
\subsection{Meson spectrum}

The simplest question one may ask regarding hadrons is how the spectrum depends on the  number of colours. As already discussed, all mesons are expected to become stable particles at large $N_c$, and thus they may be treated as asymptotic states, rather than resonances. 

\begin{figure}[h!]
\centering 
\includegraphics[width=0.5\textwidth]{plots/mesons.pdf}
\caption{Large $N_c$ meson spectrum in different channels, with $J^{PC}$ as provided in the x-labels. We compare the results from standard large-$N_c$ approach from Ref. \cite{Bali:2013kia}, to the ones obtained using the Twisted Eguchi-Kawai reduction~\cite{Perez:2020vbn}. In the case of Ref. \cite{Perez:2020vbn}, the smaller errorbars are statistical, whereas the larger ones include the systematics of chiral and continuum extrapolation. \label{fig:mesons}}
\end{figure}

The most extensive study of the meson spectrum was performed in Ref. \cite{Bali:2013kia}. It was done using the quenched approximation at a fixed lattice spacing. Although it is known that quenching alters some $1/N_c$ corrections, the strict large $N_c$ limit is the same for both the dynamical and quenched theory (with fundamental fermions). More recently, large $N_c$ results from reduced models \cite{Perez:2020fqn,Perez:2020vbn} were made public. We compare the results of the two approaches in Fig. \ref{fig:mesons}. As can be seen, there is a reasonable overall agreement, with differences of at most $2\sigma$. It must be noted that while Ref. \cite{Bali:2013kia} has better statistical accuracy, the results may be affected by significant discretization errors\footnote{A continuum extrapolation is available for $SU(7)$ \cite{Bali:2013fya}.}. In contrast, those of Ref. \cite{Perez:2020vbn} are in the continuum limit.



Another interesting observation is the one of Ref. \cite{Nogradi:2019iek}. The authors found that the ration $M_\rho/F_\pi$ in the chiral limit seemed independent on $N_f$ in $N_c=3$:
\begin{equation}
\frac{M_\rho}{F_\pi} \Bigg \rvert^{N_f=2-6}_{N_c=3} = 7.95(15),
\end{equation}
whereas at large $N_c$, the following result is found \cite{Bali:2013kia}:
\begin{equation}
\sqrt{\frac{N_c}{3}} \frac{M_\rho}{F_\pi} \Bigg \rvert_{N_c\to \infty} = 7.08(10). \label{eq:rhoNc}
\end{equation}
The comparison of the previous equation indicates that both, $1/N^2_c$ and $N_f/N_c$ effects are rather small in this ratio. This is consistent with the result of Ref.~\cite{Nogradi:2019auv}, where the authors summarized the dependence on the gauge group and fermionic representation of the same quantity using all existing lattice data~\cite{Appelquist:2018yqe,Fodor:2016pls,Bali:2013kia,Ayyar:2017qdf,Drach:2017btk,Amato:2018nvj,Bennett:2019cxd,Bennett:2019jzz}. They pointed out that it is almost constant up to a trivial factor that depends on the dimensionality of the fermion fields---$\sqrt{N_c}$ for $SU(N_c)$. We should also mention that the result in eq.~(\ref{eq:rhoNc}) is in the  ballpark of predictions from a resonance chiral theory \cite{Ledwig:2014cla}.


\begin{figure*}[th]
\begin{subfigure}{.5\textwidth}
  \centering
 
  \includegraphics[width=1\linewidth]{plots/baryons.pdf}  
  \caption{}
  \label{fig:baryonsm0}
\end{subfigure} 
\begin{subfigure}{.5\textwidth}
  \centering
 
  \includegraphics[width=1\linewidth]{plots/baryons2.pdf}  
  \caption{}
  \label{fig:baryonsB}
\end{subfigure}
\caption{$1/N_c$-scaling of the constants $r_1 m_0$ (top) and $r_1 B$ (bottom) in eq.~(\ref{eq:baryonnf2}) for the quenched data in Ref. \cite{DeGrand:2013nna}, and $N_f=2$ in Ref. \cite{DeGrand:2016pur}. The simulations at different $N_c$ have been matched at approximately fixed $M_\pi / M_\rho$. } 
\label{fig:baryons}
\end{figure*}

\subsection{Baryons}



A series of publications have studied the scaling of baryon masses with the number of colours: first, in the quenched approximation~\cite{DeGrand:2012hd,DeGrand:2013nna,Cordon:2014sda}, and more recently with dynamical quarks ($N_f=2$)~\cite{DeGrand:2016pur}. These have been carried out at a single lattice spacing. In a generic $SU(N_c)$ theory, the spin of the baryon can take half-integer for odd $N_c$ (fermionic baryons, as in QCD), or integer values for even $N_c$ (bosonic baryons). 


It has been long known~\cite{Jenkins:1993zu} that baryon masses depend on their angular momentum. This is the so-called hyperfine splitting, which for the case of two degenerate quarks takes the form:
\begin{equation}
M_B(N_c, J) = N_c m_0(N_c) + \frac{J(J+1)}{N_c} B +\cdots, \label{eq:baryonnf2}
\end{equation}
with $m_0, B$ being $O(N_c^0)$ constants. Note that they can have a subleading $N_c$ dependence, for instance, for $m_0$:
\begin{equation}
m_0 = m_0^{(0)} + \frac{m_0^{(1)} }{N_c} + \hdots,
\end{equation}
and similarly for $B$. Generalizations of the eq.~(\ref{eq:baryonnf2}) that incorporate the strange quark also exist in the literature~\cite{Dai:1995zg,Jenkins:1995td}.  The constants in eq.~(\ref{eq:baryonnf2}) can be isolated by taking appropriate combinations, for instance:
\begin{align}
\begin{split}
 m_0 &=  \frac{1}{4 N_c} \left[  5 M(N_c, 1/2) - M(N_c, 3/2)     \right],  \\
B &= \frac{3}{N_c} \left[ M(N_c, 1/2) -M(N_c, 3/2)     \right].
\end{split}
\end{align}




The general conclusions of Refs. \cite{DeGrand:2012hd,DeGrand:2013nna} is that the qualitative expectations of large $N_c$ are satisfied by the data, including those of the $SU(3)$-flavour breaking due to the strange quark. In Ref. \cite{Cordon:2014sda}, combined chiral and $N_c$ fits of the mass and hyperfine splitting were performed, using a consistent expansion in Baryon Chiral Perturbation Theory. That study was able to constrain some of the subleading low-energy constants (LECs).

As an example, we show in Fig. \ref{fig:baryons} the scaling of $m_0$ and $B$ with the number of colours for the quenched and dynamical baryons at approximately fixed Sommer scale $r_1$ \cite{Bernard:2000gd} and $M_\pi/M_\rho$. As can be seen from the plots, quenched and $N_f=2$ results for $r_1 m_0$ agree at fixed $N_c$, and in the large $N_c$ extrapolation. For $r_1 B$, uncertainties are larger, and some quenching effects may be appreciated.  We stress that relative discretization errors may be large, and so, this comparison must be taken as qualitative.
 


It is also worth mentioning that there have been lattice studies of baryons in the context of BSM physics. These have focused on studying a $SU(4)$ gauge theory with both two fundamental and two sextet fermions~\cite{Appelquist:2014jch,DeGrand:2015lna,Ayyar:2018zuk}. This model represents a slight simplification of the asymptotically-free composite-Higgs Ferretti model \cite{Ferretti:2013kya,Ferretti:2014qta}.




\subsection{Chiral and $N_c$ dependence of light meson observables}


In Ref. \cite{Hernandez:2019qed}, a first-principles calculation of the $N_c$ scaling of the low-energy couplings (LECs) of the chiral Lagrangian has been presented. The LECs are extracted from the chiral dependence of meson masses and decay constants. The lattice setup of this work is $N_f=4$, with $O(a)$-improved Wilson fermions in the sea, and mixed action with twisted mass in the valence. This has the advantage of automatic $O(a)$ improvement, avoiding the need of non-perturbative renormalization factors.


First the authors consider each value of $N_c$ separately. They perform a chiral fit of the data points to extract the LO and NLO low-energy constants appearing in the NLO chiral predictions \cite{Bijnens1,Bijnens2,Bijnens3} of $F_\pi$ and $M_\pi$: 
\begin{align}
F_\pi &= {F} \Bigg[1   - {\frac{N_f}{2} \frac{M_\pi^2}{(4 \pi F_\pi)^2}\log \frac{M_\pi^2}{\mu^2}}     +4 \frac{M_\pi^2}{F_\pi^2} L_F\Bigg], 
\label{eq:Fpi1}
 \\
\frac{M^2_\pi}{\mu} &= 2B \Bigg[1  + {\frac{1}{N_f} \frac{M_\pi^2}{(4 \pi F_\pi)^2}\log \frac{M_\pi^2}{\mu^2}}  +8\frac{M_\pi^2}{F_\pi^2} L_M\Bigg]. \label{eq:Mpi1} 
\end{align} 
Here, $B$ and $F$ are the chiral condensate\footnote{We note that in the setup of this work, the value of the bare twisted mass can be used for chiral fits as quark mass. The resulting $B$ is therefore also bare.} and decay constant in the chiral limit; $\mu$ is the quark mass, and $L_M$, $L_F$ are combinations the standard LEC:
\begin{equation}
L_F = L_5^r  +  N_f L_4^r , \ \ \ L_M = 2L_8^r - L_5^r + N_f (2 L_6^r  - L_4^r ).
\end{equation}
The scaling with the number of colours of these quantities is known: $F^2, L_5$ and $L_8$ are $O(N_c)$, while $B, L_4$ and $L_6$ are $O(N_c^0)$. The results of these fits are shown in Figs. \ref{fig:fpiscaling} and \ref{fig:massscaling}. One can see that the scaling is well described by leading and subleading $N_c$ corrections for $N_c=4-6$, while there seems to be significant $1/N_c^2$ corrections for $N_c=3$ in the case of  $F$ and $L_F$. In the case of $B$ and $L_M$, there is no sign of $1/N_c^2$ effects. 





As explained in sec.~\ref{subsec:chpt}, the chiral regime at large $N_c$ requires the addition of the $\eta'$, and a combined power counting of the usual momentum/mass expansion with that in $1/N_c$. Using the standard power-counting \cite{DiVecchia:1980yfw,PhysRevD.21.3388,Witten:1980sp,Kawarabayashi:1980dp,HerreraSiklody:1996pm,Kaiser:2000gs}
\begin{equation}
\mathcal{O}(\delta) \sim  \mathcal{O}(m_q) \sim  \mathcal{O}(p^2) \sim  \mathcal{O}(M_\pi^2) \sim  \mathcal{O}(1/N_c),
\end{equation}
the $\mathcal O (\delta^2)$ predictions  are\footnote{A technical detail in chiral fits is the choice of the renormalization scale. In Ref. \cite{Hernandez:2019qed}, the choice is $\mu^2 = \frac{3}{N_c} (4 \pi F_\pi)^2 $.}:
\begin{eqnarray}
&  F_\pi &=\sqrt{N_c} \left( {F}_0+ {{F}_1\over N_c} + {{ F}_2 \over N_c^2} \right) \Bigg[1    - {\frac{N_f}{2} \frac{M_\pi^2}{(4 \pi F_\pi)^2}\log \frac{M_\pi^2}{\mu^2}} \nonumber \\
&+&4 \frac{M_\pi^2}{F_\pi^2} \Big( N_c {L}_F^{(0)} + {L}_F^{(1)} \Big) + N_c^2 K_F^{(0)} \left({M_\pi^2\over F_\pi^2}\right)^2 \\&+& \ {\mathcal O}(\delta^3)  \Bigg] , \nonumber
\label{eq:Fpi}
\end{eqnarray}
and 
\begin{eqnarray}
&M^2_\pi &= 2  m \left( { B}_0+ {{ B}_1\over N_c} + {{B}_2 \over N_c^2} \right)  \Bigg[ \nonumber  \\
&1&+ {\frac{1}{N_f} \frac{M_\pi^2}{(4 \pi F_\pi)^2}\log \frac{M_\pi^2}{\mu^2}} - {\frac{1}{N_f} \frac{M_{\eta'}^2}{(4 \pi F_\pi)^2}\log \frac{M_{\eta'}^2}{\mu^2}}\\ & +&8\frac{M_\pi^2}{F_\pi^2} \Big( N_c {L}_M^{(0)} + {L}_M^{(1)} \Big) + N_c^2 K_M^{(0)} \left({M_\pi^2\over F_\pi^2}\right)^2 +{\mathcal O}(\delta^3) \Bigg] \nonumber  .\label{eq:Mpi}
\end{eqnarray}
Here, $F_i, B_i, L_M^{(i)}$ and $L_F^{(i)}$ are the coefficients of the $1/N_c$ expansion of the corresponding LECs. $K_{F,M}$ are combinations of LECs that contribute at  $\mathcal{O}(M_\pi^4)$.\footnote{For further details see \cite{Guo:2015xva}. }. A global chiral fit of all $N_c$ data to these expressions is shown in Fig. \ref{fig:chiral2}. The ChPT predictions seem to describe data well. 
\begin{figure}[ht]
\begin{subfigure}{.5\textwidth}
  \centering
 
  \includegraphics[width=.999\linewidth]{plots/fpiscaling.pdf}  
  \caption{LECs of the decay constant.}
  \label{fig:fpiscaling}
\end{subfigure} 
\begin{subfigure}{.5\textwidth}
  \centering
 
  \includegraphics[width=.999\linewidth]{plots/mpiscaling.pdf}  
  \caption{LECs of the meson mass.}
  \label{fig:massscaling}
\end{subfigure}
\caption{$1/N_c$ scaling of the LO and NLO low-energy constants extracted from fits to eqs. ~(\ref{eq:Fpi1}) and (\ref{eq:Mpi1}) at a fixed value of $N_c$. Source: Ref. \cite{Hernandez:2019qed}.}  
\label{fig:chiral1}
\end{figure}

\begin{figure*}[ht]
\begin{subfigure}{.5\textwidth}
  \centering
 
  \includegraphics[width=.999\linewidth]{plots/fpinew.pdf}  
  \caption{}
  \label{fig:fpichiral}
\end{subfigure} 
\begin{subfigure}{.5\textwidth}
  \centering
 
  \includegraphics[width=.999\linewidth]{plots/mpinew.pdf}  
  \caption{}
  \label{fig:mpichiral}
\end{subfigure}
\caption{Simultaneous chiral and $N_c$ fits of the decay constant (left) and the pion mass (right). Bootstrap samples are shown as ``clouds'' around the central value. The fit equations are linearized versions of eqs.~(\ref{eq:Mpi}) and (\ref{eq:Fpi}). More detail on the fitting strategy is give in the original article, Ref. \cite{Hernandez:2019qed}.  } 
\label{fig:chiral2}
\end{figure*}

An interesting observation of Ref. \cite{Hernandez:2019qed} is that by studying the first coefficients in the $1/N_c$ expansion, one can infer the values of some quantities at different value of $N_f$, since the leading corrections come from fermion loops and are therefore of the form $N_f/N_c$ (the non-planar gluonic corrections start at $1/N_c^2$). The simplest example is the decay constant in the chiral limit, whose leading $N_f,N_c$ dependence is:
\begin{equation}
\frac{F}{\sqrt{N_c}} = F_0 +  \widetilde F_1 \frac{N_f}{N_c} + \cdots.
\end{equation}
Since $\widetilde F_1$ is trivially related by a factor $N_f$ to one of the fit parameters in eq.~(\ref{eq:Fpi}), the authors are able to quote inferred results for $N_f=2$ and $3$:
\begin{align}
\begin{split}
& F^{N_c=3, N_f=2} = 81(7)  \text{ MeV},  \\
& F^{N_c=3, N_f=3} = 68(7)  \text{ MeV},\label{eq:FNf2}
\end{split}
\end{align}
which are in good agreement with phenomenological and lattice determinations---see Ref. \cite{Aoki:2019cca}. The same strategy is followed for the LECs, and the chiral condensate.

We should also mention that the chiral behaviour of $F_\pi$ in $N_f=2$ simulations (and the equivalent for vector meson) was also studied at in Ref. \cite{DeGrand:2016pur}, albeit no chiral fits were performed.

Another quantity that may be explored are meson scattering amplitudes. Although they vanish in the exact large $N_c$ limit, it is interesting to study the subleading $N_c$ behaviour, and the interplay with the quark mass dependence. For instance, the isospin-2 leading order ChPT prediction is\footnote{For the scattering length, we are using the convention $k \cot \delta_0 = -1/a_0 + O(k^2)$.}
\begin{equation}
M_\pi a_0^{I=2} = \frac{M_\pi^2}{16 \pi F_\pi^2} \sim O\left(\frac{1}{N_c} \right), \label{eq:chpta0}
\end{equation} 
that is consistent with the $N_c$-scaling discussed in sec.~\ref{subsec:chpt}.

 In finite volume, scattering amplitudes may be obtained with the Lüscher formalism~\cite{Luscher:1986pf}, and generalizations thereof (see Ref. \cite{Briceno:2017max} for a review). In the so-called threshold (or $1/L$) expansion, valid for weakly interacting systems, it is possible to relate the finite-volume energy shift of the two-particle ground state to the scattering length by to the scattering length, see Ref. \cite{Luscher:1986pf}.


Recently,  preliminary results   for the isospin-2 scattering length at $N_c=3-6$ have been presented~\cite{Romero-Lopez:2019gqt}. As $N_c$ grows, two different effects compete. The signal for the energy shift becomes weaker as the scattering length decreases, but the statistical error gets reduced as a result of the increase of gluonic degrees of freedom. In Ref. ~\cite{Romero-Lopez:2019gqt}  $a_0^{I=2}$ at $N_c>3$ was obtained with good accuracy, and a remarkable agreement with leading order ChPT [see eq. (\ref{eq:chpta0})] was found. 

Additional channels, such as the isospin-0 ($\sigma$) channel may represent an interesting problem to study in the large $N_c$ context. It has been argued that the $N_c$-scaling may shed light on the nature of the $\sigma$ resonance~\cite{Pelaez:2010er,Nebreda:2011cp,Bernard:2010fp,RuizdeElvira:2017aet}. Another very interesting application would be the study of tetraquarks with different flavour content in the planar limit \cite{PhysRevLett.110.261601,PhysRevD.88.036016}. 

Finally, we should mention the results for the $\rho$\footnote{The $\rho$ resonance has been treated as a stable state in Refs. \cite{Bali:2013kia,Nogradi:2019iek,DeGrand:2016pur}. This is rigorous for heavier pion masses, but implies systematic errors for lighter pions.}  and the isospin-2 channels in $SU(2)$\cite{Janowski:2019svg,Arthur:2014zda}. Note that in this case the symmetry breaking pattern for $N_c=2$ is different than the one in $N_c>3$, and so effective field theory expectations differ.




\subsection{Topological susceptibility with dynamical quarks }

The topological susceptibility is an observable that has a very different behaviour in the pure Yang-Mills theory and in QCD with light quarks. In the former, discussed in sec. \ref{subsec:topsus1}, $\chi_{YM}$ depends only weakly in the number of colours; whereas in the latter, it vanishes in the chiral limit for all values of $N_c$.  It has been predicted \cite{DiVecchia:1980yfw,Leutwyler:1992yt,Crewther:1977ce} that the topological susceptibility mass-dependence is as follows:
\begin{equation}
\frac{1}{\chi(M_\pi)} = \frac{2N_f}{F^2_\pi M_\pi^2 } + \frac{1}{\chi_{YM}}. \label{eq:topsusdyn}
\end{equation}
In the previous equation, the pure gauge (quenched) limit is achieved by letting $M_\pi \to \infty$, whereas at small pion mass, one simply has $\chi(M_\pi) \propto M_\pi^2$.

\begin{figure}[h!]
\centering 
\includegraphics[width=0.5\textwidth]{plots/topsusdyn.pdf}
\caption{Topological susceptibility as a function of the pion mass in $N_f=2$ QCD. The solid line is the functional form in eq.~(\ref{eq:topsusdyn}), using the continuum results of Refs. \cite{Ce:2015qha,Bruno:2014ova}. The dashed lines are the expectations in the two regimes (quenched and light pions). The various coloured points are the results of Ref. \cite{DeGrand:2020utq} with $N_c=3-5$. \label{fig:topsusdyn}}
\end{figure}


The topological susceptibility with dynamical fermions has been studied for $N_c=3$ \cite{Durr:2001ty,Chiu:2011dz,Bruno:2014ova,Aoki:2007pw}.  The most recent result \cite{Bruno:2014ova}, includes a continuum limit extrapolation. In the present year, a ``pilot study'' with varying $N_c=3-5$~\cite{DeGrand:2020utq} was also published.

Combining the results in the Yang-Mills theory \cite{Ce:2015qha}, with the $N_f=2$ ones \cite{Bruno:2014ova} one can produce an interpolating curve between the two regimes based on eq.~(\ref{eq:topsusdyn}). This can be compared to the results of Ref.~\cite{DeGrand:2020utq} with $N_c=3-5$. As can be seen, qualitative (but not quantitative) agreement is observed. One possibility to explain the disagreement are discrezation effects, which were found in Ref. \cite{Bruno:2014ova}  to be relevant with the Wilson fermionic action. Finite volume effects may also be significant in Ref.~\cite{DeGrand:2020utq}.










\subsection{The $\Delta I =1/2$ rule}




In the last few years, impressive progress has been achieved in the lattice determination of the $K\rightarrow \pi\pi$ amplitudes, and the related  CP violation observable $\epsilon'/\epsilon$~\cite{Boyle:2012ys,Bai:2015nea,Blum:2015ywa,Ishizuka:2018qbn,Abbott:2020hxn}.  The $\Delta I=1/2$ enhancement seems to be reproduced by  the latest simulations at the physical point in $2+1$ simulations with a heavy charm \cite{Abbott:2020hxn}. 

In an earlier work \cite{Boyle:2012ys}, an analysis of the various contributions to $K \to \pi\pi$   suggested that the main source of the enhancement comes from the current-current operators, eqs.~(\ref{eq:currento}). It emerges from a strong cancellation of the isospin-2 amplitude, as a result of a negative relative sign between the colour-connected and colour-disconnected contractions, which scale differently in large $N_c$. The scaling in $N_c$ of these amplitudes allows therefore to disentangle the two contributions rigorously, and has been studied in Refs. \cite{Donini:2016lwz,Donini:2020qfu}.

In these studies  an old strategy~\cite{Giusti:2004an,Giusti:2006mh} was revisited in the context of large $N_c$, where a light active charm, mass-degenerate with the up quark, was included, simplifying enormously the calculation. In this scenario, only the two current-current operators, $Q^\pm$, fully describe the $\Delta S = 1$ transitions. A matching to $N_f=4$ chiral perturbation theory, allows to extract the low-energy couplings, $g^\pm$, representing the strength of these interactions in the chiral Lagrangian, eq.~(\ref{eq:chpt4}). The couplings can be extracted from the measurement of the simpler $K \to \pi$ amplitudes, $A^\pm$ in the chiral limit. The determination of $g^\pm$ gives then an indirect estimate of the $K \to \pi \pi$ isospin amplitudes.

\begin{figure}[h!]
\begin{subfigure}{.5\textwidth}
  \centering
 
  \includegraphics[width=.999\linewidth]{plots/AmpAp.pdf}  
  \caption{}
  \label{fig:ammap}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
  \centering
 
  \includegraphics[width=.999\linewidth]{plots/AmmAp.pdf}  
  \caption{}
  \label{fig:ampap}
\end{subfigure}
\caption{Half-sum and half-difference of the amplitudes $A^\pm$ as a function of $N_c$ for three different cases: (i) quenched results in blue, (ii) dynamical results at a pion similar to the quenched case (red), and (iii) dynamical results at a lighter pion mass (orange). Error bars include only statistical errors. Source: Refs. \cite{Donini:2016lwz,Donini:2020qfu}. } 
\label{fig:ktopi1}
\end{figure}

\begin{figure*}[ht]
\begin{subfigure}{.5\textwidth}
  \centering
 
  \includegraphics[width=.999\linewidth]{plots/plotplus.pdf}  
  \caption{}
  \label{fig:plotplus}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
  \centering
 
  \includegraphics[width=.999\linewidth]{plots/plotprod.pdf}  
  \caption{}
  \label{fig:plotprod}
\end{subfigure}
\caption{Chiral extrapolation of $A^+$ and the product $A^+A^-$. Empty squares for $N_c=3$ indicate a finer lattice spacing. Solid lines indicate a simultaneous chiral and $N_c$ fit. Dashed lines represent the chiral extrapolation of the data points with $N_c = 3$. Source: Ref. \cite{Donini:2020qfu},} 
\label{fig:ktopi2}
\end{figure*}

The expected large $N_c$ scaling of the amplitudes $A^\pm$ has been presented in eq.~(\ref{eq:rnc}).
By means of appropriate linear combinations, one can isolate the (anti-)correlated coefficients:
\begin{align}
\frac{A^- + A^+}{2} &= 1 + \tilde c \frac{1}{N_c^2}  + \tilde d \frac{N_f}{N_c^3} , \\ \frac{A^- - A^+}{2} &=  \tilde a \frac{1}{N_c}+\tilde b \frac{N_f}{N_c^2}.
\end{align}
The $N_c$ scaling of these combinations was analysed in Ref. \cite{Donini:2020qfu} in three different scenarios: (i) quenched results ($N_f = 0$) at a heavy pion mass $\sim 570$ MeV, (ii) dynamical results ($N_f = 4$) at a heavy pion mass $\sim 560$ MeV, and (iii) dynamical results ($N_f = 4$) at a lighter pion mass $\sim 360$ MeV. This can be seen in Fig. \ref{fig:ktopi1} and Table \ref{tab:fitsNc}. The main conclusion from this study is that all coefficients turned out to be of the natural size. Importantly, the sign of the $\tilde a$ and  $\tilde b$ coefficients is the same and negative. This implies that both terms contribute to reduce the $A^+$ amplitude and enhance, in a correlated way, the amplitude $A^-$. Moreover, it can be seen that the mass dependence for the $N_f = 4$ results seems to affect mostly the coefficient $\tilde a$, and goes also in the direction of enhancing the ratio $A^-/A^+$ towards the chiral limit.



\begin{table}[h!]
\centering
\begin{tabular}{c|c|c|c|c}
\multicolumn{5}{c}{ Half-difference} \\ \hline
    \rule{0pt}{10.5pt}    Case            & $M_\pi$      & $\tilde{a}$ & $\tilde{b}$ & $\chi^2/\text{d.o.f.}$ \\ \hline\hline
$N_f=0$& $570$ MeV  &  -1.55(2)           &      ---       &8.8/6   \\ \hline
$N_f=4$& $560$  MeV &  -1.03(13)           &   -1.44(13)       & 6.6/2     \\ \hline
$N_f=4$& $360$ MeV &     -1.49(15)        &   -1.32(18)      & 0.3/2      \\ \hline
\end{tabular}

\vspace{0.5cm}

\begin{tabular}{c|c|c|c|c}
\multicolumn{5}{c}{ Half-sum} \\  \hline
  \rule{0pt}{10.5pt}    Case           & $M_\pi$      &$\tilde{c}$ & $\tilde{d}$& $\chi^2/\text{d.o.f.}$ \\ \hline\hline 
$N_f=0$& $570$   MeV&     2.1(1)       &     ---      & 3.5/6  \\ \hline
$N_f=4$& $560$   MeV&\  \ 1.2(3) \  \       &\  \ 2.2(3) \ \ &    1.3/2     \\ \hline
$N_f=4$& $360$  MeV& \  \  2.4(4) \   \    &\  \  1.6(4) \ \ &  3.2/2      \\ \hline
\end{tabular}

\caption{Summary of results for the $1/N_c$ fits to the half-sum and half-difference of the amplitudes $A^\pm$. Errors are only statistical.}
\label{tab:fitsNc}
\end{table}


The determination of the couplings $g^\pm$ requires an extrapolation to the chiral limit. The ChPT prediction for these amplitudes is \cite{Hernandez:2006kz,Kambor:1989tz}:
\begin{equation}
A^\pm = g^\pm \left[ 1 + 3 \left( \frac{M_\pi}{4 \pi F_\pi}\right)^2\log \frac{M_\pi^2}{\Lambda_\pm^2} \right]. 
\end{equation}
The chiral fits are shown in Fig. \ref{fig:ktopi2} for the $A^+$ amplitude (Fig. \ref{fig:plotplus}) and the product of the two $A^+ A^-$ (Fig. \ref{fig:plotprod}). The result for $N_c=3$ is:
\begin{equation}
\frac{g^-}{g^+} = 22(5),
\end{equation}
where the error is only statistical. Finally, an indirect estimate of the $K \to \pi \pi$ isospin amplitudes can be derived using the LO ChPT prediction in eq.~(\ref{eq:A0overA2}), and also the NLO
one derived in Ref. \cite{Donini:2020qfu}. The final estimate of Ref. \cite{Donini:2020qfu} for this ratio in the theory with a light charm is
\begin{equation}
\frac{A_0}{A_2} \Bigg \rvert_{N_f=4, N_c=3} = 24(5)_\text{stat}(7)_\text{sys}. \label{eq:A0A2}
\end{equation}

The main conclusion from this work is that the large enhancement observed in the $\Delta I=1/2$ rule seems consistent with coefficients in the $1/N_c$ expansion that are of the natural size, but with an important effect coming from the quark loops.  In addition, the result in eq.~(\ref{eq:A0A2}) suggests that the enhancement may indeed be largely dominated by intrinsic QCD effects, and not from the intermediate scale set by the charm quark mass. 




\section{Introduction}
\label{sec:intro}

\input{intro}



\section{Non-perturbative predictions at large $N_c$}
\label{sec:largencpheno}
\input{phenoNc}

\section{Lattice QCD at large $N_c$}
\label{sec:largenclattice}
\input{latticeNc}


\subsection{Scale setting and $N_c$-dependence}
\label{sec:scale}
\input{scalesetting}


\section{Yang-Mills theories at large $N_c$}
\label{sec:yangmills}
\input{yangmills}

\section{Hadronic quantities at large $N_c$  }
\label{sec:hadrons}
\input{hadrons}


\section{Concluding remarks}
\input{conclu}




\vspace*{1.5cm}
\noindent
\centerline{\textbf{Acknowledgements}}
\vspace*{0.1cm}

We warmly thank T. DeGrand, M. Garc{\'i}a-P{\'e}rez, M. Garc{\'i}a Vera, C. Pena, A. Ramos and R. Sommer for useful discussions. 

We acknowledge support from the Generalitat Valenciana grant PROMETEO/2019/083, the European project H2020-MSCA-ITN-2019//860881-HIDDeN, and the Spanish project FPA2017-85985-P.

The work of FRL has also received funding from the EU Horizon 2020 research and innovation program under the Marie Sk{\l}odowska-Curie grant agreement No. 713673 and La Caixa Foundation (ID 100010434). FRL also acknowledges financial support from Generalitat Valenciana through the plan GenT program (CIDEGENT/2019/040).




\vspace*{.5cm}
 \bibliographystyle{epj}
 

\subsection{Mesons}
A similar analysis can be done for mesons. Considering an operator with the quantum numbers of a meson
\begin{eqnarray}
O_M^a(x) = {1\over \sqrt{N_c}} \bar{\psi}(x) T^a \Gamma \psi(x),
\label{eq:normM}
\end{eqnarray}
the large $N_c$ counting gives
\begin{eqnarray}
\langle O_M(x_1)...O_M(x_n)\rangle_c \sim N_c^{1-{n\over 2}},
\label{eq:McorNc}
\end{eqnarray}
and therefore, if there are mesonic states in the spectrum, they are stable and non-interacting in the 't Hooft limit. The operator $O_M$ creates a meson from the vacuum with 
amplitude $N_c^0$, while the connected three-point function scales as $\sim 1/\sqrt{N_c}$, and the four-point function like $\sim N_c^{-1}$. The tower of resonances must also be infinite for the same argument outlined before for the gluons. 

From eqs.~(\ref{eq:normM}) and (\ref{eq:McorNc}), it follows 
\begin{align}
F^2_\pi &\propto  \langle A_\mu^a(x) A_\mu^a(y) \rangle \propto   N_c,\;\;\ \nonumber\\
 \Sigma &= -\langle  \bar{\psi}(x) \psi(x) \rangle \  \propto  N_c,
\end{align}
while the scattering length
\begin{eqnarray}
a_M &\propto &{ |\langle O_M^1 O_M^2 O_M^3 O_M^4 \rangle_c|\over  |\langle 0| O_M |M\rangle|^4} \propto N_c^{-1}.
\label{eq:aM}
\end{eqnarray}

\subsection{Chiral Perturbation Theory}
\label{subsec:chpt}

Spontaneous chiral symmetry breaking survives in the 't Hooft limit~\cite{Coleman:1980mx}, and therefore the lightest states in the spectrum are 
the light pseudoscalar mesons. Their mass is ${\mathcal O}(N_c^0)$, assuming the quark 
mass does not scale with $N_c$, since to leading order in the quark mass:
\begin{eqnarray}
M^2_\pi =  {\Sigma\over F_\pi^2} (m_u+ m_d).
\end{eqnarray}
Chiral perturbation theory (ChPT) and the chiral Lagrangian  \cite{Gasser:1984gg}
describe accurately meson interactions at low energies as an expansion in momentum and
quark mass:
\begin{eqnarray}
{\mathcal L}_{\chi} = {\mathcal L}^{(2)}_{\chi}+ {\mathcal L}^{(4)}_{\chi}+...,
\end{eqnarray}
with 
\begin{eqnarray}
{\mathcal L}^{(2)}_{\chi} &=& {F^2 \over 4} {\rm Tr}[\partial_\mu U^\dagger \partial^\mu U] -{\Sigma \over 2} {\rm Tr}[M_q U +h.c.],\nonumber\\
{\mathcal L}^{(4)}_{\chi} &=& \sum_{i=1,10} L_i {\mathcal O}_i[U,\partial_\mu, M_q],
\end{eqnarray}
where $M_q$ is the quark mass matrix, $U= \exp( i {\sqrt{2} \pi\over F})$ and  $\pi$ represents the pions, in the adjoint representation of $SU(N_f)$. The operators ${\mathcal O}_i$ are  Lorentz and chirally invariant  combinations of $U$, with four derivatives or two quark mass matrices. The $L_i$ are the famous low-energy couplings (LECs)~\cite{Gasser:1984gg}.
The expansion parameter in ChPT is
\begin{eqnarray}
   \left(M_\pi\over 4 \pi F_\pi\right)^2 \sim  \left(p \over 4 \pi F_\pi\right)^2  \sim  {1\over N_c},
\end{eqnarray}
and becomes smaller and smaller as $N_c \rightarrow \infty$. However, the range of validity to the chiral effective theory does not  increase: the failure of the chiral expansion will be abrupt when the energy reaches the mass of the heavy resonances, $\Lambda_\chi$. These masses are expected to scale as $N_c^0$, and therefore remain constant as we approach $N_c\rightarrow \infty$.  One may typically consider $\Lambda_\chi \sim M_\rho$. 

  An additional simplification of the chiral Lagrangian comes from the fact that only a subset of the low-energy couplings \cite{Gasser:1984gg} are leading in $N_c$, ie. $L_i \propto {\mathcal O}(N_c)$. These are the ones that correspond to chiral operators (${\mathcal O}_i$) with a single flavour trace. This is because according to the general rules, diagrams with more than one fermion loop are subleading. More concretely, for $N_f=3$ \cite{Gasser:1984gg,Peris:1994dh}:
\begin{eqnarray}
L_1,L_2,L_3,L_5,L_8,L_9,L_{10} &\propto & {\mathcal O}(N_c), \nonumber\\
2 L_1-L_2, L_4,L_6,L_7 &\propto & {\mathcal O}(1).
\end{eqnarray}
Phenomenological approaches have estimated the leading $N_c$ behaviour of these couplings by assuming that the chiral theory matches onto a theory of free resonances, the resonant chiral theory \cite{Ecker:1988te}. The low-energy couplings result from the exchange of heavier resonances and can be extracted in terms of the measured spectrum, and imposing the correct large momentum behaviour of certain correlation functions. For a review see \cite{Pich:2002xy}. 
  
It is well-known that the standard chiral Langrangian needs to be extended to include the $\eta'$ \cite{DiVecchia:1980yfw,PhysRevD.21.3388,Witten:1980sp,Kawarabayashi:1980dp,Gasser:1984gg,Leutwyler:1996sa,HerreraSiklody:1996pm,Kaiser:2000gs}, since the singlet meson becomes degenerate with the remaining pseudoscalar mesons in the planar limit. 
The $\eta'$ mass receives a contribution from the $U_A(1)$ anomaly, that scales as $N_c^{-1}$. 
The Witten-Veneziano (WV) formula \cite{Witten:1979vv,Veneziano:1979ec} (see also \cite{Giusti:2001xh,Seiler:2001je,Giusti:2004qd,Luscher:2004fu,DelDebbio:2004ns}) relates this contribution with the topological susceptibility computed in a theory without quarks, ie. in Yang-Mills (YM):
\begin{eqnarray}
M_{\eta'}^2  \equiv {2 N_f \over F^2_\pi} \int d^4 x \langle q(x) q(0)\rangle_{YM}  \left[1+ {\mathcal O}(N_c^{-1})\right],
\label{eq:W}
\end{eqnarray}
where 
\begin{eqnarray}
q(x) \equiv {\lambda \over 32 \pi^2 N_c} {\rm Tr}[F_{\mu\nu}(x) {\tilde F}^{\mu\nu}(x)] ,
\end{eqnarray}
and we have considered massless quarks. The topological susceptibility measures the $\theta$ dependence of the vacuum energy, which should vanish for massless quarks. The WV  relation can be deduced  from requiring that the $\eta'$ contribution to the $\theta$ dependence exactly cancels the leading  gluonic contribution, or from the anomalous chiral Ward identity.  This is one example where the diagrammatic analysis leads to the wrong conclusion: the leading $N_c$ scaling of the  topological susceptibility is cancelled by a  naively subleading one. 

The presence of the $\eta'$ in the chiral Lagrangian makes it necessary to correlate the large $N_c$ and  the momentum/mass expansions. The pion manifold is now $U(N_f)$, and a consistent power counting  \cite{HerreraSiklody:1996pm,Kaiser:2000gs}  ensures the same scaling of the $\eta'$ mass and the remaining pseudoscalar mesons:
\begin{eqnarray}
{\mathcal O}(M_\pi^2) \sim {\mathcal O}(M_{\eta'}^2) \sim {\mathcal O}\left({1\over N_c}\right).
\end{eqnarray}


\subsection{Baryons}

Baryons are trickier in the 't Hooft limit, since their mass has a non-trivial scaling. Nevertheless their properties can also be understood in a systematic expansion in $1/N_c$ \cite{Witten:1979kh,Dashen:1993jt,Jenkins:1993zu,Carone:1993dz,Dashen:1994qi,Dai:1995zg}. By simple arguments, it can be seen that the baryon mass scales with $N_c$. We recall that in QCD a colour singlet can be formed by combining three quarks in the fundamental representation: $\mathbf 3 \otimes\mathbf 3 \otimes \mathbf 3 \to \mathbf 1$. In a general $SU(N_c)$ theory, one would however need $N_c$ fundamental quarks to create a baryon\footnote{An alternative would be including fermions in the antisymmetric representation of $SU(N_c)$, so that large-$N_c$ baryons may still be formed by three fermions \cite{Corrigan:1979xf}.}. This motivates the simple expectation for the baryon mass: $M_B = m_0 N_c + O(N_c^0)$. Moreover, the baryon--baryon to n-meson amplitude scales like $\sqrt{N_c}^{1-n}$. 

Spin-flavour symmetry results in  consistency conditions that  allow to understand for example the baryon hyperfine splitting \cite{Jenkins:1993zu}:
 \begin{eqnarray}
{ M_B\over N_c} =  m_0+ B {J (J+1)\over N_c^2} + C \left({J (J+1)\over N_c^2}\right)^2+...,
 \end{eqnarray}
where $J$ is the baryon spin, and $m_0, B$ and $C$ are $O(N_c^0)$ constants.  For a detailed review on baryons in large $N_c$ 
 see \cite{Manohar:1998xv}.



\subsection{Exotics}
An interesting and timely question is that of exotic states, such as tetraquarks.  The standard lore  \cite{Witten:1979kh,Coleman:1980nk} used to be that tetraquarks cannot exist in 
the 't Hooft limit. The argument however has been revised recently\footnote{Note the mismatch between the arXiv and published versions of Ref. \cite{PhysRevLett.110.261601}. Here we refer to the latter. } \cite{PhysRevLett.110.261601, PhysRevD.88.036016}. The most recent conclusion  is that the large $N_c$ scaling does not allow to establish whether these particles exist or not in the spectrum, however if they do they must be infinitely narrow, just like glueballs and mesons, for the same reasons outlined above. An analysis of the expected scaling of the decay width of tetraquarks with different flavour content can be found in\cite{PhysRevD.88.036016}.  

 It is interesting to analyse the old argument supporting the non-existence of exotics \cite{Witten:1979kh,Coleman:1980nk}. It relies on the fact that the two point correlation function of two tetraquark operators (constructed from two quark bilinears) has a factorized contribution, leading in $N_c$, corresponding to the propagation of two non-interacting mesons. However,  the tetraquark pole, if it exists, should appear in the connected part of the correlator\cite{PhysRevLett.110.261601} which is subleading in $N_c$. In other words the factorized and connected terms might describe the dynamics of different processes, one is not necessarily the subleading $N_c$ correction of the other, and both can in principle survive the 't Hooft limit. It might be a challenge in practice to isolate such exotic states from a two meson state, since  the overlap of any operator with the right quantum numbers with the two meson state dominates over that with the exotic one.

\subsection{Factorization}

More generally, large $N_c$ implies the factorization of expectation values of products of singlet operators {in the pure gauge theory}:
\begin{eqnarray}
\langle O_1 O_2 \rangle = \langle O_1 \rangle \langle O_2 \rangle\left[ 1 + {\mathcal O}\left({1\over N^2_c}\right) \right].
\label{eq:fact}
\end{eqnarray}
This property underlies for example the OZI rule and the EK reduction. It should be stressed however that the disconnected and connected parts of any such 
observable could represent different physics.


\subsection{Weak interactions}

Finally, we discuss hadronic weak interactions in the planar limit. One of the most famous failures of the large $N_c$ approximation is the $\Delta I=1/2$ rule, that is the large hierarchy observed in the kaon  decay amplitudes to two pions in the two possible states of isospin $I=0,2$:
\begin{eqnarray}
T[K^0 \rightarrow (\pi\pi)_I ] = i A_I e^{i\delta_I}, \;\;\;\; \left|{A_0\over A_2}\right| \sim 22.
\end{eqnarray}
These amplitudes were established as benchmark lattice calculations since the early days \cite{Brower:1984ta}, but remain very challenging. Only very recently\cite{Boyle:2012ys,Bai:2015nea,Blum:2015ywa,Ishizuka:2018qbn,Abbott:2020hxn}, convincing evidence has been presented that the $\Delta I=1/2$ rule is reproduced near the physical point of $2+1$ simulations.  It is well known that the $\Delta I=1/2$ rule plays a key role in the Standard Model prediction of the direct CP violating ratio $\epsilon'/\epsilon$. 

Surprisingly large $N_c$ misses completely the $\Delta I=1/2$ enhancement \cite{Fukugita:1977df,Chivukula:1986du}. This is easy to see by comparing the large $N_c$ scaling of the decays $K^+\rightarrow \pi^+\pi^0$ and $K^0\rightarrow \pi^0\pi^0$,  which in terms of the isospin amplitudes are:
\begin{eqnarray}
-i T[K^0 \rightarrow \pi^0 \pi^0] &=& \sqrt{2\over 3} A_2 e^{i \delta_2} -{1\over \sqrt{3}} A_0 e^{i \delta_0} \propto {\mathcal O}(N_c^{-{1\over2}}),\nonumber\\
-i T[K^+ \rightarrow \pi^+ \pi^0] &=& {\sqrt{3}\over 2} A_2 e^{i \delta_2} \propto {\mathcal O}(N_c^{1\over 2}),
\label{eq:Ts}
\end{eqnarray}
where the $N_c$ scaling can be easily deduced by inspection of the diagrams, see Fig.~\ref{fig:deltaI}, contributing to each amplitude\footnote{We have included the normalization of each meson state by $\sqrt{N_c}$.}.  This implies that, at the leading order in large $N_c$, the neutral $K^0$ does not decay and therefore from eq.~(\ref{eq:Ts}) it follows: 
\begin{eqnarray}
\left|{A_0 \over  A_2}\right|_{N_c\rightarrow \infty} = \sqrt{2}. 
\end{eqnarray}
One possible explanation of this discrepancy is the fact that the dynamics involves multiple scales: the $W$ mass, the charm mass and hadronic scales, and maybe subleading 
$1/N_c$ corrections could be enhanced by large logarithms. 


 \begin{figure}[h!]
 \begin{center}
\[\begin{array}{cc}
\raisebox{-.45\height}{\includegraphics[scale=1]{plots/Ktopipicharged.pdf}} & \hspace{0.3cm} { \mathcal{O} \Big( N_c^{1/2}\Big)} \\ \\
\raisebox{-.43\height}{\includegraphics[scale=1]{plots/Ktopipineutral.pdf}} &\hspace{0.3cm} {\mathcal O}\Big( N_c^{-1/2}\Big)\\ 
\end{array}\]
\caption{\label{fig:deltaI} Leading diagrams in the $N_c$ scaling of  $K^+ \to \pi^+ \pi^0$ (top), and $K^0 \to \pi^0 \pi^0$ (bottom). }
\end{center}
\end{figure}

At energies below the $W$-boson mass, we can integrate it out and represent the weak interactions in the Fermi theory. 
 For example the CP-conserving $\Delta S=1$ transitions the Hamiltonian density takes the simple form:
\begin{eqnarray}
\label{eq:heffs1}
{\mathcal H}^{N_f=4}_{\Delta S=1} =\sqrt{2}G_{\rm F}V_{us}^*V_{ud}   ( k^+ \, \bar{Q}^+(x)+k^- \, \bar{Q}^-(x) )\,, 
\label{eq:hw4}
\end{eqnarray}
with $\bar{Q}^\pm = Z_Q^\pm Q^\pm$ and 
\begin{eqnarray}
Q^\pm =& (\bar{s}_L \gamma_\mu u_L)(\bar{u}_L \gamma^\mu d_L) \pm (\bar{s}_L \gamma_\mu d_L)(\bar{u}_L \gamma^\mu u_L)  \nonumber\\
&-[u\leftrightarrow c].
\label{eq:currento}
\end{eqnarray}
We have neglected quark masses and included all the operators compatible with the ${\rm SU}(4)_{\rm L} \times {\rm SU}(4)_{\rm R}$ flavour symmetries. $Q^+$ transforms as the $(84,1)$, while $Q^-$ transforms as the $(20,1)$.  Only $\bar{Q}^+$ contributes to the $A_2$ amplitude, and therefore a hierarchy in the matrix elements of the two operators is needed to explain the rule. This effective Hamiltonian should be valid down to energies above the charm mass. 
The Wilson coefficients $k^\pm$ do indeed get large logarithms from the $M_W \rightarrow \mu$ \cite{Gaillard:1974nj,Altarelli:1974exa}, but it is not enough to explain the large hierarchy, since $k^-(m_c)/k^+(m_c) \sim 2$. 

Below the charm threshold 
the theory matches onto the  $N_f=3$ theory, that has the form \cite{Buchalla:1995vs}
\begin{eqnarray}
\label{eq:heffs2}
{\mathcal H}^{N_f=3}_{\Delta S=1} =\sqrt{2}G_{\rm F}V_{us}^*V_{ud} \sum_{\sigma=1,...6} k_\sigma Q_\sigma, 
\label{eq:hw3}
\end{eqnarray}
where the operators in the list include all operators that transform as $(27,1)$ or $(8,1)$,  ignoring QED effects. Among the latter, we find the famous penguin operators such as 
\begin{eqnarray}
Q_6 =  (\bar{s}_L \gamma_\mu d_L)\sum_{q=u,d,s} (\bar{q}_R \gamma^\mu q_R).
\end{eqnarray}
The standard lore for a long time was that the matrix elements of the penguin operators were enhanced and, since they are octets that do not contribute to  $A_2$, this fact could be at the origin of the $\Delta I=1/2$ \cite{Vainshtein:1976nd}. This hypothesis does not seem to be confirmed by the last lattice results, where the current operators of eq.~(\ref{eq:currento}) give the dominant contribution to the real part of the isospin amplitudes, as anticipated long ago in Ref. \cite{Bardeen:1986uz}. Instead it has been argued that the enhancement originates in a strong cancellation of the colour-connected and colour-disconnected contributions of the current operator matrix elements \cite{Boyle:2012ys}. Such contributions have a different $N_c$ scaling, and therefore indicate anomalously large $N_c$ corrections. 

At sufficiently low energies and quark masses, a further simplification is possible since the effective hamiltonians, eqs.~(\ref{eq:hw4}) or (\ref{eq:hw3}) can be matched to the corresponding $N_f=4$ (if the charm is considered light) or $N_f=3$ (for a heavy charm) chiral effective theories.  At the leading order of the chiral expansion, the structure of the chiral weak hamiltonian in the $N_f=4$ theory is \cite{Giusti:2004an,Giusti:2006mh}
\begin{eqnarray}
\label{eq:heffs3}
{\mathcal H}^{N_f=4}_{\rm ChPT} =\sqrt{2}G_{\rm F}V_{us}^*V_{ud}   ( g^+ \, {\mathcal Q}^++g^- \, {\mathcal Q}^-)\,, 
\label{eq:chpt4}
\end{eqnarray}
with
\begin{eqnarray}
{\mathcal Q}^\pm &=& {F_\pi^4\over 4} \left[(U \partial_\mu U^\dagger)_{us} (U \partial_\mu U^\dagger)_{du}\pm (U \partial_\mu U^\dagger)_{ds} (U \partial_\mu U^\dagger)_{uu} \right. \nonumber\\
&-&\left. (u\rightarrow c)\right] .
\label{eq:currentochpt}
\end{eqnarray}
It is straightforward to evaluate the ratio of amplitudes in terms of the low-energy couplings $g^\pm$ to lowest order in ChPT:
\begin{eqnarray}
{A_0\over A_2} ={1\over 2 \sqrt{2}} \left(1+ 3 {g^-\over g^+}\right). \label{eq:A0overA2}
\end{eqnarray}
A large ratio $g^-/g^+$ could explain the $\Delta I=1/2$ rule, an effect that would be unrelated to the charm threshold.

These low-energy couplings can be determined from matching appropriate correlation functions in ChPT and lattice QCD\cite{Giusti:2004an,Giusti:2006mh}. The simplest choices are the ratios:
\begin{eqnarray}
R^\pm  = \kern-1.0em
\lim_{ \substack{z_0-x_0\to\infty \\ y_0-z_0\to \infty}}
\frac{\sum_{{\mathbf x},{\mathbf y}} \langle P(y) Q^\sigma(z) P(x) \rangle}
{\sum_{{\mathbf x,\mathbf y}}  \langle P(y) A_0(z) \rangle  \langle P(x) A_0(z) \rangle}, \label{eq:ratios}
\end{eqnarray}
with $A_0(x) \equiv \bar{\psi}^i(x)\gamma_0 \gamma_5 \psi^j(x)$ and $P(x)= \bar{\psi}^i(x) \gamma_5 \psi^j(x)$, with appropriately chosen flavours $i,j$. 

Including the renormalization factors and the Wilson coefficients, we get the physical amplitudes, $A^\pm$,  that  can be matched to the corresponding observables in the chiral theory\cite{Giusti:2004an,Giusti:2006mh}. At lowest order the matching condition is 
\begin{eqnarray}
\lim_{M_\pi \rightarrow 0} A^\pm = g^\pm. 
\end{eqnarray}

The $\pm$ three-point functions are the sum of a colour-disconnected diagram and a colour-connected one, that scale differently with $N_c$ , see Fig.~\ref{fig:condis}.
 The relative sign is $\mp$ in the $A^\pm$ amplitudes.
 \begin{figure}[h!]
 \begin{center}
\renewcommand{\arraystretch}{2}
\[\begin{array}{lll}
 \raisebox{-.5\height}{\includegraphics[scale=0.8]{plots/dis.pdf}} & \mp &\raisebox{-.5\height}{\includegraphics[scale=0.8]{plots/con.pdf}}\\
 \end{array}\]
 \caption{\label{fig:condis} Colour-disconnected (left) and colour-connected (right) contributions to the three-point function in $R^\pm$. }
\end{center}
\end{figure}
The standard diagrammatic $N_c$ counting results in the following scaling:
\begin{eqnarray}
A^\pm = 1 \pm \tilde a {1 \over N_c}\pm \tilde b {N_f \over N^2_c}+\tilde c {1 \over N^2_c}+ \tilde d {N_f \over N^3_c}+\cdots, \label{eq:rnc}
\end{eqnarray}
where the coefficients $\tilde a- \tilde d$ are 
independent of $N_c$ and $N_f$, and  a {\it natural} expectation is that they are ${\mathcal O}(1)$. Diagrams\footnote{Additional planar exchanges of gluons being of the same order contribute in all cases.} contributing to ${\tilde c}$ and ${\tilde d}$ are
shown in Fig.~\ref{fig:disnlo} while those contributing to $\tilde a$ and $\tilde b$ are in Fig.~\ref{fig:connlo}
 \begin{figure}[h]
 \begin{center}
\renewcommand{\arraystretch}{2}
\[\begin{array}{ccc}

{\tilde c}: & \raisebox{-.5\height}{\includegraphics[scale=0.7]{plots/disnlo.pdf}} &{\mathcal O}(N_c^0) \\
{\tilde d}: & \raisebox{-.5\height}{\includegraphics[scale=0.7]{plots/disnlonf.pdf}} & {\mathcal O}\left({N_f\over N_c}\right)\\
\end{array}\]
\caption{\label{fig:disnlo} $N_c, N_f$ scaling of various contributions to the colour-disconnected contraction of the three-point function.}
\renewcommand{\arraystretch}{2}
\[\begin{array}{ccc}
\tilde a:   & \raisebox{-.5\height}{\includegraphics[scale=0.7]{plots/conlo2.pdf}} & {\mathcal{O}}(N_c)\\
\tilde b:&  \raisebox{-.5\height}{\includegraphics[scale=0.7]{plots/connlonf.pdf}} & \mathcal{O}\left( N_f \right)\\
\end{array}\]
\caption{\label{fig:connlo} $N_c, N_f$ scaling of various contributions to the colour-connected contraction of the three-point function. }
\end{center}
\end{figure}

We find that the leading correction, ${\mathcal O}(1/N_c)$ together with the leading effects of dynamical quarks,  ${\mathcal O}(N_f/N_c^2)$, are fully anticorrelated in the ratios.
 This analysis does not predict the sign of the  different terms, i.e., the sign of the $\tilde a-\tilde d$ coefficients, only the (anti)-correlation between the two amplitudes. In particular,  a negative sign of $\tilde a$ and $\tilde b$ results in an enhancement of the ratio $A^-/A^+$.  The scaling of eq.~(\ref{eq:rnc}) is valid for all quark masses, but the different coefficients could be have a different quark mass dependence. 

The lattice can in principle measure all these coefficients and determine whether they have the natural size, ${\mathcal O}(1)$, and quantify to what extent the anti-correlation is enough to explain the $\Delta I=1/2$ rule.


A similar matching to chiral perturbation theory could also be done in the $N_f=3$ flavour theory, where the leading order weak Hamiltonian has also two operators with the same chiral structure, corresponding  to the $27$-plet and octet operators\cite{Bernard:1985wf}. The case of a heavy charm is however much more demanding, not only the list of operators to consider on the lattice is much longer, but it involves the computation of all-to-all propagators. 

There is a long list of phenomenological analyses that have used large-$N_c$ inspired methods to estimate the size of these couplings, in an analogous way to the strong couplings, $L_i$.  Examples are the dual QCD approach \cite{Bardeen:1986vp,Bardeen:1986uz,Buras:2014maa}, the resonance chiral theory \cite{Ecker:1988te} or the chiral quark model \cite{Antonelli:1995nv}, etc. In all cases, the relevance of subleading corrections in $N_c^{-1}$ has been demonstrated, and argued to be related to a strong scale dependence. Incorporating these systematically is not easy and the error resulting from these approaches is therefore quite uncertain. For a review see \cite{Cirigliano:2011ny}.





\subsection{Large $N_c$ factorization}

One of the simplest yet powerful results in the large $N_c$ limit is the factorization of observables, eq.~(\ref{eq:fact}). This result is derived from a perturbative analysis, but can be tested non perturbatively in lattice simulations at different values of $N_c$. Recently, such study  was carried out in Refs. \cite{Vera:2018lnx,GarciaVera:2017xif}.

\begin{figure}[h!]
\centering 
\includegraphics[width=0.5\textwidth]{plots/factorization1.pdf}
\caption{ Large $N_c$ extrapolations of $G_W (1)$ in the continuum (solid line) and at finite lattice spacing (dashed line). Source: Ref. \cite{Vera:2018lnx}.  \label{fig:f1}}
\end{figure}


In order to test the factorization hypothesis, the authors use various quantities, e.g., a square spatial Wilson loop at the point $(x_0,{\mathbf x})$ with size $R_c$ and Wilson-flowed to time $t$:
\begin{align}
\begin{split}
W(c) &= \langle W(t, x_0, \vec x, R_c)  \rangle , \\ W^{sq}(c) &= \langle W^2(t, x_0, \vec x, R_c)  \rangle,
\end{split}
\end{align}
The flow time is a fraction of $t_0$, eq.~(\ref{eq:t0Nc}),  $t =c t_0$, and the size is chosen as \footnote{The radius of a square Wilson loop is always an integer. In this work, the authors interpolate between integers using the methods described in Ref. \cite{Vera:2017dxr}.} $R_c = \sqrt{8 c t_0}$. Note that the authors use open boundary conditions, to avoid topology freezing, and $x_0$ must be taken far from the boundary. The ratio
\begin{equation}
G_W(c) = \frac{W^{sq}(c) - W^2(c)}{W^2(c)}, \label{eq:Gw}
\end{equation}
according to the factorization property, 
is expected to be of order $1/N_c^2$. The results for $G_W(1)$ in this study are shown in Fig. \ref{fig:f1}. As can be seen, the authors find very good consistency with the factorization hypothesis both at a finite lattice spacing (fixed $t_0/a^2$), and in the continuum limit. For other gluonic observables, the authors also find the expected large $N_c$ scaling and confirm the factorization hypothesis. 

The dependence of $G_W(c)$ on the size of the loop is also studied. For this, they define a new observable $\hat G_W(\xi)$, that depends on the Wilson loop 
\begin{equation}
\widehat W(\xi) = \langle W(t_0/2, x_0, \vec x, R_\xi)  \rangle   \label{eq:hatGw}
\end{equation}
with varying radius, $R_\xi = \xi \sqrt{4 t_0}$. The authors fit the $1/N_c$ scaling as:
\begin{equation}
\widehat{G}_W(\xi) = b_0 + b_1 \frac{1}{N_c^2} + b_2 \frac{1}{N_c^4}. \label{eq:fitGNc}
\end{equation}
As expected, $b_0$ is always compatible with zero. In contrast, the results for the coefficient of the subleading corrections , shown in Fig. \ref{fig:f2}, seem to grow exponentially with the size of the Wilson loop.

This shows that the coefficient of the subleading corrections can be large, but this is not necessarily a signal of failure of the large $N_c$ expansion. This is because the leading 
and subleading contributions can be related to different physics scales. While the disconnected and leading term is controlled by the static potential, the connected part could get 
contributions from other physics scales.
\begin{figure}[h!]
\centering 
\includegraphics[width=0.5\textwidth]{plots/factorization2.pdf}
\caption{ Dependence of the  corrections to factorization with the size of the Wilson loop, $\xi$ [see eq.~(\ref{eq:hatGw})]. The $b_i$ parameters are those of eq.~(\ref{eq:fitGNc}). For each parameter, we include a quadratic interpolating line. Source: Ref. \cite{Vera:2018lnx}.  \label{fig:f2}}
\end{figure}



\subsection{Glueballs at large $N_c$}
\begin{figure*}[t!]
\centering 
\includegraphics[width=1\textwidth]{plots/glueballs.pdf}
\caption{ Spectrum of glueballs at finite lattice spacing in the large $N_c$, and $N_c=3$ from Ref. \cite{Lucini:2010nv}. Some of the $SU(3)$ levels were not determined in Ref. \cite{Lucini:2010nv}. As a comparison, we also include the continuum limit results for $SU(3)$ from Ref. \cite{Athenodorou:2020ani}. \label{fig:glueballs}}
\end{figure*}

The glueball spectrum is a very interesting nontrivial manifestation of non perturbative dynamics in the pure Yang-Mills theory. Describing glueballs in the pure gauge theory may help us finding their analogues in full QCD, assuming their mix with $\bar{q} q$ states is not large. Moreover, the lowest spectrum of a pure $SU(N_c)$ that weekly interacts with the SM particles has been proposed as a candidate for dark matter \cite{Soni:2016gzf}---see also Ref. \cite{Kribs:2016cew}.

The dependence of the glueball spectrum on the number of colours has been subject of study for many years \cite{Teper:1998kw,Lucini:2001ej,Lucini:2004my,Meyer:2004jc,Meyer:2004gx,Lucini:2010nv,Amato:2015ipe}, including the $\theta$-dependence\footnote{$\theta$ is the coefficient of the topological term in the Lagrangian.} \cite{DelDebbio:2006yuf}. Early calculations determined the continuum result for the lowest-lying glueballs, with $J^{PC}=0^{++}, 2^{++}$, in the large $N_c$ limit \cite{Lucini:2004my}. Up to date, the most extensive $SU(N_c)$ glueball spectroscopy result is that of Ref. \cite{Lucini:2010nv}, with the caveat that no continuum limit was performed. In Fig. \ref{fig:glueballs}, we compare the large $N_c$ results of Ref. \cite{Lucini:2010nv} to the $N_c=3$ ones of the same work, both at similar value of the lattice spacing. As can be seen, $1/N_c^2$ corrections are quite small, but sometimes resolvable. We also include in the comparison  the continuum limit results for $SU(3)$ in Ref. \cite{Athenodorou:2020ani}. There are signs of significant discretization effects in $SU(3)$ for the states that have been measured with good statistical precision. It would be interesting to compute the large $N_c$ result including a continuum extrapolation.




Some of the states in the low-lying glueball spectrum correspond to two-particle scattering states, as discussed in Ref. \cite{Meyer:2004vr}. These levels have been omitted in Fig. \ref{fig:glueballs}. In the large $N_c$ limit, their energy is simply given by the non-interacting one, as glueball interactions are suppressed with $N_c$. Still, at large but finite $N_c$ these levels can be used to study interactions among glueballs. So far, these studies have only been attempted for $SU(2)$ using the HAL QCD method \cite{Yamanaka:2019gak,Yamanaka:2019yek,Yamanaka:2019aeq}. It would be interesting to perform a conclusive Lüscher analysis of two-glueball interactions, and its $N_c$-dependence.

{It is also worth mentioning that the large $N_c$ limit of $Sp(2N_c)$ and $SU(N_c)$ coincides, as shown in Ref. \cite{Lovelace:1982hz}. Thus, the $Sp(2 N_c)$ results in Refs. \cite{Bennett:2020hqd,Bennett:2020qtj} are also relevant for the large $N_c$ limit of QCD.  A confirmation of the expectation, supported by the data, is that the ratio of masses for the tensor and scalar glueball seems to be independent of the gauge group. In particular, it agrees for $Sp(\infty)$ and $SU(\infty)$.}




\subsection{Topological susceptibility}
\label{subsec:topsus1}
The topological susceptibility is the first momentum of the distribution of the topological charge, $Q$:
\begin{equation}
\chi_{YM} = \lim_{V \to \infty} \frac{1}{V} \langle Q^2 \rangle.
\end{equation}
As we have seen in sec.~\ref{sec:largencpheno}  the  Witten-Veneziano equation, eq.~(\ref{eq:W}), relates this observable in Yang-Mills  to the mass of the $\eta'$ meson in the planar limit of QCD. Studying the size of this subleading $1/N_c$  is important to quantify how far from the planar limit  QCD is.

\begin{figure}[h!]
\centering 
\includegraphics[width=0.5\textwidth]{plots/topsus.pdf}
\caption{Scaling of the topological susceptibility with the number of colours in the pure Yang-Mills theory. The dashed line is a linear fit in $1/N_c^2$ including up to $N_c=4$. Source: Ref. \cite{Ce:2016awn} for $N_c>3$ and Ref. \cite{Ce:2015qha} at $N_c=3$. \label{fig:topsus}}
\end{figure}

The first computation of the topological susceptibility in the large $N_c$ limit was attempted two decades ago, in Ref. \cite{Lucini:2004yh}. One of the main difficulties is the critical slowing down \cite{Schaefer:2010hu}, that is, the very long autocorrelation time in topological observables as one approaches the continuum limit using periodic boundary conditions\footnote{In a recent article \cite{Bonanno:2020hht}, a novel algorithm has been proposed to mitigate this problem.}. This is already present in $N_c=3$, and it becomes more severe for $N_c>3$. 

The use of open boundary conditions \cite{Luscher:2011kk} significantly improves this situation, and has been an essential ingredient in the recent computation of  $\chi_{YM}$ in the continuum limit \cite{Ce:2015qha,Ce:2016awn}. In Fig. \ref{fig:topsus}, we show the continuum result of $\chi_{YM}$ as a function of the number of colours. The large-$N_c$ result reported by the authors is
\begin{equation}
\lim_{N_c \to \infty} t_0^2 \chi_{YM} = 7.03(13) \cdot 10^{-4},
\end{equation}
which differs from the $N_c=3$ result by only about $O(5\%)$. As pointed out by the authors, the small size of $1/N_c^2$ effects explains why the $N_c=3$ result is already able to predict the bulk result of the $\eta'$ mass. A lattice test of the Witten-Veneziano equation, computing directly the $\eta'$ mass, has also been carried out in Ref. \cite{Cichy:2015jra} for $N_c=3$.


The large-$N_c$ topological susceptibity has also been studied through the $\theta$-dependence  in Refs. \cite{DelDebbio:2002xa,Bonati:2016tvi,Kitano:2020mfk}, and in lower dimensional theories  in Refs.~\cite{Bonati:2019ylr,Bonanno:2018xtd}.
