#  Associated Higgs boson production with $Z$ boson
in the minimal $U(1)_{X}$ extended Standard Model

Arindam<EMAIL_ADDRESS>Department of Physics,
Osaka University, Toyonaka, Osaka 560-0043, Japan  Nobuchika
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Alabama,
Tuscaloosa, Alabama 35487, USA

###### Abstract

The minimal $U(1)_{X}$ extension of the Standard Model (SM) is a simple and
well-motivated extension of the SM, which supplements the SM with the seesaw
mechanism for naturally generating the light neutrino masses and offers
various interesting phenomenologies. In the model, the $U(1)_{X}$ charge of
each SM field is characterized by the $U(1)_{X}$ charge of the SM Higgs
doublet with a free parameter $x_{H}$. Due to the $U(1)_{X}$ charge of the
Higgs doublet, the Higgs boson has a trilinear coupling with $Z$ boson and the
$U(1)_{X}$ gauge boson ($Z^{\prime}$) for $x_{H}\neq 0$. With this coupling, a
new process for the associated Higgs boson production with a $Z$ boson arises
through a $Z^{\prime}$ boson in the $s$-channel at high energy colliders. In
this paper, we calculate the associated Higgs boson production at high energy
proton-proton and electron-positron colliders and show interesting effects of
the new $Z^{\prime}$ boson mediated process, which can be tested in the
future.

††preprint: OU-HET-1068

##  I Introduction

The discovery of the Higgs boson Chatrchyan:2012xdj ; Aad:2012tfa  of the
Standard Model (SM) is definitely a very important milestone in the history of
particle physics. It ends the quest of searching for the last SM particle
after a long time and then further solidifies the plinth of the SM. Although
the observation of the Higgs boson ensures its existence and its role for the
electroweak symmetry breaking, more precise measurements of the Higgs boson
properties are necessary for verifying the SM Higgs sector and/or for
revealing new physics beyond the SM (BSM).

A simple extension of the SM can be possible by an additional $U(1)$ gauge
symmetry Appelquist:2002mw  which predicts a neutral BSM gauge boson
($Z^{\prime}$). This extra gauge boson has been searched for at the Tevatron
Aaltonen:2008vx ; Abazov:2010ti , LEP Chiappetta:1996km ; Barger:1996kr ;
Lynch:2000md  and LHC Aad:2019fac ; CMS:2019tbu  with final state
dileptons/dijes. With the precision experiments by the LEP experiment, the
strong constraints on the $Z-Z^{\prime}$ mixing has been studied in
Carena:2004xs ; Petriello:2008pu . The collider phenomenology of the
$Z^{\prime}$ boson has been intensively studied (see, for example,
Rizzo:1985kn ; Nandi:1986rg ; Baer:1987eb ; Barger:1987xw ; Gunion:1987jd ;
Hewett:1988xc ; Barger:1988sq ; Feldman:2006wb ; Langacker:2008yv ; Lee:2008cn
; Salvioni:2009mt ; Barger:2009xg ; Li:2013ava ) with the final state leptons
and quarks. Apart from the pair fermion productions, another important process
to study the $Z^{\prime}$ boson is an associated production of the SM Higgs
boson ($h$) with the SM $Z$ boson, such as $pp\rightarrow
Z^{\prime}\rightarrow Zh$ at the LHC, which can arise from a trilinear
coupling $Z^{\prime}-Z-h$ if the SM Higgs doublet has an extra $U(1)$ gauge
charge.

To study the associated Higgs boson production, we consider in this paper the
minimal $U(1)_{X}$ extension of the SM Oda:2015gna ; Das:2016zue , where the
particle content is the same as the minimal $B-L$ (baryon minus lepton number)
model Mohapatra:1980qe ; Marshak:1979fm ; Wetterich:1981bx ; Masiero:1982fi ;
Buchmuller:1991ce  while the $U(1)_{X}$ charge of each SM particle is defined
as a linear combination of its $U(1)_{Y}$ hyper-charge and $U(1)_{B-L}$
charge. In the presence of three right-handed neutrinos (RHSs), this model is
free from all the gauge and the mixed gauge-gravitational anomalies. After the
breaking of $U(1)_{X}$ and electroweak symmetries, the seesaw mechanism works
to generate light neutrino masses naturally. The minimal $U(1)_{X}$ model
provides interesting high energy collider phenomenologies. For the current LHC
bound from the search for the $Z^{\prime}$ boson resonance with dilepton final
states, see, for example, Okada:2016tci ; Okada:2017dqs ; Okada:2018tgy . A
suitable choice of the $U(1)_{X}$ charges leads to a significant enhancement
of the RHN pair production cross section from the $Z^{\prime}$ boson which can
increase the potential of the RHN discovery at the future high energy
colliders through the prompt RHN decay Das:2017flq ; Das:2017deo ; Das:2019pua
; Choudhury:2020cpm  and the displaced vertex search for a long-lived RHN
Das:2018tbd ; Das:2019fee ; Chiang:2019ajm . The main point of this paper is
to show interesting effects of the $Z^{\prime}$ mediated process on the
associated Higgs boson production at high energy colliders.

We arrange this paper as follows: In Sec. II we describe the minimal
$U(1)_{X}$ model and the interactions relevant to our analysis. In Sec. III we
calculate the $Zh$ production cross sections at the proton-proton and
electron-positron colliders and show the effects of $Z^{\prime}$ boson
mediated process. We conclude this paper in Sec. IV.

##  II Minimal gauged U(1) extension of the Standard Model

| SU(3)c | SU(2)L | U(1)Y | U(1)X
---|---|---|---|---
$q^{\alpha}_{L}$ | 3 | 2 | $1/6$ | $(1/6)x_{H}+(1/3)x_{\Phi}$
$u^{\alpha}_{R}$ | 3 | 1 | $2/3$ | $(2/3)x_{H}+(1/3)x_{\Phi}$
$d^{\alpha}_{R}$ | 3 | 1 | $-1/3$ | $-(1/3)x_{H}+(1/3)x_{\Phi}$
$\ell^{\alpha}_{L}$ | 1 | 2 | $-1/2$ | $(-1/2)x_{H}-x_{\Phi}$
$e^{\alpha}_{R}$ | 1 | 1 | $-1$ | $-x_{H}-x_{\Phi}$
$H$ | 1 | 2 | $-1/2$ | $(-1/2)x_{H}$
$N^{\alpha}_{R}$ | 1 | 1 | $0$ | $-x_{\Phi}$
$\Phi$ | 1 | 1 | $0$ | $+2x_{\Phi}$
Table 1:  Particle content of the minimal $U(1)_{X}$ model, where $x_{H}$ and
$x_{\Phi}$ are real parameters, and $\alpha=1,2,3$ is the three generation
flavor index. Since the $U(1)_{X}$ gauge coupling is a free parameter, we fix
$x_{\Phi}=1$ without loss of generality.

The simple gauged $U(1)$ extension of the SM is based on the gauge group
$SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{X}$. The particle
content of the model is listed in Tab. 1. We consider a conventional case that
the $U(1)_{X}$ charge assignment is flavor-universal Appelquist:2002mw . In
addition to the $U(1)_{X}$ gauge boson $Z^{\prime}$, three RHNs and one
$U(1)_{X}$ Higgs field ($\Phi$) are introduced. The particle content of the
model is basically the same as the one of the minimal $B-L$ model while the
$U(1)_{X}$ charge of each SM field is generalized to a linear combination of
its $U(1)_{Y}$ and $B-L$ charges with real parameters $x_{H}$ and $x_{\Phi}$.
In the presence of three RHNs, this model is anomaly-free. Since the
$U(1)_{X}$ gauge coupling is a free parameter, we can set $x_{\Phi}=1$ without
loss of generality. The minimal $B-L$ model is realized in the minimal
$U(1)_{X}$ model as the limit of $x_{H}\rightarrow 0$.

With the RHNs, the extended part of the SM Yukawa sector can be written as

$\displaystyle\mathcal{L}_{Y}\supset-\sum_{\alpha,\beta=1}^{3}Y_{D}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}HN_{R}^{\beta}-\frac{1}{2}\sum_{\alpha=1}^{3}Y_{N}^{\alpha}\Phi\overline{N_{R}^{\alpha\,C}}N_{R}^{\alpha}+\rm{H.c.},$
(1)

where $C$ stands for the charge-conjugation. The first and second terms in the
right-hand side are the Dirac-type and Majorana-type Yukawa interactions,
respectively. Here, we work in the bases where $Y_{N}$ is flavor-diagonal.
After the $U(1)_{X}$ symmetry breaking by a vacuum expectation value (VEV) of
the Higgs field $\Phi$, the Majorana masses for the RHNs are generated, while
the neutrino Dirac mass terms are generated from the SM Higgs doublet VEV.
With these mass terms, the seesaw mechanism works to naturally generate the
light SM neutrino masses.

The (renormalizable) Higgs potential of the model is given by

$\displaystyle V\
=m_{\Phi}^{2}(\Phi^{\dagger}\Phi)+\lambda_{\Phi}(\Phi^{\dagger}\Phi)^{2}+\
m_{H}^{2}(H^{\dagger}H)+\lambda_{H}(H^{\dagger}H)^{2}+\lambda^{\prime}(H^{\dagger}H)(\Phi^{\dagger}\Phi)\,.$
(2)

For simplicity, we assume $\lambda^{\prime}$ is negligibly small and hence the
scalar potential can be analyzed separately for $\Phi$ and $H$ as a good
approximation. By choosing the parameters suitably, we can realize the
$U(1)_{X}$ and electroweak symmetry breaking at the potential minimum and
parametrize the Higgs fields in the unitary gauge as follows:

$\displaystyle\quad\Phi\ =\ \frac{1}{\sqrt{2}}(v_{\Phi}+\phi)\,\quad{\rm
and}\,\quad H\ =\ \frac{1}{\sqrt{2}}\begin{pmatrix}v+h\\\ 0\end{pmatrix}\,,$
(3)

where $v_{\Phi}$ and $v\simeq 246$ GeV are the Higgs VEVs, and $\phi$ and $h$
are physical Higgs bosons. After the symmetry breaking, the mass of
$Z^{\prime}$ boson, the Majorana masses for the RHNs and the neutrino Dirac
masses are generated as

$\displaystyle m_{Z^{\prime}}\ $ $\displaystyle=$ $\displaystyle\
g_{X}\sqrt{4v_{\Phi}^{2}+\frac{1}{4}x_{H}^{2}v^{2}}\ \simeq\
2g_{X}v_{\Phi}\quad(v_{\Phi}\gg v),$ (4) $\displaystyle m_{N_{i}}\ $
$\displaystyle=$ $\displaystyle\ \frac{Y^{i}_{N}}{\sqrt{2}}v_{\Phi},$ (5)
$\displaystyle m_{D}^{ij}\ $ $\displaystyle=$ $\displaystyle\
\frac{Y_{D}^{ij}}{\sqrt{2}}v,$ (6)

where $g_{X}$ is the $U(1)_{X}$ gauge coupling. Here, we have used the LEP
LEP:2003aa , Tevatron Carena:2004xs  and LHC Amrith:2018yfb  constraints which
indicate $v_{\Phi}^{2}\gg v^{2}$.

We now consider the $Z^{\prime}$ decay. Since we focus on the $Z^{\prime}$
boson mass in the multi-TeV range in this paper, we neglect the SM fermion
masses and for this case, the partial decay width formulas are listed in
Das:2017flq . In the following analysis, we assume the RHNs are all heavier
than the $Z^{\prime}$ boson, for simplicity. Since the SM Higgs doublet has
the $U(1)_{X}$ charge $-\frac{1}{2}x_{H}$, its gauge interactions leads to the
trilinear coupling $Z^{\prime}-Z-h$. The relevant part of the Lagrangian is
given by

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\Big{|}\Big{\\{}-\frac{i}{2}g_{Z}Z_{\mu}-ig_{X}Z_{\mu}^{\prime}(-\frac{1}{2}x_{H})\Big{\\}}\frac{1}{\sqrt{2}}(v+h)\Big{|}^{2}$
(7) $\displaystyle=$
$\displaystyle\frac{1}{8}\Big{(}g_{Z}^{2}Z_{\mu}Z^{\mu}+g_{X}^{2}x_{H}^{2}Z^{\prime}_{\mu}{Z^{\prime}}^{\mu}-2g_{Z}\Big{(}g_{X}x_{H}\Big{)}Z^{\prime}_{\mu}Z^{\mu}\Big{)}v^{2}\Big{(}1+2\frac{h}{v}+\frac{h^{2}}{v^{2}}\Big{)}$
$\displaystyle\supset$
$\displaystyle-\frac{1}{2}g_{Z}(g_{X}x_{H})vhZ^{\mu}Z^{\prime}_{\mu}$
$\displaystyle=$ $\displaystyle-
M_{Z}\Big{(}g_{X}x_{H}\Big{)}hZ^{\mu}Z^{\prime}_{\mu}.$

As we expect, the trilinear coupling is proportional to $g_{X}x_{H}$. We then
calculate the partial decay width of the process $Z^{\prime}\rightarrow Zh$ to
be

$\displaystyle\Gamma(Z^{\prime}\rightarrow Zh)$ $\displaystyle=$
$\displaystyle\frac{{g_{X}}^{2}{x_{H}}^{2}}{48\pi}M_{Z^{\prime}}\sqrt{\lambda[a,b,c]}\left(\lambda[a,b,c]+12b\right)\simeq\frac{{g_{X}}^{2}{x_{H}}^{2}}{48\pi}M_{Z^{\prime}},$
(8)

where $\lambda[x,y,z]=x^{2}+y^{2}+z^{2}-2xy-2yz-2zx$, and $a=1$,
$b=\left(\frac{M_{Z}}{M_{Z^{\prime}}}\right)^{2}\ll 1$ and
$c=\left(\frac{m_{h}}{M_{Z^{\prime}}}\right)^{2}\ll 1$, respectively, with the
$Z$ boson mass $M_{Z}=91.2$ GeV and the Higgs boson mass $m_{h}=125$ GeV.

t

Figure 1:  Left Panel: Branching ratios of the $Z^{\prime}$ into di-electron
(blue, solid), up-type di-quark (black, dashed) and down-type di-quark
(purple, dot dashed) quarks as a function of $x_{H}$. Right Panel: Comparison
between the branching rations, ${\rm BR}(Z^{\prime}\rightarrow 2e)$ (red,
dashed) and ${\rm BR}(Z^{\prime}\rightarrow Zh)$ (blue, solid) as a function
of $x_{H}$. We have fixed $M_{Z^{\prime}}=4$ TeV.

In the left panel of Fig. 1, we show the branching ratios to pairs of up-type
quarks, down-type quarks and charged leptons, respectively, as a function of
$x_{H}$, for $M_{Z^{\prime}}=4$ TeV. We can see the dramatic change of the
branching ratios. In the right panel, we show the branching ratio to the $Zh$
mode as a function of $x_{H}$, along with the dilepton mode. Since the most
severe bound on $Z^{\prime}$ boson production at the LHC is obtained from the
final state dilepton search, we are particularly interested in the ${\rm
BR}(Z^{\prime}\rightarrow Zh)$ value compared with the ${\rm
BR}(Z^{\prime}\rightarrow 2e)$ value. We can see that the
$Z^{\prime}\rightarrow Zh$ mode dominates over the $Z^{\prime}\rightarrow 2e$
mode for $-3\lesssim x_{H}\lesssim-0.8$. The ratio of the two branching
rations is shown in Fig. 2 as a function of $x_{H}$ for $M_{Z^{\prime}}=4$
TeV. The ratio exhibits its maximum at $x_{H}\simeq-1.2$, where ${\rm
BR}(Z^{\prime}\rightarrow Zh)\simeq 4\times{\rm BR}(Z^{\prime}\rightarrow
2e)$. For $M_{Z^{\prime}}^{2}\gg M_{Z}^{2},m_{h}^{2}$, this ratio is
approximately expressed as

$\displaystyle\frac{{\rm BR}(Z^{\prime}\rightarrow Zh)}{{\rm
BR}(Z^{\prime}\rightarrow
2e)}\simeq\frac{x_{H}^{2}}{\frac{5}{2}x_{H}^{2}+6x_{H}+4}.$ (9)

Figure 2: Ratio of the branching ratios of $Z^{\prime}$ decaying to $Zh$ to
$2e$ as a function of $x_{H}$ for $M_{Z^{\prime}}=4$ TeV.

To conclude this section, for readers convenience, we give the Higgs boson
coupling with a pair of $Z$ bosons in the SM from the $SU(2)_{L}\otimes
U(1)_{Y}$ gauge interactions of the SM Higgs doubelt. The relevant part of the
Lagrangian is given by

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\Big{|}-\frac{i}{2}g_{z}Z_{\mu}\frac{1}{\sqrt{2}}(v+h)\Big{|}^{2}$
(10) $\displaystyle=$
$\displaystyle\frac{g_{z}^{2}}{8}Z_{\mu}Z^{\mu}(v^{2}+2vh+h^{2})$
$\displaystyle\supset$ $\displaystyle\frac{M_{Z}^{2}}{v}hZ_{\mu}Z^{\mu}.$

Through the interactions in Eqs. (8) and (10) in the minimal $U(1)_{X}$ model,
the Higgs boson can be produced in association with a $Z$ boson at the high
energy colliders.

##  III $Zh$ production at the colliders

In this section, we study the $Zh$ production at the high energy colliders,
namely, the proton-proton collider and the electron-positron collider. At both
colliders, the final state $Zh$ is produced through the $Z$ and $Z^{\prime}$
bosons in the $s$-channel. For the $Z^{\prime}$ boson with mass in the multi-
TeV range, there is an essential difference between the $Z^{\prime}$ boson
phenomenologies at the two colliders. The energy of the proton-proton collider
is high energy to produce the $Z^{\prime}$ boson resonance, so that we focus
on the $Zh$ production through the resonant $Z^{\prime}$ boson production and
its subsequent decay to $Zh$ (for this process, see the right Feynman diagram
in Fig. 3). On the other hand, the energy of the electron-positron collider
will not be high enough, and the $Z^{\prime}$ boson effect on the $Zh$
production is mainly from the interference between the $Z$ and $Z^{\prime}$
mediated processes (see the left Feynman diagram in Fig. 3).

Figure 3: Production mechanism of the SM Higgs boson in association with a $Z$
boson at the proton-proton hadron collider (left) and the electron-positron
collider (right).

###  III.1 Proton-proton collider

We first consider the $Zh$ production at the proton-proton collider. As
previously mentioned, we focus on the $Zh$ production from the resonant
$Z^{\prime}$ boson production and its subsequent decay to $Zh$. Since $g_{X}$
is constrained to be small from the LHC constraint, the total $Z^{\prime}$
boson decay width is vary narrow. In tis case, we can use the narrow width
approximation to evaluate the $Z^{\prime}$ boson production cross section at
the proton-proton collider. In this approximation, we express the total
$Z^{\prime}$ boson production cross section by

$\displaystyle\sigma(pp\rightarrow Z^{\prime})\ =\ 2\sum_{q,\,\bar{q}}\int
dx\int dy\,f_{q}(x,Q)\,f_{\bar{q}}(y,Q)\,\hat{\sigma}(\hat{s}),$ (11)

where$f_{q}$ ($f_{\bar{q}}$) is the parton distribution function (PDF) for a
quark (anti-quark), $\hat{s}=xys$ is the invariant mass squared of the
colliding partons (quarks) with a center-of-mass energy $\sqrt{s}$ of the
proton-proton collider, and the $Z^{\prime}$ boson production cross section at
the parton level is expressed as

$\displaystyle\hat{\sigma}(\hat{s})\ =\
\frac{4\pi^{2}}{3}\frac{\Gamma(Z^{\prime}\rightarrow
q\bar{q})}{m_{Z^{\prime}}}\,\delta(\hat{s}-m_{Z^{\prime}}^{2}).$ (12)

Here, $\Gamma(Z^{\prime}\rightarrow q\bar{q})$ is the $Z^{\prime}$ boson
partial decay width to $q\bar{q}$. For the decay modes to the up-type and
down-type quarks are explicitly given by

$\displaystyle\Gamma(Z^{\prime}\rightarrow u\bar{u})$ $\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{288\pi}(8+20x_{H}+17x_{H}^{2})M_{Z^{\prime}},$
$\displaystyle\Gamma(Z^{\prime}\rightarrow d\bar{d})$ $\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{288\pi}(8-4x_{H}+5x_{H}^{2})M_{Z^{\prime}}.$
(13)

In our analysis, we employ CTEQ6L Pumplin:2002vw  for the PDF with a
factorization scale $Q=M_{Z^{\prime}}$. We then simply express the $Zh$
production cross section by

$\displaystyle\sigma(pp\rightarrow Z^{\prime}\rightarrow
Zh)\simeq\sigma(pp\rightarrow Z^{\prime})\times{\rm BR}(Z^{\prime}\rightarrow
Zh).$ (14)

As we have investigated in the previous section, ${\rm
BR}(Z^{\prime}\rightarrow Zh)$ highly depends on $x_{H}$ and, in particular,
it is enhanced for $x_{H}\simeq-1.2$ compared to ${\rm
BR}(Z^{\prime}\rightarrow 2e)$.

Figure 4: The dilepton production cross section as a function of the
$Z^{\prime}$ mass $(M_{Z^{\prime}})$ for three different values of $x_{H}$ at
different center of mass energies of proton-proton collider. We have fixed the
$x_{H}$ to be $0$ (blue, solid), $-\frac{4}{3}$ (blue, dashed) and $1$ (blue,
dotted).

Figure 5:  The $Zh$ production cross section (blue, dashed) from the
$Z^{\prime}$ boson decay as a function of $Z^{\prime}$ mass $(M_{Z^{\prime}})$
(left column) and $x_{H}$ (right column). The corresponding dilepton
production cross sections (blue, solid) are shown for comparison.  Figure 6:
The prospective upper bound on $g_{X}$ for $M_{Z^{\prime}}=7.5$ TeV by the HL-
LHC experiment with a 3000/fb luminosity.

We are now ready to show the results for our production cross section
calculations. In addition to the current LHC energy of $\sqrt{s}=13$ TeV, we
have examined various center-of-mass energies for the future hadron collider
projects, such as $\sqrt{s}=27$ TeV, $33$ TeV, $87$ TeV and $100$ TeV. We
first show in Fig. 4 the dilepton production cross section as a function of
$M_{Z^{\prime}}$ for three different $x_{H}$ values and various $\sqrt{s}$
values. Since the cross section is proportional to ${g_{X}}^{2}$ in the narrow
width approximation, we have shown the cross section normalized by
${g_{X}}^{2}$. Comparing the results shown in the top-left panel with the
final ATLAS results on the $Z^{\prime}$ boson resonance search at the LHC
Run-2 Aad:2012tfa , we can read off the upper bound on $g_{X}$ as a function
of $M_{Z^{\prime}}$. For example, $g_{X}\lesssim 0.1$ for $M_{Z^{\prime}}=4$
TeV and $x_{H}\sim-1$. See Das:2019fee  for details. In Fig. 5, the left
column shows the resultant $Zh$ production cross section as a function of
$M_{Z^{\prime}}$ for the three $x_{H}$ values and various $\sqrt{s}$ values,
while the resultant $Zh$ production cross section as a function of $x_{H}$ for
$M_{Z^{\prime}}=4$ TeV is shown in right column. In Fig. 5, we have also shown
the dilepton cross section results for comparison.

Finally in this subsection, we discuss the prospect of the future $Z^{\prime}$
boson search limit by the High-Luminosity LHC (HL-LHC) experiment with its
goal integrated luminosity of 3000/fb. We refer the simulation result
presented in the ATLAS Technical Design Report ATL-TDR  for the prospective
upper bound on the cross section $\sigma(pp\rightarrow Z^{\prime}\rightarrow
e^{+}e^{-})<4\times 10^{-5}$ pb for $M_{Z^{\prime}}=7.5$ TeV. We can interpret
this prospective bound as the upper bound on $g_{X}$ as a function of $x_{H}$.
Our result is shown in Fig. 6. As we can see from Fig. 1, ${\rm
BR}(Z^{\prime}\rightarrow 2e)$ exhibits its minimum at $x_{H}\simeq-1.2$ and
hence the weakest HL-LHC constraint is obtained for $x_{H}\simeq-1.2$ as shown
in Fig. 6.

###  III.2 Electron-positron collider

Next we consider the $Zh$ production at the electron-positron collider. Since
the $Z^{\prime}$ boson is too heavy to be produced at the electron-positron
collider, the $Z^{\prime}$ boson mediated process affects the $Zh$ production
through the interference with the SM $Zh$ production through the $Z$ boson.

The interaction Lagrangian between the electron and $Z$ boson is given by

$\displaystyle\mathcal{L}_{int}^{Z}=g_{Z}\overline{e}\gamma^{\mu}\Big{(}C_{V}+C_{A}\gamma_{5}\Big{)}eZ_{\mu},$
(15)

where

$\displaystyle C_{V}$ $\displaystyle=$ $\displaystyle
g_{Z}\left(-\frac{1}{4}+\sin^{2}\theta_{W}\right),$ $\displaystyle C_{A}$
$\displaystyle=$ $\displaystyle\frac{1}{4}g_{Z}.$ (16)

The interaction between $Z^{\prime}$ and the electron is given by

$\displaystyle\mathcal{L}_{int}^{Z^{\prime}}=\overline{e}\gamma^{\mu}\Big{(}C_{V}^{\prime}+C_{A}^{\prime}\gamma_{5}\Big{)}eZ_{\mu}^{\prime}$
(17)

where

$\displaystyle C_{V}^{\prime}$ $\displaystyle=$ $\displaystyle
g_{X}\left(-\frac{3}{4}x_{H}-1\right),$ $\displaystyle C_{A}^{\prime}$
$\displaystyle=$ $\displaystyle g_{X}\left(-\frac{1}{4}x_{H}\right).$ (18)

We then calculate the differential cross section of the $Zh$ production at the
electron-positron collider to be

$\displaystyle\frac{d\sigma}{d\cos\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{32\pi}\sqrt{\frac{E_{Z}^{2}-M_{Z}^{2}}{s}}\Big{[}\Big{|}C_{Z}\Big{|}^{2}\Big{(}C_{V}^{2}+C_{A}^{2}\Big{)}+|C_{Z}^{\prime}|^{2}\Big{(}{C^{\prime}_{V}}^{2}+{C^{\prime}_{A}}^{2}\Big{)}$
(19) $\displaystyle+$
$\displaystyle\Big{(}C_{Z}^{\ast}C_{Z}^{\prime}+C_{Z}{C_{Z}^{\prime}}^{\ast}\Big{)}\Big{(}C_{V}C_{V}^{\prime}+C_{A}{C_{A}^{\prime}}\Big{)}\Big{]}\times\Big{\\{}1+\cos^{2}\theta+\frac{E_{Z}^{2}}{M_{Z}^{2}}\Big{(}1-\cos^{2}\theta\Big{)}\Big{\\}},$

where

$\displaystyle E_{Z}$ $\displaystyle=$
$\displaystyle\frac{s+M_{Z}^{2}-m_{h}^{2}}{2\sqrt{s}},$ $\displaystyle C_{Z}$
$\displaystyle=$ $\displaystyle
2\Big{(}\frac{M_{Z}^{2}}{v}\Big{)}\frac{1}{s-M_{Z}^{2}+i\Gamma_{Z}M_{Z}},$
$\displaystyle C_{Z}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{-M_{Z}g_{X}x_{H}}{s-{M_{Z}^{\prime}}^{2}+i\Gamma_{Z^{\prime}}M_{Z^{\prime}}}.$
(20)

Figure 7: $Zh$ production cross section at the electron-positron collider in
the SM (blue, solid) and the minimal $U(1)_{X}$ model (red, dashed) as a
function of the center-of-mass energy $(\sqrt{s})$ of the collider. The
interference between the $Z$ and $Z^{\prime}$ mediated processes is involved
in the $U(1)_{X}$ case. Here, we have fixed $M_{Z^{\prime}}=7.5$ TeV. Figure
8: The deviation of the $Zh$ production cross section in the $U(1)_{X}$ model
from the SM at the electron-positron collider as a function of the center-of-
mass energy for fixed $x_{H}=-\frac{4}{3}$ (red, dotted) $-1.2$ (blue, solid)
and $-0.8$ (blue, dashed).

Figure 9: The deviation of the $Zh$ production cross section in the $U(1)_{X}$
model form the SM for different $x_{H}$ values and center-of-mass energy for
the electron-positron collider, $\sqrt{s}=500$ GeV (top, left), $1$ TeV (top,
right) and $3$ TeV (bottom), respectively. We have fixed $M_{Z^{\prime}}=7.5$
TeV.

The $Zh$ production cross section is deviated from its SM prediction by the
contribution from the $Z^{\prime}$ boson mediated process. To evaluate this
deviation which may be observed at the future electron-positron collider, we
calculate the $Zh$ production cross section by setting the model parameters
from the HL-LHC prospective bound, namely, $M_{Z^{\prime}}=7.5$ TeV and
$g_{X}={g_{X}}^{\rm max}$ as a function of $x_{H}$ shown in Fig. 6. In Fig. 7,
we show the $Zh$ production cross section for $x_{H}=-1.2$ as a function of
the center-of-mass energy $\sqrt{s}$ of the electron-positron collider (blue,
solid) along with the SM prediction in the limit of
$M_{Z^{\prime}}\rightarrow\infty$ (red, dashed). The interference between the
$Z$ and $Z^{\prime}$ boson mediated processes reduces the $Zh$ production
cross section from its SM prediction. As expected, as the collider energy is
raised, the deviation becomes larger.

Let us define the deviation of the $Zh$ cross section of the minimal
$U(1)_{X}$ model ($\sigma_{U(1)_{X}}$) from the SM predication ($\sigma_{SM}$)
by

$\displaystyle
Deviation[\%]=\left|1-\frac{\sigma_{U(1)_{X}}}{\sigma_{SM}}\right|\times
100\,\%.$ (21)

For three different values of $x_{H}$, we show in Fig. 8 the deviation as a
function of the collider energy. With its very clean environment, the
electron-positron collider allows us to measure the $Zh$ production cross
section very precisely, say, an ${\cal O}(1\,\%)$ level of precision. In Fig.
8, we can see that for $x_{H}=-1.2$ the deviation from the SM cross section
can be tested at a 1 TeV electron-positron collider. In Fig. 9, we show the
deviations as a function of $x_{H}$ for three different collider energies. The
largest deviation is obtained for $x_{H}\simeq-1.2$.

##  IV Conclusion

The minimal $U(1)_{X}$ model is a simple and well-motivated BSM which provides
the origin of the neutrino mass by the seesaw mechanism with three RHNs, which
are necessary for cancelling the $U(1)_{X}$ related anomalies. The $U(1)_{X}$
gauge boson $Z^{\prime}$ has been searched by the high energy collider
experiments with the final state dilepton and dijets. In this paper, we have
noticed that due to the $U(1)_{X}$ charge $-x_{H}/2$ of the SM Higgs doublet
in the minimal $U(1)_{X}$ model, the SM Higgs boson has a trilinear coupling
of $Z^{\prime}-Z-h$ which affects the associated Higgs boson production with a
$Z$ boson at high energy colliders. We have calculated the $Zh$ production
cross section at high energy proton-proton and electron-positron colliders and
have shown interesting effects of the new $Z^{\prime}$ boson mediated process.
The $Z^{\prime}$ boson contribution dramatically changes as a function of
$x_{H}$, and we have found that $x_{H}\simeq-1.2$ exhibits the maximum effect
on the $Zh$ production. The precise measurement of the $Zh$ production cross
section at the future high energy collider experiments can reveal the
$Z^{\prime}$ boson effect, independently of the $Z^{\prime}$ boson search with
the dilepton and dijet.

## Acknowledgement

The work was supported in part by Japan Society for the Promotion of Science
(JSPS), Grant-in-Aid for Scientific Research, No. 18F18321 (A. Das) and by the
United States Department of Energy grant DE-SC0012447 (N. Okada).

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# The Open Cluster Chemical Abundances and Mapping Survey: VII.
APOGEE DR17 [C/N]-Age Calibration

Taylor Spoo Department of Physics and Astronomy, Texas Christian University,
TCU Box 298840
Fort Worth, TX 76129, USA (t.spoo, p.frinchaboy, n.myers<EMAIL_ADDRESS>Jamie Tayar NASA Hubble Fellow  Institute for Astronomy, University of Hawai’i
at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Department of
Astronomy, University of Florida, 211 Bryant Space Science Center,
Gainesville, FL 32611, USA  Peter M. Frinchaboy Department of Physics and
Astronomy, Texas Christian University, TCU Box 298840
Fort Worth, TX 76129, USA (t.spoo, p.frinchaboy, n.myers<EMAIL_ADDRESS>Katia Cunha  Observatório Nacional, Rua General José Cristino, 77, Rio de
Janeiro, RJ 20921-400, Brazil  Steward Observatory, University of Arizona, 933
North Cherry Avenue, Tucson, AZ 85721-0065, USA  Natalie Myers Department of
Physics and Astronomy, Texas Christian University, TCU Box 298840
Fort Worth, TX 76129, USA (t.spoo, p.frinchaboy, n.myers<EMAIL_ADDRESS>John Donor  Department of Physics and Astronomy, Texas Christian University,
TCU Box 298840
Fort Worth, TX 76129, USA (t.spoo, p.frinchaboy, n.myers<EMAIL_ADDRESS>Steven R. Majewski  Department of Astronomy, University of Virginia,
Charlottesville, VA 22904-4325, USA  Dmitry Bizyaev  Apache Point Observatory
and New Mexico State University, P.O. Box 59, Sunspot, NM, 88349-0059, USA
Sternberg Astronomical Institute, Moscow State University, Moscow, Russia  D.
A. García-Hernández Instituto de Astrofísica de Canarias, Vía Láctea S/N,
38205 La Laguna, Tenerife, Spain  Universidad de La Laguna, Departamento de
Astrofísica, 30206 La Laguna, Tenerife, Spain  Henrik Jönsson Materials
Science and Applied Mathematics, Malmö University, SE-205 06 Malmö, Sweden
Richard R. Lane Centro de Investigación en Astronomía, Universidad Bernardo
O’Higgins, Avenida Viel 1497, Santiago, Chile  Kaike Pan  Apache Point
Observatory and New Mexico State University, P.O. Box 59, Sunspot, NM,
88349-0059, USA  Penélope Longa-Peña  Centro de Astronomía, Universidad de
Antofagasta, Avenida Angamos 601, Antofagasta 1270300, Chile  A. Roman-Lopes
Department of Astronomy, Universidad La Serena, La Serena, Chile

(Received December 23, 2021; Revised March 1, 2022; Accepted March 10, 2022)

###### Abstract

Large scale surveys open the possibility to investigate Galactic evolution
both chemically and kinematically, however, reliable stellar ages remain a
major challenge. Detailed chemical information provided by high-resolution
spectroscopic surveys of the stars in clusters can be used as a means to
calibrate recently developed chemical tools for age-dating field stars. Using
data from the Open Cluster Abundances and Mapping (OCCAM) survey, based on the
SDSS/APOGEE-2 survey, we derive a new empirical relationship between open
cluster stellar ages and the carbon-to-nitrogen ([C/N]) abundance ratios for
evolved stars, primarily those on the red giant branch. With this calibration,
[C/N] can be used a chemical clock for evolved field stars to investigate the
formation and evolution of different parts of our Galaxy. We explore how
mixing effects at different stellar evolutionary phases, like the red clump,
affect the derived calibration. We have established the [C/N]-age calibration
for APOGEE DR17 giant star abundances to be $\log[Age({\rm yr})]_{\rm
DR17}=10.14\,(\pm 0.08)+2.23\,(\pm 0.19)\,{\rm[C/N]}$, usable for
$8.62\leq\log(Age[{\rm yr}])\leq 9.82$, derived from a uniform sample of 49
clusters observed as part of APOGEE DR17 applicable primarily to metal-rich,
thin and thick disk giant stars. This measured [C/N]-age APOGEE DR17
calibration is also shown to be consistent with astereoseismic ages derived
from Kepler photometry.

Open star clusters (1160), Galactic abundances (2002), Chemical abundances
(224), Abundance Ratios (11)

††journal: AJ††facilities: Du Pont (APOGEE), Sloan (APOGEE), Spitzer, WISE,
2MASS, Gaia††software: Astropy (Astropy Collaboration et al., 2013, 2018)

##  1 Introduction

To understand our Galaxy’s present-day structure, we need to learn how the
Milky Way formed and evolved over time. To do this we need to be able to
determine the ages of individual stars. However, ages are difficult to measure
accurately for stars outside of star clusters. A common method to determine
stellar ages is to compare key parameters, whether directly observed (e.g.,
colors and apparent magnitudes, the latter combined with parallaxes) or less
directly inferred (e.g., $\log{g}$, $T_{\rm eff}$), to those predicted from
stellar evolution models, which allows accurate relative aging of stars, but
this strategy is complicated due to degeneracies in these parameters caused by
chemical differences. Fortunately, several large-scale spectroscopic surveys
systematically collecting high-resolution data, such as the Apache Point
Observatory Galactic Evolution Experiment survey (APOGEE, Abdurro’uf et al.,
2021), the Gaia-ESO public spectroscopic survey (Gilmore et al., 2012), and
the GALactic Archaeology with HERMES survey (GALAH, Martell et al., 2017), are
providing detailed and precise chemical abundances for multiple chemical
elements spanning millions of stars across various Galactic stellar
populations, including star clusters. Apart from providing essential
information to break the age-dating degeneracies, these spectroscopic surveys
provide additional key parameters, such as Li-depletion (e.g., Gao et al.,
2018; Magrini et al., 2021, and references therein) or [C/N] abundance ratios
(e.g., Bertelli Motta et al., 2017; Hasselquist et al., 2019; Casali et al.,
2019), that are alternative, chemistry-based diagnostics of stellar age.

A natural ally in calibrating [C/N] ages is asteroseimology, now being
generated for large numbers of stars through space missions like CoRoT
(Auvergne et al., 2009), Kepler (Borucki et al., 2010), and TESS (Ricker et
al., 2015). Asteroseismology uses high-precision photometry to measure
frequency modes in stars that are dependant on the structure, and therefore
mass of the star. Ages for red giant branch (RGB) stars can be derived using
asteroseimic photometric measurements from Kepler and TESS because mass of RGB
stars correlates with age. For example, utilizing combined data from Kepler
and the APOGEE survey, the APOKASC catalog (Pinsonneault et al., 2018) has
measured masses, and thereby ages, for $\sim 7000$ evolved stars.

Such masses and ages have been shown previously to correlate with [C/N]
(Martig et al., 2016; Ness et al., 2016). However, to obtain their exact
absolute calibration is challenging (Gaulme et al., 2016; Huber et al., 2017;
Zinn et al., 2019), as is controlling for physical (e.g. mass loss) and
systematic differences. Thus, it is worthwhile searching for independent
strategies for [C/N]-age calibration.

Star clusters have long been one of the key testbeds of our understanding of
stellar evolution and stellar physics, empirically demonstrating what
properties stars of the same chemistry and age should have. In particular,
open clusters are useful objects for calibrating the relationship between age
and [C/N], because star clusters are the most reliable age-datable tracers,
and open clusters also have ages that span the history of the Galactic disk,
from currently forming stars up to systems 9-10 Gyr old. Clusters ages can be
accurately determined, using the best measured and understood stars in the
cluster, by the location of the main sequence turn off (MSTO) on a HR-diagram.

Carbon and nitrogen abundances in evolved stars on the RGB will change due to
the first dredge up, where material from regions that have previously
undergone nuclear burning are brought to the surface. The changes in the
balance of carbon and nitrogen in the surface is dependant on stellar mass
(Masseron & Gilmore, 2015; Martig et al., 2016; Ness et al., 2016). With open
cluster studies, we can therefore empirically calibrate a relationship between
stellar ages, exploiting the precise ages determined by cluster MSTO
isochrone-fitting, and precise [C/N] abundances measured for the RGB stars in
these systems.

Casali et al. (2019) has investigated this relationship using a compilation of
nearly 40 open clusters using a combined data set from the Gaia-ESO survey and
the APOGEE Data Release 14 (DR14) survey. However, these results relied on a
smaller number of clusters studied heterogeneously, which opens the
possibility of systematics that could bias the results.

With the availability of much more APOGEE data via SDSS DR17 — including the
addition of Southern Hemisphere stars and clusters — we are in a position to
improve upon the Casali et al. (2019) effort to calibrate empirically a
relationship between [C/N] abundances of RGB stars and their ages using a
homogeneous dataset of chemical abundance measurements for open cluster RGB
stars deriving entirely from APOGEE spectroscopy. Using the ages determined by
Cantat-Gaudin et al. (2020) for clusters that are in APOGEE/SDSS DR17 as
defined by the Open Cluster Chemical Abundances and Mapping (OCCAM) Survey
(Myers et. al., submitted.), we calibrate a relationship between [C/N] and
stellar age for stars that have experienced the first dredge-up. We separate
the RGB and red clump (RC) stars to investigate the effects that stellar
evolution may have on the calibration for stellar age. We also compare our
results to those from previous work, in particular from the APOKASC
astroseismology method for determining stellar ages. These improvements in the
[C/N]-age calibration will help increase the power of large-scale, high
resolution stellar spectroscopic surveys like APOGEE to explore Galactic
chemodynamical evolution using a reliable means to benchmark ages for vast
numbers of stars.

##  2 Data & Analysis

###  2.1 SDSS/APOGEE Survey

This study makes use of the Apache Point Observatory Galactic Evolution
Experiment surveys (APOGEE-1 & -2; Majewski et al., 2017, Majewski et al., in
prep) which are part of the Sloan Digital Sky Survey III & IV (Eisenstein et
al., 2011; Blanton et al., 2017), respectively. Data were taken with the 2.5-m
Sloan Foundation Telescope (Gunn et al., 2006) at the Apache Point Observatory
and the 2.5-m du Pont telescope (Bowen & Vaughan, 1973) at the Las Campanas
Observatory, using the APOGEE-North and -South spectrographs (Wilson et al.,
2019), respectively. APOGEE Target selection is described in Zasowski et al.
(2013, 2017), Beaton et al. (2021), and Santana et al. (2021), with open
cluster targeting further detailed in Frinchaboy et al. (2013) and Donor et
al. (2018).

The APOGEE spectra were reduced using the APOGEE reduction pipeline (Nidever
et al., 2015), and key parameters were derived automatically via the APOGEE
Stellar Parameters and Chemical Abundances Pipeline (ASPCAP; García Pérez et
al., 2016). As the basis of the spectral analysis large grids of synthetic
spectra are calculated (Zamora et al., 2015) using stellar atmospheric models
(Mészáros et al., 2012) and an updated line list with astrophysically tuned
and vetted atomic and molecular data (Smith et al. 2021; see also Shetrone et
al. 2015). Many aspects of these parts of the data analysis have been updated
during the years of the survey and are described in Holtzman et al. (2015,
2018), Jönsson et al. (2018, 2020), Smith et al. (2021), and Holtzman et al.,
(in prep.). These continuing analysis updates result in the possibility that
the determined stellar parameters and abundances may change for a particular
star from one data release to another. In DR17, ASPCAP reports stellar
parameters and chemical abundances for more than 20 elements. This includes
the abundances of carbon and nitrogen, which can be more challenging to
measure from optical spectra. However, APOGEE is uniquely suited to reliably
measuring the C, N, and O abundances, derived through the combined analysis of
their contributions to numerous molecular bands of CN, CO, and OH. These
particular molecular lines are fully described in Smith et al. (2021).

####  2.1.1 SDSS/APOGEE Survey DR17

In this work, we use data from the latest, and final, data release of the
APOGEE-2 survey. The APOGEE-2 Data Release (DR17; Abdurro’uf et al., 2021)
includes all observations from both APOGEE spectrographs taken from August
2011 to January 2021 and has $\sim$734,000 stars. A full description of the
APOGEE DR17 data quality and parameter limitations will be presented in
Abdurro’uf et al. (2021) and Holtzman et al., (in prep). The most significant
update in the ASPCAP analysis code for DR17 compared to that used in previous
data releases is that the library of synthetic spectra were calculated using
the Synspec code (e.g., Hubeny et al., 2021). Synspec allows for the Non-Local
Thermodynamic Equilibrium (non-LTE) line formation in a plane parallel
geometry. However, in DR17 only the elements Na, Mg, K and Ca were computed in
non-LTE (Osorio et al., 2020), while the remaining elements studied by APOGEE,
including C and N, were still computed in LTE. That said, the determined C and
N abundances — and hence [C/N] — can be different in DR17 from prior data
releases, even in cases where the same spectra are analyzed, because the DR14
and DR16 APOGEE data releases relied on the Turbospectrum code (Plez, 2012)
which, while not accommodating non-LTE populations to be specified, does
compute atmospheres within a spherical geometry (Gustafsson et al., 2008). For
the interested reader, the choice of the model atmospheres and compositions
between the various options is described more fully in Abdurro’uf et al.
(2021) and Holtzman et al., in prep.

####  2.1.2 OCCAM Survey Catalog

This work uses data from the Open Cluster Chemical Analysis and Mapping
(OCCAM, Frinchaboy et al., 2013; Cunha et al., 2015; Donor et al., 2018,
2020). The DR17 OCCAM data set (Myers et al., submitted) includes APOGEE data
from 153 open clusters with a total number of 2061 stars determined to be
likely members. From this parent catalog, Myers et al. select only 94 clusters
determined to be “highly reliable” for use in studying Galactic chemical
trends, and we refer the reader to Donor et al. (2018, 2020) for a full
description of member selection and explanation of how the “highly reliable”
sample was selected. We utilize only data from the “highly reliable” OCCAM
DR17 value-added catalog (VAC111https://www.sdss.org/dr17/data_access/value-
added-catalogs/?vac_id=open-cluster-chemical-abundances-and-mapping-catalog.)

###  2.2 Cluster Member Sample Selection

To select stars having reliable C and N measurements for our analysis, we used
ASPCAP bit-wise flags for removing unreliable C and N abundances, along with a
requirement that the spectra from which they were derived have a signal-to-
noise ratio (S/N) greater than 70. The bit-wise flags are fully described in
Jönsson et al. (2020) and Holtzman et al, in prep. In addition, we cross-
matched our sample with the SB2 Catalog (Kounkel et al., 2021) to check
whether any stars in our sample were double-lined spectroscopic binaries
(SB2s), and found none to be SB2s.

We also made a cut to our sample based on OCCAM membership probabilities in
radial velocity, proper motion, and metallicity. To provide a more reliable
sample, we selected with a tighter requirement of $>30$% membership
probability in radial velocity, proper motion, and metallicity.

In addition, we selected stars that lie within a radius two times larger than
the radius containing half the members ($r_{50}$) as determined by Cantat-
Gaudin et al. (2020). These combined criteria provide a well-vetted and pure
sample that compares well to other cluster membership catalogs (e.g., Cantat-
Gaudin et al., 2020). Because [C/N] abundances only correlate with age for
evolved stars, we applied a surface gravity cut to ensure we were obtaining
evolved stars that have experienced the first dredge-up (FDU). Based on a
combination of stellar models and the APOKASC data, we included only stars
with $\log g<3.3$, which should represent the post dredge-up stars and the
ages studies here. Next, from our uniform sample, we used the APOGEE DR17 RC
Catalog (Bovy et al., 2014, Bovy et al.in prep) to flag likely RC stars within
our sample; this will allow us to investigate RC mixing effects that could
skew the age calibration. Likely RC stars included in the APOGEE DR17 RC
Catalog were selected based on their location within color, metallicity,
surface gravity and effective temperature space, with details fully described
in Bovy et al. (2014).

The sample selection criteria adopted are listed in Table 1. After applying
this criteria, our sample is comprised of 75 clusters given in Table 2, which
includes: cluster names, cluster ages as reported by Cantat-Gaudin et al.
(2020), DR17 median [Fe/H] abundances, and median [C/N] abundances for the
member stars with good measurements. Median elemental abundance uncertainties
were calculated using the standard error of the mean.

Table 1: Stellar Data Quality Selection Criteria

Source | Parameter | Selection
---|---|---
APOGEE Data Quality Cuts
APOGEE/DR17 | STARFLAG - Stellar Parameters | $!=16$
APOGEE/DR17 | ASPCAP C Flag | $!=21$
APOGEE/DR17 | ASPCAP N Flag | $!=22$
APOGEE/DR17 | ASPCAP Flag - Chemistry | $!=23$
APOGEE/DR17 | VSCATTER (km s-1) | $<1$
APOGEE/DR17 | SNREV | $>70$
Cluster Membership Cuts
(Myers et al., submitted) | RV_PROB | $>0.3$
(Myers et al., submitted) | FEH_PROB | $>0.3$
(Myers et al., submitted) | PM_PROB | $>0.3$
(Cantat-Gaudin et al., 2020) | $\sqrt{((l_{*}-l_{CG})\cos(b_{CG}))^{2}+(b_{*}-b_{CG})^{2}}$ |  $<2\times$ r50
Stellar Evolutionary Cuts
APOGEE/DR17 | LOGG | $<3.3$

Note. — The table summarizes the various selection criteria applied to the
stars within clusters common between Cantat-Gaudin et al. (2020) and the OCCAM
survey VAC (Myers et al., submitted) to ensure that only cluster stars whose
properties are well measured and for which the [C/N] ratios should be
correlated with age. STARFLAG and ASPCAP bit-wise flags are described in:
https://www.sdss.org/dr17/irspec/apogee-bitmasks/ .

Table 2: Mean [C/N] DR17 Abundance - Full Sample Clusteraa Bold denotes clusters in the final calibration sample.  |  $\log$(Age)bb All ages come from Cantat-Gaudin et al. (2020) and the $\log(Age)$ errors are equal to 0.1. | [Fe/H] | [C/N] | DR17
---|---|---|---|---
name | (yr) | (dex) | (dex) | Memb.
Berkeley 17 | 9.86 |  $-$0.183 | $-0.164\pm 0.010$ | 8
Berkeley 18 | 9.64 |  $-$0.385 | $-0.332\pm 0.015$ | 18
Berkeley 19 | 9.34 |  $-$0.361 | $-0.517\pm 0.000$ | 1
Berkeley 2 | 8.77 |  $-$0.208 | $-0.366\pm 0.048$ | 6
Berkeley 20 | 9.68 |  $-$0.432 | $-0.372\pm 0.000$ | 1
Berkeley 21 | 9.33 |  $-$0.269 | $-0.355\pm 0.006$ | 3
Berkeley 29 | 9.49 |  $-$0.527 | $-0.280\pm 0.009$ | 2
Berkeley 31 | 9.45 |  $-$0.426 | $-0.278\pm 0.001$ | 2
Berkeley 33 | 8.37 |  $-$0.243 | $-0.560\pm 0.000$ | 1
Berkeley 53 | 8.99 |  $-$0.121 | $-0.517\pm 0.025$ | 6
Berkeley 53 | 8.99 |  $-$0.121 | $-0.517\pm 0.025$ | 6
Berkeley 66 | 9.49 |  $-$0.215 | $-0.302\pm 0.008$ | 5
Berkeley 71 | 8.94 |  $-$0.232 | $-0.519\pm 0.029$ | 5
Berkeley 75 | 9.23 |  $-$0.412 | $-0.504\pm 0.039$ | 3
Berkeley 85 | 8.62 |  $+$0.064 | $-0.420\pm 0.009$ | 10
Berkeley 9 | 9.14 |  $-$0.180 | $-0.624\pm 0.000$ | 1
Berkeley 91 | 8.80 |  $+$0.070 | $-0.327\pm 0.000$ | 1
Berkeley 98 | 9.39 |  $-$0.044 | $-0.376\pm 0.043$ | 5
Czernik 20 | 9.22 |  $-$0.177 | $-0.400\pm 0.049$ | 4
Czernik 21 | 9.41 |  $-$0.326 | $-0.280\pm 0.008$ | 2
Czernik 23 | 8.43 |  $-$0.329 | $-0.571\pm 0.000$ | 1
Czernik 30 | 9.46 |  $-$0.387 | $-0.348\pm 0.017$ | 2
ESO 211 03 | 9.11 |  $+$0.107 | $-0.447\pm 0.031$ | 4
FSR 0494 | 8.95 |  $-$0.021 | $-0.515\pm 0.015$ | 5
FSR 0496 | 9.31 |  $-$0.137 | $-0.429\pm 0.000$ | 1
FSR 0667 | 8.85 |  $-$0.034 | $-0.544\pm 0.000$ | 1
FSR 0716 | 8.94 |  $-$0.385 | $-0.398\pm 0.000$ | 1
FSR 0937 | 9.08 |  $-$0.371 | $-0.439\pm 0.032$ | 2
FSR 1113 | 9.44 |  $-$0.347 | $-0.237\pm 0.000$ | 1
Haffner 4 | 8.66 |  $-$0.182 | $-0.482\pm 0.000$ | 1
IC 1369 | 8.46 |  $-$0.112 | $-0.560\pm 0.024$ | 3
IC 166 | 9.12 |  $-$0.095 | $-0.488\pm 0.037$ | 13
King 15 | 8.47 |  $-$0.128 | $-0.457\pm 0.000$ | 1
King 2 | 9.61 |  $-$0.184 | $-0.292\pm 0.000$ | 1
King 5 | 9.01 |  $-$0.159 | $-0.464\pm 0.013$ | 4
King 7 | 8.35 |  $-$0.162 | $-0.366\pm 0.073$ | 7
King 8 | 8.92 |  $-$0.239 | $-0.469\pm 0.000$ | 1
Melotte 71 | 8.99 |  $-$0.151 | $-0.478\pm 0.020$ | 3
NGC 1193 | 9.71 |  $-$0.345 | $-0.255\pm 0.026$ | 3
NGC 1245 | 9.08 |  $-$0.089 | $-0.480\pm 0.017$ | 25
NGC 136 | 8.41 |  $-$0.266 | $-0.457\pm 0.000$ | 1
NGC 1664 | 8.71 |  $-$0.060 | $-0.564\pm 0.000$ | 1
NGC 1798 | 9.22 |  $-$0.279 | $-0.429\pm 0.017$ | 10
NGC 1817 | 9.05 |  $-$0.157 | $-0.402\pm 0.045$ | 5
NGC 1857 | 8.40 |  $-$0.176 | $-0.414\pm 0.013$ | 2
NGC 188 | 9.85 |  $+$0.064 | $-0.244\pm 0.010$ | 32
NGC 1907 | 8.77 |  $-$0.126 | $-0.506\pm 0.046$ | 3
NGC 1912 | 8.47 |  $-$0.169 | $-0.435\pm 0.016$ | 2
NGC 2158 | 9.19 |  $-$0.246 | $-0.342\pm 0.013$ | 41
NGC 2204 | 9.32 |  $-$0.280 | $-0.423\pm 0.021$ | 20
NGC 2243 | 9.64 |  $-$0.458 | $-0.338\pm 0.037$ | 6
NGC 2304 | 8.96 |  $-$0.141 | $-0.400\pm 0.009$ | 2
NGC 2324 | 8.73 |  $-$0.221 | $-0.517\pm 0.037$ | 4
NGC 2420 | 9.24 |  $-$0.205 | $-0.337\pm 0.012$ | 16
NGC 2447 | 8.76 |  $-$0.114 | $-0.548\pm 0.016$ | 3
NGC 2479 | 8.99 |  $-$0.049 | $-0.436\pm 0.000$ | 1
NGC 2682 | 9.63 |  $+$0.000 | $-0.368\pm 0.011$ | 19
NGC 4337 | 9.16 |  $+$0.222 | $-0.491\pm 0.012$ | 6
NGC 6705 | 8.49 |  $+$0.096 | $-0.440\pm 0.016$ | 12
NGC 6791 | 9.80 |  $+$0.313 | $-0.165\pm 0.010$ | 41
NGC 6811 | 9.03 |  $-$0.051 | $-0.521\pm 0.012$ | 6
NGC 6819 | 9.35 |  $+$0.040 | $-0.352\pm 0.019$ | 42
NGC 6866 | 8.81 |  $+$0.010 | $-0.565\pm 0.030$ | 2
NGC 7062 | 8.63 |  $-$0.023 | $-0.569\pm 0.000$ | 1
NGC 752 | 9.07 |  $-$0.064 | $-0.442\pm 0.010$ | 5
NGC 7789 | 9.19 |  $-$0.032 | $-0.421\pm 0.011$ | 65
Ruprecht 147 | 9.48 |  $+$0.116 | $-0.395\pm 0.030$ | 3
Ruprecht 82 | 8.66 |  $-$0.037 | $-0.702\pm 0.000$ | 1
Teutsch 10 | 8.79 |  $-$0.350 | $-0.638\pm 0.000$ | 1
Teutsch 12 | 8.92 |  $-$0.200 | $-0.569\pm 0.031$ | 4
Teutsch 51 | 8.83 |  $-$0.332 | $-0.524\pm 0.011$ | 3
Teutsch 84 | 9.02 |  $+$0.200 | $-0.506\pm 0.000$ | 1
Tombaugh 2 | 9.21 |  $-$0.371 | $-0.605\pm 0.070$ | 4
Trumpler 20 | 9.27 |  $+$0.105 | $-0.509\pm 0.017$ | 25
Trumpler 5 | 9.63 |  $-$0.449 | $-0.274\pm 0.018$ | 8
Table 3: Full calibration cluster sample stellar data from APOGEE DR17.

Cluster | 2MASS ID | $T_{eff}$ | $\log(g)$ | [Fe/H] | [C/N] | RC? aa ‘Y’ is an identified red clump star using the APOGEE DR17 red clump value-added catalog (Bovy et al., 2014).
0‘N’ denotes stars not identified as part of the catalog and considered red
giant branch stars.
---|---|---|---|---|---|---
name |  | (K) | (dex) | (dex) | (dex) | label
Berkeley 17 | 2M05195385$+$3035095 | 4665 $\pm$ 09 | 2.60 $\pm$ 0.03 |  $-$0.12 $\pm$ 0.01 |  $-$0.23 $\pm$ 0.02 | N
Berkeley 17 | 2M05202118$+$3035544 | 4799 $\pm$ 09 | 2.47 $\pm$ 0.02 |  $-$0.15 $\pm$ 0.01 |  $-$0.16 $\pm$ 0.02 | N
Berkeley 17 | 2M05202905$+$3032414 | 4783 $\pm$ 13 | 2.84 $\pm$ 0.03 |  $-$0.20 $\pm$ 0.01 |  $-$0.18 $\pm$ 0.03 | N
Berkeley 17 | 2M05203121$+$3035067 | 4820 $\pm$ 09 | 2.49 $\pm$ 0.02 |  $-$0.19 $\pm$ 0.01 |  $-$0.14 $\pm$ 0.02 | N
Berkeley 17 | 2M05203650$+$3030351 | 4444 $\pm$ 06 | 1.98 $\pm$ 0.02 |  $-$0.17 $\pm$ 0.01 |  $-$0.15 $\pm$ 0.02 | N
Berkeley 17 | 2M05203799$+$3034414 | 4307 $\pm$ 06 | 1.93 $\pm$ 0.02 |  $-$0.18 $\pm$ 0.01 |  $-$0.16 $\pm$ 0.02 | N
Berkeley 17 | 2M05204143$+$3036042 | 4824 $\pm$ 09 | 2.49 $\pm$ 0.03 |  $-$0.22 $\pm$ 0.01 |  $-$0.14 $\pm$ 0.02 | Y
Berkeley 17 | 2M05204488$+$3038020 | 4807 $\pm$ 08 | 2.43 $\pm$ 0.02 |  $-$0.21 $\pm$ 0.01 |  $-$0.21 $\pm$ 0.02 | Y
Berkeley 18 | 2M05211671$+$4533170 | 4220 $\pm$ 06 | 1.43 $\pm$ 0.02 |  $-$0.40 $\pm$ 0.01 |  $-$0.32 $\pm$ 0.02 | N
Berkeley 18 | 2M05214927$+$4525225 | 5126 $\pm$ 19 | 2.65 $\pm$ 0.04 |  $-$0.33 $\pm$ 0.01 |  $-$0.26 $\pm$ 0.05 | Y
Berkeley 18 | 2M05215476$+$4526226 | 4309 $\pm$ 06 | 1.56 $\pm$ 0.03 |  $-$0.40 $\pm$ 0.01 |  $-$0.35 $\pm$ 0.02 | N
$\cdots$

Note. — This table is available in its entirety in machine-readable form in
the online journal. A portion is shown here for guidance regarding its form
and content.

###  2.3 Systematics between DR14, DR16, & DR17

During the course of the APOGEE survey, the APOGEE team has made improvements
to the automated analysis pipeline. For each APOGEE data release there have
been changes and improvements to, for example, the atomic and molecular line
lists, stellar atmosphere models, implementation of the code, etc, as
discussed in §2.1. In order to compare our work with results from other
studies that used previous APOGEE data releases (e.g., Casali et al., 2019,
which partial used DR14 data), we briefly investigate systematic differences
between the carbon and nitrogen abundance results derived from the different
APOGEE data releases.

The APOGEE-2 Data Release 14 (DR14; Abolfathi et al., 2018) included
observations from August 2011 to July 2016 and has $\sim 263,000$ stars. DR14
APOGEE data includes data from the APOGEE 1 and first two years of the
APOGEE-2 North survey, all taken from the Apache Point Observatory. A full
description of the APOGEE DR14 data quality and parameter limitations is
presented in Holtzman et al. (2018); Jönsson et al. (2018). The APOGEE-2 Data
Release 16 (DR16; Ahumada et al., 2019) included observations from August 2011
to August 2018 and has $\sim 473,000$ stars. DR16 APOGEE data included data
from the APOGEE 1 and first four years of the APOGEE-2 North survey taken from
the Apache Point Observatory, plus the first year of APOGEE-2 South data taken
from the Las Campanas Observatory. A full description of the APOGEE DR16 data
quality and parameter limitations is presented in and Jönsson et al. (2020).

We find a global offset of [Fe/H] between DR14 and DR17 as well as for DR16
and DR17 to be 0.07 dex and 0.02 dex, respectively. This investigation for
differences between previous releases for metallicity ([Fe/H]), $\alpha$
elements (O, Mg, Si, S, Ca, Ti), iron-peak elements (V, Cr, Mn, Co, Ni), and
other elements (Na, Al, K) is discussed in detail in Donor et al. (2020) and
Myers et al., (submitted).

The top panels of Figure 1 compare the C and N abundances for star in the 75
cluster sample that have abundances in both DR14 and DR17. The median change
for C is measured to be $-0.029$ dex with a scatter of $0.047$ dex and the
median change for N is measured to be $+0.081$ dex with a scatter of $0.052$
dex. Similarly, the bottom panels of Figure 1 compare the C and N abundances
for star in the 75 cluster sample that have abundances in both DR16 and DR17.
The median change for C is measured to be $+0.065$ dex with a scatter of
$0.039$ dex and the median change for N is measured to be $-0.013$ dex with a
scatter of $0.051$ dex. These systematic offsets between the compared data
releases are due to changes within the SDSS data reduction pipeline, as
discussed in §2.1.1 This simple analysis demonstrates that it is not
appropriate or straightforward to inter-compare [C/N]-age relations derived
from difference APOGEE data releases, unless one accounts for these simple,
though systematic, variations in APOGEE carbon and nitrogen abundances over
time.

Figure 1: C and N abundance comparisons between different APOGEE data
releases: DR14 to DR17 (top panels), and DR16 to DR17 (bottom labels). A
representative error bar is shown in each panel. The horizontal black line
represents the median change. A grey shaded region is shown in each plot to
represent the scatter on the median.

##  3 Results & Discussion

###  3.1 Final Cluster Sample

After application of the various selection criteria summarized in Table 1, our
sample is comprised of 75 clusters with 577 member stars. Figure 2 shows the
[C/N] abundance ratios versus cluster age for that sample. To ensure our
calibration is reliable, we excluded clusters that contained only one member
(those shown with a red diamond), as well as those having $\log(Age[{\rm
yr}])<8.5$ (dark blue squares). As apparent in Figure 2, clusters younger than
$\log(Age[{\rm yr}])=8.5$ do not seem to follow the same trend in [C/N] and
$\log(Age[{\rm yr}])$.

Other than possible observational issues related to the analysis of young
stars (Baratella et al., 2020; Spina et al., 2020), there are likely two
effects at play that affect the more massive evolved stars ($M\geq
2.5M_{\odot}$) in these young clusters that are not present in the lower mass
evolved stars of older clusters. The first is that massive stars spend less
time on the first ascent RGB and proportionally more time in the core He-
burning phase (Iben, 1967a; Danchi et al., 2006). These young clusters
therefore are likely dominated by stars that have already undergone a weakly
degenerate or nondengeneate helium ignition and it is not yet clear whether
such an event could drive additional mixing that would alter their surface
abundances. Secondly, in more massive stars the first dredge up reaches deeper
(e.g., Iben, 1964, 1965, 1967a, 1967b) into regions where the CNO cycle was
significant in prior evolutionary phases. This means that the material being
dredged up has a fixed ratio of carbon and nitrogen, which is at an
equilibrium set by the nuclear reaction times of the CNO cycle (Dearborn et
al., 1976; Dearborn & Eggleton, 1976). Since this ratio is not changing with
mass, the relationship between the [C/N] ratio after the dredge up and the
stellar mass becomes much weaker, making it less sensitive as an age
diagnostic. Given these two caveats, we have excluded these stars from fits in
this paper, but encourage more careful studies of stars in this regime in the
future.

After application of these additional membership and cluster age cuts the
sample used for our [C/N]-age calibration (Fig. 3) is comprised of 49 clusters
(530 stars), covering an age range of $8.62\leq\log(Age[{\rm yr}])\leq 9.82$
and a metallicity range of $-0.53\leq{\rm[Fe/H]}\leq 0.31$. The individual
stars from the calibration sample are listed in Table 3. Table 2 shows these
calibration clusters in bold.

Figure 2:  The [C/N] versus $\log(Age[{\rm yr}])$ distribution for clusters
common to the OCCAM Survey and Cantat-Gaudin sample, which includes 75
clusters. Young clusters, $\log(Age)\leq 8.5$, are represented with navy
squares. Clusters marked with a red diamond are those with only one stellar
member after cuts are applied. Clusters with 2 to 4 stellar members are
represented by orange triangles, and those with 5 or more stellar members are
represented by cyan circles.

###  3.2 The DR17 [C/N] Abundance/Age Calibration

In log-log space the relationship between stellar age and [C/N] appears to be
linear; our best fit is given by

${\log[Age({\rm yr})]_{\rm DR17}}=10.14\,(\pm 0.08)+2.23\,(\pm
0.19)\,{\rm[C/N]}$ (1)

and yields a Pearson coefficient of $R=0.79$. Our Pearson coefficient is
comparable to that found by Casali et al. (2019) ($R=0.85$), although we
compute a slightly different offset and a weaker slope.

Figure 3:  The [C/N] versus $\log(Age[{\rm yr}])$ distribution for the final
sample, composed of clusters common to the OCCAM Survey and Cantat-Gaudin
sample, totalling 49 clusters. Clusters with 2 to 4 stellar members are
represented by orange triangles, and those with 5 or more stellar members are
by cyan circles.

The linear fit given by Equation 1 and shown in Figure 3 uses Monte Carlo (MC)
resampling as described in the fits from Donor et al. (2020). We computed 500
iterations of a linear fit, including uncertainties in the [C/N] abundance
ratios and age, and took the mean slope and y-intercept for our final fit,
where the respective errors are the standard deviation of the fit.

The discrepancy between our calibration and that of Casali et al. (2019) is
partially due to the different APOGEE data sets being used. Casali et al.
(2019) used not only DR14 data, but also Gaia-ESO data, and this comparison is
shown in Figure 1. Therefore, Figure 4 again reveals the shift in abundance
measurements between DR17 and DR14 that was used for the Casali et al. (2019)
clusters.

Figure 4:  Median [C/N] ratio of open cluster star members, as derived in DR14
(green circles) and DR17 (purple squares), versus $\log(Ages[{\rm yr}$]),
where these ages are provided by Cantat-Gaudin et al. (2020). The green dot-
dashed line and purple dashed line represent the linear fit for DR14 and DR17,
respectively. The plot shows a systematic shift from DR14 data to DR17 data.

###  3.3 Common Calibration for RC and RGB Stars?

Stars eventually transition from H-burning on the RGB to He-burning on the RC.
Particularly when the He-flash is explosive, stars may experience significant
mixing that could alter their surface abundances (e.g. Schwab, 2020), and
therefore the relationship between [C/N] and age. Using the APOGEE-RC catalog
(Bovy et al., 2014, Bovy et al.in prep), we have isolated RC stars from our
full sample; once the RC stars are removed, our sample consists of 48 clusters
(400 stars) covering an age range of $8.62\leq\log(Age[{\rm yr}])\leq 9.89$
and a metallicity range of $-0.53\leq{\rm[Fe/H]}\leq 0.31$. A new linear fit
to this sample yields

${\log[Age({\rm yr})]_{\rm DR17,RGB}}=10.14\,(\pm 0.07)+2.22\,(\pm
0.19)\,{\rm[C/N]}$ (2)

with a Pearson coefficient of 0.81, as shown in Figure 5 (left). This
particular calibration yields only a slight change in slope from the
previously found trend (Eqn. 1), from $2.23(\pm 0.19)$ to $2.22(\pm 0.19)$,
but more importantly, the correlation becomes more significant as the Pearson
coefficient goes from $0.79$ to $0.81$.

We then created a sample comprised solely of RC stars to investigate whether
RC stars follow a similar linear trend of [C/N] abundances and age to that by
RGB stars. These stars yield a $log(Age)$-[C/N] relationship that also follows
a linear trend with a Pearson coefficient of 0.79:

${\log[Age({\rm yr})]_{\rm DR17,RC}}=10.19\,(\pm 0.09)+2.29\,(\pm
0.24)\,{\rm[C/N]}$ (3)

Figure 5 (right) shows the relationship between the [C/N] and log(Age[Gyr]) in
our RC cluster sample. The latter is comprised of 21 clusters (78 stars)
covering an age range of $8.62\leq\log(Age[{\rm yr}])\leq 9.86$ and a
metallicity range of $-0.45\leq{\rm[Fe/H]}\leq 0.35$. We find that, within the
current uncertainties, the relationship for only RGB and only RC stars are the
same. We therefore find no evidence for additional significant mixing during
the He-flash that is affecting the surface C and N abundances for RC stars.

Figure 5:  (left) [C/N] abundances versus log($Age[{\rm yr}$]) for our cluster
sample using only RGB stars from 46 clusters common to the OCCAM Survey and
Cantat-Gaudin dataset. (right) The [C/N] abundance versus $log(Age[{\rm yr}])$
for clusters containing only RC stars in common between the OCCAM Survey and
Cantat-Gaudin dataset, which includes 21 clusters. Labels and calculation of
calibrations are similar to Figure 3

##  4 Comparison to Previous Work

The recent [C/N]-age calibration by Casali et al. (2019) made use of APOGEE
data processed with DR14 analysis software. Moreover, for their calibration
Casali et al. (2019) used both Gaia-ESO and APOGEE spectroscopic data along
with a literature compilation for cluster ages. In contrast, our analysis here
makes use of a uniform sample comprised of only APOGEE spectroscopic data, all
reduced with the most up-to-date, DR17 reduction of those data, and the most
recent Cantat-Gaudin et al. (2020) uniformly determined cluster ages. Thus, it
would not be surprising to find significant differences between the Casali et
al. (2019) relationship and that derived here. However, surprisingly, their
relationship, $log[Age(yr)]=10.54(\pm 0.06)+2.61(\pm 0.10)[C/N]$, is in fair
agreement with our own.

As another means to verify the reliability of our methodology, we can compare
predicted stellar ages from our [C/N] calibration to ages independently
derived using asteroseismology. For this purpose, we exploit the APOKASC
catalog (Pinsonneault et al., 2018). However, because the chemistry used by
APOKASC is based on APOGEE DR16 data, we generate age calibrations with
respect to DR16 data using the same process we used for our DR17
calibrations.Tthe calibrations for the full, RGB, and RC samples are found to
be:

${\log[Age({\rm yr})]_{{\rm DR16,all}}}=9.92\,(\pm 0.05)+1.94\,(\pm
0.15)\,{\rm[C/N]}$ (4) ${\log[Age({\rm yr})]_{{\rm DR16,RGB}}}=9.89\,(\pm
0.06)+1.81\,(\pm 0.16)\,{\rm[C/N]}$ (5) ${\log[Age({\rm yr})]_{{\rm
DR16,RC}}}=9.88\,(\pm 0.07)+1.94\,(\pm 0.25)\,{\rm[C/N]}$ (6)

We apply these DR16 full, RGB, and RC sample calibrations and the DR17 full,
RGB, and RC sample calibrations (Eqns. 1, 2, and 3, respectively) on field
stars within the APOKASC sample and compare the [C/N]-based stellar ages using
our calibrations to the astroseismology-based stellar ages determined by
APOKASC. We find the [C/N] based ages are more consistent with the
asteroseismic ages when the newer DR17 calibration is used (see Figure 6),
which suggests that the inferred [C/N] values have improved with time and that
the newer DR17 data should be preferred.

Figure 6:  A comparison between our DR16-based age calibration (left) and
DR17-based age calibration (right) to asteroseismically determined ages from
the APOKASC catalog (Pinsonneault et al., 2018). The grey-gradient shaded
regions represent bins of field stars. The green circles, blue triangles, and
red diamonds represent cluster member stars common to both samples for NGC
6791, NGC 6819, and NGC 6811, respectively. The horizontal dashed lines are
the Cantat-Gaudin et al. (2020) determined ages for each cluster. The diagonal
dashed grey line represents the one-to-one relationship. The solid black line
is the linear fit to the main trend of the field stars. Mean representative
error bars are shown in each panel.

For further verification, we apply our calibrations to open clusters that are
also in the APOKASC sample: NGC 6791, NGC 6819, and NGC 6811. We find that our
calculated ages from the DR17 calibration are closer to the ages determined in
Cantat-Gaudin et al. (2020) for all three clusters. The spread in the
[C/N]-calibration ages is due to the uncertainty of [C/N] in the individual
cluster members, but such a spread is expected because our calibration is
based on the average [C/N] abundance of the cluster. Comparison with these
asteroseismic results suggests that [C/N] based ages can be trusted to 10% for
99.79% of giants, as shown in Figure 6.

On the upper giant branch, above the so-called “red bump”, extra convective
mixing can occur, particularly in low-metallicity ([Fe/H]$\lesssim-0.5$) stars
(Gratton et al., 2000; Shetrone et al., 2019). In principle, this would
significantly impact the ages inferred for these stars, making them appear to
be younger than they actually are. For this reason, we caution again
extrapolating our results too far outside the calibration metallicity range
$-0.53\lesssim{\rm[Fe/H]}\lesssim 0.31$, particularly for stars on the upper
red giant branch.

##  5 Conclusions

The [C/N] age-dating technique, applicable to RGB stars, has brought a
powerful and versatile tool that can be applied to huge numbers of Galactic
field stars explored by large-scale spectroscopic surveys with which these two
chemical species can be measured, such as APOGEE DR17. In this work, we have
used open cluster ages to calibrate [C/N] as a chemical clock for evolved
stars, specifically for the SDSS/APOGEE survey.

  1. 1.

This calibration, based on APOGEE DR17, provides the following relation for
all evolved stars that experience the first dredge-up (e.g., APOGEE DR17 stars
with $\log g<3.3$):

${\log[{\it Age}({\rm yr})]}=2.23\,(\pm 0.19)\,{\rm[C/N]}+10.14\,(\pm 0.08)$
(7)

This calibration is found to be independent of metallicity of the range
explored, $-0.5\leq[Fe/H]\leq+0.3$ with our sample. The calibration is
consistent, though slightly different, with the [C/N]-age calibration using
DR14 and Gaia-ESO by Casali et al. (2019), which is primarily due to
differences in the C and N abundance determination between DR14 and DR17. We
also determined the calibration for DR16 which is presented in §4.

  2. 2.

We show that the relationship between age and [C/N] for massive ($M\geq
2.5M_{\odot}$) young (log$(Age)<8.5$) giants does not follow the same trend as
the relationship calibrated for older, lower mass giants here. We therefore
caution that our relationship should not be extrapolated to this regime.

  3. 3.

We find that RC stars can be fit with the same calibration as RGB stars.
Therefore, we see no evidence for significant extra mixing that affects C and
N on the upper RGB or during the He-flash for low-mass metal rich giants.
These results align with findings of Shetrone et al. (2019). With this in
mind, we caution the reader not to extrapolate the calibration to lower
metallicities ([Fe/H] $\lesssim-0.5$), where potential extra-mixing might have
a significant impact in measured elemental abundances.

  4. 4.

Comparison with asteroseismic results suggests that [C/N] based ages can be
trusted to 10% for 99.79% of giants, opening up an exciting future for the
estimation of precise and accurate ages for hundreds of thousands of evolved
stars across the Galaxy.

TS, PMF, NM, and JD acknowledge support for this research from the National
Science Foundation (AST-1715662). JT and PMF acknowledge this work was
performed at the Aspen Center for Physics, which is supported by National
Science Foundation grant PHY-1607611. JT acknowledges support for this work
was provided by NASA through the NASA Hubble Fellowship grant No.51424 awarded
by the Space Telescope Science Institute, which is operated by the Association
of Universities for Research in Astronomy, Inc., for NASA, under contract
NAS5-26555 was supported in part by the National Science Foundation under
Grant No. NSF PHY-1748958. KC acknowledges support for this research from the
National Science Foundation (AST-0907873). SRM acknowledges support from
National Science Foundation award AST-1908331. DAGH acknowledges support from
the State Research Agency (AEI) of the Spanish Ministry of Science, Innovation
and Universities (MCIU) and the European Regional Development Fund (FEDER)
under grant AYA2017-88254-P. Funding for the Sloan Digital Sky Survey IV has
been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy
Office of Science, and the Participating Institutions. SDSS-IV acknowledges
support and resources from the Center for High Performance Computing at the
University of Utah. The SDSS website is www.sdss.org. SDSS-IV is managed by
the Astrophysical Research Consortium for the Participating Institutions of
the SDSS Collaboration including the Brazilian Participation Group, the
Carnegie Institution for Science, Carnegie Mellon University, Center for
Astrophysics — Harvard & Smithsonian, the Chilean Participation Group, the
French Participation Group, Instituto de Astrofísica de Canarias, The Johns
Hopkins University, Kavli Institute for the Physics and Mathematics of the
Universe (IPMU) / University of Tokyo, the Korean Participation Group,
Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik
Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-
Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für
Extraterrestrische Physik (MPE), National Astronomical Observatories of China,
New Mexico State University, New York University, University of Notre Dame,
Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State
University, Shanghai Astronomical Observatory, United Kingdom Participation
Group, Universidad Nacional Autónoma de México, University of Arizona,
University of Colorado Boulder, University of Oxford, University of
Portsmouth, University of Utah, University of Virginia, University of
Washington, University of Wisconsin, Vanderbilt University, and Yale
University. This work has made use of data from the European Space Agency
(ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia
Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research made use of
Astropy, a community-developed core Python package for Astronomy (Astropy
Collaboration et al., 2013, 2018).

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Overview of the ATLAS ALFA detectors, performance and physics analysis

P. J. Bussey

SUPA-School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ,
U.K.
On behalf of the ATLAS Forward Detector group.

> An overview is presented of the ALFA forward detector system in the ATLAS
> detector at the LHC, CERN. Details of the construction are given, with
> summaries of the resulting analysis.

> PRESENTED AT
>
>
>
>
> DIS2023: XXX International Workshop on Deep-Inelastic Scattering and Related
> Subjects,
> Michigan State University, USA, 27-31 March 2023
>

The ATLAS experiment is a multi-purpose particle detection system located at
the interaction point IP1 in the LHC at CERN. It comprises a central detector,
which covers a large range of particle pseudorapidities centred on zero, and
two forward detector systems, which are able to measure protons that emerge at
vertical angles close to the beamline. Figure 1 shows the layout of the ALFA
stations around 240 m from the interaction point [1]; they are inserted
vertically and a second Roman Pot system (AFP), inserted horizontally, is also
in place and measures the deflection and energy loss of emerging protons that
are horizontally deflected by the dipole magnets in the beamline [2].

Figure 1: Schematic view of the ATLAS detector and the positions of the ALFA
stations relative to the beamline elements. The upper diagram is a plan view;
the lower diagram is a side projection indicating the paths of typical
emerging vertically scattered proton pairs.

The ALFA system was originally proposed to measure the LHC luminosity at the
ATLAS interaction point (IP). Since there are superior ways of measuring
luminosity, the ALFA system has been exploited primarily to measure
elastically scattered proton pairs, together with doubly-diffractive events
with two very forward protons accompanied by an exclusive centrally produced
hadronic system. In order to measure protons scattered through extremely small
angles, special LHC beam optics settings are required in dedicated runs with
high values of $\beta^{\star}$ at the IP, so as to give parallel-to-point
focussing at the ALFA stations. These runs are quite short, with low
luminosity, and are not compatible with the normal high-luminosity running of
the LHC.

Further details of the construction of the ALFA detectors are shown in Figures
2 and 3 [1, 6]. In each station, detectors are mounted in two Roman Pot
systems that are inserted close to the beamline from above and below. Protons
are detected by means of arrays of scintillating fibres. Three types of fibre
arrays are employed. Two of these arrays have the fibres oriented at $\pm
45^{\circ}$ relative to the vertical, in order to measure corresponding
coordinates denoted as $u$ and $v$. The third type of detector consists of
detector planes with horizontal fibres that are capable of measuring a subset
of protons that also pass through both the upper and lower detector systems.
These measurements are used to provide information on the relative positions
of the upper and lower modules while running. One overlap detector is included
in each of the upper and lower ALFA modules. A relative alignment accuracy of
5-10 $\mu$m was achieved. After the relative alignment was optimised, a global
vertical position uncertainty remains of $\pm 22\;\mu$m.

Figure 2: (Left) Configuration of the detectors and readout in one ALFA Roman
Pot; (right) layout of the $u$ and $v$ measuring detectors in the two Roman
pots of one station, together with the overlap detectors.

Figure 3:  (Left) mounting of an ALFA station in a dedicated section of the
LHC beam pipe. (Right) schematic diagram of the trigger and readout system for
ALFA.

The scintillating fibres have a square cross-section of side 0.5 mm, and are
read out by means of Multi-Anode Photomultiplier Tubes. Trigger counters,
which are 3 mm thick plastic scintillator tiles, are located between the
overlap detectors and the main detectors in each of the upper and lower
modules. The scintillating fibres have limited radiation tolerance, which
itself restricts their operation to low-luminosity running of the LHC.

Figure 3 (right) shows schematically the design of the front-end electronics
of the system. A first-level trigger signal is supplied from each ALFA module.
In particular, so-called “elastic” proton-proton combinations, as indicated by
Arm1 or Arm2 in Figure 1 may be triggered on, and also “anti-elastic”
combinations which are defined as signals from modules (A1 or A3) and (A5 or
A7) , or (A2 or A4) and (A6 or A8). The anti-elastic combinations had a high
rate and were prescaled.

Figure 4: (Left) event reconstruction efficiency of the ALFA system for proton
detection for a series of runs; (right) reconstructed proton trajectory pairs
in the A and C sides of the ATLAS system.

The reconstruction efficiencies of the system were measured by a tag-and-probe
method: well measured protons on one side of ATLAS were used as tags to
measure the efficiency for measuring a proton on the other side. The results
are illustrated in Figure 4 [6], which also illustrates the well correlated
nature of proton pairs in mainly elastic events detected in the two sections
of the arms.

A series of precise measurements of elastic proton-proton scattering have been
published [3, 4, 5]. A first measurement of exclusive pion pairs in double-
diffractive proton scattering has also been published [6]. While of limited
statistical accuracy, the observed signal in the elastic channel has very
little background after a set of cleaning cuts is made on the data (Figure 5).

Figure 5: The total vertical momentum of the outgoing $pp\pi^{+}\pi^{-}$
system with two ALFA proton tags and an exclusive central pion pair.

Further data for exclusive process are in the process of being analysed. The
final ALFA operation is scheduled in the LHC Run 3. This will enable more
precision on the elastic measurements and higher precision on exclusive
processes, to enable detailed analysis of Pomeron-Pomeron scattering final
states to be performed.

## References

  * [1] S. Abdel Khalek et al. JINST 11 (2016) P11013
  * [2] A. Sopczak, these proceedings.
  * [3] ATLAS Collaboration, Nucl. Phys. B 889 (2014) 486
  * [4] ATLAS Collaboration, Phys. Lett. B 761 (2016) 158
  * [5] ATLAS Collaboration, EPJ C 83 (2023) 441
  * [6] ATLAS Collaboration, EPJ C 83 (2023) 627

Copyright CERN for the benefit of the ATLAS Collaboration. CC-BY-4.0 license.