11affiliationtext: Department of Mathematics, City College of New York, CUNY,
New York, NY22affiliationtext: Department of Mathematics, Baruch College,
CUNY, New York, NY33affiliationtext: Department of Mathematics, CUNY Graduate
Center, New York, NY44affiliationtext: Department of Mathematics, University
of Bristol, UK

# Evidence of Random Matrix Corrections for the Large Deviations of Selberg’s
Central Limit Theorem

E. Amzallag  L.-P. Arguin  E. Bailey  K. Hui  R. Rao

(April 16, 2021)

###### Abstract

Selberg’s central limit theorem states that the values of $\log|\zeta(1/2+{\rm
i}\tau)|$, where $\tau$ is a uniform random variable on $[T,2T]$, is
distributed like a Gaussian random variable of mean $0$ and standard deviation
$\sqrt{\frac{1}{2}\log\log T}$. It was conjectured by Radziwiłł that this
breaks down for values of order $\log\log T$, where a multiplicative
correction $C_{k}$ would be present at level $k\log\log T$, $k>0$. This
constant should be equal to the leading asymptotic for the $2k^{th}$ moment of
$\zeta$, as first conjectured by Keating and Snaith using random matrix
theory. In this paper, we provide numerical and theoretical evidence for this
conjecture. We propose that this correction has a significant effect on the
distribution of the maximum of $\log|\zeta|$ in intervals of size $(\log
T)^{\theta}$, $\theta>0$. The precision of the prediction enables the
numerical detection of $C_{k}$ even for low $T$’s of order $T=10^{8}$. A
similar correction appears in the large deviations of the Keating-Snaith
central limit theorem for the logarithm of the characteristic polynomial of a
random unitary matrix, as first proved by Féray, Méliot and Nikeghbali.

##  1 Introduction and Main Results

###  1.1 Introduction

The large values of the Riemann zeta function $\zeta:\mathbb{C}\to\mathbb{C}$
on the critical line ${\rm Re}(s)=1/2$ play an important role in number
theory. There are several conjectures describing its purported behavior. For
example, the moment conjecture gives the precise asymptotics of the moments of
the function on the interval $[T,2T]$ (see for example [keasna00a, ivic85,
titchmarsh86]):

######  Conjecture 1 (Moment Conjecture).

For $k\geq 0$, as $T\rightarrow\infty$,

$\frac{1}{T}\int_{0}^{T}|\zeta(1/2+{\rm i}t)|^{2k}{\rm d}t\sim C_{k}(\log
T)^{k^{2}}.$ (1)

Lower bounds agreeing with the predicted leading exponent are known
unconditionally (see [radsou13, heasou20] and the earlier works of [heabro81,
ram94]). Consistent upper bounds on the level of the leading exponent are
known unconditionally for $0\leq k\leq 2$ (from the work of [hearadsou19]),
and for all $k\geq 0$ conditionally on the Riemann hypothesis [har13b, sou09].
This paper is predominantly about the constants $C_{k}$.

At the level of the constant, Conjecture 1 is only proved in the case $k=1$,
by Hardy and Littlewood with $C_{1}=1$, and the case $k=2$ by Ingham with
$C_{2}=\frac{1}{(2\pi)^{2}}$ [harlit18, ing26]. The constants $C_{k}$, $k>0$,
have been conjectured in [keasna00a] using random matrix theory to be of the
following form

$C_{k}=a_{k}\cdot f_{k},$ (2)

where

$\displaystyle a_{k}$ $\displaystyle=\prod_{p\text{
primes}}\left(1-\frac{1}{p}\right)^{k^{2}}\sum_{m=0}^{\infty}\left(\frac{\Gamma(k+m)}{m!\Gamma(k)}\right)^{2}p^{-m},$
(3)

and

$\displaystyle f_{k}$
$\displaystyle=\frac{\mathcal{G}^{2}(1+k)}{\mathcal{G}(1+2k)}.$ (4)

Here, $\mathcal{G}$ denotes the Barnes $\mathcal{G}-$function. An alternative
approach using Dirichlet series yields the same conjecture [diagolhof03].

As pointed out in [rad11], the constants $C_{k}$ should also appear in the
large deviations of Selberg’s central limit theorem. The theorem asserts that
for $\sigma^{2}_{T}=\frac{1}{2}\log\log T$

$\mathbf{P}\Big{(}\log|\zeta(1/2+{\rm i}\tau)|>\sigma_{T}\cdot
V\Big{)}\sim\int_{V}^{\infty}\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}{\rm
d}x,\quad V\in\mathbb{R}.$ (5)

However, for $V$ of the order of the variance, the proposed correction is:

######  Conjecture 2 (Radziwiłł’s Conjecture111There is a small typo in the
statement of the conjecture in the original paper where $V\sim k\sqrt{\log\log
T}$. A factor of $2$ in the square root is missing.).

If $\tau$ is uniformly distributed on $[T,2T]$ and
$\sigma^{2}_{T}=\frac{1}{2}\log\log T$, then for $V\sim k\sqrt{2\log\log T}$
we have for $k>0$,

$\mathbf{P}\Big{(}\log|\zeta(1/2+{\rm i}\tau)|>\sigma_{T}\cdot V\Big{)}\sim
C_{k}\int_{V}^{\infty}\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}{\rm d}x.$ (6)

The conjecture is plausible since if one biases the choice of $\tau$ by the
value $|\zeta(1/2+{\rm i}\tau)|^{2k}$, the left-hand side of (6) becomes:

$\mathbf{P}\Big{(}\log|\zeta(1/2+{\rm i}\tau)|>\sigma_{T}\cdot
V\Big{)}=\mathbf{E}[|\zeta(1/2+{\rm
i}\tau)|^{2k}]\cdot\widetilde{\mathbf{P}}\Big{(}\log|\zeta(1/2+{\rm
i}\tau)|>\sigma_{T}\cdot V\Big{)},$ (7)

where $\widetilde{\mathbf{P}}$ is defined by $\frac{{\rm
d}\widetilde{\mathbf{P}}}{{\rm d}\mathbf{P}}=\frac{|\zeta(1/2+{\rm
i}\tau)|^{2k}}{\mathbf{E}[|\zeta(1/2+{\rm i}\tau)|^{2k}]}$. The distribution
of $\log|\zeta(1/2+{\rm i}\tau)|$ has been recently proved to be Gaussian
under $\widetilde{\mathbf{P}}$ as in Selberg’s theorem, and with the same
variance [faz21]. The constant $C_{k}$ appears naturally from the asymptotics
(1) of the moments.

###  1.2 Results

The main objective of this paper is to provide more evidence that the moment
correction $C_{k}$, predicted by random matrix theory, should be present in
(6). On the theoretical side, it was proved by Féray, Méliot, and Nikeghbali
that a similar correction naturally appears in the large deviations of the
Keating-Snaith central limit theorem for the characteristic polynomial of the
circular unitary ensemble (CUE):

######  Theorem 1.1 (Theorem 7.5.1 [FerMelNik2016]).

Let $P_{N}(\theta)=\det(I-e^{{\rm i}\theta}U)$ be the characteristic
polynomial of an $N\times N$ random matrix $U$ sampled under the Haar measure
$\mathbb{P}_{\mathcal{U}(N)}$ on the unitary group $\mathcal{U}(N)$. Write
$Q_{2}(N)$ for the second cumulant of $\log|P_{N}(\theta)|$. Then we have for
$V\sim k\sqrt{2\log N}$, $k\geq 0$, and any $\theta\in[0,2\pi)$

$\mathbb{P}_{\mathcal{U}(N)}\Big{(}\log|P_{N}(\theta)|>\sqrt{Q_{2}(N)}\cdot
V\Big{)}\sim
f_{k}\int_{V}^{\infty}\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}{\rm
d}x,\quad\text{ where }f_{k}=\dfrac{\mathcal{G}^{2}(1+k)}{\mathcal{G}(1+2k)}.$

The statement of Theorem 1.1 is precisely the random matrix analogue of (6).
Using the usual dictionary (see, e.g., [keasna00a, katsar99b, cfkrs05]), one
compares the unitary characteristic polynomial $P_{N}(\theta)$ with
$\zeta(1/2+{\rm i}t)$, and by comparing densities of eigenvalues and zeros of
zeta, the matrix size $N$ corresponds to a height $\log(t/2\pi)$.

Large deviations in various ranges were also considered by Hughes et al.
[hugkeaoco01]. In that paper, they show that $\log|P_{N}(\theta)|/A(N)$
satisfies a large deviation principle, for various ranges of $A(N)$. The
appropriate range in the context of Theorem 1.1 is _moderate_ deviations:
$\sqrt{\log{N}}\ll A(N)\ll N$, and it is shown in [hugkeaoco01] that the rate
function is either quadratic or linear, depending on the precise growth of
$A(N)$. Similar results have also been proved in the general context of
$\beta$-ensembles and Wigner matrices in [DorEic2013, DorEic2013Cumu]. Theorem
1.1 differs in that it examines a very particular form of $A(N)$ and derives
the resulting precise constant multiple of the Gaussian. The proof of Theorem
1.1 was done in [FerMelNik2016] in the general context for mod-$\phi$
convergence. For completeness, we provide the detailed computation following
the result of Keating & Snaith in Section 2.

The statement of Theorem 1.1 equally holds for the imaginary part of $\log
P_{N}(\theta)$, with the appropriate change to the leading order
correction222In this case, one would instead find that
$f_{k}=\mathcal{G}(1-k)\mathcal{G}(1+k)$, the leading coefficient of the
$2k^{th}$ moment of $e^{{\rm i}{\rm Im}\log P_{N}(\theta)}$.. Additionally,
the statement of Theorem 1.1 can be generalized to other classical compact
groups, where the distribution should again be Gaussian but with a different
correction $g_{k}$ (corresponding to the relevant matrix group moment), see
for example [FerMelNik2016]. From such calculations one could deduce
conjectures akin to Conjecture 2 for symplectic and orthogonal families of
$L$-functions (cf. [cfkrs05]).

Beyond the theoretical evidence from Theorem 1.1, we also provide numerical
evidence of the presence of the correction $C_{k}$. We choose to investigate
its effect on the maximum of the real part of the logarithm of the zeta
function in short intervals, instead of directly testing (6). The reason for
this is that the presence of the correction leads to a very precise refinement
of the Fyodorov-Hiary-Keating conjecture for the maximum of
$\log|\zeta(1/2+{\rm i}\tau+{\rm i}h)|$ for $h$ in a short interval,
[fyohiakea12, fyokea14]. The conjecture was originally stated for mesoscopic
intervals, that is, intervals of size $2\pi(\log T)^{\theta}$ for $\theta\leq
0$. The statement was adapted in [argouirad19] to macroscopic intervals of
size $2\pi(\log T)^{\theta}$, for $\theta>0$, but fell short of capturing the
order one terms. It turns out that the correction $C_{k}$ has a measurable
effect on the recentering of the maximum. More precisely, we propose the
following refinement:

######  Conjecture 3.

Consider a fixed $\theta>0$. If $\tau$ is uniformly distributed on $[T,2T]$,
then we have

$\max_{|h|\leq\pi(\log T)^{\theta}}\log|\zeta(1/2+{\rm
i}(\tau+h))|=\sqrt{1+\theta}\log\log T-\frac{1}{4\sqrt{1+\theta}}\log\log\log
T+\mathcal{G}_{\theta,T},$ (8)

where $(\mathcal{G}_{\theta,T},T\geq 1)$ is a sequence of random variables
converging in distribution to a Gumbel random variable $\mathcal{G}_{\theta}$
with $\mathbf{P}(\mathcal{G}_{\theta}\leq x)=\exp(-e^{-\frac{1}{\beta}(x-m)})$
and parameters

$\displaystyle\beta$ $\displaystyle=\beta(\theta)=\frac{1}{2\sqrt{1+\theta}}$
(9) $\displaystyle m$ $\displaystyle=m(\theta)=(0.06537\dots)+\beta^{2}\log
C_{\sqrt{1+\theta}}-\frac{\beta^{2}}{2}\Big{(}\log(1+\theta)-\log(4\pi)\Big{)}.$

As can be seen from the leading order of (8), the relevant regime of large
deviation at a given $\theta$ is $\sqrt{1+\theta}\log\log T$. Together with
Conjecture 2, this leads naturally to the choice $k=\sqrt{1+\theta}$ for
$C_{k}$ in (9). The precise numerical constant appearing in the definition of
$m$ in (9) is the Meissel-Mertens constant divided by $4$, see Equation (33).

The upshot of Conjecture 3 is a very precise prediction to order one for the
maximum of $\log|\zeta(1/2+{\rm i}(\tau+h))|$, including very good control of
the finite-size effects, that can be compared to the numerical data. The high
precision of the conjecture to order one is the saving grace here, as the
factors $\log\log T$, $\log\log\log T$, and $\mathcal{G}_{\theta,T}$ in (8)
remain essentially of the same order for all testable $T$’s (around
$T=10^{23}$ seems to be the current computational limit). In particular, this
spares us some of the difficulty of testing the moment conjecture, see
[hiaodl12].

We chose to test Equations (8) and (9) at $T=10^{7},10^{8}$ and $10^{9}$,
where it is not costly to amass a good sample size for many $\theta$’s for
$0\leq\theta\leq 3$. A snapshot of the results for the empirical mean of the
maximum are given in Figure LABEL:fig:_table and Figures LABEL:fig:_mean_7,
LABEL:fig:_mean_8 and LABEL:fig:_mean_9. The main conclusion there is that the
correction $C_{k}$ is necessary to fit the data. Details on the numerical
experiments are given in Section LABEL:sect:_num.

The case $\theta=0$ is special as Conjecture 3 is not expected to hold. It was
proposed in [fyohiakea12, fyokea14] that the subleading order should instead
be $-\frac{3}{4}\log\log T$. One then would expect the empirical mean to lie
_lower_ than the prediction (8). In addition, the fluctuations should not be
exactly Gumbel but a randomly shifted Gumbel. The effect of the random shift
is such that the right tail of the distribution of the recentered maximum is
not exponential, as for a pure Gumbel, but should be heavier: of the form
$ye^{-\beta y}$. This would in effect _increase_ the contribution of the
fluctuations to the mean. The two above corrections seem hard to observe
numerically. One problem is that they are competing effects, which may
mutually cancel. Secondly, there is the systematic problem that the standard
deviation of the maximum is fairly large at $\theta=0$, as can be seen for
example in Figure LABEL:fig:_mean_9. Theoretical progress to settle the
Fyodorov-Hiary-Keating conjecture has recently been made in [naj18, abbrs19,
Har19, argbourad20]. A continuous smoothing of the subleading order between
$-\frac{1}{4\sqrt{1+\theta}}\log\log T$ to $-\frac{3}{4}\log\log T$ as
$\theta\downarrow 0$ has been proposed in [argdubhar21] by taking
$\theta\sim(\log\log T)^{-\alpha}$, $0<\alpha<1$. This gives a subleading
order of $-\frac{(1+2\alpha)}{4}\log\log T$. Again, this interpolation seems
hard to capture numerically as the standard deviation of the maximum is large
for small $\theta$.

The paper is structured as follows. We provide a proof of Theorem 1.1 in
Section 2, following the work of Keating and Snaith [keasna00a]. The details
on how the theoretical prediction based on Conjecture 3 is generated are given
in Section 3. The conjecture is derived in Section 3.1 using basic extremal
value theory, assuming Conjecture 2 and reasonable properties of $\zeta$. We
comment on the control of the finite-size effects and on the numerical
computations of the $C_{k}$’s for all $0\leq k\leq 2$ in Section 3.2.
Numerical experiments are discussed in Section LABEL:sect:_num.

Acknowledgements. We thank M. Radziwiłł for valuable comments on the first
version of the paper. We are also grateful to P. Bourgade and A. Nikeghbali
for pointing out to us that Theorem 1.1 first appeared in [FerMelNik2016].
L.-P. A. gratefully acknowledges the support from the grant NSF CAREER
DMS-1653602. K. H. and R. R. were also financially supported in part by this
grant. E. B. thanks the Heilbronn Institute for Mathematical Research for
support.

##  2 Proof of Theorem 1.1

For completeness, we provide the proof of Theorem 1.1 following Keating &
Snaith [keasna00a]. Recall that we write $P_{N}(\theta)=\det(I-e^{{\rm
i}\theta}U)$ for the characteristic polynomial of a random $N\times N$ unitary
matrix. Let $V\sim k\sqrt{2\log N}$. We will show that

$\mathbb{P}_{\mathcal{U}(N)}\left(\frac{\log|P_{N}(\theta)|}{\sqrt{Q_{2}(N)}}>V\right)\sim
f_{k}\int_{V}^{\infty}\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}{\rm d}x,$ (10)

where

$f_{k}=\frac{\mathcal{G}^{2}(1+k)}{\mathcal{G}(1+2k)},$ (11)

and $Q_{2}(N)$ is the second cumulant of $\log|P_{N}(\theta)|$. Keating and
Snaith calculated the asymptotic form of the cumulants $Q_{j}(N)$,

$\displaystyle Q_{1}(N)$ $\displaystyle=0$ (12) $\displaystyle Q_{2}(N)$
$\displaystyle=\frac{1}{2}\log
N+\frac{1}{2}(\gamma+1)+\frac{1}{24N^{2}}-\frac{1}{80N^{4}}+\mathcal{O}\left(\frac{1}{N^{6}}\right)$
(13) $\displaystyle Q_{m}(N)$
$\displaystyle=(-1)^{m}\frac{2^{m-1}-1}{2^{m-1}}\left(\Gamma(m)\zeta(m-1)-\frac{(m-3)!}{N^{m-2}}\right)+\mathcal{O}\left(\frac{1}{N^{m-1}}\right),$
(14)

for $m\geq 3$.

Let the probability density function of $\log|P_{N}(\theta)|$ be

$\rho_{N}(x)=\mathbb{E}_{\mathcal{U}(N)}[\mathbbm{1}\\{\log|P_{N}(\theta)|=x\\}],$
(15)

where the average is taken over the unitary group with respect to Haar
measure. Additionally define the moment generating function for
$\log|P_{N}(\theta)|$ to be

$M_{N}(s)=\sum_{j=0}^{\infty}\frac{\mathbb{E}_{\mathcal{U}(N)}\left[(\log|P_{N}(\theta)|)^{j}\right]}{j!}s^{j}.$
(16)

Using the Selberg integral, Keating and Snaith [keasna00a] determined a finite
$N$ formula for $M_{N}(s)$, valid for all real $\theta$ and ${\rm Re}(s)>-1$

$M_{N}(s)=\mathbb{E}_{\mathcal{U}(N)}[|P_{N}(\theta)|^{s}]=\prod_{j=1}^{N}\frac{\Gamma(j)\Gamma(j+s)}{\Gamma^{2}(j+s/2)}.$
(17)

Due to the rotational invariance of the Haar measure on $\mathcal{U}(N)$, the
right hand side of (17) is independent of $\theta$. From (17), one deduces
that, as $N\rightarrow\infty$, $M_{N}(s)\sim f_{k}N^{k^{2}}$, where $f_{k}$ is
given by (11).

Define

$\tilde{\rho}_{N}(x)=\sqrt{Q_{2}(N)}\rho_{N}(\sqrt{Q_{2}(N)}x),$ (18)

the probability density function of $\log|P_{N}(\theta)|$ rescaled. Keating
and Snaith determined (cf. [keasna00a], Equation (53)) that

$\tilde{\rho}_{N}(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}\left(1+\sum_{m=3}^{\infty}\frac{A_{m}(N)}{Q_{2}(N)^{\frac{m}{2}}}\sum_{p=0}^{m}\binom{m}{p}x^{p}\mathcal{E}(m,p)\right),$
(19)

where

$\mathcal{E}(m,p)=\begin{cases}{\rm i}^{m-p}(m-p-1)!!,&m-p\text{ even},\\\
0,&m-p\text{ odd}\end{cases},$ (20)

and where the terms $A_{m}(N)$ are determined combinatorially from the
cumulants and Equation (18). For example,

$\displaystyle A_{3}(N)$ $\displaystyle=\frac{Q_{3}(N)}{3!}$ $\displaystyle
A_{4}(N)$ $\displaystyle=\frac{Q_{4}(N)}{4!}$ $\displaystyle A_{5}(N)$
$\displaystyle=\frac{Q_{5}(N)}{5!}$ $\displaystyle A_{6}(N)$
$\displaystyle=\frac{Q_{6}(N)}{6!}+\frac{1}{2!}\frac{Q_{3}(N)^{2}}{(3!)^{2}}.$

As $N\rightarrow\infty$, $A_{m}(N)$ approaches a constant, see [keasna00a].
Evaluating Equation (19) at $x=\tfrac{k\log N}{\sqrt{Q_{2}(N)}}$ then gives

$\tilde{\rho}_{N}\left(\frac{k\log
N}{\sqrt{Q_{2}(N)}}\right)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}\frac{k^{2}\log^{2}N}{Q_{2}(N)}}\left(1+\sum_{m=3}^{\infty}\frac{A_{m}(N)}{Q_{2}(N)^{\frac{m}{2}}}\sum_{p=0}^{m}\binom{m}{p}\left(\frac{k\log
N}{\sqrt{Q_{2}(N)}}\right)^{p}\mathcal{E}(m,p)\right).$ (21)

Note that $x\sim k\sqrt{2\log N}$ as $N\rightarrow\infty$. Also, for large
$N$, the sum over $p$ is dominated at $p=m$, hence

$\displaystyle\tilde{\rho}_{N}\left(\frac{k\log N}{\sqrt{Q_{2}(N)}}\right)$
$\displaystyle\sim\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}\frac{k^{2}\log^{2}N}{Q_{2}(N)}}\left(1+\sum_{m=3}^{\infty}A_{m}(N)\left(\frac{k\log
N}{Q_{2}(N)}\right)^{m}\right)$ (22)
$\displaystyle\sim\frac{1}{\sqrt{2\pi}}e^{-k^{2}\log
N}e^{k^{2}(\gamma+1)}\left(1+\sum_{m=3}^{\infty}A_{m}(N)(2k)^{m}\right),$ (23)

using Equation (13).

Finally, recall from the comment following Equation (17) that

$f_{k}=\lim_{N\rightarrow\infty}\frac{M_{N}(2k)}{N^{k^{2}}}.$ (24)

Then, since the cumulants $Q_{m}(N)$ are related to the Taylor coefficients of
$\log M_{N}(s)$ (cf. Equation (16)),

$M_{N}(s)=\exp\left(\sum_{m=1}^{\infty}\frac{Q_{m}(N)}{m!}s^{m}\right),$ (25)

we have

$\displaystyle\frac{M_{N}(2k)}{N^{k^{2}}}$
$\displaystyle=\exp\left(2k^{2}Q_{2}(N)-k^{2}\log
N+\sum_{m=3}^{\infty}\frac{Q_{m}(N)}{m!}(2k)^{m}\right)$ (26)
$\displaystyle\sim\exp\left(k^{2}(\gamma+1)+\sum_{m=3}^{\infty}\frac{Q_{m}(N)}{m!}(2k)^{m}\right)$
(27)
$\displaystyle=\exp\left(k^{2}(\gamma+1)\right)\left(1+\sum_{m=3}^{\infty}A_{m}(N)(2k)^{m}\right).$
(28)

The final line follows by expanding the the exponential with the infinite sum
in the exponent in the Taylor series, and then grouping terms according to
power of $2k$. The fact that this results in the weights $A_{m}(N)$ follows
immediately from the combinatorial definition, see the discussion following
(20) and [keasna00a]. Hence $f_{k}$ is given by Equation (28), and the result
follows by substituting the expression into (23).

##  3 Derivation of Prediction

###  3.1 Derivation of Conjecture 3

In this section, we derive Conjecture 3 based on the following assumptions:

######  Assumption.

For $\tau$ a uniform random variable on $[T,2T]$, the stochastic process

$(|\zeta(1/2+{\rm i}(\tau+h))|,|h|\leq\pi(\log T)^{\theta})$

satisfies the following:

  1. 1.

Discretization: the maximum over the interval $[-\pi(\log T)^{\theta},\pi(\log
T)^{\theta}]$ can be reduced to the maximum over a discrete set
$\mathcal{H}_{\theta,T}$ corresponding to the midpoints between the zeros of
$|\zeta(1/2+{\rm i}(\tau+h))|$ on the interval.

  2. 2.

Independence: The variables $|\zeta(1/2+{\rm i}(\tau+h))|$,
$h\in\mathcal{H}_{\theta,T}$, are independent.

  3. 3.

Large deviations of Selberg’s central limit theorem: Equation (6) holds with
$k=\sqrt{1+\theta}$.

Assumption 1 is reasonable as the maximum should be achieved between two
zeros. This can be rigorously established, see [argouirad19]. Assumption 2
cannot be exact, but it is likely a very good approximation. Indeed, the
correlation between the values at $h$ and $h^{\prime}$ decays very fast with
the distance: as $|h-h^{\prime}|^{-1}$, [argouirad19]. Assumption 3 is the one
to be tested. The choice of $k$ comes from the expected leading order of the
maximum being $\sqrt{1+\theta}\log\log T$.

From these assumptions, the derivation of the distribution of the maximum is a
standard computation in extreme value theory. However, we shall need very good
control of the finite-size effects to compare with numerics, so we include the
details. The finite-size effects are discussed in the next section.

The number of zeros $\mathcal{N}(t)$ on $[0,t]$ is known to a very good level
of precision thanks to the Riemann-von Mangoldt formula, see for example
[tsang84, ivic85],

$\mathcal{N}(t)=\frac{t}{2\pi}\log\frac{t}{2\pi e}+\frac{1}{\pi}{\rm
Im}\log\zeta(1/2+{\rm i}t)+\mathcal{O}(1).$ (29)

This implies that the number of zeros in the interval $[\tau-\pi(\log
T)^{\theta},\tau+\pi(\log T)^{\theta}]$ is

$\displaystyle N_{\theta,T}$ $\displaystyle=\mathcal{N}(\tau+\pi(\log
T)^{\theta})-\mathcal{N}(\tau-\pi(\log T)^{\theta})$ (30) $\displaystyle=(\log
T)^{\theta}\log\frac{\tau}{2\pi e}+\mathcal{O}((\log T)^{\theta})$
$\displaystyle=(\log T)^{1+\theta}+\mathcal{O}((\log T)^{\theta}).$

This will be the approximation for the cardinality of the discrete set
$\mathcal{H}_{\theta,T}$. The above implies

$\log N_{\theta,T}=(1+\theta)\log\log T+\mathcal{O}((\log T)^{-1}).$ (31)

Focussing on Assumption 3, the right tail of the distribution of
$\log|\zeta(1/2+{\rm i}\tau)|$ is expected to be Gaussian with multiplicative
correction $C_{k}$ and variance

$\sigma^{2}_{T}=\frac{1}{2}\sum_{p\leq T}\frac{1}{p}.$ (32)

In [rad11], Conjecture 2 is stated for $\sigma_{T}^{2}=\frac{1}{2}\log\log T$.
Whereas it is true that $\frac{1}{2}\sum_{p\leq T}\frac{1}{p}=(1+{\rm
o}(1))\log\log T$, the ${\rm o}(1)$-term is needed for precise numerics.
Mertens’ second theorem asserts that, see for example [Van17],

$\sum_{p\leq T}\frac{1}{p}=\log\log
T+B+\mathcal{O}\left(\frac{e^{-\sqrt{0.175\log T}}}{(\log T)^{3/4}}\right),$
(33)

where $B=0.26149\dots$ is the Meissel-Mertens contant. Hence, from (32), the
standard deviation is asymptotically

$\sigma_{T}=\sqrt{\tfrac{1}{2}\log\log T}+\frac{B}{2\sqrt{2\log\log
T}}+\mathcal{O}(e^{-\sqrt{0.1\log T}}).$ (34)

We are now ready to derive Conjecture 3. We use the shorthand notation
$N=N_{\theta,T}$, $\sigma=\sigma_{T}$ and $C=C_{\sqrt{1+\theta}}$ (the moment
coefficient). Under Assumptions 1, 2, and 3, we have for any $Y>0$ with
$\sigma Y\sim\sqrt{1+\theta}\log\log T$,

$\mathbf{P}\left(\max_{|h|\leq\pi(\log T)^{\theta}}\log|\zeta(1/2+{\rm
i}(\tau+h))|>\sigma\
Y\right)=\left(1-\frac{NC\int_{Y}^{\infty}\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}{\rm
d}x}{N}\right)^{N}.$ (35)

First, note that

$\left(1-\frac{a}{n}\right)^{n}=e^{-a}\cdot\left(1-\mathcal{O}\left(\frac{a^{2}}{n}\right)\right),$
(36)

so that

$\mathbf{P}\left(\max_{|h|\leq\pi(\log T)^{\theta}}\log|\zeta(1/2+{\rm
i}(\tau+h))|>\sigma\
Y\right)\sim\exp\left(-NC\int_{Y}^{\infty}\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}{\rm
d}x\right).$ (37)

The correct level of the maximum is obtained by choosing $Y$ for which the
numerator in Equation (35) is of order one. With this mind, consider
$Y^{\star}$ the solution to the equation

$NC\int_{Y^{\star}}^{\infty}\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}{\rm
d}x=1.$ (38)

Write $\sigma Y=\sigma\ Y^{\star}+y$, $y\in\mathbb{R}$. With this notation,
Equation (37) becomes

$\mathbf{P}\left(\max_{|h|\leq\pi(\log T)^{\theta}}\log|\zeta(1/2+{\rm
i}(\tau+h))|>\sigma\ Y^{\star}+y\right)\sim\exp\left(-G(y)\right),$ (39)

where

$G(y)=\dfrac{\int_{Y^{\star}+y/\sigma}^{\infty}e^{-\frac{x^{2}}{2}}{\rm
d}x}{\int_{Y^{\star}}^{\infty}e^{-\frac{x^{2}}{2}}{\rm d}x}.$ (40)

The quantity $\sigma Y^{\star}$ is the deterministic shift in Equation (9). To
see this, recall that standard Gaussian estimates give the asymptotics

$\left(1-\frac{1}{Y^{2}}\right)\frac{1}{Y}\frac{e^{-\frac{Y^{2}}{2}}}{\sqrt{2\pi}}\leq\int_{Y}^{\infty}\frac{e^{-\frac{y^{2}}{2}}}{\sqrt{2\pi}}{\rm
d}y\leq\frac{1}{Y}\frac{e^{-\frac{Y^{2}}{2}}}{\sqrt{2\pi}},\quad Y\geq 1.$
(41)

In other words, it is possible to write

$\int_{Y}^{\infty}\frac{e^{-\frac{y^{2}}{2}}}{\sqrt{2\pi}}{\rm
d}y=\left(1-\mathcal{O}\left(\frac{1}{Y^{2}}\right)\right)\frac{1}{Y}\frac{e^{-\frac{Y^{2}}{2}}}{\sqrt{2\pi}}.$
(42)

Combining (38) and (42) gives the following equation for $Y^{\star}$:

$\frac{NC}{\sqrt{2\pi}}\frac{e^{-\frac{(Y^{\star})^{2}}{2}}}{Y^{\star}}=1+\mathcal{O}\left(\frac{1}{(Y^{\star})^{2}}\right).$
(43)

The solution can be approximated recursively. A first approximation omitting
$\frac{1}{Y^{\star}}=e^{-\log Y^{\star}}$ yields $Y^{\star}\approx\sqrt{2\log
N}$. Writing $Y^{\star}=\sqrt{2\log N}+\delta$ in (43) gives an equation for
$\delta$:

$\frac{\delta^{2}}{2}+\delta\sqrt{2\log N}+\frac{1}{2}\log\log
N+\frac{1}{2}\log 2+\log\left(1+\frac{\delta}{\sqrt{2\log N}}\right)=\log
C-\frac{1}{2}\log 2\pi.$ (44)

It is straightforward to solve this by expanding the logarithm to get

$\displaystyle Y^{\star}$ $\displaystyle=\sqrt{2\log N}+\delta$ (45)
$\displaystyle=\sqrt{2\log N}-\frac{1}{2\sqrt{2\log N}}\Big{(}\log\log
N+\log{4\pi}-2\log C\Big{)}+\mathcal{O}\left(\frac{\log\log N}{\log
N}\right).$ (46)

It remains to note that Equation (30) and Equation (34) imply

$\displaystyle\sigma\sqrt{2\log N}$ $\displaystyle=\sqrt{1+\theta}\ \log\log
T+\sqrt{1+\theta}\ \frac{B}{2}+{\rm o}(1),$ (47)
$\displaystyle\frac{\sigma}{2\sqrt{2\log N}}$
$\displaystyle=\frac{1}{4\sqrt{1+\theta}}\left(1+\mathcal{O}((\log\log
T)^{-1})\right).$ (48)

After multiplication by $\sigma$, the first two terms in Equation (46) give
the leading correction for the maximum of $\log|\zeta|$:

$\sqrt{1+\theta}\log\log T-\frac{1}{4\sqrt{1+\theta}}\log\log\log T.$

The remaining terms with $\log C$, $B$, $\log(1+\theta)$ and $\log 4\pi$
amount to the deterministic shift $m$ in Equation (9).

It remains to study the fluctuations around $Y^{\star}$. This is done by
computing the asymptotics of the function $G$, defined by (40). Using the
Gaussian estimate (41) again yields

$G(y)\sim
e^{-\frac{Y^{\star}}{\sigma}y}\cdot\frac{e^{-\frac{y^{2}}{(2\sigma^{2})}}}{1+\frac{y}{\sigma
Y^{\star}}}.$ (49)

Equation (9) follows by noting that $e^{-y^{2}/(2\sigma^{2})}\sim 1$ by (34),
and $1+\frac{y}{\sigma Y^{\star}}\sim 1$ for any fixed $y$ by (46). Moreover,
Equations (30), (34) and (46) imply

$\frac{Y^{\star}}{\sigma}\sim 2\sqrt{1+\theta}.$ (50)

Putting this back in (49) gives the parameter $\beta$ of the Gumbel
distribution in Equation (9).

###  3.2 Discussion of the Finite-Size Effects

In this section, we explain how the numerical predictions of Conjecture 3 are
obtained, including the computations of the coefficients $C_{k}$.

According to Conjecture 3, the mean of $\max_{|h|\leq\pi(\log
T)^{\theta}}\log|\zeta(1/2+{\rm i}(\tau+h))|$ consists of two terms: the
deterministic recentering

$\sqrt{1+\theta}\log\log T-\frac{1}{4\sqrt{1+\theta}}\log\log\log T,$ (51)

and the expectation of the Gumbel random variable $\mathcal{G}_{\theta,T}$,
which in the limit is

$\mathbf{E}[\mathcal{G}_{\theta}]=m+\beta\gamma,$ (52)

where $\gamma$ is the Euler constant $\gamma=0.577\dots$, and with the
parameters $m$ and $\beta$ given in Equation (9).

To give an idea of the orders of magnitude in the problem considered, note
that for $T=10^{8}$, we have

$\log\log 10^{8}=2.78\dots,\quad\log\log\log 10^{8}=1.07\dots\ .$ (53)

Scaling the simulations up to to $T=10^{23}$ wouldn’t result in much
additional precision, since there

$\log\log 10^{23}=3.97\dots,\quad\log\log\log 10^{23}=1.39\dots\ .$ (54)

This modest gain would come at a substantial computational time cost. Despite
the curse of iterated logarithms, it is possible to derive accurate numerical
predictions thanks to a precise control of the lower order terms and their
finite-size corrections.

The computation of the deterministic shift $\sigma Y^{\star}$ follows the
treatment in the last section. For the number of zeros $N_{\theta,T}$, we rely
on Equation (30) and take

$N_{\theta,T}\approx(\log T)^{\theta}\log\frac{T}{2\pi e}.$ (55)

The error term in (30) is of order $(\log T)^{-1}$, which is comparatively
small. For the variance $\sigma_{T}^{2}$, note that the constant
$B=0.26149\dots$ is fairly close to the value of $\log\log T$ in view of (53).
For this reason, we use Equation 34 in the computation. One might expect a
quadratic correction to $\sum_{p\leq T}\frac{1}{p}$ of the form $\sum_{p\leq
T}\frac{1}{8p^{2}}\approx\frac{1}{32}$, due to the expansion of the Euler
product. However, this is negligible for our purpose. The approximation of the
product by the exponential in Equation 36 also comes at a low cost, since the
multiplicative error $(1-\mathcal{O}(a^{2}/n))$ is evaluated at $a=1$, by
design in (38), and at $n=N_{\theta,T}\approx(\log T)^{\theta}\geq 18.4$. This
multiplicative error becomes of small shift in the exponential, allowing us to
discard it. The Gaussian estimate (43) is not quite precise enough for these
fine numerics. Indeed, this leads to Equation (46), with an error
$\mathcal{O}\left(\frac{\log\log N_{\theta,T}}{\log N_{\theta,T}}\right)$.
This error remains substantial at any $T$’s that are computationally
reachable. For this reason, we approximate $Y^{\star}$ directly by numerically
solving Equation (38). This takes care of all finite-size corrections to
$\sigma Y^{\star}$.

It remains to evaluate the mean of the Gumbel random variable
$\mathcal{G}_{\theta,T}$. As can be seen from Equation (49), the function $G$
converges to $e^{-2\sqrt{1+\theta}y}$, albeit very slowly. The term
$e^{-y^{2}/2\sigma^{2}}$ is particularly problematic since $\sigma$ is of the
order $\sqrt{\log\log T}$. To take care of this, instead of using
$\beta\gamma$ in the mean of a Gumbel (cf. Equation (52)), we simply evaluate
the mean by working directly with the function $G(y)$ in Equation (40). The
mean of the recentered random variable $\max_{|h|\leq\pi(\log
T)^{\theta}}\log|\zeta(1/2+{\rm i}(\tau+h))|-\sigma Y^{\star}$ can then be
evaluated using the cumulative distribution function $1-\exp(-G(y))$. In
Equation (50), we also use $\frac{\sigma^{2}}{\sigma Y^{\star}}$ for $\beta$
instead of the limiting value $(2\sqrt{1+\theta})^{-1}$. For $\theta=3$, these
considerations yield a mean of $0.17\dots$ compared to $0.14\dots$ for the
limiting function $e^{-2\sqrt{1+\theta}y}$. In the same way, the variance of
$\max_{|h|\leq\pi(\log T)^{\theta}}\log|\zeta(1/2+{\rm i}(\tau+h))|$ in the
limit should be the one of a Gumbel with parameter $\beta$, i.e.,

$\text{Var}(\mathcal{G}_{\theta})=\frac{\beta^{2}\pi^{2}}{6}.$ (56)

The finite-size effects of computing the variance with the CDF $1-\exp(-G(y))$
are tiny compared to using (56), so we use Equation (56) for simplicity.

Finally, we turn to computing the moment coefficient $C_{k}$ (which recall
appears in the definition of $m$, see (9)). For our purpose, we are interested
in the value of $C_{k}=a_{k}f_{k}$ given by Equation (2) for $k\in(1,2)$,
since $k=\sqrt{1+\theta}$ and for our numerics we take $\theta\in(0,3)$.
Certain evaluations of $a_{k}$ and $f_{k}$ appear in the literature: see for
example [hiaodl12], where they compute $C_{k}$ for the first few integers $k$:

$k$ | $C_{k}=a_{k}f_{k}$
---|---
$1$ | $1$
$2$ | $\frac{1}{2\pi^{2}}\approx$5.066\text{\times}{10}^{-2}$$
$3$ | $5.708\text{\times}{10}^{-6}$
$4$ | $2.465\text{\times}{10}^{-13}$
Table 1: Values of the leading order coefficient in the moment conjecture (1).
The exact values for $k=1,2$ are due to Hardy and Littlewood, and Ingham
respectively [harlit18, ing26]. The (truncated) numerical values for higher
$k$ can be found in [hiaodl12].

The numerical values of $C_{k}=a_{k}f_{k}$ presented in Section
LABEL:sec:numerics were computed333Computations were completed in SageMath
[sagemath], version 9.1, using Python 3.7. by evaluating (3) and (4). Figure
LABEL:fig:coeff_plot plots $C_{k}$ for $\theta\in(0,3)$ (i.e. $k\in(1,2)$).


# AILTTS: Adversarial Learning of Intermediate Acoustic Feature
for End-to-End Lightweight Text-to-Speech

###### Abstract

The quality of end-to-end neural text-to-speech (TTS) systems highly depends
on the reliable estimation of intermediate acoustic features from text inputs.
To reduce the complexity of the speech generation process, several non-
autoregressive TTS systems directly find a mapping relationship between text
and waveforms. However, the generation quality of these system is
unsatisfactory due to the difficulty in modeling the dynamic nature of
prosodic information. In this paper, we propose an effective prosody predictor
that successfully replicates the characteristics of prosodic features
extracted from mel-spectrograms. Specifically, we introduce a generative
model-based conditional discriminator to enable the estimated embeddings have
highly informative prosodic features, which significantly enhances the
expressiveness of generated speech. Since the estimated embeddings obtained by
the proposed method are highly correlated with acoustic features, the time-
alignment of input texts and intermediate features is greatly simplified,
which results in faster convergence. Our proposed model outperforms several
publicly available models based on various objective and subjective evaluation
metrics, even using a relatively small number of parameters.

Index Terms: Text-to-speech, single-stage, generative adversarial networks,
feature estimation, alignment

##  1 Introduction

The advancement of neural networks has led to significant quality improvement
in text-to-speech (TTS) systems. Most neural TTS models consist of a two-stage
pipeline architecture: an acoustic model that converts text or phonemes to
intermediate features such as mel-spectrograms, and a neural vocoder that
synthesizes waveforms by utilizing the intermediate features as condition.
Depending on the structural differences that occur while converting text into
intermediate features, we can classify neural TTS models as either an
autoregressive (AR) or a non-autoregressive (non-AR). AR-based methods [1, 2,
3] successfully generate high quality speech, but their inference speed is low
because they cannot be implemented in parallel. To solve the low inference
speed problem of AR-based methods, various types of non-AR TTS models have
been proposed [4, 5, 6, 7].

Recently, non-AR TTS models have been extended to contain a single-stage end-
to-end structure that incorporates the generative model-based neural vocoder
into the training process [5, 8, 9, 10]. This structure trains entire text-to-
speech conversion chains under a unified framework. Thus, it enables the
flexible usage of intermediate latent domain feature characteristics instead
of constraining them to track the characteristics of signal processing-based
features such as the mel-spectrogram. In this unified framework, it is more
important to reliably estimate trainable intermediate acoustic features from
text inputs because the features must have timing information corresponding to
the duration of each text. In addition, they are required for conditioning the
generative model-based waveform synthesis module. Utilizing the trainable
characteristic of intermediate features, several unified TTS models that
exclude acoustic feature decoding have been proposed [5, 8, 10]. However, the
synthesized speech quality of such models is unsatisfactory (e.g., monotonous
and unnatural) because: 1) it is difficult to train intermediate features to
sufficiently represent prosodic variations due to their structural limitations
while guiding acoustic features, and 2) the following neural vocoder can not
be trained to fully utilize the intermediate features from the entire
utterance due to a GPU memory constraint; thus, its receptive field size
should be very limited.

In this paper, we propose a method for reliably estimating informative
intermediate acoustic features from text inputs. We solve the problems of the
insufficient prosodic variation and limited receptive field of the neural
vocoder by strengthening the role of the trainable intermediate features. We
adopt an effective prosody predictor that powerfully models the complicated
distribution of prosody-related embeddings in the reference mel-spectrogram
using an adversarial training process. We also show that the obtained prosody
embeddings are very effective to perform the likelihood-based internal time
alignment between text and intermediate acoustic features. Our alignment
method converges quickly during training and always shows robust alignment
performance. Our experiments confirm that our method significantly enhances
the expressiveness and quality of generated speech. The generated audio
samples for the experiments are also available at our demo
page111https://hcyspeech.github.io/AILTTS_demo/.

This paper is organized as follows. Section 2 analyzes the main concepts and
problems of current single-stage models. Section 3 proposes our approach to
achieve high-quality speech generation, Section 4 presents our experimental
results, and Section 5 concludes the paper.

Figure 1: Overall architecture of the training (left) and inference stages of
the proposed model (right). The gray box represents the Lite-FFT block, a
lightweight transformer-based encoder. The red arrow represents the training
target. The conditional discriminator for the prosody predictor is omitted in
the figure for brevity.

##  2 Background

Since the length of intermediate acoustic features differs over time according
to the phonetic variation of input text, we need to predict the duration of
each text as well as its acoustic characteristics. In common non-AR TTS
systems, text-level embeddings are expanded at each time duration to match
their time scale with target intermediate acoustic features. With this
objective in mind, [4] adopted an external signal processing-based aligner
[11] to obtain reference time duration, and [6, 12] proposed likelihood-based
aligners that internally estimate the duration by selecting the most probable
path from a learned alignment map.

Also, to efficiently synthesize waveforms with neural vocoders, we must
provide rich and dynamic prosody information to intermediate acoustic
features. From this perspective, several single-stage models that perform the
aforementioned tasks without explicitly decoding target mel-spectrograms have
been proposed. VITS [9] adopts variational inference with normalizing flows to
model the complicated probabilistic distribution of the latent domain
intermediate features. Other single-stage models [5, 8, 10] simplify the above
process by utilizing their own trainable intermediate latent features;
however, their generation quality is not on par with two-stage models [5, 8].

In this paper, we use LiteTTS [10] as a baseline because it has low complexity
and relatively high synthesis quality. LiteTTS adopts a domain transfer
technique that approximates text inputs to clone prosody-related information
extracted from the target mel-spectrogram during training. Text and prosody-
related embeddings are combined first, and then intermediate features are
derived by upsampling following reference durations given by the external
aligner [11]. However, since 1) they simply minimize the L1 distance between
the target and predicted prosody-related embeddings, and 2) the following
neural vocoder can not be trained receiving the entire intermediate features,
the output quality is somewhat over-smoothed and unsatisfactory. To overcome
these problems without increasing model parameters, we propose an adversarial
training method with a conditional discriminator to reinforce the estimation
of the complex distribution of prosody embedding included in the intermediate
feature. We also adopt a likelihood-based internal aligner conditioned on the
prosody embedding, which robustly and quickly performs alignment without using
an external aligner.

##  3 Method

Fig 1 presents a block diagram of the proposed model built from the LiteTTS
[10] baseline system. Our model consists of three main modules: an
intermediate prosody embedding ($h_{pr}$) estimation module that generates the
target of the prosody predictor, a prosody predictor that predicts $h_{pr}$
from text (phonetic) embeddings ($h_{ph}$), and an internal aligner including
a duration predictor whose target is a learned alignment between text and
acoustic features. To simplify the architecture with low complexity, we adopt
a lightweight transformer-based encoder [10] for the phoneme encoder, prosody
encoder, and prosody predictor.

###  3.1 Prosody predictor with conditional discriminator

We generate the intermediate acoustic feature $I$ by upsampling the combined
phoneme-scale embeddings $h_{ph}+h_{pr}$ following the duration given by the
internal aligner. Here, the prosody embedding $h_{pr}$ is calculated through
mapping the prosody encoder output $h^{\prime}_{pr}$ to phoneme-scale by using
attention with $h_{ph}$. The main purpose of the prosody predictor is to
generate an embedding that can be successfully replace the target prosody
embedding $h_{pr}$ from input $h_{ph}$.

Considering the dynamic nature of the prosody embedding, we leverage an
effective generative model architecture that includes various types of
discriminators. The proposed discriminator distinguishes the target prosody
embedding $h_{pr}$ from the predicted embedding $h_{pred}$ and utilizes
phonetic information as conditional data. We adopt a projection-based
conditional discriminator [13] by conditioning the phonetic embedding $h_{ph}$
as shown in Fig. 2. We apply feature matching loss [14] between the generated
and target feature maps to stabilize the training of generative adversarial
network (GAN). In Fig. 2, the feature maps are obtained as the output of all
the 1-D convolution layers prior to the $PostConv1D$ layer. Seven feature maps
are extracted in total, one from $PreConv1D$ layer and six from three residual
1-D convolutional blocks.

Since audio waveforms are generated by the vocoder using intermediate acoustic
features, we designed the conditional discriminator as follows: 1) we use
upsampled embeddings instead of original phoneme-scale embeddings as input to
the discriminator to match the time-scale with intermediate acoustic features,
and 2) since the intermediate features have short-term dependencies lower than
the receptive field of the vocoder, we also carefully design the conditional
discriminator to have the same receptive field size as the vocoder, which
enables the discriminator to efficiently capture the diverse patterns of the
input prosody embeddings. Based on the generative and discriminative loss of
least-squares GAN [15], we define the total prosody predictor loss by
considering both reconstruction loss $\mathcal{L}_{recon}$ and feature
matching loss $\mathcal{L}_{fm}$:

$\mathcal{L}_{G}=\mathbb{E}_{({H}_{pred},{H}_{ph})}[(D(H_{pred},H_{ph})-1)^{2}]+\mathcal{L}_{recon}+\mathcal{L}_{fm},$
(1)
$\displaystyle\mathcal{L}_{D}=\mathbb{E}_{({H}_{pr},{H}_{pred},{H}_{ph})}[(D({H}_{pr},{H}_{ph})-1)^{2}$
$\displaystyle+(D({H}_{pred},{H}_{ph}))^{2}],$ (2)
$\mathcal{L}_{recon}=\left\|{H}_{pred}-{H}_{pr}\right\|_{1},\,\mathcal{L}_{fm}=\sum_{i=1}^{7}\left\|F^{i}_{pred}-F^{i}_{pr}\right\|_{1},$
(3)

where $H_{(\cdot)}$ denotes the upsampled embeddings, and $F^{i}$ denotes the
$i$-th feature map of the discriminator.

###  3.2 Prosody conditioned internal aligner

To learn the time alignment between phonetic and intermediate acoustic
features without using an external aligner, we adopt a likelihood-based method
[6, 12]. That is, we maximize the log-likelihood of monotonic alignments from
a probability matrix calculated by the $L2$-distance between two encoded
features with different time scales. Then, phoneme durations (binary matrix)
are obtained by selecting the most probable path from the probability matrix.
The gap between the two matrices is reduced by minimizing their KL-divergence.

Our proposed aligner utilizes the phoneme-scale embeddings $h_{ph}$ \+
$h_{pr}$ for the phonetic features and the mel-spectrogram for the acoustic
features. Since $h_{pr}$ contains local acoustic information mapped into
phoneme-level by a formerly applied attention module, it becomes much easier
for the aligner to learn the time relationship between two features than when
using only $h_{ph}$. Following [4], we jointly train the duration predictor
which accepts the phonetic embedding $h_{ph}$ with a stop gradient. We show
that the accuracy of our text-to-acoustic feature alignment increases when we
use our proposed prosody embedding as a condition, which also improves the
accuracy of the duration predictor.

Figure 2: Architecture of the proposed conditional discriminator and the
auxiliary predictor*. Both the main and conditioning branches of the
conditional discriminator consist of 1-D convolutional layers designed to have
the same receptive field size.

###  3.3 Final training loss

The total loss for training our model is defined as follows:

$\mathcal{L}_{total}=\mathcal{L}_{var}+\mathcal{L}_{align}+\mathcal{L}_{pred}+\mathcal{L}_{voc}+\mathcal{L}_{aux},$
(4)

where $\mathcal{L}_{var}$ includes pitch and energy prediction losses applied
to ${h^{\prime}}_{pr}$ in [10], and $\mathcal{L}_{align}$ includes loss
functions for the internal aligner, including the duration predictor.
$\mathcal{L}_{pred}$ and $\mathcal{L}_{voc}$ represent total losses from the
prosody predictor and the vocoder, respectively, in accordance with their GAN
structure. $\mathcal{L}_{aux}$ is defined by the $L1$ loss between the target
mel-spectrogram and the predicted mel-spectrogram obtained by the output of
the auxiliary predictor.

####  3.3.1 Auxiliary predictor

The role of the auxiliary predictor is to further provide acoustic information
to the intermediate feature. Its architecture is almost identical to that
marked ‘*’ in Fig. 2, with the exception that the input is the intermediate
feature. The output channel of the $PostConv1D$ layer is set to the dimension
of the mel-spectrogram instead of 128, and layer normalization is applied to
the last stage of every residual 1-D convolutional block. In this way, we also
match the receptive field with the vocoder because its input is also the
intermediate feature, which enables to efficiently provide acoustic
information. Since the auxiliary predictor is only used for the training
stage, the computational complexity and memory usage are not increased.

##  4 Experiments

Table 1: The comparisons of MOS with 95% confidence intervals, number of parameters and inference speed. Model | MOS | #Params. | CPU inference (kHz) | Real-time
---|---|---|---|---
Ground Truth | $4.80\pm 0.17$ | - | - | -
AILTTS | $3.93\pm 0.23$ |  $13.4$M | $121.47$ | $\times 5.51$
LiteTTS w/ UnivNet-c16 | $3.71\pm 0.24$ |  $13.4$M | $121.26$ | $\times 5.50$
Tacotron 2 + UnivNet-c16 | $3.93\pm 0.24$ |  $29.4$M | $34.07$ | $\times 1.54$
GlowTTS + UnivNet-c16 | $3.57\pm 0.28$ |  $30.0$M | $196.74$ | $\times 8.92$
|  |  |  |

###  4.1 Experimental Setup

Dataset and Model Settings We evaluated our model on the LJSpeech dataset
[16]. We converted text sequences into phoneme sequences with an open-source
tool222https://github.com/Kyubyong/g2p. To calculate spectrograms, the FFT
size, window size, hop size were set to 1024, 1024, and 256, respectively,
regarding the sampling rate of 22050Hz. Mel-spectrograms were obtained by
applying an 80 band mel filter bank. We adopted Univnet-c16 [17] as a vocoder,
which has beneficial lightweight properties by using a location-variable
convolution (LVC) technique [18]. The dimension of all hidden embeddings was
set to 256, and the receptive field of the vocoder, auxiliary predictor, and
the conditional discriminator was set to 19.

Training Configuration and Inference We trained our model with the Adam [19]
optimizer using $\beta_{1}=0.8,\beta_{2}=0.99$ under the learning rate of
$2e-4$ and its decaying factor of $0.999$. The batch size was set to $24$. The
entire model was trained in a fully end-to-end manner, except for the prosody
and duration predictors. Since the training objectives of two modules
(${h}_{pr}$ and phoneme duration) do not converge during the early stage, they
are jointly trained with other modules after 300k steps. For the prosody
predictor loss, only reconstruction loss was applied for the first 50k steps
(until 350k steps) without applying feature matching loss and adversarial loss
to stabilize the training of the prosody predictor. During inference, the
prosody embedding and duration values predicted from the above two modules
were used to generate the intermediate acoustic feature.

Models for Comparison We compared our model with both single-stage and two-
stage models. We first experimented with LiteTTS by replacing the HiFi-GAN V3
[20] vocoder with the UnivNet-c16 vocoder for a fair comparison. For the two-
stage models, we adopted Tacotron 2 and Glow-TTS as baselines, two popular AR
and non-AR TTS models, respectively, combined with the UnivNet-c16 vocoder.
Here, we trained the vocoder using 80-dim mel-spectrogram as input, and
utilized pretrained weights for the acoustic models333UnivNet-c16:
https://github.com/mindslab-ai/univnet
Tacotron 2: https://github.com/NVIDIA/tacotron2
Glow-TTS: https://github.com/jaywalnut310/glow-tts.

###  4.2 Results

Audio Quality Table 1 summarizes the mean opinion scores (MOS) and number of
required parameters. We randomly selected 15 test audio samples to measure the
quality of synthesized speech. Our proposed AILTTS outperforms two non-AR
baselines, LiteTTS ($+0.22$ MOS) and Glow-TTS ($+0.36$ MOS), with the small
number of parameters ($13.4$M). Also, our model achieves high synthesis
quality comparable to the AR model, Tacotron 2. The results show that our
proposed adversarial learning of intermediate acoustic feature and the prosody
conditioned aligner effectively enhance the generation quality.

Inference Speed We evaluated inference speed using an Intel Core i5 Quad-Core
2.0 GHz CPU by averaging values from 80 test sentences. As shown in Table 1,
our proposed model has no degradation in terms of inference speed compared to
LiteTTS.

###  4.3 Analysis

Figure 3: Comparison of the convergence rate between a conventional method
(using ${h}_{ph}$ as input to the aligner) and a prosody conditioning method
(using ${h}_{ph}+{h}_{pr}$ as input to the aligner). For each iteration, the
MCD-DTW values from 10 test samples are averaged.

Ablation Study We conducted a CMOS (comparison MOS) evaluation on the test set
to determine whether the proposed two methods enhance the performance of the
model. We removed the conditional discriminator in the first experiment. For
the second experiment, we replaced the input of the internal aligner
${h}_{ph}+{h}_{pr}$ with the phonetic embedding ${h}_{ph}$. The CMOS results
for ablation studies are shown in Table 2. Based on the first ablation
experiment, adopting the generative model (GAN) to the prosody predictor
relieves the over-smoothing problem that occurs when using L1 loss only, and
enhances the expressiveness of generated speech. As shown in the second
ablation experiment, it is demonstrated that providing phoneme-level acoustic
information to the aligner helps the model to obtain robust alignment and
results in improvement of generation quality. As a result, we conclude that
both methods improve the model’s generation of various prosody characteristics
of speech, and increase the perceptual naturalness of the generated audio.

Alignment convergence We also compared the alignment convergence rate of the
two experiments from the ablation study of the prosody conditioned aligner. To
check this, we calculated the mean mel-cepstral distance (MCD) between
synthesized and ground-truth audio samples after synchronizing them in time
with dynamic time warping (DTW) [21]. As shown in Fig 3, the MCD-DTW values of
the proposed aligner rapidly decrease during the early stage of training
(prior to 4k iterations), and continuously show higher performance (lower MCD-
DTW values) than the baseline in every iteration. As we also demonstrated in
the second ablation study, we verify that our method has robust alignment
performance and rapidly converges in the learning process.

Table 2: The CMOS results of ablation studies. Model | CMOS
---|---
AILTTS | $0$
without Conditional Discriminator | $-0.214$
without Prosody conditioned Aligner | $-0.871$
|

##  5 Conclusion

In this paper, we propose an adversarial learning method to estimate
intermediate acoustic features, which plays a significant role in generating
natural and expressive speech in the single-stage lightweight TTS system.
Equipped with a prosody conditioned aligner, our proposed model outperforms
the Glow-TTS based model ($+0.36$ MOS) and LiteTTS ($+0.22$ MOS) while
preserving lightweight properties. As future work, we will design a generative
model-based stochastic prosody predictor to further extend prosody
controllability. In addition, we will extend our proposed aligner to have a
capability of generating multi-speaker’s speech signals by conditioning
speaker-dependent features.

##  6 Acknowledgments

This work is supported by Clova Voice, NAVER Corp., Seongnam, Korea.

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$\displaystyle\int_{p,q}\bigg{\\{}-K^{\prime}(p^{2})\bigg{\\}}\frac{3\lambda^{2}}{4!}\sum_{4~{}perm~{}(i,j,k)}\big{\\{}h(p_{i}+p_{j}+p_{k})h(p_{\alpha}+p+q)h(q)-h(p+q)h(q)\big{\\}}$
$\displaystyle\frac{1}{2}\frac{\lambda^{2}}{4!}\int_{p,q}K^{\prime}(p)h(p+q)h(q)$
$\displaystyle-\frac{1}{3}\frac{\lambda^{2}}{4!}\sum_{i=1}^{4}\big{\\{}K^{\prime}(p_{i}^{2})\big{\\}}\big{\\{}\int_{q,k}h(q)\left[h(p_{i}+q+k)h(k)-h(q+k)h(k)\right]\big{\\}}$
$\displaystyle+\frac{\lambda^{2}}{4!}\sum_{i=1}^{4}K^{\prime}(p_{i}^{2})\int_{p,q}\big{\\{}-K^{\prime}(q)\big{\\}}\big{\\{}h(q+k)h(k)-h(k)h(k)\big{\\}}$
$\displaystyle+\frac{3\lambda^{2}}{4!}\sum_{i=1}^{4}h(p_{i})\int_{p,q}\big{\\{}-K^{\prime}(q)\big{\\}}\big{\\{}h(q+k)h(k)-h(k)h(k)\big{\\}}$
$\displaystyle+\frac{2}{4!}\sum_{i=1}^{4}K^{\prime}(p_{i}^{2})A_{II}(p_{i})\lambda$
$\displaystyle+\big{\\{}-2(4-D)-\sum_{i=1}^{4}p_{i}.\frac{d}{dp_{i}}\big{\\}}\frac{1}{4!}\bigg{\\{}B_{VI}^{(2)}(p_{1},p_{2},p_{3},p_{4})|_{1}\bigg{\\}}+\big{\\{}-\sum_{i=1}^{4}p_{i}.\frac{d}{dp_{i}}\big{\\}}\frac{1}{4!}\bigg{\\{}B_{VI}^{(2)}(p_{1},p_{2},p_{3},p_{4})|_{2}\bigg{\\}}=0$
(D.124)

$A_{II}^{(1)}(p_{i})=-\frac{\lambda}{3}\int_{p,q}[h(p_{i}+p+q)h(p)h(q)-h(p+q)h(p)h(q)]-\lambda\int_{p}K^{\prime}(p)\\{h(p+q)h(q)-h(q)h(q)\\}$.

On LHS of (D.2) we are left with

$\displaystyle
3(4-D)\frac{1}{4!}B_{VI}^{(2)}(p_{1},p_{2},p_{3},p_{4})|_{1}+(4-D)\frac{1}{4!}B_{VI}^{(2)}|_{2}(p_{1},p_{2},p_{3},p_{4})|_{2}$

#### Equation for $B_{VII}^{(2)}(p_{1},p_{2},p_{3},p_{4})$

Considering (D.108c),

$\displaystyle\int_{p,q}\frac{6\lambda^{2}}{4!}\big{\\{}-K^{\prime}(p)\big{\\}}\sum_{3~{}perm~{}(i,j)}h(p_{i}+p_{j}+p)\big{\\{}h(p_{i}+p_{j}+q)h(q)-h(q)h(q)\big{\\}}$
$\displaystyle+\big{\\{}-2(4-D)-p_{i}.\frac{d}{dp_{i}}\big{\\}}\frac{1}{4!}B_{VII}^{(2)}(p_{1},p_{2},p_{3},p_{4})=0$
(D.126)

On LHS of (D.2) we are left with

$\displaystyle 3(4-D)\frac{1}{4!}B_{VII}^{(2)}(p_{1},p_{2},p_{3},p_{4})$
(D.127)

#### Equation for $B_{VIII}^{(2)}(p_{1},p_{2},p_{3},p_{4})$

Collecting (D.113e) and (D.115) we get (because of the expected structure of
$B_{VIII}^{(2)}$ as $\frac{\eta}{\epsilon}$ we consider the term
$\epsilon\lambda\frac{\partial B_{IX}}{\partial\lambda}$ from RHS of (D.2)),

$\displaystyle-\frac{2}{4!}\sum_{i=1}^{4}\big{\\{}K^{\prime}(p_{i}^{2})\big{\\}}\big{\\{}\frac{\eta
p_{i}^{2}}{2\epsilon}\big{\\}}+\frac{1}{4!}\eta\sum_{i=1}^{4}p_{i}^{2}h(p_{i})+\bigg{\\{}(4-D)-\sum_{i=1}^{4}p_{i}.\frac{\partial}{\partial
p_{i}}\bigg{\\}}\frac{1}{4!}B_{IX}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=\\{\epsilon-6F\lambda\\}\frac{1}{4!}B_{IX}(p_{1},p_{2},p_{3},p_{4})+\frac{1}{4!}\epsilon\lambda\frac{\partial
B_{IX}(p_{1},p_{2},p_{3},p_{4})}{\partial\lambda}$

We ignore $\lambda B_{IX}^{(2)}$ oe $\epsilon B_{IX}^{(2)}$ terms being higher
order and get

$\displaystyle-\frac{2}{4!}\sum_{i=1}^{4}\big{\\{}K^{\prime}(p_{i}^{2})\big{\\}}\big{\\{}\frac{\eta
p_{i}^{2}}{2\epsilon}\big{\\}}+\frac{1}{4!}\eta\sum_{i=1}^{4}p_{i}^{2}h(p_{i})+\bigg{\\{}-\sum_{i=1}^{4}p_{i}.\frac{\partial}{\partial
p_{i}}\bigg{\\}}\frac{1}{4!}B_{IX}=\frac{1}{4!}\epsilon\lambda\frac{\partial
B_{IX}}{\partial\lambda}$ (D.128)

And on LHS of (D.2) we are left with

$\displaystyle(\epsilon-6\lambda
F)\frac{1}{4!}B_{IX}(p_{1},p_{2},p_{3},p_{4})$ (D.129)

#### Equation for $B_{IX}^{(2)}(p_{1},p_{2},p_{3},p_{4})$

At last we collect the terms (D.116b), (D.117b), (D.117f) and (D.117h) to get,

$\displaystyle\\{4-D-\sum_{i=1}^{4}p_{i}.\frac{d}{dp_{i}}\\}\frac{1}{4!}B^{(2)}_{X}(p_{1},p_{2},p_{3},p_{4})+2(4-D)\frac{1}{4!}B_{II}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=\left\\{\epsilon-6F\lambda+\epsilon-3F\lambda^{2}\right\\}\frac{1}{4!}B_{II}^{(1)}(p_{1},p_{2},p_{3},p_{4})$
(D.130)

We ignore the term $\epsilon\frac{1}{4!}B_{X}^{(2)}(p_{1},p_{2},p_{3},p_{4})$
and get the following euqations:

$\displaystyle-\sum_{i=1}^{4}p_{i}.\frac{d}{dp_{i}}B^{(2)}_{X}(p_{1},p_{2},p_{3},p_{4})-9F\lambda^{2}\sum_{3~{}perm~{}(i,j)}\int_{q}\big{\\{}h(p_{i}+p_{j}+q)h(q)-h(q)h(q)\big{\\}}=0$
(D.131)

To solve this, we use $\bar{p}=\frac{p}{\Lambda}$. In this noatation
$-\sum_{i=1}^{4}\bar{p_{i}}.\frac{d}{dp_{i}}B^{(2)}_{X}(\bar{p}_{1},\bar{p}_{2},\bar{p}_{3},\bar{p}_{4})$
can be written as,

$\displaystyle-\sum_{i=1}^{4}\bar{p}_{i}.\frac{d}{d\bar{p}_{i}}B^{(2)}_{X}\bigg{(}\frac{p_{1}}{\Lambda},\frac{p_{2}}{\Lambda},\frac{p_{3}}{\Lambda},\frac{p_{4}}{\Lambda}\bigg{)}=\Lambda.\frac{d}{d\Lambda}B^{(2)}_{X}\bigg{(}\frac{p_{1}}{\Lambda},\frac{p_{2}}{\Lambda},\frac{p_{3}}{\Lambda},\frac{p_{4}}{\Lambda}\bigg{)}$

So the solution is given by,

$\displaystyle\frac{1}{4!}B^{(2)}_{X}(\frac{p_{1}}{\Lambda},\frac{p_{2}}{\Lambda},\frac{p_{3}}{\Lambda},\frac{p_{4}}{\Lambda})=\frac{9F\lambda^{2}}{4!}\sum_{3~{}perm~{}(i,j)}\int_{\Lambda}^{\infty}\int_{\bar{q}}\frac{d\Lambda^{\prime}}{\Lambda^{\prime}}\bigg{\\{}h\bigg{(}\frac{p_{i}}{\Lambda^{\prime}}+\frac{p_{j}}{\Lambda^{\prime}}+\bar{q}\bigg{)}h\left(\bar{q}\right)-h\left(\bar{q}\right)h\left(\bar{q}\right)\bigg{\\}}$
(D.132)

In LHS we are left with

$\displaystyle\frac{1}{4!}(4-D)B_{X}^{(2)}(p_{1},p_{2},p_{3},p_{4})$ (D.133)

##  Appendix E Relevant operator at sub-leading operator

###  E.1 The $\phi^{6}$ equation to find
$D^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})$

$\phi^{6}$ equation is given by ( we donot have to consider
$\beta(\lambda)\frac{\partial\Delta S}{\partial\lambda}$ part because there is
no $D^{(1)}(p_{1},p_{2},p_{3},p_{4})$ in this case).

$\displaystyle-\frac{4}{6!}\sum_{10~{}perm~{}(i,j,k)}\big{\\{}-K^{\prime}(p_{i}+p_{j}+p_{k})\big{\\}}\big{\\{}\lambda\big{\\}}\big{\\{}\underbrace{B(p_{i})+B(p_{j})+B(p_{k})}_{1}+\underbrace{B(p_{i}+p_{j}+p_{k})}_{2}\big{\\}}$
$\displaystyle-\frac{2}{6!}\sum_{i=1}^{4}\big{\\{}-K^{\prime}(p_{i}^{2})\big{\\}}\underbrace{\big{\\{}A(p_{i})\big{\\}}V_{6}^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})}_{3}$
$\displaystyle+\frac{1}{6!}\bigg{(}6-2D-\sum_{i=1}^{4}p_{i}.\frac{\partial}{\partial
p_{i}}\bigg{)}D^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})$
$\displaystyle=\frac{d_{2}^{(0)}}{6!}D^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})$
(E.134)

$d_{2}^{(0)}=2$

We collect the terms marked ’2’ to find first kind of
$D^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})$.

$\displaystyle-\frac{4}{6!}\sum_{10~{}perm~{}(i,j,k)}\big{\\{}-K^{\prime}(p_{i}+p_{j}+p_{k})\big{\\}}\big{\\{}\lambda\big{\\}}\big{\\{}B(p_{i}+p_{j}+p_{k})\big{\\}}$
$\displaystyle+\frac{1}{6!}\bigg{(}6-2D-\sum_{i=1}^{4}p_{i}.\frac{\partial}{\partial
p_{i}}\bigg{)}D^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})$
$\displaystyle=\frac{d_{2}^{(0)}}{6!}D_{I}^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})$
(E.135)

Similarly collecting the terms marked as ’1’ and ’3’ we get the following
equation,

$\displaystyle-\frac{4}{6!}\sum_{10~{}perm~{}(i,j,k)}\big{\\{}-K^{\prime}(p_{i}+p_{j}+p_{k})\big{\\}}\big{\\{}\lambda\big{\\}}\big{\\{}B(p_{i})+B(p_{j})+B(p_{k})\big{\\}}$
$\displaystyle-\frac{2}{6!}\sum_{i=1}^{4}\big{\\{}-K^{\prime}(p_{i}^{2})\big{\\}}\big{\\{}A(p_{i})\big{\\}}V_{6}^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})$
$\displaystyle=\frac{2}{6!}D_{II}^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})$
(E.136)

###  E.2 The $\phi^{4}$ equation to determine
$B^{(2)}(p_{1},p_{2},p_{3},p_{4})$

The $\phi^{4}$ equation is given by,

$\displaystyle\frac{1}{6!}\int_{p}\big{\\{}-K^{\prime}(p)\big{\\}}\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}D^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})-\frac{2}{4!}\sum_{i=1}^{4}\big{\\{}-K^{\prime}(p_{i})\big{\\}}\big{\\{}A^{(0)}(p)+A^{(1)}(p)\big{\\}}\big{\\{}\lambda+U_{4}^{(2)}(p_{1},p_{2},p_{3},p_{4})\big{\\}}$
$\displaystyle-\frac{2}{4!}\sum_{i=1}^{4}\big{\\{}-K^{\prime}(p_{i})\big{\\}}\big{\\{}U_{2}^{(1)}(p_{i})+U_{2}^{(2)}(p_{i})\big{\\}}\big{\\{}\sum_{i=1}^{4}B^{(1)}(p_{i})\big{\\}}+\frac{1}{4!}\bigg{(}4-D-\sum_{i=1}^{4}p_{i}.\frac{\partial}{\partial
p_{i}}\bigg{)}B^{(2)}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle+(4-D)B^{(1)}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=\frac{d_{2}^{(1)}}{4!}B^{(1)}(p_{1},p_{2},p_{3},p_{4})+\frac{d_{2}^{(0)}}{4!}B^{(2)}(p_{1},p_{2},p_{3},p_{4})+\frac{1}{4!}\\{\epsilon\lambda+\beta_{1}^{(1)}(\lambda)\\}\frac{\partial}{\partial\lambda}B^{(1)}(p_{1},p_{2},p_{3},p_{4})$
(E.137)

Where

$U_{4}^{(2)}(p_{1},p_{2},p_{3},p_{4})=\underbrace{-\lambda^{2}\sum_{3~{}perm~{}(i,j)}\mathcal{F}(p_{i}+p_{j})}_{U_{4}^{I}}+\underbrace{\frac{F\lambda^{2}}{2}\sum_{i=1}^{4}h(p_{i})}_{U_{4}^{II}}$

$\mathcal{F}(p_{i}+p_{j})=\frac{1}{2}\int_{k}\\{h(p_{i}+p_{j}+k)h(k)-h(k)h(k)\\}$

$d_{2}^{(1)}=-F\lambda$

$B^{(1)}(p_{1},p_{2},p_{3},p_{4})=-\lambda\sum_{i=1}^{4}h(p_{i})$

####  Calculation of
$\frac{\delta}{\delta\phi(p)}\frac{\delta}{\delta\phi(-p)}D^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})$

a)

$\displaystyle\frac{\delta^{2}}{\delta\phi(p)\delta\phi(-p)}D_{I}^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})\phi(p_{5})\phi(p_{6})$
$\displaystyle=\frac{\delta^{2}}{\delta\phi(p)\delta\phi(-p)}\big{\\{}\sum_{10~{}perm~{}(i,j,k)}h(p_{i}+p_{j}+p_{k})\big{\\}}\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})\phi(p_{5})\phi(p_{6})$
$\displaystyle=30\times\big{\\{}\sum_{l=1}^{4}h(p_{l})h(p_{l})\big{\\}}\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})$
(E.138a)
$\displaystyle+60\times\sum_{3~{}perm~{}(i,j)}\big{\\{}h(p_{i}+p_{j}+p)h(p_{i}+p_{j}+p)\big{\\}}\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})$
(E.138b)

b)

$\displaystyle\frac{\delta^{2}}{\delta\phi(p)\delta\phi(-p)}D_{II}^{(2)}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})\phi(p_{5})\phi(p_{6})$
$\displaystyle\frac{\delta^{2}}{\delta\phi(p)\delta\phi(-p)}\sum_{10~{}perm~{}(i,j,k)}h(p_{i}+p_{j}+p_{k})\sum_{l=1}^{6}h(p_{l})\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})\phi(p_{5})\phi(p_{6})$
$\displaystyle=30\times\sum_{i=1}^{4}h(p_{i})2h(p)\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})$
(E.139a)
$\displaystyle+30\times\sum_{i=1}^{4}h(p_{i})\sum_{j=1}^{4}h(p_{j})\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})$
(E.139b)
$\displaystyle+60\times\sum_{3~{}perm~{}(i,j)}h(p_{i}+p_{j}+p)\big{\\{}2h(p)\big{\\}}\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})$
(E.139c)
$\displaystyle+60\times\sum_{3~{}perm~{}(i,j)}h(p_{i}+p_{j}+p)\sum_{k=1}^{4}h(p_{k})\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})$
(E.139d)

#### Equation for $B_{I}^{(2)}(p_{1},p_{2},p_{3},p_{4})$

Collecting (E.138b) and (E.139c), we get the following equations:

$\displaystyle\lambda^{2}\int_{p}\big{\\{}-K^{\prime}(p)\big{\\}}[2\sum_{3~{}perm~{}(i,j)}h(p_{i}+p_{j}+p)h(p_{i}+p_{j}+p)+4\sum_{3~{}perm~{}(i,j)}h(p_{i}+p_{j}+p)h(p)]$
$\displaystyle\bigg{\\{}(4-D)-p_{i}.\frac{d}{dp_{i}}\bigg{\\}}B_{I}^{(2)}(p_{1},p_{2},p_{3},p_{4})=d_{2}^{(0)}B_{I}^{(2)}(p_{1},p_{2},p_{3},p_{4})$
(E.140)

#### Equation for $B_{II}^{(2)}(p_{1},p_{2},p_{3},p_{4})$

We take $\beta_{1}^{(1)}(\lambda)\frac{\partial}{\partial\lambda}B^{(1)}$ term
from RHS. Collecting (E.139d) and the term with $U_{4}^{I}$ in the second line
of (E.2) we get the following equation,

$\displaystyle
2\lambda^{2}\sum_{3~{}perm~{}(i,j)}h(p_{i}+p_{j}+p)\big{\\{}-K^{\prime}(p)\big{\\}}\sum_{l=1}^{4}h(p_{l})-3\lambda^{2}F\sum_{l=1}^{4}h(p_{l})$
$\displaystyle+2\sum_{i=1}^{4}K^{\prime}(p_{i})A^{(0)}(p_{i})\bigg{\\{}-\frac{\lambda^{2}}{2}\int_{k}\sum_{3~{}perm~{}(i,j)}[h(p_{i}+p_{j}+k)h(k)-h(k)h(k)]\bigg{\\}}$
$\displaystyle+\bigg{\\{}(4-D)-p_{i}.\frac{d}{dp_{i}}\bigg{\\}}B_{II}^{(2)}(p_{1},p_{2},p_{3},p_{4})=d_{2}^{(0)}B_{II}^{(2)}(p_{1},p_{2},p_{3},p_{4})$

#### Equation for $B_{III}^{(2)}(p_{1},p_{2},p_{3},p_{4})$

Collecting (E.138a), (E.139b) and the term containing $U_{4}^{II}$ we get

$\displaystyle\int_{p}\big{\\{}-K^{\prime}(p)\big{\\}}\big{\\{}\sum_{l=1}^{4}h(p_{l})h(p_{l})+\sum_{i=1}^{4}h(p_{i})\sum_{j=1}^{4}h(p_{j})\big{\\}}$
$\displaystyle+2\sum_{i=1}^{4}K^{\prime}(p_{i}^{2})\big{\\{}A^{(0)}(p_{i})U_{4}^{(2)}|_{2}+\lambda
A^{(1)}(p_{i})\big{\\}}$
$\displaystyle+2\sum_{i=1}^{4}K^{\prime}(p_{i})U_{2}^{(1)}(p_{i})\sum_{j=1}^{4}B^{(1)}(p_{i})$
$\displaystyle+\big{(}4-D-p_{i}.\frac{d}{dp_{i}}\big{)}B_{III}^{(2)}(p_{1},p_{2},p_{3},p_{4})=d_{2}^{(0)}B_{III}^{(0)}(p_{1},p_{2},p_{3},p_{4})$

We have,

$\displaystyle A^{(0)}(p)=1~{};A^{(1)}(p)=F\lambda
h(p)~{};U_{2}^{(1)}(p)=-\frac{\lambda F}{2-\epsilon}~{};B^{(1)}(p)=-\lambda
h(p)$

So the equation for $B_{III}^{(2)}(p_{1},p_{2},p_{3},p_{4})$ becomes,

$\displaystyle F\big{\\{}\sum_{l=1}^{4}h(p_{l})h(p_{l})+\sum_{i\neq
j}h(p_{i})h(p_{j})\big{\\}}+F\sum_{i\neq j}K^{\prime}(p_{i}^{2})h(p_{j})$
$\displaystyle 2F\sum_{i\neq
j}K^{\prime}(p_{i})h(p_{j})+4F\sum_{i=1}^{4}K^{\prime}(p_{i})h(p_{i})+\big{\\{}4-D-p_{i}.\frac{d}{dp_{i}}\big{\\}}B_{III}^{(2)}(p_{1},p_{2},p_{3},p_{4})=d_{2}^{(0)}B_{III}^{(2)}(p_{1},p_{2},p_{3},p_{4})$
(E.141)

####  E.2.1 Cancellation

Note that last term in LHS and third term in RHS of (E.2) cancels. Also the
term (E.139a) cancels with the term
$d_{2}^{(1)}B^{(1)}(p_{1},p_{2},p_{3},p_{4})$.

###  E.3 The $\phi^{2}$ equation to determine $A^{(2)}(p)$

$\phi^{2}$ equation is given by,

$\displaystyle\int(-K^{\prime}(q))\frac{\delta^{2}}{\delta\phi(q)\phi(-q)}\big{\\{}B^{(2)}(p_{1},p_{2},p_{3},p_{4})\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})\big{\\}}$
$\displaystyle+\left[A(p)-p^{2}A^{\prime}(p)-2\big{\\{}-K^{\prime}(p^{2})\big{\\}}U_{2}(p)A(p)\right]\phi(p)\phi(-p)$
$\displaystyle+\eta\frac{K(p^{2})(1-K(p^{2}))}{p^{2}}\frac{p^{2}}{K(p^{2})}\phi(p)\phi(-p)-\frac{\eta}{2}A^{(0)}(p)\phi(p)\phi(-p)=\frac{d_{2}}{2}A(p)+(\epsilon\lambda+\beta(\lambda))\frac{\partial}{\partial\lambda}\frac{A^{(2)}(p)}{2}$
(E.142)

$\displaystyle U_{2}^{1}(p)=-\frac{\lambda
F}{2-\epsilon};~{}~{}U_{2}^{(2)}(p)=-\lambda^{2}G(p)-\frac{\lambda^{2}F^{2}}{4}h(p)$

Where

$G(p)=\frac{1}{3}\int_{q,k}\frac{h(q)}{2}[h(p+q+k)h(k)-h(k)h(k)]-\frac{1}{3}\int_{q}\frac{h(q)}{2}[h(q+k)h(k)-h(k)h(k)]+\frac{\eta
p^{2}}{2\epsilon}-\frac{1}{2-2\epsilon}\bigg{\\{}\frac{2}{3}\beta^{(1)}v_{2}^{(1)}+\int_{q}f(q)\mathcal{F}(q)\bigg{\\}}$

$\displaystyle\beta^{(1)}=-\int_{q}f(q)h(q)\rightarrow_{\epsilon\rightarrow
0}-F;v_{2}^{(1)}=-\int_{q}f(q)h(q)\rightarrow_{\epsilon\rightarrow
0}-\frac{F}{2}$

A.

$\displaystyle 1.~{}~{}~{}$
$\displaystyle\frac{\delta^{2}}{\delta\phi(q)\delta\phi(-q)}B_{I}^{(2)}(p_{1},p_{2},p_{3},p_{4})\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})$
$\displaystyle=\frac{\delta^{2}}{\delta\phi(q)\delta\phi(-q)}\frac{1}{4!}\sum_{3~{}perm~{}(i,j)}\int_{k}\big{\\{}h(p_{i}+p_{j}+k)h(k)h(k)\big{\\}}\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})$
$\displaystyle=$ $\displaystyle\int_{q,k}\big{\\{}h(p+q+k)h(k)h(k)$ (E.143)
$\displaystyle+\frac{1}{2}h(k)h(k)h(k)$ (E.144)

$\displaystyle 2.~{}~{}~{}$
$\displaystyle\frac{\delta^{2}}{\delta\phi(q)\delta\phi(-q)}\big{\\{}B_{II}^{(2)}(p_{1},p_{2},p_{3},p_{4})\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})\big{\\}}$
$\displaystyle=$
$\displaystyle\frac{\delta^{2}}{\delta\phi(q)\delta\phi(-q)}\frac{1}{4!}\frac{1}{2}\int_{k}\sum_{3~{}perm~{}(i,j)}\big{\\{}h(p_{i}+p_{j}+k)h(k)-h(k)h(k)\big{\\}}\sum_{l=1}^{4}h(p_{l})\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})$
$\displaystyle=$
$\displaystyle\int_{q,k}\big{\\{}h(p+q+k)h(k)-h(k)h(k)\big{\\}}\big{\\{}h(p)\big{\\}}$
(E.145) $\displaystyle+$
$\displaystyle\int_{q,k}\big{\\{}h(p+q+k)h(k)-h(k)h(k)\big{\\}}\big{\\{}h(q)\big{\\}}$
(E.146)

$\displaystyle 3.~{}~{}~{}$
$\displaystyle\int_{q}\big{\\{}-K^{\prime}(q)\big{\\}}\frac{\delta^{2}}{\delta\phi(q)\delta\phi(-q)}B_{III}^{(2)}(p_{1},p_{2},p_{3},p_{4})\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})$
$\displaystyle=\int_{q}\big{\\{}-K^{\prime}(q)\big{\\}}\frac{\delta^{2}}{\delta\phi(q)\delta\phi(-q)}\frac{1}{4!}(-F)\big{\\{}\sum_{i\neq
j}\frac{1}{2}h(p_{i})h(p_{j})+\sum_{l=1}^{4}h^{2}(p_{l})\big{\\}}\phi(p_{1})\phi(p_{2})\phi(p_{3})\phi(p_{4})$
$\displaystyle=\int_{q}\big{\\{}-K^{\prime}(q)\big{\\}}\frac{-F}{4!}\left[6\times
2\bigg{\\{}\frac{2h^{2}(p)+2h^{2}(q)+8h(p)h(q)}{2}\bigg{\\}}+6\times
2\big{\\{}2h^{2}(p)+2h^{2}(q)\big{\\}}\right]$
$\displaystyle=\frac{3\lambda^{2}F}{2}\int K^{\prime}(q)h^{2}(q)$ (E.147)
$\displaystyle-\frac{3}{2}\lambda^{2}F^{2}h^{2}(p)$ (E.148)
$\displaystyle-\lambda^{2}F^{2}h(p)$ (E.149)

$\displaystyle B.~{}~{}~{}$ $\displaystyle
2K^{\prime}(p^{2})\big{\\{}U_{2}^{(2)}(p)A^{(0)}(p)+U_{2}^{(1)}A^{(1)}(p)\big{\\}}$
$\displaystyle=2K^{\prime}(p^{2})\frac{-\lambda F}{2-\epsilon}$ (E.150)
$\displaystyle=2K^{\prime}(p^{2})\frac{-\lambda^{2}F^{2}}{4}h(p)$ (E.151)
$\displaystyle=-2K^{\prime}(p^{2})A^{(0)}(p)\frac{1}{3}\left[\int_{q,k}\frac{h(q)}{2}\big{\\{}h(p+q+k)h(k)-h(q+k)h(k)\big{\\}}\right]$
(E.152) $\displaystyle-2K^{\prime}(p^{2})A^{(0)}(p)\frac{\eta
p^{2}}{2\epsilon}$ (E.153)
$\displaystyle+2K^{\prime}(p^{2})A^{(0)}(p)\frac{1}{2-2\epsilon}\big{\\{}\frac{2}{3}\beta^{(1)}v_{2}^{(1)}\big{\\}}$
(E.154)
$\displaystyle+2K^{\prime}(p^{2})A^{(0)}(p)\frac{1}{2-2\epsilon}\big{\\{}\int_{q}f(q)\mathcal{F}(q)\big{\\}}$
(E.155) $\displaystyle+2K^{\prime}(p^{2})\big{\\{}-\frac{\lambda
F}{2-\epsilon}\big{\\}}\big{\\{}\lambda Fh(p)\big{\\}}$ (E.156)

$\displaystyle C.~{}~{}~{}$
$\displaystyle-\eta\frac{K(p^{2})(1-K(p^{2}))}{p^{2}}\frac{p^{2}}{K(p^{2})}-\frac{\eta}{2}A^{(0)}(p)$
$\displaystyle=-\eta~{}p^{2}h(p)\phi(p)\phi(-p)$ (E.157)
$\displaystyle-\frac{\eta}{2}A^{(0)}(p)$ (E.158)

#### Equation for $A_{I}^{(2)}(p)$

We collect (E.145) and (E.152) to write the following equation,

$\displaystyle\int_{q,k}\big{\\{}-K^{\prime}(q)\big{\\}}\big{\\{}h(p+q+k)h(k)-h(q+k)h(k)\big{\\}}\big{\\{}h(p)\big{\\}}$
$\displaystyle-2K^{\prime}(p^{2})A^{(0)}(p)\frac{1}{3}\left[\int_{q,k}\frac{h(q)}{2}\big{\\{}h(p+q+k)h(k)-h(q+k)h(k)\big{\\}}\right]$
$\displaystyle+A_{I}^{(2)}(p)-p^{2}\frac{\partial A_{I}^{(2)}(p)}{\partial
p^{2}}=d_{2}\frac{A_{I}^{(2)}(p)}{2}$ (E.159)

On LHS of (E.3) we are left with,

$\displaystyle\int_{q,k}\big{\\{}-K^{\prime}(q)\big{\\}}h(p)\big{\\{}h(q+k)h(k)-h(k)h(k)\big{\\}}$
(E.160)

#### Equation for $A_{II}^{(2)}(p)$

We collect (E.143) and (E.146) to write

$\displaystyle\int_{q,k}\big{\\{}-K^{\prime}(q)\big{\\}}\big{\\{}h(p+q+k)h(k)h(k)-h(q+k)h(k)h(k)\big{\\}}+\int_{q,k}\big{\\{}-K^{\prime}(q)\big{\\}}\big{\\{}h(p+q+k)h(k)-h(q+k)h(k)\big{\\}}\big{\\{}h(q)\big{\\}}$
$\displaystyle+A_{II}^{(2)}(p)-p^{2}\frac{\partial A_{II}^{(2)}(p)}{\partial
p^{2}}=d_{2}^{(0)}\frac{A_{II}^{(2)}(p)}{2}$ (E.161)

On LHS of (E.3) we are left with

$\displaystyle\int_{q,k}\big{\\{}-K^{\prime}(q)h(q+k)h(k)h(k)\big{\\}}+\int_{q,k}\big{\\{}-K^{\prime}(q)h(q)\big{\\}}\big{\\{}h(q+k)h(k)-h(k)h(k)\big{\\}}$
(E.162)

#### Equation for $A_{III}^{(2)}(p)$

We collect (E.150), (E.154) and (E.149) to get,

$\displaystyle-\lambda^{2}F^{2}h(p)+2K^{\prime}(p^{2})\big{\\{}-\frac{\lambda
F\epsilon}{4}\big{\\}}+2K^{\prime}(p^{2})A^{(0)}(p)\frac{1}{2-2\epsilon}\big{\\{}\frac{2}{3}\beta^{(1)}v_{2}^{(1)}\big{\\}}+A_{III}^{(2)}(p)-p^{2}\frac{\partial}{\partial
p^{2}}A_{III}^{(2)}(p)$
$\displaystyle=d_{2}^{(0)}\frac{A_{III}^{(2)}(p)}{2}+(d_{2}^{(1)})\frac{A^{(1)}(p)}{2}+\\{\epsilon\lambda+\beta(\lambda)\\}{1\over
2}\frac{\partial A}{\partial\lambda}$ (E.163)

Where $d_{2}^{(1)}=-\lambda F$ and $A^{(1)}(p)=\lambda Fh(p)$.

#### Equation for $A_{IV}^{(2)}(p)$

Collecting (E.151), (E.156) and (E.148) we get,

$\displaystyle-\frac{3}{2}\lambda^{2}F^{2}h^{2}(p)+2K^{\prime}(p^{2})\frac{-\lambda^{2}F^{2}}{4}h(p)+2K^{\prime}(p^{2})\big{\\{}-\frac{\lambda
F}{2-\epsilon}\big{\\}}\big{\\{}\lambda
Fh(p)\big{\\}}+A_{IV}^{(2)}(p)-p^{2}\frac{\partial}{\partial
p^{2}}A_{IV}^{(2)}(p)=d_{2}^{(0)}\frac{A_{IV}^{(2)}(p)}{2}$ (E.164)

#### Equation for $A_{V}^{(2)}(p)$

We collect (E.155) and (E.144) to get the following equation,

$\displaystyle
2K^{\prime}(p^{2})A^{(0)}(p)\frac{1}{2-2\epsilon}\big{\\{}\int_{q}f(q)\mathcal{F}(q)\big{\\}}+\int_{q,k}\big{\\{}-K^{\prime}(q)\big{\\}}h(p)\big{\\{}h(q+k)h(k)-h(k)h(k)\big{\\}}$
$\displaystyle+A_{V}^{(2)}(p)-p^{2}\frac{\partial}{\partial
p^{2}}A_{V}^{(2)}(p)=d_{2}^{(0)}\frac{A_{V}^{(2)}(p)}{2}$ (E.165)

#### Equation for $A_{VI}^{(2)}(p)$

We collect 5th term of (4.1.3), (E.153) to get the following equation

$\displaystyle\eta\frac{K(p^{2})(1-K(p^{2}))}{p^{2}}\frac{p^{2}}{K(p^{2})}-2K^{\prime}(p^{2})A^{(0)}(p)\frac{\eta
p^{2}}{2\epsilon}+A_{VI}^{(2)}(p)-\frac{1}{2}p.\frac{\partial}{\partial
p}A_{VI}^{(2)}(p)$
$\displaystyle=d_{2}^{(0)}\frac{A_{VI}(p)}{2}+d_{2}^{(1)}\frac{A_{VI}^{(2)}(p)}{2}+\epsilon\lambda\frac{\partial}{\partial\lambda}\frac{A_{VI}^{(2)}(p)}{2}$
(E.166)

##  Appendix F Evaluation of Integrals

$\displaystyle\int_{p,q}h\left(\frac{p+q}{\Lambda}\right)h\left(\frac{q}{\Lambda}\right)h\left(\frac{p}{\Lambda}\right)h\left(\frac{p}{\Lambda}\right)$
$\displaystyle=$
$\displaystyle\int_{p,q}\frac{K\left(\frac{p+q}{\Lambda_{0}}\right)-K\left(\frac{p+q}{\Lambda}\right)}{(p+q)^{2}}~{}~{}\frac{K\left(\frac{q}{\Lambda_{0}}\right)-K\left(\frac{q}{\Lambda}\right)}{(q)^{2}}~{}~{}\frac{K\left(\frac{p}{\Lambda_{0}}\right)-K\left(\frac{p}{\Lambda}\right)}{(p)^{2}}~{}~{}\frac{K\left(\frac{p}{\Lambda_{0}}\right)-K\left(\frac{p}{\Lambda}\right)}{(p)^{2}}$

We evaluate the integral for $K(x)=e^{-x^{2}}$.

$\displaystyle=\int_{p,q}\frac{e^{-\frac{(p+q)^{2}}{\Lambda_{0}^{2}}}-e^{-\frac{(p+q)^{2}}{\Lambda^{2}}}}{(p+q)^{2}}~{}~{}\frac{e^{-\frac{q^{2}}{\Lambda_{0}^{2}}}-e^{-\frac{q^{2}}{\Lambda^{2}}}}{(q)^{2}}~{}~{}\frac{e^{\frac{p^{2}}{\Lambda_{0}^{2}}}-e^{\frac{p^{2}}{\Lambda^{2}}}}{(p)^{2}}~{}~{}\frac{e^{\frac{p^{2}}{\Lambda_{0}^{2}}}-e^{\frac{p^{2}}{\Lambda^{2}}}}{(p)^{2}}$

Now we apply Schwinger parametrization.

$\displaystyle\int_{p,q}\int_{x,y,u,v=\frac{1}{\Lambda^{2}}}^{\frac{1}{\Lambda_{0}^{2}}}e^{-(p+q)^{2}x}e^{-q^{2}y}e^{-p^{2}u}e^{-p^{2}v}$

Now we do q inetgral first. We Complete the square on q and change integration
varibale q. After that we do p inetegral. Also we change $x,y$ as
$x\rightarrow\frac{1}{x},~{}y\rightarrow\frac{1}{y}$. At the end we take
$\Lambda_{0}\rightarrow\infty$.

$\displaystyle\int_{x,y,u,v}\int_{p,q}\frac{1}{(x+y)^{2}}e^{-q^{2}}e^{-p^{2}\left(\frac{xy}{x+y}+u+v\right)}$
$\displaystyle=F^{2}\int_{x,y,u,v=\frac{1}{\Lambda^{2}}}^{\frac{1}{\Lambda_{0}^{2}}}\frac{1}{\big{\\{}1+(p+q)(u+v)\big{\\}}^{2}}$
$\displaystyle=F^{2}\bigg{(}\frac{1}{2}\\{\log
2\\}^{2}-\frac{1}{2}\\{\log\frac{2\Lambda_{0}^{2}}{\Lambda^{2}}\\}^{2}+\\{\log\frac{\Lambda_{0}^{2}}{\Lambda^{2}}\\}^{2}+2\log
2-6\log 3+2\log\frac{\Lambda_{0}^{2}}{\Lambda^{2}}$
$\displaystyle+\frac{1}{4}\\{\log
4\\}^{2}-\frac{1}{2}\\{\log\frac{2\Lambda_{0}^{2}}{\Lambda^{2}}\\}^{2}+\frac{1}{4}\\{\log\frac{4\Lambda_{0}^{2}}{\Lambda^{2}}\\}^{2}-8\log
2+5\log 5-\log\frac{\Lambda_{0}^{2}}{\Lambda^{2}}$
$\displaystyle+\frac{1}{4}\\{\log\frac{4\Lambda^{2}}{\Lambda_{0}^{2}}\\}^{2}-\frac{1}{2}\\{\log
2\\}^{2}+\frac{1}{4}\\{\log 4\\}^{2}+4\log 2-6\log 3+5\log 5\bigg{)}$

So,

$\displaystyle\int_{p,q}\left[\big{\\{}-K^{\prime}(p^{2})\big{\\}}h(p)h(p+q)h(q)+\big{\\{}-K^{\prime}(p^{2})\big{\\}}h(q)h(p+q)h(q)\right]$
$\displaystyle=F^{2}\left(\frac{1}{2}-\log
2+\frac{1}{2}\log\frac{\Lambda_{0}^{2}}{\Lambda^{2}}\right)$ (F.167)

Using same procedure we can find all other integrals of this kind.

##  Appendix G Useful Mathematical identities

In this section we give various mathematical identities about the functions
$h(p)$, $\mathcal{F}(p)$, $F_{3}(p)$ etc which were used in the main text to
find the composite operators.

$h(p)=\frac{1-K(p^{2})}{p^{2}}~{}~{}f(p)=-2K^{\prime}(p)$

$\displaystyle-p.\frac{\partial}{\partial p}h(p)=-f(p)+2h(p)$ (G.168)

$\mathcal{F}(p)={1\over 2}\int_{q}\left\\{h(p+q)h(q)-h(q)h(q)\right\\}$

$\displaystyle\left(p.\frac{d}{dp}+\epsilon\right)\mathcal{F}(p)=\int_{q}f(q)\left\\{h(p+q)-h(q)\right\\}$
(G.169)

$\bar{F}_{3}(p)=\int_{q,k}h(p+q+k)h(q)h(k)$
$,~{}~{}F_{3}(p)=\bar{F}_{3}(p)-\bar{F}_{3}(0)=\int_{q}2h(q)\left[F(p+q)-F(q)\right]$

$\displaystyle\left(-\frac{p}{2}.\frac{d}{dp}+1\right)\bar{F_{3}}(p)=-6\int_{q,k}f(q)h(p+q+k)h(k)$
(G.170)

$\displaystyle\left(p.\frac{d}{dp}-2+2\epsilon\right)F_{3}(p)=3\int_{q,k}f(k)h(q)\left[h(q+k+p)-h(q+k)\right]$
(G.171)

$\bar{H}_{3}(p)=\int_{q}h(p+q)h(q)h(q)$

$\displaystyle\left(p.\frac{\partial}{\partial
p}+2+\epsilon\right)\bar{H}_{3}(p)=\int_{q}f(p)h(p+q)^{2}+2\int_{q}f(q)h(q)h(p+q)$
(G.172)

$I_{4}(p_{i}+p_{j};p_{i})=\bar{I}_{4}(p_{i}+p_{j};p_{i})-\bar{I}_{4}(0;0)=\sum_{6~{}perm~{}(i,j)}\int_{p,q}\big{\\{}h(p_{i}+p_{j}+q)h(p+q+p_{i})h(p)h(q)-h(p+q)h(p)h(q)h(q)\big{\\}}$

$\displaystyle\left(-\sum_{l=1}^{4}p_{l}.\frac{d}{dp_{l}}-2\epsilon\right)\bar{I}_{4}(p_{i}+p_{j};p_{i})=-2\int_{p,q}f(p)\left[h(p_{i}+p_{j}+p)+h(p_{i}+p_{j}+q)\right]h(p_{i}+p+q)h(q)$
(G.173)

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