(W^{-}\rightarrow l^{-}\bar{\nu})/dy_{W}}.$ (1)

The $W$ boson charge asymmetry $A_{y_{W}}$ provides a direct probe of the
relative $u$ and $d$ quark distributions as functions of the initial-state
parton momentum fractions ($x$ values) since $y_{W}$ is strongly correlated
with the $x$ values, which can generally be expressed as
$x_{1,2}=(M_{W}/\sqrt{s})e^{\pm y}$ with $M_{W}$ is the $W$ boson mass and
$\sqrt{s}$ is the center-of-mass energy. However, there is an experimental
limitation regarding the $A_{y_{W}}$ since the $p_{T}$ and $y_{W}$ of the $W$
boson cannot be directly reconstructed due to the unknown longitudinal
momentum of the decay neutrino. Despite this limitation, the same information
can still be accessed by measuring the charge asymmetry from the decay lepton.
The charge asymmetry can readily be measured as a function of the decay lepton
pseudorapidity $\eta_{l}$, which is indeed correlated with the $y_{W}$, in the
analogous form of

$A_{\eta_{l}}=\frac{d\sigma(W^{+}\rightarrow
l^{+}\nu)/d\eta_{l}-d\sigma(W^{-}\rightarrow
l^{-}\bar{\nu})/d\eta_{l}}{d\sigma(W^{+}\rightarrow
l^{+}\nu)/d\eta_{l}+d\sigma(W^{-}\rightarrow l^{-}\bar{\nu})/d\eta_{l}}.$ (2)

The lepton charge asymmetry $A_{\eta_{l}}$ corresponds to the convolution of
the original $A_{y_{W}}$ variable and the $V-A$ (vector–axial vector)
asymmetry of the $W$ boson, which implies its anisotropic decay into the
lepton and the neutrino. In a similar way, the $A_{\eta_{l}}$ variable can
provide substantial constraints on the ratio of $u$ and $d$ quark distribution
functions in the proton as a function of $x$ values of the partons. This
variable can also be benefited for discriminating among various PDF models
that predict different shapes of valence and sea quark distributions.

The $W$ boson production asymmetries were measured before mostly in terms of
the $A_{\eta_{l}}$ variable in p$\rm{\bar{p}}$ collisions by the CDF and D0
Collaborations at the Tevatron Abe:1998rv ; Abazov:2007pm ; Abazov:2008qv ;
Aaltonen:2009ta ; Abazov:2013rja ; D0:2014kma . The asymmetries were measured
at the LHC by using the $A_{\eta_{l}}$ variable in the central lepton
pseudorapidity region $|\eta_{l}|\leq$ 2.5 by the ATLAS and CMS Collaborations
at different center-of-mass energies up to 8 TeV Aad:2011dm ;
Chatrchyan:2012xt ; Chatrchyan:2013mza ; Khachatryan:2016pev ; Aaboud:2016btc
; Aad:2019bdc ; Aaboud:2018nic ; Aad:2019rou . The $A_{\eta_{l}}$ variable was
also measured by the LHCb Collaboration at the LHC up to 8 TeV Aaij:2012vn ;
Aaij:2014wba ; Aaij:2015zlq ; Aaij:2016qqz  in the forward region 2.0
$\leq\eta_{l}\leq$ 4.5 extending beyond the ATLAS and CMS detector coverage.
The entire LHC measurements have probed the inclusive $W$ boson cross sections
along with the $A_{\eta_{l}}$ variable in the range $10^{-4}<x<1$ which are
clearly important to provide valuable inputs on determining accurate PDFs also
at very small and large $x$ values. In all these measurements complementing in
the $\mu$ and $e$ decay channels in terms of the wide $\eta_{l}$ region
probed, data were compared with various theoretical predictions including
fixed-order perturbative QCD calculations at next-to-leading order (NLO) and
next-to-NLO (NNLO) accuracies, convolved with different PDF models.

The $W$ boson charge asymmetries are being determined for kinematic phase
spaces specified by the decay lepton transverse momentum $p^{l}_{T}$
threshold. The $p^{l}_{T}$ is correlated with the $W$ boson transverse
momentum $p^{W}_{T}$, and hence, impacts in both measurements and predictions
of the charge asymmetry. In the measurements, the $p^{l}_{T}$ threshold value
is chosen to match with available detector triggering conditions and to have
an efficient event reconstruction for a pure signal data sample with
sufficiently high statistics. Thereby, the measurements are subject to use a
$p^{l}_{T}$ threshold value which depends on event triggering and
reconstruction requirements. Nevertheless in the theoretical calculations,
various $p^{l}_{T}$ thresholds (including also the ones used in the
measurements) can be used alternatively to test the impact on the charge
asymmetry. Furthermore, theoretical predictions can be repeated with
increasing thresholds in the low-$p^{l}_{T}$ region to select only a subset of
phase space where the $\eta_{l}$ gets closer to the $y_{W}$. This also
facilitates to test charge asymmetry predictions in a more constrained phase
space in different ranges of the $\eta_{l}$ and $y_{W}$ allowing also a finer
probe of the dependence to $x$ values.

In this work we present the predicted charge asymmetries corresponding to the
$W^{\pm}$ boson production processes $pp\rightarrow W^{+}+X\rightarrow
l^{+}\nu+X$ and $pp\rightarrow W^{-}+X\rightarrow l^{-}\bar{\nu}+X$. The
predictions are obtained in the fiducial phase space encompassing both central
and forward regions 0 $\leq\eta_{l}\leq$ 4.5 at both 8 TeV and 13 TeV. The
predictions at NNLO accuracy as a function of the $\eta_{l}$ from various PDF
models are compared with the CMS and LHCb $pp$ collision data at 8 TeV. The
predictions are further obtained as functions of the $\eta_{l}$ and $y_{W}$ at
NNLO accuracy as well as in bins of the $p^{W}_{T}$ through resummation at
next-to-next-to-next-to-leading logarithm (N3LL) which is matched to NNLO,
i.e., NNLO+N3LL accuracy at 13 TeV. Various thresholds in low-$p^{l}_{T}$
region, $p^{l}_{T}>$ 20, 25, 30, and 40 GeV, are used to enable testing the
potential impact on the charge asymmetry. Specifically the 13 TeV predictions
are reported by aiming to assess the correlations among the increasing
low-$p^{l}_{T}$ thresholds and the charge asymmetry distributions for the
$A_{\eta_{l}}$, $A_{y_{W}}$, and also in bins of the $p^{W}_{T}$.

##  2 Methodology

###  2.1 Computational setup

The charge asymmetry calculations that are based on differential cross
sections are performed by using the MATRIX framework Grazzini:2017mhc ;
Catani:2009sm  which is interfaced with the RadISH program Bizon:2017rah ;
Monni:2016ktx , together by the computational framework MATRIX+RadISH (v1.0.1)
Kallweit:2020gva . The fixed-order calculations of differential cross sections
at NNLO in the QCD perturbation theory is achieved with the MATRIX framework
which implements the transverse momentum _$q_{T}$_ -subtraction method
Catani:2007vq ; Catani:2012qa . In the _$q_{T}$_ -subtraction approach,
infrared divergences of the real radiation contributions are extracted by
using the infrared subtraction terms in the perturbative expansion. These
divergences are regulated by employing a fixed cut-off value $r_{cut}=$ 0.0015
(0.15%) for the slicing parameter $r$, where it is defined as $r=p_{T}/m$ in
terms of the $p_{T}$ distribution and invariant mass $m$ for a system of
colorless particles. The resummation of large logarithmic contributions, which
is needed for accurate prediction of differential cross sections as a function
of the $p^{W}_{T}$, is achieved with the formalism of the RadISH program. The
RadISH code enables high-accuracy resummation for the $p^{W}_{T}$ distribution
through N3LL which is matched to the NNLO QCD calculations by the MATRIX. In
addition, the OpenLoops tool Cascioli:2011va ; Denner:2016kdg  is utilized
through an automated interface to acquire all the spin- and color-correlated
tree-level and one-loop scattering amplitudes in the computations. In the
setup, the Fermi constant $G_{F}$ input scheme is used where leptons (both
$\mu$ and $e$) and light quarks are treated massless. The default MATRIX setup
is used for the SM input parameters relevant to the inclusive $W$ boson
process that are all based on the following $W$ boson mass and $G_{F}$ values
of

$M_{W}=80.385\hskip 2.84544pt\rm{GeV},\hskip
2.84544pt\it{G_{F}}=\rm{1.16639}\times\rm{10^{-5}}\hskip
2.84544pt\rm{GeV}^{-2}.$ (3)

To this end, the QCD calculations of differential cross sections for the
charge asymmetry predictions require inclusion of knowledge of PDFs. The
evaluation of PDFs from data files is carried out by exploiting the LHAPDF
(v6.2.0) framework Buckley:2014ana  in the computations. Various PDF sets are
used in the calculations, where all are based on a constant strong coupling
$\alpha_{s}=$ 0.118. Particularly the NNLO PDF sets MMHT2014 Harland-
Lang:2014zoa , CT14 Dulat:2015mca , NNPDF3.1 Ball:2014uwa , and PDF4LHC15
Butterworth:2015oua  are used in the calculations.

###  2.2 Fiducial requirements

The calculations for both the differential cross section and charge asymmetry
predictions are performed in a realistic fiducial phase space of the $W$ boson
and its decay lepton. The fiducial phase space is defined to be in line with
the reference CMS Khachatryan:2016pev  and LHCb Aaij:2015zlq  8 TeV
measurements. The leptons (either $\mu$ or $e$) are required to have
transverse momentum $p^{l}_{T}>$ 25 GeV ($p^{l}_{T}>$ 20 GeV) and to lie in
the $\eta_{l}$ region $0\leq\eta_{l}\leq 2.4$ ($2.0\leq\eta_{l}\leq 4.5$) for
the validation of the predictions with the reference CMS (LHCb) results at 8
TeV. The leptons are required to have transverse momentum $p^{l}_{T}>$ 20 GeV
and to lie in the $\eta_{l}$ region encompassing both central and forward
acceptances $0\leq\eta_{l}\leq 4.5$ at 13 TeV. In addition, the requirements
$p^{l}_{T}>$ 25, 30, and 40 GeV are all used to assess the correlations of
these increasing thresholds with the predicted charge asymmetries in the
entire acceptance region $0\leq\eta_{l}\leq 4.5$ concerning the 13 TeV
predictions. Leptons are treated massless in the computational setup, thereby
the predictions of differential cross sections in the $\mu$ channel are the
same as in the $e$ channel. No requirements are strictly imposed for the $W$
boson transverse mass and the missing transverse energy due to the neutrino
while these requirements can make more sense depending on experimental
measurement. Moreover, no explicit requirement is applied for the final-state
hadronic jet(s) in terms of jet definition criteria and selection cuts.

###  2.3 Theoretical uncertainties

Theoretical calculations of the cross sections in perturbative QCD expansions
in the $\alpha_{s}$ depend on the choices for the renormalization $\mu_{R}$
and factorization $\mu_{F}$ scales. In this paper, the central values for the
$\mu_{R}$ and $\mu_{F}$ scales are fixed to the $W$ boson mass $\mu_{R}$ =
$\mu_{F}$ = $M_{W}$ = 80.385 GeV. Similarly the central value for the
resummation scale $x_{Q}$ is set to the $W$ boson mass $x_{Q}$ = $M_{W}$ =
80.385 GeV when resummation of large logarithmic corrections is also
considered in the calculations. Theoretical uncertainties due to the choices
for the central scale values or shortly scale uncertainties correspond to
missing higher-order corrections in the perturbative (and resummed)
calculations. Scale uncertainties are estimated by varying independently the
$\mu_{R}$ and $\mu_{F}$ by a factor of 2 up and down around their central
values. The seven-point variation method is employed, that is all possible
combinations in the variations are considered while imposing the constraint
$0.5\leq\mu_{R}/\mu_{F}\leq 2.0$. On the other hand, the nine-point variation
method is used when perturbative calculations include also matching to
resummation, that is the envelope of the seven-point variation while keeping
$x_{Q}$ at its central value and the two-point variation of $x_{Q}$ around its
central value by a factor of 2 in either direction for the central values of
the $\mu_{R}$ and $\mu_{F}$ scales. The PDF uncertainties due to different
parametrization of the PDF models are estimated by following the prescription
of the PDF4LHC working group Butterworth:2015oua ; Buckley:2014ana . The
$\alpha_{s}$ uncertainty is also estimated by varying the $\alpha_{s}$ value
by $\pm 0.001$ around 0.118. Thereafter, total theoretical uncertainties of
the predictions are obtained by summing quadratically scale, PDF, and
$\alpha_{s}$ uncertainties. Total theoretical uncertainties in the predicted
distributions are presented symmetrically by using the larger values from
estimated up and down uncertainties in a conservative consideration.

##  3 Comparisons with the 8 TeV data

The predictions are compared with the 8 TeV data from the reference CMS
measurement Khachatryan:2016pev  for the differential cross section and
$A_{\eta_{l}}$ distributions. The NNLO predictions are obtained in the $\mu$
decay mode with the fiducial requirement $p^{\mu}_{T}>$ 25 GeV in the central
region $0\leq\eta_{\mu}\leq 2.4$. The $\eta_{\mu}$ bin ranges are used from
the CMS measurement as (0.00, 0.20), (0.20, 0.40), (0.40, 0.60), (0.60, 0.80),
(0.80, 1.00), (1.00, 1.20), (1.20, 1.40), (1.40, 1.60), (1.60, 1.85), (1.85,
2.10), and (2.10, 2.40) to enable direct comparisons. Total theoretical
uncertainties are included from quadratic sum of scale, PDF, and $\alpha_{s}$
uncertainties for the predicted distributions. Total experimental
uncertainties are included by summing statistical, systematic, and luminosity
uncertainties in quadrature for the measured differential cross sections,
while summing statistical and systematic uncertainties in quadrature for the
measured asymmetry. The NNLO predictions from the PDF sets MMHT2014, CT14,
NNPDF3.1, and PDF4LHC15 are compared with the CMS data distributions. The
predicted differential cross section distributions for the $W^{+}$ and $W^{-}$
processes are compared with the data in Fig. 1. The predictions using
different PDF sets are observed to be in good agreement with each other and
the data within uncertainties. The prediction using CT14 shows better
agreement with the data, where the predictions using PDF sets other than CT14
slightly deviate from the data in only a small number of bins up to a few
percent. The predicted $A_{\eta_{\mu}}$ distributions are compared with the
data as shown in Fig. 2. Apart from a few exceptions, the predictions describe
the CMS data consistently within uncertainties throughout the entire
$\eta_{\mu}$ ranges. The predicted $A_{\eta_{\mu}}$ distribution from CT14
describes the data slightly better over the predictions using the other PDF
sets. It can also be observed that sensitivity to discriminate among various
PDF sets is enhanced in the $A_{\eta_{\mu}}$ variable in comparison to the
differential cross sections. Moreover, the predicted results from various PDF
sets for both the differential cross section and $A_{\eta_{\mu}}$
distributions are observed to be in agreement with the corresponding NNLO
predictions by the FEWZ program Li:2012wna  in the CMS paper. Difference
between the MATRIX predictions and the FEWZ ones are generally up to
$\sim$1–2% within the quoted theoretical uncertainties.

Figure 1: The predicted differential cross section distributions for the
$W^{+}$ (top) and $W^{-}$ (bottom) processes as a function of the $\eta_{\mu}$
and their comparisons with the CMS data at 8 TeV. The predictions are obtained
at NNLO accuracy by using MMHT2014, CT14, NNPDF3.1, and PDF4LHC15 PDF sets.
The predictions include total theoretical uncertainties from quadratic sum of
scale, PDF, and $\alpha_{s}$ uncertainties, while the data include total
experimental uncertainty. In the lower panels, the ratios of the predictions
to the data for the differential cross section distributions are also
displayed. Figure 2: The NNLO predictions for the muon charge asymmetry
$A_{\eta_{\mu}}$ variable from MMHT2014, CT14, NNPDF3.1, and PDF4LHC15 PDF
sets as a function of the $\eta_{\mu}$. The predictions are compared with the
CMS data in the central region $0\leq\eta_{\mu}\leq 2.4$ at 8 TeV. The
predictions include total theoretical uncertainties from quadratic sum of
scale, PDF, and $\alpha_{s}$ uncertainties, while the data include total
experimental uncertainty. In the lower panel, the ratios of the predictions to
the data for the $A_{\eta_{\mu}}$ are also displayed.

The NNLO predictions are also compared with the 8 TeV data from the reference
LHCb measurement Aaij:2015zlq  which was performed in the $\mu$ decay mode in
the forward acceptance region. Fiducial phase space requirement of
$p^{\mu}_{T}>$ 20 GeV in the forward region $2.0\leq\eta_{\mu}\leq 4.5$ is
imposed to compare with the LHCb data for the $A_{\eta_{\mu}}$ variable. Bin
edges for the $\eta_{\mu}$ are used identically from the LCHb measurement as
(2.00, 2.25), (2.25, 2.50), (2.50, 2.75), (2.75, 3.00), (3.00, 3.25), (3.25,
3.50), (3.50, 4.00), and (4.00, 4.50). Total theoretical and experimental
uncertainties are included to the central results for the predictions and
data, respectively. Comparisons of the predicted $A_{\eta_{\mu}}$
distributions from various PDF sets with the data are shown in Fig. 3. The
predictions are generally in good agreement with the data within uncertainties
throughout the entire $\eta_{\mu}$ ranges. The prediction using CT14 tends to
be slightly more consistent with the data over the results obtained using
other PDF sets. The predicted results from all the PDF sets show no
significant deviation from the FEWZ NNLO predictions that are presented in the
LHCb measurement.

To conclude here, the NNLO calculations are validated with the data for the
predicted distributions of the differential cross sections and $A_{\eta_{l}}$
variable in the $\mu$ decay mode at 8 TeV. The predictions exhibit no
significant deviations from the CMS and LHCb data within the quoted
uncertainties in both central and forward regions $0\leq\eta_{l}\leq 2.4$ and
$2.0\leq\eta_{l}\leq 4.5$. Predicted results obtained by using CT14 PDF set
reproduce data more consistently among several PDF sets that are being tested.
The predictions are also observed to be in agreement with the FEWZ NNLO
results reported in the CMS and LHCb measurements. These 8 TeV comparisons
encourage the extension of the NNLO calculations by the MATRIX+RadISH
framework to 13 TeV, the current center-of-mass energy of the LHC, where the
impact of several $p^{l}_{T}$ thresholds to the $W$ boson charge asymmetry
variables can be assessed further. Validation of the NNLO calculations in the
$e$ decay mode for the forward region $2.0\leq\eta_{e}\leq 4.25$ by using the
8 TeV LHCb data was reported before in Ref. Ocalan2021 .

Figure 3: The NNLO predictions for the $A_{\eta_{\mu}}$ variable from
MMHT2014, CT14, NNPDF3.1, and PDF4LHC15 PDF sets as a function of the
$\eta_{\mu}$. The predictions are compared with the LHCb data in the forward
region $2.0\leq\eta_{\mu}\leq 4.5$ at 8 TeV. The predictions include total
theoretical uncertainties, while the data include total experimental
uncertainty. In the lower panel, the ratios of the predictions to the data for
the $A_{\eta_{\mu}}$ are also displayed.

##  4 The $A_{\eta_{l}}$ and $A_{y_{W}}$ predictions at 13 TeV

The 13 TeV charge asymmetry predictions from the perturbative QCD calculations
of $W^{+}$ and $W^{-}$ boson differential cross sections are reported in this
section. The predictions are obtained at NNLO accuracy for the $A_{\eta_{l}}$
variable, where $l$ is either $\mu$ or $e$, by employing $p^{l}_{T}>$ 20, 25,
30, and 40 GeV thresholds in both central and forward phase space regions
$0\leq\eta_{l}\leq 4.5$. Total theoretical uncertainties are estimated using
the procedure as described in Sec. 2.3. The CT14 PDF set at NNLO accuracy is
used in the calculations. Bin edges for the $\eta_{l}$ are used identically
from the 8 TeV CMS measurement for the central region and are chosen for
broader ranges to ensure more stable numerical results in the forward region
as (0.00, 0.20), (0.20, 0.40), (0.40, 0.60), (0.60, 0.80), (0.80, 1.00),
(1.00, 1.20), (1.20, 1.40), (1.40, 1.60), (1.60, 1.85), (1.85, 2.10), (2.10,
2.40), (2.40, 2.70), (2.70, 3.00), (3.00, 3.50), (3.50, 4.00), and (4.00,
4.50). The predicted $A_{\eta_{l}}$ distributions from different
low-$p^{l}_{T}$ thresholds are shown in Fig. 4. The predicted $A_{\eta_{l}}$
numerical values corresponding to Fig. 4 are also given in Table 1. The
$A_{\eta_{l}}$ distributions increase towards to $\eta_{l}$ bin 3.00–3.50
where they start to turn down for lower values through very forward bins. The
$A_{\eta_{l}}$ distribution clearly exhibits dependency on the minimum value
of the $p^{l}_{T}$ in both central and forward regions. The $A_{\eta_{l}}$
values decrease in going from a lower $p^{l}_{T}$ threshold to a higher
$p^{l}_{T}$ threshold in the central region, while this correlation is
reversed in the most forward two bins 3.50–4.50. This is to say lepton charge
asymmetry is higher in the central region while it decreases more rapidly in
the most forward region when using a lower $p^{l}_{T}$ threshold.

Figure 4: The 13 TeV predicted distributions for the $A_{\eta_{l}}$ variable
from different low-$p^{l}_{T}$ thresholds $p^{l}_{T}>$ 20, 25, 30, and 40 GeV
in bins of the $\eta_{l}$. The NNLO predictions are obtained in both central
and forward regions by using the CT14 PDF set. Total theoretical uncertainties
are also included for the distributions.

The charge asymmetry predictions are also obtained directly for the
$A_{y_{W}}$ variable at NNLO accuracy as a function of the $y_{W}$, where
$y_{W}$ is calculated from the rapidities of the decay lepton and neutrino.
Similarly, fiducial phase space requirements and $y_{W}$ bin edges for the
$A_{y_{W}}$ predictions are used from the $A_{\eta_{l}}$ predictions at 13
TeV. The predicted $A_{y_{W}}$ distributions from different low-$p^{l}_{T}$
thresholds are compared as shown in Fig. 5. The predicted numerical values
from Fig. 5 are also provided in Table 2. The $A_{y_{W}}$ distributions
increase consistently towards higher ranges of the $y_{W}$ regardless of the
$p^{l}_{T}$ threshold. Contrary to the $A_{\eta_{l}}$ variable, the
$A_{y_{W}}$ variable does not discriminate clearly among the predictions from
different $p^{l}_{T}$ thresholds in the central region. However the prediction
with $p^{l}_{T}>$ 40 GeV tends to be slightly lower than other predictions in
the central region. The distributions start to increase more rapidly for a
lower threshold in the forward bins, where the distribution with $p^{l}_{T}>$
40 GeV threshold is predicted to be the most lower one in those bins. The
$A_{y_{W}}$ distribution increases the most with lowest threshold $p^{l}_{T}>$
20 GeV in the forward bins. The correlation between $A_{\eta_{l}}$ and
$A_{y_{W}}$ variables become more apparent in the forward region when the
distribution shapes approach each other in the presence of increasing values
of the $p^{l}_{T}$. Therefore, higher $p^{l}_{T}$ threshold relates
$A_{\eta_{l}}$ variable to $A_{y_{W}}$ variable increasingly more in the
forward region. The $A_{\eta_{l}}$ distribution with the highest $p^{l}_{T}$
threshold also approaches to the $A_{y_{W}}$ distribution more in the central
region. This observation can be supported by the explanation that the average
angle between the $W$ boson and the decay lepton decreases when $p^{l}_{T}$ is
increased. As a result, the correlation between $A_{\eta_{l}}$ and $A_{y_{W}}$
variables is enhanced. The $A_{\eta_{l}}$ distribution using a higher
threshold in the low-$p^{l}_{T}$ region probes the $A_{y_{W}}$ more by
allowing a finer dependence on the $x$ values, where a unique set of inputs
for the PDF determination can be obtained. Finally, the total theoretical
uncertainty of the $A_{y_{W}}$ prediction at NNLO is compared with the total
experimental uncertainty of a recent measurement preformed at the LHC
Sirunyan:2020oum , in which the W boson asymmetry is reported for the $y_{W}$
at 13 TeV. The theoretical uncertainties, which are in the range $\sim$2–14%
in the presence of the threshold $p^{l}_{T}>$ 25 from Table 2, are found to be
smaller than or at most comparable to the total experimental uncertainty of
the $A_{y_{W}}$ measurement in the $0\leq y_{W}\leq 2.5$ range. The total
experimental uncertainty increases towards higher ranges in the central
$y_{W}$ region, and therefore, $A_{y_{W}}$ measurements are anticipated to be
challenging in the forward region $2.0\leq y_{W}\leq 4.5$ in terms of
experimental precision to be achieved.

Figure 5: The 13 TeV predicted distributions for the $A_{y_{W}}$ variable from different low-$p^{l}_{T}$ thresholds $p^{l}_{T}>$ 20, 25, 30, and 40 GeV in bins of the $y_{W}$. The NNLO predictions are obtained in both central and forward regions by using the CT14 PDF set. Total theoretical uncertainties are also included for the distributions. Table 1: The predicted values for the lepton charge asymmetry (in percent) $A_{\eta_{l}}$(%) at NNLO accuracy by using CT14 PDF sets at 13 TeV. The predictions are reported for different $p^{l}_{T}$ thresholds in bins of the $\eta_{l}$. The predictions include total theoretical uncertainties. $\eta_{l}$ |  $p^{l}_{T}>$ 20 GeV |  $p^{l}_{T}>$ 25 GeV |  $p^{l}_{T}>$ 30 GeV |  $p^{l}_{T}>$ 40 GeV
---|---|---|---|---
0.00–0.20 | 11.93$\pm$0.4 | 07.43$\pm$0.4 | 05.39$\pm$0.7 | 04.09$\pm$0.9
0.20–0.40 | 11.28$\pm$0.4 | 09.00$\pm$0.5 | 05.39$\pm$0.6 | 03.89$\pm$0.4
0.40–0.60 | 11.37$\pm$0.3 | 09.25$\pm$0.6 | 05.40$\pm$0.8 | 04.17$\pm$0.6
0.60–0.80 | 12.02$\pm$0.5 | 09.39$\pm$0.5 | 05.74$\pm$0.6 | 05.44$\pm$0.5
0.80–1.00 | 12.24$\pm$0.5 | 11.30$\pm$0.4 | 06.55$\pm$0.7 | 05.64$\pm$0.7
1.00–1.20 | 13.45$\pm$0.6 | 12.07$\pm$0.5 | 08.96$\pm$0.7 | 05.87$\pm$0.5
1.20–1.40 | 17.31$\pm$0.6 | 14.37$\pm$0.6 | 08.50$\pm$0.6 | 07.48$\pm$0.9
1.40–1.60 | 17.94$\pm$0.6 | 15.78$\pm$0.7 | 09.74$\pm$0.8 | 08.14$\pm$0.9
1.60–1.85 | 19.70$\pm$1.0 | 19.63$\pm$0.8 | 11.93$\pm$0.7 | 09.05$\pm$0.7
1.85–2.10 | 21.25$\pm$0.8 | 19.61$\pm$0.6 | 12.87$\pm$0.6 | 10.17$\pm$0.5
2.10–2.40 | 25.16$\pm$0.9 | 22.60$\pm$0.7 | 17.71$\pm$0.8 | 12.37$\pm$0.6
2.40–2.70 | 25.50$\pm$0.7 | 22.53$\pm$0.8 | 19.40$\pm$0.8 | 16.29$\pm$0.9
2.70–3.00 | 24.93$\pm$0.8 | 26.17$\pm$0.8 | 20.57$\pm$0.8 | 16.92$\pm$0.8
3.00–3.50 | 20.42$\pm$1.0 | 21.79$\pm$1.1 | 21.21$\pm$0.9 | 19.37$\pm$0.9
3.50–4.00 | 07.72$\pm$0.9 | 11.22$\pm$1.1 | 12.62$\pm$0.8 | 20.25$\pm$1.0
4.00–4.50 | -14.93$\pm$1.1 | -12.27$\pm$1.1 | -01.82$\pm$1.0 | 14.09$\pm$1.2
Table 2: The predicted values for the $W$ boson charge asymmetry (in percent) $A_{y_{W}}$(%) at NNLO accuracy by using CT14 PDF sets at 13 TeV. The predictions are reported for different $p^{l}_{T}$ thresholds in bins of the $y_{W}$. The predictions include total theoretical uncertainties. $y_{W}$ |  $p^{l}_{T}>$ 20 GeV |  $p^{l}_{T}>$ 25 GeV |  $p^{l}_{T}>$ 30 GeV |  $p^{l}_{T}>$ 40 GeV
---|---|---|---|---
0.00–0.20 | 06.93$\pm$0.3 | 04.38$\pm$0.4 | 04.46$\pm$0.3 | 01.55$\pm$0.9
0.20–0.40 | 03.08$\pm$0.3 | 04.28$\pm$0.5 | 04.52$\pm$0.5 | 02.04$\pm$1.0
0.40–0.60 | 03.03$\pm$0.5 | 05.42$\pm$0.4 | 05.13$\pm$0.4 | 03.66$\pm$1.3
0.60–0.80 | 04.30$\pm$0.2 | 04.23$\pm$0.5 | 02.05$\pm$0.6 | 03.64$\pm$1.5
0.80–1.00 | 06.27$\pm$0.3 | 05.60$\pm$0.8 | 04.57$\pm$0.3 | 03.61$\pm$1.7
1.00–1.20 | 03.44$\pm$0.4 | 05.27$\pm$0.4 | 03.00$\pm$0.3 | 01.66$\pm$0.8
1.20–1.40 | 07.12$\pm$0.5 | 06.14$\pm$0.5 | 06.12$\pm$0.4 | 04.95$\pm$1.4
1.40–1.60 | 06.19$\pm$0.5 | 07.20$\pm$0.4 | 06.80$\pm$0.5 | 05.07$\pm$1.5
1.60–1.85 | 09.64$\pm$0.4 | 08.13$\pm$0.5 | 08.51$\pm$0.4 | 06.61$\pm$1.5
1.85–2.10 | 11.32$\pm$0.7 | 10.56$\pm$0.7 | 07.93$\pm$0.4 | 07.47$\pm$1.4
2.10–2.40 | 13.00$\pm$0.5 | 12.90$\pm$0.5 | 12.30$\pm$0.6 | 09.75$\pm$1.4
2.40–2.70 | 14.52$\pm$0.8 | 16.40$\pm$0.5 | 12.92$\pm$0.7 | 12.83$\pm$1.6
2.70–3.00 | 18.42$\pm$0.8 | 20.03$\pm$0.7 | 16.99$\pm$0.9 | 14.85$\pm$1.5
3.00–3.50 | 30.57$\pm$0.9 | 26.57$\pm$0.9 | 25.42$\pm$0.8 | 23.59$\pm$1.9
3.50–4.00 | 48.97$\pm$1.7 | 45.50$\pm$1.8 | 41.06$\pm$1.5 | 34.40$\pm$2.5
4.00–4.50 | 71.84$\pm$2.6 | 68.87$\pm$2.9 | 64.80$\pm$2.7 | 55.50$\pm$2.3

##  5 The predictions in the $p^{W}_{T}$ bins at 13 TeV

The charge asymmetry predictions are provided in the previous section for the
variables that are defined in terms of the $\eta_{l}$ and $y_{W}$. The charge
asymmetry can also be predicted in terms of the $p^{W}_{T}$ by using an
analogous definition with regards to Eqns. 1 and 2 as

$A_{p^{W}_{T}}=\frac{d\sigma(W^{+}\rightarrow
l^{+}\nu)/dp^{W}_{T}-d\sigma(W^{-}\rightarrow
l^{-}\bar{\nu})/dp^{W}_{T}}{d\sigma(W^{+}\rightarrow
l^{+}\nu)/dp^{W}_{T}+d\sigma(W^{-}\rightarrow l^{-}\bar{\nu})/dp^{W}_{T}}$ (4)

to acquire more insight into the $W$ boson production asymmetry in the
presence of different $p^{l}_{T}$ thresholds. The $A_{p^{W}_{T}}$ variable can
reveal additional information for the impact of using various $p^{l}_{T}$
thresholds on the predicted asymmetry in bins of the $p^{W}_{T}$. The
$A_{p^{W}_{T}}$ predictions at 13 TeV are first obtained at NNLO to test
dependency to low-$p^{l}_{T}$ thresholds of $p^{l}_{T}>$ 20, 25, 30, and 40
GeV. The predictions are obtained for the region $0\leq\eta_{l}\leq 4.5$ in
the $p^{W}_{T}$ range 0–150 GeV as shown in Fig. 6. The predicted numerical
values at NNLO from Fig. 6 are also provided in Table 3. The $A_{p^{W}_{T}}$
distributions consistently decrease towards higher bin ranges of the
$p^{W}_{T}$ at all $p^{l}_{T}$ thresholds except for the highest bin, however
the distribution is more flat at the lowest threshold of 20 GeV. The
$A_{p^{W}_{T}}$ distribution obviously decreases in going from the prediction
using a lower threshold to the prediction using a higher threshold in
$p^{l}_{T}$ throughout the entire $p^{W}_{T}$ region. Therefore, the
increasing $p^{l}_{T}$ threshold has an impact to yield lower $A_{p^{W}_{T}}$
values in both central and forward regions. These results already emphasize
that the $p^{W}_{T}$ and $p^{l}_{T}$ are closely correlated kinematical
variables.

Next the $A_{p^{W}_{T}}$ distributions at 13 TeV are obtained by means of the
matched calculation of the resummation with the fixed-order NNLO to have more
accurate predictions. The $p^{W}_{T}$ distribution is affected by soft and
collinear gluon radiation at low values, where the fixed-order calculations
are unable to sufficiently account for it. Thereby, resummation of large
logarithmic corrections are also needed to model $p^{W}_{T}$ more accurately
at low values. The differential cross sections of W+ and $W^{-}$ bosons as a
function of the $p^{W}_{T}$ are predicted by means of the matched prediction
at NNLO+N3LL accuracy to achieve more reliable $A_{p^{W}_{T}}$ predictions.
The $A_{p^{W}_{T}}$ distributions that are predicted at NNLO+N3LL accuracy by
employing different $p^{l}_{T}$ thresholds are shown in Fig. 7. In Table 4,
the predicted numerical values from Fig. 7 are also supplied. Similarly, the
predicted values decrease continuously with increasing $p^{l}_{T}$ threshold
apart from the highest bin, as being also tested at a higher accuracy of
NNLO+N3LL. The highest threshold yields the lowest $A_{p^{W}_{T}}$ values
through the entire $p^{W}_{T}$ ranges. Therefore, it is shown that the
$A_{p^{W}_{T}}$ predictions also depend on the $p_{T}$ threshold applied on
the decay lepton which can be attributed to the V–A structure of the $W$ boson
couplings to fermions.

To this end, experimental uncertainty of normalized differential cross section
as a function of the $p^{W}_{T}$ is checked from the CMS measurement
Khachatryan:2016nbe  to anticipate a comparison between experimental
uncertainty that can be achieved and the theoretical uncertainty estimated at
NNLO(+N3LL) accuracy for the $A_{p^{W}_{T}}$ at 13 TeV. Experimental
uncertainty including both systematic and statistical components is
$\sim$1.2–6.0%, whereas the total theoretical uncertainty ranges in 2.1–9.0%
(1.5–6.0%) in the NNLO (NNLO+N3LL) predictions for differential cross section
of the $p^{W}_{T}$ in the range 0-150 GeV. Therefore, experimental uncertainty
is anticipated to be lower than (comparable to) the theoretical uncertainty in
the NNLO (NNLO+N3LL) predictions of the $A_{p^{W}_{T}}$ at 13 TeV.

Figure 6: The 13 TeV predicted distributions for the $A_{p^{W}_{T}}$ variable based on different low-$p^{l}_{T}$ thresholds $p^{l}_{T}>$ 20, 25, 30, and 40 GeV in bins of the $p^{W}_{T}$. The predictions at NNLO accuracy are obtained in both central and forward regions by using the CT14 PDF set. Total theoretical uncertainties are also included for the distributions. Figure 7: The 13 TeV predicted distributions for the $A_{p^{W}_{T}}$ variable based on different low-$p^{l}_{T}$ thresholds $p^{l}_{T}>$ 20, 25, 30, and 40 GeV in bins of the $p^{W}_{T}$. The merged predictions at NNLO+N3LL accuracy are obtained in both central and forward regions by using the CT14 PDF set. Total theoretical uncertainties are also included for the distributions. Table 3: The predicted values for the charge asymmetry (in percent) $A_{p^{W}_{T}}$(%) at NNLO accuracy by using CT14 PDF set at 13 TeV. The predictions are reported for different $p^{l}_{T}$ thresholds in bins of the $p^{W}_{T}$. The predictions include total theoretical uncertainties. $p^{W}_{T}$ |  $p^{l}_{T}>$ 20 GeV |  $p^{l}_{T}>$ 25 GeV |  $p^{l}_{T}>$ 30 GeV |  $p^{l}_{T}>$ 40 GeV
---|---|---|---|---
0–25 | 16.04$\pm$0.7 | 16.15$\pm$0.6 | 14.63$\pm$0.6 | 13.04$\pm$0.6
25–30 | 15.00$\pm$1.9 | 13.06$\pm$1.8 | 11.78$\pm$1.8 | 09.94$\pm$1.8
30–40 | 13.67$\pm$1.9 | 12.90$\pm$1.9 | 12.24$\pm$2.0 | 08.58$\pm$1.8
40–60 | 13.50$\pm$2.1 | 11.53$\pm$2.1 | 08.88$\pm$1.9 | 06.72$\pm$2.0
60–100 | 12.79$\pm$2.2 | 10.66$\pm$2.2 | 09.20$\pm$2.2 | 06.12$\pm$2.0
100–150 | 13.67$\pm$2.6 | 12.03$\pm$2.5 | 11.25$\pm$2.5 | 07.52$\pm$2.4
Table 4: The predicted values for the charge asymmetry (in percent) $A_{p^{W}_{T}}$(%) at NNLO+N3LL accuracy by using CT14 PDF set at 13 TeV. The predictions are reported for different $p^{l}_{T}$ thresholds in bins of the $p^{W}_{T}$. The predictions include total theoretical uncertainties. $p^{W}_{T}$ |  $p^{l}_{T}>$ 20 GeV |  $p^{l}_{T}>$ 25 GeV |  $p^{l}_{T}>$ 30 GeV |  $p^{l}_{T}>$ 40 GeV
---|---|---|---|---
0–25 | 15.97$\pm$0.5 | 15.97$\pm$0.5 | 14.62$\pm$0.5 | 13.13$\pm$0.8
25–30 | 15.35$\pm$0.7 | 14.27$\pm$0.6 | 12.38$\pm$0.7 | 10.32$\pm$0.7
30–40 | 14.34$\pm$0.5 | 14.05$\pm$0.5 | 12.58$\pm$0.5 | 09.13$\pm$0.6
40–60 | 14.15$\pm$0.5 | 12.82$\pm$0.5 | 09.79$\pm$0.4 | 07.40$\pm$0.8
60–100 | 13.40$\pm$1.1 | 11.72$\pm$1.1 | 09.75$\pm$1.0 | 06.70$\pm$1.3
100–150 | 13.81$\pm$1.7 | 12.26$\pm$1.7 | 11.25$\pm$1.7 | 07.77$\pm$2.0

##  6 Conclusion

In this paper a dedicated study of the theoretical predictions for the $W$
boson charge asymmetries in $pp$ collisions is presented. The predictions are
obtained with the inclusion of NNLO corrections in perturbative QCD for both
the central and forward phase space regions $0\leq\eta_{l}\leq 4.5$. This
phase space region for the charge asymmetries allows to probe the relative $u$
and $d$ quark distributions in the proton at very small and large $x$ values.
The predictions that are based on various PDF models are compared with the CMS
and LHCb data for the lepton charge asymmetry $A_{\eta_{l}}$ in the muon decay
mode at 8 TeV. The $A_{\eta_{l}}$(%) distributions are generally observed to
be in good agreement with the 8 TeV data within the quoted uncertainties. The
distributions using the CT14 PDF set are shown to reproduce the data slightly
better over the other PDF sets that are being tested. These comparisons
enabled the justification of the predictions and encouraged the extension of
the study to 13 TeV which is the current center-of-mass energy of $pp$
collisions at the LHC.

The predicted distributions are presented for the charge asymmetries
$A_{\eta_{l}}$ and $A_{y_{W}}$ at NNLO accuracy at 13 TeV. Various increasing
low-$p^{l}_{T}$ thresholds $p^{l}_{T}>$ 20, 25, 30, and 40 GeV in the region
$0\leq\eta_{l}\leq 4.5$ are used to assess the impact on the predicted
asymmetries. The $A_{\eta_{l}}$ distributions from different $p^{l}_{T}$
thresholds increase in the central region and decrease in the forward region.
The $A_{\eta_{l}}$ distribution exhibits clear dependency to $p^{l}_{T}$,
where it increases more in correlation with decreasing threshold in the
central region up to the $\eta_{l}$ bin 3.0–3.5. After this bin, the
$A_{\eta_{l}}$ distribution decreases more with lower threshold in the very
forward range 3.5–4.5. The predicted $A_{\eta_{l}}$ is observed to be the
lowest (highest) in the central (forward) region with the highest threshold
$p^{l}_{T}>$ 40 GeV that is being tested. Furthermore, the charge asymmetry
distributions that are obtained directly with the $A_{y_{W}}$ are observed to
increase continuously towards higher ranges of the $y_{W}$. The $A_{y_{W}}$ is
predicted to be larger at higher $y_{W}$ bins which can be attributed to
increasing ratio of the $u$ and $d$ quark distribution functions while probing
the valence quarks more in those higher bins. Contrary to the $A_{\eta_{l}}$,
the $A_{y_{W}}$ distribution does not strongly discriminate among different
thresholds in most of the central ranges. The $A_{y_{W}}$ distribution
increases more in correlation with lower threshold in the forward $y_{W}$
bins, where it is lower with the $p^{l}_{T}>$ 40 GeV threshold. In the
$A_{\eta_{l}}$ and $A_{y_{W}}$ predictions, it has been clearly shown that
$A_{\eta_{l}}$ distribution gets closer to the $A_{y_{W}}$ distribution in the
presence of the highest threshold $p^{l}_{T}>$ 40 GeV in both the central and
forward regions. This observation is in support of the point that the average
angle between the $W$ boson and the decay lepton is decreased with a higher
$p^{l}_{T}$ requirement, and as a result, the correlation between the
$A_{\eta_{l}}$ and $A_{y_{W}}$ is enhanced accordingly.

The 13 TeV distributions are also reported for the charge asymmetry in bins of
the $p^{W}_{T}$, $A_{p^{W}_{T}}$, at both NNLO and NNLO+N3LL, where the
accuracy is remarkably improved in the matched predictions. The predicted
distributions decrease continuously towards the higher $p^{W}_{T}$ ranges
apart from the very last bin at 100–150 GeV. The $A_{p^{W}_{T}}$ distribution
is observed to exhibit clear dependency to $p^{l}_{T}$ threshold as
anticipated. It has been shown that the distribution remains more flat at the
lowest threshold of 20 GeV while it decreases more in correlation with the
increasing threshold. Based on the results presented, it was shown that the
$A_{p^{W}_{T}}$ can be used as an alternate probe for the $W$ boson charge
asymmetries.

Finally, the study shows the potential impact of the $p^{l}_{T}$ dependence in
the $W$ boson leptonic decay in terms of the charge asymmetries. The predicted
results can further be used for improving the existing constraints on the
ratio of $u$ and $d$ quark distribution functions in the range $10^{-4}<x<1$,
investigating into differences among PDF models, contribution in general to
accurate PDF determinations.

## Conflict of interest

The author declares that he has no conflict of interest.

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