CERN-TH-2021-229//MIT-CTP/5383

Inferring nuclear structure from heavy isobar collisions using _Trajectum_

Govert Nijs1$\star$ and Wilke van der Schee2$\dagger$

1 Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA

2 Theoretical Physics Department, CERN, CH-1211 Genève 23, Switzerland

⋆<EMAIL_ADDRESS>†<EMAIL_ADDRESS>
## Abstract

Nuclei with equal number of baryons but varying proton number (isobars) have
many commonalities, but differ in both electric charge and nuclear structure.
Relativistic collisions of such isobars provide unique opportunities to study
the variation of the magnetic field, provided the nuclear structure is well
understood. In this work we simulate collisions using several state-of-the-art
parametrizations of the ${}^{96}_{40}$Zr and ${}^{96}_{44}$Ru isobars and show
that a comparison with the exciting STAR measurement [1] of ultrarelativistic
collisions can uniquely identify the structure of both isobars. This not only
provides an urgently needed understanding of the structure of the Zirconium
and Ruthenium isobars, but also paves the way for more detailed studies of
nuclear structure using relativistic heavy ion collisions.

##  1 Introduction

The structure of nuclei is traditionally studied using low energy experiments
[2, 3] or global fits of low-energy data using Density Functional Theory (DFT,
[4]). Nuclear structure, however, also leaves characteristic imprints on the
shape of the quark-gluon plasma (QGP) that is formed shortly after an
ultrarelativistic nuclear collision. The resulting anisotropic flow
coefficients of the hadrons produced can then in turn be used to infer the
shape of this QGP [5, 6, 7].

The recent isobar run of collisions of both ${}^{96}_{40}$Zr and
${}^{96}_{44}$Ru at the Relativistic Heavy Ion Collider (RHIC) in New York had
a special motivation to search for the chiral magnetic effect (CME, [8]). The
idea here is that ${}_{44}^{96}$Ru has four extra protons and hence generates
a 10% larger magnetic field. All other things equal, the ratio of observables
between ${}_{40}^{96}$Zr and ${}_{44}^{96}$Ru would provide unique signatures
of the magnetic field, whereby comparing like-sign and unlike-sign flow in the
direction of the magnetic field can provide convincing evidence for the CME in
the presence of a chiral imbalance caused by fluctuations.

Two crucial features allowed the recent isobar run at RHIC to give results at
an unprecedented precision. First of all the two runs were done over the same
time span under equal detector settings. While for many measurements
systematic uncertainties are often sizeable, the similarity of the two isobars
makes that in the ratio almost all these systematics cancel to a high level.
Secondly, in [1] a special effort was made to unambiguously detect the CME by
a blinded analysis of ${}_{40}^{96}$Zr and ${}_{44}^{96}$Ru collisions, where
even the experimentalists did not know which dataset corresponded to which
isobar.

Unblinding the analysis did not provide evidence for the CME, but surprisingly
${}_{44}^{96}$Ru had a higher particle production than ${}_{40}^{96}$Zr (by up
to 10%) as well as larger elliptic flow ($v_{2}$, by up to 3%), albeit having
a smaller triangular flow ($v_{3}$, by 7% for central collisions). These
relatively large differences indicate a difference in nuclear structure whose
observable effects are much larger than those induced by the 10% higher
electric charge of ${}_{44}^{96}$Ru. This could have been anticipated from
nuclear structure analyses [9, 10, 11, 12, 13], though we note significant
differences between studies. It is the purpose of this work to utilise these
different studies in the state-of-the-art heavy ion code _Trajectum_ and show
how recent results from DFT can fully explain the STAR findings (see also [14,
15, 16, 17] for a similar study using the transport model AMPT for $v_{2}$ and
$v_{3}$).

The hydrodynamic postdiction of the STAR measurements using isobar collisions
paves the way for a more elaborate analysis that can also include a magnetic
field and potentially a CME. Similar techniques, including the study of
extremely ultracentral elliptic flow, can also improve our understanding of
other nuclei, such as the characteristically ellipsoidal Uranium nucleus
collided at RHIC [18, 19] or the slightly less deformed Xenon nucleus collided
at the LHC [20]. We furthermore provide predictions regarding the mean
transverse momentum in isobar collisions (see also [21]).

The aim of this work is hence twofold. First, there is an urgent need to
clarify the differences between the Ruthenium and Zirconium isobars, which is
essential for progress to establish a baseline prediction which can
subsequently be used to identify effects of the magnetic field in heavy ion
collisions. Second, the isobars provide the perfect playground to use heavy
ion collisions and its QGP formation to probe low energy nuclear structure and
hence connect with experiments and theory at lower energy relevant for fields
as far away as neutron stars (see e.g. [22, 23]).

nucleus |  $R_{p}\,$[fm] |  $\sigma_{p}\,$[fm] |  $R_{n}\,$[fm] |  $\sigma_{n}\,$[fm] | $\beta_{2}$ | $\beta_{3}$ |  $\sigma_{\text{AA}}\,$[b]
---|---|---|---|---|---|---|---
${}_{44}^{96}$Ru(1) | 5.085 | 0.46 | 5.085 | 0.46 | 0.158 | 0 | 4.628
${}_{40}^{96}$Zr(1) | 5.02 | 0.46 | 5.02 | 0.46 | 0.08 | 0 | 4.540
${}_{44}^{96}$Ru(2) | 5.085 | 0.46 | 5.085 | 0.46 | 0.053 | 0 | 4.605
${}_{40}^{96}$Zr(2) | 5.02 | 0.46 | 5.02 | 0.46 | 0.217 | 0 | 4.579
${}_{44}^{96}$Ru(3) | 5.06 | 0.493 | 5.075 | 0.505 | 0 | 0 | 4.734
${}_{40}^{96}$Zr(3) | 4.915 | 0.521 | 5.015 | 0.574 | 0 | 0 | 4.860
${}_{44}^{96}$Ru(4) | 5.053 | 0.48 | 5.073 | 0.49 | 0.16 | 0 | 4.701
${}_{40}^{96}$Zr(4) | 4.912 | 0.508 | 5.007 | 0.564 | 0.16 | 0 | 4.829
${}_{44}^{96}$Ru(5) | 5.053 | 0.48 | 5.073 | 0.49 | 0.154 | 0 | 4.699
${}_{40}^{96}$Zr(5) | 4.912 | 0.508 | 5.007 | 0.564 | 0.062 | 0.202 | 4.871
Table 1: Woods-Saxon parameters and inelastic nucleus-nucleus cross sections
for the five cases used in this work, for both ${}_{44}^{96}$Ru and
${}_{40}^{96}$Zr. The $p$ and $n$ labels denote the different WS distributions
used for protons and neutrons, respectively. The $\beta_{n}$-values are the
same for both protons and neutrons. Values for the cross-sections of ZrZr and
RuRu collisions are also given.

##  2 The model

In this work we use the publicly available _Trajectum_ 1.2 framework [24, 25,
26]111The _Trajectum_ code can be found at
https://sites.google.com/view/govertnijs/trajectum. using the maximum
likelihood settings and UrQMD for the hadronic gas phase [27, 28] as in [26].
_Trajectum_ uses the TRENTo model [29] to compute the initial state, where the
positions of the nucleons within the nucleus are given by a Woods-Saxon (WS)
distribution,

$P(r,\theta)\propto\left(1+\exp\left(\frac{r-R(\theta)}{\sigma}\right)\right)^{-1},$
(1)

where
$R(\theta)=R\cdot\left(1+\beta_{2}Y_{2}^{0}(\theta)+\beta_{3}Y_{3}^{0}(\theta)\right)$,
with $Y_{n}^{0}$ spherical harmonics, and the radius $R$, skin depth $\sigma$,
quadrupole $\beta_{2}$ and octupole deformation $\beta_{3}$ parameters. We use
separate WS distributions for protons and neutrons, which allows us to
incorporate effects such as a neutron skin. As in [26], TRENTo also includes a
hard-core repulsion implemented through a minimal inter-nucleon distance
$d_{\text{min}}$.

In this work, we have used five different cases for the WS distribution, where
for both ${}_{40}^{96}$Zr and ${}_{44}^{96}$Ru the parameters are given in
Tab. 1. Cases 1 and 2 correspond to cases 1 and 2 from [9, 1]. Case 1 from [9]
is in turn based on $e$–A scattering experiments [2, 3], and case 2 from [9]
is produced from calculations based on the finite-range liquid-drop
macroscopic model [30]. Cases 3 and 4 come from the top and bottom rows of
Tab. 2 from [31], which are based on recent DFT. The DFT calculations produce
more information than can be captured by the WS parameterization, so in [31]
fits were performed to approximate the DFT result by a WS distribution. This
has been done assuming $\beta_{2}=0$ (case 3) and $\beta_{2}=0.16$ (case 4).
Case 5 is the same as 4, but with $\beta_{2}$ and $\beta_{3}$ taken from [16]
(see also [2, 32]; this $\beta_{3}$ in particular was motivated by the STAR
measurement [1]). Cases 3 to 5 have fitted protons and neutrons separately,
which leads to a significant neutron skin for ${}_{40}^{96}$Zr. It is
important that in this case the size of the nucleus does not correspond to its
charge radius (as in case 1 and 2), but we note that in our computation we
treat protons and neutrons equally (see also Appendix A). In this way, case 5
captures both the neutron skin effect, which leads to ${}_{44}^{96}$Ru being
slightly smaller than ${}_{40}^{96}$Zr, as well as nuclear deformations
motivated by experimental measurements.

Figure 1: We show $R(\theta)$ (left) as in (1) for our 5 different WS
distributions of ${}_{44}^{96}$Ru (left half of plot) and ${}_{40}^{96}$Zr
(right half of plot), where $z=0$ and nuclei have azimuthal symmetry around
the $y$-axis. Here the $R$ parameter has been averaged from $R_{p}$ and
$R_{n}$ according to the number of protons and neutrons in each nucleus. We
also show 3D models (right) of the nuclei.

Fig. 1 illustrates the different sizes and shapes of ${}_{44}^{96}$Ru (left)
and ${}_{40}^{96}$Zr (right). Since the WS distribution is separate for
protons and neutrons, in Fig. 1 we use averaged values for the radius $R$,
weighted according to the number of protons and neutrons in each nucleus. For
${}_{44}^{96}$Ru, it can be seen that cases 1, 4 and 5 are more elliptical
than 2 and 3, which is caused by the larger value for $\beta_{2}$ in cases 1,
4 and 5. For ${}_{40}^{96}$Zr, in addition to an elliptical shape, one can
also see a large triangular deformation for case 5, which is due to the non-
zero value for $\beta_{3}$.

Having specified the shape of the isobars we simulate collisions using
_Trajectum_. We stress, however, that we use settings tuned to PbPb collisions
at LHC energies, and hence no perfect agreement should be expected at RHIC
collision energies. Furthermore, we reduced the normalization of the initial
energy density from 18.3 to 7.5, which is loosely tuned to RHIC entropy
densities (we find that we underestimate the multiplicity by 18%). For all
cases for both ${}_{44}^{96}$Ru and ${}_{40}^{96}$Zr we simulated 0.5M hydro
evolutions, except for case 5, where we have 5M hydro events each to be able
to probe the 0.1% most central collisions 222The total computational cost was
about 200k CPU hours, _Trajectum_ output files can be found at
http://wilkevanderschee.nl/public-codes..

##  3 Multiplicity and STAR centrality classes

One of the potentially surprising features of the STAR [1] result is that the
multiplicity in Ruthenium collisions is higher than in Zirconium. This can
potentially be explained by Ruthenium’s smaller size and hence smaller cross
section (see Tab. 1). To compare the exact multiplicities to theoretical
computations is however subtle due to various reasons.

Figure 2: We show the STAR $N_{\rm trk,raw}^{\rm off}$ distributions [1]
together with _Trajectum_ results as in Fig. 4 together with centrality
classes computed by _Trajectum_ (case 5, colored lines) and STAR (dashed).
Several corrections have been applied to the theoretical curves, including a
correction to the multiplicity (1.21) due to insufficient tuning to RHIC
energies, a normalization correction (1.31) to account for STAR not detecting
every collision, a correction for detector acceptance (following [33]) and in
the ratio a correction to the experimentally chosen normalizations of
${}_{44}^{96}$Ru and ${}_{40}^{96}$Zr in the ratio. After these corrections
all cases 3 through 5 have satisfactory agreement. Figure 3: We show ratios of
multiplicity (left) and elliptic flow (right) using the ${}_{44}^{96}$Ru and
${}_{40}^{96}$Zr centrality classes defined in [1]. Note that we divide
${}_{44}^{96}$Ru (blue) and ${}_{40}^{96}$Zr (red) by itself and hence just
illustrate the centrality class effect. To emphasize that centrality is in
this case a label for a more sophisticated definition we put “centrality” in
quotation marks on the x-axis. Statistical uncertainties are estimated
conservatively by adding up the original fractional uncertainties in
quadrature. In reality both quantities use the same event sample and by
comparing the Ru and Zr independent samples it can be seen that significant
cancellations occur.

Ideally an experiment would know the exact luminosity and total hadronic cross
section, such that events can be accurately binned into centrality classes. In
practice the luminosity is poorly known and at low multiplicity not all events
are detected, which is the reason why _Trajectum_ gives a much higher
probability for events with fewer than 20 tracks in the detector as seen in
Fig. 2. The centrality classes in this Figure hence have to be computed using
theoretical modelling, which is done here using _Trajectum_ for the
theoretical results for both ${}_{44}^{96}$Ru and ${}_{40}^{96}$Zr separately
and done using an MC Glauber fitting procedure by experiments [1] (see also
[34]). Secondly, the distributions by STAR are raw tracks, which have to be
corrected for detector effects (without these corrections it is difficult to
fit the multiplicity distribution [35]). Here we apply the inverse correction
on _Trajectum_ results using the linear interpolation as in [1] (original in
[33]) and we apply pseudorapidity $|\eta|<0.5$ and $p_{T}>0.2\,$GeV cuts for
all charged hadrons.

After these corrections all in all _Trajectum_ gives an excellent description
of the STAR multiplicity in the regime where it is applicable, provided the
_Trajectum_ multiplicity is multiplied by 1.21 (this factor arises since the
_Trajectum_ norm is only loosely tuned to collisions at RHIC energies) and the
normalization of the $N_{\rm trk,raw}^{\rm off}$ distribution changed to 1.31
(this factor captures the fact the STAR curves are normalized and that STAR
cannot detect some low multiplicity events).

For the present work we are mostly interested in the isobar ratio. This ratio
has an extra subtlety, again related to the normalizations. The experimental
curves can only be reliably measured for high multiplicity events and hence a
prescription has to be made for the normalization. Here we follow STAR and
normalize the models such that the normalization restricted to $N_{\rm
trk,raw}^{\rm off}>50$ matches between the model and the data. Since the
multiplicity distributions for ${}_{44}^{96}$Ru and ${}_{40}^{96}$Zr are
rather different this leads to non-trivial correction factors. For the STAR
distributions this gives a correction factor of 1.00857 with respect to the
normalized distributions over the full range. For the _Trajectum_ postdictions
we simply integrate the $N_{\rm trk,raw}^{\rm off}>50$ distribution and
subsequently take the ratio, which leads to correction factors of respectively
0.99844, 1.0005, 0.97943, 0.9792 and 0.9777. We then apply both the
experimental and model corrections to the theoretical curves to allow the full
comparison in Fig. 2 (note that the STAR ratio is simply the ratio of the full
normalized raw distributions).

All cases 3 through 5 do a good job reproducing the STAR multiplicity
distribution, which illustrates the fact that the multiplicity distribution is
mostly determined by the size of the nuclei and not so much by their shape. It
is somewhat curious that case 2 performs well in the central region, even
though this is not the case for the MC Glauber simulation done in [1]. This
could indicate that for the ratio it is necessary to do a full hydrodynamic
evolution as opposed to a simple MC Glauber simulation.

Once we identified the multiplicity distribution it is possible to bin our set
of events in centrality classes. For centrality classes that contain low
multiplicity events there are essentially two methods to deal with the fact
that a centrality definition (involving an integer number of detected
particles satisfying some cuts) does not necessarily align with the chosen
centrality classes. The first method, commonly used by experiments, is to
compute the observable measured for each integer multiplicity separately. When
measuring in a centrality class one would give a particular integer an
appropriate weight if it happens to lie on the centrality class boundary.

A second method, employed by _Trajectum_ , is to group events in the
centrality classes, whereby events across a centrality class are simply
assigned to either class in a properly chosen fraction. Of course for each
event such an assignment is arbitrary (in _Trajectum_ it depends on the order
of event generation), but since each event is random this arbitrariness does
not affect the final observable.

Of course, another option is to conveniently choose centrality classes where
the boundaries exactly correspond to the centrality definition chosen. This is
for instance done in the STAR paper [1], such that it is possible to choose
for instance 70.04–79.93% class for ${}_{44}^{96}$Ru and 70.00–80.88% for
${}_{40}^{96}$Zr. In principle there is nothing wrong with such a choice, but
when taking a ratio it is important to be careful and remember that one is
dividing unequal centrality classes. In _Trajectum_ it is possible to take
arbitrary classes, so it is possible to divide such unequal classes, to be
able to do a full comparison with the experimental data.

For the multiplicity, being strongly centrality dependent, this is an
important consideration, as illustrated in Fig. 3 (left). In fact it is the
dominant effect going from the 60–70% class to the 70–80% class. For elliptic
flow ($v_{2}\\{2\\}$, shown in Fig. 3 (right)) on the other hand the
observable is not strongly dependent on centrality and we did not find an
effect within our statistical uncertainties.

##  4 Flow observables and transverse momentum

Figure 4: We show $v_{2}\\{2\\}{}$ (left), $v_{3}\\{2\\}{}$ (middle) and
$\langle p_{T}\rangle{}$ (right) for ${}_{44}^{96}$Ru (top), ${}_{40}^{96}$Zr
(middle) and their ratio (bottom) for all five cases of Tab. 1 together with
STAR data [1]. Note that _Trajectum_ is only tuned to LHC energies and hence
an absolute agreement is not expected. Case 5 is the only case with an
octupole deformation $\beta_{3}$, which leads to a qualitative agreement for
the $v_{3}\\{2\\}{}$ ratios and full consistency for the $v_{2}\\{2\\}{}$
ratios. All theoretical uncertainties are statistical only (gray).

Fig. 4 shows the elliptic flow $v_{2}\\{2\\}$ (left), the triangular flow
$v_{3}\\{2\\}$ (middle) and the mean transverse momentum $\langle
p_{T}\rangle$ (right) of charged hadrons for both ${}_{44}^{96}$Ru and
${}_{40}^{96}$Zr for all five cases and compared to STAR data where available
[1]. For the absolute values we show charged particle $v_{2}\\{2\\}$, while
for the ratio RuRu/ZrZr we include all particles to increase statistics
(charged particle $v_{2}\\{2\\}$ is about 1.5% smaller than the inclusive
$v_{2}\\{2\\}$, but this effect cancels in the ratio). The absolute
postdictions of $v_{2}\\{2\\}$ are slightly too high, but we stress again that
_Trajectum_ has only been tuned to LHC energies. More important is the perfect
agreement of case 5 for the RuRu/ZrZr ratio of $v_{2}\\{2\\}$, capturing both
the increase in very central collisions (0–5%), as well as the increase
towards peripheral collisions. The trend for the central $v_{3}\\{2\\}$ ratio
is also reproduced, but not quantitatively. We note here that case 5 is the
only case that includes a $\beta_{3}$ deformation, which was fitted to these
STAR data based on AMPT simulations [16]. We see that the $\beta_{3}$
deformation is indeed essential to explain the central $v_{3}\\{2\\}$, but
likely $\beta_{3}=0.202$ is an overestimate of the deformation (see also Fig.
1 and see Appendix B for a more quantitative analysis of $\beta_{3}$).

In the right panel we show the mean transverse momentum, for which STAR data
is not yet available (see however [21]). All cases 3 through 5 have similar
ratios, which confirms the intuition that mean transverse momentum is not
sensitive to the deformations of the nucleus, but that the difference in
neutron skin does matter. The effect of the neutron skin is further studied in
Appendix A.

Figure 5: For case 5 we show ratios of $v_{2}\\{2\\}{}$ (left),
$v_{3}\\{2\\}{}$ (middle) and $\delta p_{T}/\langle p_{T}\rangle{}$ (right)
together with preliminary STAR data [36] as a function of $N_{\rm
trk,raw}^{\rm off}$. This allows to go to extremely ultracentral collisions,
up to centrality classes of 0.01% (see colored lines), which are much more
sensitive to deformations in the nuclear shape. Indeed the $v_{2}\\{2\\}{}$
and $v_{3}\\{2\\}{}$ deviate more than in Fig. 4, but nevertheless they are as
well described by _Trajectum_ as in Fig. 4. The STAR data contains 0.5%
preliminary systematic uncertainty.

##  5 Ultracentral collisions

As is clear from both Fig. 1 and 4 it is interesting to study extremely
ultracentral collisions. These have about $N_{\rm trk,raw}^{\rm off}>300$ and
correspond to collisions where the two colliding nuclei are completely
aligned: there are no spectator nucleons that fly on without interactions.
Importantly, this implies that the deformations of the nuclei translate rather
precisely into the deformation of the QGP formed, which can then be extracted
from the elliptic and triangular flow. Indeed Fig. 5 shows a much stronger
effect on the ratio of $v_{2}\\{2\\}$ and $v_{3}\\{2\\}$ as we increase
$N_{\rm trk,raw}^{\rm off}$. As in Fig. 4 the $v_{2}\\{2\\}$ dependence is
exactly reproduced, whereas $v_{3}\\{2\\}$ is overestimated.

Lastly, we show the fluctuations of the mean transverse momentum in Fig. 5
(right) (see also [26] for similar predictions for PbPb collisions). This
observable is statistically difficult, especially to obtain the isobar ratio
precisely, but the overall trend seems to be consistent with preliminary STAR
data [36].

Figure 6: It is an interesting question if our $v_{2}\\{2\\}$ (top) and
$v_{3}\\{2\\}$ (bottom) could have been anticipated by initial geometric
differences of $\epsilon_{n}\\{n\\}$, as in [7]. We show such comparisons for
ZrZr/RuRu (case 5, left) and for a Zr-only ratio, where we divide the case of
$\beta_{3}=0.202$ with the case $\beta_{3}=0$ (right).

##  6 Initial eccentricity and hydro response

Part of the fit of $v_{2}\\{2\\}{}$ and $v_{3}\\{2\\}{}$ could be anticipated
by the changes in the initial geometry of the QGP, as shown in Fig. 6. Indeed,
in QGP studies the $v_{n}\\{2\\}$ distributions usually closely follow the
spatial anisotropy distributions $\epsilon_{n}\\{2\\}$ so that
$v_{n}\\{2\\}=\kappa_{n}\epsilon_{n}\\{2\\}$ (see [7]). The coefficients
$\kappa_{n}$ have a strong dependence on the shear viscosity (larger shear
viscosity leads to reduced anisotropic flow $v_{n}$), but a priori it is
unclear if $\kappa_{n}$ depends on the isobar in question. Fig. 6 (left) shows
indeed that it does, since the $v_{n}\\{2\\}$ and $\epsilon_{n}\\{2\\}$
isobars are qualitatively different for peripheral collisions (note that for
central collisions we find better agreement). This implies that $\kappa_{n}$
is different for ${}_{44}^{96}$Ru and ${}_{40}^{96}$Zr, which likely can be
attributed to their different sizes.

This interpretation is further strengthened by Fig. 6 (right), which shows a
ratio of the two simulations of ${}_{40}^{96}$Zr with case 5 and case 5 with
$\beta_{3}=0$ (see also Appendix B). In this case the $v_{n}\\{2\\}$ and
$\epsilon_{n}\\{2\\}$ ratios are consistent. Interestingly, the change in
$\beta_{3}$ has a strong effect also on the ellipticity $\epsilon_{2}\\{2\\}$.
We also note that known non-linearities [7] between $v_{n}\\{2\\}$ and
$\epsilon_{n}\\{2\\}$ do not play a large role here, since indeed
$v_{n}\\{2\\}$ and $\epsilon_{n}\\{2\\}$ change indistinguishably within our
statistical uncertainties.

Even though $\kappa_{n}$ depends on the viscosities we stress that all our
isobar ratios do not (strongly) depend on the viscosities, since the
viscosities lead to a similar change in $\kappa_{n}$ for both ${}_{44}^{96}$Ru
and ${}_{40}^{96}$Zr (see also [21] for similar observations using AMPT for
$\langle p_{T}\rangle{}$). We show this by explicit computation in Appendix B.
More generally, it is expected that the dependence of observables on other
model dependencies such as $d_{\rm min}$ in the initial state or other pre-
hydrodynamic parameters mostly cancel when taking a ratio of observables from
the two isobars.

##  7 Discussion

All of our results agree qualitatively with AMPT studies [17, 37] and as such
our study is an important confirmation of those conclusions using a true
hydrodynamic evolution. One surprising finding, however, is the non-trivial
$v_{3}\\{2\\}{}$ isobar ratio for central collisions even in the absence of an
octupole deformation $\beta_{3}$. This can be seen for all cases in Fig. 4,
and also for a high statistics verification in Appendix B. It is hence non-
trivial to estimate the $\beta_{3}$ value that would fit the measured
$v_{3}\\{2\\}{}$ ratio. Moreover, as shown in Section 6 $\beta_{3}$ also
strongly affects $v_{2}\\{2\\}{}$, which indicates that a full global analysis
[38, 25, 39] may be needed to obtain even more accurate results. Importantly,
a full hydrodynamic analysis seems necessary as just focusing on the
$\epsilon_{n}$ initial anisotropies does not capture the $v_{n}\\{2\\}$
dependence quantitatively (see Section 6).

In one way the current work provides an accurate analysis of the ‘baseline’
observables for the ${}_{44}^{96}$Ru and ${}_{40}^{96}$Zr isobar run performed
at RHIC. The true motivation for this run was to study the effects of magnetic
field, which will only be included in _Trajectum_ in the future. The first
step for the isobar run is however to obtain an accurate baseline including
the shapes of ${}_{44}^{96}$Ru and ${}_{40}^{96}$Zr, which is provided in this
work. Importantly, we have seen that this baseline is interesting in its own
right, which bridges the fields of relativistic heavy ions and nuclear
structure.

## Acknowledgements

We are very grateful to Chunjian Zhang for providing STAR data and for his
explanations. We thank Jiangyong Jia, Rongrong Ma, Rosi Reed, Mike Sas and
Prithwish Tribedy for interesting discussions, and Giuliano Giacalone for a
careful reading of the manuscript. GN is supported by the U.S. Department of
Energy, Office of Science, Office of Nuclear Physics under grant Contract
Number DE-SC0011090.

Figure 7: We show the effect of using single radius and skin depth parameters
in the WS distribution for both neutrons and protons for case 4. These are
refits to the results from DFT, though they are close to taking the weighted
average of the corresponding proton and neutron WS parameters. All results
agree within our statistical uncertainties.

##  Appendix A Averaging WS proton and neutron values

The values for WS parameters in Tab. 1 treat protons and neutrons separately
for cases 3 through 5. Recall that these values were obtained by fitting to a
DFT result. Tab. 2 from [31] also contains fits which do not treat protons and
neutrons separately, but instead describe both using a single WS distribution.
It is important that this single WS distribution still takes the neutron skin
into account in the sense that this WS distribution is not fitted to the
charge radius, but instead in a sense averaged from the proton and neutron
distributions. This procedure leads to $R=5.065$, $\sigma=0.485$ for
${}_{44}^{96}$Ru and $R=4.961$, $\sigma=0.544$ for ${}_{40}^{96}$Zr. This is
not exactly the same as simply taking a weighted average of $R$ and $\sigma$
over the separate proton and neutron WS distributions, which would have given
$R=5.064$, $\sigma=0.485$ for ${}_{44}^{96}$Ru and $R=4.967$, $\sigma=0.541$
for ${}_{40}^{96}$Zr, but close enough to justify the term average. In Fig. 7,
we compare the flow and mean transverse momentum for case 4 with separate
proton and neutron distributions to the corresponding results with the
‘averaged’ WS distribution. As the results are compatible with each other
within the statistical uncertainties, we conclude that for these nuclei the
averaging does not have a big effect on the observables shown, but as is clear
from comparing case 3 with case 1 and 2 the neutron skin by itself is
important.

Figure 8: We show the effect of turning off $\beta_{3}$ on $v_{2}\\{2\\}$
(left), $v_{3}\\{2\\}$ (middle) and hadron $\langle p_{T}\rangle$ (right) for
case 5. As expected $\beta_{3}$ has a strong effect on $v_{3}\\{2\\}{}$,
although it can be seen that even without $\beta_{3}$ there is a nontrivial
baseline $v_{3}\\{2\\}$ ratio (estimated as 0.98 by averaging case 1 through 4
in Fig. 4). Importantly, we see that $\beta_{3}$ has an important effect also
on $v_{2}\\{2\\}{}$. Figure 9: We show the effect on $v_{2}\\{2\\}{}$ (left),
$v_{3}\\{2\\}{}$ (middle) and $\langle p_{T}\rangle{}$ of halving and doubling
the shear viscosity (dashed and dotted) as well as doubling the bulk viscosity
(dashed-dotted). As expected the viscosities have significant effects on all
observables, but it can be seen that in the isobar ratio these effects cancel.

##  Appendix B Hydrodynamic response to $\beta_{3}$ and the viscosities.

Given that case 5 does a much better job than case 4 at describing the
$v_{2}\\{2\\}$ ratio between RuRu and ZrZr, an interesting question is to what
extent this is due to the octupole deformation $\beta_{3}$. Since cases 4 and
5 differ in both $\beta_{2}$ and $\beta_{3}$, it is not immediately obvious
how the effects from $\beta_{2}$ and $\beta_{3}$ can be disentangled. To study
this in some more detail, in Fig. 8, we show $v_{2}\\{2\\}$ (left),
$v_{3}\\{2\\}$ (middle) and charged hadron $\langle p_{T}\rangle$ (right) for
case 5, as well as a modified case 5 where $\beta_{3}$ has been set to zero
(we use simulations with 5M and 2.5M events respectively). Indeed $\beta_{3}$
has a strong effect on $v_{3}\\{2\\}{}$ for central collisions, but even with
$\beta_{3}=0$ the isobar ratio is significantly below unity. Interestingly,
even though $\beta_{3}$ is a triangular deformation, its effect on
$v_{2}\\{2\\}$ is also sizable, on the order of a couple percent for more
central collisions. Given that $\beta_{2}$ and $\beta_{3}$ interplay in their
influence on $v_{2}\\{2\\}$ and $v_{3}\\{2\\}$, this indeed shows, as remarked
in the discussion, that decreasing $\beta_{3}$ to improve the fit of
$v_{3}\\{2\\}$ would also necessitate a refit of $\beta_{2}$ to ensure that
$v_{2}\\{2\\}$ remains in agreement with experimental data.

In Fig. 9 we show explicitly that the results on the isobar ratio do not
depend on the values of the shear and bulk viscosities (see also [17] for
similar observations using AMPT). This implies that while $\kappa_{n}$ depends
on the viscosities this dependence cancels in the ratio. In particular it
implies that $\kappa_{n}$ consists of terms that depend on the size of the
nucleus (which differs between the isobars) and the viscosities (which are
independent of the nuclei colliding), but does not (sensitively) depend on a
combination of the two.

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# On $Q$-polynomial distance-regular graphs with a linear dependency involving
a $3$-clique

Mojtaba Jazaeri  Department of Mathematics, Shahid Chamran University of
Ahvaz, Ahvaz, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 2$. Let $E$
denote a primitive idempotent of $\Gamma$ with respect to which $\Gamma$ is
$Q$-polynomial. Assume that there exists a $3$-clique $x,y,z$ such that
$E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. In this paper, we
classify all the $Q$-polynomial distance-regular graphs $\Gamma$ with the
above property. We describe these graphs from multiple points of view.

###### Key words and phrases:

Distance-regular graph, $Q$-polynomial, classical parameters, regular near
polygon.

###### 1991 Mathematics Subject Classification:

Primary: 05E30. Secondary: 05C25.

##  1\. Introduction

This paper is about a certain kind of finite undirected graph, said to be
distance-regular [1, § 4.1(A)], [5, § 2]. There is a well known property for a
distance-regular graph, called the $Q$-polynomial property [1, § 4.1(E)], [5,
§ 11]. In this paper, we classify a certain type of $Q$-polynomial distance-
regular graph. In our treatment, the following concepts will be relevant. We
will consider the concepts of classical parameters [5, § 18], negative type
[9], the cosine sequence [5, § 4], and regular near $2D$-gons [1, § 6.4].
Before we state our main results, we give some background about these
concepts.

For the rest of this section, let $\Gamma$ denote a distance-regular graph
with diameter $D\geq 2$. The concept of classical parameters was introduced in
[1, § 6.1]. If $\Gamma$ has classical parameters, then the intersection
numbers of $\Gamma$ are given by attractive formulas in terms of four
parameters $(D,b,\alpha,\sigma)$ [1, § 6.1(1a,1b)]. See [1, § 6], [2, §
3.1.1], [5, § 18] for some results about classical parameters.

Assume that $\Gamma$ has classical parameters $(D,b,\alpha,\sigma)$. It is
known that $b$ is an integer not equal to $0$ or $-1$ (cf. [1, Proposition
6.2.1]). The graph $\Gamma$ is said to have negative type whenever $b<-1$ [9,
§ 1]. Distance-regular graphs with negative type are investigated in [2, §
5.2], [9], [10].

Associated with each eigenvalue of $\Gamma$, there is a sequence of scalars
$\\{\sigma_{i}\\}_{i=0}^{D}$ called the cosine sequence. For $0\leq i\leq D$,
the scalar $\sigma_{i}$ can be interpreted as an angle cosine, see (5) below.
It is known that the cosine sequence satisfies a $3$-term recurrence, see [5,
§ 14] and (2), (3), (4) below. See [1, § 8.1], [2, § 2.5], [5, § 4] for some
results about the cosine sequence.

A regular near $2D$-gon is a type of distance-regular graph with an attractive
geometric structure [1, § 6.4]. The regular near $2D$-gons were first
introduced in [4]. An example of the regular near $2D$-gons are the dual polar
graphs [1, § 9.4]. See [3] for a detailed study of the regular near $2D$-gons
that are $Q$-polynomial. Other studies of the regular near $2D$-gons can be
found in [1, § 6.6], [2, § 9.6], [8].

Paul Terwilliger has stated the following problem (cf. [7, Problem 1]).

######  Problem 1.1.

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 2$. Let $E$
denote a primitive idempotent of $\Gamma$ with respect to which $\Gamma$ is
$Q$-polynomial. Assume that there exists a $3$-clique $x,y,z$ such that
$E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. Investigate the
combinatorial meaning of this condition.

In this paper, we investigate the $Q$-polynomial distance-regular graphs with
the given property. The following is our main result.

######  Theorem 1.2.

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 2$. Let $E$
denote a primitive idempotent of $\Gamma$ with respect to which $\Gamma$ is
$Q$-polynomial. Then the following are equivalent:

  1. (i)

There exists a $3$-clique $x,y,z$ such that $E\hat{x},E\hat{y},E\hat{z}$ are
linearly dependent;

  2. (ii)

The graph $\Gamma$ has classical parameters
$(D,b,\alpha,\sigma)=(D,-2,\alpha,2+\alpha-\alpha[_{1}^{D}])$ and $E$ is for
the eigenvalue $\frac{b_{1}}{b}-1$;

  3. (iii)

The graph $\Gamma$ is a regular near $2D$-gon of order $(2,t)$ and $E$ is for
the eigenvalue $-t-1$;

  4. (iv)

The intersection number $a_{1}=1$ and for every $3$-clique $x,y,z$ we have
$E\hat{x}+E\hat{y}+E\hat{z}=0$;

  5. (v)

The graph $\Gamma$ is listed below and $E$ is for the minimal eigenvalue of
$\Gamma$:

     * •

The unique regular near $4$-gon of order $(2,1)$,

     * •

The unique regular near $4$-gon of order $(2,2)$,

     * •

The unique regular near $6$-gon of order $(2,8)$,

     * •

The unique regular near $6$-gon of order $(2,11)$,

     * •

The unique regular near $6$-gon of order $(2,14)$,

     * •

The dual polar graph $A_{2D-1}(2)$;

  6. (vi)

The cosine sequence $\\{\sigma_{i}\\}_{i=0}^{D}$ for $E$ satisfies
$\sigma_{i}=(-\frac{1}{2})^{i}$, where $0\leq i\leq D$.

This paper is organized as follows. In Section 2, we give some basic facts
that will be used to state and prove our main results. In Section 3, first we
state and prove a sequence of lemmas and propositions. Next we use these
results to prove Theorem 1.2.

##  2\. Preliminaries

From now on, let $\Gamma$ be a connected graph with vertex set $X$ and
diameter $D\geq 2$. For $x,y\in X$, let $d(x,y)$ denote the path-length
distance between $x$ and $y$. Pick $x,y\in X$ and write $d(x,y)=i$. Let
$b_{i}$ denote the number of neighbors of $x$ at distance $i+1$ from $y$,
$a_{i}$ denote the number of neighbors of $x$ at distance $i$ from $y$, and
$c_{i}$ denote the number of neighbors of $x$ at distance $i-1$ from $y$. The
graph $\Gamma$ is called distance-regular whenever $a_{i}$, $b_{i}$, and
$c_{i}$ are independent of $x,y$ and depend only on $i$. For the rest of this
paper, assume that $\Gamma$ is distance-regular. Note that $\Gamma$ is regular
with valency $k=b_{0}$, and that

(1) $k=a_{i}+b_{i}+c_{i}\;\;\;\;(0\leq i\leq D),$

where $b_{D}=0$ and $c_{0}=0$. The sequence

$\\{b_{0},b_{1},\ldots,b_{D-1};c_{1},c_{2},\ldots,c_{D}\\}$

is called the intersection array of $\Gamma$. Pick $x\in X$. For $0\leq i\leq
D$, let $k_{i}$ denote the number of vertices in $X$ at distance $i$ from $x$.
Note that $k_{0}=1$ and $k_{1}=k$. By a routine counting argument, we find
$k_{i}c_{i}=k_{i-1}b_{i-1}$ for $1\leq i\leq D$. It follows that $k_{i}$ is
independent of choice of $x$.

Let $V=\mathbb{R}^{|X|}$ denote the vector space over $\mathbb{R}$, consisting
of the column vectors with coordinates indexed by $X$ and all entries in
$\mathbb{R}$. We endow $V$ with a bilinear form $\langle~{},~{}\rangle$ that
satisfies $\langle u,v\rangle=u^{t}v$ for $u,v\in V$, where $t$ denotes
transpose. We abbreviate $\langle u,u\rangle$ by $\|u\|^{2}$. We note that
$\|u\|^{2}\geq 0$, with equality if and only if $u=0$. For $x\in X$, let
$\hat{x}$ denote a vector in $V$ that has $x$-coordinate $1$ and all other
coordinates $0$. Observe that the vectors $\\{\hat{x}\mid x\in X\\}$ form an
orthonormal basis for $V$.

Let $Mat_{X}(\mathbb{R})$ denote the $\mathbb{R}$-algebra consisting of the
matrices with rows and columns indexed by $X$ and all entries in $\mathbb{R}$.
Let $A\in Mat_{X}(\mathbb{R})$ denote the adjacency matrix of $\Gamma$. Then
the Bose-Mesner algebra of $\Gamma$ is the subalgebra of $Mat_{X}(\mathbb{R})$
generated by $A$. The Bose-Mesner algebra of $\Gamma$ has a basis
$\\{E_{i}\\}_{i=0}^{D}$ such that $E_{0}=|X|^{-1}J$,
$E_{i}E_{j}=\delta_{i,j}E_{i}$ ($0\leq i,j\leq D$), and
$\sum_{i=0}^{D}E_{i}=I$, where $I$ is the identity matrix, and $J$ is the all
one matrix (cf. [5, p. 4]). Following [5], we call $\\{E_{i}\\}_{i=0}^{D}$ the
primitive idempotents of $\Gamma$. The primitive idempotent $E_{0}$ is called
trivial. For $B,C\in Mat_{X}(\mathbb{R})$, define the matrix $B\circ C\in
Mat_{X}(\mathbb{R})$ with entries

$(B\circ C)_{y,z}=B_{y,z}C_{y,z}\;\;\;\;(y,z\in X).$

The operation $\circ$ is called entrywise multiplication. Recall that the
Bose-Mesner algebra of $\Gamma$ is closed under entrywise multiplication (see
[5, § 5]). By [5, Eqn.(8)], there exist real numbers $q_{i,j}^{h}$ ($0\leq
h,i,j\leq D$) such that

$E_{i}\circ E_{j}=\frac{1}{|X|}\sum_{h=0}^{D}q_{i,j}^{h}E_{h}\;\;\;\;\;(0\leq
i,j\leq D).$

The parameters $q_{i,j}^{h}$ are called the Krein parameters. These parameters
are nonnegative and this property is called Krein condition [5, § 5]. Because
$\\{E_{i}\\}_{i=0}^{D}$ is a basis for the Bose-Mesner algebra of $\Gamma$,
there exist real numbers $\\{\theta_{i}\\}_{i=0}^{D}$ such that

$A=\sum_{i=0}^{D}\theta_{i}E_{i}.$

The scalars $\theta_{i}$ $(0\leq i\leq D)$ are mutually distinct (see [1, §
4.1(B)]). For $0\leq i\leq D$, we have $AE_{i}=\theta_{i}E_{i}$. Therefore
$\theta_{i}$ is an eigenvalue of $A$, and $E_{i}V$ is the corresponding
eigenspace. By the eigenvalues of $\Gamma$, we mean the scalars $\theta_{i}$
$(0\leq i\leq D)$. Let $m_{i}$ denote the dimension of $E_{i}V$. Then
$\|E_{i}\hat{x}\|^{2}=|X|^{-1}m_{i}$ for all $x\in X$ (cf. [5, Lemma
4.1(ii)]).

Let $E$ denote a primitive idempotent of $\Gamma$, and let $\theta$ denote the
corresponding eigenvalue. We define a sequence of scalars
$\\{\sigma_{i}\\}_{i=0}^{D}$ such that

(2) $\displaystyle\sigma_{0}=1,\qquad\sigma_{1}=\frac{\theta}{k},$ (3)
$\displaystyle\theta\sigma_{i}=c_{i}\sigma_{i-1}+a_{i}\sigma_{i}+b_{i}\sigma_{i+1}\;\;\;\;1\leq
i\leq D-1.$

By [5, Lemma 4.8], we have

(4) $\displaystyle\;\;c_{D}\sigma_{D-1}+a_{D}\sigma_{D}=\theta\sigma_{D}.$

The sequence $\\{\sigma_{i}\\}_{i=0}^{D}$ is called the cosine sequence for
$E$ (or $\theta$). This name is motivated by the following result. Pick
$x,y\in X$ and write $d(x,y)=i$. By [5, Lemma 4.1(iii)] we have

(5) $\sigma_{i}=\frac{\langle
E\hat{x},E\hat{y}\rangle}{\|E\hat{x}\|\|E\hat{y}\|}.$

Observe that $\sigma_{i}$ is the cosine of the angle between $E\hat{x}$ and
$E\hat{y}$.

The graph $\Gamma$ is called $Q$-polynomial (with respect to the given
ordering $\\{E_{i}\\}_{i=0}^{D}$ of the primitive idempotents) whenever the
following holds for $0\leq h,i,j\leq D$ (cf. [5, Definition 11.1]):

  1. (i)

$q_{i,j}^{h}=0$ if one of $h,i,j$ is greater than the sum of the other two,

  2. (ii)

$q_{i,j}^{h}\neq 0$ if one of $h,i,j$ is equal to the sum of the other two.

In this case, we say that $\Gamma$ is $Q$-polynomial with respect to $E$,
where $E=E_{1}$. Recall that if $\Gamma$ is $Q$-polynomial with respect to
$E$, then the elements of $\\{\sigma_{i}\\}_{i=0}^{D}$ are mutually distinct
(cf. [1, Proposition 8.1.3]).

Recall that our distance-regular graph $\Gamma$ is said to have classical
parameters $(D,b,\alpha,\sigma)$ whenever the intersection array satisfies

(6) $c_{i}=[^{i}_{1}](1+\alpha[^{i-1}_{1}])\;\;\;\;\;(0\leq i\leq D),$ (7)
$b_{i}=([^{D}_{1}]-[^{i}_{1}])(\sigma-\alpha[^{i}_{1}])\;\;\;\;\;(0\leq i\leq
D),$

where $[^{i}_{1}]=[^{i}_{1}]_{b}=1+b+\cdots+b^{i-1}$ for $1\leq i\leq D$. Note
that by a convention in [1, § 6.1(2)], we have $[^{0}_{1}]=0$. Recall that if
$\Gamma$ has classical parameters, then $\Gamma$ is $Q$-polynomial with
respect to the following ordering of the eigenvalues of $\Gamma$ (cf. [5,
Theorem 18.2, Lemma 18.3]):

(8) $\theta_{i}=\frac{b_{i}}{b^{i}}-[^{i}_{1}]\;\;\;\;\;(0\leq i\leq D).$

##  3\. Main result

In this section, we concentrate on Problem 1.1.

######  Lemma 3.1.

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 2$. Let $E$
denote a nontrivial primitive idempotent of $\Gamma$. Let $x,y,z$ denote a
$3$-clique in $\Gamma$. Then the following items hold.

  1. (i)

The matrix of inner products of $E\hat{x},E\hat{y},E\hat{z}$ is $|X|^{-1}mC$,
where $m$ is the rank of $E$ and

$C=\left[\begin{array}[]{ccc}\sigma_{0}&\sigma_{1}&\sigma_{1}\\\
\sigma_{1}&\sigma_{0}&\sigma_{1}\\\ \sigma_{1}&\sigma_{1}&\sigma_{0}\\\
\end{array}\right].$

  2. (ii)

The eigenvalues of $C$ are $1-\sigma_{1}$, $1-\sigma_{1}$, and
$1+2\sigma_{1}$.

  3. (iii)

We have $1>\sigma_{1}\geq-\frac{1}{2}$.

  4. (iv)

$\sigma_{1}=-\frac{1}{2}$ if and only if $E\hat{x},E\hat{y},E\hat{z}$ are
linearly dependent. In this case, $E\hat{x}+E\hat{y}+E\hat{z}=0$.

###### Proof.

(i): By (5).
(ii): Use linear algebra and recall that $\sigma_{0}=1$.
(iii): By construction $C$ is positive semidefinite, so its eigenvalues are
nonnegative. We have $\sigma_{1}\neq 1$ since $E$ is nontrivial.
(iv): By linear algebra, we have $\sigma_{1}=-\frac{1}{2}$ if and only if $0$
is an eigenvalue of $C$ if and only if $C$ is singular if and only if
$E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. Assume that this is the
case. Then

$\|E\hat{x}+E\hat{y}+E\hat{z}\|^{2}=3|X|^{-1}m(1+2\sigma_{1})=0.$

It follows that $E\hat{x}+E\hat{y}+E\hat{z}=0$. ∎

######  Lemma 3.2.

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 2$. Let $E$
denote a primitive idempotent of $\Gamma$ with respect to which $\Gamma$ is
$Q$-polynomial. Assume that there exists a $3$-clique $x,y,z$ such that
$E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. Then:

  1. (i)

For $E$ the corresponding eigenvalue $\theta$ is equal to $-\frac{k}{2}$, and
this is the minimal eigenvalue of $\Gamma$.

  2. (ii)

$E\hat{x}+E\hat{y}+E\hat{z}=0$ for all $3$-cliques $x,y,z$.

  3. (iii)

$\sigma_{i}=(-\frac{1}{2})^{i}$ for $0\leq i\leq D$.

  4. (iv)

$a_{i}=c_{i}$ for $0\leq i\leq D$.

  5. (v)

$a_{1}=1$.

###### Proof.

(i): We have $\theta=-\frac{k}{2}$ because $\theta=k\sigma_{1}$ and
$\sigma_{1}=-\frac{1}{2}$ by Lemma 3.1(iv). Moreover, $-\frac{k}{2}$ is the
minimal eigenvalue of $\Gamma$ by Lemma 3.1(iii).
(ii): Immediate from Lemma 3.1(iv).
(iii): We use induction on $i$. The result holds for $i=0$ since
$\sigma_{0}=1$. The result holds for $i=1$ by (i) and (2). For the rest of
this proof, assume that $i\geq 2$. By induction, we assume that
$\sigma_{j}=(-\frac{1}{2})^{j}$ for $0\leq j\leq i-1$. We show that
$\sigma_{i}=(-\frac{1}{2})^{i}$. We have $E\hat{x}+E\hat{y}+E\hat{z}=0$ by
(ii). Let $w\in X$ be a vertex at distance $i$ from $x$ and $i-1$ from $y$.
Define $r=d(z,w)$. We show that $r=i$. By the triangle inequality, $r=i-1$ or
$r=i$. Taking the inner product of $E\hat{w}$ and
$E\hat{x}+E\hat{y}+E\hat{z}$, we find $\sigma_{i-1}+\sigma_{i}+\sigma_{r}=0$
by (5). Suppose that $r=i-1$. Then $0=2\sigma_{i-1}+\sigma_{i}$. This implies
that $\sigma_{i}=-2\sigma_{i-1}=\sigma_{i-2}$, for a contradiction. Therefore
$r=i$. We have $0=\sigma_{i-1}+2\sigma_{i}$. This implies that
$\sigma_{i}=-\frac{1}{2}\sigma_{i-1}=(-\frac{1}{2})^{i}$ as desired.
(iv): If we substitute the data of (i) and (iii) in (3), then we have
$k=4c_{i}-2a_{i}+b_{i}$, where $1\leq i\leq D-1$. This implies that
$a_{i}=c_{i}$ using (1). Moreover, we have $4c_{D}-2a_{D}=k$ by (4) and the
mentioned substitution. By this and $a_{D}+c_{D}=k$ we obtain $a_{D}=c_{D}$.
(v): Immediate from (iv) since $c_{1}=1$. ∎

######  Definition 3.3.

Let $\Gamma$ be a distance-regular graph with diameter $D\geq 2$. For $x\in
X$, let $\Gamma(x)$ denote the set of neighbors of $x$ in $\Gamma$. The
induced subgraph on $\Gamma(x)$ is called the first subconstituent or local
graph of $\Gamma$ with respect to $x$. If $\Gamma(x)$ is a disjoint union of
cliques, then each clique has size $a_{1}+1$ and there are $\frac{k}{a_{1}+1}$
cliques. $\Gamma$ is said to be locally a disjoint union of cliques whenever
$\Gamma(x)$ is a disjoint union of cliques for all $x\in X$. Assume that
$\Gamma$ is locally a disjoint union of cliques. Then $\Gamma$ is said to have
order $(s,t)$, where $s=a_{1}+1$ and $t+1=\frac{k}{a_{1}+1}$. If $s\geq 2$,
then $\Gamma$ is called thick (cf. [2, p. 35]).

######  Definition 3.4.

Let $\Gamma$ be a distance-regular graph with diameter $D\geq 2$. Then
$\Gamma$ is called regular near $2D$-gon whenever $\Gamma$ is locally a
disjoint union of cliques and $a_{i}=a_{1}c_{i}$ for $1\leq i\leq D$ (see [2,
p. 35]).

######  Proposition 3.5.

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 2$. Let $E$
denote a primitive idempotent of $\Gamma$ with respect to which $\Gamma$ is
$Q$-polynomial. Assume that there exists a $3$-clique $x,y,z$ such that
$E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. Then $\Gamma$ is regular
near $2D$-gon.

###### Proof.

We have $a_{1}=1$ by Lemma 3.2(v). Therefore $\Gamma$ is locally a disjoint
union of $2$-cliques. Moreover, $a_{i}=c_{i}$ for $0\leq i\leq D$ by Lemma
3.2(iv). This completes the proof. ∎

######  Definition 3.6.

(cf. [6])  Let $\Gamma$ be a distance-regular graph with diameter $D\geq 2$.
For $2\leq i\leq D$, a kite of length $i$ is $4$-tuple $xyzw$ of vertices of
$\Gamma$ such that $x,y,z$ are mutually adjacent and $w$ is at distance
$d(x,w)=i$, $d(y,w)=i-1$, and $d(z,w)=i-1$.

######  Remark 3.7.

Let $\Gamma$ be a distance-regular graph with diameter $D\geq 2$. If $\Gamma$
is kite-free, then it has no kite of length $2$ and therefore $\Gamma$ is
locally a disjoint union of cliques. Let $\Gamma$ be regular near $2D$-gon.
Then every $3$-clique lies in a unique maximal clique in $\Gamma$.
Furthermore, for a given $x\in X$ and maximal clique $C$ of $\Gamma$, there is
a unique vertex $y\in C$ that is closest to $x$ (cf. [1, § 6.4]). This implies
that $\Gamma$ is kite-free. It follows that the distance-regular graph
$\Gamma$ is regular near $2D$-gon if and only if $\Gamma$ is kite-free and
$a_{i}=a_{1}c_{i}$ for $1\leq i\leq D$.

######  Lemma 3.8.

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 2$. Let $E$
denote a primitive idempotent of $\Gamma$ with respect to which $\Gamma$ is
$Q$-polynomial. Assume that there exists a $3$-clique $x,y,z$ such that
$E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. Then $\Gamma$ has
classical parameters $(D,b,\alpha,\sigma)$, where

$b=-2,\qquad\qquad\sigma=2+\alpha-\alpha[_{1}^{D}].$

Moreover, $\alpha=-1-c_{2}$.

###### Proof.

Using Lemma 3.2(iii) and from [1, Theorem 8.4.1], $\Gamma$ has classical
parameters $(D,b,\alpha,\sigma)$ with $b=-2$. This implies that
$\alpha=-1-c_{2}$ by (6). We have $b_{0}=k$ and therefore $k=\sigma[_{1}^{D}]$
by (7). Moreover, $b_{1}=k-2$ by Lemma 3.2(v). By substituting
$k=\sigma[_{1}^{D}]$ in $b_{1}=k-2$ and using (7), we have
$\sigma=2+\alpha-\alpha[_{1}^{D}]$. This completes the proof. ∎

######  Lemma 3.9.

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 2$. Let $E$
denote a primitive idempotent of $\Gamma$ with respect to which $\Gamma$ is
$Q$-polynomial. Assume that there exists a $3$-clique $x,y,z$ such that
$E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. Then $1\leq c_{2}\leq 5$.

###### Proof.

By construction $c_{2}\geq 1$. We show that $c_{2}\leq 5$. Pick $x,y\in X$
with $d(x,y)=2$. Note that $|\Gamma(x)\cap\Gamma(y)|=c_{2}$. Also note that
two distinct vertices in $\Gamma(x)\cap\Gamma(y)$ are at distance $2$, because
$a_{1}=1$ by Lemma 3.2(v). Define

$u=E\hat{x}+E\hat{y},$

and

$v=\sum_{z\in\Gamma(x)\cap\Gamma(y)}E\hat{z}.$

By the Cauchy-Schwarz inequality,

(9) $\langle u,v\rangle^{2}\leq\langle u,u\rangle\langle v,v\rangle.$

Using the data in (5) and Lemma 3.2, we obtain

(10) $\langle u,v\rangle=-c_{2}m|X|^{-1},$ (11) $\langle
u,u\rangle=\frac{5m}{2}|X|^{-1},$ (12) $\langle
v,v\rangle=\frac{(c_{2}^{2}+3c_{2})m}{4}|X|^{-1},$

where $m$ denotes the rank of $E$. Evaluating (9) using (10), (11), and (12),
we obtain $c_{2}(5-c_{2})\geq 0$. By this, we have $c_{2}\leq 5$. This
completes the proof. ∎

######  Proposition 3.10.

Let $\Gamma$ denote a distance-regular graph with $D=2$. Let $E$ denote a
primitive idempotent of $\Gamma$ with respect to which $\Gamma$ is
$Q$-polynomial. Assume that there exists a $3$-clique $x,y,z$ such that
$E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. Then $\Gamma$ is
isomorphic to one of the following graphs.

  * •

The unique regular near $4$-gon of order $(2,1)$,

  * •

The unique regular near $4$-gon of order $(2,2)$,

  * •

The unique regular near $4$-gon of order $(2,4)$.

###### Proof.

The graph $\Gamma$ is regular near $4$-gon by Proposition 3.5. Moreover,
$a_{1}=1$ by Lemma 3.2(v). The regular near $4$-gons with $a_{1}=1$ are
classified in [1, p. 30(Examples)]. The result follows from that
classification. ∎

######  Proposition 3.11.

Let $\Gamma$ denote a distance-regular graph with diameter $D=3$. Let $E$
denote a primitive idempotent of $\Gamma$ with respect to which $\Gamma$ is
$Q$-polynomial. Assume that there exists a $3$-clique $x,y,z$ such that
$E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. Then $\Gamma$ is
isomorphic to one of the following graphs.

  * •

The unique regular near $6$-gon of order $(2,8)$,

  * •

The unique regular near $6$-gon of order $(2,11)$,

  * •

The unique regular near $6$-gon of order $(2,14)$,

  * •

The dual polar graph $A_{5}(2)$.

###### Proof.

By Lemma 3.9 we have $1\leq c_{2}\leq 5$. For each choice of $c_{2}$, we
compute the intersection array using Lemma 3.8 and (6),(7). The results are in
the following table.

$c_{2}$ | Intersection array
---|---
$1$ | $\\{18,16,16;1,1,9\\}$
$2$ | $\\{24,22,20;1,2,12\\}$
$3$ | $\\{30,28,24;1,3,15\\}$
$4$ | $\\{36,34,28;1,4,18\\}$
$5$ | $\\{42,40,32;1,5,21\\}$

Assume that $c_{2}=1$. Then $\Gamma$ exists and is unique by [1, p. 427].
Assume that $c_{2}=2$. Then $\Gamma$ exists and is unique by [1, p. 427].
Assume that $c_{2}=3$. Then $\Gamma$ exists and is unique by [1, p. 428].
Assume that $c_{2}=4$. Then $\Gamma$ does not exist. Indeed the intersection
array is not feasible for the following reason. By Lemma 3.2(i), $\theta=-18$
is an eigenvalue of $\Gamma$ and by the Bigg’s formula [2, Theorem 2.8] the
multiplicity of $\theta$ is not integer. Using Lemma 3.2(iii) and the
intersection array of $\Gamma$, the multiplicity of $\theta$ is

$\frac{\sum_{i=0}^{3}k_{i}}{\sum_{i=0}^{3}k_{i}\sigma_{i}^{2}}=$
$\frac{819}{1+36(\frac{1}{4})+306(\frac{1}{16})+476(\frac{1}{64})}=22.4.$

Assume that $c_{2}=5$. Then $\Gamma$ exists and is unique by [1, p. 428]. This
completes the proof. ∎

######  Proposition 3.12.

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 4$. Let $E$
denote a primitive idempotent of $\Gamma$ with respect to which $\Gamma$ is
$Q$-polynomial. Assume that there exists a $3$-clique $x,y,z$ such that
$E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent, and $c_{2}=1$. Then
$\Gamma$ does not exist.

###### Proof.

By Lemma 3.8 with $c_{2}=1$, we find that $\Gamma$ has classical parameters
$(D,-2,-2,2[_{1}^{D}])$. The graph $\Gamma$ does not exist by [3, Corollary
5.4]. ∎

######  Proposition 3.13.

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 4$ and
$c_{2}\geq 2$. Let $E$ denote a primitive idempotent of $\Gamma$ with respect
to which $\Gamma$ is $Q$-polynomial. Assume that there exists a $3$-clique
$x,y,z$ such that $E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. Then
$c_{2}=5$ and $\Gamma$ is the dual polar graph $A_{2D-1}(2)$.

###### Proof.

The graph $\Gamma$ has classical parameters $(D,b,\alpha,\sigma)$, where
$b=-2$, by Lemma 3.8. Therefore $\Gamma$ is the dual polar graph $A_{2D-1}(2)$
by [10, Theorem B] because $a_{1}=1$ by Lemma 3.2(v). Moreover, $c_{2}=5$ by
Lemma 3.8 because $\alpha=-6$ by [2, Tbl. 1]. This completes the proof. ∎

Now we can prove Theorem 1.2.

Proof of Theorem 1.2. (i) $\Rightarrow$ (ii): By Lemma 3.8, $\Gamma$ has
classical parameters
$(D,b,\alpha,\sigma)=(D,-2,\alpha,2+\alpha-\alpha[_{1}^{D}])$. Moreover, the
corresponding eigenvalue for $E$ is equal to $-\frac{k}{2}$ by Lemma 3.2(i).
We have $-\frac{k}{2}=\frac{b_{1}}{b}-1$ by Lemma 3.2(v) and $b=-2$.
Therefore, $E$ is for the eigenvalue $\frac{b_{1}}{b}-1$.
(ii) $\Rightarrow$ (iii): Using (1) along with (6), (7), we find that
$a_{i}=c_{i}$ for $1\leq i\leq D$. In particular $a_{1}=c_{1}=1$, so $\Gamma$
has no $2$-kites. By these comments and Remark 3.7, we see that $\Gamma$ is a
regular near $2D$-gon. With reference to Definition 3.3, we see that $\Gamma$
has order $(s,t)$, where $s=a_{1}+1=2$ and
$t=\frac{k}{a_{1}+1}-1=\frac{k}{2}-1$. Note that
$\frac{b_{1}}{b}-1=-\frac{k}{2}=-t-1$. The result follows.
(iii) $\Rightarrow$ (iv): The intersection number $a_{1}=1$ because $\Gamma$
is regular near $2D$-gon of order $(2,t)$. Moreover, the corresponding
eigenvalue for $E$ is equal to $-\frac{k}{2}$ since $t=\frac{k}{2}-1$. This
implies that $\sigma_{1}=-\frac{1}{2}$ by (2) and the result follows by Lemma
3.1(iv) and Lemma 3.2(ii).
(iv) $\Rightarrow$ (v): We refer to the table in Remark 3.14. First assume
that $D=2$. Then $c_{2}=2,3,5$ and $\Gamma$ is the unique regular near $4$-gon
of order $(2,t)$, where $t=1,2,4$, by Proposition 3.10. Moreover, the
intersection array of the unique regular near $4$-gon of order $(2,4)$ is the
same as intersection array of the dual polar graph $A_{3}(2)$ (cf. [1, Theorem
9.4.3]). Next assume that $D=3$. Then $c_{2}=1,2,3,5$ and $\Gamma$ is the
unique regular near $6$-gon of order $(2,t)$, where $t=8,11,14$, or the dual
polar graph $A_{5}(2)$ by Proposition 3.11. Next assume that $D\geq 4$. Then
by Proposition 3.12 and Proposition 3.13, $c_{2}=5$ and $\Gamma$ is the dual
polar graph $A_{2D-1}(2)$.
(v) $\Rightarrow$ (vi): It is easily checked that for each of the cases listed
in (v), the cosine sequence of $E$ satisfies $\sigma_{i}=(-\frac{1}{2})^{i}$
for $0\leq i\leq D$.
(vi) $\Rightarrow$ (i): We have
$-\frac{k}{2}\sigma_{1}=\sigma_{0}+a_{1}\sigma_{1}+(k-a_{1}-1)\sigma_{2}$ by
(3) and using (1). This implies that
$\frac{k}{4}=1-\frac{a_{1}}{2}+\frac{k-a_{1}-1}{4}$ and therefore $a_{1}=1$.
Thus the result follows by Lemma 3.1(iv). This completes the proof.

######  Remark 3.14.

In the following tables, we bring out some properties of the distance-regular
graphs listed in item (v) of Theorem 1.2.

Name of graph $D$ $\\{b_{i}\\}_{i=0}^{D-1}$ $\\{c_{i}\\}_{i=1}^{D}$ Regular
near $4$-gon of order $(2,1)$ $2$ $4,2$ $1,2$ Regular near $4$-gon of order
$(2,2)$ $2$ $6,4$ $1,3$ Regular near $6$-gon of order $(2,8)$ $3$ $18,16,16$
$1,1,9$ Regular near $6$-gon of order $(2,11)$ $3$ $24,22,20$ $1,2,12$ Regular
near $6$-gon of order $(2,14)$ $3$ $30,28,24$ $1,3,15$ Dual polar graph
$A_{2D-1}(2)$ $D$ see (7) see (6)

$t$ from Theorem Minimal Classical Reference 1.2(iii) eigenvalue parameters
$1$ $-2$ $(2,-2,-3,-4)$ [1, p. 30 (Examples)] $2$ $-3$ $(2,-2,-4,-6)$ [1, p.
30 (Examples)] $8$ $-9$ $(3,-2,-2,6)$ [1, p. 427] $11$ $-12$ $(3,-2,-3,8)$ [1,
p. 427] $14$ $-15$ $(3,-2,-4,10)$ [1, p. 428] $3[^{D}_{1}]^{2}-2[^{D}_{1}]-1$
$2[^{D}_{1}]-3[^{D}_{1}]^{2}$ $(D,-2,-6,6[^{D}_{1}]-4)$ [1, Thm. 9.4.3]

### Acknowledgements

Mojtaba Jazaeri is indebted to Paul Terwilliger for his supervision in
obtaining and editing the results of this paper.

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