The remaining terms can be simplified into the following expression

$\begin{split}\hbar^{2}&\partial_{t}Z(t)=\int_{X}\mu_{X}\;\;2Ve^{-\frac{1}{\hbar}S_{t}}\;[2tQ^{2}(V)+t^{2}Q(\partial_{a}V\partial_{b}V\Pi^{ab})]\\\
&+\int_{X}\mu_{X}\;\;Ve^{-\frac{1}{\hbar}S_{t}}\;(-1)^{|\phi^{a}|}\partial_{b}V\Pi^{ab}(-2t\partial_{a}S-4t^{2}\partial_{a}(Q^{c}\partial_{c}V)-2t^{3}\partial_{a}(\partial_{c}V\partial_{d}V\Pi^{cd})).\end{split}$
(5.14)

Let us carefully look at terms at different orders in $t$ in equation (5.14).
The linear order simplifies into

$\begin{split}2t\int_{X}\mu_{X}\;\;Ve^{-\frac{1}{\hbar}S_{t}}[2Q^{2}(V)-(-1)^{|\phi^{a}|}\partial_{b}V\Pi^{ab}\partial_{a}S]=0.\end{split}$
(5.15)

An expression (5.15) vanishes due to the on-shell refinement (3.69) of the on-
shell supersymmetry algebra for the vector field $Q$.

The quadratic order in $t$-expansion can be further simplified

$\begin{split}&2t^{2}\int_{X}\mu_{X}\;\;Ve^{-\frac{1}{\hbar}S_{t}}\;[Q(\partial_{a}V\partial_{b}V\Pi^{ab})-2(-1)^{|\phi^{a}|}\partial_{b}V\Pi^{ab}\partial_{a}(Q^{c}\partial_{c}V)]\\\
&=2t^{2}\int_{X}\mu_{X}\;\;Ve^{-\frac{1}{\hbar}S_{t}}\;[(-1)^{|\phi^{a}|+|\phi^{b}|}\partial_{a}V\partial_{b}VQ^{c}\partial_{c}\Pi^{ab}-2(-1)^{|\phi^{a}|}\partial_{b}V\Pi^{ab}\partial_{a}Q^{c}\partial_{c}V]=0.\end{split}$
(5.16)

The last equality in (5.16) follows from the equality (3.72).

The cubic term in the $t$-expansion can be further simplified

$\begin{split}-2t^{3}&\int_{X}\mu_{X}\;\;e^{-\frac{1}{\hbar}S_{t}}\;(-1)^{|\phi^{a}|}V\partial_{b}V\Pi^{ab}\partial_{a}(\partial_{c}V\partial_{d}V\Pi^{cd})\\\
&=4t^{3}\int_{X}\mu_{X}\;\;e^{-\frac{1}{\hbar}S_{t}}\;(-1)^{|\phi^{c}||\phi^{a}|+|\phi^{c}|}V\partial_{a}\partial_{c}V\partial_{b}V\Pi^{ab}\partial_{d}V\Pi^{cd}\\\
&\qquad-2t^{3}\int_{X}\mu_{X}\;\;e^{-\frac{1}{\hbar}S_{t}}\;(-1)^{|\phi^{a}|}\partial_{c}V\partial_{d}V\cdot
V\partial_{b}V\Pi^{ab}\partial_{a}\Pi^{cd}=0.\end{split}$ (5.17)

The term in the second line with two derivatives vanishes due to the symmetry
of the sign factors. In particular, we can change the labels
$(a\leftrightarrow c)$ and rearrange the expression to get the same expression
with a minus sign

$\begin{split}&(-1)^{|\phi^{c}||\phi^{a}|+|\phi^{c}|}V\partial_{a}\partial_{c}V\partial_{b}V\Pi^{ab}\partial_{d}V\Pi^{cd}=(-1)^{|\phi^{c}||\phi^{a}|+|\phi^{a}|}V\partial_{c}\partial_{a}V\partial_{d}V\Pi^{cd}\partial_{b}V\Pi^{ab}\\\
&\qquad\qquad=-(-1)^{|\phi^{c}||\phi^{a}|+|\phi^{c}|}V\partial_{a}\partial_{c}V\partial_{b}V\Pi^{ab}\partial_{d}V\Pi^{cd}.\end{split}$
(5.18)

The last term in (5.17) vanishes due to the (3.73) relation for bivector. We
showed that all orders in $t$-expansion of (5.14) vanish, hence the proof of
the theorem is complete. ∎

Comparing two proofs we observe that the BV formalism is way more efficient in
proving localization-like formulas! However, the existence of the second proof
allows us to formulate and proof the new localization theorem without any
reference to the BV formalism. Given a refined form (3.69) of the on-shell
supersymmetry algebra and as long as the bivector satisfies the definition
3.31 the corresponding quantum physical system is subject to the localization
theorem 5.2!

###  5.3 Localization in a theory of superpotential

Let us recast an example 2.13 of supersymmetric deformation using the BV
localization formalism. For a BV description of the off-shell theory with
superpotential (4.4) we set ${\varepsilon}=1$, $\bar{{\varepsilon}}=0$ and
choose Lagrangian submanifold given by the odd function $V_{H}=-t\psi H(x)$.
Note that the function is identical to the function (2.49). Using symplectic
form (4.3) the corresponding Lagrangian submanifold is given by

$\psi^{\ast}=\frac{\partial
V_{H}}{\partial\psi}=-tH(x),\;\;\;x^{\ast}=\frac{\partial V_{H}}{\partial
x}=-tH^{\prime}(x)\psi.$ (5.19)

The BV action restricted to the Lagrangian sub-manifold (5.19)

$\mathcal{S}\big{|}_{{\cal
L}_{H}}=-\frac{1}{4}F^{2}+F(W(x)+tH(x))+\psi\bar{\psi}(W^{\prime}(x)+tH^{\prime}(x)).$
(5.20)

The BV action is the calssical action for the off-shell theory with deformed
superpotential $W+tH$, identical to the deformation (2.50).

We can use the same Lagrangian submanifold (5.19) to evaluate the restriction
of the BV action (4.12) for the on-shell theory of superpotential

$\mathcal{S}_{W}|_{\mathcal{L}_{H}}=\psi\bar{\psi}W^{\prime}(x)+(W+tH)^{2}+tH^{\prime}(x)\psi\bar{\psi}=S_{W+tH}.$
(5.21)

The result of restriction is identical to the classical action for the on-
shell theory with deformed superpotential $W+tH$. Moreover the deformation is
identical to the off-shell case.

We can use the deformation invariance of the partition function and a limit
$t\to\infty$ to localize the partition function to zeroes of $H$! In
particular, for the on-shell supersymmetric system with superpotential
$W=x+\frac{1}{2}gx^{3}$ we can choose $H=x$ to remove the $g$-dependence of
partition function.

Note that there is a slight restriction on choice of function $H$: functions
$H(x)$ and $W(x)$ should have the same behavior at $x\to\pm\infty$. An extra
restriction is due to non-compactness of the body for integration
supermanifold in partition function, which is a real line in our example.

###  5.4 Localization for the quadratic deformation model

In case of quadratic deformation model (2.60) we can evaluate partition
function perturbatively using BV action to the particular order in coupling
$g$. The BV action (4.24) up to quadratic corrections in $g$ is

$\begin{split}\mathcal{S}&=x^{2}+gx^{4}+\psi\bar{\psi}(1+2gx^{2})+({\varepsilon}\bar{\psi}+\bar{{\varepsilon}}\psi)x^{\ast}+(2x+4gx^{3})(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})\\\
&\qquad+(1-2gx^{2})(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})^{2}-\frac{4gx}{3}(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})^{3}-\frac{g}{3}(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})^{4}+\mathcal{O}(g^{2}).\end{split}$
(5.22)

Let us choose $\bar{{\varepsilon}}=0,\;\;{\varepsilon}=1$ and the linear
Lagrangian submanifold

$\psi^{\ast}=\frac{\partial V}{\partial\psi}=-tx,\;\;\;x^{\ast}=\frac{\partial
V}{\partial x}=-t\psi.$ (5.23)

The restriction of the BV action to the Lagrangian submanifold

$\begin{split}\mathcal{S}\Big{|}_{{\cal
L}}&=x^{2}+gx^{4}+\psi\bar{\psi}(1+2gx^{2})+t\psi\bar{\psi}+2x(tx)\\\
&\qquad+(1-2gx^{2})(tx)^{2}-\frac{4gx}{3}(tx)^{3}-\frac{g}{3}(tx)^{4}+\mathcal{O}(g^{2}).\end{split}$

The simplified version

$\begin{split}\mathcal{S}\Big{|}_{{\cal
L}}&=x^{2}(1+t)^{2}+(1+t)\psi\bar{\psi}+\frac{4}{3}(1+t)gx^{4}+2gx^{2}\psi\bar{\psi}-\frac{1}{3}g(t+1)^{4}x^{4}+\mathcal{O}(g^{2}).\end{split}$
(5.24)

We can perform the measure-preserving variable change

$x\to
x(1+t),\;\;\psi\to\psi(1+t),\;\;\bar{\psi}\to\bar{\psi},\;\;\;\mu_{X}=dxd\psi
d\bar{\psi}\to\mu_{X},$ (5.25)

so the classical action simplifies into

$\begin{split}\mathcal{S}\Big{|}_{{\cal
L}}&=x^{2}+\psi\bar{\psi}-\frac{1}{3}gx^{4}+\frac{4}{3(1+t)^{3}}gx^{4}+\frac{2g}{(1+t)^{3}}x^{2}\psi\bar{\psi}+\mathcal{O}(g^{2}).\end{split}$
(5.26)

The leading order $g$-correction to the partition function can be expressed
using the super-Gaussian averages (2.65) in the form

$Z_{t}=\int_{\mathcal{L}}\sqrt{\mu}\;\;e^{-\mathcal{S}\Big{|}_{{\cal
L}}}=\left\langle
e^{\frac{1}{3}gx^{4}-\frac{4}{3(1+t)^{3}}gx^{4}-\frac{2g}{(1+t)^{3}}x^{2}\psi\bar{\psi}+\mathcal{O}(g^{2})}\right\rangle.$
(5.27)

The Gaussian averages for the leading order correction

$\begin{split}\partial_{g}\Big{|}_{g=0}Z_{t}&=\left(\frac{1}{3}-\frac{4}{3(1+t)^{3}}\right)\langle
x^{4}\rangle-\frac{2}{(1+t)^{3}}\langle\psi\bar{\psi}x^{2}\rangle\\\
&=\frac{3}{4}\left(\frac{1}{3}-\frac{4}{3(1+t)^{3}}\right)+\frac{1}{2}\cdot\frac{2}{(1+t)^{3}}=\frac{1}{4}.\\\
\end{split}$ (5.28)

The large $t$-limit of the BV action on a Lagrangian submanifold simplifies to

$\begin{split}\lim_{t\to\infty}\mathcal{S}\Big{|}_{{\cal
L}}&=x^{2}+\psi\bar{\psi}-\frac{1}{3}gx^{4}+\mathcal{O}(g^{2})\end{split}$
(5.29)

while the partition function matches with the perturbative computation (2.68)

$\lim_{t\to\infty}Z_{t}=\left\langle
e^{\frac{1}{3}gx^{4}+\mathcal{O}(g^{2})}\right\rangle=1+\frac{1}{3}g\langle
x^{4}\rangle+\mathcal{O}(g^{2})=\left(1+\frac{1}{4}g+\mathcal{O}(g^{2})\right).$
(5.30)

For more general function $V=-tH(x)\psi$, the Larangian submanifold is

$\psi^{\ast}=-tH(x),\;\;\;x^{\ast}=-tH^{\prime}(x)\psi.$ (5.31)

The BV action becomes

$\begin{split}\mathcal{S}\Big{|}_{{\cal
L}_{H}}&=x^{2}+gx^{4}+\psi\bar{\psi}(1+2gx^{2})+2txH(x)+tH^{\prime}(x)\psi\bar{\psi}\\\
&\qquad+(1-2gx^{2})t^{2}H^{2}(x)-\frac{4}{3}gt^{3}xH^{3}(x)-\frac{1}{3}gH(x)^{4}t^{4}+\mathcal{O}(g^{2}).\end{split}$
(5.32)

We can rewrite the BV action

$\begin{split}\mathcal{S}\Big{|}_{{\cal
L}_{H}}&=(x+tH(x))^{2}+\psi\bar{\psi}(1+tH^{\prime}(x)+2gx^{2})+\frac{4}{3}gx^{3}(x+tH(x))-\frac{1}{3}g(x+H(x)t)^{4}+\mathcal{O}(g^{2}).\end{split}$

Using the change of variables $y=x+tH(x)$, $\psi\to(1+tH^{\prime}(x))\psi$,
which preserves the integration measure for partition function
$\mu_{X}=dxd\psi d\bar{\psi}$ we can rewrite

$\begin{split}\mathcal{S}\Big{|}_{{\cal
L}_{H}}&=y^{2}+\psi\bar{\psi}+\frac{2gx^{2}}{x+tH(x)}\psi\bar{\psi}+\frac{4g}{3}x^{3}y-\frac{1}{3}gy^{4}+\mathcal{O}(g^{2}).\end{split}$

In the limit $t\to\infty$ the value of $x$ for fixed $y$ will go to zero and
the BV action simplifies to the same expression (5.29).

###  5.5 BV localization for the theories with bivector

######  Conjecture 5.3.

The partition function for the quantum physical system with on-shell
supersymmetry of bivector type (definition 3.31) simplifies to the Gaussian
integral.

In this section we present the supporting evidence for this conjecture. For
the case a single supermultiplet the BV action for the on-shell supersymmetric
system of bivector type takes the following schematic form

$\mathcal{S}=S(x,\psi)+{\varepsilon}\bar{\psi}x^{\ast}-{\varepsilon}f(x)\psi^{\ast}+{\varepsilon}^{2}\Pi(x)\psi^{\ast}\psi^{\ast}.$
(5.33)

The definition of an on-shell model with BV bivector implies

$[Q,\Pi\\}=0\Longrightarrow\bar{\psi}\partial_{x}\Pi=0,$ (5.34)

hence $\Pi(x)=\Pi$ is constant! Using the linear Lagrangian submanifold, with
generator $V=-t\psi x$, for the BV integral gives as the BV action

$\mathcal{S}\Big{|}_{\mathcal{L}_{V}}=S(x,\psi)+t\psi\bar{\psi}+txf(x)+t^{2}\Pi
x^{2}.$ (5.35)

The BV integral for the partition function in the limit $t\to\infty$ becomes
the Gaussian integral

$Z=\lim_{t\to\infty}\int_{\mathbb{R}^{2|1}}dxd\psi
d\bar{\psi}\;\;e^{-S(x,\psi)-t\psi\bar{\psi}-txf(x)-t^{2}\Pi x^{2}}=\int
dy\;e^{-\Pi y^{2}}=\frac{\sqrt{\pi}}{\sqrt{\Pi}}.$ (5.36)

Alternatively, we can use $V_{f}=-t\psi f(x)$ to ensure that the
$\mathcal{O}(t)$ part of the integral is also Gaussian

$\mathcal{S}\Big{|}_{\mathcal{L}_{f}}=S(x,\psi)+tf^{\prime}(x)\bar{\psi}\psi+tf(x)^{2}+t^{2}\Pi
f(x)^{2}.$ (5.37)

The partition function

$\begin{split}Z&=\lim_{t\to\infty}\int_{\mathbb{R}^{2|1}}dxd\psi
d\bar{\psi}\;\;e^{-S(x,\psi)-tf^{\prime}(x)\bar{\psi}\psi-tf^{2}(x)-t^{2}\Pi
f^{2}(x)}\\\ &=\lim_{t\to\infty}t\int
dy\;e^{-(t+t^{2}\Pi)y^{2}}=\lim_{t\to\infty}\frac{\sqrt{\pi}}{\sqrt{\Pi}+t^{-1}}=\frac{\sqrt{\pi}}{\sqrt{\Pi}}\end{split}$
(5.38)

is identical to the case (5.36) with linear Lagrangian submanifold choice.

###  5.6 Localization considerations

For our refined classification from section let us formulate a conjecture(s)
about the supersymmetric partition function simplification.

  1. 1.

Linear. Conventional off-shell localization in BV formalism. The
supersymmetric partition function localizes to zeroes of the supersymmetric
vector field.

  2. 2.

Quadratic. We can choose a linear Lagrangian submanifold with a parameter to
make the antifields parametrically large, so that the BV integral will be
dominated by the Gaussian integral with bivector representing the
corresponding quadratic form. Surprisigly many of the on-shell supersymmetic
models from physics [5, 20] belong to this class.

  3. 3.

Polynomial. We can choose a linear Lagrangian submanifold with a parameter to
make the antifields parametrically large, so that the BV integral will be
dominated by polynomial polyvector terms. This might be useful for the theory
of polynomial maps and polynomial inverses in context of the quantum field
theory formulation for the Jacobian conjecture [21].

  4. 4.

Formal series. We can choose a linear Lagrangian submanifold with a parameter
to make the antifields parametrically large so that the BV integral will be
dominated by the polyvector terms. In section 6.4 we show that the
corresponding partition function is the same as the Nicolai transform for the
original supersymmetric sytem. The Nicolai map is often limited to the
quadratic actions in fermions with recent attempts to improve for the quartic
actions [22]. The BV constructions does not have such limitations, so it can
serve as a generalization of the Nicolai map construction to the non-quadratic
actions in fermions.

  5. 5.

Obstruction. No solution to the master equation, so we cannot use the BV
integration methods.

##  6 Supersymmetry for one-dimensional integrals

In this section we recast the one-dimensional integrals as supersymmetric
partition functions and study the corresponding BV actions.

###  6.1 Exponential integral

Let us consider a textbook exponential integral

$I_{h}=\int_{\mathbb{R}}dx\;e^{-\frac{1}{\hbar}h(x)}.$ (6.1)

We can evaluate this integral using the saddle point approximation as a formal
series in $\hbar$

$I_{h}=\hbar^{\frac{1}{2}}\sum_{x_{0}:h^{\prime}(x_{0})=0}e^{-\frac{1}{\hbar}V(x_{0})}\;\;\sqrt{\frac{2\pi}{h^{\prime\prime}(x_{0})}}(1+\mathcal{O}(\hbar)).$
(6.2)

We can introduce the odd variables $\psi,\bar{\psi}$ to turn an integral (6.1)
into a supersymmetric partition function

$I_{h}(\hbar)=\hbar\int_{\mathbb{R}^{1|2}}dxd\psi
d\bar{\psi}\;e^{-\frac{1}{\hbar}(h(x)+\psi\bar{\psi})}=\int_{X}\mu_{X}\;e^{-\frac{S}{\hbar}}=Z_{h}$
(6.3)

for a supresymmetric system on $X=\mathbb{R}^{1|2}$ with action

$S_{h}(x,\psi,\bar{\psi})=h(x)+\psi\bar{\psi},$ (6.4)

and integration measure

$\mu_{X}=dxd\psi d\bar{\psi}.$ (6.5)

The action (6.4) is invariant under the the odd vector fields

$Q=\bar{\psi}\frac{\partial}{\partial
x}-h^{\prime}\frac{\partial}{\partial\psi},\;\;\;\bar{Q}=\psi\frac{\partial}{\partial
x}+h^{\prime}\frac{\partial}{\partial\bar{\psi}}.$ (6.6)

Note that the odd vector fields (6.6) are similar to the vector fields for the
superpotential case (4.8). The vector fields (6.6) preserve the integration
measure, i.e.

$\hbox{div}_{\mu_{X}}Q=\hbox{div}_{\mu_{X}}\bar{Q}=0.$ (6.7)

######  Remark 6.1.

The zeroes of the vector field (6.6) are critical points of $h$ and match with
the saddle points of the integral (6.1).

The vector fields (6.6) form the $d=0\;\;\mathcal{N}=2$ on-shell supersymmetry
algebra, moreover the refined version of the on-shell condition is similar to
the superpotential case (4.11). In particular,

$\begin{split}[Q,Q\\}&=-2h^{\prime\prime}(x)\bar{\psi}\frac{\partial}{\partial\psi}=-2h^{\prime\prime}(x)\frac{\partial
S_{h}}{\partial\psi}\frac{\partial}{\partial\psi},\;\;\;[\bar{Q},\bar{Q}\\}=2h^{\prime\prime}(x)\psi\frac{\partial}{\partial\bar{\psi}}=-2h^{\prime\prime}(x)\frac{\partial
S_{h}}{\partial\bar{\psi}}\frac{\partial}{\partial\bar{\psi}},\\\
[\bar{Q},Q\\}&=h^{\prime\prime}(x)\left(\bar{\psi}\frac{\partial}{\partial\bar{\psi}}-\psi\frac{\partial}{\partial\psi}\right)=h^{\prime\prime}(x)\frac{\partial
S_{h}}{\partial\psi}\frac{\partial}{\partial\bar{\psi}}+h^{\prime\prime}(x)\frac{\partial
S_{h}}{\partial\bar{\psi}}\frac{\partial}{\partial\psi}.\end{split}$ (6.8)

Hence by proposition 3.30 we have an approximate solution classical master
equation solution

$\mathcal{S}_{h}=h(x)+\psi\bar{\psi}-{\varepsilon}h^{\prime}(x)\psi^{\ast}+\bar{{\varepsilon}}h^{\prime}(x)\bar{\psi}^{\ast}+(\bar{{\varepsilon}}\psi+{\varepsilon}\bar{\psi})x^{\ast}+\frac{1}{2}h^{\prime\prime}(x)(\psi^{\ast}{\varepsilon}-\bar{\psi}^{\ast}\bar{{\varepsilon}})^{2}.$
(6.9)

We can extend an approximate BV action (6.9) to the full solution of the
quantum master equation

$\mathcal{S}_{h}=h(x)+\psi\bar{\psi}-{\varepsilon}h^{\prime}(x)\psi^{\ast}+\bar{{\varepsilon}}h^{\prime}(x)\bar{\psi}^{\ast}+(\bar{{\varepsilon}}\psi+{\varepsilon}\bar{\psi})x^{\ast}+\sum_{k=2}^{\infty}\frac{h^{(k)}(x)}{k!}(\bar{\psi}^{\ast}\bar{{\varepsilon}}-\psi^{\ast}{\varepsilon})^{k}.$
(6.10)

We will present the detailed proof for the more general case in proposition
6.4. The sum over polyvectors is just a Taylor expansion for
$h(x+\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})$, hence we
can simplify the BV action into

$\mathcal{S}_{h}=h(x+\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})+\psi\bar{\psi}+(\bar{{\varepsilon}}\psi+{\varepsilon}\bar{\psi})x^{\ast}.$
(6.11)

The classical action (6.4) is a trivial solution to the quantum master
equation because it has trivial antifield dependence.

######  Proposition 6.2.

The BV action (6.11) is a canonical BV transformation of (6.4).

###### Proof.

Let us consider a diffeomorphism

$\Phi:\;\;x\mapsto
x+\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast},\;\;\psi\mapsto\psi-{\varepsilon}x^{\ast},\;\;\;\bar{\psi}\mapsto\bar{\psi}+\bar{{\varepsilon}}x^{\ast},$
(6.12)

which preserves the odd symplectic form, i.e.

$\begin{split}\Phi^{\ast}\omega&=d(x+\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})\wedge
dx^{\ast}-d(\psi-{\varepsilon}x^{\ast})\wedge
d\psi^{\ast}-d(\bar{\psi}+\bar{{\varepsilon}}x^{\ast})\wedge
d\bar{\psi}^{\ast}\\\ &=dx\wedge dx^{\ast}-d\psi\wedge
d\psi^{\ast}-d\bar{\psi}\wedge d\bar{\psi}^{\ast}.\end{split}$ (6.13)

The classical action (6.4) in the new variables becomes the BV action (6.11).
∎

######  Remark 6.3.

We are yet to understand the significance of the BV action being related to
the classical action by a BV canonical transformation. We suspect that there
might be potential implications for the BV localization, but an additional
investigation is required.

###  6.2 Saddle point integral

We can represent the generic saddle point integral

$\int_{\mathbb{R}}dx\;f(x)e^{-\frac{1}{\hbar}h(x)}=\hbar\int_{\mathbb{R}^{1|2}}dxd\psi
d\bar{\psi}\;e^{-\frac{1}{\hbar}\left(h(x)+f(x)\psi\bar{\psi}\right)}$ (6.14)

as an on-shell supersymmetric system on $X=\mathbb{R}^{1|2}$ with the action

$S=h(x)+f(x)\psi\bar{\psi}$ (6.15)

and measure $\mu_{X}=dxd\psi d\bar{\psi}$. The supersymmetry vector fields are

$Q=\bar{\psi}\frac{\partial}{\partial
x}-\frac{h^{\prime}(x)}{f(x)}\frac{\partial}{\partial\psi},\;\;\;\bar{Q}=\psi\frac{\partial}{\partial
x}+\frac{h^{\prime}(x)}{f(x)}\frac{\partial}{\partial\bar{\psi}}.$ (6.16)

The supersymmetry algebra closes on-shell and we can write down a refined
version, which is similar to the exponential case (6.8) with the different
overall factor. The overall factor can be restored from a single commutator,
for example

$[Q,Q\\}=-2\frac{h^{\prime\prime}(x)f(x)-h^{\prime}(x)f^{\prime}(x)}{f(x)^{2}}\bar{\psi}\frac{\partial}{\partial\psi}=-2\frac{h^{\prime\prime}(x)f(x)-h^{\prime}(x)f^{\prime}(x)}{f(x)^{3}}\frac{\partial
S}{\partial\psi}\frac{\partial}{\partial\psi}$ (6.17)

Hence the approximate solution to the classical master equation has bivector

$\Pi^{(2)}=\frac{1}{2}({\varepsilon}\psi^{\ast}-\bar{{\varepsilon}}\bar{\psi}^{\ast})^{2}\frac{1}{f}\frac{d}{dx}\frac{h^{\prime}(x)}{f(x)}.$
(6.18)

We can construct the higher order terms in antifields to arrive into all order
solution.

######  Proposition 6.4.

The solution to the quantum master equation to all orders in antifields is

$\mathcal{S}=f(x)\psi\bar{\psi}+(\bar{{\varepsilon}}\psi+{\varepsilon}\bar{\psi})x^{\ast}+\sum_{k=0}^{\infty}\frac{(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})^{k}}{k!}\left(\frac{1}{f}\frac{d}{dx}\right)^{k}h(x).$
(6.19)

###### Proof.

We will prove the proposition using induction in $k$. The classical action,
supersymmetry vector fields (6.16) and the BV bivector (6.18) represent the
$k=0,1,2$ terms in the sum. We assume that for all $k\leq n$ the polyvectors
are of the form

$\Pi^{(k)}=\frac{(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})^{k}}{k!}\left(\frac{1}{f}\frac{d}{dx}\right)^{k}h(x),$
(6.20)

The BV bracket between any pair of polyvectors in the form (6.20) is trivial,
i.e. $\\{\Pi^{(k)},\Pi^{(l)}\\}=0$ and $\Delta_{\mu}\Pi^{(k)}=0$. Hence the
equation (3.81) for the polyvector $\Pi^{(n+1)}$ becomes a linear problem

$D_{S}\Pi^{(n+1)}=-\\{\mathcal{Q},\Pi^{(n)}\\}=(\bar{{\varepsilon}}\psi+{\varepsilon}\bar{\psi})\frac{d}{dx}\Pi^{(n)}.$
(6.21)

The right hand side of the linear problem (6.21) is independent on $x^{\ast}$
hence we can use the homotopy (3.144) to evaluate

$\begin{split}\Pi^{(n+1)}&=-K\\{\mathcal{Q},\Pi^{(n)}\\}=\frac{(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})^{n+1}}{(n+1)!}\left(\frac{1}{f}\frac{d}{dx}\right)^{n+1}h(x).\end{split}$
(6.22)

Our result for $\Pi^{(n+1)}$ matches with the assumptions of induction, hence
the proof is complete. ∎

###  6.3 Supersymmetric systems from polynomial class

Proposition 6.4 describes the BV action for the most general on-shell
supersymmetric system with a single $d=0$ $\mathcal{N}=2$ on-shell
supermultiplet. The structure of the antifield dependence for the BV action is
determined by pair of function $f(x)$ and $h(x)$. For a specific choice we can
generate BV action with polynomial dependence on antifields.

######  Lemma 6.5.

For a degree $n$ polynomial $P_{n}$ and functions $f(x)=W^{\prime}(x)$,
$h(x)=P_{n}(W(x))$ the BV action (6.19) is a polynomial of degree $n$ in
antifields.

###### Proof.

Proposition 6.4 gives us an explicit form of the polyvectors, while the the BV
action for $f(x)=W^{\prime}(x)$ and $h(x)=P_{n}(W(x))$ simplifies into

$\mathcal{S}=W^{\prime}(x)\psi\bar{\psi}+(\bar{{\varepsilon}}\psi+{\varepsilon}\bar{\psi})x^{\ast}+P_{n}(W(x)+\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast}).$
(6.23)

Since $P_{n}(x)$ is a polynomial of degree $n$, then the BV action is also a
polynomial of degree $n$ in antifields. ∎

The classical action

$S=P_{n}(W(x))+W^{\prime}(x)\psi\bar{\psi}$ (6.24)

is a generalization of the on-shell superpotential system from section 2.4.
Note that for our choice of $f(x)$ and $h(x)$ the supersymmetry vector fields
(6.16) do not have poles. In particular

$Q=\bar{\psi}\frac{\partial}{\partial
x}-P_{n}^{\prime}(W(x))\frac{\partial}{\partial\psi},\;\;\;\bar{Q}=\psi\frac{\partial}{\partial
x}+P_{n}^{\prime}(W(x))\frac{\partial}{\partial\bar{\psi}}.$ (6.25)

The partition function for the supersymmetric model (6.24) simplifies, using
the change of variables $y=W(x)$, i.e.

$Z=\frac{1}{\hbar}\int
dx\;W^{\prime}\;e^{-\frac{1}{\hbar}P_{n}(W(x))}=\frac{1}{\hbar}\int
dy\;e^{-\frac{1}{\hbar}P_{n}(y)}.$ (6.26)

Note that the partition function depends only on the relative homology class
for the curve defined by the function $W(x)$. Also, similarly to the
superpotential case, it is invariant under the deformations of the function
$W(x)$.

The partition function (6.26) emerges in the $t\to\infty$ limit of the BV
localization for the Lagrangian submanifold generated by the linear function
$V=-tx\psi$.

$Z=\lim_{t\to\infty}\int_{{\cal
L}_{V}}\sqrt{\mu}\;\;e^{-\frac{1}{\hbar}\mathcal{S}}.$ (6.27)

###  6.4 Nicolai map

Let us adopt the Nicolai map construction [23, 24] to the case of $d=0$
supersymmetric theory with a single supermultiplet. The partition function for
the most general action can be simplified by integrating out the fermionic
(odd) varibles

$Z=\int_{\mathbb{R}^{1|2}}dxd\psi
d\bar{\psi}\;\;e^{-\frac{1}{\hbar}(h(x)+f(x)\psi\bar{\psi})}=\frac{1}{\hbar}\int_{\mathbb{R}}dx\;f(x)\;e^{-\frac{1}{\hbar}h(x)}.$
(6.28)

We can treat $f(x)$ as a Jacobian for the coordinate transformation to the new
variable

$y=\int^{x}f(x^{\prime})dx^{\prime}$ (6.29)

and express the partition function in the form

$Z=\frac{1}{\hbar}\int_{\mathbb{R}}dy\;\;e^{-\frac{1}{\hbar}\tilde{h}(y)},\;\;\;\tilde{h}(y)=h(x(y)).$
(6.30)

The partition function (6.30) is an integral over just even variables with the
standard measure $dy$ for a classical theory with action $\tilde{h}(y)$.
However, an explicit form for the action $\tilde{h}(y)$ requires us to invert
the coordinate transform (6.29). In our discussion for BV localization we
assumed $f(x)=W^{\prime}(x)$, hence the coordinate transform simplifies to
$y=W(x)$ while the Nicolai map becomes $\tilde{h}(W(x))=h(x)$.

The Nicolai map is usually constructed perturbatively in coupling constant,
what can be illustrated by our example of quadratic deformation for the on-
shell theory.

######  Example 6.6.

Let us perform the Nicolai transform for the partition function (2.62). The
integration over fermions produces a factor $(1+2gx^{2})$ which we use as the
Jacobian of the coordinate transformation, i.e.

$dy=(1+2gx^{2})dx=d\left(x+\frac{2g}{3}x^{3}\right).$ (6.31)

The perturbative inverse to the leading order in $g$ is

$x=y-\frac{2g}{3}y^{3}+\mathcal{O}(g^{2}).$ (6.32)

The action for the purely even system

$\begin{split}\tilde{h}(y)&=h(x(y))=x^{2}+gx^{4}=y^{2}-\frac{1}{3}gy^{4}+\mathcal{O}(g^{2}).\end{split}$
(6.33)

The leading order partition function after the Nicolai transform matches with
result of the BV localization (5.30) for the same system using the limiting
Lagrangian submanifold generated by the linear function.

An example 6.6 is a consequence of a more general relation.

######  Proposition 6.7.

The BV integral for the quantum master equation solution (6.19) over the
linear Lagrangian submanifold generated by $V=-t\psi x$ at the $t\to\infty$
limit is identical to the Nicolai transform for partition function (6.30),
i.e.

$\int_{{\cal
L}_{V}}\sqrt{\mu}\;e^{-\frac{1}{\hbar}\mathcal{S}}=\frac{1}{\hbar}\int_{\mathbb{R}}dy\;\;e^{-\frac{1}{\hbar}\tilde{h}(y)}.$
(6.34)

###### Proof.

In section 6.1 we observed that the sum over $k$ in the solution to the
quantum master equation

$\mathcal{S}=f(x)\psi\bar{\psi}+x^{\ast}(\bar{{\varepsilon}}\psi+{\varepsilon}\bar{\psi})+\sum_{k=0}^{\infty}\frac{(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})^{k}}{k!}\left(\frac{1}{f}\frac{d}{dx}\right)^{k}h(x).$
(6.35)

when $f(x)=1$ becomes a Taylor expansion and leads to the simplification of
the BV action. For $f(x)\neq 1$ we can perform a similar summation if we
introduce a new function $W(x)$ such that $f(x)=W^{\prime}(x)$. The
derivatives simplify into

$\frac{1}{f}\frac{d}{dx}=\frac{1}{W^{\prime}(x)}\frac{d}{dx}=\frac{d}{dW}.$
(6.36)

The sum over $k$ is the exponential operator, describing the shift in
$W$-variable

$\sum_{k=0}^{\infty}\frac{(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})^{k}}{k!}\left(\frac{1}{f}\frac{d}{dx}\right)^{k}=e^{(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})\frac{d}{dW}}.$
(6.37)

Let us use the function $\tilde{h}(x)$ from the Nicolai map construction
(6.30), then the BV action can be written in the following form

$\mathcal{S}=W^{\prime}(x)\psi\bar{\psi}+(\bar{{\varepsilon}}\psi+{\varepsilon}\bar{\psi})x^{\ast}+\tilde{h}(W(x)+\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast}).$
(6.38)

We can perform the BV integral for ${\varepsilon}=1,\;\;\bar{{\varepsilon}}=0$
over the linear Lagrangian submanifold generated by

$V=-t\psi x.$ (6.39)

The restriction of BV action

$\mathcal{S}\Big{|}_{\mathcal{L}_{V}}=W^{\prime}(x)\psi\bar{\psi}+t\psi\bar{\psi}+\tilde{h}(W(x)+tx).$
(6.40)

The $t\to\infty$ limit of the BV integral

$Z=\int_{\mathbb{R}^{1|2}}dxd\psi
d\bar{\psi}\;e^{-\frac{1}{\hbar}(t\psi\bar{\psi}+\tilde{h}(W(x)+tx))}=\frac{1}{\hbar}\int_{\mathbb{R}}tdx\;e^{-\frac{1}{\hbar}\tilde{h}(W(x)+tx)}=\frac{1}{\hbar}\int_{\mathbb{R}}dy\;e^{-\frac{1}{\hbar}\tilde{h}(y)}.$
(6.41)

∎

The BV integral for $t=0$ equals to the original partition function for on-
shell supersymmetric theory, hence we have an equality

$Z=\int_{\mathbb{R}^{1|2}}dxd\psi
d\bar{\psi}\;\;e^{-\frac{1}{\hbar}(h(x)+f(x)\psi\bar{\psi})}=\frac{1}{\hbar}\int_{\mathbb{R}}dy\;e^{-\frac{1}{\hbar}\tilde{h}(y)}.$
(6.42)

An equality (6.42) is the the $d=0$ version of Nicolai map construction.

###  6.5 BV action from superspace

In section 6.3 we described a generalization (6.23) of the on-shell
superpotential model (4.12), with the corresponding BV action being polynomial
type. There is a further generalization of the quantum master equation
solution, parametrized by the pair of function $W$ and $G$.

######  Proposition 6.8.

For a pair of functions $W(x)$ and $G(x)$ there is a solution to the quantum
master equation

$\mathcal{S}=W^{\prime}(x)\psi\bar{\psi}+({\varepsilon}\bar{\psi}+\bar{{\varepsilon}}\psi)x^{\ast}+G(W(x)+\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast}).$
(6.43)

###### Proof.

The non-trivial part of BV brackets

$\begin{split}\\{\mathcal{S},\mathcal{S}\\}&=\\{W^{\prime}(x)\psi\bar{\psi}+({\varepsilon}\bar{\psi}+\bar{{\varepsilon}}\psi)x^{\ast},G(W(x)+\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})\\}\\\
&=\\{W^{\prime}(x)\psi\bar{\psi}+({\varepsilon}\bar{\psi}+\bar{{\varepsilon}}\psi)x^{\ast},W(x)+\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast}\\}G^{\prime}(W(x)+\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})=0.\end{split}$

The last equality follows from the individual BV brackets

$\begin{split}&\\{({\varepsilon}\bar{\psi}+\bar{{\varepsilon}}\psi)x^{\ast},W(x)\\}=-W^{\prime}(x)({\varepsilon}\bar{\psi}+\bar{{\varepsilon}}\psi),\\\
&\\{W^{\prime}(x)\psi\bar{\psi},\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast}\\}=W^{\prime}(x)({\varepsilon}\bar{\psi}+\bar{{\varepsilon}}\psi),\\\
&\\{({\varepsilon}\bar{\psi}+\bar{{\varepsilon}}\psi)x^{\ast},{\varepsilon}\psi^{\ast}-\bar{{\varepsilon}}\bar{\psi}^{\ast}\\}=({\varepsilon}\bar{{\varepsilon}}-\bar{{\varepsilon}}{\varepsilon})x^{\ast}=0.\end{split}$
(6.44)

The BV action (6.43) satisfies the $\Delta_{\mu}\mathcal{S}=0$ for the
standard Berezinian $\mu=dxdx^{\ast}d\psi
d\psi^{\ast}d\bar{\psi}d\bar{\psi}^{\ast}$ in rather trivial way due to
antifield derivatives of it being independent on the corresponding fields.

∎

In section 4.3 we observed that the BV action of quadratic type (4.12) for the
on-shell theory of superpotential is related to the BV action of the linear
type for the off-shell theory of superpotential (4.4) by the BV induction
(3.40). We can further generalize the BV induction to the family of solutions
(6.43) parametrized by the pair of functions $W$ and $G$.

The first step is to generalize the off-shell supersymmetric actions
constructed using the superfield formalism from section 2.2 by adding an
additional function $K$ for an auxiliary field $F$. In particular

$S=-\int d\theta
d\bar{\theta}\left(\hat{x}K(\mathfrak{\bar{D}}\mathfrak{D}\hat{x})+H(\hat{x})\right)=FK(F)+H^{\prime}(x)F+H^{\prime\prime}(x)\psi\bar{\psi}.$
(6.45)

######  Remark 6.9.

Note that the superspace action (6.45) is the $d=0$ superspace version of the
higher derivative theory.

We replace $H^{\prime}(x)=W(x)$ and use the proposition 3.25 to write down the
corresponding solution to the quantum master equation

$\mathcal{S}=FK(F)+W(x)F+W^{\prime}(x)\psi\bar{\psi}+({\varepsilon}\bar{\psi}+\bar{{\varepsilon}}\psi)x^{\ast}+F(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast}).$
(6.46)

######  Proposition 6.10.

The two families of solutions to quantum master equation (6.43) and (6.46) are
related by the BV induction (3.40), while the functions $G$ and $K$ are
related by the Laplace-like transform

$\mathcal{C}(\hbar)e^{-\frac{1}{\hbar}G(y,\hbar)}=\int
dF\;e^{-\frac{1}{\hbar}\left(FK(F)+yF\right)}.$ (6.47)

###### Proof.

The BV induction (3.40) for the Lagrangian submanifold $F^{\ast}=0$ gives us
an integral

$\int
dF\;\exp-\frac{1}{\hbar}\left(FK(F)+W(x)F+W^{\prime}(x)\psi\bar{\psi}+({\varepsilon}\bar{\psi}+\bar{{\varepsilon}}\psi)x^{\ast}+F(\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast})\right)$
(6.48)

The result of an integration gives us an induced BV action

$\mathcal{S}_{ind}=W^{\prime}(x)\psi\bar{\psi}+({\varepsilon}\bar{\psi}+\bar{{\varepsilon}}\psi)x^{\ast}+G(W(x)+\bar{{\varepsilon}}\bar{\psi}^{\ast}-{\varepsilon}\psi^{\ast}),$
(6.49)

where we introduced a function $G(y,\hbar)$ defined by

$\mathcal{C}(\hbar)\cdot e^{-\frac{1}{\hbar}G(y,\hbar)}=\int
dF\;e^{-\frac{1}{\hbar}\left(FK(F)+yF\right)}.$ (6.50)

Hence the proof is complete. ∎

The Laplace-like transform (6.47) for the linear function $K$ gives a
quadratic function $G$ which is $\hbar$-independent. While for the higher
degree polynomial function $F(K)$ the corresponding function $G(y,\hbar)$ is
not a polynomial function with a complicated $\hbar$-dependence. Similarly,
the polynomial BV actions ($G$ is polynomial function) even though can be
induced from the BV description of the off-shell supersymmetric system, but
the off-shell action for such system will have a very complicated auxiliary
field dependence, the function $K(F,\hbar)$ is not a polynomial and has a non-
trivial $\hbar$-dependence.

######  Remark 6.11.

In the classical ($\hbar\to 0$) limit the Laplace-like transform (6.47)
becomes the Legendre transform. The Legendre transform in general will give us
a non-trivial $\hbar$-dependence for the $G(y,\hbar)$. However in some cases
the Legendre transform can be 1-loop exact, i.e. the $G(y,\hbar)$ is at most
linear in $\hbar$. This phenomenon was studied and completely classified for
the case of one variable by Kontsevich and Odesskii [25].

## Acknowledgments

We are grateful to Yasha Neiman and Pavel Mnev for many discussions on the
topics presented in this paper. The work A.L. is supported by the Basic
Research Program of the National Research University Higher School of
Economics and by Shanghai Institute for Mathematics and Interdisciplinary
Sciences. The work of V.L. is supported by the Arnold Fellowship at the London
Institute for Mathematical Sciences.

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11institutetext: Massachusetts Institute of Technology, Cambridge MA, USA
22institutetext: NSF AI Institute for Artificial Intelligence and Fundamental
Interactions (IAIFI) 33institutetext: TU Dortmund University, Dortmund,
Germany 44institutetext: CERN, Meyrin, Switzerland 55institutetext: Meta AI
(FAIR)

# Applications of Lipschitz neural networks to the Run 3 LHCb trigger system

Blaise Delaney 1122<EMAIL_ADDRESS>Nicole Schulte 33 Gregory Ciezarek
44 Niklas Nolte 55 Mike Williams 1122 Johannes Albrecht 33

###### Abstract

The operating conditions defining the current data taking campaign at the
Large Hadron Collider, known as Run 3, present unparalleled challenges for the
real-time data acquisition workflow of the LHCb experiment at CERN. To address
the anticipated surge in luminosity and consequent event rate, the LHCb
experiment is transitioning to a fully software-based trigger system. This
evolution necessitated innovations in hardware configurations, software
paradigms, and algorithmic design. A significant advancement is the
integration of monotonic Lipschitz neural networks into the LHCb trigger
system. These deep learning models offer certified robustness against detector
instabilities, and the ability to encode domain-specific inductive biases.
Such properties are crucial for the inclusive heavy-flavour triggers and, most
notably, for the topological triggers designed to inclusively select
$b$-hadron candidates by exploiting the unique kinematic and decay topologies
of beauty decays. This paper describes the recent progress in integrating
Lipschitz neural networks into the topological triggers, highlighting the
resulting enhanced sensitivity to highly displaced multi-body candidates
produced within the LHCb acceptance.

##  1 Introduction

The LHCb detector LHCb:detector  located at the Large Hadron Collider at CERN
is a single-arm forward spectrometer instrumented to achieve acceptance in the
pseudorapidity range, $2<\eta<5$. The primary goal of the LHCb experiment is
the discovery of Beyond the Standard Model (BSM) physics through the analysis
of heavy-flavour processes, with particular focus placed on investigating
$b$-hadron decays. Since its inception, however, the LHCb physics programme
has grown substantially to include, among other endeavours, the search for
feebly interacting dark-portal candidates produced in the LHCb geometrical
acceptance LHCb:dark-photon-i ; LHCb:dark-photon-ii ; LHCb:lowmass_mumu ;
LHCb:dark-boson-i ; LHCb:dark-boson-ii .

Throughout its Run 3 data taking campaign, the LHCb experiment is tasked with
operating under unprecedented conditions, namely a nominal instantaneous
luminosity, $\mathcal{L}=2\times 10^{33}\leavevmode\nobreak\
\textrm{cm}^{-2}\textrm{s}^{-1}$, amounting to a five-fold increase on the
data taking conditions present in Runs 1 and 2. In order to cope with the
challenging detector occupancy of Run 3, the LHCb experiment has pioneered the
deployment of a redesigned, fully software-based trigger system for real-time
data acquisition. In this paradigm, the full detector readout and event
building is enacted at an incoming 30 MHz rate of visible proton-proton ($pp$)
bunch crossings. A two-staged trigger system is deployed to select events of
interest with a 100 kHz output rate suitable for storage.

The core objective of the LHCb trigger is therefore data-volume reduction
within the real-time data storage constraints. This process translates to a
reduction by a factor of approximately 400 of the input 4 TB/s bandwidth, at
nominal instantaneous luminosity, to achieve an output bandwidth of 10 GB/s
LHCb:UpgradeI . Figure 1 provides a schematic representation of the LHCb data
flow in Run 3. Following the full detector readout, the GPU-enabled first
trigger stage, the High Level Trigger 1 (HLT1), delivers partial event
reconstruction from charged-track information, resulting in a data-volume
reduction by a factor of approximately 20. Subsequently, the data is passed to
a buffer stage to achieve real-time alignment and calibration, thus enabling
the CPU-based High Level Trigger 2 (HLT2). This trigger stage operates
selection algorithms exploiting offline-quality, full-event reconstruction
observables.

To achieve the real-time rate reduction required by the Run 3 operating
conditions, the LHCb trigger exploits a combination of expert systems and
machine learning solutions. Lipschitz monotonic neural networks (NNs) find
optimal application in the latter, specifically in the inclusive algorithms
deployed to select heavy-flavour particles.

Figure 1: Schematic representation of the LHCb data flow during the Run 3 data
taking period. The incoming rate of 30 MHz of non-empty bunch crossings is
processed by the full detector readout and thereafter passed through the two-
tiered trigger system. The selected data is stored and further processed
offline for end-user analysis. Taken from LHCB-FIGURE-2020-016 .

##  2 Applications of Lipschitz neural networks to the LHCb trigger

Implementing trigger selections is a critical step in High Energy Physics data
acquisition, as discarded events are irreversibly lost. No margin for error is
therefore afforded to the event-selection protocols implemented in the trigger
system. Moreover, effective classification algorithms must be able to process
event-level information within the memory and compute limits imposed by real-
time data acquisition. The application of monotonic Lipschitz NNs to the LHCb
trigger meets such mission-critical requirements whilst delivering additional
desirable certified benefits: namely, robustness and interpretability.

In this work, robustness signifies mitigated sensitivity to both simulation
inaccuracies and detector instabilities during data taking. Such conditions
may be achieved by constraining the gradient of the function approximated by a
deep learning classifier with respect to the input features. The models
deployed in the LHCb experiment realise this result by bounding the Lipschitz
constant of the learnt decision-boundary function. This approach effectively
sets a strict upper limit on the classifier-response variation resulting from
experimental instabilities of limited magnitude. Robustness is thus essential
for the estimators deployed in the trigger system, serving to reduce
sensitivity to resolution and calibration effects conditioning the input
features in real time. Additionally, robustness simplifies the evaluation of
trigger-related systematic uncertainties in end-user physics measurements.

A complementary advantage is offered by enforcing a monotonic estimator
response with respect to a set of input features. In essence, this procedure
expresses domain-specific inductive bias. Deploying provably monotonic
networks in the trigger system enhances the selective retention of interesting
outlier candidates absent from training samples and, therefore, not learnt by
the trigger models. Specifically, in the context of the LHCb heavy-flavour
triggers, such architectures enable enhanced sensitivity to potential yet-
undiscovered feebly interacting BSM states produced within the LHCb
acceptance.

By adopting the deep learning models introduced in ref. Kitouni:2021fkh ,
robustness and certified monotonic response with respect to a set of input
features are realised through a minimal set of architecture-level constraints.
Such conditions enable highly expressive classifiers capable of inference
within the bandwidth-rate constraints set by the LHCb Run 3 trigger
operations. Consequently, Lipschitz monotonic NNs supersede the decision-tree
implementations of the inclusive heavy-flavour triggers deployed in previous
data taking campaigns Gligorov:2012qt ; Likhomanenko:2015aba . The topological
triggers, presented in the following section, exemplify the advantages
obtained through this methodological advancement.

##  3 The LHCb topological triggers

The topological triggers are designed to inclusively select $b$-hadron decay
processes based on the offline-quality information available at the HLT2
trigger stage. Decays of beauty particles display a distinct topology: owing
to their forward boost in the detector, they exhibit a long lifetime
Workman:2022ynf , causing them to traverse distances of
$\mathcal{O}(1\,\textrm{cm})$ before decaying within the detector acceptance.
Additionally, owing to their relatively high mass, beauty hadrons typically
exhibit sizeable transverse momentum, $p_{T}$. Combined, both conditions make
for a distinct experimental signature. The topological triggers thus aim to
select displaced secondary decay-vertex candidates reconstructed in the
pairwise combination of final-state charged tracks.

Two variations of the Run 3 topological architectures have been incorporated
in the LHCb online software stack, henceforth referred to as the two- and
three-body topological triggers. These are designed to identify beauty
secondary-vertex candidates reconstructed in the combination of two or three
charged particles, respectively. Such a suite of topological triggers thus
facilitates the identification of multi-particle beauty decays. The two-body
composites are reconstructed from final-state tracks compatible with
originating from a well-resolved secondary-vertex candidate appreciably
displaced from the $pp$ collision point. The three-body candidates, in turn,
are reconstructed by adding a companion track to the two-body object and
imposing similar vertex-fit and kinematic criteria as for the two-body
counterparts. Notably, the topological triggers record the full-event level
information, writing to the so-called _Full Stream_ , depicted in Figure 1. As
a result, $n$-body $b$-hadron decays, with $n>3$, may be successfully
reconstructed by combining additional tracks persisted in the event with the
trigger-selected two- and three-body candidates. Crucially, this design
maximises the selection efficiency on signal whilst meeting the HLT2 output-
rate requirements.

The HLT2 topological triggers are tasked with rejecting several sources of
background: combinatorial and soft-QCD processes; interactions of particles
with the LHCb detector material, exhibiting softer $p_{T}$ than the beauty
signal, and a comparatively larger flight distance from the $pp$ interaction
point; fake particles, typically referred to as _ghosts_ , erroneously
inferred from tracking-level information and exhibiting high $p_{T}$, as a
linear trajectory within the detector translates to the maximal possible
momentum reconstructed by the tracking system; and charm decays, rendered
challenging by the high charm production cross section, approximately
$\mathcal{O}(10)$ times higher than the beauty counterpart, and topologies
compatible with the signal, albeit with comparatively reduced lifetimes
Workman:2022ynf .

Each topological trigger exploits a two-staged selection to achieve effective
signal isolation. Firstly, a cut-based prefilter is devised to discard prompt
and soft background candidates. Subsequently, a deep learning classifier is
implemented as a monotonic Lipschitz neural network. The two- and three-body
models share the same architecture complexity, amounting to four hidden layers
comprising 16, 32, 64, and 32 internal nodes, respectively. This design choice
mitigates inference-time consumption during real-time trigger operations.
Model training is enacted through an offline Python-based pipeline exploiting
the PyTorch paszke2019pytorch  software package. Thereafter, the network
weights are ported to the LHCb trigger software stack for event-based
inference in real time.

Table 1: Features used to train the two-body classifier, along with the monotonicity requirements and the operations enacted to rescale the feature distributions to a range of order unity. The operators $\cdot\\{\\}$ and $\sum$ run over the final-state tracks, and are evaluated on a per-candidate basis. The shorthand notations GeV and $\log$ signify rescaling from units of $\textrm{MeV}/c^{(2)}$ to $\textrm{GeV}/c^{(2)}$ and evaluating the natural logarithm of the observable, respectively. Feature | Monotonicity | Rescaling to $\mathcal{O}(1)$ procedure
---|---|---
$\min\\{p_{T}\\}$ | Increasing | GeV
$\sum{p_{T}}$ | Increasing | GeV
$\min\\{\chi^{2}_{\textrm{IP}}\\}$ | Increasing | $\log$
$\max\\{\chi^{2}_{\textrm{IP}}\\}$ | — | $\log$
Composite $p_{T}$ | Increasing | GeV
Composite $\max({\textrm{DOCA}})$ | — | —
Composite $m_{\textrm{corr}}$ | — | GeV
Composite flight distance $\chi^{2}$ | — | $\log$
Composite $\chi^{2}_{\textrm{vtx}}$ | — | —

Both topological triggers are optimised by considering a suite of exclusive
simulations representative of the LHCb beauty physics programme. In this way,
the respective selection criteria and architecture complexity are optimised to
attain sensitivity to the characteristic decay topologies and kinematics of
$b$-hadron decays. The exclusive signal Monte Carlo (MC) simulations are
combined to contribute the same number of decay-vertex candidates to the
inclusive signal sample. This, in turn, is used both to optimise the cut-based
prefilter and train the NNs. Crucially, this procedure prevents potential
biases in the form of heightened sensitivity to a subset of signal channels,
by construction. The background is modelled by an inclusive minimum-bias MC
sample, generated to represent the average content of a $pp$ collision.
Notably, events containing beauty candidates are filtered from the minimum-
bias sample, making it a suitable inclusive proxy of the aforementioned
background processes.

Table 2: Features used to train the three-body classifier, along with the monotonicity requirements and the operations enacted to rescale the feature distributions to a range of order unity. The operators $\cdot\\{\\}$ and $\sum$ run over all final-state tracks, and are evaluated on a per-candidate basis. Conversely, the $\cdot_{\textrm{2body}}\\{\\}$ operations run over the two-body children only. The shorthand notations GeV and $\log$ signify rescaling from units of $\textrm{MeV}/c^{(2)}$ to $\textrm{GeV}/c^{(2)}$ and evaluating the natural logarithm of the observable, respectively. Feature | Monotonicity | Rescaling to $\mathcal{O}(1)$ procedure
---|---|---
$\min\\{p_{T}\\}$ | Increasing | GeV
$\sum{p_{T}}$ | Increasing | GeV
$\min_{\textrm{2body}}\\{p_{T}\\}$ | Increasing | GeV
$\sum_{\textrm{2body}}{p_{T}}$ | Increasing | GeV
$\min\\{\chi^{2}_{\textrm{IP}}\\}$ | Increasing | $\log$
$\max\\{\chi^{2}_{\textrm{IP}}\\}$ | — | $\log$
Three-body $p_{T}$ | Increasing | GeV
Two-body child $p_{T}$ | Increasing | GeV
Three-body $\max({\textrm{DOCA}})$ | — | —
Two-body child $\max({\textrm{DOCA}})$ | — | —
Two-body child $\chi^{2}_{\textrm{IP}}$ | — | $\log$
Two-body child $m_{\textrm{corr}}$ | — | GeV
Three-body $m_{\textrm{corr}}$ | — | GeV
Three-body flight distance $\chi^{2}$ | — | $\log$
Two-body child flight distance $\chi^{2}$ | — | $\log$
Three-body $\chi^{2}_{\textrm{vtx}}$ | — | $\log$
Two-body child $\chi^{2}_{\textrm{vtx}}$ | — | —

The NNs are trained on feature sets maximising the post-prefilter signal-to-
background divergence whilst maintaining low pairwise feature correlation. The
two- and three-body feature sets are presented in Tables 1 and 2. Crucially,
the adopted inputs must capture the decay topology and kinematics of multi-
body beauty composite candidates. To this end, the NNs are trained on a suite
of kinematic and geometric observables comprising the candidate transverse
momentum, $p_{T}$; the decay-vertex fit quality, $\chi^{2}_{\textrm{vtx}}$;
flight distance $\chi^{2}$; the charged-tracks distance of closest approach
(DOCA); the multi-body corrected mass value, $m_{\textrm{corr}}$ mcorr ; and
the IP $\chi^{2}$, the impact parameter $\chi^{2}$ with respect to the primary
vertex of the multi-body decay-vertex objects. Similar kinematic and geometric
features are extracted from the charged final-state tracks.

The features are subjected to a preprocessing stage constraining each input
observable distribution to a range of $\mathcal{O}(1)$. Additionally, a
$5\sigma$ window is imposed about the mean of each feature distribution. Input
values exceeding the retention boundaries are mapped onto the last value
accepted in the per-feature selection interval. Combined, these preprocessing
steps stabilise the performance of the classifier without depleting the
statistical population of the training samples. Moreover, comparing similarly
ranged distributions facilitates the choice of the Lipschitz-constant bound,
$\lambda$, of the learnt decision functions.

Separate upper bounds are imposed on the two- and three-body Lipschitz
constants, respectively. These are individually selected to maximise
reconstruction efficiency on the multi-body signal candidates whilst
maintaining compatibility of the resolution expected of the LHCb detector. In
this way, the Lipschitz-bound assignments prevent significant variations in
the classification score produced by the estimators, in the limit of the
inputs varying within their respective resolution scale.

Figure 2: Selection Efficiency on simulated $B^{0}\to
D^{*-}\tau^{+}\nu_{\tau}$ candidates reconstructed from two final-state
tracks. Top: performance delivered by a feed-forward, unconstrained network.
Bottom: efficiency values obtained by the two-body monotonic Lipschitz
network. The normalised $B^{0}$ logarithmic flight distance $\chi^{2}$
distribution is shown in the shaded grey histogram. Binwise efficiencies for
response cuts yielding global trigger-on-signal efficiency values,
$\varepsilon^{TOS}$, at the $60\%$-$90\%$ level are marked. The Lipschitz
bound, $\lambda$, is shown where relevant.

Finally, a monotonically increasing response is enforced with respect to a
subset of features, identified independently for the two- and three-body
classifiers. Monotonicity is imposed in the IP $\chi^{2}$ and the $p_{T}$
budget of the composite candidates and their constituents. This conservative
choice prevents the introduction of unwanted biases in selection efficiency
due to monotonicity. Furthermore, this approach delivers enhanced sensitivity
to highly displaced outliers compared to a baseline unconstrained model with
identical network complexity, as demonstrated by Figure 2. The per-candidate
selection efficiency delivered by the two-body topological trigger is
evaluated for simulated $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ decays, where the
$\tau$ lepton decays hadronically, excluded by the training MC samples. The
efficiency extraction is performed in bins of the signal logarithmic
$\chi^{2}$ fit-quality of the partially reconstructed $B^{0}$ candidate flight
distance with respect to the $pp$ collision point, taken to be a proxy for the
decay-vertex candidate lifetime. Compared to a performance attained by the
baseline unconstrained NN, the topological trigger delivers comparatively
higher selection efficiencies for highly displaced outliers. Concurrently, it
delivers compatible performance in the remainder of the observed range. Such a
trend is evident when requiring thresholds on the respective network responses
delivering overall efficiency values at the $60\%$-, $70\%$-, $80\%$-, and
$90\%$-level on signal. The efficiency drop evident at very high displacement
is due to the fact that monotonicity is enforced in a subset of the input
features. Consequently, all else being equal, the distributions of other
features can be much more background-like at large flight distance, leading to
a marginal efficiency drop in the high tail of the flight distance $\chi^{2}$
projection. Nevertheless, the results presented in Figure 2 demonstrate the
capacity of the two-body topological trigger to efficiently select signal
$b$-hadron decays and enhance in sensitivity to highly displaced candidates.
This property bolsters the capacity to retain outlier multi-body objects, and
thus feebly interacting BSM candidates produced in the LHCb acceptance.

##  4 Summary and Discussion

LHCb is pioneering the deployment of a fully software-based trigger in the LHC
Run 3 data taking campaign. Such a paradigm shift facilitates the adoption of
Lipschitz monotonic NNs in the LHCb inclusive heavy-flavour triggers. These
models are particularly well suited to the real-time selection of events
containing decay-vertex candidates exhibiting topologies and kinematics
compatible with heavy-flavour decays.

The utility of monotonic Lipschitz NNs is exemplified by the topological
triggers deployed in the highest level of the LHCb Run 3 trigger system, HLT2.
The preliminary results presented in this contribution demonstrate capacity to
efficiently select two- and three-body signal candidates. Enforcing a
monotonically increasing response with respect to a subset of input features
yields enhanced sensitivity to highly displaced decay-vertex candidates, as
evaluated on a probe decay channel. From a physics standpoint, this
performance strengthens sensitivity to beauty candidates at high lifetime and,
notably, to potential BSM states produced within the LHCb detector.
Furthermore, the Lipschitz bound applied to the learnt decision function
provides certified protection against instabilities during data taking,
thereby easing the evaluation of the relevant systematic uncertainties in end-
user measurements.

The combined output of the topological triggers dominates the HLT2 bandwidth
allocation. The threshold imposed on the classifier response for real-time
inference on data, in turn, is fixed by the maximum bandwidth allocated to the
HLT2 trigger stage when writing data to storage. Optimization studies are
currently underway to improve the classification power of these triggers. The
aim is to maximise selection efficiency for standard-candle beauty decay modes
whilst satisfying the HLT2 output-rate constraints.

Finally, as robustness and monotonicity are ideal inductive biases in
experimental particle physics, investigations are progressing to deploy
Lipschitz NNs in the Run 3 LHCb tracking, lepton identification and ghost-
rejection algorithms.

##  5 Acknowledgements

The authors would like to thank the LHCb computing and simulation teams for
their support and for producing the simulated LHCb samples used in the paper.
The authors would also like to thank the LHCb online, DPA and RTA team for
providing most of the software that this project was built upon and for all
the support and guidance that was given. BD and MW were supported by NSF grant
PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental
Interactions, http://iaifi.org/). BD and MW were also supported by U.S. NSF
grant PHY-2209181. NS and JA acknowledge funding from the German Science
Foundation DFG, within the Collaborative Research Center SFB1491 “Cosmic
Interacting Matters - From Source to Signal.” JA acknowledges funding from the
German Federal Ministry of Education and Research (BMBF, grant no. 05H21PECL1)
within ErUM-FSP T04.

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