# Flavor dependence of jet quenching in heavy-ion collisions

Shan-Liang Zhang<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory
of Nuclear Science, Institute of Quantum Matter, South China Normal
University, Guangzhou 510006, China.  Guangdong-Hong Kong Joint Laboratory of
Quantum Matter, Southern Nuclear Science Computing Center, South China Normal
University, Guangzhou 510006, China.  Enke Wang<EMAIL_ADDRESS>Guangdong
Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter,
South China Normal University, Guangzhou 510006, China.  Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing Center,
South China Normal University, Guangzhou 510006, China.  Hongxi Xing
<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of Nuclear Science,
Institute of Quantum Matter, South China Normal University, Guangzhou 510006,
China.  Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern
Nuclear Science Computing Center, South China Normal University, Guangzhou
510006, China.  Ben-Wei Zhang<EMAIL_ADDRESS>Institute of Particle
Physics and Key Laboratory of Quarks and Lepton Physics (MOE), Central China
Normal University, Wuhan 430079, China

( )

###### Abstract

We investigate the flavor dependence of jet quenching, by performing a
systematic analysis of medium modifications on the inclusive jet,
$\gamma$+jet, and $b$-jet in Pb+Pb collisions at the LHC. Our results from
MadGraph+PYTHIA and LBT exhibit excellent agreement with experimental
measurements of the inclusive jet, $\gamma$+jet and $b$-jet simultaneously
both in p+p and Pb+Pb collisions. We then utilize a Bayesian data-driven
method to extract systematically the flavor-dependent jet energy loss
distributions from experimental data, where the gluon, light quark and
$b$-quark initiated energy loss distributions are well constrained. It is
shown that the quark-initiated jet energy loss distribution shows weaker
centrality and $p_{T}$ dependence than the gluon-initiated one. We demonstrate
the impacts of the slope of initial spectra, color-charge as well as parton
mass dependent jet energy attenuation on the $\gamma/b$-jet suppression
observed in heavy-ion collisions.

###### pacs:

13.87.-a; 12.38.Mh; 25.75.-q

##  I Introduction

The understanding of strongly interacting nuclear matter at extremely high
temperature and energy density is one of the primary subjects in the study of
high-energy nuclear collisions at Relativistic Heavy Ion Collider
(RHIC)Adcox:2001jp ; Adler:2002xw ; Adler:2002tq  and the Large Hadron
Collider (LHC) Aad:2010bu ; Chatrchyan:2012gt ; Chatrchyan:2013kwa ;
Aad:2014bxa . Jet quenching has long been identified as a very powerful tool
to investigate the phase transition from hadron gas to the the quark-gluon
plasma (QGP) with deconfined quarks and gluons Wang:1991xy ; Wang:1998ww , and
numerous studies have shown that parton energy loss in the QGP may lead to the
suppression of the single inclusive hadron/jet spectra Adcox:2001jp ;
Adler:2002xw ; Adler:2002tq ; Qin:2007rn ; Chen:2011vt ; Buzzatti:2011vt ;
Majumder:2011uk ; Aamodt:2010jd ; CMS:2012aa ; Qiu:2019sfj , the shift of
$\gamma$/Z+hadron/jet correlations Wang:1996yh ; Renk:2006qg ; Zhang:2009rn ;
Qin:2009bk ; Adare:2009vd ; Abelev:2009gu ; Chen:2017zte ; Dai:2012am ;
Neufeld:2010fj ; Kang:2017xnc ; Chatrchyan:2012gt ; Luo:2018pto ;
Zhang:2018urd ; Sirunyan:2017jic ; Yang:2021qtl ; Zhang:2021oki ;
Zhang:2022bhq  and dihadron transverse momentum asymmetry Zhang:2007ja ;
stardihadron ; Ayala:2009fe ; Ayala:2011ii , the modification of jet internal
structures ATLAS:2019dsv ; Zhang:2021sua ; Chang:2019sae ; Chang:2019sae ;
KunnawalkamElayavalli:2017hxo ; Kang:2017mda ; Ma:2013uqa ; Chatrchyan:2014ava
; Aaboud:2017bzv ; Vitev:2008rz , as well as the azimuthal anisotropy
($v_{2}$) of hadrons and jets STAR:2003wqp ; CMS:2017xgk ; ATLAS:2018ezv ;
He:2022evt  with the large transverse momentum ($p_{T}$) in nucleus-nucleus
(A+A) collisions, by comparison with those in proton-proton (p+p) collisions
Qin:2015srf ; CMS:2021vui ; Apolinario:2022vzg .

The interaction between an energetic parton and the QGP is sensitive to the
colour charge and the mass of the parton, while medium-induced gluon radiation
is expected to be enhanced for gluon due to its larger color factor, and to be
suppressed for heavy quarks by the dead-cone effect relative to that for light
quarks Dokshitzer:2001zm ; Zhang:2003wk ; Djordjevic:2003qk ; Armesto:2003jh .
A separate determination of quark and gluon jet energy loss could play a
significant role in revealing the fundamental color structures of the QGP and
testing the color representation dependence of the jet-medium interaction
Frye:2017yrw ; Gras:2017jty . This however proves difficult, as the final
state hadronic observables are a mixture of quark and gluon contributions. A
clean method for identifying quark or gluon energy loss remains a challenge,
despite many past attempts such as the multivariate analysis of jet
substructure observables Chien:2018dfn , the proposal of using the averaged
jet charge Chen:2019gqo ; CMS:2020plq ; Li:2019dre  and electroweak gauge
boson tagged jet Dai:2012am ; Neufeld:2010fj ; Kang:2017xnc ; Luo:2018pto ;
Zhang:2018urd ; CMS:2017eqd ; Yang:2021qtl ; Yan:2020zrz ; Zhang:2021oki ;
Aaboud:2019oac .

One recent important measurement by the ATLAS Collaboration, i.e. the nuclear
modification factor for $\gamma$-tagged and $b$-tagged jets ATLAS:2022cim ;
ATLAS:2022fgb , shows quite different modification pattern from that of single
inclusive jets ATLAS:2018gwx . It is reported that the $\gamma$-tagged jets
$R_{AA}$ ATLAS:2022cim  are much higher and show a weaker centrality
dependence than inclusive jet $R_{AA}$ ATLAS:2018gwx  , indicating a sensitive
observation of color factor dependence of jet-medium interaction. In addition,
the ratio of $R_{AA}$ between $\gamma$-tagged jet and inclusive jets are above
most of the theoretical model calculations ATLAS:2022cim , which challenges
the implemented color charge dependence of energy loss in these models.
Likewise, systematic difference between $b$-jets and inclusive jets $R_{AA}$
are also observed ATLAS:2022fgb , suggesting a role for mass and colour charge
effects in partonic energy loss in heavy-ion collisions. Those differences may
arise from not only the inclusive jet mixture of quarks and gluons, where
gluon lose more energy, but also the slope of their initial spectrum
He:2018xjv . Therefore, it is necessary to have explicit knowledge of model-
independent but flavor-dependent jet energy loss distributions, which can help
to constrain jet quenching model uncertainties and to identify the transport
properties of QGP Zhang:2022rby .

The purpose of this work is to extract the flavor-dependent jet energy loss
distributions by performing a systematic study of the medium suppression of
the inclusive jet, $\gamma$+jet, and $b$-jet in Pb+Pb collisions relative to
that in p+p in a unified framework simultaneously. In the numerical
calculation a Monte Carlo event generator MadGraph5+PYTHIA8 Alwall:2014hca ,
which can perform the next-to-leading order (NLO) matrix element (ME) matched
to the resummation of parton shower (PS) calculations, is employed to simulate
initial hard partons with shower partons and jet cross sections, and the
framework of Linear Boltzmann Transport (LBT) Monte Carlo model Li:2010ts ;
Wang:2013cia ; He:2015pra ; Cao:2016gvr ; Cao:2017hhk  is applied to study the
interaction and propagation of these hard partons in the hot/dense QGP medium.
Specifically, a Bayesian data-driven analysis He:2018gks  of the nuclear
modification factors of inclusive jet ATLAS:2018gwx , $\gamma$+jet
ATLAS:2022cim , and $b$-jet ATLAS:2022fgb  is performed to quantitatively
extract the flavor dependent jet energy loss distributions. We study the
relative contributions from the slope of initial spectra, color-charge as well
as parton mass dependent jet energy attenuation to the $\gamma/b$-jet
suppression in heavy-ion collisions at the same time.

The remainder of the paper is organized as follows. In Sec. II we first
introduce the framework. With a systematic study of the inclusive jet,
$\gamma$+jet, and $b$-jet productions in p+p collisions using MadGarph+Pythia,
and their medium alterations in Pb+Pb collisions within LBT, a Bayesian data-
driven analysis of nuclear modification factors of these processes is
performed to quantitatively extract flavor dependent jet energy loss
distributions in Sec. III. Finally, a summary is presented in Sec.IV.

##  II Framework

In order to study the flavor dependence of jet energy loss, we express the
final observable of the nuclear modification factor $R_{AA}$ in a given
centrality in terms of the flavor dependent $R_{AA}^{i,C}$,

$\begin{split}R_{AA}^{C}&=\frac{\sum_{i}R_{AA}^{i,C}d\sigma^{i}_{pp}}{\sum_{i}d\sigma^{i}_{pp}}=R_{AA}^{g,C}+\sum_{i\neq
g}(R_{AA}^{i,C}-R_{AA}^{g,C})f_{i},\end{split}$ (1)

where the superscripts $i$ and $C$ stand for the parton flavor and centrality,
respectively. $d\sigma^{i}_{pp}$ is the differential cross section for parton
$i$ initiated jet in p+p collisions,
$f_{i}=d\sigma^{i}_{pp}/\sum_{i}d\sigma_{pp}^{i}$ is the fraction of the total
jet cross section from the parton $i$ initiated one.

In our analysis, the flavor and centrality dependent nuclear modification
factor $R_{AA}^{i,C}$ is assumed to be factorized as the convolution of its
cross section in p+p collisions and the corresponding parton energy loss
distribution He:2018xjv ; He:2018gks

$R_{AA}^{i,C}(p_{T})=\frac{\int d\Delta p_{T}d\sigma_{pp}^{i}(p_{T}+\Delta
p_{T})\otimes W_{AA}^{i,C}(x)}{d\sigma_{pp}^{i}(p_{T})},$ (2)

where $x=\Delta p_{T}/\langle\Delta p_{T}\rangle$ is the scaled variable with
$\Delta p_{T}$ the amount of energy loss and $\langle\Delta p_{T}\rangle$ the
averaged jet energy loss, which can be parametrized as $\langle\Delta
p_{T}\rangle=\beta_{i}(p_{T})^{\gamma_{i}}\log(p_{T})$ following Refs.
He:2018xjv ; Zhang:2021sua . In Eq. (2), $W_{AA}^{i,C}$ is the scaled energy
loss distribution of parton $i$ in a given centrality class $C$ of A+A
collisions and can be assumed as:

$W^{i}_{AA}(x)=\frac{\alpha_{i}^{\alpha_{i}}x^{\alpha_{i}-1}e^{-\alpha_{i}x}}{\Gamma(\alpha_{i})}$
(3)

where $\Gamma$ is the standard Gamma-function, and the above functional form
can be empirically interpreted as the energy loss distribution resulting from
$\alpha_{i}$ number of jet-medium scattering in the medium.

In this setup, for each parton flavor $i$, the scaled jet energy loss
distributions $W_{AA}^{i}(x)$ can be determined by three parameters,
$\alpha_{i},\beta_{i},\gamma_{i}$.

According to this flavor decomposition, one can extract
$\alpha_{i},\beta_{i},\gamma_{i},$ for each parton flavor $i$ to determine the
flavor and centrality dependent jet energy loss distributions $W_{AA}^{i}(x)$
through a global analysis by combining the simulations of p+p cross section
and the measurements of nuclear modification factor $R_{AA}$ for jet related
observables.

We apply an advanced statistical tool, i.e. Bayesian analysis, for this
purpose. Such a method has been successfully employed to extract the bulk and
heavy quark transport coefficients Xu:2017obm , as well as the inclusive jet
He:2018gks  and gluon Zhang:2022rby  energy loss distributions in heavy-ion
collisions. The process can be summarized as

$P(\theta|data)=\frac{P(\theta)P(data|\theta)}{P(data)},$ (4)

where $P(\theta|data)$ is the posterior distribution of parameters $\theta$
given the experimental data, $P(\theta)$ is the prior distribution of
$\theta$, $P(data|\theta)$ is the Gaussian likelihood between experimental
data and the output for any given set of parameters and $P(data)$ is the
evidence. Uncorrelated uncertainties in experimental data are used in the
evaluation of the Gaussian likelihood. To estimate the posterior distribution
given by Eq. 4, the Markov chain Monte Carlo (MCMC) process is carried out
using the Methropolis-Hasting algorithm Andrieu . A uniform prior distribution
$P(\theta)$ in the region
$[\alpha_{i},\beta_{i},\gamma_{i}]\in[(0,10),(0,8),(0,0.8)]$ is used for the
Bayesian analysis. We first run $1\times 10^{6}$ burn-in MCMC steps to allow
the chain to reach equilibrium, and then generate $1\times 10^{6}$ MCMC steps
in parameter space.

##  III Results and Discussions

###  III.1 Cross sections in p+p and in Pb+Pb

In our analysis, we consider three different observables, i.e. the inclusive
jet, $\gamma$+jet and $b$-jet, to study the flavor dependence of jet energy
loss distribution. We simulate $d\sigma^{i}_{pp}$ using a Monte Carlo event
generator MadGraph5+PYTHIA8 Alwall:2014hca , which combines the NLO matrix
element (ME) with the matched parton shower (PS). Furthermore, those shower
partons are reconstructed using the anti-$k_{t}$ algorithm Cacciari:2008gp
implemented in the FastJet Cacciari:2011ma . In order to compare with the
$b$-jet measurements, we define $b$-jet to be the one that contains at least
one $b$-quark (or $\bar{b}$-quark) with momentum $p_{T}>5$ GeV/c and a radial
separation from the reconstructed jet axis $\Delta R<0.3$. In ATLAS
measurements ATLAS:2018gwx ; ATLAS:2022cim ; ATLAS:2022fgb , the jets are
accepted in the rapidity range of $|y|<2.8$ for inclusive jet and
$\gamma$+jet, $|y|<2.1$ for $b$-jet. Besides, for $\gamma$+jet event, $\gamma$
is required to have $p_{T}^{\gamma}>$ 50 GeV/$c$, and a cut
$\Delta\phi_{\gamma j}>\pi/2$ is imposed to select the back-to-back
$\gamma$+jet pairs. In our simulations, we implement correspondingly the same
kinematic cuts adopted by experiments.

Figure 1: (Color online) Up: Transverse momentum distributions of: (a)
inclusive jet, (b) $\gamma$-tagged jet, and (c) $b$-jet simulated by
MadGraph+Pythia8 (lines) and the comparison with experimental data (samples)
ATLAS:2018gwx ; ATLAS:2022cim ; ATLAS:2022fgb  in p+p collisions. The inset in
(a) is the ratio of $\gamma$-tagged jet (blue solid) and $b$-jet (red dashed)
to inclusive jet cross section. Middle: fraction of quark (Dashed blue line)
and gluon (Solid red line) initiated jet of : (d) inclusive jet, (e)
$\gamma$-tagged jet, and (f) $b$-jet as a function of jet $p_{T}$ in p+p
collisions. Bottom: nuclear modification factor of: (g) inclusive jet, (h)
$\gamma$-tagged jet, and (i) $b$-jet calculated by LBT (lines) and the
comparison with experimental data (samples) ATLAS:2018gwx ; ATLAS:2022cim ;
ATLAS:2022fgb  in Pb+Pb collisions.

In the top panel of Fig. 1, we plot the differential cross section of: (a)
inclusive jet, (b) $\gamma$+jet, and (c) $b$-jet as a function of jet
transverse momentum $p_{T}$ obtained from MadGraph+Pythia8 simulation at 5.02
TeV in p+p collisions. Through the comparison with experimental data
ATLAS:2018gwx ; ATLAS:2022cim ; ATLAS:2022fgb , one can see clearly that the
simulations give very well descriptions of all experimental data. Notice that
the inset of Fig. 1(a) is the scaled ratio of $\gamma$+jet (blue solid) and
$b$-jet (red dashed) cross section to that of inclusive jet. In Fig. 1(a-c),
one can see that the inclusive jet spectrum is much steeper than $\gamma$+jet,
while $b$-jet have similar slope as the inclusive jet, consistent with the
results of Refs. ATLAS:2022cim ; ATLAS:2022fgb .

In order to study the flavor dependence of jet energy attenuation in heavy ion
collisions, we present the relevant contributions in terms of jet flavor,
which is defined as the flavor of the hard parton that fragments into the
final observed jet. In the middle panel of Fig. 1, we show the fraction from
quark- and gluon- initiated jet in: (d) inclusive jet, (e) $\gamma$+jet, and
(f) $b$-jet as a function of jet $p_{T}$. One can see that for inclusive jet,
the contribution from gluon (quark) initiated jet dominates in low (large)
$p_{T}$ region, and gradually decreases (increases) with increasing $p_{T}$.
While for $\gamma$+jet, the quark initiated jet dominates ($\sim 80\%$) in the
whole $p_{T}$ region. For $b$-jet, it can be generated either from the initial
hard scattering or from the parton showers via gluon and quark splitting. In
the first case, it is the $b$-quark that initiates the $b$-jet, the relevant
contribution is shown by $b$-quark in Fig. 1(f). In heavy-ion collisions, the
medium modification to such $b$-jet has direct connection to the heavy quark
energy loss Zhang:2003wk ; Djordjevic:2003qk ; Armesto:2003jh ; Dai:2022sjk .
On the other hand, the medium modification on the latter two cases (with gluon
and quark splitting) would resemble that of a massive quark or gluon jets. As
can be seen, $b$-jet from gluon initiated contributes about $40\%$ to the
cross section in the whole $p_{T}$ region, while the light quark initiated
contribution goes up with increasing $p_{T}$.

Figure 2: (Color online) Distributions of and the correlations between the
Bayesian-extracted parameters for gluon (left) and quark (right) initiated jet
energy loss via fitting to $R_{AA}$ of inclusive jet and $\gamma$-tagged jet
in central 0-10$\%$ Pb+Pb collisions at 5.02 TeV ATLAS:2018gwx ; ATLAS:2022cim
.

To benchmark the medium effect which bridges the initial flavor origin and the
final observed jet attenuation in A+A collisions. We present in the bottom
panel of Fig. 1 the nuclear modification factor $R_{AA}$ evaluated as a
function of jet $p_{T}$ for: (g) inclusive jet, (h) $\gamma$+jet, and (i)
$b$-jet, and compared with ATLAS data ATLAS:2022fgb ; ATLAS:2018gwx ;
ATLAS:2022cim . The theory curves are obtained from LBT model Li:2010ts ;
Wang:2013cia ; He:2015pra ; Cao:2016gvr ; Cao:2017hhk , which includes both
elastic Li:2010ts ; Wang:2013cia ; He:2015pra  and inelastic scatterings
Cao:2016gvr ; Cao:2017hhk ; Guo:2000nz ; Zhang:2003yn ; Zhang:2003wk ;
Zhang:2004qm  for jet shower and recoil medium partons. Our results from LBT
with $\alpha_{s}=0.18$, which is the only parameter in LBT that controls the
strength of parton interaction, show excellent agreements with the
experimental data for inclusive jet ATLAS:2018gwx , $\gamma$+jet ATLAS:2022cim
and $b$-jet ATLAS:2022fgb . As can be seen from the figure, the nuclear
modification factor $R_{AA}$ for $\gamma$+jet is larger than that of inclusive
jet. This is attributed to their different quark and gluon origins and the
slope of the reference spectra in p+p collisions. The nuclear modification
factor $R_{AA}$ for $b$-jet is also larger than that of inclusive jet in low
$p_{T}$ region, while the difference disappear at large $p_{T}$ region, which
should be a mixed effect of color charge and parton mass dependence of jet
quenching in medium.

###  III.2 Colour-charge dependence of $R_{AA}$

In Fig. 2, we present the distributions of the final-extracted parameters for
gluon (left) and quark (right) initiated jet energy loss as well as their
correlations, via Bayesian-fitting to the ATLAS data ATLAS:2018gwx ;
ATLAS:2022cim  on $R_{AA}$ of inclusive jets and $\gamma$-tagged jets in
0-10$\%$ Pb+Pb collisions at 5.02 TeV simultaneously. As can be seen,
$\beta_{i}$ and $\gamma_{i}$, which reflect the average energy loss, are
strongly correlated and well constrained for quark and gluon initiated jet.
The mean value as well as its standard deviation of those final extracted
parameters for gluon and charm quark energy loss distribution are summarized
in Table 1.

Figure 3: (Color online) (a) Data-driven Bayesian fitted nuclear modification
factor $R_{AA}$ of inclusive jet (orange) and $\gamma$-tagged jet (gray) and
the comparison to experimental data at ATLAS:2018gwx ; ATLAS:2022cim . (b)
Data-driven extracted nuclear modification factor of quark (blue) and gluon
(red) initiated jet. (c) Fraction of jet average energy loss of light quark
(blue) and gluon (quark) initiated jet, (d) scaled energy loss distributions
$W_{AA}^{i}(x)$ of quark (blue) and gluon (red) initiated jet .

The final fitted nuclear modification factor $R_{AA}$ of inclusive jet and
$\gamma$-tagged jet as well as the comparison to experimental data
ATLAS:2018gwx ; ATLAS:2022cim  in 0-10$\%$ centrality at 5.02 TeV are shown in
Fig. 3.(a), and data-driven extracted nuclear modification factor of quark-
and gluon- initiated inclusive jet are shown in Fig. 3.(b). The corresponding
bands are results with one sigma deviation from the average fits of $R_{AA}$.
Data-driven extracted average energy loss fraction $\langle\Delta
p_{T}\rangle/p_{T}$ and scaled energy loss distributions $W_{AA}(x)$ of quark
and gluon initiated jet are also presented in Fig. 3.(c) and Fig. 3.(d). As
can be seen, average energy loss of gluon and quark jet is well constrained in
$p_{T}<150$ GeV/$c$, but is weaker constrained at high $p_{T}$ due to large
experimental data errors and the scarcity of $\gamma$-tagged jet experimental
data at such high $p_{T}$. The quark-initiated jets lose less fraction of its
energy and shows a weaker dependence on the jet $p_{T}$ compared to gluon-
initiated jets due to its color factor as expected. Since jet showers also
contain gluons even if they are initiated by a hard quark, the net energy loss
of a gluon-tagged jet is always larger than that of a quark-tagged jet but the
ratio is smaller than 9/4 from the naive leading order estimation Wang:1998bha
; Liu:2006sf ; Chen:2008vha .

Figure 4: (Color online) (a) The reference $R_{AA}^{\text{ref}}$ (magenta) and
the comparison with $R_{AA}$ of $\gamma$+jet (grey) and inclusive jet (orange)
jet in 0-10$\%$ centrality at 5.02 TeV, and the comparison with experimental
data ATLAS:2022cim . (b) The relative contribution fraction from large quark
fraction to the less suppression of $\gamma$+jet $R_{AA}$ compared to
inclusive jet $R_{AA}$.

Fig. 3.(a) shows that $\gamma$-tagged jet $R_{AA}$ is less suppressed compared
to that for inclusive jet, which is a mix effect of the slope of initial
spectra and parton color-charge in p+p collisions. To clarify the relative
contributions from the color-charge effect and the initial parton spectra
between $\gamma$-tagged jet and inclusive jet, we calculate an artificial
reference $R_{AA}^{\text{ref}}$ following Eq.(1), by assuming the inclusive
jet production has the same fraction of quark jet as $\gamma$+jet. This
reference $R_{AA}^{\text{ref}}$ is shown by magenta lines in Fig. 4.(a). The
difference between $R_{AA}^{\text{ref}}$ and inclusive jet $R_{AA}$ (denoted
as “$R_{AA}^{\text{jet}}$”) should be attributed largely to the different
color-charge effect between quark-medium and gluon-medium interactions, while
the distinction between $R_{AA}^{\text{ref}}$ and $\gamma$+jet $R_{AA}$
(denoted as “$R_{AA}^{\gamma\text{+jet}}$”) should be attributed mostly to the
slope of reference spectra in p+p.

Fig. 4.(b) shows the relative contribution fraction from large quark fraction,
evaluated as
$f^{\text{flavor}}=(R_{AA}^{\text{ref}}-R_{AA}^{\text{jet}})/(R_{AA}^{\gamma\text{+jet}}-R_{AA}^{\text{jet}})$,
to the less suppression of $\gamma$+jet $R_{AA}$ compared to inclusive jet
$R_{AA}$. The increased quark jet fraction in inclusive jet production give
the dominant contributions to the difference of $R_{AA}$ between $\gamma$+jet
and inclusive jet at $p_{T}>60$ GeV/$c$. Then $1-f^{\text{flavor}}$
characterized approximately the relative contribution from the slope of
reference spectra, which plays a dominated role in the suppression at low
$p_{T}$. Besides, the distinction between $\gamma$+jet $R_{AA}$ and inclusive
jet $R_{AA}$ will diminish with increasing $p_{T}$, because quark-initiated
jets contribute a lion’s share to the yields of both $\gamma$+jet and the
inclusive jet at very large $p_{T}$, which can be verified with the upcoming
high precision measurements at the LHC.

###  III.3 Centrality dependence of $R_{AA}$

Figure 5: (Color online) Data-driven fitted nuclear modification factor $R_{AA}$ of the inclusive jet $R_{AA}$ ATLAS:2018gwx  and $\gamma$+jet $R_{AA}$ ATLAS:2022cim  in 10-30$\%$, 30-80$\%$ centrality bins and predictions of inclusive jet $R_{AA}$ in 10-20$\%$, 20-30$\%$, 30-40$\%$ and 0-20$\%$ centrality bins as well as the comparison with experimental data ATLAS:2018gwx .  Table 1: Parameters [$\alpha_{i}$, $\gamma_{i}$, $\beta_{i}$] of quark and gluon jet energy loss distribution from Bayesian fits to experimental data ATLAS:2018gwx ; ATLAS:2022cim  on inclusive jet and $\gamma$+jet suppressions at 5.02 TeV.  |  | $\alpha_{i}$ | $\beta_{i}$ | $\gamma_{i}$
---|---|---|---|---
0-10$\%$ | gluon |  5.44$\pm$2.15 |  1.46$\pm$0.22 |  0.25$\pm$0.03
quark |  0.47$\pm$0.06 |  1.09$\pm$0.21 |  0.24$\pm$0.04
10-30$\%$ | gluon |  1.48$\pm$0.45 |  1.65$\pm$0.32 |  0.21$\pm$0.03
quark |  3.96$\pm$1.05 |  1.47$\pm$0.13 |  0.11$\pm$0.02
30-80$\%$ | gluon |  4.84$\pm$2.72 |  0.89$\pm$0.14 |  0.14$\pm$0.03
quark |  2.28$\pm$0.88 |  1.07$\pm$0.07 |  0.07$\pm$0.01

Moreover, we extract the centrality dependent quark and gluon jet energy loss
distributions before exploring parton-mass effect on jet quenching motivated
by two reasons. First, $\gamma$-tagged jet $R_{AA}$ ATLAS:2022cim  shows a
weaker dependence on centrality compared to inclusive jet ATLAS:2018gwx ,
indicating that gluon-initiated jets may show a distinct centrality dependence
with quark-initiated jets. Second, the experimental data of $\gamma$+jet
$R_{AA}$ ATLAS:2022cim , inclusive jet $R_{AA}$ ATLAS:2018gwx  and $b$-jet
$R_{AA}$ ATLAS:2022fgb  are in different centrality bins. We need centrality
dependent quark and gluon jet energy loss distributions to fit $\gamma$+jet
$R_{AA}$, inclusive jet $R_{AA}$ and $b$-jet $R_{AA}$ simultaneously.

Figure 6: (Color online) The centrality dependence of final fitted gluon jet
(red), quark jet (blue) and inclusive jet (green) $R_{AA}$ in Pb+Pb collisions
at 5.02 TeV.

As a matter of fact, there are no experimental data of inclusive jet $R_{AA}$
and $\gamma$+jet $R_{AA}$ in the same centrality class except in central
0-10$\%$ centrality. For inclusive jet measurements, the existing measurements
are provided for centrality bins 0-10$\%$, 10-20$\%$, 20-30$\%$, 30-40$\%$,
40-50$\%$,50-60$\%$, 60-70$\%$, 70-80$\%$ ATLAS:2018gwx , while for
$\gamma$+jet $R_{AA}$, it is limited to 0-10$\%$, 10-30$\%$, 30-80$\%$
ATLAS:2022cim . In order to take full advantage of the existing measurements
for inclusive jet $R_{AA}$ in different centrality bins, we generate the
inclusive jet $R_{AA}$ as well as the corresponding errors in 10-30$\%$ and
30-80$\%$ centrality bins according to $R_{AA}^{C}=\sum_{c\in
C}P^{c}R_{AA}^{c}$, where $P^{c}=N^{c}_{bin}/\sum_{c}N^{c}_{bin}$ is the
probability of finding jet events in a given centrality bin following Ref.
Xing:2019xae . With such an extension, we can perform a simultaneous fit for
both inclusive jet $R_{AA}$ and $\gamma$+jet $R_{AA}$ in 10-30$\%$ and
30-80$\%$ centralities. In Fig. 5, we present the data-driven fitted nuclear
modification factor $R_{AA}$ of inclusive jet ATLAS:2018gwx  and $\gamma$+jet
ATLAS:2022cim  in 10-30$\%$ and 30-80$\%$ centralities and the comparison with
experimental data at 5.02 TeV. All final spectra based on Eq. (1) and Eq. (2)
are in nice agreement with the experimental data. The corresponding mean value
as well as its standard deviation of those final extracted parameters for
gluon and light quark energy loss distribution are summarized in Table 1.

Meanwhile, we obtain $R_{AA}$ for quark-initiated jets and gluon-initiated
jets in 10-30$\%$, 30-80$\%$ centrality. Combined with the flavor dependent
$R_{AA}$ in 0-10$\%$ as extracted in the previous section (Fig. 3.(b)), we
obtain the centrality dependence of final fitted gluon-initiated jet, quark-
initiated jet and inclusive jet $R_{AA}$. In Fig. 6, we show the centrality
dependence of final fitted gluon jet (red), quark jet (blue) and inclusive jet
(green) $R_{AA}$ in Pb+Pb collisions in the region $100<p_{T}<112$ GeV/$c$ by
step lines. One finds that the quark-initiated jet has weaker dependence on
the centrality than that for gluon-initiated jet.

Next, we can fit the centrality dependent $R_{AA}$ of quark- and gluon-
initiated jet via a simple parametrization $h^{i}(C)=a_{i}C^{2}+b_{i}C+c_{i}$,
with $C$ stands for the centrality. The best fit curves of $h^{i}(C)$ are
shown in Fig. 6 by dashed and dotted lines, and the corresponding best-fit
parameter values are presented in Table. 2. Notice that the extrapolation to
peripheral collisions ($>80\%$) is greater than one and can not be trusted, a
reasonable identification of jet energy loss distribution for peripheral
collisions would require a corresponding extension of experimental
measurements. If we ignore the $p_{T}$ dependence of $h^{i}(C)$,
$R_{AA}^{i,C}$ for any centrality $C$ can be simply obtained by
$R_{AA}^{i,C}=h^{i}(C)*R_{AA}^{i,rc}/h^{i}(rc)$, where $rc$ stands for
reference centrality. Based on Eq. (1) and the above extracted centrality
dependent quark and gluon jet $h^{i}(C)$, the predication of inclusive jet
$R_{AA}$ in 0-20$\%$, 10-20$\%$, 20-30$\%$, 30-40$\%$ are presented in Fig. 5.
One can see that our extracted centrality dependence of quark and gluon jet
energy loss distributions can describe the experimental data $R_{AA}$
ATLAS:2018gwx  very well.

Table 2: The best-fitted Parameters [$a_{i}$, $b_{i}$, $c_{i}$] for centrality dependent quark and gluon jet energy loss distributions.  |  $a_{i}\ (\times 10^{-5})$ |  $b_{i}\ (\times 10^{-3})$ |  $c_{i}$
---|---|---|---
Quark |  5.58$\pm$1.09 |  1.65$\pm$0.57 |  0.58$\pm$0.0048
Gluon |  2.20$\pm$1.57 |  7.37$\pm$0.78 |  0.34$\pm$0.0056
Table 3: Parameters [$\alpha_{i}$, $\gamma_{i}$, $\beta_{i}$] for gluon, quark
and $b$-quark energy loss in 0-20$\%$ centrality from fitting to $\gamma$+jet
$R_{AA}$ ATLAS:2022cim , inclusive jet $R_{AA}$ ATLAS:2018gwx  and $b$-jet
$R_{AA}$ ATLAS:2022fgb  at $\sqrt{s}=5.02$ TeV simultaneously.  (0-20$\%$)
5.02 TeV
---
|  $\alpha_{i}$ |  $\beta_{i}$ |  $\gamma_{i}$
Gluon |  3.62$\pm$1.40 |  1.52$\pm$0.31 |  0.232$\pm$0.035
Quark |  3.0$\pm$0.85 |  1.45$\pm$0.12 |  0.142$\pm$0.019
$b$-quark |  4.5$\pm$2.69 |  0.93$\pm$0.28 |  0.138$\pm$0.06

###  III.4 Parton-mass dependence of $R_{AA}$

Finally, with the extracted centrality dependent quark and gluon energy loss
distributions, we also extract $b$-jet energy loss in the same framework based
on Eqs. (1) and (2) through fitting to the experimental data of $b$-jet
$R_{AA}$ ATLAS:2022fgb , inclusive jet $R_{AA}$ in 0-20$\%$ centrality
ATLAS:2018gwx  and $\gamma$-tagged jet $R_{AA}$ in 0-10$\%$ ATLAS:2022cim
simultaneously. The final fitted nuclear modification factor $R_{AA}$ of
$b$-jet (lime green lines), inclusive jet (magenta lines) and $\gamma$-tagged
jet (yellow lines) as well as the comparison with experimental data
ATLAS:2018gwx ; ATLAS:2022cim ; ATLAS:2022fgb  are shown in Fig. 7(a). The
corresponding bands are results with one sigma deviation from the average fits
of $R_{AA}$. Meanwhile, Fig. 7(b) shows the extracted nuclear modification
factor $R_{AA}$ for $b$-quark initiated (green, denoted as “$R_{AA}^{b}$”),
light-quark (denoted as “$R_{AA}^{\text{quark}}$”) and gluon (denoted as
“$R_{AA}^{\text{gluon}}$”) initiated $b$-jet in 0-20$\%$ centrality, with the
corresponding parameters for gluon, quark and $b$-quark energy loss
distribution summarized in Table.3. The final extracted light-quark and gluon
initiated jet energy loss distributions are consistent with our previous
results in the same centrality, while $b$-quark initiated jets is less
suppressed compared to light-quark initiated jets due to its large mass in low
$p_{T}$ region as expected.

To explore the underlying $b$-jet suppression mechanism in heavy-ion
collisions, we also present in Fig. 7(c) the ratio of $b$-quark initiated jets
$R_{AA}$ to light quark initialed jet $R_{AA}$ as
$R_{AA}^{b}/R_{AA}^{\text{quark}}$, and also in Fig. 7(d) the ratio of $b$-jet
$R_{AA}$ (denoted as “$R_{AA}^{b\text{-jet}}$”) to inclusive jet $R_{AA}$
$R_{AA}^{b\text{-jet}}/R_{AA}^{\text{jet}}$ extracted from global analysis and
the comparison to the experimental measurements ATLAS:2022fgb . Our numerical
results can describe the experimental data within large uncertainties
ATLAS:2022fgb . These ratios are greater than unity and goes down with
increasing $p_{T}$, indicating that the parton mass effect is reduced with
increasing $p_{T}$ Huang:2013vaa . However, the mass effect for $b$-jet could
persist to large $p_{T}$, even at $p_{T}\sim 300$ GeV/$c$, and is consistent
with the current data and a model based on strong coupling (via the AdS/CFT
correspondence) Horowitz:2007su , in contrast to Ref. Huang:2013vaa ;
Xing:2019xae  in which mass effects are expected to be small at $p_{T}>70$
GeV/$c$. Those disagreements may be explained by two reasons. For one thing,
due to the subdominant contributions and the limited $b$-jet $R_{AA}$ data
points with large uncertainties, especially at large $p_{T}$, which have weak
constraints on $b$-quark initialed jet. From there, $b$-quark initiated jet
energy loss distributions is weakly constrained at present. For another, this
may be attributed to the mixture of mass effect and color effect, as we may
show below.

Figure 7: (Color online) (a): final fitted nuclear modification factor
$R_{AA}$ of $b$-jets (lime green lines), inclusive jet (magenta lines) and
$\gamma$-tagged jet (yellow lines) and the comparison with experimental data
ATLAS:2018gwx ; ATLAS:2022cim ; ATLAS:2022fgb . (b): the data-driven extracted
$R_{AA}$ of gluon (red), light quark (blue), and $b$-quark (green) initiated
jets. (c): The ratio of $R_{AA}^{b}/R_{AA}^{quark}$ from global analysis. (d):
the ratio of $R_{AA}^{b\text{-jet}}/R_{AA}^{\text{jet}}$ extracted from global
analysis and the comparison to the experimental measurements. The quark mass
effect (green) and less gluon fraction effect (yellow) to the ratio of
$R_{AA}$ are also presented.

For further demonstration of $b$-quark mass effect on the suppression of
$b$-jet, we show in Fig. 7(d) (green lines) the ratio of $b$-jet $R_{AA}$ to
inclusive jet $R_{AA}$, assuming $b$-jet has the same fraction of gluon
initiated jet as inclusive jet (denoted as
“$f^{b\text{-jet}}=f^{\text{jet}}$”). The difference between this ratio and
$R_{AA}^{b\text{-jet}}/R_{AA}^{\text{jet}}$ should be attributed to the
$b$-quark mass effect. One can see that the deviation between $b$-jet and the
inclusive jet is significantly reduced with increasing $p_{T}$. The mass
effect roughly give considerable contributions to the ratio of
$R_{AA}^{b\text{-jet}}/R_{AA}^{\text{jet}}$ and are expected to be small at
$p_{T}\sim 300$ GeV/$c$.

To further illustrate the color-charge effect on the suppression of $b$-jet,
we also calculated the ratio of $b$-jet $R_{AA}$ to inclusive jet $R_{AA}$,
assuming $b$-quark jet lose the same fraction of energy as light-quark
initiated jet (denoted as “$R_{AA}^{b}=R_{AA}^{\text{quark}}$”), as shown by
yellow lines in Fig. 7(d). The difference between this ratio and
$R_{AA}^{b\text{-jet}}/R_{AA}^{\text{jet}}$ should be attributed to the
different gluon and quark fraction. As can be seen, those ratio is
significantly enhanced and also show a downward tendency with increasing
$p_{T}$, indicating that, less gluon initiated jet contribution also lead to
the less suppression of $b$-jet compared to inclusive jet in heavy-ion
collisions, especially in low $p_{T}$ region. Furthermore, the contribution
from gluon initialed jet to inclusive jet production is greater than that to
$b$-jet in the $p_{T}<300$ GeV/$c$ region as shown in Fig. 1. Thus $b$-jet
$R_{AA}$ is moderately larger than inclusive jet $R_{AA}$. Therefore, we can
see that color charge effect and mass effect gives almost the same
contributions to $b$-jet suppression respectively in heavy-ion collisions.

##  IV Summary

We have carried out a systematic investigation of parton color-charge and
parton mass dependence of nuclear modification factor by a systematic study of
the medium modifications on three full jet observables: the inclusive jet,
$\gamma$+jet, and $b$-jet, in Pb+Pb collisions relative to that in p+p at the
LHC. Our results from MadGraph+PYTHIA and LBT give very nice descriptions of
the experimental data for these three jet observables both in p+p and Pb+Pb.
Then a Bayesian data-driven method is applied to extract the model-independent
but flavor-dependent jet energy loss distributions. Fitting to those
experimental data simultaneously, gluon, quark and $b$-quark initiated jet
energy loss can be well constrained. It is seen that the energy loss of quark-
initiated jets shows a weaker centrality dependence and weaker $p_{T}$
dependence compared to that of gluon-initiated jets. We find that large quark-
initiated jet fraction underlies $\gamma$+jet suppression at large $p_{T}$,
while the flat spectra give the dominate contribution to $\gamma$+jet
suppression at low $p_{T}$. However, the $b$-initiated jet is less suppressed
compared to light-quark initiated jet. We demonstrate that the quark mass
effect and color charge effect have comparable impacts to the ratio
$R_{AA}^{b\text{-jet}}/R_{AA}^{\text{jet}}$, though their influence may
decrease significantly at $p_{T}\sim 300$ GeV/$c$. Such a systematic
extraction of jet energy loss distributions can help constrain model
uncertainties and pave the way to precise predictions of the properties of the
hot QCD medium created in relativistic heavy-ion collisions111When finalizing
this paper, the authors notice a very recent parallel study of extracting the
flavor dependence of parton energy loss Xing:2023ciw , but from the nuclear
modifications of various hadron species instead of jet observables presented
in our work..

Acknowledgments: This research is supported by Natural Science Foundation of
China with Project Nos. 12035007, 12022512, 12147131, Guangdong Major Project
of Basic and Applied Basic Research No. 2020B0301030008. S.Z. is further
supported by the MOE Key Laboratory of Quark and Lepton Physics (CCNU) under
Project No. QLPL2021P01.

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(3.11). Then there exists a constant $\omega\in\mathbb{R}$ such that
$-\mathrm{J}_{1}+\omega$ admits bounded $\mathcal{H}^{\infty}$-calculus on
${\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}})\times\mathrm{L}^{q}(\mathbb{R}^{2})^{2}$.

###### Proof.

We denote by $\mathrm{A}:=\mathrm{A}^{\mathrm{ocn}}_{0}$,
$\mathrm{B}:=-\mathrm{L}_{\mu}\mathrm{A}^{\mathrm{H}}$,
$\mathrm{C}:=-{\rho}_{\mathrm{ocn}}{C}_{\mathrm{ocn}}{R}_{\mathrm{ocn}}\partial_{\nu_{\Gamma_{o}}}$
and $\mathrm{D}:=\mathrm{A}^{\mathrm{H}}$. First, we consider the operator
matrix $\tilde{\mathrm{J}_{1}}$ given by

(8.3)
$\tilde{\mathrm{J}}_{1}:=\begin{pmatrix}\mathrm{A}^{\mathrm{ocn}}_{0}-\mu&-\mathrm{L}_{\mu}\mathrm{A}^{\mathrm{H}}\\\
-{\rho}_{\mathrm{ocn}}{C}_{\mathrm{ocn}}{R}_{\mathrm{ocn}}\partial_{\nu_{\Gamma_{o}}}&\mathrm{A}^{\mathrm{H}}\end{pmatrix}=\begin{pmatrix}\mathrm{A}&\mathrm{B}\\\
\mathrm{C}&\mathrm{D}\end{pmatrix}$

with diagonal domain. We verify the assumptions of [1, Corollary 5.7] for
$\tilde{\mathrm{J}}_{1}$.

Thanks to Subsection 7.1, the operator matrix $\tilde{\mathrm{J}}_{1}$ is
diagonally dominant, i.e., it satisfies [1, Assumption 3.1]. Further, we know
from Subsection 7.1 that
$\mathrm{C}\in\mathcal{L}((D(-\mathrm{A})^{\gamma}),{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}}))$
for all $\gamma\in(0,\frac{1}{2}(1+\frac{1}{q}))$.

It remains to verify [1, Assumption 5.1(+)]. Let
$\delta\in(0,\nicefrac{{1}}{{q}})$. By Subsection 5.1(d) and Subsection
6.1(d), we know that

$\displaystyle D(\mathrm{A}^{1+\delta})$
$\displaystyle\hookrightarrow\mathrm{H}^{{2+2\delta},q}({\Omega}_{\mathrm{ocn}})^{2}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}}),$
$\displaystyle
D(\mathrm{A}^{\delta})=\mathrm{H}^{{2\delta},q}({\Omega}_{\mathrm{ocn}})^{2}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}}),$
$\displaystyle D(\mathrm{D}^{1+\delta})$
$\displaystyle=\mathrm{H}^{{2+2\delta},q}(\mathbb{R}^{2})^{2},$ $\displaystyle
D(\mathrm{D}^{\delta})=\mathrm{H}^{{2\delta},q}(\mathbb{R}^{2})^{2}.$

As in the proof of Subsection 7.1, it follows that
$\mathrm{C}D(\mathrm{A}^{1+\delta})\subset\mathrm{H}^{2\delta}(\mathbb{R}^{2})^{2}=D(\mathrm{D}^{\delta})$.
From Subsection 7.1 and $\mathrm{B}=\mathrm{L}_{\mu}\mathrm{D}$ it follows
that
$\mathrm{B}D(\mathrm{D}^{1+\delta})=\mathrm{L}_{\mu}D(\mathrm{D}^{\delta})=\mathrm{L}_{\mu}\mathrm{H}^{{2\delta},q}(\mathbb{R}^{2})^{2}\subset\mathrm{H}^{{2\delta},q}(\Omega)^{2}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}})=D(\mathrm{A}^{\delta})$.
Since the operators $\mathrm{B}$ and $\mathrm{C}$ are closed, the estimates in
[1, Assumption 5.1(+)] follow by the closed graph theorem.

Now, [1, Corllary 5.7] implies that $\tilde{\mathrm{J}}_{1}-\omega$, given by
(8.3), admits a bounded $\mathcal{H}^{\infty}$-calculus on
${\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}})\times\mathrm{L}^{q}(\mathbb{R}^{2})^{2}$
for some constant $\omega\in\mathbb{R}$. By Subsection 7.1 and Subsection 7.1,
the operator
$\mathrm{L}_{\mu}{\rho}_{\mathrm{ocn}}{C}_{\mathrm{ocn}}{R}_{\mathrm{ocn}}\partial_{\nu_{\Gamma_{o}}}$
is relatively $(-\mathrm{A}^{\mathrm{ocn}})^{\gamma}$-bounded for all
$\gamma\in(0,\frac{1}{2}(1+\frac{1}{q}))$. In addition, by Subsection 7.1 and
Subsection 7.2, the operators $\mathrm{L}_{\mu}\mathrm{N}_{\mu}$ and
$\mathrm{N}_{\mu}$ are relatively
$(-\mathrm{A}^{\mathrm{H}})^{\theta}$-bounded for all
$\theta\in(0,\nicefrac{{1}}{{2}})$. Therefore, the claim follows from results
in [32]. ∎

We conclude the main theorem of this section.

######  Theorem 8.1.

Let $p,q\in(1,\infty)$ such that
$\nicefrac{{1}}{{p}}+\nicefrac{{1}}{{q}}<\nicefrac{{1}}{{2}}$ and
$(v_{\mathrm{atm},0},v_{\mathrm{ocn},0},v_{0})\in V$. Then there exists a
constant $\omega\in\mathbb{R}$ such that the operator matrix
$-\mathcal{A}+\omega$, with $\mathcal{A}$ given in (4.19), admits a bounded
$\mathcal{H}^{\infty}$-calculus on $\mathcal{X}_{0}$.

###### Proof.

In the sequel, the constants $\omega\in\mathbb{R}$ may change from line to
line. It follows from [43, Theorem 8.22] that $-\mathrm{J}_{2}+\omega$ admits
a bounded $\mathcal{H}^{\infty}$-calculus on
$\mathrm{L}^{q}(\mathbb{R}^{2})\times\mathrm{L}^{q}(\mathbb{R}^{2})$ for some
$\omega\in\mathbb{R}$. Combining this with Subsection 8.1, we deduce that
$-\mathrm{diag}({\mathrm{J}_{1},\mathrm{J}_{2}})+\omega$ with diagonal domain
admits a bounded $\mathcal{H}^{\infty}$-calculus on
${\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}})\times\mathrm{L}^{q}(\mathbb{R}^{2})^{2}\times\mathrm{L}^{q}(\mathbb{R}^{2})\times\mathrm{L}^{q}(\mathbb{R}^{2})$
for some $\omega\in\mathbb{R}$. Further, we know from Subsection 5.1 that
$\mathrm{B}^{\prime}$ is relatively
$(-\mathrm{A}^{\mathrm{H}}+\tilde{\omega})^{\frac{1}{2}}$-bounded, and it
follows from [32, Proposition 13.1] and (8.2) that $-\mathrm{J}+\omega$ admits
a bounded $\mathcal{H}^{\infty}$-calculus on
${\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}})\times\mathrm{L}^{q}(\mathbb{R}^{2})\times\mathrm{L}^{q}(\mathbb{R}^{2})\times\mathrm{L}^{q}(\mathbb{R}^{2})$
for some $\omega\in\mathbb{R}$.

Invoking Subsection 6.1(a), we conclude from (8.1) that
$-\tilde{\mathcal{A}}+\omega$ admits a bounded $\mathcal{H}^{\infty}$-calculus
on $\mathcal{X}_{0}$. The claim finally follows from Subsection 8.1. ∎

Again, we collect some interesting and useful functional analytic properties.
We recall the spaces $\mathcal{Y}_{\beta}$ and $\mathcal{Y}_{\theta}$ from
(5.5).

######  Corollary .

Let $p,q\in(1,\infty)$ such that
$\nicefrac{{1}}{{p}}+\nicefrac{{1}}{{q}}<\nicefrac{{1}}{{2}}$ and
$(v_{\mathrm{atm},0},v_{\mathrm{ocn},0},v_{0})\in V$. Then there exists
$\omega_{0}\in\mathbb{R}$ such that for $\omega>\omega_{0}$, the following
properties hold true.

  1. (a)

The operator $-\mathcal{A}+\omega$ has an $\mathcal{R}$-bounded
$\mathcal{H}^{\infty}$-calculus on $\mathcal{X}_{0}$ with angle
$\Phi_{-\mathrm{A}_{\mathrm{b.c.},q}+\mu}<\nicefrac{{\pi}}{{2}}$.

  2. (b)

It holds that $-\mathcal{A}+\omega\in\mathcal{BIP}(\mathcal{X}_{0})$, i.e.,
the operator $-\mathcal{A}+\omega$ has bounded imaginary powers on
$\mathcal{X}_{0}$.

  3. (c)

The operator $-\mathcal{A}+\omega$ has the property of maximal
$\mathrm{L}^{p}[0,\infty)$-regularity in $\mathcal{X}_{0}$.

  4. (d)

The operator $\mathcal{A}-\omega$ generates a bounded analytic
$\mathrm{C}_{0}$-semigroup of angle $\nicefrac{{\pi}}{{2}}$ on
$\mathcal{X}_{0}$, and $\mathcal{A}$ generates an analytic semigroup of angle
$\nicefrac{{\pi}}{{2}}$ on $\mathcal{X}_{0}$.

  5. (e)

For any $\theta\in[0,1]$ with
$\theta\notin\\{\nicefrac{{1}}{{2q}},\nicefrac{{1}}{{2}}+\nicefrac{{1}}{{2q}}\\}$,
the fractional power domains coincide with the complex interpolation spaces,
i.e.,

$\displaystyle D((-\mathcal{A}+\omega)^{\theta})$
$\displaystyle\simeq[\mathcal{X}_{0},D(-\mathcal{A})]_{\theta}\hookrightarrow\mathcal{Y}_{\beta}.$

  6. (f)

For any $\theta\in[0,1]$ with
$\theta\notin\\{\nicefrac{{1}}{{2q}},\nicefrac{{1}}{{2}}+\nicefrac{{1}}{{2q}}\\}$,
the real interpolation spaces satisfy

$(\mathcal{X}_{0},D(-\mathcal{A}))_{\theta,p}\hookrightarrow\mathcal{Y}_{\theta}.$

  7. (g)

The Riesz transform $\nabla(-\mathcal{A}+\omega)^{\nicefrac{{1}}{{2}}}$ is
bounded on $\mathcal{X}_{0}$.

###### Proof.

For (a), we observe that the space $\mathcal{X}_{0}$ has property $(\alpha)$
as a Cartesian product of closed subspaces of $\mathrm{L}^{q}$-spaces, which
in turns has this property, so we can deduce assertion (a) from Subsection 6.1
as in the proof of Subsection 5.1 in the situation of Hibler’s operator.

The verification (b) and (d) is also completely analogous as for the
respective properties in Subsection 5.1.

Concerning (c), we observe that $\mathcal{X}_{0}$ is in particular a UMD-
space, so the assertion is obtained e.g. by the considerations in [44, Section
4.5] and [15, Theorem 4.4].

The first part of property (e) follows from [15, Theorem 2.5]. Note that for
$\tilde{\mathcal{A}}$, we obtain from Subsection 5.1(e) and Subsection 6.1(e)
that the complex interpolation spaces are given by

$\displaystyle\tilde{\mathcal{X}_{\theta}}$
$\displaystyle:=[\mathcal{X}_{0},D(-\tilde{\mathcal{A}})]_{\theta}$
$\displaystyle=({\mathrm{H}}_{\mathrm{b.c.}}^{{2\theta},q}({\Omega}_{\mathrm{atm}})^{2}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{atm}}))\times({\mathrm{H}}_{\mathrm{b.c.}}^{{2\theta},q}({\Omega}_{\mathrm{ocn}})^{2}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}}))\times\mathrm{H}^{{2\theta},q}(\mathbb{R}^{2})^{2}\times\mathrm{H}^{{2\theta},q}(\mathbb{R}^{2})\times\mathrm{H}^{{2\theta},q}(\mathbb{R}^{2}).$

Now, Appendix A implies that the complex interpolation spaces for
$\mathcal{A}$ are given by

$\displaystyle[\mathcal{X}_{0},D(-\mathcal{A})]_{\theta}=\\{({v}_{\mathrm{atm}},{v}_{\mathrm{ocn}}+\mathrm{L}_{\mu}{u}_{\mathrm{ice}},{u}_{\mathrm{ice}},h,a):({v}_{\mathrm{atm}},{v}_{\mathrm{ocn}},{u}_{\mathrm{ice}},h,a)\in\tilde{\mathcal{X}}_{\beta}\\}$
$\displaystyle\hookrightarrow({\mathrm{H}}_{\mathrm{b.c.}}^{{2\theta},q}({\Omega}_{\mathrm{atm}})^{2}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{atm}}))\times(({\mathrm{H}}_{\mathrm{b.c.}}^{{2\theta},q}({\Omega}_{\mathrm{ocn}})^{2}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}}))+\mathrm{L}_{\mu}(\mathrm{H}^{{2\theta},q}(\mathbb{R}^{2})^{2}))\times\mathrm{H}^{{2\theta},q}(\mathbb{R}^{2})^{4}.$

Finally, Subsection 7.1 yields

$({\mathrm{H}}_{\mathrm{b.c.}}^{{2\theta},q}({\Omega}_{\mathrm{ocn}})^{2}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}}))+\mathrm{L}_{\mu}(\mathrm{H}^{{2\theta},q}(\mathbb{R}^{2})^{2})\hookrightarrow\mathrm{H}^{{2\theta},q}({\Omega}_{\mathrm{ocn}})^{2}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}}).$

Property (f) follows from analogous arguments to property (e), replacing
complex by real interpolation and Bessel potential by Besov spaces.

The last assertion is a consequence of the shape of the corresponding
fractional power domain obtained in (e). ∎

###  8.2. Lipschitz estimates

We establish Lipschitz estimates from $\mathcal{X}_{\gamma}$ to
$\mathcal{X}_{0}$ of the operator matrix $\mathcal{A}$ given by (4.19) and the
nonlinearity $\mathcal{F}$ given by (4.20). To account for the fact that we
only have maximal regularity up to translation, for $\omega\in\mathbb{R}$, we
set $\mathcal{A}_{\omega}:=\mathcal{A}-\omega$ as well as
$\mathcal{F}_{\omega}:=\mathcal{F}-\omega$. Thanks to Subsection 8.1(f), we
may prove the Lipschitz estimates component-wise.

######  Proposition .

Let $p,q\in(1,\infty)$ such that
$\nicefrac{{1}}{{p}}+\nicefrac{{1}}{{q}}<\nicefrac{{1}}{{2}}$ and
$z_{0}=(v_{\mathrm{atm},0},v_{\mathrm{ocn},0},v_{0})\in V$. Then there exists
$r_{0}>0$ and a constant $L>0$ such that
$\overline{B}_{\mathcal{X}_{\gamma}}(z_{0},r_{0})\subset V$ and

$\displaystyle\|\mathcal{A}_{\omega}(z_{1})-\mathcal{A}_{\omega}(z_{2})\|_{\mathcal{X}_{0}}$
$\displaystyle\leq
L\|z_{1}-z_{2}\|_{\mathcal{X}_{\gamma}}\|w\|_{\mathcal{X}_{1}},$
$\displaystyle\|\mathcal{F}_{\omega}(z_{1})-\mathcal{F}_{\omega}(z_{2})\|_{\mathcal{X}_{0}}$
$\displaystyle\leq L\|z_{1}-z_{2}\|_{\mathcal{X}_{\gamma}}$

for all $z_{1},z_{2}\in\overline{B}_{\mathcal{X}_{\gamma}}(z_{0},r_{0})$ and
all $w\in\mathcal{X}_{1}$.

###### Proof.

We observe that $z_{0}\in V$ especially satisfies that $z_{0}\in W$, and that
$\mathcal{X}_{0}$ and $\mathcal{Y}_{0}$ coincide. The assertion of this
proposition then follows by concatenating Subsection 5.2 and Subsection 6.2,
and by exploiting Subsection 8.1(f) with the concrete choice
$\theta=1-\nicefrac{{1}}{{p}}$. We remark that the assumptions on $p$ and $q$
imply in particular that $\theta>\nicefrac{{1}}{{2}}+\nicefrac{{1}}{{2q}}$. ∎

##  9\. Local wellposedness

This section is dedicated to showing local strong well-posedness of the
coupled system.

###### Proof of Theorem 3.1.

First, we observe that

$\mathrm{C}_{\mathrm{c}}^{\infty}(\overline{{\Omega}_{\mathrm{atm}}})^{2}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{atm}})\times\mathrm{C}_{\mathrm{c}}^{\infty}(\overline{{\Omega}_{\mathrm{ocn}}})^{2}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}({\Omega}_{\mathrm{ocn}})\times\mathrm{C}_{\mathrm{c}}^{\infty}(\mathbb{R}^{2})^{4}\subset\mathcal{X}_{1}$

is dense in $\mathcal{X}_{0}$. Moreover, it follows from Subsection 8.1(c)
that $-\mathcal{A}_{\omega}$ has the property of maximal
$\mathrm{L}^{p}$-regularity on $\mathcal{X}_{0}$, and Subsection 8.2 yields
that suitable Lipschitz estimates are valid for $\mathcal{A}_{\omega}$ and
$\mathcal{F}_{\omega}$. The assertion then follows from the local existence
theorem for quasilinear evolution equations, see [44, Theorem 5.1.1], and the
fact that finding a solution to (4.22) is equivalent to solving (3.5) by the
reformulation in Section 4. ∎

##  Appendix A Interpolation theory: isomorphisms and layer domains

The aim of this section is twofold: On the one hand, we show that real and
complex interpolation are compatible with isomorphisms of semigroups. On the
other hand, we investigate real and complex interpolation of spaces with
boundary conditions as well as properties of the trace and the normal
derivative in the situation of infinite layer domains.

Let $X_{0}$ and $X_{1}$ be Banach spaces such that $X_{1}\hookrightarrow
X_{0}$ is dense. We denote by $X_{\beta}:=[X_{0},X_{1}]_{\beta}$ the (complex)
interpolation spaces, where $[\cdot,\cdot]_{\beta}$ denotes the complex
interpolation functor for $\beta\in(0,1)$. Similarly, we denote by
$X_{\alpha,r}:=(X_{0},X_{1})_{\alpha,r}$ the real interpolation spaces, where
$(\cdot,\cdot)_{\alpha,r}$ denotes the real interpolation functor for
$\alpha\in(0,1)$ and $r\in(1,\infty)$. Moreover, we denote by $X_{\gamma}$ the
trace space, and it is well known that
$X_{\gamma}=(X_{0},X_{1})_{1-\nicefrac{{1}}{{p}},p}$.

Thanks to the functoriality of the interpolation, we obtain that the real and
complex interpolation is compatible with isomorphisms of semigroups.

######  Lemma .

Consider the Banach spaces $X_{0}$ and $Y_{0}$ and an isomorphism
$\mathrm{S}\in\mathcal{L}(X_{0},Y_{0})$, i.e.,
$\mathrm{S}^{-1}\in\mathcal{L}(Y_{0},X_{0})$. Let $\mathrm{A}\colon
X_{1}\subset X_{0}\to X_{0}$ be generator of a strongly continuous semigroup
on $X$. Then $\mathrm{B}:=\mathrm{S}\mathrm{A}\mathrm{S}^{-1}$ with
$Y_{1}:=\mathrm{S}X_{1}$ is the generator of a strongly continuous semigroup
on $Y_{0}$. Besides, we obtain the following relations between the
interpolation and trace spaces:

  1. (i)

the complex interpolation spaces $X_{\beta}$ and $Y_{\beta}$ associated with
$\mathrm{A}$ and $\mathrm{B}$, respectively, satisfy

$Y_{\beta}=\mathrm{S}X_{\beta}.$

  2. (ii)

the real interpolation spaces $X_{\alpha,r}$ and $Y_{\alpha,r}$ associated
with $\mathrm{A}$ and $\mathrm{B}$, respectively, satisfy

$Y_{\alpha,r}=\mathrm{S}X_{\alpha,r}.$

  3. (iii)

the trace spaces $X_{\gamma}$ and $Y_{\gamma}$ associated with $\mathrm{A}$
and $\mathrm{B}$, respectively, satisfy

$Y_{\gamma}=\mathrm{S}X_{\gamma}.$

###### Proof.

The first result follows from [19, Section II.2.1]. Since
$Y_{1}=\mathrm{S}X_{1}$ and $\mathrm{S}$ is an isomorphism between $X_{0}$ and
$Y_{0}$, we obtain that $\mathrm{S}|_{X_{1}}$ is an isomorphism between
$X_{1}$ and $Y_{1}$. Therefore, $\mathrm{S}$ is an isomorphism in the category
of Banach interpolation couples, see [2, Section I.2.1]. As real and complex
interpolation functors are functors, we deduce that

β $\displaystyle\colon X_{\beta}=[X_{0},X_{1}]_{\beta}\quad\longrightarrow$
$\displaystyle Y_{\beta}=[Y_{0},Y_{1}]_{\beta},$
$\displaystyle(\mathrm{S}\cdot,\mathrm{S}\cdot)_{\alpha,r}$
$\displaystyle\colon X_{\alpha,r}=(X_{0},X_{1})_{\alpha,r}\to$ $\displaystyle
Y_{\alpha,r}=(Y_{0},Y_{1})_{\alpha,r}$

are isomorphisms in the category of Banach spaces, i.e., they are bounded
linear bijections. For the sake of simplicity, these isomorphisms are again
denoted by $\mathrm{S}$. Now, we obtain

$\mathrm{S}X_{\beta}=Y_{\beta}\qquad\text{ and
}\qquad\mathrm{S}X_{\alpha,r}=Y_{\alpha,r}.$

Finally, assertion (iii) is the special case $\alpha=1-\nicefrac{{1}}{{p}}$
and $r=p$ of assertion (ii). ∎

Next, we discuss the shape of real and complex interpolation spaces on layer
domains. To this end, for $-\infty<a<b<\infty$, we consider
$\Omega=\mathbb{R}^{2}\times(a,b)$. For simplicity, we only consider the
scalar-valued case, and we remark that
${\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega)$ is also supposed to be scalar-
valued in this section. We then get the following result.

For $s\in(0,2]$ and $p,q\in(1,\infty)$, the Besov space with boundary
conditions $\mathrm{B}^{{s}}_{qp,\mathrm{b.c.}}(\Omega)$ is defined
analogously as the Bessel potential space in (4.9).

######  Lemma .

Let $p,q\in(1,\infty)$, and for $s\in[0,2]$, let
${\mathrm{H}}_{\mathrm{b.c.}}^{{s},q}(\Omega)$ and
${\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega)$ be as in (4.9) and (3.8),
respectively, and let $\mathrm{B}^{{s}}_{qp,\mathrm{b.c.}}(\Omega)$ be the
Besov space with boundary conditions defined analogously as the Bessel
potential space in (4.9). Then for $\theta\in(0,1)$ with
$\theta\neq\nicefrac{{1}}{{2q}},\nicefrac{{1}}{{2}}+\nicefrac{{1}}{{2q}}$, it
holds that

$\displaystyle\left[{\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega),{\mathrm{H}}_{\mathrm{b.c.}}^{{2},q}(\Omega)\cap{\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega)\right]_{\theta}$
$\displaystyle={\mathrm{H}}_{\mathrm{b.c.}}^{{2\theta},q}(\Omega)\cap{\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega),$
$\displaystyle\left({\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega),{\mathrm{H}}_{\mathrm{b.c.}}^{{2},q}(\Omega)\cap{\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega)\right)_{\theta,p}$
$\displaystyle=\mathrm{B}^{{2\theta}}_{qp,\mathrm{b.c.}}(\Omega)\cap{\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega).$

###### Proof.

The idea is to reduce the problem to the case of the half-space by means of a
partition of unity and a retraction-co-retraction argument. More precisely,
let $\chi_{b}\colon[a,b]\to[0,1]$ be a smooth cut-off function with the
property that it equals $1$ close to $b$ and vanishes at $a$. We then set
$\chi_{a}:=1-\chi_{b}$, so it holds that $\chi_{a}+\chi_{b}\equiv 1$ on
$[a,b]$. Setting
$\varphi_{a}=\frac{\chi_{a}}{\sqrt{\chi_{a}^{2}+\chi_{b}^{2}}}$ and
$\varphi_{b}=\frac{\chi_{b}}{\sqrt{\chi_{a}^{2}+\chi_{b}^{2}}}$, we obtain

(A.1) $\varphi_{a}^{2}+\varphi_{b}^{2}\equiv 1$

on $[a,b]$, and we observe that $\varphi_{a}$ and $\varphi_{b}$ are well-
defined and smooth by virtue of $\chi_{a}+\chi_{b}\equiv 1$.

For $s\in[0,2]$, we introduce the spaces
$\mathrm{H}^{{s},q}_{\mathrm{b.c.},a}(\Omega_{a})$ and
$\mathrm{B}^{{s}}_{qp,{\mathrm{b.c.},a}}(\Omega_{a})$, defined on the half-
space $\Omega_{a}=\mathbb{R}^{2}\times(a,\infty)$, as well as
$\mathrm{H}^{{s},q}_{\mathrm{b.c.},b}(\Omega_{b})$ and
$\mathrm{B}^{{s}}_{qp,{\mathrm{b.c.},b}}(\Omega_{b})$, defined on the half-
space $\Omega_{b}=\mathbb{R}^{2}\times(-\infty,b)$. We remark that Dirichlet
or Neumann boundary conditions are considered on the respective boundary
$\mathbb{R}^{2}\times\\{a\\}$ or $\mathbb{R}^{2}\times\\{b\\}$, and we
emphasize that each of the spaces has pure Dirichlet or pure Neumann boundary
conditions on the whole boundary $\mathbb{R}^{2}\times\\{a\\}$ or
$\mathbb{R}^{2}\times\\{b\\}$. As in (4.9), the incorporation of the boundary
conditions depends on the parameter: If $0\leq s<\nicefrac{{1}}{{q}}$, the
respective spaces do not have boundary conditions, if
$\nicefrac{{1}}{{q}}<s<1+\nicefrac{{1}}{{q}}$, only Dirichlet boundary
conditions are incorporated into the spaces, and if
$1+\nicefrac{{1}}{{q}}<s\leq 2$, Dirichlet or Neumann boundary conditions are
included.

For $s\in[0,\infty)$ and $p\in(1,\infty)$, we set

$\displaystyle\mathrm{R}\colon\mathrm{H}^{{s},q}_{\mathrm{b.c.},a}(\Omega_{a})\times\mathrm{H}^{{s},q}_{\mathrm{b.c.},b}(\Omega_{b})$
$\displaystyle\to{\mathrm{H}}_{\mathrm{b.c.}}^{{s},q}(\Omega),$ $\displaystyle
u\mapsto\varphi_{a}u_{a}+\varphi_{b}u_{b},\text{ and }$
$\displaystyle\mathrm{S}\colon{\mathrm{H}}_{\mathrm{b.c.}}^{{s},q}(\Omega)$
$\displaystyle\to\mathrm{H}^{{s},q}_{\mathrm{b.c.},a}(\Omega_{a})\times\mathrm{H}^{{s},q}_{\mathrm{b.c.},b}(\Omega_{b}),$
$\displaystyle v\mapsto(\varphi_{a}v,\varphi_{b}v).$

We remark that $\mathrm{R}$ and $\mathrm{S}$ are linear and continuous, and
they preserve the boundary conditions by construction.

For $v\in{\mathrm{H}}_{\mathrm{b.c.}}^{{s},q}(\Omega)$, the identity (A.1)
yields that $\mathrm{R}\mathrm{S}v=v$, so $\mathrm{R}$ is indeed a retraction,
with corresponding co-retraction $\mathrm{S}$. We first concentrate on the
complex interpolation case. Exploiting [49, Theorem 1.2.4], we deduce that

$\begin{aligned}
{}_{\theta}&=\left[\mathrm{R}\left(\mathrm{L}^{q}(\Omega_{a})\times\mathrm{L}^{q}(\Omega_{b})\right),\mathrm{R}\left(\mathrm{H}^{{2},q}_{\mathrm{b.c.},a}(\Omega_{a})\times\mathrm{H}^{{2},q}_{\mathrm{b.c.},b}(\Omega_{b})\right)\right]_{\theta}\\\
&=\mathrm{R}\left(\left[\mathrm{L}^{q}(\Omega_{a}),\mathrm{H}^{{2},q}_{\mathrm{b.c.},a}(\Omega_{a})\right]_{\theta}\times\left[\mathrm{L}^{q}(\Omega_{b}),\mathrm{H}^{{2},q}_{\mathrm{b.c.},b}(\Omega_{b})\right]_{\theta}\right)\end{aligned}.$

By [3, Theorem VIII.2.4.8], it follows that for $i\in\\{a,b\\}$, we get

$\left[\mathrm{L}^{q}(\Omega_{i}),\mathrm{H}^{{2},q}_{\mathrm{b.c.},i}(\Omega_{i})\right]_{\theta}:=\left\\{\begin{aligned}
&\\{v\in\mathrm{H}^{{2\theta},q}(\Omega_{i})\colon
v|_{\Gamma_{\mathrm{D},i}}=0,\,\partial_{z}v|_{\Gamma_{\mathrm{N},i}}=0\\},&&\text{
for }\nicefrac{{1}}{{2}}+\nicefrac{{1}}{{2q}}<\theta<1,\\\
&\\{v\in\mathrm{H}^{{2\theta},q}(\Omega_{i})\colon
v|_{\Gamma_{\mathrm{D},i}}=0\\},&&\text{ for
}\nicefrac{{1}}{{2q}}<\theta<\nicefrac{{1}}{{2}}+\nicefrac{{1}}{{2q}},\\\
&\mathrm{H}^{{2\theta},q}(\Omega_{i}),&&\text{ for
}0<\theta<\nicefrac{{1}}{{2q}}.\end{aligned}\right.$

We stress that either $\Gamma_{\mathrm{D},i}=\mathbb{R}^{2}\times\\{i\\}$ and
$\Gamma_{\mathrm{N},i}=\emptyset$ or $\Gamma_{\mathrm{D},i}=\emptyset$ and
$\Gamma_{\mathrm{N},i}=\mathbb{R}^{2}\times\\{i\\}$, $i=a,b$. Consequently, we
get

$\left[\mathrm{L}^{q}(\Omega),{\mathrm{H}}_{\mathrm{b.c.}}^{{2},q}(\Omega)\right]_{\theta}=\mathrm{R}\left(\left[\mathrm{L}^{q}(\Omega_{a}),\mathrm{H}^{{2},q}_{\mathrm{b.c.},a}(\Omega_{a})\right]_{\theta}\times\left[\mathrm{L}^{q}(\Omega_{b}),\mathrm{H}^{{2},q}_{\mathrm{b.c.},b}(\Omega_{b})\right]_{\theta}\right)={\mathrm{H}}_{\mathrm{b.c.}}^{{2\theta},q}(\Omega),$

where ${\mathrm{H}}_{\mathrm{b.c.}}^{{2\theta},q}(\Omega)$ takes the shape as
in (4.9). Finally, [49, Theorem 1.17.1] implies that

$\left[{\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega),{\mathrm{H}}_{\mathrm{b.c.}}^{{2},q}(\Omega)\cap{\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega)\right]_{\theta}=\left[\mathrm{L}^{q}(\Omega),{\mathrm{H}}_{\mathrm{b.c.}}^{{2},q}(\Omega)\right]_{\theta}\cap{\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega)={\mathrm{H}}_{\mathrm{b.c.}}^{{2\theta},q}(\Omega)\cap{\mathrm{L}}_{\overline{\sigma}}^{q}(\Omega).$

For the real interpolation case, the interpolation theoretic arguments from
[49] remain valid for real interpolation, so we can use a completely analogous
strategy, this time relying on [3, Theorem 2.4.5] for the shape of the real
interpolation spaces with boundary conditions. ∎

We also investigate the trace and the normal derivative in the situation of a
layer domain. As $\Gamma=\partial\Omega$ is the disjoint union of
$\mathbb{R}^{2}\times\\{a\\}$ and $\mathbb{R}^{2}\times\\{b\\}$, it is
possible to identify a function $f\colon\Gamma\to\mathbb{C}$ with its values
on the lower boundary $\mathbb{R}^{2}\times\\{a\\}$ and the upper boundary
$\mathbb{R}^{2}\times\\{b\\}$. Thus, for $s>\nicefrac{{1}}{{q}}$, we identify
the boundary space $\mathrm{H}^{{s-\nicefrac{{1}}{{q}}},q}(\Gamma)$ associated
to $\mathrm{H}^{{s},q}(\Omega)$ with
$\mathrm{H}^{{s-\nicefrac{{1}}{{q}}},q}(\mathbb{R}^{2})\times\mathrm{H}^{{s-\nicefrac{{1}}{{q}}},q}(\mathbb{R}^{2})$.
The following result is implied e.g. by [3, Theorem VIII.1.3.1], establishing
the result for the half-space, and a suitable partition of unity procedure
similarly as in the proof of Appendix A.

######  Lemma .

Let $q\in(1,\infty)$.

  1. (a)

For $s>\nicefrac{{1}}{{q}}$, it holds that
$\mathrm{tr}\colon\mathrm{H}^{{s},q}(\Omega)\to\mathrm{W}^{{s-\nicefrac{{1}}{{q}}},q}(\Gamma)$
is linear, continuous and surjective.

  2. (b)

If $s>1+\nicefrac{{1}}{{q}}$, then
$\partial_{z}\colon\mathrm{H}^{{s},q}(\Omega)\to\mathrm{W}^{{s-1-\nicefrac{{1}}{{q}}},q}(\Gamma)$
is linear, continuous and surjective.

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