Real Part of Twisted-by-Grading Spectral Triples

Real Part of Twisted-by-Grading Spectral Triples††This paper is a contribution
to the Special Issue on Noncommutative Manifolds and their Symmetries in
honour of Giovanni Landi. The full collection is available at
https://www.emis.de/journals/SIGMA/Landi.html

Manuele FILACI † and Pierre MARTINETTI ‡

M. Filaci and P. Martinetti

† Università di Genova – Dipartimento di Fisica and INFN sezione di Genova,
Italy<EMAIL_ADDRESS>
‡ Università di Genova – Dipartimento di Matematica and INFN sezione di
Genova, Italy<EMAIL_ADDRESS>
Received September 03, 2020, in final form October 23, 2020; Published online
October 29, 2020

After a brief review on the applications of twisted spectral triples to
physics, we adapt to the twisted case the notion of _real part_ of a spectral
triple. In particular, when one twists a usual spectral triple by its grading,
we show that – depending on the $KO$ dimension – the real part is either
twisted as well, or is the intersection of the initial algebra with its
opposite. We illustrate this result with the spectral triple of the standard
model.

noncommutative geometry; twisted spectral triple; standard model

58B34; 46L87; 81T75

_In honor of Giovanni Landi for its sixtieth birthday_

##  1 Introduction

Twisted spectral triples have been defined by Connes and Moscovici in [13] in
order to adapt the theory of spectral triples to type $III$ algebras. Later,
it turned out that twists also have applications to models of high energy
physics in noncommutative geometry [16], paving the way to models beyond the
standard model.

Most of the properties of twisted spectral triples relevant for physics (in
particular those regarding the real structure and gauge transformation) have
been developed by Gianni Landi together with one of the authors, in a couple
of papers [19, 20]. This note first presents a short review of some of these
results. Then we show how the notion of _real part_ of a spectral triple
(defined in [7]) easily adapts to the twisted case (Proposition 3.1). We then
focus on twisted spectral triples obtained by twisting usual spectral triples
by their grading. We investigate the behaviour of the real part, stressing the
dependence on the $KO$-dimension. More precisely, as shown in Proposition 3.3,
the real part of the twisted spectral triple is either the twist of the real
part of the initial triple, or the intersection of the algebra with its
opposite. In the last section, we illustrate this result with the spectral
triple of the standard model.

##  2 Minimal twist by grading

This section is a (partial) review of the work of one of the authors with
Gianni Landi, regarding twisted spectral triples [19, 20]. We begin with some
considerations on the use of spectral triples in physical models of
fundamental interactions, and how the discovery of the Higgs boson in 2012 has
motivated some of us to use rather _twisted_ spectral triples, as defined
earlier by Connes and Moscovici [13]. This led to the _twist-by-grading_
procedure, that has been somehow “touched” in [16] and clearly formalised in
[19].

###  2.1 Motivation

In noncommutative geometry [11], the standard model of fundamental
interactions is described by the product (in the sense of spectral triples) of
a $4$-dimensional closed spin manifold ${\mathcal{M}}$ with an internal
geometry that encodes the gauge degrees of freedom. The latter is given by the
finite-dimensional real algebra

${\mathcal{A}}_{\text{SM}}={\mathbb{C}}\oplus{\mathbb{H}}\oplus
M_{3}({\mathbb{C}})$

acting on the finite-dimensional Hilbert space
${\mathcal{H}}_{F}={\mathbb{C}}^{32n}$ ($n$ is the number of generations of
fermions), together with a selfadjoint operator $D_{F}$ on ${\mathcal{H}}_{F}$
that encodes the Yukawa coupling of fermions. We refer the reader to the
literature for the details, in particular the raison d’être of this algebra
(see [7] for the original paper and, e.g., [10] for a recent review). In
brief, the algebra is such that its group of unitary elements
($u\in{\mathcal{A}_{\text{SM}}}$ such that $u^{*}u=uu^{*}=\mathbb{I}$) yields
back – modulo a unimodularity condition – the gauge group ${\rm
U}(1)\times{\rm SU}(2)\times{\rm SU}(3)$ of the standard model.

The product of spectral triples is

${\mathcal{A}}={C^{\infty}({\mathcal{M}})}\otimes{\mathcal{A}}_{\text{SM}},\qquad{\mathcal{H}}=L^{2}({\mathcal{M}},S)\otimes{\mathcal{H}}_{F},\qquad
D={\not{\partial}}\otimes\mathbb{I}+\gamma^{5}\otimes D_{F},$ (1)

where ${C^{\infty}({\mathcal{M}})}$ is the algebra of smooth functions on
${\mathcal{M}}$, acting by multiplication of the space
$L^{2}({\mathcal{M}},S)$ of square integrable spinors with111As usual we use
Einstein convention for the summation on indices repeated in alternate up/down
positions. ${\not{\partial}}=-{\rm i}\gamma^{\mu}\nabla_{\mu}$ the Dirac
operator associated with the spin structure of ${\mathcal{M}}$.

Fermions are elements of ${\mathcal{H}}$, bosons are connection $1$-forms
which, in noncommutative geometry, are of the form

$A=a^{i}[D,b_{i}],\qquad a^{i},b_{i}\in{\mathcal{A}}.$ (2)

For the spectral triple

$\big{(}{C^{\infty}({\mathcal{M}})},L^{2}({\mathcal{M}},S),{\not{\partial}}\big{)}$
(3)

of a spin manifold, equation (2) gives back the usual $1$-forms. For the
product of geometry (1), it gives the gauge bosons of the standard model
together with the Higgs field. In noncommutative geometry, the latter is thus
obtained on the same footing as the other gauge bosons, that is as a
connection $1$-form.

Furthermore, the mass of the Higgs boson is not a free parameter and can be
computed as a function of the parameters of the model (that is, the entries of
the matrix $D_{F}$). From the beginning of the model in the 90’s, the
prediction has always been around $m_{H}\simeq 170~{}\text{GeV}$. After the
discovery of the Higgs boson with $m_{H}\simeq 125~{}\text{GeV}$, several ways
have been explored to accomodate the correct mass (e.g., [1, 2, 3, 8, 9], see
[10] for a recent review).

Most of them start from the following observation: in the spectral triple of
the standard model, there is a part of the operator $D_{F}$ – the one that
contains the Majorana mass of the neutrinos – that commutes with the algebra
and, as such, does not contribute to the bosonic content of the model via (2).
However, it turns out that by turning this Majorana mass (which is a constant
$k_{R}$) into a field, say $\sigma$, then one obtains precisely the kind of
scalar field proposed in particle physics to stabilize the electroweak vacuum.
In addition, by altering the running of the renormalisation group, this extra
scalar field makes the computation of the mass of the Higgs boson compatible
with its experimental value [6].

The point is then to understand how to turn the constant $k_{R}$ into a field
$\sigma$ within the framework of noncommutative geometry. Various scenarios
have been proposed, one of them consists in viewing the total Hilbert space
${\mathcal{H}}$ of the product (1) as a $(32n\times 4)$-dimensional space
[15], allowing the algebra to act non trivially on the spinorial degrees of
freedom. By doing so, still following the classification of all possible
internal algebras in [4, 5], one is able to consider an internal algebra
bigger than ${\mathcal{A}}_{\text{SM}}$ – called _grand algebra_ in [15] –
that no longer commutes with the part of $D_{F}$ that contains $k_{R}$, making
the latter contribute to the bosonic content of the theory.

However, as a side effect, one gets that the commutator of
${\not{\partial}}\otimes\mathbb{I}$ with this grand algebra is no longer
bounded. This is in contradiction with one of the basic requirement of
spectral triples (namely that $[D,a]$ should be bounded for any
$a\in{\mathcal{A}}$). This kind of problem has already been encountered when
one deals with conformal maps on the canonical spectral triple (3) of a
manifold. A solution, as explained by Connes and Moscovici in [13], consists
in twisting the spectral triple. Quite remarkably (given that twists were
originally motivated by purely mathematical reasons), this solution also works
for the standard model.

###  2.2 Twisted spectral triple

A twisted spectral triple [13] is given by an involutive algebra
${\mathcal{A}}$ acting on a Hilbert space ${\mathcal{H}}$, a selfadjoint
operator $\operatorname{\mathcal{D}}$ on ${\mathcal{H}}$ with compact
resolvent, together with an automorphism $\rho$ of ${\mathcal{A}}$ such that
the _twisted commutator_

$[\operatorname{\mathcal{D}},a]_{\rho}:=\operatorname{\mathcal{D}}a-\rho(a)\operatorname{\mathcal{D}}$

is bounded for any $a\in{\mathcal{A}}$.

A _grading_ is a selfadjoint operator $\Gamma$ on ${\mathcal{H}}$ that squares
to $\mathbb{I}$, commutes with the algebra and anti-commutes with $D$. A _real
structure_ is an anti-linear isometry $J$ (that is $J^{*}J=\mathbb{I}$) on
$\mathcal{H}$, that squares to $+\mathbb{I}$ or $-\mathbb{I}$ and commutes or
anticommutes with the Dirac operator and the grading:

$JD=\epsilon^{\prime}DJ,\qquad J\Gamma=\epsilon^{\prime\prime}\Gamma J.$

The choice of these various signs determines the $KO$-dimension of the
spectral triple. Notice that since $J^{2}=\pm\mathbb{I}$, one has that $J$ is
surjective hence unitary: $J^{*}=J^{-1}$.

As in the non-twisted case, the real structure is asked to implement a
representation of the opposite algebra ${\mathcal{A}}^{\circ}$, identifying
$a^{\circ}$ with $Ja^{*}J^{-1}$. This action commutes with the one of
${\mathcal{A}}$, yielding the _order zero condition_

$[a,b^{\circ}]=0\qquad\forall\,a,b\in\operatorname{\mathcal{A}}.$ (4)

This condition is the same as in the non-twisted case. However, it was shown
in [19] that another important condition, the _first order condition_ , needs
to be modified in the twisted context, yielding the following _twisted firt-
order condition_ (originally introduced in [16]):

$[[\operatorname{\mathcal{D}},a]_{\rho},b^{\circ}]_{\rho_{0}}=0\qquad\forall\,a,b\in\operatorname{\mathcal{A}},$
(5)

where $\rho^{\circ}$ is the automorphism of ${\mathcal{A}}^{\circ}$ defined by

$\rho^{\circ}(a^{\circ}):=\big{(}\rho^{-1}(a)\big{)}^{\circ}.$ (6)

One uses $\rho^{-1}$ instead of $\rho$ because the twisting automorphism in
[13] is not asked to be a $*$ automorphism but rather to satisfy

$\rho(a^{*})=\big{(}\rho^{-1}(a)\big{)}^{*}.$

###  2.3 Twist by grading

Given a real spectral triple $({\mathcal{A}},{\mathcal{H}},D)$, requiring that
there exists a non-trivial automorphism $\rho$ such that $[D,a]_{\rho}$ is
bounded for any $a$ puts severe constraints on the algebra (actually
$a-\rho(a)$ must be bounded for any $a$ [19, Lemma 3.1]). So in order to
introduce a twist one needs to modify some of the elements of the triple.
Since ${\mathcal{H}}$ and $D$ encode the fermionic sector of the standard
model, and the point is to generate a new scalar field (no new fermions), it
makes sense to look for a minimal modification of the spectral triple, letting
the Hilbert space and the Dirac operator untouched. Playing only with the
representation does not allow much freedom, so one should be allowed to
enlarge the algebra, in agreement with the grand algebra idea mentioned in the
introduction. We call this procedure a _minimal twist_.

As a matter of fact, to twist the standard model, one considers as a grand
algebra twice the algebra (1), that is

${\mathcal{A}}\otimes\mathbb{R}^{2},$ (7)

with each of the copies of ${\mathcal{A}}$ acting independently on the $+1$,
$-1$ eigenspaces of the grading operator $\Gamma$. Namely the representation
$\pi$ of (7) on ${\mathcal{H}}$ is

$\pi(a,a^{\prime})=\frac{1}{2}(\mathbb{I}+\Gamma)a+\frac{1}{2}(\mathbb{I}-\Gamma)a^{\prime}\qquad\forall\,a,a^{\prime}\in{\mathcal{A}},$
(8)

where $\mathbb{I}$ is the identity operator on ${\mathcal{H}}$. The twisting
automorphism is simply the exchange of the two components of (7):

$\rho(a,a^{\prime})=(a^{\prime},a).$ (9)

This construction is generic: given any real graded spectral triple
$({\mathcal{A}},{\mathcal{H}},D)$ (with ${\mathcal{A}}$ a complex algebra to
fix the notations), we call its _twist by grading_ the twisted spectral triple

$\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2},{\mathcal{H}},D\big{)}_{\rho}$

with representation (8) and twist (9). This is a real graded twisted spectral
triple, with the same grading $\Gamma$ and real structure $J$ (hence same
$KO$-dimension) as the initial spectral triple [19, Proposition 3.8].

The flip (9) coincides with the inner automorphism of
$\mathcal{B}(\mathcal{H})$ induced by the unitary operator

$\displaystyle{\mathcal{R}}=\begin{pmatrix}0&\mathbb{I}_{+}\\\
\mathbb{I}_{-}&0\end{pmatrix},$

where $\mathbb{I}_{\pm}$ are the identity operators on the subspace
${\mathcal{H}}_{\pm}$ of the grading $\Gamma$. To be able to define the
fermionic action in a twisted context, we required in [14] a compatibility
condition between the automorphism $\rho$ and the real structure, namely one
asks that

$J{\mathcal{R}}=\epsilon^{\prime\prime\prime}{\mathcal{R}}J\qquad\text{for}\quad\epsilon^{\prime\prime\prime}=\pm
1.$ (10)

Using the same notation $\rho$ to denote the extension of the flip to the
whole of $\mathcal{B}(\mathcal{H})$,

$\rho(\mathcal{O}):=\mathcal{R}\mathcal{O}\mathcal{R}^{\dagger}\qquad\forall\,\mathcal{O}\in\mathcal{B}(\mathcal{H}),$

the compatibility condition (10) amounts to

$\rho(a^{\circ})=\left(\rho(a)\right)^{\circ}.$ (11)

##  3 Real part

We now come to the original content of this paper, which is to compute the
real part of a twisted-by-grading spectral triple

$\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2},{\mathcal{H}},D\big{)}_{\rho}.$
(12)

Being graded, such a spectral triple necessarily has an even $KO$-dimension.
We show below that in $KO$-dimension $0$, $4$ the real part of (12) is the
twist by grading of the real part of the initial spectral triple
$({\mathcal{A}},{\mathcal{H}},D)$, whereas in $KO$-dimension $2$, $6$, the
real part is the intersection ${\mathcal{A}}\cap{\mathcal{A}}^{\circ}$ of the
algebra with its opposite.

###  3.1 Real part of a twisted spectral triple

The _real part_ of a real spectral triple $({\mathcal{A}},{\mathcal{H}},D)$ –
as defined in [7] – is the spectral triple
$({\mathcal{A}}_{J},{\mathcal{H}},D)$ where ${\mathcal{A}}_{J}$ is the
subalgebra of ${\mathcal{A}}$ generated by the elements that commute with the
real structure. This definition easily generalizes to the twisted case, thanks
to following proposition which is a twisted version of [12, Proposition 1, p.
125].

######  Proposition 3.1.

Let $\operatorname{\left(\right)}_{\rho}$ be a real twisted spectral triple
with real structure $J$. Then, the following holds:

  1. $1.$

The equality

Algorithm 2

defines an involutive commutative real subalgebra of the center of .

  2. $2.$

If the twisting automorphism $\rho$ is induced by a unitary operator
compatible with the real structure in the sense of (10), then $($ is a real
twisted spectral triple.

  3. $3.$

Any $a\in$ twist-commutes with the algebra generated by the sums
$a^{i}[\operatorname{\mathcal{D}},b_{i}]_{\rho}$ for $a^{i}$, $b_{i}$ in .

######  Proof 3.2.

The proof is a straightforward adaptation of the non-twisted case.

1\. By construction is a real subalgebra of (but not a complex one, being $J$
antilinear). Since $J$ is unitary, and remembering that the usual rule of
adjoint for a product of operators also holds in the antilinear case as soon
as the product involves an even number of antilinear operators, one has

$\big{(}JaJ^{-1}\big{)}^{*}=Ja^{*}J^{-1}\qquad\forall\,\in$ Algorithm 10

Since for $a\in$ one has $JaJ^{-1}=a$, one gets

$Ja^{*}J^{-1}=a^{*},$ (13)

meaning $a^{*}\in$, that is is a involutive algebra. To show that it is
contained in the center of , notice that (13) implies

$a^{\circ}=a^{*}\qquad\forall\,a\in$ (14)

so that the order zero condition (4) yields $[b,a^{*}]=0$ for any $b\in$.

2\. Being a subalgebra of , the only point one needs to check is the stability
of $\mathcal{A}_{J}$ under the automorphism $\rho$, that is

$\rho(a)\in{\mathcal{A}}_{J}\qquad\forall\,a\in{\mathcal{A}}_{J}.$

This is guaranteed by the compatibility (10) of the real structure with
$\rho$: for $a\in{\mathcal{A}}_{J}$ one has

$J\rho(a)J^{-1}=J\mathcal{R}a\mathcal{R}^{\dagger}J^{-1}=(\epsilon^{\prime\prime\prime})^{2}\mathcal{R}\big{(}JaJ^{-1}\big{)}\mathcal{R}^{\dagger}=\mathcal{R}a\mathcal{R}^{\dagger}=\rho(a).$

The definition of the twisted commutator yields

$\big{[}a^{i}[\operatorname{\mathcal{D}},b_{i}]_{\rho},a^{\circ}\big{]}_{\rho^{\circ}}=a^{i}\big{[}[\operatorname{\mathcal{D}},b_{i}]_{\rho},a^{\circ}\big{]}_{\rho^{\circ}}+\big{[}a^{i},\rho^{\circ}(a^{\circ})\big{]}[\operatorname{\mathcal{D}},b_{i}]_{\rho}.$

The first term on the right hand side is zero by the twisted first order
condition (5), the second vanishes because of the order zero condition (4),
remembering that $\rho^{\circ}(a^{\circ})=(\rho(a))^{\circ}$ (this follows
from (6) because the flip automorphism is its own inverse). By (LABEL:eq:8)
one then has that $a_{i}[\operatorname{\mathcal{D}},b_{i}]_{\rho}$ twist-
commutes with any $a^{*}$, hence the result.

This definitions makes sense also when ${\mathcal{A}}$ is a real algebra, as
for the standard model.

Notice that the compatibility condition (10) between the real structure and
the twisting automorphism is important to guarantee the stability of
${\mathcal{A}}_{J}$ under $\rho$. Whether this condition is necessary in order
to be able to define the real part of a twisted spectral triple should be
further investigated [17].

###  3.2 Real part of a twisted-by-grading

How the real part behaves under the twist-by-grading heavily depends on the
$KO$-dimension. In $KO$-dimension $0$, $4$, the grading commutes with the real
structure and we show below that the real part of the twist-by-grading is the
twist-by-grading of the real part. In $KO$-dimension $2$, $6$, the grading
anticommutes with the real structure (as for the standard model which has
$KO$-dimension $2$) and the real part is the intersection of the algebra with
its opposite.

######  Proposition 3.3.

Let $({\mathcal{A}},{\mathcal{H}},D)$ be a real graded spectral triple with
real part ${\mathcal{A}}_{J}$. If its twist-by-grading
$\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2},{\mathcal{H}},D\big{)}_{\rho}$ is
compatible with the real structure, then its real part is either
${\mathcal{A}}_{J}\otimes\mathbb{C}^{2}$ in $KO$-dimension $0$, $4$, or it is
${\mathcal{A}}\cap{\mathcal{A}}^{\circ}$ in $KO$-dimension $2$, $6$.

######  Proof 3.4.

In $KO$-dimension $0$, $4$ the grading and the real structure commute (i.e.,
$\epsilon^{\prime\prime}=1$) so that

$\displaystyle[J,\pi(a,a^{\prime})]=\frac{1}{2}[J,(\mathbb{I}+\Gamma)a]+\frac{1}{2}[J,(\mathbb{I}-\Gamma)a^{\prime}]=\frac{1}{2}[J,a+a^{\prime}]+\frac{1}{2}[J,\Gamma(a-a^{\prime})]$
(15)

reduces to

$\frac{1}{2}[J,a+a^{\prime}]+\frac{1}{2}\Gamma[J,a-a^{\prime}].$

If $a$, $a^{\prime}$ are in ${\mathcal{A}}_{J}$, this is zero, showing that

${\mathcal{A}}_{J}\otimes\mathbb{R}^{2}\subset\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J}.$
(16)

To show the opposite inclusion, assume that
$(a,a^{\prime})\in\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J}$.
Hence (15), rewritten as

$\frac{1}{2}(\mathbb{I}+\Gamma)[J,a]+\frac{1}{2}(\mathbb{I}-\Gamma)[J,a^{\prime}]$

is zero. Multiplying by $\mathbb{I}+\Gamma$ and $\mathbb{I}-\Gamma$ one
obtains

$(\mathbb{I}+\Gamma)[J,a]=0,\qquad(\mathbb{I}-\Gamma)[J,a^{\prime}]=0.$ (17)

By compatibility of the real structure with the twist, $(a^{\prime},a)$ also
belongs to $\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J}$ so that

$(\mathbb{I}+\Gamma)[J,a^{\prime}]=0,\qquad(\mathbb{I}-\Gamma)[J,a]=0.$ (18)

Combining (17), (18), one gets $[J,a]=[J,a^{\prime}]=0$, that is $a$ and
$a^{\prime}$ are in ${\mathcal{A}}_{J}$. In other terms,
$\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J}\subset{\mathcal{A}}_{J}\otimes\mathbb{R}^{2}$.
Together with (16), this shows the first statement of the proposition.

In $KO$-dimension $2$, $6$, grading and real structure anti-commute
($\epsilon^{\prime\prime}=-1$) so that

$\displaystyle[J,\pi(a,a^{\prime})]$
$\displaystyle=\frac{1}{2}(J(\mathbb{I}+\Gamma)a-(\mathbb{I}+\Gamma)aJ+J(\mathbb{I}-\Gamma)a^{\prime}-(\mathbb{I}-\Gamma)a^{\prime}J)$
$\displaystyle=\frac{1}{2}((\mathbb{I}-\Gamma)Ja-(\mathbb{I}+\Gamma)aJ+(\mathbb{I}+\Gamma)Ja^{\prime}-(\mathbb{I}-\Gamma)a^{\prime}J)$
$\displaystyle=\frac{1}{2}((\mathbb{I}-\Gamma)(Ja-a^{\prime}J)+(\mathbb{I}+\Gamma)(Ja^{\prime}-aJ)).$
(19)

For $(a,a^{\prime})$ in
$\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J}$, this is zero. The
same is true for $(a^{\prime},a)$ by the invariance of
$\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J}$ by the twist.
Multiplying by $\mathbb{I}\pm\Gamma$ one thus obtains

$\displaystyle(\mathbb{I}-\Gamma)(Ja-a^{\prime}J)=0,\qquad(\mathbb{I}+\Gamma)(Ja^{\prime}-aJ)=0,$
$\displaystyle(\mathbb{I}-\Gamma)(Ja^{\prime}-aJ)=0,\qquad(\mathbb{I}+\Gamma)(Ja-a^{\prime}J)=0.$

Combining these two sets of equations yields $Ja^{\prime}=aJ$ and
$Ja=a^{\prime}J$, so that

$a^{\prime}=JaJ^{-1}=(a^{*})^{\circ}.$ (20)

Therefore $a=Ja^{\prime}J^{-1}$ is in ${\mathcal{A}}^{\circ}$, but also in
${\mathcal{A}}$ by hypothesis. In other terms, any element of
$\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J}$ is of the type
$(a,(a^{*})^{\circ})$ with $a\in{\mathcal{A}}\cap{\mathcal{A}}^{\circ}$.

Conversely, by definition of the opposite algebra any
$a\in{\mathcal{A}}\cap{\mathcal{A}}^{\circ}$ is equal to
$b^{\circ}=Jb^{*}J^{-1}$ for some $b\in{\mathcal{A}}$. Then
$(a^{*})^{\circ}=JaJ^{-1}=b^{*}$ is in ${\mathcal{A}}$, so that
$(a,(a^{*})^{\circ})$ is in ${\mathcal{A}}\otimes{\mathbb{C}}^{2}$. Inserting
in (19) one gets $[J,\pi(a,(a^{*})^{\circ})]=0$, meaning that
$(a,(a^{*})^{\circ})\in\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J}$.

Therefore
$\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J}\simeq{\mathcal{A}}\cap{\mathcal{A}}^{\circ}$,
hence the second statement of the proposition.

Obviously ${\mathcal{A}}_{J}$ is in ${\mathcal{A}}\cap{\mathcal{A}}^{\circ}$,
so the real part of the initial triple is contained in the real part of the
twist-by-grading, whatever the $KO$-dimension. However
$\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J}$ may equal
${\mathcal{A}}_{J}$ only in $KO$-dimension $2$, $6$, and only if the
intersection ${\mathcal{A}}\cap{\mathcal{A}}^{\circ}$ reduces to
${\mathcal{A}}_{J}$. In that case, $a^{\prime}$ in (20) coincides with $a$ and
the flip $\rho$ is the identity automorphism. The spectral triple defined in
the second point of Proposition 3.1 is then a usual (non-twisted) spectral
triple. This actually happens with the standard model as discussed in the next
section.

###  3.3 The twisted standard model and its real part

We now compute the real part of the twist-by-grading of the Standard Model,
working with one generation of fermions only, that is $n=1$. Following [7],
the $32$ degrees of freedom of the finite-dimensional Hilbert space
${\mathcal{H}}_{F}$ are labelled by a multi-index $CI\alpha$ where

  * $\bullet$

$C=0,1$ is for particle $(C=0)$ or anti-particle $(C=1)$;

  * $\bullet$

$I=0,i$ with $i=1,2,3$ is the lepto-colour index: $I=0$ means lepton, while
$I=1,2,3$ are for the quarks, which exists in three colors;

  * $\bullet$

$\alpha=\dot{a},a$ with $a=1,2$ is the flavour index:

$\displaystyle\dot{1}=\nu_{R},\qquad\dot{2}=e_{R},\qquad 1=\nu_{L},\qquad
2=e_{L}\qquad\text{for leptons ($I=0$)},$
$\displaystyle\dot{1}=u_{R},\qquad\dot{2}=d_{R},\qquad 1=q_{L},\qquad
2=d_{L}\qquad\text{for quarks ($I=i)$}.$

To deal with the twist, it is convenient to label the degrees of freedom of
$L^{2}({\mathcal{M}},S)$ by two extra-indices $s\dot{s}$ where

  * $\bullet$

$s=r,l$ is the chirality index;

  * $\bullet$

$\dot{s}=\dot{0},\dot{1}$ denotes particle ($\dot{0}$) or anti-particle part
($\dot{1}$).

The grading is

$\Gamma=\gamma^{5}\otimes\gamma_{F},$

where $\gamma^{5}=\left(\begin{smallmatrix}\delta_{\dot{s}}^{\dot{t}}&0\\\
0&-\delta_{\dot{s}}^{\dot{t}}\end{smallmatrix}\right)_{s}^{t}$ is the product
of the four Euclidean Dirac matrices on ${\mathcal{M}}$ while $\gamma_{F}$
takes value $+1$ on right particles and left antiparticles, and $-1$ on left
particles and right antiparticles.

An element $a$ of the double algebra (7) is a pair of elements of
${\mathcal{A}}$, namely

$a=(c,c^{\prime},q,q^{\prime},m,m^{\prime})$ (21)

with

$c,c^{\prime}\in C^{\infty}({\mathcal{M}},{\mathbb{C}}),\qquad q,q^{\prime}\in
C^{\infty}({\mathcal{M}},{\mathbb{H}}),\qquad m,m^{\prime}\in
C^{\infty}({\mathcal{M}},M_{3}({\mathbb{C}})).$

Following the twist by grading procedure of Section 2.3, its action on
$\operatorname{\mathcal{H}}$ is given by the $128\times 128$ matrix

$a=\left(\begin{matrix}Q&\\\ &M\end{matrix}\right)_{C}^{D}$ (22)

in which the element $(c,q,m)\in{\mathcal{A}}$ acts on the $+1$ eigenspace of
${\mathcal{H}}$ (that is $s=r$ with $C=0$, $\alpha=\dot{a}$ or $C=1$,
$\alpha=a$, and $s=l$ with $C=0$, $\alpha=a$ or $C=1$, $\alpha=\dot{a}$) while
the element $(c^{\prime},q^{\prime},m^{\prime})\in{\mathcal{A}}$ acts on the
$-1$ eigenspace of ${\mathcal{H}}$ (that is $s=r$ with $C=0$, $\alpha=a$ or
$C=1$, $\alpha=\dot{a}$ and $s=l$ with $C=0$, $\alpha=\dot{a}$ or $C=1$,
$\alpha=a$). Explicitly,

$Q=\left(\begin{matrix}Q_{r}&\\\ &Q_{l}\end{matrix}\right)_{s}^{t},\qquad
M=\left(\begin{matrix}M_{r}&\\\ &M_{l}\end{matrix}\right)_{s}^{t},$

are $64\times 64$ matrices whose blocks are the $32\times 32$ matrices

$\displaystyle
Q_{s}=\delta_{\dot{s}}^{\dot{t}}\left(\begin{matrix}\delta_{I}^{J}{\sf
c}_{s}&\\\
&\delta_{I}^{J}q_{\overline{s}}\end{matrix}\right)_{\alpha}^{\beta},\qquad
M_{s}=\delta_{\dot{s}}^{\dot{t}}\left(\begin{matrix}\delta_{\dot{a}}^{\dot{b}}{\sf
m}_{\overline{s}}&\\\ &\delta_{a}^{b}{\sf
m}_{s}\end{matrix}\right)_{\alpha}^{\beta},$

where $s=r,l$, $\overline{s}$ denotes the opposite chirality of $s$ and we
define the $2\times 2$ and $4\times 4$ matrices

$\displaystyle{\sf c}_{s}=\left(\begin{matrix}c_{s}&\\\
&\overline{c_{s}}\end{matrix}\right)_{a}^{b},\qquad{\sf
m}_{s}=\left(\begin{matrix}c_{s}&\\\ &m_{s}\end{matrix}\right)_{I}^{J},$

whose components are the elements of $a$:

$\displaystyle c_{r}=c,\qquad c_{l}=c^{\prime},\qquad q_{r}=q,\qquad
q_{l}=q^{\prime},\qquad m_{r}=m,\qquad m_{l}=m^{\prime},$ (23)

identifying the quaternions $q$, $q^{\prime}$ with their usual representation
as complex $2\times 2$ matrices.

Note that this representation is not the one used in [18] (where the twisting
operator is not the grading), neither the one of [16] (in which only the
electroweak sector of the theory had been twisted). A extensive discussion on
the various ways of twisting an almost commutative geometry, and the physical
consequence for the standard model, is in preparation [17].

######  Proposition 3.5.

The real part of the twist-by-grading of the standard model is the $($non-
twisted$)$ spectral triple
$\big{(}C^{\infty}({\mathcal{M}},\mathbb{R}),{\mathcal{H}},D\big{)}$.

######  Proof 3.6.

For any $a$ as in (22), the conjugation by the real structure amounts to
exchanging $Q$ with $M$ and taking the complex conjugate (similar proof as in
[18, Proposition 3.1])

$JaJ^{-1}=\begin{pmatrix}\bar{M}&\\\ &\bar{Q}\end{pmatrix}.$ (24)

The twist is the exchange of the primed quantities with the unprimed ones.
From (23), this amounts to exchanging $Q_{r}$, $M_{r}$ with $Q_{l}$, $M_{l}$.
This operation extends as an inner automorphism of $\mathcal{B}(\mathcal{H})$,
and commutes with the conjugation by the real structure as described above. In
other terms,

$J\rho(a)J^{-1}=\rho\big{(}JaJ^{-1}\big{)},$

which shows that the twist of the standard model is compatible with the real
structure in the sense of (11). Therefore, from Proposition 3.1, the real part
of the twist-by-grading of the standard model is

$\big{(}\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J},{\mathcal{H}},D\big{)}_{\rho}.$
(25)

By definition, $a$ in (21) being in ${\mathcal{A}}_{J}$ is equivalent to
$a=JaJ^{-1}$. From (24) this is equivalent to $Q=\bar{M}$ that is, in
components,

$\left(\begin{matrix}\delta_{I}^{J}\left(\begin{matrix}c_{s}&\\\
&\overline{c_{s}}\end{matrix}\right)_{a}^{b}&\\\
&\delta_{I}^{J}\left(q_{\overline{s}}\right)_{a}^{b}\end{matrix}\right)_{\alpha}^{\beta}=\left(\begin{matrix}\delta_{a}^{b}\left(\begin{matrix}\overline{c_{\overline{s}}}&\\\
&\left(\overline{m_{\overline{s}}}\right)_{i}^{j}\end{matrix}\right)_{I}^{J}&\\\
&\delta_{a}^{b}\left(\begin{matrix}\overline{c_{s}}&\\\
&\left(\overline{m_{s}}\right)_{i}^{j}\end{matrix}\right)_{I}^{J}\end{matrix}\right)_{\alpha}^{\beta}.$

In the first line, imposing the Kronecker $\delta$’s in the $a$, $b$ indices
(on the right), and on the $I$, $J$ indices (on the left) force to identify

$c_{s}=\overline{c_{s}}:=\lambda\in
C^{\infty}({\mathcal{M}},\mathbb{R})\qquad\text{and}\qquad
m_{\overline{s}}=c_{\overline{s}}\mathbb{I}_{3}.$

Then the equality between the left and the right hand terms yields
$\lambda=\overline{c_{\overline{s}}}$, hence

$c_{\overline{s}}=\lambda,\qquad m_{\overline{s}}=\lambda\mathbb{I}_{3}.$

The second line of the matricial equation then gives

$q_{s}=q_{\overline{s}}=\lambda\mathbb{I}_{2}.$

Going back to (23), one thus obtains

$a=(\lambda,\lambda,\lambda\mathbb{I}_{2},\lambda\mathbb{I}_{2},\lambda\mathbb{I}_{3},\lambda\mathbb{I}_{3}),$
(26)

hence

$\big{(}{\mathcal{A}}\otimes{\mathbb{C}}^{2}\big{)}_{J}\simeq
C^{\infty}({\mathcal{M}},\mathbb{R}).$

Since the twist $\rho$ leaves (26) invariant, the spectral triple (25) is
actually non twisted.

The real part of the twist-by-grading of the standard model is the same as the
non-twisted one (computed in [12, Section 14.1]). This is in agreement with
Proposition 3.3: one has that $a=b^{\circ}$ for some $b=(R,N)$, if and only if
$Q=\bar{N}$ and $M=\bar{R}$. By the same analysis as above, this shows that
$a=b=\lambda\mathbb{I}$ that is, from (26), $a\in{\mathcal{A}}_{J}$. Hence
${\mathcal{A}}\cap{\mathcal{A}}^{\circ}={\mathcal{A}}_{J}$.

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# Is active motion beneficial for target search with resetting in a thermal
environment?

P. S. Pal  School of Physics, Korea Institute for Advanced Study, Seoul 02455,
Korea  Jong-Min Park  Asia Pacific Center for Theoretical Physics, Pohang,
37673, Republic of Korea  Department of Physics, Postech, Pohang 37673,
Republic of Korea  Arnab Pal  The Institute of Mathematical Sciences, CIT
Campus, Taramani, Chennai 600113, India  Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar, Mumbai 400094, India  Hyunggyu Park
Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455,
Korea  Jae Sung Lee  School of Physics, Korea Institute for Advanced Study,
Seoul 02455, Korea

###### Abstract

Stochastic resetting has recently emerged as an efficient target-searching
strategy in various physical and biological systems. The efficiency of this
strategy depends on the type of environmental noise, whether it is thermal or
telegraphic (active). While the impact of each noise type on a search process
has been investigated separately, their combined effects have not been
explored. In this work, we explore the effects of stochastic resetting on an
active system, namely a self-propelled run-and-tumble particle immersed in a
thermal bath. In particular, we assume that the position of the particle is
reset at a fixed rate with or without reversing the direction of self-
propelled velocity. Using standard renewal techniques, we compute the mean
search time of this active particle to a fixed target and investigate the
interplay between active and thermal fluctuations. We find that the active
search can outperform the Brownian search when the magnitude and flipping rate
of self-propelled velocity are large and the strength of environmental noise
is small. Notably, we find that the presence of thermal noise in the
environment helps reduce the mean first passage time of the run-and-tumble
particle compared to the absence of thermal noise. Finally, we observe that
reversing the direction of self-propelled velocity while resetting can also
reduce the overall search time.

stochastic thermodynamics, stochastic resetting

##  I Introduction

Target search is essential for the survival of biological organisms. Notable
examples include searching for energy sources, food, shelter, and suitable
mates. Searching processes are also prevalent in cells. For instance,
transcript factors search for target sequences along DNA strands through a
process known as facilitated diffusion Winter ; Berg ; Benichou , which
involves one-dimensional diffusion and stochastic jumps to other locations.
These searching processes are basically conducted by agents that
stochastically wander their environment until they reach the target.
Therefore, the searching time of the processes depends on their strategies how
to wander around for searching. The representative measure for this searching
time is the mean first passage time (FPT). Thus, one of the primary concerns
in studies related to the searching problem is to find an optimal wandering
strategy that minimizes the mean FPT.

In the past decade, stochastic resetting has emerged as an efficient searching
strategy that reduces the mean FPT evans2011 ; gupta2014 ; evans2019 ;
majumdar2015 ; eule2015 ; pal-ness ; mendez2016 ; basu2019 ; olsen2024 ;
reuveni2016 ; pal2017 ; pal2019prl ; kusmirez2014 ; chechkin2018 ; belan2018 .
Stochastic resetting is a renewal process where a dynamic system is
interrupted at random times and reset to a predetermined configuration.
Initially applied to a one-dimensional freely diffusing Brownian particle
searching for a specific target, it has been demonstrated that stochastic
resetting eliminates the possibility of the particle straying far from the
target, resulting in a finite mean FPT. In contrast, the mean FPT for a free
diffusing particle without resetting diverges. Subsequently, researchers have
extensively studied various aspects of stochastic resetting, including
properties of its nonequilibrium steady state evans2011 ; pal-ness , methods
to further accelerate search processes evans2011 ; reuveni2016 ; pal2017  and
thermodynamic prospect of the search process psp2023 . Beyond its applications
in physics, stochastic resetting also plays a crucial role in accelerating
various biological processes. Examples include kinetic proofreading BarZiv ;
Murugan , protein-folding process Bhaskaran ; Hyeon ; Chakrabarti , and
chemical reaction process unbinding . See evans2019 ; pal22 ; pal23 ; pal23_sm
for an extensive review on the topic.

However, the impact of stochastic resetting on biological organisms is unclear
since their dynamics are often more complex than simple diffusion. In fact,
many biological searching agents consume energy and exhibit self-propelled
motion, known as active particles. These active particles operate in a
nonequilibrium regime and therefore do not follow the fluctuation-dissipation
relation. Examples of active particles include motile cells, motor proteins
Alberts-book , and artificial active Brownian particles such as self-propelled
colloids Howse07 ; Zheng13 , nano-rotors Nourhani13 , vibrated granular
particles Feitosa04 ; Kumar11 . Various models have been proposed to describe
the motions of active particles, such as run-and-tumble particles (RTP)
Schnitzer93 ; Malakar18 , active Ornstein-Uhlenbeck particles (AOUP) Bonilla19
, and active Brownian particles (ABP) Romanczuk12 . These active motions give
rise to unique thermodynamic properties, which have never been observed in
passive systems Krishnamurthy ; Zakine ; JSLee2020 ; JSLee2022 .

For examining the impact of active particle on the search process, in this
study, we focus on a target-searching process by RTP particles under
stochastic resetting dynamics. Earlier researches in this area have primarily
concentrated on the dynamics of RTP without considering the influence of the
environment Evans18 . This RTP-only dynamics can be modeled by substituting
the thermal random noise with telegraphic noise Rosenau93 . In reality,
however, active particles usually exist in environments that provide thermal
noise and influence their dynamics. Hence, it is important to study the
combined effects of the active motion and thermal noise to gain a
comprehensive understanding of search dynamics in biological systems.

Here, we study the dynamics of a RTP that moves freely in one-dimensional
space and is reset to a predetermined position at random times with a constant
rate in the presence of thermal noise. Employing generic renewal techniques,
we compute the moment generating function of FPT, then evaluate mean FPT and
finally, investigate the combined effect of thermal noise and active motion on
this target-search process. The results clearly verify that active motion can
reduce the mean FPT compared to that of Brownian motion within a certain range
of parameters. More specifically, our comprehensive analysis reveals that an
active particle can outperform a passive Brownian particle when the magnitude
and flipping rate of the self-propelled speed are high and the strength of
thermal noise is low. Additionally, we demonstrate that two other factors, the
presence of a thermal environment and reversing the direction of the self-
propelled velocity during resetting, are helpful in reducing the search time
of the RTP.

The rest of this paper is organized as follows: In Section II, we provide a
detailed description of our system setup, including the governing dynamics of
the RTP and resetting strategy used in the search process. Section III
introduces the moment generating function used to calculate the mean FPT. The
results are presented in Section IV, which is divided into three subsections:
Section IV.1 demonstrates the effect of active motion on mean FPT, Section
IV.2 discusses the effects of thermal noise and reversal of active-motion
direction during resetting on the mean FPT, and Section IV.3 presents
comprehensive analysis results on when an RTP outperforms a Brownian particle
as a search agent. Finally, we summarize our findings in Section V.

##  II Model

We consider a RTP that diffuses freely in one-dimensional space. Run-and-
tumble dynamics is modeled by an active speed $f$ that randomly reverses its
direction at a flipping rate $r_{\rm F}$. The particle is situated in a
thermal environment characterized by a diffusion constant $D$, which is
connected to the environmental temperature $T$ by the relation
$D=\gamma^{-1}k_{\rm B}T$, where $k_{\rm B}$ is the Boltzmann constant. Then,
the motion of the particle is described by the following Langevin equation:

$\displaystyle\dot{x}$ $\displaystyle=\sigma(t)f+\xi(t),$ (1)

where $\xi(t)$ is a Gaussian white noise with zero mean and
$\langle\xi(t)\xi(t^{\prime})=2D\delta(t-t^{\prime})$. Here, $\sigma(t)$
denotes the instantaneous direction of the active speed, which is given by a
telegraphic or dichotomous noise that alternates between two values, $+1$ and
$-1$, at a rate of $r_{\rm F}$. The particle begins from an initial position
$x(0)=x_{0}$ with a specified direction $\sigma(0)=\sigma_{0}$. During its
motion, the RTP’s position is instantaneously reset to $x_{\rm R}$ at random
times with a resetting rate of $r_{\rm R}$. When the particle’s position is
reset, the direction of the active speed may flip with a probability of $\eta$
Evans18 .

When the position $x(t)$ and the direction $\sigma(t)$ of the RTP’s active
speed at time $t$ are given, their subsequent values after an infinitesimal
time interval $\Delta t$ are determined by one of the following four cases:
(i) the position is reset, and the direction of active speed is reversed with
probability $q_{1}=r_{\rm R}\eta\Delta t$, (ii) the position is reset, but the
direction of active speed remains unchanged with probability $q_{2}=r_{\rm
R}(1-\eta)\Delta t$, (iii) the motion of the particle follows Eq. (1) without
direction flipping of active speed with probability $q_{3}=(1-r_{\rm R}\Delta
t)(1-r_{\rm F}\Delta t)$, and (iv) the motion of the particle follows Eq. (1)
with direction flipping of active speed with probability $q_{4}=(1-r_{\rm
R}\Delta t)r_{\rm F}\Delta t$. In short,

$\displaystyle x(t+\Delta t)=\left\\{\begin{array}[]{llll}x_{\rm R}\hskip
5.69046pt\&\hskip 5.69046pt\sigma(t)f\rightarrow-\sigma(t)f,&\text{with prob.
$q_{1}$}\\\ x_{\rm R}\hskip 5.69046pt\&\hskip
5.69046pt\sigma(t)f\rightarrow\sigma(t)f,&\text{with prob. $q_{2}$}\\\
x(t)+\sigma(t)f\Delta t+\xi(t)\Delta t,&\text{with prob. $q_{3}$}\\\
x(t)-\sigma(t)f\Delta t+\xi(t)\Delta t,&\text{with prob. $q_{4}$
.}\end{array}\right.$ (6)

The diffusion and resetting process continues until the particle reaches the
target position, which is set to the origin without loss of generality. It is
worth noting that previously introduced models can be derived as specific
cases of our model by taking proper limits. For instance, when $f\rightarrow
0$, our model reduces to the original stochastic-resetting model evans2011 .
In another limit $D\rightarrow 0$, our model simplifies to one with only
telegraphic noise Evans18 .

##  III Moment generating functions of First Passage Time

FPT depends on RTP’s initial configuration, including its starting position
$x_{0}$ and initial direction $\sigma_{0}$. Therefore, we define the
conditional FPT $t_{\rm F}^{\sigma_{0}}$, representing the first time the
particle reaches the target (the origin) starting from $x_{0}$ with an initial
direction $\sigma_{0}=\pm 1$. Following the Brownian functional method curr-sc
; Singh ; Dubey , one can express the moment generating function of FPT,
conditioned on the given initial position $x_{0}$ and initial direction
$\sigma_{0}$ as

$\displaystyle Q_{\sigma_{0}}(p|x_{0})=\int_{0}^{\infty}dt^{\sigma_{0}}_{\rm
F}e^{-pt^{\sigma_{0}}_{\rm F}}P_{\sigma_{0}}(t^{\sigma_{0}}_{\rm
F}|x_{0})\equiv\langle e^{-pt^{\sigma_{0}}_{\rm F}}\rangle,$ (7)

where $P_{\sigma_{0}}(t^{\sigma_{0}}_{\rm F}|x_{0})$ is the probability
density function of $t^{\sigma_{0}}_{\rm F}$ for the process with $x(0)=x_{0}$
and $\sigma(0)=\sigma_{0}$. Then, the $m$th moment of $t_{\rm F}^{\sigma_{0}}$
can be evaluated as

$\langle(t_{\rm
F}^{\sigma_{0}})^{m}\rangle=(-1)^{m}\partial_{p}^{m}Q_{\sigma_{0}}(p|x_{0})|_{p\rightarrow
0}.$ (8)

Now we derive the differential equation for $Q_{\sigma_{0}}(p|x_{0})$.
Consider a trajectory of RTP whose FPT is denoted by $t^{\sigma_{0}}_{\rm F}$.
We divide the trajectory into an initial infinitesimal segment with duration
$\Delta t$ and the remaining part with duration $t^{\sigma_{0}}_{\rm F}-\Delta
t$. During the initial infinitesimal segment, one of the four events can occur
with respective probabilities $q_{i}$ as presented in Eq. (6). Then, the
moment generating function $Q_{\sigma_{0}}(p|x_{0})$ can be written as

$\displaystyle Q_{\sigma_{0}}(p|x_{0})$ $\displaystyle=e^{-p\Delta t}\langle
e^{-p(t^{\sigma_{0}}_{\rm F}-\Delta t)}\rangle$ $\displaystyle=e^{-p\Delta
t}\bigg{\langle}Q_{-\sigma_{0}}(p|x_{\rm R})q_{1}+Q_{\sigma_{0}}(p|x_{\rm
R})q_{2}\bigg{\rangle}$ $\displaystyle+e^{-p\Delta
t}\bigg{\langle}Q_{\sigma_{0}}(p|x_{0}^{\prime})q_{3}+Q_{-\sigma_{0}}(p|x_{0}^{\prime\prime})q_{4}\bigg{\rangle},$
(9)

where $x_{0}^{\prime}=x_{0}+\sigma_{0}f\Delta t+\xi(0)\Delta t$ and
$x_{0}^{\prime\prime}=x_{0}-\sigma_{0}f\Delta t+\xi(0)\Delta t$. By expanding
Eq. (III) and keeping the terms up to the order of $\Delta t$, we obtain the
following backward differential equation of the moment generating functions:

$\displaystyle\textbf{L}(\partial_{x_{0}})\textbf{Q}(p|x_{0})=-r_{\rm
R}\textbf{HQ}(p|x_{\rm R}),$ (10)

where the moment generating functions are expressed as the vector
$\textbf{Q}(p|x)=(Q_{+}(p|x),Q_{-}(p|x))^{\textsf{T}}$ with T denoting matrix
transpose. The matrix $\textbf{L}(\hat{O})$ and H are given by

$\displaystyle\textbf{L}(\hat{O})=\begin{pmatrix}\mathcal{L}_{+}(\hat{O})&r_{\rm
F}\\\ r_{\rm
F}&\mathcal{L}_{-}(\hat{O})\end{pmatrix}~{}~{}\textrm{and}~{}~{}\textbf{H}=\begin{pmatrix}1-\eta&\eta\\\
\eta&1-\eta\end{pmatrix},$ (11)

respectively, where $\mathcal{L}_{\pm}(\hat{O})\equiv D\hat{O}^{2}\pm
f\hat{O}-g$ and $g\equiv p+r_{\rm R}+r_{\rm F}$. To solve Eq. (10), we need to
establish appropriate boundary conditions for $Q_{\sigma_{0}}(p|x_{0})$. The
first condition is that when $x_{0}\rightarrow 0$, the RTP reaches the target
instantly. The second condition requires that the moment generating functions
converge to a finite value as $x\rightarrow\pm\infty$ due to instantaneous
resetting. These boundary conditions can be expressed as:

$\displaystyle Q_{\pm}(p|x_{0}\rightarrow
0)=1,~{}~{}Q_{\pm}(p|x_{0}\rightarrow\pm\infty)<\infty.$ (12)

Note that $Q_{\sigma_{0}}(p|x_{0})$ is symmetric in $x_{0}$ when $x_{0}$ is
set to $x_{\rm R}$. In this case, it is sufficient to focus on the positive
range of $x_{0}$.

We solve the differential equation (10) analytically using the boundary
conditions (12). In particular, when the initial position and the resetting
position are the same, i.e., $x_{0}=x_{\rm R}$, the solutions can be expressed
in the following compact form:

$\displaystyle\textbf{Q}(p|x_{\rm R})=\bigg{[}\mathbb{1}+r_{\rm
R}(\mathbb{1}-\textbf{T}_{x_{\rm
R}})\textbf{L}_{0}^{-1}\textbf{H}\bigg{]}^{-1}\textbf{T}_{x_{\rm
R}}\textbf{I},$ (13)

where $\mathbb{1}$ denotes the identity matrix,
$\textbf{L}_{0}=\textbf{L}(0)$, $\textbf{I}=(1,1)^{\textsf{T}}$, and
$\textbf{T}_{x}=\textbf{F}\textbf{E}_{x}\textbf{F}^{-1}$. Here, the matrices F
and $\textbf{E}_{x}$ are defined in Eq. (A). The derivation of Eq. (13) can be
found in Appendix A.

##  IV Results

###  IV.1 Unconditional mean first passage time

To evaluate the mean conditional FPT, we can use Eq. (8) and the expressions
of moment generating functions from Eq. (13) so that

$\displaystyle\langle t^{\pm}_{\rm F}\rangle=-\partial_{p}Q_{\pm}(p|x_{\rm
R})|_{p\rightarrow 0}.$ (14)

In real experiments, the initial direction of active speed is not fixed, but
is determined probabilistically. Let $p_{\pm}$ represent the probabilities
that the initial direction of active speed is $\sigma_{0}=\pm 1$; then the
unconditional mean FPT can be calculated as

$\displaystyle\langle t_{\rm F}\rangle=p_{+}\langle t^{+}_{\rm
F}\rangle+p_{-}\langle t^{-}_{\rm F}\rangle.$ (15)

If the active particle is unbiased, it is reasonable to set $p_{+}=p_{-}=1/2$.
This assumption is maintained throughout the rest of this paper. Hereafter,
the unconditional mean FPT with $p_{\pm}=1/2$ will be referred to simply as
mean FPT.

Figure 1: Plot of mean FPT $\langle t_{\rm F}\rangle$ as a function of
resetting rate $r_{\rm R}$ for different combinations of flipping rate $r_{\rm
F}$ and active speed $f$ at $\eta=0$ and $D=1/2$. Dots and solid curves
represent simulation data and analytic results, respectively.

The full analytical expression for mean FPT is too lengthy to effectively
handle and comprehend for physical insights. Instead, we illustrate the
behavior of the mean FPT through various plots and discuss the impact of
active motion on it. Figure 1 shows the plot of $\langle t_{\rm F}\rangle$
versus $r_{\rm R}$ for different combinations of flipping rate $r_{\rm F}$ and
active speed $f$ at $D=1/2$. The various dots represent simulation data
averaged over $10^{5}$ realizations of the explicit dynamics in Eq. (6) and
the solid curves indicate analytical results. For the simulation, $\Delta t$
in Eq. (6) is set to be $10^{-5}$. The agreement between them validates our
analysis. For this plot, reversing the direction of active speed during
positional reset is not considered, i.e., $\eta$ is set to zero. The effect of
a finite $\eta$ will be discussed in Section IV.2. The value of $x_{\rm R}$ is
consistently set to $1$ throughout this paper.

The black solid curve in Fig. 1 shows mean FPT of the stochastic-resetting
search using a Brownian particle. This original process is identical to our
model with $f=0$, resulting in the mean FPT denoted by $\langle t_{\rm
F}^{f=0}\rangle$ evans2011 :

$\langle t_{\rm F}^{f=0}\rangle=\frac{1}{r_{\rm R}}\left(e^{x_{\rm
R}\sqrt{r_{\rm R}/D}}-1\right).$ (16)

Compared to this original dynamics, when $r_{\rm F}=1$, mean FPT is shorter
than $\langle t_{\rm F}^{f=0}\rangle$ and decreases as $f$ increases. This
clearly demonstrates that active motion can reduce mean FPT compared to that
of a Brownian particle. However, active motion is not always beneficial for
enhancing search performance. When $r_{\rm F}=0.1$, mean FPT becomes longer
than $\langle t_{\rm F}^{f=0}\rangle$ and increases as $f$ rises. Regardless
of magnitude of $f$ and $r_{\rm F}$, mean FPT exhibits a nonmonotonic behavior
with respect to $x_{\rm R}$, implying that it is minimized at an optimal
resetting rate $r_{\rm R}^{*}$. In particular, the optimal resetting rate for
the $f=0$ case is denoted by $r_{{\rm R},f=0}^{*}$.

Figure 2: Plot of mean FPT as a function of active speed $f$ for different
values of flipping rate $r_{\rm F}$ at $\eta=0$ and $D=1/2$. For all cases,
resetting rate is set to $r_{{\rm R},f=0}^{*}$, which minimizes mean FPT of
the original dynamics using a Brownian particle. Black solid line denotes mean
FPT of the dynamics with $f=0$. Dashed curves represent the asymptotic
behaviors of $\langle t_{\rm F}\rangle$ calculated from Eq. (18).

To better understand the effect of active motion on mean FPT, we investigate
dependence of $\langle t_{\rm F}\rangle$ on active speed and flipping rate at
a fixed resetting rate. Figure 2 shows the plot of $\langle t_{\rm F}\rangle$
versus $f$ for different values of $r_{\rm F}$ at $\eta=0$ and $D=1/2$. For
all curves in this plot, we set the resetting rate to $r_{\rm R}=r_{{\rm
R},f=0}^{*}$ to demonstrate when $\langle t_{\rm F}\rangle$ is lower than the
minimum mean FPT achieved by a Brownian particle. This minimum value, $\langle
t_{\rm F}^{f=0}\rangle|_{r_{\rm R}=r_{{\rm R},f=0}^{*}}$, is indicated by the
black line. As mentioned earlier, when active speed approaches zero, i.e.,
$f\rightarrow 0$, our model simplifies to the original model regardless of
$r_{\rm F}$ value. This is evident in Fig. 2, where all curves converge to the
black line for small $f$. In Appendix B, we demonstrate that the leading term
of mean FPT in the small $f$ expansion is independent of flipping rate and is
expressed by

$\displaystyle\langle t_{\rm F}\rangle=\langle t_{{\rm
F},f=0}\rangle+{O}(f^{2}).$ (17)

In addition, we obtain the expression of mean FPT in the large $f$ limit as
follows:

$\displaystyle\langle t_{\rm F}\rangle=\frac{\sqrt{1+\frac{2r_{\rm F}}{r_{\rm
R}}}}{2(r_{\rm F}+\eta r_{\rm R})}+x_{\rm R}\bigg{(}1+\sqrt{1+\frac{2r_{\rm
F}}{r_{\rm R}}}\bigg{)}\frac{1}{f}+{O}(f^{-2}).$ (18)

Details of the derivation are provided in Appendix B. This asymptotic behavior
of $\langle t_{\rm F}\rangle$ in the large $f$ regime is represented by the
dashed curves in Fig. 2. The dashed curves demonstrate that mean FPT saturates
to a certain value at large $f$. This saturation value, given by $[2(r_{\rm
F}+\eta r_{\rm R})]^{-1}\sqrt{1+2r_{\rm F}/r_{\rm R}}$, is a monotonically
decreasing function of $r_{\rm F}$ for a fixed $r_{\rm R}$, indicating a
higher $r_{\rm F}$ accelerates the search process. For moderate values of $f$,
mean FPT decreases monotonically for high $r_{\rm F}$, while it exhibits
nonmonotonic behavior for low $r_{\rm F}$. Nontheless, within this moderate
range of $f$, mean FPT tends to decrease monotonically as $r_{\rm F}$
increases. Overall, increasing $r_{\rm F}$ highly seems to be a crucial factor
to reduce $\langle t_{\rm F}\rangle$ below $\langle t_{{\rm F},f=0}\rangle$
across the entire range of $f$.

This behavior of mean FPT can be understood in the following way. Active
motion is characterized by two factors: run and tumble. Here, ‘run’ or
ballistic motion is governed by the parameter $f$, while ‘tumble’ or
stochasticity is controlled by the parameter $r_{\rm F}$. For a searching
process where $x_{\rm R}=x_{0}>0$ and $\eta=0$, we first consider the case of
low $r_{\rm F}$. At $f=0$, mean conditional FPTs with different initial
directions are equal, i.e., $\langle t_{\rm F}^{+}\rangle=\langle t_{\rm
F}^{-}\rangle$. However, if we increase $f$ from zero, the particle exhibits
near-ballistic motion due to a finite $f$ with infrequent flipping. As a
result, $\langle t_{\rm F}^{+}\rangle$ increases as the particle continues to
move away from the target, while $\langle t_{\rm F}^{-}\rangle$ decreases as
it approaches the target. In the large $f$ regime, it is straightforward to
see that $\langle t_{\rm F}^{-}\rangle\rightarrow x_{\rm R}/f$. Moreover,
$\langle t_{\rm F}^{+}\rangle$ approaches $1/r_{\rm F}+x_{\rm R}/f$, where the
first term, $r_{\rm F}^{-1}$, denotes the duration before the initial $+1$
direction of active speed flips into $-1$ direction, and the second term,
$x_{\rm R}/f$, represents the time required to reach the target with $-1$
direction from $x_{\rm R}$. Thus, the mean unconditional FPT $\langle t_{\rm
F}\rangle=(\langle t_{\rm F}^{+}\rangle+\langle t_{\rm F}^{-}\rangle)/2$
approaches $1/(2r_{\rm F})+x_{\rm R}/f$, which is consistent with Eq. (18) for
$r_{\rm F}\ll r_{\rm R}$. Therefore, for low value of $r_{\rm F}$, $\langle
t_{\rm F}\rangle$ roughly increases from the value of $\langle t_{\rm
F}^{f=0}\rangle$ to $1/(2r_{\rm F})$ as shown in Fig. 2.

Now, we consider the case with high $r_{\rm F}$. In this case, active motion
is more stochastic rather than ballistic. This stochastic tumbling of active
speed effectively enhances the ‘diffusivity’ of the particle, enabling it to
explore the search area more rapidly. This effective diffusivity increases as
$f$ increases. As a result, increasing $f$ expedites the search process and
results in reducing FPT. This explains why we can achieve a lower mean FPT
value compared to $\langle t_{\rm F}^{f=0}\rangle|_{r_{\rm R}=r_{{\rm
R},f=0}^{*}}$ at high $r_{\rm F}$.

###  IV.2 Effects of thermal noise and finite $\eta$

Figure 3: Effects of thermal noise and $\eta$ on mean FPT of RTP. $\langle
t_{\rm F}\rangle$ is plotted as a function of $r_{\rm R}$ for different values
of $\eta$ and $D$. (a) and (b) show plots with $r_{\rm F}=0.1$ and $r_{\rm
F}=1$, respectively. For both plots, $f=1$ is used. The solid curves represent
results with $\eta=0$, where there is no flipping direction of active speed
during reset. The dashed curves show results with $\eta=1/2$, indicating that
positional reset is accompanied by flipping direction of active speed with a
probability $1/2$. Different colors represent different values of $D$, with
black indicating dynamics without thermal noise (i.e., $D=0$).

We also explore the effects of thermal noise and finite $\eta$ on mean FPT.
For a low flipping rate of $r_{\rm F}=0.1$, the results are plotted in Fig.
3(a), which shows the mean FPT as a function of $r_{\rm R}$ for various $D$
and $\eta$. In the figure, solid curves represent $\langle t_{\rm F}\rangle$
with $\eta=0$ for different $D$ values. Among these, the black solid curve
indicates mean FPT for RTP without thermal noise, i.e., the $D=0$ case. The
analytic expression of mean FPT for RTP with $D=0$, denoted by $\langle t_{\rm
F}^{D=0}\rangle$, is given by Evans18

$\displaystyle\langle t_{\rm
F}^{D=0}\rangle=\left\\{\begin{array}[]{ll}-\frac{1}{2r_{\rm
F}R}+\frac{e^{\chi\sqrt{R(1+R)}}-R^{2}+R\sqrt{R(1+R)}}{2r_{\rm
F}R[1+R-\sqrt{R(1+R)}]},&\eta=0\\\ -\frac{1}{2r_{\rm
F}R}+\frac{e^{\chi\sqrt{R(1+R)}}}{2r_{\rm
F}R[1+R-\sqrt{R(1+R)}]},&\eta=\frac{1}{2}\\\ \end{array}\right.$ (21)

where $R\equiv r_{\rm R}/(2r_{\rm F})$ and $\chi\equiv 2r_{\rm F}x_{\rm R}/f$.
Compared to $\langle t_{\rm F}^{D=0}\rangle$, mean FPT with a finite $D$ is
always lower and decreases as $D$ increases as the figure shows. Dashed curves
in Fig. 3(a) represent mean FPT with $\eta=1/2$ for different $D$ values. As
also seen from Eq. (21), $\langle t_{\rm F}^{D=0}\rangle$ with $\eta=1/2$ is
lower than that with $\eta=0$. Moreover, increasing $D$ from $D=0$ further
reduces mean FPT. Similar to the case of a low flipping rate, for a high
flipping rate $r_{\rm F}=1$, $\langle t_{\rm F}\rangle$ decreases as $D$ or
$\eta$ increases, as shown in Fig. 3(b).

These results indicate that thermal noise and flipping of active speed during
reset expedite the searching process. We can understand these behaviors as
follows: First, increasing $D$ raises diffusivity, thus helping the particle
explore the searching area more rapidly. Consequently, this results in a
reduced mean FPT. This principle is also valid for the original model using a
Brownian particle, where mean FPT, expressed as (16), decreases as $D$
increases. Second, an increase in $\eta$ is effectively similar to raising
$r_{\rm F}$, which also aids in reducing mean FPT, as discussed in Section
IV.1. This effect is evident from the asymptotic expression of mean FPT in the
large $f$ regime in Eq. (18); the saturated mean FPT, $[2(r_{\rm F}+\eta
r_{\rm R})]^{-1}\sqrt{1+2r_{\rm F}/r_{\rm R}}$, is a monotonically decreasing
function of $\eta$. Particularly, the impact of increasing $\eta$ is more
pronounced for low $r_{\rm F}$, as seen in the expression and the plots shown
in Figs. 3(a) and 3(b).

###  IV.3 Optimal resetting rate and beneficiality of active motion

In this section, we provide a comprehensive analysis of when an RTP
outperforms a Brownian particle as a search agent. Suppose there exists a
Brownian particle in a thermal environment with a diffusion constant $D$ and
an RTP with active speed $f$ and flipping rate $r_{\rm F}$ in the same thermal
bath. If mean FPT of the RTP is lower than that of the Brownian particle, one
can say that using an RTP is more beneficial for searching a target than a
Brownian particle. For a given particle, however, mean FPT varies depending on
$r_{\rm R}$. Specifically, mean FPT is a nonmonotonic (convex) function of the
resetting rate, and thus, it is minimized at an optimal value of $r_{\rm
R}=r_{\rm R}^{*}$. The optimal resetting rate for RTP $r_{\rm R}^{*}$ is a
function of $f$, $r_{\rm F}$, and $D$, whereas the optimal resetting rate for
a Brownian particle $r_{{\rm R},f=0}^{*}$ depends solely on $D$. Therefore, it
is reasonable to compare the minimum mean FPTs evaluated at $r_{\rm R}^{*}$
for RTP and $r_{{\rm R},f=0}^{*}$ for Brownian particle, to quantitatively
assess the outperformance of RTP. In this context, as a measure for discerning
the degree of benefit of using an active particle for searching a target, we
introduce the concept of _beneficiality_ of active motion
$\mathcal{B}=\mathcal{B}(D,f,r_{\rm F})$ as a logarithmic ratio between the
minimum mean FPTs of an RTP $\langle t_{\rm F}\rangle|_{r_{\rm R}=r_{\rm
R}^{*}}$ and a Brownian particle $\langle t_{\rm F}^{f=0}\rangle|_{r_{\rm
R}=r_{{\rm R},f=0}^{*}}$:

$\mathcal{B}=\log\bigg{[}\frac{\langle t_{\rm F}\rangle|_{r_{\rm R}=r_{\rm
R}^{*}}}{\langle t_{\rm F}^{f=0}\rangle|_{r_{\rm R}=r_{{\rm
R},f=0}^{*}}}\bigg{]}.$ (22)

Therefore, when $\mathcal{B}>0$, active motion provides no benefit for
searching compared to a Brownian particle. Conversely, when $\mathcal{B}<0$,
an active particle outperforms a Brownian particle.

Figure 4: Panels (a), (b), and (c) depict the contour plots of the optimal
resetting rate $r_{\rm R}^{*}$ as a function of diffusion coefficient $D$ and
active speed $f$ for three different values of flipping rate $r_{\rm F}=$
$0.1$ , $0.5$, and $1$, respectively. Panel (d), (e), and (f) are the contour
plots of beneficiality $\mathcal{B}$ as a function of $D$ and $f$ for three
different values of flipping rate $r_{\rm F}=$ $0.1$ , $0.5$, and $1$,
respectively. Black dashed lines denote isolines where $\mathcal{B}=0$.

To evaluate $\mathcal{B}$, we begin by estimating the optimal resetting rate
$r_{\rm R}^{*}$ numerically from data of $\langle t_{\rm F}\rangle$ as a
function of $r_{\rm R}$ for a wide range of combinations of $D$, $f$, and
$r_{\rm F}$. Figures 4(a), 4(b), and 4(c) present two-dimensional contour
plots showing how $r_{\rm R}^{*}$ varies with $D$ and $f$ for three different
values of $r_{\rm F}$: $0.1$, $0.5$, and $1$. These plots also include
isolines where $r_{\rm R}^{*}(D,f,r_{\rm F})$ remains constant. Overall, the
tendency is for $r_{\rm R}^{*}$ to increase as either $D$ or $f$ increases.

Using this estimated $r_{\rm R}^{*}$, we evaluate $\mathcal{B}$. Figures
4(d)-(f) show contour plots of $\mathcal{B}$ versus $D$ and $f$ for different
values of $r_{\rm F}$. The black dashed lines represent isolines where
$\mathcal{B}=0$. These isolines divide the $D$-$f$ space into two regions: (i)
the $\mathcal{B}>0$ region, where active motion provides no benefit, and (ii)
the $\mathcal{B}<0$ region, where active particle outperforms a Brownian
particle.

From these observation, we can deduce the overall tendencies of $\mathcal{B}$
as follows. First, the region with $\mathcal{B}<0$ expands as $r_{\rm F}$
increases, suggesting that higher $r_{\rm F}$ is advantageous for enhancing
search performance. Second, active motion loses its benefit as $D$ increases.
This results from the fact that the decrease in $\langle t_{\rm
F}^{f=0}\rangle|_{r_{\rm R}=r_{{\rm R},f=0}^{*}}$ caused by increasing $D$
outweighs the decrease in $\langle t_{\rm F}\rangle|_{r_{\rm R}=r_{\rm
R}^{*}}$. This highlights the importance of environment; active motion is most
beneficial in low-temperature environments. Third, increasing $f$ usually aids
in reducing mean FPT, except in cases with high $D$ and low $r_{\rm F}$. In
summary, high $r_{\rm F}$ and large $f$ in an environment with low $D$ are
best conditions for achieving maximum benefit from an RTP.

##  V Conclusion

We have investigated the target-search problem using an active particle
governed by run-and-tumble dynamics in the presence of a thermal noise under
the instantaneous stochastic resetting mechanism. Our model is a
generalization of two previously studied stochastic resetting models: (i)
simple diffusive search in a thermal bath evans2011  and (ii) simple run-and-
tumble search in a non-thermal environment Evans18 . Most of the earlier
results can be retrieved from ours when appropriate limits are taken. From our
model, we find that active motion significantly modifies the search dynamics
either in reducing or prolonging the mean FPT. To understand the overall
combined influence of active motion and thermal noise on mean FPT, we have
introduced the function ‘beneficiality’ as a measure for discerning the degree
of benefit of using an active particle for searching a target. Extensive
calculations verify that large $f$, high $r_{\rm F}$, and low $D$ are the best
choices for an active particle to outperform a Brownian particle as a search
agent. We also demonstrate that the thermal environment plays a significant
role in reducing mean FPT. In addition to active motion and thermal noise, the
search process by an RTP can also be made more effective by reversing the
direction of the active speed during position resetting.

It is worth noting that our model enables the conducting of target-searching
experiments using RTP in a thermal environment. Previous studies solely
considered RTP without thermal noise Evans18 , making it experimentally
difficult to prepare such conditions because thermal noise is indispensable in
mesoscopic systems. Moreover, in the real world, especially in biological
systems, search processes can take place in more complex environments.
Therefore, it is crucial to investigate the combined effect of active motion
and environmental influences on searching dynamics. In this regard, our
approach and the proposed measure of ‘beneficiality’ will be a good starting
point and basis for future research in this direction.

##  VI Acknowledgement

P.S.P. and J.-M.P. equally contributed to this work. The authors acknowledge
Korea Institute for Advanced Study for providing computing resources [KIAS
Center for Advanced Computation Linux Cluster System]. This research was
supported by NRF Grant No. 2017R1D1A1B06035497 (H.P.), and individual KIAS
Grants No. PG064902 (J.S.L.), No. PG085601 (P.S.P), and No. QP013601 (H.P.) at
Korea Institute for Advanced Study. AP gratefully acknowledges research
support from the DST-SERB Start-up Research Grant Number SRG/2022/000080 and
the DAE, Govt. of India.

##  Appendix A Solution of Eq. (10)

Solution of Eq. (10) can be attained by dividing $\textbf{Q}(p|x_{0})$ into
the homogeneous and inhomogeneous parts as

$\displaystyle\textbf{Q}(p|x_{0})=\textbf{Q}^{\rm h}(p|x_{0})+\textbf{Q}^{\rm
inh}(p|x_{\rm R}).$ (A1)

Then, the inhomogeneous part satisfies the following equation:

$\displaystyle\textbf{L}_{0}\textbf{Q}^{\rm inh}(p|x_{\rm R})$
$\displaystyle=-\textbf{HR}~{}\implies~{}\textbf{Q}^{\rm inh}(p|x_{\rm
R})=-\textbf{L}_{0}^{-1}\textbf{HR},$ (A2)

where $\textbf{L}_{0}\equiv\textbf{L}(0)$ and $\textbf{R}\equiv r_{\rm
R}\textbf{Q}(p|x_{\rm R})$. The trial solution for the homogeneous part
$\textbf{Q}^{\rm h}(p|x_{0})=C\bm{\Phi}(p)e^{mx_{0}}$, where $C$ is a constant
independent of $p$ and $x_{0}$, satisfies

$\displaystyle\textbf{L}(\partial_{x_{0}})C\bm{\Phi}(p)e^{mx_{0}}=\textbf{L}(m)C\bm{\Phi}(p)e^{mx_{0}}=0.$
(A3)

Nonzero solutions of Eq. (A3) requires $\text{det}[\textbf{L}(m)]=0$ which is
expressed as

$\displaystyle(Dm^{2}-g)^{2}-f^{2}m^{2}-r_{\rm F}^{2}$ $\displaystyle=0.$ (A4)

The four solutions of Eq. (A4) are

$\displaystyle m_{1,\pm}$
$\displaystyle=\pm\frac{\sqrt{2\ell+2\sqrt{\ell^{2}-4D^{2}(g^{2}-r_{\rm
F}^{2})}}}{2D},$ $\displaystyle m_{2,\pm}$
$\displaystyle=\pm\frac{\sqrt{2\ell-2\sqrt{\ell^{2}-4D^{2}(g^{2}-r_{\rm
F}^{2})}}}{2D},$ (A5)

where $\ell\equiv 2gD+f^{2}$. Note that all $m_{i,\pm}$ ($i=1,2$) are real
numbers. First, $m_{1,\pm}$ are real because

$\displaystyle\ell^{2}-4D^{2}(g^{2}-r_{\rm F}^{2})=f^{4}+4gDf^{2}+4D^{2}r_{\rm
F}^{2}>0.$

$m_{2,\pm}$ are also real numbers because

$\displaystyle\ell^{2}-\sqrt{\ell^{2}-4D^{2}(g^{2}-r_{\rm
F}^{2})}^{2}=4D^{2}(g^{2}-r_{\rm F}^{2})>0,$ (A6)

where $g=p+r_{\rm R}+r_{\rm F}$ with $p,r_{\rm R},r_{\rm F}>0$. The
corresponding eigenvectors are given by

$\displaystyle\textbf{L}(m)\bm{\Phi}(p)=0~{}\implies~{}\bm{\Phi}_{i,\pm}(p)=\begin{pmatrix}F_{i,\pm}-fm_{i,\pm}\\\
F_{i,\pm}+fm_{i,\pm}\end{pmatrix},$ (A7)

where $F_{i,\pm}\equiv Dm_{i,\pm}^{2}-g-r_{\rm F}$. Hence, the general
homogeneous solution is expressed as

$\displaystyle\textbf{Q}^{\rm h}(p|x_{0})$
$\displaystyle=\sum_{i=1}^{2}[C_{i,+}\bm{\Phi}_{i,+}(p)e^{m_{i,+}x_{0}}+C_{i,-}\bm{\Phi}_{i,-}(p)e^{m_{i,-}x_{0}}].$
(A8)

The boundary condition $Q_{\pm}(p|x_{0}\rightarrow\infty)<\infty$ ensures
$C_{i,+}=0$ for $i=1,2$ as $m_{i,+}>0$. As a result, the expression of the
homogeneous part of the moment generating function can be rewritten as

$\displaystyle\textbf{Q}^{\rm h}(p|x_{0})=\textbf{FE}_{x_{0}}\textbf{C},$ (A9)

where F, $\textbf{E}_{x}$, and C are defined as follows:

$\displaystyle\textbf{F}\equiv\begin{pmatrix}F_{1,-}-fm_{1,-}&F_{2,-}-fm_{2,-}\\\
F_{1,-}+fm_{1,-}&F_{2,-}+fm_{2,-}\end{pmatrix},$
$\displaystyle\textbf{E}_{x}\equiv\begin{pmatrix}e^{m_{1,-}x}&0\\\
0&e^{m_{2,-}x}\end{pmatrix},~{}~{}\textbf{C}\equiv\begin{pmatrix}C_{1,-}\\\
C_{2,-}\end{pmatrix}.$ (A10)

Therefore, the solution of the moment generating function is

$\displaystyle\textbf{Q}(p|x_{0})=\textbf{FE}_{x_{0}}\textbf{C}+\textbf{Q}^{\rm
inh}(p|x_{\rm R}).$ (A11)

Using the boundary condition $Q_{\pm}(p|x_{0}\rightarrow 0)=1$, we have

$\displaystyle\textbf{C}=\textbf{F}^{-1}(\textbf{I}-\textbf{Q}^{\rm inh}),$
(A12)

where $\textbf{I}=\begin{pmatrix}1,1\end{pmatrix}^{\textsf{T}}$. Hence, Eq.
(A11) is rewritten as

$\displaystyle\textbf{Q}(p|x_{0})=\textbf{T}_{x_{0}}\textbf{I}+(\mathbb{1}-\textbf{T}_{x_{0}})\textbf{Q}^{\rm
inh},$ (A13)

where $\textbf{T}_{x}=\textbf{FE}_{x}\textbf{F}^{-1}$. Plugging
$\textbf{Q}(p|x_{\rm R})=\textbf{R}/r_{\rm R}$ and $\textbf{Q}^{\rm
inh}(p|x_{\rm R})=-\textbf{L}_{0}^{-1}\textbf{HR}$ in Eq. (A2) into Eq. (A13),
we have

$\displaystyle\textbf{R}=\bigg{[}\frac{1}{r_{\rm
R}}\mathbb{1}+(\mathbb{1}-\textbf{T}_{x_{\rm
R}})\textbf{L}_{0}^{-1}\textbf{H}\bigg{]}^{-1}\textbf{T}_{x_{\rm
R}}\textbf{I}.$ (A14)

Then, the inhomogeneous part of the moment generating function is

$\displaystyle\textbf{Q}^{\rm inh}(p|x_{\rm
R})=-\textbf{L}_{0}^{-1}\textbf{H}\bigg{[}\frac{1}{r_{\rm
R}}\mathbb{1}+(\mathbb{1}-\textbf{T}_{x_{\rm
R}})\textbf{L}_{0}^{-1}\textbf{H}\bigg{]}^{-1}\textbf{T}_{x_{\rm
R}}\textbf{I}.$ (A15)

By substituting this $\textbf{Q}^{\rm inh}(p|x_{\rm R})$ expression into Eq.
(A13), we finally arrive at

$\displaystyle\textbf{Q}(p|x_{0})=\textbf{T}_{x_{0}}\textbf{I}-\textbf{K}_{x_{0}}\bigg{[}\frac{1}{r_{\rm
R}}\mathbb{1}+\textbf{K}_{x_{\rm R}}\bigg{]}^{-1}\textbf{T}_{x_{\rm
R}}\textbf{I},$ (A16)

where
$\textbf{K}_{x}\equiv(\mathbb{1}-\textbf{T}_{x})\textbf{L}_{0}^{-1}\textbf{H}$.
For simplicity, if we choose $x_{0}=x_{\rm R}$, the moment generating function
simplifies as

$\displaystyle\textbf{Q}(p|x_{\rm R})$ $\displaystyle=r_{\rm R}\textbf{R}$
$\displaystyle=\bigg{[}\mathbb{1}+r_{\rm R}(\mathbb{1}-\textbf{T}_{x_{\rm
R}})\textbf{L}_{0}^{-1}\textbf{H}\bigg{]}^{-1}\textbf{T}_{x_{\rm
R}}\textbf{I}.$ (A17)

##  Appendix B Derivation of approximate expressions of mean FPT

Although the complete expression of mean FPT can be achieved from Eq. (14), it
is too lengthy to handle and interpret for physical insights. Therefore, in
this section, we present mean FPT in two limiting cases: small and large $f$.
First, in the limit of small $f$, $m_{i,-}$ simplifies as

$\displaystyle m_{1,-}$ $\displaystyle=-\sqrt{\frac{g+r_{\rm
F}}{D}}+O(f^{2}),$ $\displaystyle m_{2,-}$
$\displaystyle=-\sqrt{\frac{g-r_{\rm F}}{D}}+O(f^{2}).$ (B1)

Furthermore, F and $\textbf{E}_{x_{\rm R}}$ in the small $f$ limit are given
by

$\displaystyle\textbf{F}=\begin{pmatrix}f\sqrt{\frac{g+r_{\rm F}}{D}}&-2r_{\rm
F}+f\sqrt{\frac{g-r_{\rm F}}{D}}\\\ -f\sqrt{\frac{g+r_{\rm F}}{D}}&-2r_{\rm
F}-f\sqrt{\frac{g-r_{\rm F}}{D}}\end{pmatrix}+O(f^{2}),$
$\displaystyle\textbf{E}_{x_{\rm R}}=\begin{pmatrix}e^{-\sqrt{\frac{g+r_{\rm
F}}{D}}x_{\rm R}}&0\\\ 0&e^{-\sqrt{\frac{g-r_{\rm F}}{D}}x_{\rm
R}}\end{pmatrix}+O(f^{2}).$ (B2)

Then, from Eqs. (14) and (A), the conditional mean FPT for small $f$ becomes

$\displaystyle\langle t_{\rm F}^{\pm}\rangle=\frac{1}{r_{\rm
R}}\left(e^{x_{\rm R}\sqrt{\frac{r_{\rm
R}}{D}}}-1\right)\pm\mathcal{A}f+O(f^{2}),$ (B3)

where $\mathcal{A}$ is defined as

$\displaystyle\mathcal{A}\equiv\frac{\left(e^{x_{\rm R}\sqrt{\frac{2r_{\rm
F}+r_{\rm R}}{D}}}-e^{x_{\rm R}\sqrt{\frac{r_{\rm R}}{D}}}\right)(2r_{\rm
F}+r_{\rm R})}{2r_{\rm F}\sqrt{Dr_{\rm R}}\left(2r_{\rm F}e^{x_{\rm
R}\sqrt{\frac{2r_{\rm F}+r_{\rm R}}{D}}}+r_{\rm R}\right)}.$ (B4)

Hence, in the limit of small $f$, the unconditional mean FPT approaches that
of a freely diffusing Brownian particle as expected:

$\displaystyle\langle t_{\rm F}\rangle=\frac{1}{2}(\langle t_{\rm
F}^{+}\rangle+\langle t_{\rm F}^{-}\rangle)=\frac{1}{r_{\rm R}}\left(e^{x_{\rm
R}\sqrt{\frac{r_{\rm R}}{D}}}-1\right)+O(f^{2}).$ (B5)

Next, in the limit of large $f$ (or equivalently small $f^{-1}$), $m_{i,-}$
reduces to

$\displaystyle m_{1,-}$ $\displaystyle=-\frac{f}{D}-\frac{g}{f}+O(f^{-2}),$
$\displaystyle m_{2,-}$ $\displaystyle=-\frac{\sqrt{g^{2}-r_{\rm
F}^{2}}}{f}+O(f^{-2}).$ (B6)

In addition, F and $\textbf{E}_{x_{\rm R}}$ in the large $f$ limit are
expressed as

$\displaystyle\textbf{F}=\begin{pmatrix}2g-r_{\rm
F}+\frac{2}{D}f^{2}&-g-r_{\rm F}+\sqrt{g^{2}-r_{\rm F}^{2}}\\\ -r_{\rm
F}&-g-r_{\rm F}-\sqrt{g^{2}-r_{\rm F}^{2}}\end{pmatrix}+O(f^{-2}),$
$\displaystyle\textbf{E}_{x_{\rm R}}=\begin{pmatrix}0&0\\\ 0&1-\frac{x_{\rm
R}\sqrt{g^{2}-r_{\rm F}^{2}}}{f}\end{pmatrix}+O(f^{-2}).$ (B7)

Thus, the conditional mean FPT in this limit is given by

$\displaystyle\langle t_{\rm F}^{+}\rangle$
$\displaystyle=\frac{\sqrt{1+\frac{2r_{\rm F}}{r_{\rm R}}}}{r_{\rm F}+\eta
r_{\rm R}}+x_{\rm R}\bigg{(}1+\sqrt{1+\frac{2r_{\rm F}}{r_{\rm
R}}}\bigg{)}\frac{1}{f}+O(f^{-2}),$ $\displaystyle\langle t_{\rm
F}^{-}\rangle$ $\displaystyle=x_{\rm R}\bigg{(}1+\sqrt{1+\frac{2r_{\rm
F}}{r_{\rm R}}}\bigg{)}\frac{1}{f}+O(f^{-2}).$ (B8)

Finally, we arrive at the expression of the unconditional mean FPT for large
$f$ as

$\displaystyle\langle t_{\rm F}\rangle$ $\displaystyle=\frac{1}{2}(\langle
t_{\rm F}^{+}\rangle+\langle t_{\rm F}^{-}\rangle)$
$\displaystyle=\frac{\sqrt{1+\frac{2r_{\rm F}}{r_{\rm R}}}}{2(r_{\rm F}+\eta
r_{\rm R})}+x_{\rm R}\bigg{(}1+\sqrt{1+\frac{2r_{\rm F}}{r_{\rm
R}}}\bigg{)}\frac{1}{f}+O(f^{-2}).$ (B9)

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