# On general approach to Bessenrodt-Ono type inequalities and log-concavity
property

Krystian Gajdzica, Piotr Miska and Maciej Ulas  Institute of Mathematics
Faculty of Mathematics and Computer Science
Jagiellonian University in Cracow, Łojasiewicza 6, 30-348 Kraków, Poland
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###### Abstract.

In recent literature concerning integer partitions one can find many results
related to both the Bessenrodt-Ono type inequalities and log-concavity
property. In this note we offer some general approach to this type of
problems. More precisely, we prove that under some mild conditions on an
increasing function $F$ of at most exponential growth satisfying the condition
$F(\mathbb{N})\subset\mathbb{R}_{+}$, we have $F(a)F(b)>F(a+b)$ for
sufficiently large positive integers $a,b$. Moreover, we show that if the
sequence $(F(n))_{n\geq n_{0}}$ is log-concave and
$\limsup_{n\rightarrow+\infty}F(n+n_{0})/F(n)<F(n_{0})$, then $F$ satisfies
the Bessenrodt-Ono type inequality.

###### Key words and phrases:

integer partition, log-concavity, Bessenrodt-Ono type inequality, sub-
exponential function.

###### 2020 Mathematics Subject Classification:

Primary 11P82, 11P84; Secondary 05A17.

The research of the authors was supported by a grant of the National Science
Centre (NCN), Poland, no. UMO-2019/34/E/ST1/00094

##  1\. Introduction

Let $\mathbb{N}$ be the set of non-negative integers, $\mathbb{N}_{+}$ the set
of positive integers and $\mathbb{R}_{+}=(0,+\infty)$. Moreover, for an
integer $n_{0}\geq 2$ we put $N_{\geq n_{0}}=\\{n\in\mathbb{N}:\;n\geq
n_{0}\\}$.

For given $A\subset\mathbb{N}_{+}$ and $n\in\mathbb{N}$ by $p_{A}(n)$ we
denote the number of partitions of $n$ with parts in the set $A$. As usual, we
put $p_{A}(0)=1$. It is well known that the ordinary generating function for
the sequence $(p_{A}(n))_{n\in\mathbb{N}}$ takes the form

$\sum_{n=0}^{\infty}p_{A}(n)x^{n}=\prod_{a\in A}\frac{1}{1-x^{a}}.$

If $A=\mathbb{N}_{+}$ then we simply write $p(n)$ instead of
$p_{\mathbb{N}_{+}}(n)$. In this case the partition function is the famous
Euler partition function thoroughly studied by Ramanujan and many others. The
number of papers devoted to various properties of $p(n)$, or more generally,
for $p_{A}(n)$ for various choice of the set $A$ is enormous. The standard
reference is the book of Andrews [2] (see also [3] for less advanced
approach).

A few years ago there emerged a broad research devoted to log-behavior of
partition statistics. DeSalvo and Pak [11] were the first who initiated this
kind of study. More precisely, they investigated the log-concavity of the
partition function $p(n)$. Let us recall that a sequence
$(\omega_{n})_{n\in\mathbb{N}}$ of real numbers is said to be log-concave if
the inequality

$\omega_{n}^{2}>\omega_{n-1}\omega_{n+1}$

is valid for all sufficiently large values of $n$. On the other hand, it is
called log-convex if the inverse inequality holds.

Using analytic methods DeSalvo and Pak [11] reproved the Nicolas’ theorem [24]
which states that the sequence $(p(n))_{n\geq 26}$ is log-concave.

Now, there is a wealth of literature devoted to the log-concavity property for
other variations of the partition function. Since we are unable to present a
complete list of the papers concerning the topic, let us recall at least a few
of them. For instance, Bringmann, Kane, Rolen and Tripp [6] investigated the
case of the $k$-colored partition function. On the other hand, Dawsey and
Masri [9] examined the Andrews spt-function in that direction. Further, Engel
[12] proved that the overpartition function $\overline{p}(n)$ is log-concave
for every $n\geq 2$. Gajdzica [13] discovered a similar phenomenon for the
$A$-partition function when $A$ is finite. O’Sullivan [25] investigated the
number of partitions into powers and proved a conjecture of Ulas [28]. One can
also mention that Ono, Pujahari and Rolen [26] showed that the plane partition
function satisfies the log-concavity property as well.

However, the research devoted to log-behavior of partition functions is not
only bounded by the log-concavity or log-convexity properties. Another
interesting phenomenon is the so-called Bessenrodt-Ono inequality. More
precisely, in 2016, Bessenrodt and Ono [5] showed that for $a,b\geq 2$ and
$a+b>9$ we have

(1.1) $p(a)p(b)>p(a+b).$

Their proof is based on the asymptotic estimates due to Lehmer [20]. It is
worth noting that there are two alternative approaches to derive the result.
Alanazi, Gagola and Munagi [1] showed the Bessenrodt-Ono inequality in
combinatorial manner by constructing appropriate injections between some sets
of partitions. Heim and Neuhauser [16], on the other hand, presented a proof
which is based on the induction on $a+b$.

Moreover, the inequality (1.1) can be effectively used in practice. For
instance, one can apply it to determine the value of

$\max{p(n)}=\max{\left\\{p(\boldsymbol{\lambda}):\boldsymbol{\lambda}\text{ is
a partition of }n\right\\}},$

where $p(\boldsymbol{\lambda})$ denotes the extended partition function
defined as $p(\boldsymbol{\lambda}):=\prod_{i=1}^{j}p(\lambda_{i})$ for
$\bf{\lambda}=(\lambda_{1},\lambda_{2},\ldots,\lambda_{j})$. For additional
information, we refer the reader to [5].

A lot of works concerning the Bessenrodt-Ono type inequality were published.
For example, Beckwith and Bessenrodt [4] found out similar properties for the
$k$-regular partition function. Chern, Fu and Tang [8] considered the
$k$-colored partition function. Dawsey and Masri [9] discovered the analogous
phenomenon for the Andrews spt-function. Heim, Neuhauser and Tröger [18]
examined the issue for the plane partition function. Moreover, Heim and
Neuhauser widely generalized the property and investigated the so-called
polynomization of the Bessenrodt-Ono inequality for a couple of partition
functions [14, 15, 17, 18]. Hou and Jagadeesan [19], and Males [23] discovered
the analogues of the Bessenrodt-Ono inequality for the so-called partition
rank function.

Actually, there are many more properties related to the log-behavior of
partition statistics. We do not discuss them here, but focus on some general
criteria for both Bessenrodt-Ono type inequalities and log-concavity problems.
It turns out thata plenty of the aforementioned results are proved using some
asymptotic estimates. Therefore, it would be convenient to posses a few
results which assert that a function with an appropriate growth is log-concave
or fulfills the Bessenrodt-Ono inequality. Essentially, these goals were the
main motivations for our investigation.

Let us describe the content of the paper in some details. In Section 2 we get
a general criterion on the sequence $(F(n))_{n\in\mathbb{N}}$, which
guarantees that it satisfies the Bessenrodt-Ono type inequality. In other
words, for all sufficiently large $a,b\in\mathbb{N},a\geq b$ we have
$F(a)F(b)>F(a+b)$. In particular, as an application we reprove recent result
of Heim and Neuhauser which says that the plane partition function satisfies
the Bessenrodt-Ono type inequality. In Section 3 we present some conditions
guaranteeing asymptotic log-concavity of the sequence
$(F(n))_{n\in\mathbb{N}}$. As an application we reprove recent result of
DeSalvo and Pak [11, Theorem 1.1]. Finally, in the last section, under some
mild condition on the sequence $(F(n))_{n\in\mathbb{N}}$ we show that log-
concavity implies the Bessenrodt-Ono type inequality. We also prove that the
opposite implication is not true. More precisely, we show that for each
$m\in\mathbb{N}_{\geq 2}$ the $m$-ary partition function $b_{m}(n)=p_{A}(n)$,
where $A=\\{m^{i}:\;i\in\mathbb{N}\\}$, satisfies the Bessenrodt-Ono type
inequality but is not log-concave.

##  2\. Bessenrodt-Ono type inequality holds for a class of sub-exponential
functions

In this section we offer a general approach to Bessenrodt-Ono type
inequalities. More precisely, we prove the following general result.

######  Theorem 2.1.

Let $c_{1},c_{2},f,F:\mathbb{N}_{+}\to\mathbb{R}_{+}$ be given and such that
the inequalities

$\displaystyle c_{1}(n)e^{f(n)}<F(n)<c_{2}(n)e^{f(n)}$

are valid for all positive integers $n\geq N_{0}$ for some
$N_{0}\in\mathbb{N}_{+}$. Suppose further that the following conditions hold:

  1. (1)

$\exists g:\mathbb{N}_{+}\to\mathbb{R}_{+},N_{1}\in\mathbb{N}_{+}\text{
}\forall a\geq b\geq N_{1}:f(a)+f(b)-f(a+b)\geq g(b)$;

  2. (2)

$\exists h:\mathbb{N}_{+}\to\mathbb{R}_{+},N_{2}\in\mathbb{N}_{+}\text{
}\forall a\geq b\geq N_{2}:c_{2}(a+b)/c_{1}(a)\leq h(b)$;

  3. (3)

$\exists N_{3}\in\mathbb{N}_{+}\text{ }\forall n\geq N_{3}:g(n)\geq\log
h(n)-\log c_{1}(n)$.

Then, the inequality

$\displaystyle F(a)F(b)>F(a+b)$

is satisfied for all $a,b\geq\max\\{N_{0},N_{1},N_{2},N_{3}\\}$.

###### Proof.

Let us fix functions $c_{1},c_{2},f,F$ and $g$ as in the statement. Since both
of the inequalities

$\displaystyle F(a)F(b)>c_{1}(a)c_{1}(b)e^{f(a)+f(b)}$

and

$\displaystyle F(a+b)<c_{2}(a+b)e^{f(a+b)}$

are true for any positive integers $a,b\geq N_{0}$, it is enough to prove that
the following inequality

$\displaystyle e^{f(a)+f(b)-f(a+b)}\geq\frac{c_{2}(a+b)}{c_{1}(a)c_{1}(b)}$

is satisfied for all sufficiently large values of $a$ and $b$. From ($1$), it
follows that the inequality

$\displaystyle f(a)+f(b)-f(a+b)\geq g(b)$

holds for every $a\geq b\geq N_{1}$. On the other hand, ($2$) implies that the
inequality

$\displaystyle\frac{c_{2}(a+b)}{c_{1}(a)}\leq h(b)$

is valid for any $a\geq b\geq N_{2}$. Hence, it is enough to show that the
following

$\displaystyle e^{g(b)}\geq h(b)/c_{1}(b)$

holds for all large values of $b$ — but that is a direct consequence of ($3$),
as required. ∎

Although the above theorem is very easy we show that it is strong enough to be
applied to classical examples of partition functions.

######  Example 2.2.

Let us consider the partition function $p(n)$. It follows from the Bessenrodt
and Ono’s paper [5] that the inequalities

$\displaystyle\frac{\sqrt{3}}{12n}\left(1-\frac{1}{\sqrt{n}}\right)e^{\frac{\pi}{6}\sqrt{24n-1}}<p(n)<\frac{\sqrt{3}}{12n}\left(1+\frac{1}{\sqrt{n}}\right)e^{\frac{\pi}{6}\sqrt{24n-1}}$

hold for every positive integer $n$. Therefore, we will use Theorem 2.1 with
$F(n):=p(n)$,

$\displaystyle f(n):=\frac{\pi}{6}\sqrt{24n-1},\hskip
8.5359ptc_{1}(n):=\frac{\sqrt{3}}{12n}\left(1-\frac{1}{\sqrt{n}}\right)\hskip
8.5359pt\text{and}\hskip
8.5359ptc_{2}(n):=\frac{\sqrt{3}}{12n}\left(1+\frac{1}{\sqrt{n}}\right).$

Now, if we assume that $a\geq b\geq 1$, then we have that

$\displaystyle f(a)+f(b)-f(a+b)$
$\displaystyle=\frac{\pi}{6}\cdot\frac{2\sqrt{24a-1}\sqrt{24b-1}-1}{\sqrt{24a-1}+\sqrt{24b-1}+\sqrt{24(a+b)-1}}$
$\displaystyle>\frac{\pi}{6}\cdot\frac{\sqrt{24a-1}\sqrt{24b-1}}{\sqrt{24a-1}+\sqrt{24b-1}}-\frac{1}{24}$
$\displaystyle\geq\frac{\pi}{12}\sqrt{24b-1}-\frac{1}{24}=:g(b)$

On the other hand, one can derive that

$\displaystyle\frac{c_{2}(a+b)}{c_{1}(a)}$
$\displaystyle=\frac{12a}{12(a+b)}\left(1+\frac{1}{\sqrt{a+b}}\right)\left(1+\frac{1}{\sqrt{a}-1}\right)<1+\frac{2}{\sqrt{a}-1}\leq
2=:h(b)$

whenever $a\geq 9$ and $b\leq a$. Hence, it is enough to observe that

$\displaystyle e^{g(n)}\geq 8\sqrt{3}n\left(1+\frac{1}{\sqrt{n}-1}\right)$

is true for all sufficiently large values of $n$. In fact, one can show that
the above is valid for every $n\geq 15$. Thus, Theorem 2.1 implies that the
Bessenrodt-Ono inequality

$\displaystyle p(a)p(b)>p(a+b)$

holds for all $a\geq b\geq 15$.

######  Example 2.3.

Let us recall that the number $pp(n)$ of plane partitions of $n$ can be
computed via the generating function (obtained by MacMahon [21])

$\sum_{n=0}^{\infty}pp(n)x^{n}=\prod_{n=1}^{\infty}\frac{1}{(1-x^{n})^{n}}.$

From Wright’s formula [30] one can deduce the existence of
$\alpha,\beta,\gamma,N>0$ such that for all $n>N$ the following inequalities

$\displaystyle\alpha
n^{-\frac{25}{36}}\left(1-\frac{\beta}{\sqrt{n}}\right)e^{\gamma
n^{2/3}}<pp(n)<\alpha
n^{-\frac{25}{36}}\left(1+\frac{\beta}{\sqrt{n}}\right)e^{\gamma n^{2/3}}$

hold. We set $F(n):=pp(n)$,

$\displaystyle f(n):=\gamma n^{2/3},\hskip 8.5359ptc_{1}(n):=\alpha
n^{-\frac{25}{36}}\left(1-\frac{\beta}{\sqrt{n}}\right)\hskip
8.5359pt\text{and}\hskip 8.5359ptc_{2}(n):=\alpha
n^{-\frac{25}{36}}\left(1+\frac{\beta}{\sqrt{n}}\right),$

and apply Theorem 2.1. For every $a\geq b\geq N$, we have that

$\displaystyle f(a)+f(b)-f(a+b)$
$\displaystyle=\gamma\left(a^{\frac{2}{3}}+b^{\frac{2}{3}}-a^{\frac{2}{3}}\left(1+\frac{b}{a}\right)^{\frac{2}{3}}\right)$
$\displaystyle\geq\gamma\left(a^{\frac{2}{3}}+b^{\frac{2}{3}}-a^{\frac{2}{3}}\left(1+\frac{2b}{3a}\right)\right)=\gamma\left(b^{\frac{2}{3}}-\frac{2b}{3a^{1/3}}\right)\geq\frac{\gamma}{3}b^{\frac{2}{3}},$

where the first inequality is a consequence of Bernoulli’s inequality. On the
other hand, it is not difficult to see that

$\displaystyle\frac{c_{2}(a+b)}{c_{1}(a)}=\left(1+\frac{b}{a}\right)^{-\frac{25}{36}}\left(1+\frac{\beta}{\sqrt{a+b}}\right)\left(1+\frac{\beta}{\sqrt{a}-\beta}\right)<1+\frac{2\beta}{\sqrt{a}-\beta}\leq
2\beta+1$

whenever $a\geq b\geq\max\\{N,(\beta+1)^{2}\\}$. Now, it is straightforward to
deduce that

$\displaystyle
e^{\frac{\gamma}{3}n^{\frac{2}{3}}}\geq\frac{2\beta+1}{\alpha}n^{\frac{25}{36}}\left(1+\frac{\beta}{\sqrt{n}-\beta}\right)$

is satisfied for all but finitely many values of $n$. In conclusion, we get
that $pp(n)$ fulfills the Bessenrodt-Ono type inequality for all large
parameters $a$ and $b$.

##  3\. Log-concavity property holds for a class of sub-exponential functions

Let us recall that a sequence $(F(n))_{n\in\mathbb{N}}$ (or just a function
$F:\;\mathbb{N}\rightarrow\mathbb{R}_{+}$) is log-concave if

$F(n)^{2}>F(n-1)F(n+1)$

for all $n\geq n_{0}$, where $n_{0}$ is some positive integer.

In this section we get a general result which under mild conditions on the
growth of the function $F:\mathbb{N}\to\mathbb{R}_{+}$ guarantees that $F$ is
log-concave.

######  Theorem 3.1.

Let $c_{1},c_{2},f,F:\mathbb{N}\to\mathbb{R}_{+}$ be given and such that the
inequalities

$\displaystyle c_{1}(n)e^{f(n)}<F(n)<c_{2}(n)e^{f(n)}$

are valid for all positive integers $n\geq N_{0}$ for some
$N_{0}\in\mathbb{N}$. Suppose further that the following conditions hold:

  1. (1)

$\exists N_{1}\in\mathbb{N}_{+}\text{ }\exists
h:\mathbb{N}_{+}\to\mathbb{R}_{+}\text{ }\forall n\geq N_{1}:h(n)\leq
2f(n)-f(n-1)-f(n+1)$;

  2. (2)

$\exists N_{2}\in\mathbb{N}_{+}\text{ }\forall n\geq
N_{2}:c_{2}(n+1)c_{2}(n-1)/c_{1}^{2}(n)\leq e^{h(n)}$.

Then, for all the values of $n\geq\max\\{N_{0},N_{1},N_{2}\\}$ the inequality

$\displaystyle F^{2}(n)>F(n-1)F(n+1)$

is true.

###### Proof.

Let $c_{1},c_{2},f,F,h$ be as in the statement of the theorem. It follows that
both of the inequalities

$\displaystyle F^{2}(n)>c_{1}^{2}(n)e^{2f(n)}$

and

$\displaystyle F(n-1)F(n+1)<c_{2}(n-1)c_{2}(n+1)e^{f(n-1)+f(n+1)}$

are valid for every $n\geq N_{0}$. Therefore, it is enough to show that the
inequality

$e^{h(n)}\geq\frac{c_{2}(n-1)c_{2}(n+1)}{c_{1}^{2}(n)}$

is true for all sufficiently large values of $n$, but this is exactly $(2)$.
Hence, we conclude that the inequality

$F^{2}(n)>F(n-1)F(n+1)$

is satisfied for all $n\geq\max\\{N_{0},N_{1},N_{2}\\}$, as required. ∎

######  Example 3.2.

For every positive integer $n\geq 37$, we have that

(3.1) $\displaystyle c_{1}(n)e^{\mu(n)}<p(n)<c_{2}(n)e^{\mu(n)},$

where

$\displaystyle\mu(n)$ $\displaystyle:=\frac{\pi}{6}\sqrt{24n-1},$
$\displaystyle c_{1}(n)$
$\displaystyle:=\frac{\sqrt{12}}{24n-1}\left(1-\frac{1}{\mu(n)}-\frac{1}{\mu^{3}(n)}\right),$
$\displaystyle c_{2}(n)$
$\displaystyle:=\frac{\sqrt{12}}{24n-1}\left(1-\frac{1}{\mu(n)}+\frac{1}{\mu^{3}(n)}\right).$

The inequalities (3.1) follow directly from Chen, Jia and Wang [7, Lemma 2.2]
and some numerical computations carried out in Wolfram Mathematica [29].

Now, we want to apply Theorem 3.1 to deduce the log-concavity for $p(n)$. At
first, let us observe that the generalized binomial theorem asserts that for
every $|j|<|n|$ the following inequalities

(3.2) $\displaystyle t_{-j}(n)<\sqrt{n+j}<t_{+j}(n)$

are true, where

$\displaystyle
t_{-j}(n)=n^{\frac{1}{2}}+\frac{1}{2}jn^{-\frac{1}{2}}-\frac{1}{8}j^{2}n^{-\frac{3}{2}}-2\cdot|j|^{3}n^{-\frac{5}{2}}$

and

$\displaystyle
t_{+j}(n)=n^{\frac{1}{2}}+\frac{1}{2}jn^{-\frac{1}{2}}-\frac{1}{8}j^{2}n^{-\frac{3}{2}}+2\cdot|j|^{3}n^{-\frac{5}{2}}.$

Therefore, we have that

$\displaystyle 2\mu(n)-\mu(n-1)-\mu(n+1)$
$\displaystyle=\frac{\sqrt{24}\pi}{6}\left(2\sqrt{n-\frac{1}{24}}-\sqrt{n-\frac{25}{24}}-\sqrt{n+\frac{23}{24}}\right)$
$\displaystyle\geq\frac{\sqrt{24}\pi}{6}\left(\frac{1}{4}n^{-\frac{3}{2}}-\frac{55588}{13824}n^{-\frac{5}{2}}\right).$

On the other hand, one can also calculate that

$\displaystyle\frac{c_{2}(n-1)c_{2}(n+1)}{c_{1}^{2}(n)}$
$\displaystyle=\left(1+\frac{24^{2}}{(24n-1)^{2}-24^{2}}\right)^{\frac{5}{2}}$
$\displaystyle\times\frac{\left(\mu^{3}(n-1)-\mu^{2}(n-1)+1\right)\left(\mu^{3}(n+1)-\mu^{2}(n+1)+1\right)}{\left(\mu^{3}(n)-\mu^{2}(n)-1\right)^{2}}$
$\displaystyle\leq\left(1+\frac{24^{2}}{(24n-1)^{2}-24^{2}}\right)^{\frac{5}{2}}\times\left(1+\frac{24\sqrt{6}}{7\pi^{3}}n^{-\frac{3}{2}}\right),$

where the last inequality is a consequence of (3.2) and some elementary but
tiresome computations. Hence, it is enough to check under what conditions on
$n$ the inequality

$\displaystyle
1+\frac{\sqrt{24}\pi}{6}\left(\frac{1}{4}n^{-\frac{3}{2}}-\frac{55588}{13824}n^{-\frac{5}{2}}\right)\geq\left(1+\frac{24^{2}}{(24n-1)^{2}-24^{2}}\right)^{\frac{5}{2}}\times\left(1+\frac{24\sqrt{6}}{7\pi^{3}}n^{-\frac{3}{2}}\right)$

is true. One can verify that it holds for all $n\geq 94$. Therefore, if we
examine the positivity of $p^{2}(n)-p(n-1)p(n+1)$ for every $1\leq n\leq 93$,
then we obtain [11, Theorem 1.1].

##  4\. Log-concavity (usually) implies Bessendrodt-Ono type inequality

In this section we show a strict connection between log-concavity property and
the Bessenrodt-Ono type inequality.

######  Theorem 4.1.

Let $f:\mathbb{N}_{+}\to\mathbb{R}_{+}$ be fixed. Assume further that there
exists $n_{0}\in\mathbb{N}_{+}$ such that $(f(n))_{n\geq n_{0}}$ is log-
concave and

$\displaystyle\limsup_{n\to\infty}\frac{f(n+n_{0})}{f(n)}<f(n_{0}).$

Then, the inequality

$\displaystyle f(a)f(b)>f(a+b)$

holds for all sufficiently large numbers $a,b\in\mathbb{N}_{+}$.

###### Proof.

At first, let us observe that the log-concavity property implies that the
inequality

(4.1) $\displaystyle\frac{f(n)}{f(n-1)}>\frac{f(n+1)}{f(n)}$

is valid for each $n\geq n_{0}$. Thus, it is enough to check the validity of
the inequality in

$\displaystyle f(a+b)$
$\displaystyle=\frac{f(a+b)}{f(a+b-1)}\times\cdots\times\frac{f(a+n_{0}+1)}{f(a+n_{0})}f(a+n_{0})$
$\displaystyle<f(a)\frac{f(b)}{f(b-1)}\times\cdots\times\frac{f(n_{0}+1)}{f(n_{0})}f(n_{0})=f(a)f(b)$

for all sufficiently large $a$ and $b$. However, (4.1) points out that

$\displaystyle\frac{f(n_{0}+i+1)}{f(n_{0}+i)}>\frac{f(a+n_{0}+i+1)}{f(a+n_{0}+i)}$

is true for any $i=0,1,\ldots,b-n_{0}-1$. Therefore, the task boils down to
proving that the inequality

$\displaystyle\frac{f(a+n_{0})}{f(a)}<f(n_{0})$

is true for all large parameters $a$ and $b$, but it is a direct consequence
of the leftover assumption from the statement. This ends the proof. ∎

One can easily notice that besides Theorem 4.1, the above proof implies the
following property.

######  Proposition 4.2.

Let $f:\mathbb{N}_{+}\to\mathbb{R}_{+}$ be fixed. Assume further that there
exists $n_{0}\in\mathbb{N}_{+}$ such that $(f(n))_{n\geq n_{0}}$ is log-
concave and the inequality

$\displaystyle f(n)f(n_{0})>f(n+n_{0})$

is valid for every $n\geq n_{0}$. Then, we have that

$\displaystyle f(a)f(b)>f(a+b)$

is satisfied for all $a,b\geq n_{0}$.

######  Example 4.3.

Suppose that we know that the classical partition function satisfies

$\displaystyle p(26)p(a)>p(a+26)$

for every $a\geq 26$. Then, from [11, Theorem 1.1] and Proposition 4.2 we get
that the inequality

$\displaystyle p(a)p(b)>p(a+b)$

holds for all $a,b\geq 26$.

######  Theorem 4.4.

Let $f:\mathbb{N}\to\mathbb{R}_{+}$ be fixed and such that $f(0)\geq 1$.
Suppose further that the sequence $\left(f(n)\right)_{n=0}^{\infty}$ is log-
concave for every positive integer $n$. Then

$\displaystyle f(a)f(b)>f(a+b)$

is true for all positive integers $a$ and $b$.

###### Proof.

The log-concavity property and the assumption that $f(0)\geq 1$ imply that

$\displaystyle f(a+b)$
$\displaystyle=\frac{f(a+b)}{f(a+b-1)}\times\frac{f(a+b-1)}{f(a+b-2)}\times\cdots\times\frac{f(a+1)}{f(a)}f(a)$
$\displaystyle<f(a)\frac{f(b)}{f(b-1)}\times\frac{f(b-1)}{f(b-2)}\times\cdots\times\frac{f(1)}{f(0)}f(0)=f(a)f(b)$

holds for all positive integers $a$ and $b$. This completes the proof. ∎

######  Example 4.5.

Since the sequence $(p(n))_{n\geq 26}$ is log-concave by [11, Theorem 1.1] and
we have $p(26)=2436>1$, Theorem 4.4 implies that the sequence
$(p(n+26))_{n\in\mathbb{N}}$ satisfies the Bessenrodt-Ono type inequality.

######  Example 4.6.

Let us set $q(n):=F_{2n}$, where $F_{j}$ denotes the $j$-th Fibonacci number.
From Cassini identity we deduce that

$q^{2}(n)-q(n+1)q(n-1)=F_{2n}^{2}-F_{2n-2}F_{2n+2}=1>0.$

Thus, the sequence $(q(n))_{n\in\mathbb{N}}$ is log-concave. However, it does
not satisfy the Bessenrodt-Ono type inequality. Indeed, one can easily check
that $\phi^{2n}/\sqrt{5}-1<F_{2n}<\phi^{2n}/\sqrt{5}$, where
$\phi=(1+\sqrt{5})/2$ is the golden mean. Thus, for $a+b\geq 3$ we have

$q(a+b)-q(a)q(b)=F_{2(a+b)}-F_{2a}F_{2b}>\frac{\sqrt{5}-1}{5}\phi^{2(a+b)}-1>0.$

As a consequence, despite the fact that the sequence $(q(n))_{n\in\mathbb{N}}$
is log-concave, it does not satisfy the Bessenrodt-Ono type inequality. In
conclusion, we see that the assumption $f(0)\geq 1$ in Proposition 4.4 is
necessary.

On the other hand, let us note that for any fixed $j\in\mathbb{N}_{+}$ the
sequence $(q(n+j))_{n\in\mathbb{N}}$ satisfies the Bessenrodt-Ono type
inequality.

Theorem 4.1 asserts that a positive log-concave sequence usually satisfies the
Bessenrodt-Ono inequality. Therefore, there appears a natural question whether
the inverse statement is true. In general, it is not the case as the following
example shows.

######  Example 4.7.

For an arbitrary positive integer $m\geq 2$, the $m$-ary partition function
$b_{m}(n)$ is just $p_{A}(n)$ for $A=\\{m^{i}:\;i\in\mathbb{N}\\}$, i.e.,
$b_{m}(n)$ is the number of representations of $n$ as sums of powers of $m$.
Let us recall that $b_{m}(0)=1,b_{m}(mn+i)=b_{m}(mn)$ for $i=1,\ldots,m-1$ and
$b_{m}(mn)=b_{m}(mn-1)+b_{m}(n)$.

The sequence $(b_{m}(n))_{n\in\mathbb{N}}$ is not log-concave. Indeed, if we
have that $n\equiv-1\mkern 4.0mu({\operator@font mod}\mkern 6.0mum)$, then
$b_{m}(n-1)=b_{m}(n)<b_{m}(n+1)$.

On the other hand, the following is true.

######  Theorem 4.8.

The sequence $(b_{m}(n))_{n\in\mathbb{N}}$ satisfies the Bessenrodt-Ono type
inequality for every $m\geq 2$.

###### Proof.

In order to get the statement we apply Mahler’s theorem (see [10, 22, 27]),
which states that

$\displaystyle\log b_{m}(n)\sim\frac{\left(\log n\right)^{2}}{2\log m}.$

In particular, it means that there are some functions
$c_{1},c_{2}:\mathbb{N}_{+}\to\mathbb{R}$ such that

$\displaystyle\lim_{n\to\infty}c_{1}(n)=\lim_{n\to\infty}c_{2}(n)=1,$

and the inequalities

$\displaystyle c_{1}(n)e^{\frac{\left(\log n\right)^{2}}{2\log
m}}<b_{m}(n)<c_{2}(n)e^{\frac{\left(\log n\right)^{2}}{2\log m}}$

hold for all $n\geq N$ for some positive integer $N$.

To get the result we apply Theorem 2.1. Let us put $f(n):=(\log n)^{2}/(2\log
m)$. Without loss of generality we assume that $a\geq b$. Then we set
$a=\lambda b$ for some $\lambda\geq 1$. We have that

$\displaystyle\log^{2}a+\log^{2}b-\log^{2}(a+b)$
$\displaystyle=\log^{2}(\lambda b)+\log^{2}b-\log^{2}((\lambda+1)b)$
$\displaystyle=\log^{2}b+2(\log
b)\log\frac{\lambda}{\lambda+1}+\log^{2}\lambda-\log^{2}(\lambda+1)$
$\displaystyle>\left(\log b-2\log 2\right)\log b-\frac{3}{2},$

where the last inequality follows from elementary calculations. Therefore, one
can put

$\displaystyle g(n):=\frac{1}{2\log m}\left(\left(\log n-2\log 2\right)\log
n-\frac{3}{2}\right).$

To find out an appropriate function $h(n)$ from Theorem 2.1, we observe that
for some $M\in\mathbb{N}$ and all $n\geq M$, the inequalities

$\displaystyle c_{1}(n)\geq\frac{1}{2}\hskip 5.69046pt\text{and}\hskip
5.69046ptc_{2}(n)\leq 2$

are true. Hence, if $a\geq b\geq M$, then

$\displaystyle\frac{c_{2}(a+b)}{c_{1}(a)}\leq 4.$

Now, it is clear that the following

$\displaystyle\frac{1}{2\log m}\left(\left(\log n-2\log 2\right)\log
n-\frac{3}{2}\right)\geq 3\log 2\geq\log 4-\log c_{1}(n)$

is valid for all but finitely many values of $n$. In consequence, Theorem 2.1
guarantees that the sequence $(b_{m}(n))_{n\in\mathbb{N}}$ satisfies the
Bessenrodt-Ono type inequality, as required. ∎

## Acknowledgments

The first author would like to thank Bernhard Heim for fruitful discussion and
valuable suggestions. Moreover, he is also supported by a grant from the
Faculty of Mathematics and Computer Science under the Strategic Program
Excellence Initiative at the Jagiellonian University in Kraków.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-LHCb-DP-2019-004

Real-time discrimination of photon pairs using machine learning at the LHC

Sean Benson1, Adrián Casais Vidal2, Xabier Cid Vidal2, Albert Puig Navarro3

1Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
2Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
3Physik-Institut, Universität Zürich, Zürich, Switzerland

ALP–mediated decays and other as-yet unobserved $B$ decays to di-photon final
states are a challenge to select in hadron collider environments due to the
large backgrounds that come directly from the $pp$ collision. We present the
strategy implemented by the LHCb experiment in 2018 to efficiently select such
photon pairs. A fast neural network topology, implemented in the LHCb real-
time selection framework achieves high efficiency across a mass range of
4–20$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. We discuss implications and future
prospects for the LHCb experiment.

Submitted to SciPost Physics

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

##  1 Introduction

The $\gamma\gamma$ final state is interesting for a variety of reasons. On one
hand, the decay of a ${B}^{0}_{s}$ or a ${B}^{0}$ meson to two photons remains
unobserved and is described by an annihilation topology. It is sensitive to
contributions from physics beyond the Standard Model of Particle Physics (SM)
including a fourth generation [1], an extended Higgs sector [2] and SUSY [3].
Previous measurements by the Belle and BaBar collaborations have set limits of
${\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow\gamma\gamma$})<3.1\times
10^{-6}$ at $90\,\%$ confidence level (CL) [4] and
${\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow\gamma\gamma$})<3.3\times
10^{-7}$ at $90\,\%$ CL [5], which are significantly above the SM predictions
of
${\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow\gamma\gamma$})\sim(2-37)\times
10^{-7}$ and
${\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow\gamma\gamma$})\sim(1-10)\times
10^{-8}$ [6].

Undiscovered particles known as Axion-Like Particles (ALPs) could also be
accessed through this final state. ALPs are pseudo Nambu-Goldstone bosons,
associated to spontaneously broken approximate symmetries, which appear in
several models and can solve many of the SM problems [7]. Probing very small
couplings of ALPs to the SM sets indirect constraints on the New Physics (NP)
scale. For the type of model addressed in Ref. [7], ALPs couple to gluons
(which allows their production at the LHC) or photons (which can be used for
their detection) in the SM sector. The mass of ALPs can be arbitrarily below
the NP scale. In particular, for ALPs with a mass in the range between $5$ and
$10\,$$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, the LHCb experiment, described
in Ref. [8], has unique sensitivity for their discovery provided they can be
selected by its trigger algorithms [7]. For masses below 5
$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, LHCb may have sensitivity through
other decay channels, as described in Ref. [9].

The maximum rate at which events can be read out of the LHCb detector is
imposed by the front-end electronics and corresponds to $\sim 1$ MHz. In order
to determine which events are kept, hardware triggers based on field-
programmable gate arrays are used with a fixed latency of
$4\,\,\upmu\text{s}$. Information from the ECAL, HCAL, and muon stations is
used in separate hardware-level (L0) triggers. As explained in Ref. [10], the
strategy at L0 for the photon pairs under study here is similar to that of
other radiative decays and relies on the Photon or Electron channels, based on
inputs from the ECAL. All events selected by L0 are transferred to the High
Level Trigger (HLT). The HLT is a software application, executed on an event
filter farm, that is implemented in the same Gaudi framework [11] as the
software used for the offline reconstruction. It consists of two levels: an
initial selection of high energy and/or displaced single- or double-particle
signatures (HLT1) and a second level (HLT2), in which full offline
reconstruction is available, allowing for more complex searches to be
performed [12].

The study of these purely neutral modes at LHCb is challenging, but the use of
$\gamma\rightarrow e^{+}e^{-}$ conversions in the detector material, which
happens for around $25\,\%$ of photons, provides additional information to
reduce the background levels. With offline selections already in place since
Run 1, this paper describes the trigger strategy adopted in Run 2, where a set
of trigger selections were introduced to select the $\gamma\gamma$ signature
for the case of zero, one, and two photon conversions, labelled as _0CV_ ,
_1CV_ and _2CV_ , respectively. An additional label _LL_ and _DD_ is used to
distinguish photon conversions reconstructed as:

  * •

Long tracks (_LL_): when all possible tracking information from every tracking
station is available, implying that the parent particle decayed within about 1
metre of the $pp$ interaction point

  * •

Downstream tracks (_DD_): using only information from tracking stations
different to the vertex locator, implying that the parent particle decayed
after this.

The work presented in this paper builds on the strategy first introduced in
2015 to select only the $B^{0}_{({s})}$ decay [10], in which a selection was
put in place for _0CV_ , _1CV LL_ , _1CV DD_ , and _2CV_ (which includes both
_LL_ and _DD_ combinations). The new approaches to select candidates in a
wider mass range are described for the _0CV_ , _1CV LL_ , and _1CV DD_
topologies. The _2CV_ selection remains as is described in Ref. [10]. Section
2 describes the method by which our signal decays are simulated, Section 3
describes the HLT1 strategy. Section 4 describes the HLT2 strategy,
performance, and corresponding implementation. The prospects for the current
dataset collected by LHCb in addition to that expected to be collected by the
upgraded LHCb detector (during Run 3 of the LHC and beyond) are discussed in
Section 5.

##  2 Simulating the signal ${B}^{0}_{s}$ and ALP decays

In order to simulate ${{B}^{0}_{s}}\rightarrow\gamma\gamma$ decays, $pp$
collisions are generated using Pythia [13, *Sjostrand:2007gs] with a specific
LHCb configuration [15]. Decays of hadronic particles are described by EvtGen
[16], in which final-state radiation is generated using Photos [17]. The
interaction of the generated particles with the detector, and its response,
are implemented using the Geant4 toolkit [18, *Agostinelli:2002hh] as
described in Ref. [20]. This allows accurate simulation of the different
topologies of the decay, as photons may interact with the detector itself and
subsequently decay to electron-positron pairs.

For the ALP signal, samples are generated using MadGraph v2.6 [21], with
parameters taken from the ALP model described in Ref. [22]. The hadronization
is performed by Pythia, and the rest of the simulation steps are identical to
simulating the ${{B}^{0}_{s}}\rightarrow\gamma\gamma$ decay. Three ALP masses
are simulated: 5$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
10$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, and
15$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$.

After the detector response has been simulated, the trigger reconstruction and
associated selection requirements are simulated using data taking conditions
similar to those of the 2017 LHCb running period, with mean interactions per
bunch crossing of 1.3 and center-of-mass-energy of 13 TeV.

##  3 HLT1 strategy

The first trigger software level is required to reduce the input rate by a
factor 10 from the output of the hardware trigger level. In order to achieve
this, a search is made primarily for high transverse momentum tracks or tracks
with a high impact parameter with respect to the primary vertex. A detailed
description of the HLT1 selections for 2018 can be found in Ref. [10]: in the
case of photon conversions with at least one electron reconstructed as a long
track, a generic inclusive single-track, MVA-based trigger selection is used
[12]; in all other cases, the HLT1 strategy relies on a custom reconstruction
approach using information from the calorimeter collected in the hardware
trigger stage. This latter reconstruction technique, imposed by the limited
time available for decisions in HLT1, applies simple approximations to
calculate the mass of the photon pair with a minimum $E_{\mathrm{T}}$
requirement in a negligible time using only their energies as calculated in
L0.

Two selections with a different set of requirements are used: a first one,
referred to as HLT1($B$), focused on the selection of $B$ decays, and a second
one, not included in Ref. [10], with stricter $E_{\mathrm{T}}$ requirements
and with a wider mass range, which extends the reach to ALP masses above the
$B$ mass window (HLT1(ALP)). Their requirements are given in Table 1; it is
worth noting that the $E_{\mathrm{T}}$ requirements of the HLT1(ALP) are very
close to the saturation of the LHCb ECAL when using L0 energy
clusters,111Calorimeter clusters in L0 are made up of $2\times 2$ calorimeter
cells. As a consequence, the mass that can be reconstructed at L0 level is
limited by the saturation of these 4 cells of the cluster. This limitation
will not be present in the final analysis, as the offline reconstruction
allows to build larger clusters. and therefore the true mass reach is higher
than the requirement given in the table. The efficiency of the two selections
relative to all candidates passed by the L0 hardware trigger is given in Table
2.

Table 1: Selection applied in the Hlt1B2GammaGamma and Hlt1B2GammaGammaHighMass HLT1 trigger selection. Energies and masses given here are computed with $2\times 2$ cell clusters.  Requirement | HLT1($B$) | HLT1(ALP)
---|---|---
$E_{\mathrm{T}}(\gamma)~{}[\\!\text{\,Ge\kern-1.00006ptV}]$ | $>3.5$ | $>5$
$E_{\mathrm{T}}(\gamma_{1})+E_{\mathrm{T}}(\gamma_{2})~{}[\\!\text{\,Ge\kern-1.00006ptV}]$ | $>8$ | $>11$
$M(\gamma_{1}\gamma_{2})~{}[\\!\text{\,Ge\kern-1.00006ptV\\!/}c^{2}]$ | $[3.5,6.0]$ | $[6.0,11.0]$
$p_{\mathrm{T}}(\gamma_{1}\gamma_{2})~{}[\\!\text{\,Ge\kern-1.00006ptV\\!/}c]$ | $>2$ | $>5$
Table 2: Percentage efficiency relative to all candidates accepted by the Photon and Electron channels of the L0 hardware trigger for the ${B}^{0}_{s}$ and ALP samples, combining all the $\gamma$ reconstruction modes.  Efficiency (%) | HLT1($B$) | HLT1(ALP)
---|---|---
${{B}^{0}_{s}}\rightarrow\gamma\gamma$ | $3.3\pm 0.2$ | -
ALP 5 GeV | $6.2\pm 0.5$ | $0.6\pm 0.2$
ALP 10 GeV | $5.3\pm 0.3$ | $6.7\pm 0.4$
ALP 15 GeV | $3.8\pm 0.3$ | $10.5\pm 0.5$

##  4 HLT2 strategy

The second trigger software level performs a more complete event
reconstruction. In HLT2, over 400 multibody decay signatures are searched for
in parallel, with candidate selection information equivalent in quality to
that used by analysts offline.

###  4.1 Training sample preparation

The first step in designing an HLT2 strategy is to collect representative
samples of signal and background in order to train the neural network (NN)
classifiers. For the case of the signal, i.e. ${B}^{0}_{s}$ and ALP decays,
simulated data is used, which is generated as described in Sec. 2. In order to
describe the background, proton-proton collisions collected by the LHCb
collaboration during 2017 are used, in which the high level trigger selected
candidates randomly.

Since the intention is to implement a NN inside the trigger software, and in
order to generate the largest possible NN training sample, a set of loose
requirements are applied to the samples mentioned above. These are inspired by
those applied in the analysis of other radiative decays (e.g. those in
Refs.[23, 24, 25]). In summary,

  * •

photons reconstructed as calorimeter energy clusters are required to have an
energy above $6\,\text{\,Ge\kern-1.00006ptV}$ and a transverse energy with
respect of the beam direction above $3\,\text{\,Ge\kern-1.00006ptV}$;

  * •

photons reconstructed as electron pairs are required to have a transverse
momentum in excess of $2\,\text{\,Ge\kern-1.00006ptV}$, have a mass below
$60\,\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and be displaced with respect to
the $pp$ collision;

  * •

the sum of transverse momentum of the two photons must be in excess of $6.5$,
$5.5$ and $5\,\text{\,Ge\kern-1.00006ptV\\!/}c$ for the _0CV_ , _1CV_ and
_2CV_ cases, respectively; and

  * •

diphoton candidates are required to have a combined transverse momentum above
$3\,\text{\,Ge\kern-1.00006ptV\\!/}c$ and, in the case of _2CV_ , to form a
good vertex.

In the training of the NN, a random subset of the
${{B}^{0}_{s}}\rightarrow\gamma\gamma$ sample is taken, such that efficiencies
of the ALP signal (for the 3 different masses under consideration) and
${B}^{0}_{s}$ decay remain similar. The yields of the samples are provided in
Table 3.

Table 3: Sample sizes for the signal decays and background after reconstruction and trigger requirements. Sample | _1CV LL_ | _1CV DD_ | _0CV_
---|---|---|---
${{B}^{0}_{s}}\gamma\gamma$ | 9940 | 17368 | 36844
ALPs | 229 | 420 | 233
Background | 228 | 393 | 457

###  4.2 Multi-layer perceptron training and implementation

A multilayer perceptron (MLP) was chosen for the NN updates. A Scikit-learn
[26] implementation was preferred due to the relative simplicity of the
topology that allows for quick evaluation in real-time environments in
association with the NNDrone package [27]. Due to the small training dataset,
a LBFGS solver method [28], which shows better results under these conditions,
was used. To avoid overtraining a small ($\mathcal{O}(10^{-5}$))
regularization parameter was chosen. A simple hidden layer structure was
implemented: three neurons in the first and two in the latter. The small
dimension was primarily set to avoid large time complexity. Finally, for
simplicity, the logistic activation function, which is also used in the output
layers, was chosen for the hidden layers.

The model is created as

⬇

1from sklearn.neural_network import MLPClassifier

2classifier = MLPClassifier(activation=’logistic’

3 , alpha=1e-05, batch_size=’auto’,

4 beta_1=0.9, beta_2=0.999, early_stopping=False,

5 epsilon=1e-08, hidden_layer_sizes=(3, 2),

6 learning_rate=’constant’, learning_rate_init=0.001,

7 max_iter=200, momentum=0.9,

8 nesterovs_momentum=True, power_t=0.5, random_state=1,

9 shuffle=True, solver=’lbfgs’, tol=0.0001,

10 validation_fraction=0.1, verbose=False, warm_start=False)

In order to train the MLP for each topology, different variables are found to
have different separation powers. We select those providing significant
discrimination between signal and background. The following variables are used
as a feature for one or more of the NN models:

  * •

The transverse momentum of the parent ${B}^{0}_{s}$ or ALP candidate (X
$p_{\mathrm{T}}$).

  * •

The impact parameter significance of the ${B}^{0}_{s}$ or ALP candidate with
respect to the best primary vertex (X IP $\chi^{2}$), defined as the
difference in $\chi^{2}$ of a given PV reconstructed with and without the
considered particle.

  * •

The invariant mass of the electron-positron combination in a photon
conversion.

  * •

The probability that the photon candidate is not a ${\pi}^{0}$ meson based on
a combination of calorimeter information [29] ($\gamma$ prob).

  * •

Asymmetry of the $p_{\mathrm{T}}$ of the two photon candidates
($p_{\mathrm{T}}$ asym).

  * •

The ratio of the candidate ECAL energy deposit between the $2\times 2$ and
$3\times 3$ clusters ($\gamma$ Calo E49).

  * •

The output of a multi-variate classifier [29] trained using various inputs
corresponding to calorimeter shape variables ($\gamma$ shower shape).

The signal and background distributions for each topology are given in
Appendix A. It should be remarked that, as mentioned above, our classifiers
use as inputs the outputs of two other different classifiers, one designed
generically to identify calorimeter photons ($\gamma$ shower shape) and one to
differentiate photons and ${\pi}^{0}$ mesons ($\gamma$ prob). While, ideally,
one could integrate the features used in these classifiers into our own one,
making use of the ability of algorithms such as Deep Neural Networks as
feature extractors, it was decided against this given the high level of
complexity in the training of both $\gamma$ shower shape and $\gamma$ prob.
The first accounts for the expected deposit shapes in different regions of the
calorimeter and excludes the possibility that these are originated by a
charged track. The second also relies on the energy value and the calorimeter
zone comparing the expected energy deposit of a photon with respect to that of
a ${\pi}^{0}$. Although both classifiers were trained to maximize its
sensitivity for $B$ meson decays, potential extensions of this work could
involve integrating all the features into a single classifier to be trained
specifically with our signal samples.

The samples are split into training and test data, where half of the data are
used for training and the other half used for testing. Optimizations of this
splitting, such as the use of techniques like k-folding [30], will be explored
in future versions of this algorithm to maximize the size of the training and
test samples. The output distributions of the trained models for both the test
and training samples are given in Fig. 1. Good agreement can be seen in all
training and test comparisons, meaning the models show few signs of
overtraining. To quantify this, p-values obtained with the Kolmogorv-Smirnov
test [31] can be found in Table 4.

Figure 1: Output probability of the classifier for the different topologies. Upper left: _0CV_ , upper right: _1CV DD_ , lower _1CV LL_ Topology | Signal p-value | Background p-value
---|---|---
_0CV_ | 0.25 | 0.63
_1CV DD_ | 0.99 | 0.99
_1CV LL_ | 0.48 | 0.37
Table 4: Kolmogorov-Smirnov p-values for the comparison of the classifier’s
distribution between the test and training samples for the different
topologies

###  4.3 Performance

The performance of each of the models shown in terms of ROC curves is provided
in Appendix B. Ultimately, the efficiency of the models depends on the chosen
working point. This is driven by resource requirements. The chosen working
point of each of the models is shown in Table 5, along with the rejections and
efficiencies per sample, per decay topology.

Table 5: Percentage efficiency for the ${B}^{0}_{s}$ and ALP samples relative to the reconstructed and loosely selected samples. |  | _1CV LL_ | _1CV DD_ | _0CV_
---|---|---|---|---
Efficiency (%) per channel | ${{B}^{0}_{s}}\rightarrow\gamma\gamma$ | 67 | 31 | 68
ALP 5 GeV | 52 | 67 | 71
ALP 10 GeV | 55 | 50 | 50
ALP 15 GeV | 62 | 64 | 46
Rejection | 85 | 90 | 85
MLP requirement | $>0.70$ | $>0.85$ | $>0.80$

###  4.4 Implementation in the real-time software stack

In order to apply the neural networks in the C++ software stack used to
perform event selection in real time, a conversion must take place so that the
models trained in the Python implementation are reproduced.

In order to do this, the NNDrone framework [27] is used to convert the network
weights and bias parameters to JSON format. This allows reducing the
processing time by an order of magnitude while keeping the same classifier
performance. The weights and parameters are then read in by a dedicated tool
which uses the JSON parameters to initialise a LWTNN network [32] that
reproduces the original model exactly. The reconstructed LWTNN model has been
validated to produce identical outputs from the same input values.

##  5 Prospects

As explained in Section 1, the trigger strategy described in this paper is
essential to improve the sensitivity of LHCb to $\gamma\gamma$ final states,
especially after 2018. In this section, we describe briefly the prospects for
the two benchmark analyses studied: the search for the rare decay
$B^{0}_{({s})}\rightarrow\gamma\gamma$ and that of an ALP decaying to a pair
of photons. It should be noted that the same final state could also be
sensitive to other models, such as those including a composite Higgs together
with light scalars [33].

During the full LHCb Run 2 data-taking, selections providing high efficiencies
for the reconstruction of $B^{0}_{({s})}\rightarrow\gamma\gamma$ decays were
included (see Tab. 2 and Tab. 5). In the same regard, the efficiencies for an
ALP with a mass close to that of the $B_{({s})}$ meson is also very high for
that period. For the case of an ALP with a mass between $\sim 6$
$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ and $\sim 12$
$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, only data collected in 2018 with the
strategy described in this document provides significant sensitivity. A
similar strategy to that used in 2018 is expected to be used during Run 3 of
the LHC, in which LHCb will have been upgraded [34]. Concerning the trigger,
the new design [35] including the removal of the first hardware level, should
provide similar or higher efficiencies.

The efficiencies reported in this document, together with the fraction of
triggered data provided in Ref. [10], are used to roughly estimate the
expected sensitivity in both analyses. A $\gamma\gamma$ invariant mass
resolution of $\sim 2.5$% is assumed for these studies. Additional offline
discrimination against the background can be achieved by using similar but
more powerful classifiers than those available in HLT2. This additional
discrimination is based on the use of larger training samples and variables
whose reconstruction is too slow to be performed in real time. Additional
background rejections of $\sim 90\,\%$ can then be achieved with associated
signal efficiencies of $\sim 60\,\%$ for all the photon reconstruction
categories included in this document.

For the ${{B}^{0}_{s}}\\!\rightarrow\gamma\gamma$ decay an upper limit
${\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow\gamma\gamma$})\lesssim
10^{-5}$ at 90 % CL could be achieved using the Run 2 LHCb dataset. This is
around two times the Belle limit, currently the most stringent. Assuming
similar efficiencies and backgrounds in Run 3 of the LHC, a simple projection
yields
${\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow\gamma\gamma$})\lesssim
4\times 10^{-6}$. This assumption might not hold if the ECAL performance is
affected by the larger occupancy expected in Run 3. If a more optimistic
background discrimination is assumed (95% background rejection for the same
signal efficiency) upper limits of
${\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow\gamma\gamma$})\lesssim
6\times 10^{-6}$ and
${\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow\gamma\gamma$})\lesssim
2\times 10^{-6}$ could be achieved using the Run 2 and Run 3 LHCb datasets,
respectively. A discovery, should the SM prediction hold, would probably need
to wait until Run 4 or a potential Phase-II LHCb upgrade [36].

Concerning a search for ALPs, estimations agree with those of Ref.[7]. For the
reasons explained previously, the sensitivity with the current dataset is more
limited for an ALP with a mass in the $\sim 6-12$
$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ range. In the most sensitive region,
decay constants below $\sim 0.3\text{\,Te\kern-1.00006ptV}$ could be excluded
using the LHCb Run 2 dataset. Keeping the same efficiency as in 2018 for Run 3
would provide an increase to $\sim 0.4\text{\,Te\kern-1.00006ptV}$ for the
$\sim 4-12\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ mass range. No other
experiment is expected to contribute to the measurement in this mass range in
near future.

##  6 Summary

Di-photon selections were first implemented in the LHCb trigger in 2015
focusing on the ${B}^{0}_{s}$ decay. In this paper, we have detailed the
selection modifications required to expand the search region to allow
sensitivity to undiscovered ALPs in the 2018 trigger. In order to do this and
remain within resource budgets, neural network models have been introduced
using the NNDrone framework to ensure fast evaluation. This is the first time
such models have been used to directly select multibody candidates in real
time.

## Acknowledgements

We would like to thank the LHCb simulation team for their extensive support
and for generating the simulated samples used in this work. We also thank the
LHCb RTA team for supporting this publication and reviewing this work. We
express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).

##  Appendix A Neural network feature distributions

The signal and background distributions for each topology are shown in Figs.
2, 3, and 4, respectively, where “signal” refers to the combination of all
signal modes (i.e., ${{B}^{0}_{s}}\rightarrow\gamma\gamma$ and the ALP at the
three mass points of reference).

Figure 2: Signal and background distributions of information used to train the
_1CV LL_ classifier. Variables are explained in the text.

Figure 3: Signal and background distributions of information used to train the
_1CV DD_ classifier. Variables are explained in the text.

Figure 4: Signal and background distributions of information used to train the
_0CV_ classifier. Variables are explained in the text.

##  Appendix B ROC curve performances of the NN models

The performance of each of the models is shown in terms of ROC curves in Fig.
5, which display the signal efficiency versus background rejection power.

Figure 5: ROC curves for the test data using the different topologies. 0CV NN
(left) , 1CV DD NN (center),1CV LL NN (right) .

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