1

# Enrichment by supernovae in globular clusters with multiple populations

Jae-Woo Lee1 Young-Woon Kang1 Jina Lee1 & Young-Wook Lee2

###### Abstract

The most massive globular cluster in the Milky Way, $\omega$ Centauri, is
thought to be the remaining core of a disrupted dwarf galaxy[1, 2], as
expected within the model of hierarchical merging[3, 4]. It contains several
stellar populations having different heavy elemental abundances supplied by
supernovae[5] — a process known as metal enrichment. Although M22 appears to
be similar to $\omega$ Cen[6], other peculiar globular clusters do not[7, 8].
Therefore $\omega$ Cen and M22 are viewed as exceptional, and the presence of
chemical inhomogeneities in other clusters is seen as ‘pollution’ from the
intermediate-mass asymptotic-giant-branch stars expected in normal globular
clusters[9]. Here we report Ca abundances for seven globular clusters and
compare them to $\omega$ Cen. Calcium and other heavy elements can only be
supplied through numerous supernovae explosions of massive stars in these
stellar systems[10], but the gravitational potentials of the present-day
clusters cannot preserve most of the ejecta from such explosions[11]. We
conclude that these globular clusters, like $\omega$ Cen, are most probably
the relics of more massive primeval dwarf galaxies that merged and disrupted
to form the proto-Galaxy.

Department of Astronomy and Space Science, ARCSEC, Sejong University, Seoul
143-747, Korea

Center for Space Astrophysics, Yonsei University, Seoul 120-749, Korea

The Sejong/ARCSEC Ca uvby survey program was initiated in 2006 to investigate
the homogenous metallicity scale for globular clusters and to obtain the
complete metallicity distribution function of the Galactic bulge using the
$hk$ index [= $(Ca-b)-(b-y)$][12]. The Ca filter in the $hk$ index measures
ionized calcium H and K lines, which have been frequently used to calibrate
metallicity scale for globular clusters[13, 14]. The utility of the $hk$ index
is that it is known to be about three times more sensitive to metallicity than
the $m_{1}$ index is for stars more metal-poor than the Sun and it has half
the sensitivity of the $m_{1}$ index to interstellar reddening[12]. During the
last three years, we have used more than 85 nights of CTIO 1.0-m telescope
time for this project. The telescope was equipped with an STA 4k $\times$ 4k
CCD camera, providing a plate scale of 0.289 arcsec/pixel and a field of view
of 20 $\times$ 20 arcmin. All of our targets accompanied with standards were
observed under the photometric weather conditions and most of targets were
repeatedly visited between separate runs. The photometry of our targets and
standards were analyzed using DAOPHOT II, ALLSTAR, and ALLFRAME[15, 16].

In the course of metallicity calibration of red giant branch (RGB) stars in
GCs, we found that many GCs show split in the RGB in their $hk$ versus $V$
color-magnitude diagrams (Figs 1 and 2). The prime examples are M22 and
NGC1851. In particular, the double RGB sequence in M22 is very intriguing. The
differential reddening effect and the contamination from the off cluster
populations cannot explain the double RGB sequences in M22 (see Supplementary
Information). It has been debated for decades whether this cluster is
chemically inhomogeneous or not, but the recent high resolution spectroscopic
study of 17 RGB stars in the cluster suggests that it contains chemically
inhomogeneous subpopulations[6]. The bimodality in the $m_{1}$ index of M22
RGB stars was also known, but it has been argued that it is most likely due to
the bimodal CN abundances, where CN absorption strengths strongly affect the
$m_{1}$ index, not due to the bimodal distribution of heavy elements in the
cluster[17, 18, 19]. The star-to-star light elemental abundance (C, N, O, Na,
Mg and Al) variations have been known for decades and they are now generally
believed to be resulted from chemical pollutions by intermediate-mass
asymptotic giant branch stars[9] or fast rotating massive stars[20]. However,
it should be emphasized that our $hk$ measurements for RGB stars in M22,
NGC1851 and other GCs show discrete or bimodal distributions in calcium
abundance, which cannot be supplied by intermediate-mass asymptotic giant
branch stars or fast rotating massive stars.

As shown in Fig 3, the difference in calcium, silicon, titanium and iron
abundances between the calcium weak (Ca-w hereafter) group with smaller $hk$
index and the calcium strong (Ca-s hereafter) group with larger $hk$ index in
M22 and NGC1851 suggests that they are indeed chemically distinct[24, 22, 21,
23]. (It is not shown in the figure but europium also has a bimodal abundance
distribution in M22, in the sense that the Ca-s group has a higher europium
abundance.) As for the origin of chemical inhomogeneity in globular clusters,
at least four viable chemical enrichment mechanisms have been proposed up to
date. They are, in the order of time required to emerge; (i) fast rotating
massive stars, (ii) Type II supernovae, (iii) intermediate-mass asymptotic
giant branch stars, and (iv) Type Ia supernovae. If the current understanding
of supernovae explosions is correct, only Type Ia and II supernovae can supply
the heavy elements such as calcium and iron[10]. To explain the discrete
calcium abundances seen in M22 and NGC1851, however, the contribution from
Type Ia supernovae can be ruled out for two reasons. First, the longer
timescale ($\geq$ 1 – 2 Gyr) before the onset of Type Ia supernova explosions,
which would produce detectable age spread between two populations; and second,
the enhanced $\alpha$-elemental abundances, indicative of absence of
contributions from Type Ia supernovae[10]. Qualitatively, the differences in
elemental abundances between the two stellar populations in M22 and NGC1851
can be naturally explained by invoking chemical enrichment by Type II
supernovae, where $\alpha$-elements (silicon, calcium, and titanium) and
$r$-process element (europium) are dominantly produced. However, our results
do not necessarily imply that Type II supernovae are solely responsible for
the chemical enrichment in M22 and NGC1851, since all four above-mentioned
mechanisms may be involved. We emphasize that the crux of our results is the
undeniable evidence for Type II supernovae contribution to chemical enrichment
of some globular clusters, in sharp contrast to the widely accepted idea of
chemical pollution only by intermediate-mass asymptotic giant branch or fast
rotating massive stars, with which the chemical enhancement of the $\alpha$\-
and $r$-process elements in the second generation of the stars cannot be
easily explained.

More than half of 37 globular clusters in our sample shows discrete or broad
distributions of the $hk$ index in their RGB sequences. In Fig 2, We show
color-magnitude diagrams for some of exemplary globular clusters in the order
of $hk$ widths of RGB sequences at $V_{HB}$, the $V$ magnitude level at the
horizontal branch: $\omega$ Cen, M22, NGC1851, NGC2808, M4, M5, NGC6752 and
NGC6397 (see also Supplementary Table 3 and Figs 6 – 13). NGC2808 is known to
have multiple main-sequences but no multiple RGB sequences have been reported
to date. Our new results show that NGC2808 shows at least two discrete RGB
sequences with a large spread in calcium abundance. Similarly, M5 has very
broad $hk$ index in the RGB sequence and NGC6752 shows discrete RGB sequences.
It is interesting to note that all the globular clusters with signs of
multiple stellar populations have relatively extended horizontal branch, while
the globular clusters with normal horizontal branch (e.g. NGC6397 in Fig 2 and
Supplementary Fig 13) show no spread or split in RGB. This is consistent with
the suggestion that the extended horizontal branch is a signal of the presence
of multiple stellar populations in globular clusters[25].

The overwhelming problem of the chemical enrichment by Type II supernovae in
globular clusters is that their ejecta are considered to be too energetic to
be retained by less massive systems like typical Galactic globular clusters
($\leq$ a few times 105 $M_{\odot}$)[11]. Our results therefore suggest that
M22, NGC1851 and other globular clusters with RGB split were much more massive
in the past, unless the current understanding of supernovae explosions is in
great error. Perhaps, these globular clusters were once nuclei of dwarf-
galaxy-like fragments and then accreted and dissolved in the Milky Way, as is
widely accepted for $\omega$ Cen[1, 2, 26]. Recent calculations suggest that a
massive ($\geq$ a few times $10^{6}$ $M_{\odot}$) star cluster embedded in a
proto-dwarf galaxy could accrete gas from its host dwarf galaxy which may
cause the formation of the second generation stars, producing multiple stellar
populations[27]. Note that this scenario is also suggesting that the globular
clusters with multiple stellar populations would be the remaining cores of the
proto-galactic building blocks. This idea is supported by the recent
investigations of the extended horizontal branch globular clusters (i.e.
globular clusters with signatures of multiple stellar populations), which has
shown that extended horizontal branch globular clusters are clearly distinct
from the normal globular clusters in orbital kinematics and mass[25].
Extensive photometric surveys for fainter stars in these globular clusters, as
well as spectroscopic surveys for stars in double RGB sequences, would
undoubtedly help to shed more light into the discovery reported here.

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is linked to the online version of the paper at www.nature.com/nature.

J.-W. L. thanks A. Walker for providing the CTIO Ca filter transmission curve,
D. Yong for NGC1851 spectroscopic data before publication and A. Yushchenko
for discussions on spectrum synthesis. Support for this work was provided by
the National Research Foundation of Korea to the Astrophysical Research Center
for the Structure and Evolution of the Cosmos (ARCSEC). This work was based on
observations made with the CTIO 1.0-m telescope, which is operated by the
SMARTS consortium.

J.-W. L. performed observations, data analysis, interpretation, model
simulations and writing of the manuscript. Y.-W. K. participated in
observation planning, J. L. performed a part of observations and data
analysis. Y.-W. L. performed interpretation and writing of the manuscript. All
authors discussed the results and commented on the manuscript.

Reprints and permissions information is available at www.nature.com/reprints.
The authors declare that they have no competing financial interests.
Correspondence should be addressed to J.-W. L<EMAIL_ADDRESS>or
Y.-W. L. (ywlee2@yonsei.ac.kr).

Figure 1: Color-magnitude diagrams for M22. a, $V$ versus $b-y$; b, $V$ versus
$hk$. In b, note the distinct and discrete double RGB sequences in M22. This
cannot be due to differential reddening effect across the cluster or the
contamination from the off cluster field but, is due to the difference in
calcium abundance, which was synthesized in supernovae, between the two RGB
sequences. The number ratio between the Ca-w group with smaller $hk$ index and
the Ca-s group with larger $hk$ index is about 70:30. Black arrows in each
panel denote reddening vectors.  Figure 2: Color-magnitude diagrams for
$\omega$ Cen, M22, NGC1851, NGC2808, M4, M5, NGC6752 and NGC6397. Note that,
while the distributions of the RGB sequences in the $b-y$ color are relatively
narrow, those in the $hk$ index are either discrete or broad. This is evidence
for the multiple stellar populations with distinct calcium abundances. Among
these globular clusters, NGC6397 appears to be the only normal globular
cluster with simple population (i.e. coeval and monometallic).  Figure 3:
Differences in chemical compositions between double RGB sequences in M22 and
NGC1851. a, b, Black ‘plus’ signs denote stars in M22 with proper motion
membership probabilities $P$ $\geq$ 90%; blue filled diamonds and red filled
circles denote RGB stars studied with high-resolution spectroscopy in the Ca-w
and the Ca-s groups, respectively[23, 24]. The green solid line denotes the
fiducial sequence of RGB stars and $\Delta hk$ denotes the difference in the
$hk$ index against the fiducial sequence. The double RGB sequences persist in
proper motion member stars. c – f, Comparisons of elemental abundances between
the Ca-w and the Ca-s groups in M22. Solid lines denote the mean values, and
dashed lines denote standard deviations of each group. The Ca-s group has
higher $\alpha$-elements (Si, Ca, and Ti) and iron abundances, which must be
supplied by numerous Type II supernova explosions. g, h, Black ‘plus’ signs
denote stars in NGC1851; blue filled diamonds and red filled circles denote
RGB stars studied with high-resolution spectroscopy in the Ca-w and the Ca-s
groups, respectively[21, 22]. i – l, As c – f but for NGC1851.

Supplementary Information

##  1 The CTIO Ca Filter System

###  1.1 The Filter Transmission

The Ca filter system was designed to include Ca II H and K lines at $\lambda$
3968 and 3933 Å, respectively, with a full-width half maximum (FWHM) of
approximately 90 Å. The CN band absorption strengths at $\approx$ $\lambda$
3885 Å are often very strong in stellar spectra and the lower limit of the Ca
filter is set to avoid contamination by the CN band12. The Ca filter used at
Cerro Tololo Inter-American Observatory (CTIO) has a similar FWHM,
approximately 90 Å, but its passband is shifted approximately 15 Å to the
longer wavelength (Alistair Walker, private communication) compared to that in
Anthony-Twarog _et al._ 12 In Supplementary Figure 1, we show the transmission
functions of the both Ca filters. In the figure, we also show synthetic
spectra for the CN normal and the CN strong RGB stars in typical intermediate
metallicity globular clusters (GCs) as an illustration. The effect of the CN
band on the $hk$ index is negligible as will be discussed below.

###  1.2 The Central Wavelength Drift of the Ca Filter

The CTIO Ca filter transmission function shown in Supplementary Figure 1 is
that measured with a collimated beam. It is known that the passband of the
narrow band interference filter depends on the angle of the incidence beam
following,

$\lambda=\lambda_{0}\left(1-\frac{\sin^{2}\beta}{n^{*2}}\right)^{1/2},$ (1)

where $\lambda_{0}$ is the wavelength of peak transmittance at normal
incidence, $\beta$ is the angle of incidence of the collimated beam on the
filter and $n^{*}$ is the effective refractive index of the filter[28].
Therefore, when the Ca filter is used with a fast telescope, the filter
passband can be significantly different from that shown in Supplementary
Figure 1. The CTIO 1-m telescope used for our survey is a slow telescope with
$f$/10.5 and the effect resulted from the angular dependency of a converging
beam is expected to be very small. Assuming $\beta$ $\approx$ 1/21 radians for
the converging beam at the CTIO 1-m telescope and $n^{*}$ $\approx$ 1.4 for
the CTIO Ca filter, the peak wavelength of the Ca filter will be shifted by
2.3 Å to the shorter wavelength. Given the much larger FWHM of the CTIO Ca
filter, this may contribute small effect. We investigate contributions to the
$hk$ index resulted from the shifted Ca passband using synthetic spectra for
typical intermediate metallicity RGB stars in our GCs. Our calculations
integrating over the filter transmission curve show that this effect
contributes no larger than 0.011 mag to our $hk$ measurements, in the sense
that the shifted Ca passband to the shorter wavelength gives slightly larger
$hk$ values. We emphasize that, since our results are based on a single
instrument setup (the same telescope, filters and the CCD camera) during the
observations of our science targets and the photometric standards, this effect
is expected to be cancelled out during our photometric calibrations. Also
importantly, our main results presented here rely on the split or the spread
in the $hk$ index of RGB stars of an individual GC. Therefore, the shifted Ca
passband affects similar degree to the $hk$ index of the RGB stars in a GC and
does not contribute to the apparent RGB split or the spread in the $hk$ index
of an individual GC.

###  1.3 Effect of radial motions and internal velocity dispersions of GCs

The mean radial motion of 139 GCs in our Milky Way Galaxy[29] is $|v_{r}|$ =
110 km/s, equivalent to the wavelength shift by $|\Delta\lambda|$ $\approx$
1.4 Å at $\lambda$ 3950Å. We calculate the contribution due to the mean radial
motion of GCs to the $hk$ index using the shifted CTIO Ca passband and
synthetic spectra for typical intermediate metallicity GC RGB stars. We find
that the net effect is negligibly small, $|\Delta hk|$ $<$ 0.006 mag. Among
our eight GCs, NGC1851 has the largest radial velocity, $v_{r}$ = 321 km/s,
equivalent to the wavelength shift by $\Delta\lambda$ = 4.2 Å to the longer
wavelength at $\lambda$ 3950Å. We calculate the contribution due to the radial
motion of NGC1851 using the shifted CTIO Ca passband and the red-shifted
synthetic spectra with a fixed CN abundance for the cluster. We obtain $\Delta
hk$ $\approx$ 0.015 mag, in the sense that the red-shift is resulted in a
slightly larger $hk$ index. As we discussed above, the difference in the $hk$
index due to the high radial motion of NGC1851 does not affect our results
presented here, since the $hk$ indices of RGB stars in NGC1851 will be
affected by similar degree and the mean radial motion of the cluster does not
produce an apparent split or a spread in the $hk$ index. Perhaps, this effect
may become important in the inter-cluster comparisons, which is beyond the
scope of our study.

What concerns us most about the high radial velocities of some GCs, in
particular for red-shift, is the potential contamination by the strong CN band
absorption features at $\lambda$ 3885 Å as shown in Supplementary Figure 1.
For example, NGC1851 has a bimodal CN distribution and some RGB stars show
very strong CN band absorption strengths[30]. Due to its high radial velocity
away from us (i.e. red-shifted), the CN band absorption features in the CN-
strong RGB stars may affect the $hk$ index and, subsequently, may produce an
apparent RGB split of the cluster as shown in Figure 2 or Supplementary Figure
8. We calculate the CN band contributions using the shifted CTIO Ca passband
and the red-shifted synthetic spectra for the CN-normal and the CN-strong RGB
stars (see discussion below). Our calculations suggest that the net effect is
negligibly small, $\Delta hk$ $\leq$ 0.003 mag, and the high radial velocity
of NGC1851 combined with a bimodal CN distributions does not produce the RGB
split in the $hk$ index.

We also investigate the effect of the internal velocity dispersion of an
individual GC. Assuming $\sigma_{LOS}$ = 15 km/s, equivalent to
$\Delta\lambda$ $\leq$ 0.2 Å at $\lambda$ 3950Å, we obtain $\Delta hk$ $\leq$
0.001 mag following the same method described above, and the effect from the
internal velocity dispersions of GCs does not affect our results.

###  1.4 Summary of uncertainties on the $hk$ index

Supplementary Table 1 summarizes the uncertainties in our $hk$ index
measurements relevant to the CTIO Ca passband. (The variations in the $hk$
index due to differences in elemental abundances of GC RGB stars will be
discussed below.) As discussed above, the effects due to the shifted CTIO Ca
passband and the radial motions of GCs do not affect our results, since both
effects contribute similar degree to the $hk$ indices among RGB stars in a GC.
(i.e. They only affect the zero point of the $hk$ index and they do not affect
the $\Delta hk$ distributions). In addition to our photometric measurement
errors which will be discussed below, the effects due to the internal velocity
dispersions of GCs and the differential interstellar Ca II absorption (see
discussion below) can affect our $hk$ index measurements. However, their
contributions to our $hk$ measurements are no larger than 0.022 mag and they
do not affect our main conclusion presented here. Therefore, our results
strongly suggest that the split or the spread in the $hk$ index of RGB stars
in GCs are related to the variations in elemental, in particular calcium,
abundances among RGB stars in a GC, which will be discussed below.

##  2 The Double RGB Sequences of M22

###  2.1 Differential Reddening Effect on the Double RGB Sequences in M22

The continuous interstellar extinction by the interstellar dust and the
discrete interstellar line extinction by the interstellar Ca II atoms may
affect our main results. We considered both effects and will discuss that the
RGB split of M22 in the $hk$ index is indeed due to the difference in calcium
abundances between two stellar populations in M22 and other explanations are
highly unlikely. Also both effects tend to produce spreads in RGB sequences
rather than the distinct and discrete RGB sequences of GCs reported here.

The differential continuous reddening across the cluster can thicken the
apparent RGB sequence of GCs in broad-band optical photometry[31]. In contrast
to other color indices being used in broad-band photometry, the $hk$ index is
known to be insensitive to interstellar reddening12, $E(hk)/E(b-y)$ = $-$0.16
and $E(hk)/E(B-V)$ = $-$0.12. The difference in the $hk$ index between the two
RGB sequences in M22 is about 0.2 mag at the magnitude level of the horizontal
branch. If this $hk$ split is only due to differential reddening effect, we
would expect even larger separation of the two RGB sequences in the $b-y$
color and the $V$ magnitude. The reddening correction value in the $b-y$
color, $E(b-y)$, for the Ca-s group is about $-$1.25 mag, equivalent to
$E(B-V)$ = $-$1.69 mag assuming $E(b-y)/E(B-V)$ = 0.74, making the RGB stars
in the Ca-s group too hot to be RGB stars (see Supplementary Figure 2 – d). At
the same time, the extinction correction value in the $V$ magnitude is $-$5.24
mag, assuming $A_{V}$ = 3.1$\times E(B-V)$, for the Ca-s RGB stars. Applying
this large extinction correction makes the RGB stars in the Ca-s group too
bright to be members of M22 (see Supplementary Figure 2 – e & f). We emphasize
that both the Ca-w and the Ca-s groups are proper motion members of the
cluster as shown in Figure 3. Note also that the reddening vector (see Figure
1 or Supplementary Figure 4) is almost parallel to the slopes of HB and RGB in
the $hk$ versus $V$ CMD, and thus the differential reddening can not produce
the RGB split. Therefore, continuous differential reddening effect can be
completely ruled out to explain the observed bimodal RGB sequences in M22.
Similarly, the interstellar reddening toward NGC1851 is very small, $E(B-V)$ =
0.02 mag[29], but the $hk$ split in RGB stars of the cluster is as large as
0.2 mag (see Supplementary Figure 8), which can not be explained by
differential continuous reddening effect22.

The previous study for the GCs showed that the equivalent width of the
interstellar Ca II K absorption line strength can be as large as several times
100 mÅ[32]. The interstellar Ca II atom is thought to be heavily depleted on
to dust in denser clouds[33], which may cause small-scale differential
discrete reddening effect across M22 and other GCs studied here (see also
Andrew _et al._[34] for the small-scale variations of interstellar Na I D
lines111Note that the number of interstellar Na I atoms appears to be about a
factor of ten larger than that of interstellar Ca II atoms[33]. toward the
less extincted globular cluster M92). We generate synthetic spectrum to
surrogate interstellar Ca II H & K absorption lines. We adopt a gaussian line
profile with a FWHM of 1 Å, equivalent to $\Delta v_{r}$ $\approx$ 76 km/s,
and we assign equivalent widths of 350 mÅ and 650 mÅ for the interstellar Ca
II H & K lines, respectively. Our synthetic spectrum is shown in Supplementary
Figure 1 – (c). Assuming they are linear part of the curve of growth[35], the
column density of the interstellar Ca II can be estimated as

$N({\rm Ca~{}II})=1.13\times 10^{20}\frac{EW}{\lambda^{2}f},$ (2)

where $EW$ and $\lambda$ are the equivalent width and wavelength in Å and $f$
is the oscillator strength. Using the oscillator strengths of 0.681 and 0.341
for Ca II H & K, respectively, the column density for interstellar Ca II is
$\log N$(Ca II) $\approx$ 12.8 cm-2, equivalent to $\Delta E(B-V)$ $\approx$
0.32 mag[33]. If this large amount of small-scale interstellar Ca II variation
exists among our GCs studied here, how much will it affect our $hk$ index
measurements? We calculate the $Ca$ magnitudes with and without the
interstellar Ca II variations using the shifted CTIO Ca transmission function.
The difference in the $Ca$ magnitude (i.e. in the $hk$ index since the
interstellar Ca II H & K lines do not affect $b$ or $y$ passbands) is only
0.010 mag and, therefore, the differential discrete reddening effect due to
the variations in the interstellar Ca II abundances can be completely ruled
out to explain the GC RGB splits in the $hk$ index.

###  2.2 The Spatial Distributions

In Supplementary Figure 3, we show the spatial distributions of Ca-w and Ca-s
RGB stars in M22. As can be seen in the figure, each population does not show
any spatially patched features, supporting our results that differential
reddening is not responsible for the RGB split in M22.

###  2.3 Contamination from the Milky Way’s Bulge Population

M22 is located in the direction of the Milky Way’s bulge and the contamination
from the bulge population may affect our results. However, this is very
unlikely, since the proper motion member RGB stars show discrete double RGB
sequences as shown in Figure 3. In addition, the bulge RGB stars are located
farther from the Sun, more metal-rich and suffering from heavier interstellar
reddening than those in M22 are. In Supplementary Figure 4, we compare M22
CMDs with those of two bulge fields (NGC6528 and OGLEII - 12). As can be seen
in the figure, the RGB stars in the bulge are fainter and redder than those in
M22 are and the contamination from the Milky Way’s bulge population does not
affect our results.

###  2.4 Effects of Metal Contents and Helium Abundances on the M22 RGB

As shown in Figure 3, the stars in the Ca-s group are about 0.2 dex more
metal-rich than those in the Ca-w group. It is suspected that this large
metallicity spread may produce any detectable discrepancy in stellar
evolutionary sequences, in particular for RGB sequence, between two stellar
populations based on broad-band photometry. To explore metallicity effect on
the RGB sequence, we compare $BV$ CMD by Monaco _et al._[31] with the latest
$Y^{2}$ isochrones (Version 3, Yi _et al._ in preparation). In Supplementary
Figure 5, we show model isochrones for [Fe/H] = $-$1.6 and $-$1.4 with the
helium abundance of $Y$ = 0.23 and the age of 11 Gyr, using the reddening
value and the distance modulus for the cluster from Harris[29]. Although the
split in the RGB sequences of two model isochrones is noticeable, the
discrepancy in the RGB sequence does not appear to cause a serious problem to
explain the $BV$ CMD by Monaco _et al._[31] Note that the $V$ magnitude
difference in the sub-giant branch between two model isochrones can be as
large as 0.2 mag, apparently consistent with recent HST/ACS observations of
the cluster[36].

As inferred from the extended HB (EHB) morphology of M22, the second
generation of the stars is expected to have enhanced helium abundance by
$\Delta Y\approx$ 0.05[37]. Since the new version of $Y^{2}$ isochrones
provides models with enhanced helium abundances, we investigate the effect of
helium abundance on the evolutionary sequence. As illustrated in Supplementary
Figure 5 – (c), the discrepancy between two stellar populations alleviates due
to the opposite effect of metal contents and helium abundances on the RGB
temperature. Since the second generation of the stars in M22 shows signs of
the chemical enrichment by Type II supernovae and intermediate-mass asymptotic
giant branch (AGB) stars, the second generation of stars may be slightly
younger than the first generation. Assuming the age difference of 1 Gyr
between two stellar populations, two model isochrones are in excellent
agreement except for bright RGB sequence, where the observed number of stars
is small.

It is intriguing to note that the number ratio between the Ca-w and the Ca-s
RGB stars (70:30) found here is very similar to those found between (1) the
two stellar groups with different [Fe/H] and [$s$-process/Fe] ratios6, (2) the
brighter SGB and the fainter SGB stars[36], and (3) the two groups of HB stars
with the bluer HB being less populated. The population synthesis models (Han
_et al._ 2009, in preparation) suggest that this can be naturally reproduced
by the enhanced metal and helium abundances in the second generation of stars.

##  3 The _hk_ and Metallicity Distributions of GCs

###  3.1 Observations

In Supplementary Table 2, we show the journal of observations for eight GCs.
They were observed under the photometric weather conditions and, for most
cases, the median seeing was about 1.5 – 1.6 arcsec during our observations.
Note that the RGB stars in NGC2808 are roughly 21 times fainter than those in
NGC6397 in the Ca passband for a fixed magnitude, while our total integration
time for NGC2808 is only about three times longer than that for NGC6397 in the
Ca passband. Statistically, the lack of total integration time for NGC2808
will be resulted in a $\approx$ 2.6 times larger $Ca$ measurement error than
that expected for NGC6397 at a given $Ca$ magnitude, particularly for fainter
stars. Although our survey relied on a rather small telescope through a rather
narrow filter at a rather blue wavelength222Fortunately, the CCD camera used
in our survey has rather high quantum efficiency (QE) at shorter wavelength
with QE $\approx$ 0.686 at $\lambda$ 3800Å and $\approx$ 0.770 at $\lambda$
4000Å. (see http://www.astronomy.ohio-state.edu/Y4KCam/OSU4K/index.html#DQE).,
our investigations of the multiple stellar populations of GCs are focused on
bright RGB stars, where the numbers of photon in the Ca passband are enough so
that the measurement errors, including propagation errors during the
photometric calibrations, are less than 0.020 mag (see Supplementary Table 3).

###  3.2 Color Distributions of Bright RGB Stars

Here, we investigate the $b-y$ color and the $hk$ index distributions of RGB
stars brighter than $V-V_{\rm HB}$ = 1.0 mag. We derive lower order ($\approx$
4 – 5) polynomial fits, which are forced to pass through the peak $b-y$ colors
or the peak $hk$ indices of given magnitude bins, as fiducial sequences for
eight GCs and then we calculate differences in the $b-y$ color, $\Delta(b-y)$,
or the $hk$ index, $\Delta hk$, with respect to fiducial sequences of each GC.
We show our results in Supplementary Figures 6 – 13. In the figures, the blue
horizontal bars denote the mean measurement errors including propagation
errors during the photometric calibrations with a 2$\sigma_{*}$ range ($\pm$
1$\sigma_{*}$) for individual stars at given magnitude bins.

In Supplementary Table 3, we show comparisons of the observed FWHMs of RGB
stars and the measurement errors at the magnitude level of horizontal branch
stars, $V_{HB}+0.5$ $\geq$ $V$ $\geq$ $V_{HB}-0.5$, in each GC. To calculate
the FWHMs of RGB stars in GCs, we used the following relation,

${\rm FWHM(RGB)}\approx 2.3548\times\sigma_{\Delta},$ (3)

where $\sigma_{\Delta}$ is the standard deviation of RGB stars in the
$\Delta(b-y)$ or the $\Delta hk$ distributions in a fixed magnitude bin. Note
that the FWHM of RGB stars in the $hk$ index is slightly larger than the $hk$
separation between the double populations such as in M22, NGC1851, NGC2808, M4
and NGC6752. The mean measurement errors, $\sigma_{*}(b-y)$ or
$\sigma_{*}(hk)$, given in the table include propagation errors during the
photometric calibrations and they are those for individual stars. Therefore,
the mean measurement errors for an individual population in GCs,
$\sigma_{p}(b-y)$ or $\sigma_{p}(hk)$, will be given by
$\approx\sigma_{*}(b-y)$/$\sqrt{n_{p}}$ or $\sigma_{*}(hk)$/$\sqrt{n_{p}}$,
where $n_{p}$ ($\geq$ 20) is the number of stars in each population. Note
that, while the FWHMs of most GCs have much larger values [$\geq$
8$\sigma_{*}(hk)$] than the measurement errors for individual stars in the
$hk$ index, the FWHM of NGC6397 is comparable in size to the measurement error
in the $hk$ index, consistent with the idea that NGC6397 is the only normal GC
with a simple stellar population (i.e. coeval and monometallic) in our sample.
Also note that NGC6397 shows similar degree of the full RGB widths in the
$\Delta(b-y)$ and the $\Delta hk$ distributions (see Supplementary Table 3 and
Supplementary Figure 13).

The last two columns of Supplementary Table 3, $E(b-y)_{1/2}$ and
$E(hk)_{1/2}$, are for the contributions due to the continuous differential
reddening effect assuming a 50% variation in the total interstellar reddening
across each GC. The observed FWHMs of GCs in the $(b-y)$ color and in the $hk$
index can not be explained simultaneously, similar to what shown in
Supplementray Figure 2. Therefore, the continuous differential reddening
effect can be ruled out to explain the differences between the observed
FWHMs(RGB) and the measurement errors in GCs.

###  3.3 $\Delta hk$ as a probe of multiple stellar populations in GCs

For $\omega$ Cen, M22 and NGC1851 (see Figure 3 and Supplementary Figure 15),
when the two subpopulations are defined by our $hk$ index (or our $\Delta hk$
distribution), we can also see the clear division in spectroscopic elemental
abundances. Similarly, we will discuss that the split or the spread in the
$\Delta hk$ distributions of RGB stars in other clusters can provide a
powerful method to probe the multiple stellar populations in GCs.

The calcium abundance is the major factor that determines the $hk$ index or
the $\Delta hk$ distribution of RGB stars in a GC (see discussion below) and
Type II supernovae are responsible for the calcium enrichment in a GC. As
discussed, however, our results do not imply that Type II supernovae are
solely responsible for the chemical enrichment in GCs. In an attempt to
explain the observed large star-to-star lighter elemental abundance variations
(in particular O and Na) in GCs, chemical pollution by intermediate mass AGB
stars9 or fast rotating massive (FRM) stars20 has been widely accepted. It
should be reminded that, however, neither AGB nor FRM scenarios can explain
the chemical enrichment of the $\alpha$\- and $r$-process elements in the
second generation of the stars. It is most likely that all three
aforementioned mechanisms (and perhaps including Type Ia supernovae) are
required to explain elemental abundance patterns found in GCs. In addition to
the chemical enrichment by Type II supernovae, which is the main results
presented here, if the second generation of the stars in some of our GCs have
experienced the chemical pollution by intermediate-mass AGB or FRM stars, the
lighter elemental abundances, such as oxygen and sodium, between the two
generatrions of stars must have been different. Furthermore, the variations in
[O/Fe] and [Na/Fe] can be as large as 1 dex in some GCs7 and the differences
in the oxygen and sodium abundances are easily detectable compared to those in
the heavy elements, such as calcium and iron.

During the last few years, tremendous amount of effort has been directed at
spectroscopic study of RGB stars in GCs, in particular, using the multi-object
spectrograph mounted at VLT. Among our eight GCs, NGC2808[38], M4[39] and
NGC6752[40] have been studied using moderately high resolution spectra for
more than 100 RGB stars. In Supplementary Figure 14, we show comparisons of
$\Delta hk$ versus O, Na and Fe distributions of the clusters. In panel (a),
we show the plot of $V-V_{\rm HB}$ versus $\Delta hk$ for NGC2808 RGB stars.
In the figure, the plus signs denote the RGB stars with known [O/Fe] and
[Na/Fe] ratios[38]. From the $\Delta hk$ distribution of RGB stars shown in
panel (b), we define the boundary at $\Delta hk$ = $-$0.05 mag (the vertical
dashed line) assuming that NGC2808 has two major stellar populations as shown
in panel (b) or Supplementary Figure 9. Similar to the procedure employed in
M22 and NGC1851 (see Figure 3), we define the Ca-w group with smaller $hk$
index and the Ca-s group with larger $hk$ index and they are denoted by the
blue and the red plus signs, respectively, in panel (a). In panels (c), (d)
and (e), we show the [O/Fe], [Na/Fe] and [Fe/H] distributions for each group,
where the shaded histograms outlined with blue color are for the Ca-w group
and the blank histrograms outlined with red color are for the Ca-s group. The
Ca-w group has a higher mean oxygen and a lower mean sodium abundances, while
the Ca-s group has a lower mean oxygen and a higher mean sodium abundances,
indicative of the presence of the proton-capture process at high temperature
between the two formation epochs presumably via intermediate-mass AGB or FRM
stars, where oxygen is depleted by the CNO cycle while sodium is enriched from
the 22Ne + 1H $\rightarrow$ 23Na reaction. Our results strongly suggest that
they are truly different stellar populations and not related to, for example,
our photometric measurement errors and differential reddening effect: the Ca-w
group is the first generation of stars while the Ca-s group is the second
generation of stars enriched by Type II supernovae (e.g. calcium) and
intermediate-mass AGB or FRM stars (e.g. sodium). Although the difference in
the [Fe/H] distributions between the two groups does not appear to be as
compelling as those in the [O/Fe] and the [Na/Fe] distributions, the Ca-w
group has a slightly lower mean metallicity than the Ca-s group does. We
performed non-parametric Kolmogorov-Smirnov (K-S) tests to see if the [Fe/H]
distributions of the two populations in NGC2808 are drawn from the same parent
population. Our calculation shows that the probability of being drawn from
identical stellar populations is 5.5% for NGC2808, suggesting that they have
different parent populations.

The same results can be found in M4 and NGC6752. From the comparisons of the
[O/Fe] and the [Na/Fe] distributions between the Ca-w and the Ca-s groups, it
can be seen that the Ca-w groups are the first generations of stars while the
Ca-s groups are the second generations of stars in the clusters. We also
performed K-S tests for the [Fe/H] distribution of M4, we obtained that the
probability of being drawn from identical parent populations is 5.5% for M4,
indicating that each subpopulation in M4 has different parent populations.

###  3.4 Recalibration of [Fe/H]hk Based on RGB Stars in $\omega$ Cen and
Metallicity Distributions of Eight GCs

In our previous study for NGC1851, we showed that the $hk$ index traces the
calcium abundance and, furthermore, it can provide a very powerful method to
distinguish multiple stellar populations in GCs22. However, it can be seen
that the full range of $\Delta hk$ increases with the luminosity of RGB stars
(i.e. different temperature or surface gravity), in particular, in $\omega$
Cen and M22. Due to the temperature dependency on the $hk$ index versus
metallicity relation, the $\Delta hk$ distributions cannot be directly
translated into the absolute metallicity scale. Therefore, we calculate the
photometric metallicity, [Fe/H]hk, of individual RGB stars in eight GCs using
the [Fe/H] relations on the $hk_{0}$ versus $(b-y)_{0}$ plane12,22.

Recently, Johnson _et al._ 5 studied elemental abundances, including calcium,
of large sample of RGB stars in $\omega$ Cen using moderately high resolution
spectra (R $\approx$ 18,000). Since $\omega$ Cen contains multiple stellar
populations with very broad metallicity range, $\Delta$[Fe/H] $\approx$ 1.5
dex, comparisons of our results of RGB stars in $\omega$ Cen with those of
Johnson _et al._ may provide an wonderful opportunity to assess our
photometric metallicity scale using the $hk$ index, [Fe/H]hk. In Supplementary
Figure 15, we show elemental abundances of 40 RGB stars in $\omega$ Cen
studied by Johnson _et al._ as a function of $\Delta hk$. As shown in the
figure, [Ca/H] and [Fe/H] appear to be well correlated with $\Delta hk$,
indicating that $\Delta hk$ can truly be treated as the relative calcium
abundance or metallicity indicators for RGB stars with similar luminosities in
a GC. We also show plots of [Fe/H]spec versus [Fe/H]hk and [Ca/H]spec versus
[Fe/H]hk for 32 RGB stars with sufficiently high signal-to-noise ratios
($\geq$ 100). We derive linear fits to each relation and we find

$\rm{[Fe/H]}_{\rm
spec}=0.533\rm{[Fe/H]}_{hk}-0.775~{}~{}~{}~{}~{}~{}(\sigma=0.087\rm{dex}),$
(4)

and

$\rm{[Ca/H]}_{\rm
spec}=0.587\rm{[Fe/H]}_{hk}-0.403~{}~{}~{}~{}~{}~{}(\sigma=0.106\rm{dex}).$
(5)

We recalibrate our photometric metallicity using the equation (4),
[Fe/H]hk,corr, and we derive metallicity distribution functions (MDFs) for
eight GCs. During our calculations of MDFs, we use RGB stars with $-$2.0
$\leq$ $V$ $-$ $V_{\rm HB}$ $\leq$ $-$0.5 mag in order to minimize
contamination from off-cluster field and red-clump populations. We show our
results in Supplementary Figure 15. As expected from the $\Delta hk$
distributions, the signs of multiple stellar populations persist in our MDFs
for most GCs.

Finally, cautions are advisable on our MDFs of GCs. Our metallicity scale is
not on the traditional Zinn & West14 scale, therefore our MDFs for GCs can be
different from those from other photometric or spectroscopic studies. RGB
stars in $\omega$ Cen of Johnson _et al._ have different individual elemental
abundances, which were not taken into consideration in our [Fe/H]spec versus
[Fe/H]hk or [Ca/H]spec versus [Fe/H]hk relations. Furthermore, each GCs may
have slightly different elemental abundance ratios and our calibrated
photometric metallicities for GCs would be affected. However, it should be
emphasized that the crux of our results is the split or the spread in the $hk$
index in the RGB stars of individual GCs, which is insensitive to other
elemental abundances except calcium as will be discussed below.

##  4 The Influence of Elemental Abundances on the _hk_ Index

The realistic modeling of the resonance Ca II H & K lines requires proper
understanding of stellar atmospheres, including chromospheres, and
hydrodynamic non-local thermodynamic equilibrium treatments, which have posed
difficult problems for decades[41]. Here, we demonstrate that the calcium
abundance is the major factor that determines the $hk$ index of RGB stars
using 1-dimensional plane-parallel stellar atmospheres[42].

###  4.1 Calcium

Using the model atmosphere for the RGB star at the magnitude level of the
horizontal branch with $T_{\rm eff}$ = 4750 K, $\log g$ = 2.0 (in cgs unit),
$v_{\rm turb}$ = 2.0 km/s, [Fe/H] = $-$1.6, we calculate synthetic spectra for
[Ca/Fe] = 0.25, 0.30, 0.35, 0.40, 0.45 and we show some of our synthetic
spectra in Supplementary Figure 17. We convolve the filter transmission
functions with synthetic spectra and we obtain the calcium abundance
sensitivity on the $hk$ index, $\partial(hk)$/$\partial$[Ca/H] $\approx$ 0.422
mag/dex. Note that this result is based on the fixed model parameters, such as
$T_{\rm eff}$, $\log g$, $v_{\rm turb}$, and [Fe/H], except calcium abundance.
To interpret observed $\Delta hk$ between the two stellar populations in a GC
in terms of different calcium abundances, proper atmospheric parameters should
be taken into consideration. For example, the two stellar groups with
different [Fe/H] and [$s$-process/Fe] ratios in M22 by Marino _et al._ 6 have
slightly different elemental abundances and temperatures. The stars in the
metal-poor group by Marino _et al._ have $\langle$[Fe/H]$\rangle$ = $-$1.82,
$\langle$[Ca/Fe]$\rangle$ = +0.25 and those in the metal-rich group have
$\langle$[Fe/H]$\rangle$ = $-$1.68, $\langle$[Ca/Fe]$\rangle$ = +0.35. The
mean temperature of the stars in the metal-poor group is $\sim$ 100 $\pm$ 42 K
hotter than those in the metal-rich group. Using these atmospheric parameters,
we obtain $\Delta hk$ = 0.121 $\pm$ 0.072 mag, which is apparently consistent
with the double peaks in the $\Delta hk$ distribution of M22 within the error
as shown in Supplementary Figure 7.

###  4.2 Helium

As shown in Supplementary Figure 5, helium is very important in stellar
structure and evolution. Very unfortunately, however, there is no direct
method to measure helium abundances of stars in GCs. As we discussed, all the
GCs with signs of multiple stellar populations have relatively EHB, for
example, the second generation of the stars in M22 is expected to have
enhanced helium abundance by $\Delta Y$ $\approx$ 0.05 inferred from its EHB
morphology. Using the model atmospheres with enhanced helium abundance ($Y$
$\approx$ 0.35, equivalent to $\Delta Y$ $\approx$ 0.10) by
Castelli333http://wwwuser.oat.ts.astro.it/castelli/grids.html, we obtain
$\Delta hk$ $\approx$ $-$0.002 mag, in the sense that the $hk$ index decreases
as helium abundance increases, and thus the effect of enhanced helium
abundance on the $hk$ index is negligible. Given the cool temperatures of RGB
stars in GCs, the helium enhancement by $\Delta Y$ = 0.05 – 0.10 does not
appear to be important.

###  4.3 CNO

It is well-known fact that many GCs show large star-to-star elemental
abundance variations. In particular, almost all GCs show variations in the CNO
abundances resulted from the internal evolutionary mixing accompanied with the
CNO-cycle or the primordial pollution by intermediate-mass AGB stars to the
second generation of the stars[43, 44]. In spite of their high abundances, the
CNO abundances are hard to measure in the optical wavelength mainly due to the
lack of atomic transitions. On the other hand, in the form of molecules, the
CNO can affect the $hk$ index, in particular the CN band at $\lambda$ 3885 Å
as shown in Supplementary Figure 1. Both carbon and nitrogen contribute in the
formation of CN molecules. The typical RGB stars in GCs show an
anticorrelation between the CN band and the CH band strengths and a
correlation between the CN band and the NH band strengths, indicating that the
nitrogen controls the CN band strength[45]. Our results suggest that the
variations in the CNO abundances do not affect the $hk$ index significantly.
We obtain $\Delta hk$ $\approx$ $-$0.007, +0.002, $-$0.004 for
$\Delta$[C,N,O/Fe] = +1.0 dex, respectively, and their influence on the $hk$
index appears to be negligible.

###  4.4 Aluminium

It is also well-known fact that many GCs show large star-to-star aluminium
variations by more than $\Delta$[Al/Fe] $\approx$ 1.0 dex, presumably resulted
from the proton-capture process at high temperature or the primordial
pollution by intermediate-mass AGB stars to the second generation of the
stars[46, 44]. The resonance lines of Al I at $\lambda$ 3944.01 and 3961.52 Å
are often very strong (see Supplementary Figure 17) and it may affect our
conclusions that the $hk$ index traces calcium abundances of RGB stars in GCs.
We obtain $\Delta hk$ $\approx$ 0.013 mag for $\Delta$[Al/Fe] = +1.0 dex. The
effect of the variations in aluminium abundances on the $hk$ index is
insignificant compared to our observations, by more than a factor of ten. The
insignificant influence of aluminium on the Ca II H & K lines was also
confirmed by others for the globular cluster NGC675217.

###  4.5 $\alpha$-elements

The $\alpha$-elements (O, Ne, Mg, Si, S, Ar, Ca, and Ti) are quite abundant
and they are major donors to the H- opacity in RGB stars. However, their
influence, except for Ca, on the $hk$ index appears to be small. We obtained
$\Delta hk$ $\approx$ $-$0.008 mag for +0.3 dex variation in $\alpha$-elements
excluding calcium.

###  4.6 _s_ -process elements

RGB stars in GCs show large star-to-star $s$-process elemental abundance
variations presumably resulted from the primordial pollution by intermediate-
mass AGB stars to the second generation of the stars9,[44]. For example,
globular clusters M22 and NGC1851 show bimodal $s$-process elemental abundance
distributions with $\Delta$[$s$-process/Fe] $\geq$ 0.5 dex. We obtained
$\Delta hk$ $\approx$ +0.008 mag for $\Delta$[$s$-process/Fe] = +0.5 dex, and
thus their influence on the $hk$ index is small.

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1.5

|  Uncertainty on the $hk$ index | Note
---|---|---
Photometry |  $\leq$ 0.020 mag | random
Shifted CTIO $Ca$ passband |  $<$ 0.011 mag | systematic
Mean radial motion of GCs |  $<$ 0.006 mag | systematic
Internal velocity dispersion |  $<$ 0.001 mag | random
Interstellar Ca II absorption |  $<$ 0.010 mag | random
total |  $\leq$ 0.024 mag |
total (random) |  $\leq$ 0.022 mag |
Supplementary Table 1: Summary of uncertainties relevant to the $hk$ index.

ID | $V_{HB}$ | $E(B-V)$ | Exposure Time (sec) |  | Obs. Pos. | Date
---|---|---|---|---|---|---
|  |  | $Ca$ | $u$ | $v$ | $b$ | $y$ |  | RA | DEC | (MM/YY)
$\omega$ Cen | 14.53 | 0.12 | 12,860 | …… | 4,890 | 2,130 | 1,300 |  | 13:26:44 |  $-$47:26:28 | 05/07, 02/08
M22 | 14.15 | 0.34 | 8,100 | 2,400 | 1,200 | 2,530 | 1,500 |  | 18:36:29 |  $-$23:55:34 | 07/08, 08/08
NGC1851 | 16.09 | 0.02 | 19,100 | 12,300 | 7,400 | 7,100 | 3,810 |  | 5:14:14 |  $-$40:01:49 | 02/08, 08/08
NGC2808 | 16.22 | 0.22 | 10,820 | …… | 3,600 | 4,960 | 3,080 |  | 9:11:57 |  $-$64:49:24 | 05/07
M4 | 13.45 | 0.36 | 8,400 | 5,400 | 5,570 | 2,920 | 2,060 |  | 16:23:33 |  $-$26:30:47 | 05/07, 08/08
M5 | 15.07 | 0.03 | 9,660 | 3,900 | 2,100 | 4,010 | 2,390 |  | 15:18:29 | 2:04:03 | 05/07, 08/08
NGC6752 | 13.70 | 0.04 | 7,500 | 2,400 | 1,800 | 2,400 | 1,200 |  | 19:10:57 |  $-$60:00:20 | 07/08, 08/08
NGC6397 | 12.87 | 0.18 | 3,560 | 3,560 | 2,140 | 1,355 | 930 |  | 17:40:52 |  $-$53:36:06 | 08/06, 09/07
Supplementary Table 2: Journal of observations for eight GCs. Only one field
has been observed for a particular GC and the coordinates are given in columns
(9) and (10).

ID | FWHM(RGB) |  | Measurement errors |  | Differential Reddening
---|---|---|---|---|---
| $(b-y)$ | $hk$ |  | $\sigma_{*}(b-y)$ | $\sigma_{*}(hk)$ |  | $E(b-y)_{1/2}$ | $E(hk)_{1/2}$
$\omega$ Cen | 0.079 | 0.534 |  | 0.012 | 0.020 |  | 0.089 | 0.014
M22 | 0.050 | 0.216 |  | 0.004 | 0.008 |  | 0.252 | 0.041
NGC1851 | 0.035 | 0.182 |  | 0.006 | 0.013 |  | 0.015 | 0.002
NGC2808 | 0.042 | 0.159 |  | 0.008 | 0.019 |  | 0.163 | 0.026
M4 | 0.037 | 0.119 |  | 0.005 | 0.010 |  | 0.266 | 0.043
M5 | 0.025 | 0.105 |  | 0.006 | 0.013 |  | 0.022 | 0.004
NGC6752 | 0.022 | 0.090 |  | 0.004 | 0.007 |  | 0.030 | 0.005
NGC6397 | 0.024 | 0.034 |  | 0.007 | 0.012 |  | 0.133 | 0.022
Supplementary Table 3: Comparisons of the observed FWHMs of RGB stars and the
measurement errors at the magnitude level of horizontal branch stars,
$V_{HB}+0.5$ $\geq$ $V$ $\geq$ $V_{HB}-0.5$. The measurement errors are those
for individual stars and, therefore, measurement errors for individual
subpopulation in GCs will be given by $\approx\sigma_{*}(b-y)$/$\sqrt{n_{p}}$
or $\sigma_{*}(hk)$/$\sqrt{n_{p}}$, where $n_{p}$ ($\geq$ 20) is the number of
stars in each subpopulation in GCs. Note that, while the FWHMs of most GCs
have much larger values [$\geq$ 8$\sigma_{*}(hk)$] than the measurement errors
for individual stars in the $hk$ index, the FWHM of NGC6397 is comparable in
size to the measurement error in the $hk$ index, consistent with the idea that
NGC6397 is the only normal GC in our sample (see Supplementary Figures 6 –13).
The last two columns, $E(b-y)_{1/2}$ and $E(hk)_{1/2}$, denote contributions
due to the differential reddening effect assuming a 50% variation in the total
interstellar reddening of each GC, with which observed FWHMs of GCs in the
$(b-y)$ color and in the $hk$ index can not be explained simultaneously.

1

Supplementary Figure 1: (a) A comparison of $Ca$ filter transmission functions
between that in Anthony-Twarog _et al._ 12 (the black line) and that for the
CTIO 1-m telescope (the blue line). Both filters have similar FWHMs,
approximately 90 Å, but the passband for the CTIO 1-m telescope is shifted
approximately 15 Å to the longer wavelength. (b) Synthetic spectra for the CN
normal (the blue line) and the CN strong (the red line) RGB stars. The CN band
at $\lambda$ 3885 Å lies on the lower tail of the $Ca$ filter but the
contamination from the CN band is insignificant. (c) Synthetic spectra for the
interstellar Ca II $H$ & $K$ lines. We adopt equivalent widths of 350 mÅ and
650 mÅ for the interstellar Ca II $H$ & $K$ lines, respectively, with a
gaussian line profile with a FWHM of 1 Å (equivalent to $\Delta v_{r}$
$\approx$ 76 km/s). In the inset of the figure, the red line denotes the
velocity profile for the interstellar Ca II $H$ line and the blue line for the
interstellar Ca II $K$ line. This large amount of discrete interstellar
absorption contributes only 0.010 mag to our results.

Supplementary Figure 2: (a & b) Blue crosses and red crosses denote RGB stars
in the Ca-w and the Ca-s groups, respectively, with proper motion membership
probabilities $P$ $\geq$ 90%. (c) RGB stars in the Ca-s group are shifted by
$\Delta hk$ = $-$0.20 mag to match with those in the Ca-w group, assuming the
RGB split in M22 is due to differential reddening. The reddening correction
value of $\Delta hk$ = $-$0.20 mag for the Ca-s group is equivalent to
$E(B-V)$ = $-$1.69. (d) The RGB stars in the Ca-s group are shifted by
$\Delta(b-y)$ = $-$1.25 mag, assuming $E(b-y)/E(B-V)$ = 0.74, and two RGB
sequences do not agree, in the sense that RGB stars in the Ca-s group is too
hot to be in the RGB phase. (e & f) After applying reddening correction in $V$
(= $-$5.24 mag), assuming $A_{V}$ = 3.1$\times E(B-V)$. The RGB stars in the
Ca-s group become too bright to be members of M22, inconsistent with the
proper motion study of the cluster.

Supplementary Figure 3: Spatial distribution of the Ca-w (blue dots) and the
Ca-s (red dots) groups in M22. Note the absence of spatially patched features,
indicating that differential reddening is not responsible for the RGB split in
M22.

Supplementary Figure 4: Color-magnitude diagrams for M22 and two bulge fields
(NGC6528 and OGLEII-12). The black dots represent M22, the red dots and the
blue dots denote NGC6528 and OGLEII-12, respectively. The stars in the Milky
Way bulge are fainter and redder than those in M22 are and they do not affect
the double RGB sequences in M22. Black arrows indicate reddening vectors,
assuming $E(B-V)$ = 0.34 for M22[29].

Supplementary Figure 5: (a) The color-magnitude diagram by Monaco _et al._[31]
(b) Model isochrones for [Fe/H] = $-$1.6 (a blue line), and $-$1.4 (a red
line), $Y$ = 0.23, and 11 Gyr. (c) Model isochrones for [Fe/H] = $-$1.6,
$Y$=0.23, 11 Gyr (a blue line) and [Fe/H] = $-$1.4, $Y$=0.28, 11 Gyr (a red
line). (d) Model isochrones for [Fe/H] = $-$1.6, $Y$=0.23, 11 Gyr (a blue
line) and [Fe/H] = $-$1.4, $Y$=0.28, 10 Gyr (a red line).

Supplementary Figure 6: The $(b-y)$ and $hk$ distributions of RGB stars in
$\omega$ Cen. The red lines denote fiducial sequences of the cluster, which
are forced to pass through the peak $(b-y)$ colors or $hk$ indices of given
magnitude bins. The $\Delta(b-y)$ and the $\Delta hk$ are differences in the
$(b-y)$ color and the $hk$ index, respectively, of each RGB stars from the
fiducial sequences. The blue horizontal bars indicate measurement errors with
a 2$\sigma_{*}$ range ($\pm$ 1$\sigma_{*}$) of individual stars at given
magnitude bins. From the $hk$ distribution, at least five distinct
populations, whose $hk$ splits are much larger than the measurement errors,
can be found in $\omega$ Cen.

Supplementary Figure 7: The $(b-y)$ and $hk$ distributions of RGB stars in
M22. The blue horizontal bars indicate measurement errors with a 2$\sigma_{*}$
range ($\pm$ 1$\sigma_{*}$) of individual stars at given magnitude bins. Two
distinct and discrete populations can be found in M22. At the magnitude of HB,
the $hk$ split between two populations is larger than 25$\times\sigma_{*}(hk)$
or 250$\times\sigma_{p}(hk)$, where $\sigma_{*}(hk)$ and $\sigma_{p}(hk)$
denote measurement errors for individual stars and populations in the $hk$
index, respectively.

Supplementary Figure 8: The $(b-y)$ and $hk$ distributions of RGB stars in
NGC1851. The blue horizontal bars indicate measurement errors with a
2$\sigma_{*}$ range ($\pm$ 1$\sigma_{*}$) of individual stars at given
magnitude bins. Two discrete populations can be found in NGC1851. At the
magnitude of HB, the $hk$ split between two populations is larger than
11$\times\sigma_{*}(hk)$ or 55$\times\sigma_{p}(hk)$.

Supplementary Figure 9: The $(b-y)$ and $hk$ distributions of RGB stars in
NGC2808. The blue horizontal bars indicate measurement errors with a
2$\sigma_{*}$ range ($\pm$ 1$\sigma_{*}$) of individual stars at given
magnitude bins. At least two discrete populations can be found in NGC2808. At
the magnitude of HB, the $hk$ split between two major populations is larger
than 5$\times\sigma_{*}(hk)$ or 50$\times\sigma_{p}(hk)$.

Supplementary Figure 10: The $(b-y)$ and $hk$ distributions of RGB stars in
M4. The blue horizontal bars indicate measurement errors with a 2$\sigma_{*}$
range ($\pm$ 1$\sigma_{*}$) of individual stars at given magnitude bins. Two
discrete populations can be found in M4. At the magnitude of HB, the $hk$
split between two populations is larger than 10$\times\sigma_{*}(hk)$ or
45$\times\sigma_{p}(hk)$.

Supplementary Figure 11: The $(b-y)$ and $hk$ distributions of RGB stars in
M5. The blue horizontal bars indicate measurement errors with a 2$\sigma_{*}$
range ($\pm$ 1$\sigma_{*}$) of individual stars at given magnitude bins. The
RGB sequence of the cluster shows a large spread in the $hk$ index, indicative
of heterogeneous calcium abundances. At the magnitude of HB, the FWHM of RGB
stars in M5 is larger than 8$\times\sigma_{*}(hk)$.

Supplementary Figure 12: The $(b-y)$ and $hk$ distributions of RGB stars in
NGC6752. The blue horizontal bars indicate measurement errors with a
2$\sigma_{*}$ range ($\pm$ 1$\sigma_{*}$) of individual stars at given
magnitude bins. Two discrete populations can be found in NGC6752. At the
magnitude of HB, the $hk$ split between two populations is larger than
10$\times\sigma_{*}(hk)$ or 70$\times\sigma_{p}(hk)$.

Supplementary Figure 13: The $(b-y)$ and $hk$ distributions of RGB stars in
NGC6397. The blue horizontal bars indicate measurement errors with a
2$\sigma_{*}$ range ($\pm$ 1$\sigma_{*}$) of individual stars at given
magnitude bins. It is the only normal GC in Figure 2. Note the similar degree
of the RGB widths in $\Delta(b-y)$ and $\Delta hk$ and the similar degree of
RGB widths as the measurement errors.

Supplementary Figure 14: (a) A plot of $V-V_{\rm HB}$ versus $\Delta hk$ for
RGB stars in NGC2808. The blue and red plus signs denote the Ca-w and the Ca-s
RGB stars. The dashed line denotes the boundary between the two groups at
$\Delta hk$ = $-$0.05 mag. (b) The $\Delta hk$ distribution of NGC2808 RGB
stars. (c) The [O/Fe] distributions of the two RGB populations in NGC2808. The
shaded histogram outlined with blue color is for the Ca-w group and the blank
histrogram outlined with red color is for the Ca-s group. (d) The [Na/Fe]
distributions. (e) The [Fe/H] distributions. (f–j) Same as (a–e), but for M4
RGB stars with the boundary at $\Delta hk$ = $-$0.08 mag. (k–o) Same as (a–e),
but for NGC6752 RGB stars with the boundary at $\Delta hk$ = $-$0.04 mag.

Supplementary Figure 15: (a) A plot of $V-V_{HB}$ versus $\Delta hk$ for 40
RGB stars in $\omega$ Cen of Johnson et al.5 (b – e) Elemental abundances of
40 RGB stars in $\omega$ Cen as functions of $\Delta hk$. (f – g) Comparisons
of our photometric metallicity, [Fe/H]hk, with spectroscopic metallicity,
[Fe/H]spec, and calcium abundance, [Ca/H]spec. The linear fits to the data are
shown with red lines.

Supplementary Figure 16: Metallicity distribution functions for eight GCs
derived from the $hk$ index. In the figure, [Fe/H]hk,corr is our recalibrated
photometric metallicity using the equation (4). Note that we only use bright
RGB stars in order to minimize contamination from off-cluster field and red-
clump populations. For most GCs, signs of multiple stellar populations persist
in our MDFs.

Supplementary Figure 17: Comparisons of synthetic spectra for [Fe/H] = $-$1.6,
Teff = 4750 K, $\log$ g = 2.0. (Upper panel) The red line denotes synthetic
spectrum for [Ca/Fe] = 0.25 dex and the blue line denotes that for [Ca/Fe] =
0.45 dex. We adopt the fixed aluminium abundance of [Al/Fe] = 0.50 dex. (Lower
panel) The red line denotes the synthetic spectrum for [Al/Fe] = 0.00 dex and
the blue line denotes that for [Al/Fe] = 1.00 dex. We adopt the fixed calcium
abundance of [Ca/Fe] = 0.30 dex. The effect of aluminium contamination on the
$hk$ index appears to be negligible.


# On eigenvalues of symmetric matrices with PSD principal submatrices

Khazhgali Kozhasov  Khazhgali Kozhasov, Technische Universität Braunschweig,
Institut für Analysis und Algebra, Universitätsplatz 2, 38106 Braunschweig,
Germany<EMAIL_ADDRESS>
###### 2010 Mathematics Subject Classification:

14P10, 15A18, 15B57, 90C22

  * Abstract. We investigate convexity properties of the set of eigenvalue tuples of $n\times n$ real symmetric matrices, whose all $k\times k$ (where $k\leq n$ is fixed) minors are positive semidefinite. It is proven that the set $\lambda({\mathcal{S}}^{n,k})$ of eigenvalue vectors of all such matrices is star-shaped with respect to the nonnegative orthant $\mathbb{R}^{n}_{\geq 0}$ and not convex already when $(n,k)=(4,2)$.

##  1\. Introduction and statement of results

Let $\operatorname{Sym}_{n}$ denote the space of $n\times n$ real symmetric
matrices. A matrix $X\in\operatorname{Sym}_{n}$ is called _positive
semidefinite_ (or, simply, PSD), if $\bm{v}^{T}X\bm{v}\geq 0$ holds for all
column vectors $\bm{v}\in\mathbb{R}^{n}$. Equivalently,
$X\in\operatorname{Sym}_{n}$ is PSD if all its eigenvalues are nonnegative.
For $k\in\\{1,\dots,n\\}$ let ${\mathcal{S}}^{n,k}$ denote the set of those
matrices in $\operatorname{Sym}_{n}$, whose all $k\times k$ principal
submatrices are positive semidefinite. Elements of ${\mathcal{S}}^{n,k}$ are
known as _$k$ -locally positive semidefinite matrices_, they have been
systematically studied in [1] and [2]. Note that ${\mathcal{S}}^{n,n}$
consists of all positive semidefinite matrices in $\operatorname{Sym}_{n}$.
The set ${\mathcal{S}}^{n,k}$ is a closed convex cone containing
${\mathcal{S}}^{n,n}$ and, moreover,
${\mathcal{S}}^{n,k}\supsetneq{\mathcal{S}}^{n,k+1}$ for any
$k\in\\{1,\dots,n-1\\}$. The dual cone to
${\mathcal{S}}^{n,k}\subset\operatorname{Sym}_{n}$ consists of PSD matrices of
factor width $k$, which were introduced and investigated in [3, 7, 5].
Matrices of factor width $2$ are also known as scaled diagonally dominant
matrices, see [3]. One of the motivations to study properties of locally PSD
matrices came from empirical observations that some optimization problems (see
[8, 6]), where constraints are given by PSD matrices, have close optimal
values to those of their relaxations, where constraints are relaxed to be
$k$-locally PSD.

Given a matrix $X\in\operatorname{Sym}_{n}$, let
$\lambda(X)=(\lambda_{1}(X),\dots,\lambda_{n}(X))$,
$\lambda_{1}(X)\geq\dots\geq\lambda_{n}(X)$, denote the vector of its ordered
eigenvalues and for any permutation on $n$ elements $\sigma\in S_{n}$ let
$\lambda(X)^{\sigma}=(\lambda_{\sigma_{1}}(X),\dots,\lambda_{\sigma_{n}}(X))$
be the vector of permuted eigenvalues. In this work we are interested in the
set

$\displaystyle\lambda({\mathcal{S}}^{n,k})\ =\
\left\\{\lambda(X)^{\sigma}\,:\,X\in{\mathcal{S}}^{n,k},\ \sigma\in
S_{n}\right\\}$

of all possible (permuted) eigenvalue vectors of $k$-locally PSD matrices. As
observed in [2], the set $\lambda({\mathcal{S}}^{n,k})$ is contained in _the
closed hyperbolicity cone_

$\displaystyle H(e^{n}_{k})\ =\
\\{\lambda\in\mathbb{R}^{n}\,:\,e^{n}_{k}(\lambda+t(1,\dots,1))=0\,\Rightarrow
t\leq 0\\}$

of the $k$-th elementary symmetric polynomial $e^{n}_{k}(\lambda)=\sum_{1\leq
i_{1}<\dots<i_{k}\leq n}\lambda_{i_{1}}\cdot\ldots\cdot\lambda_{i_{k}}$.
Moreover, the inclusion $\lambda({\mathcal{S}}^{n,k})\subseteq H(e^{n}_{k})$
is strict for $2<k<n-1$, for $(n,k)=(4,2)$ [2, Cor. 3] and, surprisingly, one
has equality $\lambda({\mathcal{S}}^{n,n-1})=H(e^{n}_{n-1})$ for $k=n-1$ (see
[2, Thm. $2.2$]). The authors of [2] pointed out that it is unknown whether
the set $\lambda({\mathcal{S}}^{n,k})$ of eigenvalue vectors of matrices in
${\mathcal{S}}^{n,k}$ is convex for any $k$ and $n$. We give a negative answer
to this question, showing that $\lambda({\mathcal{S}}^{4,2})\subsetneq
H(e^{4}_{2})$ is not convex.

######  Theorem 1.1.

The set

$\displaystyle\lambda({\mathcal{S}}^{4,2})\ =\
\\{(\lambda_{\sigma_{1}}(X),\lambda_{\sigma_{2}}(X),\lambda_{\sigma_{3}}(X),\lambda_{\sigma_{4}}(X))\,:\,X\in{\mathcal{S}}^{4,2},\
\sigma\in S_{4}\\}$

of eigenvalue vectors of $2$-locally PSD matrices in $\operatorname{Sym}_{4}$
is not convex.

######  Remark 1.2.

Specifically, we prove that $(4,4,-1,-1)$ is not in
$\lambda({\mathcal{S}}^{4,2})$, though it lies on the segment joining the
vector $(4,4,0,-2)\in\lambda({\mathcal{S}}^{4,2})$ with its permuted copy.

Even though the set $\lambda({\mathcal{S}}^{n,k})$ is not convex in general,
it is star-shaped with respect to the nonnegative orthant
$\mathbb{R}^{n}_{\geq 0}=\lambda({\mathcal{S}}^{n,n})$.

######  Proposition 1.3.

For all $\lambda\in\lambda({\mathcal{S}}^{n,k})$,
$\lambda^{+}\in\mathbb{R}^{n}_{\geq 0}$ the sum $\lambda+\lambda^{+}$ is in
$\lambda({\mathcal{S}}^{n,k})$.

We now discuss one application of this result. It is shown in [2] that every
point on the boundary of $H(e^{4}_{2})$ with exactly one negative entry is an
eigenvalue vector of some matrix in ${\mathcal{S}}^{4,2}$. We observe that the
same holds for all points in $H(e^{4}_{2})$ with this property.

######  Corollary 1.4.

Any $\lambda\in H(e^{4}_{2})$ with at most $1$ negative entry is an eigenvalue
vector of some matrix in ${\mathcal{S}}^{4,2}$.

In the rather trivial case $k=1$ the set
$\lambda({\mathcal{S}}^{n,1})=H(e^{n}_{1})$ is a closed half-space.

######  Theorem 1.5.

For any $n\geq 1$ we have

$\displaystyle\lambda({\mathcal{S}}^{n,1})\ =\
\left\\{\lambda\in\mathbb{R}^{n}\,:\,e^{n}_{1}(\lambda)=\sum_{j=1}^{n}\lambda_{j}\geq
0\right\\}$

##  2\. Some open questions

By its definition, the set $\lambda({\mathcal{S}}^{n,k})$ consists of permuted
eigenvalue vectors of $k$-locally PSD matrices. It is also natural to consider
the subset

$\displaystyle\lambda_{\leq}({\mathcal{S}}^{n,k})\ =\
\\{\lambda(X)\,:\,X\in{\mathcal{S}}^{n,k}\\}$

of ordered eigenvalue vectors, that is, the intersection of
$\lambda({\mathcal{S}}^{n,k})$ with half-spaces cut out by inequalities
$\lambda_{i}\geq\lambda_{i+1}$, $i=1,\dots,n-1$. Our proof of Theorem 1.1 does
not imply that $\lambda_{\leq}({\mathcal{S}}^{4,2})$ is non-convex, see Remark
1.2. It would be interesting to understand whether
$\lambda_{\leq}({\mathcal{S}}^{n,k})$ is non-convex for some $(n,k)$. When
$(n,k)=(n,n-1)$ the set $\lambda_{\leq}({\mathcal{S}}^{n,k}))$ is a convex
cone by [2, Thm. $2.2$] and when $(n,k)=(n,1)$ it is the convex polyhedral
cone

$\displaystyle\lambda_{\leq}({\mathcal{S}}^{n,1})\ =\
\left\\{\lambda\in\mathbb{R}^{n}\,:\,\lambda_{1}\geq\dots\geq\lambda_{n},\
\sum_{j=1}^{i}\lambda_{j}\geq 0\ \textrm{for all}\ i=1,\dots,n\right\\},$

see the proof of Theorem 1.5 below.

Nonzero vectors in the hyperbolicity cone $H(e^{n}_{k})$ can have at most
$n-k$ negative entries. Generalizing [2, Lemma $2$] one can show that if a
vector $\lambda\in H(e^{n}_{k})$ has $n-k$ negative entries, then the other
$k$ entries must be positive. One interesting question is to understand
whether for any $(n,k)$ there exist $k$-locally PSD matrices with $n-k$
negative eigenvalues. In the cases $(n,n-1)$ and $(n,1)$ [2, Thm. $2.2$] and
Theorem 1.5 imply a positive answer to this question. Also, one can see that
the real symmetric matrix

$\displaystyle X\ =\ \begin{pmatrix}346&84&-16&-98\\\ 84&240&77&156\\\
-16&77&30&70\\\ -98&156&70&170\end{pmatrix}$

is in ${\mathcal{S}}^{4,2}$ and has two negative eigenvalues.

See also Remark 4.2 for an interesting polynomial optimization problem.

##  3\. Acknowledgements

We would like to thank Grigoriy Blekherman, whose talk at the POEMA 3rd
Workshop inspired this work. We wish to thank Boulos El Hilany for his helpful
insight with the SageMath computation.

##  4\. Proofs

We need the following elementary fact that we prove first.

######  Lemma 4.1.

If $(\lambda_{1},\dots,\lambda_{n^{\prime}})$ is in
$\lambda({\mathcal{S}}^{n^{\prime},k})$, then
$(\lambda_{1},\dots,\lambda_{n^{\prime}},\underbrace{0,\dots,0}_{n-n^{\prime}})$
is in $\lambda({\mathcal{S}}^{n,k})$.

###### Proof.

Let $(\lambda_{1},\dots,\lambda_{n^{\prime}})=\lambda(X^{\prime})$, where
$X^{\prime}\in{\mathcal{S}}^{n^{\prime},k}$. Note that the $n\times n$ matrix

$\displaystyle X\ =\ \begin{pmatrix}X^{\prime}&{\Huge 0}\\\ {\Huge 0}&{\Huge
0}\end{pmatrix}$

is $k$-locally PSD, since its principal submatrices $X|_{S}$,
$S\subseteq\\{1,\dots,n\\}$, $|S|=k$, read

$\displaystyle X|_{S}\ =\ \begin{pmatrix}X^{\prime}|_{S^{\prime}}&{\Huge 0}\\\
{\Huge 0}&{\Huge 0}\end{pmatrix},$

where $S^{\prime}=S\cap\\{1,\dots,n^{\prime}\\}$. It is easy to see that
$(\lambda_{1},\dots,\lambda_{n^{\prime}},0,\dots,0)=\lambda(X)$. ∎

  * Proof of Theorem 1.1.

Let us observe that the vector $\lambda=(4,4,-1,-1)$ belongs to the cone
$H(e^{4}_{2})$. Indeed, for any $t\geq 0$ the value

$\displaystyle e^{4}_{2}(4+t,4+t,-1+t,-1+t)\ $ $\displaystyle=\
(4+t)^{2}+(t-1)^{2}+4(4+t)(t-1)$ $\displaystyle=\ 1+18t+6t^{2}$

is positive and hence the vector $\lambda+t(1,1,1,1)$ lies in $H(e^{4}_{2})$.
Furthermore, $\lambda$ is the mid-point of the segment joining
$\lambda^{\prime}=(4,4,-2,0)$ and $\lambda^{\prime\prime}=(4,4,0,-2)$. Note
that $\lambda^{\prime}$ is obtained by appending a zero to the vector
$(4,4,-2)\in H(e^{3}_{2})=\lambda({\mathcal{S}}^{3,2})$, where equality of
sets is due to [2, Thm. $2.2$]. Thus, Lemma 4.1 implies that
$\lambda^{\prime}$ (and hence also its permuted copy $\lambda^{\prime\prime}$)
belongs to $\lambda({\mathcal{S}}^{4,2})$. We show that $\lambda$ is not in
$\lambda({\mathcal{S}}^{4,2})$. Note that $\lambda=\lambda(X)$ for some matrix
$X\in\operatorname{Sym}_{4}$ if and only if we can write

(4.1) $\begin{aligned} X\ =\ V^{T}\Lambda V\ &=\
\begin{pmatrix}\bm{v}^{T}_{1}\,|\,\bm{v}^{T}_{2}\,|\,\bm{v}^{T}_{3}\,|\,\bm{v}^{T}_{4}\end{pmatrix}\begin{pmatrix}4&&&{\huge
0}\\\ &4&&\\\ &&-1&\\\ {\huge 0}&&&-1\end{pmatrix}\begin{pmatrix}\bm{v}_{1}\\\
\bm{v}_{2}\\\ \bm{v}_{3}\\\ \bm{v}_{4}\end{pmatrix}\\\ &=\
\begin{pmatrix}\begin{subarray}{c}4v_{11}^{2}+4v_{21}^{2}\\\
-v_{31}^{2}-v_{41}^{2}\end{subarray}&\begin{subarray}{c}4v_{11}v_{12}+4v_{21}v_{22}\\\
-v_{31}v_{32}-v_{41}v_{42}\end{subarray}&\begin{subarray}{c}4v_{11}v_{13}+4v_{21}v_{23}\\\
-v_{31}v_{33}-v_{41}v_{43}\end{subarray}&\begin{subarray}{c}4v_{11}v_{14}+4v_{21}v_{24}\\\
-v_{31}v_{34}-v_{41}v_{44}\end{subarray}\\\
\begin{subarray}{c}4v_{11}v_{12}+4v_{21}v_{22}\\\
-v_{31}v_{32}-v_{41}v_{42}\end{subarray}&\begin{subarray}{c}4v_{12}^{2}+4v_{22}^{2}\\\
-v_{32}^{2}-v_{42}^{2}\end{subarray}&\begin{subarray}{c}4v_{12}v_{13}+4v_{22}v_{23}\\\
-v_{32}v_{33}-v_{42}v_{43}\end{subarray}&\begin{subarray}{c}4v_{12}v_{14}+4v_{22}v_{24}\\\
-v_{32}v_{34}-v_{42}v_{44}\end{subarray}\\\
\begin{subarray}{c}4v_{11}v_{13}+4v_{21}v_{23}\\\
-v_{31}v_{33}-v_{41}v_{43}\end{subarray}&\begin{subarray}{c}4v_{12}v_{13}+4v_{22}v_{23}\\\
-v_{32}v_{33}-v_{42}v_{43}\end{subarray}&\begin{subarray}{c}4v_{13}^{2}+4v_{23}^{2}\\\
-v_{33}^{2}-v_{43}^{2}\end{subarray}&\begin{subarray}{c}4v_{13}v_{14}+4v_{23}v_{24}\\\
-v_{33}v_{34}-v_{43}v_{44}\end{subarray}\\\
\begin{subarray}{c}4v_{11}v_{14}+4v_{21}v_{24}\\\
-v_{31}v_{34}-v_{41}v_{44}\end{subarray}&\begin{subarray}{c}4v_{12}v_{14}+4v_{22}v_{24}\\\
-v_{32}v_{34}-v_{42}v_{44}\end{subarray}&\begin{subarray}{c}4v_{13}v_{14}+4v_{23}v_{24}\\\
-v_{33}v_{34}-v_{43}v_{44}\end{subarray}&\begin{subarray}{c}4v_{14}^{2}+4v_{24}^{2}\\\
-v_{34}^{2}-v_{44}^{2}\end{subarray}\end{pmatrix}\end{aligned},$

where $V\in O(4)$ is an orthogonal matrix, whose rows are eigenvectors
$\bm{v}_{1},\bm{v}_{2},\bm{v}_{3},\bm{v}_{4}\in\mathbb{R}^{4}$ of $X$. The
representation (4.1) is unique up to the choice of an orthonormal basis
$\\{\bm{v}_{1},\bm{v}_{2}\\}$ (respectively, $\\{\bm{v}_{3},\bm{v}_{4}\\}$) of
the eigenspace of $X$ corresponding to the eigenvalue $4$ (respectively,
$-1$). In particular, performing orthogonal changes of bases in
$L=\mathrm{{Span}}\\{\bm{v}_{1},\bm{v}_{2}\\}$ and in
$L^{\perp}=\mathrm{{Span}}\\{\bm{v}_{3},\bm{v}_{4}\\}$ we can put
$v_{12}=v_{34}=0$. Now, the matrix (4.1) with $v_{12}=v_{34}=0$ is $2$-locally
PSD if and only if the polynomial system

$\displaystyle 4v_{11}^{2}+4v_{21}^{2}-v_{31}^{2}-v_{41}^{2}-m_{1}\ =\ 0,$
$\displaystyle 4v_{22}^{2}-v_{32}^{2}-v_{42}^{2}-m_{2}\ =\ 0,$ $\displaystyle
4v_{13}^{2}+4v_{23}^{2}-v_{33}^{2}-v_{43}^{2}-m_{3}\ =\ 0,$ $\displaystyle
4v_{14}^{2}+4v_{24}^{2}-v_{44}^{2}-m_{4}\ =\ 0,$
$\displaystyle(4v_{11}^{2}+4v_{21}^{2}-v_{31}^{2}-v_{41}^{2})(4v_{22}^{2}-v_{32}^{2}-v_{42}^{2})-(4v_{21}v_{22}-v_{31}v_{32}-v_{41}v_{42})^{2}-m_{12}\
=\ 0,$
$\displaystyle(4v_{11}^{2}+4v_{21}^{2}-v_{31}^{2}-v_{41}^{2})(4v_{13}^{2}+4v_{23}^{2}-v_{33}^{2}-v_{43}^{2})-(4v_{11}v_{13}+4v_{21}v_{23}-v_{31}v_{33}-v_{41}v_{43})^{2}-m_{13}\
=\ 0,$
$\displaystyle(4v_{11}^{2}+4v_{21}^{2}-v_{31}^{2}-v_{41}^{2})(4v_{14}^{2}+4v_{24}^{2}-v_{44}^{2})-(4v_{11}v_{14}+4v_{21}v_{24}-v_{41}v_{44})^{2}-m_{14}\
=\ 0,$
$\displaystyle(4v_{22}^{2}-v_{32}^{2}-v_{42}^{2})(4v_{13}^{2}+4v_{23}^{2}-v_{33}^{2}-v_{43}^{2})-(4v_{22}v_{23}-v_{32}v_{33}-v_{42}v_{43})^{2}-m_{23}\
=\ 0,$
$\displaystyle(4v_{22}^{2}-v_{32}^{2}-v_{42}^{2})(4v_{14}^{2}+4v_{24}^{2}-v_{44}^{2})-(4v_{22}v_{24}-v_{42}v_{44})^{2}-m_{24}\
=\ 0,$
$\displaystyle(4v_{13}^{2}+4v_{23}^{2}-v_{33}^{2}-v_{43}^{2})(4v_{14}^{2}+4v_{24}^{2}-v_{44}^{2})-(4v_{13}v_{14}+4v_{23}v_{24}-v_{43}v_{44})^{2}-m_{34}\
=\ 0,$ $\displaystyle v_{11}^{2}+v_{21}^{2}+v_{31}^{2}+v_{41}^{2}-1=0,$
$\displaystyle v_{22}^{2}+v_{32}^{2}+v_{42}^{2}-1=0,$ $\displaystyle
v_{13}^{2}+v_{23}^{2}+v_{33}^{2}+v_{43}^{2}-1=0,$ $\displaystyle
v_{14}^{2}+v_{24}^{2}+v_{44}^{2}-1=0,$ $\displaystyle
v_{21}v_{22}+v_{31}v_{32}+v_{41}v_{42}=0,$ $\displaystyle
v_{11}v_{13}+v_{21}v_{23}+v_{31}v_{33}+v_{41}v_{43}=0,$ $\displaystyle
v_{11}v_{14}+v_{21}v_{24}+v_{41}v_{44}=0,$ $\displaystyle
v_{22}v_{23}+v_{32}v_{33}+v_{42}v_{43}=0,$ $\displaystyle
v_{22}v_{24}+v_{42}v_{44}=0,$ $\displaystyle
v_{13}v_{14}+v_{23}v_{24}+v_{43}v_{44}=0,$

has a real solution with nonnegative
$m_{1},m_{2},m_{3},m_{4},m_{12},m_{13},m_{14},m_{23},m_{24},m_{34}$. These
variables represent principal minors of $X$ of sizes $1\times 1$ and $2\times
2$, and the last $10$ equations encode orthogonality of the matrix $V$. Let
$I\vartriangleleft\mathbb{Q}[v_{ij},m_{i},m_{ij}]$ be the ideal generated by
the $20$ polynomials of the system. Using $\mathrm{{SageMath}}$ [10] we find
out that the elimination ideal $J=I\cap\mathbb{Q}[m_{i},m_{ij}]$ of $I$ with
respect to the variables $v_{ij}$ is generated by polynomials

$\displaystyle L\ =$ $\displaystyle
m_{12}+m_{13}+m_{14}+m_{23}+m_{24}+m_{34}-1,$ $\displaystyle f_{1}\ =$
$\displaystyle 3m_{1}+m_{23}+m_{24}+m_{34}-5,$ $\displaystyle f_{2}\ =$
$\displaystyle 3m_{2}+m_{13}+m_{14}+m_{34}-5,$ $\displaystyle f_{3}\ =$
$\displaystyle 3m_{3}-m_{13}-m_{23}-m_{34}-4,$ $\displaystyle f_{4}\ =$
$\displaystyle 3m_{4}-m_{14}-m_{24}-m_{34}-4,$ $\displaystyle F\ =$
$\displaystyle-2816-256m_{13}-256m_{14}-256m_{23}-256m_{24}-224m_{34}-50m_{34}^{3}$
$\displaystyle+192m_{14}m_{24}+256m_{13}^{2}+192m_{13}m_{14}+256m_{14}^{2}+192m_{13}m_{23}+160m_{14}m_{23}+256m_{23}^{2}$
$\displaystyle+160m_{13}m_{24}+192m_{23}m_{24}+256m_{24}^{2}+96m_{13}m_{34}+96m_{14}m_{34}+96m_{23}m_{34}+96m_{24}m_{34}+249m_{34}^{2}$
$\displaystyle+64m_{13}^{2}m_{14}+64m_{13}m_{14}^{2}+64m_{13}^{2}m_{23}+112m_{13}m_{14}m_{23}+128m_{14}^{2}m_{23}+64m_{13}m_{23}^{2}$
$\displaystyle+128m_{14}m_{23}^{2}+128m_{13}^{2}m_{24}+112m_{13}m_{14}m_{24}+64m_{14}^{2}m_{24}+112m_{13}m_{23}m_{24}+112m_{14}m_{23}m_{24}$
$\displaystyle+64m_{23}^{2}m_{24}+128m_{13}m_{24}^{2}+64m_{14}m_{24}^{2}+64m_{23}m_{24}^{2}+128m_{13}^{2}m_{34}+240m_{13}m_{14}m_{34}+128m_{14}^{2}m_{34}$
$\displaystyle+240m_{13}m_{23}m_{34}+206m_{14}m_{23}m_{34}+128m_{23}^{2}m_{34}+206m_{13}m_{24}m_{34}+240m_{14}m_{24}m_{34}$
$\displaystyle+240m_{23}m_{24}m_{34}+128m_{24}^{2}m_{34}+78m_{13}m_{34}^{2}+78m_{14}m_{34}^{2}+78m_{23}m_{34}^{2}+78m_{24}m_{34}^{2}$
$\displaystyle+16m_{13}^{2}m_{14}m_{23}+16m_{13}m_{14}^{2}m_{23}+16m_{13}m_{14}m_{23}^{2}+25m_{14}^{2}m_{23}^{2}+16m_{13}^{2}m_{14}m_{24}+16m_{13}m_{14}^{2}m_{24}$
$\displaystyle+16m_{13}^{2}m_{23}m_{24}+14m_{13}m_{14}m_{23}m_{24}+16m_{14}^{2}m_{23}m_{24}+16m_{13}m_{23}^{2}m_{24}+16m_{14}m_{23}^{2}m_{24}+25m_{13}^{2}m_{24}^{2}$
$\displaystyle+16m_{13}m_{14}m_{24}^{2}+16m_{13}m_{23}m_{24}^{2}+16m_{14}m_{23}m_{24}^{2}+16m_{13}^{2}m_{14}m_{34}+16m_{13}m_{14}^{2}m_{34}$
$\displaystyle+16m_{13}^{2}m_{23}m_{34}+82m_{13}m_{14}m_{23}m_{34}+50m_{14}^{2}m_{23}m_{34}+16m_{13}m_{23}^{2}m_{34}+50m_{14}m_{23}^{2}m_{34}$
$\displaystyle+50m_{13}^{2}m_{24}m_{34}+82m_{13}m_{14}m_{24}m_{34}+16m_{14}^{2}m_{24}m_{34}+82m_{13}m_{23}m_{24}m_{34}+82m_{14}m_{23}m_{24}m_{34}$
$\displaystyle+16m_{23}^{2}m_{24}m_{34}+50m_{13}m_{24}^{2}m_{34}+16m_{14}m_{24}^{2}m_{34}+16m_{23}m_{24}^{2}m_{34}+25m_{13}^{2}m_{34}^{2}+66m_{13}m_{14}m_{34}^{2}$
$\displaystyle+25m_{14}^{2}m_{34}^{2}+66m_{13}m_{23}m_{34}^{2}+100m_{14}m_{23}m_{34}^{2}+25m_{23}^{2}m_{34}^{2}+100m_{13}m_{24}m_{34}^{2}+66m_{14}m_{24}m_{34}^{2}$
$\displaystyle+66m_{23}m_{24}m_{34}^{2}+25m_{24}^{2}m_{34}^{2}+50m_{13}m_{34}^{3}+50m_{14}m_{34}^{3}+50m_{23}m_{34}^{3}+50m_{24}m_{34}^{3}+25m_{34}^{4}.$

Note that $L$ and $F$ depend only on the variables $m_{ij}$. In particular, if
$(m_{i},m_{ij})$ is a nonnegative zero of $J$, then $(m_{ij})$ is a
nonnegative zero of $L=F=0$. Vice versa, if $L(m_{ij})=F(m_{ij})=0$ for some
nonnegative $m_{ij}$, setting

$\displaystyle m_{1}$ $\displaystyle=\ \frac{1}{3}(m_{12}+m_{13}+m_{14}+4)\geq
0,$ $\displaystyle m_{2}$ $\displaystyle=\
\frac{1}{3}(m_{12}+m_{23}+m_{24}+4)\geq 0,$ $\displaystyle m_{3}$
$\displaystyle=\ \frac{1}{3}(m_{13}+m_{23}+m_{34}+4)\geq 0,$ $\displaystyle
m_{4}$ $\displaystyle=\ \frac{1}{3}(m_{14}+m_{24}+m_{34}+4)\geq 0,$

we recover a nonnegative zero of $J$.

Finally, we prove that $L$ and $F$ have no common nonnegative zeros. For this
it is enough to show that $F$ (which does not depend on $m_{12}$) has no zeros
in the unit simplex

$\displaystyle\Delta\ =\
\\{(m_{13},m_{14},m_{23},m_{24},m_{34})\in\mathbb{R}^{5}_{\geq
0}\,:\,m_{13}+m_{14}+m_{23}+m_{24}+m_{34}\leq 1\\}.$

Since $0\leq m_{ij}^{4}\leq m^{3}_{ij}\leq m_{ij}^{2}\leq m_{ij}^{1}\leq 1$
hold for points in $\Delta$, term-wise estimates give

$\displaystyle F\ =$
$\displaystyle-2816-256(m_{13}+m_{14}+m_{23}+m_{24}+m_{34})-50m_{34}^{3}-7m_{34}^{2}$
$\displaystyle+32m_{34}+192(m_{14}m_{24}+m_{13}m_{14}+m_{13}m_{23}+m_{23}m_{24})+256(m_{13}^{2}+m_{14}^{2}+m_{23}^{2}+m_{24}^{2}+m_{34}^{2})$
$\displaystyle+160(m_{14}m_{23}+m_{13}m_{24})+96(m_{13}m_{34}+m_{14}m_{34}+m_{23}m_{34}+m_{24}m_{34})$
$\displaystyle+64(m_{13}^{2}m_{14}+m_{13}m_{14}^{2}+m_{13}^{2}m_{23}+m_{13}m_{23}^{2}+m_{14}^{2}m_{24}+m_{23}^{2}m_{24}+m_{14}m_{24}^{2}+m_{23}m_{24}^{2})$
$\displaystyle+112(m_{13}m_{14}m_{23}+m_{13}m_{14}m_{24}+m_{13}m_{23}m_{24}+m_{14}m_{23}m_{24})$
$\displaystyle+128(m_{14}^{2}m_{23}+m_{14}m_{23}^{2}+m_{13}^{2}m_{24}+m_{13}m_{24}^{2}+m_{13}^{2}m_{34}+m_{14}^{2}m_{34}+m_{23}^{2}m_{34}+m_{24}^{2}m_{34})$
$\displaystyle+240(m_{13}m_{14}m_{34}+m_{13}m_{23}m_{34}+m_{14}m_{24}m_{34}+m_{23}m_{24}m_{34})$
$\displaystyle+206(m_{14}m_{23}m_{34}+m_{13}m_{24}m_{34})+78(m_{13}m_{34}^{2}+m_{14}m_{34}^{2}+m_{23}m_{34}^{2}+m_{24}m_{34}^{2})$
$\displaystyle+16(m_{13}^{2}m_{14}m_{23}+m_{13}m_{14}^{2}m_{23}+m_{13}m_{14}m_{23}^{2}+m_{13}^{2}m_{14}m_{24}+m_{13}m_{14}^{2}m_{24}+m_{13}^{2}m_{23}m_{24}$
$\displaystyle+m_{14}^{2}m_{23}m_{24}+m_{13}m_{23}^{2}m_{24}+m_{14}m_{23}^{2}m_{24}+m_{13}m_{14}m_{24}^{2}+m_{13}m_{23}m_{24}^{2}+m_{14}m_{23}m_{24}^{2}+m_{13}^{2}m_{14}m_{34}$
$\displaystyle+m_{13}m_{14}^{2}m_{34}+m_{13}^{2}m_{23}m_{34}+m_{13}m_{23}^{2}m_{34}+m_{14}^{2}m_{24}m_{34}+m_{23}^{2}m_{24}m_{34}+m_{14}m_{24}^{2}m_{34}+m_{23}m_{24}^{2}m_{34})$
$\displaystyle+25(m_{14}^{2}m_{23}^{2}+m_{13}^{2}m_{24}^{2}+m_{13}^{2}m_{34}^{2}+m_{14}^{2}m_{34}^{2}+m_{23}^{2}m_{34}^{2}+m_{24}^{2}m_{34}^{2}+m_{34}^{4})$
$\displaystyle+82(m_{13}m_{14}m_{23}m_{34}+m_{13}m_{14}m_{24}m_{34}+m_{13}m_{23}m_{24}m_{34}+m_{14}m_{23}m_{24}m_{34})+14m_{13}m_{14}m_{23}m_{24}$
$\displaystyle+50(m_{14}^{2}m_{23}m_{34}+m_{14}m_{23}^{2}m_{34}+m_{13}^{2}m_{24}m_{34}+m_{13}m_{24}^{2}m_{34}+m_{13}m_{34}^{3}+m_{14}m_{34}^{3}+m_{23}m_{34}^{3}+m_{24}m_{34}^{3})$
$\displaystyle+66(m_{13}m_{14}m_{34}^{2}+m_{13}m_{23}m_{34}^{2}+m_{14}m_{24}m_{34}^{2}+m_{23}m_{24}m_{34}^{2})+100(m_{14}m_{23}m_{34}^{2}+m_{13}m_{24}m_{34}^{2})$
$\displaystyle\leq$
$\displaystyle-2816-256m-50m_{34}^{3}-7m_{34}^{2}+32m_{34}+192(m_{14}+m_{13}+m_{23}+m_{24})+256m$
$\displaystyle+160(m_{14}+m_{13})+96(m_{13}+m_{14}+m_{23}+m_{24})+64(m_{13}^{2}+m_{14}^{2}+m_{23}^{2}+m_{24}^{2})$
$\displaystyle+112(m_{13}+m_{14}+m_{23}+m_{24})+128(m_{14}^{2}+m_{23}^{2}+m_{13}^{2}+m_{24}^{2})+240(m_{14}+m_{13}+m_{24}+m_{23})$
$\displaystyle+206(m_{23}+m_{24})+78(m_{13}+m_{14}+m_{23}+m_{24})+16(m_{13}m_{14}m_{23}+m_{13}m_{14}m_{24}$
$\displaystyle+m_{13}m_{23}m_{24}+m_{14}m_{23}m_{24}+m_{13}m_{14}m_{34}+m_{13}m_{23}m_{34}+m_{14}m_{24}m_{34}+m_{23}m_{24}m_{34})$
$\displaystyle+25(m_{14}^{2}+m_{13}^{2}+m_{23}^{2}+m_{24}^{2}+m_{34}^{2})+82(m_{13}+m_{14}+m_{23}+m_{24})+14$
$\displaystyle+50(m_{14}^{2}+m_{23}^{2}+m_{13}^{2}+m_{24}^{2}+m_{34}^{2})+66(m_{14}+m_{13}+m_{24}+m_{23})+100(m_{14}+m_{13})$
$\displaystyle\leq$
$\displaystyle-2816+32+192+160+96+64+112+128+240+206+78+16+25+82+14$
$\displaystyle+50+66+100\ =\ -1155\ <\ 0,$

where $m=m_{13}+m_{14}+m_{23}+m_{24}+m_{34}\leq 1$. It follows that $F$ takes
only negative values on $\Delta$. The above implies that no matrix
$X=V^{T}\Lambda V$ as in (4.1) can be $2$-locally PSD and hence
$\lambda=(4,4,-1,-1)\notin\lambda({\mathcal{S}}^{4,2})$. ∎

######  Remark 4.2.

Note that the absense of nonnegative roots of $L=F=0$ implies that the minimum
$\ell_{*}$ of the linear function $\ell=m_{13}+m_{14}+m_{23}+m_{24}+m_{34}$ on
$\\{F=0\\}\cap\mathbb{R}^{5}_{\geq 0}$ is greater than $1$. Numerical solving
of a bunch of constrained optimization problems suggests that the minimum is
$\ell_{*}=3.2$ and it is attained at
$(m_{13},m_{14},m_{23},m_{24},m_{34})=(0,0,0,0,3.2)$. It would be interesting
to find an algebraic certificate of nonnegativity (see e.g. [9] and [4]) of
say $\ell-2$ on the semialgebraic set $\\{F=0\\}\cap\mathbb{R}^{5}_{\geq 0}$.
This would give an alternative proof of emptiness of
$\\{L=F=0\\}\cap\mathbb{R}^{5}_{\geq 0}$ (cf. the proof of Theorem 1.1).

Proposition 1.3 essentially follows from properties of the PSD cone
${\mathcal{S}}^{n,n}$.

  * Proof of Proposition 1.3.

Let $X=V^{T}\Lambda V\in{\mathcal{S}}^{n,k}$ be a spectral decomposition of a
$k$-locally PSD matrix, where $V$ is an orthogonal matrix and $\Lambda$ is the
diagonal matrix given by $\lambda\in\lambda({\mathcal{S}}^{n,k})$. Due to the
orthogonal invariance of the PSD cone ${\mathcal{S}}^{n,n}$, the matrix
$X^{+}=V^{T}\Lambda^{+}V\in{\mathcal{S}}^{n,n}\subset{\mathcal{S}}^{n,k}$ is
in particular $k$-locally PSD, where $\Lambda^{+}$ is the diagonal matrix
given by the nonnegative vector $\lambda^{+}\in\mathbb{R}^{n}_{\geq
0}\subset\lambda({\mathcal{S}}^{n,k})$. By convexity of ${\mathcal{S}}^{n,k}$

$\displaystyle X+X^{+}\ =\ V^{T}(\Lambda+\Lambda^{+})V$

is in ${\mathcal{S}}^{n,k}$ and hence
$\lambda+\lambda^{+}\in\lambda({\mathcal{S}}^{n,k})$. ∎

  * Proof of Corollary 1.4.

Let $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})\in
H(e^{4}_{2})$ satisfy
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$. If
$\lambda_{4}\geq 0$, trivially $\lambda\in\mathbb{R}^{4}_{\geq
0}\subset\lambda({\mathcal{S}}^{4,2})$. Let then $\lambda_{3}\geq
0>\lambda_{4}$. Note that for
$t=e^{4}_{2}(\lambda)/(\lambda_{1}+\lambda_{2}+\lambda_{3})\geq 0$ the vector
$\lambda-t(0,0,0,1)$ lies on the boundary of $H(e^{4}_{2})$ and it has exactly
one negative entry $\lambda_{4}-t$. By [2, Thm. $8.1$] the vector
$\lambda-t(0,0,0,1)$ is in $\lambda({\mathcal{S}}^{4,2})$. Thus, by
Proposition 1.3 the original vector $\lambda=(\lambda-t(0,0,0,1))+t(0,0,0,1)$
is also in $\lambda({\mathcal{S}}^{4,2})$ because
$t(0,0,0,1)\in\mathbb{R}^{4}_{\geq 0}$. ∎

Our proof of Theorem 1.5 is based on the following classical result.

######  Theorem 4.3 (Schur-Horn theorem).

Let $x,\lambda\in\mathbb{R}^{n}$ be two vectors with $x_{1}\geq\dots\geq
x_{n}$ and $\lambda_{1}\geq\dots\geq\lambda_{n}$. There exists a symmetric
matrix $X\in\operatorname{Sym}_{n}$ with diagonal entries $x_{1},\dots,x_{n}$
and with eigenvalues $\lambda_{1},\dots,\lambda_{n}$ if and only if for all
$i=1,\dots,n-1$

$\displaystyle\sum_{j=1}^{i}x_{j}\ \leq\
\sum_{j=1}^{i}\lambda_{j}\quad\textrm{{and}}\quad\sum_{j=1}^{n}x_{j}\ =\
\sum_{j=1}^{n}\lambda_{j}.$

  * Proof of Theorem 1.5.

Note that the statement is equivalent to

(4.2) $\displaystyle\lambda({\mathcal{S}}^{n,1})\ =\
\left\\{\lambda^{\sigma}\in\mathbb{R}^{n}\,:\,\sigma\in S_{n},\
\lambda_{1}\geq\dots\geq\lambda_{n},\ \sum_{j=1}^{i}\lambda_{j}\geq 0\
\textrm{{for all}}\ i=1,\dots,n\right\\}.$

Indeed, if $\sum_{j=1}^{i}\lambda_{j}<0$ for some $i<n$, then
$\lambda_{i}\geq\lambda_{i+1}\geq\dots\geq\lambda_{n}$ must be negative.
Therefore, $\sum_{j=1}^{n}\lambda_{j}\leq\sum_{j=1}^{i}\lambda_{j}<0$ and
hence no permuted vector $\lambda^{\sigma}$, $\sigma\in S_{n}$, can satisfy
$\sum_{j=1}^{n}\lambda_{j}^{\sigma}=\sum_{j=1}^{n}\lambda_{j}\geq 0$. The
opposite direction is trivial. Thus, we need to prove that ordered vectors in
$\lambda({\mathcal{S}}^{n,1})$ are exactly those that satisfy inequalities in
(4.2).

Let $\lambda=\lambda(X)$, where the matrix $X\in{\mathcal{S}}^{n,1}$ has
ordered diagonal entries $x_{1}\geq\dots\geq x_{n}$. By Theorem 4.3 we have
that $\sum^{i}_{j=1}\lambda_{j}\geq\sum^{i}_{j=1}x_{j}\geq 0$ holds for
$i=1,\dots,n$.

Let now $\lambda\in\mathbb{R}^{n}$ be an ordered nonzero vector that satisfies
inequalities from (4.2). There exists an index $r\in\\{1,\dots,n\\}$ so that
$\lambda_{1}\geq\dots\geq\lambda_{r}>0\geq\lambda_{r+1}\geq\dots\geq\lambda_{n}$.
Let us consider a vector $x\in\mathbb{R}^{n}$ defined by
$x_{1}=\dots=x_{r}=\frac{1}{r}\sum^{n}_{j=1}\lambda_{j}\geq 0$ and
$x_{r+1}=\dots=x_{n}=0$. For $i=1,\dots,r-1$ we have

$\displaystyle\sum_{j=1}^{i}x_{j}\ $ $\displaystyle=\
\frac{i}{r}\sum_{j=1}^{n}\lambda_{j}\ \leq\
\frac{i}{r}\sum_{j=1}^{r}\lambda_{j}\ =\
\sum_{j=1}^{i}\lambda_{j}+\frac{i-r}{r}\sum_{j=1}^{i}\lambda_{j}+\frac{i}{r}\sum_{j=i+1}^{r}\lambda_{j}$
$\displaystyle\leq\
\sum_{j=1}^{i}\lambda_{j}+\frac{i(i-r)}{r}\lambda_{i}+\frac{i}{r}\sum_{j=i+1}^{r}\lambda_{j}\
\leq\
\sum_{j=1}^{i}\lambda_{j}+\frac{i}{r}\sum_{j=i+1}^{r}(\lambda_{j}-\lambda_{i})\
\leq\sum_{j=1}^{i}\lambda_{j},$

where ordering
$\lambda_{1}\geq\dots\geq\lambda_{r}>0\geq\lambda_{r+1}\geq\dots\geq\lambda_{n}$
is used in several places. For $i=r,\dots,n$ we have

$\displaystyle\sum_{j=1}^{i}x_{j}\ =\ \sum^{n}_{j=1}\lambda_{j}\ \leq\
\sum_{j=1}^{i}\lambda_{j},$

where we use negativity of $\lambda_{r+1},\dots,\lambda_{n}<0$. These
inequalities, the equality $\sum_{j=1}^{n}x_{j}=\sum_{j=1}^{n}\lambda_{j}$ and
Theorem 4.3 imply that $\lambda_{1},\dots,\lambda_{n}$ are eigenvalues of some
matrix $X\in\operatorname{Sym}_{n}$ with diagonal entries
$x_{1},\dots,x_{n}\geq 0$, that is, the vector $\lambda=\lambda(X)$ is in
$\lambda({\mathcal{S}}^{n,1})$. ∎

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