#  Two Particles with Zero-Range Interaction in a Magnetic Field

Johannes Kirscher  Theoretical Physics Division, School of Physics and
Astronomy, The University of Manchester, Manchester M13 9PL, UK  Brian C.
Tiburzi  Department of Physics, The City College of New York, New York, NY
10031, USA  Graduate School and University Center, The City University of New
York, New York, NY 10016, USA

###### Abstract

Energy levels are investigated for two charged particles possessing an
attractive, momentum-independent, zero-range interaction in a uniform magnetic
field. A transcendental equation governs the spectrum, which is characterized
by a collective Landau-level quantum number incorporating both center-of-mass
and relative degrees of freedom. Results are obtained for a system of one
charged and one neutral particle, with the interaction chosen to produce a
bound state in vanishing magnetic field. Beyond deriving the weak-field
expansion of the energy levels, we focus on non-perturbative aspects. In the
strong-field limit, or equivalently for a system in the unitary limit, a
single bound level with universal binding energy exists. By contrast, excited
states are resonances that disappear into the continuum as the magnetic field
is raised beyond critical values. A hyperbola is derived that approximates the
number of bound levels as a function of the field strength remarkably well.

##  I Introductory Remarks

The study of quantum systems in external fields is central to many
disciplines, with applications ranging from the study of ultracold gases of
trapped atoms [1] to quantum field theories in condensed matter, nuclear, and
particle physics [2]. In numerical gauge-theory calculations, external fields
present a practical computational method to study properties of light nuclei
using lattice quantum chromodynamics (LQCD). For an overview, see [3, 4, 5]
and references therein. The first LQCD results for light nuclei in external
magnetic fields [6, 7, 8] point to an intriguing possibility. The energies of
a subset of two-nucleon systems in large magnetic fields appear to approach
binding thresholds, which suggests an analogue of a Feshbach resonance [9, 10,
11]. While such resonances are extensively studied in atomic systems [12], the
LQCD results hint at a similar effect in two-nucleon systems. The magnetic
field, moreover, could provide a lever with which to tune two-nucleon systems
simultaneously to threshold, arriving at a point of conformal symmetry [13].

Unlike atomic systems, the magnetic fields required to alter strongly
interacting systems are exceptionally large, ${\mathcal{O}}(10^{19})$ Gauss.
Fields of this size are routine in LQCD computations, due to quantization
conditions required of uniform fields on a torus [14]. In astrophysics, strong
magnetic fields have been conjectured to occur in the interiors of magnetars
[15, 16, 17]. Such extreme fields, have been estimated to exist in non-central
heavy-ion collisions [18, 19], moreover, for which considerable work has
explored the implications for QCD matter [20].

In the present work, we investigate a two-body problem in uniform magnetic
fields. For atomic systems, there is substantial work on Coulomb interactions
in an external magnetic field, _e.g._ [21]. The analogous study of QCD bound
states has begun recently; examples include quarkonium in external magnetic
fields [22, 23], as well as modification of the quark-antiquark potential
computed from LQCD [24]. For two protons in a magnetic field, the center-of-
mass and relative motion can be separated. In this system, the effect of a
sufficiently large magnetic field has been proven to result in a bound di-
proton state [25]. The argument hinges on magnetic confinement of the protons
into narrow cyclotron orbits. In such a scenario, the short-range nuclear
interaction results in an attractive one-dimensional potential well, which
always supports a bound state.111In atomic systems, a related argument was
made to prove the existence of bound negative ions in large magnetic fields
[26]. For this case, the atomic polarizability leads to the attractive
interaction with the electron. To address the neutron-proton system in
magnetic fields, however, one needs to confront the non-separability of the
center-of-mass and relative degrees of freedom, and we take a first step in
this work.

General results for charged particles with translationally invariant
interactions in a magnetic field were established in [27]. In particular, a
quantum number exists that collectively incorporates the center-of-mass and
relative degrees of freedom, and each level is infinitely degenerate. This
provides a striking generalization of the Landau problem to all values of the
magnetic field. We exhibit these results by solving for the energy levels of
two charged particles interacting with a contact potential. The choice of an
attractive, momentum-independent, zero-range interaction is motivated by the
pion-less effective theory of few-nucleon systems [28, 29, 30, 31].222
Throughout this initial study, we assume the particles are scalars. Treatment
of the two-nucleon problem requires including spin and magnetic moments, which
is left to future work.  The zero-range interaction between particles produces
valuable simplifications that facilitate determining the spectrum. The
energies of this system are found to obey a transcendental equation; moreover,
through a judicious form of renormalization, the inverse solution (magnetic
field as a function of energy) is obtained simply by quadrature.

We begin our treatment with a general discussion in Sec. II, where we exhibit
the quantum number incorporating both center-of-mass and relative degrees of
freedom. In Sec. III, we focus on a system consisting of one charged and one
neutral particle, for which the spectrum is obtained as a function of the
magnetic field. Perturbative expansions are derived about the small-field and
unitary limits. The number of bound levels supported as a function of the
magnetic field is shown to obey a simple equation. A summary of results is
given in Sec. IV, while various technical details are provided in Apps. A and
B.

##  II General Discussion

The system under consideration is that of two particles with charges $q_{a}$
and masses $m_{a}$, where $a=1$, $2$. The particles are in a uniform magnetic
field and interact through an attractive, zero-range potential in three
spatial dimensions. The Hamiltonian for this system has the form

$H=H^{(0)}-c\,{\delta}(\bm{r}),$ (1)

where $\bm{r}={\bm{r}}_{1}-{\bm{r}}_{2}$ is the relative coordinate between
the particles. Without the two-body potential,333 A regulator is needed to
define the zero-range potential in Eq. (1); hence, $c$ is a running coupling.
We demonstrate in Eqs. (17) and (19) that the running of $c$ determined in
zero magnetic field yields regulator-independent equations for the spectrum
even when the field is present; i.e., the magnetic field does not affect the
interaction at zero range.  the Hamiltonian for two particles in a magnetic
field is additive

$H^{(0)}=\sum_{a=1}^{2}H^{(0)}_{a}=\sum_{a=1}^{2}\frac{\big{[}{\bm{p}}_{a}-q_{a}{\bm{A}}({\bm{r}}_{a})\big{]}^{2}}{2m_{a}},$
(2)

where ${\bm{A}}$ is the vector potential that specifies the magnetic field.
For arbitrary values of the charges and masses of the two particles, there is
no general separation between the relative and center-of-mass motion.444
Solutions for the separable case of equal charge-to-mass ratios do emerge as a
limit of the general treatment above, see Footnote 9. A simpler method for
solving such systems, however, is to enforce factorization of the relative and
center-of-mass wave-functions from the outset. Additionally, the special case
where the total charge of the system vanishes requires a separate treatment.
At the quantum level, the eigenvalues of $H$ are neither the sum of relative
and center-of-mass contributions, nor are they the sum of single-particle
contributions.

A bound state with energy ${\mathcal{E}}$ appears as a solution to the
homogeneous Lippmann-Schwinger integral equation

$\displaystyle\Psi({\bm{r}}_{1},{\bm{r}}_{2})=-c\int
d{\bm{R}}\,G^{(0)}({\bm{r}}_{1},{\bm{r}}_{2};{\bm{R}},{\bm{R}}|{\mathcal{E}})\Psi({\bm{R}},{\bm{R}}),\,\,$
(3)

where $G^{(0)}$ is the non-interacting two-particle Green’s function

$\displaystyle
G^{(0)}({\bm{r}}^{\prime}_{1},{\bm{r}}^{\prime}_{2};{\bm{r}}_{1},{\bm{r}}_{2}|{\mathcal{E}})=\big{\langle}{\bm{r}}^{\prime}_{1},{\bm{r}}^{\prime}_{2}\big{|}\frac{1}{{\mathcal{E}}-H^{(0)}+i{\epsilon}}\big{|}{\bm{r}}_{1},{\bm{r}}_{2}\big{\rangle},\quad\,\,\,$
(4)

and the integration is over the center-of-mass coordinate ${\bm{R}}$. By
virtue of the contact potential, the integration over the relative separation
amounts to an evaluation of the wave-function in the integrand at $\bm{r}=0$.
While center-of-mass and relative coordinates are not completely decoupled in
$H^{(0)}$, partial factorization of the wave-function is possible. Assuming a
non-vanishing charge for the system $Q\equiv q_{1}+q_{2}$, the center-of-mass
wave-function is specified by the momentum parallel to the magnetic field,
along with one other quantum number.555 More formally, we can simultaneously
specify two components of the so-called pseudo-momentum [27]. Parallel to the
magnetic field, the pseudo-momentum is simply the total momentum $P_{z}$. The
$x$-component of pseudo-momentum is $-\sqrt{QB}\,\xi$, where $\xi$ appears in
Eq. (11).  This quantum number can be chosen as an additional component of the
total momentum, for which the asymmetric gauge is natural. Therefore, we adopt

${\bm{A}}({\bm{r}})=\left(-By,0,0\right),$ (5)

leading to the magnetic field ${\bm{B}}=B\bm{\hat{z}}$. Accordingly, the $x$\-
and $z$-components of the total momentum ${\bm{P}}={\bm{p}}_{1}+{\bm{p}}_{2}$,
are good quantum numbers. For any vector ${\bm{v}}$, we define a related
vector $\bm{\widetilde{v}}$, with two non-vanishing components
$\bm{\widetilde{v}}=\left(v_{x},0,v_{z}\right)$, so that, _e.g._ ,
$\bm{\widetilde{P}}$ refers to the good components of ${\bm{P}}$.

The two-particle wave-function can be written in the partially factorized form

$\Psi({\bm{r}}_{1},{\bm{r}}_{2})=e^{i\bm{\widetilde{P}}\cdot\bm{\widetilde{R}}}\,\,\Psi_{P_{x}}(Y,\bm{r}),$
(6)

in which $\Psi_{P_{x}}(Y,\bm{r})$ reflects the coupled center-of-mass and
relative motion. The $\bm{\widetilde{P}}$-projected non-interacting Green’s
function is obtained through the Fourier decomposition

$\displaystyle
G^{(0)}({\bm{r}}^{\prime}_{1},{\bm{r}}^{\prime}_{2};{\bm{r}}_{1},{\bm{r}}_{2}|{\mathcal{E}})$
$\displaystyle=$
$\displaystyle\int\frac{d\bm{\widetilde{P}}}{(2{\pi})^{2}}\,e^{i\bm{\widetilde{P}}\cdot\left(\bm{\widetilde{R}}^{\prime}-\bm{\widetilde{R}}\right)}\qquad\qquad$
(7) $\displaystyle\phantom{sp}\times
G^{(0)}_{P_{x}}(Y^{\prime},\bm{r}^{\prime};Y,\bm{r}|E),\,\,$

where the shifted energy $E={\mathcal{E}}-\frac{P_{z}^{2}}{2M}$ serves to
remove all $P_{z}$-dependence from the projected Green’s function. Projecting
Eq. (3) onto states of good $\bm{\widetilde{P}}$, leads to the wave equation

$\displaystyle\Psi_{P_{x}}(Y,\bm{r})=-c\int
dY^{\prime}G^{(0)}_{P_{x}}(Y,\bm{r};Y^{\prime},{\bm{0}}|E)\Psi_{P_{x}}(Y^{\prime},{\bm{0}}).\quad\,\,\,$
(8)

As with Eq. (3), the wave-function appearing in the integrand is evaluated at
zero relative separation owing to the contact interaction. Knowledge of this
restricted wave-function $\Psi_{P_{x}}(Y^{\prime},{\bm{0}})$ thus determines
the value for $\bm{r}\neq{\bm{0}}$ by convolution with the non-interacting
Green’s function.

The restricted equation

$\displaystyle\Psi_{P_{x}}(Y,{\bm{0}})=-c\int
dY^{\prime}G^{(0)}_{P_{x}}(Y,{\bm{0}};Y^{\prime},{\bm{0}}|E)\Psi_{P_{x}}(Y^{\prime},{\bm{0}}),\quad\quad$
(9)

is obtained by evaluating Eq. (8) at $\bm{r}={\bm{0}}$, and suffices to
determine the energy $E$. Simplifications arise because the configuration
space is restricted to both particles being at the same point, for which there
remains only center-of-mass motion of a system having charge $Q$. This is
manifest in the functional form of the restricted non-interacting Green’s
function

$G^{(0)}_{P_{x}}(Y,{\bm{0}};Y^{\prime},{\bm{0}}|E)=\sqrt{QB}\,\,g^{(0)}({\xi};{\xi}^{\prime}|E),$
(10)

which has a simple form in terms of the dimensionless center-of-mass
coordinates

${\xi}=\sqrt{QB}\,(Y-Y_{0})\quad\text{and}\quad{\xi}^{\prime}=\sqrt{QB}\,(Y^{\prime}-Y_{0}),$
(11)

where $Y_{0}=\frac{P_{x}}{QB}$ is the $y$-coordinate of the center of
cyclotron motion of the restricted system. Written as a function of the
variables ${\xi}$ and ${\xi}^{\prime}$, the restricted Green’s function
$g^{(0)}$ has no dependence on $P_{x}$. Consequently, the energy determined
from Eq. (9) is independent of $P_{x}$, which confirms that each $E$ is
infinitely degenerate.

For arbitrary charges and masses of the two particles, explicit computation
establishes that $g^{(0)}$ is diagonal in the basis of Hermite functions666
The dimensionless Hermite functions employed above form an orthonormal basis,
where the $n$-th member is given by
$\psi_{n}({\xi})=H_{n}({\xi})\,e^{-\frac{1}{2}{\xi}^{2}}\Big{/}\sqrt{2^{n}\,n!\sqrt{{\pi}}},$
with $H_{n}({\xi})$ as the corresponding Hermite polynomial.

$g^{(0)}({\xi};{\xi}^{\prime}|E)=\sum_{n=0}^{\infty}g^{(0)}_{n}(E)\,\psi_{n}({\xi})\,\psi_{n}({\xi}^{\prime}).$
(12)

Projecting Eq. (9) onto the $n$-th Hermite function, we obtain the equation

$\frac{1}{c}+g^{(0)}_{n}(E)=0,$ (13)

that determines $E$ for each value of $n$. This quantum number is the Landau
level of the center of mass only when the two particles are restricted to the
same point. With $\bm{r}\neq{\bm{0}}$, the Green’s function
$G^{(0)}_{P_{x}}(Y,\bm{r};Y^{\prime},{\bm{0}}|E)$ is not diagonal in the basis
of Hermite functions; accordingly, factorization of $\Psi_{P_{x}}(Y,\bm{r})$
does not result from Eq. (8). This reflects that the quantum number $n$
incorporates collective behavior of both center-of-mass and relative degrees
of freedom.

##  III Determining the Energies:
An Example

As an example, we solve for the spectrum of a system consisting of one charged
and one neutral particle. Accordingly, we take $q_{1}=Q$ and $q_{2}=0$. To
exhibit various properties of the solution, we keep the masses $m_{1}$ and
$m_{2}$ different.

###  III.1 Equations for the Spectrum

For this system, computation of the restricted, non-interacting Green’s
function, Eq. (10), is detailed in App. A. The calculation produces the
coefficients $g_{n}^{(0)}(E)$ of Eq. (12) expressed in terms of a Laplace
transform. The equation for the bound-state energy, Eq. (13), thus takes the
form

$\displaystyle\frac{1}{c(t_{0})}-\int_{t_{0}}^{\infty}dt\,e^{Et}\,g^{(0)}_{n}(t)=0,$
(14)

where we introduce a small-time cutoff $t_{0}$ to regulate the divergence at
$t=0$.777 Identical results are obtained using more conventional
regularization schemes, such as dimensional regularization (using $d-2$
spatial dimensions longitudinal to the magnetic field) with the power-
divergence subtraction scheme [32, 33].  In Eq. (14), the coefficients are

$\displaystyle g^{(0)}_{n}(t)$ $\displaystyle=$
$\displaystyle\left(\frac{{\mu}}{2\pi
t}\right)^{\frac{3}{2}}\frac{M}{m_{1}}\,\frac{\sinh\frac{\Omega(t)\,t}{2}}{\sinh\frac{{\omega}t}{2}}\,e^{-\left(n+\frac{1}{2}\right){\Omega}(t)\,t},$
(15)

and have been written in terms of the charged particle’s cyclotron frequency
${\omega}=QB/m_{1}$, as well as a time-dependent frequency $\Omega(t)$ defined
through the relation

$\coth\frac{{\Omega}(t)\,t}{2}=\coth\frac{{\omega}t}{2}+\frac{m_{2}}{m_{1}}\left(\frac{{\omega}t}{2}\right)^{-1}.$
(16)

This time-dependent frequency is a curious feature governing the restricted
cyclotron motion of the center of mass.

To renormalize Eq. (14), we inspect the small-time behavior of the integrand.
In this limit, there is no magnetic field dependence, and we find the zero-
field result

$\frac{1}{c(t_{0})}-\int_{t_{0}}^{\infty}dt\,e^{E_{0}t}\left(\frac{{\mu}}{2\pi
t}\right)^{\frac{3}{2}}=0,$ (17)

where $E_{0}<0$ denotes the energy in zero magnetic field. The $t_{0}\to 0$
divergence of the integral in Eq. (14) thus subtracts the identical divergence
in $\frac{1}{c(t_{0})}$. A renormalized form of Eq. (14) defined for $t_{0}=0$
is therefore

$\displaystyle\int_{0}^{\infty}dt\left[e^{Et}\,g^{(0)}_{n}(t)-e^{E_{0}t}\left(\frac{{\mu}}{2\pi
t}\right)^{\frac{3}{2}}\right]=0.$ (18)

Any real energy $E$ below the two-particle dissociation threshold, see Eq.
(22), that solves this equation corresponds to an infinitely degenerate bound
state with quantum number $n$. Below, we utilize this equation to obtain the
small-field expansion.

An alternate equation for the spectrum is obtained by realizing that Eq. (17)
can be rewritten as

$\displaystyle\frac{1}{c(t_{0})}-\int_{t_{0}}^{\infty}dt\,e^{Et}\left(\frac{{\mu}}{2{\pi}t}\right)^{\frac{3}{2}}=\frac{{\mu}}{2{\pi}}\left[\sqrt{-2{\mu}E}-\sqrt{-2{\mu}E_{0}}\right]$
(19)

with $E$ corresponding to a general energy, and small $t_{0}$ understood for
the integral. Using Eq. (19) for the running coupling, the renormalized and
analytically continued equation for the energy $E$ can be cast in the form

$\displaystyle b^{-\frac{1}{2}}={\mathcal{I}}_{n}(\chi,x),\quad$ (20)

where the function ${\mathcal{I}}_{n}(\chi,x)$ is defined in App. B. Appearing
above, $x$ is the mass ratio $x=m_{1}/M$, the magnetic field in dimensionless
units is defined as

$b=\frac{QB}{-2{\mu}E_{0}},$ (21)

and the energy $E$ is written in terms of $\chi(b)$ in the form

$E=\frac{{\omega}}{2}-\chi\frac{QB}{2\mu}.$ (22)

This parametrization of the energy accounts for the fact that the threshold
for two-particle decay is set by the energy of the lowest Landau level of the
charged particle. With the sign of the second term in Eq. (22), $\chi$ is the
binding energy in units of $\frac{QB}{2\mu}$. Notice that Eq. (20) readily
gives the inverse function $b(\chi)$ by simply evaluating
${\mathcal{I}}_{n}(\chi,x)$, which is a one-dimensional integral. In
accordance with regularization independence, the energies determined
numerically from Eqs. (18) and (20) agree.

###  III.2 Small-Field Limit

For small magnetic fields, one can treat ${\omega}t\ll 1$,888 This treatment
seems invalid when $t\gtrsim\omega^{-1}$, however, the integrand appearing in
Eq. (18) is exponentially small in this domain.  for which the time-dependent
frequency in Eq. (16) becomes approximately time independent and equal to the
cyclotron frequency ${\Omega}$ of the center of mass

${\Omega}(t)\approx\frac{QB}{M}\equiv{\Omega}.$ (23)

Retaining only this linear magnetic field dependence, the equation for the
energies, Eq. (18), becomes

$\int_{0}^{\infty}\frac{dt}{t^{3/2}}\left(e^{\left[E-(n+\frac{1}{2}){\Omega}\right]t}-e^{E_{0}t}\right)=0,$
(24)

and leads to the requirement $E=E_{0}+(n+\frac{1}{2}){\Omega}$. For small
magnetic fields, the energies are ordinary Landau levels of the center of
mass.

Figure 1: Behavior of the lowest level in small magnetic fields. For the case
of equal masses, the ratio of the $n{=}0$ energy level $E$ to that in
vanishing magnetic field $E_{0}$ is plotted (solid curve) as a function of the
magnetic field, using the center-of-mass cyclotron frequency $\Omega$ in Eq.
(23). Also plotted (dashed curves) are the perturbative approximations in
powers of $\Omega/E_{0}$ from Eq. (25). While the state becomes less bound,
note that $E=0$ does not correspond to threshold, see Eq. (22).

To obtain corrections to the leading-order result in small fields, we write
${\Delta}E=E-\left[E_{0}+(n+\frac{1}{2}){\Omega}\right]$, and iteratively
solve for ${\Delta}E$ using Eq. (18). To third-order in the magnetic field, we
find

$\displaystyle E$ $\displaystyle=$ $\displaystyle
E_{0}\Big{[}1+\Big{(}n+\frac{1}{2}\Big{)}\frac{{\Omega}}{E_{0}}-\frac{1}{48}\Big{(}4+\frac{m_{2}}{m_{1}}\Big{)}\frac{m_{2}}{m_{1}}\Big{(}\frac{{\Omega}}{E_{0}}\Big{)}^{2}$
(25)
$\displaystyle\phantom{sp}-\frac{1}{16}\Big{(}n+\frac{1}{2}\Big{)}\frac{m_{2}}{m_{1}}\Big{(}\frac{{\Omega}}{E_{0}}\Big{)}^{3}+{\mathcal{O}}(B^{4})\Big{]}.$

In particular, the term at second order is independent of the quantum number
$n$, and allows us to identify the magnetic polarizability of the bound state.
Using the customary definition of the second-order energy shift
${\Delta}E=-\frac{1}{2}4{\pi}{\beta}_{M}B^{2}+\cdots$, we obtain the
polarizability

${\beta}_{M}=\frac{{\alpha}}{24}\left(4+\frac{m_{2}}{m_{1}}\right)\frac{m_{2}}{m_{1}}\,\frac{(Q/e)^{2}}{E_{0}M^{2}},$
(26)

where ${\alpha}=e^{2}/4{\pi}$ is the fine-structure constant. The magnetic
polarizability is diamagnetic ${\beta}_{M}<0$, due to the absence of
paramagnetic spin contributions.

In Fig. 1, we exhibit the small-field behavior of the $n=0$ energy level
determined from Eq. (20). For simplicity, we take the equal mass case
$m_{1}=m_{2}$. Additionally shown is the small-field behavior determined
analytically from Eq. (25). The perturbative expansion lines up well with the
numerically determined energy, and appears to work beyond the
$\Omega/|E_{0}|\ll 1$ regime.

###  III.3 Unitary Limit

To determine the behavior of the binding energy in the unitary limit $E_{0}\to
0$, we turn to Eq. (20). With $b\gg 1$, the binding energy in the unitary
limit $\chi^{(0)}_{\infty}$ is obtained as the root of the equation

$\displaystyle{\mathcal{I}}_{n}(\infty)\equiv{\mathcal{I}}_{n}\Big{(}\chi^{(0)}_{\infty},x\Big{)}=0.$
(27)

For $n>0$, there are no roots; only the $n{=}0$ level is bound for a system at
unitarity. The binding energy in this regime, moreover, is independent of the
short-range structure of the interaction. The bound state is thus a universal
feature, albeit, in a limited sense. In dimensionless units, the unitary-limit
binding energy still depends upon the mass ratio between the charged and
neutral particles. This mass-ratio dependence is a consequence of the coupling
between relative and center-of-mass motion. For identical masses
($x{=}\tfrac{1}{2}$), we obtain the numerical value
$\chi^{(0)}_{\infty}=0.081588$. Only when the charged particle is much more
massive than the neutral one ($x{\to}1$) does the system become unbound
$\chi^{(0)}_{\infty}=0$. In the opposite limit, the binding energy of a system
with a relatively light charge ($x{\to}0$) attains, by contrast, its largest
value $\chi_{\infty}^{(0)}=0.605444$, which is twice the root of the Hurwitz
zeta-function, see Eq. (39). Curiously, this value also appears as the binding
energy of the spin-zero di-proton system in a strong magnetic field [25].999
We remark that the spectrum of that system near unitarity can be determined
from
$b^{-\frac{1}{2}}=-\frac{1}{\sqrt{2}}\zeta\left(\frac{1}{2},\frac{\chi}{2}\right)$,
which happens to be the $x\ll 1$ limit of Eq. (20).  In other words, a single
charge that interacts resonantly with a fixed force center (our model for
$x{\to}0$) has the same ground-state energy as two particles with equal
charge-to-mass ratios that interact via long-range Coulomb repulsion in
addition to the zero-range attraction.

Figure 2: Behavior of the lowest level near unitarity. For the case of equal
masses, $\chi$, the binding energy in units of $\frac{QB}{2{\mu}}$, is plotted
(solid curve) as a function of the inverse magnetic field, using $b$ defined
in Eq. (21). Also plotted (dashed curves) are the successive approximations in
Eq. (28) about the unitary limit $b\gg 1$. For this level, the threshold
$\chi{=}0$ is never reached.

Away from unitarity, the binding energy is no longer universal as it depends
on $E_{0}$ through the $b^{-\frac{1}{2}}$ corrections. To obtain these
corrections, we write the binding energy in the expansion

$\chi=\chi^{(0)}_{\infty}+\frac{1}{\sqrt{b}}\,\chi^{(1)}_{\infty}+\frac{1}{b}\,\chi^{(2)}_{\infty}+{\mathcal{O}}\big{(}b^{-\frac{3}{2}}\big{)},$
(28)

and then solve for the higher-order terms using Eq. (20) iteratively. This
procedure results in

$\displaystyle\chi^{(1)}_{\infty}$ $\displaystyle=$ $\displaystyle
1\Big{/}{\mathcal{I}}^{\prime}_{0}(\infty)=0.388235,$
$\displaystyle\chi^{(2)}_{\infty}$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}\,{\mathcal{I}}^{\prime\prime}_{0}(\infty)\Big{/}[{\mathcal{I}}^{\prime}_{0}(\infty)]^{3}=0.567853,\quad$
(29)

where derivatives of ${\mathcal{I}}_{n}(\chi,x)$ are with respect to $\chi$,
and the evaluation is at the location of the root in Eq. (27), namely at
$\chi=\chi^{(0)}_{\infty}$. The formulas apply to any mass ratio, however, the
numerical values quoted are exclusively for the equal-mass case
($x{=}\frac{1}{2}$). As the corrections away from unitarity are positive, the
magnetic field always increases the ground-state binding energy and thus does
not allow crossing to a virtual state.

In Fig. 2, we show the approach to unitarity of the $n=0$ energy level
determined from Eq. (20). For simplicity, we again take the equal-mass case.
Additionally shown is the behavior near the unitary limit obtained from Eq.
(28), which does well in describing the magnetic-field dependence of the
lowest energy level in this regime.

###  III.4 Binding Thresholds and the Number of Levels

The binding threshold is reached when $\chi{=}0$. As remarked above, the
lowest level does not reach threshold; it remains bound in the unitary limit,
which can alternately be viewed as the large-field limit with $E_{0}$ held
fixed. The excited levels, however, reach threshold at critical values of the
magnetic field.101010 Note that for all values of the magnetic field, excited
levels are unstable to radiative decay to the ground level. Such decays
require electrodynamics, which is beyond the scope of this investigation.
According to Eq. (20), the threshold for single-particle production occurs
when the magnetic field has the value

$\displaystyle b_{c}=\big{[}{\mathcal{I}}_{n}(0,x)\big{]}^{-2},$ (30)

provided ${\mathcal{I}}_{n}(0,x)>0$. For the first excited level and with
equal masses for the particles, we find

$b_{c}=3.87554,\,\,\,\text{for}\,\,n{=}1\,\,\text{and}\,\,x{=}\tfrac{1}{2}.$
(31)

For the higher-lying levels, the critical value of the magnetic field
decreases monotonically. Consequently, we have the single-particle decay
threshold at $b_{c}\ll 1$, for $n\gg 1$. Viewed as a function of the level
number $n$, the critical field $b_{c}(n)$ can be obtained analytically in the
large-$n$ limit. The analysis detailed in App. B produces the approximate
formula

$b_{c}=\frac{4}{2n-1}\left[1+{\mathcal{O}}(n^{-2})\right],$ (32)

for the equal mass case. Surprisingly, the formula works at $3\%$ already for
the $n{=}1$ level. Results improve with increasing $n$; the approximate
critical field for $n{=}2$ differs by $0.6\%$ from the exact value.

Figure 3:  For the case of equal masses, the number of energy levels $N$ is
plotted (solid curve) versus the magnetic field $b$ in Eq. (21). Each step
down is due to a level reaching the binding threshold. The ground level never
reaches the single-particle decay threshold, for which a horizontal dashed
line depicts the large-$b$ value $N=1$. Additionally shown (dashed curve) is
the approximate function $N(b)$ given in Eq. (33).

The number of energy levels supported in a given magnetic field can be
obtained from knowledge of the critical field for each value of $n$. If we
start at large $b$, the number of levels is $N=1$ until the field is lowered
below the value in Eq. (31). Below this value, the $n{=}1$ level has not
crossed threshold, consequently $N=2$. Continuing to lower the magnetic field,
successive critical fields will be crossed and the number of levels increased
by one each time. The number of levels obtained by counting thresholds is
shown in Fig. 3 up to $N=50$. By identifying $N=n+1$ and simply inverting the
formula for the critical field Eq. (32), we obtain an approximate formula for
the number of levels as a function of the magnetic field

$N=\tfrac{2}{b}+\tfrac{3}{2}.$ (33)

The figure shows that this simple formula accounts for the number of bound
levels remarkably well.

##  IV Summary

Systems of two interacting charged particles in a magnetic field present an
analytically challenging non-separable problem. When the particles have an
attractive, momentum-independent, zero-range interaction, we confirm in Sec.
II that the bound states have infinite degeneracy, and are characterized by a
quantum number that incorporates both center-of-mass and relative degrees of
freedom. In small magnetic fields, this quantum number is the Landau level of
the center of mass, as exhibited by Eq. (25), which is the small-field
expansion of the energy of a system consisting of one charged and one neutral
particle. Near the unitary limit, there is dramatically different behavior for
the ground and excited levels of this system. The lowest level is bound, with
a universal binding energy determined by Eq. (27). The excited states
disappear into the continuum at critical values of the magnetic field, which
are determined by Eq. (30). The number of bound levels as a function of the
magnetic field is well described by the remarkably simple formula in Eq. (33).

A striking feature of this two-body system in a magnetic background is that
its bound-state spectrum is identical to that of an effective one-body
problem. The effective interaction in this restricted space accounts for the
background field and zero-range interaction. The closed-form correlation
function, Eq. (15), is suggestive of an effective time-dependent Hamiltonian.
In this way, complexity of the non-separable system manifests as non-trivial
time-dependence of the restricted system. Nonetheless, renormalization can be
carried out by hand, thus enabling straightforward determination of spectral
properties. In particular, Eq. (20) shows that the inverse of the solution,
namely the magnetic field as a function of the binding energy, only requires
evaluation of a one-dimensional integral. The number of systems possessing
such almost-in-closed-form solutions can be enlarged by generalizing and
extending the problem solved here. Of particular interest would be a complete
treatment of the two-nucleon system.

###### Acknowledgements.

We thank M. Birse for comments and discussions.

##  Appendix A Green’s Function

The Green’s function $G^{(0)}$ in Eq. (4) is computed from a Laplace transform
using the inverse operator

$\frac{1}{E-H^{(0)}+i{\epsilon}}=-\int_{0}^{\infty}dt\,e^{Et}\,e^{-H^{(0)}t},$
(34)

where analytic continuation is necessary when $E>0$. We refer to $t$ as time;
although, strictly speaking, it is imaginary time. As $H^{(0)}_{1}$ and
$H^{(0)}_{2}$ in Eq. (2) commute, $G^{(0)}$ factorizes into the product of
single-particle Green’s functions

$\displaystyle
G^{(0)}({\bm{r}}^{\prime}_{1},{\bm{r}}^{\prime}_{2};{\bm{r}}_{1},{\bm{r}}_{2}|t)$
$\displaystyle=$
$\displaystyle\prod_{a=1}^{2}\,\big{\langle}{\bm{r}}^{\prime}_{a}\big{|}e^{-H^{(0)}_{a}t}\big{|}{\bm{r}}_{a}\big{\rangle}.$
(35)

For the gauge chosen in Eq. (5), the Green’s function of the charged particle
takes the form

$\displaystyle\big{\langle}{\bm{r}}^{\prime}_{1}\big{|}e^{-H^{(0)}_{1}t}\big{|}{\bm{r}}_{1}\big{\rangle}$
$\displaystyle=$
$\displaystyle\int\frac{d\bm{\tilde{p}}_{1}}{(2\pi)^{2}}e^{i\bm{\tilde{p}}_{1}\cdot(\bm{\tilde{r}}^{\prime}_{1}-\bm{\tilde{r}}_{1})}e^{-\frac{p_{1,z}^{2}}{2m_{1}}t}$
(36)
$\displaystyle\qquad\times\sqrt{QB}\,\,\big{\langle}{\xi}^{\prime}_{1},\,{\omega}t\,\big{|}{\xi}_{1},0\big{\rangle},$

and has been written in terms of the Green’s function of the simple harmonic
oscillator. With dimensionless conjugate variables ${\xi}$ and $p_{\xi}$, the
oscillator Hamiltonian appears as
${\mathcal{H}}=\frac{1}{2}\big{(}p_{\xi}^{2}+{\xi}^{2}\big{)}$, for which the
corresponding propagator we define as
$\langle{\xi}^{\prime},s\,|{\xi},0\rangle=\langle{\xi}^{\prime}|e^{-{\mathcal{H}}\,s}|{\xi}\rangle$.
In Eq. (36), ${\omega}$ is the charged particle’s cyclotron frequency, and the
dimensionless charged-particle coordinates are

$\displaystyle{\xi}^{\prime}_{1}=\sqrt{QB}\left(y^{\prime}_{1}-\frac{p_{1,x}}{QB}\right)\,\text{and}\,\,\,{\xi}_{1}=\sqrt{QB}\left(y_{1}-\frac{p_{1,x}}{QB}\right).$

The coordinate-space Green’s function of the neutral particle can be written
as a Fourier transform

$\displaystyle\big{\langle}{\bm{r}}^{\prime}_{2}\big{|}e^{-H^{(0)}_{2}t}\big{|}{\bm{r}}_{2}\big{\rangle}$
$\displaystyle=$
$\displaystyle\int\frac{d{\bm{p}}_{2}}{(2{\pi})^{3}}e^{i{\bm{p}}_{2}\cdot({\bm{r}}^{\prime}_{2}-{\bm{r}}_{2})}e^{-\frac{{\bm{p}}_{2}^{2}}{2m_{2}}t}.$
(38)

Following Sec. II, we project Eq. (35) onto good $\bm{\widetilde{P}}$, and
restrict the computation of the matrix element to vanishing relative
separation $\bm{r}^{\prime}{=}\bm{r}{=}{\bm{0}}$. The resulting time-
dependent, non-interacting Green’s function $g^{(0)}({\xi};{\xi}^{\prime}|t)$
is both $P_{x}$ independent and diagonal in the basis of Hermite functions.
The diagonal matrix elements are the $g_{n}^{(0)}(t)$ coefficients appearing
in Eq. (15).

##  Appendix B The Function ${\mathcal{I}}_{n}(\chi,x)$

The function ${\mathcal{I}}_{n}$ appearing in Eq. (20) is defined as

${\mathcal{I}}_{n}(\chi,x)=-\tfrac{1}{\sqrt{2}}\,{\zeta}\left(\tfrac{1}{2},\tfrac{\chi+x}{2}\right)-{\mathcal{J}}_{n}(\chi,x),$
(39)

with ${\zeta}(\frac{1}{2},z)$ as a Hurwitz zeta-function, and where

$\displaystyle{\mathcal{J}}_{n}(\chi,x)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}ds\frac{e^{(1-x-\chi)s}}{\sqrt{4\pi}s^{3/2}}\Big{[}\,{\mathcal{F}}_{n}(x,s)-\frac{s}{\sinh
s}\Big{]}.\quad\,$ (40)

Appearing in the above integrand is ${\mathcal{F}}_{n}(x,s)$, which has the
explicit form

${\mathcal{F}}_{n}(x,s)=\frac{\sinh\frac{\Omega(s)\,s}{2}}{x\,\sinh[(1-x)s]}\,e^{-\left(n+\frac{1}{2}\right)\Omega(s)\,s},$
(41)

using $\coth\frac{\Omega(s)\,s}{2}=\coth[(1-x)s]+(xs)^{-1}$. The integral
defining ${\mathcal{J}}_{n}(\chi,x)$ converges in the infrared ($s\to\infty$)
provided $\chi\geq 0$. The definition of this function, moreover, is such that
$\lim_{x\to 0}{\mathcal{J}}_{n}(\chi,x)=0$.

To investigate the critical field $b_{c}$ in Eq. (30) as a function of the
level number, we differentiate Eq. (20) with respect to $n$ at $\chi=0$, which
produces

$\frac{1}{b_{c}^{3/2}}\frac{db_{c}}{dn}=\int_{0}^{\infty}ds\frac{e^{(1-x)s}}{\sqrt{\pi}s^{3/2}}\frac{d}{dn}{\mathcal{F}}_{n}(x,s).$
(42)

Scaling $s\to 2\,s/n$, and treating $n\gg 1$ leads to the approximate formula

$\frac{1}{b_{c}^{3/2}}\frac{db_{c}}{dn}=-\int_{0}^{\infty}\frac{\,ds\,e^{-s}}{\sqrt{2\pi
ns}}\,e^{\frac{s}{2n}}\left[1+{\mathcal{O}}(n^{-2})\right],$ (43)

for the case of equal masses. The exponential factor $e^{\frac{s}{2n}}$ need
not be expanded in powers of $n^{-1}$, as the integral can be performed with
this factor treated exactly. Solving the above differential equation for
$b_{c}$ subject to the condition $\lim_{n\to\infty}b_{c}(n)=0$ results in Eq.
(32).

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]Also Institut Universitaire de France, 103 boulevard Saint-Michel, 75005
Paris, France

# Kaluza-Klein gluon + jets associated production at the Large Hadron Collider

A. M. Iyer<EMAIL_ADDRESS>Department of Theoretical Physics,
Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400
005, India  F. Mahmoudi  [<EMAIL_ADDRESS>Univ Lyon, Univ Lyon 1, ENS de
Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230 Saint-
Genis-Laval, France  Theoretical Physics Department, CERN, CH-1211 Geneva 23,
Switzerland  N. Manglani<EMAIL_ADDRESS>Department of Physics,
University of Mumbai, Kalina, Mumbai 400098, India  Shah and Anchor Kutchhi
Engineering College, Mumbai 400088, India.  K. Sridhar
<EMAIL_ADDRESS>Department of Theoretical Physics, Tata Institute
of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India

###### Abstract

The Kaluza-Klein excitations of gluons offer the exciting possibility of
probing bulk Randall-Sundrum (RS) models. In these bulk models either a
custodial symmetry or a deformation of the metric away from AdS is invoked in
order to deal with electroweak precision tests. Addressing both these models,
we suggest a new channel in which to study the production of KK-gluons
($g_{KK}$): one where it is produced in association with one or more hard
jets. The cross-section for the $g_{KK}+$ jets channel is significant because
of several contributing sub-processes. In particular, the 1-jet and the 2-jet
associated processes are important because at these orders in QCD the $qg$ and
the $gg$ initial states respectively come into play. We have performed a
hadron-level simulation of the signal and present strategies to effectively
extract the signal from what could potentially be a huge background. We
present results for the kinematic reach of the LHC Run-II for different
$g_{KK}$ masses in bulk-RS models.

###### pacs:

73.21.Hb, 73.21.La, 73.50.Bk

††preprint: CERN-TH-2016-002††preprint: TIFR/TH/16-01

##  I The Bulk Randall-Sundrum Model

The Randall-Sundrum model (RS model) Randall:1999ee  is a five-dimensional
model with a warped metric and was proposed as a solution to the gauge-
hierarchy problem. In this five-dimensional model, the fifth dimension $y$ is
compactified on an $S^{1}/Z^{2}$ orbifold of radius, $R$. At the fixed points
of the orbifold, $y=0$ and $y=\pi R\equiv L$, two branes– the UV and the IR
brane respectively are located.

One starts with a warped metric given as:

$ds^{2}=e^{-2A(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^{2}$ (1)

and using a five-dimensional gravity action with a bulk cosmological constant
$\Lambda$ one can show that the solutions to the Einstein equation imply

$A(y)=\pm k|y|$ (2)

where $k^{2}\equiv-\Lambda/12M^{3}$ with $M$ being the Planck scale.

The Standard Model (SM) fields are all IR-localised and only the graviton
fields access the bulk. The factor of $v/M\sim 10^{-16}$ (where $v$ is the
vacuum expectation value of the SM Higgs field) is generated by choosing a
value of $kL\sim 30$ thereby stabilising the gauge hierarchy.

The problem, however, is that the suppression that one obtains for the Higgs
vev materialises for all fields localised on the IR brane. Thus, mass scales
which suppress dangerous higher-dimensional operators responsible for proton
decay or neutrino masses also become small which spells a disaster for the RS
model. One way out of this is to realise that to solve the gauge-hierarchy
problem one needs to only localise the Higgs on the IR brane but all the other
SM fields could be in the bulk Davoudiasl:1999tf ; Pomarol:1999ad ;
Gherghetta:2000qt ; Grossman:1999ra . In fact, even the Higgs need not be
sharply localised on the IR brane but only somewhere close to it. In this way,
it is possible to make viable variations of the RS model, now collectively
known as Bulk RS models. These models yield a bonus: localising fermions at
different positions in the bulk gives different overlaps of their profiles
with the Higgs field, which is localised on or close to the IR brane. This
gives rise in a natural way to the Yukawa-coupling hierarchy Gherghetta:2000qt
.

The AdS/CFT correspondence when worked out for a slice of AdS spacetime also
provides both a motivation for and an understanding of bulk models. (For a
review, see Ref. Gherghetta:2010cj .) In fact, this correspondence shows that
the dual of the RS model with all the SM fields on the IR brane is a fully
composite SM in four dimensions – a theory which we know does not survive in
the face of extant experimental constraints. But when the SM fields are
localised at different positions in the bulk, one ends up with a partially
composite SM where the compositeness is primarily in the Higgs, the top and in
the KK sector. Such a model is, in fact, a viable model as we will discuss in
the following.

It is possible to construct a bulk-extension of the SM by having the gauge and
fermion fields in the bulk, the Higgs localised on or near the IR brane and
with a suitable mechanism to make the model successfully confront constraints
from electroweak precision measurements Gherghetta:2010cj . Such a model has
interesting features. Other than providing a framework for addressing the
question of fermion mass hierarchy, it also naturally results in small mixing
angles in the Cabibbo-Kobayashi-Maskawa (CKM) matrix, gauge-coupling
universality and suppression of flavour-changing neutral currents to
experimentally acceptable values Burdman:2003nt ; Huber:2003tu ;
Casagrande:2008hr ; Bauer:2009cf ; Agashe:2004cp .

Electroweak precision tests provide very strong constraints on bulk models.
If, for example, only gauge bosons propagate in the bulk but the fermions are
localised on the IR brane then the couplings of the gauge boson Kaluza-Klein
(KK) modes to the IR-localised fermions yield unacceptably large contributions
to $T$ and $S$ and this gives a lower bound of 25 TeV on the mass of the first
KK mode of the gauge boson. Of course, one way to relax this bound is to
localise the fermions in the bulk and especially the light fermions close to
the UV brane and this significantly reduces the constraint coming from the
$S$-parameter. But the $T$ parameter constraints require further attention.
One way to handle this Agashe:2003zs ; Agashe:2006at  is by enlarging the
gauge symmetry in the bulk to $SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times
U(1)_{y}$ which is an enlarged custodial symmetry which is broken on the IR
brane to recover the SM gauge group. It turns out that the corrections to the
$T$ parameter coming from the dangerous KK gauge boson sector can be tamed
using this custodial symmetry and by a judicious choice of the fermion
representations under the extended gauge group it is also possible to rein in
the non-oblique $Z\rightarrow b\bar{b}$ corrections and eventually the bound
on the lightest KK gauge boson mode comes down to about 3 TeV
Davoudiasl:2009cd ; Iyer:2015ywa .

An alternate proposal to address the issue of the $T$ parameter Cabrer:2010si
; Cabrer:2011fb  uses a deformed metric near the IR brane along with moving
the Higgs scalar into the bulk. For this setup, the function $A(y)$ in Eq. (2)
is then modified to

$A(y)=ky-\frac{1}{\nu^{2}}\log(1-\frac{y}{y_{s}})$ (3)

The UV brane, similar to the RS setup, is located at $y=0$. The IR brane is
however located at $y=y_{1}$ with the position of the singularity ($y=y_{s}$)
located behind the IR brane at $y_{s}=y_{1}+\Delta$, where
$\Delta\sim\frac{1}{k}$. $y_{1}$ is determined by demanding the solution to
the hierarchy problem which requires $A(y_{1})\sim 35$. The limit
$\nu\rightarrow 0$ reverts to the original RS geometry in Eq. (2). The
deformation of the metric actually causes the Higgs field to be moved further
away from the IR brane but the gauge boson KK modes are moved by the
deformation towards the IR brane. This differential action causes the overlap
of the Higgs and KK gauge boson modes to be reduced and that relaxes the
bounds coming from the $T$ parameter and the mass of first KK gauge boson mode
in this model can be as small as 1.5 TeV Iyer:2015ywa .

##  II Kaluza-Klein gluons and collider searches

In typical Bulk RS models, the lightest KK excitations are those of the gauge
bosons and searches for these are likely to be the most promising probes of
such a model. Of these, the KK gluons, because of their larger couplings, are
the most interesting though there are interesting signals from KK excitations
of electroweak gauge bosons and fermions.

In the custodial models, the couplings of the $g_{KK}$ to the SM states
Agashe:2006hk  are parametrised in terms of the parameter
$\xi\equiv\sqrt{kL}\sim 5$ and relative to the QCD coupling $g_{s}$ are given
as:

$g^{q\bar{q}g_{KK}}\approx{1\over\xi}g_{s},\hskip
15.0ptg^{Q\bar{Q}^{3}g_{KK}}\approx 1g_{s},\hskip
15.0ptg^{t_{R}\bar{t}_{R}g_{KK}}\approx\xi g_{s},\hskip 15.0ptg^{ggg_{KK}}=0,$
(4)

These denote the coupling of the first Kaluza-Klein mode of the gluon to light
quarks, to the third-generation left-handed doublet, to the right-handed top
quark and to the gluon, respectively. Note that the $g_{KK}$ couples
predominantly to the right-handed top quark and, consequently, the
$g_{KK}\rightarrow t\bar{t}$ branching ratio is more than 90%. For the
deformed metric the couplings are similar except for overall factors to be
discussed later.

Also one sees that the coupling of the $g_{KK}$ to the zero-mode Standard
Model gluons vanishes because of the orthonormality of the Kaluza-Klein modes.
This means that, at leading order, $g_{KK}$ production takes place via
annihilation of light quarks, to which the coupling of the $g_{KK}$ is
suppressed and, consequently, the cross-section is small especially since
electroweak precision constraints require the mass of the $g_{KK}$ to be not
less than 2-3 TeV. The produced $g_{KK}$ decays into a $t\bar{t}$ pair and
this tiny cross-section has to compete with a huge QCD $t\bar{t}$ production
background. The fact that this is a resonant cross-section helps somewhat but
then the $g_{KK}$ has a very large width, owing to its strong coupling to the
tops, and so the resonant bump is not sharp but rather smeared out. The fact
that the $g_{KK}$ couples chirally to the tops is, however, an advantage and a
forward-backward asymmetry to pick out the signal may be used. However, in a
$pp$ machine like the LHC this is not easy. Finally, since the $t$ and the
$\bar{t}$ come from the decay of a heavy object not less than 2-3 TeV in mass,
they are highly boosted. These boosted top jets can effectively be used as
signal discriminant Agashe:2006hk ; Lillie:2007yh .

Nonetheless, given the smallness of the cross-section it becomes important to
look for other production mechanisms for the $g_{KK}$, especially the ones
which have gluon initial states. The associated production of $g_{KK}$ with a
$t\bar{t}$ pair leading to distinctive four-top final states (with two boosted
tops) is a process that has been studied with this in mind Guchait:2007jd ;
Lillie:2007hd ; Pomarol:2008bh ; Servant:2010zza . The production of $g_{KK}$
through top loops has also been considered Allanach:2009vz  though the loop-
contributions are greatly suppressed. Model independent analysis for the
searches of colour octet states (colorons) were considered in Chivukula:2011ng
; Chivukula:2013xla  111The phenomenology of such models have been studied in
Chivukula:2013xka ; Chivukula:2014rka ; Chivukula:2015kua ..

##  III Associated jet production

In this paper, we are proposing the production of a $g_{KK}$ with associated
jets (a light quark or gluon jet) in order to complement the leading order
$g_{KK}$ production and increase the sensitivity for the $g_{KK}$ at the LHC.
At the parton level, the most important contribution is the one where $g_{KK}$
is produced in association with a single hard parton. Top row in Fig. 1 shows
few of the contributing diagrams for this process. As this appears at an order
of $\alpha_{s}$ higher than the leading-order $g_{KK}$ production, one would
think that the cross-section is smaller. Effects of PDF do partly compensate
for the suppression due to $\alpha_{s}$. In addition, the associated jet
production process has both $q\bar{q}$ and $qg$ initial states and a larger
number of sub-process contributions. Consequently, the cross-section for this
process is comparable to the leading-order $g_{KK}$ production process. Table
1 and 2 gives a comparison for the LO cross-section with the cross-section
with associated partons for the RS and deformed RS model respectively.

$g_{KK}$ mass(GeV) | LO Cross-section(fb) | Associated production (AP)(fb) | AP/LO
---|---|---|---
2500 | 169.1 | 107.6 | 0.636
3000 | 52.53 | 33.28 | 0.634
3500 | 17.4 | 10.99 | 0.632
4000 | 5.993 | 3.723 | 0.621
4500 | 2.096 | 1.277 | 0.609
Table 1: Comparison for cross-sections for mass range in Normal RS  $g_{KK}$ mass(GeV) | LO Cross-section(fb) | Associated production (AP)(fb) | AP/LO
---|---|---|---
1500 | 1127 | 695.5 | 0.617
2000 | 257.8 | 162.1 | 0.629
2500 | 71.42 | 45.51 | 0.637
3000 | 22.19 | 14.09 | 0.635
3500 | 7.362 | 4.617 | 0.627
Table 2: Comparison for cross-sections for mass range in Deformed RS

We would like to emphasise that, unlike in Refs. Mathews:2004xp ;
Mathews:2005bw ; Allanach:2009vz , we are not doing the full NLO analysis here
since we are concentrating on hard scattering processes and include only real
emission.

The $g_{KK}$ produced via the diagrams shown in Fig. 1 will decay
predominantly to tops. The tops so produced will decay to a $b$ and a $W$
which will finally result in a multi-jet final state with the associated jet
being one of these jets. A detailed analysis of the signal and background,
including hadronisation and jet reconstruction is presented in the following
section. We will, however, make a couple of points here before proceeding to
the discussion of the detailed analysis.

In trying to include final states with more than one jet, we are going to
higher orders in QCD perturbation theory. One may wonder then whether we need
to include virtual corrections and worry about soft and collinear
singularities. We do not have to include these corrections because we are
ensuring hard jets in our process. Moreover, some part of the infrared issues
are handled by the showering in PYTHIA Sjostrand:2007gs  (albeit in a model-
dependent way) and by the matching of the hard amplitude computation with the
results of PYTHIA that we have taken into account in a careful manner. If
there is any residual doubt about our calculational procedure, it is best to
bear the analogy with the Drell-Yan process in mind. The leading-order Drell-
Yan process, like leading-order $g_{KK}$ production, is a $q\bar{q}$-initiated
process. When we consider the high-$p_{T}$ Drell-Yan process, in which we
consider a single jet recoiling against the lepton-antilepton pair, then as in
our case, the $qg$ process kicks in. If we were to ask for two hard jets to be
produced in association with the lepton-antilepton pair, then the $gg$ process
also comes in. In computing these hard-scattering amplitudes contributing to
high-$p_{T}$ Drell-Yan, we do not have to worry about the soft or collinear
singularities because we have ensured that the produced jets are hard. Our
process is similar, at the level of the partonic processes, to high-$p_{T}$
Drell-Yan (and differs from it only in the kinematics peculiar to the
production of a heavy particle like the KK gluon) and we are, therefore,
justified in neglecting the soft and collinear singularities in our
calculation.

Figure 1: The subprocesses contributing to $g_{KK}$ production in association
with partons. Top row is the $g_{KK}$ production in association with a single
parton and the bottom row is the production in association with two partons.

Additionally, we calculated the squared matrix-elements for the subprocesses
shown in Fig. 1 explicitly and computed the cross-sections, using a parton-
level Monte Carlo. This provided us a quick estimate of the cross-section and
justified our intuition about its magnitude. It also provided us a rough
comparison with the more detailed analysis that we did using MADGRAPH
Alwall:2014hca  and PYTHIA Sjostrand:2007gs .

Again, while we postpone the detailed discussion of the choice of kinematic
cuts to the next section, it is useful to note here some features of the
kinematics, arising due to the production of a very massive object at the LHC
energies. We are interested in producing a $g_{KK}$, at least 1.5 TeV or so in
mass, in association with a jet. Even at the highest energies at the LHC now
available the sub-process centre-of-mass energy will be sufficient only to
produce the $g_{KK}$ with small momentum with the $p_{T}$-balancing associated
jet also, therefore, possessing a $p_{T}$ that is not very large. When the
$g_{KK}$ decays into the $t\bar{t}$ pair, the $t$ and $\bar{t}$ are produced
almost back-to-back. But the $t$ and the $\bar{t}$ so produced will be highly
boosted and give rise to very collimated hadronic decay products. The small
magnitude of the $g_{KK}$ $p_{T}$ also implies that the recoiling parton
carries small net $p_{T}$ which also poses a challenge in trying to
distinguish it from the other hadronic environment. The resultant final state
has a $g_{KK}+$ several jets which could arise from partonic sub-processes
with more than one associated parton. We therefore also take into account the
cross-section due to processes with additional partons. The process with two
additional partons arises also from $gg$ initial states (in addition to the
$q\bar{q}$ and $qg$ channels) which contribute at the lower orders. The bottom
row in Fig. 1 shows the production of $g_{KK}$ in association with two
partons. The cross-section is again not very suppressed. Processes with larger
number of partons in association will follow the usual perturbation theory
pattern and are expected to be suppressed. Further processes with more than
two partons will have a tendency to produce softer jets which may not pass the
cuts. We have checked the cross-section for $g_{KK}$ with one, two and three
additional partons in MADGRAPH and find the above expectation to be borne out.
We therefore include the contribution of one parton and two parton sub-
processes to the signal cross-section.

##  IV Signal and Background simulations

The signal is characterised by a massive RS KK-gluon ($g_{KK}$) produced in
association with hard partons. The $g_{KK}$ then decays dominantly into a
$\bar{t}t$ pair leading to the following signal topology

$p~{}p\rightarrow g_{KK}+a~{}\quad\text{or}\quad~{}p~{}p\rightarrow
g_{KK}+a+b$ (5)

where $a,b$ are partons and $g_{KK}\rightarrow t\bar{t}$. The requirement of
an additional hard parton does not change the signal. This process is at the
next to leading order in comparison to following leading order (LO) topology

$p~{}p\rightarrow g_{KK}\rightarrow\bar{t}t$ (6)

In comparison the $g_{KK}$ produced in a LO process will be almost at rest.
The parton-level amplitudes for both the signal and the background were
generated using MADGRAPH Alwall:2014hca  at 13 TeV centre of mass energy using
parton distribution function NNLO1 Ball:2012cx . The model files have been
generated using FEYNRULES Alloul:2013bka . The signal events were generated by
adding the following amplitudes:

$\mathcal{M}_{signal}=\mathcal{M}(pp\rightarrow
g_{KK})+\mathcal{M}(pp\rightarrow g_{KK}+a)+\mathcal{M}(pp\rightarrow
g_{KK}+a+b)$ (7)

The most dominant backgrounds are $t\bar{t}+$ jets and QCD. The events for the
former is generated by adding the following amplitudes at the parton level:

$\displaystyle\mathcal{M}_{bg}=\mathcal{M}(pp\rightarrow
t\bar{t})+\mathcal{M}(pp\rightarrow t\bar{t}+a)+\mathcal{M}(pp\rightarrow
t\bar{t}+a+b)$ (8)

The QCD background is simulated by adding the following amplitudes at the
parton level

$\displaystyle\mathcal{M}_{bg}=\mathcal{M}(pp\rightarrow
a+b)+\mathcal{M}(pp\rightarrow a+b+c)$ (9)

The associated partons for the $t\bar{t}$ and the signal are required to have
a minimum transverse momentum of 50 GeV. A softer cut enhances the background
cross-section and hence not desirable. The signal cross-section on the other
hand is not drastically affected as the associated parton is expected to have
a fairly large transverse momentum from the recoiling $g_{KK}$.

We need the ‘hard jet’ from the matrix elements in the second and third terms
in the right hand side of Eq. (7). The ‘soft jets’ are modelled by emissions
off $pp\rightarrow g_{KK}$ in the first term generated by PYTHIA 8
Sjostrand:2007gs . This was done by matching the MADGRAPH output to PYTHIA 8
using MLM Mangano:2002ea  matching scheme. Subsequently, after showering and
hadronisation using PYTHIA 8 we apply the following selection criteria to
extract the signal:

Jet Selection: Assuming hadronic decay of the top, the final state is
characterised by at least 6 partons (including 2 $b$ partons). Jets are
reconstructed from these partons by employing FASTJET Cacciari:2006sm ;
Cacciari:2011ma  using the $anti-k_{T}$ Cacciari:2008gp  clustering algorithm
and setting the jet radius parameter $R=0.4$. We accept only those jets which
satisfy $|\eta|<2.5$ and $p_{T}>50$ GeV.

Event Selection: Leptons are required to satisfy $p_{T}>20$ GeV and
$\eta<2.5$. Since we are considering a hadronic decay channel for both the
tops, only events with no leptons are accepted. In order to reconstruct the
$g_{KK}$ mass from the top decay products, each event is further required to
have a minimum of 3 jets. An additional condition on a maximum of 5 jets is
imposed which is helpful to reduce qcd background Due to the heavy mass of the
$g_{KK}$, the decay products of $t$ are likely to be reconstructed in a single
jet. Alternatively, one could have imposed the requirement of a minimum $4$
normal jets 222Jets which are not identified as $b-$ jets. and 2 $b-$ jets.
However the requirement of a minimum of two $b$ jets in each event results in
depletion of signal events due to $b$ tagging efficiency. As a result, we do
not attempt to segregate the $b-$ like jets to those coming from the $W-$
decay and $g_{KK}$ mass is reconstructed from the 3 leading jets in the event.
Fig. 2 shows the invariant mass of 3 leading jets in each event demonstrating
a distinct peaking behaviour at 3 TeV which is the mass chosen for the
$g_{KK}$.

---
Figure 2: Reconstructed mass of the $g_{KK}$ from the three leading jets where
the mass of $g_{KK}$ is chosen to be 3 TeV.

Background rejection: For the signal, the top pairs and hence its decay
products are likely to be highly boosted owing to large $g_{KK}$ mass. As a
result the leading jet ($j_{0}$) is likely to have a very large $p_{T}$ in
comparison to that of the background. Fig. 3 shows the $p_{T}$ distribution
for the leading jet for the background (blue and green) and the signal (red).
We give a hard cut of 1100 GeV on the leading jet. To increase the statistics
for the background in this high $p_{T}$ region, background events are
simulated near the tail end of the $p_{T}$ spectrum. This is implemented by
demanding the sum of $p_{T}$ of the associated partons for background
simulation is 1250 GeV. Further, each event is required to have a minimum of
two a maximum of 8 jets. Inspite of such a hard cut on the leading jet,
contamination due to QCD processes is non-zero due to its significant
production cross-section. Traditional kinematic cuts prove insufficient to get
a significant signal sensitivity.

---
Figure 3: $p_{T}$ distribution for the leading jet for the signal (red) and
background (blue) for $m_{g_{KK}}=3$ TeV.

In this case, studying the jet substructure is extremely useful in eliminating
the background to a great extent. It was pointed out earlier that the leading
jet for the signal is primarily composed of top decay products. As a result it
has a three-lobed structure. This is in sharp contrast to QCD which is one-
lobed. A useful way to quantify the number of lobes for a given jet is by
considering a variable called N-subjettiness Thaler:2010tr ; Thaler:2011gf
defined as:

$\tau_{N}=\frac{\sum_{k}p_{T_{k}}\times\text{min}\left(\Delta R_{1k},\Delta
R_{2k}\ldots\Delta R_{Nk}\right)}{\sum p_{T_{k}}\times R}$ (10)

where $k$ runs over the jet constituents and $p_{T_{k}}$ is the transverse
momentum for the $k-th$ constituent. In the above definition, assuming there
are $N$ candidate subjets, $\Delta R_{lk}$ is the distance in the rapidity-
azimuth plane between the $k-th$ constituent and the $l-th$ candidate subjet.
For a jet with $N-$ distinct lobes of energy, all the radiation inside the jet
will be aligned along their direction, which is the same as the direction of
the candidate subjet. As a result each constituent of the jet will be
clustered with one of the subjets and one can expect the
$\text{min}\left(\Delta R_{1k},\Delta R_{2k}\ldots\Delta R_{Nk}\right)$ to be
closer to zero. Thus, $\tau_{N}\rightarrow 0$ for $N-$ lobe configuration
while $\tau_{N-1}>\tau_{N}$ for $N>1$. In this case $\tau_{N+1}$ is expected
to be comparable to $\tau_{N}$. For QCD $\tau_{1}$ is small while for the
signal $\tau_{3}$ is small. A useful application is to consider ratios
$\tau_{N+1,N}=\tau_{N+1}/\tau_{N}$. For the scenario under consideration, we
evaluate $\tau_{32}$ and $\tau_{31}$. Both these ratios are expected to be
smaller for the signal than QCD as shown in Fig. 4.

|
---|---
Figure 4: Subjettiness ratio $\tau_{31}$ on the left and $\tau_{32}$ on the
right for both signal and background.

One of the most dominant background to the signal is $t\bar{t}+$ jets
production via SM processes due to its significant production cross-section.
It is therefore essential to suppress this background significantly to have
sufficient signal over background efficiency for luminosities attainable in
the near future. We select $\tau_{31}<0.35$ and $\tau_{32}<0.35$ for the
leading jet. Further we also impose a cut of $\tau_{21}<0.6$ on the sub-
leading jet.

Results:

Table 3 gives the summary of the number of events passing various cuts at each
level for both the signal and background. These results correspond to
$m_{g_{KK}}=3$ TeV. The cuts are optimised for this particular mass. As
explained earlier events with zero leptons are accepted to facilitate hadronic
decay of the top. A hard cut on the transverse momentum of the leading jet
($j_{0}$) drastically reduces the $t\bar{t}$ background without affecting the
signal significantly. With this set of cuts, $\sim 5\sigma$ sensitivity can be
obtained for a minimum luminosity of 90 fb-1 for $m_{g_{KK}}$ = 3 TeV.

Sr.No | Cuts | QCD($p^{sum}_{T}>1250$) |  $t\bar{t}jj$($p^{sum}_{T}>1250$) | Signal events
---|---|---|---|---
1 | Given number of Events | $10^{7}$ | $10^{5}$ | 10000
2 | Cross-section(fb) | $720\times 10^{3}$ | 132 | 94.2
3 | $n_{lepton}=0$ | 9203793 | 39386 | 3122
4 |  $p^{j_{0}}_{T}>$1100 GeV | 233520 | 14051 | 1363
5 | Subjettiness cut | 262 | 218 | 265
6 |  $|m_{g_{KK}}-3000|<80$ GeV | 48 | 55 | 112
Table 3: Cut flow table for $m_{g_{KK}}=3$ TeV.

We repeat the analysis above for different masses of the $g_{KK}$ and we
follow exactly the same pattern of cuts as in Table 3. The background events
are simulated differently for different masses of $g_{KK}$ since the fifth
line in Table 3 uses the cut around the mass of the KK-gluon and presents out
results for both normal and deformed RS model. For the normal RS model we
assume 92% branching fraction into $t\bar{t}$ pair, while for the deformed
model we assume 83% branching fraction Carmona:2011ib . Left plot in Fig. 5
presents the minimum luminosity required for a $5\sigma$ signal sensitivity
for the different masses for both normal RS and deformed RS models. Owing to
constraints from precision electroweak data we do not consider masses below
2.5 TeV for normal RS model. Due to their larger production cross-section, the
lower masses (indicated by blue points in the figure) have better sensitivity
in terms of early discovery prospects.

Lower masses can be admitted in a deformed RS model. For the deformed model we
choose $\nu=0.4$ while the curvature radius is chosen to be $L_{1}=0.2/k$
Carmona:2011ib ; deBlas:2012qf . This scenario however suffers from reduced
production cross-section owing to the smaller coupling of the $g_{KK}$ to the
lighter quarks which is roughly $0.13g_{s}$.

For both scenarios, we use a very hard cut on the transverse momentum of the
leading jet, $p_{T}^{j_{0}}>1100$ GeV in Table 3. Since the $p_{T}$ of the
leading jet is $\sim m_{g_{KK}}/2$, this cut is more effective for the heavier
masses as compared to the lighter masses. While this depletes majority of the
signal points for 2 TeV KK gluon, this is helpful in depleting the background
to a great extent.

|
---|---
Figure 5: Minimum luminosity required for a 5$\sigma$ sensitivity for normal
RS (blue) and deformed RS (orange). The right plot shows the production cross-
section for the different masses.

##  V Conclusion:

Search for KK excitations of the gluons in warped framework is interesting due
to their relatively large production cross-section in comparison to other KK
states. Their masses are, however, strongly constrained due to limits from
electroweak precision data. This necessitates the need for an effective
strategy to probe relatively heavy states in the Run II of LHC. We consider a
process where the $g_{KK}$ is produced in association with jets. We consider a
simple set of cuts to extract the signal from the background. The signal is
characterised by highly collimated leading jets owing to the massive nature of
$g_{KK}$. Cuts as strong as 900 GeV is imposed on the $p_{T}$ of the leading
jet without adversely affecting the number of signal sensitivity. We present
results for both the normal RS model and the deformed RS model. In normal RS,
$g_{KK}$ as heavy as 3 TeV can be probed in the Run II of the LHC with
luminosities $\sim 90$ fb-1 while masses as heavy as 4 TeV can be accessed in
the HL-LHC. For the deformed case masses $\sim 2.5$ TeV are accessible in the
current run of LHC. The sensitivity to 3 TeV masses can be probed in the HL-
LHC.

While this simple cut based analysis is highly effective, it would be
interesting to find alternate strategies so that one can explore the heavy
mass regime more efficiently. The observation of highly collimated decay
products of either top coming from the $g_{KK}$ provides strong motivation to
study the jet-substructure in greater detail and constitutes work for the
future.

##  VI Acknowledgements

We would like to thank Monoranjan Guchait for the extensive discussions and
immensely useful inputs on the collider analysis. We are also grateful to him
for the going through the manuscript in detail and for his suggestions on
improving the quality of the draft. AI would like to thank Amit Chakraborty
for discussions on simulations. NM would like to thank the Department of
Theoretical Physics, TIFR for computational resources. We would also like to
acknowledge the contributions of Sophie Renner to the initial stages of this
work.

The authors would like to dedicate this paper to the memory of Guido
Altarelli.

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