\section{One loop: the Hagedorn temperature}
\label{sec:oneloop}

The first effect of the interaction is to move the Hagedorn temperature. This was computed by Spradlin and Volovich \cite{SpradlinVolovich}. They replaced thermal diagrams by the trace of the dilatation operator over gauge-invariant operators, below $T_H$. We use the same operator at fixed holonomy, where their counting of necklaces gives way to Wick contractions in a twisted ensemble, so we set this out in some detail. We then record what integrability has since added.

\subsection{The partition function as a trace over operators}

Thermal field theory on $S^3\times S^1$ is usually done with Feynman diagrams, with fields expanded in spherical harmonics and propagators periodic in Euclidean time, or antiperiodic for fermions. AMMPV worked that way. There is a second way, special to conformal theories.

The theory on $\mathbb{R}\times S^3$ is equivalent to the theory on $\mathbb{R}^4$ by the map $r=e^\tau$. States on the sphere are local operators at the origin, and the Hamiltonian on the sphere is the dilatation operator $D$, whose eigenvalues are scaling dimensions. The partition function at fixed holonomy is then
\begin{equation}
Z[U]=\Tr\big(x^{D}\,\hat U\big),\qquad e^{-S_{\rm eff}[U]}=e^{-N^2\sum_n|u_n|^2/n}\;Z[U],
\label{eq:ZU}
\end{equation}
where the trace runs over all words in the letters of table~\ref{tab:letters} with no singlet constraint, $\hat U$ rotates their colour indices, and the prefactor is the Haar measure. At zero coupling $D=D_0$ counts the engineering dimension and \eqref{eq:ZU} reproduces \eqref{eq:Zfree}.

In perturbation theory the dilatation operator has an expansion. We write it as
\begin{equation}
D=D_0+g^2D_2+g^4D_4+\ldots,\qquad g^2=\frac{\lambda}{16\pi^2}=\frac{g_{\rm YM}^2N}{16\pi^2}.
\label{eq:Dexp}
\end{equation}
The subscript counts powers of $g_{\rm YM}$, so $D_2$ is the one-loop operator and $D_4$ the two-loop one. The coupling $g^2$ is the one that is standard in the integrability literature. All three of $D_0$, $D_2$, $D_4$ commute with $D_0$. That is a choice of basis that is always possible in degenerate perturbation theory, and we will make it explicit in section~\ref{sec:computation}.

Because the pieces commute with $D_0$, the expansion of \eqref{eq:ZU} is the elementary expansion of an exponential. To first order,
\begin{equation}
-S_{\rm eff}[U]=-S^{(0)}_{\rm eff}[U]-\beta g^2\,\vev{D_2}_U+O(g^4),
\label{eq:S1}
\end{equation}
where $\vev{\,\cdot\,}_U$ is the average in the free ensemble twisted by $U$. Explicitly, $\vev{X}_U$ is the ratio of $\Tr(x^{D_0}\hat UX)$ to $\Tr(x^{D_0}\hat U)$. The ensemble is Gaussian, so the average is computed by Wick's theorem. The elementary contraction is the thermal propagator of a letter $A$ of energy $E_A$, which in the background $U$ is a sum over the number of times $n$ the letter winds around the thermal circle,
\begin{equation}
\sum_{n\ge1}(\pm1)^{n+1}\,x^{nE_A}\;U^n\otimes U^{-n},
\label{eq:twisted}
\end{equation}
with the lower sign for fermions. The factor $U^n\otimes U^{-n}$ acts on the two colour indices of the adjoint letter. When all contractions are done, the factors of $U$ are strung together into traces along the index loops of the diagram, and one reads off a polynomial in the $u_n$.

This summarises the method. A term in $S_{\rm eff}$ of order $g^{2k}$ is the thermal average of an operator built from $D_2,\ldots,D_{2k}$, with twisted propagators. No time integrals and no integrals over the sphere are left.

\subsection{The complete one-loop dilatation operator}

Beisert found $D_2$ on all the letters of the theory \cite{BeisertComplete,BeisertStaudacher}. It acts on two letters at a time. The symmetry algebra $psu(2,2|4)$ acts on the space of single letters $V_F$, and the product of two copies decomposes into irreducible modules labelled by a non-negative integer,
\begin{equation}
V_F\otimes V_F=\bigoplus_{j=0}^\infty V_j .
\end{equation}
The module $V_0$ contains the half-BPS combinations, $V_1$ the antisymmetric ones, $V_2$ contains the Konishi operator, and so on. Symmetry requires $D_2$ to be a function of $j$ only, and the function is a harmonic number. In the planar limit, acting on a single trace of $L$ letters,
\begin{equation}
g^2D_2=\frac{\lambda}{8\pi^2}\,\cH,\qquad \cH=\sum_{i=1}^LH_{i,i+1},\qquad H_{12}=\sum_{j=0}^\infty2h(j)\,P_j,\qquad h(j)=\sum_{k=1}^j\frac1k .
\label{eq:Beisert}
\end{equation}
Here $P_j$ projects the pair of neighbouring letters onto $V_j$. The Konishi operator is two letters in $V_2$ and has two links, so its anomalous dimension is $2\times2h(2)\times\lambda/8\pi^2=3\lambda/4\pi^2$, which is the known value.

Away from the planar limit the same two-letter kernel is dressed with colour in a definite way. Each letter is removed from wherever it sits by a derivative $\check W=\partial/\partial W$ and the transformed letter is put back, with the colour indices contracted in commutators,
\begin{equation}
\cH=-\frac1N\sum_j h(j)\,(P_j)^{AB}_{CD}\;{:}\tr\,[W_A,\check W^C][W_B,\check W^D]{:}\; .
\label{eq:Hfull}
\end{equation}
Acting inside one trace this reproduces \eqref{eq:Beisert} plus terms of order $1/N$ that split the trace into two. Acting on two traces it can join them. We call \eqref{eq:Hfull} the complete one-loop operator, and from now on $\cH$ means \eqref{eq:Hfull}. It is convenient to absorb the temperature and write
\begin{equation}
\beta g^2D_2=G\,\cH,\qquad G=2g^2\beta=\frac{\beta\lambda}{8\pi^2}.
\label{eq:G}
\end{equation}
The combination $G$ is the natural expansion parameter of the thermal problem.

\subsection{The shift of the Hagedorn temperature}

We need the average $\vev{\cH}_U$. It is of order $N^2$, and its expansion in moments begins
\begin{equation}
\frac{1}{N^2}\vev{\cH}_U=d_1(x)\,|u_1|^2+c_3(x)\,\big(u_1^2u_{-2}+{\rm c.c.}\big)+\ldots
\label{eq:H-avg}
\end{equation}
The function $d_1(x)$ is the coefficient we want. It comes from contractions in which both letters wind once around the thermal circle. The kernel is then evaluated on the diagonal, and one finds $d_1(x)=\sum_j2h(j)\,\chi_j(x)$, where $\chi_j(x)=\Tr_{V_j}x^{D_0}$ is the partition function of the module $V_j$. Spradlin and Volovich computed the characters and did the sum in closed form. In terms of $y=\sqrt x$,
\begin{equation}
d_1(x)=\frac{2(1+y)^2}{(1-y)^6}\Big[-(1-4y+y^2)^2\ln(1-y^2)-y^2(1-8y+2y^2)\Big].
\label{eq:d1}
\end{equation}
It begins $33\,x^2$. That is the contribution of pairs of scalars.

The expression simplifies at the Hagedorn point. The polynomial multiplying the logarithm is the square of $1-4y+y^2$, which is the polynomial whose root defines $y_H$. The logarithm drops out. Using $y_H^2=4y_H-1$ one finds $(1+y_H)^2=6y_H$, $(1-y_H)^2=2y_H$ and $1-8y_H+2y_H^2=-1$, and therefore
\begin{equation}
d_1(x_H)=\frac{2\cdot6y_H\cdot y_H^2}{8y_H^3}=\frac32 .
\label{eq:d1H}
\end{equation}

With \eqref{eq:S1}, \eqref{eq:G} and \eqref{eq:H-avg} the mass of the Polyakov loop to this order is
\begin{equation}
m_1^2=1-z(x)+G\,d_1(x)+O(g^4).
\label{eq:m1-oneloop}
\end{equation}
The Hagedorn temperature is where it vanishes. Near $\beta_0$ we have $1-z\simeq\tfrac32(\beta-\beta_0)$ by \eqref{eq:slope}, so $\tfrac32(\beta_H-\beta_0)+2g^2\beta_0\cdot\tfrac32=0$. The two factors of $\tfrac32$ cancel, and
\begin{equation}
T_H=T_H^{(0)}\big(1+2g^2+O(g^4)\big)=T_H^{(0)}\Big(1+\frac{\lambda}{8\pi^2}+\ldots\Big).
\label{eq:TH1}
\end{equation}
The interaction raises the Hagedorn temperature. This is consistent with strong coupling, where $T_H$ grows like $\lambda^{1/4}$.

The coefficient $c_3$ in \eqref{eq:H-avg} is the cubic coupling $\kappa=Gc_3$ of \eqref{eq:Sgeneral}. It comes from contractions in which the windings of the two letters leave three index loops with holonomies $U$, $U$ and $U^{-2}$. Spradlin and Volovich, who integrate the holonomy out, computed two of the three traces of the one-loop operator that enter it. The corresponding coefficient at fixed holonomy is $f_{11}$ of Mussel and Yacoby \cite{MusselYacoby} for general adjoint matter, and $C_{1,1,-2}$ of AMMPV \cite{AMMPV2} for pure Yang--Mills theory. It does not enter the Hagedorn temperature, but it enters $b$ through \eqref{eq:b-from-btilde}.

\subsection{Two loops and beyond, by integrability}

At the next order $m_1^2$ receives two contributions, one from $D_4$ and one from the square of $D_2$. Writing
\begin{equation}
\frac1{N^2}\vev{\cH^2}_{U,\rm conn}=C_{41}(x)|u_1|^2+\ldots,\qquad
\frac1{N^2}\vev{D_4}_U=d_4(x)|u_1|^2+\ldots
\label{eq:C41d4}
\end{equation}
for the connected average of the square and the average of the two-loop operator, the expansion of the exponential gives
\begin{equation}
m_1^2=1-z+G\,d_1+g^4\big(\beta\,d_4-2\beta^2C_{41}\big)+O(g^6).
\label{eq:m1-twoloop}
\end{equation}
The planar $D_4$ is not known in closed form on all letters. For the two-scalar $su(2)$ sector it is \cite{BKS}, and the two-loop Hagedorn temperature in that sector was obtained from it \cite{GNS}.

The complete answer came from integrability. The Hagedorn temperature is determined by the planar spectrum, and the planar spectrum of $\cN=4$ super Yang--Mills theory is governed by an integrable system \cite{MinahanZarembo,BeisertStaudacher,IntReview}. Harmark and Wilhelm turned the thermodynamic Bethe ansatz, and later the quantum spectral curve \cite{QSC}, into an equation for $T_H$ at any coupling \cite{HW1,HW2,HW3}. Its weak-coupling expansion is
\begin{equation}
T_H=T_H^{(0)}\big(1+2g^2-11.50399\ldots g^4+98.048\ldots g^6-\ldots\big),
\label{eq:HW}
\end{equation}
where the two-loop coefficient is known analytically,
\begin{equation}
-11.50399\ldots=\frac{1}{T_H^{(0)}}\Big[-\frac{86}{\sqrt3}+\frac{24\ln12}{\ln(2+\sqrt3)}\Big].
\end{equation}
The one-loop term agrees with \eqref{eq:TH1}. We will use the two-loop term in section~\ref{sec:checks} as an exact target for our Hamiltonian, by inverting \eqref{eq:m1-twoloop} for $d_4(x_H)$.

None of this determines $b$, for the following reason. The mass $m_1^2$ is the coefficient of $\tr\,U\,\tr\,U^\dagger$. In the twisted Wick expansion it comes from words in which a single index loop separates two regions of colour, and that is a single-trace, planar quantity. The coefficient of $(\tr\,U)^2(\tr\,U^\dagger)^2$ needs four index loops carrying holonomy. In $\vev{D_4}_U$ these arise only from the parts of $D_4$ that cut a trace in two or sew two traces together. Those are the first $1/N$ corrections to the dilatation operator. No sign of integrability has been found in them, and, as far as we know, at two loops they have been given only in the $su(2)$ sector \cite{BKS} and, as a conjecture, in the $su(1,1|2)$ sector \cite{Zwiebel}; see the review \cite{Kristjansen}. The full set of letters contributes at the Hagedorn temperature, and so a direct computation is needed.