# QGSJET-III model of high energy hadronic interactions: II. Particle
production and extensive air shower characteristics

Sergey Ostapchenko
Universität Hamburg, II Institut für Theoretische Physik, 22761 Hamburg,
Germany

Abstract The hadronization procedure of the QGSJET-III Monte Carlo (MC)
generator of high energy hadronic interactions is discussed. Selected results
of the model, regarding production spectra of secondary particles, are
presented in comparison to experimental data and to the corresponding
predictions of the QGSJET-II-04 MC generator. The model is applied to
calculations of basic characteristics of extensive air showers initiated by
cosmic ray interactions in the atmosphere and the results are compared to
predictions of other MC generators of cosmic ray interactions.

##  1 Introduction

All current Monte Carlo (MC) generators of high energy hadronic interactions
rely on the qualitative picture of quantum chromodynamics (QCD) in the sense
that such interactions are assumed to be mediated by parton cascades
developing between the colliding hadrons (nuclei). Particle production occurs
when the coherence of (some of) these parton cascades is broken by the
scattering process and final $s$-channel partons convert into hadrons. Since
such a hadronization is essentially a nonperturbative process, the
corresponding treatment relies on phenomenological approaches. The most
popular ones are string fragmentation procedures implying a formation of
strings of color field between final partons. With such partons flying apart,
the tension of a string rises until it breaks up, with the color field being
neutralized via a creation of quark-antiquark and diquark-antidiquark pairs
from the vacuum, ending up with a formation of colorless parton clusters
identified with hadrons [1].

Needless to say, modeling such a process involves a substantial number of
adjustable parameters and requires a thorough calibration with various
experimental data. Nevertheless, this does not necessarily spoil the
predictive power of hadronic interaction models since the mechanism is assumed
to be universal. E.g., moving to higher energies or replacing the interacting
hadrons, say, protons, by other hadrons or by nuclei, one arrives to a
different parton configuration. However, the hadronization procedure remains
unaffected by such changes, hence, the conversion of partons into hadrons can
be described using one and the same approach, regardless the type and
kinematics of the interaction. This allows one to test both the validity of an
interaction model itself and the quality of the hadronization mechanism
employed and, in case both perform satisfactorily enough, to use the complete
model for predictive calculations, e.g., regarding the properties of so-called
extensive air showers (EAS) initiated by interactions of high energy cosmic
rays in the atmosphere (see [2] for a recent review).

In the particular case of the QGSJET-III MC generator [3], the treatment of
the hadronization follows closely the one of the original QGSJET model [4, 5],
with a number of technical improvements, as discussed in Section 2. In Section
3, a comparison with selected accelerator data will be presented, followed in
Section 4 by a discussion of basic properties of proton-nitrogen interactions,
relevant for EAS predictions. In turn, Section 5 is devoted to calculations of
extensive air shower characteristics. The conclusions will be given in Section
6.

##  2 Hadronization procedure of the QGSJET-III model

As described in [3], the procedure for simulating a particular hadronic
(nuclear) collision event in the QGSJET-III model consists of two parts.
First, a “macro-configuration” for an inelastic interaction is defined, i.e.,
the complete “net” of “elementary” parton cascades giving rise to secondary
hadron production and being described by cut Pomerons is reconstructed, based
on the corresponding partial cross sections, and the $t$\- and $s$-channel
evolution of perturbative parton cascades corresponding to parton virtualities
$|q^{2}|>Q_{0}^{2}$, $Q_{0}^{2}$ being the “infrared” cutoff of the model, is
treated explicitly.111The modeling of the final (s-channel) parton emission
follows closely the procedure described in [6], imposing the final transverse
momentum cutoff $p^{2}_{t,{\rm cut(f)}}=0.15$ GeV2. At the second step, one
considers a formation of two strings of color field per cut Pomeron, stretched
between constituent partons of the interacting hadrons (nuclei) or/and between
the final $s$-channel partons resulting from the treatment of perturbative
parton cascades, and models the breakup and hadronization of those strings by
means of a string fragmentation procedure.

While the 4-momenta of final $s$-channel partons resulting from perturbative
parton cascades are already defined at the first step, this is not the case
for constituent partons. For a given projectile or target hadron $h$, to which
$2n$ strings are attached, the sharing of the, respectively, light cone plus
(LC+) or light cone minus (LC-) momentum fractions $x^{\pm}$ between
constituent partons (string “ends”) is performed according to the distribution

$\displaystyle f_{h}^{\rm
LC}(x_{1}^{\pm},...,x_{2n}^{\pm})\propto\left[\prod_{i=1}^{2n-2}(x_{i}^{\pm})^{-\alpha_{\rm
sea}}\right]$
$\displaystyle\times\;(x_{2n-1}^{\pm})^{-\alpha_{\mathbb{R}}}\,(x_{2n}^{\pm})^{-\alpha_{\rm
r}}\,\delta\\!\left(1-\sum_{j=1}^{2n}x_{j}^{\pm}\right).$ (1)

Here the exponent $\alpha_{\rm sea}$ is used for sea (anti)quarks, considering
only light $u$ and $d$ (anti)quarks as string ends, while the small $x$ limit
of a valence (anti)quark distribution follows the Regge behavior ($\propto
x_{2n-1}^{-\alpha_{\mathbb{R}}}$), with
$\alpha_{\mathbb{R}}=\alpha_{\rho}(0)=0.5$. Similarly, the LC momentum
distribution of the remaining valence parton [(anti)quark for $h$ being a
meson or (anti)diquark for a baryon] follows the corresponding Regge behavior
($\propto x_{2n}^{-\alpha_{\rm r}}$) [7, 8, 9]:

$\displaystyle\alpha_{u}=\alpha_{d}=\alpha_{\mathbb{R}}$ (2)
$\displaystyle\alpha_{ud}=\alpha_{uu}+1=2\alpha_{N}(0)-\alpha_{\mathbb{R}}$
(3) $\displaystyle\alpha_{s}=\alpha_{\phi}(0)$ (4)
$\displaystyle\alpha_{us}=\alpha_{ds}=\alpha_{ud}-\alpha_{\mathbb{R}}+\alpha_{\phi}(0)\,,$
(5)

the corresponding Regge intercepts being compiled in Table 1.

$\alpha_{\rho}(0)$ | $\alpha_{\phi}(0)$ | $\alpha_{N}(0)$ | $\alpha_{\pi}(0)$ | $\alpha_{f}(0)$
---|---|---|---|---
0.5 | 0 | 0 | 0 | 0.7
Table 1: Intercepts of selected Regge trajectories.

Regarding the LC momentum distribution of constituent sea quarks, in
principle, it should also be characterized by the Regge behavior ($\propto
x_{i}^{-\alpha_{\mathbb{R}}}$) in the low $x$ limit [7], which was adopted in
the QGSJET [4, 5] and QGSJET-II [10, 11] models. However, here we treat the
respective exponent as an adjustable parameter, $0.5\leq\alpha_{\rm sea}<1$.
The reason is that we consider Pomeron-Pomeron interactions, introducing a
minimal Pomeron rapidity “size” $\xi$ [3], for the Pomeron asymptotics to be
applicable. Therefore, the deviation of $\alpha_{\rm sea}$ from
$\alpha_{\mathbb{R}}$ is assumed to take effectively into account a formation
of small rapidity gaps $y_{\rm gap}<\xi$, not treated explicitly by the model.

In turn, transverse momenta of string ends are sampled according to the
distribution:

$\displaystyle
f_{h}^{\perp}(\vec{p}_{t_{1}},...,\vec{p}_{t_{2n}})\propto\exp\\!\left(-\sum_{i=1}^{2n-1}\frac{p_{t_{i}}^{2}}{\gamma_{\perp}}-\frac{p_{t_{2n}}^{2}}{\gamma_{\perp}^{h}}\right)$
$\displaystyle\times\;\delta^{(2)}\\!\left(1-\sum_{j=1}^{2n}\vec{p}_{t_{j}}\right),$
(6)

the corresponding parameters being listed in Table 2.

$\alpha_{\rm sea}$ | $w^{\pi}_{p}$ | $w^{\pi}_{\pi}$ | $w_{K}^{\pi}$ | $w^{f}_{p}$ | $w^{f}_{\pi}$ | $w^{f}_{K}$ | $\gamma_{\perp}$ | $\gamma_{\perp}^{p}$ | $\gamma_{\perp}^{\pi}$ | $\gamma_{\perp}^{K}$ | $p^{2}_{t,{\rm cut(f)}}$
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  | GeV2 | GeV2 | GeV2 | GeV2 | GeV2
0.65 | 0.28 | 0.37 | 0.11 | 0.08 | 0.15 | 0 | 0.15 | 0.72 | 0.2 | 0.2 | 0.15
Table 2: Parameters of the energy-momentum sharing and hadronization procedures of the QGSJET-III model (excluding the parameters of the string fragmentation procedure). $\Lambda$ | $p^{2}_{t,{\rm thr}}$ | $a_{ud}$ | $a_{us}=a_{ds}$ | $a_{s}$ | $\gamma_{ud}=\gamma_{us}=\gamma_{ds}$ | $\gamma_{u}=\gamma_{d}$ | $\gamma_{s}$ | $w_{\eta}$ | $w_{\rho}$
---|---|---|---|---|---|---|---|---|---
| GeV2 |  |  |  | GeV | GeV | GeV |  |
1.55 | 0.55 | 0.085 | 0.038 | 0.125 | 0.33 | 0.21 | 0.28 | 0.11 | 0.33
Table 3: Parameters of the string fragmentation procedure of the QGSJET-III
model.

The fragmentation of a string into secondary hadrons is performed considering
iteratively an emission of a hadron from either string end (see, e.g. [12]),
with the hadron LC momentum fraction $x$ (LC+ or LC- for a projectile or
target side end, respectively) and the transverse momentum $p_{t}$ in the
center-of-mass (c.m.) frame of the string being sampled according to the
distribution:

$\displaystyle f_{qq^{\prime}\rightarrow h}^{\rm fragm}(x,\vec{p}_{t})\propto
x^{1-\alpha_{q}-\alpha_{q^{\prime}}}(1-x)^{\Lambda-\alpha_{q^{\prime}}}e^{-\frac{p_{t}}{\gamma_{q^{\prime}}}}\\!,$
(7)

using Eqs. (2-5) for the relations between the parameters $\alpha_{q}$ and the
intercepts of the corresponding Regge trajectories. Here the large $x$ limit
is defined by the probability to slow down the antiparton [(anti)quark or
(anti)diquark] $\bar{q}^{\prime}$ from the vacuum-created pair, while the
small $x$ limit is obtained by slowing down both the original parton $q$ and
the parton $q^{\prime}$ attached to it to form the final hadron
($q+q^{\prime}\rightarrow h$) [9] (c.f. Fig. 1).

Figure 1: Schematic view of the fragmentation of $ud$ diquark into a proton:
both large $x$ and small $x$ limits of the proton LC momentum distribution
correspond to a large rapidity $y$ separation between the $u$ and $\bar{u}$ of
the vacuum-created $u\bar{u}$ pair. Left: the large $x$ limit ($1-x\propto
e^{-y}$) is obtained by slowing down the $\bar{u}$ antiquark. Right: the small
$x$ limit ($x\propto e^{-y}$) corresponds to slowing down both the $u$ quark
and the original $ud$ diquark.

Various parton-antiparton pairs $\bar{q}^{\prime}q^{\prime}$ created from the
vacuum are sampled according to the corresponding probabilities
$a_{q^{\prime}}$, relative to $\bar{u}u$ and $\bar{d}d$ pairs
($a_{u}=a_{d}=1/2$). The iterative string fragmentation is continued until the
mass of the string reminder falls below a specified threshold222We use $M_{\rm
thr}^{qq^{\prime}}=\sum_{i=1}^{3}\sqrt{m_{h_{i}}^{2}+p^{2}_{t,{\rm thr}}}$,
where $h_{1}$, $h_{2}$, and $h_{3}$ correspond to the minimal mass 3-hadron
final state for the types $q$ and $q^{\prime}$ of the string end partons.
$M_{\rm thr}$, followed by a modeling of a two particle decay of the remaining
string. The string fragmentation parameters are compiled in Table 3.

The above-discussed treatment takes into account the production of final
“stable” hadrons: pions, kaons, nucleons, lambda and sigma baryons, including
the corresponding antiparticles, where relevant. Contributions of decays of
short-lived resonances are assumed here to be accounted for in the stable
hadron yields via the duality principle. The exceptions are the formation of
$\Delta^{++}$ and $\Delta^{-}$ resonances (and the corresponding
antiparticles) by a fragmentation of constituent $uu$ and $dd$ (anti)diquarks
as well as the production of $\rho$ and $\eta$ mesons, which is also treated
explicitly. Regarding the latter, the probability to produce an $\eta$ meson,
relative to $\pi^{0}$, is chosen as $w_{\eta}=1/9$ from the quark counting
rules. Similarly, we use $w_{\rho}=1/3$ for the corresponding probability of
$\rho$ mesons, relative to pions, assuming that $\sim 50$% of final pions
originate from $\rho$ decays. For brevity, we restrain here from discussing
other technicalities of the hadronization procedure.

Figure 2: $x_{\rm F}$-dependence of invariant cross sections in c.m. frame,
for different $p_{t}$ values (as indicated in the plots), for proton
production in $pp$ (left) and $p$C (right) collisions at 158 GeV/c, as
calculated using the QGSJET-III (solid lines) and QGSJET-II-04 (dotted lines)
models, compared to NA49 data [16, 17] (points).

In addition to the basic procedure described above, a treatment of the pion
exchange contribution to forward hadron production, i.e., the Reggeon-Reggeon-
Pomeron ($\mathbb{RRP}$) process, with $\mathbb{R}=\pi$, has been considered
in the QGSJET-III model333A simplified treatment of the pion exchange process
for pion-proton and pion-nucleus collisions had earlier been implemented in
the QGSJET-II-04 model [13]. [14]. As discussed in more detail in [14], due to
the low intercept of the pion Regge trajectory, $\alpha_{\pi}(0)\simeq 0$, the
process does not lead to a formation of a large rapidity gap between the
“leading”, i.e., most energetic, secondary hadron and the remaining hadronic
system, with the consequence that absorptive corrections to the corresponding
“bare” triple-Reggeon contribution are essentially the same as for the single
cut Pomeron process. This motivated us to sample the $\mathbb{RRP}$ process,
for $\mathbb{R}=\pi$, as a fixed fraction, with probability $w^{\pi}_{h}$, of
the single cut Pomeron contribution, for a given hadron $h$. The hadronization
procedure remains practically unaltered in such a case, except that the
leading hadron is sampled according to the respective $x$\- and
$p_{t}$-distribution for the $\pi$-exchange process [14], while the pair of
strings is attached here to the quark and antiquark of the virtual pion,
instead of constituent partons of the initial hadron $h$. In other words,
after the production of the leading hadron, the rest of the final state
corresponds to an inelastic interaction444Since absorptive corrections “push”
the $\pi$-exchange process towards large impact parameters, the contribution
of an elastic scattering of the virtual pion is strongly suppressed and can be
neglected [14]. of the virtual pion with the target (in case $h$ is a
projectile hadron), with the respectively reduced c.m. energy for the
collision. Added in a similar way is the $f$-Reggeon exchange process
($\mathbb{RRP}$ contribution, with $\mathbb{R}=f$), with $\alpha_{f}(0)=0.7$
[15] and with the corresponding probabilities $w^{f}_{h}$.

##  3 Results for secondary particle production

Regarding applications of MC event generators to EAS simulations, of primary
importance is an accurate enough description of forward hadron production.
Here comes a significance of experimental measurements performed at fixed
target energies, where forward spectra of secondary particles have been
studied in great detail.

We start in Fig. 2 with invariant cross sections in c.m. frame for proton
production in $pp$ and $p$C collisions at 158 GeV/c, comparing the
corresponding results of the QGSJET-III model to the ones of QGSJET-II-04 and
to measurements by the NA49 experiment. The same comparison regarding Feynman
$x$ ($x_{\rm F}$) distributions of charged pions, neutrons, and antiprotons,
also for $pp$ and $p$C interactions at 158 GeV/c, is presented in Figs. 3, 4,
and 5;

Figure 3: $x_{\rm F}$-distributions of charged pions (in c.m. frame) in $pp$
(left) and $p$C (right) collisions at 158 GeV/c, as calculated using the
QGSJET-III model, compared to NA49 data [18, 19] (points): $\pi^{+}$ – solid
lines and filled squares, $\pi^{-}$ – dashed lines and open squares. The
corresponding results of the QGSJET-II-04 model are shown by dotted and dash-
dotted lines for $\pi^{+}$ and $\pi^{-}$, respectively. Figure 4: $x_{\rm
F}$-distributions of neutrons (in c.m. frame) in $pp$ (left) and $p$C (right)
collisions at 158 GeV/c, as calculated using the QGSJET-III (solid lines) and
QGSJET-II-04 (dotted lines) models, compared to NA49 data [16, 17] (points).
Partial contributions of the pion exchange process in the QGSJET-III model are
shown by dash-dotted lines. Figure 5: $x_{\rm F}$-distributions of antiprotons
(in c.m. frame) in $pp$ (left) and $p$C (right) collisions at 158 GeV/c, as
calculated using the QGSJET-III (solid lines) and QGSJET-II-04 (dotted lines)
models, compared to NA49 data [16, 17] (points). Figure 6: $x_{\rm
F}$-distributions of charged kaons (in c.m. frame) in $pp$ collisions at 158
GeV/c, as calculated using the QGSJET-III model, compared to NA49 data [20]
(points): $K^{+}$ – solid line and filled squares, $K^{-}$ – dashed line and
open squares. The corresponding results of the QGSJET-II-04 model are shown by
dotted and dash-dotted lines for $K^{+}$ and $K^{-}$, respectively.

the partial contribution to neutron production due to the pion exchange
mechanism in the QGSJET-III model is shown separately in Fig. 4. In turn,
$x_{\rm F}$ distributions of secondary charged kaons, for $pp$ collisions at
158 GeV/c, calculated using the QGSJET-III and QGSJET-II-04 models, are
compared to NA49 data in Fig. 6. Regarding the production of $p$, $n$,
$\pi^{\pm}$, and $K^{\pm}$, we observe relatively small differences between
QGSJET-III and QGSJET-II-04, both models being in a satisfactory agreement
with the measurements. This is, however, not the case for secondary
antiprotons: the predicted $x_{\rm F}$ distributions are substantially harder
than observed by the experiment, especially, in the case of the QGSJET-III
model.555This issue has been discussed in some detail in [21].

Further, in Fig. 7,

Figure 7: $x_{\rm F}$-dependence of production cross sections in c.m. frame
for charged hadrons ($h^{\pm}$) in $\pi^{+}p$ (left) and $K^{+}p$ (right)
collisions at 250 GeV/c, as calculated using the QGSJET-III model, compared to
NA22 data [22] (points): $h^{+}$ – solid lines and filled squares, $h^{-}$ –
dashed lines and open squares. The corresponding results of the QGSJET-II-04
model are shown by dotted and dash-dotted lines for $h^{+}$ and $h^{-}$,
respectively.

we compare $x_{\rm F}$ spectra of positively and negatively charged secondary
hadrons, for $\pi^{+}p$ and $K^{+}p$ collisions at 250 GeV/c, calculated using
the QGSJET-III and QGSJET-II-04 models, to measurements by the NA22 experiment
[22]. Additionally, in Fig. 8, we compare the results of the two models for
momentum distributions of various secondary hadrons produced in $\pi^{-}C$
collisions at 158 and 350 GeV/c to recent measurements of the NA61 experiment
[23]. For both models, we observe an acceptable agreement with the data both
for the charged particle spectra in Fig. 7 and for the pion distributions in
Fig. 8. However, charged kaon production in $\pi^{-}C$ collisions appears to
be underestimated by QGSJET-III by $\sim 30$%, compared to the NA61 data. This
is somewhat surprising since no such a “kaon deficit” was observed for $pp$
collisions (c.f. Fig. 6). Even more striking is the considerable
underestimation by the QGSJET-III and QGSJET-II-04 models of proton and
antiproton production in $\pi^{-}C$ collisions, compared to the NA61
measurements. While these discrepancies may, in principle, be attributed to
potential deficiencies of the hadronization procedures of the models, it is
noteworthy that no such a striking underestimation of secondary proton and
antiproton yields is observed if we compare the predictions of QGSJET-III for
$p$ and $\bar{p}$ production in $\pi^{-}p$ collisions at 360 GeV/c to the data
of the LEBC-EHS experiment [24], as one can see in Fig. 9.

Figure 8: Momentum distributions in laboratory frame of charged pions (top
panels), charged kaons (middle panels), protons and antiprotons (bottom
panels), produced in $\pi^{-}C$ collisions at 158 GeV/c (left) and 350 GeV/c
(right), as calculated using the QGSJET-III model, compared to NA61 data [23]
(points). Solid lines and filled squares correspond to positively charged
hadrons while dashed lines and open squares refer to negatively charged
hadrons. The corresponding results of the QGSJET-II-04 model are shown by
dotted and dash-dotted lines for positively and negatively charged hadrons,
respectively. Figure 9: $x_{\rm F}$-distributions of protons and antiprotons
(in c.m. frame) in $\pi^{-}p$ collisions at 360 GeV/c, as calculated using the
QGSJET-III model, compared to LEBC-EHS data [24] (points): $p$ – solid line
and filled squares, $\bar{p}$ – dashed line and open squares. The
corresponding results of the QGSJET-II-04 model are shown by dotted and dash-
dotted lines for $p$ and $\bar{p}$, respectively.

As discussed in [13], of particular importance for model predictions regarding
EAS muon content is a satisfactory description of forward production of
$\rho$-mesons in pion-air collisions, which is dominated by the pion exchange
process. In Fig. 10,

Figure 10: $x_{\rm F}$-distributions of $\rho^{0}$-mesons (in c.m. frame) in
$\pi^{-}C$ collisions at 158 GeV/c (solid line) and 350 GeV/c (dashed line),
as calculated using the QGSJET-III model, compared to NA61 data [25] (points).
Partial contribution of the pion exchange process for $\pi^{-}C$ collisions at
158 GeV/c is shown by a dash-dotted line.

we compare $x_{\rm F}$ distributions of $\rho^{0}$ mesons produced in
$\pi^{-}C$ collisions at 158 and 350 GeV/c, calculated using the QGSJET-III
model, to the respective data of the NA61 experiment, while Fig. 11 contains a
comparison of the model predictions

Figure 11: $x_{\rm F}$-dependence of production cross sections in c.m. frame
for $\rho^{0}$ (left) and $\rho^{+}$ (right) mesons in $\pi^{+}p$ collisions
at 250 GeV/c, as calculated using the QGSJET-III model, compared to NA22 data
[26] (points). Partial contributions of the pion exchange process are shown by
dash-dotted lines.

for the spectra of $\rho^{0}$ and $\rho^{+}$ mesons produced in $\pi^{+}p$
interactions at 250 GeV/c to measurements of the NA22 experiment. Both in Fig.
10 and in Fig. 11, we show also the partial contribution of the pion exchange
mechanism to $\rho$ meson production, which clearly dominates the forward
parts of the spectra. Overall, the agreement is quite good, notably, when
comparing to the recent measurements of NA61.

Coming now to measurements at the Large Hadron Collider (LHC), those mostly
provided information on hadron production in $pp$, $pA$, and $AA$ collisions,
at central ($y\sim 0$) rapidities in c.m. frame. A comparison of the
pseudorapidity density of charged hadrons, $dN^{\rm ch}_{pp}/d\eta$, in $pp$
collisions at $\sqrt{s}=0.9$, 2.36, 7, and 13 TeV, calculated using the
QGSJET-III and QGSJET-II-04 models, to the respective ATLAS data666The
experimental event selection requires at least one charged secondary hadron of
$p_{t}>0.5$ GeV at $|\eta|<2.5$. [27, 28] is presented in Fig. 12.

Figure 12: Pseudorapidity distributions of charged hadrons (with $p_{t}>0.5$
GeV/c) in c.m. frame, produced in $pp$ collisions at different $\sqrt{s}$
(from top to bottom: 13, 7, 2.36, and 0.9 TeV), as calculated using the
QGSJET-III (solid lines) and QGSJET-II-04 (dotted lines) models, compared to
ATLAS data [27, 28] (points).

Since these experimental data have been used to tune the QGSJET-III model, the
overall satisfactory agreement comes at no surprise. Much less trivial is the
agreement of the predictions of both models to the results of a combined
measurement777The experimental event selection requires at least one charged
secondary hadron at $5.3<|\eta|<6.5$. by the CMS and TOTEM experiments of
$dN^{\rm ch}_{pp}/d\eta$ over a wide $\eta$ range, in $pp$ collisions at
$\sqrt{s}=8$ TeV [29], see Fig. 13.

Figure 13: Pseudorapidity distribution of charged hadrons in c.m. frame,
produced in $pp$ collisions at $\sqrt{s}=8$ TeV, as calculated using the
QGSJET-III (solid line) and QGSJET-II-04 (dotted line) models, compared to a
simultaneous measurement by CMS and TOTEM [29] (points).

As discussed in [30], this set of experimental data allows one to test very
basic model assumptions regarding the momentum distributions of constituent
partons.

We also show in Fig. 14

Figure 14: Pseudorapidity distributions of charged hadrons in c.m. frame,
produced in $p$Pb collisions at different $\sqrt{s}$, as calculated using the
QGSJET-III model, compared to CMS data [31] (points): $\sqrt{s}=5$ TeV – solid
line and filled squares, $\sqrt{s}=8$ TeV – dashed line and open squares. The
corresponding results of the QGSJET-II-04 model are shown by dotted and dash-
dotted lines for $\sqrt{s}=5$ TeV and $\sqrt{s}=8$ TeV, respectively.

the calculated pseudorapidity density of charged hadrons, $dN^{\rm ch}_{p{\rm
Pb}}/d\eta$, for proton-lead collisions at $\sqrt{s}=5$ and 8 TeV, compared to
the data of the CMS experiment [31]. Here the results of the QGSJET-III model
agree with the data substantially better, compared to the ones of QGSJET-
II-04.

$M_{X}$ range, GeV | $3.1-7.7$ | $7.7-380$ | $380-1150$ | $1150-3100$ | $3.1-3100$
---|---|---|---|---|---
TOTEM [32] | $1.83\pm 0.35$ | $4.33\pm 0.61$ | $2.10\pm 0.49$ | $2.84\pm 0.40$ | $11.10\pm 1.66$
QGSJET-II-04 | 1.88 (0) | 3.21 (0.07) | 1.06 (0.57) | 5.91 (5.53) | 12.06 (6.17)
QGSJET-III | 1.41 (0.04) | 3.19 (0.24) | 1.51 (0.44) | 6.38 (4.97) | 12.49 (5.69)
Table 4: Predictions of the QGSJET-II-04 and QGSJET-III models for cross
sections of SD-like events (in mb) at $\sqrt{s}=7$ TeV, for different ranges
of mass $M_{X}$ of diffractive states produced, compared to TOTEM data [32].
Values in brackets correspond to the respective contributions of
nondiffractive events.

However, regarding the relevance to EAS calculations, of primary significance
is the energy-dependence of forward hadron production. In that respect, of
particular importance are diffraction studies at LHC. In Table 4, we compare
the predictions of the QGSJET-III and QGSJET-II-04 models, regarding partial
cross sections for single diffraction-like (SD-like) events at $\sqrt{s}=7$
TeV, for different intervals of the diffractive state mass $M_{X}$, to the
respective data of the TOTEM experiment [32]. While the cumulative cross
section for the full range $3.1<M_{X}<3100$ GeV, studied by TOTEM, is well in
agreement between both models and the data, the predicted rates of SD-like
events for intermediate diffractive masses, $7.7<M_{X}<1150$ GeV, are somewhat
lower than observed by TOTEM. On the other hand, the production of large mass
states, $1150<M_{X}<3100$ GeV, is overestimated in both models by more than a
factor of two, compared to the TOTEM measurements. Taking into account that
the event rate in this mass interval is dominated by a formation of random
rapidity gaps in nondiffractive hadron production (see the corresponding
values in brackets in Table 4), primarily, by $\mathbb{RRP}$ processes (see,
e.g. [33] for the corresponding discussion), this may indicate a certain
overestimation of the $\mathbb{RRP}$ contributions or/and some deficiencies of
the hadronization procedures of both models.

Regarding the so-called inelasticity, i.e., the relative energy loss of most
energetic secondary hadrons, important constraints come from measurements of
forward neutron production by the LHCf experiment [34, 35]. When interpreting
those experimental results, it is important to take into consideration the
contribution of the $t$-channel pion exchange [36, 37, 38]. Here the data of
the NA49 experiment on neutron production in $pp$ and $p$C collisions at 158
GeV/c [16, 17] (c.f. Fig. 4) can be combined with those of LHCf at LHC
energies in order to test the implementation of the pion exchange mechanism in
the QGSJET-III model [14], notably, regarding the predicted energy-dependence
over a wide c.m. energy range, $17<\sqrt{s}<13000$ GeV. Comparing the
calculated neutron energy spectra to the observations in Figs. 15-16,

Figure 15: Neutron energy spectra (in c.m. frame) in $pp$ collisions at
$\sqrt{s}=13$ TeV, calculated using the QGSJET-III (solid lines) and QGSJET-
II-04 (dotted lines) models, for different pseudorapidity intervals (as
indicated in the plots), in comparison to LHCf data [35] (points). Partial
contributions of the pion exchange process in the QGSJET-III model are shown
by dash-dotted lines. Figure 16: Same as in Fig. 15, for $\sqrt{s}=7$ TeV.

we see a satisfactory agreement of the model with the measurement at
$\sqrt{s}=13$ TeV, while there is a certain underestimation of the neutron
yield at $\sqrt{s}=7$ TeV, compared to LHCf data. The latter is actually
surprising, given the small difference between the two c.m. energies for LHCf
studies.

Another important benchmark from the LHCf experiment concerns the measured
forward spectra of neutral pions [39], which is also of importance for
astrophysical studies based on $\gamma$-rays (see, e.g. [40, 41] for the
corresponding discussion). The results of the QGSJET-III and QGSJET-II-04
models are confronted to the LHCf data in Fig. 17.

Figure 17: Invariant momentum spectra of neutral pions in c.m. frame,
$E\,d^{3}n_{pp}^{\pi^{0}}/dp^{3}$, for different $p_{t}$ selections (as
indicated in the plots), for $pp$ collisions at $\sqrt{s}=7$ TeV, as
calculated using the QGSJET-III (solid histograms) and QGSJET-II-04 (dotted
histograms) models, compared to the data of the LHCf experiment [39] (points).

While both models demonstrate a satisfactory agreement with the measurements,
somewhat disturbing is a certain underestimation of $\pi^{0}$ production at
$p_{z}\sim 1$ TeV by the QGSJET-III model, which may be related to its softer
momentum distribution of constituent partons [larger $\alpha_{\rm sea}$, c.f.
Eq. (1)].

##  4 Basic characteristics of proton-nitrogen collisions

Before proceeding with the model application to extensive air shower modeling,
it is worth considering its predictions for some basic characteristics of
hadron-nucleus collisions, relevant for EAS predictions.

The first quantity to consider is the inelastic proton-air cross section
$\sigma^{\rm inel}_{p-{\rm air}}$ which defines the mean free path (m.f.p.)
$\lambda_{p}=m_{\rm air}/\sigma^{\rm inel}_{p-{\rm air}}$ ($m_{\rm air}$ being
the average mass of air nuclei) of a primary CR proton in the atmosphere and
impacts thereby many EAS characteristics, notably, the EAS maximum depth
$X_{\max}$ corresponding to the position of the maximum of charged particle
profile of an air shower. The energy-dependence of $\sigma^{\rm inel}_{p-{\rm
air}}$, calculated using the QGSJET-III, QGSJET-II-04, EPOS-LHC [42], and
SIBYLL-2.3 [21] models, is plotted in Fig. 18.

Figure 18: Energy-dependence of $\sigma^{\rm inel}_{p-{\rm air}}$, calculated
using the QGSJET-III, QGSJET-II-04, EPOS-LHC, and SIBYLL-2.3 models – solid,
dotted, dashed, and dash-dotted lines, respectively.

Not surprisingly, there is a good agreement between the predictions of QGSJET-
III and QGSJET-II-04, as well as with the ones of the other two models
considered, since more or less the same set of accelerator data on the total
and elastic proton-proton cross sections had been used for the relevant
parameter tuning in all the cases. The small differences between the results
for $\sigma^{\rm inel}_{p-{\rm air}}$, all obtained using the Glauber-Gribov
formalism [43, 44], may stem from a choice of the nuclear ground state wave
function or/and from a model treatment of the inelastic screening effects.
With the most steep energy rise of $\sigma^{\rm inel}_{p-{\rm air}}$ being
predicted by EPOS-LHC, its cross section is $\simeq 5$% higher at
$E_{0}=10^{10}$ GeV than the one of QGSJET-III. Yet the potential impact of
that difference on $X_{\max}$ at the highest energies is $<3$ g/cm2.

Next, we consider the rate of the inelastic diffraction, which may also impact
$X_{\max}$ predictions noticeably because of its typically small inelasticity
$K_{\rm inel}$, i.e., the relative energy loss of leading (most energetic)
secondary nucleons. This is particularly so for target diffraction
characterized at very high energies by a tiny inelasticity $K_{\rm inel}\simeq
M_{X}^{2}/s$, $M_{X}$ being the diffractive state mass. The effect of such
“quasi-elastic” collisions is equivalent to a reduction of the inelastic
proton-air cross section, giving rise to an increase of m.f.p of primary CR
protons:

$\lambda_{p}\rightarrow\lambda_{p}(1+\sigma^{\rm diffr}_{p-{\rm
air}}/\sigma^{\rm inel}_{p-{\rm air}})\,.$ (8)

In Fig. 19,

Figure 19: Energy-dependence of the probability of diffractive-like
interactions, for $p\,^{14}$N collisions, calculated using different models.
The notations for the lines are the same as in Fig. 18.

we compare the energy-dependence of the rate of diffractive-like interactions,
for proton-nitrogen collisions, i.e., the fraction of events characterized by
$K_{\rm inel}<0.1$, predicted by different models. It is worth reminding that,
apart from diffractive interactions, sizable contributions to this rate come
from a formation of random rapidity gaps in nondiffractive collisions,
notably, by $\mathbb{RRP}$ processes. While the results of QGSJET-III and
QGSJET-II-04 agree with each other within 10%, the overall spread of the model
predictions reaches here 30%, stemming both from different approaches to the
treatment of diffraction and from differences regarding the hadronization
procedure, the latter potentially having a strong impact on the probability to
create random rapidity gaps. Yet, since such a variation of the rate of
diffractive-like interactions can produce only $\simeq 3$% change of
$\lambda_{p}$, the corresponding impact on $X_{\max}$ is limited by $\simeq 2$
g/cm2 [c.f. Eq. (8)].

Further we consider the inelasticity $K_{\rm inel}$ of general proton-nitrogen
collisions, which impacts strongly the energy loss of leading secondary
nucleons in the cause of the nuclear cascade development, influencing
significantly the predicted $X_{\max}$ (see, e.g. [45]). The energy dependence
of $K_{\rm inel}$, predicted by different models, is plotted in Fig. 20.

Figure 20: Energy-dependence of the inelasticity of $p\,^{14}$N collisions,
calculated using different models. The notations for the lines are the same as
in Fig. 18.

Despite large differences between QGSJET-II-04 and QGSJET-III, both regarding
the new developments in the latter, notably, the treatment of higher twist
corrections to hard parton processes [3, 46] and the implementation of the
pion exchange process [14], and concerning the values of important model
parameters (e.g., the parameter $\alpha_{\rm sea}$ governing the momentum
distribution of constituent partons), their predictions for $K_{\rm inel}$ are
remarkably identical, both for the energy dependence and for the absolute
values. A qualitatively similar behavior of $K_{\rm inel}$ is predicted by
EPOS-LHC. In contrast, in case of the SYBYLL-2.3 model, the inelasticity
depends rather weakly on the collision energy, which is related to very basic
model assumptions regarding the structure of constituent parton Fock states
and impacts strongly the model results for $X_{\max}$ [30].

Regarding the EAS muon content, the simple qualitative model of Heitler [47]
implies a correlation of this observable with the number of “stable” secondary
hadrons [(anti)nucleons, kaons, and charged pions], i.e., those which have
significant chances to interact in the atmosphere, before decaying. More
precisely, since the cascade nature of extensive air showers enhances the
importance of forward hadron production, the relevant quantity is the second
moment of the distribution $dn_{\rm stable}/dx_{E}$ of the energy fraction
$x_{E}=E/E_{0}$ of such hadrons [48] (see also [49, 50] for recent studies),
i.e., the total fraction of the parent hadron energy taken by all stable
hadrons,

$\langle x_{E}\,n_{\rm
stable}(E_{0})\rangle=\int_{0}^{1}\\!dx_{E}\,x_{E}\,\frac{dn_{\rm
stable}(E_{0},x_{E})}{dx_{E}}\,.$ (9)

In Fig. 21, we compare the energy-dependence of this quantity

Figure 21: Energy-dependence of $\langle x_{E}\,n_{\rm stable}\rangle$, for
$\pi^{+}\,{}^{14}$N collisions, calculated using different models. The
notations for the lines are the same as in Fig. 18.

for pion-nitrogen collisions, calculated using the QGSJET-III, QGSJET-II-04,
EPOS-LHC, and SIBYLL-2.3 models. While there are only minor differences
between QGSJET-III and QGSJET-II-04 for the predicted $\langle x_{E}\,n_{\rm
stable}\rangle$, the results of the other two models are quite different,
which is related to a copious forward production of, respectively,
(anti)baryons and $\rho$-mesons in the EPOS-LHC and SIBYLL-2.3 models.888See,
however, [51] for the corresponding criticism.

Overall, the results of the QGSJET-III and QGSJET-II-04 models, plotted in
Figs. 18–21, appear to be rather similar to each other, suggesting that EAS
predictions of the two models should not differ significantly.

##  5 Predictions for EAS characteristics

In Fig. 22,

Figure 22: Dependence on primary energy of the maximum depth $X_{\max}$ (left)
and of its fluctuations $\sigma(X_{\max})$ (right) of proton-initiated EAS,
calculated using the QGSJET-III, QGSJET-II-04, EPOS-LHC, and SIBYLL-2.3 models
– solid, dotted, dashed, and dash-dotted lines, respectively.

predictions of the QGSJET-III, QGSJET-II-04, EPOS-LHC, and SIBYLL-2.3 models
for the energy-dependence of the maximum depth $X_{\max}$ and of its
fluctuation, $\sigma_{X_{\max}}$, of proton-induced extensive air showers are
compared. As anticipated in Section 4, the results of QGSJET-III and QGSJET-
II-04 are rather similar to each other, their $X_{\max}$ values differing by
less than 10 g/cm2. Such an agreement is very impressive, given the
substantial differences with the predictions of the other two models. In turn,
there is a very good agreement between all the models, regarding the
calculated values of $\sigma_{X_{\max}}$. This is not surprising since the
quantity is mainly defined by the inelastic proton-air cross section (see,
e.g. [52]), while the impact of uncertainties regarding the rate of inelastic
diffraction does not exceed few g/cm2 [53].

Further, the calculated energy-dependence of the muon number $N_{\mu}$ (at sea
level) of proton-initiated EAS, for muon energies $E_{\mu}>1$ GeV, is compared
between the models in Fig. 23.

Figure 23: Dependence on primary energy of the muon number $N_{\mu}$ (at sea
level) of proton-initiated EAS, for $E_{\mu}>1$ GeV, calculated using
different models. The meaning of the lines is the same as in Fig. 22.

Here again, the results of QGSJET-III and QGSJET-II-04 agree with each other
within 5%. On the other hand, higher $N_{\mu}$ predicted by EPOS-LHC and
SIBYLL-2.3 comes at no surprise, given their larger values for $\langle
x_{E}\,n_{\rm stable}\rangle$ (c.f. Fig. 21 and the corresponding discussion
in Section 4), being due to an enhanced forward production of, respectively,
(anti)baryons and $\rho$-mesons in those models.

Finally, plotted in Fig. 24 is

Figure 24: Dependence on primary energy of the maximal muon production depth
$X^{\mu}_{\max}$ ($E_{\mu}>1$ GeV) of proton-initiated EAS, calculated using
different models. The meaning of the lines is the same as in Fig. 22.

the energy-dependence of the maximal muon production depth $X^{\mu}_{\max}$,
i.e., the depth in the atmosphere corresponding to a maximal number of muons
being produced by hadron decays, calculated using the different models. As
discussed in [54], this quantity is very sensitive to various aspects of pion-
air collisions and, thus, can be used to test and constrain the respective
model treatment, based on CR data. Even here, despite the fact that the energy
dependence of hadron production in pion-nucleus collisions is weakly
constrained by available accelerator data, the agreement between the results
of QGSJET-III and QGSJET-II-04 is surprisingly good. On the other hand, the
substantially larger values of $X^{\mu}_{\max}$, predicted by the EPOS-LHC and
SIBYLL-2.3 models appear to be in a strong contradiction to the corresponding
measurements by the Pierre Auger Observatory [55].

##  6 Conclusions

In this work, we discussed in some detail the hadronization procedure of the
QGSJET-III MC generator and presented selected results of that generator,
regarding secondary hadron production, in comparison to experimental data and
to the corresponding results of the previous model version, QGSJET-II-04.
Overall, the quality of agreement with the data for QGSJET-III remained at
approximately the same level, as in the case of QGSJET-II-04, certain
improvements being mostly related to the treatment of the pion exchange
process in hadron-proton and hadron-nucleus collisions.

Comparing the predictions of different MC generators, regarding basic
characteristics of proton-induced extensive air showers, we observed an
outstanding agreement between the results of QGSJET-III and QGSJET-II-04, in
spite of an implementation of new theoretical approaches and a number of
technical improvements in the new model. This may indicate that the treatment
of relevant aspects of hadronic interaction physics is already sufficiently
constrained by available accelerator data. On the other hand, in view of
rather different results of the other MC generators, regarding, e.g., the air
shower maximum depth or EAS muon content, one may rise a question whether the
similarity of extensive air shower predictions of QGSJET-III and QGSJET-II-04
is rather a consequence of the underlying theoretical approaches shared by the
two models or/and common deficiencies regarding the treatment of certain
aspects of the interaction physics. Therefore, a general analysis of potential
model uncertainties for EAS predictions is a warranted task. Such an analysis
will be presented elsewhere [50].

### Acknowledgments

This work was supported by Deutsche Forschungsgemeinschaft (project number
465275045).

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# A Nakano Carrier Theorem for Polynomials

Christopher Boyd, Raymond A. Ryan and Nina Snigireva

###### Abstract

00footnotetext: Keywords: Banach lattice; Homogeneous polynomial; Orthogonally
additive; Order continuity; Carrier.00footnotetext: MSC(2020): 46B42; 46G25.

We use a localisation technique to study orthogonally additive polynomials on
Banach lattices. We derive alternative characterisations for orthogonal
additivity of polynomials and orthosymmetry of $m$-linear mappings. We prove
that an orthogonally additive polynomial which is order continuous at one
point is order continuous at every point and we give an example to show that
this result does not extend to regular polynomials in general. Finally, we
prove a Nakano Carrier theorem for orthogonally additive polynomials,
generalising a result of Kusraev.

##  1 Introduction

The purpose of this paper is to prove a carrier theorem for homogeneous
polynomials on Banach lattices that generalises the classical Nakano Carrier
Theorem for linear functionals.

Let us first recall the definitions of the null ideal and the carrier of a
linear functional. Let $E$ be a Banach lattice and let $\varphi\in
E^{\prime}$. The null ideal (or absolute kernel) of $\varphi$ is

$N_{\varphi}=\bigl{\\{}x\in E:|\varphi|(|x|)=0\bigr{\\}}\,.$

The carrier of $\varphi$ is the disjoint complement of its null ideal:

$C_{\varphi}=N_{\varphi}^{\perp}=\bigl{\\{}x\in E:x\perp y\text{ for every
}y\in N_{\varphi}\bigr{\\}}\,.$

Let us now state the Nakano Carrier Theorem, see for example [2, 21]. We will
give the statement of this theorem for Banach lattices, though it is valid in
a more general setting.

######  Nakano Carrier Theorem.

Let $\varphi$ and $\psi$ be order bounded linear functionals on the Banach
lattice $E$, one of which is order continuous. Then $\varphi\perp\psi$ if and
only if $C_{\varphi}\perp C_{\psi}$.

A natural question is how can we extend this to $m$-homogeneous polynomials on
Banach lattices. Each $m$-homogeneous polynomial has an associated symmetric
$m$-linear mapping on $E^{m}$. This complicates the definition of a carrier
for an $m$-homogeneous polynomial as it is not clear if it should be a subset
of $E$ or $E^{m}$. On the other hand, every $m$-homogeneous polynomial can be
considered as a linear functional on a suitable tensor product. From this
point of view, the carrier could be defined as a subset of a tensor product.
However, the natural tensor product, which is the Fremlin tensor product, does
not behave well with respect to order continuity. Namely, there is no perfect
correspondence between order continuity of a multilinear map and the
associated linear functional on the tensor product [13].

In this paper, we will see that if we restrict our attention to the class of
orthogonally additive polynomials then the carrier is a natural, well behaved
object. Let us note that the orthogonally additive polynomials correspond to
orthosymmetric multilinear maps and that Kusraev [15] has proved a carrier
theorem for a special class of orthosymmetric multilinear maps. Here we take a
different approach. In particular, our approach involves localisation to
principal ideals. This allows us to consider orthogonally additive polynomials
on $C(K)$ which in turn allows us to view these polynomials as measures.

As in the linear case, order continuity of polynomials will play an important
role in the proof of the nonlinear version of Nakano’s theorem. So we will
discuss order continuous polynomials in detail. In particular, we will show
that for orthogonally additive polynomials order continuity at the origin
implies order continuity at every point. We will also, construct a
counterexample to show that this is not the case for regular polynomials.

For further information on polynomials we refer the reader to [10] and for the
theory of Banach lattices to [3, 17].

##  2 Orthogonally additive polynomials and their properties

Let $E$ be a Banach lattice. Recall that a regular $m$-linear form $A\colon
E^{m}\to\mathbb{R}$ is said to be orthosymmetric if $A(x_{1},\dots,x_{m})=0$
whenever some pair $x_{i}$, $x_{j}$ of the arguments are disjoint, i.e.
$|x_{i}|\wedge|x_{j}|=0$. Note that it suffices to check this condition for
positive arguments, since $|x_{i}|\perp|x_{j}|$ if and only if $x_{i}^{\pm}$
are disjoint from $x_{j}^{\pm}$. Bu and Buskes [6] showed that every
orthosymmetric $m$-linear form is symmetric. Recall as well that a regular
$m$-homogeneous polynomial $P=\widehat{A}$ is said to be orthogonally additive
if $P(x+y)=P(x)+P(y)$ whenever $x$ and $y$ are disjoint. Bu and Buskes [6]
showed that this is equivalent to the associated symmetric $m$-linear mapping
$A$ being orthosymmetric.

Orthosymmetric $m$-linear mappings on $C(K)^{m}$ and orthogonally additive
$m$-homogeneous polynomials on $C(K)$ have very useful integral
representations which we will use throughout this paper. If we work in a
principal ideal $E_{a}$ generated by a positive element $a$, then we can
identify $E_{a}$ with the Banach lattice $C(K_{a})$ for some compact Hausdorff
space $K_{a}$ [14] and make use of these integral representations.

######  Theorem 1 ([9, 18]).

Let $P$ be an orthogonally additive $m$-homogeneous polynomial on $C(K)$ and
$A$ be the associated orthosymmetric $m$-linear mapping on $C(K)^{m}$. Then

$P(x)=\int_{K}x(t)^{m}\,d\mu(t)$

and therefore

$A(x_{1},\dots,x_{m})=\int_{K}x_{1}(t)\cdots x_{m}(t)\,d\mu(t),$

for some regular Borel measure $\mu$ on $K$.

Let us note that the above theorem was proved in [18] and also, with a shorter
proof, in [9], for orthogonally additive $m$-homogeneous polynomials. The
corresponding result for orthosymmetric $m$-linear mappings follows easily
from the integral formulae for polynomials. For completeness, we will now give
a short proof of this fact. Using the polarisation formula, we have

$\displaystyle A(x_{1},\dots,x_{m})$
$\displaystyle=\frac{2^{m}}{m!}\sum_{\varepsilon_{i}=\pm
1}\varepsilon_{1}\cdots\varepsilon_{m}P(\varepsilon_{1}x_{1}+\cdots+\varepsilon_{m}x_{m})$
$\displaystyle=\frac{2^{m}}{m!}\sum_{\varepsilon_{i}=\pm
1}\varepsilon_{1}\cdots\varepsilon_{m}\int_{K}(\varepsilon_{1}x_{1}+\cdots+\varepsilon_{m}x_{m})^{m}\,d\mu$
$\displaystyle=\int_{K}\frac{2^{m}}{m!}\sum_{\varepsilon_{i}=\pm
1}\varepsilon_{1}\cdots\varepsilon_{m}(\varepsilon_{1}x_{1}+\cdots+\varepsilon_{m}x_{m})^{m}\,d\mu$
$\displaystyle=\int_{K}x_{1}\cdots x_{m}\,d\mu.$

The above theorem allowed us to show that there is a Banach lattice isometric
isomorphism from the space $\mathcal{M}(K)$ of regular Borel measures on a
compact Hausdorff space $K$, equipped with the variation norm, to the space
$\mathcal{P\\!}_{o}(^{m}{C(K)})$ of orthogonally additive polynomials on
$C(K)$, equipped with the regular norm [4].

######  Theorem 2 ([4]).

Let $K$ be a compact, Hausdorff space and let $m$ be a positive integer. Let
$I_{m}\colon\mathcal{M}(K)\to\mathcal{P\\!}_{o}(^{m}{C(K)})$ be given by

$(I_{m}\mu)(x)=\int_{K}x^{m}\,d\mu\,.$

Then $I_{m}$ is a Banach lattice isometric isomorphism from $\mathcal{M}(K)$
onto $\mathcal{P\\!}_{o}(^{m}{C(K)})$.

Orthosymmetry can be characterised using functional equations with no
restriction (such as disjointness) on the arguments. We begin by generalising
the result for the bilinear case which was proved by Kusraev [16].

######  Proposition 1 ([16]).

A regular bilinear form $A$ on $E\times E$ is orthosymmetric if and only if

$A(x,y)=A(x\vee y,x\wedge y)$ (1)

for all $x,y\in E$.

Now let $A$ be an orthosymmetric $3$-linear form. Let $x,y,z\in E$. By
orthosymmetry with respect to the first two variables, we have
$A(x,y,z)=A(x\vee y,x\wedge y,z)$. Now using orthosymmetry in the first and
third variables,

$\displaystyle A(x,y,z)$ $\displaystyle=A(x\vee y\vee z,x\wedge y,(x\vee
y)\wedge z)$ $\displaystyle=A(x\vee y\vee z,x\wedge y,(x\wedge z)\vee(y\wedge
z)\,,$

by the distributivity of the lattice operations. Finally, we use orthosymmetry
with respect to the second and third variables to get

$A(x,y,z)=A(x\vee y\vee z,(x\wedge y)\vee(x\wedge z)\vee(y\wedge z),(x\wedge
y)\wedge((x\wedge z)\vee(y\wedge z)))\,.$

Now

$\displaystyle(x\wedge y)\wedge((x\wedge z)\vee(y\wedge z))$
$\displaystyle=((x\wedge y)\wedge(x\wedge z))\vee((x\wedge y)\wedge(y\wedge
z))$ $\displaystyle=x\wedge y\wedge z$

So we get the following identity that is satisfied by every orthosymmetric
$3$-linear form:

$A(x,y,z)=\\\ A(x\vee y\vee z,(x\wedge y)\vee(x\wedge z)\vee(y\wedge
z),x\wedge y\wedge z)$ (2)

for all $x,y,z\in E$. Conversely, if $A$ satisfies this, then clearly it is
orthosymmetric.

One interesting consequence of this identity is that if $A$ is an
orthosymmetric $3$-linear form, then in order to have $A(x,y,z)=0$, it is not
necessary for some pair of $x,y,z$ to be disjoint. It suffices for them to be
“triply disjoint”, in the sense that $|x|\wedge|y|\wedge|z|=0$.

We now consider the general case. The pattern is clear, but we need some
notation to avoid very cumbersome formulae. Let $x=(x_{1},\dots,x_{m})$ be an
$m$-tuple in $E^{m}$. We define

$\displaystyle J_{1}^{m}(x)$ $\displaystyle=\bigvee_{i=1}^{m}x_{i}$
$\displaystyle J_{2}^{m}(x)$ $\displaystyle=\bigvee_{\stackrel{{\scriptstyle
1\leq i_{1},i_{2}\leq m}}{{i_{1}<i_{2}}}}\bigl{(}x_{i_{1}}\wedge
x_{i_{2}}\bigr{)}$ $\displaystyle J_{3}^{m}(x)$
$\displaystyle=\bigvee_{\stackrel{{\scriptstyle 1\leq i_{1},i_{2},i_{3}\leq
m}}{{i_{1}<i_{2}<i_{3}}}}\bigl{(}x_{i_{1}}\wedge x_{i_{2}}\wedge
x_{i_{3}}\bigr{)}$ $\displaystyle\vdots$ $\displaystyle J_{m}^{m}(x)$
$\displaystyle=x_{1}\wedge\dots\wedge x_{m}$

Note that, if $x_{1},\dots,x_{m}$ are real numbers, then
$J_{1}^{m}(x),\dots,J_{m}^{m}(x)$ is the decreasing rearrangement of
$x_{1},\dots,x_{m}$. So we have

$\prod_{i=1}^{m}x_{i}=\prod_{i=1}^{m}J_{i}^{m}(x)\,.$ (3)

The next result shows that orthosymmetric multilinear forms are characterised
by a similar identity. The $3$-linear case suggests that a natural way to
prove this result would be to use induction. However, it turns out that an
easier approach is to use a localisation technique.

######  Proposition 2.

Let $E$ be a Banach lattice. A regular $m$-linear form $A$ on $E^{m}$ is
orthosymmetric if and only if

$A(x_{1},\dots,x_{m})=A(J_{1}^{m}(x),\dots,J_{m}^{m}(x))$ (4)

for every $x=(x_{1},\dots,x_{m})\in E^{m}$.

###### Proof.

Clearly if $A$ satisfies $A(x_{1},\dots,x_{m})=A(J_{1}(x),\dots,J_{m}(x))$
then $A$ is orthosymmetric. To prove the converse we will use a localisation
technique. Let $a=|x_{1}|\vee\dots\vee|x_{m}|$. The required identity can now
be easily established on $E_{a}$ since $E_{a}$ can be identified with
$C(K_{a})$ for a suitable compact set $K_{a}$. Hence, using Theorem 1 and
equation (3), we have

$\displaystyle A(x_{1},\dots,x_{m})$ $\displaystyle=\int_{K}x_{1}(t)\dots
x_{m}(t)\,d\mu(t)$ $\displaystyle=\int_{K}J_{1}^{m}(x(t))\dots
J_{m}^{m}(x(t))\,d\mu(t)$ $\displaystyle=\int_{K}J_{1}^{m}(x)(t)\dots
J_{m}^{m}(x)(t)\,d\mu(t)$ $\displaystyle=A(J_{1}^{m}(x),\dots,J_{m}^{m}(x))$

∎

We now consider orthogonally additive polynomials. Schwanke [19] showed that
it suffices to verify the orthogonal additivity property of a regular
homogeneous polynomial for positive arguments.

######  Lemma 1 ([19]).

Let $P$ be an $m$-homogeneous polynomial on a Riesz space $E$. If $P$ is
orthogonally additive on the positive cone of $E$, then $P$ is orthogonally
additive.

Let us now show that in order for a homogeneous polynomial to be orthogonally
additive, it suffices for it to be orthogonally additive on the positive and
negative parts of $x$ in $E$.

######  Lemma 2.

An $m$-homogeneous polynomial $P$ on a Riesz space $E$ is orthogonally
additive if and only if

$P(x)=P(x^{+})+(-1)^{m}P(x^{-})$ (5)

for every $x\in E$.

###### Proof.

The condition is clearly necessary. Conversely, suppose that $P$ satisfies
this condition. Let $x,y$ be disjoint positive elements of $E$. Taking
$z_{t}=x-ty$ for $t>0$, we have $z_{t}^{+}=x$ and $z_{t}^{-}=ty$ and so
$P(z_{t})=P(x-ty)=P(x)+(-1)^{m}t^{m}P(y)$. We also have that

$P(z_{t})=P(x-ty)=\sum_{k=0}^{m}\binom{m}{k}(-1)^{k}t^{k}A(x^{m-k},y^{k}).$

These two identities for $P(z_{t})$ imply that $A(x^{m-k},y^{k})=0$ for
$0<k<m$. Therefore

$P(x+y)=\sum_{k=0}^{m}\binom{m}{k}A(x^{m-k},y^{k})=P(x)+P(y)$

and hence $P$ is orthogonally additive on $E_{+}$. ∎

We recall that a Riesz space is uniformly complete if every principal ideal is
complete in the unit norm and that every Banach lattice is uniformly complete.
Buskes and Schwanke [7], improving on a result of Kusraeva, showed that a
regular $m$-homogeneous polynomial $P=\widehat{A}$ on a uniformly complete
Riesz space $E$ is orthogonally additive if and only if

$P\bigl{(}(x_{1}^{m}+\dots+x_{k}^{m})^{1/m}\bigr{)}=P(x_{1})+\dots+P(x_{k})$

for every $k\in\mathbb{N}$, $x_{1},\dots,x_{k}\in E_{+}$, and in addition

$P\bigl{(}(x_{1}\dots x_{m})^{1/m}\bigr{)}=A(x_{1},\dots,x_{m})$

for all $x_{1},\dots,x_{m}\in E_{+}$. The expressions that appear on the left
hand side of these formulae are defined by the Krivine functional calculus.
Recently, Schwanke [19] showed that just one of these conditions suffices.

######  Proposition 3 ([19]).

Let $P=\widehat{A}$ be a regular $m$-homogeneous polynomial on a uniformly
complete Riesz space $E$. The following are equivalent:

  1. (a)

$P$ is orthogonally additive.

  2. (b)

$P\bigl{(}(x_{1}^{m}+\dots+x_{k}^{m})^{1/m}\bigr{)}=P(x_{1})+\dots+P(x_{k})$
for every $k\in\mathbb{N}$, $x_{1},\dots,x_{k}\in E_{+}$.

  3. (c)

$P\bigl{(}(x^{m}+y^{m})^{1/m}\bigr{)}=P(x)+P(y)$
for every $x,y\in E_{+}$.

  4. (d)

$P\bigl{(}(x_{1}\dots x_{m})^{1/m}\bigr{)}=A(x_{1},\dots,x_{m})$
for all $x_{1},\dots,x_{m}\in E_{+}$.

A real valued function $f$ on a vector lattice $E$ is said to be a valuation
if $f(x\vee y)+f(x\wedge y)=f(x)+f(y)$ for all $x,y\in E$. Buskes and Roberts
[8] showed that an $m$-homogeneous polynomial on a Riesz space is orthogonally
additive if and only if it is a valuation. We give an alternative proof that
demonstrates the power of the localisation technique.

######  Proposition 4 ([8]).

A regular $m$-homogeneous polynomial $P$ on a Banach lattice $E$ is
orthogonally additive if and only if

$P(x\vee y)+P(x\wedge y)=P(x)+P(y)$ (6)

for all $x,y\in E_{+}$.

###### Proof.

This condition is clearly sufficient. Conversely, suppose $P$ is orthogonally
additive and let $x,y\in E_{+}$. Restricting to the principal ideal generated
by $a=x\vee y$, we have an integral representation for $P$ on $E_{a}\cong
C(K)$ given by a regular Borel measure $\mu$. Thus

$\displaystyle P(x\vee y)+P(x\wedge y)$ $\displaystyle=\int_{K}(x\vee
y)(t)^{m}+(x\wedge y)(t)^{m}\,d\mu$ $\displaystyle=\int_{K}(x(t)\vee
y(t))^{m}+(x(t)\wedge y(t))^{m}\,d\mu$
$\displaystyle=\int_{K}x(t)^{m}+y(t)^{m}\,d\mu=P(x)+P(y)$

∎

We note that this result has an obvious generalisation to an identity relating
to any $k$-tuple $(x_{1},\dots,x_{k})$ of positive elements, namely,

$\sum_{i=1}^{k}P(J_{i}x)=\sum_{i=1}^{k}P(x_{i})\,.$ (7)

Finally, let us note that the orthogonally additive $m$-homogeneous
polynomials form a band in the Banach lattice of regular $m$-homogeneous
polynomials [8]. In particular, a regular $m$-homogeneous polynomial $P$ is
orthogonally additive if and only if $|P|$ is orthogonally additive.

##  3 Localisation of multilinear mappings and polynomials to principal ideals

We now develop further the technique of localisation used in the previous
section. We will consider which properties of multilinear mappings and
polynomials are preserved by localisation to principal ideals. In particular,
we are interested whether order continuity of multilinear maps and polynomials
is preserved by localisation.

Let $E_{1},\dots,E_{m},F$ be Banach lattices. Recall that $A\colon E_{1}\times
E_{2}\times\dots\times E_{m}\to F$ is separately order continuous if it is
order continuous in each variable when the other $m-1$ variables are kept
fixed. Recall also that $A\colon E_{1}\times E_{2}\times\dots\times E_{m}\to
F$ is jointly order continuous if

$A(x^{1}_{\alpha_{1}},\dots,x^{m}_{\alpha_{m}})\stackrel{{\scriptstyle
o}}{{\to}}A(x_{1},\dots,x_{m})\quad\mbox{whenever
$x^{1}_{\alpha_{1}}\stackrel{{\scriptstyle
o}}{{\to}}x_{1},\dots,x^{m}_{\alpha_{m}}\stackrel{{\scriptstyle
o}}{{\to}}x_{m}$.}$

Shotaev [20] has shown that for bilinear maps, separate order continuity is
equivalent to joint order continuity. We have extended this result to
$m$-linear mappings in [5]. Let us note that a net $(x_{\alpha})_{\alpha}$ in
$E$ converges in order to $x$ if there exists a net $(y_{\alpha})_{\alpha}$,
decreasing to $0$, and an index $\alpha_{o}$ so that $|x_{\alpha}-x|\leq
y_{\alpha}$ for all $\alpha\geq\alpha_{o}$ (see [1]). Hence for all
$\alpha\geq\alpha_{o}$ we have that

$\displaystyle|x_{\alpha}|$ $\displaystyle=|x_{\alpha}-x+x|$
$\displaystyle\leq|x_{\alpha}-x|+|x|$ $\displaystyle\leq y_{\alpha}+|x|$
$\displaystyle\leq y_{\alpha_{o}}+|x|.$

This tells us that if a net $(x_{\alpha})_{\alpha}$ converges in order then we
can find $\alpha_{o}$ so that $(x_{\alpha})_{\alpha\geq\alpha_{o}}$ is bounded
in order and therefore a norm bounded subset of $E$.

Let $A\colon E^{m}\to F$ be an $m$-linear mapping and let $a\in E_{+}$. We
will denote by $A_{a}$ the restriction of $A$ to $(E_{a})^{m}$. The next
theorem shows which properties, including order continuity, of $A$ are
characterised by restricting to principal ideals.

######  Theorem 3.

Let $E$, $F$ be Banach lattices, with $F$ Dedekind complete and let $A,B\colon
E^{m}\to F$ be $m$-linear mappings. Then

  * (a)

$A$ is bounded if and only if $A_{a}$ is bounded on $E^{m}_{a}$ for every
$a\in E_{+}$.

  * (b)

$A$ is positive (respectively regular) if and only if $A_{a}$ is positive
(respectively regular) on $E^{m}_{a}$ for every $a\in E_{+}$.

  * (c)

Suppose that $A$ is regular. Then $A$ is jointly order continuous if and only
if $A_{a}\colon(E_{a})^{m}\to F$ is jointly order continuous for every $a\in
E_{+}$.

  * (d)

Suppose that $A$ is regular. Then

$|A|_{a}=|A_{a}|,\quad(A\vee B)_{a}=A_{a}\vee B_{a},\quad(A\wedge
B)_{a}=A_{a}\wedge B_{a}$

for every $a\in E_{+}$. In particular, $A\perp B$ if and only if $A_{a}\perp
B_{a}$ for every $a\in E_{+}$.

###### Proof.

(a) If $A$ is a bounded $m$-linear mapping, then clearly $A_{a}$ is a bounded
$m$-linear mapping on $E_{a}^{m}$ for every $a\in E_{+}$.

To prove the converse, suppose that $A$ is unbounded. Then there exist
sequences $(x^{1}_{n})$, $\dots$, $(x^{m}_{n})$ in the unit ball of $E$ such
that $\|A(x^{1}_{n},\dots,x^{m}_{n})\|\geq n^{m+1}$ for every $n$. The
sequences defined by $y^{i}_{n}=x^{i}_{n}/n$ converge to zero and so they have
subsequences with the same indexing set that converge relatively uniformly to
zero. In other words, the subsequences lie in some principal ideal $E_{a}$ and
they converge to zero for the principal ideal norm. Therefore $A_{a}$ is
unbounded for some $a\in E_{+}$, a contradiction.

(b) Let us first note that an $m$-linear mapping $A$ is positive if and only
if $A_{a}$ is positive on every principal ideal $E_{a}$.

Suppose that every $A_{a}$ is regular. To show that $A$ is regular, by [6], we
have to show that the following supremum exists for each $m$-tuple of positive
elements of $E$:

$|A|(x_{1},\dots,x_{m})=\sup\Bigl{\\{}\sum|A(u^{1}_{i_{1}},\dots,u^{m}_{i_{m}})|:u^{i}\in\pi(x_{i}),1\leq
i\leq m\Bigr{\\}}.$

Note that if $x_{1}\vee\dots\vee x_{m}\leq a$, then this supremum defines
$|A_{a}|$ at the $m$-tuple $(x_{1},\dots,x_{m})$ and its value is independent
of $a$, since replacing $a$ by a bigger element does not add any additional
partitions of $x_{1},\dots,x_{m}$. The converse is obvious.

(c) Observe that if $(x_{\alpha})$ is an order convergent net in $E$, then
there is an index $\alpha_{0}$ and a positive element $a$ in $E$ such that the
net $(x_{\alpha}:\alpha\geq\alpha_{0})$ is order convergent in $E_{a}$.
Suppose now that $A_{a}$ is jointly order continuous at $(x_{1},\dots,x_{m})$.
Taking $\alpha_{i}\geq\alpha^{i}_{o}$ for $i=1,\dots,m$, we have that

$A(x^{1}_{\alpha_{1}},\dots,x^{m}_{\alpha_{m}})=A_{a}(x^{1}_{\alpha_{1}},\dots,x^{m}_{\alpha_{m}})\longrightarrow
A_{a}(x_{1},\dots,x_{m})=A(x_{1},\dots,x_{m}),$

showing that $A$ is jointly order continuous at $(x_{1},\dots,x_{m})$. The
converse direction is trivial.

(d) The argument in Part (b) also shows that

$|A_{a}|=|A|_{a}$

for every $a\in E_{+}$ with similar identities for $A^{+}$ and $A^{-}$. By the
same reasoning, we have formulae such as $(A\vee B)_{a}=A_{a}\vee B_{a}$. And
in particular, disjointness is also determined on the principal ideals:
$A\perp B$ if and only if $A_{a}\perp B_{a}$ for every $a\in E_{+}$. ∎

To state the corresponding results for $m$-homogeneous polynomials, let us
recall that a scalar valued $m$-homogeneous polynomial, $P$, is said to be
order continuous at $x$ if $P(x_{\alpha})$ converges to $P(x)$ whenever
$(x_{\alpha})_{\alpha}$ is a net in $E$ which converges in order to $x$.
Moreover, it is easy to see that an $m$-homogeneous polynomial is order
continuous if and only if its symmetric $m$-linear generator is jointly order
continuous.

######  Proposition 5.

Let $E$, $F$ be Banach lattices, with $F$ Dedekind complete and let $P,Q\colon
E\to F$ be $m$-homogeneous polynomials. Then

  * (a)

$P$ is bounded if and only if $P_{a}$ (the restriction of $P$ to $E_{a}$) is
bounded on $E_{a}$ for every $a\in E_{+}$.

  * (b)

$P$ is positive (respectively regular) if and only if $P_{a}$ is positive
(respectively regular) on $E_{a}$ for every $a\in E_{+}$.

  * (c)

Suppose that $P$ is regular. Then $P$ is order continuous if and only if
$P_{a}$ is order continuous on $E_{a}$ for every $a\in E_{+}$.

  * (d)

Suppose that $P$ and $Q$ are regular. Then

$|P|_{a}=|P_{a}|,\quad(P\vee Q)_{a}=P_{a}\vee Q_{a},\quad(P\wedge
Q)_{a}=P_{a}\wedge Q_{a}$

for every $a\in E_{+}$. In particular, $P\perp Q$ if and only if $P_{a}\perp
Q_{a}$ for every $a\in E_{+}$.

##  4 Order continuous polynomials

In this section we will focus on order continuous polynomials and discuss when
order continuity at one point implies order continuity at every point. Let us
begin by showing that it is possible for a polynomial on a Banach lattice $E$
to be order continuous at $0$ yet not to be order continuous at every point of
$E$. To see this suppose that $E$ is a Banach lattice which admits an order
continuous functional $\varphi$ and a norm continuous linear functional $\psi$
which is not order continuous. For example, take $E=C[0,1]\bigoplus
L_{1}[0,1]$, since there are no order continuous linear functionals on
$C[0,1]$ and every bounded linear functional on $L_{1}[0,1]$ is order
continuous.

Define the $m$-homogeneous polynomial $P$ by $P(x)=\varphi^{m-1}(x)\psi(x)$.
Let us first observe that $P$ is order continuous at $0$. To see this let
$(y_{\alpha})_{\alpha}$ be a net in $E$ which converges to $0$ in order. As we
have already observed, we can assume without loss of generality that
$(y_{\alpha})_{\alpha}$ is a (norm) bounded subset of $E$. Therefore, we can
find $B>0$ so that $|\psi(y_{\alpha})|\leq B$ for all $\alpha$. Let
$\varepsilon>0$. As $(y_{\alpha})_{\alpha}$ converges to $0$ in order and
$\varphi$ is order continuous we can find $\alpha_{0}$ so that
$|\varphi(y_{\alpha})|^{m-1}<\frac{\varepsilon}{B}$ for all
$\alpha>\alpha_{o}$. Then for all $\alpha>\alpha_{o}$ we have

$|P(y_{\alpha})|=|\varphi(y_{\alpha})|^{m-1}|\psi(y_{\alpha})|\leq\frac{\varepsilon}{B}B=\varepsilon$

which proves that $P$ is order continuous at $0$.

As $\psi$ is not order continuous, for any $x_{o}$ there exists a net
$(x_{\alpha})_{\alpha}$ which converges to $x_{o}$ in order but for which
$\psi(x_{\alpha})\not\to\psi(x_{o})$. Let us suppose that
$\varphi(x_{o})\not=0$. As $\varphi$ is order continuous, we can find $A>0$
and $\alpha_{1}$ so that $|\varphi(x_{\alpha})|^{m-1}>A$, all
$\alpha>\alpha_{1}$.

We consider two cases. In the first we assume that $C=|\psi(x_{o})|\not=0$. As
$\psi(x_{\alpha})\not\to\psi(x_{o})$ we can find $t>0$, a subnet
$(x_{\alpha_{\beta}})_{\beta}$ of $(x_{\alpha})_{\alpha}$ and $\beta_{2}$ so
that $|\psi(x_{\alpha_{\beta}})-\psi(x_{o})|>t$ for all $\beta>\beta_{2}$. As
$\varphi(x_{\alpha_{\beta}})$ converges to $\varphi(x_{o})$ we can find
$\beta_{3}$ so that
$|\varphi^{m-1}(x_{\alpha_{\beta}})-\varphi^{m-1}(x_{o})|<\frac{tA}{2C}$ for
all $\beta>\beta_{3}$. Let us choose $\beta_{1}$ so that
$\alpha_{\beta_{1}}>\alpha_{1}$ and set
$\beta_{0}=\max\\{\beta_{1},\beta_{2},\beta_{3}\\}$. Then for
$\beta>\beta_{o}$ we have

$\displaystyle|P(x_{\alpha_{\beta}})-P(x_{o})|$
$\displaystyle=|\varphi^{m-1}(x_{\alpha_{\beta}})\psi(x_{\alpha_{\beta}})-\varphi^{m-1}(x_{o})\psi(x_{o})|$
$\displaystyle=|\varphi^{m-1}(x_{\alpha_{\beta}})\psi(x_{\alpha_{\beta}})-\varphi^{m-1}(x_{\alpha_{\beta}})\psi(x_{o})-(\varphi^{m-1}(x_{o})\psi(x_{o})-\varphi^{m-1}(x_{\alpha_{\beta}})\psi(x_{o}))|$
$\displaystyle\geq\left||\varphi^{m-1}(x_{\alpha_{\beta}})\psi(x_{\alpha_{\beta}})-\varphi^{m-1}(x_{\alpha_{\beta}})\psi(x_{o})|-|\varphi^{m-1}(x_{o})\psi(x_{o})-\varphi^{m-1}(x_{\alpha_{\beta}})\psi(x_{o})|\right|$
$\displaystyle=\left||\varphi(x_{\alpha_{\beta}})|^{m-1}|\psi(x_{\alpha_{\beta}})-\psi(x_{o})|-|\psi(x_{o})||\varphi^{m-1}(x_{\alpha_{\beta}})-\varphi^{m-1}(x_{o})|\right|$
$\displaystyle\geq tA-C\frac{tA}{2C}$ $\displaystyle=\frac{tA}{2}$

proving that $P$ is not order continuous at $x_{o}$.

Now let us suppose that $C=|\psi(x_{o})|=0$. Then we can find $t>0$ and
$\beta_{2}$ so that $|\psi(x_{\alpha_{\beta}})|>t$ for all $\beta>\beta_{2}$.
Let $\beta_{o}=\max\\{\beta_{1},\beta_{2}\\}$. Then for $\beta>\beta_{o}$ we
have

$|P(x_{\alpha_{\beta}})-P(x_{o})|=|\varphi(x_{\alpha_{\beta}})|^{m-1}|\psi(x_{\alpha_{\beta}})|>tA$

and therefore $P$ is not order continuous at $x_{o}$.

While a regular homogeneous polynomial which is order continuous at one point
of a Banach lattice need not be order continuous at every point we can still
say something about the set of points at which a regular polynomial is order
continuous.

######  Proposition 6.

Let $E$ be a Banach lattice and $P$ a regular $m$-homogeneous polynomial on
$E$. Let $C_{o}(P)$ denote the set of all points $x$ in $E$ for which $P$ is
order continuous at $x$. Then $C_{o}(P)$ is a norm closed subset of $E$.

###### Proof.

Let $(u_{n})_{n}$ be a sequence in $C_{o}(P)$ which converges to some point
$u$ in $E$. Let $\varepsilon>0$. Let $(x_{\alpha})_{\alpha}$ be a net in $E$
which is convergent to $0$ in order. Replacing $(x_{\alpha})_{\alpha}$ with a
suitable tail, we may assume without loss of generality, that
$(x_{\alpha})_{\alpha}$ is bounded. As $P$ is uniformly (norm) continuous on
bounded sets there is $n_{o}$ in $\mathbb{N}$ so that
$|P(u)-P(u_{n})|<\varepsilon/3$ and
$|P(u+x_{\alpha})-P(u_{n}+x_{\alpha})|<\varepsilon/3$ for all $n\geq n_{o}$
and all $\alpha$. Since $u_{n_{o}}$ belongs to $C_{o}(P)$, $P$ is order
continuous at $u_{n_{o}}$. Therefore, we can find $\alpha_{o}$ so
$|P(u_{n_{o}}+x_{\alpha})-P(u_{n_{o}})|<\varepsilon/3$ whenever
$\alpha>\alpha_{o}$. Then for $\alpha>\alpha_{o}$ we have

$\displaystyle|P(u+x_{\alpha})-P(u)|$
$\displaystyle\leq|P(u+x_{\alpha})-P(u_{n_{o}}+x_{\alpha})|+|P(u_{n_{o}}+x_{\alpha})-P(u_{n_{o}})|+|P(u_{n_{o}})-P(u)|$
$\displaystyle<\frac{\varepsilon}{3}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}$
$\displaystyle=\varepsilon.$

Hence we have that $P$ is order continuous at $u$. ∎

This allows us to show that order continuity of a homogeneous polynomial at
any point implies order continutiy at $0$.

######  Corollary 1.

Let $E$ be a Banach lattice and $P$ is a regular $m$-homogeneous polynomial on
$E$. If $P$ is order continuous at some point of $E$ then $P$ is order
continuous at $0$.

###### Proof.

Let us suppose that $P$ is order continuous at $x$. By homogeneity it follows
that $P$ is order continuous at $\lambda x$ for every $\lambda$ in
$\mathbb{R}$. Letting $\lambda$ tend to $0$ the above Proposition gives us
that $P$ is order continuous at $0$. ∎

Let us now look at order continuity of orthogonally additive polynomials. We
will show that in contrast to regular polynomials in general, order continuity
of an orthogonally additive homogeneous polynomial at any point implies order
continuity at every point. We begin by looking at order continuity on
$C(K)$-spaces. In this setting, we will make use of the following lemma.

######  Lemma 3.

If $(x_{\alpha})_{\alpha}$ is a net in $C(K)$ which converges in order to $x$
then the net $(x^{m}_{\alpha})_{\alpha}$ converges in order to $x^{m}$.

###### Proof.

Suppose that $(x_{\alpha})_{\alpha}$ is a net in $C(K)$ which converges in
order to $x$. Replacing $(x_{\alpha})_{\alpha}$ with
$(x_{\alpha})_{\alpha\geq\alpha_{o}}$ we may assume without loss of generality
that $(x_{\alpha})_{\alpha}$ is order bounded. Let
$B=\sup_{\alpha}\|x_{\alpha}\|$.

First suppose that $x=0$. Then we have

$|x_{\alpha}^{m}|\leq B^{m-1}|x_{\alpha}|$

giving us that $(x_{\alpha}^{m})_{\alpha}$ converges to $0$ in order.

Now suppose that $x\neq 0$. In this case we have

$\displaystyle|x_{\alpha}^{m}-x^{m}|$
$\displaystyle=\left|(x_{\alpha}-x)\left(\sum_{j=0}^{m-1}x_{\alpha}^{m-1-j}x^{j}\right)\right|$
$\displaystyle\leq|x_{\alpha}-x|\sum_{j=0}^{m-1}|x_{\alpha}|^{m-1-j}|x|^{j}$
$\displaystyle\leq|x_{\alpha}-x|\sum_{j=0}^{m-1}\|x_{\alpha}\|^{m-1-j}|\|x\|^{j}1_{K}$
$\displaystyle\leq mB^{m-1}|x_{\alpha}-x|,$

from this we conclude that $(x_{\alpha}^{m})_{\alpha}$ converges to $x^{m}$ in
order. ∎

In order to state our next result, we need to recall the definition of a
normal measure. A regular Borel measure on a compact Hausdorff space $K$ is
said to be normal if $\mu(B)=0$ for every closed, nowhere dense subset $B$ of
$K$. It is easy to see that $\mu$ is normal if and only if the positive
measure $|\mu|$ is normal. Our next theorem is in the spirit of the following
result of Dixmier [11] which relates order continuous linear functionals and
normal measures.

######  Theorem 4 ([11]).

Let $K$ be a compact Hausdorff space. A bounded linear functional on $C(K)$ is
order continuous if and only if the (unique) regular Borel measure that
represents it is a normal measure.

We now have the following theorem for polynomials.

######  Theorem 5.

Let $P$ be an orthogonally additive $m$-homogeneous polynomial on $C(K)$. Then
$P$ is order continuous at $0$ if and only if its representing measure, $\mu$,
is normal.

###### Proof.

We can assume without loss of generality that $P$, and hence $\mu$, is
positive. Let $B$ be a closed nowhere dense subset of $K$. Using Urysohn’s
Lemma we find a bounded decreasing net $(x_{\alpha})_{\alpha}$ of nonnegative
continuous functions on $K$, indexed by the collection of open subsets of $K$
containing $B$, that converges pointwise to $\chi_{B}$.

Then the infimum of the set $\\{x_{\alpha}\\}$ is the zero function. To see
this, suppose there exists a nonnegative function $y$ such that $0<y\leq
x_{\alpha}$ for every $\alpha$. Applying this at each point, we see that
$0<y\leq\chi_{B}$. But now the open set $\\{t\in K:y(t)<1\\}$ is contained in
$B$, a contradiction, since $B$ is nowhere dense.

Therefore, the zero function is the greatest lower bound of $\\{x_{\alpha}\\}$
as claimed and so $x_{\alpha}\searrow 0$. It follows that
$x_{\alpha}^{1/m}\searrow 0$ and since $P$ is order continuous at $0$ we have

$\mu(B)=\int_{K}\chi_{B}\,d\mu\leq\int_{K}x_{\alpha}\,d\mu=\int_{K}(x^{1/m}_{\alpha})^{m}\,d\mu=P(x_{\alpha}^{1/m})\to
0\,,$

and so $\mu(B)=0$. Therefore $\mu$ is a normal measure.

To prove the converse, suppose that $(x_{\alpha})_{\alpha}$ is a net in $C(K)$
which converges in order to $0$. By Lemma 3 and since $\mu$ is a normal
measure, we have that

$P(x_{\alpha})=\int_{K}x_{\alpha}^{m}\,d\mu\longrightarrow 0.$

Hence $P$ is order continuous at $0$.

∎

The next proposition is a consequences of the above theorem.

######  Proposition 7.

Let $P$ be an orthogonally additive $m$-homogeneous polynomial on $C(K)$. Then
$P$ is order continuous at $0$ if and only if $P$ is order continuous at every
point of $C(K)$.

###### Proof.

Suppose that $P$ is order continuous at $0$ then Theorem 5 gives us that its
representing measure, $\mu$, is normal. Let $(x_{\alpha})_{\alpha}$ be a net
in $C(K)$ which converges in order to $x$. Using Lemma 3 and the fact that
$\mu$ is a normal measure, we have that

$P(x_{\alpha})=\int_{K}x_{\alpha}^{m}\,d\mu\longrightarrow\int_{K}x^{m}\,d\mu=P(x),$

proving that $P$ is order continuous at every point of $C(K)$.

The converse is trivial. ∎

Let us extend this result to arbitrary Banach lattices.

######  Proposition 8.

Let $E$ be a Banach lattice and $P$ be an orthogonally additive
$m$-homogeneous polynomial on $E$ which is order continuous at some point of
$E$. Then $P$ is order continuous at every point of $E$.

###### Proof.

By Corollary 1, $P$ is order continuous at $0$. Let $a\in E_{+}$. We first
observe that $P_{a}$ is orthogonally additive on $E_{a}$. Let $(y_{\alpha})$
be net in $E_{a}$ which converges to $0$ in order in $E_{a}$. Since $E_{a}$ is
an ideal in $E$, it follows that $(y_{\alpha})_{\alpha}$ converges to $0$ in
order in $E$. So we have that $P_{a}(y_{\alpha})=P(y_{\alpha})$ converges to
$0$, proving that $P_{a}$ is order continuous at $0$. From Proposition 7, it
follows that $P_{a}$ is order continuous at each point of $E_{a}$. Finally, by
Proposition 5 (c), we have that $P$ is order continuous at each point of $E$.

∎

##  5 A carrier theorem for polynomials

Let $E$, $F$ be Banach lattices, with $F$ Dedekind complete. Following Kusraev
[15], we define the null ideal $N(A)$ of a regular symmetric $m$-linear
mapping $A\colon E^{m}\to F$ by

$N(A)=\left\\{x\in E\,:\,|A|(|x|,\dots,|x|)=0\right\\}.$

Now if $P$ is the regular $m$-linear polynomial associated with $A$ then we
define the null ideal $N(P)$ of $P$ by

$N(P)=\left\\{x\in E\,:\,|P|(|x|)=0\right\\}=\left\\{x\in
E\,:\,|A|(|x|,\dots,|x|)=0\right\\}=N(A).$

Then the carrier $C(P)$ of $P$ and the carrier $C(A)$ of $A$ are defined by

$C(P)=N(P)^{\perp}=N(A)^{\perp}=C(A).$

Let us recall that if $P$ is a regular $m$-homogeneous polynomial on $C(K)$,
then it corresponds to a regular Borel measure $\mu$ on $K^{m}$. More
precisely, Fremlin [12] proved the following result, which readily extends to
regular $m$-homogeneous polynomials on $C(K)$: If $P$ is a regular
$2$-homogeneous polynomial on $C(K)$ then there exists a regular Borel measure
$\mu$ on $K^{2}$ such that

$P(x)=\int_{K^{2}}x(s)x(t)\,d\mu(s,t)$

for all $x\in C(K)$.

For the linear functional on $C(K^{m})$ defined by $\mu$, the the null ideal
of $\mu$ is given by

$N(\mu)=\bigl{\\{}x\in C(K^{m}):\int_{K^{m}}|x|\,d|\mu|=0\bigr{\\}}\,.$

We note that since $N(P)$ is a subset of $C(K)$ and $N(\mu)$ is a subset of
$C(K^{m})$, we cannot relate the null ideal of $P$ directly to the null ideal
of the corresponding measure $\mu$.

However, if $P$ is orthogonally additive then the null ideal of $P$ coincides
with null ideal of the corresponding measure $\mu$ as given by Theorem 1. We
note that in this case the null ideal of $P$ is given by

$N(P)=\left\\{x\in C(K):\int|x|^{m}\,d|\mu|=0\right\\}$

and the null ideal of $\mu$ is given by

$N(\mu)=\bigl{\\{}x\in C(K):\int_{K}|x|\,d|\mu|=0\bigr{\\}}\,.$

So clearly $N(P)=N(\mu)$.

Next let us recall that the support of a regular Borel measure $\mu$ on a
compact set $K$, denoted by $\operatorname{supp}\mu$, is the smallest closed
set $S$ such that $|\mu|(K\setminus S)=0$.

######  Proposition 9.

Let $P\colon C(K)\to\mathbb{R}$ be an orthogonally additive $m$-homogeneous
polynomial and $\mu$ the corresponding regular Borel measure on $K$. Then we
have

$N(P)=\left\\{x\in C(K):x=0\quad|\mu|{\rm-a.e}.\right\\}.$

Therefore,

$C(P)=\bigl{\\{}x\in
C(K):\operatorname{supp}{x}\subset\operatorname{supp}{\mu}\bigr{\\}}.$

###### Proof.

It suffices to note that

$\displaystyle N(P)$ $\displaystyle=\left\\{x\in C(K):|P|(|x|)=0\right\\}$
$\displaystyle=\left\\{x\in C(K):\int|x|^{m}(t)\,d|\mu|(t)=0\right\\}$
$\displaystyle=\left\\{x\in C(K):x=0\quad|\mu|{\rm-a.e.}\right\\}.$

We now have that

$C(P)=N(P)^{\perp}=\bigl{\\{}x\in
C(K):\operatorname{supp}{x}\subset\operatorname{supp}{\mu}\bigr{\\}}.$

∎

As we have noted earlier, $N(P)=N(\mu)$. We now observe that

$C(P)=N(P)^{\perp}=N(\mu)^{\perp}=C(\mu).$

We can now state a Nakano carrier theorem for orthogonally additive
polynomials on $C(K)$.

######  Theorem 6.

Let $P$ and $Q$ be orthogonally additive $m$-homogeneous polynomials on
$C(K)$. If $P$ is order continuous then $P\perp Q$ if and only if $C(P)\perp
C(Q)$.

###### Proof.

Let $\mu_{P}$ and $\mu_{Q}$ be the representing measures for $P$ and $Q$
respectively as given by Theorem 1. Recall that Theorem 5 and Proposition 7
give us that if $P$ is order continuous if and only if $\mu_{p}$ is normal.
Now by Theorem 2, we have $P\perp Q$ if and only if $\mu_{P}\perp\mu_{Q}$.
Nakano’s theorem for linear functionals gives us that $\mu_{P}\perp\mu_{Q}$ if
and only if $C(\mu_{P})\perp C(\mu_{Q})$. By definition $C(\mu_{P})=C(P)$ and
$C(\mu_{Q})=C(Q)$ and the result follows. ∎

To extend the above result to arbitrary Banach lattices, we need the following
lemma.

######  Lemma 4.

Let $E$, $F$ be Banach lattices, with $F$ Dedekind complete and let $P,Q\colon
E\to F$ be regular $m$-homogeneous polynomials. Then

$N(P_{a})=N(P)\cap E_{a}\quad\text{and}\quad C(P_{a})=C(P)\cap E_{a}\,,$

for every $a\in E_{+}$. In particular, $C(P)\perp C(Q)$ if and only if
$C(P_{a})\perp C(Q_{a})$ for every $a\in E_{+}$.

###### Proof.

Suppose that $x\in N(P_{a})$. Then $x\in E_{a}$. Now if we consider $x$ as an
element of $E$ and use Proposition 5 (d), we have

$|P|(|x|)=|P|_{a}(|x|)=|P_{a}|(|x|)=0$

Hence $x\in N(P)$ and so $N(P_{a})\subset N(P)\cap E_{a}$.

Next let $x\in N(P)\cap E_{a}$. Then

$|P_{a}|(|x|)=|P|_{a}(|x|)=|P|(|x|)=0$

Hence $x\in N(P_{a})$ and so $N(P)\cap E_{a}\subset N(P_{a})$.

For the second identity, suppose that $x\in C(P)\cap E_{a}$. Let $y\in
N(P_{a})$. Then $y\in N(P)$ and so $x\perp y$. That is, $x\in
N(P_{a})^{\perp}=C(P_{a})$.

To establish the reverse inclusion, suppose that $x\in C(P_{a})$ but $x\not\in
C(P)$. Then there exists $y\in N(P)$ such that $|x|\wedge|y|>0$. Since
$|x|\wedge|y|\leq|x|$, we have $|x|\wedge|y|\in E_{a}$. Hence

$|P_{a}|(|x|\wedge|y|)=|P|_{a}(|x|\wedge|y|)=|P|(|x|\wedge|y|)\leq|P|(|y|)=0$

and so $|x|\wedge|y|\in N(P_{a})$. We conclude that $x\perp(|x|\wedge|y|)$.
However, $|x|\wedge(|x|\wedge|y|)=|x|\wedge|y|\neq 0$, a contradiction.

Next we prove that $C(P)\perp C(Q)$ if and only if $C(P_{a})\perp C(Q_{a})$
for every $a\in E_{+}$. So suppose that $C(P)\perp C(Q)$. Since by the above
$C(P_{a})\subseteq C(P)$ and $C(Q_{a})\subseteq C(Q)$, we see that
$C(P_{a})\perp C(Q_{a})$.

Now assume that $C(P_{a})\perp C(Q_{a})$ for every $a$ in $E_{+}$. Then
$C(P)\cap E_{a}\perp C(Q)\cap E_{a}$ for every $a\in E_{+}$. Let $x\in C(P)$
and $y\in C(Q)$. Now take any $a>|x|\wedge|y|$. Then $x,y\in E_{a}$. So $x\in
C(P)\cap E_{a}=C(P_{a})$ and $y\in C(Q)\cap E_{a}=C(Q_{a})$. Hence, by our
assumption $x\perp y$ and so $C(P)\perp C(Q)$.

∎

Combining Lemma 4 and Theorem 6, we have the following Nakano carrier theorem
for orthogonally additive polynomials.

######  Theorem 7.

Let $P$ and $Q$ be orthogonally additive $m$-homogeneous polynomials on a
Banach lattice $E$. If $P$ is order continuous then $P\perp Q$ if and only if
$C(P)\perp C(Q)$.

###### Proof.

By Proposition 5 (d), we have $P\perp Q$ if and only if $P_{a}\perp Q_{a}$ for
every $a\in E_{+}$. By part (c) of Proposition 5 we also have that $P$ is
order continuous if and only if $P_{a}$ is order continuous for every $a\in
E_{+}$. Now Theorem 6 gives us that $P_{a}\perp Q_{a}$ if and only if
$C(P_{a})\perp C(Q_{a})$ for every $a\in E_{+}$. Finally Lemma 4 tells us that
$C(P_{a})\perp C(Q_{a})$ for every $a\in E_{+}$ if and only if $C(P)\perp
C(Q)$. ∎

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Christopher Boyd, School of Mathematics & Statistics, University College
Dublin, Belfield, Dublin 4, Ireland.
e-mail<EMAIL_ADDRESS>
Raymond A. Ryan, School of Mathematics, Statistics and Applied Mathematics,
National University of Ireland Galway, Ireland.
e-mail<EMAIL_ADDRESS>
Nina Snigireva, School of Mathematics & Statistics, University College Dublin,
Belfield, Dublin 4, Ireland.
e-mail<EMAIL_ADDRESS>