text:
\section{Introduction}
\label{section: method}

Knowledge about condensed-matter properties of quantum chromodynamics (QCD) in thermodynamic equilibrium is required for the
interpretation of
experimental and observational data in cosmology, high-energy nuclear physics and the physics of neutron stars. While tremendous
progress has been made for the case of high temperature and small baryon densities using direct simulations in lattice QCD
\cite{Borsanyi:2010cj,Borsanyi:2013bia,Bazavov:2014pvz}, much less is known for the case of small temperature and large densities.
The reason for this shortcoming is that the so-called sign problem prohibits the direct simulation
of QCD (which is an SU$(3)$ gauge theory with $n_f=2+1$ fundamental fermionic degrees of freedom) at large density using established
importance sampling techniques.  While established techniques fail, several recent techniques have been studied that at least
in principle could permit one to calculate  thermodynamic properties from first principles in QCD at large density. These
techniques include
Lefschetz thimbles \cite{Cristoforetti:2012su}, complex Langevin \cite{Aarts:2013uxa,Sexty:2013ica,Aarts:2014bwa,Aarts:2014fsa},
strong coupling expansion \cite{deForcrand:2014tha}, and hadron resonance gas plus perturbative QCD (``HRG+pQCD'' in the following)
\cite{Laine:2006cp,Huovinen:2009yb,Kurkela:2009gj}.
In this work, we propose a series of `control studies'
in QCD-like theories (in particular two-color QCD with two fundamental flavors and four-color QCD with two flavors in the two-index,
antisymmetric representation), which---despite not corresponding to the actual theory of strong interactions realized in nature---have
the advantage
of not suffering from a sign problem, and are thus amenable to direct simulations using established lattice-QCD techniques. We then
proceed to calculate thermodynamic properties in these QCD-like theories in one of the above non-traditional approaches (HRG+pQCD,
Ref.~\cite{Laine:2006cp,Huovinen:2009yb,Kurkela:2009gj}), which effectively makes predictions for possible future lattice-QCD
studies that can be used to validate or falsify this HRG+pQCD\ approach.  Since two- and four-color QCD are qualitatively similar to
three-color QCD, we furthermore expect the level of agreement between lattice
QCD and HRG+pQCD\ in the two- or four-color cases to be roughly comparable to the three-color QCD case, thus offering an indirect
validation of non-traditional methods for QCD at large densities.

The paper is organized as follows.
In Section \ref{sec: pQCD} we review the HRG+pQCD\ method and give the equation of state in pQCD by stating the pressure $P$ as a function
of temperature $T$ at baryon chemical potential $\mu = 0$ and $P$ as a function of $\mu$ at $T = 0$.  This section is essentially
a compilation of what has been derived in the
literature.  In Section \ref{sec: HRG} we compute the HRG pressure as a function of $T$ or $\mu$.  The former case is simple and is
derived quickly, whereas the latter is derived in more detail, especially for the theories where the baryons of the theory are bosons.
This section also contains an explanation of how the hadrons in the theories listed above are computed.  Section \ref{sec: matching}
contains a description of how we perform the matching between these two asymptotic equations of state, and in Section
\ref{sec: results} we discuss our results.

\section{pQCD equation of state}
\label{sec: pQCD}

In this work, we are interested in calculating the pressure $P$
along the $\mu = 0$ and $T = 0$ axes in the theories $(N , n_{f}) = (2 , 2) ,
(3 , 3) , \text{ and } (4 , 2)$ with quarks in the fundamental representation (fundamental) and $(4 , 2)$ with quarks in the two-index,
antisymmetric representation (antisymmetric).  In order to constrain the pressure of these
theories, we derive the asymptotic behavior for both low and high $T$ or $\mu$ and then
match these behaviors using basic thermodynamics.  At high $T$ or high $\mu$, the equation of state can be calculated using (resummed)
pQCD and at low $T$ or $\mu$, the equation of state of the theory is to good approximation \cite{Borsanyi:2010cj,Borsanyi:2013bia,
Bazavov:2014pvz,Langelage:2008dj} that of a HRG (a noninteracting collection of the hadrons of that theory).
In the intermediate regime, the equations of state can be constructed by matching the high/low energy asymptotic behavior using the
criterion that the pressure $P$ must increase as a function of $T$ or as a function of $\mu$ (see \cite{Vuorinen:2003fs}). More details
of the matching procedure will be discussed below. Throughout this paper, Boltzmann's constant $k$, Planck's reduced constant $\hbar$,
and the speed of light $c$ will be set equal to one.

The high-$T$, pQCD equation of state can be calculated by following the equations and procedure of Kajantie \emph{et al.}
\cite{Kajantie:2002wa,Kajantie:2003ax} and Vuorinen \cite{Vuorinen:2003fs}
with the resummation modifications described by Blaizot \emph{et al.} \cite{Blaizot:2003iq} (cf.\ Ref.~\cite{Haque:2014rua} for a
different approach to the resummed pQCD equation of state).  We first define the following group-theory terms to be used in
all future pQCD expressions:
\begin{align}
	C_{A} &= N, \\
	d_{A} &= N^{2} - 1,
\end{align}
and
\begin{alignat}{2}
	C_{\text{fundamental}} &= \frac{N^{2} - 1}{2 N}, \qquad \qquad &C_{\text{antisymmetric}} &= \frac{(N - 2)(N + 1)}{N}, \\
	T_{\text{fundamental}} &= \frac{n_{f}}{2}, &T_{\text{antisymmetric}} &= \frac{(N - 2)}{2} n_{f} , \\
	d_{\text{fundamental}} &=  N n_{f}, &d_{\text{antisymmetric}} &= \frac{N (N - 1)}{2} n_{f} .
\end{alignat}
In all of the expressions that follow, we let group-theory terms with a subscript $R$ denote the fermionic group-theory
invariants, which must be replaced by the corresponding fundamental or antisymmetric representation group-theory invariants
above as needed.  In terms of these group-theory terms, the pQCD pressure at $\mu = 0$ in these theories can be written
\begin{equation}
	P_{\text{pQCD}}(T) = P_{\text{sb}}(T) + P_{\text{hard}}(T) + P_{\text{EQCD}}(T).
\label{eq: pQCD T}
\end{equation}
Here, the $P_{\text{sb}}$ the Stefan-Boltzmann pressure given by
\begin{equation}
	P_{\text{sb}}(T) = \frac{\pi^{2} T^{4}}{45} \left( d_{A} + \frac{7}{4} d_{R} \right).
\label{eq: pQCD sb}
\end{equation}
To 3-loop order, $P_{\text{hard}}$ is given by Braaten and Nieto \cite{Braaten:1995jr} as
\begin{align}
	P_{\text{hard}}(T) &= \frac{\pi^{2} d_{A}}{9} T^{4} \bigg\{
			- \left(C_{A} + \frac{5}{2} T_{R} \right) \frac{\alpha_{s}}{4 \pi} \nonumber \\
		& + \bigg( C_{A}^{2} \left[ 48 \ln \frac{\Lambda_{E}}{4 \pi T} - \frac{22}{3} \ln \frac{\overline{\Lambda}}{4 \pi T}
			+ \frac{116}{5} + 4 \gamma + \frac{148}{3} \frac{\zeta'(-1)}{\zeta(-1)}
			- \frac{38}{3} \frac{\zeta'(-3)}{\zeta(-3)} \right]
				\nonumber \\
		& + C_{A} T_{R} \left[ 48 \ln \frac{\Lambda_{E}}{4 \pi T} - \frac{47}{3} \ln \frac{\overline{\Lambda}}{4 \pi T}
			+ \frac{401}{60} -\frac{37}{5} \ln 2 + 8 \gamma + \frac{74}{3} \frac{\zeta'(-1)}{\zeta(-1)}
			- \frac{1}{3} \frac{\zeta'(-3)}{\zeta(-3)} \right] \nonumber \\
		& + T_{R}^{2} \left[ \frac{20}{3} \ln \frac{\overline{\Lambda}}{4 \pi T} + \frac{1}{3}
			-\frac{88}{5} \ln 2 + 4 \gamma + \frac{16}{3} \frac{\zeta'(-1)}{\zeta(-1)} -
			\frac{8}{3} \frac{\zeta'(-3)}{\zeta(-3)} \right] \nonumber \\
		& + C_{R} T_{R} \left[ \frac{105}{4} - 24 \ln 2 \right]\bigg) \left( \frac{\alpha_{s}}{4 \pi} \right)^{2} \bigg\},
\label{eq: pQCD hard}
\end{align}
where $\Lambda_{E}$ is the factorization scale between the hard and soft modes, and $\alpha_{s}$ is the strong coupling constant
squared over $4 \pi$ in the $\overline{\text{MS}}$ renormalization scheme at
the scale $\overline{\Lambda} = \sqrt{(2 \pi T)^{2} + (\mu)^{2}}$.  This is given
by \cite{vanRitbergen:1997va,Kurkela:2009gj}
\begin{equation}
	\alpha_{s}(\overline{\Lambda}) = \frac{4 \pi}{\beta_{0} L} \left(1 - \frac{\beta_{1}}{\beta_{0}^{2}} \frac{\ln L}{L} \right),
		\qquad L = \ln \left( \overline{\Lambda}^{2} / \Lambda^{2}_{\overline{\text{MS}}} \right),
\end{equation}
with
\begin{equation}
	\beta_{0} = \frac{11}{3} C_{A} - \frac{4}{3} T_{R}, \quad \beta_{1} = \frac{34}{3} C_{A}^{2} - 4 C_{R} T_{R}
			- \frac{20}{3} C_{A} T_{R},
\end{equation}
where $\Lambda_{\overline{\text{MS}}}$ is the $\overline{\text{MS}}$ renormalization point (to be set later).  In all our results,
we set $\Lambda_{E} = \overline{\Lambda}$ and vary $\overline{\Lambda}$ about the aforementioned value by a factor of two
(cf.\ the end of Section \ref{sec: matching}). Finally,
$P_{\text{EQCD}}$ is given by
\begin{align}
	P_{\text{EQCD}}(T) = \frac{d_{A}}{4 \pi} T \bigg( \frac{1}{3} m_{E}^{3} - \frac{C_{A}}{4 \pi} \left( \ln \frac{\Lambda_{E}}{2 m_{E}} + \frac{3}{4} \right)
		g_{E}^{2} m_{E}^{2} - \left( \frac{C_{A}}{4 \pi} \right)^2 \left( \frac{89}{24} - \frac{11}{6} \ln 2 + \frac{1}{6} \pi^{2} \right)
		g_{E}^{4} m_{E} \bigg),
\label{eq: pQCD EQCD}
\end{align}
where
\begin{align}
	m_{E}^{2} &= \frac{4 \pi}{3} \alpha_{s} T^{2} \bigg\{ C_{A} + T_{R} \nonumber \\
		&+ \Big[ C_{A}^{2} \left( \frac{5}{3} + \frac{22}{3} \gamma + \frac{22}{3} \ln \frac{\overline{\Lambda}}{4 \pi T}
			\right) + C_{A} T_{R} \left( 3 - \frac{16}{3} \ln 2 + \frac{14}{3} \gamma + \frac{14}{3} \ln
			\frac{\overline{\Lambda}}{4 \pi T} \right)
				\nonumber \\
		&+ T_{R}^{2} \left( \frac{4}{3} - \frac{16}{3} \ln 2 - \frac{8}{3} \gamma
			- \frac{8}{3} \ln \frac{\overline{\Lambda}}{4 \pi T} \right)
			- 6 C_{R} T_{R} \Big] \left( \frac{\alpha_{s}}{4 \pi} \right) \bigg\},
\end{align}
and
\begin{equation}
	g_{E}^2 = 4 \pi \alpha_{s} T.
\end{equation}

The $T = 0$, pQCD equation of state is more straightforward in the sense that resummation of the strict perturbative series is not
required. The result is given in Ref.~\cite{Vuorinen:2003fs} by
\begin{align}
	P_{\text{pQCD}}(\mu) & = \frac{1}{4 \pi^{2}} \Bigg( \sum_{f} \mu_{f}^{4} \bigg\{ \frac{d_{R}}{3 n_{f}} - d_{A}
			\left( \frac{2 T_{R}}{n_{f}} \right)
			\left(
			\frac{\alpha_{s}}{4 \pi} \right) - d_{A} \left( \frac{2 T_{R}}{n_{f}} \right)
			\left( \frac{\alpha_{s}}{4 \pi} \right)^{2} \bigg[
			\frac{2}{3}(11 C_{A} - 4 T_{R}) \ln \frac{\overline{\Lambda}} {\mu_{f}} + \frac{16}{3} \ln 2 \nonumber \\
		& + \frac{17}{4} \left( \frac{C_{A}}{2} - C_{R} \right) + \frac{1}{36}(415 - 264 \ln 2) C_{A} - \frac{8}{3} \left(
			\frac{11}{6} - \ln 2 \right) T_{R} \bigg] \bigg\} \nonumber \\
		& - d_{A} \left( \frac{2 T_{R}}{n_{f}} \right)\left( \frac{\alpha_{s}}{4 \pi} \right)^{2} \left\{ \left( 2 \ln
			\frac{\alpha_{s}}{4 \pi} - \frac{22}{3} +
			\frac{16}{3} \ln 2 \, (1 - \ln 2) + \delta + \frac{2 \pi^{2}}{3} \right) (\bm{\mu}^{2})^{2}
			+ F(\bm{\mu}) \right\} \Bigg) \nonumber \\
		& + \mathcal{O} (\alpha_{s}^{3} \ln \alpha_{s}),
\label{eq: pQCD mu}
\end{align}
where the sum is over all the quark flavors in the theory, $\mu_{f}$ is the $f$-quark chemical potential, $\bm{\mu}^{2} = \sum_{f}
\mu_{f}^{2}$, and
\begin{align}
	F(\bm{\mu}) & = -2 \bm{\mu}^{2} \left( \frac{2 T_{R}}{n_{f}} \right)\sum_{f} \mu_{f}^{2} \ln \frac{\mu_{f}^{2}}{\bm{\mu}^{2}}
			+ \frac{2}{3} \left( \frac{2 T_{R}}{n_{f}} \right)^{2} \sum_{f > g}
			\bigg\{ (\mu_{f} - \mu_{g})^{2} \ln \frac{|\mu_{f}^{2} - \mu_{g}^{2}|}{\mu_{f} \mu_{g}} \nonumber \\
		& + 4 \mu_{f} \mu_{g} (\mu_{f}^{2} + \mu_{g}^{2}) \ln \frac{(\mu_{f} + \mu_{g})^{2}}{\mu_{f} \mu_{g}} -
			(\mu_{f}^{4} - \mu_{g}^{4}) \ln \frac{\mu_{f}}{\mu_{g}} \bigg\},
\label{eqn: F-function}
\end{align}
with the constant $\delta$ having the value $\delta = −0.85638320933$.  In what follows we always set
all of the quark chemical potentials equal to each other, so that $\mu_{f} = \mu / N_{b}$ for each flavor $f$, where $N_{b}$ is the
number of quarks in a baryon.  Note that this means that some of the terms in \eqref{eqn: F-function} do not contribute.

\section{Hadron resonance gas equation of state}
\label{sec: HRG}

The low-$T$ pressure in these theories is given by considering the system to be a free gas of hadrons.  Moreover, the statistics of
the hadrons may be ignored, so that the distribution functions may all be assumed to be Boltzmann factors.  In that case, we have
\begin{equation}
	P_{\text{HRG}}(T) = T \sum_{i \in H} {g_{i}} \int\! \frac{\mathrm{d}^{3}p}{(2 \pi)^{3}} e^{- \sqrt{\mathbf{p}^{2} + m_{i}^{2}} / T}
		= T^{4} \sum_{i \in H} \frac{g_{i}}{2 \pi^{2}} \left( \frac{m_{i}}{T} \right)^{2} K_{2} \! \left( \frac{m_{i}}{T} \right),
\label{eq: HRG}
\end{equation}
where here the sum is over the hadron spectrum of the theory; $g_{i}$ and $m_{i}$ are the degeneracy and the mass of the $i$th
particle, respectively; $\mathbf{p} = | \vec{p} \,|$; and $K_{2}$ is a modified Bessel function of the second kind.

The low-$\mu$ pressure can be calculated in a similar way, but in this case the statistics of the particles cannot be ignored.  For
$T = 0$ and $\mu > 0$, the only particles that contribute to the partition function are particles containing no antiquarks,
which we denote by $B$. In all
the fundamental theories, these are simply the baryons, whereas for the antisymmetric theory there are more particles fitting this
description (see
below).  As such, in this section we shall refer to all the particles in $B$ as baryons.
Taking the $T \to 0$ limit in the fermionic-baryon ($\eta = 1$) or bosonic-baryon ($\eta = -1$) case yields
\begin{align}
	P_{\text{HRG}}(\mu) & = \lim_{T \to 0} T \sum_{i \in B} g_{i} \eta \int\! \frac{\mathrm{d}^{3}p}{(2 \pi)^{3}} \ln \left(1 + \eta \,
			e^{\left(\mu r_{i} - \sqrt{\mathbf{p}^{2} + m_{i}^{2}}\right) / T} \right) \nonumber \\
		& = \sum_{i \in B} g_{i} \eta \int\! \frac{\mathrm{d}^{3}p}{(2 \pi)^{3}} \ln \left[ \lim_{T \to 0} \left(1 + \eta \,
			e^{\left(\mu r_{i} - \sqrt{\mathbf{p}^{2} + m_{i}^{2}}\right) / T} \right)^{\! T} \right] \nonumber \\
		& =  \sum_{i \in B} g_{i} \eta \int_{0}^{\sqrt{(\mu r_{i})^{2} - m_{i}^{2}}} \frac{\mathbf{p}^{2} \mathrm{d} \mathbf{p}}
			{2 \pi^{2}} \left(\mu r_{i} - \sqrt{\mathbf{p}^{2} + m_{i}^{2}}\right) \theta( \mu r_{i} - m_{i}) \nonumber \\
		& = \eta \sum_{i \in B} \frac{g_{i}}{48 \pi^{2}} \left[ \mu r_{i} \sqrt{(\mu r_{i})^{2} - m_{i}^{2}}
			(2 (\mu r_{i})^{2} - 5 m_{i}^{2}) + 3 m_{i}^{4} \cosh^{-1} \left( \frac{m_{i}}{\sqrt{(\mu r_{i})^{2}
			- m_{i}^{2}}}
			\right) \right] \theta( \mu r_{i} - m_{i}),
\label{eq: fermion mu}
\end{align}
where here $\theta$ is the Heaviside step function and $r_{i} = N_{i} / N_{b}$, with $N_{i}$ being the number of quarks in the $i$th
particle.  This result is correct for the fermionic-baryon case, but this formula gives
negative $P$ in the bosonic-baryon case when $\mu > \min_{i \in B} m_{i} / r_{i}$.  This is because, in the bosonic case, a condensate
of the $i$th baryon forms when $\mu = m_{i} / r_{i}$. (This has been numerically investigated in the two-color case
by Hands \emph{et al.} \cite{Hands:2006ve} and analytically by Kogut \emph{et al.} \cite{Kogut:2000ek} in all QCD-like theories
with pseudoreal fermions.)  In fact, in the completely-noninteracting case it is nonsensical for $\mu$
to exceed $\min_{i \in B} m_{i} / r_{i}$.  Since the hadrons in these theories are composite particles, they are not truly
noninteracting, and we can have $\mu > \min_{i \in B} m_{i} / r_{i}$.

To make sense of this case, we consider the bosons as a (complex) quantum field $\Phi$ with a $|\Phi|^{4}$ repulsive interaction.
For simplicity, we consider each baryon to be an independent field, and we examine the case of a scalar field (degeneracies may easily
be incorporated at the end).  A single baryon then has the Lagrangian density (in the mostly minus convention)
\begin{equation}
	\mathscr{L} = (\partial_{\mu} \Phi^{\dagger})(\partial^{\mu} \Phi) - m^{2} \Phi^{\dagger} \Phi - \lambda (\Phi^{\dagger} \Phi)
		(\Phi^{\dagger} \Phi),
\end{equation}
with $\lambda > 0$.  Following Kapusta and Gale \cite{Kapusta:2006pm}, we introduce a baryon chemical potential $\mu r$ and explicitly
factor out the zero momentum mode
\begin{equation}
	\Phi = \xi + \chi,
\end{equation}
where $\xi \in \mathbb{R}$ is a constant and the constant Fourier component of $\chi$ satisfies $\chi_{n = 0}(\mathbf{p} = 0) = 0$.  One may
think of $\xi$ as the condensate field and $\chi$ as the fluctuations about the vacuum state.  We also write the fluctuations in terms
of the normalized real and imaginary parts
\begin{equation}
	\chi = \frac{1}{\sqrt{2}} (\chi_{1} + i \chi_{2}).
\end{equation}
In terms of these new variables, the Euclidean Lagrangian density becomes
\begin{align}
	\mathscr{L} & = -\frac{1}{2} \left( \frac{\partial \chi_{1}}{\partial \tau} - i \mu r \chi_{2} \right)^{2}
			- \frac{1}{2} \left( \frac{\partial \chi_{2}}{\partial \tau} + i \mu r \chi_{1} \right)^{2}
			- \frac{1}{2} \nabla^{2} \chi_{1} - \frac{1}{2} \nabla^{2} \chi_{2} \nonumber \\
		& -\frac{1}{2}(6 \lambda \xi^{2} + m^{2}) \chi_{1}^{2} -\frac{1}{2}(2 \lambda \xi^{2} + m^{2}) \chi_{2}^{2}
			- U(\xi) + \mathscr{L}_{\text{I}},
\label{eq: lagrangian}
\end{align}
where $\tau$ is the Euclidean time, $\mathscr{L}_{\text{I}}$ contains interacting terms in $\chi$ (which we henceforth ignore), and
\begin{equation}
	U(\xi) = (m^{2} - (\mu r)^{2}) \xi^{2} + \lambda \xi^{4}.
\label{eq: vacuum}
\end{equation}
We thus see from \eqref{eq: vacuum} that for $\mu r < m$ the state $\xi_{0} = 0$ is the stable vacuum and \eqref{eq: lagrangian}
describes a system of particles and antiparticles of equal masses.  However, for $\mu r > m$, the stable vacuum becomes
\begin{equation}
	\xi_{0}^{2} = \frac{(\mu r)^{2} - m^{2}}{2 \lambda},
\end{equation}
and \eqref{eq: lagrangian} describes a collection of two particles with differing masses:
$\overline{m}_{1}^{2} = 3 (\mu r)^{2} - 2 m^{2}$ and $\overline{m}_{2}^{2} = (\mu r)^{2}$ respectively. Because of the chemical
potential, the dispersion relation of the latter is gapless, and Goldstone's theorem is satisfied.  At zero temperature, the
pressure is simply
\begin{equation}
	P_{\text{HRG}}(\mu) = U(\xi)\big|_{\xi = \xi_{0}} = \frac{1}{4 \lambda} ((\mu r)^{2} - m^{2})^{2} \theta(\mu r - m).
\end{equation}
This gives us the dependence of the pressure on $\mu$, but we still have not set the coupling constant $\lambda$.  We set it as
follows.  According to \eqref{eq: pQCD mu}, we see that the Fermi--Dirac pressure for a quark in the theory $(N , n_{f})$ with
representation $R$ becomes
\begin{equation}
	P_{\text{fd}} = \frac{d_{R}}{12 \pi^{2}} \left( \frac{\mu r}{N_{b}} \right)^{4},
\end{equation}
so that for a single degree of freedom (recalling that a fermionic quark has two degrees of freedom) one has
\begin{equation}
	P_{\text{fd}} = \frac{1}{24 \pi^{2} N_{b}^{4}} (\mu r)^{4}.
\end{equation}
Thus, in order for $P_{\text{HRG}} \to P_{\text{fd}}$ when $\mu \to \infty$, we must have for a single scalar baryon
\begin{equation}
	P_{\text{HRG}}(\mu) = \frac{1}{24 \pi^{2} N_{b}^{4}} ((\mu r)^{2} - m^{2})^{2} \theta(\mu r - m),
\end{equation}
and so for a theory with bosonic baryons we have
\begin{equation}
	P_{\text{HRG}}(\mu) = \sum_{i \in B} \frac{g_{i}}{24 \pi^{2} N_{b}^{4}} ((\mu r_{i})^{2} - m_{i}^{2})^{2}
			\theta(\mu r_{i} - m_{i}).
\label{eq: bosonic mu}
\end{equation}

\subsection{Determining the hadron spectrum}

For the three-color $(N , n_{f}) = (3 , 3)$ fundamental case, we use the real world spectrum of hadrons up to 2.25 GeV
\cite{Agashe:2014kda}.  For the two- and four-color theories with two fundamental quarks and the four-color theory with two
antisymmetric quarks, we determine the hadrons using group-theoretic
arguments and Fermi statistics (in the case of objects composed of quarks only).  We explicitly ignore the glueballs in these theories
because they tend to be more massive than the lightest hadrons \cite{Meyer:2004jc}. For the two- and four-color theories, we set the
scale using the string tension $\sqrt{\sigma}$ and the relation between the string tension and
the $\overline{\text{MS}}$ renormalization scale $\Lambda_{\overline{\text{MS}}}$ given in Ref.~\cite{Lucini:2008vi}.
However, the ratios $\Lambda_{\overline{\text{MS}}} / \sqrt{\sigma}$
given in the aforementioned reference are for the pure-gauge theories.  To remedy this, we scale these ratios by
$\Lambda^{N = 3}_{\overline{\text{MS}}}(n_{f}\!=\!2) / \Lambda^{N = 3}_{\overline{\text{MS}}}(n_{f}\!=\!0)$, determined from
Ref.~\cite{Fritzsch:2012wq}.  These lead to the values
\begin{equation}
	\Lambda^{N = 2}_{\overline{\text{MS}}}(n_{f}\!=\!2) / \sqrt{\sigma} = 1.032 \quad \text{and} \quad
		\Lambda^{N = 4}_{\overline{\text{MS}}}(n_{f}\!=\!2) / \sqrt{\sigma} = 0.723
\end{equation}
for the fundamental theories. For the three-color theory, we use $\Lambda_{\overline{\text{MS}}} = 0.378 \text{ GeV}$,
as in \cite{Kurkela:2009gj}.

For the four-color antisymmetric theory, we were unable to locate a result for $\Lambda^{N = 4}_{\overline{\text{MS}}}(n_{f}\!=\!2)
/\sqrt{\sigma}$
from the lattice in the literature.  Since some of the group-theory terms for the antisymmetric theory scale more strongly with
the number of colors than the corresponding terms in the fundamental theory, it seems reasonable to expect that
$\Lambda_{\overline{\text{MS}}}$
will scale differently with the number of quark flavors in the antisymmetric theory than in the fundamental theory.  Moreover,
it would be most accurate to view $\Lambda_{\overline{\text{MS}}} / \sqrt{\sigma}$ as a free parameter in our HRG+pQCD\ scheme that
must be determined independently from the lattice.  In light of these considerations, we have decided to use both the pure-glue value
\cite{Lucini:2008vi}
\begin{equation}
		\Lambda^{N = 4}_{\overline{\text{MS}}} / \sqrt{\sigma} = 0.527,
\end{equation}
and the previously-given value of $\Lambda^{N = 4}_{\overline{\text{MS}}}(n_{f}\!=\!2) /\sqrt{\sigma}$ that we use for the four-color
fundamental theory for the four-color antisymmetric theory, with the expectation that the true value will lie somewhere near
this range.

For both the two- and four-color cases, the mesons are taken to be the analogues of the flavorless mesons that exist
in the real world (up to a mass of about 2 GeV) whose masses are written in multiples of the string tension $\sigma_{\text{SU}(3)}
= (420 \text{ MeV})^{2}$.  In the two-color case, we mainly use the analogues of the real-world mesons, substituting the two-color
masses calculated by Bali \emph{et al.} \cite{Bali:2013kia} when available. (We also note here that the $\mu$-dependence of the
two-color spectrum has been studied numerically in Ref.~\cite{Hands:2007uc} and analytically in Ref.~\cite{Kogut:2000ek}, though
we do not need this $\mu$-dependence for our HRG+pQCD\ scheme.)

We now discuss in some detail how the non-meson objects in these three cases are determined.  For convenience and as a summary of
these sections, we list tables for all of the particles that we have included in the SU(2) and SU(4) cases in Appendix
\ref{sec: particle tables}.

\subsubsection{Two-color case}

In two-color QCD, the baryons are composed of two quarks with the added simplicity that the masses are degenerate with the corresponding
mesons made from the same quarks \cite{Lewis:2011zb}.  Thus, the mass spectrum of the baryons is the same as the mass spectrum of
mesons.  However, there are fewer mesons than baryons, for there is an additional constraint imposed by Fermi statistics in the case
of the baryons. Since we may view the two massless quarks as part of an isospin doublet, one sees that exchanging the two internal
quarks in a baryon causes the wavefunction to become multiplied by
\begin{equation}
	(-1)^{1 + L + S + I}.
\end{equation}
In this equation, $L$ is the angular momentum quantum number, $S$ is the spin, and $I$ is the isospin, with the additional $1$ due to the
fact that the quarks are in an antisymmetric color singlet.  We thus see that for even $L$
the spin and isospin must be equal ($S = 0$ implies $I = 0$ and $S = 1$ implies $I = 1$), and for odd $L$ they must be the opposite
in order
to have a totally antisymmetric wavefunction. (Even though the composite baryon is itself a boson in two-color QCD, it is still a
multiparticle state of fundamental fermions.) This information is enough to determine the set of hadrons in \eqref{eq: HRG}.

\subsubsection{Four-color fundamental case}

Baryons in four-color QCD with fundamental fermions consist of four quarks.  In this case, to determine the masses $M$ we use the
large-$N$ expansion
\begin{equation}
	M(J) = N A + \frac{J (J + 1)}{N} B,
\label{eq: large N masses}
\end{equation}
where $J$ is the total angular momentum of the baryon, and $A$, $B$ are constants independent of $N$
\cite{Adkins:1983ya,Jenkins:1993zu}.  As pointed out
by DeGrand \cite{DeGrand:2012hd} and demonstrated by Appelquist \emph{et al.} \cite{Appelquist:2014jch}, a term independent
of $N$ could be used for better agreement. However, we have no way to set the value of that term and thus do not include it.

We find the possible values of $J$ beginning with the ground-state baryons of zero orbital angular momentum.  Since we still have a
isospin doublet of massless, spin-one-half quarks, we only need the group-theory expression
\begin{equation}
	2 \otimes 2 \otimes 2 \otimes 2 = 5_{S} \oplus 3_{M} \oplus 3_{M} \oplus 3_{M} \oplus 1_{A} \oplus 1_{A},
\end{equation}
where the $5_{S}$ state is fully symmetric, the $3_{M}$ states are are symmetric in three of the four quarks and antisymmetric in the
other, and the $1_{A}$ states are pairwise antisymmetric. Since, again, the quarks are in an antisymmetric color singlet, it must be the
case that they are in a \emph{symmetric} combination of spin and flavor.  This means that there is a spin-2 quintet, a spin-1
triplet, and a spin-0 singlet of ground-state baryons.

We may also determine the first excited states in this simple manner by realizing that for this four-body problem there are three
relevant
orbital-angular-momentum quantum numbers and the first excited state corresponds to when exactly one of them is one.  In order to
still be in a completely antisymmetric state, either the spin or the flavor state must now be in one of the $3_{M}$ states while the other
must be in a $5_{S}$ state.  This means that there are a quintet of particles with $S = 1$ and a triplet of particles with $S = 2$.
Combining these with an orbital angular momentum $L = 1$ yields three baryonic quintets with $J = 0, 1, 2$ and three baryonic
triplets with $J = 1, 2, 3$. We did not determine the baryons for any higher excited states.

\subsubsection{Four-color antisymmetric case}

The hadron spectrum in the four-color theory with two antisymmetric quarks consists of two-quark objects: mesons and diquarks;
four-quark objects: tetraquarks, di-mesons, and diquark-mesons; and six-quark baryons
\cite{DeGrand:2014U,DeGrand:2014cea,Bolognesi:2006ws}.
Since the antisymmetric representation is real, the arguments of Ref.~\cite{Lewis:2011zb} carry through here and one may conclude that
all two-quark objects with the same quark content have degenerate masses and that the same holds for the four-quark objects.
In addition, the four-quark objects have a mass equal to the sum of their constituent two-quark-object masses \cite{DeGrand:2014U}.
Because of this mass degeneracy,
we need not determine how all of the four-quark-object degrees of freedom break up into spin and isospin multiplets; rather, we may
simply combine the two-quark-object degrees of freedom in every possible way.  One major difference from
the two-color case, however, is that in the four-color theory with antisymmetric quarks the color-singlet state for diquarks
is \emph{symmetric}.
This means that the spin-isospin locking in this theory is the opposite of the locking in the two-color theory.  That is, for odd $L$
the spin and isospin must be equal ($S = 0$ implies $I = 0$ and $S = 1$ implies $I = 1$), and for even $L$ they must be opposite.
As for the six-quark baryons, we again use the large-$N$ expression
\eqref{eq: large N masses}, but with $N$ replaced by $N_{b} = 6$, the number of quarks in the baryon.  We include only the
ground-state baryons, where isospin and spin are locked as $I = J =$ 3, 2, 1, and 0 \cite{DeGrand:2014U}.

\subsection{Chiral symmetry breaking and the Nambu--Goldstone bosons}
\label{sec: ChSB}

The lowest-mass particles in all of the aforementioned theories are precisely zero at zero quark mass.  This can be understood in
terms of
the pattern of chiral symmetry breaking in these theories \cite{Peskin:1980gc}.  Consider an SU($N$) gauge theory with
$n_{f}$ massless fermions.  For fermions in a complex representation (such as in the cases $N \ge 3$ with fundamental fermions),
the Lagrangian density possesses the symmetry $\text{U}(n_{f}) \otimes \text{U}(n_{f})$,
corresponding to the separate left- and right-handed chiral symmetries, and for real representations (such as any $N$ with
adjoint fermions or $N = 4$ with antisymmetric fermions) or pseudoreal representations (such as in $N = 2$ with fundamental
fermions), the Lagrangian density possesses the
larger symmetry $\text{U}(2 n_{f})$.  In all of these cases, the axial U(1) symmetry is broken by an anomaly, and
the remaining symmetries are spontaneously broken in the following ways. For fermions in a complex representation:
\begin{equation}
	\text{SU}(n_{f}) \otimes \text{SU}(n_{f}) \to \text{SU}(n_{f});
\end{equation}
for fermions in a real representation:
\begin{equation}
	\text{SU}(2 n_{f}) \to \text{O}(2 n_{f});
\end{equation}
and for fermions in a pseudoreal representation:
\begin{equation}
	\text{SU}(2 n_{f}) \to \text{Sp}(2 n_{f}).
\end{equation}
(See Refs.~\cite{Peskin:1980gc,Kogut:2000ek} for more details.)  The generators of the broken symmetries become massless
Nambu--Goldstone bosons.  Since SU($n_{f}$) has $n_{f}^{2} - 1$ generators,  O($n_{f}$) has $n_{f} (n_{f} - 1) / 2$ generators,
and Sp($n_{f}$) has $n_{f} (n_{f} + 1) / 2$ generators, we see that in the three-color, three fundamental-quark case there
will be 8 Nambu--Goldstone bosons
(a meson octet); in the four-color, two antisymmetric-quark case there will be 9 Nambu--Goldstone bosons (a triplet each of mesons,
diquarks, and antidiquarks); and in the two-color, two fundamental-quark case there will be 5 Nambu--Goldstone bosons (a triplet of
mesons, a diquark, and an antidiquark).

In addition, recall that if the quarks in these theories are not precisely massless, then the massless Nambu--Goldstone bosons will become
instead small-mass, pseudo-Nambu--Goldstone bosons.  In our spectra, we are free to vary the mass of these lightest particles to see what
effects this will have on the equation of state of the theories.  This is especially interesting for lattice practitioners.  We discuss
this further in Section \ref{sec: results}.

\section{Matching the pQCD and HRG equations of state}
\label{sec: matching}

To match the two asymptotic equations of state, we employ the same technique on the $T$ axis as on the $\mu$ axis.  As such, let us
introduce the symbol $F$ to stand for either $T$ or $\mu$ so that we may discuss the matching in full generality.

To perform the matching, we assume that at the phase-transition point the pressures of the two phases are equal;
and we use the thermodynamic constraints that the pressure of a system must increase with $F$
\begin{equation}
	P(F + \Delta F) \ge P(F),
\label{eq: pressure increase}
\end{equation}
and that above a phase-transition point, the physical phase is the one with the higher pressure.  We also add a bag constant $B$ to the pQCD
pressure so that
\begin{equation}
	P_{\text{pQCD}}(F) = P_{\text{pQCD}}^{0}(F) + B,
\end{equation}
where $P_{\text{pQCD}}^{0}$ is given by either \eqref{eq: pQCD T}-\eqref{eq: pQCD EQCD} or \eqref{eq: pQCD mu}.  In the plots that follow,
we solve the following set of two equations with two unknowns (for a given $\overline{\Lambda}$):
\begin{align}
	P_{\text{HRG}}(F_{0}) &= P_{\text{pQCD}}(F_{0} , B_{0}),
\label{eq: matching A} \\
	\frac{\mathrm{d} P_{\text{HRG}}(F)}{\mathrm{d} F} \bigg|_{F = F_{0}} &= \frac{\partial P_{\text{pQCD}} (F , B_{0})}{\partial F}
		\bigg|_{F = F_{0}}.
\label{eq: matching B}
\end{align}
The second of these equations amounts to assuming that the phase transition is of second order. Less restrictive schemes can be implemented
as well, such as truncating the HRG and pQCD equations of state far from the transition region and interpolating using the thermodynamic
constraint \eqref{eq: pressure increase} as well as other phenomenological assumptions (see Refs.~\cite{Kurkela:2014vha,Kojo:2014rca} for examples of this). By
varying $\overline{\Lambda}$ between $\pi T$ and $4 \pi T$ for the case $F = T$ and between $\mu / 2$ and $2 \mu$ in the case $F = \mu$,
\eqref{eq: matching A}-\eqref{eq: matching B} allows us to obtain a region of possible equations of state in the $(F , P)$ plane
for each theory.

\section{Results}
\label{sec: results}

In Figure~\ref{fig: Three Color Agree}, we overlay the bands for the pressure and trace anomaly $\epsilon-3 P$ (with $\epsilon$
the energy density) at $\mu = 0$ that we calculate in the
three-color, three-massless-quark case with lattice data from the Budapest--Marseille--Wuppertal Collaboration \cite{Borsanyi:2013bia}
and the HotQCD Collaboration \cite{Bazavov:2014pvz} in their respective regions of validity.  We observe  that
the lattice data agree reasonably well with the band resulting from our HRG+pQCD\ calculation, both for the pressure as well as for the
trace anomaly.

In Figure~\ref{fig: Overlay EoS}, we  show the HRG+pQCD\ pressure and trace-anomaly bands at $\mu = 0$
for all four theories with the $T$ axis scaled by the critical temperature $T_{c}$, which we define to be the
average of the matching temperatures for the upper and lower edge of our pQCD band to the HRG equation of state. $T_c$ should
thus be regarded as an estimate of the confinement-deconfinement critical temperature. We list the explicit
values obtained in HRG+pQCD\ in Table \ref{tab: Tc MuC}. We see that once the temperature axis has been scaled by $T_c$,
all the theories show similar behavior both for the pressure and trace anomaly, a phenomenon that is well-known from pure-gauge
theories \cite{Panero:2009tv}.  The differences at low temperatures are due to the different numbers of Nambu--Goldstone bosons
with zero quark mass in the two-color and four-color theories (see Section~\ref{sec: ChSB} or Appendix~\ref{sec: particle tables})
and the fact that in the real world there are only pseudo-Nambu--Goldstone bosons. We tested this by increasing the mass of the
lightest (now pseudo-) Nambu--Goldstone bosons, which qualitatively changed the shape of the pressure curves until they matched that
of the real-world, three-color theory.

In Figure~\ref{fig: Overlay EoS mu},
we show the pressure and trace-anomaly bands at $T = 0$ for all four theories with the $\mu$ axis scaled by the critical
chemical potential $\mu_{c}$, again, defined to be the average of the matching chemical potential of the upper and lower edge of our
pQCD band to the HRG equation of state. The value of $\mu_c$ should be regarded as an estimate for the confinement-deconfinement
transition, whereas the critical chemical potential for the onset transition would be given by the smallest value of
$m_{i} / r_{i}$, to use the notation of Section~\ref{sec: HRG}.  In the fundamental theories, this value of $m_{i} / r_{i}$
corresponds to the lightest baryon mass. Similar to the $\mu=0$ case, the $\mu\neq 0, T=0$ results show similar
trends when scaled appropriately. Again, the different behaviors at low $\mu/\mu_{c}$ are due to the fact that there are
Nambu--Goldstone bosons composed solely of quarks in the two-color fundamental and four-color antisymmetric theories. Again,
this was tested by increasing the masses of the lightest particles.

The values of $T_{c} /\sqrt{\sigma}$ and $\mu_{c} / \sqrt{\sigma}$ for our HRG+pQCD\ calculations are given in Table \ref{tab: Tc MuC}.
While the results suggests that the $T_c$ values for the different theories are within 20 percent of each other, the extracted $\mu_c$
values span a much broader range.

We stress that in the four-color antisymmetric
case with $\Lambda_{\overline{\text{MS}}} / \sqrt{\sigma}=0.723$, we were unable to carry out our matching procedure at $\mu = 0$ in the chiral limit.
We found that in this case, the HRG pressure rose too sharply and never intersected the pQCD pressure-band. Thus, we have
only plotted the $\Lambda_{\overline{\text{MS}}} / \sqrt{\sigma} = 0.527$ results for the four-color antisymmetric theory in our figures.
We feel this is justified for a few reasons. First of all, the values of $\mu_{c}$ are equal within uncertainties
for the two different values of $\Lambda_{\overline{\text{MS}}} / \sqrt{\sigma}$. Secondly, in the case where $\Lambda_{\overline{\text{MS}}} / \sqrt{\sigma} = 0.723$, we were able to carry out our HRG+pQCD\
matching procedure when we increased the mass of the lightest bosons (the pion mass). By varying the pion mass, we were able to
extrapolate to the chiral limit, obtaining a value of $\mu_{c} / \sqrt{\sigma} = 0.3$, which agrees with the value found for
$\Lambda_{\overline{\text{MS}}} / \sqrt{\sigma} = 0.527$. In light of this agreement, and in light of how the four-color antisymmetric theory was the only theory where
our matching was strained, we conjecture that the true value of $\Lambda_{\overline{\text{MS}}} / \sqrt{\sigma}$ in this case is closer to the pure-glue value than it
is in the real-world, three-color case. We point out that this prediction could be tested in future lattice-gauge-theory calculations.

Finally, we have calculated the speed of sound $c_{s}$ at $T = 0$ in all four QCD-like theories using our HRG+pQCD\ scheme, shown
in Figure~\ref{fig: Overlay SpdSnd}.  We note that in some cases, $c_s$ exceeds the speed of light, and thus these particular
matching results from HRG+pQCD\ should be considered unphysical (a standard constraint when using cold-nuclear-matter equations of
state). Nevertheless, our results indicate that it is generally possible to obtain physical equations of state wherein $c_{s}^{2}
> 1 / 3$ for all fundamental QCD-like theories. This finding could be of interest because restricting $c_{s}^2 < 1 / 3$ has
previously been noted to be in tension with astrophysical observations \cite{Bedaque:2014sqa}. Again, we point out that this
is a property which could be tested in future lattice-gauge-theory calculations.

\begin{figure}[h]
\centering
\includegraphics[width = 0.49\linewidth]{Figure_1a.pdf}
\includegraphics[width = 0.49\linewidth]{Figure_1b.pdf}
\caption{ Normalized pressure (left) and trace anomaly (right) at $\mu = 0$ for the
 three-color, three-massless-quark case from HRG+pQCD\ in comparison to lattice-QCD data from the Budapest--Marseille--Wuppertal
 Collaboration
 \cite{Borsanyi:2013bia} and the HotQCD Collaboration \cite{Bazavov:2014pvz}.}
\label{fig: Three Color Agree}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width = 0.49\linewidth]{Figure_2a.pdf}
\includegraphics[width = 0.49\linewidth]{Figure_2b.pdf}
\caption{Normalized pressure (left) and trace anomaly (right) at $\mu = 0$ for the
 two-color, three-color, four-color fundamental, and four-color antisymmetric theories in HRG+pQCD.  Note that the $T$ axis has been scaled by
 the critical temperature (see main text).}
\label{fig: Overlay EoS}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width = 0.49\linewidth]{Figure_3a.pdf}
\includegraphics[width = 0.49\linewidth]{Figure_3b.pdf}
\caption{Normalized pressure (left) and trace anomaly (right) at $T = 0$ for the
 two-color, three-color, four-color fundamental, and the four-color antisymmetric theories in HRG+pQCD.  Note that the $\mu$ axis
 has been scaled by the critical chemical potential (see main text).}
\label{fig: Overlay EoS mu}
\end{figure}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\em \textbf{Group}, \textbf{Representation}, $\bm{n_{f}}$ & $\,\bm{T_{c} / \sqrt{\sigma}}\,$ & $\,\bm{\mu_{c} / \sqrt{\sigma}}\,$ \\
\hline
SU(2), fundamental, 2 & 0.400 & 3.24 \\
\hline
SU(3), fundamental, 3 & 0.47 & 2.382 \\
\hline
SU(4), fundamental, 2 & 0.44 & 2.853 \\
\hline
SU(4), antisymmetric, 2 ($\Lambda_{\overline{\text{MS}}} / \sqrt{\sigma} = 0.527$) & 0.29 & 5.09 \\
\hline
SU(4), antisymmetric, 2 ($\Lambda_{\overline{\text{MS}}} / \sqrt{\sigma} = 0.723$) & no matching & 5.0 \\
\hline
\end{tabular}
\end{center}
\caption{The ratios $T_{c} / \sqrt{\sigma}$ and $\mu_{c} / \sqrt{\sigma}$ for the theories analyzed in this paper. Errors are
given by the number of significant figures.}
\label{tab: Tc MuC}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[width = 0.49\linewidth]{Figure_4.pdf}
\caption{Deconfinement-transition temperature $T_{c}$ as a function of pion mass for the four-color, antisymmetric theory in
 the $\mu = 0$, $\Lambda_{\overline{\text{MS}}} / \sqrt{\sigma} = 0.723$ case. The straight line is a fit to the results
 where matching could be performed, and defines the extrapolation to the chiral limit (see main text).}
\label{fig: MassExtrap}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width = 0.49\linewidth]{Figure_5.pdf}
\caption{The speed of sound squared at $T = 0$ for the
 two-color, three-color, four-color fundamental, and the four-color antisymmetric theories in HRG+pQCD. Note that the $\mu$ axis
 has been scaled by the critical chemical potential (see main text). }
\label{fig: Overlay SpdSnd}
\end{figure}

\FloatBarrier

\section{Conclusions}

We have calculated the equation of state at non-zero temperatures and densities in a first-principles approach: by matching physics
from the hadron resonance gas at low energies to perturbative QCD at high energies for two-,\linebreak three-, and four-color `QCD'.
In particular, our work provides predictions for results in future lattice studies at zero temperature and non-zero chemical potential
for two-color QCD with two fundamental fermions and four-color QCD with two flavors of fermions in the two-index, antisymmetric
representation. While some aspects of our study are systematically improvable, we expect the current HRG+pQCD\ results to be sufficiently
robust that a direct comparison with future lattice-QCD studies in the two- and four-color cases could validate or rule out the HRG+pQCD\
method, depending on the quantitative agreement. In the case of agreement, one could thus also reasonably expect HRG+pQCD\ results to be
quantitatively accurate in the physically-relevant, three-color-QCD case.

To make our results accessible, we have made them electronically available \cite{Paul:website}.

\acknowledgements

We thank Gert Aarts,  Tom DeGrand, Simon Hands, Yuzhi Liu, Marco Panero, Andreas Schmitt, and  Aleksi Vuorinen for helpful discussions
and suggestions.  This work was funded by the Department of Energy, DoE award No.\ DE-SC0008132.

redpajama_set_name: RedPajamaArXiv


text:
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Tue, April 21, 2020, 8:00 PM PDT
Doors at 7:00 PM
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Adelmo Fornaciari was born in Roncocesi on 25th September 1955. He later moved to Forte dei Marmi in Tuscany where, between 1970 & 1978 he formed his first R&B bands: 'Le Nuove Luci', 'Sugar & Daniel' and 'Sugar & Candies'. By that time he was already known as Zucchero, a nickname given to him by one of his school teachers.
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'Baila (sexy thing)' the first single from the new album went to #1 in Italy upon release (July 2001) and also topped the charts in Switzerland (GOLD) and in Spain (for the first time ever!). The new album 'Shake' recorded in the Sausalito US and Real World UK studios was completed in Italy and released in September in Europe. It was certified IFPI PLATINUM for sales exceeding 1 million copies in Europe alone. The album includes an incredible duet with John Lee Hooker to whom the record was dedicated. A long SOLD OUT SHAKE TOUR followed.
In November 2003 Zucchero was invited by Brian May and Dave Stewart to take part in the mega-concert "46664 give 1 minute of your life to AIDS" an event organised by Dave Stewart and Queen to support Nelson Mandela's campaign. Zucchero sang "Everybody's got to learn sometime" accompanied by Brian May, Roger Taylor, Sharon Corr and then Queen's marvellous medley "I want it all" and "We are the Champions" with Anastasia. And finally, he joined all the other artists for the grand finale to sing "46664 (long walk of freedom)."
Zucchero was also the only 46664 Ambassador to appear at the "Festival 46664 Comunidad de Madrid" event which took place in the Spanish capital. He also performed at the "46664 Arctic" concert which took place in Tromsø, Norway in June 2004.
Zucchero's 2004 international hit album "Zu & Co." featured duets with many music greats, including Miles Davis, Paul Young, Ronan Keating, Sheryl Crow, The Cranberries's Dolores O'Riordan, B.B. King, John Lee Hooker, Maná, Brian May, and Solomon Burke. The international launch of the album took place at London's Royal Albert Hall with a special concert featuring Brian May, Mousse T., Jenny Bae, Tina Arena, Fher from Mana, Paul Young, Dolores O'Riordan, Solomon Burke, Cheb Mami, Ronan Keating, Luciano Pavarotti, Eric Clapton, and Irene Fornaciari. The album sold millions of copies throughout Europe and the rest of the world and entered the Billboard charts after its release in the U.S. in the summer of 2005.
In July 2005, Zucchero took part in the Live 8 concerts in both Rome and Paris.
The next album to be released in September 2006 was entitled "Fly". Produced by Don Was, it includes the singles "Bacco perbacco", "Cuba Libre" and "Occhi" and features collaborations with artists such as Ivano Fossati and Jovanotti. The album track "L'amore è nell'aria" ("Love Is In The Air"), borrows instrumental backing from the b-side "Broken" by UK rock band Feeder, but it is a new recording based on the Feeder original rather than a sample taken directly from the recording. The album immediately entered all European charts debuting at No. 1 in many countries.
Sam Moore's album "Overnight Sensational" was released including a duet of "You Are So Beautiful" by Zucchero and the legendary soul man, accompanied by Billy Preston (Hammond organ), Eric Clapton (guitar) and Robert Randolph (pedal steel). This version of "You Are So Beautiful" from the Sam Moore album was nominated for the 49th Grammy Awards in the "Traditional R & B Vocal" category.
The beginning of 2007 saw Zucchero promoting "Fly", throughout Europe guesting on major television and radio shows, followed by a worldwide tour which opened in Paris and ran until the end of the year, playing more than 150 concerts (almost all of them sold-out) all over Europe, Canada, the US and Latin America. The tour was a huge success both in terms of audience response and critical acclaim, and Zucchero played legendary places and venues like the Opera in Paris, the Arena in Verona, the Valle dei Templi in Sicily, the Carnegie Hall in New York, the House of Blues in Los Angeles, and many others…
On November 26th "All the best", a collection of Zucchero Sugar Fornaciari's greatest hits, was released simultaneuosly all over the world.
Following this release, Zucchero went on a 2 year worldwide tour, playing more then 200 gigs in 5 continents.
In November 2008, the album "Live in Italy" was released. It was a double package, 2 CDs + 2 DVDs of recorded and filmed footage capturing the magic atmosphere of the concerts held in Milan's San Siro Stadium and the Arena in Verona. Again, the album was a huge, worldwide success.
Then in 2009, following the terrible earthquake which shook the central Italian town of L'Aquila on April 24th, Zucchero and other major Italian artists joined forces to record a single called "Domani 21-04-09" (Tomorrow 21-04-09), with all proceeds from the sale of this single going to the rebuilding and restoration of the "Alfredo Casella" Academy of Music and the Stabile Theatre of Abruzzo in L'Aquila.
At the VII Ischia Global Film & Music Fest Zucchero and Paul Haggis presented Sting with an award, and later on during the gala Zucchero gave a surprize performance with Sting and Andrea Bocelli. On the second day of the event it was Zucchero's turn to receive an award, presented to him by Sting and his wife Trudie Styler.
Zucchero was invited to the festivities for Nelson Mandela's 91st birthday and to take part in the "46664 Make An Imprint" concert held at New York's Radio City Music Hall. Other artists performing were Aretha Franklin, Stevie Wonder, Will.I.Am, Josh Groban, Carla Bruni, Dave Stewart, Wyclef Jean, Alicia Keys, Angelique Kidjo, Whoopi Goldberg, Forest Whitaker, Morgan Freeman and many more. Zucchero, who was introduced to the audience as "the biggest rockstar in Italy", sang "You Are So Beautiful" and then joined the other stars to sing the chorus accompanying Stevie Wonder's rendition of "Happy Birthday".
A terrible disaster at Viareggio railway station in Tuscany (June 30th 2009) led Zucchero and promoter Mimmo d'Alessandro to organize a "Concert For Viareggio – Lest We Forget". Accompanied by a house band formed by Massimo Marcolini (musical director and guiitarist), Andrea Cozzani (bass), Guido Carli (drums), Giuseppe Zito (keyboards), Eros Cristiani (keyboards), Joe Damiani (percussion), Sara Grimaldi (vocalist), Stefania Orrico (vocalist), Mario Schilirò (guitar), Derek Wilson (drums), James Thompson (sax) and Massimo Greco (trumpet). Zucchero sang "Diavolo In Me" with Solomon Burke, "Baila (Sexy Thing)", "Dio è Morto" with Bebbe Carletti from The Nomadi, "Dune Mosse" with Jeff Beck on guitar, "Every Breath You Take" with Sting, "Fragile" with Sting and Jeff Beck, "Così Celeste" with the Puccini Festival Choir and "Miserere" performed as a virtual duet with Luciano Pavarotti on the mega screens.
November 2010 saw the release of 'Chocabeck', already hailed as one of the best, if not the best album ever written by Zucchero. Recorded across the Globe including stints in Los Angeles, Bolgheri, and London the album featured production from some of music's best known alumni including Don Was (The Rolling Stones, Bob Dylan, and Roy Orbison) and Brendan O'Brien (AC/DC, Bruce Springsteen, Pearl Jam). Additional production also came from people who were at the helm of the incredibly successful last Coldplay record (Viva la Vida or Death and All His Friends) messrs Jon Hopkins and Davide Rossi. Featuring some of Zucchero's most poignant and touching lyrics he also managed to enlist the assistance of his good friends Bono (who the lyrics for the track 'Someone Else's Tears' ((Il Suono Della Domenica)) and Brian Wilson (who recorded backing vocals for the title track). With this record Zucchero opened up a new chapter in his illustrious and coveted career.
'Chocabeck' was also the name of Zucchero's acclaimed 'Chocabeck World Tour' that saw him to play more than one hundred concerts in Europe, North America and Latin America.
In summer 2012 Zucchero recorded in Havana 'La Sesión Cubana', an album that featured collaborations from some of Cuba's most important musicians. The album was preceded by the first worldwide single "Guantanamera (Guajira)", which was produced by Zucchero and Don Was, and mixed by Michael Brauer. 'La sesión cubana' combined seven brand new songs with six newly arranged Zucchero hits ("Baila", "Un Kilo", "Così Celeste", "Cuba Libre", "Everybody's got to learn sometime" and "L'Urlo"). The new recordings consisted of the first single "Guantanamera (Guajira)", adapted by Zucchero into Italian, original compositions "Love is all around" and "Sabor a ti", and covers "Never is a moment", "Nena", "Pana" and "Ave Maria no morro". "Nena" and "Pana" are in Spanish, the second one a duet with Bebe, and "Ave Maria no morro", sung in Portuguese, was a duet with one of the most prestigious singers of contemporary Brazilian music, Djavan.
Zucchero described his album in his own words: My music, as everybody knows, comes from blues, from soul, from gospel. This project is a trip from New Orleans to Cuba via Mexicali. Latin, Cuban, tex-mex sounds and rhythms melt together dancing. "Hasta l'asta siempre".
The album was launched with a mega concert event at the University of Arts of Cuba with the collaboration of the Cuban Culture minister, the Cuban Institute of Music and the Italian embassy in Havana. Zucchero performed in front of 80,000 people.
In 2013, Zucchero was invited to take part at the "Los Angeles, Italia – Film, Fashion & Art Fest" the night before the annual Academy Awards. At the Grammy Museum, he received the "L.A. Italian Excellence Award 2013'' . The night after, at the Chinese Theatre there was a the premiere of the recording of the Havana concert that took place in December.
In the same year he brought his tour to Australia. After 3 sold-out dates at the Arena of Verona, La Sesiòn Cubana World Tour 2013 arrived in Europe with more than 60 dates including a 4th Arena di Verona show.
Zucchero took part at the flash-mob wanted and organized by Bono in London called "Agit8", where he sang "Love is all around" transformed for the occasion to "Love is still around" and then, by popular demand, "Everybody's got to learn something".
The Italian and International version of the cd+dvd live "Una rosa blanca" that contained the recording of legendary Cuba concert was released in December 2013.
The following year Zucchero embarked on his Americana Tour 2014, which lasted 2 months and consisted of more than 40 dates around the United States and Canada. During the triumphal Madison Square Garden concert, Zucchero invited on stage Sting, Sam Moore, Fher, of Mana, Elisa, Jovanotti, Fiorella Mannoia, Chris Botti, Monica Messinger, Andrea Griminelli and Irene Fornaciari.
During the Americana Tour 2014, Randy Jackson, Corrado Rustici, Don Was, Jimmy LaFave and Andy J. Forest all made guest appearances.
Zucchero was chosen as the headliner of th sold out 18 city German "Night of the promsms" arena tour.
April 29th 2016 saw the release of Black Cat (Universal Music), a new studio album which Zucchero presented live on the Black Cat Tour 2016. The tour kicked off at the Verona Arena on September 16th with the extraordinary "10 + 1" shows, and between October 1st and November 15th 2016 it became the "Black Cat World Tour", enjoying enormous success among an international audience. On November 25th the "BLACK CAT DELUXE EDITION" was released, a box set containing 2 CDs and 1 DVD to mark the great success of ZUCCHERO and his latest studio album "Black Cat".
In February 2017 Zucchero was, for the first time, a special guest at the Festival of Sanremo during the final evening of Italy's most famous song contest. In March, the "Black Cat World Tour" began again, this time taking in North America, Australia, New Zealand, Japan, Europe and South America, as Zucchero presented to the world the songs from "Black Cat" as well as all his greatest hits. In Italy, the "Black Cat World Tour" reached the city of Verona, where Zucchero set an extraordinary record of completing 22 concerts at the Verona Arena within a 12-month period (September 16-17-18-20-21-23-24-25-26-27-28 2016, May 1-2-3-4-5 2017 and September 20-21,22,23,24,25 2017).
On the "Black Cat World Tour", Zucchero played to over 1 million people in 611 cities, 67 countries and 5 continents.
November 2017 saw the release of "WANTED - THE BEST COLLECTION", celebrating the best of over 30 years of music and featuring live material from the Verona Arena shows, a docufilm with backstage footage as well as interviews, other special content and three brand new songs.
In February 2018 Zucchero played some of the most important indoor arenas in Italy during his "WANTED ITALIAN TOUR 2018", and all ten dates were sold out! Still in Italy, on July 3rd and 4th, Zucchero performed for the first time in Piazza San Marco, the first pop/rock artist to do so for seven years. The summer then took Zucchero on an international tour entitled "Summer Events and Festivals 2018", and he played at some of the most important festivals in Europe, including "British Summer Time" in Hyde Park, London, the Cornbury Music Festival in Oxfordshire, the Bospop Festival in the Netherlands, as well as events in Croatia, France, Bulgaria, Macedonia, Germany, Switzerland and Austria, in other words, pretty much the whole of Europe.
In October, Zucchero was awarded the TENCO ARTISTA PRIZE 2018 (the most prestigious recognition for Italian singer-songwriters) during the 42nd edition of the TENCO AWARDS.
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