Note About D-Branes in Carrollian Background

J. Klusoň 111Email addresses<EMAIL_ADDRESS>(J. Klusoň)

Department of Theoretical Physics and Astrophysics, Faculty of Science,
Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

This paper is devoted to the construction of stable and unstable Dp-branes in
generalized Carroll space-times. We also study tachyon solutions on the world-
volume of unstable Dp-brane in Carrollian background and discuss their
physical interpretations.

##  1 Introduction and Summary

Recently the analysis of small speed of light expansion of general relativity
was performed in very interesting paper [1]. The main importance of this work
is that this expansion was performed with the help of the perspective of non-
Lorentzian geometry that allows elegant formulation of Carroll expansion of
general relativity (GR). This formulation has its inspiration in the expansion
of GR for large speed of light that leads to geometrical formulation of non-
relativistic expansion of GR that was performed in [2, 3, 4]. Then systematic
speed of light expansion was performed in [1] along these lines when small
speed of light expansion leads to Carroll geometry with higher order
geometrical fields. At the leading order we have well known duality between
Newton-Cartan geometry and Carroll geometry [19, 20].

Another interesting question is to study strings and p-branes in this
Carrollian background. In fact, Carrollian strings and particles were
previously analysed in [5, 6, 9], see also [7, 8]. These objects were defined
by specific limiting procedure of the world-volume (world-sheet) fields that
label positions of these objects in the target space-time together with
rescaling of p-brane(string) tensions. It was shown in [5] that the dynamics
of Carroll particle is trivial which was also confirmed in case of the string
in [6]. Further, in [6] $p-$brane Carrollian limit was defined as rescaling
$p+1$ longitudinal target space coordinates $x^{\mu}=cX^{\mu}$ and taking
limit $c\rightarrow 0$. In this way an action for Carroll p-brane was found
and it was also shown that its dynamics is trivial. This analysis was further
generalized to the case of Carrollian limit of non-BPS Dp-brane that was
performed in [9].

The goal of this paper is to apply formalism developed in [1] to the case of
Carrollian k-brane limit and find corresponding action for Dp-brane action in
this background. Interestingly, we show that it is not necessary to have
correlation between the number $k$ and $p$ which is in sharp contrast with the
non-relativistic p-brane limit [21, 22, 23, 24] where in order to cancel
divergence that arises through non-relativistic limit in the kinetic term of
p-brane we should have specific form of the $p+1$-gauge form that couples to
world-volume of p-brane. In case of Carrollian $k-$brane limit we find
corresponding Dp-brane action in canonical formalism and we discuss its form
in dependence on its world-volume dimensions and number of dimensions where
Carrollian limit is performed.

As the next step we extend this analysis to the case of non-BPS Dp-branes [25,
26, 27] that includes tachyon field on its world-volume together with the
gauge field. We find the form of non-BPS Dp-brane action in Carrollian k-brane
limit of gravity and study their equations of motion. We firstly consider
tachyon kink solution following very nice analysis performed in [10] and
further extended in [12, 13]. In case of non-BPS Dp-brane in Carroll
background the situation is slightly different since the action is written in
canonical form but we again show that the tachyon kink solution corresponds to
the stable lower dimensional D(p-1)-brane in $k-$brane Carrollian background
which is nice consistency check of the tachyon condensation. Another
interesting situation occurs when we consider tachyon at its vacuum state. It
was argued in case of relativistic non-BPS Dp-brane in [15, 16, 17, 11, 18]
that this vacuum solution with non-zero electric flux corresponds to the gass
of relativistic strings. The same situation occurs in case of the Carroll non-
BPS Dp-brane in flat space-time as was shown in [9]. We generalize this
analysis to the case of non-BPS Dp-brane in $k-$brane Carrollian background
and we show that the tachyon vacuum solution describes dynamics of fundamental
string in the same background.

Let us outline our results. We introduced $k-$brane Carrollian limiting
procedure as generalization of the small speed of light expansion of general
relativity that was performed in [1]. Then we studied p-branes in this
background and we showed that can be defined for any values of $k$ and $p$. We
determined corresponding actions and discuss conditions under which they can
be written in Lagrangian form. Then we studied unstable Dp-branes in $k-$
brane Carrollian background. We determined corresponding equations of motion.
We argued that the tachyon kink solution describes D(p-1)-brane in $k-$brane
Carrollian background. Finally we also analysed tachyon vacuum solution and we
argued that again corresponds to the gas of fundamental strings in $k-$brane
Carrollian background.

This paper is organized as follows. In the next section (2) we review basic
facts about PUL expansion of metric and generalize it to the case of $k-$brane
Carrollian limit. Then in section (3) we study p-brane in this background and
discuss conditions when it is possible to determine its Lagrangian form. In
section (4) we generalize this analysis to the case of unstable Dp-brane and
solve corresponding equations of motion.

##  2 $k-$Brane Carrollian Limit

In this section we review and generalize small speed of light expansion of
general relativity introduced in [1] which was based on pre-ultra local (PUL)
expansion of vielbein. As was shown there the leading order expansion of PUL
variables leads to Carroll geometry. Explicitly, in terms of metric
$g_{\mu\nu}$ and its inverse $g^{\mu\nu}$ the small speed of light expansion
is the PUL parametrization

$g_{\mu\nu}=-c^{2}T_{\mu}T_{\nu}+\Pi_{\mu\nu}\ ,\quad
g^{\mu\nu}=-\frac{1}{c^{2}}V^{\mu}V^{\nu}+\Pi^{\mu\nu}\ ,$ (1)

where one form and vector $T_{\mu}$ and $V^{\mu}$ were introduced together
with spatial tensors $\Pi_{\mu\nu}$ and $\Pi^{\mu\nu}$ that obey following
orthogonality and completeness conditions

$T_{\mu}V^{\mu}=-1\ ,\quad T_{\mu}\Pi^{\mu\nu}=0\ ,\quad\Pi_{\mu\nu}V^{\nu}=0\
,\quad-V^{\mu}T_{\nu}+\Pi^{\mu\rho}\Pi_{\rho\nu}=\delta^{\mu}_{\nu}\ .$ (2)

This PUL parametrization corresponds to the splitting of tangent space into
temporal and spatial components.

Few years ago the Carroll limit of strings and p-branes was studied in [6]
where the generalized $k-$brane and string Carroll limit was introduced.
Roughly speaking they correspond to taking Carrollian limit along $(k+1)-$
directions which is generalization of standard Carroll limit that is defined
as small speed of light expansion along time-like direction only. Following
analysis presented in [1] we would like to introduce PUL parametrization in
case of $k-$brane Carroll limit as well. Explicitly, we consider following
expansion

$g_{\mu\nu}=c^{2}T_{\mu}^{\ I}T_{\nu}^{\ J}\eta_{IJ}+\Pi_{\mu\nu}\ ,\quad
g^{\mu\nu}=\frac{1}{c^{2}}V^{\mu}_{\ I}V^{\nu}_{\ J}\eta^{IJ}+\Pi^{\mu\nu}\ ,$
(3)

where we have longitudinal forms $T_{\mu}^{\ I},V^{\mu}_{\ I}$ with
$I=0,1,\dots,k$ and where $\eta_{IJ}=\eta^{IJ}=\mathrm{diag}(-1,1,\dots,1)$

Inserting (3) into $g_{\mu\nu}g^{\nu\rho}=\delta_{\mu}^{\rho}$ we obtain

$V^{\mu}_{\ I}\Pi_{\mu\nu}=0\ ,\quad T_{\mu}^{\ I}\Pi^{\mu\nu}=0\ $ (4)

and also

$T_{\mu}^{\ I}T_{\nu}^{\ J}\eta_{IJ}V^{\nu}_{\ K}V^{\rho}_{\
L}\eta^{KL}+\Pi_{\mu\nu}\Pi^{\nu\rho}=\delta_{\mu}^{\rho}\ .$ (5)

To proceed further we generalize first condition in (2) in the following way

$T_{\mu}^{\ I}V^{\mu}_{\ J}=-\delta^{I}_{J}\ $ (6)

so that (5) takes the form

$-T_{\mu}^{\ I}V^{\rho}_{\ I}+\Pi_{\mu\nu}\Pi^{\nu\rho}=\delta_{\mu}^{\rho}\ $
(7)

that can be interpreted as generalization of the fourth equation in (2).

As the next step we formulate generalized PUL expansion of vielbein
components. Recall that the metric can be written as $g_{\mu\nu}=E_{\mu}^{\
A}E_{\nu}^{\ B}\eta_{AB},A,B=0,1,\dots,D$ while the inverse metric as
$g^{\mu\nu}=\Theta^{\mu}_{\ A}\Theta^{\mu}_{\ B}\eta^{AB}$. Then, following
[1], we write

$\displaystyle E_{\mu}^{\ I}=cT_{\mu}^{\ I}\ ,\quad I=0,1,\dots,k\ ,\quad
E_{\mu}^{\ a}=E_{\mu}^{\ a}\ ,a=k+1,\dots,D\ ,$ $\displaystyle\Theta^{\mu}_{\
I}=-\frac{1}{c}V^{\mu}_{\ I}\ ,\quad\Theta^{\mu}_{\ a}=\Theta^{\mu}_{\ a}\ ,$

where

$\Pi_{\mu\nu}=E_{\mu}^{\ a}\delta_{ab}E_{\nu}^{\ b}\
,\quad\Pi^{\mu\nu}=E^{\mu}_{\ a}\delta^{ab}E^{\nu}_{\ b}\ .$ (9)

Then by definition we obtain

$\displaystyle T_{\mu}^{\ I}\Theta^{\mu}_{\ J}=-\delta^{I}_{J}\ ,\quad
T_{\mu}^{\ I}\Theta^{\mu}_{\ a}=0\ ,\quad E_{\mu}^{\ a}V^{\mu}_{\ I}=0\ ,\quad
E_{\mu}^{\ a}\Theta^{\mu}_{\ b}=\delta^{a}_{b}\ .$

Let us further presume that these variables are analytic in $c^{2}$ so that we
can write them as

$\displaystyle V^{\mu}_{\ I}=v^{\mu}_{\ I}+c^{2}M^{\mu}_{\
I}+\mathcal{O}(c^{4})\ ,$ $\displaystyle T_{\mu}^{\ I}=\tau_{\mu}^{\
I}+\mathcal{O}(c^{2})\ ,$ $\displaystyle\Theta^{\mu}_{\ a}=\theta^{\mu}_{\
a}+c^{2}\pi^{\mu}_{\ a}+\mathcal{O}(c^{4})\ ,$
$\displaystyle\Pi^{\mu\nu}=h^{\mu\nu}+c^{2}\Phi^{\mu\nu}+\mathcal{O}(c^{4})\
,$ $\displaystyle E_{\mu}^{\ a}=e_{\mu}^{\ a}+\mathcal{O}(c^{2})\ ,$
$\displaystyle\Pi_{\mu\nu}=h_{\mu\nu}+\mathcal{O}(c^{2})\ .$

Now inserting expansion (2) into (2) we obtain

$\displaystyle\tau_{\mu}^{\ I}\theta^{\mu}_{\ a}=0\ ,\quad e_{\mu}^{\
a}v^{\mu}_{\ I}=0\ ,\quad\tau_{\mu}^{\ I}v^{\mu}_{\ J}=-\delta_{I}^{J}\ ,$
$\displaystyle e_{\mu}^{\ a}\theta^{\mu}_{\ b}=\delta^{a}_{b}\
,\quad-\tau_{\mu}^{\ I}v^{\nu}_{\
I}+h_{\mu\rho}h^{\rho\nu}=\delta_{\mu}^{\nu}\ .$

In the same way as in [1] we could study properties of these more general PUL
expansions but we leave it for future work. Instead we focus on the analysis
of p-branes in these background.

##  3 p-Brane in $k-$brane Carrollian Background

In this section we will study p-brane action in $k-$brane Carrollian
background. Recall that p-brane is $p+1$-dimensional object whose action has
the form

$S=-\tilde{\tau}_{p}\int d^{p+1}\xi\sqrt{-\det g_{\alpha\beta}}\ ,$ (13)

where
$g_{\alpha\beta}=g_{\mu\nu}\partial_{\alpha}x^{\mu}\partial_{\beta}x^{\nu}$ is
pull-back of target space metric $g_{\mu\nu}$ to $p+1$-dimensional world-
volume of p-brane that is parametrized by coordinates $\xi^{\alpha}\
,\quad\alpha,\beta,\dots=0,1,\dots,p$. Further,
$x^{\mu}(\xi),\mu,\nu,\dots=0,1,\dots,D$ are world-volume fields that define
embedding of p-brane into target space-time and
$\partial_{\alpha}x^{\mu}\equiv\frac{\partial x^{\mu}}{\partial\xi^{\alpha}}$.
Finally $\tilde{\tau}_{p}$ is $p+1$-brane tension. In order to define $k-$
Carroll limit we consider an PUL expansion of the metric

$g_{\mu\nu}=c^{2}\tau_{\mu}^{\ I}\tau_{\nu}^{\ J}\eta_{IJ}+h_{\mu\nu}\ ,$ (14)

where $I,J=0,1,\dots,k$. Then the induced metric has the form

$g_{\alpha\beta}=c^{2}\tau_{\alpha}^{\ I}\tau_{\beta}^{\
J}\eta_{IJ}+h_{\alpha\beta}\ .$ (15)

As the next step we introduce canonical form of p-brane action

$S=\int
d^{p+1}\xi(p_{\mu}\partial_{0}x^{\mu}-\tilde{N}\mathcal{H}_{\tau}-\tilde{N}^{i}\mathcal{H}_{i})\
,$ (16)

where $\mathcal{H}_{\tau}$ and $\mathcal{H}_{i}$ are following first class
constraints

$\displaystyle\mathcal{H}_{\tau}=p_{\mu}g^{\mu\nu}p_{\nu}+\tilde{\tau}_{p}^{2}\det
g_{ij}\ ,\quad g_{ij}=\partial_{i}x^{\mu}g_{\mu\nu}\partial_{j}x^{\nu}\ ,$
$\displaystyle\mathcal{H}_{i}=p_{\mu}\partial_{i}x^{\mu}\ ,\quad
i,j=1,\dots,p\ .$

Now we are ready to proceed to the formulation of p-brane action in $k-$brane
Carrollian background. Note that the PUL expansion of the inverse metric has
the form

$g^{\mu\nu}=\frac{1}{c^{2}}V^{\mu}_{\ I}V^{\nu}_{\ J}\eta^{IJ}+\Pi^{\mu\nu}\
.$ (18)

Due to the prefactor $c^{-2}$ (18) we see that the action (16) is finite when
we rescale $\tilde{N}$ as

$\tilde{N}=c^{2}N\ .$ (19)

However then it is also clear from the Hamiltonian constraint that we should
rescale $\tilde{\tau}_{p}$ as

$\tilde{\tau}_{p}=\frac{1}{c}\tau_{p}$ (20)

while we do not rescale $\tilde{N}^{i}$ so that $\tilde{N}^{i}=N^{i}$. As a
result we obtain Dp-brane action in the $k-$Carrollian background

$\displaystyle S=\int
d^{p+1}\xi(p_{\mu}\partial_{0}x^{\mu}-N(p_{\mu}v^{\mu}_{\
I}\eta^{IJ}v^{\nu}_{\ J}p_{\nu}+\tau_{p}^{2}\det
h_{ij})-N^{i}p_{\mu}\partial_{i}x^{\mu})\ ,$

where $h_{ij}=h_{\mu\nu}\partial_{i}x^{\mu}\partial_{j}x^{\nu}$. Note that
this canonical form of p-brane action can be formulated in any $k-$brane
Carrollian background.

Let us now determine corresponding Lagrangian density. Note that the
Hamiltonian has the form

$H=\int d^{p}\xi\mathcal{H}\ ,\quad\mathcal{H}=N(p_{\mu}v^{\mu}_{\
I}\eta^{IJ}v^{\nu}_{\ J}p_{\nu}+\tau_{p}^{2}\det
h_{ij})+N^{i}p_{\mu}\partial_{i}x^{\mu}\ .$ (22)

Using this Hamiltonian we obtain

$\partial_{0}x^{\mu}=\left\\{x^{\mu},H\right\\}=2Nv^{\mu}_{\
I}\eta^{IJ}v^{\nu}_{\ J}p_{\nu}+N^{i}\partial_{i}x^{\mu}\ $ (23)

and hence Lagrangian density is equal to

$\displaystyle\mathcal{L}=p_{\mu}\partial_{0}x^{\mu}-\mathcal{H}=Np_{\mu}v^{\mu}_{\
I}\eta^{IJ}v^{\nu}_{\ J}p_{\nu}-N\tau_{p}^{2}\det h_{ij}=$
$\displaystyle=\frac{1}{4N}(\partial_{0}x^{\mu}-N^{i}\partial_{i}x^{\mu})\tau_{\mu}^{\
I}\eta_{IJ}\tau_{\nu}^{\
J}(\partial_{0}x^{\nu}-N^{i}\partial_{i}x^{\nu})-N\tau_{p}^{2}\det h_{ij}\ ,$

using

$\displaystyle v^{\nu}_{\ J}p_{\nu}=-\frac{1}{2N}\eta_{JK}\tau_{\mu}^{\
K}(\partial_{0}x^{\mu}-N^{i}\partial_{i}x^{\mu})\ .$

Finally we should solve the equations of motion for $N^{i},N$. For $N^{i}$ it
has the form

$\tau_{i0}-\tau_{ij}N^{j}=0\ ,$ (26)

where we defined $\tau_{\alpha\beta}$ as

$\tau_{\alpha\beta}=\tau_{\alpha}^{\ I}\eta_{IJ}\tau_{\beta}^{\ J}\ .$ (27)

Next step depends on the question whether $\tau_{ij}$ is non-singular matrix
or not. In this section we will presume that $\tau_{ij}$ is non-singular
matrix and denote its inverse as $\tilde{\tau}^{ij}$ so that

$\tau_{ij}\tilde{\tau}^{jk}=\delta_{i}^{k}\ .$ (28)

Then $N^{i}$ is equal to

$N^{i}=\tilde{\tau}^{ij}\tau_{j0}\ .$ (29)

Further, the equation of motion for $N$ has the form

$-\frac{1}{4N^{2}}(\tau_{00}-2N^{i}\tau_{i0}+\tau_{ij}N^{i}N^{j})-\tau_{p}^{2}\det
h_{ij}=0\ .$ (30)

Inserting (29) into the equation above we find equation for $N$ that has
solution

$\displaystyle N=\frac{1}{2\tau_{p}}\sqrt{-\frac{\det\tau_{\alpha\beta}}{\det
h_{ij}\det\tau_{ij}}}\ .$

Finally inserting (29) and (3) into Lagrangian density (3) we finally get

$\displaystyle\mathcal{L}=-\tau_{p}\sqrt{-\det\tau_{\alpha\beta}}\sqrt{\frac{\det
h_{ij}}{\det\tau_{ij}}}\ .$

This is final form of the Lagrangian density for Dp-brane in $k-$Carroll
background that was derived on presumption that matrices $\tau_{ij}$ and
$h_{ij}$ are non-singular. Let us discuss these conditions in more details.

As the first case we discuss the special situation when $k=0$.

###  3.1 $k=0$

For $k=0$ the equation of motion for $N^{i}$ has the form

$-\tau_{i}\tau_{0}+\tau_{i}\tau_{j}N^{j}=0$ (33)

that for non-zero $\tau_{i}$ implies

$\tau_{i}N^{i}=\tau_{0}\ .$ (34)

Then inserting back to the Lagrangian density we get

$\displaystyle\mathcal{L}=\frac{1}{N}(-\tau_{0}^{2}+2N^{i}\tau_{i}\tau_{0}-N^{i}\tau_{i}N^{j}\tau_{j})-N\tau_{p}^{2}\det
h_{ij}=-N\tau_{p}^{2}\det h_{ij}$

and we see that there is no dynamics at all which is in agreement with the
particle Carroll limit studied in [6].

###  3.2 General $k$

Now we discuss in more details conditions under the matrices $\tau_{ij}$ and
$h_{ij}$ are non-singular. Recall that $\tau_{ij}=\tau_{i}^{\
I}\eta_{IJ}\tau_{j}^{\ J}$. Now $\tau_{i}^{\ I}$ is $p\times(k+1)$ matrix that
has the rank $\mathrm{min}(p,k+1)$. Further, $\eta_{IJ}$ is the matrix of the
rank $k+1$. Then $\tau_{ij}$, which is $p\times p$ matrix has either rank
$k+1$(for $k+1<p$) or $p$(for $p<k+1$). Let us discuss these two possibilities
in more details.

  * •

$\mathbf{k+1<p}$

Then the matrix $\tau_{ij}$ is singular and we cannot solve equations of
motion for $N^{i}$. Further, $h_{ij}=e_{i}^{\ a}e_{j}^{\ b}\delta_{ab}$ where
$e_{i}^{\ a}$ is $p\times(D-k)$ matrix that is non-singular for $p\leq D-k$.
If we combine these two conditions together we get $2k<D-1$ and hence
$k<\frac{D-1}{2}$. On the other hand the matrix $h_{ij}$ is singular for
$p>D-k$. In this case the Lagrangian density takes the form

$\mathcal{L}=\frac{1}{N}(\partial_{0}x^{\mu}-N^{i}\partial_{i}x^{\mu})\tau_{\mu}^{\
I}\eta_{IJ}\tau_{\nu}^{\ J}(\partial_{0}x^{\nu}-N^{i}\partial_{i}x^{\nu})\\\ $

that has effectively the form of massless Carroll Dp-brane. For example, in
case of the string Carroll limit when $k=1$ and when $D=9$ we get that this
corresponds to massless D8-brane.

  * •

$\mathbf{k+1\geq p}$

In this case the matrix $\tau_{ij}$ is non-singular. Further, $h_{ij}$ is non-
singular for $p\leq(D-K)$. This is exactly situation that leads to the
Lagrangian density (3). On the other hand when $D-K<p$ we find that $h_{ij}$
is singular and Lagrangian density has the form

$\mathcal{L}=-\frac{1}{N}(\partial_{0}x^{\mu}-N^{i}\partial_{i}x^{\mu})\tau_{\mu}^{\
I}\eta_{IJ}\tau_{\nu}^{\ J}(\partial_{0}x^{\nu}-N^{i}\partial_{i}x^{\nu})\ .$
(36)

Now we can solve equations of motion for $N^{i}$ that gives

$\tau_{0i}=N^{j}\tau_{ji}$ (37)

that can be solved for $N^{j}$ introducing inverse matrix $\tilde{\tau}^{ij}$.
Inserting back to the Lagrangian density (36) we obtain

$\mathcal{L}=-\frac{1}{N}(\tau_{00}-\tau_{0i}\tilde{\tau}^{ij}\tau_{j0})=-\frac{1}{N}\frac{\det\tau_{\alpha\beta}}{\det\tau_{ij}}\
.$ (38)

From this Lagrangian density we immediately see that the dynamics is trivial
since the equation of motion for $N$ implies that $\det\tau_{\alpha\beta}=0$
that is satisfied when $\partial_{0}x^{\mu}=0$ due to the fact that
$\tau_{ij}$ is non-singular.

##  4 Carroll Limit of Non-BPS Dp-Brane

In this section we consider non-BPS Dp-brane in $k-$brane Carrollian
background. For simplicity we will consider the case of zero NSNS two form and
RR background. In this case the tachyon effective action has the form

$S=-\tilde{\tau}_{non,p}\int
d^{p+1}e^{-\phi}V(T)\sqrt{-\det(g_{\alpha\beta}+\tilde{l}_{s}^{2}\tilde{F}_{\alpha\beta}+\partial_{\alpha}T\partial_{\beta}T)}\
,$ (39)

where

$g_{\alpha\beta}=g_{\mu\nu}\partial_{\alpha}x^{\mu}\partial_{\beta}x^{\nu}\
,\quad\tilde{F}_{\alpha\beta}=\partial_{\alpha}\tilde{A}_{\beta}-\partial_{\beta}\tilde{A}_{\alpha}\
.$ (40)

Further, $T$ is the tachyon field defined on the world-volume of unstable Dp-
brane, $V(T)$ is its potential when we presume that $V$ is even function with
three extremes, where $T_{0}=0$ is unstable maximum with $V(T_{0})=V_{max}$
while $\pm\infty$ are global minima of the potential with $V(\pm\infty)=0$.
Finally $\tilde{\tau}_{non,p}$ is tension for non-BPS Dp-brane and $\phi$ is
dilaton which generally depends on $x^{\mu}$.

As in previous section we begin with the canonical form of the action (39)
that has the form

$\displaystyle S=\int
d^{p+1}\xi(p_{\mu}\partial_{0}x^{\mu}+\tilde{\pi}^{i}\partial_{0}\tilde{A}_{i}-$
$\displaystyle-\tilde{N}(p_{\mu}g^{\mu\nu}p_{\nu}+p_{T}^{2}+\tilde{\pi}^{i}(g_{ij}+\partial_{i}T\partial_{j}T)\tilde{\pi}^{j}+\tilde{\tau}_{non,p}^{2}V^{2}e^{-2\phi}\det(g_{ij}+\tilde{l}_{s}^{2}\tilde{F}_{ij}+\partial_{i}T\partial_{j}T))-$
$\displaystyle-\tilde{N}^{i}(p_{\mu}\partial_{0}x^{\mu}+\tilde{F}_{ij}\tilde{\pi}^{j}+p_{T}\partial_{i}T)-\tilde{\pi}^{i}\partial_{i}\tilde{A}_{0})\
,$

where $p_{T}$ is momentum conjugate to $T$ and where $\tilde{\pi}^{i}$ is
momentum conjugate to $\tilde{A}_{i}$. Let us again consider metric components
in the form

$g_{\mu\nu}=c^{2}\tau_{\mu}^{\ I}\tau_{\nu}^{\ J}\eta_{IJ}+h_{\mu\nu}\ ,\quad
g^{\mu\nu}=\frac{1}{c^{2}}V^{\mu}_{\ I}V^{\nu}_{\ J}\eta^{IJ}+\Pi^{\mu\nu}\ ,$
(42)

so that the action has the form

$\displaystyle S=\int
d^{p+1}\xi(p_{\mu}\partial_{0}x^{\mu}+\tilde{\pi}^{i}\partial_{0}\tilde{A}_{i}-$
$\displaystyle-\tilde{N}(p_{\mu}(\frac{1}{c^{2}}V^{\mu}_{\ I}V^{\nu}_{\
J}\eta^{IJ}+\Pi^{\mu\nu})p_{\nu}+\tilde{\pi}^{i}(c^{2}\tau_{i}^{\
I}\tau_{j}^{\
J}\eta_{IJ}+h_{ij}+\partial_{i}T\partial_{j}T)\tilde{\pi}^{j}+p_{T}^{2}$
$\displaystyle+\tilde{\tau}_{non,p}^{2}e^{-2\phi}V^{2}\det(c^{2}\tau_{i}^{\
I}\tau_{j}^{\
J}\eta_{IJ}+h_{ij}+\tilde{l}_{s}^{2}\tilde{F}_{ij}+\partial_{i}T\partial_{j}T))-$
$\displaystyle-\tilde{N}^{i}(p_{\mu}\partial_{0}x^{\mu}+\tilde{F}_{ij}\tilde{\pi}^{j}+p_{T}\partial_{i}T)-\tilde{\pi}^{i}\partial_{i}\tilde{A}_{0})\
.$

First of all we see that we have to scale $\tilde{N}$ as

$\tilde{N}=c^{2}N$ (44)

in order to keep action finite. Then however to keep non-trivial gauge field
we should scale the momenta $\tilde{\pi}^{i}$ as

$\tilde{\pi}^{i}=\frac{1}{c}\pi^{i}\ ,\quad\tilde{A}_{\alpha}=cA_{\alpha}\ .$
(45)

Finally if we rescale $\tilde{\tau}_{p}$ as
$\tilde{\tau}_{non,p}=\frac{1}{c}\tau_{non,p}$ we should be careful with
rescaling $\tilde{l}_{s}$. Recall that
$\tilde{\tau}_{non,p}\sim\tilde{l}_{s}^{-(p+1)}$ so that we should scale
$\tilde{l}_{s}$ as

$\tilde{l}_{s}=c^{\frac{1}{p+1}}l_{s}\ .$ (46)

Then in the limit $c\rightarrow 0$ we find that the action takes the form

$\displaystyle S=\int
d^{p+1}\xi(p_{\mu}\partial_{0}x^{\mu}+\pi^{i}\partial_{0}A_{i}+p_{T}\partial_{0}T-$
$\displaystyle-N(p_{\mu}v^{\mu}_{\ I}\eta^{IJ}v^{\nu}_{\
J}p_{\nu}+\pi^{i}(h_{ij}+\partial_{i}T\partial_{j}T)\pi^{j}+\tau_{non,p}^{2}e^{-2\phi}V^{2}\det(h_{ij}+\partial_{i}T\partial_{j}T))-$
$\displaystyle-N^{i}(p_{\mu}\partial_{i}x^{\mu}+F_{ij}\pi^{j}+p_{T}\partial_{i}T)-\pi^{i}\partial_{i}A_{0})\
.$

This is final form of non-BPS Dp-brane in $k-$brane Carrollian background. In
the next section we focus on the canonical equations of motion and discuss
their solutions.

###  4.1 Equations of Motion and Their Solutions

Now we determine equations of motion that follow from the Hamiltonian density
of the action (4) that has the form

$\displaystyle H=\int d^{p}\xi\mathcal{H}\
,\quad\mathcal{H}=N\mathcal{H}_{\tau}+N^{i}\mathcal{H}_{i}-\pi^{i}\partial_{i}A_{0}\
,\quad\mathcal{H}_{i}=p_{\mu}\partial_{i}x^{\mu}+F_{ij}\pi^{j}+p_{T}\partial_{i}T\
,$ $\displaystyle\mathcal{H}_{\tau}=p_{\mu}v^{\mu}_{\ I}\eta^{IJ}v^{\nu}_{\
J}p_{\nu}+\pi^{i}(h_{ij}+\partial_{i}T\partial_{j}T)\pi^{j}+\tau_{non,p}^{2}e^{-2\phi}V^{2}\det(h_{ij}+\partial_{i}T\partial_{j}T)\
.$

With the help of this Hamiltonian we obtain following canonical equations of
motion

$\displaystyle\partial_{0}T=\left\\{T,H\right\\}=N^{i}\partial_{i}T\ ,$
$\displaystyle\partial_{0}p_{T}=\left\\{p_{T},H\right\\}=-2N\tau_{non,p}^{2}e^{-2\phi}V\frac{dV}{dT}\det(h_{ij}+N\partial_{i}T\partial_{j}T)+$
$\displaystyle+2\tau_{non,p}^{2}\partial_{i}[Ne^{-2\phi}V^{2}(\mathbf{A}^{-1})^{ij}\partial_{j}T\det\mathbf{A}]+\partial_{i}[N^{i}p_{T}]+2\partial_{i}[N\pi^{j}\partial_{j}T\pi^{i}]\
,$ $\displaystyle\partial_{0}x^{\mu}=\left\\{x^{\mu},H\right\\}=2Nv^{\mu}_{\
I}v^{\nu}_{\ J}\eta^{IJ}p_{\nu}+N^{i}\partial_{i}x^{\mu}\ ,$
$\displaystyle\partial_{0}A_{i}=\left\\{A_{i},H\right\\}=2Nh_{ij}\pi^{j}-F_{ij}N^{j}+\partial_{i}A_{0}\
,$
$\displaystyle\partial_{0}p_{\mu}=\left\\{p_{\mu},H\right\\}=-Np_{\rho}\partial_{\mu}(v^{\mu}_{\
I}v^{\nu}_{\ J}\eta^{IJ})p_{\nu}-\partial_{i}[N^{i}p_{\mu}]$
$\displaystyle+2\partial_{i}[N\pi^{i}h_{\mu\nu}\partial_{j}x^{\nu}\pi^{j}]-N\pi^{i}\partial_{i}x^{\rho}\partial_{\mu}h_{\rho\sigma}\partial_{j}x^{\sigma}-\tau_{non,p}^{2}\partial_{\mu}[e^{-2\phi}]V^{2}\det(h_{ij}+\partial_{i}T\partial_{j}T)+$
$\displaystyle+2\tau_{non,p}^{2}\partial_{i}[Ne^{-2\phi}V^{2}h_{\mu\nu}\partial_{j}x^{\nu}\left(\mathbf{A}^{-1}\right)^{ji}\det\mathbf{A}_{ij}]-\tau_{non,p}^{2}e^{-2\phi}NV^{2}\partial_{i}x^{\rho}\partial_{\mu}h_{\rho\sigma}\partial_{j}x^{\sigma}\left(\mathbf{A}^{-1}\right)^{ji}\det\mathbf{A}\
,$
$\displaystyle\partial_{0}\pi^{i}=\left\\{\pi^{i},H\right\\}=\partial_{k}[N^{k}\pi^{i}]-\partial_{l}[N^{i}\pi^{l}]\
,$

where we introduced matrix $\mathbf{A}_{ij}=h_{ij}+\partial_{i}T\partial_{j}T$
and $\left(\mathbf{A}^{-1}\right)^{jk}$ is its inverse
$\mathbf{A}_{ij}\left(\mathbf{A}^{-1}\right)^{jk}=\delta_{i}^{k}$.

Let us solve the equations of motion (4.1) with the ansatz corresponding to
the tachyon kink solution. Note that the tachyon kink solution was previously
studied in the context of relativistic non-BPS Dp-branes in [10, 11, 12, 13],
for review and extensive list of references, see [14]. We apply similar
procedure in case of non-BPS Dp-brane in $k-$ brane Carrollian background
where however we solve canonical equations of motion (4.1) instead of
Lagrangian ones that were analysed in [10, 11, 12, 13].

Explicitly, we presume that the tachyon has the profile

$T=f(a(y-t(\xi^{\hat{i}})))\ ,$ (50)

where we select one world-volume coordinate $\xi^{p}=y$ and where
$\hat{i}=1,\dots,p-1$. Note that we presume that $T$ does not depend on
$\xi^{0}$. We further presume that all world-volume fields depend on
$\xi^{\hat{\alpha}}$ only so that

$x^{\mu}(\xi^{\alpha})=\tilde{x}^{\mu}(\xi^{\hat{\alpha}})\ ,\quad
A_{\hat{i}}(\xi^{\alpha})=\tilde{a}_{\hat{i}}(\xi^{\hat{\alpha}})\ ,\quad
A_{y}=0\ ,\quad A_{0}(\xi^{\alpha})=\tilde{a}_{0}(\xi^{\hat{\alpha}})\ ,$ (51)

where $\hat{\alpha}=0,1,\dots,p-1$. The function $f(u)$ that appears in (50)
satisfies conditions

$f(u)=-f(-u)\ ,\quad f^{\prime}(u)>0\ \forall\ u\ ,\quad
f(\pm\infty)=\pm\infty$ (52)

but is otherwise an arbitrary function of its argument [10].

From the reasons that will be clear below we suggest following ansatz for
conjugate momenta

$p_{\mu}(\xi^{\alpha})=\frac{\tau_{non,p}}{\tau_{p-1}}af^{\prime}V\tilde{p}_{\mu}(\xi^{\hat{\alpha}})\
,\quad\pi^{\hat{i}}(\xi^{\alpha})=\frac{\tau_{non,p}}{\tau_{p-1}}af^{\prime}V\tilde{\pi}^{\hat{i}}(\xi^{\hat{\alpha}})\
,$ (53)

where the factor $\frac{\tau_{non,p}}{\tau_{p-1}}$ was included through
dimensional reasons. We also presume that $N$ has the form

$N(\xi^{\alpha})=\frac{\tau_{p-1}}{\tau_{non,p}}\tilde{n}(\xi^{\hat{\alpha}})\frac{1}{af^{\prime}V}\
,$ (54)

where again the factor $\frac{\tau_{p-1}}{\tau_{non,p}}$ was include through
dimensional reasons.

Inserting (50) into equation of motion for $T$ we get

$0=N^{y}af^{\prime}-N^{\hat{i}}\partial_{\hat{i}}tf^{\prime}a\\\ $

that can be solved for $N^{y}$ as

$N^{y}=\tilde{n}^{\hat{i}}\partial_{\hat{i}}t\ ,\quad
N^{\hat{i}}(\xi^{\alpha})\equiv\tilde{n}^{\hat{i}}(\xi^{\hat{\alpha}})\ .$
(55)

Further, the equation of motion for $N^{y}$ has the form

$\partial_{y}x^{\mu}p_{\mu}+\partial_{y}Tp_{T}=0$ (56)

that, since $\partial_{y}T\neq 0$ and since $\partial_{y}x^{\mu}=0$ for the
ansatz (51) implies that $p_{T}=0$.

Let us further consider equation of motion for $\pi^{\hat{i}}$ that for the
ansatz (51) and (53) has the form

$\displaystyle
af^{\prime}V\partial_{0}\tilde{\pi}^{\hat{i}}=\partial_{y}[af^{\prime}V]N^{y}\tilde{\pi}^{\hat{i}}+\partial_{\hat{k}}[af^{\prime}V]N^{\hat{k}}\tilde{\pi}^{\hat{i}}+af^{\prime}V\partial_{\hat{k}}[N^{\hat{k}}\tilde{\pi}^{\hat{i}}]-$
$\displaystyle-\partial_{\hat{j}}[af^{\prime}V]\tilde{\pi}^{\hat{j}}N^{\hat{i}}-af^{\prime}V\partial_{\hat{j}}[N^{\hat{i}}\tilde{\pi}^{\hat{j}}]-\partial_{y}\pi^{y}N^{\hat{i}}=$
$\displaystyle=af^{\prime}V\left(\partial_{\hat{k}}[\tilde{n}^{\hat{k}}\tilde{\pi}^{\hat{i}}]-\partial_{\hat{j}}[\tilde{n}^{\hat{i}}\tilde{\pi}^{\hat{j}}]\right)+\partial_{y}[af^{\prime}V]\partial_{\hat{j}}t\tilde{\pi}^{\hat{j}}\tilde{n}^{i}-\partial_{y}\pi^{y}\tilde{n}^{i}\
,$

where we used (55) together with the fact that

$\partial_{\hat{i}}(af^{\prime}V)=-\partial_{y}(af^{\prime}V)\partial_{\hat{i}}t\
$ (58)

as follows from the form of the tachyon field given in (50). Now in order to
cancel term proportional to $\partial_{y}[af^{\prime}V]$ we suggest that
$\pi^{y}$ is equal to

$\pi^{y}=af^{\prime}V\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}t\ .$ (59)

With this ansatz the equation (4.1) reduces into

$af^{\prime}V\left[\partial_{0}\tilde{\pi}^{\hat{i}}-\left(\partial_{\hat{k}}[\tilde{n}^{\hat{k}}\tilde{\pi}^{\hat{i}}]-\partial_{\hat{j}}[\tilde{n}^{\hat{i}}\tilde{\pi}^{\hat{j}}]\right)\right]=0\
.$ (60)

Now $af^{\prime}V$ goes to zero in the limit $a\rightarrow\infty$ for $y\neq
t(\xi^{\hat{i}})$. This can be easily seen when we note that $V(T)\sim e^{-T}$
for large $T$ [10]. On the other hand for $y=t(\xi^{\hat{i}})$ we get that
$af^{\prime}V$ is finite at $y=t(\xi^{\hat{i}})$ and hence in order to obey
equation of motion for $\pi^{i}$ we have to impose condition that

$\partial_{0}\tilde{\pi}^{\hat{i}}-\partial_{\hat{k}}[\tilde{n}^{\hat{k}}\tilde{\pi}^{\hat{i}}]+\partial_{\hat{j}}[\tilde{n}^{\hat{i}}\tilde{\pi}^{\hat{j}}]=0$
(61)

that are equations of motion for $\tilde{\pi}^{\hat{i}}$ on the world-volume
of D(p-1)-brane in $k-$brane Carrollian background. In more details, following
limiting procedure performed in case of non-BPS Dp-brane we obtain canonical
action for D(p-1)-brane in $k-$brane Carrollian background in the form

$\displaystyle S=\int
d^{p-1}\xi(\tilde{p}_{\mu}\partial_{0}\tilde{x}^{\mu}+\tilde{\pi}^{\hat{i}}\partial_{0}\tilde{a}_{\hat{i}}-\tilde{n}\tilde{\mathcal{H}}-\tilde{n}^{\hat{i}}\tilde{\mathcal{H}}_{i}-\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}\tilde{a}_{0})\
,$ $\displaystyle\tilde{\mathcal{H}}=\tilde{p}_{\mu}v^{\mu}_{\
I}\eta^{IJ}v^{\nu}_{\
J}\tilde{p}_{\nu}+\tilde{\pi}^{\hat{i}}h_{\hat{i}\hat{j}}\tilde{\pi}^{\hat{j}}+\tau_{p-1}^{2}e^{-2\phi}\det
h_{\hat{i}\hat{j}}\
,\quad\tilde{\mathcal{H}}_{\hat{i}}=\tilde{p}_{\mu}\partial_{\hat{i}}\tilde{x}^{\mu}+\tilde{F}_{\hat{i}\hat{j}}\tilde{\pi}^{\hat{j}}\
.$

The equations of motion that follow from this action have the form

$\partial_{0}\tilde{x}^{\mu}=2\tilde{n}v^{\mu}_{\ I}v^{\nu}_{\
J}\eta^{IJ}\tilde{p}_{\nu}+\tilde{n}^{\hat{i}}\partial_{\hat{i}}\tilde{x}^{\mu}\
,$ (63)
$\partial_{0}\tilde{a}_{\hat{i}}=2\tilde{n}h_{\hat{i}\hat{j}}\tilde{\pi}^{\hat{j}}-\tilde{F}_{\hat{i}\hat{j}}\tilde{n}^{j}+\partial_{\hat{i}}\tilde{a}_{0}\
,$ (64)

and

$\displaystyle\partial_{0}\tilde{p}_{\mu}=-\tilde{n}\tilde{p}_{\rho}\partial_{\mu}(v^{\mu}_{\
I}v^{\nu}_{\
J}\eta^{IJ})p_{\nu}-\partial_{\hat{i}}[\tilde{n}^{\hat{i}}p_{\mu}]$
$\displaystyle+2\partial_{\hat{i}}[\tilde{n}\tilde{\pi}^{\hat{i}}h_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}\tilde{\pi}^{\hat{j}}]-\tilde{n}\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}\tilde{x}^{\rho}\partial_{\mu}h_{\rho\sigma}\partial_{\hat{j}}\tilde{x}^{\sigma}-\tau_{p-1}^{2}\partial_{\mu}[e^{-2\phi}]\det
h_{\hat{i}\hat{j}}+$
$\displaystyle+2\tau_{p-1}^{2}\partial_{\hat{i}}[\tilde{n}e^{-2\phi}h_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}h^{\hat{j}\hat{i}}\det
h_{\hat{i}\hat{j}}]-\tau_{p-1}^{2}e^{-2\phi}\tilde{n}\partial_{\hat{i}}\tilde{x}^{\rho}\partial_{\mu}h_{\rho\sigma}\partial_{\hat{j}}\tilde{x}^{\sigma}h^{\hat{j}\hat{i}}\det
h_{\hat{i}\hat{j}}\ ,$

and finally

$\partial_{0}\tilde{\pi}^{\hat{i}}=\partial_{\hat{k}}[\tilde{n}^{\hat{k}}\tilde{\pi}^{\hat{i}}]-\partial_{\hat{k}}[\tilde{n}^{\hat{i}}\tilde{\pi}^{\hat{k}}]\
.$ (66)

We see that the equation (4.1) is obeyed when the conjugate momenta
$\tilde{\pi}^{\hat{i}}$ obey the equations of motion (66). In other words, the
dynamics of conjugate momenta $\tilde{\pi}^{\hat{i}}$ at the core of kink
localized at $y=t(\xi^{\hat{i}})$ is governed equations of motion that follow
from the variation of D(p-1)-brane action in $k-$brane Carrollian background.

As the next step we should study equation of motion for $\pi^{y}$. Firstly
inserting (53) and (59) together with (55) into right side of the equation of
motion for $\pi^{y}$ we obtain

$\displaystyle
af^{\prime}V(\partial_{\hat{k}}[\tilde{n}^{\hat{k}}\pi^{\hat{i}}]-\partial_{\hat{k}}[\tilde{\pi}^{\hat{k}}\tilde{n}^{\hat{i}}])\partial_{\hat{i}}t\
,$ (67)

where we also used (58). On the other hand time derivative of (59) gives

$\partial_{0}\pi^{y}=af^{\prime}V\partial_{0}\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}t\
.$ (68)

If we combine (68) with (67) we get that the equation of motion for $\pi^{y}$
is obeyed on condition when $\tilde{\pi}^{\hat{i}}$ solves the equations of
motion (66) which proves consistency of our ansatz.

As the next step we proceed to the equations of motion for $T$ and $p_{T}$.
Note that for the ansatz (50) and (51) we have

$\det(h_{ij}+\partial_{i}T\partial_{j}T)=\det h_{\hat{i}\hat{j}}a^{2}f^{\prime
2}\ .$ (69)

Then after some calculations we find that right side of the equation of motion
for $p_{T}$ is identically zero

$\displaystyle-2\tilde{n}e^{-2\phi}\frac{dV}{dT}af^{\prime}\det
h_{\hat{i}\hat{j}}+$
$\displaystyle\partial_{y}[\tilde{n}e^{-2\phi}V^{2}a^{2}f^{\prime
2}(\frac{1}{a^{2}f^{\prime
2}}+\partial_{\hat{m}}th^{\hat{m}\hat{n}}\partial_{\hat{n}}t)\det
h_{\hat{i}\hat{j}}]-\partial_{y}[\tilde{n}e^{-2\phi}V^{2}a^{2}f^{\prime
2}\partial_{\hat{m}}th^{\hat{m}\hat{i}}\partial_{\hat{i}}t\det
h_{\hat{i}\hat{j}}]+$
$\displaystyle+\partial_{\hat{i}}[\tilde{n}e^{-2\phi}V^{2}a^{2}f^{\prime
2}\partial_{\hat{m}}th^{\hat{m}\hat{i}}\det
h_{\hat{i}\hat{j}}]-\partial_{\hat{i}}[\tilde{n}e^{-2\phi}V^{2}a^{2}f^{\prime
2}h^{\hat{i}\hat{j}}\partial_{\hat{j}}t\det h_{\hat{i}\hat{j}}]=0$

for the ansatz (50) and (51). In more details, we used the fact that the
matrix has the form

$\left(\mathbf{A}^{-1}\right)=\left(\begin{array}[]{cc}h^{\hat{i}\hat{j}}&h^{\hat{i}\hat{m}}\partial_{\hat{m}}t\\\
\partial_{\hat{m}}th^{\hat{m}\hat{j}}&\frac{1}{a^{2}f^{\prime
2}}+\partial_{\hat{m}}th^{\hat{m}\hat{n}}\partial_{\hat{n}}t\\\
\end{array}\right)$ (71)

which, together with (58) also implies

$\displaystyle\partial_{i}[Ne^{-2\phi}V^{2}\left(\mathbf{A}^{-1}\right)^{ij}\partial_{j}T\det\mathbf{A}_{ij}]=\frac{\tau_{p-1}}{\tau_{non,p}}\partial_{y}[\tilde{n}e^{-2\phi}Vaf^{\prime
2}(\frac{1}{a^{2}f^{\prime
2}}+\partial_{\hat{m}}th^{\hat{m}\hat{n}}\partial_{\hat{n}}t)\det
h_{\hat{i}\hat{j}}]-$
$\displaystyle-\frac{\tau_{p-1}}{\tau_{non,p}}\partial_{y}[\tilde{n}e^{-2\phi}Vaf^{\prime}\partial_{\hat{m}}th^{\hat{m}\hat{i}}\partial_{\hat{i}}t\det
h_{\hat{i}\hat{j}}]+$
$\displaystyle+\frac{\tau_{p-1}}{\tau_{non,p}}\partial_{\hat{i}}[\tilde{n}e^{-2\phi}Vaf^{\prime}\partial_{\hat{m}}th^{\hat{m}\hat{i}}\det
h_{\hat{i}\hat{j}}]-\frac{\tau_{p-1}}{\tau_{non,p}}\partial_{\hat{i}}[\tilde{n}e^{-2\phi}Vaf^{\prime}h^{\hat{i}\hat{j}}\partial_{\hat{j}}t\det
h_{\hat{i}\hat{j}}]\ .$

We also used the fact that for (50),(51) and (58)
$\partial_{i}[N\pi^{i}\partial_{j}T\pi^{j}]$ is equal zero. Since $p_{T}$ is
zero we finally find that the equations of motion for $T$ is obeyed for the
configuration of fields defined above. Note that $t(\xi^{\hat{i}})$ is
completely free which is manifestation of spatial diffeomorphism invariance of
the Hamiltonian formalism. Let us further discuss equations of motion for
$x^{\mu}$ that for the ansatz (50),(51),(53) and (54) reduces into

$\displaystyle
af^{\prime}V\left(\partial_{0}\tilde{x}^{\mu}-2\tilde{n}v^{\mu}_{\
I}v^{\nu}_{\
J}\eta^{IJ}\tilde{p}_{\nu}-\tilde{n}^{\hat{i}}\partial_{\hat{i}}\tilde{x}^{\nu}\right)=0$

and we see that this equation is obeyed at the core of the kink
$y=t(\xi^{\hat{i}})$ when the expression in the bracket vanishes which is
nothing else then the equation of motion (63).

Further, inserting ansatz given above to the equations of motion for
$A_{\hat{i}}$ we obtain

$af^{\prime}V\left(\partial_{0}\tilde{a}_{i}-2\tilde{n}h_{\hat{i}\hat{j}}\tilde{\pi}^{\hat{j}}+F_{\hat{i}\hat{k}}\tilde{n}^{\hat{k}}-\partial_{\hat{i}}\tilde{a}_{0}\right)=0$
(74)

that is again obeyed at the core of the kink when the expression in the
bracket that corresponds to (64), is zero. In case of the equations of motion
for $A_{y}$ we get that it is identically obeyed for $A_{y}=0$.

Let us now proceed to the equation of motion for $p_{\mu}$. Again, inserting
ansatz (50),(51),(53) together with (55) and using (58) we find that it takes
the form

$\displaystyle\frac{\tau_{non,p}}{\tau_{p-1}}af^{\prime}V\left(-\partial_{0}\tilde{p}_{\mu}-\tilde{n}\tilde{p}_{\rho}\partial_{\mu}(v^{\rho}_{\
I}v^{\nu}_{\
J}\eta^{IJ})\tilde{p}_{\nu}+\partial_{\hat{i}}[\tilde{n}\tilde{\pi}^{\hat{i}}h_{\mu\nu}\partial_{\hat{j}}x^{\nu}\tilde{\pi}^{\hat{j}}]\right.$
$\displaystyle-\tilde{n}\tilde{\pi}^{\hat{i}}\partial_{\hat{i}}\tilde{x}^{\rho}\partial_{\mu}h_{\rho\sigma}\partial_{\hat{j}}\tilde{x}^{\sigma}\tilde{\pi}^{\hat{j}}+2\tau_{p-1}^{2}\partial_{\hat{i}}[\tilde{n}e^{-2\phi}h_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}h^{\hat{j}\hat{i}}\det
h_{\hat{i}\hat{j}}]-$
$\displaystyle\left.-\tau_{p-1}^{2}\tilde{n}\partial_{\mu}[e^{-2\phi}]\det
h_{\hat{i}\hat{j}}-\tau_{p-1}^{2}\tilde{n}e^{-2\phi}\partial_{\hat{i}}\tilde{x}^{\rho}\partial_{\mu}h_{\rho\sigma}\partial_{\hat{j}}x^{\sigma}h^{\hat{j}\hat{i}}\det
h_{\hat{i}\hat{j}}\right)=0\ ,$

where we used the fact that

$\displaystyle\tau_{non,p}^{2}\partial_{i}[Ne^{-2\phi}V^{2}h_{\mu\nu}\partial_{j}x^{\nu}\left(\mathbf{A}^{-1}\right)^{ji}\det\mathbf{A}]=2\tau_{non,p}\tau_{p-1}Vaf^{\prime}\partial_{\hat{i}}[\tilde{n}e^{-2\phi}h_{\mu\nu}\partial_{\hat{j}}\tilde{x}^{\nu}h^{\hat{j}\hat{i}}\det
h_{\hat{i}\hat{j}}]\ ,$
$\displaystyle\partial_{i}[N\pi^{i}h_{\mu\nu}\partial_{\hat{j}}x^{\nu}\pi^{\hat{j}}]=\frac{\tau_{non,p}}{\tau_{p-1}}af^{\prime}V\partial_{\hat{i}}[\tilde{n}\tilde{\pi}^{\hat{i}}h_{\mu\nu}\partial_{\hat{j}}x^{\nu}\tilde{\pi}^{\hat{j}}]\
.$

In summary, we showed that fields given in (50),(51) and (53) solve equations
of motion for non-BPS Dp-brane in $k-$brane Carrollian background on
conditions when the fluctuations around kink solution obey canonical equations
of motion for D(p-1)-brane in the same background.

###  4.2 Fundamental String Solutions

Let us now discuss solutions when the tachyon field on the world-volume of
unstable Dp-brane is in its vacuum state corresponding to
$V(T_{min})=\frac{dV}{dT}(T_{min})=0$. This situation was extensively studied
in case of relativistic tachyon effective action [15, 16, 17, 9, 11, 18] and
in case of the flat Carroll Dp-bran in [9]. Our analysis will closely follow
[15] and also [9].

To begin with we again write equations of motion for Dp-brane in $k-$brane
Carrollian background at the tachyon vacuum state. In this case the equations
of motion simplify considerably

$\displaystyle\partial_{0}x^{\mu}=2NV^{\mu}_{\ I}V^{\nu}_{\
J}\eta^{IJ}p_{\nu}+N^{i}\partial_{i}x^{\mu}\ ,$
$\displaystyle\partial_{0}p_{\mu}=-Np_{\rho}\partial_{\mu}(v^{\rho}_{\
I}v^{\nu}_{\ J}\eta^{IJ})p_{\nu}$
$\displaystyle+2\partial_{i}[N\pi^{i}h_{\mu\nu}\partial_{j}x^{\nu}\pi^{j}]-N\pi^{i}\partial_{i}x^{\rho}\partial_{\mu}h_{\rho\sigma}\partial_{j}x^{\sigma}-\partial_{i}[N^{i}p_{\mu}]\
,$
$\displaystyle\partial_{0}\pi^{i}=\partial_{k}[N^{k}\pi^{i}]-\partial_{l}[N^{i}\pi^{l}]\
,$
$\displaystyle\partial_{0}A_{i}=2Nh_{ij}\pi^{j}-F_{ij}N^{j}+\partial_{i}A_{0}\
,$ $\displaystyle p_{\mu}v^{\mu}_{\ I}v^{\nu}_{\
J}\eta^{IJ}p_{\nu}+\pi^{i}h_{ij}\pi^{j}=0\ ,\quad
p_{\mu}\partial_{i}x^{\mu}+F_{ij}\pi^{j}=0\ .$

We would like to show that these equations of motion can be interpreted as
equations of motion for fundamental string in $k-$brane Carrollian background.
To begin with let us consider string action in the canonical form

$S=\int d\tau
d\sigma(K_{\mu}\partial_{\tau}Z^{\mu}-\tilde{\lambda}_{\tau}\mathcal{K}_{\tau}-\tilde{\lambda}_{\sigma}\mathcal{K}_{\sigma})\
,$ (78)

where

$\mathcal{K}_{\tau}=K_{\mu}g^{\mu\nu}K_{\nu}+\tilde{\tau}_{F}^{2}\partial_{\sigma}Z^{\mu}\partial_{\sigma}Z^{\nu}g_{\mu\nu}\
,\quad\mathcal{K}_{\sigma}=K_{\mu}\partial_{\sigma}Z^{\mu}\ ,$ (79)

and where $K_{\mu}(\tau,\sigma)$ are momenta conjugate to
$Z^{\mu}(\tau,\sigma)$ and where the world-sheet is labeled by $\tau$ and
$\sigma$. Let us consider Carroll expansion of the metric

$g_{\mu\nu}=c^{2}T_{\mu}^{\ \tilde{I}}T_{\nu}^{\
\tilde{J}}\eta_{\tilde{I}\tilde{J}}+h_{\mu\nu}\ ,\quad
g^{\mu\nu}=\frac{1}{c^{2}}V^{\mu}_{\ \tilde{I}}V^{\nu}_{\
\tilde{J}}\eta^{\tilde{I}\tilde{J}}+\Pi^{\mu\nu}\ ,$ (80)

where $\tilde{I},\tilde{J}$ vary from $0,\dots,l$ where generally $l$ is
different from $k$. Then we also rescale $\tilde{N}$ and $\tilde{\tau}_{F}$ as

$\tilde{\lambda}_{\tau}=c^{2}\lambda_{\tau}\
,\quad\tilde{\lambda}_{\sigma}=\lambda_{\sigma}\
,\quad\tilde{\tau}_{F}=\frac{1}{c}\tau_{F}\ .$ (81)

As a result we obtain an action for string in $l-$brane Carrollian background
in the form

$\displaystyle S=\int d\tau
d\sigma(K_{\mu}\partial_{0}Z^{\mu}-\lambda_{\tau}(K_{\mu}v^{\mu}_{\
\tilde{I}}v^{\nu}_{\
\tilde{J}}\eta^{\tilde{I}\tilde{J}}K_{\nu}+\tau_{F}^{2}h_{\sigma\sigma})-\lambda_{\sigma}K_{\mu}\partial_{\sigma}Z^{\mu})\
.$

From this action we derive corresponding equations of motion

$\displaystyle\partial_{\tau}Z^{\mu}=2\lambda_{\tau}v^{\mu}_{\
\tilde{I}}v^{\nu}_{\
\tilde{J}}\eta^{\tilde{I}\tilde{J}}K_{\nu}+\lambda_{\sigma}\partial_{\sigma}Z^{\mu}\
,$
$\displaystyle-\partial_{\tau}K_{\mu}=-\lambda_{\tau}K_{\rho}\partial_{\mu}(v^{\rho}_{\
\tilde{I}}\eta^{\tilde{I}\tilde{J}})K_{\nu}-\tau_{F}^{2}\lambda_{\tau}\partial_{\sigma}Z^{\rho}\partial_{\mu}h_{\rho\sigma}\partial_{\sigma}Z^{\sigma}+$
$\displaystyle+2\tau_{F}^{2}\partial_{\sigma}[\lambda_{\tau}\partial_{\sigma}Z^{\nu}h_{\nu\mu}]+\partial_{\sigma}[\lambda_{\sigma}K_{\mu}]\
,$ $\displaystyle K_{\mu}v^{\mu}_{\ \tilde{I}}v^{\nu}_{\
\tilde{J}}\eta^{\tilde{I}\tilde{J}}K_{\nu}+\tau_{F}^{2}h_{\sigma\sigma}=0\
,\quad K_{\mu}\partial_{\sigma}Z^{\mu}=0\ .$

In order to interpret the equations of motion for non-BPS Dp-brane at the
tachyon vacuum (4.2) as the equations of motion for Carrollian string we
immediately see that the range of $I$ should coincide with $\tilde{I}$ so that
$k=l$. Then we proceed in similar way as in [9]. First of all we introduce
projector

$\triangle^{i}_{\
j}=\delta^{i}_{j}-\frac{\pi^{i}h_{jk}\pi^{k}}{\pi^{m}h_{mn}\pi^{n}}$ (84)

that obeys

$\triangle^{i}_{\ j}\pi^{j}=0\ ,\quad\triangle^{i}_{\ j}\triangle^{j}_{\
k}=\triangle^{i}_{\ k}\ .$ (85)

Then we split $N^{i}$ as

$N^{i}=N^{j}\delta_{j}^{i}=N^{j}(\triangle_{j}^{i}+\frac{\pi^{i}h_{jk}\pi^{k}}{\pi^{m}h_{mn}\pi^{n}})=N^{i}_{\bot}+\pi^{i}N_{II}\
,$ (86)

where by definition

$\pi^{i}h_{ij}N^{j}_{\bot}=0\ .$ (87)

As the next step we introduce dimensionless vector $\omega^{i}$ through the
relation

$\pi^{i}=\tau_{non,p}\omega^{i}\ .$ (88)

and search for solution when $\pi^{i}$ is constant. In fact, this ansatz
clearly solves equation $\partial_{i}\pi^{i}=0$ while the equations of motion
for $\pi^{i}$ take the form

$\displaystyle
0=\partial_{0}\pi^{i}=\partial_{k}N^{k}_{\bot}\pi^{i}-\partial_{\sigma}N^{i}_{\bot}\tau_{non,p}\
,$

where we introduced $\sigma$ coordinate through the relation

$\partial_{\sigma}\equiv\omega^{i}\partial_{i}\ .$ (90)

We solve the equation (4.2) above by imposing $N^{k}=\mathrm{const}$. Note
however that $N_{II}$ is still non-specified so that there is still spatial
dimensional invariance along $\sigma$ direction. We further presume that all
world-volume fields depend on $\sigma$ only. Then when we perform
identification

$\pi_{\mu}(\xi^{0},\sigma)|_{\xi^{0}=\tau}=\frac{\tau_{non,p}}{\tau_{F}}K_{\mu}(\tau,\sigma)\
,\quad x^{\mu}(\xi^{0},\sigma)|_{\xi^{0}=\tau}=Z^{\mu}(\tau,\sigma)$ (91)

we see that the equations of motion for non-BPS Dp-brane at the tachyon vacuum
reduces to the equations of motion for string in $k-$brane Carrollian
background. Clearly the dynamics of this string is again trivial however this
result shows consistency of the tachyon vacuum condensation in case of non-BPS
Dp-brane in $k-$brane Carrollian background.

Acknowledgement:
This work is supported by the grant “Integrable Deformations” (GA20-04800S)
from the Czech Science Foundation (GACR).

## References

  * [1] D. Hansen, N. A. Obers, G. Oling and B. T. Søgaard, _“Carroll Expansion of General Relativity,”_ [arXiv:2112.12684 [hep-th]].
  * [2] D. Van den Bleeken, _“Torsional Newton–Cartan gravity from the large c expansion of general relativity,”_ Class. Quant. Grav. 34 (2017) no.18, 185004 doi:10.1088/1361-6382/aa83d4 [arXiv:1703.03459 [gr-qc]].
  * [3] D. Hansen, J. Hartong and N. A. Obers, _“Action Principle for Newtonian Gravity,”_ Phys. Rev. Lett. 122 (2019) no.6, 061106 doi:10.1103/PhysRevLett.122.061106 [arXiv:1807.04765 [hep-th]].
  * [4] D. Hansen, J. Hartong and N. A. Obers, _“Non-Relativistic Gravity and its Coupling to Matter,”_ JHEP 06 (2020), 145 doi:10.1007/JHEP06(2020)145 [arXiv:2001.10277 [gr-qc]].
  * [5] E. Bergshoeff, J. Gomis and L. Parra, _“The Symmetries of the Carroll Superparticle,”_ J. Phys. A 49 (2016) no.18, 185402 doi:10.1088/1751-8113/49/18/185402 [arXiv:1503.06083 [hep-th]].
  * [6] B. Cardona, J. Gomis and J. M. Pons, _“Dynamics of Carroll Strings,”_ JHEP 07 (2016), 050 doi:10.1007/JHEP07(2016)050 [arXiv:1605.05483 [hep-th]].
  * [7] E. Bergshoeff, J. M. Izquierdo and L. Romano, _“Carroll versus Galilei from a Brane Perspective,”_ JHEP 10 (2020), 066 doi:10.1007/JHEP10(2020)066 [arXiv:2003.03062 [hep-th]].
  * [8] D. Roychowdhury, _“Carroll membranes,”_ JHEP 10 (2019), 258 doi:10.1007/JHEP10(2019)258 [arXiv:1908.07280 [hep-th]].
  * [9] J. Kluson, _“Carroll Limit of Non-BPS Dp-Brane,”_ JHEP 05 (2017), 108 doi:10.1007/JHEP05(2017)108 [arXiv:1702.08685 [hep-th]].
  * [10] A. Sen, _“Dirac-Born-Infeld action on the tachyon kink and vortex,”_ Phys. Rev. D 68 (2003), 066008 doi:10.1103/PhysRevD.68.066008 [arXiv:hep-th/0303057 [hep-th]].
  * [11] J. Kluson, _“Note about unstable D-branes with dynamical tension,”_ Phys. Rev. D 94 (2016) no.4, 046004 doi:10.1103/PhysRevD.94.046004 [arXiv:1605.09510 [hep-th]].
  * [12] J. Kluson, _“Tachyon kink on non-BPS Dp-brane in the general background,”_ JHEP 10 (2005), 076 doi:10.1088/1126-6708/2005/10/076 [arXiv:hep-th/0508239 [hep-th]].
  * [13] J. Kluson, _“Note about tachyon kink in nontrivial background,”_ JHEP 08 (2005), 032 doi:10.1088/1126-6708/2005/08/032 [arXiv:hep-th/0506250 [hep-th]].
  * [14] A. Sen, _“Tachyon dynamics in open string theory,”_ Int. J. Mod. Phys. A 20 (2005), 5513-5656 doi:10.1142/S0217751X0502519X [arXiv:hep-th/0410103 [hep-th]].
  * [15] A. Sen, _“Open and closed strings from unstable D-branes,”_ Phys. Rev. D 68 (2003), 106003 doi:10.1103/PhysRevD.68.106003 [arXiv:hep-th/0305011 [hep-th]].
  * [16] C. j. Kim, H. B. Kim, Y. b. Kim and O. K. Kwon, _“Electromagnetic string fluid in rolling tachyon,”_ JHEP 03 (2003), 008 doi:10.1088/1126-6708/2003/03/008 [arXiv:hep-th/0301076 [hep-th]].
  * [17] G. W. Gibbons, K. Hori and P. Yi, _“String fluid from unstable D-branes,”_ Nucl. Phys. B 596 (2001), 136-150 doi:10.1016/S0550-3213(00)00716-1 [arXiv:hep-th/0009061 [hep-th]].
  * [18] O. K. Kwon and P. Yi, _“String fluid, tachyon matter, and domain walls,”_ JHEP 09 (2003), 003 doi:10.1088/1126-6708/2003/09/003 [arXiv:hep-th/0305229 [hep-th]].
  * [19] C. Duval, G. W. Gibbons, P. A. Horvathy and P. M. Zhang, _“Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time,”_ Class. Quant. Grav. 31 (2014), 085016 doi:10.1088/0264-9381/31/8/085016 [arXiv:1402.0657 [gr-qc]].
  * [20] J. Hartong, _“Gauging the Carroll Algebra and Ultra-Relativistic Gravity,”_ JHEP 08 (2015), 069 doi:10.1007/JHEP08(2015)069 [arXiv:1505.05011 [hep-th]].
  * [21] J. Kluson, _“Note about Hamiltonian formalism for Newton–Cartan string and p-brane,”_ Eur. Phys. J. C 78 (2018) no.6, 511 doi:10.1140/epjc/s10052-018-5993-8 [arXiv:1712.07430 [hep-th]].
  * [22] C. D. A. Blair, D. Gallegos and N. Zinnato, _“A non-relativistic limit of M-theory and 11-dimensional membrane Newton-Cartan geometry,”_ JHEP 10 (2021), 015 doi:10.1007/JHEP10(2021)015 [arXiv:2104.07579 [hep-th]].
  * [23] J. Kluson, _“Stable and unstable D $p$-branes in $p$-brane Newton–Cartan background,”_ J. Phys. A 54 (2021) no.21, 215401 doi:10.1088/1751-8121/abf768 [arXiv:2003.14037 [hep-th]].
  * [24] J. Klusoň and P. Novosad, _“Non-Relativistic M2-Brane,”_ JHEP 06 (2019), 072 doi:10.1007/JHEP06(2019)072 [arXiv:1903.12450 [hep-th]].
  * [25] A. Sen, _“Supersymmetric world volume action for nonBPS D-branes,”_ JHEP 10 (1999), 008 doi:10.1088/1126-6708/1999/10/008 [arXiv:hep-th/9909062 [hep-th]].
  * [26] E. A. Bergshoeff, M. de Roo, T. C. de Wit, E. Eyras and S. Panda, _“T duality and actions for nonBPS D-branes,”_ JHEP 05 (2000), 009 doi:10.1088/1126-6708/2000/05/009 [arXiv:hep-th/0003221 [hep-th]].
  * [27] J. Kluson, _“Proposal for nonBPS D-brane action,”_ Phys. Rev. D 62 (2000), 126003 doi:10.1103/PhysRevD.62.126003 [arXiv:hep-th/0004106 [hep-th]].


Study of the semi-boosted topology in top anti-top quark pair spectra

J. Pácalt

Regional Center of Advanced Technologies and Materials
Joint Laboratory of Optics of Palacký University and Institute of Physics AS
CR, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc,
Czech Republic

The aim of this contribution is to present results of adding the semi-boosted
transition region between the resolved and boosted top quark pair
reconstruction in the $\ell$ \+ jets channel. This region is often omitted due
to more complex reconstruction, which usually leads to a smaller efficiency.
The reconstruction is performed for all three topologies for comparison and
combination purposes. Hypothetical particle $Z^{\prime}$ was used as a probe
to study the increase in the number of events in the studied energy region.

> PRESENTED AT
>
>
>
>
> $12^{\mathrm{th}}$ International Workshop on Top Quark Physics
> Beijing, People’s Republic of China, September 22–27, 2019

##  1 Introduction

This contribution aims to explore possibilities of adding an almost non-used
semi-boosted topology into reconstruction procedures for studies of top anti-
top quark pair spectra in the semi-leptonic decay channel ($\ell$+jets). The
top anti-top quark pair decay channels are named according to the way the $W$
boson from the decay of top quark decays either to a lepton and a neutrino
(leptonic decay), or to a pair of a quark and an anti-quark (hadronic decay).
In the semi-leptonic top anti-top decay channel, one $W$ boson decays
leptonically and the other one hadronically. Two topologies are commonly used
at present, resolved and boosted, but in between them lies a region, where the
resolved topology transits to the boosted one and neither of those has a full
reconstruction efficiency. This semi-boosted region of energies is often
omitted from analyses.

##  2 Reconstruction of events and topologies

The resolved topology considers all reconstructed elements (jets and leptons)
entering the to quark reconstruction angularly separated from each other, in
contrary to the boosted topology, which uses collimated products (jets) coming
from the hadronically decaying top quark, reconstructed as one object (large
jet). The semi-boosted topology is a transition between those two topologies,
where the b-tagged 111At Large Hadron Collider is the b-tagged jet defined as
a jet with a secondary vertex, which was identified by the algorithm as a
decay of the $B$-hadron inside the jet. The Delphes package uses an efficiency
formula from the real detector for the determination of the flavor of quark,
this also simulates the misidentification of the tagging algorithm. jet coming
from the hadronically decaying top quark is not merged together with the two
jets from the W boson decay, which are collimated, or one jet coming from W
boson decay is separated while the other is collimated with the b-tagged jet
(semi-boosted mixed topology). Reconstruction of the leptonically decaying top
quark is the same for all topologies by using the lepton with high a
transverse momentum, $E_{\mathrm{T,miss}}$ (missing energy in transverse
plane, which is usually carried away by neutrinos) and the b-tagged jet
closest to the lepton by imposing the condition $m_{\ell\nu}=m_{\mathrm{W}}$.
The situation for the boosted and semi-boosted topologies is shown in Figure 1
a) and b), respectively. Tagging of large jets as coming either from the W
boson or the top quark decay is done by using a substructure variable
$\tau_{32}$ [1] for the top quark tagging and $\tau_{21}$ for the W boson
tagging, and a mass window for the large jet, the chosen regions are shown in
Figure 1 c) and d), respectively.

---
a)
b)

|
---|---
c) | d)

Figure 1: Schematic of the studied boosted (a) and semi-boosted (b)
topologies. Tagging regions of large jets as top quark candidates in the
$\tau_{32}$ and large jet mass plane (c); and W boson candidates in the
$\tau_{21}$ and large jet mass plane (d).

Migration of events between topologies between particle and detector levels
and migration of events for a particular kinematic variable are shown in
Figure 2.

|
---|---
Figure 2: The migration of events between the resolved (R), semi-boosted mixed
(SB-mix, SBM), semi-boosted (SB) and boosted (B) topologies between particle
and detector levels (left), and the migration matrix for the transverse
momentum of the reconstructed top anti-top quark pair for sample with
$m_{\mathrm{Z^{\prime}}}=700$ GeV (right).

The reconstruction algorithm is set to first try the boosted reconstruction,
if the event does not pass the criteria, then both semi-boosted reconstruction
algorithms are tried. If the event is not selected by any of them, resolved
reconstruction algorithm is attempted. The event is discarded if all three
reconstruction algorithms fail.

##  3 Samples and resolution

The simulated samples of the process $pp\rightarrow Z^{\prime}\rightarrow
t\bar{t}$ at $\sqrt{s}=14$ TeV were prepared by the MadGraph generator version
2.6.4 [2] interfaced with Pythia8 [3] for the parton showering and
hadronization simulation; and then processed by the Delphes package version
3.4.1 [4] for a simplified parameterized detector simulation with different
values of $Z^{\prime}$ mass to cover the effective range of energies, where
the semi-boosted topology is relevant. This range begins around 500 GeV and
ends around 1000 GeV. The prepared samples were used as input for a private
framework based on ROOT 6.16 [5]. Fraction of events as function of the
reconstructed mass of the top anti-top quark pair for all the topologies for a
sample with $m_{\mathrm{Z^{\prime}}}=700$ GeV is shown in Figure 3.

|
---|---
Figure 3: Fraction of events in resolved, semi-boosted mixed, semi-boosted and
boosted topologies at particle level (left) and detector level (right) for the
sample with $m_{\mathrm{Z^{\prime}}=700}$ GeV.

There is an obvious difference between the resolution at the particle and
detector levels. The invariant mass of the reconstructed top anti-top quark
pair mass was fitted by a Gaussian function and the resolution, in terms of
fitted $Z^{\prime}$ mass width, for different samples and topologies is shown
as both absolute and relative values with respect to $Z^{\prime}$ mass of the
studied sample in Figure 4. The relative resolution in the semi-boosted case,
especially at the detector level, is similar to the resolution of resolved and
boosted topologies.

|
---|---
Figure 4: Resolution of reconstructed $m_{\mathrm{t\bar{t}}}$ for topologies
as function of the $Z^{\prime}$ mass.

##  4 Unfolding, closure test and the migration matrix properties

Reconstruction was also put to test through the unfolding procedure. The Fully
Bayesian Unfolding (FBU) procedure [6] was used to correct for influences of
the detector. Unfolding can be described by the following formula

$\hat{T}_{\mathrm{i}}=f_{\mathrm{i,eff}}M_{\mathrm{ij}}^{-1}f_{\mathrm{j,acc}}(D_{\mathrm{j}}-B_{\mathrm{j}}),$
(1)

where $\hat{T}_{\mathrm{i}}$ is the unfolded particle level spectrum,
$D_{\mathrm{j}}$ is the detector level spectrum, $B_{\mathrm{j}}$ is the
background spectrum contributing to the detector level spectrum,
$M_{\mathrm{ij}}$ is the corresponding migration matrix between detector and
particle levels and the inversion symbolizes the unfolding procedure; and
$f_{\mathrm{j,acc}}$ and $f_{\mathrm{i,eff}}$ are acceptance and efficiency
correction factors. The example of the unfolded posterior, migration matrix
and example of the detector, particle and unfolded spectra are shown in Figure
5.

|
---|---

---

Figure 5: An example of the posterior in a selected bin (top left), the
migration matrix (top right) between the detector and the particle level; and
the unfolded spectrum for $p_{\mathrm{T,toph}}$ (bottom) for the sample with
$m_{\mathrm{Z^{\prime}}}=700$ GeV for the semi-boosted topology, where the
ratio is with respect to the particle level spectrum and the yellow band is
RMS of the posterior in the given bin.

The unfolding procedure is working well, this can be seen from the achieved
closure test in the Figure 5, where the deviation of the unfolded spectrum
from the particle level spectrum is under 3%.

##  5 Conclusion

This contribution presented a study of possible efficiency enhancement of top
anti-top quark pair analyses by adding semi-boosted topologies in
consideration. The semi-boosted resolution is similar to other topologies in
given $Z^{\prime}$ mass region and it helps to increase the number of events
by $60-70$%, this could be beneficial especially to boosted analyses. The next
step will be to find out the way to combine results from different topologies.

##  6 Acknowledgements

This article was supported by MSMT projects GACR 19-21484S and LTT-17018 and
by Palacky University project IGA_PrF_2019_008.

## References

  * [1] Jesse Thaler and Ken Van Tilburg.  Identifying boosted objects with n-subjettiness.  Journal of High Energy Physics, 2011(3), Mar 2011.
  * [2] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro.  The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations.  JHEP, 07:079, 2014.
  * [3] Torbjörn Sjöstrand, Stefan Ask, Jesper R. Christiansen, Richard Corke, Nishita Desai, Philip Ilten, Stephen Mrenna, Stefan Prestel, Christine O. Rasmussen, and Peter Z. Skands.  An introduction to pythia 8.2.  Computer Physics Communications, 191:159–177, Jun 2015.
  * [4] J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi.  DELPHES 3, A modular framework for fast simulation of a generic collider experiment.  JHEP, 02:057, 2014.
  * [5] Rene Brun and Fons Rademakers.  ROOT - An Object Oriented Data Analysis Framework.  Nucl. Inst. & Meth. in Phys. Res. A, 389:81–86, 1997.
  * [6] Georgios Choudalakis.  Fully bayesian unfolding.  https://arxiv.org/abs/1201.4612v4, 2012.