LTH1264

Anomalous dimensions at large charge for $U(N)\times U(N)$ theory in three and
four dimensions
I<EMAIL_ADDRESS>and D.R.T<EMAIL_ADDRESS>

Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK

Recently it was shown that the scaling dimension of the operator $\phi^{n}$ in
$\lambda(\bar{\phi}\phi)^{2}$ theory may be computed semiclassically at the
Wilson-Fisher fixed point in $d=4-\epsilon$, for generic values of $\lambda
n$, and this was verified to two loop order in perturbation theory at leading
and subleading $n$. This result was subsequently generalised to operators of
fixed charge $Q$ in $O(N)$ theory and verified up to four loops in
perturbation theory at leading and subleading $Q$. More recently, similar
semiclassical calculations have been performed for the classically scale-
invariant $U(N)\times U(N)$ theory in four dimensions, and verified up to two
loops, once again at leading and subleading $Q$. Here we extend this
verification to four loops. We also consider the corresponding classically
scale-invariant theory in three dimensions, similarly verifying the leading
and subleading semiclassical results up to four loops in perturbation theory.

##  1 Introduction

The investigation of renormalizable scalar field theories with scale invariant
self-interactions has attracted renewed attention in recent years. Such
theories have been central to the study of conformal field theory; in
particular, the development of the theory of critical phenomena was
illuminated by the study of quartic ($\phi^{4}$) interactions in
$d=4-\epsilon$ dimensions in the crucial work of Wilson [1, 2] and Wilson and
Fisher [3]. Critical indices for the theory are determined via the
renormalisation constants of the theory.

In recent years, considerable progress [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] has
been made for this class of theories, using a semi-classical expansion in the
path integral formulation of the theory333A similar approach was also pursued
for $\phi^{6}$ theories for $d=3-\epsilon$ and $\phi^{3}$ theories for
$d=6-\epsilon$ in Refs. [14, 15, 16]. Note, however, that Ref. [15] does not
perform any of the kind of perturbative analysis which has been featured in 3
and 4 dimensions.. In particular, this approach gives a useful handle on
$n$-point amplitudes for large $n$.

In Ref [8] the anomalous dimension of the $\phi^{n}$ operator was considered
in the $O(N)$-invariant $g(\phi^{2})^{2}$ theory with an $N$-dimensional
scalar multiplet $\phi$, for large $n$ and fixed $gn^{2}$. In Ref. [9] the
scaling dimension of the same operator in the $U(1)$-invariant
$\lambda(\bar{\phi}\phi)^{2}$ theory (corresponding to the special case $N=2$)
was computed at the Wilson-Fisher fixed point $\lambda_{*}$ as a semiclassical
expansion in $\lambda_{*}$, for fixed $\lambda_{*}n$. Subsequently this was
generalised in Ref. [10] to the case of an operator of charge $Q$ in the
$O(N)$-invariant theory. In Ref. [9], the $U(1)$ result was compared with
perturbation theory up to two loops, and in Refs. [10], [17] the check was
performed for the $O(N)$ theory up to three and four loops respectively. Most
recently, the classically scale-invariant $U(N)\times U(N)$ model has been
investigated in Ref. [11]. Here once again the leading and non-leading terms
in a large charge expansion have been derived by a semi-classical calculation,
and compared with perturbative results up to two loop order. In the current
work we extend this check to three and four loops in perturbation theory, and
further perform a similar analysis and check for the classically scale-
invariant $U(N)\times U(N)$ theory in three dimensions.

The paper is organised as follows: In Section 2 we describe the semiclassical
calculation in the $U(N)\times U(N)$ case, following Ref. [11]. Then in
Section 3 we compare the result of this calculation with perturbative
calculations up to and including 4 loops. This represents a significant
extension of previous calculations. In Section 4 we extend these semiclassical
results to the corresponding classically scale-invariant theory in three
dimensions, and perform a perturbative check through four loops.

##  2 The $U(N)\times U(N)$ model

Our discussion in this section follows closely that of Ref. [11], where the
semiclassical calculation was first performed for the $U(N)\times U(N)$ model.
In this model we have an $n\times n$ complex matrix $H$, and the Lagrangian is
given by

${\cal
L}=\hbox{Tr}\left(\partial^{\mu}H^{\dagger}\partial_{\mu}H\right)+u\hbox{Tr}\left(H^{\dagger}HH^{\dagger}H\right)+v\left[\hbox{Tr}\left(H^{\dagger}H\right)\right]^{2}.$
(2.1)

Note that one qualitative difference with earlier models considered in this
context is that this is a multi-coupling theory. As usual we work within
dimensional regularisation with $d=4-\epsilon$ and with the divergences for
this theory cancelled by replacing the couplings by bare counterparts given to
leading (one-loop) order by

$\displaystyle\alpha_{y}^{B(1)}=$
$\displaystyle\alpha_{y}+\frac{1}{\epsilon}\left[\frac{2}{N}\alpha_{y}^{2}(N^{2}+4)+16\alpha_{y}\alpha_{h}+\frac{24}{N}\alpha_{h}^{2}\right],$
$\displaystyle\alpha_{h}^{B(1)}=$
$\displaystyle\alpha_{h}+\frac{1}{\epsilon}\left[\frac{12}{N}\alpha_{y}\alpha_{h}+8\alpha_{h}^{2}\right],$
(2.2)

where

$\alpha_{h}=\frac{uN}{16\pi^{2}},\quad\alpha_{y}=\frac{2vN}{16\pi^{2}}.$ (2.3)

In principle of course the field $H$ is also replaced by a bare counterpart,
but for this theory this only becomes necessary from two-loop order. For the
purposes of the semiclassical calculation presented here, it is mostly
sufficient to work with the renormalised couplings in order to avoid a
proliferation of indices; we shall point out the single instance where we need
to reinstate the bare couplings. The $\beta$-functions are given by[18]

$\displaystyle\beta_{\alpha_{y}}=$
$\displaystyle-\epsilon\alpha_{y}+\frac{2}{N}\alpha_{y}^{2}(N^{2}+4)+16\alpha_{y}\alpha_{h}+\frac{24}{N}\alpha_{h}^{2}+\ldots,$
$\displaystyle\beta_{\alpha_{h}}=$
$\displaystyle-\epsilon\alpha_{h}+\frac{12}{N}\alpha_{y}\alpha_{h}+8\alpha_{h}^{2}+\ldots,$
(2.4)

where we display only the one-loop contributions corresponding to the one-loop
bare couplings in Eq. (2.2). The conformal fixed point is defined by
$\alpha_{h}=\alpha_{h}^{*}$, $\alpha_{y}=\alpha_{y}^{*}$, with
$\alpha_{h}^{*}$, $\alpha_{y}^{*}$ satisfying

$\beta_{\alpha_{h}}=\beta_{\alpha_{y}}=0.$ (2.5)

We do not need to give $\alpha_{h}^{*}$, $\alpha_{y}^{*}$ explicitly, but as
pointed out in Ref. [11], the fixed point is complex for $N>\sqrt{3}$, which
may be associated with “walking” behaviour of the couplings near the fixed
point.

We initially work in general $d$. The semiclassical computation is performed
by mapping the theory at the conformal fixed point via a Weyl transformation
to a cylinder $\mathbb{R}\times S^{d-1}$, where $S^{d-1}$ is a sphere of
radius $R$, giving

$\displaystyle S_{\rm{cyl}}=$ $\displaystyle\int
d^{d}x\sqrt{g}[\hbox{Tr}\left(\partial^{\mu}H^{\dagger}\partial_{\mu}H\right)+u\hbox{Tr}\left(H^{\dagger}HH^{\dagger}H\right)+v\left[\hbox{Tr}\left(H^{\dagger}H\right)\right]^{2}$
$\displaystyle+m^{2}\hbox{Tr}\left(H^{\dagger}H\right)],$ (2.6)

where $g$ denotes the metric determinant and
$m^{2}=\left(\frac{d-2}{2R}\right)^{2}$ is the conformal coupling required by
Weyl invariance. We consider a stationary configuration parametrised by

$H_{0}(\tau)=e^{iM\tau}B$ (2.7)

where $M$, $B$ are constant $n\times n$ matrices. We parametrise $M$ and $B$
by

$\displaystyle B=$ $\displaystyle\hbox{diag}\\{b,b,0\ldots 0\\},$
$\displaystyle M=$ $\displaystyle-i\hbox{diag}\\{\mu,-\mu,0,\ldots 0\\},$
(2.8)

and consider an operator of charge $Q$ given by

$T_{Q}=(H_{11}H^{*}_{22})^{2J},$ (2.9)

where $Q=4J$. Note that our definition of $M$ in Eq. (2.7) differs by a factor
of 2 from that in Ref. [11], resulting in our definitions of $\mu$ also
differing by 2. The parameter $\mu$ may be regarded as a chemical potential,
satisfying

$\displaystyle J=$ $\displaystyle V\mu b^{2},$ $\displaystyle\mu^{2}=$
$\displaystyle 2(u+2v)b^{2}+m^{2},$ (2.10)

where $V$ is the cylinder volume. In four dimensions this results in the
equation for $\mu$

$R\mu=\frac{3^{\frac{1}{3}}+x^{\frac{2}{3}}}{3^{\frac{2}{3}}x^{\frac{1}{3}}},$
(2.11)

where

$x=36{\cal J}+\sqrt{1296{\cal J}^{2}-3}$ (2.12)

with

${\cal J}=2J\frac{\alpha_{h}+\alpha_{y}}{N}.$ (2.13)

For simplicity we give in Eq. (2.11) the result for $d=4$. An operator with
scaling dimension $\Delta$ corresponds to a state with energy
$E=\frac{\Delta}{R}$, which may be computed from a fixed-charge path integral
on the cylinder. The fixed charge path integral is equivalent to an
unconstrained path integral with the addition of boundary terms to the
cylinder lagrangian $S_{\rm{cyl}}$, to give an effective action
$S_{\rm{eff}}$. The scaling dimension $\Delta_{T_{Q}}$ of the operator $T_{Q}$
is expanded in a charge expansion as

$\Delta_{T_{Q}}=Q\left(\frac{d}{2}-1\right)+\gamma_{T_{Q}}=\Delta_{LO}+\Delta_{NLO}+\ldots,$
(2.14)

where $\gamma_{T_{Q}}$ is the anomalous dimension of the operator $T_{Q}$. The
leading contribution $\Delta_{LO}$ may be written as $\Delta_{LO}=RE_{LO}$ in
terms of the classical energy $E_{LO}$, obtained by substituting the classical
solution Eqs. (2.7), (2.8) into $S_{\rm{eff}}$, with the boundary terms now
resulting in an additional $4\mu^{2}b^{2}$. The leading quantum correction
$\Delta_{NLO}=RE_{NLO}$ is obtained from a functional determinant, which may
be derived from the modes associated with the quadratic fluctuations around
the classical solution. The dispersion relations for the modes
$\omega_{i}(l)$, and their corresponding multiplicities $g_{i}(N)$, are given
by

$\displaystyle\omega_{1}(l)=$ $\displaystyle\sqrt{J_{l}^{2}+\mu^{2}},\quad
g_{1}=4N-8,$ $\displaystyle\omega_{2}(l)=$
$\displaystyle\sqrt{J_{l}^{2}+\mu^{2}-a\tilde{\mu}^{2}},\quad
g_{2}=2(N-2)^{2},$ $\displaystyle\omega_{3,4}(l)=$
$\displaystyle\sqrt{J_{l}^{2}+\mu^{2}}\mp\mu,\quad g_{3,4}=2N-3,$
$\displaystyle\omega_{5,6}(l)=$
$\displaystyle\sqrt{J_{l}^{2}+\mu^{2}+2a\tilde{\mu}^{2}}\pm\mu,\quad
g_{5,6}=1,$ $\displaystyle\omega_{7,8}(l)=$
$\displaystyle\sqrt{J_{l}^{2}+3\mu^{2}-m^{2}\pm\sqrt{4J_{l}^{2}\mu^{2}+(3\mu^{2}-m^{2})^{2}}},\quad
g_{7,8}=1,$ $\displaystyle\omega_{9,10}(l)=$
$\displaystyle\left(J_{l}^{2}+2\mu^{2}+a\tilde{\mu}^{2}\pm\sqrt{4\mu^{2}J_{l}^{2}+(2\mu^{2}+a\tilde{\mu}^{2})^{2}}\right)^{\frac{1}{2}},\quad
g_{9,10}=1,$ (2.15)

where

$a=\frac{u}{u+2v}=\frac{\alpha_{h}}{\alpha_{h}+\alpha_{y}},$ (2.16)

and where

$\tilde{\mu}^{2}=\mu^{2}-m^{2}$ (2.17)

and

$J_{l}^{2}=\frac{l(l+d-2)}{R^{2}}$ (2.18)

is the eigenvalue of the Laplacian on the sphere. Then using Eq. (2.10), the
result may be written in the form

$\Delta_{LO}=\frac{Q}{4}\left[3R\mu+\frac{1}{R\mu}\right],$ (2.19)

where $R\mu$ is given by Eq. (2.11). Its expansion for small $\alpha_{h}Q$,
$\alpha_{y}Q$ takes the form

$\displaystyle\Delta_{LO}=$ $\displaystyle
Q\Bigl{[}1+\frac{Q(\alpha_{h}+\alpha_{y})}{N}-2\frac{Q^{2}(\alpha_{h}+\alpha_{y})^{2}}{N^{2}}+8\frac{Q^{3}(\alpha_{h}+\alpha_{y})^{3}}{N^{3}}$
$\displaystyle-42\frac{Q^{4}(\alpha_{h}+\alpha_{y})^{4}}{N^{4}}+\ldots\Bigr{]},$
(2.20)

where we have displayed the expansion up to fourth order, corresponding to
four loops in perturbation theory.

As mentioned earlier, the mapping to the cylinder (along with other technical
simplifications[9]) relies on conformal invariance. At leading order, the
Lagrangian Eq. (2.1) is Weyl invariant for any values of the couplings, and so
Eq. (2.19) is valid for arbitrary couplings; but now to proceed to next-to-
leading order, we must assume that we are at the conformal fixed point
$\alpha_{h}=\alpha_{h}^{*}$, $\alpha_{y}=\alpha_{y}^{*}$. We then find that
$\Delta_{NLO}$ is given by

$\Delta_{NLO}=\frac{1}{2}\sum_{l=0}^{\infty}\sigma_{l}$ (2.21)

where

$\displaystyle\sigma_{l}=Rn_{l}\sum_{i=1}^{10}g_{i}(N)\omega_{i}(l).$ (2.22)

Here

$n_{l}=\frac{(2l+d-2)\Gamma(l+d-2)}{\Gamma(l+1)\Gamma(d-1)}$ (2.23)

is the multiplicity of the Laplacian on the $d$-dimensional sphere, and
$\omega_{i}(l)$, $g_{i}(N)$ are the modes and their multiplicities defined as
in Eqs. (2.15), but now evaluated at the fixed point. For the small
$\alpha_{h}^{*}Q$, $\alpha_{y}^{*}Q$ computation, we need to isolate the
divergent contribution in the sum in Eq. (2.21). We use the large-$l$
expansion of $\sigma_{l}$,

$\sigma_{l}=\sum_{n=1}^{\infty}c_{n}l^{d-n}$ (2.24)

with

$\displaystyle c_{1}=$ $\displaystyle 2N^{2},\quad c_{2}=6N^{2},$
$\displaystyle c_{3}=$
$\displaystyle\frac{1}{\alpha_{h}^{*}+\alpha_{y}^{*}}[(R\mu^{*})^{2}(N^{2}\alpha_{y}^{*}+4N\alpha_{h}^{*}+\alpha_{y}^{*})+N^{2}(6\alpha_{h}^{*}+5\alpha_{y}^{*})-4N\alpha_{h}^{*}-\alpha_{y}^{*}],$
$\displaystyle c_{4}=$
$\displaystyle\frac{1}{\alpha_{h}^{*}+\alpha_{y}^{*}}[(R\mu^{*})^{2}(N^{2}\alpha_{y}^{*}+4N\alpha_{h}^{*}+\alpha_{y}^{*})+N^{2}(2\alpha_{h}^{*}+\alpha_{y}^{*})-4N\alpha_{h}^{*}-\alpha_{y}^{*}],$
$\displaystyle c_{5}=$
$\displaystyle-\frac{[(R\mu^{*})^{2}-1]^{2}N}{8(\alpha_{h}^{*}+\alpha_{y}^{*})^{2}}G(\alpha^{*}_{h},\alpha^{*}_{y})(1-\gamma\epsilon)$
$\displaystyle-\frac{\epsilon}{24(\alpha_{h}^{*}+\alpha_{y}^{*})^{2}}\Bigl{\\{}N^{2}[9(R\mu^{*})^{4}\alpha_{y}^{*2}+2(R\mu^{*})^{2}\alpha_{y}^{*}(5\alpha_{h}^{*}+2\alpha_{y}^{*})$
$\displaystyle+\frac{1}{5}(62\alpha_{h}^{*2}+74\alpha_{h}^{*}\alpha_{y}^{*}-3\alpha_{y}^{*2})]$
$\displaystyle+4[(R\mu^{*})^{2}-1]\alpha_{h}^{*}N[9(R\mu^{*})^{2}(\alpha_{h}^{*}+2\alpha_{y}^{*})+13\alpha_{h}^{*}+16\alpha_{y}^{*}]$
$\displaystyle+2[(R\mu^{*})^{2}-1][9(R\mu^{*})^{2}(6\alpha_{h}^{*2}+3\alpha_{h}^{*}\alpha_{y}^{*}+2\alpha_{y}^{*2})+18\alpha_{h}^{*2}+14\alpha_{h}^{*}\alpha_{y}^{*}+11\alpha_{y}^{*2}]\Bigr{\\}}$
$\displaystyle+{\cal O}(\epsilon^{2}),$ (2.25)

where for future convenience we define

$G(\alpha_{h},\alpha_{y})=\frac{2}{N}[(N^{2}+4)\alpha_{y}^{2}+2(4N+3)\alpha_{y}\alpha_{h}+4(N+3)\alpha_{h}^{2}].$
(2.26)

The ${\cal O}(\epsilon^{2})$ in $c_{5}$ refers to powers of $\epsilon$
appearing explicitly; neglecting the fact that the fixed point values
$\alpha_{h}^{*}$, $\alpha_{y}^{*}$ are themselves ${\cal O}(\epsilon)$. We
also compute $\mu^{*}$ using Eq. (2.11) but similarly evaluated at the
conformal fixed point. We can write Eq. (2.21) in the form

$\Delta_{NLO}=\rho+\frac{1}{2}\sum_{l=1}^{\infty}\overline{\sigma}_{l}.$
(2.27)

Here

$\rho=\frac{1}{2}\sum_{n=1}^{5}c_{n}\sum_{l=1}^{\infty}l^{d-n}+\frac{1}{2}\sigma_{0}$
(2.28)

and

$\overline{\sigma}_{l}=\sigma_{l}-\sum_{n=1}^{5}c_{n}l^{d-n}$ (2.29)

We have now isolated the divergent contributions in $\rho$ in order to
evaluate them separately, as explained in Refs. [9] and [10]. The sum over
$\overline{\sigma}_{l}$ is convergent and may be evaluated directly in $d=4$.
The terms with $c_{1-5}$ in Eq. (2.28) are evaluated using

$\sum_{l=1}^{\infty}l^{s}=\zeta(-s)$ (2.30)

where $\zeta(x)$ is the Riemann $\zeta$-function. The terms in Eq. (2.28) with
$n=1\ldots 4$ may be evaluated setting $d=4$ and using

$\zeta(-s)=(-1)^{s}\frac{B_{s+1}}{s+1},\quad s=0,1,2\ldots;$ (2.31)

while for the term with $n=5$ we use

$\zeta(1+\epsilon)=\frac{1}{\epsilon}+\gamma.$ (2.32)

The pole here in $\epsilon$ cancels against the poles arising from the bare
couplings $\alpha_{h}^{B(1)}$, $\alpha_{y}^{B(1)}$ in the leading-order energy
$E_{LO}$ (this being the occasion mentioned earlier where we need to
reintroduce them). This follows from the fact that $G(\alpha_{h},\alpha_{y})$
in Eq. (2.26) satisfies

$\alpha_{h}^{B(1)}+\alpha_{y}^{B(1)}=\alpha_{h}+\alpha_{y}+\frac{1}{\epsilon}G(\alpha_{h},\alpha_{y}),$
(2.33)

after substituting the expressions for the bare couplings in Eq. (2.2). We
then obtain a finite result as $\epsilon\rightarrow 0$. The terms in $\gamma$
clearly also cancel in this limit. After some calculation we find

$\displaystyle\rho=$
$\displaystyle-\frac{1}{16(\alpha_{h}^{*}+\alpha_{y}^{*})^{2}}\\{3[4(N+3)\alpha_{h}^{*2}+2(4N+3)\alpha_{h}^{*}\alpha_{y}^{*}+(N^{2}+4)\alpha_{y}^{*2}](R\mu^{*})^{4}$
$\displaystyle+2[12(N-1)\alpha_{h}^{*2}+2(2N^{2}+4N-1)\alpha_{h}^{*}\alpha_{y}^{*}+3N^{2}\alpha_{y}^{*2}](R\mu^{*})^{2}$
$\displaystyle+[4(4N^{2}-9N-3)\alpha_{h}^{*2}+2(12N^{2}-20N-7)\alpha_{h}^{*}\alpha_{y}^{*}+(7N^{2}-12)\alpha_{y}^{*2}]\\}$
$\displaystyle+\frac{1}{2}\sigma_{0},$ (2.34)

with $\omega_{i}(l)$ given by Eq. (2.15). Performing the sum over
$\overline{\sigma}_{l}$, putting everything together, and expanding in
$\alpha_{h}^{*}Q$, $\alpha_{y}^{*}Q$ up to fourth order (again corresponding
to four loops in perturbation theory), we obtain [11]

$\displaystyle\Delta_{NLO}=$
$\displaystyle-\frac{Q}{N}\frac{2(2N+7)\alpha_{h}^{*2}+(8N+9)\alpha_{h}^{*}\alpha_{y}^{*}+(N^{2}+5)\alpha_{y}^{*2}}{\alpha_{y}^{*}+\alpha_{h}^{*}}$
$\displaystyle-\frac{Q^{2}}{N^{2}}\left[2\left(2N-1)\right)\alpha_{h}^{*2}+4\left(2N-3\right)\alpha_{h}^{*}\alpha_{y}^{*}+\left(N^{2}-3\right)\alpha_{y}^{*2}\right]$
$\displaystyle+\frac{Q^{3}}{N^{3}}\Bigl{\\{}4\left[4(N+9)\alpha_{h}^{*3}+6(2N+7)\alpha_{y}^{*}\alpha_{h}^{*2}+6(2N+3)\alpha_{h}^{*}\alpha_{y}^{*2}+(N^{2}+7)\alpha_{y}^{*3}\right]\zeta_{3}$
$\displaystyle+2\left[4(N-8)\alpha_{h}^{*2}+2(4N-25)\alpha_{y}^{*}\alpha_{h}^{*}+(N^{2}-18)\alpha_{y}^{*2}\right](\alpha_{h}^{*}+\alpha_{y}^{*})\Bigr{\\}}$
$\displaystyle-\frac{Q^{4}}{N^{4}}\Bigl{\\{}\Bigl{[}4(24N+197)\alpha_{h}^{*4}+8(48N+209)\alpha_{y}^{*}\alpha_{h}^{*3}+48(12N+25)\alpha_{y}^{*2}\alpha_{h}^{*2}$
$\displaystyle+8(3N^{2}+36N+56)\alpha_{y}^{*3}\alpha_{h}^{*}+2(12N^{2}+65)\alpha_{y}^{*4}\Bigr{]}\zeta_{3}$
$\displaystyle+10\Bigl{[}2(2N+53)\alpha_{h}^{*4}+16(N+11)\alpha_{y}^{*}\alpha_{h}^{*3}+12(2N+11)\alpha_{y}^{*2}\alpha_{h}^{*2}$
$\displaystyle+8(2N+7)\alpha_{y}^{*3}\alpha_{h}^{*}+(N^{2}+15)\alpha_{y}^{*4}\Bigr{]}\zeta_{5}$
$\displaystyle+8(4N-69)\alpha_{h}^{*4}+8(16N-233)\alpha_{y}^{*}\alpha_{h}^{*3}+2(4N^{2}+80N-1187)\alpha_{y}^{*2}\alpha_{h}^{*2}$
$\displaystyle+8(2N^{2}+8N-169)\alpha_{y}^{*3}\alpha_{h}^{*}+4(2N^{2}-73)\alpha_{y}^{*4}\Bigr{\\}}+\ldots$
(2.35)

Writing $\Delta_{S}=\Delta_{LO}+\Delta_{NLO}$ for our full semiclassical
result, we find from Eqs. (2.20), (2.35)

$\displaystyle\Delta_{S}=$ $\displaystyle
Q+\frac{Q}{N}\bigg{\\{}Q(\alpha^{*}_{h}+\alpha^{*}_{y})$
$\displaystyle-\frac{2(2N+7)\alpha_{h}^{*2}+(8N+9)\alpha_{h}^{*}\alpha_{y}^{*}+(N^{2}+5)\alpha_{y}^{*2}}{\alpha_{y}^{*}+\alpha_{h}^{*}}\bigg{\\}}$
$\displaystyle-\frac{Q^{2}}{N^{2}}\bigg{\\{}2Q(\alpha^{*}_{h}+\alpha^{*}_{y})^{2}$
$\displaystyle+\left[2\left(2N-1)\right)\alpha_{h}^{*2}+4\left(2N-3\right)\alpha_{h}^{*}\alpha_{y}^{*}+\left(N^{2}-3\right)\alpha_{y}^{*2}\right]\bigg{\\}}$
$\displaystyle+\frac{Q^{3}}{N^{3}}\bigg{\\{}8Q(\alpha^{*}_{h}+\alpha^{*}_{y})^{3}$
$\displaystyle+4\left[4(N+9)\alpha_{h}^{*3}+6(2N+7)\alpha_{y}^{*}\alpha_{h}^{*2}+6(2N+3)\alpha_{h}^{*}\alpha_{y}^{*2}+(N^{2}+7)\alpha_{y}^{*3}\right]\zeta_{3}$
$\displaystyle+2\left[4(N-8)\alpha_{h}^{*2}+2(4N-25)\alpha_{y}^{*}\alpha_{h}^{*}+(N^{2}-18)\alpha_{y}^{*2}\right](\alpha_{h}^{*}+\alpha_{y}^{*})\bigg{\\}}$
$\displaystyle-\frac{Q^{4}}{N^{4}}\bigg{\\{}42Q(\alpha^{*}_{h}+\alpha^{*}_{y})^{4}$
$\displaystyle+\Bigl{[}4(24N+197)\alpha_{h}^{*4}+8(48N+209)\alpha_{y}^{*}\alpha_{h}^{*3}+48(12N+25)\alpha_{y}^{*2}\alpha_{h}^{*2}$
$\displaystyle+8(3N^{2}+36N+56)\alpha_{y}^{*3}\alpha_{h}^{*}+2(12N^{2}+65)\alpha_{y}^{*4}\Bigr{]}\zeta_{3}$
$\displaystyle+10\Bigl{[}2(2N+53)\alpha_{h}^{*4}+16(N+11)\alpha_{y}^{*}\alpha_{h}^{*3}+12(2N+11)\alpha_{y}^{*2}\alpha_{h}^{*2}$
$\displaystyle+8(2N+7)\alpha_{y}^{*3}\alpha_{h}^{*}+(N^{2}+15)\alpha_{y}^{*4}\Bigr{]}\zeta_{5}$
$\displaystyle+8(4N-69)\alpha_{h}^{*4}+8(16N-233)\alpha_{y}^{*}\alpha_{h}^{*3}+2(4N^{2}+80N-1187)\alpha_{y}^{*2}\alpha_{h}^{*2}$
$\displaystyle+8(2N^{2}+8N-169)\alpha_{y}^{*3}\alpha_{h}^{*}+4(2N^{2}-73)\alpha_{y}^{*4}\bigg{\\}}+\ldots$
(2.36)

Finally, we note that the $U(1)$ results of Ref. [9] may be recovered by
setting $\alpha_{h}=0$, $N=1$, $Q=n$, and
$\alpha_{y}=\frac{\lambda}{32\pi^{2}}$.

##  3 The diagrammatic calculation

In this section we carry out the perturbative calculation of the anomalous
dimension $\gamma_{T_{Q}}$ on the left-hand side of Eq. (2.14), to confirm the
result displayed in Eq. (2.36); namely the semiclassical result at leading and
next-to-leading order in $Q$ up to four-loop level.

Figure 1: One- and two-loop diagrams for $\gamma_{T_{Q}}$

The one-loop contribution to $\gamma_{T_{Q}}$ comes solely from the diagram
depicted in Fig. 1(a). Strictly the propagators should be directional, with
the two ends corresponding to $H$, $H^{\dagger}$ respectively; but we have
suppressed the directional arrows for convenience. Each of our graphs in
general might thus correspond to several distinct ways of assigning the
arrows. Conversely, at three and higher loops one can draw a small number of
graphs for which there is no way of assigning the arrows; we have naturally
suppressed these. In fact, at least up to four loops the set of contributing
graphs turns out to be exactly the same as in the $U(1)$ and $O(N)$ cases. It
is not obvious to us why this is so, though possibly it can be traced back to
the equivalence mentioned in Ref. [11] between the $U(N)\times U(N)$ results
for $\alpha_{h}=0$, and those for the $O(2N^{2})$ theory.

We give the diagrammatic results in the form of a table, where the first
column labels the diagram; the second column contains the product of the
appropriate symmetry factor and a coupling-dependent term; and the third
column contains the simple pole residues for the Feynman diagram. The one, two
and three-loop results are all listed in Table 1. For the one-loop diagram,
the coupling-dependent term is given by

$A_{1}=\frac{2Q^{2}(\alpha_{h}+\alpha_{y})}{N}-\frac{2Q(2\alpha_{h}+\alpha_{y})}{N},$
(3.1)

and the one-loop contribution to $\gamma^{(1)}_{T_{Q}}$ is therefore given by

$\gamma^{(1)}_{T_{Q}}=\frac{Q^{2}(\alpha_{h}+\alpha_{y})}{N}-\frac{Q(2\alpha_{h}+\alpha_{y})}{N}.$
(3.2)

As mentioned before, the derivation of the semiclassical result relied on
working at the conformal fixed point. However, surprisingly, at two, three and
four loops we will see that the functional forms of the semiclassical and
perturbative results agree for general $\alpha_{h}$, $\alpha_{y}$ and not just
on substitution of $\alpha_{h}=\alpha_{h}^{*}$, $\alpha_{y}=\alpha_{y}^{*}$,
with $\alpha_{h}^{*}$, $\alpha_{y}^{*}$ defined by Eq. (2.5). It is only at
one loop where the agreement only holds at the fixed point. Specifically, the
${\cal O}(Q\alpha_{h}^{0}\epsilon)$, ${\cal O}(Q\alpha_{y}^{0}\epsilon)$ and
${\cal O}(Q\alpha_{h})$, ${\cal O}(Q\alpha_{y})$ terms appearing at the
classical and one-loop perturbative levels in
$Q\left(\frac{d}{2}-1\right)+\gamma^{(1)}_{T_{Q}}$ on the left-hand side of
Eq. (2.14) (as given in Eq. (3.2)) only agree with those in
$\Delta_{LO}+\Delta_{NLO}$ on the right-hand side of Eq. (2.14) (as obtained
from Eq. (2.36)) after substituting the fixed point values satisfying Eq.
(2.5). It is easy to see from Eq. (2.4) that the fixed point values satisfy

$\epsilon(\alpha_{y}^{*}+\alpha_{h}^{*})=G(\alpha^{*}_{h},\alpha^{*}_{y}),$
(3.3)

where $G(\alpha_{h},\alpha_{y})$ is defined in Eq. (2.26) (this identity is of
course related to Eq. (2.33)). Using this identity, it is straightforward to
check that the two sides of Eq. (2.14) agree up to the one-loop level[11]. In
this case, specialising to the fixed point has induced a mixing between the
classical and one-loop ${\cal O}(Q)$ terms. However this mixing only occurs
between the classical and one-loop results, due to the explicit ${\cal
O}(\epsilon)$ terms on the left-hand side of Eq. (2.14).

The leading ${\cal O}(Q^{3})$ two-loop contribution to $\gamma_{T_{Q}}$ comes
purely from the diagram depicted in Fig. 1(b) (with three lines emerging from
the $T_{Q}$ vertex), while the next-to-leading ${\cal O}(Q^{2})$ contributions
are generated by this diagram together with those in Fig. 1(c) (with two lines
emerging from the $T_{Q}$ vertex). The two-loop coupling-dependent terms in
Table 1 are given by

$\displaystyle B_{1}=$ $\displaystyle
4(\alpha_{h}+\alpha_{y})^{2}\frac{Q^{3}}{N^{2}}-4(3\alpha_{y}^{2}+8\alpha_{y}\alpha_{h}+6\alpha_{h}^{2})\frac{Q^{2}}{N^{2}}+\ldots,$
(3.4) $\displaystyle B_{2}=$ $\displaystyle
2[2(2N+5)\alpha_{h}^{2}+4(2N+1)\alpha_{y}\alpha_{h}+(N^{2}+3)\alpha_{y}^{2}]\frac{Q^{2}}{N^{2}}+\ldots,$
(3.5)

where the ellipses in the coupling-dependent terms here and later represent
terms of lower order in $Q$. Consequently the leading and next-to-leading
contributions to $\gamma^{(2)}_{T_{Q}}$ are given by

$\displaystyle\gamma^{(2)}_{T_{Q}}=$
$\displaystyle-\frac{2Q^{3}(\alpha_{h}+\alpha_{y})^{2}}{N^{2}}$
$\displaystyle-\frac{Q^{2}}{N^{2}}\left[2\left(2N-1\right)\alpha_{h}^{2}+4\left(2N-3\right)\alpha_{h}\alpha_{y}+\left(N^{2}-3\right)\alpha_{y}^{2}\right]+\ldots$
(3.6)

in accord with the semiclassical results in Eq. (2.36). As emphasised earlier,
this agreement holds for general $\alpha_{h}$, $\alpha_{y}$ and not just at
the conformal fixed point. This is because at two and higher loops, in
contrast to what we saw at one loop, specialising to the fixed point values
$\alpha_{h}=\alpha_{h}^{*}$, $\alpha_{y}=\alpha_{y}^{*}$ satisfying Eq. (2.5)
does not induce any mixing between leading or next-to-leading terms at
different loop orders. Therefore if Eq. (2.14) holds at the fixed point, it
must also hold in general. This point was already made in Ref. [11], though
there the argument was used to predict the full (i.e. beyond next-to-leading
order) two-loop anomalous dimension - see the Conclusions for further
discussion of this.

Figure 2: Three-loop diagrams for $\gamma_{T_{Q}}$ contributing at leading $n$
Figure 3: Three-loop diagrams for $\gamma_{T_{Q}}$ contributing at next-to-
leading $n$

The leading ${\cal O}(Q^{4})$ three-loop contributions to $\gamma_{T_{Q}}$
come purely from the diagrams depicted in Fig. 2 (with four lines emerging
from the $T_{Q}$ vertex), while the next-to-leading ${\cal O}(Q^{3})$
contributions are generated by these diagrams together with those in Fig. 3
(with three lines emerging from the $T_{Q}$ vertex).

Graph | Symmetry and | Simple Pole
---|---|---
| Coupling Factors |
1(a) | $\frac{1}{4}A_{1}$ | $2$
1(b) | $-\frac{1}{4}B_{1}$ | $\frac{1}{2}$
1(c) | $-\frac{1}{4}B_{2}$ | $\frac{1}{2}$
2(a) | $\frac{1}{16}C_{1}$ | $-\frac{2}{3}$
2(b) | $\frac{1}{16}C_{1}$ | $\frac{2}{3}$
2(c) | $\frac{1}{4}C^{\prime}_{1}$ | $\frac{4}{3}$
3(a) | $\frac{1}{4}C_{2}$ | $-\frac{2}{3}$
3(b) | $\frac{1}{4}C_{2}$ | $\frac{4}{3}$
3(c) | $\frac{1}{8}C_{3}$ | $-\frac{2}{3}$
3(d) | $\frac{1}{4}C^{\prime}_{3}$ | $\frac{4}{3}$
3(e) | $\frac{1}{6}C_{4}$ | $4\zeta_{3}$
Table 1: Three-loop results from Figs. 2, 3

The simple pole contributions from individual three-loop diagrams may be
extracted from Ref. [19] and are listed in Table 1, together with the
corresponding symmetry factors, and a coupling-dependent term $C_{1-4}$. The
latter are given by

$\displaystyle C_{1}=$ $\displaystyle
8\frac{Q^{4}}{N^{3}}(\alpha_{h}+\alpha_{y})^{3}-8\frac{Q^{3}}{N^{3}}(3\alpha_{h}+2\alpha_{y})(4\alpha_{h}^{2}+6\alpha_{h}\alpha_{y}+3\alpha_{y}^{2})+\ldots,$
$\displaystyle C_{1}^{\prime}=$ $\displaystyle
C_{1}-4\frac{Q^{3}}{N^{3}}\alpha_{h}^{2}\alpha_{y}+\ldots,$ $\displaystyle
C_{2}=$ $\displaystyle 2\frac{Q}{N}(\alpha_{h}+\alpha_{y})B_{2}+\ldots,$
$\displaystyle C_{3}=$ $\displaystyle
8\frac{Q^{3}}{N^{3}}(\alpha_{h}+\alpha_{y})(2\alpha_{h}^{2}+2\alpha_{h}\alpha_{y}+\alpha_{y}^{2})+\ldots,$
$\displaystyle C^{\prime}_{3}=$ $\displaystyle
C_{2}-4\frac{Q^{3}}{N^{3}}\alpha_{h}^{2}\alpha_{y}+\ldots,$ $\displaystyle
C_{4}=$ $\displaystyle
2\frac{Q^{3}}{N^{3}}[N^{2}\alpha_{y}^{3}+4N\alpha_{h}(\alpha_{h}^{2}+3\alpha_{h}\alpha_{y}+3\alpha_{y}^{2})$
$\displaystyle+36\alpha_{h}^{3}+42\alpha_{h}^{2}\alpha_{y}+18\alpha_{h}\alpha_{y}^{2}+7\alpha_{y}^{3}]+\ldots,$
(3.7)

where $B_{2}$ is defined in Eq. (3.5). We shall see shortly that relations
like that between $C_{2}$ and $B_{2}$ occur more frequently at the next loop
order. A necessary but not sufficient condition for a relation of this kind
seems to be that a diagram containing the lower-loop quantity (Fig. 1(c) in
this case, as we see from Table 1) may be formed by removing one of the lines
emerging from the $T_{Q}$ vertex (the “lozenge”) in a diagram containing the
higher-loop quantity (Fig. 3(a) or 3(b) in this case). The line removed may
incorporate a four-point vertex with two external lines, in which case an
extra external line is added at the end of the removed line; otherwise, when
the line is removed, the external line at its end is also removed. We
emphasise that this relation only holds true at leading order in $Q$; and
indeed a similar but somewhat trivial relation holds between the leading terms
in $C_{1}$, $C_{1}^{\prime}$ and $B_{1}$ (defined in Eq. (3.5)), and also
between $B_{1}$ and $A_{1}$ in Eq. (3.2). When the three-loop contributions
from the table are added and multiplied by a loop factor of 3, the leading and
non-leading three-loop contributions to $\gamma_{T_{Q}}$ are found to be

$\displaystyle\gamma^{(3)}_{T_{Q}}=$ $\displaystyle
8\frac{Q^{4}(\alpha_{h}+\alpha_{y})^{3}}{N^{3}}$
$\displaystyle+\frac{Q^{3}}{N^{3}}\Bigl{\\{}4\left[4(N+9)\alpha_{h}^{3}+6(2N+7)\alpha_{y}\alpha_{h}^{2}+6(2N+3)\alpha_{h}\alpha_{y}^{2}+(N^{2}+7)\alpha_{y}^{3}\right]\zeta_{3}$
$\displaystyle+2\left[4(N-8)\alpha_{h}^{2}+2(4N-25)\alpha_{y}\alpha_{h}+(N^{2}-18)\alpha_{y}^{2}\right](\alpha_{h}+\alpha_{y})\Bigr{\\}}+\ldots$
(3.8)

once again in accord with the semiclassical results in Eqs. (2.36), for
general $\alpha_{h}$, $\alpha_{y}$.

The leading ${\cal O}(Q^{5})$ four-loop contributions to $\gamma_{T_{Q}}$ come
purely from the diagrams depicted in Fig. 4 (with five lines emerging from the
$T_{Q}$ vertex), while the next-to-leading ${\cal O}(Q^{4})$ contributions are
generated by these diagrams together with those in Fig. 5 (with four lines
emerging from the $T_{Q}$ vertex).

Figure 4: Four-loop diagrams for $\gamma_{T_{Q}}$ contributing at leading $n$ Graph | Symmetry and | Simple Pole
---|---|---
| Coupling Factors |
4(a) | $-\frac{1}{16}D_{1}$ | $-\frac{5}{6}$
4(b) | $-\frac{1}{16}D_{1}$ | $\frac{11}{6}$
4(c) | $-\frac{1}{8}D_{2}$ | $-\frac{2}{3}$
4(d) | $-\frac{1}{8}D_{2}$ | $\frac{2}{3}$
4(e) | $-\frac{1}{4}D_{3}$ | $\frac{5}{2}$
4(f) | $-\frac{1}{16}D_{4}$ | $-\frac{1}{2}$
Table 2: Four-loop results from Fig. 4 Figure 5: Four-loop diagrams for $\gamma_{T_{Q}}$ contributing at next-to-leading $n$ Figure 6: Four-loop diagrams for $\gamma_{T_{Q}}$ contributing at next-to-leading $n$ (continued) Graph | Symmetry and | Simple Pole
---|---|---
| Coupling Factors |
5(a) | $-\frac{1}{16}E_{1}$ | $\frac{1}{6}(11-6\zeta_{3})$
5(b) | $-\frac{1}{8}E_{1}$ | $-\frac{1}{2}$
5(c) | $-\frac{1}{8}E_{1}$ | $-\frac{1}{6}(5-6\zeta_{3})$
5(d) | $-\frac{1}{8}E_{1}$ | $-\frac{5}{6}$
5(e) | $-\frac{1}{8}E_{1}$ | $-\frac{2}{3}$
5(f) | $-\frac{1}{4}E_{1}$ | $\frac{5}{2}$
5(g) | $-\frac{1}{4}E_{1}$ | $-\frac{2}{3}$
5(h) | $-\frac{1}{16}E_{1}$ | $\frac{1}{2}(1-2\zeta_{3})$
5(i) | $-\frac{1}{16}E_{2}$ | $\frac{1}{2}(1-2\zeta_{3})$
5(j) | $-\frac{1}{32}E_{2}$ | $-2(1-\zeta_{3})$
5(k) | $-\frac{1}{8}E_{2}$ | $-\frac{2}{3}$
Table 3: Four-loop results from Fig. 5 Graph | Symmetry and | Simple Pole
---|---|---
| Coupling Factors |
6(a) | $-\frac{1}{8}E_{2}$ | $-\frac{1}{6}(5-6\zeta_{3})$
6(b) | $-\frac{1}{8}E_{3}$ | $\frac{1}{6}(11-6\zeta_{3})$
6(c) | $-\frac{1}{8}E_{3}$ | $-\frac{1}{6}(5-6\zeta_{3})$
6(d) | $-\frac{1}{4}E_{3}$ | $\frac{1}{2}(5-4\zeta_{3})$
6(e) | $-\frac{1}{4}E_{3}$ | $\frac{5}{2}$
6(f) | $-\frac{1}{4}E_{4}$ | $\frac{3}{2}(2\zeta_{3}-\zeta_{4})$
6(g) | $-\frac{1}{4}E_{4}$ | $\frac{3}{2}(2\zeta_{3}+\zeta_{4})$
6(h) | $-\frac{1}{8}E_{5}$ | $-(1-2\zeta_{3})$
6(i) | $-\frac{1}{4}E_{5}$ | $\frac{1}{2}(5-4\zeta_{3})$
6(j) | $-\frac{1}{8}E_{6}$ | $10\zeta_{5}$
6(k) | $-\frac{1}{64}E_{7}$ | $-2(1-\zeta_{3})$
Table 4: Four-loop results from Fig. 6

The simple pole contributions from the four-loop diagrams in Fig. 4 were
readily evaluated using standard techniques (see for instance Ref. [20]).
Those from Figs. 5, 6 may be extracted from Ref. [19]. The contributions from
the four-loop diagrams in Figs. 4, 5 and 6 are listed in Tables 2, 3 and 4
respectively, together with the corresponding symmetry factor and coupling-
dependent factor $D_{1-4}$ or $E_{1-9}$. The latter are listed below:

$\displaystyle D_{1}=$ $\displaystyle
16\frac{Q^{5}}{N^{4}}(\alpha_{h}+\alpha_{y})^{4}-32\frac{Q^{4}}{N^{4}}(\alpha_{h}+\alpha_{y})(10\alpha_{h}^{3}+21\alpha_{h}^{2}\alpha_{y}+17\alpha_{h}\alpha_{y}^{2}+5\alpha_{y}^{3})+\ldots,$
$\displaystyle D_{2}=$ $\displaystyle
D_{1}+8\frac{Q^{4}}{N^{4}}\alpha_{h}^{2}\alpha_{y}(2\alpha_{h}+\alpha_{y})+\ldots,$
$\displaystyle D_{3}=$ $\displaystyle
D_{1}-4\frac{Q^{4}}{N^{4}}\alpha_{h}^{2}\alpha_{y}^{2}+\ldots,$ $\displaystyle
D_{4}=$ $\displaystyle
D_{1}+16\frac{Q^{4}}{N^{4}}\alpha_{h}^{2}\alpha_{y}(2\alpha_{h}+\alpha_{y})+\ldots,$
(3.9)

and furthermore

$\displaystyle E_{1}=$ $\displaystyle
2\frac{Q}{N}(\alpha_{h}+\alpha_{y})C_{2}+\ldots,$ $\displaystyle E_{2}=$
$\displaystyle 2\frac{Q}{N}(\alpha_{h}+\alpha_{y})C_{3}+\ldots,$
$\displaystyle E_{3}=$ $\displaystyle
2\frac{Q}{N}(\alpha_{h}+\alpha_{y})C^{\prime}_{3}+\ldots,$ $\displaystyle
E_{4}=$ $\displaystyle 2\frac{Q}{N}(\alpha_{h}+\alpha_{y})C_{4}+\ldots,$
$\displaystyle E_{5}=$ $\displaystyle
4\frac{Q^{4}}{N^{4}}(8\alpha_{h}^{4}+20\alpha_{y}\alpha_{h}^{3}+25\alpha_{y}^{2}\alpha_{h}^{2}+16\alpha_{y}^{3}\alpha_{h}+4\alpha_{y}^{4})+\ldots,$
$\displaystyle E_{6}=$ $\displaystyle
2\frac{Q^{4}}{N^{4}}[2(2N+53)\alpha_{h}^{4}+16\alpha_{y}(N+11)\alpha_{h}^{3}+12(2N+11)\alpha_{y}^{2}\alpha_{h}^{2}$
$\displaystyle+8(2N+7)\alpha_{y}^{3}\alpha_{h}+\alpha_{y}^{4}(N^{2}+15)]+\ldots,$
$\displaystyle E_{7}=$ $\displaystyle
16\frac{Q^{4}}{N^{4}}(2\alpha_{h}^{4}+4\alpha_{y}\alpha_{h}^{3}+6\alpha_{y}^{2}\alpha_{h}^{2}+4\alpha_{y}^{3}\alpha_{h}+\alpha_{y}^{4})+\ldots,$

where $C_{2}$, $C_{3}$, $C^{\prime}_{3}$ and $C_{4}$ are as defined in Eq.
(3.7). As anticipated earlier, we see more examples of relations between the
coupling-dependent quantities at different loop orders. It is easy to check
that the corresponding diagrams follow the rule of thumb described previously.
Furthermore, similar trivial relations hold between the leading terms in
$D_{1-4}$ in Eq. (3.9) and those in $C_{1}$, $C^{\prime}_{1}$ in Eq. (3.7).
When the four-loop results in the tables are added and multiplied by a loop
factor of 4, the leading and non-leading four-loop contributions to
$\gamma_{T_{Q}}$ are found to be

$\displaystyle\gamma^{(4)}_{T_{Q}}=$
$\displaystyle-42\frac{Q^{5}(\alpha_{h}+\alpha_{y})^{4}}{N^{4}}$
$\displaystyle-\frac{Q^{4}}{N^{4}}\Bigl{\\{}\Bigl{[}4(24N+197)\alpha_{h}^{4}+8(48N+209)\alpha_{y}\alpha_{h}^{3}+48(12N+25)\alpha_{y}^{2}\alpha_{h}^{2}$
$\displaystyle+8(3N^{2}+36N+56)\alpha_{y}^{3}\alpha_{h}+2(12N^{2}+65)\alpha_{y}^{4}\Bigr{]}\zeta_{3}$
$\displaystyle+10\Bigl{[}2(2N+53)\alpha_{h}^{4}+16(N+11)\alpha_{y}\alpha_{h}^{3}+12(2N+11)\alpha_{y}^{2}\alpha_{h}^{2}$
$\displaystyle+8(2N+7)\alpha_{y}^{3}\alpha_{h}+\alpha_{y}^{4}(N^{2}+15)\Bigr{]}\zeta_{5}$
$\displaystyle+8(4N-69)\alpha_{h}^{4}+8(16N-233)\alpha_{y}\alpha_{h}^{3}+2(4N^{2}+80N-1187)\alpha_{y}^{2}\alpha_{h}^{2}$
$\displaystyle+8(2N^{2}+8N-169)\alpha_{y}^{3}\alpha_{h}+4(2N^{2}-73)\alpha_{y}^{4}\Bigr{\\}}+\ldots$
(3.11)

once again in accord with the semiclassical results in Eqs. (2.36), for
general $\alpha_{h}$, $\alpha_{y}$.

##  4 The $d=3$ calculation

In this section we shall consider the classically scale-invariant $U(N)\times
U(N)$ theory in three dimensions. Once again we shall show how the leading and
next-to-leading terms in the charge expansion may be obtained from a
semiclassical computation, and verify by comparison with perturbative results.
Much of the formalism is a natural adaptation of the corresponding results in
four dimensions.

The Lagrangian is given by

$\displaystyle{\cal L}=$
$\displaystyle\hbox{Tr}\left(\partial^{\mu}H^{\dagger}\partial_{\mu}H\right)+u\hbox{Tr}\left(H^{\dagger}HH^{\dagger}HH^{\dagger}H\right)$
$\displaystyle+v\left[\hbox{Tr}\left(H^{\dagger}H\right)\right]^{3}+w\hbox{Tr}\left(H^{\dagger}HH^{\dagger}H\right)\hbox{Tr}\left(H^{\dagger}H\right).$
(4.1)

We define for convenience

$\alpha_{h}=\frac{uN}{64\pi^{2}},\quad\alpha_{y}=\frac{4vN}{64\pi^{2}},\quad\alpha_{w}=\frac{2wN}{64\pi^{2}}.$
(4.2)

Note that $\alpha_{h}$ and $\alpha_{y}$ are defined differently from the case
of four dimensions. Also, in $d=3$ the $\beta$-function starts at two-loop
order and the model is conformally invariant up to ${\cal O}(\alpha_{h,y,w})$
so that at leading and next-to-leading order our results are valid away from
the conformal fixed point. As in the $d=4$ case, we map to a cylinder and
consider a stationary solution given by Eqs. (2.7), (2.8). The relation
between $J$ and $\mu$ is as in Eq. (2.10) but $\mu$ is now given by

$\mu^{2}=3(u+4v+2w)b^{4}+m^{2},$ (4.3)

with of course now a different value for
$m^{2}=\left(\frac{d-2}{2R}\right)^{2}$ with $d=3$. Consequently we now find

$R\mu=\frac{1}{2\sqrt{2}}\sqrt{1+\sqrt{1+{\cal J}}}$ (4.4)

where

${\cal J}=48Q^{2}\frac{\alpha_{h}+\alpha_{y}+\alpha_{w}}{N}.$ (4.5)

The scaling dimension $\Delta_{T_{Q}}$ is once again expanded as in Eq.
(2.14), but now we find on computing the classical energy by substituting Eqs.
(2.7) and (2.8) into Eq. (4.1) and using Eqs. (2.10) and (4.3)

$\Delta_{LO}=RE_{LO}=\frac{Q}{12}\left[8R\mu+\frac{1}{R\mu}\right],$ (4.6)

where $R\mu$ is given by Eq. (4.4). $\Delta_{LO}$ now has an expansion

$\Delta_{LO}=Q\left(\frac{1}{2}+Q^{2}\frac{\alpha_{h}+\alpha_{w}+\alpha_{y}}{N}-9Q^{4}\frac{(\alpha_{h}+\alpha_{w}+\alpha_{y})^{2}}{N^{2}}+\ldots\right).$
(4.7)

The modes $\omega_{i}(l)$ and their corresponding multiplicities $g_{i}(N)$
are given by

$\displaystyle\omega_{1}(l)=$ $\displaystyle\sqrt{J_{l}^{2}+\mu^{2}},\quad
g_{1}=4N-8,$ $\displaystyle\omega_{2}(l)=$
$\displaystyle\sqrt{J_{l}^{2}+\mu^{2}-\tilde{a}\tilde{\mu}^{2}},\quad
g_{2}=2(N-2)^{2},$ $\displaystyle\omega_{3,4}(l)=$
$\displaystyle\sqrt{J_{l}^{2}+\mu^{2}}\mp\mu,\quad g_{3,4}=2N-3,$
$\displaystyle\omega_{5,6}(l)=$
$\displaystyle\sqrt{J_{l}^{2}+\mu^{2}+4\overline{a}\tilde{\mu}^{2}}\pm\mu,\quad
g_{5,6}=1,$ $\displaystyle\omega_{7,8}(l)=$
$\displaystyle\sqrt{J_{l}^{2}+4\mu^{2}-2m^{2}\pm
2\sqrt{J_{l}^{2}\mu^{2}+(2\mu^{2}-m^{2})^{2}}},\quad g_{7,8}=1,$
$\displaystyle\omega_{9,10}(l)=$
$\displaystyle\left(J_{l}^{2}+2\mu^{2}+2\overline{a}\tilde{\mu}^{2}\pm
2\sqrt{\mu^{2}J_{l}^{2}+(\mu^{2}+\overline{a}\tilde{\mu}^{2})^{2}}\right)^{\frac{1}{2}},\quad
g_{9,10}=1,$ (4.8)

where $\tilde{\mu}$ is defined in Eq. (2.17), and

$\displaystyle\overline{a}=$
$\displaystyle\frac{u+\frac{2}{3}w}{u+4v+2w}=\frac{\alpha_{h}+\frac{1}{3}\alpha_{w}}{\alpha_{h}+\alpha_{y}+\alpha_{w}},$
$\displaystyle\tilde{a}=$
$\displaystyle\frac{u+\frac{4}{3}w}{u+4v+2w}=\frac{\alpha_{h}+\frac{2}{3}\alpha_{w}}{\alpha_{h}+\alpha_{y}+\alpha_{w}},$
(4.9)

Notice that the modes in the $d=3$ case in Eq. (4.8) may be simply obtained
from those for the $d=4$ case in Eq. (2.15), by replacing $a$ (defined in Eq.
(2.16)) in $\omega_{2}(l)$ by $\tilde{a}$ (defined in Eq. (4.9)) and $a$ in
$\omega_{5,6}(l)$ and $\omega_{9,10}(l)$ by $2\overline{a}$ (again defined in
Eq. (4.9)).

The next-to-leading contribution to the anomalous dimension is again given by
Eqs. (2.21), (2.22) with the multiplicity $n_{l}$ given by Eq. (2.23). In the
large-$l$ expansion of Eq. (2.24), we now have

$\displaystyle c_{1}=c_{2}=$ $\displaystyle 4N^{2},$ $\displaystyle c_{3}=$
$\displaystyle\frac{1}{6(\alpha_{h}+\alpha_{w}+\alpha_{y})}\Bigl{\\{}[4(\alpha_{w}+3\alpha_{y})(R\mu)^{2}+6\alpha_{h}+5\alpha_{w}+3\alpha_{y}]N^{2}$
$\displaystyle+4(3\alpha_{h}+2\alpha_{w})[4(R\mu)^{2}-1]N+2[4(R\mu)^{2}-1](6\alpha_{h}+2\alpha_{w}+3\alpha_{y})\Bigr{\\}},$
$\displaystyle c_{4}=$
$\displaystyle\epsilon\left(\frac{1}{3}N^{2}-\frac{1}{2}c_{3}\right)+{\cal
O}(\epsilon^{2}).$ (4.10)

We explicitly isolate the divergent contribution to the sum over $l$ in a
similar expression to Eq. (2.27):

$\Delta_{NLO}=\rho+\frac{1}{2}\sum_{l=1}^{\infty}\overline{\sigma}_{l}$ (4.11)

but now valid away from the conformal fixed point and with

$\overline{\sigma}_{l}=\sigma_{l}-\sum_{n=1}^{4}c_{n}l^{d-n}$ (4.12)

and

$\rho=\frac{1}{2}\sum_{n=1}^{4}c_{n}\sum_{l=1}^{\infty}l^{d-n}+\frac{1}{2}\sigma_{0}$
(4.13)

where the $\omega_{i}(l)$ are as given in Eq. (4.8). Here once again the
divergent parts have been isolated and the sums over $l$ in Eq. (4.13) may now
be performed. There is a slight subtlety here. With now $d=3-\epsilon$ the sum
over $\frac{1}{l^{d-n}}$ for $n=4$ leads using Eqs. (2.30), (2.32) to a pole
in $\epsilon$ which multiplied by $c_{4}$ in Eq. (4.10) leaves a finite
remainder. Using also Eq. (2.31), we obtain

$\rho=-\frac{1}{2}c_{3}+\frac{1}{2}\sigma_{0},$ (4.14)

The sum over $\overline{\sigma}_{l}$ is then finite and we may set $d=3$,
whereupon $c_{4}$ vanishes from $\overline{\sigma}_{l}$. Expanding in small
$\alpha_{h}Q$, $\alpha_{y}Q$, $\alpha_{w}Q$, we find

$\displaystyle\Delta_{NLO}=$
$\displaystyle-\frac{Q^{2}}{N}(6\alpha_{h}+3\alpha_{y}+4\alpha_{w})$
$\displaystyle-\frac{1}{8}\pi^{2}\frac{Q^{4}}{N^{2}}[(3\alpha_{y}+\alpha_{w})^{2}N^{2}+4(3\alpha_{h}+2\alpha_{w})(3\alpha_{h}+6\alpha_{y}+4\alpha_{w})N$
$\displaystyle+12(27\alpha_{h}^{2}+6\alpha_{y}^{2}+8\alpha_{w}^{2}+28\alpha_{h}\alpha_{w}+15\alpha_{h}\alpha_{y}+11\alpha_{y}\alpha_{w})]$
$\displaystyle+2\frac{Q^{4}}{N^{2}}(72\alpha_{h}^{2}+36\alpha_{y}^{2}+46\alpha_{w}^{2}+114\alpha_{h}\alpha_{w}+99\alpha_{h}\alpha_{y}+81\alpha_{y}\alpha_{w})+\ldots$
(4.15)

Figure 7: Two-loop diagram corresponding to $\gamma_{T_{Q}}$in $d=3$

We now perform the perturbative check of Eqs. (4.7), (4.15) up to four-loop
order (note that in odd dimensions, divergences only appear at even loop order
in dimensional regularisation). The two-loop diagram is shown in Fig. 7 and
the relevant four-loop diagrams in Fig. 8 (Fig. 8(a) providing leading and
next-to-leading contributions in $Q$, and Figs. 8(b), (c) giving the remaining
next-to-leading order contributions). As before, the lozenge represents the
location of the $T_{Q}$ vertex.

Figure 8: Four-loop diagrams corresponding to $\gamma_{T_{Q}}$ in $d=3$ Graph | Symmetry and | Simple Pole
---|---|---
| Coupling Factors |
7 | $\frac{1}{36}F$ | $2$
8(a) | $-\frac{1}{144}G_{1}$ | $4$
8(b) | $-\frac{1}{64}G_{2}$ | $\pi^{2}$
8(c) | $-\frac{1}{72}G_{3}$ | $4$
Table 5: Two-loop and four-loop results from Figs. 7, 8

The coupling-dependent factors in Table 5 are given by

$\displaystyle F=$ $\displaystyle
9\left[\frac{Q^{3}}{N}(\alpha_{h}+\alpha_{y}+\alpha_{w})-\frac{Q^{2}}{N}(6\alpha_{h}+4\alpha_{y}+3\alpha_{w})\right],$
$\displaystyle G_{1}=$ $\displaystyle
9\Bigl{[}9\frac{Q^{5}}{N^{2}}(\alpha_{h}+\alpha_{y}+\alpha_{w})^{2}$
$\displaystyle-2\frac{Q^{4}}{N^{2}}(90\alpha_{h}^{2}+45\alpha_{y}^{2}+56\alpha_{w}^{2}+138\alpha_{h}\alpha_{w}+117\alpha_{h}\alpha_{y}+99\alpha_{y}\alpha_{w})\Bigr{]},$
$\displaystyle G_{2}=$ $\displaystyle
2\frac{Q^{4}}{N^{2}}[(3\alpha_{y}+\alpha_{w})^{2}N^{2}+4(3\alpha_{h}+2\alpha_{w})(3\alpha_{h}+6\alpha_{y}+4\alpha_{w})N$
$\displaystyle+6(54\alpha_{h}^{2}+12\alpha_{y}^{2}+16\alpha_{w}^{2}+56\alpha_{h}\alpha_{w}+30\alpha_{h}\alpha_{y}+22\alpha_{y}\alpha_{w})],$
$\displaystyle G_{3}=$ $\displaystyle
9\frac{Q^{4}}{N^{2}}(18\alpha_{h}^{2}+9\alpha_{y}^{2}+10\alpha_{w}^{2}+24\alpha_{h}\alpha_{w}+18\alpha_{h}\alpha_{y}+18\alpha_{y}\alpha_{w}).$
(4.16)

One might guess at a similar relation between leading-order coupling-dependent
factors at successive loops as we found in four dimensions, this time with a
relative factor of $9\frac{Q^{2}}{N}(\alpha_{h}+\alpha_{y}+\alpha_{w})$; but
more evidence would be required to be convincing. After adding the
contributions from the diagrams and multiplying by a loop factor of two or
four as appropriate, we find

$\displaystyle\gamma_{T_{Q}}=$
$\displaystyle\frac{Q^{3}}{N}(\alpha_{h}+\alpha_{y}+\alpha_{w})-Q^{2}(6\alpha_{h}+3\alpha_{y}+4\alpha_{w})$
$\displaystyle-9\frac{Q^{5}}{N^{2}}(\alpha_{h}+\alpha_{y}+\alpha_{w})^{2}$
$\displaystyle-\frac{1}{8}\pi^{2}\frac{Q^{4}}{N^{2}}[(3\alpha_{y}+\alpha_{w})^{2}N^{2}+4(3\alpha_{h}+2\alpha_{w})(3\alpha_{h}+6\alpha_{y}+4\alpha_{w})N$
$\displaystyle+12(27\alpha_{h}^{2}+6\alpha_{y}^{2}+8\alpha_{w}^{2}+28\alpha_{h}\alpha_{w}+15\alpha_{h}\alpha_{y}+11\alpha_{y}\alpha_{w})]$
$\displaystyle+2\frac{Q^{4}}{N^{2}}(72\alpha_{h}^{2}+36\alpha_{y}^{2}+46\alpha_{w}^{2}+114\alpha_{h}\alpha_{w}+99\alpha_{h}\alpha_{y}+81\alpha_{y}\alpha_{w})+\ldots$
(4.17)

We easily see that Eq. (2.14) is satisfied with $\Delta_{LO}$ and
$\Delta_{NLO}$ as given by Eqs. (4.7), (4.15). Note that the agreement is for
general values of the couplings and not only at the fixed point, even at
lowest perturbative order. This is because there is no additional mixing for
leading and non-leading $Q$ at the fixed point where $\alpha_{h,y,w}={\cal
O}(\epsilon)$; for instance the leading and non-leading terms at lowest order
(two loops) are ${\cal O}(Q^{3})$ and ${\cal O}(Q^{2})$, whereas the classical
terms are ${\cal O}(Q)$. We note that the $U(1)$ results of Ref. [14], [16]
may be recovered by setting $\alpha_{h}=\alpha_{w}=0$, $N=1$, $Q=n$, and
$\alpha_{y}=\frac{\lambda^{2}}{192\pi^{2}}$.

##  5 Conclusions

We have extended the investigations of Ref. [11] into large charge operators
in scalar $U(N)\times U(N)$ theory. However we have proceeded in a slightly
different direction; the authors of Ref. [11] used their leading order (LO)
and next-to-leading order (NLO) semiclassical result, combined with a known
two-loop result for a particular charge ($Q=2$) to deduce the next-to-next-to-
leading order (NNLO) two-loop terms for general $Q$, hence obtaining the full
two-loop result for general $Q$. We have taken the approach of using an
operator of general charge $Q$ and thereby performing a perturbative check of
the LO and NLO semiclassical results up to four-loop order, for general $Q$.
This was facilitated by the fact that many of the relevant three-loop and
four-loop Feynman diagrams had already been evaluated, in Ref. [19]. This in
turn was because the NLO graphs at three loops contained a $T_{Q}$ vertex
(corresponding to the fixed-charge operator) with three internal lines, and
the LO graphs at three loops and NLO graphs at four loops contained a $T_{Q}$
vertex with four internal lines. These types of diagrams had already occurred
and been evaluated in the computation of the $\beta$-function for the theory
(which of course involves only four-point vertices and therefore graphs with
four or fewer internal lines). One could pursue the approach of Ref. [11] and
use the LO and NLO semiclassical results to try to deduce lower-order (in $Q$)
perturbative results, but beyond two loops it seems (due to the increasing
numbers of contributions from different powers of $Q$ involved) one would
still need to supplement this information with additional perturbative
computations in order to obtain the full perturbative result.

We have also extended the analysis of Ref. [11] to a classically scale-
invariant $U(N)\times U(N)$ theory in three dimensions. It is interesting to
see how much of the formalism is rather similar. We were able to adapt
perturbative results from Ref.[16] to the present theory; we only performed
the comparison with the semiclassical results up to four-loop order, but the
results of Ref. [16] would permit a six-loop check if desired.

In Ref. [12], more general theories such as $U(N)\times U(M)$ and also more
general fixed-charge operators are considered. In this case the NLO result
cannot be computed analytically and so there is probably little to be gained
by a detailed perturbative comparison to high orders. It would be interesting
to know if the $d=3$ NLO semiclassical contribution is equally intractable to
analytic computation in these scenarios.

We have not considered the large $\alpha_{h,y}Q$ (for $d=4$) or large
$\alpha_{h,y,w}Q$ (for $d=3$) expansions in the current work, but these should
be straightforward if required, following the method of Refs. [9], [14];
though the dependence of the modes on multiple couplings might present
technical problems in the curve-fitting required.

The marriage of semi-classical methods with perturbation theory has continued
to prove a productive method of probing classically scale invariant theories
involving scalar field self-interactions in 3, 4 and 6 dimensions. Obviously,
scale invariant theories of particular significance that await exploration are
gauge theories, both QCD (in $d=4$) and Chern Simons theories (in $d=3$). Also
of interest would be the Thirring model (quartic fermion interactions in
$d=2$). The massless Thirring model is exactly solvable inasmuch as the
$nn$-points field correlation is known, which would make for an interesting
test case. Regarding the massive model, the mass spectrum of the model and the
scattering matrix have been evaluated. An explicit formula for the
correlations is not known; so, again, exploration from the perspective used
here would be worthwhile.

We also plan to perform a perturbative analysis of the kind we have presented
here for $\phi^{3}$ interactions in six dimensions, where in a recent
interesting article[21] semiclassical methods have been used to check the
equivalence of cubic and quartic $O(N)$ theories, as conjectured in Refs.
[22], [23].

## Acknowledgements

We are grateful to Francesco Sannino and Chen Zhang for very helpful
correspondence. DRTJ thanks the Leverhulme Trust for the award of an Emeritus
Fellowship. This research was supported by the Leverhulme Trust, STFC and by
the University of Liverpool.

## References

  * [1] K.G. Wilson, “Renormalization group and critical phenomena. I. Renormalization Group and the Kadanoff scaling picture”, Phys. Rev. B 4, 3174 (1971).
  * [2] K.G. Wilson, “Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior”, Phys. Rev. B 4, 3184 (1971).
  * [3] K.G. Wilson and Michael E. Fisher, “Critical exponents in 3.99 dimensions”, Phys. Rev. Lett. 28, 240 (1972).
  * [4] D.T. Son, “Semiclassical approach for multiparticle production in scalar theories”, Nucl.Phys. B477, 378 (1996).
  * [5] S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, “On the CFT operator spectrum at large global charge, J. High Energy Phys. 12 (2015) 071, [arXiv:1505.01537 [hep-th]].
  * [6] L. Alvarez-Gaume, O. Loukas, D. Orlando and S. Reffert, “Compensating strong coupling with large charge”, J. High Energy Phys. 04 (2017) 059, [arXiv:1610.04495 [hep-th]].
  * [7] L. Alvarez-Gaume, D. Orlando and S. Reffert, “Large charge at large N”, J. High Energy Phys. 12 (2019) 142, [arXiv:1909.12571 [hep-th]].
  * [8] G. Arias-Tamargo, D. Rodriguez-Gomez and J.G. Russo, “The large charge limit of scalar field theories and the Wilson-Fisher fixed point at $\epsilon=0$,” J. High Energy Phys. 10 (2019) 201, [arXiv:1908.11347 [hep-th]].
  * [9] G. Badel, G. Cuomo, A. Monin and R. Rattazzi, “The epsilon expansion meets semiclassics”, J. High Energy Phys. 11 (2019) 110 [arXiv:1909.01269[hep-th]].
  * [10] O. Antipin, J. Bersini, F. Sannino, Z. Wang and C. Zhang, “Charging the $O(N)$ model,” Phys. Rev. D 102, 045011 (2020) [arXiv:2003.13121 [hep-th]].
  * [11] O. Antipin, J. Bersini, F. Sannino, Z. Wang and C. Zhang, “Charging the walking $U(N)\otimes U(N)$ Higgs theory as a complex CFT,” Phys. Rev. D 102, 125033 (2020) [arXiv:2006.10078 [hep-th]].
  * [12] O. Antipin, J. Bersini, F. Sannino, Z. W. Wang and C. Zhang, “Untangling scaling dimensions of fixed charge operators in Higgs Theories,” Phys. Rev. D 103, 125024 (2021) [arXiv:2102.04390 [hep-th]].
  * [13] L. Alvarez-Gaume, D. Orlando and S. Reffert, “Selected topics in the large quantum number expansion”, arXiv:2008.03308 [hep-th].
  * [14] G. Badel, G. Cuomo, A. Monin and R. Rattazzi, “Feynman diagrams and the large charge expansion in $3-\varepsilon$ dimensions,” Phys. Lett. B 802, 135202 (2020) [arXiv:1911.08505 [hep-th]].
  * [15] Guillermo Arias-Tamargo, Diego Rodriguez-Gomez and Jorge G. Russo, “On the UV completion of the $O(N)$ model in $6-\epsilon$ dimensions: a stable large-charge sector”, J. High Energy Phys. 09 (2020) 064 [arXiv:2003.13772 [hep-th]]
  * [16] I. Jack and D.R.T. Jones, “Anomalous dimensions for $\phi^{n}$ in scale invariant $d=3$ theory.”, Phys. Rev. D 102, 085012 (2020) [arXiv:2007.07190 [hep-th]]
  * [17] I. Jack and D. R. T. Jones, “Anomalous dimensions at large charge in $d=4$ O(N) theory,” Phys. Rev. D 103, 085013 (2021) [arXiv:2101.09820 [hep-th]].
  * [18] R. D. Pisarski and D. L. Stein, “Critical behavior of linear $\phi^{4}$ models with $G\times G^{\prime}$ symmetry,” Phys. Rev. B 23, 3549 (1981)
  * [19] D. I. Kazakov, O. V. Tarasov and A. A. Vladimirov, “Calculation of critical exponents by quantum field theory methods,” Sov. Phys. JETP 50, 521 (1979) JINR-E2-12249.
  * [20] H. Kleinert and V. Schulte-Frohlinde, “Critical properties of $\phi^{4}$-theories”, World Scientific (2001) .
  * [21] O. Antipin, J. Bersini, F. Sannino, Z. W. Wang and C. Zhang, [arXiv:2107.02528 [hep-th]].
  * [22] L. Fei, S. Giombi and I. R. Klebanov, Phys. Rev. D 90, 025018 (2014) [arXiv:1404.1094 [hep-th]].
  * [23] L. Fei, S. Giombi, I. R. Klebanov and G. Tarnopolsky, Phys. Rev. D 91, 045011 (2015) [arXiv:1411.1099 [hep-th]].


$\pm$ 1.18 2.7 $\pm$ 1.15 55–60% 7.77 8.11 13.25 $\pm$ 6.69 14.3 $\pm$ 5.65
0.318 $\pm$ 0.16 0.612 $\pm$ 0.225 0.447 $\pm$ 0.208 1.24 $\pm$ 1.07 2.11
$\pm$ 1.04 60–65% 8.11 8.45 9.988 $\pm$ 5.67 11.4 $\pm$ 5.13 0.24 $\pm$ 0.14
0.654 $\pm$ 0.228 0.443 $\pm$ 0.22 1.11 $\pm$ 0.954 1.64 $\pm$ 0.922 65–70%
8.45 8.78 7.576 $\pm$ 4.75 9.111 $\pm$ 4.56 0.182 $\pm$ 0.11 0.692 $\pm$ 0.23
0.431 $\pm$ 0.236 1 $\pm$ 0.84 1.29 $\pm$ 0.811 70–75% 8.78 9.11 5.774 $\pm$
3.9 7.305 $\pm$ 3.94 0.139 $\pm$ 0.094 0.726 $\pm$ 0.231 0.413 $\pm$ 0.255
0.91 $\pm$ 0.738 1.01 $\pm$ 0.699 75–80% 9.11 9.47 4.453 $\pm$ 3.18 5.906
$\pm$ 3.34 0.107 $\pm$ 0.076 0.757 $\pm$ 0.233 0.391 $\pm$ 0.276 0.832 $\pm$
0.633 0.798 $\pm$ 0.599 80–85% 9.47 9.86 3.465 $\pm$ 2.55 4.822 $\pm$ 2.78
0.0833 $\pm$ 0.061 0.787 $\pm$ 0.233 0.362 $\pm$ 0.295 0.774 $\pm$ 0.553 0.636
$\pm$ 0.51 85–90% 9.86 10.3 2.703 $\pm$ 2 3.953 $\pm$ 2.23 0.065 $\pm$ 0.048
0.818 $\pm$ 0.231 0.325 $\pm$ 0.311 0.721 $\pm$ 0.457 0.509 $\pm$ 0.43 90–95%
10.3 11 2.116 $\pm$ 1.52 3.261 $\pm$ 1.7 0.0509 $\pm$ 0.037 0.852 $\pm$ 0.223
0.277 $\pm$ 0.319 0.679 $\pm$ 0.377 0.409 $\pm$ 0.36 95–100% 11 19.1 1.582
$\pm$ 1.06 2.629 $\pm$ 1.15 0.038 $\pm$ 0.025 0.902 $\pm$ 0.2 0.191 $\pm$
0.304 0.639 $\pm$ 0.292 0.316 $\pm$ 0.294 0–100% 0 19.1 47.93 $\pm$ 59 32.51
$\pm$ 31.7 1.15 $\pm$ 1.4 0.537 $\pm$ 0.3 0.346 $\pm$ 0.233 1.61 $\pm$ 1.54
5.7 $\pm$ 6.2

Table 21: Various quantities for CuCu collisions at $\sqrt{s_{\rm NN}}=0.2$
TeV obtained with the improved MCG model for centrality classes defined by
slicing the impact parameter ($b$) distribution. The mean and standard
deviation of each quantity (denoted as rms) are given. Deformed profile for Cu
was used.


dedicated to my family and

to all the people

to whom I owe more than I can express …


# Time series forecasting with high stakes: A field study of the air cargo
industry

Abhinav Garg  FLYR Inc.San FranciscoCA<EMAIL_ADDRESS>and  Naman Shukla
FLYR Inc.GainesvilleFL<EMAIL_ADDRESS>
###### Abstract.

Time series forecasting in the air cargo industry presents unique challenges
due to volatile market dynamics and the significant impact of accurate
forecasts on generated revenue. This paper explores a comprehensive approach
to demand forecasting at the origin-destination (O&D) level, focusing on the
development and implementation of machine learning models in decision-making
for the air cargo industry. We leverage a mixture of experts framework,
combining statistical and advanced deep learning models to provide reliable
forecasts for cargo demand over a six-month horizon. The results demonstrate
that our approach outperforms industry benchmarks, offering actionable
insights for cargo capacity allocation and strategic decision-making in the
air cargo industry. While this work is applied in the airline industry, the
methodology is broadly applicable to any field where forecast-based decision-
making in a volatile environment is crucial.

Time Series Forecasting, Mixture of Experts Framework, Revenue Management,
Seasonal Trends, Deep Learning, Meta-Learning, Case Study

††booktitle: The 10th Mining and Learning from Time Series Workshop: From
Classical Methods to LLMs

##  1\. Introduction

The increasing availability and complexity of time series data have amplified
the need for robust forecasting techniques across various industries,
including finance, healthcare, and transportation (Lim and Zohren, 2021; Sezer
et al., 2020; Torres et al., 2021). In the air cargo sector, accurate
forecasting is critical for maximizing revenue generated through each unit of
capacity used by cargo bookings (Kasilingam, 1997; Billings et al., 2003).
Unlike passenger sales, air cargo bookings are notoriously volatile, are prone
to be cancelled, and are influenced by many factors such as economic
conditions, seasonal events, and disruptions affecting global trade (Kupfer et
al., 2017). Additionally, cargo bookings are often part of longer-term
customer commitments, which help airlines plan their network schedules and
budgeting efforts. Finally, since there are significantly fewer cargo bookings
per flight as compared to passenger bookings, the importance of making the
right decision (informed by an accurate forecast) is increased. This paper
presents a field study on time series forecasting in the air cargo industry,
focusing on the challenges and solutions associated with mid-to-long-term
demand prediction.

Recent advancements in sensing technologies and data analytics have paved the
way for more sophisticated forecasting models (Heng et al., 2009; Anguita and
Olariaga, 2023). However, the air cargo industry faces unique challenges,
including irregular booking patterns, varying shipment types, and the need for
high accuracy in forecasts. Traditional forecasting methods often fail to
address these complexities, necessitating the development of specialized
machine-learning approaches.

In this study, we introduce a comprehensive forecasting framework that
combines multiple statistical and machine learning models to predict cargo
demand at the origin and destination level. Our approach involves a detailed
data preparation process, integrating various data sources to capture the full
spectrum of factors affecting cargo demand. We employ a mixture of experts
framework, which allows us to select the best-performing model for each sample
based on historical performance, thereby enhancing the overall prediction
accuracy.

Our research is conducted within the context of a major air cargo carrier,
providing a real-world application of the proposed method, and validated
across diverse market segments. We evaluate the performance of our models
using key metrics such as root mean square error (RMSE) and weighted
normalised root mean square (WnRMSE), demonstrating significant improvements
over existing industry benchmark forecasting techniques. The findings of this
study offer valuable insights for applied teams in various industries,
enabling more informed decision-making leading towards better adoption of
forecasting models.

Our contributions are as follows:

  * •

We present an applied real-world case study where time-series forecasting is
crucial in decision-making. Additionally, we describe the modeling process to
make machine learning solution adoption smoother by the end user.

  * •

We propose a practically usable framework of a mixture of experts. Also, our
training methodology of meta-learning on machine learning models outperformed
industry benchmarks.

  * •

We share results from our implementation backtested on real data from a major
airline carrier, aiming to make high-stakes decisions that significantly
impact the majority of the annual revenue.

###  1.1. Background

In the context of airline cargo, mid-term refers to a forecasting period of up
to six months ahead, effectively covering everything beyond the typical cargo
booking window of up to 30 days before the flight departure. Mid-term
forecasting is crucial because it allows stakeholders to have a proper idea of
upcoming trends and demands, enabling them to make informed decisions on mid-
term allotment requests. Mid-term allotment contracts in air cargo are
agreements between airlines and shippers or freight forwarders to allocate a
certain amount of cargo space on flights over a defined period, typically
ranging from a few months to a couple of years (Sales and Scholte, 2023;
Hellermann, 2006; Kasilingam, 1997). These contracts facilitate more accurate
demand forecasting and reduce risks associated with fluctuating cargo volumes
and market rates (Pun and Klabjan, 2010; Amaruchkul et al., 2011). For
shippers and freight forwarders, these contracts provide guaranteed access to
cargo space, enhancing service reliability and customer satisfaction,
especially during peak seasons or unexpected demand surges (Popescu et al.,
2010; Zivcic, 2022). The contracts typically detail the volume of cargo space,
routes, duration, and pricing terms, with negotiations often considering
historical cargo volumes, projected future needs, and current market
conditions (Amaruchkul et al., 2011). Despite the advantages, challenges arise
from the dynamic nature of global trade, economic fluctuations, and unexpected
events like the COVID-19 pandemic, which impact cargo volumes and market
rates. Flexibility and adaptive strategies are essential in managing these
contracts amidst such uncertainties (Becker and Dill, 2007; Dewulf, 2014).
Overall, mid-term allotment contracts are pivotal in ensuring stability and
predictability in the air cargo industry, benefiting both airlines and their
customers by providing a structured approach to capacity allocation and
revenue management (Feng et al., 2015; Wada et al., 2017).

###  1.2. Related Work

Forecasting techniques have significantly evolved, with traditional methods
like ARIMA and exponential smoothing remaining popular due to their simplicity
and effectiveness in stable environments (De Gooijer and Hyndman, 2006).
Recent advancements in computational capabilities have led to the development
of more sophisticated models, particularly neural network-based approaches.
These models, including convolutional neural networks (CNNs) and recurrent
neural networks (RNNs), have demonstrated superior performance in capturing
complex, non-linear relationships in data, thereby enhancing forecasting
accuracy (Kim and Kim, 2019; Siami-Namini et al., 2018). Additionally,
attention-based neural network architectures have also shown some promise in
time series forecasting (Shih et al., 2019; Lim et al., 2021).

Applications of such techniques in airline revenue management have picked up
recently. For instance, application of machine learning-based solutions for
dynamic pricing and passenger flow estimation (Koc and Arslan, 2021; Yu, 2022)
in passenger business. Recent research on gravity models, dynamic linear
model, fuzzy regression forecasting models and simplified variants of neural
networks have shown promising results in air cargo industry (Alexander and
Merkert, 2021; Rodriguez et al., 2020; Chou et al., 2011; Chen et al., 2012).
Historically, there have been fewer publications on air cargo demand
forecasting, despite its considerable importance, especially when compared to
the extensive research on air passenger demand forecasting (Anguita and
Olariaga, 2023).

##  2\. Decision factors

In this section, we explain the two primary factors that are responsible for
mid-term allotment decision-making: demand estimation and, airline network and
capacity.

###  2.1. Demand estimation

Estimating the upcoming mid-term demand is crucial for airline cargo carriers
as it allows them to allocate capacity efficiently between long-term stable
contracts and short-term volatile bookings, which typically arrive within 30
days. Demand for air cargo is influenced by various factors including the
origin and destination, product category, global events (i.e., COVID-19 and
Suez Canal blockade), and macro trends (Kupfer et al., 2017). For instance,
countries like Columbia and Ethiopia are significant exporters of flowers,
resulting in a peak in cargo volumes before Valentine’s Day and Mother’s Day.
Similarly, there is a notable increase in import volumes of electronics and
technology components from Asia to Europe and North America during Black
Friday and Cyber Monday. Moreover, macroeconomic policies and trends play a
critical role in influencing both the demand and supply of products.

###  2.2. Network and capacity

Passenger airlines with significant revenue from air cargo typcially operate a
hub-and-spoke network model (Brueckner et al., 1992), where flights are routed
through a central hub to various destinations. This model facilitates
efficient passenger transfers and operational management. In contrast, air
cargo logistics often prioritize the origin and destination points without
significant regard to intermediary stops, focusing instead on cost-
effectiveness and efficient routes for their shipments. To optimize the
overall logistics, a comprehensive network-wide consideration of capacity is
essential. This includes evaluating forward-looking schedules and sellable
capacities, encompassing both passenger and cargo flights for each flight leg
(Levina et al., 2011). Such a holistic approach ensures that both passenger
and cargo needs are met efficiently, maximizing the use of available resources
and improving overall service delivery.

##  3\. Forecasting framework

We want to predict the total demand in terms of expected shipped weight,
volume, and revenue for each week for each origin and destination (O&D) for
the next 6 months. Airline carriers may have a lot of cargo destinations that
they fly to and therefore, the possible O&D combinations for forecasting could
be in the order of tens of thousands. To make the problem simpler, we only
focus on O&D that contribute to the top 90% of cargo revenue for the airline.
We call this group of O&Ds the significant cluster. This reduces the problem
space to a few 1000s of O&Ds. Through exploratory data analysis, we found that
most of the variability and important patterns in the data are captured by the
high-frequency signals at the departure date level. Therefore in our approach,
we make predictions at the O&D - departure date level, and then aggregate them
up to the O&D - departure week level.

###  3.1. Models

We experimented with eight different statistical models for baseline analysis.
Additionally, the industry-wide benchmark, used by the majority of airlines,
relies on year-over-year predictions. We used Croston Classic, Historic
Average, Auto ETS, Window Average, Dynamic Optimized Theta (DOT), Simple
Exponential Smoothing (SES), Holt-Winters, and Seasonal Naive for statistical
baseline (Croston, 1972; Jain and Mallick, 2017; Fioruci et al., 2015;
Chatfield, 1978; Hyndman and Athanasopoulos, 2018). With their strengths and
weaknesses, these methods collectively aim to capture a broad range of trend
and seasonality patterns in the time series that help set a baseline for
building advanced models. Furthermore, our study focused on the implementation
of three machine learning model architectures.

####  3.1.1. DNN-LADD

We implemented a simplified deep neural network (DNN) based primarily on fully
connected layers assuming independent and identically distributed data points
as inputs. Each timestamp point is considered independent of others without
considering the sequential nature of the time series. The only time series
considered is a small period that ”Looks Around the Departure Dates” (LADD).
These features help capture the static information in the time series and any
seasonal and calendar events that the airline cares about. The LADD features
are currently set to be equal to harmonic features (Panofsky and Brier, 1958)
and are encoded using a bidirectional LSTM (Hochreiter and Schmidhuber, 1997)
as illustrated in the figure 1. All categorical features go through their
embedding layers. These embeddings and encodings along with the dense features
are concatenated together before being fed to the fully connected layers. The
primary hypothesis for implementing this architecture is to capture patterns
based on context rather than driven by traditional time series concepts,
leveraging the flexibility of neural networks to model complex relationships
within the data.

Figure 1. Illustration of look around departure date (LADD) method using the
bidirectional LSTM

####  3.1.2. NBEATS

This is a state-of-the-art deep-learning-based time-series forecasting model
proven to work well on short-horizon forecasting (Oreshkin et al., 2019). It
takes a time-series input to predict the entire forecasting horizon in one
shot. The model tries to sequentially learn the residual from the inputs with
the use of stacks. By default, NBEATS is a univariate time-series model, that
is, it only takes time-series being forecasted as input. To incorporate
additional input features, we modified its architecture to add context in the
form of encodings concatenated together to the input vector. We observed that
this model performs better when patterns are seasonal and struggles when data
is sparse or highly context-driven such as capacity or events. LADD and
embedding features are learned the same way as the DNN-LADD model. Dense
features are embedded using an attention block (Vaswani et al., 2017).

####  3.1.3. TFT

Temporal Fusion Transformer (TFT) is one of the transformational methods in
time-series forecasting utilizing the transformer architecture (Lim et al.,
2021). Unlike NBEATS, TFT is designed to support input features and can learn
from static metadata, time-varying past inputs, and time-varying a priori
known future inputs. It incorporates LSTM layers and a multi-head self-
attention transformer block to enable efficient information flow, enhanced by
skip connections. However, being a transformer-based architecture, this model
is computationally expensive and requires significant memory to train,
especially for long-horizon forecasting.

###  3.2. Meta learning

Meta-learning is an advanced approach in artificial intelligence aimed at
training systems to effectively learn new tasks or adapt to novel situations
based on past experiences, rather than merely mastering specific tasks. This
paradigm, often described as ”learning to learn,” enables a system to acquire
new concepts or tasks with minimal additional training quickly. Given the
variability in data sizes across different Origin-Destination (O&D) pairs,
traditional machine learning models can easily overfit to O&Ds with larger
datasets. Meta-learning addresses this issue by implementing a 2-step
optimization process rather than standard gradient descent, as detailed in the
Model-Agnostic Meta-Learning (MAML) framework (Finn et al., 2017).

In this approach, each O&D is treated as a distinct task, and gradients for
model updates are derived by moving toward the local minima for a set of O&Ds.
This is achieved through a stochastic gradient descent update for a particular
O&D on a model copy in the inner loop using the support set. Concurrently, the
gradients for the main model updates are gathered using a query set from the
updated model copy in the outer loop. During inference, the model undergoes
fine-tuning using a subset of each O&D’s training data for a few steps before
making predictions. A significant advantage of this meta-learning approach is
its applicability to O&Ds with limited historical data, such as the bottom 10%
of O&Ds. By fine-tuning the model with minimal steps, accurate predictions can
be made without extensive retraining, provided the data supports the creation
of such O&Ds.

Algorithm 1 O&D Meta-Learning

1:$p(\mathcal{D})$: distribution over O&D

2:$\alpha,\beta$: step size hyperparameters

3:Randomly initialize $\theta$

4:while not done do

5: Sample batch of O&D: $\mathcal{D}_{i}\sim p(\mathcal{D})$

6: for all $\mathcal{D}_{i}$ do

7: Evaluate $\nabla_{\theta}\mathcal{L}_{\mathcal{D}_{i}}(f_{\theta})$ using
support set

8: Compute adapted parameters with gradient descent:
$\theta^{\prime}_{i}=\theta-\alpha\nabla_{\theta}\mathcal{L}_{\mathcal{D}_{i}}(f_{\theta})$

9: end for

10: Update
$\theta\leftarrow\theta-\beta\nabla_{\theta}\sum_{\mathcal{D}_{i}\sim
p(\mathcal{D})}\mathcal{L}_{\mathcal{D}_{i}}(f_{\theta^{\prime}_{i}})$ using
query set

11:end while

In the context of the meta-learning algorithm, let $p(\mathcal{D})$ represent
the distribution over tasks, where each O&D (Origin-Destination pair) is
considered a separate task. The step size hyperparameters, $\alpha$ and
$\beta$, are employed in the inner and outer loop updates respectively,
governing the magnitude of parameter adjustments during gradient descent. The
model parameters are denoted by $\theta$, while $\theta^{\prime}_{i}$ refers
to the adapted parameters for a specific task $\mathcal{D}_{i}$ after the
inner loop update. The loss function for a task $\mathcal{D}_{i}$ is
represented by $\mathcal{L}_{\mathcal{D}_{i}}$, which evaluates the model’s
performance on that task. Finally, $f_{\theta}$ signifies the model
parameterized by $\theta$, encapsulating the function learned by the meta-
learning algorithm. These variables collectively enable the systematic
optimization and fine-tuning of the model across diverse tasks as mentioned in
algorithm 1.

###  3.3. Mixture of experts

For each O&D pair $\mathcal{D}_{i}$, the model $f_{\theta}$ with the lowest
loss value $\mathcal{L}$ on the validation set is selected, as denoted in
equation 1. The selected model $f_{\theta_{\mathcal{D}}^{*}}$ is then used to
make predictions on the test set for the corresponding O&D pair. By selecting
the best-performing model for each O&D pair based on the validation loss, the
overall prediction error across all O&D pairs is minimized. This approach
leverages the strengths of different models, acknowledging that no single
model consistently performs best across all O&D pairs. The implementation of
this framework involves training each model on the training set, evaluating
their performance on the validation set to compute the loss for each O&D pair,
and then selecting and using the model with the lowest loss for making
predictions on the test set. This process ensures that the forecasting
approach is tailored to the specific characteristics of each O&D pair,
ultimately aiming to minimize the overall prediction error through a data-
driven model selection process.

(1)
$f_{\theta_{\mathcal{D}_{i}}^{*}}=\arg\min_{f_{\theta}\in\mathcal{F}}\mathcal{L}_{\mathcal{D}_{i}}(f_{\theta})$

We chose the mixture of experts approach to provide the most accurate and
reliable forecasts for air cargo analysts, thereby enabling them to make well-
informed decisions. This method allows us to leverage baseline forecasts for
inference, particularly in scenarios where the existing modeling approaches
fall short, ensuring comprehensive coverage across various origin and
destination pairs. Furthermore, this approach is apt for applications with
low-frequency inference due to high computational complexity and hence
affordable to mid-term forecasts as it needs to be updated only a few times a
month. Therefore, the mixture of experts framework is a practically viable
solution, aligning with the operational needs.

##  4\. Results

Table 1. Model performance and win ratio on significant cluster Model Name |  | Model
---
Type
Win ratio | nRMSE |  | WnRMSE
---
TFT | ML | 0.245 | 1.493 | 1.042
Auto ETS | Stat | 0.160 | 1.459 | 1.016
NBEATS | ML | 0.151 | 1.440 | 1.000
DOT | Stat | 0.087 | 1.447 | 1.006
DNN-LADD | ML | 0.084 | 1.618 | 1.092
Holt-Winters | Stat | 0.070 | 1.698 | 1.178
SES | Stat | 0.068 | 1.001 | 1.037
Window Average | Stat | 0.051 | 1.521 | 1.044
Historic Average | Stat | 0.044 | 1.626 | 1.094
Croston Classic | Stat | 0.027 | 1.489 | 1.030
Seasonal Naive | Stat | 0.006 | 1.898 | 1.210

The target variable chosen for all experiments is the total shipped cargo
weight in kilograms on a given date for a given O&D pair. All models are
trained from April 2019 to March 2023, and validated on a period starting from
April 2023 to September 2023 111The exact dates and revenue figures cannot be
included due to proprietary, privacy, and sensitivity restrictions.

The primary metrics chosen for evaluation are root mean squared error (RMSE),
unweighted normalized root mean squared error (nRMSE), and weighted normalized
root mean squared error (WnRMSE). The WnRMSE formulation is described in
equation 2.

(2) $\textrm{WnRMSE}=\sum_{OD}w_{OD}\times\text{nRMSE}_{OD}$

where
$w_{OD}=\frac{sum(\text{weight})_{OD}}{\sum_{OD}sum(\text{weight})_{OD}},\textrm{
nRMSE}_{OD}=\frac{\text{RMSE}_{OD}}{avg(\text{weight})_{OD}}$

RMSE, by definition, is biased towards high-value data points, leading to
higher magnitude values for bigger O&Ds in the airline network and potentially
skewing network-wide RMSE values. Normalized RMSE (nRMSE) addresses this bias
by normalizing RMSE values by the average of the target variable. Therefore,
we calculate nRMSE for each O&D to ensure fair performance comparison across
the network. We use a weighted mean of the individual nRMSE values to obtain
the network-wide error, where weights are the total shipped weight of each O&D
pair over the evaluation period. This approach provides a performance
assessment of the overall network with proportionate relevance of O&Ds.

Individual model performance is presented in Table 1. Here, average nRMSE and
WnRMSE are recorded for each model across the entire validation period. The
win ratios are calculated for the significant cluster due to resource
constraints. Advanced techniques like TFT and NBEATS are 2 of the top 3 best
performing models within the significant cluster. Also, statistical methods
like Auto ETS effectively capture seasonal trends, resulting in improved
performance.

Table 2. Aggregated metrics for clustered O&Ds O&D’s |  | Sample
---
Size
| Share of
---
Revenue
nRMSE |  | WnRMSE
---
Significant cluster | ¿2000 | 0.90 | 1.04 | 0.52
Top 100 | 100 | 0.37 | 0.34 | 0.30
101-500 | 400 | 0.30 | 0.56 | 0.51
501-1000 | 500 | 0.12 | 0.75 | 0.69
Above 1001 | ¿1000 | 0.11 | 1.32 | 0.99

To do further analysis, we ranked the entire dataset based on weight
contribution. A significant O&D cluster is a subset of the dataset that
contributes to 90% of the total revenue across the airline network in the year
before the validation period. This significant cluster contains 2000+ O&D
pairs. Aggregated metrics are reported in Table 2. Both RMSE and WnRMSE
indicate performance degrades towards the tail end of the ranked O&D.
Additionally, the significant cluster has low WnRMSE. This implies that the
forecast can capture trends where variability averages out due to volume as
compared to a more stochastic tail end. In table 3, we use the same dataset
slices and evaluate the winning ratios of the machine learning methods as
compared to (a) statistical baseline and (b) industry benchmark i.e. year-
over-year predictions. Although, the winning percentage for machine learning
models is about 50% as compared to the baseline, we outperformed the industry
benchmark overall. Hence, the utility of a mixture of experts is relevant as
it provides a fallback option where machine learning models are not performing
better than the statistical baselines.

Table 3. Win ratios of ML models for clustered O&Ds O&D’s |  | Statistical
---
baseline
| Industry
---
benchmark
Significant cluster | 0.49 | 0.94
Top 100 | 0.42 | 0.96
101-500 | 0.49 | 0.96
501-1000 | 0.46 | 0.93
Above 1001 | 0.50 | 0.94

##  5\. Future work

In the future, we intend to extend this work to other target variables that
are as crucial as shipped weight for the airline in its decision-making. These
include shipped volume (in $m^{3}$), chargeable weight (normalized metric
between gross weight and volumetric weight), yield, and/or revenue.

Furthermore, the proposed mixture of experts framework currently misses out on
integrating information from models other than the best performing one on a
particular O&D pair, limiting its predictive power. Therefore, we propose
extending the framework to a two-step forecasting process. In the first step,
we train simpler models such as statistical benchmarks, decision stumps, and
simple DNN networks. The predictions from these models would serve as
additional features for advanced deep learning models like NBEATS and TFT,
enhancing overall forecast accuracy. Integrating information from these
simpler models allows the deep learning model to focus on complex, context-
driven scenarios without being burdened with learning the simple trend and
seasonality patterns already captured by the statistical models in the first
step.

Lastly, we would like to build an attribution system to measure the revenue
uplift from decisions made using improved forecasting techniques. Thereby,
validating the overall approach for time series forecast in the air cargo
industry.

##  6\. Conclusion

The field study presented in this paper demonstrates the efficacy of advanced
machine learning models in improving mid-term demand forecasting for the air
cargo industry. Our experiments show that machine learning models perform
better than statistical baseline models. Additionally, mixture of experts
which combines both statistical and machine learning models, significantly
outperforms machine learning only approaches. The real-world application of
these models within a major air cargo carrier underscores their practical
utility and potential for broader adoption in other volatile market
environments. Future research will focus on extending this framework to
additional target variables and enhancing the predictive power by
incorporating a two-step forecasting process. This work lays a strong
foundation for further advancements in the adoption of machine learning
techniques in air cargo demand forecasting, ultimately contributing to more
efficient and informed decision-making processes in the industry.

###### Acknowledgements.

##  7\. Acknowledgments

We sincerely and gratefully acknowledge our airline partners for their
continuing support.

## References

  * (1)
  * Alexander and Merkert (2021) David W Alexander and Rico Merkert. 2021.  Applications of gravity models to evaluate and forecast US international air freight markets post-GFC.  _Transport Policy_ 104 (2021), 52–62.
  * Amaruchkul et al. (2011) Kannapha Amaruchkul, William L Cooper, and Diwakar Gupta. 2011.  A note on air-cargo capacity contracts.  _Production and Operations Management_ 20, 1 (2011), 152–162.
  * Anguita and Olariaga (2023) Juan Gerardo Muros Anguita and Oscar Díaz Olariaga. 2023.  Air cargo transport demand forecasting using ConvLSTM2D, an artificial neural network architecture approach.  _Case Studies on Transport Policy_ 12 (2023), 101009.
  * Becker and Dill (2007) Bjoern Becker and Nadja Dill. 2007.  Managing the complexity of air cargo revenue management.  _Journal of Revenue and Pricing Management_ 6 (2007), 175–187.
  * Billings et al. (2003) John S Billings, Adriana G Diener, and Benson B Yuen. 2003.  Cargo revenue optimisation.  _Journal of Revenue and Pricing Management_ 2 (2003), 69–79.
  * Brueckner et al. (1992) Jan K Brueckner, Nichola J Dyer, and Pablo T Spiller. 1992.  Fare determination in airline hub-and-spoke networks.  _The RAND Journal of Economics_ (1992), 309–333.
  * Chatfield (1978) Chris Chatfield. 1978.  The Holt-winters forecasting procedure.  _Journal of the Royal Statistical Society: Series C (Applied Statistics)_ 27, 3 (1978), 264–279.
  * Chen et al. (2012) Shu-Chuan Chen, Shih-Yao Kuo, Kuo-Wei Chang, and Yi-Ting Wang. 2012.  Improving the forecasting accuracy of air passenger and air cargo demand: the application of back-propagation neural networks.  _Transportation Planning and Technology_ 35, 3 (2012), 373–392.
  * Chou et al. (2011) Tsung-Yu Chou, Gin-Shuh Liang, and Tzeu-Chen Han. 2011.  Application of fuzzy regression on air cargo volume forecast.  _Quality & Quantity_ 45, 6 (2011), 1539–1550.
  * Croston (1972) John D Croston. 1972.  Forecasting and stock control for intermittent demands.  _Journal of the Operational Research Society_ 23, 3 (1972), 289–303.
  * De Gooijer and Hyndman (2006) Jan G De Gooijer and Rob J Hyndman. 2006.  25 years of time series forecasting.  _International journal of forecasting_ 22, 3 (2006), 443–473.
  * Dewulf (2014) Wouter Dewulf. 2014.  _The strategy of air cargo operators: about carpet sellers and cargo stars_.  Ph. D. Dissertation. University of Antwerp.
  * Feng et al. (2015) Bo Feng, Yanzhi Li, and Huaxiao Shen. 2015.  Tying mechanism for airlines’ air cargo capacity allocation.  _European Journal of Operational Research_ 244, 1 (2015), 322–330.
  * Finn et al. (2017) Chelsea Finn, Pieter Abbeel, and Sergey Levine. 2017.  Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks. In _Proceedings of the 34th International Conference on Machine Learning_ _(Proceedings of Machine Learning Research, Vol. 70)_ , Doina Precup and Yee Whye Teh (Eds.). PMLR, 1126–1135.  https://proceedings.mlr.press/v70/finn17a.html
  * Fioruci et al. (2015) José Augusto Fioruci, Tiago Ribeiro Pellegrini, Francisco Louzada, and Fotios Petropoulos. 2015.  The optimised theta method.  _arXiv preprint arXiv:1503.03529_ (2015).
  * Hellermann (2006) Rolf Hellermann. 2006.  _Capacity options for revenue management: theory and applications in the air cargo industry_. Vol. 575.  Springer Science & Business Media.
  * Heng et al. (2009) Hong-jun Heng, Bing-zhong Zheng, and Ya-jing Li. 2009.  Study of SVM-based air-cargo demand forecast model. In _2009 International Conference on Computational Intelligence and Security_ , Vol. 2. IEEE, 53–55.
  * Hochreiter and Schmidhuber (1997) Sepp Hochreiter and Jürgen Schmidhuber. 1997.  Long short-term memory.  _Neural computation_ 9, 8 (1997), 1735–1780.
  * Hyndman and Athanasopoulos (2018) Rob J Hyndman and George Athanasopoulos. 2018.  _Forecasting: principles and practice_.  OTexts.
  * Jain and Mallick (2017) Garima Jain and Bhawna Mallick. 2017.  A study of time series models ARIMA and ETS.  _Available at SSRN 2898968_ (2017).
  * Kasilingam (1997) Raja G Kasilingam. 1997.  Air cargo revenue management: Characteristics and complexities.  _European Journal of Operational Research_ 96, 1 (1997), 36–44.
  * Kim and Kim (2019) Taewook Kim and Ha Young Kim. 2019.  Forecasting stock prices with a feature fusion LSTM-CNN model using different representations of the same data.  _PloS one_ 14, 2 (2019), e0212320.
  * Koc and Arslan (2021) Ismail Koc and Emel Arslan. 2021.  Dynamic ticket pricing of airlines using variant batch size interpretable multi-variable long short-term memory.  _Expert Systems with Applications_ 175 (2021), 114794.
  * Kupfer et al. (2017) Franziska Kupfer, Hilde Meersman, Evy Onghena, and Eddy Van de Voorde. 2017.  The underlying drivers and future development of air cargo.  _Journal of Air Transport Management_ 61 (2017), 6–14.
  * Levina et al. (2011) Tatsiana Levina, Yuri Levin, Jeff McGill, and Mikhail Nediak. 2011.  Network cargo capacity management.  _Operations Research_ 59, 4 (2011), 1008–1023.
  * Lim et al. (2021) Bryan Lim, Sercan Ö Arık, Nicolas Loeff, and Tomas Pfister. 2021.  Temporal fusion transformers for interpretable multi-horizon time series forecasting.  _International Journal of Forecasting_ 37, 4 (2021), 1748–1764.
  * Lim and Zohren (2021) Bryan Lim and Stefan Zohren. 2021.  Time-series forecasting with deep learning: a survey.  _Philosophical Transactions of the Royal Society A_ 379, 2194 (2021), 20200209.
  * Oreshkin et al. (2019) Boris N Oreshkin, Dmitri Carpov, Nicolas Chapados, and Yoshua Bengio. 2019.  N-BEATS: Neural basis expansion analysis for interpretable time series forecasting.  _arXiv preprint arXiv:1905.10437_ (2019).
  * Panofsky and Brier (1958) H.A. Panofsky and G.W. Brier. 1958.  _Some Applications of Statistics to Meteorology_.  Mineral Industries Extension Services, College of Mineral Industries, Pennsylvania State University.  https://books.google.com/books?id=UyRRAAAAMAAJ
  * Popescu et al. (2010) Andrea Popescu, Pinar Keskinocak, and I al Mutawaly. 2010.  _Intermodal transportation: Moving freight in a global economy_ (2010), 209–237.
  * Pun and Klabjan (2010) Chan Seng Pun and Diego Klabjan. 2010.  Air Cargo Allotment Planning.  (2010).
  * Rodriguez et al. (2020) Yesid Rodriguez, Wilmer Pineda, and Oscar Diaz Olariaga. 2020.  Air traffic forecast in post-liberalization context: a Dynamic Linear Models approach.  _Aviation_ 24, 1 (2020), 10–19.
  * Sales and Scholte (2023) Michael Sales and Sebastiaan Scholte. 2023.  _Air cargo management: Air freight and the global supply chain_.  Routledge.
  * Sezer et al. (2020) Omer Berat Sezer, Mehmet Ugur Gudelek, and Ahmet Murat Ozbayoglu. 2020.  Financial time series forecasting with deep learning: A systematic literature review: 2005–2019.  _Applied soft computing_ 90 (2020), 106181.
  * Shih et al. (2019) Shun-Yao Shih, Fan-Keng Sun, and Hung-yi Lee. 2019.  Temporal pattern attention for multivariate time series forecasting.  _Machine Learning_ 108 (2019), 1421–1441.
  * Siami-Namini et al. (2018) Sima Siami-Namini, Neda Tavakoli, and Akbar Siami Namin. 2018.  A comparison of ARIMA and LSTM in forecasting time series. In _2018 17th IEEE international conference on machine learning and applications (ICMLA)_. IEEE, 1394–1401.
  * Torres et al. (2021) José F Torres, Dalil Hadjout, Abderrazak Sebaa, Francisco Martínez-Álvarez, and Alicia Troncoso. 2021.  Deep learning for time series forecasting: a survey.  _Big Data_ 9, 1 (2021), 3–21.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  _Advances in neural information processing systems_ 30 (2017).
  * Wada et al. (2017) Masato Wada, Felipe Delgado, and Bernardo K Pagnoncelli. 2017.  A risk averse approach to the capacity allocation problem in the airline cargo industry.  _Journal of the Operational Research Society_ 68 (2017), 643–651.
  * Yu (2022) Jiangni Yu. 2022.  Short-term airline passenger flow prediction based on the attention mechanism and gated recurrent unit model.  _Cognitive Computation_ 14, 2 (2022), 693–701.
  * Zivcic (2022) Rudolf Zivcic. 2022.  Applicability of Dynamic Pricing Concept for Air Cargo.  (2022).