# Isovector Meson Masses from QCD Sum Rules

Nasrallah F. Nasrallah(a), Karl Schilcher(b,c)
(a) Faculty of Science, Lebanese University, Tripoli 1300, Lebanon
(b) Institut für Physik, Johannes Gutenberg-Universität
Staudingerweg 7, D-55099 Mainz, Germany
(c)Centre for Theoretical Physics and Astrophysics
University of Cape Town, Rondebosch 7700, South Africa
Supported in part by NRF (South Africa) and Alexander von Humbold Foundation
(Germany)

###### Abstract

We present a calculation of the masses of the isovector mesons ( vector,
scalar and pseudoscalar including the established recurrences) using a new
method of finite energy QCD sum rules. The method is based on the idea of
choosing a suitable integration kernel which minimizes the occurring integral
over the cut in the complex energy (squared) plane. We obtain remarkably
stable results in a wide range $R$, where $R$ is the radius of the integration
contour. The sum rule predictions agree with the experimental values within
the expected accuracy showing that QCD describes single resonances.

KEYWORDS: Sum Rules, QCD, meson masses.

##  1 QCD sum rules and meson masses

The mass of the $\rho$-meson was first calculated from QCD and the operator
product expansion (OPE) in the pioneering paper of Shifman, Vainshtein and
Zakharov [1]. The calculation is based on QCD sum-rules of the Borel (or
Laplace) type. Although the results were at the time rather spectacular, it
was soon recognized [2] that, apart from the arbitrariness of the integration
kernel, they suffer from instabilities related to the specific choice of the
Borel variable and the assumption of a constant continuum of the isovector
spectral function. There are two arbitrary parameters, the Borel parameter,
called $M_{0}^{2}$, and the onset of the continuum. Note, in the original
paper the authors based their analysis on a small QCD coupling, corresponding
to a scale $\Lambda_{\text{QCD}}\approx 100$ MeV compared to the modern value
$\Lambda_{\text{QCD}}\approx 350$ MeV).

A more stringent approach is based on finite energy sum rules [3]. We
developed this idea further and used it in numerous applications [4]. Here we
will use our approach to calculate all relevant hadron masses, starting with
the $\rho$-meson [5]

$m_{\rho}=0.775.25\pm.0.00026\text{ GeV\ }m_{\rho}^{2}=0.6010\text{ GeV}^{2}$

Consider the isovector current

$j_{\mu}=\frac{1}{2}(\bar{u}\gamma_{\mu}u-\bar{d}\gamma_{\mu}d)$ (1)

with the quantum numbers of the $\rho$. The relevant spectral function is
given by the absorptive part of the correlator

$\displaystyle\Pi_{\mu\nu}(q)$ $\displaystyle=$ $\displaystyle i\int
d^{4}xe^{iqx}\left\langle 0\left|T\,j_{\mu}(x)j_{\nu}(0)\right|0\right\rangle$
(2) $\displaystyle=$ $\displaystyle(q_{\mu}q_{\nu}-g_{\mu\nu}q^{2})\Pi(q^{2})$
(3)

Phenomenologically the spectral functions built up by the $\rho$ and the
higher resonances $\rho^{\prime}(1450)$, $\rho^{\prime\prime}(1700)$,….
Neglecting its width, the experimental spectral function of the $\rho$ is
given by

$\rho^{\text{exp}}(t)=\frac{m_{\rho}^{2}}{g_{\rho}^{2}}\delta(t-m_{\rho}^{2})=\frac{1}{\pi}\text{Im}\Pi^{\text{exp}}(t)$
(4)

where $g_{\rho}$ is defined by

$\left\langle
0\left|j_{\mu}(0)\right|\rho(p,s)\right\rangle=\frac{m_{\rho}^{2}}{g_{\rho}}\varepsilon_{\mu}$

with $g_{\rho}=4.97\pm 0.07$ as determined from the leptonic decay of the rho-
meson. The spectral function of Eq.(4) corresponds to an amplitude

$\Pi^{\text{exp}}(t)=-\frac{m_{\rho}^{2}}{g_{\rho}^{2}}\frac{1}{(t-m_{\rho}^{2})}\text{.}$
(5)

To lowest non-trivial order, the corresponding QCD expression is

$\Pi^{\text{QCD}}(t)=-\frac{1}{8\pi^{2}}(1+a_{s})L+\frac{\left\langle
m_{u}\bar{u}u+m_{d}\bar{d}d\right\rangle}{2t^{2}}+\frac{\left\langle
a_{s}GG\right\rangle}{24t^{2}}+\frac{112\pi}{81t^{3}}(\sqrt{\alpha_{s}}\left\langle\bar{q}q\right\rangle)^{2}+...$
(6)

where $a_{s}=\alpha_{s}/\pi$ is the strong coupling at the scale $\mu$ and$\
L\equiv\ln\frac{-t}{\mu^{2}}$. We take $a_{s}=0.1$ for $\mu$ of order $3\text{
GeV}^{2}$ to $4\text{ GeV}^{2}$ as measured in $\tau$-decay. The variation of
$a_{s}$ in this region is of higher order.

To next order in QCD and the $\overline{MS}$ scheme, the perturbative part of
the vector correlator is given by [6]

$8\pi^{2}\Pi^{\text{QCD}}=-\left[1+L+aL+a^{2}(F_{3}L+\frac{\beta_{1}}{4}L^{2})\right]+...$
(7)

where

$\beta_{1}=-\frac{1}{2}(11-\frac{2}{3}n_{f}),\ \ \ \ F_{3}=1.9857-0.1153n_{f}$

The basis of FESR is Cauchy’s theorem applied to the contour of Fig. 1

Fig. 1: Integration contour of FESR

which implies

$-\frac{m_{\rho}^{2}}{g_{\rho}^{2}}P(m_{\rho}^{2})=\frac{1}{\pi}\int\limits_{\text{cut}}^{R}dt\,P(t).Im\Pi^{\text{exp}}(t)+\frac{1}{2\pi
i}\oint\limits_{\left|t\right|=R}dt\,P(t)\Pi^{\text{QCD}}(t)$ (8)

where $P(t)$ is an entire function, e. g. a polynomial. Over the circle of
large radius $R$ the correlator $\Pi(t)$ has been replaced by its QCD
expression. The principal unknown in Eq.(8) is the integral over the cut, i.
e. over the higher vector-isovector resonances with mass $m_{\rho_{i}}^{2}\leq
R$. To minimize this integral (before neglecting it), a judicious choice of
the weight-function $P(t).$ has to be made. With the classic choice [1]
$P(t)=\exp(-t/M_{0}^{2})$ the Borel variable $M_{0}^{2}$ cannot be chosen too
large because it would minimize the contribution of the $\rho$-meson. Also
$M_{0}^{2}$ cannot be too small because the unknown condensates in Eq.(6)
would explode. It was hoped in [1] that a region of stability at an
intermediate $M_{0}^{2}$ can be found. This can be shown to be not the case
[2]. In our FESR approach we take $P(t)$ to be a polynomial

$P(t).=\sum_{n=0}^{n_{\text{max}}}c_{n}t^{n}\text{ .}$ (9)

It is clear that the order $n_{\text{max}}$ cannot be chosen arbitrarily high
because of the contribution of unknown condensates. We choose a polynomial
$P_{1}(t)$ which vanishes at the mass $m_{1}=(1465\pm 25)$ MeV [5]
($m_{1}^{2}=2.15\text{ GeV}^{2}$) of the first resonance and at the
integration radius $R$. Explicitly, we take

$\displaystyle P_{1}(t,R).$ $\displaystyle=$
$\displaystyle(1-\frac{t}{m_{1}^{2}})(1-\frac{t}{R})$ (10) $\displaystyle=$
$\displaystyle 1-a_{1}t-a_{2}t^{2}\text{ \ where \
}a_{1}(R)=\frac{1}{m_{1}^{2}}+\frac{1}{R}\text{, \
}a_{2}(R)=-\frac{1}{m_{1}^{2}R}$ (11)

When necessary we take $R\gtrsim 3$ GeV2 ($\sim m_{\tau}^{2}$) as we know from
the $\tau$-decay analysis [7] that global duality is valid there. Later we
will need $P_{1}(m_{\rho}^{2}=0.6\text{ GeV}^{2}$, $R=3\text{
GeV}^{-2}$$)=0.576$ and $a_{1}(R=3\text{ GeV}^{2}$) $=0.799\,$ GeV-2 and
$a_{2}(R=3\text{ GeV}^{2}$$)=$ $-0.155$ GeV-4.

If we neglect the contribution of the physical continuum for $0\leq t\leq R$
and use Cauchy’s theorem we arrive at the sum-rule

$\displaystyle\frac{m_{\rho}^{2}}{g_{\rho}^{2}}P_{1}(m_{\rho}^{2},R)$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\oint\limits_{\left|t\right|=R}dt\,P_{1}(t,R)\Pi^{\text{QCD}}(t)$ (12)
$\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}(1+a_{s})\int\limits_{0}^{R}dt\,P_{1}(t,R).-a_{1}\frac{\left\langle
m_{u}\bar{u}u+m_{d}\bar{d}d\right\rangle}{2}-a_{1}\frac{\left\langle
a_{s}GG\right\rangle}{24}+....$

As a check of our method we plot the integral appearing in Eq.(12). As can be
seen from Fig. 2.

$I_{0}(R)=\int_{0}^{R}(1-\frac{t}{m_{1}^{2}})(1-\frac{t}{R})dt=\frac{R}{2}(1-\frac{1}{3}\frac{R}{m_{1}^{2}})\text{
}$ (13)

Fig. 2: The integrals $I_{0}(R)$ (solid line) and $I_{1}(R)$ (dashed line)

If we look for stability of the result for $m_{\rho}$ we have to restrict
$2.5\text{ GeV}^{2}$$\leq R\leq 3.5\text{ GeV}^{2}$, a typical result being
$I_{0}(3GeV^{2}$$)=0.801\text{ GeV}^{2}$. On the one hand this is just the
region where we expect perturbative QCD to be valid, on the other hand we do
not expect higher resonances to contribute significantly to the sum rule in
this region. Our results do not depend on the precise choice of $R$ as an be
seen from Fig. 2 as long as $2.5\text{ GeV}^{2}$ $\leq R\leq 3.5\text{
GeV}^{2}$. We will use Eq.(12) in this region in the chiral limit

$\displaystyle\frac{m_{\rho}^{2}}{g_{\rho}^{2}}P_{1}(m_{\rho}^{2},R)$
$\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}(1+a_{s})\int\limits_{0}^{R}dt\,P_{1}(t,R).-\frac{a_{1}}{24}\left\langle
a_{s}GG\right\rangle$ (14)
$\displaystyle\frac{m_{\rho}^{2}}{g_{\rho}^{2}}0.583$ $\displaystyle=$
$\displaystyle\frac{1.1}{8\pi^{2}}0.801\text{ GeV}^{2}-\frac{0.799\text{
GeV}^{-2}}{24}\left\langle a_{s}GG\right\rangle$ $\displaystyle\text{for
}R\sim 3\text{ GeV}^{2}$

Experimentally, $g_{\rho}$ can be determined from its leptonic decay width

$\Gamma(\rho\rightarrow
e_{+}e_{-})=\frac{1}{3}\alpha^{2}m_{\rho}\frac{4\pi}{g_{\rho}^{2}}=7.04\pm
0.06\text{ KeV}$

$\frac{g_{\rho}^{2}}{4\pi}=1.96\pm 0.02$ (16)

Neglecting the gluon condensate and setting $m_{\rho}^{2}=0.6\text{ GeV}^{2}$,
Eq. (1) gives

$\frac{g_{\rho}^{2}}{4\pi}=\frac{0.6\text{
GeV}^{2}}{4\pi}\left(\frac{1}{0.583}\frac{1.1\times 0.801\text{
GeV}^{2}}{8\pi^{2}}\right)^{-1}=2.494$

This result is consistent with the experimental one Eq.(16) considering that
the former involves the difference of two large numbers. The error cannot be
estimated reliably as it arises mainly from the Ansatz of a narrow
$\rho$-resonance for the spectral function.

The error due to $\Pi_{\text{pert}}^{\text{QCD}}$ is small. To order $a^{2}$
($=(\alpha_{s}/\pi)^{2}$) the correlator is given in Eq.(7). The relevant
integrals can be found in [8].The error due to neglected higher order
perturbative terms turns out to be of order 2% to 4%, significantly smaller
than the further errors to be discussed below.

##  2 The Isovector Vector Mesons

We propose a sum rule method that is optimally suited to calculate all
resonance masses from QCD. We define a polynomial $P_{i}(t)$

$P_{i}(t)=\left(1-\frac{t}{m_{i}^{2}}\right)\left(1-\frac{t}{R}\right)$

which vanishes at the mass $m_{i}$ and at the integration radius $R$. For
example $m_{1}=(1465\pm 25)$ MeV [5] ($m_{1}^{2}=2.15\text{ GeV}^{2}$ ) is the
first resonance recurrence. In order to get the mass of the $\rho$-meson we
take the first moment integral

$\displaystyle\frac{m_{\rho}^{4}}{g_{\rho}^{2}}P_{1}(m_{\rho}^{2},R)$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\oint\limits_{\left|t\right|=R}dt\,t\,P_{1}(t,R).\Pi^{\text{QCD}}(t)$ (17)
$\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}(1+a_{s})\int\limits_{0}^{R}dt\,t\,P_{1}(t,R).-\frac{\left\langle
m_{u}\bar{u}u+m_{d}\bar{d}d\right\rangle}{2}-\frac{\left\langle
a_{s}GG\right\rangle}{24}+....$

Consider the integral

$I_{1}(R,m_{1}^{2})=\int_{0}^{R}(1-\frac{t}{m_{1}^{2}})(1-\frac{t}{R})\,t\,dt=\frac{R^{2}}{12}(2-\frac{R}{m_{1}^{2}})$
(18)

For $R=3.0\text{ GeV}^{2}$ and $m_{1}=1.465$ GeV the result is $I_{1}(3\text{
GeV}^{2}$$)=0.451\text{ GeV}^{4}$. The result is still stable for $2.5\text{
GeV}^{2}$ $\leq R\leq 3.5\text{ GeV}^{2}$, see Fig.2. With the standard values
of the condensates

$\left\langle a_{s}GG\right\rangle=0.013\ \text{GeV}^{4}\ \ ,\ \ \left\langle
m_{u}\bar{u}u+m_{d}\bar{d}d\right\rangle=-1.67\times 10^{-4}\text{ GeV}^{4}$

and choosing $R=3.0\text{ GeV}^{2}$ Eqs.(12) and (17) give

$m_{\rho}=0.73\text{ GeV}$

Our choice (Eq.10) for $P_{1}(t)$ provides a good damping for the contribution
of the continuum in the interval $2.5\text{ GeV}^{2}$$\leq t\leq 3.5\text{
GeV}^{2}$: The contribution of the resonances $\rho(1450)$ and $\rho(1700)$
(almost) vanishes and that of the $\rho(1580)$ is shrunk by a factor of
$P_{1}(1.58^{2}\text{ GeV}^{2})$ $/P_{1}(0.6\text{ GeV}^{2}$$)=-5.7\times
10^{-2}$. This renders the contribution of the continuum negligible.

It is nevertheless worthwhile to assess the influence of the variation of
$P(t)$ in the result for $m_{\rho}$ in order to estimate the error inherent in
the method.

One choice would be

$P(t)=\left(1-\frac{t}{2.1\text{ GeV}^{2}}\right)\left(1-\frac{t}{2.5\text{
GeV}^{2}}\right)\left(1-\frac{t}{2.89\text{ GeV}^{2}}\right)$

yielding $m_{\rho}=0.71$ GeV. Or

$P(t)=\left(1-\frac{t}{2.5\text{ GeV}^{2}}\right)^{2}$

which gives $m_{\rho}=0.79$ GeV. We have tried several other polynomials, so
we give finally

$m_{\rho}=(0.74\pm 0.04)\text{ GeV}$ (19)

Taking an additional moment yields the mass $m_{1}$ of the $\rho_{1}(1450)$.
In addition to Eqs.(12) and (17) we have, neglecting the higher condensates

$\frac{m_{\rho}^{6}}{g_{\rho}^{2}}P_{1}(m_{\rho}^{2},R)=\frac{(1+a_{s})}{8\pi^{2}}\int_{0}^{R}dt\,t^{2}P_{1}(t,R)$
(20)

The mass $m_{1}$ can be determined by the two ratios

$\displaystyle m_{\rho}^{2}$ $\displaystyle=$ $\displaystyle\frac{\text{rhs of
Eq.}(\ref{1.13})\text{ }}{\text{rhs of
Eq.}(\ref{1.7})}=\frac{\int_{0}^{R=3GeV^{2}}dt\,t^{2}P_{1}(t,R)}{\int_{0}^{R=3GeV^{2}}dt\,tP_{1}(t,R)}$
$\displaystyle=$ $\displaystyle\frac{\text{rhs of Eq.}(\ref{1.7})\text{
}}{\text{rhs of
Eq.}(\ref{1.6})}=\frac{\int_{0}^{R=3GeV^{2}}dt\,tP_{1}(t,R)}{\int_{0}^{R=3GeV^{2}}dt\,P_{1}(t,R)}$

all at $R=3.0\text{ GeV}^{2}$. The mass $m_{1}$ is determined by the equating
the two ratios using Eqs. (13), (18) and

$I_{2}(R,m_{1}^{2})=\int_{0}^{R}(1-\frac{t}{m_{1}^{2}})(1-\frac{t}{R})t^{2}dt=\frac{R^{3}}{60}(5-3\frac{R}{m_{1}^{2}})$
(21)

From

$\frac{I_{1}}{I_{0}}=\frac{I_{2}}{I_{1}}$

we obtain

$m_{1}=m_{\rho}(1450)=1.42\pm 0.10\text{ GeV}$

The error is again estimated by varying $R$ by $10\%$.

Figure 3: Alternative integration contour

We can proceed further and consider the integral over the contour indicated in
Fig. 3 with the kernel

$P_{3}(t,R)=\left(1-\frac{t}{m_{3}^{2}}\right)\left(1-\frac{t}{R}\right)$ (22)

In a calculation of $m_{i}$ we choose $m_{i-1}^{2}$ as the lower limit of
integration because the contour starts there. In addition we have a check of
the consitency of the full set of isovector meson mass determination. $R$ (and
$a_{s})$ is again chosen in the stability region ($R\sim 3.5\text{ GeV}^{2}$).

We have to assume here that global duality is for $t\geq m_{1}^{2}=2.15\text{
GeV}^{2}$.

Neglecting all condensates the mass of the $\rho_{2}=\rho(1580)$ is obtained
from the sum rules

$\displaystyle\frac{m_{2}^{2}}{g_{\rho_{2}}^{2}}P_{3}(m_{2}^{2},R$
$\displaystyle=$ $\displaystyle
3.5)=\frac{(1+a_{s})}{8\pi^{2}}\int_{m_{1}^{2}}^{3.5\text{
GeV}^{2}}dt\,P_{3}(t,R)$ (23)
$\displaystyle\frac{m_{2}^{4}}{g_{\rho_{2}}^{2}}P_{3}(m_{2}^{2},R$
$\displaystyle=$ $\displaystyle
3.5)=\frac{(1+a_{s})}{8\pi^{2}}\int_{m_{1}^{2}}^{3.5\text{
GeV}^{2}}dt\,tP_{3}(t,R)$ (24)

taking the mass $m_{3}=m_{\rho}(1700)$ (entering via $P_{3}$) as given. The
ratio of the above integrals gives

$m_{2}^{2}=\frac{\int_{m_{1}^{2}}^{3.5\text{
GeV}^{2}}dt\,tP_{3}(t,R)}{\int_{m_{1}^{2}}^{3.5\text{
GeV}^{2}}dt\,P_{3}(t,R)}$ (25)

in the stability region ($R\sim 3.5\text{ GeV}^{2}$)

$m_{2}=m_{\rho}(1580)=(1.50\pm 0.05)\text{ GeV}$ (26)

The error is obtained by varying $R$ by $\pm 10\%$.

Similarly we can obtain $m_{3}=m_{\rho}(1700)$ from the kernel

$P_{4}(t,R)=\left(1-\frac{t}{m_{4}^{2}}\right)\left(1-\frac{t}{R}\right)$

and integrating over the contour of Fig. 3. Assuming QCD duality is valid from
$m_{2}^{2}=1.465^{2}=2.146\text{ GeV}^{2}$the corresponding sum rule reads

$m_{3}^{2}=\frac{\int_{m_{2}^{2}}^{R}dt\,tP_{4}(t,R)}{\int_{m_{2}^{2}}^{R}dt\,P_{4}(t,R)}$
(27)

For $m_{4}=2.150$ GeV the stability region being $R\sim 4.5\text{ GeV}^{2}$.
The result is

$m_{3}=1.71\text{ GeV}$

The mass $m_{3}$ can also be obtained from $P_{3}(t,R=3.5\text{
GeV}^{2}$$)=\left(1-\frac{t}{m_{3}^{2}}\right)\left(1-\frac{t}{R}\right)$ from

$m_{2}^{2}=\frac{\int_{m_{1}^{2}}^{R=3.5\text{
GeV}^{2}}dt\,t^{2}P_{3}(t,R)}{\int_{m_{1}^{2}}^{R=3.5\text{
GeV}^{2}}dt\,tP_{3}(t,R)}=\frac{\int_{m_{1}^{2}}^{R=3.5\text{
GeV}^{2}}dt\,tP_{3}(t,R)}{\int_{m_{1}^{2}}^{R=3.5\text{
GeV}^{2}}dt\,P_{3}(t,R)}$

The equality of the ratios gives

$m_{3}=1.74\text{ GeV}$ (28)

Combining the two results we obtain

$m_{3}=m_{\rho}(1700)=1.73\pm 0.05\text{ GeV}$

We finally proceed to calculate $m_{4}=m_{\rho}(2150)$ using the kernel

$P_{4}(t,R)=\left(1-\frac{t}{m_{4}^{2}}\right)\left(1-\frac{t}{R}\right)$

in the stability region $R\sim 6.0\text{ GeV}^{2}$. The mass $m_{4}$ follows
from the sum rule

$m_{3}^{2}=\frac{\int_{m_{2}^{2}}^{6.0\text{
GeV}^{2}}dt\,tP_{4}(t,R)}{\int_{m_{2}^{2}}^{6.0\text{
GeV}^{2}}dt\,P_{4}(t,R)}=\frac{\int_{m_{2}^{2}}^{6.0\text{
GeV}^{2}}dt\,t^{2}P_{4}(t,R)}{\int_{m_{2}^{2}}^{6.0\text{
GeV}^{2}}dt\,tP_{4}(t,R)}\text{ \ all at }R=6.0\text{ GeV}^{2}$ (29)

Equality of the ratios give

$m_{4}=m_{\rho}(2150)=2.18\pm 0.09\text{ GeV}$

The error is obtained by varying $R$ by 10%.

We collect our results together with the experimental numbers from [5] in a
table.

Table of Results:

Resonance | Result for the mass in GeV | Experimental value in GeV
---|---|---
$\rho(770)$ | $0.74\pm 0.04$ | $0.77511\pm 0.00034$
$\rho(1450)$ | $1.42\pm 0.10$ | $1.465\pm 0.025$
$\rho(1570)$ | $1.50\pm 0.05$ | $1.570\pm 0.036\pm 0.062$
$\rho(1700)$ | $1.73\pm 0.05$ | $1.720\pm 0.020$
$\rho(2150)$ | $2.18\pm 0.09$ | $2.155\pm 0.021$

We conclude that the QCD sum rules can predict the masses of all established
higher $\rho$ recurrences. QCD describes in this case a single resonance.

##  3 The Isovector Pseudoscalars

The following isovector pseudoscalars have been observed $\pi$ with
$m_{\pi}\approx 0$ GeV, $\pi_{1}(1300)$ with $m_{1}=1.3$ $\pm 0.1$ GeV,
$\pi_{2}(1810)$ with $m_{2}=$ $1.81\pm 0.01$ GeV, $\pi_{3}(2370)$ with
$m_{3}=2.360\pm 25$ GeV [5].

We start with the correlator

$\Pi(q)=i\int d^{4}xe^{iqx}\left\langle
0\left|T\,j_{5}(x)j_{5}(0)\right|0\right\rangle$ (30)

of the pseudoscalar current

$j_{5}=i\overline{q}\gamma_{5}q\qquad
j_{5}=\frac{1}{2m_{q}}\partial^{\mu}A_{\mu}$ (31)

where $m_{q}$ is the quark mass, $q=u$ or $d$. The QCD expression for the
correlator is

$\Pi^{\text{QCD}}(t)=-\frac{3}{8\pi^{2}}(1+\frac{11}{3}a_{s})t\ln(-t)+\frac{\left\langle
m_{q}^{2}\overline{q}q\right\rangle}{t}+\frac{\left\langle
a_{s}GG\right\rangle}{8t}+...$ (32)

The scale $\mu^{2}$ only enters at order $\alpha_{s}^{2}$. The method used for
the vector mesons is repeated here. We use the kernel
$P_{2}(t,R).=(1-\frac{t}{m_{2}^{2}})(1-\frac{t}{R})$ which vanishes at
$t=m_{2}^{2}$ and $t=R$ to get the sum rule

$\displaystyle m_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}(1+\frac{11}{3}a_{s})\int\limits_{0}^{R}dt\,t^{2}\,P_{2}(t,R)$
$\displaystyle=$ $\displaystyle\frac{\int_{0}^{\text{R
}}dt\,t^{2}P_{2}(t,R)}{\int_{0}^{\text{R }}dt\,tP_{2}(t,R)-\delta}$

where

$\delta=\frac{\pi^{2}\left\langle
a_{s}GG\right\rangle}{(1+\frac{11}{3}a_{s})}=0.094\text{ GeV}^{4}$

With the stability region $R\approx 4.0\text{ GeV}^{2}$ this gives

$m_{1}=m_{\pi}(1300)=1.22~{}\text{GeV .}$ (33)

Taking an additional moment with the kernel

$P_{2}(t,m_{2})=\left(1-\frac{t}{m_{2}^{2}}\right)\left(1-\frac{t}{R}\right)\text{
\ at }R=4.2\text{ GeV}^{2},$ (34)

we obtain the consistency condition

$\frac{\int_{0}^{\text{R }}dt\,t^{3}P_{2}(t,R)}{\int_{0}^{\text{R
}}dt\,t^{2}P_{2}(t,R)}=\frac{\int_{0}^{\text{R
}}dt\,t^{2}P_{2}(t,R)}{\int_{0}^{\text{R }}dt\,t\,P_{2}(t,R)-\delta}\text{, \
all at }R=4.2\text{ GeV}^{2}$ (35)

This yields

$m_{2}=m_{\pi}(1810)=1.77\text{ GeV}$

Alternatively one can use the kernel

$P_{3}(t,R)=\left(1-\frac{t}{m_{3}^{2}}\right)\left(1-\frac{t}{R}\right)$ (36)

in the sum rule

$m_{2}^{2}=\frac{\int_{1.69}^{\text{R
}}dt\,t^{2}P_{3}(t,R)}{\int_{1.69}^{\text{R }}dt\,t\,P_{3}(t,R)-\delta}\text{
.}$ (37)

assuming optimistically that QCD duality is valid from $R=1.69\text{
GeV}^{2}$. At stability ($R\approx 7\text{ GeV}^{2}$) this gives

$m_{2}=m_{\pi}(1810)=1.74\text{ GeV}$ (38)

One can calculate the mass $m_{3}$ making use of the kernel $P_{3}(t,m_{2})$
at $R=4.4\text{ GeV}^{2}$,and imposing the condition

$\frac{\int_{m_{2}^{2}}^{7.1\text{
GeV}^{2}}dt\,t^{3}P_{3}(t,m_{3})}{\int_{m_{2}^{2}}^{7.1\text{
GeV}^{2}}dt\,t^{2}P_{3}(t,m_{3})}=\frac{\int_{m_{2}^{2}}^{7.1\text{
GeV}^{2}}dt\,t^{2}P_{3}(t,m_{3})}{\int_{m_{2}^{2}}^{7.1\text{
GeV}^{2}}dt\,P_{3}(t,m_{3})-\delta}$ (39)

yields

$m_{3}=m_{X}(2370)=2.66\text{ GeV}$

This is an argument for the isovector pseudoscalar nature of the X(2370).

Table of Results:

Resonance | Result for the mass in GeV | Experimental value in GeV
---|---|---
$\pi_{1}(1300)$ | $1.22$ | $1.300$
$\pi_{2}(1810)$ | $1.77\pm 0.04$ | $1.810$
$\pi_{3}=\pi_{3}(2370)$ | $2.66$ | $2.370$

The predictions, although qualitatively in agreement with the data, are not as
good as in the vector meson case mainly because QCD perturbation theory is
less convergent.

##  4 The isovector scalar mesons

The spectrum of the scalar mesons is $a_{0}(980)$, $a_{0}(1450)$, $X(1835)$,
$a_{0}(1950)$ where the status of the $X(1835)$ is uncertain. We start with
the correlator Eq.(30) with the scalar current

$j(x)=\bar{q}(x)q(x)$ (40)

The QCD expression is the same as given in Eq.(32) except for the negligible
$m_{q}^{2}\left\langle\bar{q}q\right\rangle$ term. The method gives

$f_{1}^{2}P_{2}(m_{1}^{2},R)=\frac{1}{8\pi^{2}}(1+\frac{11}{3}a_{s})\int\limits_{0}^{R}dt\,t\,P_{2}(t,R)-\frac{\left\langle
a_{s}GG\right\rangle}{8}+....$ (41)

and

$f_{1}^{2}m_{1}^{2}P_{2}(m_{1}^{2},R)=\frac{1}{8\pi^{2}}(1+\frac{11}{3}a_{s})\int\limits_{0}^{R}dt\,t^{2}\,P_{2}(t,R)-\frac{\left\langle
a_{s}GG\right\rangle}{8}+....$ (42)

with $P_{2}(m_{1}^{2},R)$ given by Eq. (34) with the input
$m_{2}^{2}=2.10\text{ GeV}^{2}$. At stability ($R\sim 2.7\text{ GeV}^{2}$) the
ratio of the above equations gives

$m_{1}=m_{a_{0}}(980)=0.93\text{ GeV}$ (43)

We emphasize that we assume that the correlator is given by QCD perturbation
theory is for $|t|\,\geq R=2.7\text{ GeV}^{2}$as phenomenologically supported
by the analysis of $\tau$-decay [7].

Taking an additional moment with the kernel
$P_{2}(t,R)=(1-\frac{t}{m_{2}^{2}})(1-\frac{t}{2.7GeV^{2}})$ and the condition

$m_{1}^{2}=\frac{\int_{0}^{\text{R }}dt\,t^{3}P_{2}(t,R)}{\int_{0}^{\text{R
}}dt\,t^{2}P_{2}(t,R)}=\frac{\int_{0}^{\text{R
}}dt\,t^{2}P_{2}(t,R)}{\int_{0}^{\text{R }}dt\,t\,P_{2}(t,R)-\delta}\text{, \
all at }R=2.7\text{ GeV}^{2}$ (44)

Equating the two ratios yields

$m_{2}=m_{a_{0}}(1450)=1.51\text{ GeV}$

We next use the kernel $P_{4}(t,R)=(1-\frac{t}{m_{4}^{2}})(1-\frac{t}{R})$
with $m_{4}=m_{a_{0}}(1950)$ to determine $m_{3}$:

$m_{3}^{2}=\frac{\int_{m_{2}^{2}}^{\text{R
}}dt\,t^{2}P_{4}(t,R)}{\int_{m_{2}^{2}}^{\text{R
}}dt\,t\,P_{4}(t,R)-\delta}\text{, \ all at }R=4.7\text{ GeV}^{2}$ (45)

The result is $m_{3}=m_{X}(1835)=1.80$ GeV.

$m_{4}$ can be determined by

$\frac{\int_{m_{2}^{2}}^{\text{R
}}dt\,t^{3}P_{4}(t,R)}{\int_{m_{2}^{2}}^{\text{R
}}dt\,t^{2}P_{4}(t,R)}=\frac{\int_{m_{2}^{2}}^{\text{R
}}dt\,t^{2}P_{4}(t,R)}{\int_{m_{2}^{2}}^{\text{R
}}dt\,t\,P_{4}(t,R)-\delta}\text{, \ all at }R=4.7\text{ GeV}^{2}$ (46)

with the result

$m_{4}=m_{a_{0}}(1950)=1.80\text{ GeV}$

Table of results:

Resonance | Result for the mass in GeV | Experimental value in GeV
---|---|---
$a_{0}(980)$ | $0.93$ | $0.98$
$a_{0}(1450)$ | $1.51$ | $1.45$
$X(1835)$ | $1.80$ | $1.895$
$a_{0}(1950)$ | $1.80$ | $1.930$

Conclusions: We have calculated the masses of the isovector (vector,
pseudoscalar, scalar) mesons and their recurrences with a new variant of QCD
finite energy sum rules. The method works well for all similar systems such as
the nucleon resonances. The main source of error is the zero width
approximation for the resonances. We have estimated this error by allowing the
radius entering the sum rule to vary by $\pm 10\%$. Order $\alpha_{s}$
corrections are included, order $\alpha_{s}^{2}$ are calculated and found to
be negligible. The sum rule predictions are compared with the experimental
numbers and agreement within the expected accuracy is found. It can be
concluded that QCD is applicable to single resonances and their recurrences.

Acknowledgements: This work was supported in part by the Alexander von
Humboldt Foundation (Germany), under the Research Group Linkage Programme, and
by the University of Cape Town (South Africa).

## References

  * [1] M. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979), ibid., B 147, 448 (1979).
  * [2] B. L. Ioffe, Prog. Part. Nucl. Phys. 56, 232 (2006)
  * [3] A. L. Kataev, N. V. Krasnikov and A. A. Pivovarov, Phys. Lett. B 123, 93 (1983), S. G. Gorishnii, A. L. Kataev and S. A. Larin, Phys. Lett. B 135, 457 (1984), A. L. Kataev, Phys. Atom. Nucl. 68, 567 (2005)
  * [4] For a review see C. A. Dominguez, Quantum Chromodynamics Sum Rules, Springer Briefs in Physics, Springer, Cham (2018).
  * [5] P. A. Zylaet al.(Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020) and 2021 update
  * [6] K. G. Chetyrkin, J.H. Kuehn, A. Kwiatkowski, Physics Reports 277, 189 (1996)
  * [7] A. Pich, Prog. Part. Nucl. Phys. 117, 103846 (2021)
  * [8] C.A. Dominguez and K. Schilcher, Journal of High Energy Physics 01, 093 (2007)


# Gas dynamics in dwarf galaxies as testbeds for dark matter and galaxy
evolution

Federico Lelli
INAF, Arcetri Astrophysical Observatory, Florence 50125, Italy;
E-mail<EMAIL_ADDRESS>
Dwarf galaxies are ideal laboratories to test dark matter models and
alternative theories because their dynamical mass (from observed kinematics)
largely outweighs their baryonic mass (from gas and stars). In most star-
forming dwarfs, cold atomic gas forms regularly rotating disks extending
beyond the stellar component, thus probing the gravitational potential out to
the outermost regions. Here I review several aspects of gas dynamics in dwarf
galaxies, such as rotation curves, mass models, and noncircular motions. Star-
forming dwarfs extend the dynamical laws of spiral galaxies to lower masses,
surface densities, and accelerations. The three main dynamical laws of
rotation-supported galaxies point to three distinct acceleration scales, which
play different physical roles but display the same value, within
uncertainties. The small scatter around these dynamical laws implies a tight
coupling between baryons and dark matter in galaxies, which will be better
understood with next-generation surveys that will enlarge current sample sizes
by orders of magnitude.

##  1 Introduction

Gas dynamics plays a key role in the formation and evolution of galaxies.
During the history of our Universe, cosmic gas accreated onto galaxies, cooled
down, and settled into regularly rotating disks where new stars are formed.
Gas disks are precious tools to trace the gravitational potential ($\Phi$) of
galaxies because (unlike stellar disks) the rotation support largely dominates
over the pressure support in the radial direction ($R$), thus the observed
rotation velocity is virtually equal to the circular velocity of a test
particle ($V_{c}^{2}=-R\,\nabla\Phi$). For example, the discovery of
ubiquitous flat rotation curves in spiral galaxies using H$\alpha$ [1] and H I
[2] emission-line observations was pivotal to establish the dark matter (DM)
problem.

In high-surface-brightness (HSB) and high-mass galaxies (with stellar masses
$M_{\star}>3\times 10^{9}$ M⊙), the visible baryonic matter can usually
explain the inner parts of the observed rotation curves, such as those from
H$\alpha$ observations [3, 4], while the DM effect becomes evident in the
outer regions, which can be traced with H I observations [5, 6]. In low-
surface-brightness (LSB) and low-mass galaxies ($M_{\star}\leq 3\times 10^{9}$
M⊙), instead, the DM effect is already important at small radii [7, 8, 9].
Thus, LSB and low-mass galaxies have been prime targets to test DM models
because the modeling of the baryonic mass distribution is less crucial than in
HSB and high-mass galaxies. In addition, dwarf galaxies allow extending the
study of dynamical laws down to the lowest accessible masses, surface
densities, and accelerations, where DM is expected to play a major role in
galaxy evolution according to the $\Lambda$ Cold Dark Matter ($\Lambda$CDM)
cosmological model. Dwarf galaxies pose several challenges to the $\Lambda$CDM
paradigm, which have been extensively reviewed [10, 11, 12]. This review is
focused on gas dynamics in dwarf galaxies from an empirical perspective, and
discusses only the small-scale $\Lambda$CDM challenges posed by gas emission-
line observations.

Over the past 20 years, enormous progress has been made in our understanding
of gas dynamics in dwarf galaxies thanks to dedicated surveys using H I
interferometry, H$\alpha$ Fabry-Pérot observations, and optical integral-field
spectroscopy. In my view, the key advances have been (i) the confirmation that
dwarf galaxies have slowly rising rotation curves in their inner regions,
implying the existence of DM cores in a $\Lambda$CDM context [10], (ii) the
discovery that gas-dominated dwarfs fall on the same baryonic Tully-Fisher
relation (BTFR) as star-dominated spirals, posing a fine-tuning problem for
$\Lambda$CDM models of galaxy formation [13, 14], (iii) the establishment of
new dynamical laws such as the central density relation (CDR) [15, 16] and the
radial acceleration relation (RAR) [17, 18, 19], extending from high-mass to
low-mass galaxies. In particular, the main dynamical laws of rotation-
supported galaxies (BTFR, CDR, and RAR) have been found to have small
intrinsic scatters, implying that baryons and DM must be tightly coupled on
sub-kpc scales. This review will summarize this progress but also highlight
the key scientific questions that remain to be addressed: (i) Are the observed
dynamical properties of dwarf galaxies (rotation curve shapes, DM content,
strength of noncircular motions) consistent with the expectations from
$\Lambda$CDM models of galaxy formation? (ii) Do the properties of the
dynamical laws depend on the galaxy large-scale environment? (iii) Is there
any intrinsic scatter along the dynamical laws, especially in the dwarf regime
where DM is expected to largely dominate the dynamics?

###  1.1 Taxonomy of dwarf galaxies

The dwarf galaxy classification covers more than 7 orders of magnitude in
stellar mass, from a few billions to a few hundreds solar masses [11]. This
mass range is much broader than the $\sim$2.5 dex covered by ellipticals,
lenticulars, and spirals along the Hubble sequence. This review will focus on
the mass range $M_{\star}\simeq 10^{5.5}-10^{9.5}$ M⊙ where we find both gas-
rich dwarfs (in which we can simultaneously study gas and stellar dynamics)
and gas-poor dwarfs (in which we are limited to stellar dynamics). I will not
discuss the ultra-faint dwarfs (UFDs) with $M_{\star}<10^{5.5}$ M⊙ found in
the Local Group because they are generally devoid of detectable gas [20].
Next-generation optical and H I surveys, however, may potentially discover
gas-rich and/or star-forming UFDs in the field, providing key constraint to
$\Lambda$CDM models down to the lowest possible masses [11].

The taxonomy of dwarf galaxies is complex due to historical reasons [21]. I
will use an extremely simplified nomenclature that distinguish two main types
of dwarfs based on their gas content and star-formation activity:

  * •

_Quiescent dwarfs_ have no recent (the past 10 Myr) star formation [22] and
are generally undetected by H I surveys (“gas-poor dwarfs”). This definition
includes dwarf spheroidals, dwarf ellipticals, and dwarf lenticulars.
Quiescent dwarfs are copiously found in galaxy clusters [23] and as satellites
of high-mass galaxies in groups [24, 25] but are extremely rare in isolation
[24, 26].

  * •

_Star-forming dwarfs_ form stars at a steady rate [27] and are usually
detected by H I surveys (“gas-rich dwarfs”). This definition encompasses dwarf
irregulars, star-forming LSB dwarfs, and late morphological types such as Sd,
Sm, and Im galaxies. Star-forming dwarfs are found in isolation, in galaxy
groups, and in the outskirts of galaxy clusters [24, 28].

In addition to these two main types of low-mass galaxies, I will occasionally
refer to more exotic gas-rich dwarfs:

  * •

_Starburst dwarfs_ have enhanced star-formation rates (SFRs). Various
definitions of starbursts exist in the literature; an effective one for dwarf
galaxies is that the recent SFR is at least 3 times higher than the past
average SFR [29, 30]. This definition results in typical burst durations of a
few hundreds Myrs [31, 32]. Starburst dwarfs include blue compact dwarfs and
other historical nomenclatures such as amorphous galaxies [33] and H II
galaxies [34].

  * •

_Transition dwarfs_ have some gas but no current star formation [9, 35]. They
are usually found among the least massive gas-rich dwarfs and may represent
the evolutionary link between star-forming and quiescent dwarfs [20].

  * •

_Tidal dwarf galaxies_ (TDGs) are recycled objects that form out of
collisional debris ejected from more massive galaxies during galaxy
interactions [36, 37]. It is unclear how many dwarf galaxies may have a tidal
origin with profund implications for $\Lambda$CDM cosmology [38, 39].

I will not distinguish between dwarf galaxies and so-called ultra-diffuse
galaxies (UDGs) because there is compelling evidence that the latter are a LSB
extension of the former [40, 41, 42], including both gas-rich and gas-poor
objects [43].

###  1.2 The multiphase gas content of dwarf galaxies

The inter-stellar medium (ISM) of star-forming dwarfs is mostly composed of
cold atomic gas ($T\simeq 10^{2}-10^{4}$ K), which is directly traced by the H
I emission line at 21 cm. This is a hyperfine transition of atomic hydrogen,
for which the H I luminosity can be readily converted into a H I mass (minor
corrections for H I self-absorption may possibly be necessary in edge-on disks
[44]). The atomic gas mass can then be estimated using statistical corrections
for the minor contributions of helium and heavier elements [45].

Cold molecular gas ($T\simeq 10$ K) has been historically difficult to study
in dwarf galaxies because they have low gas volume densities and low gas
metallicities, so CO lines (tracing H2) are hard to detect [46, 47]. Even when
CO is detected, the molecular gas mass remains uncertain due to the uncertain
CO-to-H2 conversion factor [48]. Indirect estimates of the H2 mass,
extrapolating the scaling relations with $M_{\star}$ and/or SFR of high-mass
galaxies, suggest that molecules make between 3% and 10% of the ISM mass of
star-forming dwarfs [45, 49, 50]. Recent CO observations with the Atacama
Large Millimeter/submillimeter Array (ALMA) have been able to detect giant
molecular clouds in low-mass galaxies, including low-metalliticy dwarfs [51],
starburst dwarfs [52], and TDGs [53]. These detections represent major
progress for understanding star formation in dwarf galaxies, but CO emission
does not seem to be a promising kinematic tracer because its distribution is
irregular and often made of distinct CO clumps at random galaxy locations.

Warm ionized gas ($T\gtrsim 10^{4}$ K), probed by optical recombination lines
(e.g., the H$\alpha$ line), provides an insignificant contribution to the ISM
mass [54]. Still, it is a precious kinematic tracer in the inner galaxy
regions because optical observations generally reach higher spatial
resolutions than radio H I interferometry [55, 56]. H$\alpha$ and H I lines
are the most widely used kinematic tracers in star-forming dwarfs.

Hot ionized gas ($T\gtrsim 10^{5}-10^{6}$ K) is difficult to probe in dwarf
galaxies because one expects weak, diffuse X-ray emission. $\Lambda$CDM models
of galaxy formation predict that low-mass galaxies should be surrounded by
large amounts of hot gas, containing a significant fraction of the
cosmological missing baryons [57, 58]. Observationally, diffuse X-ray emission
has been detected around a few starburst dwarfs but the visible hot gas mass
is $\sim$1$\%$ or less than the H I mass [59].

Figure 1: The cold gas content of nearby galaxies. Gas mass (a) and gas
fraction (b) are plotted as a function of stellar mass for H I-detected
galaxies. Yellow symbols show representative galaxies in or near the Local
Group: the Milky Way (large cross) [60], M31 (large plus) [61], M33 (large
diamond) [62, 63], the LMC (pentagon) [64, 65], the SMC (hexagon) [66], and
Leo P (octagon) [67, 68]. Grey symbols show single-dish H I data for Local
Volume galaxies within 11 Mpc (small dots) [69], high-mass SDSS galaxies
(small plusses) [70], and isolated SDSS galaxies (small crosses) [71]. The
other colored symbols show interferometric H I data from SPARC (circles) [72],
LITTLE-THINGS (squares) [73, 74], SHIELD-I (diamonds) [75], SHIELD-II (tris
down) [76], FIGGS (tris up) [77], LVHIS (stars) [78] and VLA-ANGST (crosses)
[79]. Stellar masses are heterogeneous but mostly come from $K$-band or
Spitzer [3.6] photometry using mass-to-light ratios of 0.65 and 0.50 M⊙/L⊙,
respectively. Typical 1$\sigma$ uncertainties (not shown) are of the order of
30$\%$ on stellar masses and 10$\%$ on gas masses. In panel a, the dashed line
shows the 1:1 line. In panel b, the horizontal line corresponds to $M_{\rm
gas}>M_{\star}$ while the vertical line shows our definition of a dwarf galaxy
($M_{\star}\leq 3\times 10^{9}M_{\odot}$) set by the LMC stellar mass.

In star-forming dwarfs, therefore, the mass contributions of cold molecular
gas, warm ionized gas, and hot ionized gas are generally less than 10$\%$ of
the atomic gas mass, thus smaller than the typical uncertainty in H I fluxes.
Figure 1 shows the cold gas content of star-forming galaxies over $\sim$6 dex
in stellar mass, where the cold gas mass $M_{\rm gas}$ is obtained from the
observed H I mass with statistical corrections for heavier elements and
molecules [45]. The baryon content of high-mass spirals is usually dominated
by stars ($M_{\star}>M_{\rm gas}$) while that of low-mass dwarfs can be
largely dominated by cold gas ($M_{\rm gas}>M_{\star}$), especially for
isolated galaxies [71]. The gas fraction of star-forming dwarfs can be
extremely high: up to $\sim$95$\%$ of the baryonic mass can be in gas rather
than stars. This implies that (i) star-forming dwarfs convert atomic gas into
stars in a very inefficient way; most likely the bottleneck is the conversion
from H I to H2 [50]; (ii) star-forming dwarfs have extremely long gas
depletion times ($\sim$10$-$100 Gyr); the existing H I reservoir can amply
sustain the SFR without the need of accreting external gas [49]; (iii) star-
forming dwarfs in isolation will not evolve into quiescent dwarfs in the
foreseeable future [26].

##  2 Gas dynamics

###  2.1 Gas distribution and kinematics

Spatially resolved studies of gas dynamics in dwarf galaxies were pioneered in
the 1980s and 1990s using H I interferometry [7]. During the past 20 years,
the number of dwarf galaxies with spatially resolved data has increased
significantly thanks to dedicated H I surveys (see Fig. 1), providing a broad
overview of the gas distribution and kinematics in different types of dwarfs.

The vast majority of star-forming dwarfs with $M_{\star}\simeq
10^{7.5}-10^{9.5}$ M⊙ host rotating H I disks, similarly to more massive
spiral galaxies [77, 80]. Typical dwarf “irregulars” are very regular in their
H I emission [77, 80]: the optical morphology may look complex due to star-
forming clumps but do not reflect the underlying gas kinematics. Star-forming
dwarfs with lower masses ($M_{\star}\simeq 10^{5.5}-10^{7.5}$ M⊙) can host
rotating H I disks [75, 76] but their rotation velocities ($V_{\rm
rot}\lesssim 40$ km s-1) become comparable to the typical H I velocity
dispersion ($\sigma_{\rm V}\simeq 5-10$ km s-1), so the gas disk cannot be
entirely supported by rotation. If one aims to measure their circular
velocities ($V_{\rm c}^{2}=-R\,\nabla\Phi$), the observed rotation velocities
need to be corrected for pressure support using the so-called asymmetric-drift
correction [30, 74]. In addition, small $V_{\rm rot}/\sigma_{\rm V}$ ratios
imply that the gas disks cannot be “razor thin” as generally assumed when
deriving rotation curves with 2-dimensional methods [81], so one needs to use
3-dimensional methods to model thick disks [82, 83] and possibly “disk
flares”, i.e., the outward increase of disk thickness with radius [84].
Finally, in the tiniest star-forming dwarfs with $V_{\rm rot}\simeq 10-15$ km
s-1, the rotation velocities become comparable to typical noncircular motions
($\sim$5 km s-1 as discussed later), so the H I kinematics is unavoidably more
complex [67].

Figure 2: The star-forming dwarf galaxy DDO 125. The galaxy has a stellar mass
of $\sim 2\times 10^{7}$ M⊙ [72] and an outer rotation velocity of $\sim$20 km
s-1 [85]. The figure shows the $r-$band SDSS image (a), H I column density map
(b), H I line-of-sight velocity map (c) and H I velocity dispersion map (d).
In panels a and b, the black contour corresponds to H I column densities of
$5\times 10^{19}$ cm-2, while the bar to the bottom-right corner corresponds
to $\sim$1 kpc. In panels c and d, the colorbars show the velocity scale. In
panel c, the bold contour corresponds to the systemic velocity (set to zero)
while the other contours range from $-15$ to $+15$ km s-1 in steps of 5 km
s-1. In all panels, the circle to the bottom-left corner shows the full-width
half-maximum of the H I point spread function [79].

Figure 2 shows the example of the dwarf galaxy DDO 125 [79] with
$M_{\star}\simeq 2\times 10^{7}$ M⊙ [72]. The H I observations reveal a
regularly rotating gas disk with a rotation velocity of only $\sim$20 km s-1
in the outer parts and a velocity dispersion $\sigma_{\rm V}\lesssim 10$ km
s-1. The H I disk extends further out than the stellar component. On average,
the ratio of H I to stellar size in star-forming dwarfs is $1.8\pm 0.8$ [80],
similar to spiral galaxies in the field [86, 87], with DDO 125 being in the
lower portion of the scatter. Extended H I disks allow rotation curves to be
traced beyond the stellar component, where the DM halo is expected to largely
dominate the gravitational potential.

Figure 3: H I distribution in peculiar types of dwarf galaxies. Optical images
(color scale) are overlaid with H I column density maps (contours,
corresponding to $\sim$3$\sigma$, 6$\sigma$, 12$\sigma$, and so on). In panel
a, the starburst dwarf NGC 2915 ($M_{\star}\simeq 3\times 10^{8}$ M⊙, $M_{\rm
gas}\simeq 6\times 10^{8}$ M⊙) shows a very extended H I disk with spiral arms
[183]. In panel b, the starburst dwarf I Zw 18 ($M_{\star}\simeq 0.9\times
10^{8}$ M⊙, $M_{\rm gas}\simeq 3\times 10^{8}$ M⊙) displays a long H I tail to
the South and a companion dwarf galaxy to the North [118]. In panel c, the
transition dwarf Antlia ($M_{\star}\simeq 7\times 10^{6}$ M⊙, $M_{\rm
gas}\simeq 0.7\times 10^{6}$ M⊙) displays an off-center H I distribution [79].
In panel d, the TDG NGC 7252 NW ($M_{\star}\simeq 0.9\times 10^{8}$ M⊙,
$M_{\rm gas}\simeq 7\times 10^{8}$ M⊙) is forming at the tip of a tidal tail
around a late-stage major merger [37]. In all panels, the bar to the bottom-
right corner shows the physical scale.

Peculiar types of gas-rich dwarfs show more diversity. Starburst dwarfs, which
are estimated to be $\sim$6$\%$ of local star-forming dwarfs [88], can show
either regularly rotating H I disks (Fig. 3, a) or disturbed H I distributions
and kinematics (Fig. 3, b) with similar probability [30]. The latter ones are
likely due to dwarf-dwarf interactions [89, 90], gas accretion [91, 92],
and/or stellar feedback [93, 94]. Transition dwarfs can show unsettled H I
distributions with significant off-sets between stellar and H I emission (Fig.
3, c), suggesting either gas stripping from a star-forming dwarf or gas
accretion into a quiescent dwarf [9, 78]. TDGs, which form out of collisional
debris, can host rotating H I disks (Fig. 3, d) but it is unclear whether they
are in dynamical equilibrium because they did not have enough time to complete
a single orbit since the time of their formation [37].

Spectroscopic H$\alpha$ surveys are largely complementary to H I surveys
because they can probe the inner galaxy regions with higher spatial resolution
[55]. The higher velocity dispersion of ionized gas ($\sim 15-30$ km s-1),
however, pose an intrinsic limit in tracing H$\alpha$ circular velocities in
galaxies with $M_{\star}\lesssim 10^{8}$ M⊙ [56]: the pressure support becomes
important and the asymmetric-drift correction uncertain. H$\alpha$ surveys are
also useful to study gas outflows, as discussed in the section on noncircular
motions.

Figure 4: Rotation curves of dwarf and spiral galaxies from the SPARC
database. In panel a, rotation curves are plotted on a linear scale and color-
coded by stellar mass. In panel b, they are plotted on a logaritmic scale and
color-coded by effective stellar surface density. Typical 1$\sigma$
uncertainties on the rotation velocities (not shown) are of the order of
10$\%$, excluding inclination uncertainties that affect the overall
normalization of the rotation curve [72].

###  2.2 Circular motions

In spiral galaxies, the shapes of rotation curves have long been known to
relate to the distribution of stellar light, suggesting that baryons dominate
the gravitational potential in the inner regions [3, 4, 5, 6]. Remarkably, the
same phenomenology holds in dwarf galaxies even though the DM halo is expected
to dominate the gravitational potential down to the inner parts [85, 95].
Figure 4 shows hybrid H$\alpha$+H I rotation curves for 175 disk galaxies from
the SPARC database [72], spanning $\sim$5 dex in stellar mass. Panel a shows
that the amplitude of the rotation curve systematically increases with stellar
mass: this is the backbone of the Tully-Fisher relation. Panel b shows that
the inner steepness of the rotation curve (i.e., how fast it reaches the flat
part) systematically increases with stellar surface density: this is the
backbone of the central density relation. Both relations are discussed in
detail in the section on dynamical laws.

In a $\Lambda$CDM context, one may expect that the rotation curves of dwarf
galaxies systematically vary with mass but show little variation at fixed
mass, reflecting the expected self-similarity of the underlying DM halos [96].
This is at odds with the observed rotation curves of dwarf galaxies: at fixed
mass, HSB dwarfs have steeply rising rotation curves while LSB dwarfs have
slowly rising rotation curves [97, 98, 99]. However, when the rotation curves
are expressed in units of the disk scale length [85] or of the optical radius
[100], they do show a self-similar shape to a first-order approximation. The
key problem is not the “unexpected diversity” but the remarkable regularity:
at fixed mass, the shapes of rotation curves systematically depend on the
baryon distribution (surface mass density and/or scale length), pointing to a
local baryon-DM coupling.

When high-resolution and high-sensitivity data are available, rotation curves
can display characteristic kpc-scale features that usually correspond to
analogous features in the baryonic mass profiles [95]. The best way to
visualize this connection is to build mass models that account for the
expected circular velocities from the main baryonic components: the gas and
stellar disks (bulges are generally absent in dwarf galaxies). The
gravitational contribution of a razor-thin exponential disk can be computed
analytically [101] but this approximation neglects relevant features in the
observed surface brightness profiles, such as bumps, wiggles, inner
flattenings, and so on. Thus, it is more appropriate to numerically solve the
Poisson’s equation in cylindrical symmetry [102] for
$\rho(R,z)=\Sigma(R)Z(z)$, where $\Sigma(R)$ is the observed radial density
profile (derived from near-IR images for the stars and H I maps for the gas)
and $Z(z)$ is an adopted vertical density profile (e.g., an exponential or
$sech^{2}$ distribution).

Box 1 $-$ Mass models in a $\Lambda$CDM context
N-body simulations of structure formation in a $\Lambda$CDM cosmology find
that DM halos follow the Navarro-Frenk-White (NFW) profile [184], in which the
volume density diverges to infinity towards the center (a so-called “cusp”).
The rotation curves of dwarf galaxies, instead, are best fitted with cored DM
profiles, leading to the so-called “cusp-vs-core” problem [10]. A closely
related issue is the “too-big-too-fail” problem, which considers the internal
dynamics of dwarf galaxies (both quiescent and star-forming ones) concurrently
with their observed cosmic abundance [185, 186, 187]. The figure shows a NFW
fit to the dwarf galaxy NGC 1560 [103]. The discrepancy at $R<3$ kpc
illustrates the usual cusp-vs-core problem, which can be circumvented fitting
cored DM halos [188, 189, 190] motivated by hydrodynamic simulations of galaxy
formation with strong stellar feedback [191, 192, 193]. The most interesting
discrepancy, however, is the one concerning the entire rotation-curve shape. A
smooth halo profile (NFW or any other form) cannot capture the characteristic
features in the rotation curve. The actual DM density profile must also
contain features mirroring those in the baryonic mass distribution. NFW fit to
the dwarf galaxy NGC 1560. The observed rotation curve (dots with
1$\sigma$-errorbars) is compared to the best-fit mass model (red solid line)
given by the gravitational contributions of stars (dotted line), gas (short-
dashed line), and DM (long-dashed line). The galaxy distance is fixed to 3.4
Mpc [103].

Box 2 $-$ Mass models in a Milgromian dynamics context
Milgromian dynamics (MOND) is an alternative to particle DM proposed in 1983
by Mordehai Milgrom [167, 168, 169]. MOND alters Newtonian gravity and/or
inertia at accelerations smaller than $\sim$10-10 m s-2, which are typical for
the outer parts of spiral galaxies and for every radii in dwarf galaxies. MOND
has been tested against hundreds of galaxy rotation curves [194, 171, 159].
The figure shows the example of the dwarf galaxy NGC 1560 [103]. The rotation
curve is well reproduced in MOND because its characteristic shape is mirrored
by the characteristic shape of the gas contribution. In general, MOND made
several successful predictions on galaxy scales but has problems on galaxy-
cluster scales [171, 195]. Relativistic extensions of MOND that can reproduce
gravitational lensing, the speed of gravitational waves, the cosmic microwave
background, and other cosmological observations are under active development
[196, 197].
MOND fit to the dwarf galaxy NGC 1560. The observed rotation curve (dots with
1$\sigma$-errorbars) is compared with the MOND prediction (red solid line)
based on the Newtonian contributions from stars (dotted line) and gas (dashed
line). The best-fit galaxy distance is 2.9 Mpc [103].

The importance of such features is discussed in Boxes 1 and 2, showing mass
models for the dwarf galaxy NGC 1560 [103]. The shape of the observed rotation
curve at $R\simeq 4-6$ kpc is mirrored by the shape of the gas gravitational
contribution. This is a common phenomenology in gas-rich dwarfs: mass models
that scale up the baryonic contributions by “brute force” (above acceptable
values of $M_{\star}$ and $M_{\rm gas}$) can usually reproduce the observed
rotation-curve shapes without DM [104, 105]. Rephrasing a longstanding rule
from Renzo Sancisi [95]: “for any feature in the baryonic mass profile of a
galaxy, there is a corresponding feature in the rotation curve, and vice
versa”. This is the backbone of the radial acceleration relation, discussed in
the section on dynamical laws.

###  2.3 Noncircular motions

In addition to regular rotation ($V_{\rm rot}$) and turbulent motions
($\sigma_{\rm V}$), large-scale noncircular motions may exist in gas-rich
galaxies. In a $\Lambda$CDM context, noncircular motions are expected in the
dwarf regime for several reasons: (i) DM halos are predicted to be triaxial or
prolate driving noncircular motions in the disk plane [106], (ii) the strong
stellar feedback implemented in most hydrodynamic simulations should drive gas
outflows outside the disk plane, following the path of least resistance [57,
107], and (iii) cold gas accretion from cosmic filaments may be on-going at
$z=0$ in low-mass galaxies driving radial gas flows [108].

Noncircular motions in the disk plane can be studied via harmonic
decompositions of the observed velocity maps [109]. In typical star-forming
dwarfs (excluding starburst), H I studies [110, 111, 112, 113, 114] find that
planar noncircular motions range from $\sim$1$\%$ to $\sim$10$\%$ of $V_{\rm
rot}$ and are generally smaller than $5-10$ km s-1. This is comparable to
typical uncertainties in $V_{\rm rot}$, which are usually computed considering
the difference between the approaching and receding sides of the disk [85].
Thus, noncircular motions are effectively captured in the error budget.
H$\alpha$ studies [115, 116] show that noncircular motions in the disk plane
could be stronger than those from H I studies, possibly due to the fact that
ionized gas is more closely related to the star-formation sites than atomic
gas (e.g., expanding bubbles and shells on small spatial scales). Importantly,
observed noncircular motions are not strong enough to “hide” a central DM cusp
[110, 111, 112]. Opposite conclusions are found analysing simulated dwarf
galaxies [117] because noncircular motions in simulated dwarfs appear to be
stronger than in real ones [113]. Interestingly, significant radial motions of
$\sim$5$-$15 km s-1 (corresponding to about 10$\%$ to 40$\%$ of $V_{\rm c}$)
have been detected in the disk plane of some starburst dwarfs [118, 98, 119,
120]. In general, one cannot determine the direction of these radial motions
because it is unknown which side of the disk is the nearest to the observer.
Starburst dwarfs have a central H I mass excess [30, 121], so it is likely
that the radial motions are planar inflows driven by dwarf-dwarf interactions
and/or cold gas accretion [118, 90, 91, 92].

Noncircular motions outside the disk plane are generally referred to as
“inflows” or “outflows”. For the latter ones, it is important to determine
whether the gas will escape from the galaxy potential well (blowouts) or fall
back onto the disk (galactic fountains). In typical star-forming dwarfs
(excluding starbursts), there is no observational evidence at any wavelength
for large-scale blowouts. Instead, there is evidence for thick gas layers
extending up to $\sim$2-3 kpc above the disk plane [122, 123, 124]. The thick
gas layer seems to rotate slower than the thin disk, analogously to extra-
planar gas in spiral galaxies [125, 126], but the presence of warps and flares
complicates the kinematic analysis [127]. Extraplanar gas in star-forming
dwarfs may be explained by supernovae-driven galactic fountains, creating a
small-scale gas circulation just above the disk plane [128].

In starburst dwarfs, one may expect to find ubiquitous and multiphase gas
blowouts [129] but the observational situation is complex. Regarding cold gas
($T\lesssim 10^{4}$ K), absorption-line observations of the Na doublet of six
starburst dwarfs [130] provided evidence for gas outflows in three cases but
(i) the velocities of the outflowing gas do not clearly exceed the escape
velocity and (ii) the mass of the outflowing gas cannot be easily estimated
from absortion-line data. For larger samples of starburst dwarfs, emission-
line H I observations that directly probe the cold gas mass do not show clear
evidence for gas blowouts [30, 131]. Regarding warm ionized gas ($T\simeq
10^{4}$ K), narrow-band H$\alpha$ imaging reveals diffuse emission in
$\sim$50$\%$ of starburst dwarfs, often characterized by shells and bubbles
with sizes up to 2 kpc [132]. H$\alpha$ spectroscopy shows that the velocities
of the warm ionized gas are generally below the galaxy escape velocity [133,
134, 135, 136] apart from a few extreme exceptions [137]. Larger outflow
velocities for warm ionized gas have been reported using UV absorption lines
[138, 139]. Regarding hot gas ($T\simeq 10^{6}-10^{7}$ K), several starburst
dwarfs show diffuse X-ray emission [59, 140] which likely traces blowouts
because the thermal velocity of the hot gas exceed the galaxy escape velocity.
The observed mass of the hot gas, however, is less than 1$\%$ of the ISM mass
[59].

##  3 Dynamical laws

Galaxies follow several dynamical laws, i.e., tight empirical correlations
between a baryonic quantity (from the distribution of gas and stars) and a
dynamic one (from the observed kinematics). In a $\Lambda$CDM context, the
internal dynamics of dwarf galaxies are largely dominated by the DM halo, so
it is pivotal to study the bottom-end of the dynamical laws to quantify the
relation between baryons and DM. I will focus on three main dynamical laws for
rotation-supported galaxies: the BTFR that applies at large radii, the CDR
that applies at small radii, and the RAR that applies at each and every
radius.

###  3.1 Baryonic Tully-Fisher relation

The classic Tully-Fisher (TF) relation [141] links the galaxy luminosity in
optical or near-IR bands (proxy for the stellar mass) to the width of the
spatially integrated H I line profile (proxy for the circular velocity, after
correcting for disk inclination). The classic TF relation, however, breaks
down for dwarf galaxies with $V_{\rm c}\simeq 100$ km s-1 because their gas
mass can be higher than their stellar mass [13] (see Fig. 1). The BTFR [13]
replaces the stellar mass with the total baryonic mass ($M_{\rm
bar}=M_{\star}+M_{\rm gas}$), recovering a single linear relation over $\sim$6
dex in $M_{\rm bar}$ (see Fig. 5, a).

It has been determined empirically [142, 143] that the scatter along the BTFR
is minimized using the mean circular velocity along the flat part of the
rotation curve ($V_{\rm flat}$). Other characteristic velocities, such as the
velocity $V_{2.2}$ at 2.2 exponential scale lengths ($R_{\rm d}$), give BTFRs
with larger scatters and shallower slopes. This is counter-intuitive because
baryons are dynamically important at small radii and their gravitational
contribution should approximately peak at 2.2$R_{\rm d}$ [101], so one may
expect $V_{2.2}$ to be more closely related to $M_{\rm bar}$ than $V_{\rm
flat}$. This does not occurr.

The $M_{\rm bar}-V_{\rm flat}$ relation has an orthogonal intrinsic scatter of
just $\sim$6 [143]. The residuals do not correlate with other galaxy
properties such as scale length or surface brightness, so the scatter cannot
be further decreased using a third independent variable [144, 145].
Consequently, the relation between the specific angular momentum ($j\simeq
V_{\rm flat}\times R_{\rm d}$) and the visible mass of disk galaxies [146,
147] has larger scatter than the BTFR [143, 148]. For circular velocities
smaller than $\sim$40 km s-1, a correlation between BTFR residuals and disk
scale length has been reported [149] but the data are scarce with large
uncertainties due to asymmetric-drift corrections and possibly disk-thickness
effects [74, 84]. TDGs [37] and UDGs [150] have been found to deviate from the
BTFR but the available kinematic data are very uncertain.

The $M_{\rm bar}-V_{\rm flat}$ relation has a fitted slope close to four [13,
14, 45, 143, 142], so can be expressed as $M_{\rm bar}=NV_{\rm flat}^{4}$. By
dimensional arguments, the physical units of the intercept $N$ are
acceleration times $G_{\rm N}$ (with $G_{\rm N}$ being Newton’s gravitational
constant), so the BTFR defines a purely empirical acceleration scale:

$a_{\rm BTFR}=\dfrac{x}{NG_{\rm N}}$ (1)

where $x=0.80\pm 0.05$ is a dimensionless parameter accounting for the disk
geometry of rotating galaxies. Fixing the slope to 4, a linear fit to the data
gives $a_{\rm BTFR}=(1.3\pm 0.1)\times 10^{-10}$ m s-1, where the uncertainty
on $x$ dominates the error budget. The observed properties of the BTFR were
predicted a-priori by MOND (see Box 2).

In a $\Lambda$CDM context, the relation between halo mass and halo velocity
has a slope equal to 3 [14]. If we assume that (i) $V_{\rm flat}$ is a proxy
of the halo velocity and (ii) DM halos contain all baryons available from
cosmology (i.e., we set the normalization of the theoretical $M_{\rm
bar}-V_{\rm flat}$ relation using the baryon fraction from $\Lambda$CDM fits
to the cosmic microwave background), we obtain the dashed line in Fig. 5 (a).
The data lies systematically to the left of this line, implying that the
baryons observed in galaxies are much less than the cosmic ones. The “missing
baryons” are commonly thought to reside in a hot gaseous halo around galaxies,
which is heated by stellar feedback processes [57, 58]. Thus, in a
$\Lambda$CDM context, the value of $a_{\rm BTFR}$ relates to the average
amount of missing baryons in galaxies. Given the difference between the
predicted ($\sim$3) and observed ($\sim$4) slopes, the amount of missing
baryons must systematically change with circular velocity (or halo mass),
ranging from $\sim$50$\%$ in high-mass spirals down to $\sim$99$\%$ in low-
mass dwarfs. Intriguingly, these baryon fractions are roughly consistent with
those expected from $\Lambda$CDM abundance-matching relations [151]. On the
other hand, the small intrinsic scatter and lack of curvature in the observed
BTFR are at variance with abundance-matching prescriptions [152]. In general
terms, given the small BTFR scatter, a galaxy with a given $V_{\rm f}$ (or
halo mass) must “know” exactly how many baryons to hide in order to stay in
the “right” location of the BTFR. This represents a conceptual fine-tuning
problem for $\Lambda$CDM models of galaxy formation [14].

Figure 5: Dynamical laws for dwarf and spiral galaxies. The baryonic Tully-
Fisher relation (a), the central density relation (b), and the radial
acceleration relation (c). In panel a, the dashed line shows the $\Lambda$CDM
prediction assuming that galaxies contain all available cosmological baryons.
For SPARC galaxies [143] $V_{\rm flat}$ is directly measured, while for
LITTLE-THINGS galaxies [74] and SHIELD-I galaxies [75] the outermost circular
velocity is used as a proxy for $V_{\rm flat}$. Yellow symbols show
representative galaxies in or near the Local Group: the Milky Way (cross) [60,
198], M31 (plus) [199, 61], M33 (large diamond) [62, 63], the LMC (pentagon)
[64, 65, 200], the SMC (hexagon) [66], and Leo P (octagon) [67, 68]. In panels
b and c, the dashed line corresponds to the one-to-one line. In all panels,
black lines show empirical fitting functions with only one free parameter: a
characteristic acceleration scale (given at the top). For internal
consistency, the fits are performed considering only the SPARC data (red and
blue circles corresponding to spiral and dwarf galaxies, respectively). The
errorbars correspond to 1$\sigma$ uncertainties.

###  3.2 Central density relation

The CDR is complementary to the BTFR because it involves quantities measured
at small radii [16]: the central baryonic surface density ($\Sigma_{\rm
bar,0}$) from stellar-mass maps and the central dynamical surface density
($\Sigma_{\rm dyn,0}$) from rotation curves. The root of the CDR is a relation
between the central surface brightness and the inner rotation-curve steepness
[85, 153, 154, 155], which is proportional to the square root of $\Sigma_{\rm
dyn,0}$ [156]. The CDR exploits an exact analytic formula for self-gravitating
disks [157]:

$\Sigma_{\rm dyn,0}=\dfrac{1+q_{0}}{2\pi G_{\rm
N}}\int_{0}^{\infty}\dfrac{V_{\rm c}^{2}(R)}{R^{2}}dR,$ (2)

where $q_{0}$ is the intrinsic axial ratio of the disk [16]. The CDR formally
holds for $R\rightarrow 0$ but in practice $\Sigma_{\rm dyn,0}$ is measured
within one resolution element of the kinematic data (typically a few hundred
parsecs). The mean baryonic surface density is then measured within the same
aperture. At these small radii, the stellar surface density is much larger
than the gas surface density, so the gas contribution can be neglected in
measuring $\Sigma_{\rm bar,0}$. A relation similar to the CDR exists between
the central surface brightness and the central dynamical surface density
measured from the vertical velocity dispersion of stellar disks in face-on
spirals [15].

The CDR is shown in Figure 5 (b). Differently from the BTFR, the relation is
non-linear. For surface densities above $\sim$500 M⊙ pc-2, the data are along
the one-to-one relation indicating that baryons dominate the inner dynamics.
At surface densities below $\sim$500 M⊙ pc-2, the data systematically deviate
from unity indicating that the DM effect is already significant in the inner
regions. Fixing the top-end slope to one, the CDR can be fitted by a double
power law with three free parameters: the bottom-end slope and two
characteristic surface densities, one setting the transition point from the
unity line and one setting the overall normalization [144]. The two
characteristic surface densities have approximately the same value [144], so
we will refer to both of them as $\Sigma_{\rm CDR}$ for simplicity. The
bottom-end slope of the CDR is close to $0.5-0.6$.

By dimensional arguments, surface densities can be expressed in units of
acceleration over $G_{\rm N}$, so the characteristic surface density
$\Sigma_{\rm CDR}$ implies the existence of another acceleration scale:

$a_{\rm CDR}=2\pi G_{\rm N}\Sigma_{\rm CDR},$ (3)

which is conceptually different from $a_{\rm BTFR}$ because it sets the
transition from baryon-to-DM domination in the inner galaxy regions. In a
$\Lambda$CDM context, this characteristic scale is presumably related to the
angular momentum of baryonic gas, i.e., its ability to collapse to the bottom
of the DM halo and form stars, reaching high stellar surface densities. In a
MOND context, the CDR is naturally explained [158] and can be fitted by a
function with a single free parameter: $\Sigma_{\rm CDR}$ or equivalently
$a_{\rm CDR}$. Irrespective of the physical meaning of MOND, the black line in
Fig. 5 (b) provides a practical, effective description of the CDR shape. Using
this function, we find $a_{\rm CDR}=(0.9\pm 0.3)\times 10^{-10}$ m s-2 where
the error budget is dominated by systematic uncertainties in $q_{0}$ [16].

The CDR has an observed scatter of $\sim$0.2 dex ($\sim$45%) that is largely
driven by observational uncertainties. The residuals show no correlation with
other galaxy properties like stellar mass or scale length. This is unexpected
because Newton’s shell theorem does not hold in disk galaxies, so the mass
outside a given radius contribute to the gravitational field within that
radius. In fact, Eq. 2 formally integrates $V^{2}_{\rm c}/R^{2}$ from zero to
infinity. Thus, in a Newtonian context, one may expect the residuals around
$\Sigma_{\rm dyn,0}$ to correlate with galaxy mass or size. This is not
observed.

To date there has been no attempt to reproduce the CDR in a $\Lambda$CDM
context. However, the “unexpected diversity” of the dwarf galaxy rotation
curves [96] suggests that the existence of this relation is not “natural”: one
may expect $\Sigma_{\rm dyn,0}$ to correlate more closely with stellar mass
(or halo mass) than stellar surface density, but the opposite is observed
[144].

###  3.3 Radial acceleration relation

The RAR [18, 19] relates the observed centripetal acceleration from rotation
curves ($g_{\rm obs}=V_{\rm c}^{2}/R$) to the baryonic gravitational field
($g_{\rm bar}=-\nabla\Phi_{\rm bar}$), which can be computed by solving the
Poisson’s equation for the observed distribution of stars and gas
($\nabla^{2}\Phi_{\rm bar}=4\pi G_{\rm N}\rho_{\rm bar}$). The RAR applies at
each radius in rotation-supported galaxies, so it is a spatially resolved
relation that embeds the BTFR for $R\rightarrow\infty$ and the CDR for
$R\rightarrow 0$.

The RAR is non-linear (Fig. 5, c) and displays two explicit acceleration
scales: one setting the transition value from the unity line and one setting
the overall normalization. The two acceleration scales have approximately the
same value [19], similarly to the surface-density scales in the CDR [16], so
we will refer to them as $a_{\rm RAR}$ for simplicity. Above $a_{\rm RAR}$, we
have $g_{\rm obs}=g_{\rm bar}$ because the data largely come from the inner
baryon-dominated regions of spiral galaxies. Below $a_{\rm RAR}$, we have
$g_{\rm obs}>g_{\rm bar}$ because the data come either from the outer DM-
dominated regions of spiral galaxies or from nearly every radii of dwarf
galaxies. The fact that spiral and dwarf galaxies smoothly connects in the
low-acceleration part of the RAR is highly non-trivial: the amount of DM in
the outer parts of spirals is somehow comparable to that in the inner parts of
dwarfs. The residuals around the RAR show no relation with other galaxy
properties [19] and the observed scatter is consistent with observational
uncertainties [18, 19, 159].

In a $\Lambda$CDM context, there have been several attempts to reproduce the
RAR but the results are mixed. Some hydrodynamic simulations of galaxy
formation reproduce the RAR with the observed value of $a_{\rm RAR}$ [160,
161], other simulations give a relation with too high value of $a_{\rm RAR}$
[162], while other ones display no acceleration scale at all [163, 164]. This
illustrates that, contrary to some claims [165], the RAR shape is not a
“natural” outcome of $\Lambda$CDM cosmology but depends on how different
models treat the complex interplay between baryons and DM during the formation
and evolution of galaxies. The most severe challenge for $\Lambda$CDM models,
however, is not the RAR shape but the RAR scatter, which appears to be
significantly smaller than predicted by fiducial abundance-matching
prescriptions [166].

The shape of the RAR is similar to that of the CDR but the two functional
forms are different. In a MOND framework (see Box 2), the RAR shape is given
by the derivative of the CDR shape [158]. Fitting a CDR-motivated function to
the RAR data, we obtain $a_{\rm RAR}=(1.2\pm 0.2)\times 10^{-10}$ m s-2 where
the error budget is dominated by systematic uncertainties in the conversion
from stellar light to stellar mass. Other empirical fitting functions provide
similar values of $a_{\rm RAR}$ [18, 19]. In a DM context, it is remarkable
that $a_{\rm BTFR}\simeq a_{\rm CDR}\simeq a_{\rm RAR}$ within the
uncertainties because these acceleration scales have distinct physical
meanings: $a_{\rm BTFR}$ sets the BTFR intercept and is related to the average
amount of missing baryons, $a_{\rm CDR}$ sets the transition from baryon to DM
dominated galaxies in the inner regions and is presumably related to the
baryonic angular momentum, whereas $a_{\rm RAR}$ sets the transition from
baryon-to-DM domination inside each galaxy and is related to the complex
history of gas, stellar, and DM build-up at each radius. To make an explicit
example: for $g_{\rm bar}\ll a_{\rm RAR}$ we have $g_{\rm obs}=\sqrt{a_{\rm
RAR}g_{\rm bar}}$ that gives $V^{4}_{\rm flat}=a_{\rm RAR}G_{\rm N}M_{\rm
bar}$ at large radii [19]. This is equivalent to the BTFR but it is unclear
why $a_{\rm RAR}$ (related to the internal mass distribution) should have the
same numerical value as $a_{\rm BTFR}$ (related to the average amount of
missing baryons). In a MOND context, instead, one must necessarily have
$a_{\rm BTFR}=a_{\rm CDR}=a_{\rm RAR}$ because these acceleration scales
subtend the existence of a new constant of Nature: $a_{0}$ [167, 168, 169].
Thus it is of outmost importance to understand (i) why these acceleration
scales exist, (ii) whether they have exactly the same value, (iii) which
physical processes set their values, and (iv) whether they evolve or not with
cosmic time.

##  4 Outlook

###  4.1 Next-generation kinematic surveys

The outstanding progress I reviewed here has been possible thanks to H I and
H$\alpha$ kinematic data accumulated over the past 40 years by several
generations of astronomers. This has allowed studying gas dynamics in all
known types of gas-rich dwarfs, spanning $\sim$4 dex in stellar mass. The
available kinematic samples, however, remain relatively small (a few hundreds
galaxies) and rather heterogeneous in nature. These considerations are
important when studying dynamical laws and their intrinsic scatters, which are
a key testbed for both $\Lambda$CDM models [152, 166, 57, 161] and alternative
theories [14, 19, 159, 170]. $\Lambda$CDM models, indeed, generally predict
that there must be non-negligible intrinsic scatter around the dynamical laws,
driven by the diverse evolutionary paths followed by galaxies and their DM
halos [152, 166]. This intrinsic scatter is expected to become increasingly
large towards the least massive galaxies [57, 161]. On the other hand, MOND
predicts tiny or null intrinsic scatter around these dynamical laws, depending
on the precise theory formulation [171] and on the galaxy environment [170].
Obtaining large and homogeneous kinematic samples, probing the faintest
dwarfs, is the next observational challenge.

The future looks bright. The Square Kilometer Array (SKA) observatory and its
pathfinders are expected to provide spatially resolved H I observations for
several thousand galaxies at $z\simeq 0$ [172, 173, 174], possibly detecting
dwarf galaxies with H I masses around $\sim$105 M⊙ or less [175]. These next-
generation H I surveys will be complementary to planned optical and near-IR
wide-field surveys, probing the stellar component of dwarf galaxies, for
example with the upcoming Euclid space mission [176] and the Vera C. Rubin
observatory [177]. In the coming years, therefore, we will gather large,
homogeneous samples of dwarf galaxies, enlarging the sizes of the current
samples by orders of magnitude. This will allow us to measure the intrinsic
scatter in the dynamical laws of galaxies with high accuracy and investigate
their possible environmental dependence in a robust way. In this context, it
will be beneficial to have clear a-priori predictions from $\Lambda$CDM
simulations tailored to the planned surveys, before the observations are
actually taken.

###  4.2 Next-generation modeling techniques

While statistical studies of dwarf galaxies will be crucial in the near
future, detailed high-resolution studies of individual objects will also be
game-changing. For example, the SKA observatory is expected to provide H I
data with unprecedented spatial resolutions and sensitivities, probing gas
dynamics in significant samples of dwarf galaxies at small spatial scales
(below 10 pc) and column densities ($\sim$1018 cm-2). In this context, it will
be necessary to extend existing 3D modeling techniques [178, 179, 82] to
include complex small-scale noncircular motions. Moreover, in the lowest mass
galaxies (with $V_{\rm rot}\lesssim 20$ km s-1), accurate modeling of pressure
support and disk thickness will become important [74, 76]. A promising
approach is to simultaneously model the midplane gravitational potential in
the radial direction together with that in the vertical direction assuming
hydrostatic equilibrium [84].

From the galaxy formation perspective, it will be important to compare
observed galaxies with simulated ones in a robust way. The most promising
approach is building mock observations of simulated galaxies, taking
observational effects into account [180, 181, 113, 117]. The goal will be to
assess whether $\Lambda$CDM models are able to form realistic dwarf galaxies
by reproducing diverse (yet interrelated) aspects of their gas kinematics,
such as the rotation-curve shape, the DM content, the strength of noncircular
motions, the gas velocity dispersion, and the occurrence of extraplanar gas.
Most existing models are often able to reproduce a subset of them (e.g.,
rotation curve shapes using strong stellar feedback) but fail on others (e.g.,
too strong noncircular motions [113]). This does not occur by chance. For
example, strong stellar feedback is commonly implemented to solve the cusp-vs-
core and too-big-too-fail problems (see Box 1), but may then generate too
strong noncircular motions and/or too massive gas outflows. Again, it will be
desiderable to build samples of mock-observed galaxies before the actual
observations are taken, so one can have clear and unambiguous predictions from
different galaxy formation models. This is the only way to adhere to the
scientific method [182] and push forward our understanding of galaxy formation
and evolution.

## Acknowledgements

Writing this review would have not been possible without the many enlightening
conversations with many colleagues over the years. In particular I thank Alice
Concas, Edvige Corbelli, Filippo Fraternali, Stacy McGaugh, James Schombert,
and Paolo Tozzi for their precious comments on a first draft of this
manuscript.

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