#  Geometric scaling in leading neutron events at HERA

Arjun Kumar<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

###### Abstract

This analysis provides new fits of the GBW model and the impact parameter-
dependent saturation model (bSat or IP-Sat) to the leading neutron structure
function HERA data in one pion exchange approximation. Both parametrizations
of the dipole cross section provide good descriptions of the considered data.
It is shown here for the first time that the experimental leading neutron
production HERA data exhibits geometric scaling, which in this context means
that the total $\gamma^{*}\pi^{*}$ cross section is a function of only one
dimensionless variable $\tau=Q^{2}/Q_{s}^{2}(x)$. The geometric scaling region
extends over a broad range of $Q^{2}$ and can be attributed to the presence of
a saturation boundary which manifests at $Q^{2}\geq Q^{2}_{s}$. The scaling
behaviour in leading neutron events is profoundly similar to what has been
observed for the inclusive DIS events.

##  I INTRODUCTION

Over the last few decades, deep inelastic scattering (DIS) experiments have
been essential in understanding the structure of the proton. The H1 and ZEUS
experiments at HERA measured the proton structure function at an unprecedented
precision to date. Mostly, this data is recorded in bins of the Bjorken
variable $x$ and the photon virtuality $Q^{2}$ and available in a wide
kinematic region reaching very small values of $x\sim 10^{-7}$. Though $x$ and
$Q^{2}$ are independent variables, the total cross section
$\sigma^{\gamma^{*}p}_{\text{tot}}$ extracted from this inclusive data shows a
striking feature that it depends only one dimensionless variable
$\tau=Q^{2}R_{s}^{2}(x)$ at low $x$. This was first observed by Staśto, Golec-
Biernat and Kwieciński in [1] and is commonly known as ”geometric scaling”.
This scaling behavior has a natural explanation in the dipole models with
saturation for photon virtualities smaller than the saturation scale
($Q_{s}^{2}$) but for $Q^{2}>Q_{s}^{2}$ this is not associated with saturation
physics, rather this regularity exists in solution of evolution equations as
demonstrated by Iancu, Itakura and Mclerran for BFKL equations in [2] and in
DGLAP equations with initial condition provided along the critical line
$Q^{2}=Q_{s}^{2}(x)$ as discussed in detail in [3]. Further in [4], the
authors argued that the geometric scaling can be explained with generic
boundary conditions for the standard DGLAP evolution. In general, the
geometric scaling is expected to hold for ln$Q^{2}/Q^{2}_{s}(x)<<$
ln$Q^{2}_{s}(x)/\Lambda_{QCD}^{2}$, usually referred to as the extended
geometric scaling regime. More detailed investigations of scaling behavior in
inclusive DIS events are provided in [5, 6, 7, 8, 9]. In addition, the
diffractive DIS data also exhibits similar scaling behavior [10].

In some of the DIS events, baryons carrying large fraction of the proton’s
longitudinal momentum ($x_{L}>0.3$) are produced in the far forward direction,
commonly known as the leading baryons. These kind of events have been observed
at HERA with leading neutrons, protons and photons [11, 12, 13, 14]. Recently,
the dipole framework has been extended to study the leading neutron events
employing the one pion exchange (OPE) approximation [15, 16, 17, 18, 19, 20].
In the dipole picture, the virtual photon emitted from the incoming electron
splits into quark-antiquark pair forming a color dipole which then interacts
with the target via strong interaction. In the case of leading neutrons, the
color dipole interacts with the pion cloud of the proton, and the forward
neutron comes from the proton as it splits into a neutron and a positive pion.

In our recent study [19], we showed using the saturated and non-saturated
impact parameter dependent dipole models that the leading neutron data is
insensitive to non-linear effects. This could be understood as the Bjorken $x$
value probed in this semi-inclusive measurement is considerably larger than
the Bjorken $x$ in proton DIS events, where the latter has exhibited no clear
signal for saturation. The next crucial step in this direction is to check
whether the leading neutron events exhibit geometric scaling as observed in
inclusive DIS events. This has not been tested thus far and is an important
step from a phenomenological point of view. This paper aims at investigating
the leading neutron events to find whether or not this regularity is observed
in the experimental data. Here two well known parametrisation of the dipole
models with saturation are used; the original Golec-Biernat and Wüsthoff (GBW)
model [21] and the bSat (IP-Sat) model [22, 23] which has an explicit DGLAP
evolution. To obtain the saturation scale, new fits of the leading neutron
structure function data are performed with both the models employing OPE.

Figure 1: Leading neutron production in one-pion exchange approximation in ep
collisions (a) and $\gamma^{*}\pi^{*}$ scattering cross section in dipole
model (b)

The rest of the paper is organised as follows. In the next section, a brief
outline of the leading neutron production in GBW and bSat models is given and
the fitting procedure is discussed. In section 3, the fit results, the
extracted saturation scale and the geometric scaling results are presented. In
section 4, we summarise and discuss the main conclusions of our study.

##  II LEADING NEUTRON PRODUCTION IN THE DIPOLE MODELS

The dipole framework is formulated in the target rest frame where the incoming
electron emits a photon which splits into a quark antiquark pair forming a
color dipole which subsequently interacts with the target strongly. For
leading neutron production, the dipole probes the pion cloud of the proton
where the virtual pion comes from the proton as it splits into a neutron and a
pion as illustrated in Fig. 1. In the one-pion exchange approximation, at high
energies, the differential cross section for $\gamma^{*}p\rightarrow Xn$ can
be written as [16]:

$\frac{{\rm d}^{2}\sigma(W,Q^{2},x_{L},t)}{{\rm d}x_{L}{\rm
d}t}=f_{\pi/p}(x_{L},t)~{}\sigma^{\gamma^{*}\pi^{*}}(\hat{W^{2}},Q^{2})$ (1)

where $t$ is the four-momentum transfer squared at the proton vertex, $x_{L}$
is the proton’s longitudinal momentum fraction taken by the neutron, $W$ is
the centre-of-mass energy for the photon-proton system, $\hat{W}$ is the
centre-of-mass energy for the photon-pion system with
$\hat{W}^{2}=(1-x_{L})W^{2}$, $f_{\pi/p}(x_{L},t)$ is the flux of pions
emitted by the proton and $\sigma^{\gamma^{*}\pi^{*}}$ is the cross section of
$\gamma^{*}\pi^{*}$ interactions. The $t$ variable is related to $p_{T}$, the
transverse momentum of the neutron, and $x_{L}$ as:

$t\simeq-\frac{p_{T}^{2}}{x_{L}}-(1-x_{L})\left(\frac{m_{n}^{2}}{x_{L}}-m_{p}^{2}\right)$
(2)

where $m_{n}$ and $m_{p}$ are the masses of neutron and proton respectively.
The leading neutron structure function is given by [11]:

$\displaystyle
F_{2}^{LN}(W,Q^{2},x_{L})=\Gamma(x_{L},Q^{2})F_{2}^{\pi}(W,Q^{2},x_{L})$ (3)

Here $\Gamma(x_{L},Q^{2})=\int_{t_{min}}^{t_{max}}f_{\pi/p}(x_{L},t){\rm d}t$
is the pion flux factor integrated over the _t_ -region of the measurement and
$F_{2}^{\pi}(W,Q^{2},x_{L})=\frac{Q^{2}}{4\pi^{2}\alpha_{\rm
EM}}\sigma^{\gamma^{*}\pi^{*}}(\hat{W^{2}},Q^{2})$ is the pion structure
function.

###  II.1 The pion flux

The flux factor $f_{\pi/p}(x_{L},t)$ describes the splitting of a proton into
a $\pi n$ system. Following [19, 15, 16, 17, 18], the flux factor given by:

$f_{\pi/p}(x_{L},t)=\frac{1}{4\pi}\frac{2g^{2}_{p\pi
p}}{4\pi}\frac{|t|}{(m_{\pi}^{2}+|t|)^{2}}(1-x_{L})^{1-2\alpha(t)}[F(x_{L},t)]^{2}$
(4)

where $m_{\pi}$ is the pion masss, $g^{2}_{p\pi p}/(4\pi)=14.4$ is the
$\pi^{0}pp$ coupling. $F(x_{L},t)$ is the form factor which accounts for the
finite size of the vertex. The form factor is given as:

$F(x_{L},t)=\exp\bigg{[}-R^{2}\frac{|t|+m_{\pi}^{2}}{(1-x_{L})}\bigg{]},\alpha(t)=0$
(5)

where $R=0.6~{}$GeV-1 has been determined from HERA data [24].

GBW |  $\sigma_{0}$[mb] | $\lambda$ | $x_{0}/10^{-4}$ | $R_{q}$ | $\chi^{2}/N_{\text{dof}}$
---|---|---|---|---|---
Fit 1 | 17.171 $\pm$ 2.777 | 0.223 $\pm$ 0.018 | 0.036 $\pm$ 0.024 | – | 63.26/48= 1.32
Fit 2 | 27.43 | 0.248 | 0.40 | 0.438 $\pm$ 0.005 | 64.52/50= 1.29

bSat | $A_{g}$ | $\lambda_{g}$ | C | $R_{q}$ | $\chi^{2}/N_{\text{dof}}$
---|---|---|---|---|---
Fit 3 | 1.208 $\pm$ 0.012 | 0.0600 $\pm$ 0.038 | 1.453 $\pm$ 0.024 |  – | 58.75/48 = 1.22
Fit 4 |  2.195 |  0.0829 |  2.289 | 0.520 $\pm$0.006 | 66.19/50 = 1.32

Table 1: Fit results of the GBW model and the bSat model to the leading
neutron structure function HERA data for $\beta\leq 0.01$, with
$N_{\text{p}}=51$ points for $x_{\text{L}}$${}_{\text{min}}=0.6$. Quark masses
are fixed and as given in [25].

###  II.2  GBW model

Using the optical theorem, the total $\gamma^{*}\pi^{*}$ cross section is
given by the imaginary part of the forward elastic
$\gamma^{*}\pi^{*}\rightarrow\gamma^{*}\pi^{*}$ amplitude as following:

$\displaystyle\sigma_{L,T}^{\gamma^{*}\pi^{*}}(\beta,Q^{2})=\int{\rm
d}^{2}\textbf{r}\int_{0}^{1}\frac{{\rm
d}z}{4\pi}|\Psi^{f}_{L,T}(\textbf{r},z,Q^{2})|^{2}\sigma^{\pi}(\textbf{r},\beta)$
(6)

where $\beta=\frac{Q^{2}+m_{f}^{2}}{(1-x_{L})W^{2}+Q^{2}}$ is the scaled
Bjorken variable for the photon-pion system. The photon wavefunctions are well
known quantities calculated in [23] and $\sigma^{\pi}(\textbf{r},\beta)$ is
the dipole-virtual pion cross section. The GBW model was proposed in [21], the
first successful attempt to explain the inclusive HERA data in a saturation
mechanism, the dipole cross section in GBW model given by [21]:

$\sigma^{\pi}(r,\beta)=\sigma_{0}(1-e^{r^{2}Q_{s}^{2}(\beta)/4})$ (7)

where the saturation scale $Q_{s}$ is defined as

$Q_{s}^{2}(\beta)=Q_{0}^{2}(\beta/x_{0})^{-\lambda}$ (8)

with $Q_{0}^{2}=1$ GeV2. The above cross section has an important property of
”geometric scaling” [1], which means that it depends only on one dimensionless
variable $rQ_{s}$ or $\tau$ defined as:

$\tau=Q^{2}R_{s}^{2}(\beta)=\frac{Q^{2}}{Q_{s}^{2}(\beta)}$ (9)

Here, there are three free parameters $\sigma_{0},\lambda,x_{0}$, now either
these can be fitted to the leading neutron structure function data or we could
use the fit results of the inclusive proton data and make use of the
assumption that the dipole-proton cross section and dipole-pion cross section
are universal up to normalization at small $x$ [26, 27, 19]. This means

$\sigma^{\pi}(r,\beta)=R_{q}~{}\sigma^{p}(r,\beta)$ (10)

where $R_{q}$ is determined by fit to the leading neutron structure function
data and the dipole-proton cross section is taken from usual DIS fit of the
GBW model from [28]. Both these strategies are used in this analysis and the
fit results are provided in Table 1. For fit 1 of the GBW model, the
parameters ($\sigma_{0},~{}\lambda,$ and $x_{0}$) are fitted to the leading
neutron structure function data, while for fit 2, these parameters are the
same as determined from the usual inclusive DIS data and the assumption of
hadronic universality at small $x$ between pions and protons (Eq.(10)) is used
and the only parameter is $R_{q}$, which is determined from the fit. Though
this assumption works well as shown in [19, 29, 30, 31], it does not yield a
physical saturation scale other than the dilute limit of this cross section.
Hence, we will use the saturation scale extracted from fit 1 of the GBW model.

Figure 2: Sensitivity of the quality of Fit 1 from Table 1 to the choice of
$x_{\text{L}}$${}_{\text{min}}$ in the data.

###  II.3  bSat model

Using the optical theorem, the total $\gamma^{*}\pi^{*}$ cross section is
given by:

$\displaystyle\sigma_{L,T}^{\gamma^{*}\pi^{*}}(\beta,Q^{2})=$
$\displaystyle\int{\rm d}^{2}\textbf{b}~{}{\rm
d}^{2}\textbf{r}\int_{0}^{1}\frac{{\rm
d}z}{4\pi}|\Psi^{f}_{L,T}(\textbf{r},z,Q^{2})|^{2}\times$
$\displaystyle\frac{{\rm d}\sigma^{(\pi)}_{q\bar{q}}}{{\rm
d}^{2}\textbf{b}}(\textbf{b},\textbf{r},\beta)$ (11)

The bSat model contains DGLAP equation for evolution of gluon density for
large scales and also has an explicit b-dependence. The dipole-pion cross
section in the bSat model is given by [22, 23]:

$\displaystyle\frac{{\rm d}\sigma^{(\pi)}_{q\bar{q}}}{{\rm
d}^{2}\textbf{b}}(\textbf{b},\textbf{r},\beta)=2\big{[}1-\text{exp}\big{(}-F(\beta,r^{2})T_{p}(\textbf{b})\big{)}\big{]}$
(12)

with

$\displaystyle
F(\beta,r^{2})=\frac{\pi^{2}}{2N_{C}}r^{2}\alpha_{s}(\mu^{2})\beta
g(\beta,\mu^{2}),$ (13)

Figure 3: Saturation scale of pion as a function Bjorken $x$ (left) and ratio
of saturation scale of pions to that of protons (right) in GBW model and bSat
model (at b=0 fm) at $x_{\text{L}}=0.6$ with the parameters from Fit 1 and Fit
3 in table 1 respectively.

The dipole cross section saturates for large dipole sizes $r$ and for large
gluon densities.The scale at which the strong coupling $\alpha_{s}$ and gluon
density is evaluated at is $\mu^{2}=\mu_{0}^{2}+\frac{C}{r^{2}}$ and the gluon
density at the initial scale $\mu_{0}$ is parametrised as:

$\displaystyle\beta
g(\beta,\mu_{0}^{2})=A_{g}\beta^{-\lambda_{g}}(1-\beta)^{6}$

The transverse profile of the pion is assumed to be Gaussian with
$T_{\text{p}}(\textbf{b})=\frac{1}{2\pi
B_{\pi}}\exp\bigg{(}-\frac{\textbf{b}^{2}}{2B_{\pi}}\bigg{)}$ where $B_{\pi}$
is the width of the pion. There is no available data which constrains this
parameter. The width of pion is chosen to be $B_{\pi}=2$ GeV-2, motivated from
the Belle measurements [32, 33] of hadron-pair production in a two-photon
process $\gamma^{*}\gamma\rightarrow\pi^{0}\pi^{0}$ where it was found to be
$B_{\pi}=1.33-1.96~{}$GeV-2 [34]. Also, H1 measurements [35] of the $\hat{t}$
spectrum for exclusive $\rho$ photo-production with leading neutrons in $ep$
scattering suggests $B_{\pi}\approx 2.3$ GeV-2. For more detailed discussion
on probing the transverse width of pion experimentally see [19]. Similar to
the GBW model, we follow both methods; first performing an independent fit of
the gluon density parameters $A_{g},~{}\lambda_{g}$ and $C$ to the leading
neutron structure function data and secondly making use of the assumption:

$\frac{{\rm d}\sigma^{(\pi)}_{q\bar{q}}}{{\rm
d}^{2}\textbf{b}}(\textbf{b},\textbf{r},\beta)=R_{q}~{}\frac{{\rm
d}\sigma^{(p)}_{q\bar{q}}}{{\rm
d}^{2}\textbf{b}}(\textbf{b},\textbf{r},\beta)$ (14)

where only $R_{q}$ is fitted to the leading neutron structure function data
and the dipole-proton cross section is taken from the fit of the bSat model to
the usual DIS data from [25]. Again, the saturation scale is extracted from
the fit where $A_{g},~{}\lambda_{g}$ and $C$ are fitted to the leading neutron
structure function data. The saturation scale in this case is given by [22]:

$Q_{S}^{2}(\beta,b)\equiv 2/r_{S}^{2}$ (15)

where $r_{S}$ is defined by solving $1/2=F(\beta,r_{S}^{2})T_{p}(b)$ in the
dipole amplitude.

##  III RESULTS

Figure 4: Normalised Dipole cross section (Eq. (7)) with the parameters from
Fit 1 in table 1 as a function $r$ (left) and $rQ_{s}$ (right) for different
values of $\beta$. As a result of geometric scaling, all the curves from left
plot merge into one curve in right plot.

Figure 5: Dipole cross section (Eq. (12)) with the parameters from Fit 3 in
table 1 as a function $r$ (left) and $rQ_{s}$ (right) for different values of
$\beta$. Due to the breaking of geometric scaling in bSat model, the curves
from left plot do not merge into a single curve in the right plot. Figure 6:
Comparison of the HERA data for leading neutron structure function
$F_{2}^{LN}(\beta,Q^{2},x_{\text{L}})$ with the results from the GBW model
(dashed black line) and the bSat model (solid blue line) with the parameters
of Fit 1 and 3 respectively.

Figure 7: Geometric scaling in experimental data on
$\sigma^{\gamma^{*}\pi^{*}}$(extracted from $F_{2}^{LN}$) as a function of
$\tau$ from Fit 1 of the GBW model (left) and Fit 3 of the bSat model (right)
in leading neutron events at HERA.

In Table 1, the fit results with the GBW and bSat models are shown. Fit 1
corresponds to the fit where the GBW model parameters
($\sigma_{0},~{}\lambda,$ and $x_{0}$) are fitted to the leading neutron
structure function data, while for fit 2, these parameters are the same as
determined from the usual inclusive DIS data and the assumption of hadronic
universality at small $x$ between pion and proton (Eq.(10)) is used and the
only parameter is $R_{q}$, which is determined from the fit. Similarly, fit 3
corresponds to the fit where the gluon density parameters in the bSat model
$A_{g},~{}\lambda_{g}$ and $C$ are fitted to the leading neutron structure
function data, while for fit 4, these are the same as determined from the
usual inclusive DIS data and the assumption of hadronic universality at small
$x$ between pions and protons (Eq.(14)) is used where the only parameter
$R_{q}$ is determined from the fit. The fits are performed using the MINUIT2
package [36] and the corresponding $\chi^{2}/N_{\text{dof}}$ are shown, where
$N_{\text{dof}}$ is $N_{\text{p}}$ \- (# of parameters) and $N_{\text{p}}$ is
number of data points in the fit. We see that the GBW model in its original
form, in addition to inclusive DIS data, also provide a good description of
the leading neutron data for both the scenarios. The bSat model fit having an
explicit DGLAP evolution provides the best description of the leading neutron
data in all the fits and prefers a slower evolution of the gluon density, as
well as almost half the number of gluons as compared to the inclusive DIS case
as illustrated in Fit 3. For bSat, fit 4 with hadronic universality assumption
also provides a reasonable description of the data.

In Fig. 2, the sensitivity of the fit 1 quality to the choice of the minimal
value of the proton’s longitudinal momentum fraction taken by the neutron,
$x_{\text{L}}$${}_{\text{min}}$, in the data is shown. The blue solid curve
represents the variation of $\chi^{2}/N_{\text{dof}}$ with the cutoff
$x_{\text{L}}$${}_{\text{min}}$ keeping the other parameters
($\sigma_{0},~{}\lambda,$ and $x_{0}$) fixed from fit 1, while the red dashed
curve shows the behaviour of $\chi^{2}/N_{\text{dof}}$ where all the
parameters are kept free in the fit. We see that the choice
$x_{\text{L}}$${}_{\text{min}}=0.6$ is optimal and the quality of the fit
deteriorates as the $x_{\text{L}}$${}_{\text{min}}$ is reduced further. This
is not surprising as the one pion exchange approximation(OPE) holds good only
for large momentum fractions carried by the neutron.

In Fig. 3, the saturation scale of the pion, $Q_{s}^{2}$, is plotted as a
function of Bjorken $x$ at $x_{\text{L}}=0.6$ in the left plot. The saturation
scale in the GBW model is extracted from fit 1 as defined in Eq.(8) and from
fit 3 for bSat model as defined in Eq.(15). The energy dependence of the
saturation scale of the pion is different in both the models, though both the
models predict the saturation scale to be $Q_{s}^{2}\sim 1$ GeV2 at
$x=10^{-6}$. For reference, the ratio of the saturation scale of pions to that
of the protons is also shown in the right plot. For the GBW model, we see that
the saturation scale of the virtual pion probed in the leading neutron events
is about half of the saturation scale of the proton for all $x$ values, and
the energy dependence is almost identical for both. For the bSat model, this
is not the case and the saturation scale is half of the value of the proton’s
saturation scale only at small $x$. The evolution is considerably different
for the two models. The small values of saturation scale of pions indicates
that it is less sensitive to saturation and one needs to go to higher energies
to probe non-linear effects in leading neutron events as compared to the usual
DIS events.

In Fig. 4, the normalized dipole cross section ($\sigma/\sigma_{0}$) in the
GBW model with the fit 1 parameters from Table 1 is plotted, evaluated at
different scaled Bjorken variable $\beta=10^{-2},~{}10^{-4},~{}10^{-6}$. In
the left plot, the dipole cross section is plotted as a function of dipole
size $r$ where we observe that the cross section saturates for large dipole
sizes for all values of $\beta$ as expected. In the right plot, the presence
of geometric scaling in the GBW model is illustrated as all the curves from
left plot with different $\beta$ values merge into a single curve when the
dipole cross section is plotted against the dimensionless variable $rQ_{s}$
which shows that indeed the dipole cross section is a function of a single
dimensionless variable rather than $x$ and $Q^{2}$ independently. The
geometric scaling is exact in the GBW model.

In Fig. 5, the dipole cross section $\frac{{\rm
d}\sigma^{(\pi)}_{q\bar{q}}}{{\rm d}^{2}\textbf{b}}$ in the bSat model is
plotted, with parameters from the fit 3, evaluated at different values of the
scaled Bjorken variable $\beta=10^{-2},~{}10^{-4},~{}10^{-6}$. In the left
plot, the dipole cross section is plotted as a function of dipole size $r$,
where the cross section saturates for large dipole sizes. For the bSat model,
the curves from the left plot with different $\beta$ values do not merge into
a single curve when the dipole cross section is plotted against the
dimensionless variable $rQ_{s}$ because of the breaking of geometric scaling.
This is primarily due to the explicit DGLAP evolution of the gluon density in
the bSat model where the gluon density is evaluated at a scale
$\mu^{2}=\mu_{0}^{2}+\frac{C}{r^{2}}$ which depends upon the dipole size $r$,
hence resulting in violation of scaling. The scaling is exact only for large
dipole sizes where $\mu^{2}\sim\mu_{0}^{2}$ as is seen in Fig. 5 for
$rQ_{s}>2.5$.

In Fig. 6, the predictions of the fits 1and 3 of the leading neutron structure
function $F_{2}^{LN}$ as a function of the scaled Bjorken variable, $\beta$
are presented, for different values of $x_{\text{L}}$ with varying $Q^{2}$ in
the GBW and bSat dipole models and confronted with the HERA measurements from
[11]. Both the models provide a good description of the energy dependence of
the data for $x_{\text{L}}\geq 0.6$, while for lower $x_{\text{L}}$ values the
models underestimates the data. This is because the models have been fitted
for kinematic region $x_{\text{L}}\geq 0.6$, where one pion exchange
approximation holds good. The curves for both the GBW and the bSat model are
practically on top of each other in the whole kinematic region.

In Fig. 7, the experimental data of the total cross section
$\sigma^{\gamma^{*}\pi^{*}}$ extracted from the leading neutron structure
function, $F_{2}^{LN}$, employing the one pion exchange approximation is shown
for $x_{\text{L}}>0.6$ in the phase space defined by the photon virtuality in
$6<Q^{2}<100$ GeV2 and the Bjorken $x$ values in $1.5\cdot 10^{-4}<x<3\cdot
10^{-2}$. The total cross section $\sigma^{\gamma^{*}\pi^{*}}$ is plotted
against the dimensionless variable $\tau=Q^{2}/Q_{s}^{2}(\beta)$ with
saturation scale from Eq.(8) in the GBW model in the left plot and from
Eq.(15) in the bSat model in the right plot . We observe that the experimental
data exhibit geometric scaling behavior for the events with
$x<0.01,~{}\beta<0.1$ for the saturation scale values obtained from both GBW
and bSat models. For the available data, the total cross section
$\sigma^{\gamma^{*}\pi^{*}}$ shows the 1/$\tau$ dependence at large $\tau$
which is very similar to what has been discovered for the total cross section
$\sigma^{\gamma^{*}p}$ in usual DIS events. On the right plot, a few
experimental data points at small values of $\tau$ are off from the scaling
behaviour. This is due to the different magnitude and the energy dependence of
saturation scale in the bSat model as compared to GBW model at moderate $x$
values. The left plot is the main result of this paper where the saturation
scale is extracted from the GBW model since geometric scaling is exact in the
GBW model while it breaks down in the bSat model where the dipole size affects
the evolution of strong coupling constant and gluon density and hence the
saturation scale values. It is interesting to note that the scaling region
extends to $\beta<0.1$, if we take a closer look at the geometric scaling plot
of the inclusive DIS data from [1], the experimental data points in the large
$x$ region from H1 and ZEUS also show geometric scaling and the data points
responsible for the breaking of scaling are from experiments other than HERA.
Several other investigations with the analysis of HERA data [8, 9] also
concluded that geometric scaling holds for Bjorken $x<0.1$ for inclusive DIS
events. The scaling behavior in leading neutron DIS is identical to what has
been observed for inclusive DIS events.

##  IV CONCLUSIONS AND DISCUSSION

In this work, it is shown that the dipole model phenomenology can successfully
describe the leading neutron DIS cross section with the new fits of the dipole
model to the leading neutron structure function data. Using OPE, both the GBW
and the bSat (IP-Sat) models provide a good description of the leading neutron
data. Among all the fits, the fit with bSat model having an explicit DGLAP
evolution describes the data best with a $\chi^{2}/N_{\text{dof}}=1.22$, as
depicted in Table 1. The absorptive corrections are not included in this
analysis as they are only dominant for small photon virtualities ($Q^{2}<6$
GeV2) which is outside the kinematic regime considered in this analysis and
would just affect the normalization of the pion flux. In the first part, we
have shown the presence of geometric scaling in the GBW model with the new fit
parameters of the model from the leading neutron DIS. The scaling behaviour is
not exact in the bSat model for small dipole sizes where the evolution effects
play a dominant role, while for the large dipole sizes the model shows scaling
behaviour similar to the scaling in the GBW model. The new fit results are
used to extract the saturation scale of the pion which in general is about
half of the proton’s saturation scale and the energy dependence of the pion
saturation scale is identical to that of the proton in the GBW model. The main
result of this study is presented in left plot of Fig. 7, where we have shown,
using OPE, that the experimental data on $\sigma^{\gamma^{*}\pi^{*}}$ exhibits
geometric scaling over an extended region of $Q^{2}$ and shows $1/\tau$
behaviour for large $\tau$ values. The presence of the scaling for
$Q^{2}>Q_{s}^{2}$ shows that the geometric scaling is more general than in the
saturation model and can be attributed to the presence of a saturation
boundary in the data which has its root in the evolution equations as
described in detail in [2, 3]. This is profoundly similar to what has been
previously seen in the $\sigma^{\gamma^{*}p}$ in inclusive DIS events. The
presence of geometric scaling and similar behaviour of
$\sigma^{\gamma^{*}\pi^{*}}$ as a function of $\tau$ hints toward the
universality of small $x$ structure between pions and protons, though we need
more experimental data at smaller $\beta$ values to further validate this
statement. Future colliders such as EIC [37, 38] and FCC or LHeC [39] have the
potential to probe this region of phase space and it will be interesting to
see the differences and the similarities between inclusive DIS events and
leading neutron DIS.

## ACKNOWLEDGMENTS

Sincere thanks to Tobias Toll for a careful reading of the manuscript and
valuable discussions that helped in a better understanding of this work. This
work is supported by the Department of Science & Technology, India under Grant
No. DST/ INSPIRES/03/2018/000344.

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# Preliminaries in Many-Particle Quantum Gravity. Einstein–Friedmann Spacetime

L.A. Glinka<EMAIL_ADDRESS><EMAIL_ADDRESS>Bogoliubov Laboratory of
Theoretical Physics,
Joint Institute for Nuclear Research, 141980 Dubna, Russia

###### Abstract

Preliminaries for the Many-Particle approach to quantization of the
Einstein–Hilbert theory of gravitation are presented in this paper. The
Einstein–Friedmann Spacetime is detailed discussed from this point of view,
and the Von Neumann–Araki–Woods second quantization in the Fock space of the
canonical Hamilton field equations is presented. In result the Spacetime as
system with the Bose–Einstein statistics is described and thermodynamics of
the system is computed.

Quantum Cosmology, Quantum Gravity, Boson systems, Einstein–Hilbert theory

###### pacs:

04.60.-m, 98.80.Qc, 05.30.Jp

##  I Introduction

In this paper introductory results of the Fock space formulation of Quantum
Gravity are discussed by example of Einstein–Friedmann Spacetime. Firstly the
classical Dirac–Arnowitt–Deser–Misner Hamiltonian approach to General
Relativity for the Spacetime is briefly presented, and the Wheeler–DeWitt
evolution as primary quantization of constraints is formulated. Next evolution
is separated on the canonical Hamilton field equations system, and consistent
the von Neumann–Araki–Woods quantization in the Fock space is realized. For
diagonalization the Bogoliubov–Heisenberg basis is used, and internal
logarithmic conformal field theory structure in this basis is proved. By
treatment of the Spacetime as the Bose–Einstein system thermodynamics of the
system is computed.

##  II Classical approach

###  II.1 The Dirac constraints

The flat, homogenous and isotropic Spacetime investigated by Einstein e  and
Friedmann f  is described by an interval

$ds^{2}=a^{2}(\eta)\left[(d\eta)^{2}-(dx^{i})^{2}\right],~{}~{}d\eta=N_{d}(x^{0})dx^{0},$
(1)

here $a(\eta)$ is the Friedmann conformal scale factor, $\eta$ is the
Dirac–Arnowitt–Deser–Misner conformal time d ; adm , and $N_{d}$ is lapse
function. The Hilbert action h  for (1) can be written as

$\mathcal{A}[a]=\int{dx^{0}}\left\\{p_{a}\dfrac{da}{dx^{0}}-N_{d}\left[-\dfrac{p_{a}^{2}}{4V_{0}^{2}}+\rho(a)\right]V_{0}\right\\},$
(2)

where $p_{a}$ is nontrivial canonical conjugate momentum, $\rho(a)$ is energy
density

$\displaystyle
p_{a}=-\dfrac{2V_{0}}{N_{d}}\dfrac{da}{dx^{0}},~{}~{}\rho(a)=\dfrac{a^{4}}{V_{0}}\int{d^{3}x}~{}\mathcal{H}_{M},$
(3)

and $V_{0}=\int{d^{3}}x<\infty$ is finite space volume. Variational principle
with respect to $N_{d}$ applied to action (2) leads to constraints

$\dfrac{\delta\mathcal{A}[a]}{\delta
N_{d}}=0\Rightarrow-\dfrac{p_{a}^{2}}{4V_{0}^{2}}+\rho(a)=0,$ (4)

with solution given by the Hubble law

$\int_{a_{I}}^{a}\dfrac{da}{\sqrt{\rho(a)}}=\pm|\eta-\eta_{I}|,$ (5)

where index $I$ means initial data.

###  II.2 The Hamilton field equations

One can see that the primary constraints (4) simplify to a form

$p_{a}^{2}-{\omega}^{2}(a)=0,~{}~{}\omega(a)\equiv\pm 2V_{0}\sqrt{\rho(a)},$
(6)

and that direct canonical quantization

$i\left[\hat{\mathrm{p}}_{a},a\right]=1,~{}~{}\hat{\mathrm{p}}_{a}\equiv-i\dfrac{\partial}{\partial{a}},$
(7)

leads to the Wheeler–DeWitt evolution equation w1 ; w2

$\left(\dfrac{\partial^{2}}{\partial{a^{2}}}+{\omega}^{2}(a)\right)\Psi(a)=0.$
(8)

This equation can be represented as the canonical Hamilton field equations
system

$\left\\{\begin{array}[]{ccc}\dfrac{\partial}{\partial\Pi_{\Psi}}\mathrm{H}(\Pi_{\Psi},\Psi)\\!\\!\\!\\!\\!&=&\\!\\!\\!\\!\\!\dfrac{\partial}{\partial{a}}\Psi,\vspace*{5pt}\\\
-\dfrac{\partial}{\partial\Psi}\mathrm{H}(\Pi_{\Psi},\Psi)\\!\\!\\!\\!&=&\\!\dfrac{\partial}{\partial{a}}\Pi_{\Psi},\end{array}\right.$
(9)

where considered Hamiltonian $\mathrm{H}(\Pi_{\Psi},\Psi)$ has a form

$\mathrm{H}(\Pi_{\Psi},\Psi)=\dfrac{1}{2}\left(\Pi^{2}_{\Psi}+{\omega}^{2}(a)\Psi^{2}\right).$
(10)

In further text we will use compact form of (9)

$\dfrac{\partial}{\partial{a}}\left[\begin{array}[]{c}\Psi\\\
\Pi_{\Psi}\end{array}\right]=\left[\begin{array}[]{cc}0&1\\\
-{\omega}^{2}(a)&0\end{array}\right]\left[\begin{array}[]{c}\Psi\\\
\Pi_{\Psi}\end{array}\right].$ (11)

##  III Quantization

We focus attention on the Hamilton field equations (11).

###  III.1 Von Neumann–Araki–Woods quantization

Essence of quantization constitutes general transition between classical
fields and field operators

$\left[\begin{array}[]{c}\Psi(a)\\\
\Pi_{\Psi}(a)\end{array}\right]\longrightarrow\left[\begin{array}[]{c}\mathbf{\Psi}[a(\eta)]\\\
\mathbf{\Pi_{\Psi}}[a(\eta)]\end{array}\right],$ (12)

with standard Canonical Commutation Relations (CCRs)

$\left[\mathbf{\Pi_{\Psi}}[a(\eta)],\mathbf{\Psi}[a(\eta^{\prime})]\right]=-i\delta\left(a(\eta)-a(\eta^{\prime})\right).$
(13)

Problem of CCRs representations is not new, and was investigated in ccr1  and
ccr2 . In g  was proposed the Von Neumann–Araki–Woods quantization in the Fock
space $\left(\mathcal{G},\mathcal{G}^{\dagger}\right)$ of annihilation and
creation operators

$\left[\begin{array}[]{c}\mathbf{\Psi}\\\
\mathbf{\Pi_{\Psi}}\end{array}\right]=\left[\begin{array}[]{cc}\dfrac{1}{\sqrt{2{\omega}(a)}}&\dfrac{1}{\sqrt{2{\omega}(a)}}\\\
-i\sqrt{\dfrac{{\omega}(a)}{2}}&i\sqrt{\dfrac{{\omega}(a)}{2}}\end{array}\right]\left[\begin{array}[]{c}\mathcal{G}[a(\eta)]\\\
\mathcal{G}^{\dagger}[a(\eta)]\end{array}\right],$ (14)

which is consistent with (13) by CCRs

$\displaystyle\left[\mathcal{G}[a(\eta)],\mathcal{G}^{\dagger}[a(\eta^{\prime})]\right]$
$\displaystyle=$ $\displaystyle\delta(a(\eta)-a(\eta^{\prime})),$ (15)
$\displaystyle\left[\mathcal{G}[a(\eta)],\mathcal{G}[a(\eta^{\prime})]\right]$
$\displaystyle=$ $\displaystyle 0.$ (16)

Quantization (14) applied to equations (11) gives

$\dfrac{\partial}{\partial{a}}\left[\begin{array}[]{c}\mathbf{\Psi}\\\
\mathbf{\Pi_{\Psi}}\end{array}\right]\\!=\\!\left[\begin{array}[]{cc}0&1\\\
-{\omega}^{2}(a)&0\end{array}\right]\left[\begin{array}[]{c}\mathbf{\Psi}\\\
\mathbf{\Pi_{\Psi}}\end{array}\right],$ (17)

or in terms of the Fock space

$\dfrac{\partial}{\partial{a}}\left[\begin{array}[]{c}\mathcal{G}[a(\eta)]\\\
\mathcal{G}^{\dagger}[a(\eta)]\end{array}\right]\\!=\\!\left[\begin{array}[]{cc}-i{\omega}(a)&\Delta\\\
\Delta&i{\omega}(a)\end{array}\right]\left[\begin{array}[]{c}\mathcal{G}[a(\eta)]\\\
\mathcal{G}^{\dagger}[a(\eta)]\end{array}\right],$ (18)

where
$\Delta=\partial_{a}\ln\left|\sqrt{\dfrac{\omega(a)}{\omega(a_{I})}}\right|$,
and $\omega(a_{I})=\pm 4V_{0}\sqrt{\rho(a_{I})}$ is initial value of
$\omega(a)$.

###  III.2 The Bogoliubov–Heisenberg basis

Evolution (18) should be diagonalize in order to build correct quantum theory.
We apply the Bogoliubov automorphism for Bose–Einstein systems, which lies in
accordance with considered field equations (8)

$\displaystyle\left[\begin{array}[]{cc}\mathcal{W}[a(\eta)]\\\
\mathcal{W}^{\dagger}[a(\eta)]\end{array}\right]$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}u(a)&v(a)\\\
v^{\ast}(a)&u^{\ast}(a)\end{array}\right]\left[\begin{array}[]{cc}\mathcal{G}[a(\eta)]\\\
\mathcal{G}^{\dagger}[a(\eta)]\end{array}\right],$ (26)
$\displaystyle|u(a)|^{2}-|v(a)|^{2}=1.$

This transformation preserves CCRs (15) and (16) in new basis

$\displaystyle\left[\mathcal{W}[a(\eta)],\mathcal{W}^{\dagger}[a(\eta^{\prime})]\right]$
$\displaystyle=$ $\displaystyle\delta(a(\eta)-a(\eta^{\prime})),$ (27)
$\displaystyle\left[\mathcal{W}[a(\eta)],\mathcal{W}[a(\eta^{\prime})]\right]$
$\displaystyle=$ $\displaystyle 0.$ (28)

Next we reduce evolution (18) to Heisenberg type one

$\dfrac{\partial}{\partial{a}}\left[\begin{array}[]{c}\mathcal{W}[a(\eta)]\\\
\mathcal{W}^{\dagger}[a(\eta)]\end{array}\right]=\left[\begin{array}[]{cc}-i\lambda&0\\\
0&i\lambda\end{array}\right]\left[\begin{array}[]{c}\mathcal{W}[a(\eta)]\\\
\mathcal{W}^{\dagger}[a(\eta)]\end{array}\right].$ (29)

In result we obtain $\lambda=0$,
$\mathcal{W}\equiv\mathrm{w}=\mathit{constant}$, and vacuum state $|0\rangle$
defined by relation $\mathrm{w}|0\rangle=0$. Evolution (18) is equivalent to
equations on functions $u$ and $v$

$\dfrac{\partial}{\partial{a}}\left[\begin{array}[]{c}v(a)\\\
u(a)\end{array}\right]=\left[\begin{array}[]{cc}-i{\omega}(a)&-\Delta\\\
-\Delta&i{\omega}(a)\end{array}\right]\left[\begin{array}[]{c}v(a)\\\
u(a)\end{array}\right],$ (30)

that can be solved in hyperbolic parametrization

$\displaystyle
v(a)=e^{i\theta(a)}\sinh\phi(a),~{}~{}u(a)=e^{i\theta(a)}\cosh\phi(a).$ (31)

Here functions–parameters

$\displaystyle\theta(a)=\int_{a_{I}}^{a}p_{a}da,~{}\phi(a)=\ln{\left|\sqrt{\dfrac{\omega(a_{I})}{\omega(a)}}\right|}.$
(32)

are classical phase integral and logarithmic conformal field. In this manner
we have proved that theory builded by the Von Neumann–Araki–Woods quantization
in form (14) has internal logarithmic conformal field theory structure.

##  IV Physical implications

Now some physical results will present.

###  IV.1 Density functional

We take density functional operator in standard form of occupation number
consistent with quantization (14)

$\varrho=\mathcal{G}^{\dagger}\mathcal{G},$ (33)

which in the Bogoliubov–Heisenberg basis has a following matrix representation

$\varrho=\left[\begin{array}[]{cc}|u|^{2}&-uv\\\
-u^{\ast}v^{\ast}&|v|^{2}\end{array}\right].$ (34)

One can see that the Von Neumann entropy for considered system has the
Boltzmann one form

$\mathrm{S}\equiv-\dfrac{\mathrm{tr}(\varrho\ln\varrho)}{\mathrm{tr}(\varrho)}=\ln\left\\{\dfrac{1}{2|u|^{2}-1}\right\\}.$
(35)

It creates opportunity to formulate of thermodynamics for quantized the
Einstein–Friedmann Spacetime represented by entropy (35).

###  IV.2 Thermodynamics of Spacetime

In opposite to results of paper g , in this text we propose use of (10) as the
Hamiltonian of theory. Theory constructed by this Hamiltonian is conceptually
simpler than theory presented in g , because in (10) superfluidity term is
absent. However, as it will turn out physical consequences presented below are
principally the same as in g .

The Hamiltonian (10) has a following matrix representation in the
Bogoliubov–Heisenberg basis

$\displaystyle\mathrm{H}=\left[\begin{array}[]{cc}\dfrac{|u|^{2}+|v|^{2}}{2}\omega&-uv\omega\\\
-u^{\ast}v^{\ast}\omega&\dfrac{|u|^{2}+|v|^{2}}{2}\omega\end{array}\right].$
(38)

Statistical mechanics understand internal energy as average
$\mathrm{U}=\dfrac{\mathrm{tr}(\varrho\mathrm{H})}{\mathrm{tr}\varrho}$, which
for considered case is

$\mathrm{U}=\left(\dfrac{1}{2}+\dfrac{4\mathrm{n}+3}{2\mathrm{n}+1}\mathrm{n}\right)\omega(a).$
(39)

Here we introduced quantity

$\mathrm{n}=\langle
0|\mathcal{G}^{\dagger}\mathcal{G}|0\rangle=\dfrac{1}{4}\Bigg{|}\sqrt{\left|\dfrac{\omega(a_{I})}{\omega(a)}\right|}-\sqrt{\left|\dfrac{\omega(a)}{\omega(a_{I})}\right|}\Bigg{|}^{2},$
(40)

which is number of particles produced from vacuum. Averaged number of
particles is

$\langle\mathrm{n}\rangle=2\mathrm{n}+1.$ (41)

Chemical potential is defined as
$\mu=\dfrac{\partial\mathrm{U}}{\partial\mathrm{n}}$, in this case

$\mu=\left(1+\dfrac{1}{(2\mathrm{n}+1)^{2}}-\dfrac{1}{2}\dfrac{4\mathrm{n}+1}{4\mathrm{n}^{2}+2\mathrm{n}}\sqrt{\dfrac{\mathrm{n}}{\mathrm{n}+1}}\right)\omega(a).$
(42)

Description of the system according to quantum theory principles demands using
of the thermal Gibbs ensemble. The system is characterized by CCRs (27, 28).
It leads to identification of function under logarithm in (35) with the
Bose–Einstein partition function

$\dfrac{1}{2|u|^{2}-1}\equiv\left\\{\exp\left[\dfrac{\mathrm{\mathrm{U}-\mu\mathrm{n}}}{\mathrm{T}}\right]-1\right\\}^{-1}.$
(43)

Using of relations (26), (39), (40), (42), and (43) give the formula for
temperature of considered system

$\displaystyle\mathrm{T}=\dfrac{1+\left(\dfrac{2\mathrm{n}}{2\mathrm{n}+1}\right)^{2}+\dfrac{8\mathrm{n}^{2}+8\mathrm{n}+1}{4\mathrm{n}+2}\sqrt{\dfrac{\mathrm{n}}{\mathrm{n}+1}}}{2\ln(2\mathrm{n}+2)}\omega(a).$
(44)

This formula describes temperature of the system as a function of Friedmann
conformal scale factor $a$.

##  V Summary

Above formalism points at direct way to the Quantum Cosmology understood as
many-particle open quantum system. On base of this type reasoning formulation
of physics is obvious and clear - in presented case description of the
Einstein–Friedmann Spacetime as the quantum Bose–Einstein system leads to
well-defined thermodynamics. Furthermore, for considered Spacetime open
quantum system point of view gives opportunity to formulate Quantum Gravity as
theory with internal structure of logarithmic conformal field theory.

In opinion of the author presented way creates substantially wider
opportunities for construction of similar formalism for general gravitational
fields and to research connections between geometry and physics in terms of
logarithmic conformal field theory.

## Acknowledgements

The author is especially thankful for valuable discussions to B.M. Barbashov,
I. Białynicki-Birula, A. Borowiec, S.J. Brodsky, G. `t Hooft, J. Lukierski,
V.N. Pervushin, V.B. Priezzhev, and C. Rovelli.

## References

  * (1) Einstein A., The meaning of relativity (Pricenton University Press, Princeton, 1922).
  * (2) Friedmann A.A., Z. Phys. 10, 377, (1922).
  * (3) Dirac P.A.M., Phys. Rev. 114, 924, (1959).
  * (4) Arnowitt R., Deser S., Misner C.W., Phys. Rev. 122, 997, (1961).
  * (5) Hilbert D., Gott. Nachr. 27, 395, (1915).
  * (6) Wheeler J.A., in Battelle Rencontres: 1967 Lectures in Mathematics and Physics, edited by C.M. DeWitt and J.A. Wheeler (New York, 1968), p. 242.
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# Topological modes in relativistic hydrodynamics

Yan Liua,b<EMAIL_ADDRESS>Ya-Wen Sunc,d<EMAIL_ADDRESS>aCenter for
Gravitational Physics, Department of Space Science, Beihang University,
Beijing 100191, China
bKey Laboratory of Space Environment Monitoring and Information Processing,
Ministry of Industry and Information Technology, Beijing 100191, China
cSchool of physics $\&$ CAS Center for Excellence in Topological Quantum
Computation, University of Chinese Academy of Sciences, Beijing 100049, China
dKavli Institute for Theoretical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

We show that gapless modes in relativistic hydrodynamics could become
topologically nontrivial by weakly breaking the conservation of energy
momentum tensor in a specific way. This system has topological semimetal-like
crossing nodes in the spectrum of hydrodynamic modes that require the
protection of a special combination of translational and boost symmetries in
two spatial directions. We confirm the nontrivial topology from the existence
of an undetermined Berry phase. These energy momentum non-conservation terms
could naturally be produced by an external gravitational field that comes from
a reference frame change from the original inertial frame, i.e. by fictitious
forces in a non-inertial reference frame. This non-inertial frame is the rest
frame of an accelerating observer moving along a trajectory of a helix. This
suggests that topologically trivial modes could become nontrivial by being
observed in a special non-inertial reference frame, and this fact could be
verified in laboratories, in principle. Finally, we propose a holographic
realization of this system.

##  I  Introduction

Hydrodynamics is the universal low energy theory for systems close to local
thermal equilibrium at a long distance and time. It could describe a variety
of physical systems ranging from matter at large scales in the Universe, the
quark-gluon plasma CasalderreySolana:2011us , to Weyl semimetals
Landsteiner:2014vua ; Lucas:2016omy  and graphenes Lucas:2017idv  in the
laboratory. At small momentum and frequency, perturbations of a hydrodynamic
system close to the equilibrium would produce propagating as well as diffusive
modes Kovtun:2012rj . These modes are gapless, whose poles are at $\omega={\bf
k}=0$, which reflects the fact that energy momentum is conserved.

During the last decade, topologically nontrivial quantum states have been
discovered in condensed matter physics vishwanath ; Witten:2015aoa . Later it
has been found that many classical systems have nontrivial topological states
too, including topological optical/sound systems (see e.g. topphoton ;
topphoton2 ; natphy  and references therein), which have also been observed
experimentally.

It raises the question if the gapless modes in relativistic hydrodynamics
could also become topologically nontrivial under certain conditions. In this
paper, we start from the relativistic hydrodynamics and show that after weakly
breaking conservation of energy momentum, hydrodynamic modes could become
topological semimetal-like nontrivial states that require the protection of a
special spacetime symmetry, and interestingly, these non-conservation terms
for the energy momentum tensor could come from a non-inertial reference frame
of an accelerating observer moving along a helix.

##  II Effective Hamiltonian and spectrum in relativistic hydrodynamics

We focus on the simplest hydrodynamic systems with no internal charges whose
only conserved quantity is the energy momentum tensor that satisfies
$\partial_{\mu}T^{\mu\nu}=0\,.$ Up to the first order in derivative, the
constitutive equation for the energy momentum tensor in the Landau frame is

$\displaystyle T^{\mu\nu}$ $\displaystyle=\epsilon
u^{\mu}u^{\nu}+P\Delta^{\mu\nu}-\eta\Delta^{\mu\alpha}\Delta^{\nu\beta}\big{(}\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}$
$\displaystyle-\frac{2}{3}\eta_{\alpha\beta}\partial_{\sigma}u^{\sigma}\big{)}-\zeta\Delta^{\mu\nu}\partial_{\alpha}u^{\alpha}+\mathcal{O}(\partial^{2})\,,$

where $\Delta_{\mu\nu}=\eta_{\mu\nu}+u_{\mu}u_{\nu}$, $\epsilon$, $P$ are the
energy densities and pressure and $\eta$, $\zeta$ are the shear and bulk
viscosities.

With small perturbations away from the equilibrium, the system would respond
to the perturbations and develop hydrodynamic modes. There are four eigenmodes
of the system. Two of them are the sound modes propagating in the direction of
${\bf k}=(k_{x},k_{y},k_{z})$ with the dispersion relation $\omega=\pm
v_{s}k-i\Gamma_{s}k^{2}$ , where $v_{s}=\sqrt{\frac{\partial
P}{\partial\epsilon}}$ and $\Gamma_{s}=(\frac{4}{3}\eta+\zeta)/(\epsilon+P)$.
The other two are transverse modes with
$\omega=-i\frac{\eta}{\epsilon+P}k^{2}$. To the first order in $k$,
dissipative terms disappear and the spectrums of the four modes are real,
which cross each other at $\omega={\bf k}=0$. This spectrum looks similar to
the spectrum of Dirac semimetals, except that we have two extra flat bands
here.

To change the spectrum to a topological semimetal-like one, we need to add
non-conservation terms of $T^{\mu\nu}$ into the conservation equations. As a
first step, we develop the notion of an effective Hamiltonian in
hydrodynamics. Substituting the constitutive equations for the perturbations
$\delta T^{\mu\nu}$ into $\partial_{\mu}\delta T^{\mu\nu}=0\,$, we could
rewrite the equations into the form

$i\partial_{t}\Psi=H\Psi$ (1)

where we have defined

$\Psi=\begin{pmatrix}\delta\epsilon\\\ \delta\pi^{x}\\\ \delta\pi^{y}\\\
\delta\pi^{z}\end{pmatrix}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ H=\begin{pmatrix}0&\leavevmode\nobreak\
\leavevmode\nobreak\ k_{x}&\leavevmode\nobreak\ \leavevmode\nobreak\
k_{y}&\leavevmode\nobreak\ \leavevmode\nobreak\ k_{z}\\\
k_{x}v_{s}^{2}&\leavevmode\nobreak\ \leavevmode\nobreak\
0&\leavevmode\nobreak\ \leavevmode\nobreak\ 0&\leavevmode\nobreak\
\leavevmode\nobreak\ 0\\\ k_{y}v_{s}^{2}&\leavevmode\nobreak\
\leavevmode\nobreak\ 0&\leavevmode\nobreak\ \leavevmode\nobreak\
0&\leavevmode\nobreak\ \leavevmode\nobreak\ 0\\\
k_{z}v_{s}^{2}&\leavevmode\nobreak\ \leavevmode\nobreak\
0&\leavevmode\nobreak\ \leavevmode\nobreak\ 0&\leavevmode\nobreak\
\leavevmode\nobreak\ 0\end{pmatrix}\,$ (2)

at leading order in $k$, i.e. omitting dissipative terms at
$\mathcal{O}(k^{2})$.

In this way, in analogy to the electronic systems shenbook  we have defined an
effective Hamiltonian matrix $H$, whose eigenvalues give the spectrum of
hydrodynamic modes footnote1 . The four eigenvalues of the matrix Hamiltonian
above give the sound modes $\omega=\pm
v_{s}\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$ and double copies of transverse
modes $\omega=0$. The form (2) is the “free” Hamiltonian matrix for a
conserved energy momentum tensor.

##  III Topologically nontrivial modes

To deform the spectrum of the hydrodynamic modes, we introduce non-
conservation terms for the energy momentum tensor and make sure that the non-
conservation terms are small enough to stay within the hydrodynamic limit. The
non-conservation of energy and momentum could come from a certain external
system, which couples to the hydrodynamic system under study. At this stage we
assume that the constitutive equations for $T^{\mu\nu}$ do not get modified,
and later, we will take the modifications into account and show that the
spectrum does not change up to a rescaling of parameters.

We take a 4D hydrodynamic system and introduce non-conservation terms for
$T^{\mu\nu}$ as follows

$\displaystyle\begin{split}\partial_{\mu}\delta T^{\mu t}&=m\delta
T^{tx}\,,\leavevmode\nobreak\ \leavevmode\nobreak\ \partial_{\mu}\delta T^{\mu
x}=-mv_{s}^{2}\delta T^{tt}\,,\\\ \partial_{\mu}\delta T^{\mu y}&=bv_{s}\delta
T^{tz}\,,\leavevmode\nobreak\ \leavevmode\nobreak\ \partial_{\mu}\delta T^{\mu
z}=-bv_{s}\delta T^{ty}\,,\end{split}$ (3)

where $m$ terms gap the spectrum, while $b$ terms change the momentum position
of the crossing nodes in the spectrum and we assume
$\mathcal{O}(k^{2})<\mathcal{O}(m,b)\lesssim\mathcal{O}(k)$.

Physically, (3) states that energy (momentum in the $x$ direction) is not
conserved, whose non-conservation is proportional to the momentum in the $x$
direction (energy). Later, we will show that these seemingly ad hoc non-
conservation terms naturally arise from the observation of an accelerating
observer moving in a helix, which could be tested in experiments in principle.

After substituting the fluctuations of constitutive equation into (3) we
obtain

$\begin{split}i\partial_{t}\Psi=H\Psi\end{split}$ (4)

where
$\Psi=\big{(}\delta\epsilon,\delta\pi^{x},\delta\pi^{y},\delta\pi^{z}\big{)}^{T}$
and

$H=\begin{pmatrix}0&\leavevmode\nobreak\ \leavevmode\nobreak\
k_{x}+im&\leavevmode\nobreak\ \leavevmode\nobreak\ k_{y}&\leavevmode\nobreak\
\leavevmode\nobreak\ k_{z}\\\ (k_{x}-im)v_{s}^{2}&\leavevmode\nobreak\
\leavevmode\nobreak\ 0&\leavevmode\nobreak\ \leavevmode\nobreak\
0&\leavevmode\nobreak\ \leavevmode\nobreak\ 0\\\
k_{y}v_{s}^{2}&\leavevmode\nobreak\ \leavevmode\nobreak\
0&\leavevmode\nobreak\ \leavevmode\nobreak\ 0&\leavevmode\nobreak\
\leavevmode\nobreak\ ibv_{s}\\\ k_{z}v_{s}^{2}&\leavevmode\nobreak\
\leavevmode\nobreak\ 0&\leavevmode\nobreak\ \leavevmode\nobreak\
-ibv_{s}&\leavevmode\nobreak\ \leavevmode\nobreak\ 0\end{pmatrix}\,.$ (5)

$H$ is similar to a Hermitian matrix as could be seen by redefining
$\delta\epsilon\to\frac{1}{v_{s}}\delta\epsilon$. Thus, this effective $H$ has
real eigenvalues and the factor $v_{s}$ could be ignored which could be taken
back by an inverse transformation when necessary.

The spectrum of the hydrodynamic modes for (5),

$\displaystyle\omega=\pm\frac{1}{\sqrt{2}}\sqrt{b^{2}+k^{2}+m^{2}\pm\sqrt{(k_{x}^{2}+m^{2}-b^{2})^{2}+(k_{y}^{2}+k_{z}^{2})^{2}+2(k_{y}^{2}+k_{z}^{2})(k_{x}^{2}+m^{2}+b^{2})}}\,,$

where $k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$.

Figure 1 shows this spectrum as a function of $k_{x}$ for $k_{y}=k_{z}=0$ in
three different situations: $m<b$, $m=b$ and $m>b$ as well as for
$k_{y}>0,k_{z}=0$ at $m<b$. The effect of $m$ terms is to gap the two sound
modes. The effect of $b$ terms is to lift and lower the two transverse flat
bands to symmetric positions of opposite sides of the $k$ axis. In this way,
the modes have band crossings at nonzero values of $k$ for $m<b$.

Figure 1: The spectrum of the modified hydrodynamics with dynamical equation
(3). From left up to right down: the first three plots are for $m<b$, $m=b$,
and $m>b$, respectively, and $k_{y}=k_{z}=0$. The fourth plot is for
$k_{y}>0$, $k_{z}=0$, and $m<b$. The distance between two flat bands is
determined by the value of $b$ while the gap between two curved bands is
determined by the parameter $m$.

From Fig. 1, we could see that for $m<b$, there are four band crossing nodes
at $k_{y}=k_{z}=0$ while $k_{x}\neq 0$, and for these nodes, $\omega\neq 0$.
These four nodes are still points in the expanded space of $\omega$, $k_{x}$,
$k_{y}$, and $k_{z}$ as can be seen from the fourth plot in Fig. 1. For
$m=b>0$, the system becomes critical with two nodes, and for $m>b$, the system
becomes gapped again. This behavior is qualitatively similar to the
topological phase transition of a topological semimetal Landsteiner:2015pdh .

The $m$ terms in (5) do not gap the four band crossing nodes in the $m<b$
case; however, if we have extra $m$ terms in the $y$ or $z$ directions, the
gaps will open no matter how small the $y$ or $z$ mass parameters are. The
spectrum in this case looks the same as the bottom right one in Fig. 1. In
this situation, there are no crossing nodes anymore. This means that the nodes
should be topologically nontrivial under the protection of symmetries that
forbid the $m$ terms in the $y$ and $z$ directions. We will see later that the
symmetry needed here is a special combination of translational and boost
symmetry in $y$ and $z$ directions. In this sense, the system (3) experiences
a symmetry protected topological phase transition that happens at the critical
point $m=b$.

Note that for the hydrodynamical modes, $\omega(k=0)$ is not zero anymore due
to the non-conservation of energy, i.e. energy is constantly pumped into or
out of the system. These crossing nodes at $m<b$ are dissipative when order
$k^{2}$ terms are taken into account. This is different from the $\omega=k=0$
nodes which are real poles in hydrodynamics with unbroken translational
symmetries.

The new $m$ and $b$ terms above are not dissipative so they only change the
shape of the spectrum while do not introduce any imaginary parts in the
dispersion relation. In contrast, momentum dissipation terms in, e.g.,
Hartnoll:2007ih ; Hartnoll:2016apf ; Grozdanov:2018fic  are dissipative terms.

##  IV Origin for non-conservation terms of $T^{\mu\nu}$

The simplest way to have the non-conservation terms of $T^{\mu\nu}$ in (3) is
to introduce an external rank two symmetric tensor field. A natural
possibility is for this external field to be a gravitational field
$h_{\mu\nu}$ footnote-fmunu . Then the whole metric field is
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, and the energy momentum tensor is
conserved as $\nabla_{\mu}T^{\mu\nu}=0$ in the new spacetime so that
$\partial_{\mu}T^{\mu\nu}=0$ does not hold anymore. Expanding this equation in
$h_{\mu\nu}$, we get

$\partial_{\mu}\delta T^{\mu\nu}=-\frac{1}{2}\partial_{\alpha}h\delta
T^{\alpha\nu}-\frac{1}{2}\eta^{\nu\beta}(2\partial_{\mu}h_{\alpha\beta}-\partial_{\beta}h_{\mu\alpha})\delta
T^{\mu\alpha}\,.$ (6)

Again, we have assumed that $\mathcal{O}(h_{\mu\nu})\sim\mathcal{O}(k)$ and
only kept leading order in $k$ terms footnote2 . To get the exact $m$ and $b$
terms in the effective Hamiltonian (5), there are infinite many choices for
$h_{\mu\nu}$ and the simplest choice is for $h_{\mu\nu}$ to be footnote3

$\displaystyle\begin{split}&h_{tt}=h_{xx}=mx\,,\leavevmode\nobreak\
\leavevmode\nobreak\
h_{tx}=h_{xt}=\frac{1}{2}mt(v_{s}^{2}+1)\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \\\
&h_{ty}=h_{yt}=-\frac{1}{2}bv_{s}z\,,\leavevmode\nobreak\ \leavevmode\nobreak\
h_{tz}=h_{zt}=\frac{1}{2}bv_{s}y\,.\end{split}$ (7)

This graviton field $h_{\mu\nu}$ (S2) could come from sources of massive
matter, and more interestingly, it could also come from a reference frame
transformation from the flat Minkowski metric generated by
$\tilde{x}_{\mu}=x_{\mu}+\xi_{\mu}$ with footnote4 ,

$\xi_{\mu}=\bigg{(}\frac{mxt}{2},\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{mx^{2}}{4}+\frac{mt^{2}}{4}v_{s}^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ -\frac{b}{4}v_{s}zt,\leavevmode\nobreak\
\leavevmode\nobreak\ \frac{b}{4}v_{s}yt\bigg{)}\,.$ (8)

This is an intriguing result as usually a nontrivial gravitational field could
not be transformed to a flat spacetime globally but only locally. It could be
checked that this new metric field has all the components of the Riemann
tensor vanishing at leading order, thus could be transformed to the flat
spacetime. Though equivalent to a flat spacetime, $h_{\mu\nu}$ could still be
viewed as a nontrivial gravitational field according to the equivalence
principle. This $h_{\mu\nu}$ denotes a non-inertial reference frame. This
result suggests that in a specific non-inertial frame, we could observe
hydrodynamic modes that are topologically protected even when they are
topologically trivial in the original inertial frame. This could be tested in
laboratories, in principle.

Note that with a nonzero $h_{\mu\nu}$, the constitutive equations for
$T_{\mu\nu}$ could also be written into a covariant form thus leading to extra
terms compared to the original constitutive equations. However, it can be
explicitly checked that these extra terms do not change the spectrum (3) at
all or do not change the spectrum up to a rescaling of parameters $m$, $b$,
and $v_{s}$ depending on whether the fluid is resting in the original inertial
frame or is accelerating together with the accelerating observer; i.e., the
new spectrum could be obtained by substituting the rescaling relation above to
the spectrum (3), and this does not mean that the real speed of sound changes.
More details could be found in the appendix.

##  V The exact non-inertial reference frame

We could work out the exact reference frame from the infinitesimal
transformation (8). A rest observer in the new frame $\tilde{x}^{\mu}$ has
$d\tilde{x}^{i}=0$ for $i=1,2,3$. From this we could obtain $d\tilde{t}=dt$,
$dx=-\frac{mv_{s}^{2}tdt}{2}$, $dy=\frac{bv_{s}zdt}{4}$, and
$dz=-\frac{bv_{s}ydt}{4}$ at order $O(m)$. From the last two equations above,
we have $y=R_{0}\cos\frac{bv_{s}}{4}t$ and $z=-R_{0}\sin\frac{bv_{s}}{4}t$
with appropriate choice for $t=0$ and a constant radius $R_{0}$. These
together confirm that the rest observer of the new non-inertial frame is in
fact an accelerating observer in the original inertial frame who has a
constant acceleration $a=-\frac{mv_{s}^{2}}{2}$ in the $x$ direction and a
constant angular velocity $\omega_{x}=\frac{bv_{s}}{4}$ in the $y$ and $z$
plane. As $m$ and $b$ are small parameters, the observer is moving in the non-
relativistic limit, which is consistent with $d\tilde{t}=dt$. Note that the
fluid is still a relativistic one though moving collectively non-
relativisitically.

Thus, the topological modes are those observed by an accelerating observer
moving together with the fluid along a helix in the non-relativistic limit as
shown in Fig. 2, which makes it, in principle, a realizable setting for
experimental tests of this system. Physically, the non-conservation terms for
$T^{\mu\nu}$ could be thought of as coming from the fictitious force,
including the Coriolis force, the centrifugal force as well as the inertial
force associated with the $x$ direction constant acceleration. The non-trivial
topological nodes could also be viewed as coming from these fictitious
effects. Now we have shown that the seemingly ad hoc non-conservation terms of
$T^{\mu\nu}$ could in fact be generated from a very natural non-inertial
reference frame.

Figure 2: The helix trajectory of the accelerating observer and the fluid in
the laboratory frame. The observer/fluid has both a constant acceleration and
a constant angular velocity in the $x$ direction.

Now we could work out the symmetry of the system (3) as the isometry of the
metric $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, i.e. coordinate transformations
that leave $g_{\mu\nu}$ unchanged. The symmetry could be viewed as the Lie
transformation of the Poincare symmetry generated by the vector (8). Among the
ten generators of this new isometry Liu:2020abb , two of them are responsible
for forbidding $m$ terms in the $y$ and $z$ directions and protecting the
nontrivial topological states, which are generated by $x^{\mu}\to
x^{\mu}+\epsilon^{\mu}$, where $\epsilon^{\mu}=a_{y}\chi_{y}+a_{z}\chi_{z}$
with $\chi_{y}=\left(-\frac{bzv_{s}}{4},\leavevmode\nobreak\
0,\leavevmode\nobreak\ 1,\leavevmode\nobreak\ -\frac{btv_{s}}{4}\right)$,
$\chi_{z}=\left(\frac{byv_{s}}{4},\leavevmode\nobreak\ 0,\leavevmode\nobreak\
\frac{byv_{s}}{4},\leavevmode\nobreak\ 1\right)$ and $a_{y}$, $a_{z}$ two
infinitesimal constants. $\chi_{y}$ (or $\chi_{z}$) is a special combination
of the $y$ (or $z$) direction translational symmetry and the boost symmetry of
the $t$-$y$ (or $t$-$z$) direction. Though this symmetry looks complicated,
physically it only requires the covariant conservation of momentum in the $y$
and $z$ directions in the non-inertial frame; i.e. there are no extra external
forces in the $y$ and $z$ directions.

Finally we mention another possible circumstance to have this nonzero
$h_{\mu\nu}$, which could arise in analog gravity systems, i.e. certain
materials could give rise to effective hydrodynamic equations as if there
exists a nontrivial gravitational field.

##  VI Topological invariant

For systems protected by a certain symmetry, we could calculate the
topological invariant at a high symmetric point in the momentum space, which
is $k_{y}=k_{z}=0$ in this case. There is a charge conjugation symmetry for
the solutions and we could focus on the lower two nodes in the left top plot
of Fig. 1. Here, as we are in zero effective residual spatial dimension, the
calculation of the topological invariant is different from the Berry phase or
Berry curvature for a nodal line or Weyl semimetals. For the left node at
$k_{x}=k_{1}$ in the left top plot of Fig. 1, the green solution at the left
limit $k_{x}\to k_{1-}$ and the right limit $k_{x}\to k_{1+}$ are denoted as
$|n_{1}\rangle$ and $|n_{2}\rangle$ separately. We could define a Berry phase
between the two states $e^{-i\alpha}=\frac{\langle
n_{1}|n_{2}\rangle}{|\langle n_{1}|n_{2}\rangle|}$ to denote the topological
invariant here. If the Berry phase is an undetermined one; i.e.,
$|n_{1}\rangle$ and $|n_{2}\rangle$ are orthogonal to each other, the system
would be topologically nontrivial as the two states cannot be connected
without passing through a singularity, which means the lower band and the
upper band could not be separated by small perturbations.

In (3), $|n_{1}\rangle=\frac{1}{\sqrt{2}}\big{(}0,0,-i,1\big{)}$ and
$|n_{2}\rangle=\frac{1}{\sqrt{1+\frac{1}{v_{s}^{2}}}}\Big{(}-\frac{i\sqrt{m^{2}+k_{x}^{2}}}{v_{s}(m+ik_{x})},1,0,0\Big{)}$.
Therefore $\langle n_{1}|n_{2}\rangle=0$, which means that the Berry phase is
undetermined. From the argument above the two bands cannot be separated easily
by a gap without going through a topological phase transition. Similar
behavior of an undetermined Berry phase has also happened for the holographic
nodal line semimetals Liu:2018djq ; Landsteiner:2019kxb .

This result confirms that the four nodes in Fig. 1 are topologically
nontrivial protected by a special combination of translational and boost
symmetry in the $y$ and $z$ directions. At the same time, the Berry phase
accumulated through the whole circle around this node would be trivial
indicating that it is indeed topologically trivial without the symmetry.

##  VII Transport properties

We can follow the calculations in Kovtun:2012rj ; Davison:2014lua  to compute
the heat transport for this system to uncover more observational effects. We
obtain

$\displaystyle\kappa_{xx}(\omega,k_{x})$ $\displaystyle=$
$\displaystyle-\frac{i\omega(\epsilon+P)}{T\left((k_{x}^{2}+m^{2})v_{s}^{2}+i\frac{\eta}{\epsilon+P}\omega
k_{x}^{2}-\omega^{2}\right)}\,,$ $\displaystyle\kappa_{yy}(\omega,k_{x})$
$\displaystyle=$
$\displaystyle\kappa_{zz}(\omega,k_{x})=-\frac{k_{x}^{2}\eta+i\omega(\epsilon+P)}{T\left(b^{2}v_{s}^{2}+(i\omega+\frac{\eta}{\epsilon+P}k_{x}^{2})^{2}\right)}\,,$
$\displaystyle\kappa_{yz}(\omega,k_{x})$ $\displaystyle=$
$\displaystyle-\kappa_{zy}(\omega,k_{x})=\frac{(\epsilon+P)bv_{s}}{T\left(b^{2}v_{s}^{2}+(i\omega+\frac{\eta}{\epsilon+P}k_{x}^{2})^{2}\right)}\,.$

With the formulas above, when $m=b=0$, all diagonal components of the dc heat
transport diverge. For generic $m$ and $b$, we have vanishing dc heat
transport $\kappa_{xx}(0,0),\kappa_{yy}(0,0)$ and $\kappa_{zz}(0,0)$ while
$\kappa_{yz}(0,0)=-\kappa_{zy}(0,0)=\frac{\epsilon+P}{Tbv_{s}}.$ These $m$ and
$b$ terms eliminate the unphysical divergence of dc heat transports and lead
to interesting vanishing dc heat transport behavior. An intuitive and physical
reason for this system to be a dc thermal insulator could be seen from the
spectrum, where there is an energy gap at $k=0$ in contrast to being gapless
at $k=0$ for standard hydrodynamics, as the consequence of energy non-
conservation. This is similar to the mechanism of finite dc conductivity
obtained for cases with momentum dissipation.

##  VIII $\mathcal{O}(k^{2})$ effects

$\mathcal{O}(k^{2})$ terms lead to dissipative effects and give rise to
imaginary parts of frequency in the spectrum. The $\mathcal{O}(k^{2})$ terms
make the effective Hamiltonian matrix non-Hermitian. Here, we still keep terms
at $m\sim b$ order while not $m^{2}\sim b^{2}$ order assuming that $m\sim
k^{2}$.

From the eigenvalues of Hamiltonian with $\mathcal{O}(k^{2})$ effects
included, we find that the real part has not changed while imaginary parts
appear. At the four nodes, the imaginary parts are not zero indicating that
the four nodes are dissipative in comparison to nondissipative nodes at
$\omega=0$ in the usual hydrodynamics. The imaginary part for each of the band
has a jump at the crossing nodes at $k_{y}=0$ in the $k_{x}$ axis, i.e. the
imaginary parts of the same band are different at the left and right limits of
the singular node. This behavior is similar to the behavior of the eigenstates
when calculating the Berry phase and thus provides another piece of evidence
of the existence of a symmetry protected topological singular node.

##  IX Ward identities and holographic realization

The physics of hydrodynamics has been studied extensively in holography for
strongly coupled systems Kovtun:2004de ; Son:2007vk ; Rangamani:2009xk . We
aim to construct a holographic system possessing the same non-conservation
equation of (3), thus providing an example of this system in the strongly
coupled limit.

Holographically we could also perform a coordinate transformation to get a
non-inertial frame version of AdS/CFT correspondence which has the metric
$g^{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ at the boundary. This system should have
the same spectrum of the hydrodynamic modes. As a first step for a
confirmation, we need to show that our holographic system indeed has the non-
conservation of (3). For this purpose, we will first obtain the Ward
identities for $T^{\mu\nu}$ in the non-conserved hydrodynamic system from (3)
and match these identities to those in the holographic non-inertial frame
system.

In the case that these non-conservation terms come from a gravitational field,
we could start from the covariant conservation equation
$\nabla_{\mu}T^{\mu\nu}=0$ and differentiate it with respect to
$g_{\lambda\rho}$ to obtain the Ward identities in the momentum space of the
boundary system. To the first order in $h_{\mu\nu}$, the Ward identities are

$\displaystyle
k_{\mu}G^{\mu\nu,\lambda\rho}(k)+i\Big{[}\Gamma^{(1)\mu}_{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mu\alpha}G^{\alpha\nu,\lambda\rho}(k)+\Gamma^{(1)\nu}_{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mu\alpha}G^{\mu\alpha,\lambda\rho}(k)\Big{]}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\text{contact terms}=0\,,$ (9)

where the explicit form of the contact terms is omitted. With nonzero
$h_{\mu\nu}$ several components of $\Gamma^{(1)\nu}_{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mu\alpha}$
would be nonzero and contribute extra terms to the Ward identities of
$k_{\mu}G^{\mu\nu,\lambda\rho}(k)+\text{contact terms}=0$ in hydrodynamics
systems with conserved $T^{\mu\nu}$ Policastro:2002tn .

The Ward identities (IX) could be reproduced from the holographic non-inertial
frame system where we start from the usual AdS Schwartzchild black hole and
perform coordinate transformations so that the boundary metric becomes
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$.

We have checked that the holographic Ward identities match exactly to the
hydrodynamic Ward identities (IX). The details will be presented in
Liu:2020abb , and the hydrodynamic modes and Green functions will be
systematically studied in future work.

##  X Outlook

One important application of the observation of topologically nontrivial
hydrodynamics modes at finite frequency would be to enlarge the amplitudes at
the crossing frequencies and momenta due to the doubling of modes and this
enlargement could be stable from perturbations under certain conditions.

The fact that an accelerating observer moving along a helix would see
topological hydrodynamic modes of a hydrodynamic system moving in the same
helix could, in principle, be checked in experiments, including probing the
spectrum of the modes and measuring the featured transport properties of the
system.

Finally, and most importantly, it is possible that systems other than
hydrodynamic systems, e.g., electronic/photonic systems, would also become
topologically nontrivial being observed in a certain non-inertial frame
footnote-ele , which would provide another way to obtain topologically
nontrivial materials by mechanically accelerating the detector in a
laboratory. This brings up a new interesting effect for accelerating observers
in addition to the well-known Unruh effect Unruh:1976db .

### Acknowledgments

We would like to thank Matteo Baggioli, Rong-Gen Cai, Wen-Bin Pan, Koenraad
Schalm, Qing Zhang, anonymous referees and especially Karl Landsteiner for
useful discussions. This work is supported by the National Key R&D Program of
China (Grant No. 2018FYA0305800). The work of Y.L. was also supported by the
National Natural Science Foundation of China Grant No.11875083. The work of
Y.W.S. has also been partly supported by starting grants from University of
Chinese Academy of Sciences and Chinese Academy of Sciences, and by the Key
Research Program of Chinese Academy of Sciences (Grant No. XDPB08-1), the
Strategic Priority Research Program of Chinese Academy of Sciences, Grant No.
XDB28000000.

## References

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Appendix: Topological modes in relativistic hydrodynamics

##  I Another possible origin for the rank two symmetric external tensor field

In this appendix we show the possibility for the rank two symmetric external
tensor field $f_{\mu\nu}$ to be an external symmetric tensor matter field that
couples to the energy momentum tensor of the system and contributes an
effective $f_{\mu\nu}T^{\mu\nu}$ term to the Lagrangian of the system. With
this extra term the energy momentum of the system will not be conserved as it
can be transferred to the external system.

Omitting terms at order of $\mathcal{O}(k^{2})$ or higher, we get the non-
conservation equation for $T^{\mu\nu}$

$\partial_{\mu}T^{\mu}_{\leavevmode\nobreak\
\nu}=-\frac{1}{2}T^{\rho\mu}(2\partial_{\mu}f_{\rho\nu}-\partial_{\nu}f_{\rho\mu})\,.$
(S1)

As a simple realization, we could switch on the following components of
$f_{\mu\nu}$ to get the form of effective Hamiltonian (5)

$\displaystyle\begin{split}&f_{tt}=f_{xx}=mx\,,\leavevmode\nobreak\
\leavevmode\nobreak\
f_{tx}=f_{xt}=\frac{1}{2}mt(v_{s}^{2}+1)\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \\\
&f_{ty}=f_{yt}=-\frac{1}{2}bv_{s}z\,,\leavevmode\nobreak\ \leavevmode\nobreak\
f_{tz}=f_{zt}=\frac{1}{2}bv_{s}y\,.\end{split}$ (S2)

##  II Change in the constitutive equations

In this appendix we show explicitly that after carefully considering the
modifications of the constitutive equations to the equations (3) in the main
text, the spectrum does not change qualitatively and the discussions in the
main text at the leading order in $k$ stay the same. Note that there are two
possibilities for the hydrodynamic system: accelerating in the same way as the
observer and accelerating in a different way.

We will first consider the case where the observer is in a non-inertial frame
and the fluid is at rest in the same non-inertial frame. Then we also consider
a second possibility where the observer is accelerating and the fluid is
accelerating at another constant acceleration in the original inertial frame,
and one special limit would be that the fluid is at rest in the original
inertial frame while the observer is accelerating in the helix. For both
cases, the spectrum stays the same up to a rescaling of parameters $m$, $b$
and $v_{s}$, which does not change the qualitative behavior, though there
would be a little difference in the explanation of the topological nature.

###  II.1  I. Observer and fluid accelerating in the same way

In this part we show explicitly in the case of observer and fluid accelerating
in the same way, how the change in the constitutive equations for $T^{\mu\nu}$
due to the graviton field $h^{\mu\nu}$ will not affect the spectrum that was
calculated from the constitutive equations in the flat metric except a
rescaling of the parameters $m,\leavevmode\nobreak\ b$ and $v_{s}$.

For the background metric $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, the covariant
form of the constitutive equation for the energy momentum tensor is

$T^{\mu\nu}=(\epsilon+P)u^{\mu}u^{\nu}+Pg^{\mu\nu}$ (S3)

at zeroth order in derivative, where $g_{\mu\nu}u^{\mu}u^{\nu}=-1$ and
$u^{i}=0$ for $i=x,y,z$ because the fluid is at rest in the non-inertial
frame, i.e. accelerating in the inertial frame. The corrections of
$T^{\mu\nu}$ compared to the energy momentum tensor for the flat metric
$\eta^{\mu\nu}$ is at order $O(m)\sim O(k)$ as $h^{\mu\nu}$ is of order
$O(m)$.

Now we examine carefully how these extra terms contribute to the
(non-)conservation equation (6). In (6), the extra terms on the right hand
side compared to the original equation are at order $O(k^{2})$ thus have no
extra contributions compared to original terms when modifications of the
constitutive equations were not considered. On the left hand side, in
$\partial_{\mu}\delta T^{\mu\nu}$ extra terms that have derivatives of
perturbations $\delta\epsilon,\delta P$ or $\delta u^{\mu}$ have order
$O(k^{2})$ thus do not contribute at leading order. Extra terms that might
have contributions to the conservation equation are

$\delta(\epsilon+P)\partial_{\mu}(u^{\mu}u^{\nu})+(\epsilon+P)(\delta
u^{\mu}\partial_{\mu}u^{\nu}+\delta u^{\nu}\partial_{\mu}u^{\mu})+\delta
P\partial_{\mu}g^{\mu\nu}.$ (S4)

As the only nonvanishing component for $u^{\mu}u^{\nu}$ is the $tt$ component,
it can be checked that $\partial_{\mu}(u^{\mu}u^{\nu})=0$ and $\delta
u^{\nu}\partial_{\mu}u^{\mu}=0$ as $h^{tt}$ does not depend on $t$. Thus
substituting the exact expression for $h^{\mu\nu}$, the nonzero contribution
to the (non-)conservation equation (6) only exists for the $\nu=t$ and $\nu=x$
components, each being proportional to the nonconservation term on the right
with a different factor in front: for $\nu=t$ there will be a contribution of
an extra $-\frac{m}{2}T^{0x}$ on the right and for $\nu=x$ there is an extra
$-\frac{m}{2}(1-v_{s}^{2})v_{s}^{2}T^{00}$ on the right. Thus these terms only
result in a rescaling in the corresponding parameters of $m$ and $v_{s}$, i.e.
the resulting spectrum could be obtained from rescaled parameters
$\bar{m},\bar{v_{s}},\bar{b}$ in the long expression for the spectrum below
(5) as

$\bar{m}=\frac{m}{2}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\bar{v_{s}}=v_{s}\sqrt{3-v_{s}^{2}}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \bar{b}=\frac{b}{\sqrt{3-v_{s}^{2}}}\,.$ (S5)

As $v_{s}$ cannot be larger than $1$ ($v_{s}=1/\sqrt{3}$ for a conformal field
theory and this value is expected to be the upper bound for the speed of
sound), all the rescaled parameters above are nonzero thus the rescaling does
not change the qualitative behavior of the spectrum. Besides this, there is
another similar case with a different set of $h_{\mu\nu}$ which we will
discuss in a later work.

Note that though the fluid is collectively accelerating under an external
force $\nabla_{\mu}\delta T^{\mu\nu}=0$ would still hold for perturbations of
the fluid slightly away from equilibrium. One could write out the explicit
conservation equation of energy momentum of the accelerating fluid with
external forces (that accelerate the fluid as seen from the inertial frame) as
well as gravitational force on the right side. Then we perturb this system
from the equilibrium state and external forces would not contribute to the
equations for the perturbations as they stay the same as long as the total
mass of the system does not change. Physically this is because the fluid
elements only experience internal forces as well as gravitational forces while
not external forces except at the boundary of the fluid thus
$\nabla_{\mu}\delta T^{\mu\nu}=0$ still holds for perturbations.

###  II.2 Observer accelerating and fluid rest in the inertial frame

Note that from the discussions on the origin of non-conservation terms of
$T^{\mu\nu}$ from reference frame changes, we can see that the observer is in
a non-inertial frame $g_{\mu\nu}$, and the fluid could stay at rest in the
inertial frame. In this case, the hydrodynamic system has the four velocity
$u^{\mu}=(-1,\leavevmode\nobreak\ 0,\leavevmode\nobreak\
0,\leavevmode\nobreak\ 0)$. By the reference frame transformation (8) we could
obtain the four velocity in the non-inertial reference frame
$u^{\mu}=(-1-\frac{mx}{2},\frac{mt}{2}v_{s}^{2},-\frac{bv_{s}z}{4},\frac{bv_{s}y}{4})$
up to the first order in $m,\leavevmode\nobreak\ b$. In the accelerating
frame, $\eta^{\mu\nu}$ in $T^{\mu\nu}$ of the fluid should also be replaced by
$g^{\mu\nu}$, which is

$g_{\mu\nu}=\begin{pmatrix}-1+mx&\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{mt}{2}(1+v_{s}^{2})&\leavevmode\nobreak\ \leavevmode\nobreak\
-\frac{1}{2}bv_{s}z&\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{1}{2}bv_{s}y\\\ \frac{mt}{2}(1+v_{s}^{2})&\leavevmode\nobreak\
\leavevmode\nobreak\ 1+mx&\leavevmode\nobreak\ \leavevmode\nobreak\
0&\leavevmode\nobreak\ \leavevmode\nobreak\ 0\\\
-\frac{1}{2}bv_{s}z&\leavevmode\nobreak\ \leavevmode\nobreak\
0&\leavevmode\nobreak\ \leavevmode\nobreak\ 1&\leavevmode\nobreak\
\leavevmode\nobreak\ 0\\\ \frac{1}{2}bv_{s}y&\leavevmode\nobreak\
\leavevmode\nobreak\ 0&\leavevmode\nobreak\ \leavevmode\nobreak\
0&\leavevmode\nobreak\ \leavevmode\nobreak\ 1\end{pmatrix}\,.$ (S6)

The perturbations $\delta T^{\mu\nu}$ at the leading order are

$\delta T^{\mu\nu}=(1+v_{s}^{2})\delta\epsilon
u^{\mu}u^{\nu}+(\epsilon+P)(\delta u^{\mu}u^{\nu}+u^{\mu}\delta
u^{\nu})+v_{s}^{2}\delta\epsilon g^{\mu\nu}$ (S7)

where $v_{s}=\sqrt{\frac{\partial P}{\partial\epsilon}}$ and $u^{\mu}$ in this
formula is the four velocity of the fluid in the non-inertial frame.

Note that the four velocities of the fluid is normalized as
$g_{\mu\nu}u^{\mu}u^{\nu}=-1$, thus one of the fluctuations of these four
velocities is not independent. We take the independent fluctuations of the
fluid as $\Psi=(\delta\epsilon,\delta\pi^{x},\delta\pi^{y},\delta\pi^{z})^{T}$
with $\delta\pi^{i}=-(\epsilon+P)\delta u^{i}$. Note that here $\delta\pi^{x}$
is not $\delta T^{tx}$, and the spectrum of $H$ will stay the same if we use
instead variables $(\delta\epsilon,\delta u_{x},\delta u_{y},\delta
u_{z})^{T}$ or any other set of combined variables. Plugging (S7) into the
dynamical equation (6) in the main text, we obtain equation
$i\partial_{t}\Psi=H\Psi$ with

$H=\begin{pmatrix}0&\leavevmode\nobreak\ \leavevmode\nobreak\
k_{x}-{im}&\leavevmode\nobreak\ \leavevmode\nobreak\
k_{y}&\leavevmode\nobreak\ \leavevmode\nobreak\ k_{z}\\\
(k_{x}+im)v_{s}^{2}&\leavevmode\nobreak\ \leavevmode\nobreak\
0&\leavevmode\nobreak\ \leavevmode\nobreak\ 0&\leavevmode\nobreak\
\leavevmode\nobreak\ 0\\\ k_{y}v_{s}^{2}&\leavevmode\nobreak\
\leavevmode\nobreak\ 0&\leavevmode\nobreak\ \leavevmode\nobreak\
0&\leavevmode\nobreak\ \leavevmode\nobreak\ -ibv_{s}\\\
k_{z}v_{s}^{2}&\leavevmode\nobreak\ \leavevmode\nobreak\
0&\leavevmode\nobreak\ \leavevmode\nobreak\ ibv_{s}&\leavevmode\nobreak\
\leavevmode\nobreak\ 0\end{pmatrix}\,$ (S8)

at leading order in $k$. Thus we get exactly the same spectrum as in (5). Note
that in the calculation, we have assumed that $\delta T^{\mu\nu}(\omega,k)$
could also be expanded as $\omega$ and $k$, and one has to be careful about
the Fourier transformation process where certain partial derivatives in
$\omega$ and $k$ have to be taken into account.