#  Random Matrices and Quantum Hamilton-Jacobi Method

K. Haritha and K.V.S.Shiv Chaitanya 1<EMAIL_ADDRESS>1Department of Physics, BITS Pilani, Hyderabad Campus, Jawahar Nagar,
Shamirpet Mandal, Hyderabad, India 500 078.

###### Abstract

In this paper, we start with the quantum Hamilton-Jacobi approach and show
that the underlying complex pole evolution of the Schrödinger equation is
described by the quantum action in terms of a random matrix. The wave function
is given by the random matrix probability distribution function. In literature
this is known as the famous Cole-Hopf Transformation.

##  I Introduction

The classical action first appeared in the quantum mechanics as quantization
rule given by Wilson and Sommerfeld ws , wherein the energy levels are
approximately obtained by setting the classical action variable equal to an
integral multiple of the Planck’s constant. Later on several attempts were
made to construct a quantum analogue to the classical action. Defining a
quantum action allows us to transform one set of canonically conjugate
coordinates and momenta to another set, analogous to the classical mechanics.
This is known as quantum transformation, originated from the works of Jordon
jor  and Dirac dir . Dirac defined the following quantum action $S$ as the
quantum analogue of the classical action function dir1 ; dir2  such that

$(\xi^{\prime}|\alpha^{\prime})=exp(iS/\hbar)$ (1)

where $(\xi^{\prime}|\alpha^{\prime})$ is the transformation function that
connects the $\xi$ representation to $\alpha$ representation. Starting from
the definition (1), he has shown that the quantum transformation equations
have the same form as the classical equations.

Dirac quantum action $S$ has connections to the Feynmann path integral
formulation of quantum mechanics fey ; schw ; fin . The comprehensive
examination of the quantum transformation theory is done by Schwinger in the
references schw ; fin ; schw1 . Schwinger postulated the action principle such
that one obtains operators $\delta S$ by the variation of one operator $S$ fin
; schw1 . He derived the Schrödinger equation from an ”action” principle in a
way very similar to the Hamilton derivation of the Hamilton-Jacobi equation
from the classical action principle schw1 . It is shown in the references fin
; schw1  that the Schwinger’s action principle is identical to the Dirac’s
original expression.

Leacock and Padgett developed quantum transformation theory as a
generalization of classical Hamilton-Jacobi theory qhj ; qhj1 . In the Quantum
Hamilton-Jacobi approach, we start with the Schrödinger equation,

$-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}\psi(x)+V(x)\psi(x)=E\psi(x).$ (2)

We define a function in terms of quantum action or quantum characteristic
function $S$ analogous to the classical characteristic function by the
relation

$\psi=exp\left[\frac{iS}{\hbar}\right].$ (3)

The equation (3) is known as Cole-Hopf transformation and when substituted in
(2), gives

$\left(\frac{d}{dx}S\right)^{2}-i\hbar\frac{d}{dx}\left(\frac{d}{dx}S\right)=2m(E-V(x)).$
(4)

The quantum momentum function $p$ is

$p=\frac{d}{dx}S.$ (5)

Substituting (5) in (4) gives the Quantum Hamilton-Jacobi equation for $p$ as

$p^{2}-i\hbar\frac{dp}{dx}=2m(E-V(x)),$ (6)

which is the Riccati equation. It should be clear to the readers from the
equation (6) when $\hbar\rightarrow 0$ we get the classical momentum function
$p_{c}^{2}=2m(E-V(x))$. The relation between the quantum momentum function and
the wave function is given by

$p=i\hbar\frac{d}{dx}ln\psi(x).$ (7)

It is shown by Leacock and Padgett qhj ; qhj1  that the action angle variable
gives rise to the exact quantization condition

$J(E)\equiv\frac{1}{2\pi}\oint_{C}{pdx}=n\hbar.$ (8)

We would like to point out that the transformation (3) is used in the Wentzel,
Kramers, Brillouin (WKB) approximation and is the semi-classical approach. The
$S$ defined in WKB is the classical action whereas $S$ defined in equation (3)
is the quantum action. For more details refer to appendix C of qhj1 . From the
equation (6) it is evident that the term $i\hbar\frac{dp}{dx}$ is the one that
gives rise to quantum effects and is absent in classical momentum function.
Hence, the $S$ defined in equation (3) is the effective action or the quantum
action zo . The connection between the QHJ and Dirac’s action is given in the
appendix. Readers should note that all the definitions of quantum action
defined in literature have correspondence to the Dirac’s action (1).

In this paper, we show that the quantum action $S$ is described by the random
matrix of Gaussian unitary ensemble and the Cole-Hopf transformation (3) is
the random matrix probability distribution function. We use the fact that
quantum Hamilton-Jacobi can be solved in two different ways. One way of
solving is by using the Riccati equation which leads to the action angle
variable gives rise to the exact quantization condition (8). The other way is
using the Burger equation. We show that this way of solving quantum Hamilton-
Jacobi gives quantum action. In literature, the transformation (3) is used to
map the Schrödinger equation to the Burger equation chj . It is well known
that underlying complex pole evolution of Burger equation in turn the
Schrödinger equation is described by the Kirchhoff equations kvs1

$\displaystyle\dot{x}_{i}=2\Gamma i\sum_{i\neq
j}^{n}\frac{1}{(x_{i}-x_{j})}+i\mathcal{W}(x_{i}),$ (9)

where $x_{i}$ is the position of charges and $\mathcal{W}(x_{i})$ is the
superpotential. These equations describe the evolution of $n$ point vortices
in Hydrodynamics. The solution to the stationary Kirchhoff equations (9), that
is $\dot{x}_{i}=0$, is by found by the Stieltjes electrostatic model st ; st1
. The Stieltjes electrostatic model enables the Langrangian formalism. We show
that when Stieltjes electrostatic model is used for solving quantum
Hamiltonian-Jacobi it leads to quantum action $S$. In random matrix theory the
probability distribution function of random matrix is the stationary solution
of Fokker-Planck equation. We observe that when Fokker-Planck equation is
mapped to Schrödinger equation by Cole-Hopf transformation. We also use the
fact that Stieltjes electrostatic model is used to solve the Dyson-log-gas
model in random matrix theory met . The equations obtained in Dyson-log-gas
model are the Kirchhoff equations. In the reference iopj  the authors have
shown that equations obtained in Dyson-log-gas model is obtained by minimizing
action. Hence, the claim of this paper is the Dirac action is same is the
action obtained in reference iopj  by using the Stieltjes electrostatic model.

##  II Random Matrices

In physics, random matrices’ initial application was to describe the excited
states of the atomic nuclei wig  by Wigner. Random matrix theory has a wide
variety of applications from chiral symmetry breaking in quantum
chromodynamics, quantum chaos to many-body physics qcd ; qtqd ; sas ; cs ; phi
; tar ; zaf ; guy ; sud ; dan . Here the dynamics of the underlying system is
treated as an ensemble of random matrix $H$ and is described by the
probability distribution of the eigenvalues $x_{i}$met

$.{\cal P}(x_{1},\ldots,x_{N})\,d\Lambda=C_{n}e^{-\beta Tr(H)}\,\,d\Lambda,$
(10)

here the index $\beta=1,2,4$ and

$\displaystyle
Tr(H)=-\sum_{i=1}^{N}V(x_{i})-\beta\sum_{i<j}^{N}ln|x_{i}-x_{j}|$ (11)

where $V(x_{i})$ is the potential then $\prod_{i<j}|x_{i}-x_{j}|$ is the
Vandermonde determinant, $d\Lambda=dx_{1}\ldots dx_{N}$ and $c_{n}$ is the
constant of proportionality. For details please refer to met ; rm ; ran1 ;
ran2 ; ran3 ; ran4 . The $\beta$ in equation (10) characterizes the real
parameters of the symmetry class of orthogonal, unitary and symplectic
respectively for the random matrix $H$.

As the random matrix is invariant under the symmetry group these ensembles are
called the invariant ensembles. The probability distribution function can be
rewritten as rm

${\cal
P}(x_{1},\ldots,x_{N})\,d\Lambda=c_{n}\prod_{i=1}^{N}w_{\beta}(x_{i})\prod_{i<j}|x_{i}-x_{j}|^{\beta}\,\,d\Lambda$
(12)

where $w_{\beta}(x_{i})$ is the weight function of the classical orthogonal
polynomials. The classical orthogonal polynomials are classified into three
different categories depending upon the range of the polynomials. The
polynomials in the interval $(-\infty;\infty)$ with weight function
$w=e^{-x^{2}}$ are the Hermite polynomials. In the interval $[0;\infty)$ with
weight function $w=x^{b}e^{-x}$ are the Laguerre polynomials. In the interval
$[-1;1]$ with weight function $w=(1+x)^{a}(1-x)^{b}$ are the Jacobi
polynomials.

The random matrices are used to model the real world problems and one of the
well known model is the Dyson-log-gas model. The probability distribution
function for Dyson-log-gas model is given by met ; dyson

$P(x_{1}x_{2}.....x_{n})=c_{n}e^{-\beta
H}=\frac{1}{c_{\beta}}e^{-\beta(\sum_{j=1}^{n}x_{j}^{2}+\sum_{j,k}ln(x_{j}-x_{k}))}.$
(13)

. Dyson treated this as the partition function and the random matrix $H$ as
potential energy. Then the thermodynamic equilibrium for this model is
obtained by minimizing the energy

$\frac{d}{dx_{j}}H=\sum_{1\leq j\leq k,j\neq
k}^{n}\frac{1}{x_{j}-x_{k}}-x_{j}=0.$ (14)

The equation (14) is the system of $n$ linear equations which is solved using
the Stieltjes electrostatic model met . Theorem $1$ : In Stieltjes
electrostatic model, the following Lagrangian

$\displaystyle L$ $\displaystyle=$ $\displaystyle-\sum_{1\leq i<k\leq
n}^{n}ln|x_{i}-x_{j}|-p\sum_{i=1}^{n}ln(|1-x_{i}|)-q\sum_{i=1}^{n}ln(|1+x_{i}|),$
(15)

is considered, where $x_{i}$ is the position of charges, with interaction
forces for the $n$ moving unit charges arising from a logarithmic potential
which are in between the two fixed charges $p$ and $q$ at $-1$ and $1$
respectively on a real line. Then he proved in ref st ; st1  that the
expression (15) becomes a minimum when $(x_{1},x_{2},\cdots,x_{n})$ are the
zeros of the Jacobi polynomial. For the proof refer to Szego’s book (section
6.7) sz . The zeros of the Laguerre and the Hermite polynomials admit the same
interpretation. Thus, we solve (14) using Stieltjes electrostatic model. By
using the following identity met

$\displaystyle\sum_{1\leq j\leq k,j\neq
k}^{n}\frac{1}{x_{j}-x_{k}}=\frac{f^{\prime\prime}(x_{j})}{2f^{\prime}(x_{j})}.,$
(16)

and substituting in equation (14) reduces to

$f^{\prime\prime}(x_{j})-2x_{j}f^{\prime}(x_{j})=0.$ (17)

. This system has thermodynamic equilibrium at the zeros of the Hermite
orthogonal polynomial. Thus the differential equation (17) reduces to Hermite
differential equation

$f^{\prime\prime}(x)-2x_{j}f^{\prime}(x)+2\lambda f(x)=0,$ (18)

and the solutions are the Hermite orthogonal polynomials

$f(x)=c_{n}e^{-{\frac{x^{2}}{2}}}\cdot H_{n}\left(x\right),\qquad
n=0,1,2,\ldots.$ (19)

In the next section we present a connection between the random matrices and
the self-adjoint operator.

##  III Random matrices and the self-adjoint operator

It is well known that the probability distribution function of a random matrix
is the stationary solution of the Fokker-Planck equation met . Then the the
Fokker-Planck operator is mapped to the Schrödinger equation is studied in
reference nelson . We observe the mapping of Fokker-Planck operator to
Schrödinger equation is through a by Cole-Hopf transformation.

Theorem $2$: The probability distribution function in the equation (10) is a
stationary solution of the Fokker-Planck equation met .

Proof: Differentiating the probability distribution function (10) gives

$\frac{\partial{\cal P}}{\partial x_{j}}=-\beta P\frac{dH}{dx_{j}},$ (20)

where $H$ is random matrix. in general it need not be of the form (11).
Differentiating the equation (20) again w.r.t $x_{j}$, follows that

$\sum_{j=1}^{N}\frac{1}{\beta}\frac{\partial^{2}{\cal P}}{\partial
x_{j}^{2}}+\frac{\partial}{\partial x_{j}}(H^{\prime}(x_{j}){\cal P})\\}=0.$
(21)

Since the Fokker-Planck equation is

$\frac{\partial P}{\partial t}=\sum_{j=1}^{N}\\{1/\beta\frac{\partial^{2}{\cal
P}}{\partial x_{j}^{2}}+\frac{\partial}{\partial x_{j}}(H^{\prime}(x_{j}){\cal
P})\\},$ (22)

it follows that ${\cal P}$ is a stationary solution of the Fokker-Planck
equation. This ends the proof.

Theorem $3$: The Fokker-Planck operator is self-adjoint if and only if the
drift term is the gradient of a potential nelson .

Proof: We start with the Fokker-Planck equation

$\frac{\partial P}{\partial t}=\sum_{j=1}^{N}\\{1/\beta\frac{\partial^{2}{\cal
P}}{\partial x_{j}^{2}}+\frac{\partial}{\partial x_{j}}(H^{\prime}(x_{j}){\cal
P})\\},$ (23)

By defining a new function

${\cal P}=Ne^{-\frac{\beta}{2}H(x)+\frac{1}{2}Et}\psi$ (24)

where $\beta$ is a parameter which will be identified later and $N$ is a
constant. Thus, the Fokker-Planck operator (23) reduces to

$\displaystyle\frac{1}{\beta}\frac{\partial^{2}}{\partial
x_{j}^{2}}\psi+V\psi$ $\displaystyle=$ $\displaystyle\frac{1}{2}E\psi$ (25)

which can be interpreted as the Schrödinger operator for appropriate $\beta$
with the potential defined as

$V=\frac{1}{2}\frac{\partial^{2}H(x_{j})}{\partial
x_{j}^{2}}-\frac{\beta}{4}(\frac{\partial H(x_{j})}{\partial x_{j}})^{2}.$
(26)

where $\mathcal{W}(x)=\frac{\partial H}{\partial x_{j}}$ is superpotential
defined in equation (57) and for $\beta=2$ the equation (25) reduces to the
Schrödinger type equation. Hence, from equation (26) the drift term is the
gradient of the potential. This ends the proof.

Corollary $1$: Fokker-Planck reduces to the Schrödinger equation (2) in
imaginary time formalism that is $t\rightarrow iT$ in equation (23) and the
term $H(x)-\frac{E}{2}t$ in exponential (24) is the canonical transformation
(49).

Proof: When exponential factor $e^{i(H(x)-\frac{1}{2}ET)}$ that is $\beta=-2i$
and $i\mathcal{W}(x)=\frac{\partial H}{\partial x_{j}}$ Schrödinger equation
(25) reduces to (2) with
$V=-\frac{1}{2}(\mathcal{W}^{2}(x)-\mathcal{W}^{\prime}(x))$. Thus, the term
$H(x)-\frac{E}{2}T$ in exponential (24) is the Legendre transformation (49).
Then the quantum Hamilton-Jacobi connection to random matrix is immediate and
in turn to Dirac’s action. This ends proof.

##  IV Two approaches of quantum Hamilton-Jacobi

There are two ways of approaching quantum Hamiltonian-Jacobi formalism. The
first way is by using the quantum momentum function (5) and second way is
through Stieltjes electrostatic model kvs . In the first case, it is evident
that the quantum momentum function is the solution to the Riccati equation
(6). Thus, by analytically continuing $p$ to the complex plane and knowing the
singularities of the quantum momentum function is sufficient to determine the
wave functions sree . The solutions of the Riccati equation have two types of
singularities inc , the fixed and the moving singularities. The fixed singular
points of $p(x)$ are reflected in the potential and are energy independent.
The pole structure of the Riccati equation is given in terms of quantum
momentum function sree

$p=\sum_{j=1}^{n}\frac{-i\hbar}{x-x_{j}}+Q(x).$ (27)

where $Q(x)$ is meromorphic function and $i\hbar$ are the residues of moving
poles in the quantum momentum function sree . The equation (27) represents $n$
moving poles on a real line coming from
$\sum_{j=1}^{n}\frac{-i\hbar}{x-x_{j}}$ between the two fixed poles coming
from the $Q(x_{J})$.

Theorem $4$: The equation (34) admits orthogonal polynomial solutions and this
leads to quantization provided the function $Q(x_{j})$ is a superpotential.
kvs

Proof : Starting from equation (27) and introducing the polynomial

$\displaystyle\psi(x)$ $\displaystyle=$
$\displaystyle(x-x_{1})(x-x_{2})\cdots(x-x_{n})=\prod_{k=1}^{n}(x-x_{j}),$
(28)

and substituting the equation (28) in the quantum momentum function (27) and
taking( $\hbar=1$) reads as

$p=-i\frac{f^{\prime}(x)}{f(x)}+Q(x)$ (29)

and substituting equation (29) in equation (6) one gets

$\displaystyle-f^{\prime\prime}(x)+2iQ(x)f^{\prime}(x)+[Q^{2}(x)-iQ^{\prime}(x)-E+V(x)]f(x)=0.$
(30)

Finding the polynomial solutions to the equation (30) leads to quantization.
This is equivalent to demanding that $[Q^{2}(x)-iQ^{\prime}(x)-E+V(x)]$ is a
constant. From equation (57) this is only possible if $Q(x)=i\mathcal{W}(x)$
is the superpotential. Thus, the differential equation (30) reduces to Sturm-
Liouville equation. This ends the proof.

In the second case that is by Stieltjes electrostatic model, the problem is
solved using the complex pole evolution of Burger’s equation.

Theorem $5$: The Schrödinger equation is mapped to Burger equation by the
complex Cole-Hopf transformation chj .

Proof: Substituting the complex Cole-Hopf transformation (3) in the
Schrödinger equation (2) gives the Riccati equation (6). Differentiating the
equation (6) with respect to $x$ gives

$-i\hbar\frac{\partial^{2}p}{\partial x^{2}}-2p\frac{\partial p}{\partial
x}+\frac{\partial V}{\partial x}=0$ (31)

This is Burger’s equation and $p$ is given in equation (5), this ends the
proof.

It is clear from the equation (31) quantum momentum function $p$ is also
solution to Burger’s equation. Therefore, we consider the following pole
structure quantum momentum function $p$ with $V(x)=0$, that is

$p(x)=\sum_{k=1}^{n}\frac{-i\hbar}{x-x_{k}},$ (32)

Theorem $6$: The complex pole expansions of the form (32) of the Burger’s
equation is satisfied if and only if the following system of differential
equations are satisfied or the complex poles evolve according to the system of
$n$ linear equations given by

$\frac{dx_{j}}{dt}=\sum_{1\leq j\leq n,j\neq k}^{n}\frac{1}{x_{k}-x_{j}}.$
(33)

For proof please refer to chood . The equation (33) are known as Kirchhoff
equation. Therefore, in the stationary case $\frac{dx_{j}}{dt}=0$ the equation
(33), reduces to the free particle Kirchhoff equations (9) for
$\Gamma=\hbar/2$ and the superpotential $\mathcal{W}=0$. Thus by adding a
potential term to Kirchhoff equation (33) and working in stationary case reads
as

$\sum_{1\leq j\leq n,j\neq k}^{n}\frac{-i\hbar}{x_{k}-x_{j}}+Q(x_{j})=0.$ (34)

where $Q(x_{j})$ is meromorphic function. If $x_{j}$ are eigenvalues,
$i\hbar=1$ and $Q(x_{j})=x_{j}$ then equation (34) reduces to Dyson log-gas
model (14).

Theorem $7$ : The stationary Kirchhoff equation (34) are solved using
Stieltjes electrostatic model and admits a classical polynomial solution if
$Q(x_{j})$ is superpotential. kvs1

Proof: We start with the Kirchhoff equation (34) and using (16) we get

$-f^{\prime\prime}(x_{j})+2iQ(x_{j})f^{\prime}(x_{j})=0.$ (35)

This system has thermodynamic equilibrium at the zeros of the classical
orthogonal polynomial provided the equation (35) reduces to the equation (30).
Therefore, from theorem $4$ we get $Q(x_{j})=i\mathcal{W}(x_{j})$ is the
superpotential. This ends the proof.

##  V Random matrices and Quantum Action

It should be clear to the readers from previous section that solving the
quantum Hamiltonian-Jacobi through Riccati equation (6) and the burger
equation (31) leads to the differential equation (30). In this section, we
show that solving the quantum Hamiltonian-Jacobi through the burger equation
(31) leads to Kirchhoff equations (34). These Kirchhoff equations (34) are
solved using Stieltjes electrostatic model. Hence, from theorem $1$ this model
allows a Lagrangian formalism. We show that when Stieltjes electrostatic model
is used for solving quantum Hamiltonian-Jacobi it leads to quantum action $S$.

Theorem $8$ : The quantum action or quantum characteristic function $S$,
defined in (3), is

$\displaystyle S=i\sum_{i\neq
j}^{n}ln|(x_{k}-x_{j})|+i\sum_{j}\int\mathcal{W}(x_{j})dx_{j}.$ (36)

Proof: We refer readers for proof iopj . End of proof We start by defining the
quantum momentum function $p$ to be

$p=\sum_{1\leq j\leq n,j\neq
k}^{n}\frac{-i}{x_{k}-x_{j}}+i\mathcal{W}(x_{j}).$ (37)

This is justified because the quantum momentum function $p$ is the solution to
the Riccati equation (6) and Burger equation (31). The equation (37) is
complex pole evolution of Burger equation (31) whose pole structure is
identical to quantum momentum function (27). Substituting (37) in equation
(50) and taking measure $dx=\prod_{j}dx_{j}$ iopj , we get that equation (36).
Using the definition of the quantum momentum function (5) and minimizing it

$\displaystyle p=\frac{dS}{dx_{j}}=0$ (38)

gives (34). Thus, equation (38) is minimizing quantum action $S$. Therefore,
the equation (36) is quantum action.

In the classical Hamilton-Jacobi, equation of motion is governed by the
Hamiltonian and the problem is solved by continuously transforming the
Hamiltonian from the initial state to the final state through a series of
canonical transformations. In random matrix theory, the probability
distribution function (10) is a stationary solution of Fokker-Planck equation.
From theorem $3$ and Corollary $1$ it is evident that, the Fokker-Planck
equation is mapped to Schrödinger equation by transformations (24). The
exponential in transformation (24) is a Cole-Hopf transformations. Then, it is
obvious that by a series of Cole-Hopf transformations leads to Burger equation
whose complex pole evolution is given by Kirchhoff equations (34). Integrating
these Kirchhoff equations (34) gives a random matrix (11). Thus, this picture
is exactly identical to classical Hamilton-Jacobi. Therefore, the $H(x)$ is
the quantum action.

Theorem $9$ : The quantum action (51) is the random matrix and the associated
probability distribution function is the famous Cole-Hopf transformation.
kvs1

Proof: We start with the random matrix probability distribution function (10)
taking $\beta=-i$ which reduces to Cole-Hopf transformation (3) given by

$\psi={\cal P}(x_{1},\ldots,x_{N})=C_{n}e^{iH}$ (39)

where $H$ is given by equation (11). Substituting (39) in Schrödinger equation
(2) gives (6) with the quantum momentum function defined as $p=\frac{\partial
H}{\partial x_{j}}$. If $V(x_{j})=\int\mathcal{W}(x_{j})dx_{j}$ is
superpotential in the random matrix $H$ is given by equation (11) then it
reduces to quantum action (50). It is also evident from the theorem $2$ and
$3$ that the random matrix probability distribution function (10) is the
solution to the self-adjoint operator that is the Schrödinger equation. This
ends the proof.

Theorem $10$ : If $V(x_{j})=\int\mathcal{W}(x_{j})dx_{j}$ is the
superpotential then the random matrix probability distribution (10) reduces to
(12) for $\beta=2$, that is Gaussian unitary ensemble, in terms of the weight
function and is of the form kvs1

$P(x_{o},x_{1}.....x_{n})=\sum_{i}|\psi(x_{i})|^{2}$ (40)

the quantum mechanics probability distribution is in natural units.

Proof: It is clear from theorem $9$ that the random matrix probability
distribution function (39) is Cole-Hopf transformation if
$V(x_{j})=\int\mathcal{W}(x_{j})dx_{j}$ is the superpotential in the equation
(11). In quantum mechanics, wave function for polynomial solutions are of the
form square root of weight function times the polynomial. Therefore, quantum
mechanics probability distribution in natural units (40) reduces to random
matrix probability distribution (12) for $\beta=2$, that is the Gaussian
unitary ensemble. The Vandermonde determinant (12) is written in terms of
polynomials ratm . The weight function and superpotential are related as

$\displaystyle{\cal W}(x_{j})$ $\displaystyle=$
$\displaystyle\frac{d}{dx_{j}}ln(w(x_{j}))$ (41)

where $w(x_{j})$ is the weight function. In quantum Hamilton-Jacobi the
superpotential essentially captures the fixed poles, these are computed from
the turning points of the classical potential, which are reflected in the
range of classical orthogonal polynomials in terms of weight function. This
ends the proof.

It should be noted that from Corollary $1$ we have $\beta=-2i$. Then, from
theorem $9$ and $10$ we have $\beta=i$ and $\beta=2$ respectively. Suppose, if
we use $\beta=-2i$ in theorem $9$ and rewrite potential in terms of weight
function (41) we get random matrix probability distribution (12) is consistent
with Theorem $10$. Probability distribution function in quantum mechanics (40)
ground state starts from $x_{0}$ and in the random matrix probability
distribution function $\mathcal{P}(x_{1}.....x_{n})$ starts from $x_{1}$ by
definition.

##  VI Illustrations

As a first illustration, we consider the 1-D harmonic oscillator. The harmonic
oscillator potential in natural units is $V(x_{j})=\frac{1}{2}x_{j}^{2}$ such
that the superpotential is $\mathcal{W}(x_{j})=x_{j}$ kharebook . Hence,
Kirchhoff equation (34) read as

$\sum_{1\leq j\leq n,j\neq k}^{n}\frac{-i}{x_{k}-x_{j}}+ix_{j}=0.$ (42)

Integrating these equations gives the quantum action from theorem $9$ and the
wave function (39) reduces to Dyson-gas model (13) with $\beta=i\hbar$ in
natural units $\hbar=1$. The solution to 1-D harmonic oscillator are Hermite
polynomials given by

$\psi(x)=c_{n}e^{-{\frac{x^{2}}{2}}}\cdot H_{n}\left(x\right),\qquad
n=0,1,2,\ldots.$ (43)

It is well known in literature that the equation (43) can be written as

$\displaystyle\psi(x)=c_{n}e^{-{\frac{x^{2}}{2}}}\cdot\begin{vmatrix}H_{0}(x_{0})&H_{0}(x_{1})&\ldots&H_{0}(x_{n})\\\
H_{0}(x_{0})&H_{1}(x_{1})&&\vdots\\\ \vdots&\;\qquad&\ddots&\vdots\\\
H_{0}(x_{0})&\ldots&\ldots&H_{n}(x_{n})\end{vmatrix}.$ (44)

The matrix in the equation (44) is the Vandermonde determinant and its
connection to polynomials is given reference ratm . The probability
distribution function in quantum mechanics is given by

Hence, then the probability distribution explicitly

$P(x_{o},x_{1}.....x_{n})=C_{n}e^{-X^{2}}\prod_{0\leq i\leq j\leq
n}|X_{i}-X_{j}|^{2}$ (45)

Readers should note that we have used $X$ instead of $x$, it is well known in
quantum mechanics that one obtains Hermite differential equation from
Schrödinger equation for a dimensionless variables
$X={\sqrt{\frac{m\omega}{\hbar}}}x$ as we have solved in natural units
$m=\omega=\hbar=1$. Thus the quantum bound state problems mimic the Coulomb
Dyson-gas model. It is clear from this example that weight function and the
superpotential are related.

As another illustration, we consider the Coulomb problem and the wave function
is given by

$\psi(r)=r^{l+1}exp[-\frac{1}{2}r]L^{2l+1}_{n}(r)$ (46)

where the weight function
$w_{\beta}(r_{j})^{\frac{1}{2}}=r^{l+1}exp[-\frac{1}{2}r]$ and the
$L^{2l+1}_{n}(r)$ are Laguerre polynomials. We obtain the superpotential the
Coulomb problem as follows

$\displaystyle{\cal W}(r_{j})$ $\displaystyle=$
$\displaystyle\frac{d}{dr_{j}}ln(w(r_{j}))=\frac{d}{dr_{j}}ln(r_{j}^{l+1}exp[-\frac{1}{2}r_{j}])=\frac{d}{dr_{j}}((l+1)lnr_{j}-\frac{1}{2}r_{j})=\frac{(l+1)}{r_{j}}-\frac{1}{2}.$
(47)

is the desired result. As the third illustration, the weight function for the
Jacobi polynomials is given by $w(x_{j})=(1-x_{j})^{\alpha}(1+x_{j})^{\beta}$,
then the superpotential is given as follows

$\displaystyle{\cal W}(x_{j})$ $\displaystyle=$
$\displaystyle\frac{d}{dx_{j}}ln(w(x_{j}))=\frac{1}{w(x_{j})}\frac{d}{dx_{j}}(w(x_{j}))=\frac{1}{w(x_{j})}\frac{d}{dx_{j}}([(1-x_{j})^{\alpha}(1+x_{j})^{\beta}])=\frac{\beta}{(1+x_{j})}-\frac{\alpha}{(1-x_{j})}$
(48)

is the desired result. Hence, we have proved that the random matrix
probability distribution function is the famous Cole-Hopf transformation and
the underlying complex pole evolution of the Schrödinger equation is described
quantum action in terms of random matrix.

##  VII Conclusion

In this paper, starting with the quantum Hamilton-Jacobi approach, we have
shown that the underlying complex pole evolution of the Schrödinger equation
is described quantum action in terms of random matrix. The random matrix
probability distribution function is the famous Cole-Hopf transformation. The
quantum bound state problems mimic the Coulomb Dyson-gas model. Thus the
physical systems which are modeled using random matrices in terms of the
Coulomb Dyson-gas model can use all quantum bound state potentials. Random
matrix is the heart of quantum transformation theory.

## Acknowledgments

KVSSC acknowledges the Department of Science and Technology, Govt of India
(Matrices scheme (D. O. No: MTR/2018/001046)) for financial support. Authors
acknowledges Bindu A Bambah for her encouragement and comments.

##  VIII Appendix

###  VIII.1  Dirac quantum characteristic function and quantum Hamilton-Jacobi

When the Hamiltonian does not explicitly depend on time, the Hamilton’s
principle in classical Hamilton-Jacobi is given by

$\mathcal{S}(x,t)=S(x)-Et$ (49)

where $S(x)$ does not depend on $t$ and the Hamilton’s characteristic function
and action is defined as

$S=\int pdx$ (50)

where $p=\frac{\partial S}{\partial x}$ is the momentum function. For more
details refer ro cs . substituting

$\psi=exp(\frac{i}{\hbar}\mathcal{S}(x,t))$ (51)

in the time dependent Schrödinger equation

$i\hbar{\frac{\partial}{\partial
t}}\psi(x,t)=\left[-{\frac{\hbar^{2}}{2m}}{\frac{\partial^{2}}{\partial
x^{2}}}+V(x,t)\right]\psi(x,t)\,.$ (52)

gives

$\displaystyle\frac{\hbar}{i}\frac{\partial^{2}}{\partial
x^{2}}\mathcal{S}(x,t)+\left(\frac{\partial}{\partial
x}\mathcal{S}(x,t)\right)^{2}=2m\left(-\frac{\partial}{\partial
t}\mathcal{S}(x,t)-V(x)\right).$ (53)

Substituting equation (51) in equation (53) reduces to (4). Thus, $S$ in
equation (4) is the Dirac quantum characteristic function.

###  VIII.2 Supersymmetry

In supersymmetric quantum mechanics, the superpotential $\mathcal{W}(x)$ is
defined in terms of the intertwining operators $\hat{A}$ and
$\hat{A}^{\dagger}$ as

$\hat{A}=\frac{d}{dx}+\mathcal{W}(x),\qquad\hat{A}^{\dagger}=-\frac{d}{dx}+\mathcal{W}(x),$
(54)

This allows one to define a pair of factorized Hamiltonians $H^{\pm}$ as

$\displaystyle H^{+}$ $\displaystyle=$
$\displaystyle\hat{A}^{\dagger}\hat{A}=-\frac{d^{2}}{dx^{2}}+\mathcal{V}^{+}(x)-E,$
(55) $\displaystyle H^{-}$ $\displaystyle=$
$\displaystyle\hat{A}{\hat{A}}^{\dagger}=-\frac{d^{2}}{dx^{2}}+\mathcal{V}^{-}(x)-E,$
(56)

where $E$ is the factorization energy. The partner potentials
$\mathcal{V}^{\pm}(x)$ are related to $\mathcal{W}(x)$ by

$\mathcal{V}^{\pm}(x)=\mathcal{W}^{2}(x)\mp\mathcal{W}^{\prime}(x)+E.$ (57)

## References

  * (1) A. Sommerfeld, Sitzungsber. Bayr. Akad. , p. 425 (1915); W. Wilson, Phil. Mag. 31, 156 (1916);M. Born, The Mechanics of the Atom (G. Bell, London, 1927), translation by J. W. Fisher
  * (2) P. Jordan, Z. Phys. 38, 513 (1926); 40, 809 (1927); 44, 1 (1927).
  * (3) P. A. M. Dirac, Proc. R. Soc. London 113A, 621 (1927).
  * (4) P. A. M. Dirac, Phys. Z. Sowjetunion 3, 64 (1927); Rev. Mod. Phys. 17, 195 (1945).
  * (5) P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, London, 1958), fourth edition.
  * (6) R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948); also R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path In tegrals (McGraw-Hill, New York, 1965). sJ. Schwinger, Quantum Electrodynamics (Dover, New York, 1958); this reprint volume contains papers relevant to our dis- cussion, including the first paper cited in Ref. 3, and the pa- pers cited in Refs. 5 and 8.
  * (7) J. Schwinger, Quantum Electrodynamics (Dover, New York, 1958); this reprint volume contains papers relevant to our dis- cussion, including the first paper cited in Ref. 3, and the papers cited in Refs. 5 and 8.
  * (8) R. J. Finkelstein, Nonrelati Uistic Mechanics (Benjamin- Cummings, CA 1973).
  * (9) J. Schwinger, Quantum ICinematics and Dynamics (Benjamin- Cummings, CA 1970), and references therein. (1953)
  * (10) R. A Leacock and M. J. Padgett Phys. Rev. Lett. 50, 3, (1983)
  * (11) R. A. Leacock and M. J. Padgett Phys. Rev. D28, 2491, (1983).
  * (12) E.Gozzi, Physics Letters B, 158, 6, 489, (1985)
  * (13) Sirin A. Buyukasik , Oktay K. Pashaev arXiv:1005.5059 [nlin.SI]
  * (14) K.V.S Shiv Chaitanya, Foundations of Physics, 49, 351, (2019)
  * (15) K. V. S. Shiv Chaitanya, PRAMANA journal of physics, Vol. 83, No. 1, 139, (2014).
  * (16) T.J. Stieltjes, Sur Quelques theorems d’algebre, Comptes Rendus de l’Academie des Sciences, Paris, 100 (1885), 439-440; Oeuvres Completes, Vol. 1, 440-441.
  * (17) T.J. Stieltjes, Sur Quelques theorems d’algebre, Comptes Rendus de l’Academie des Sciences, Paris, 100 (1885), 620-622; Oeuvres Completes, Vol. 1, 442-444.
  * (18) Mehta, M.L. (2004). Random Matrices, 3rd Edition, Pure and Applied Mathematics (Amsterdam), 142, Amsterdam, Netherlands: Elsevier/Academic Press.
  * (19) Wigner, E. P., Gatlinburg Conference on Neutron Physics. Report 2309:59, Oak Ridge National Laboratory, Oak Ridge, TN (1957).
  * (20) J.J.M. Verbaarschot, T. Wettig, Ann.Rev.Nucl.Part.Sci. 50, 343-410, (2000).
  * (21) Y. Alhassid, Rev.Mod.Phys.72:895-968,2000.
  * (22) B Sriram Shastry and Abhishek Dhar, J. Phys. A: Math. Gen. 34, 6197, (2001).
  * (23) Peter J. Forrester, MSJ Memoirs Volume 1, 97-181, (1998).
  * (24) Philip Choquord and Stephane Rey, European Journal of Physics, 17 2, 45 (1996)
  * (25) Tarun R Tummaru et al Physics PDFs A381,47,3917(2017)
  * (26) Zaffar Ahmed and Sudhir R. Jain, 2000, Pramana 54,3, 413
  * (27) Guy Auberson et al 2001, J Phys A: Mathematical and general 34, 4, 695.
  * (28) Sudhir R Jain and Avinash Khare, 1999, Physics PDFs A, 262, 1, 35.
  * (29) Daniel Alenso and Sudheer R Jain, 1996, Physics PDFs B, 387, 4, 812.
  * (30) Alan J. Izenman, Introduction to Random-Matrix Theory, lecture notes.
  * (31) Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni, An Introduction to Random Matrices, volume 118 of Cambridge studies in advanced mathematics. Cambridge University Press, 2009.
  * (32) Jinho Baik, Percy Deift, and Kurt Johansson. On the distribution of the length of the longest increasing subsequence of random permutations. Journal of the American Mathematical Society, 12:1119–1178, 1999.
  * (33) Percy Drift. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, volume 3 of Courant Lecture Notes in Mathematics. Courant Institute of Mathematical Sciences, 1999.
  * (34) Sudhir R. Jain, Czechoslovak Journal of Physics, 56, 9, 1021, (2006) between eigenvalues of a random matrix. Comm. Math. Phys. 19 (1970), no. 3, 235–250
  * (35) Freeman J. Dyson, 1972, J Math Phys 13 90\.
  * (36) G. Szegő, Orthogonal Polynomials, Fourth Edition, Amer. Math. Soc., Providence, 1975.
  * (37) E Nelson, Phy Rev, Vol 150, No 4, 1079, (1966).
  * (38) S. Sree Ranjani, K. G. Geojo, A. K. Kapoor, P. K. Panigrahi, Mod. Phys. Lett. A. Vol 19, No. 19, 1457, (2004).
  * (39) E. L. Ince, Ordinary Differential equation (Dover Publication Inc, New York 1956).
  * (40) Choodnovsky, D. V., Choodnovsky, G. V, Nuovo Cimento B 40:339-53 (1977)
  * (41) F. Cooper, A. Khare, U. P. Sukhatme, 2001 Supersymmetric quantum mechanics (Singapore: World Scientific Publishing Co. Ltd.)
  * (42) Giacomo Livan, Marcel Novaes, Pierpaolo Vivo, ”Introduction to Random Matrices Theory and Practice” (Springer 2017)
  * (43) Herbert Goldstein , Charles Poole Jr. , John Safko, ”Classical Mechanics”, Pearson; 3rd edition (26 June 2001).
  * (44) Giacomo Livan Marcel NovaesPierpaolo Vivo, ”Introduction to Random Matrices Theory and Practice”, SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 26).


LTH1252

Anomalous dimensions at large charge in $d=4$ $O(N)$ theory
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 $n$. In subsequent work, this result was generalised to
operators of fixed charge $\bar{Q}$ in $O(N)$ theory and verified up to three
loops in perturbation theory at leading and subleading $\bar{Q}$. Here we
extend this verification to four loops in $O(N)$ theory, once again at leading
and subleading $\bar{Q}$. We also investigate the strong-coupling regime.

##  1 Introduction

Renormalizable theories with scale invariant scalar self-interactions have
been subjects of enduring interest. In particular, the study of theories with
quartic ($\phi^{4}$) interactions in $d=4-\epsilon$ dimensions has played a
central role in the development of the theory of critical phenomena, since the
pioneering work of Wilson [1, 2] and Wilson and Fisher [3] in 1971. Study of
the renormalisation group flow of the coupling or couplings of the theories
facilitates the determination of the order of phase transitions and the
associated critical indices. For example, the theory with a single scalar
field exhibits a Wilson-Fisher fixed point (FP) where the coupling constant
$\lambda$ is $O(\epsilon)$, and this infra-red (IR) attractive FP is
associated with a second order phase transition.

Historically, the majority of work in renormalisable quantum field theories
has involved the weak coupling expansion, in other words the Feynman diagram
loop expansion. However this expansion fails or becomes ponderous at either
strong coupling or (less obviously) for $\phi^{n}$ amplitudes at large $n$.
The latter has obviously developed in importance as collider energies have
increased. Remarkable progress [4, 5, 6, 7, 8, 9, 10, 11, 12] here came with
the use of a semi-classical expansion in the path integral formulation of the
theory333An analogous analysis was pursued for $\phi^{6}$ theories for
$d=3-\epsilon$ and $\phi^{3}$ theories for $d=6-\epsilon$ in Refs. [13, 14,
15].

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 $\bar{Q}$ in the
$O(N)$-invariant theory. In Ref. [9], the $U(1)$ result was compared with
perturbation theory up to two loops, and in Ref. [10] the check was performed
for the $O(N)$ theory up to three loops. Here we proceed directly with the
$O(N)$ case, since, at least for our purposes, many salient features of the
analysis are very similar in both cases; and the results for $U(1)$ may be
recovered from those for $O(N)$, essentially by setting $N=2$. We extend the
comparison with perturbation theory up to four loops, and also discuss the
large $(g\bar{Q})$ case, generalising the large $\lambda n$ analysis of Ref.
[9].

The paper is organised as follows: In Section 2 we describe the semiclassical
calculation in the $O(N)$ case, following Ref. [10]. 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 address the large $(g\bar{Q})$ limit and compare
in detail with earlier work.

##  2 The $O(N)$ case

In the $O(N)$ case we have a multiplet of fields $\phi_{i}$, $i=1\ldots N$,
and the Lagrangian is

${\cal
L}=\frac{1}{2}\partial^{\mu}\phi_{i}\partial_{\mu}\phi_{i}+\frac{g}{4!}(\phi_{i}\phi_{i})^{2}.$
(2.1)

The $\beta$-function for this theory is well-known[18]

$16\pi^{2}\beta(g)=-\epsilon
g+\frac{g^{2}}{3}(N+8)-\frac{g^{3}}{3}(3N+14)+{\cal O}(g^{4}),$ (2.2)

and leads to an infra-red conformal fixed point

$g_{*}=\frac{3\epsilon}{N+8}+\frac{9(3N+14)}{(N+8)^{3}}\epsilon^{2}+{\cal
O}(\epsilon^{3}).$ (2.3)

As shown in Ref. [10], the fixed-charge operator of charge $\bar{Q}$ may be
taken to be

$T_{\bar{Q}}=T_{i_{1}i_{2}\ldots
i_{\bar{Q}}}\phi_{i_{1}}\phi_{i_{2}}\ldots\phi_{i_{\bar{Q}}},$ (2.4)

where $T_{i_{1}i_{2}\ldots i_{\bar{Q}}}$ is symmetric, and traceless on any
pair of indices. The scaling dimension $\Delta_{T_{\bar{Q}}}$ is expanded as

$\Delta_{T_{\bar{Q}}}=\bar{Q}\left(\frac{d}{2}-1\right)+\gamma_{T_{\bar{Q}}}=\sum_{\kappa=-1}g^{\kappa}\Delta_{\kappa}(g\bar{Q}).$
(2.5)

We initially work in general $d$. The semiclassical computation of
$\Delta_{-1}$ and $\Delta_{0}$ is performed by mapping the theory via a Weyl
transformation to a cylinder $\mathbb{R}\times S^{d-1}$, where $S^{d-1}$ is a
sphere of radius $R$; where the ${\cal R}\phi^{*}\phi$ term (${\cal R}$ being
the Ricci curvature) generates an effective $m^{2}\phi^{*}\phi$ mass term with
$m=\frac{d-2}{2R}$. This mapping process along with other technical
simplifications[9] relies on conformal invariance and therefore we now assume
that we are at the conformal fixed point in Eq. (2.3). It was shown in Ref.
[9] that stationary configurations of the action are characterised by a
chemical potential $\mu$, related to the cylinder radius $R$ by

$R\mu_{*}=\frac{3^{\frac{1}{3}}+\left[6g_{*}\bar{Q}+\sqrt{36(g_{*}\bar{Q})^{2}-3}\right]^{\frac{2}{3}}}{3^{\frac{2}{3}}[6g_{*}\bar{Q}+\sqrt{36(g_{*}\bar{Q})^{2}-3}]^{\frac{1}{3}}}$
(2.6)

The computation of the leading contribution $\Delta_{-1}$ is entirely
analogous to the $U(1)$ case and is given by

$\frac{4\Delta_{-1}(g_{*}\bar{Q})}{g_{*}\bar{Q}}=\frac{3^{\frac{2}{3}}[x+\sqrt{x^{2}-3}]^{\frac{1}{3}}}{3^{\frac{1}{3}}+[x+\sqrt{x^{2}-3}]^{\frac{2}{3}}}+\frac{3^{\frac{1}{3}}\\{3^{\frac{1}{3}}+[x+\sqrt{x^{2}-3}]^{\frac{2}{3}}\\}}{[x+\sqrt{x^{2}-3}]^{\frac{1}{3}}},$
(2.7)

where $x=6g_{*}\bar{Q}$. Its expansion for small $g_{*}\bar{Q}$ takes the form

$\frac{\Delta_{-1}(g_{*}\bar{Q})}{g_{*}}=\bar{Q}\left[1+\frac{1}{3}g_{*}\bar{Q}-\frac{2}{9}(g_{*}\bar{Q})^{2}+\frac{8}{27}(g_{*}\bar{Q})^{3}-\frac{14}{27}(g_{*}\bar{Q})^{4}+{\cal
O}\left\\{(g_{*}\bar{Q})^{5}\right\\}\right].$ (2.8)

The non-leading corrections $\Delta_{0}$ are once more given by the
determinant of small fluctuations. There are two modes corresponding to those
in the abelian case, with the dispersion relation

$\omega_{\pm}^{2}(l)=J_{l}^{2}+3\mu^{2}-m^{2}\pm\sqrt{4J_{l}^{2}\mu^{2}+(3\mu^{2}-m^{2})^{2}}$
(2.9)

where

$J_{l}^{2}=\frac{l(l+d-2)}{R^{2}}$ (2.10)

is the eigenvalue of the Laplacian on the sphere. In addition there are
$\frac{N}{2}-1$ “Type II” (non-relativistic)[16] Goldstone modes and
$\frac{N}{2}-1$ massive states with dispersion relation

$\omega_{\pm\pm}(l)=\sqrt{J_{l}^{2}+\mu^{2}}\pm\mu,$ (2.11)

with $J_{l}$ as defined in Eq. (2.10). We then find that $\Delta_{0}$ is given
by

$\Delta_{0}(g_{*}\bar{Q})=\frac{1}{2}\sum_{l=0}^{\infty}\sigma_{l}$ (2.12)

where

$\displaystyle\sigma_{l}=Rn_{l}\left\\{\omega^{*}_{+}(l)+\omega^{*}_{-}(l)+\left(\frac{N}{2}-1\right)[\omega^{*}_{++}(l)+\omega^{*}_{--}(l)]\right\\}.$
(2.13)

Here

$n_{l}=\frac{(2l+d-2)\Gamma(l+d-2)}{\Gamma(l+1)\Gamma(d-1)}$ (2.14)

is the multiplicity of the laplacian on the $d$-dimensional sphere, and
$\omega^{*}_{\pm}$, $\omega^{*}_{++}$, $\omega^{*}_{--}$ are defined as in
Eqs. (2.9), (2.11) respectively, evaluated at the fixed point with $R$,
$\mu_{*}$ related by Eq. (2.6). For the small $(g_{*}\bar{Q})$ computation, we
need to isolate the divergent contribution in the sum in Eq. (2.12). We use
the large-$l$ expansion of $\sigma_{l}$,

$\sigma_{l}=\sum_{n=1}^{\infty}c_{n}l^{d-n}$ (2.15)

with

$\displaystyle c_{1}=N,\quad c_{2}=$ $\displaystyle 3N,$ $\displaystyle
c_{3}=$ $\displaystyle\frac{1}{2}[5N-2+(N+2)(R\mu_{*})^{2}],$ $\displaystyle
c_{4}=$ $\displaystyle\frac{1}{2}[N-2+(N+2)(R\mu_{*})^{2}],$ $\displaystyle
c_{5}=$
$\displaystyle\frac{N+8}{8}(R^{2}\mu_{*}^{2}-1)^{2}\left[-1+\left(\gamma-\frac{3}{2}\right)\epsilon\right]$
$\displaystyle-\frac{29}{12}(R^{2}\mu_{*}^{2}-1)\epsilon-\left(\frac{11}{24}R^{2}\mu_{*}^{2}-\frac{1}{5}\right)N\epsilon.$
(2.16)

We can write

$\displaystyle\Delta_{0}(g_{*}\bar{Q})=$
$\displaystyle-\frac{15\mu_{*}^{4}R^{4}+6\mu_{*}^{2}R^{2}-5}{16}+\frac{1}{2}\sum_{l=1}^{\infty}\overline{\sigma}_{l}+\sqrt{\frac{3\mu_{*}^{2}R^{2}-1}{2}}$
$\displaystyle-\frac{1}{16}\left(\frac{N}{2}-1\right)[7-16R\mu_{*}+6R^{2}\mu_{*}^{2}+3R^{4}\mu_{*}^{4}],$
(2.17)

where

$\overline{\sigma}_{l}=\sigma_{l}-c_{1}l^{3}-c_{2}l^{2}-c_{3}l-c_{4}-c_{5}\frac{1}{l}.$
(2.18)

Here the divergent parts have been isolated and the sums over $l$ performed,
as explained in Refs. [9] and [10]. The sum over $\frac{1}{l^{d-n}}$ for $n=5$
leads to a pole in $\epsilon$ which cancels against the pole in the bare
coupling. The sum over $\overline{\sigma}_{l}$ is then finite and setting
$d=4$ and expanding in small $g_{*}\bar{Q}$ can be performed analytically. We
obtain

$\displaystyle\Delta_{0}=$
$\displaystyle-\frac{1}{6}(10+N)g_{*}\bar{Q}+\frac{1}{18}(6-N)(g_{*}\bar{Q})^{2}$
$\displaystyle+\frac{1}{27}[N-36+2(14+N)\zeta_{3}](g_{*}\bar{Q})^{3}$
$\displaystyle-\frac{1}{81}[4(N-73)+2(6N+65)\zeta_{3}+5(N+30)\zeta_{5}](g_{*}\bar{Q})^{4}+\ldots$
(2.19)

Adding Eqs. (2.8) and (2.19), we find [10]

$\displaystyle\frac{\Delta_{-1}(g_{*}\bar{Q})}{g_{*}}+\Delta_{0}(g_{*}\bar{Q})=$
$\displaystyle\bar{Q}+\frac{1}{6}[2\bar{Q}-(N+10)]g_{*}\bar{Q}-\frac{1}{18}[4\bar{Q}+(N-6)](g_{*}\bar{Q})^{2}$
$\displaystyle+\frac{1}{27}[8\bar{Q}+N-36+2(N+14)\zeta_{3}](g_{*}\bar{Q})^{3}$
$\displaystyle+\Bigl{\\{}-\frac{14}{27}\bar{Q}-\frac{1}{81}[4(N-73)+2(6N+65)\zeta_{3}$
$\displaystyle+5(N+30)\zeta_{5}]\Bigr{\\}}(g_{*}\bar{Q})^{4}+\ldots$ (2.20)

##  3 The diagrammatic calculation

In this section we carry out the perturbative calculation to confirm the
semiclassical result at leading and next-to-leading order in $\bar{Q})$ up to
four-loop level, as displayed in Eq. (2.20).

(a) (b) (c) Figure 1: One- and two-loop diagrams for $\gamma_{T_{\bar{Q}}}$
contributing at leading $n$. Here and elsewhere, the lozenge denotes the
$T_{\bar{Q}}$ vertex.

The one-loop contribution to $\gamma_{T_{\bar{Q}}}$ comes solely from the
diagram depicted in Fig. 1(a) and is given by

$\gamma^{(1)}_{T_{\bar{Q}}}=-\frac{1}{3}g\bar{Q}(1-\bar{Q}).$ (3.1)

As mentioned before, the derivation of the semiclassical result relied on
working at the conformal fixed point $g_{*}$. However, surprisingly, at two,
three and four loops we will see that the functional forms of the
semiclassical and perturbative results agree for general $g$ and not just on
substitution of $g=g_{*}$ with $g_{*}$ as given in Eq. (2.3). It is only at
one loop where the agreement only holds at the fixed point. Specifically, the
leading terms $\bar{Q}\left(\frac{d}{2}-1\right)+\gamma^{(1)}_{T_{\bar{Q}}}$
on the left-hand side of Eq. (2.5) (as given in Eq. (3.1)) only agree with the
${\cal O}(g^{0})$ and ${\cal O}(g)$ terms in
$\frac{\Delta_{-1}(g\bar{Q})}{g}+\Delta_{0}(g\bar{Q})$ on the right-hand side
of Eq. (2.5) (as obtained from Eq. (2.20)) after substituting
$g=g_{*}\approx\frac{3\epsilon}{N+8}$. In this case, specialising to the fixed
point has induced a mixing between the classical and one-loop ${\cal
O}(\bar{Q})$ terms.

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

$\displaystyle\gamma^{(2)}_{(b)}=$
$\displaystyle-\frac{2}{9}g^{2}\bar{Q}(\bar{Q}-1)(\bar{Q}-2),$ (3.2)
$\displaystyle\gamma^{(2)}_{(c)}=$
$\displaystyle-\frac{1}{9}g^{2}\left(3+\frac{1}{2}N\right)\bar{Q}(\bar{Q}-1),$
(3.3)

producing leading and next-to-leading terms given by

$\gamma^{(2)}_{T_{\bar{Q}}}=-\frac{1}{18}(g\bar{Q})^{2}(4\bar{Q}-6+N),$ (3.4)

in accord with the semiclassical results in Eq. (2.20). As emphasised earlier,
this agreement holds for general $g$ 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 $g=g_{*}$ as given in Eq. (2.3) does
not induce any mixing between leading or next-to-leading terms at different
loop orders. Therefore if Eq. (2.5) holds at the fixed point, it must also
hold in general. In fact the agreement was already checked at the fixed point
in Ref. [10] in the general $O(N)$ case, and in the $U(1)$ case in Ref. [9].

(a) (b) (c) Figure 2: Three-loop diagrams for $\gamma_{T_{\bar{Q}}}$
contributing at leading $n$ (a) (b) (c) (d) (e) Figure 3: Three-loop diagrams
for $\gamma_{T_{\bar{Q}}}$ contributing at next-to-leading $n$

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

Graph | Symmetry Factor | Simple Pole
---|---|---
2(a) | $\frac{1}{54}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $-\frac{2}{3}$
2(b) | $\frac{2}{27}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $\frac{4}{3}$
2(c) | $\frac{1}{54}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $\frac{2}{3}$
3(a) | $\frac{1}{27}\frac{\bar{Q}!}{(\bar{Q}-3)!}$ | $-\frac{2}{3}$
3(b) | $\frac{4}{27}\frac{\bar{Q}!}{(\bar{Q}-3)!}A$ | $-\frac{2}{3}$
3(c) | $\frac{2}{27}\frac{\bar{Q}!}{(\bar{Q}-3)!}$ | $\frac{4}{3}$
3(d) | $\frac{4}{27}\frac{\bar{Q}!}{(\bar{Q}-3)!}A$ | $\frac{4}{3}$
3(e) | $\frac{8}{27}\frac{\bar{Q}!}{(\bar{Q}-3)!}B$ | $4\zeta_{3}$
Table 1: Three-loop results from Figs. 2, 3

The simple pole contributions from individual three loop diagrams may be
extracted from Ref. [17] and are listed in Table 1, together with the
corresponding symmetry factor. A factor of $g^{3}$ is understood in each case.
The $N$-dependent factors $A$ and $B$ are given by

$A=\frac{1}{8}(N+6),\quad B=\frac{1}{16}(N+14).$ (3.5)

When added and multiplied by a loop factor of 3, the leading and non-leading
three-loop contributions to $\gamma_{T_{\bar{Q}}}$ are found to be

$\gamma^{(3)}_{T_{\bar{Q}}}=\frac{1}{27}(g\bar{Q})^{3}[8\bar{Q}+N-36+2(14+N)\zeta_{3}],$
(3.6)

once again in accord with the semiclassical results in Eqs. (2.20), for
general $g$. Equivalently, this agreement was already checked at the fixed
point in Ref. [10].

The leading ${\cal O}(\bar{Q}^{5})$ four-loop contributions to
$\gamma_{T_{\bar{Q}}}$ come purely from the diagrams depicted in Fig. 4 (with
five lines emerging from the $T_{\bar{Q}}$ vertex), while the next-to-leading
${\cal O}(\bar{Q}^{4})$ contributions are generated by these diagrams together
with those in Figs. 5 and 6 (with four lines emerging from the $T_{\bar{Q}}$
vertex).

(a) (b) (c)(d)(e)(f) Figure 4: Four-loop diagrams for $\gamma_{T_{\bar{Q}}}$ contributing at leading $n$ Graph | Symmetry Factor | Simple Pole
---|---|---
4(a) | $\frac{4}{81}\frac{\bar{Q}!}{(\bar{Q}-5)!}$ | $\frac{5}{2}$
4(b) | $\frac{2}{81}\frac{\bar{Q}!}{(\bar{Q}-5)!}$ | $-\frac{2}{3}$
4(c) | $\frac{1}{81}\frac{\bar{Q}!}{(\bar{Q}-5)!}$ | $-\frac{5}{6}$
4(d) | $\frac{1}{81}\frac{\bar{Q}!}{(\bar{Q}-5)!}$ | $\frac{11}{6}$
4(e) | $\frac{2}{81}\frac{\bar{Q}!}{(\bar{Q}-5)!}$ | $\frac{2}{3}$
4(f) | $\frac{1}{81}\frac{\bar{Q}!}{(\bar{Q}-5)!}$ | $-\frac{1}{2}$
Table 2: Four-loop results from Fig. 4 (a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k) Figure 5: Four-loop diagrams for $\gamma_{T_{\bar{Q}}}$ contributing at next-to-leading $n$ (a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k) Figure 6: Four-loop diagrams for $\gamma_{T_{\bar{Q}}}$ contributing at next-to-leading $n$ (continued) Graph | Symmetry Factor | Simple Pole
---|---|---
5(a) | $\frac{2}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $\frac{1}{6}(11-6\zeta_{3})$
5(b) | $\frac{2}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}A$ | $\frac{1}{6}(11-6\zeta_{3})$
5(c) | $\frac{4}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}A$ | $-\frac{1}{2}$
5(d) | $\frac{16}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}C$ | $10\zeta_{5}$
5(e) | $\frac{8}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}B$ | $\frac{3}{2}(2\zeta_{3}-\zeta_{4})$
5(f) | $\frac{8}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}A$ | $-\frac{2}{3}$
5(g) | $\frac{2}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}A$ | $\frac{1}{2}(1-2\zeta_{3})$
5(h) | $\frac{1}{324}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $-2(1-\zeta_{3})$
5(i) | $\frac{1}{162}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $-2(1-\zeta_{3})$
5(j) | $\frac{2}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $-(1-2\zeta_{3})$
5(k) | $\frac{1}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $\frac{1}{2}(1-2\zeta_{3})$
Table 3: Four-loop results from Fig. 5 Graph | Symmetry Factor | Simple Pole
---|---|---
6(a) | $\frac{4}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}A$ | $-\frac{1}{6}(5-6\zeta_{3})$
6(b) | $\frac{8}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}B$ | $\frac{3}{2}(2\zeta_{3}+\zeta_{4})$
6(c) | $\frac{4}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}A$ | $-\frac{5}{6}$
6(d) | $\frac{2}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $-\frac{2}{3}$
6(e) | $\frac{4}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}A$ | $-\frac{2}{3}$
6(f) | $\frac{2}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $-\frac{1}{6}(5-6\zeta_{3})$
6(g) | $\frac{2}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $-\frac{1}{6}(5-6\zeta_{3})$
6(h) | $\frac{4}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $\frac{1}{2}(5-4\zeta_{3})$
6(i) | $\frac{4}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $\frac{1}{2}(5-4\zeta_{3})$
6(j) | $\frac{8}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}A$ | $\frac{5}{2}$
6(k) | $\frac{4}{81}\frac{\bar{Q}!}{(\bar{Q}-4)!}$ | $\frac{5}{2}$
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. [18]).
Those from Figs. 5, 6 may be extracted from Ref. [17]. The contributions from
each four-loop diagram are listed in Tables 2, 3 and 4 respectively, together
with the corresponding symmetry factor. A factor of $g^{4}$ is understood in
each case, and the $N$-dependent factor $C$ is given by

$C=\frac{1}{32}(N+30).$ (3.7)

When added and multiplied by a loop factor of 4, the leading and non-leading
four-loop contributions to $\gamma_{T_{\bar{Q}}}$ are found to be

$\gamma^{(4)}_{T_{\bar{Q}}}=-\frac{1}{81}(g\bar{Q})^{4}[42\bar{Q}+4(N-73)+2(6N+65)\zeta_{3}+5(N+30)\zeta_{5}],$
(3.8)

once again in accord with the semiclassical results in Eqs. (2.20), for
general $g$.

##  4 The large $g_{*}\bar{Q}$ calculation

In this section we discuss the large $g_{*}\bar{Q}$ limit of
$\Delta_{T_{\bar{Q}}}$. The large $g_{*}\bar{Q}$ limit of $\Delta_{-1}$ as
given by Eq. (2.7) is readily obtained as

$\frac{\Delta_{-1}}{g_{*}}=\frac{3}{4g_{*}}\left[\frac{3}{4}\left(\frac{4g_{*}\bar{Q}}{3}\right)^{\frac{4}{3}}+\frac{1}{2}\left(\frac{4g_{*}\bar{Q}}{3}\right)^{\frac{2}{3}}+{\cal
O}(1)\right].$ (4.1)

We follow the procedure described in Ref. [9] for evaluating $\Delta_{0}$ by
means of an approximation to the sum over $l$ followed by a numerical fit. The
procedure involves selecting integers $N_{1}$, $N_{2}$ and picking $A\geq 1$
such that $AR\mu_{*}$ is an integer (this represents a cut-off in the
summation, beyond which we approximate it by an integral). The accuracy may be
made as great as desired by increasing $N_{1}$, $N_{2}$ and $A$. We obtain

$\Delta_{0}=\frac{N+8}{16}(R^{2}\mu^{*2}-1)^{2}\ln(AR\mu_{*})+F(R\mu_{*}),$
(4.2)

where

$F(R\mu_{*})=f_{N_{2},A}(R\mu_{*})-\frac{1}{4}\sigma_{AR\mu_{*}}+\frac{1}{2}\sum_{l=0}^{AR\mu_{*}}\sigma_{l}-\frac{1}{2}\sum_{k=1}^{N_{1}}\frac{B_{2k}}{(2k)!}\sigma^{(2k-1)}_{AR\mu_{*}},$
(4.3)

and here

$\displaystyle f_{N_{2},A}(R\mu_{*})=$
$\displaystyle\frac{1}{2}(AR\mu_{*})^{4}\sum_{n=1,n\neq
5}^{N_{2}}\frac{c_{n}}{(AR\mu_{*})^{n-1}(n-5)}$
$\displaystyle+\frac{N+8}{16}(R^{2}\mu^{*2}-1)^{2}\left(\gamma-\frac{3}{2}\right)-\frac{29}{24}(R^{2}\mu^{*2}-1)-\left(\frac{11}{48}R^{2}\mu^{*2}-\frac{1}{10}\right)N.$
(4.4)

With some help from one of the authors[19] we have corrected some typos in the
corresponding equations in Ref. [9], which were not reflected in their final
results. The function $f_{N_{2},A}(R\mu_{*})$ derives from replacing the sum
over $l$ for $l\geq AR\mu_{*}$ in Eq. (2.12) by an integral over $l$. It is
then appropriate to use the large $l$ expansion in Eq. (2.15). The integral
over $\frac{1}{l^{1+\epsilon}}$ corresponding to the $c_{5}$ term leads to a
pole term in $\epsilon$. The potential pole in $\Delta_{0}$ is cancelled by
the pole in the bare coupling, but the $O(\epsilon)$ term in $c_{5}$ in Eq.
(2.16) leads to the terms in the last line of Eq. (4.4). The details of the
procedure may be found in Ref. [9]. In Eq. (4.3), we can set $d=4$. We now
evaluate $F(R\mu_{*})$ in Eq. (4.3) numerically. We take $N_{1}=4$, $N_{2}=10$
and $A=10$, using the same numbers as Ref. [9] for comparison purposes. The
result is then fitted with an expansion in $(R\mu_{*})^{-2}$, starting from
$(R\mu_{*})^{4}$, with 4 parameters. We find that $F(R\mu_{*})$ is given by

$\displaystyle F(R\mu_{*})\sim$
$\displaystyle-(1.5559+0.2293N)(R\mu_{*})^{4}+(1.8536+0.3231N)(R\mu_{*})^{2}$
$\displaystyle-(0.4467+0.0826N)+{\cal O}((R\mu_{*})^{-2}),$ (4.5)

and this may be inserted into Eq. (4.2) to give the full result for
$\Delta_{0}$. Expanding $R\mu_{*}$ as given by Eq. (2.6) in terms of large
$g_{*}\bar{Q}$, we find

$R\mu_{*}=\left(\frac{4g_{*}\bar{Q}}{3}\right)^{\frac{1}{3}}+\frac{1}{3}\left(\frac{4g_{*}\bar{Q}}{3}\right)^{-\frac{1}{3}}+\ldots$
(4.6)

and then we obtain from Eq. (4.2)

$\displaystyle\Delta_{0}=$
$\displaystyle\left[\alpha+\frac{N+8}{48}\ln\left(\frac{4g_{*}\bar{Q}}{3}\right)\right]\left(\frac{4g_{*}\bar{Q}}{3}\right)^{\frac{4}{3}}$
$\displaystyle+\left[\beta-\frac{N+8}{72}\ln\left(\frac{4g_{*}\bar{Q}}{3}\right)\right]\left(\frac{4g_{*}\bar{Q}}{3}\right)^{\frac{2}{3}}+{\cal
O}(1),$ (4.7)

where

$\displaystyle\alpha=$ $\displaystyle-0.4046-0.0854N,$ $\displaystyle\beta=$
$\displaystyle-0.8218-0.0577N.$ (4.8)

The results for $U(1)$ should be recovered by setting $N=2$; and indeed for
$N=2$ we find Eqs. (4.5), (4.7), (4.8) agree with the corresponding results
given in Ref. [9].

Following Ref.[9] and combining Eqs. (2.3), (2.5), (4.1) and (4.7), we may
write the full scaling dimension in the form

$\displaystyle\Delta_{T_{\bar{Q}}}=$
$\displaystyle\frac{1}{\epsilon}\left(\frac{4\epsilon\bar{Q}}{N+8}\right)^{\frac{d}{d-1}}\left[\frac{3(N+8)}{16}+\epsilon\left(\alpha+\frac{3(3N+14)}{16(N+8)}\right)+{\cal
O}(\epsilon^{2})\right]$
$\displaystyle+\frac{1}{\epsilon}\left(\frac{4\epsilon\bar{Q}}{N+8}\right)^{\frac{d-2}{d-1}}\left[\frac{N+8}{8}+\epsilon\left(\beta-\frac{3N+14}{8(N+8)}\right)+{\cal
O}(\epsilon^{2})\right]+{\cal O}[(\epsilon\bar{Q})^{0}]$ (4.9)

In Ref. [15], we found that we could reproduce the coefficients in the large
$R\mu_{*}$ expansion of the $N$-dependent part of $\Delta_{0}$ (the terms
involving $\omega^{*}_{++}$ and $\omega^{*}_{--}$ in Eq. (2.13)) by an
analytic computation. This fails to work here; an analytic large-$R\mu_{*}$
expansion of $\omega^{*}_{++}$ and $\omega^{*}_{--}$ as given by Eq. (2.11)
leads to odd negative powers of $R\mu_{*}$, whereas our numeric computation in
Eq. (4.5) only contains even powers of $R\mu_{*}$. It appears that the simple
properties of $\omega^{*}_{++}$ and $\omega^{*}_{--}$ identified in Ref. [15],
in particular their expansion in powers of
$\frac{J_{l}^{2}}{R^{2}\mu_{*}^{2}}$ , are not enough for our analytic
computation to work in the $d=4$ case. A little trial and error indicates that
the fact that in $d=3$, $n_{l}\propto\frac{d}{dl}J_{l}^{2}$, may also be
crucial; but further insight is required.

##  5 Conclusions

Approaches that extend the reach of (or even transcend the need for)
perturbation theory have always been challenging, and are all the more
interesting now because of the increased importance attached to multi-leg
amplitudes, which can present formidable calculational obstacles at higher
loop orders. In this paper we have followed Refs. [9, 8, 10] in the
application of semi-classical methods to the calculation of $\phi^{n}$
amplitudes in $d=4$ renormalisable scalar theories with quartic interactions.
Ref [10] generalises this calculation of Ref. [9] from $U(1)$ to an $O(N)$
invariant interaction. Another motivation for studying this class of theories
is their (classical) scale invariance (CSI). As remarked in Ref [10], the
Standard Model (SM) is “almost” CSI. Indeed, in 1973, Coleman and Weinberg
(CW) [20] had hoped to argue that the SM might indeed be viable with the
omission of the Higgs (wrong-sign) $(\hbox{mass})^{2}$ term. This attractive
idea failed. Neglecting Yukawa couplings (which seemed reasonable at the time)
led to a Higgs mass prediction which was too small; but including the top
quark Yukawa coupling destabilised the Higgs vacuum altogether 444For a review
of some controversy over this development, see Ref. [21]. CW introduced the
idea of dimensional transmutation as a means of generating a physical mass
scale in a CSI theory. The same phenomenon has been pursued [22, 23, 24] in
the CSI form of quantum gravity [25, 26, 27, 28, 29, 30].

Our purpose here has been to compare the results of Ref [10] with
straightforward (albeit intricate) perturbation theory. Generally the results
have supported the validity of the semi-classical approximation, in its domain
of validity.

Future work might include the application of the semi-classical methods and
perturbative methods used here to the remaining class of CSI theories with
scalar self-interactions; that is $\phi^{3}$ theories in $d=6$; or even
perhaps the case of CSI quantum gravity mentioned above.

## Acknowledgements

We are grateful to Gabriel Cuomo for 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. B4 (9): 3174–3183.
  * [2] K.G. Wilson, ”Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior”, Phys. Rev. B4 (9): 3184–3205.
  * [3] K.G. Wilson and Michael E. Fisher, “Critical exponents in 3.99 dimensions”, Phys.Rev.Lett. 28 (1972) 240-243.
  * [4] D.T. Son, “Semiclassical approach for multiparticle production in scalar theories”, Nucl.Phys. B 477 (1996) 378-406.
  * [5] S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, “On the CFT Operator Spectrum at Large Global Charge, JHEP 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”, JHEP 04 (2017) 059, [arXiv:1610.04495 [hep-th]].
  * [7] L. Alvarez-Gaume, D. Orlando and S. Reffert, “Large charge at large N”, JHEP 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 $\epsilon=0$,” JHEP 10 (2019) 201, [arXiv:1908.11347 [hep-th]].
  * [9] G. Badel, G. Cuomo, A. Monin and R. Rattazzi, “The epsilon expansion meets semiclassics”, JHEP 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 (2020) 4, 045011, [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, 12, 125033, [arXiv:2006.10078 [hep-th]].
  * [12] L. Alvarez-Gaume, D. Orlando and S. Reffert, “Selected Topics in the Large Quantum Number Expansion”, arXiv:2008.03308 [hep-th].
  * [13] G. Badel, G. Cuomo, A. Monin and R. Rattazzi, “Feynman diagrams and the large charge expansion in $3-\varepsilon$ dimensions,” Phys. Lett. B 802 (2020) 135202 [arXiv:1911.08505 [hep-th]].
  * [14] 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”, JHEP 09 (2020) 064 [arXiv:2003.13772 [hep-th]]
  * [15] I. Jack and D.R.T. Jones, “Anomalous dimensions for $\phi^{n}$ in scale invariant $d=3$ theory.”, Phys.Rev. D 102 (2020) 8, 085012, [arXiv:2007.07190 [hep-th]]
  * [16] H.B. Nielsen and S. Chadha, “On how to count Goldstone bosons,” Nucl. Phys. B 105 (1976), 445-453
  * [17] D. I. Kazakov, O. V. Tarasov and A. A. Vladimirov, “Calculation of Critical Exponents by Quantum Field Theory Methods,” Sov. Phys. JETP 50 (1979), 521 JINR-E2-12249.
  * [18] H. Kleinert and V. Schulte-Frohlinde, “Critical properties of $\phi^{4}$-theories”, World Scientific (2001) .
  * [19] G. Cuomo, private communication.
  * [20] S.R. Coleman and E.J. Weinberg, “Radiative Corrections as the Origin of Spontaneous Symmetry Breaking”, Phys.Rev. D 7 (1973) 1888-1910.
  * [21] M.B. Einhorn and D.R.T. Jones, “The Effective potential, the renormalisation group and vacuum stability”, JHEP 04 (2007) 051, hep-ph/0702295 [hep-ph]
  * [22] M. B. Einhorn and D. R. T. Jones, “Naturalness and Dimensional Transmutation in Classically Scale-Invariant Gravity,” JHEP 1503 (2015) 047 [arXiv:1410.8513 [hep-th]].
  * [23] M. B. Einhorn and D. R. T. Jones, “Induced Gravity I: Real Scalar Field,” JHEP 1601 (2016) 019 [arXiv:1511.01481 [hep-th]].
  * [24] M. B. Einhorn and D. R. T. Jones, “Induced Gravity II: Grand Unification,” JHEP 1605 (2016) 185 [arXiv:1602.06290 [hep-th]].
  * [25] K. S. Stelle, “Renormalization of Higher Derivative Quantum Gravity,” Phys. Rev. D 16 (1977) 953.
  * [26] E. S. Fradkin and A. A. Tseytlin, “Renormalizable Asymptotically Free Quantum Theory of Gravity,” Phys. Lett. B 104 (1981) 377.
  * [27] E. S. Fradkin and A. A. Tseytlin, “Renormalizable asymptotically free quantum theory of gravity,” Nucl. Phys. B 201 (1982) 469.
  * [28] I. G. Avramidi and A. O. Barvinsky, “Asymptotic Freedom In Higher Derivative Quantum Gravity,” Phys. Lett. B 159 (1985) 269.
  * [29] I. G. Avramidi, “Asymptotic Behavior of the Quantum Theory of Gravity With Higher Order Derivatives,” Sov. J. Nucl. Phys. 44 (1986) 160.
  * [30] I. G. Avramidi, “Heat kernel and quantum gravity,” Lect. Notes Phys. M 64 (2000) 1.