# Tunable nonlinear Landau-Zener tunnelings in a spin-orbit-coupled spinor
Bose-Einstein condensate

Zhiqian Gui  Institute for Quantum Science and Technology, Department of
Physics, Shanghai University, Shanghai 200444, China  Jin Su
<EMAIL_ADDRESS>Department of Biomedical Engineering, Changzhi Medical
College, Changzhi 046000, China  Hao Lyu<EMAIL_ADDRESS>Quantum Systems
Unit, Okinawa Institute of Science and Technology Graduate University, Onna,
Okinawa 904-0495, Japan  Yongping Zhang<EMAIL_ADDRESS>Institute for
Quantum Science and Technology, Department of Physics, Shanghai University,
Shanghai 200444, China

###### Abstract

Nonlinear Landau-Zener tunneling is an important nonlinear phenomenon. We
propose to stimulate the nonlinear tunneling in a spin-orbit-coupled spinor
Bose-Einstein condensate. The system provides an experimentally tunable
nonlinearity as well as multiple avoided crossings with tunable gap size in
nonlinear dispersion relations. The nonlinearity generates tilted cusp and
loop structures around the avoided crossings, and the physical consequence of
these nonlinear structures is the nonlinear Landau-Zener tunneling. The spin-
momentum locking induced by the spin-orbit coupling leads to the time-resolved
observation of the nonlinear tunneling by measuring atom populations.

##  I Introduction

Landau-Zener tunneling (LZT), occurring nonadiabatically through an avoided
level crossing, is a fundamental quantum phenomenon [1, 2]. The transition
probability is determined by the celebrated Landau-Zener formula and has an
exponential feature related to the competition between the avoided crossing
gap and the sweeping rate. Due to its fundamental importance, LZT has been
experimentally implemented in a variety of systems, such as solid-state
devices and photonic systems, and a broad range of applications have been
found [3, 4, 5, 6, 7].

Highly tunable ultracold atoms provide an ideal arena to investigate quantum
dynamics and many-body physics [8]. Loading atomic Bose-Einstein condensates
(BECs) into optical lattices can generate tunable avoided crossings of Bloch
energy bands, and the acceleration of optical lattices can induce atomic LZT
through the crossings [9]. Atomic LZT has been experimentally observed in
different accelerating optical lattices [10, 11]. An intriguing application of
the LZT in experiments is to implement Stückelberg interferometry with the
help of closely related Bloch oscillations [12]. Further experimental
progresses make the observation of atomic LZT time-resolved [13, 14].

On the other hand, BECs feature nonlinearities induced by atomic interactions,
and their effects on LZT become an interesting research direction. The
nonlinearity can make the Landau-Zener transition from the lowest to the first
excited Bloch bands to be asymmetric with the reversed transition [15, 16].
When the nonlinearity dominates over the gap size of the optical-lattice-
induced avoided crossing, a loop energy structure adhering to the one band of
the crossing appears inside the gap [17, 18, 19, 20, 21, 22]. Applying an
acceleration force which pushes atoms to move along the looped avoided
crossing gives rise to nonlinear Landau-Zener tunneling (NLZT), which was
first revealed by Wu and Niu [23, 24]. When atoms reach the loop edge, there
is no further trajectory for them to go due to the hysteretic characteristic
of the loop [25]. Consequently, atoms must jump to neighboring bands, so that
the tunneling can occur. The general theory of NLZT has been provided in Ref.
[26], which indicates that the transition probability of NLZT does not follow
the Landau-Zener formula and it becomes nonexponential. Another outstanding
feature of NLZT is that the transition happens even in the adiabatic limit of
the sweeping rate due to the breakdown of adiabaticity [23, 26, 27]. The loop-
induced NLZT has been experimentally observed [28]. Its nonexponential
transition behavior has been recently confirmed in experiments using an
accelerating optical lattice [29, 30].

NLZT becomes a prototype of nonlinear tunneling physics and attracts great
research interests. For example, it plays an important role on Josephson
tunneling in a double-well potential [31], quantum control of NLZT by a
periodic driving field shows interesting properties [32, 33], and NLZT was
analyzed in the presence of higher-order dispersion [34]. NLZT has been
generalized to other nonlinear systems, such as nonlinear waveguide arrays
[35] and superfluid Fermi gas [36]. It has also been proposed that NLZT can be
induced in complicate avoided crossings involving three energy levels such as
a triple-well trapped BEC [37]. More recently, strong nearest- and next-
nearest-neighbor interactions are considered in the triple-well trapped
Rydberg-dressed BEC, and the resultant exotic loop structures generate
intriguing NLZTs [38, 39].

The experimental implementations of synthetic spin-orbit coupling in ultracold
atoms [40, 41, 42, 43, 44, 45, 46, 47] bring ultracold atomic physics into a
new era [48, 49, 50]. The spin-orbit-coupling-engineered single-particle
dispersion relation has fundamentally interesting features. It possesses
multiple energy minima which provide many possibilities for bosonic atoms to
condense [51, 52, 53, 54, 55, 56]. Therefore, the spin-orbit coupling enriches
ground-state phase diagram and makes it possible to generate exotic superfluid
phases and elementary excitations [57, 58]. For the spin-1/2 case, there is a
spin-orbit-coupled (SOC) avoided crossing in the dispersion, which can
introduce topology into ultracold atoms [59], and the existence of gap
solitons can also be hosted in the gap [60, 61]. In a remarkable experiment
[62], the avoided crossing was used to observe completely tunable LZT, since
the gap size, initial state preparations and the sweeping rate related to the
acceleration force are fully adjustable. Due to the spin-momentum locking, the
time-dependent measurement of the spin polarization makes the observation of
the LZT time-resolved. Later, the LZT through the SOC avoided crossing is used
to realize Stückelberg interferometry by the same experimental group [63]. The
experiments [62, 63] and theories [64, 65] show that a SOC BEC with negligible
nonlinearity is a perfect platform to stimulate LZT. Considering nonlinearity,
Ref. [66] demonstrates that a loop structure emerges inside the SOC avoided
crossing only when the difference between inter- and intra-component
interactions exceeds the gap size. However, the required interactions are not
experimentally preferable in the 87Rb BECs where the spin-orbit coupling is
usually implemented.

In the present paper, we reveal that a SOC spinor BEC offers an experimentally
accessible platform for studying NLZT. The SOC spinor BEC with a highly
tunable quadratic Zeeman effect has been experimentally realized in 87Rb atoms
[44]. NLZTs are completely tunable and experimentally realizable in the SOC
spin-1 BEC.

(i) Avoided crossings have rich choices in the SOC dispersion. There are
multiple SOC avoided crossings in the dispersion of the spin-1 spin-orbit
coupling. Two of them are induced by a two-photon transition, and the third
one originates from a four-photon transition. The gap sizes of these crossings
are tunable, and especially the gap of the third one may be very small, which
is benefit for NLZTs.

(ii) The nonlinearity is tunale. The emergence of loop-like structures inside
these avoided crossings only depends on the spin-spin interaction $c_{2}$ and
is completely irrelevant to the density-density interaction $c_{0}$. This
unique characteristic ensures experimental accessibility. Physics of a spinor
BEC are usually determined by the ratio of the spin-spin to density-density
interactions, which is very small unless the Feshbach resonance technique
applies [67]. For the emergence of the loop-like structures, the decoupling of
the spin-spin interactions from the density-density part makes it possible to
adjust the spin-spin interaction by changing atomic density. It is crucially
important for 87Rb spinor BECs due to their imperfection of Feshbach
resonances.

(iii) Other tunable parameters and time-resolved measurement can be realized.
The experiments [62, 63] on the conventional LZT in SOC BECs have already
witnessed tunable parameters and the time-resolved measurement of tunnelings
due to the spin-momentum locking in the SOC dispersion.

The remainder of the paper is outlined as follows. In Sec. II, we show the
nonlinear dispersion of a SOC spin-1 BEC. The tilted cusps and loop structures
can emerge around the SOC avoided crossings. These exotic structures can be
captured by an effective two-level model. Furthermore, the quadratic Zeeman
effect provides an essential tool to adjust these nonlinear structures. In
Sec. III, we study NLZTs with different well-prepared initial states in the
presence of both tiled cusp and loop structures. In Sec. IV, we show NLZTs in
the presence of only one loop structure. Finally, the conclusion follows in
Sec. V.

Figure 1: Nonlinear dispersion relation of a spin-orbit-coupled spinor BEC.
The Rabi frequency is $\Omega=0.1$, and the constant energy shift is
$c_{0}n=1$. The quadratic Zeeman shift is set as $\epsilon=0$ in (a1)-(a3),
$\epsilon=-4$ in (b1)-(b3), and $\epsilon=2$ in (c1)-(c3). The first, second
and third columns correspond to $c_{2}n=0$, $0.8$, and $-0.8$, respectively.
In the single-particle dispersions in (a1), (b1) and (c1), there are three SOC
avoided crossings. The quadratic Zeeman shift can tune the locations of the
crossings in both $k$ and $\mu$ directions. The nonlinearity $c_{2}n\neq 0$
generates a loop in the middle avoided crossing and tilted cusps in the other
two crossings. The bifurcations of the loop and cusps are determined by the
sign of $c_{2}$.

##  II Loop and tilted cusp featured nonlinear dispersion

The spin-1 spin-orbit coupling has been implemented in a spinor BEC by three
Raman lasers [44]. We consider this experimentally realizable system described
by the following Hamiltonian,

$\displaystyle H=\int
d{x}\Psi^{\dagger}\left(H_{\text{SOC}}+\frac{c_{0}}{2}\Psi^{\dagger}\Psi+\frac{c_{2}}{2}\Psi^{\dagger}\bm{F}\Psi\cdot\bm{F}\right)\Psi,$
(1)

where $\Psi=(\Psi_{1},\Psi_{2},\Psi_{3})^{T}$ is the spin-1 spinor wave
function with $\Psi_{i}$ being the wave functions of the $i$-th hyperfine
state ($i=1,2,3$), which satisfy the normalization condition $\int
dx\Psi^{\dagger}\Psi=N$ with $N$ being the total atom number.
$\bm{F}=(F_{x},F_{y},F_{z})$ are the spin-1 Pauli matrices. The spin-1
interactions include the density-density part with the coefficient $c_{0}$ and
the spin-spin part with the coefficient $c_{2}$ [67]. The single-particle SOC
Hamiltonian is [44, 53, 54, 55]

$H_{\text{SOC}}=-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}-2i\frac{\partial}{\partial x}F_{z}+\sqrt{2}\Omega F_{x}+\epsilon
F^{2}_{z}.$ (2)

Here, $\Omega$ is the Rabi frequency, which originates from two-photon
transitions induced by the Raman lasers, $\epsilon$ is the quadratic Zeeman
shift, which can be changed by tuning the frequencies of the Raman lasers. The
Hamiltonian $H$ is dimensionless. The units of momentum, length, and energy
are $\hbar k_{\mathrm{Ram}}$, $1/k_{\mathrm{Ram}}$, and
$\hbar^{2}k^{2}_{\mathrm{Ram}}/m$, respectively, where $m$ is the atom mass
and $k_{\mathrm{Ram}}=2\pi/\lambda_{\mathrm{Ram}}$ is the wave number of the
Raman lasers with $\lambda_{\mathrm{Ram}}$ being the wavelength. Due to these
units, the spin-orbit coupling strength becomes a fixed number $2$ in
$H_{\text{SOC}}$. By applying $i\partial\Psi/\partial t=\partial
H/\partial\Psi^{\dagger}$, we obtain the Gross-Pitaevskii (GP) equation which
describes dynamics of the SOC spinor BEC,

$i\frac{\partial\Psi}{\partial
t}=\left(H_{\text{SOC}}+H_{\text{int}}\left[\Psi\right]\right)\Psi,$ (3)

with the interacting part being

$H_{\text{int}}\left[\Psi\right]=c_{0}\Psi^{\dagger}\Psi+c_{2}\Psi^{\dagger}\bm{F}\Psi\cdot\bm{F}.$
(4)

NLZTs are related to nonlinear dispersion relations. To obtain the nonlinear
dispersion, we assume Eq. (3) having plane-wave solutions,

$\Psi=\sqrt{n}\exp(ikx-i\mu t)\Phi,$ (5)

where $n$ is the mean density, $k$ is the quasimomentum, and $\mu$ denotes the
chemical potential. Then the spinor
$\Phi=\left(\Phi_{1},\Phi_{2},\Phi_{3}\right)^{T}$ satisfies the stationary GP
equation,

$\mu\Phi=\left\\{H_{\text{SOC}}\left(k\right)+H_{\text{int}}\left[\Phi\right]\right\\}\Phi,$
(6)

and the normalization condition becomes
$|\Phi_{1}|^{2}+|\Phi_{2}|^{2}+|\Phi_{3}|^{2}=1$. In Eq. (6), the interacting
Hamiltonian is

$H_{\text{int}}[\Phi]=c_{0}n+c_{2}n\Phi^{\dagger}\bm{F}\Phi\cdot\bm{F}.$ (7)

Therefore, the nonlinear coefficients are dressed with the mean density $n$,
and the density-density interaction becomes a constant energy shift due to the
feature of the plane-wave solutions. The SOC Hamiltonian becomes

$H_{\text{SOC}}(k)=\frac{k^{2}}{2}+2kF_{z}+\sqrt{2}\Omega F_{x}+\epsilon
F^{2}_{z}.$ (8)

The nonlinear dispersion relation $\mu(k)$, which is the dependence of the
chemical potential on the quasimomentum, can be obtained by numerically
solving Eq. (6). The density-density interactions $c_{0}n$ only contribute to
an overall shift in the dispersion, and therefore do not have essentially
physical consequences. The nonlinear effect is solely determined by the spin-
spin interactions $c_{2}n$ which can be tuned by varying the mean density $n$
and by choosing different kinds of atoms to change the sign of $c_{2}$.

We first study the dispersions with tunable $c_{2}n$ in the absence of the
quadratic Zeeman effect ($\epsilon=0$), which are demonstrated in Figs.
1(a1)-1(a3).

(i) Figure 1(a1) shows the single-particle dispersion ($c_{2}n=0$). The
dispersion has three bands, which are symmetric with respect to $k=0$ due to
the symmetry $\mathcal{P}F_{x}$ with the parity operator
$\mathcal{P}k\mathcal{P}^{-1}=-k$. There are three SOC avoided crossings; the
left (right) one located around $k=-1$ ($k=1$) originates from the two-photon
transition that couples $\Phi_{1}$ and $\Phi_{2}$ ($\Phi_{2}$ and $\Phi_{3}$),
and therefore the gap sizes of these two crossings are proportional to the
two-photon coupling strength $\Omega$; the middle crossing around $k=0$
originates from the four-photon transition that couples $\Phi_{1}$ and
$\Phi_{3}$, so its gap size is proportional to $\Omega^{2}$ [which is
invisible in Fig. 1(a1)]. The left and right ones are symmetric with respect
to $k=0$ and have a lower energy than the middle one.

(ii) For an antiferromagnetic spin-spin interaction $c_{2}n=0.8$, the
nonlinear dispersion is depicted in Fig. 1(a2). The nonlinear dispersion is
still symmetric with respect to $k=0$. The outstanding feature is the
appearance of loop-like structures around the SOC avoided crossings. Tilted
cusp structures bifurcating from the lowest band appear in both the left and
right avoided crossings, and a loop adhering to the highest band appear in the
middle avoided crossing. We emphasize that the two cusps and single loop are
purely nonlinear effects which will lead to intriguing NLZTs.

(iii) The nonlinear dispersion for a typical ferromagnetic spin-spin
interaction $c_{2}n=-0.8$ is depicted in Fig. 1(a3). In comparison with Fig.
1(a2), both the tiled cusps and the loop bifurcate from the middle band.

The physical origination of these nonlinear structures can explain which band
they bifurcate from and why they show a cusp or loop geometry. The middle
crossing around $k=0$ originates from the coupling between $\Phi_{1}$ and
$\Phi_{3}$. By assuming $\partial\Phi_{2}/\partial t=0$ and
$\partial^{2}\Phi_{2}/\partial x^{2}=0$ in Eq. (3), we obtain
$\Phi_{2}\approx-\Omega\left(\Phi_{1}+\Phi_{3}\right)/[c_{0}n+c_{2}n(|\Phi_{1}|^{2}+|\Phi_{3}|^{2}+2\Phi_{1}\Phi_{3})]$,
which is small quantity so that $\Phi_{2}$ can be adiabatically eliminated. In
this way, the coupling between $\Phi_{1}$ and $\Phi_{3}$ can be shown
explicitly. By substituting $\Phi_{2}$ into Eq. (6) and neglecting the higher
order terms of $\Phi_{2}$, we obtain the effective two-level model,

$\mu_{\text{ntl1}}\Phi^{\prime}=H_{\text{ntl1}}\Phi^{\prime}.$ (9)

Here, $\Phi^{\prime}=(\Phi_{1},\Phi_{3})^{T}$ satisfies the normalization
condition $|\Phi_{1}|^{2}+|\Phi_{3}|^{2}=1$, and
$\mu=\mu_{\text{ntl1}}+k^{2}/2+\Omega^{\prime}+c_{0}n$ with

$\displaystyle\Omega^{\prime}$
$\displaystyle=-\frac{\Omega^{2}}{c_{0}n+c_{2}n\left(1+2\Phi_{1}\Phi_{3}\right)},$
$\displaystyle\approx-\frac{\Omega^{2}}{c_{0}n+2c_{2}n}.$

We have assumed that $2\Phi_{1}\Phi_{3}\approx|\Phi_{1}|^{2}+|\Phi_{3}|^{2}=1$
to achieve a wave-function independent $\Omega^{\prime}$. The effective two-
level Hamiltonian can be written as

$H_{\text{ntl1}}=\Omega^{\prime}\sigma_{x}+2k\sigma_{z}-c_{2}n\left(\left|\Phi_{3}\right|^{2}-\left|\Phi_{1}\right|^{2}\right)\sigma_{z},$
(10)

where $\sigma_{x,z}$ are the spin-1/2 Pauli matrices. The effective two-level
model is similar to the standard two-level nonlinear system from which the
existence and features of loop structure are analytically identified [26]. The
terms $\Omega^{\prime}\sigma_{x}+2k\sigma_{z}$ define an avoided crossing
centered at $k=0$ with the gap size being $2|\Omega^{\prime}|$. If the
nonlinearity dominates over the gap, i.e., $|c_{2}|n>|\Omega^{\prime}|$, a
loop structure appears. When the effective interaction is attractive
($c_{2}>0$), the loop bifurcates from the upper band of the avoided crossing
and grows downwards into the gap [26, 66]. When the effective interaction is
repulsive ($c_{2}<0$), the loop adheres to the lower band of the avoided
crossing and grows upwards into the gap. According to Ref. [26], the loop
extends in the region $2|k|<|(c_{2}n)^{2/3}-\Omega^{\prime 2/3}|^{3/2}$. In
Fig. 2(a), we show the nonlinear dispersion of the effective two-level model
for an attractive interaction $c_{2}n=0.8$. Generally, the dispersion of the
effective model can match that from full calculation of the spinor BEC.
However, a slight mismatch still exists due to the adiabatic elimination of
$\Phi_{2}$.

Figure 2: Nonlinear dispersion of the effective two-level model [shown by the
red-dotted line]. The blue lines are resultant dispersion from the spinor BEC
in Eq. (6). The parameters for (a) and (b) are same as those in Figs. 1(a2)
and 1(a3), respectively.

The left avoided crossing around $k=-1$ shown in Figs. 1(a1)-1(a3) originates
from the coupling between $\Phi_{1}$ and $\Phi_{2}$ and meanwhile has a
negligible occupation of $\Phi_{3}$. By setting $\Phi_{3}=0$, Eq. (6) becomes
the following effective two-level model,

$\mu_{\text{ntl2}}\Phi^{\prime\prime}=H_{\text{ntl2}}\Phi^{\prime\prime},$
(11)

with $\Phi^{\prime\prime}=(\Phi_{1},\Phi_{2})^{T}$ satisfying
$|\Phi_{1}|^{2}+|\Phi_{2}|^{2}=1$ and
$\mu=\mu_{\mathrm{ntl2}}+k^{2}/2+k+1+c_{0}n+c_{2}n/2$. The corresponding
effective Hamiltonian is

$H_{\text{ntl2}}=\Omega\sigma_{x}+\left(k+1+\frac{c_{2}n}{2}\right)\sigma_{z}+\frac{c_{2}n}{2}\left|\Phi_{1}\right|^{2}\left(I-\sigma_{z}\right),$
(12)

with $I$ being the $2\times 2$ identical matrix. In this case, the linear part
of $H_{\text{ntl2}}$ represents an avoided crossing centered at
$k=-1-c_{2}n/2$ with the gap of size $2\Omega$. The nonlinear term
$c_{2}n|\Phi_{1}|^{2}(I-\sigma_{z})/2$ is not symmetric as that in Eq. (10).
Due to the the asymmetric nonlinearity, the nonlinear structure is no longer a
loop. The non-commutation of $\Omega\sigma_{x}$ and
$-c_{2}n/2|\Phi_{1}|^{2}\sigma_{z}$ gives rise to the nonlinear cusp when
$|c_{2}|n/2>\Omega$. In comparison with the standard nonlinear two-level model
[26], we know that the effective interaction behaves as repulsive when
$c_{2}>0$ and it is effectively attractive when $c_{2}<0$. In the presence of
the repulsive (attractive) interaction, the cusp bifurcates from the lower
(higher) band of the avoided crossing. Furthermore, the cusp structure is
tilted due to the terms $k^{2}/2+k$ in the chemical potential. In Fig. 2(b),
we plot the nonlinear dispersion $\mu(k)$ calculated from the effective two-
level mode in Eq. (11) for an effectively attractive interaction
$c_{2}n=-0.8$. It shows that the effective model can describe the tiled cusp
very well.

The experimentally tunable quadratic Zeeman shift $\epsilon$ is of essential
importance to modify the nonlinear dispersion. Figures 1(b1)-1(b3) demonstrate
the dispersions for different nonlinearities with a negative quadratic Zeeman
shift $\epsilon=-4$. In the single-particle dispersion shown in Fig. 1(b1),
the negative $\epsilon$ pulls down the components $\Phi_{1}$ and $\Phi_{3}$.
As a result, the middle avoided crossing at $k=0$ [which is still invisible
due to the extremely small gap] declines and the right and left ones at $k=\pm
1$ are lifted, meanwhile the gap size of the left and right ones is obviously
smaller than that of the $\epsilon=0$ case [see Fig. 1(a1)]. Finally, the
middle one is located in the lower part compared with the right and left ones.
The left (right) one is due to the coupling between $\Phi_{2}$ and $\Phi_{3}$
($\Phi_{2}$ and $\Phi_{1}$). For the antiferromagnetic spin-spin interaction
$c_{2}n=0.8$ demonstrated in Fig. 1(b2), tiled cusps appear in the right and
left avoided crossings and a loop exists in the middle avoided crossing. Both
the cusps and the loop bifurcate from the middle band. While, for the
ferromagnetic spin-spin interaction $c_{2}n=-0.8$ [see Fig. 1(b3)], the tilted
cusps bifurcate from the highest band and the loop bifurcates from the lowest
band.

Figures 1(c1)-1(c3) describe the dispersions for different nonlinearities with
a positive quadratic Zeeman shift $\epsilon=2$. In the single-particle
dispersion shown in Fig. 1(c1), the positive $\epsilon$ pushes up $\Phi_{1}$
and $\Phi_{3}$. The left (right) avoided crossing still originates from the
coupling between $\Phi_{1}$ and $\Phi_{2}$ ($\Phi_{2}$ and $\Phi_{3}$), but
its location moves towards a larger $|k|$ around 2. The middle one is still at
$k=0$. The nonlinear dispersions in Fig. 1(c2) [$c_{2}n=0.8$] and Fig. 1(c3)
[$c_{2}n=-0.8$] show the appearance of tilted cusps and a single loop.

We can make following conclusions from Fig. 1. In a SOC spinor system, there
are three SOC avoided crossings originating from the couplings between the two
of three components. The coupling of $\Phi_{1}$ and $\Phi_{3}$ stemming from
the four-photon transition induces the middle avoided crossing which is always
located at $k=0$ with a higher order smaller gap size. The couplings of
$\Phi_{1}$ and $\Phi_{2}$ (or $\Phi_{2}$ and $\Phi_{3}$) induced by the two-
photon transitions give rise to the other two crossings located at a finite
$|k|$. They are always symmetric with respect to $k=0$. The quadratic Zeeman
shift $\epsilon$ can change the locations of the other two crossings in
quasimomentum $k$ space and tune the relative chemical potential between the
three crossings. The nonlinearity $c_{2}n\neq 0$ always generates a loop in
the middle avoided crossing and tilted cusps in the other two. The loop
bifurcates from the upper (lower) band of the middle crossing for $c_{2}n>0$
($c_{2}n<0$). The tilted cusps bifurcate from the lower (upper) band of the
other two avoided crossings for $c_{2}n>0$ ($c_{2}n<0$).

Figure 3: Nonlinear dynamics induced by the weak acceleration force
$F=0.0001$. (a1) Nonlinear dispersion relation shown in Fig. 1(a3) which
incorporates the color scale to show the tensor magnetization $\langle
F_{z}\rangle=|\Phi_{1}|^{2}-|\Phi_{3}|^{2}$ of the corresponding eigenstates.
Parameters are $\epsilon=0$, $\Omega=0.1$, $c_{0}n=1$, and $c_{2}n=-0.8$. (a2)
The zoom-in of the selected box region in (a1). (a3)-(a5) The time evolution
of populations $\rho_{i}=|\Phi_{i}|^{2}$ for the different initial states
which are labeled by the dots “(a3)”, “(a4)”, and “(a5)” in Fig. 1(a1)
respectively. (b1)-(b5) show the same quantities as (a1)-(a5), but the
nonlinear dispersion is modified by the quadratic Zeeman shift $\epsilon=-4$,
and other parameters are $\Omega=0.1$, $c_{0}n=1$, and $c_{2}n=-0.8$.

##  III NLZT in the presence of both the tilted cusps and the loop

The fundamental consequence of these nonlinear structures of tilted cusps and
loops is the NLZT, which can be observed by applying an acceleration force $F$
to move the BEC along the nonlinear dispersion [66]. In experiments, the force
may come from the gravitational force or the optical trap induced force in
momentum space, the magnitude of which can be tuned by adjusting the trap
frequency [62]. The force changes the quasimomentum of the BEC as

$k(t)=k_{\text{ini}}+Ft,$ (13)

with $k_{\text{ini}}$ being the initial quasimomentum. We stimulate nonlinear
dynamics induced by a weak acceleration force $F=0.0001$ by numerically
integrating Eq. (3) in momentum space with help from Eq. (13). The weak force
is chosen in order to avoid the excitation induced by the normal Landau-Zener
transitions.

We first study motions along the nonlinear dispersion shown in Fig. 1(a3).
This dispersion is shown again in Fig. 3(a1) which incorporates the tensor
magnetization $\langle F_{z}\rangle=|\Phi_{1}|^{2}-|\Phi_{3}|^{2}$ by the
color. The blue, green and red colors correspond to eigenstates dominated by
$\Phi_{1}$, $\Phi_{2}$, and $\Phi_{3}$ respectively. The nonlinear dynamics
induced by the weak acceleration force strongly depend on the initial states
which can be well prepared in experiments [62].

(i) The initial state is chosen as the ground state at $k=-2$ which is labeled
by the dot “(a3)” in Fig. 3(a1). The weak acceleration force pushes atoms to
move adiabatically along the lowest band, in the presence of a finite gap
between the lowest two bands [see the zoom-in dispersion shown by Fig. 3(a2)].
The spin-momentum locking of the spin-orbit coupling is represented by the
continuous color change in the lowest band. The time evolution of populations
$\rho_{i}=|\Phi_{i}|^{2}$ are demonstrated in Fig. 3(a3). The initial ground
state is mainly occupied by $\rho_{1}$. When atoms go through the left avoided
crossing, $\rho_{1}$ is smoothly swapped with $\rho_{2}$, and around the right
avoided crossing $\rho_{2}$ is smoothly swapped with $\rho_{3}$. These smooth
exchanges of populations are the signature of the adiabatic motion along the
lowest band.

(ii) The initial state is arranged at $k=-2$ in the middle band [as shown by
the dot labeled as “(a4)” in Fig. 3(a1)]. The dominant occupation in the
initial state is $\rho_{2}$. The force moves the initial state towards the
tilted cusp in the left avoided crossing. When atoms reach the edge of the
cusp [represented by the vertical dashed line “$\alpha_{1}$” in Fig. 3(a2)],
there is no further trajectory for them to go forward. They must jump to the
neighboring lowest two bands, leading to NLZT. The population $\rho_{2}$
suddenly decreases and $\rho_{1}$ dramatically increases at the time
corresponding to $\alpha_{1}$ [see Fig. 3(a4)], and after these sudden changes
both $\rho_{2}$ and $\rho_{1}$ oscillate fast. These sudden changes and fast
oscillations are the distinguished features of NLZT. After the NLZT at the
edge of the cusp, atoms cannot adiabatically follow a certain band and the
emergent fast oscillations are due to the beating between the lowest two
bands. Further acceleration pushes atoms to move along the bands represented
by green and blue colors in Fig. 3(a2). The time evolutions of $\rho_{2}$ and
$\rho_{1}$ have no obvious changes even when atoms reach the edge of the loop
[the vertical line “$\alpha_{2}$”] and the edge of the right cusp [the
vertical line “$\alpha_{3}$”]. There is no occupation of the red-colored band
shown in Fig. 3(a2). After the time corresponding to “$\alpha_{3}$”, there is
a swapping between $\rho_{2}$ and $\rho_{3}$ [see Fig. 3(a4)], which indicates
that some atoms move along the lowest band and pass the right avoided
crossing.

(iii) The initial state is arranged at $k=-2$ in the highest band [as shown by
the dot labeled as “(a5)” in Fig. 3(a1)]. The dominant occupation in the
initial state is $\rho_{3}$. The force pushes the initial state towards the
middle avoided crossing. The loop bifurcates from the middle band and there is
a very small gap between the loop and the highest band [which is invisible in
Fig. 3(a2)]. Finally, atoms adiabatically move along the highest band. Since
the population of $\rho_{2}$ is the negligible in the highest band, the time
evolution of $\rho_{2}$ illustrated in Fig. 3(a5) always vanishes. Around the
edges of middle avoided crossing at $k\approx 0.4$, the highest band has
dramatic warps due to the existence of the loop, which makes the swapping
between $\rho_{3}$ and $\rho_{2}$ not smooth [see Fig. 3(a5)].

Therefore, in order to stimulate NLZT, the initial state should be prepared in
the same band from which the nonlinear structures bifurcate. In the following,
we study NLZT in the dispersion modified by the quadratic Zeeman effect, as
shown in Fig. 1(b3). This dispersion is demonstrated in Fig. 3(b1) again with
the color scale to guide the dominant population in eigenstates. The cusp and
loop structures are zoomed-in in Fig. 3(b2).

(i) Figure 3(b3) demonstrates the time evolution of populations for the
initial state in the lowest band [labeled by the dot “(b3)” in Fig. 3(b1)].
The force pushes atoms to the edge of the loop in the middle avoided crossing
[represented by the vertical dashed line “$\beta_{2}$”] and further push leads
to NLZT. Around the NLZT, $\rho_{1}$ and $\rho_{2}$ change dramatically [see
the vertical dashed line “$\beta_{2}$” in Fig. 3(b3)]. The dramatic changes
and following fast oscillations in $\rho_{1}$ and $\rho_{2}$ signify that the
NLZT results in the nonadiabatic superposition of the blue- and red-colored
bands shown in Fig. 3(b2). In the further evolution, there is a swapping
between $\rho_{1}$ and $\rho_{2}$ [see Fig. 3(b3)], since the blue-colored
band crosses over to the green-colored band around the right avoided crossing
at $k=\beta_{3}$ in Fig. 3(b2).

(ii) Figure 3(b4) illustrates the time evolution of populations for the
initial state in the middle band [labeled by the dot “(b4)” in Fig. 3(b1)].
The results indicate that the smooth swapping between $\rho_{2}$ and
$\rho_{3}$ around $Ft=0.8$, $\rho_{3}$ and $\rho_{1}$ around $Ft=2$, and
$\rho_{1}$ and $\rho_{2}$ around $Ft=3.4$ are due to the adiabatic motions
along the band around the left, middle, and right avoided crossings. There is
no NLZT happening in this case.

(iii) Figure 3(b5) shows the time evolution of populations for the initial
state in the highest band [labeled by the dot “(b5)” in Fig. 3(b1)]. There are
two NLZTs around $Ft=\beta_{1}$ and $Ft=\beta_{3}$ which correspond to the
edges of the cusps in the left and right avoided crossings [see vertical lines
in Fig. 3(b2)]. After the first NLZT, atoms occupy the green and red colored
bands shown in Fig. 3(b2). The population of the red-colored band
adiabatically evolves along this band so that there is no physical change when
$Ft>\beta_{1}$. However, the population of the green-colored band experiences
the second NLZT when atoms reach the edge $\beta_{3}$ of the cusp in the right
avoided crossing.

##  IV NLZT in the presence of the loop

Figure 4: NLZT in the presence of only one loop in the middle avoided
crossing, with $c_{2}n=-0.05$. (a) Nonlinear dispersion with the color scale
to show the tensor magnetization of corresponding eigenstates. The inset is
the zoom-in of the looped avoided crossing. (b) The dynamics of populations
with the initial state shown by the dot in (a), which is triggered by the weak
acceleration force $F=0.0001$. Other parameters are same as in Fig. 3(a1).

We have shown the possible NLZTs induced by the nonlinear cusp and loop
structures. The appearance of these structures requires the nonlinearity
$c_{2}n$ to dominate gap sizes of avoided crossings. In the SOC spin-1 system,
the four-photon transition generates an SOC avoided crossing with the gap much
smaller than the two-photon transition induced ones. Therefore, there exists
the nonlinear regime, in which a loop appears in the four-photon-induced
avoided crossing, and there is no cusp in the two-photon-induced ones. Figure
4(a) demonstrates such a nonlinear dispersion with $c_{2}n=-0.05$, in which
the loop structure bifurcating from the middle band is zoomed-in. In order to
observe NLZT, the initial state should be prepared in the same band, which is
labeled by the dot in Fig. 4(a). The weak acceleration force pushes the
initial state to move along the middle band. When atoms pass the left avoided
crossing, the populations $\rho_{2}$ and $\rho_{1}$ are swapped [see Fig.
4(b)]. The further acceleration pushes atoms to the edge of the loop
[represented by the vertical dashed line “$\gamma$” in the inset of Fig.
4(b)]. Then the NLZT happens, which is reflected by the sudden changes of
populations $\rho_{1}$ and $\rho_{3}$, and by the following fast oscillations
in Fig. 4(b). After the transition at $Ft=\gamma$, the populations $\rho_{1}$
and $\rho_{3}$ are occupied, illustrating that atoms stay in the bands
represented by red and blue colors in Fig. 4(a). The atoms in the red-colored
band further have a swapping between $\rho_{3}$ and $\rho_{2}$ when they are
accelerated around the right avoided crossing.

The NLZT shown in Fig. 4 needs only a small nonlinearity $c_{2}n$, which is
preferable for its experimental realization in 87Rb spinor BECs.

##  V Conclusion

We have shown that a SOC spinor BEC is a tunable and versatile platform to
study NLZT. The NLZTs can be induced by nonlinear cusp and loop structures
appearing around SOC avoided crossings in the nonlinear dispersion. The
emergence of these nonlinear structures is only related to the spin-spin
interactions and does not depend on the density-density interactions. This
makes it possible to tune the nonlinearity by adjusting atomic density and to
observe NLZT in a 87Rb spinor BEC with the spin-orbit coupling. Our study
provides an interesting route to investigate nonlinear dynamics in spinor
BECs.

## Acknowledgment

We acknowledge valuable discussions with Prof. Doerte Blume and Prof. Yong Xu.
This work is supported by National Natural Science Foundation of China with
Grants No. 12374247 and No. 11974235, and Shanghai Municipal Science and
Technology Major Project (Grant No. 2019SHZDZX01-ZX04). L.H. acknowledges
support from Okinawa Institute of Science and Technology Graduate University.

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FERMILAB-FN-1143-PPD-SCD

Sensitivity to Dijet Resonances at Proton-Proton Colliders

Robert M. Harris

Fermilab, P.O. Box 500, Batavia, IL, 60510, U.S.A.

Emine Gurpinar Guler, Yalcin Guler

Konya Technical University, Road 42250, Selcuklu/Konya, Turkey

> ABSTRACT
>
>
>
>
> A significant benchmark for discovery at a proton-proton collider is the
> sensitivity to a dijet resonance, X, the intermediate state of the s-channel
> process $pp\rightarrow X\rightarrow 2\mbox{ jets}$. To probe the highest
> resonance masses, hadron collider experiments have used the classic
> technique of searching for bumps in the mass spectrum of two individually
> resolved jets. In this Snowmass 2021 study, we explore the search
> sensitivity to multiple benchmark models of dijet resonances at current and
> future proton-proton colliders. We present the expected masses for $5\sigma$
> discovery or 95% confidence level exclusion of diquarks, colorons, excited
> quarks, $W^{\prime}s$, $Z^{\prime}s$ and Randall-Sundrum gravitons,
> resulting from accumulation of integrated luminosities between 10 and
> $10^{5}$ fb-1, at proton-proton colliders operating at energies $\sqrt{s}=$
> 13, 14, 27, 75, 100, 150, 300 and 500 TeV.

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

###### Contents

  1. 1 Introduction
    1. 1.1 Dijet Resonance Models
      1. 1.1.1 Strongly Produced Resonances
      2. 1.1.2 Weakly Produced Resonances
    2. 1.2 pp Colliders
  2. 2 Methodology
    1. 2.1 QCD Background
    2. 2.2 Signal Acceptance
    3. 2.3 Check: Limits at LHC Run 2
    4. 2.4 Determination of Sensitive Cross Section and Mass
  3. 3 Mass Sensitivity
    1. 3.1 $q^{*}$ and $Z^{\prime}$
    2. 3.2 Strongly and Weakly Produced Models
    3. 3.3 HL-LHC and FCC-hh
    4. 3.4 Sensitivities of Colliders
  4. 4 Conclusions
  5. 5 Appendix: Sensitivity Tables

##  1 Introduction

Dijet resonances are an essential benchmark of discovery capability of proton-
proton colliders. The process, shown in Fig. 1, is sensitive to a variety of
new physics at the highest mass scales. Predicted by many models of new
physics that have been proposed to address fundamental questions, dijet
resonances have been searched for by every hadron collider [1]. Proton-proton
colliders are natural dijet resonance factories. Dijet resonances, X, produced
by annihilation of partons in the colliding protons, must decay to two partons
giving dijets.

Figure 1: s-channel production, via parton-parton annihilation, of a resonance
(X) that subsequently decays to pairs of partons ($q$, $\overline{q}$ or $g$),
giving a dijet in the final state.

This whitepaper estimates the sensitivity of pp colliders to this essential
process. In the remainder of this section we discuss the benchmark models and
pp collider scenarios we will consider. In section 2 we give the details of
the methodology we use to evaluate the sensitivity of the pp collider
scenarios. There we also present estimates of the QCD background, checked
against the published results from the LHC, and present similar checks for the
estimated sensitivities. In section 3 we summarize the results of the study.
We begin with examples of the discovery cross sections compared to the cross
sections of the models. We then discuss examples of the discovery sensitivity
at 5$\sigma$ significance and exclusion sensitivity at 95% confidence level,
for a few different ways to organize the models and pp collider scenarios. In
section 4 we conclude with some final observations. In the appendix, section
5, we tabulate the sensitivity results for all models and pp collider
scenarios.

###  1.1 Dijet Resonance Models

We consider multiple benchmark models, exploring all of the parton-parton
initial states available at pp colliders, and spanning a range of cross
sections from various strengths of interaction and parton density. All models
are narrow resonances, those for which the natural half-width, $\Gamma/2$, is
smaller than the effective experimental resolution. We perform lowest order
calculations, using the same code as used by ref [2], of the product of the
cross section, branching ratio, and acceptance in the narrow-width
approximation, using CTEQ6L1 parton distributions and renormalization scale
equal to the resonance mass. We classify the models into two groups according
to their production cross section: strongly and weakly produced.

Figure 2: Benchmark models of narrow resonances. Strongly produced models
(left), include scalar diquarks, colorons and excited quarks. Weakly produced
models (right), include heavy bosons W’ and Z’, and the Randall-Sundrum
graviton. The tables list the resonance’s spin, the partons within the
resonance production and decay process, and the coupling strength.

####  1.1.1 Strongly Produced Resonances

Figure 2 summarizes the models. The strongly produced models are the ones that
have larger cross section, typically of order the QCD background cross
section. The largest cross section are for the scalar diquarks [3], despite
couplings to $uu$ and $dd$ assumed to be electromagnetic strength. This is
because diquarks are produced from large parton densities, the parton
distribution functions (PDF), found for the valence $u$ and $d$ quarks inside
each of the colliding protons. Colorons are massive gluons from an extra color
interaction, where we assume a minimal mixing betweeen the two color
interactions ($\cot\theta=1$). For this choice of coupling the resonance is
narrow, and the production cross sections are identical for two canonical
models of colorons: axigluons [4] and flavor-universal colorons [5]. Also
produced from the color-like interaction, specifically a chromo-magnetic
transition, are excited quarks ($q^{*}$) predicted by models of quark
compositeness [6]. We consider mass-degenerate first generation excited quarks
($u^{*}$, $d^{*}$), standard model couplings ($f=f_{s}=f^{\prime}=1$), and set
the energy scale $\Lambda$ of quark compositeness equal to the mass of the
excited quark. While this is a conventional model that has been used by every
search at hadron colliders, we note that this model implies an even larger
effect from the contact interaction of the colliding quarks at this
compositeness scale [7]. Such an effect should be experimentally observed
earlier than a resonance, from measurements of the shape of the jet $p_{T}$,
dijet mass, or dijet angular distributions.

####  1.1.2 Weakly Produced Resonances

The weakly produced models in Fig. 2 are the ones having smaller cross
section, typically of order the electroweak processes. The new gauge bosons
$W^{\prime}$ and $Z{{}^{\prime}}$ [8], are the canonical heavy cousins to the
electroweak bosons $W$ and $Z$, respectively. They have the same couplings to
quarks and leptons as in the standard model, and hence the model is frequently
called the sequential standard model (SSM). When calculating the branching
fraction to dijets we only include decays to the five lightest quarks, so the
decays to top quarks reduces our branching fraction to jets, and we neglect
possible decays to $W$ or $Z$ bosons. A next-to-leading order correction
factor of $K=1+8\pi\alpha_{s}/9$ is applied to the leading order predictions
for the $W^{\prime}$ model and $K=1+(4\alpha_{/}6\pi)(1+4\pi^{2}/3)$ for the
$Z^{\prime}$ model. Randall–Sundrum (RS) gravitons are predicted in a model of
warped extra dimensions [9]. The value of the dimensionless coupling parameter
$k/\overline{M}_{Pl}$ is chosen to be 0.1, as is usually done for dijet
resonance searches. RS graviton cross sections are typically the lowest of all
the models considered, so we classify them as weakly produced, although we
note that the production mechanism has nothing to do with electroweak
interactions.

###  1.2 pp Colliders

We perform a comprehensive study of all scenarios for current and future pp
colliders, considering eight collision energies ($\sqrt{s}$):

  1. 1.

13 TeV: Large Hadron Collider (LHC) during Run 2 occurring from 2016 to 2018.

  2. 2.

14 TeV: LHC in the High Luminosity (HL-LHC) era.

  3. 3.

27 TeV: High Energy LHC (HE-LHC) and Fermilab site filler (FNAL-SF) proton-
proton options.

  4. 4.

75 TeV: Super Proton-Proton Collider (SPPC) and a lower energy option for the
Future Circular Collider-hadron hadron (FCC-hh).

  5. 5.

100 TeV: Median and usually assumed energy for FCC-hh.

  6. 6.

150 TeV: High energy option for FCC-hh or a similar machine.

  7. 7.

300 TeV: A Very Large Hadron Collider (VLHC) proposed at a snowmass energy
frontier workshop.

  8. 8.

500 TeV: Collider in the Sea, also discussed at snowmass energy frontier
workshop.

We consider ten integrated luminosities ($\int L\/dt$). Five general decades,
$10^{1}$-$10^{5}$ fb-1, and five baseline values which were actually used for
LHC runs or anticipated for future runs and colliders : 140 fb-1 (LHC Run 2),
200 fb-1 (LHC Run 3), 3 ab-1 (HL-LHC), 2.5 and 30 ab-1 (FCC-hh).

We determine the mass sensitivity for discovery and exclusion of each dijet
resonance model at all values of collision energy and integrated luminosity.

##  2 Methodology

In order to obtain sensitivities for six models, considering eight collision
energies and ten integrated luminosities, we require a simple and fast
methodology. We adopt and update the methodology of a prior study one of the
authors had conducted for Snowmass 1996 [10], using lowest order parton level
calculations of the QCD background and the dijet resonance signal. QCD
calculations are performed using CTEQ6L1 parton distributions with a
renormalization scale of $\mu=p_{T}/2$, and dijet resonance calculations are
performed with the same PDF and a renormalization scale equal to the resonance
mass.

Selection cuts are similar to publications from LHC [2, 11]. The dijet mass,
$m$, is calculated for the two final state partons with pseudo-rapidity
$|\eta|<2.5$. To suppress the large background from QCD t-channel scattering
of the partons, we require the pseudorapidity separation of the the two
partons to satisfy $|\Delta\eta|<1.1$, which is equivalent to selecting events
with a center-of-momentum frame scattering angle $|\cos\theta^{*}|<0.5$.

The expected number of signal and background events, inside a search window in
$m$, is calculated and used to estimate sensitivities. As in the prior
Snowmass 1996 study, the window is centered on the pole mass of the resonance,
$M$, and has a width equal to 16.4%: $0.836M<m<1.164M$. That width, originally
constructed to provide 90% acceptance for a Gaussian with width 10% of the
pole mass, turns out to accept a reasonable amount of signal for the dijet
resonance shapes we expect, discussed in section 2.2.

From the estimated number of background events in the mass window, $N_{QCD}$,
we can estimate the expected exclusion or discovery cross section.

The 95% confidence level excluded number of events is defined as

$N_{excl}=(3+\sqrt{3^{2}+4(1.64^{2})N_{QCD}})/2$ (1)

When $N_{QCD}$ is large, this equation predicts the known number of signal
events that can be excluded at 95% CL using Gaussian statistics,
$N_{excl}=1.64\sqrt{N_{QCD}}$. When $N_{QCD}=0$, it also gives the known
number of events for 95% exclusion from Poisson statistics, $N_{excl}=3$
events. We have checked that Eq. 1 also (remarkably) gives the exact Poisson
statistical answer within 0.5% accuracy for the intermediate region, at least
for $0\leq N_{QCD}\leq 10$.

The 5 $\sigma$ discovery number of events is defined as

$N_{5\sigma}=(25+\sqrt{25^{2}+4(5^{2})N_{QCD}})/2$ (2)

When $N_{QCD}$ is very large, as is usually the case, this equation predicts
the known number of events for discovery from Gaussians statistics,
$N_{5\sigma}=5\sqrt{N_{QCD}}$. When $N_{QCD}=0$, it conservatively assigns the
number of events required for discovery at $N_{5\sigma}=25$, which is the
events required to measure a cross section at $5\sigma$. Equation 2 smoothly
connects the regions of high and low statistics in the same way as Eq. 1 did.

Figure 3: Left) The QCD background as a function of dijet resonance mass, from
a lowest order parton level calculation of the cross section within a 16.4%
mass window centered on the resonance pole, is shown for all collider energies
from 13 to 500 TeV. Right) For $\sqrt{s}=13$ TeV, the lowest order parton
level QCD calculation of the differential cross section as a function of dijet
mass (histogram), is compared to published data from Ref. [2] (points), as a
check of the calculation on the left.

###  2.1 QCD Background

In Fig. 3 we show the QCD background cross section to a dijet resonance for
all scenarios of pp colliders considered, from the LHC to the Collider in the
Sea. The QCD background is calculated for resonance masses between 5% and 75%
of the collision energy, which covers the typical range for a high mass search
at a hadron collider. We note that the QCD background at a mass proportional
to the collision energy decreases smoothly with increasing collision energy.
Figure 3 also shows that the QCD calculation at $\sqrt{s}=13$ TeV agrees with
the measured CMS dijet data [2] to within about 10%. This is likely because
CMS uses the wide jet algorithm, that corresponds well to the partons within a
$2\rightarrow 2$ process. We have also explicitly chosen the renormalization
scale $\mu=p_{T}/2$ for all colliders, because it gives a larger normalization
of the QCD cross section, which is closer to the observed CMS data than the
choice $\mu=p_{T}$ used in the previous study [10]. Thus, while our
calculations are only at lowest order, our ability to predict existing pp
collider data increases our confidence in the QCD background estimates at
future pp colliders.

###  2.2 Signal Acceptance

Figure 4: Left) Experimental simulations of fully reconstructed dijet
resonance signal shapes at $\sqrt{s}=13$ TeV from CMS [12] with our search
window superimposed. Right) Simulations of dijet resonance signal shapes at
genjet level, for excited quarks with a mass equal to $0.5\sqrt{s}$, are
compared for three collision energies: $\sqrt{s}=13$, $100$ and $500$ TeV. For
both plots the wide jet algorithm is used.

The signal acceptance of our mass window for the fully reconstructed shape of
a dijet resonance is illustrated in Fig. 4, which shows signal simulations
using PYTHIA. The fully simulated and reconstructed resonance shape from ref.
[12] at $\sqrt{s}=13$ TeV, has an approximately Gaussian core from
experimental resolution, and a long tail to low values of dijet mass. The tail
is primarily caused by final state radiation falling outside the reconstructed
jet. To minimize this tail and optimize the signal significance ref. [12] used
the wide jet algorithm to reconstruct dijets, where the two jets with the
highest transverse momentum, reconstructed with the anti-$k_{T}$ algorithm and
a distance parameter of 0.4 (AK4 jets), are used as seeds for the wide jets
which contain selected AK4 jets within $\Delta
R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}<1.1$. This effectively widens the
cone size to 1.1, recovering the energy lost to jets radiated outside that
distance, with transverse momentum $p_{T}>30$ GeV inside the fiducial region
$|\eta|<2.5$. Applying our window to the fully reconstructed resonance shapes
at $\sqrt{s}=13$ TeV, as illustrated in Fig. 4, gives the acceptance for each
of the basic types of resonances: quark-quark (qq), quark-gluon (qg) and
gluon-gluon (gg).

The acceptance for any resonance type should be approximately the same at all
$\sqrt{s}$. First, this is based on the observation that the generated shape
of a dijet resonance, with pole mass equal to a fixed fraction of $\sqrt{s}$,
is indeed approximately independent of $\sqrt{s}$. Figure 4 demonstrates the
approximate invariance of the shape of a dijet resonance for the full range of
collision energies we consider. We are using particle level quantities from
the generator, with wide genjets, where we reconstruct AK4 genjets and cluster
them into wide jets if they pass the aforementioned $|\eta|$ and $\Delta R$
cuts, and a generalized transverse momentum cut scaled to the collider energy:
$p_{T}>(\sqrt{s}/13\ \mbox{TeV})30$ GeV. Second, we reason that any detector
will need to be designed to deliver an appropriate experimental resolution for
jets, and that resolution would be one that delivers at least a similar
intrinsic Gaussian core resolution as previous experiments at proton-proton
colliders. We further note that the natural range of dijet mass exploration
scales with $\sqrt{s}$ in the same was as do the genjet resonances shown in
Fig. 4. Thus, the window acceptances of the fully simulated and reconstructed
shapes of ref. [12], should be approximately applicable to any $\sqrt{s}$
value. Furthermore, the signal acceptances, shown in Fig. 5, depend more
strongly on the type of resonance (qq, qg and gg), than on the resonance mass
value for a given type, so we have used the acceptances tabulated in Fig. 5 as
a function of model for all values of $\sqrt{s}$, constant as a function of
resonance mass. See section 5 for more discussion of the accuracy of this
approximation.

###  2.3 Check: Limits at LHC Run 2

The LHC experiments provide an example pp collider scenario, a testbed for our
methodology, and an important check. In Fig. 5 we show for $\sqrt{s}=13$ TeV
the expected upper limits on the QCD background cross section within the mass
window, compared to the signal cross section within that mass window. The
signal cross section is the product of the full signal cross section,
branching fraction to dijets, angular acceptance of the jet $|\eta|<2.5$ and
$|\Delta\eta|<1.1$ cuts which is a part of our calculation, and the mass
window acceptance tabulated in the figure. The expected mass limit on each
model is equal to the mass value at which the expected upper limit on the
window cross section is equal to the signal cross section for the model. Fig.
5 explicitly demonstrates that the expected mass limit on each model of dijet
resonance is approximately equal to the published limits of the LHC
experiments [2, 11]. We conclude that the methodology of this analysis,
estimating dijet resonance sensitivity from a lowest order calculation of
events observed within a mass window, provides sufficient accuracy.

Figure 5: Limits at the LHC during Run 2. Left) Our estimation using this
analysis of the 95% CL expected limit (dashed curve) on the product of cross
section, branching fraction and acceptance, is compared with the signal
prediction for six models of dijet resonances (solid curves) at $\sqrt{s}=13$
TeV for 140 pb-1. Right) For the same models, the published 95% CL expected
mass limits from the LHC experiments (CMS [2], ATLAS [11]) are compared to the
mass limits from this analysis (Snowmass). The acceptance of the mass window
for the CMS resonance shapes in Fig. 4 are also shown for the mass value at
which the limit is found on a given model.

###  2.4 Determination of Sensitive Cross Section and Mass

The $5\sigma$ discovery cross section, the number of events required to make a
$5\sigma$ discovery divided by the integrated luminosity, and the model cross
sections times acceptance in the mass window, are calculated for all pp
collider scenarios discussed in section 1.2. The $5\sigma$ discovery masses on
a given model are those mass values for which the discovery cross section and
the model cross section are equal. This is illustrated for HL-LHC and FCC-hh
in Fig. 6, where we compare the discovery cross sections with the
corresponding signal cross sections for models of dijet resonances. The
discovery masses are the masses where the curves cross. This process is
repeated for all collider scenarios. Note that the discovery cross section is
inversely proportional to the square root of the integrated luminosity when
the background is large, and Gaussian statistics dominates, and is inversely
proportional to the integrated luminosity when there is no background and
discovery requires a fixed number of events.

The process is similar for the determination of 95% CL limits on the cross
section and mass, which is also performed for every pp collider scenario in
section 1.2. We have illustrated the process of finding these limits for the
case of the LHC in Fig. 5. The scaling behavior of the cross section limits
with integrated luminosity is the same as noted above for discovery cross
sections.

##  3 Mass Sensitivity

Figure 6:  Discovery cross sections and models. The $5\sigma$ discovery value
on the product of cross section, branching fraction and acceptance, for the
integrated luminosities shown (dashed curves), is compared with the signal
prediction for six models of dijet resonances (solid curves). Left) For HL-LHC
at $\sqrt{s}=14$ TeV. Right) For FCC-hh at $\sqrt{s}=100$ TeV.

We use the $5\sigma$ discovery mass and the 95% CL mass limits, defined in
section 2.4 and listed in Tables 1-4 in section 5, as metrics of sensitivity
to dijet resonances at proton proton colliders. They are discussed for various
combinations of our benchmark models and machine scenarios in this section.

###  3.1 $q^{*}$ and $Z^{\prime}$

Figure 7:  Sensitivity to an excited quark. Left) the $5\sigma$ discovery mass
and Right) the 95% CL mass limit, for eight values of collider $\sqrt{s}$
(colors), at ten values of integrated luminosity (solid circles). Also shown
for comparison are both types of sensitivities for $\sqrt{s}=27$ and $100$ TeV
from the analysis of Ref. [13] (open boxes), and the public expected mass
limits for $\sqrt{s}=13$ TeV from CMS [2, 12, 14, 15] (solid stars). Figure 8:
the $5\sigma$ discovery mass for an SSM $Z^{\prime}$, for eight values of
collider $\sqrt{s}$ (colors), at multiple values of integrated luminosity
(solid circles).

The sensitivity to an excited quark at proton proton colliders is shown in
Fig. 7. This is the most frequently used benchmark model for dijet resonances
at hadron colliders, and was the first model searched for by both CMS and
ATLAS when the LHC was turned on. Fig. 7 demonstrates that the sensitivity
results from this analysis are close to those of previous studies, and in
excellent agreement with limits from CMS.

We note two scaling behaviors of the sensitivity. First, the mass sensitivity
is proportional to the collision energy. This is expected as the parton
energies are proportional to the collision energies. Second, for any fixed
value of $\sqrt{s}$, the sensitivity is proportional to the logarithm of the
integrated luminosity. Again this is expected, as significant gains in
sensitivity require orders of magnitude more luminosity, due to the
exponential falloff of the parton distributions as a function of energy.

The sensitivity of a Z′ boson in the SSM at proton-proton colliders is shown
in Fig. 8. This is the most frequently used benchmark of a heavy boson which
is weakly produced. The results shown in Fig. 8 are only reported for
resonance mass values greater than $0.06\sqrt{s}$, the lowest mass for which
our window lower edge is above our QCD background minimum dijet mass of
$0.05\sqrt{s}$. Similar to excited quarks, the mass sensitivity of
$Z^{\prime}$ bosons is approximately proportional to $\sqrt{s}$ and to the
logarithm of integrated luminosity.

###  3.2 Strongly and Weakly Produced Models

Figure 9:  Sensitivity to strongly and weakly produced resonance models. The
$5\sigma$ discovery mass for four values of collider $\sqrt{s}$ (colors) as a
function of integrated luminosity for dijet resonances from (left) the large
cross section models of diquarks (boxes), colorons (Xs), and excited quarks
(circles) and from (right) the smaller cross section models of W′ SSM bosons
(crosses) $Z^{\prime}$ SSM bosons (stars) and Randall-Sundrum gravitons
(diamonds).

The sensitivity to a dijet resonance is mainly determined by it’s cross
section, which is why we present the sensitivity organized in two sets of
models in Fig. 9, strongly produced models with a cross section that is
roughly two orders of magnitude greater than weakly produced models. The
maximum vertical axis value in the two plots in Fig. 9 shows that the
discovery mass for the strongly produced resonances is roughly twice that for
the weakly produced resonances. Among the strongly produced models the
sensitivity is largest for diquarks, because the valence quark distribution of
the proton becomes the largest PDF when the momentum of the quark approaches
and exceeds one third of the momentum of the proton. The coloron requires an
anti-quark to produce it, and the excited quark requires a gluon, and the PDFs
for these partons are significantly smaller than for quarks at high momentum.
Among the weakly produced models the sensitivity is largest for $W^{\prime}$
reflecting roughly a factor of two larger cross section than $Z^{\prime}$
caused primarily by the relative SM coupling strengths of the two type of
bosons to quarks within the SM. The Randall-Sundrum graviton production via
gluons pairs is significant at low mass, but becomes negligible at very high
mass compared to quark-antiquark production, giving two slopes as a function
of integrated luminosity from these two processes and their different parton
distribution behavior. Therefore, near the turn-on of the quark-antiquark
production process, the increase in the sensitivity for a Randall-Suncrum
graviton with integrated luminosity is faster than logarithmic. None of these
characteristics distinguishing the models depend on $\sqrt{s}$, hence Fig. 9
demonstrates a very similar behavior of the discovery sensitivity among them
at four colliders, HL-LHC ($\sqrt{s}=14$ TeV), FNAL-SF or HE-LHC
($\sqrt{s}=27$ TeV), FCC-hh ($\sqrt{s}=100$ TeV) and VLHC ($\sqrt{s}=300$
TeV).

Figure 10: Comparison of the sensitivity to dijet resonances of HL-LHC and
FCC-hh for six models. Left) The $5\sigma$ discovery sensitivity as a function
of integrated luminosity. Right) The $5\sigma$ discovery and 95% CL exclusion
sensitivities for the baseline integrated luminosities of each machine.

###  3.3 HL-LHC and FCC-hh

Two important proton-proton machines to discuss and compare are the HL-LHC and
FCC-hh, both based at CERN. The HL-LHC upgrades should be completed over the
next several years, and operations are now planned for 2029-42, and hence
projections for it are important to understand the sensitivity we can expect
to achieve within the next generation. The FCC-hh, while farther off, is being
planned and is a critical part of a world-wide strategy for high energy
physics [16]. The experimental physics motivation for FCC-hh is clear from
Fig. 10. By virtue of the increased collision energy compared to LHC, with
sufficient luminosity FCC-hh can proportionally extend the mass reach for
discovery of dijet resonances. Fig. 10 demonstrates that the mass reach of
FCC-hh with 30 ab-1 is roughly six times that of HL-LHC with 3 ab-1. The
advantage of energy is realized soon after turning on the machine. With just
10 fb-1 of integrated luminosity, accumulated during the first year of data
taking, the mass reach for strongly produced resonances like a coloron or an
excited quark, is already three times that of HL-LHC with a full 3 ab-1. A
modest additional mass reach beyond the baseline can be achieved by running at
higher luminosity. For this phenomena, with 100 ab-1 at FCC-hh, a factor of
seven increase in mass reach is achieved over HL-LHC, fully proportional to
the increase in collision energy. Fig. 10 also makes clear the significant
difference in sensitivity between the three models we classify as strongly
produced, and the three that are weakly produced. We see that for FCC-hh we
would be able to discover strongly produced resonances up to about one half
the $\sqrt{s}$, and weakly produced resonances up to about a quarter of the
$\sqrt{s}$, and a little beyond those fractions of $\sqrt{s}$ for HL-LHC
because it will accumulate a high luminosity sample.

###  3.4 Sensitivities of Colliders

Figure 11: Comparison of the sensitivity to dijet resonances of five pp
Colliders at their approximate baseline integrated luminosities. The mass for
discovery at $5\sigma$, or exclusion at 95% confidence level, is listed for
six models in descending order of model cross section.

In Fig. 11 we summarize the sensitivity of the five major options for future
colliders: HL-LHC, FNAL-SF, FCC-hh at default energy $\sqrt{s}=100$ TeV, VLHC
and the Collider in the Sea. Baseline integrated luminosities for HL-LHC and
FCC-hh are pretty well established, but the baseline integrated luminosities
listed for the other collider options are only a rough guess, and are limited
by the integrated luminosity choices we made when we began the study. In the
appendix in Tables 1-4 we present the sensitivities for all eight collision
energies and ten integrated luminosities, along with a few remarks on the
range of applicability. The production strength is useful for organizing the
results, and we again note that very roughly speaking, the discovery mass
reach of a proton-proton collider is about half it’s collision energy for
strongly produced dijet resonances and about one quarter of it’s collision
energy for weakly produced dijet resonances.

##  4 Conclusions

Dijet resonances are a powerful channel for discovery at pp colliders, that
directly explores the highest mass scales of new physics in parton-parton
collisions. Dijet resonance searches are sensitive to heavy bosons and
fermions, strongly or weakly produced. We have estimated the sensitivity for
discovery or exclusion to multiple models of dijet resonances, across a wide
range of pp collision energy ($\sqrt{s}$) and integrated luminosity ($\int
L\/dt$). Sensitivity to dijet resonance production at pp colliders scales as
expected, increasing roughly linearly with $\sqrt{s}$ and logarithmically with
$\int L\/dt$. A very rough rule of thumb is that the sensitivity is
approximately half the collision energy for strongly produced models and one
quarter the collision energy for weakly produced models. These results may be
compared with those from other channels, and at other types of future
colliders.

##  5 Appendix: Sensitivity Tables

The full results of the study are shown in Tables 1-4. We have produced
results for the full matrix of eight collider energies and ten integrated
luminosities.

There are some exceptions and caveats we want to note. Combinations where
there is no entry in the table, are those for which the sensitivities (or
signal windows) are outside of the range of diparton mass for which we
calculated the QCD background: $0.05<m/\sqrt{s}<0.75$. Trivially, some of the
integrated luminosities are unlikely to be achieved at a given machine, and
are only listed for completeness, like $10^{5}$ fb-1 for $\sqrt{s}=13$ TeV.
More important, there is a maximum resonance mass for which our approximation
of constant window acceptance is reasonable. For example, for excited quarks,
the acceptance for our window in Fig. 4 is fairly flat for excited quarks with
mass $M/\sqrt{s}<0.5$. However, at higher values of resonance mass a very long
tail to lower dijet masses is significantly reducing the acceptance, cutting
the acceptance in half when the resonance mass is $M/\sqrt{s}=0.6$. We
estimate that this reduction in acceptance leads to roughly a 6% reduction in
sensitivity for excited quarks at $M/\sqrt{s}=0.6$. So, for example, the HL-
LHC discovery mass of 7 TeV for an excited quark at the baseline integrated
luminosity of 3 ab-1 in Table 1 is just at the edge of the region of flat
acceptance, $M/\sqrt{s}<0.5$, and is likely accurate. However, the HL-LHC mass
limit of 7.9 TeV at 3 ab-1 in Table 3 corresponds to $M/\sqrt{s}=0.56>0.5$,
which is outside the range of flat acceptance. We estimate this 7.9 TeV
sensitivity should be reduced by roughly 0.3 TeV to account for the estimated
reduction in acceptance. While this is still reasonable accuracy, at higher
values of integrated luminosity, well beyond the HL-LHC baseline, the quoted
sensitivity needs to be reduced significantly more. Other machines run at
ultra-high integrated luminosity, much more than their baseline, may also be
sensitive to excited quark masses where our acceptance approximation is no
longer valid and the mass sensitivities quoted would not be accurate. Similar
statements may apply to the other strongly produced models, although the
models that couple to gluons have more serious problems with acceptance at
ultra-high resonance masses.

5$\sigma$ discovery mass of strong dijet resonances (TeV)
---
$\int L\/dt$ |  pp Collider $\sqrt{s}$ (TeV)
(fb${}^{-1})$ | 13 | 14 | 27 | 75 | 100 | 150 | 300 | 500
| Scalar diquark $\rightarrow qq$
101 | 5.7 | 6.0 | 10.1 | 20.9 | 25.2 | 31.9 | 44.3 | 52.0
102 | 6.8 | 7.2 | 12.6 | 28.8 | 35.9 | 48.2 | 77.1 | 104.5
$1.4\times 10^{2}$ | 6.9 | 7.4 | 13.0 | 29.9 | 37.3 | 50.5 | 81.7 | 112.1
$2.0\times 10^{2}$ | 7.1 | 7.6 | 13.3 | 31.0 | 38.8 | 52.9 | 86.7 | 120.8
103 | 7.7 | 8.3 | 14.8 | 35.6 | 45.3 | 62.9 | 108.1 | 157.3
$2.5\times 10^{3}$ | 8.0 | 8.6 | 15.5 | 38.1 | 48.7 | 68.4 | 119.9 | 177.5
$3.0\times 10^{3}$ | 8.1 | 8.7 | 15.7 | 38.5 | 49.3 | 69.3 | 121.9 | 181.2
104 | 8.5 | 9.1 | 16.6 | 41.5 | 53.4 | 75.9 | 136.1 | 205.9
$3.0\times 10^{4}$ | 8.8 | 9.5 | 17.4 | 43.9 | 56.8 | 81.3 | 148.0 | 227.4
105 | 9.2 | 9.9 | 18.2 | 46.4 | 60.3 | 86.7 | 160.0 | 249.2
| Coloron $\rightarrow q\overline{q}$
101 | 4.5 | 4.8 | 8.0 | 17.1 | 21.0 | 27.6 | 42.8 | 57.5
102 | 5.4 | 5.7 | 9.9 | 22.6 | 28.1 | 38.2 | 62.8 | 88.7
$1.4\times 10^{2}$ | 5.5 | 5.9 | 10.2 | 23.3 | 29.2 | 39.7 | 65.8 | 93.7
$2.0\times 10^{2}$ | 5.7 | 6.0 | 10.5 | 24.1 | 30.3 | 41.4 | 69.1 | 99.1
103 | 6.2 | 6.6 | 11.7 | 27.8 | 35.2 | 48.8 | 84.0 | 123.5
$2.5\times 10^{3}$ | 6.5 | 7.0 | 12.4 | 29.9 | 37.9 | 53.1 | 92.5 | 137.4
$3.0\times 10^{3}$ | 6.6 | 7.1 | 12.6 | 30.2 | 38.5 | 53.9 | 94.2 | 140.2
104 | 7.0 | 7.5 | 13.4 | 32.8 | 42.0 | 59.2 | 105.2 | 158.6
$3.0\times 10^{4}$ | 7.3 | 7.8 | 14.2 | 35.0 | 45.0 | 63.8 | 114.8 | 175.4
105 | 7.7 | 8.2 | 14.9 | 37.4 | 48.2 | 68.8 | 125.2 | 192.9
| Excited quark $\rightarrow qg$
101 | 4.3 | 4.6 | 7.7 | 16.3 | 20.0 | 26.4 | 41.7 | 57.3
102 | 5.3 | 5.6 | 9.6 | 21.4 | 26.6 | 36.0 | 59.2 | 84.1
$1.4\times 10^{2}$ | 5.4 | 5.7 | 9.9 | 22.2 | 27.6 | 37.5 | 62.0 | 88.4
$2.0\times 10^{2}$ | 5.5 | 5.9 | 10.2 | 23.0 | 28.7 | 39.0 | 65.0 | 93.1
103 | 6.2 | 6.6 | 11.5 | 26.7 | 33.6 | 46.3 | 78.9 | 115.4
$2.5\times 10^{3}$ | 6.5 | 7.0 | 12.2 | 28.8 | 36.4 | 50.5 | 87.2 | 128.6
$3.0\times 10^{3}$ | 6.6 | 7.0 | 12.4 | 29.2 | 37.0 | 51.3 | 88.8 | 131.3
104 | 7.0 | 7.5 | 13.3 | 31.9 | 40.6 | 56.8 | 99.7 | 149.3
$3.0\times 10^{4}$ | 7.4 | 8.0 | 14.2 | 34.4 | 43.9 | 61.8 | 109.7 | 165.8
105 | 7.9 | 8.4 | 15.1 | 37.0 | 47.4 | 67.2 | 120.6 | 184.0
Table 1:  Discovery sensitivity for strongly produced dijet resonances at
proton-proton colliders. As a function of collision energy (columns) and
integrated luminosity (rows) the largest resonance mass for which a $5\sigma$
discovery can be expected is listed for three strongly produced models:
diquarks (upper), colorons (middle) and excited quarks (lower)

5$\sigma$ discovery mass of weak dijet resonances (TeV)
---
$\int L\/dt$ |  pp Collider $\sqrt{s}$ (TeV)
(fb${}^{-1})$ | 13 | 14 | 27 | 75 | 100 | 150 | 300 | 500
| W′ SSM $\rightarrow q\overline{q}^{\prime}$
101 | 2.1 | 2.2 | 3.2 | 5.3 | — | — | — | —
102 | 3.1 | 3.3 | 5.2 | 9.9 | 11.6 | 14.2 | 19.7 | —
$1.4\times 10^{2}$ | 3.3 | 3.4 | 5.5 | 10.6 | 12.6 | 15.5 | 22.0 | —
$2.0\times 10^{2}$ | 3.4 | 3.6 | 5.8 | 11.5 | 13.6 | 17.1 | 24.4 | 30.1
103 | 4.1 | 4.3 | 7.2 | 15.3 | 18.7 | 24.3 | 36.8 | 48.0
$2.5\times 10^{3}$ | 4.4 | 4.7 | 8.0 | 17.5 | 21.6 | 28.7 | 44.8 | 60.6
$3.0\times 10^{3}$ | 4.5 | 4.8 | 8.2 | 18.0 | 22.2 | 29.6 | 46.5 | 63.2
104 | 4.9 | 5.2 | 9.1 | 20.8 | 26.0 | 35.2 | 57.8 | 81.0
$3.0\times 10^{4}$ | 5.3 | 5.7 | 9.9 | 23.2 | 29.3 | 40.3 | 68.1 | 98.1
105 | 5.7 | 6.1 | 10.8 | 25.8 | 32.8 | 45.7 | 79.2 | 116.6
| Z′ SSM $\rightarrow q\overline{q}$
101 | 1.6 | 1.7 | 2.4 | — | — | — | — | —
102 | 2.5 | 2.7 | 4.1 | 7.3 | 8.6 | 10.5 | — | —
$1.4\times 10^{2}$ | 2.7 | 2.8 | 4.4 | 7.9 | 9.3 | 11.5 | — | —
$2.0\times 10^{2}$ | 2.8 | 3.0 | 4.7 | 8.7 | 10.1 | 12.7 | — | —
103 | 3.5 | 3.7 | 6.0 | 12.2 | 14.6 | 18.7 | 27.4 | 35.8
$2.5\times 10^{3}$ | 3.9 | 4.1 | 6.8 | 14.3 | 17.4 | 22.5 | 34.2 | 45.3
$3.0\times 10^{3}$ | 4.0 | 4.2 | 7.0 | 14.7 | 17.9 | 23.4 | 35.7 | 47.2
104 | 4.5 | 4.7 | 8.0 | 17.6 | 21.7 | 28.9 | 45.6 | 62.7
$3.0\times 10^{4}$ | 4.9 | 5.2 | 9.0 | 20.2 | 25.1 | 34.0 | 55.6 | 77.9
105 | 5.4 | 5.7 | 10.0 | 23.0 | 28.9 | 39.6 | 66.6 | 95.9
| RS graviton $\rightarrow q\overline{q}$,gg
101 | 1.5 | 1.6 | 2.5 | 4.9 | — | — | — | —
102 | 2.1 | 2.2 | 3.6 | 7.2 | 8.9 | 11.8 | 18.3 | —
$1.4\times 10^{2}$ | 2.2 | 2.4 | 3.8 | 7.6 | 9.3 | 12.4 | 19.6 | —
$2.0\times 10^{2}$ | 2.4 | 2.5 | 4.0 | 8.1 | 9.8 | 13.2 | 21.0 | —
103 | 2.9 | 3.1 | 5.0 | 10.5 | 12.9 | 17.1 | 27.4 | 39.2
$2.5\times 10^{3}$ | 3.2 | 3.4 | 5.7 | 12.1 | 14.8 | 19.8 | 31.6 | 45.3
$3.0\times 10^{3}$ | 3.3 | 3.5 | 5.8 | 12.4 | 15.2 | 20.4 | 32.7 | 46.5
104 | 3.8 | 4.0 | 6.7 | 14.6 | 18.1 | 24.3 | 39.9 | 56.7
$3.0\times 10^{4}$ | 4.2 | 4.5 | 7.6 | 16.9 | 21.0 | 28.4 | 46.9 | 67.7
105 | 4.6 | 4.9 | 8.5 | 19.4 | 24.2 | 33.1 | 55.7 | 80.7
Table 2:  Discovery sensitivity for weakly produced dijet resonances at
proton-proton colliders. As a function of collision energy (columns) and
integrated luminosity (rows) the largest resonance mass for which a $5\sigma$
discovery can be expected is listed for three strongly produced models: heavy
bosons $W^{\prime}$ (upper), $Z^{\prime}$ (middle) and Randall-Sundrum
graviton (lower)

95% CL mass limit of strong dijet resonances (TeV)
---
$\int L\/dt$ |  pp Collider $\sqrt{s}$ (TeV)
(fb${}^{-1})$ | 13 | 14 | 27 | 75 | 100 | 150 | 300 | 500
| Scalar diquark $\rightarrow qq$
101 | 6.7 | 7.2 | 12.5 | 28.4 | 35.3 | 47.3 | 75.5 | 101.8
102 | 7.7 | 8.2 | 14.7 | 35.3 | 44.9 | 62.2 | 106.5 | 154.4
$1.4\times 10^{2}$ | 7.8 | 8.4 | 15.0 | 36.2 | 46.1 | 64.2 | 110.6 | 161.6
$2.0\times 10^{2}$ | 7.9 | 8.5 | 15.3 | 37.2 | 47.4 | 66.3 | 115.2 | 169.6
103 | 8.5 | 9.1 | 16.5 | 41.3 | 53.0 | 75.3 | 134.7 | 203.4
$2.5\times 10^{3}$ | 8.8 | 9.4 | 17.2 | 43.3 | 55.9 | 79.8 | 144.6 | 221.4
$3.0\times 10^{3}$ | 8.8 | 9.4 | 17.3 | 43.7 | 56.5 | 80.7 | 146.6 | 225.1
104 | 9.2 | 9.8 | 18.1 | 46.2 | 60.0 | 86.2 | 158.8 | 246.8
$3.0\times 10^{4}$ | 9.5 | 10.2 | 18.8 | 48.4 | 62.9 | 91.0 | 169.2 | 265.0
105 | — | — | 19.5 | 50.5 | 65.9 | 95.7 | 180.0 | 283.8
| Coloron $\rightarrow q\overline{q}$
101 | 5.4 | 5.7 | 9.9 | 22.3 | 27.8 | 37.7 | 61.9 | 87.1
102 | 6.2 | 6.6 | 11.7 | 27.6 | 35.0 | 48.4 | 83.1 | 121.8
$1.4\times 10^{2}$ | 6.3 | 6.7 | 11.9 | 28.4 | 36.0 | 50.0 | 86.2 | 127.0
$2.0\times 10^{2}$ | 6.4 | 6.9 | 12.2 | 29.2 | 37.0 | 51.6 | 89.6 | 132.4
103 | 7.0 | 7.4 | 13.4 | 32.6 | 41.7 | 58.8 | 104.3 | 157.1
$2.5\times 10^{3}$ | 7.2 | 7.7 | 14.0 | 34.5 | 44.3 | 62.7 | 112.4 | 171.1
$3.0\times 10^{3}$ | 7.3 | 7.8 | 14.1 | 34.9 | 44.8 | 63.5 | 114.0 | 173.9
104 | 7.6 | 8.2 | 14.9 | 37.2 | 48.0 | 68.5 | 124.4 | 191.5
$3.0\times 10^{4}$ | 8.0 | 8.5 | 15.6 | 39.3 | 50.8 | 72.8 | 133.7 | 207.3
105 | 8.3 | 8.9 | 16.3 | 41.4 | 53.7 | 77.4 | 143.2 | 224.1
| Excited quark $\rightarrow qg$
101 | 5.2 | 5.6 | 9.5 | 21.2 | 26.3 | 35.6 | 58.4 | 82.7
102 | 6.1 | 6.5 | 11.4 | 26.5 | 33.3 | 45.9 | 78.1 | 113.9
$1.4\times 10^{2}$ | 6.3 | 6.7 | 11.7 | 27.2 | 34.4 | 47.4 | 81.1 | 118.8
$2.0\times 10^{2}$ | 6.4 | 6.8 | 12.0 | 28.0 | 35.5 | 49.1 | 84.3 | 124.0
103 | 7.0 | 7.5 | 13.3 | 31.7 | 40.4 | 56.4 | 98.9 | 147.9
$2.5\times 10^{3}$ | 7.4 | 7.9 | 14.0 | 33.8 | 43.1 | 60.6 | 107.2 | 161.6
$3.0\times 10^{3}$ | 7.4 | 7.9 | 14.1 | 34.2 | 43.6 | 61.4 | 108.9 | 164.4
104 | 7.9 | 8.4 | 15.0 | 36.8 | 47.2 | 66.8 | 119.8 | 182.6
$3.0\times 10^{4}$ | 8.3 | 8.8 | 15.9 | 39.2 | 50.4 | 71.6 | 129.6 | 199.2
105 | 8.7 | 9.3 | 16.8 | 41.7 | 53.8 | 76.8 | 140.2 | 217.0
Table 3:  Mass limit sensitivity for strongly produced dijet resonances at
proton-proton colliders. As a function of collision energy (columns) and
integrated luminosity (rows) the largest resonance mass for which a 95% CL
exclusion can be expected is listed for three strongly produced models:
diquarks (upper), colorons (middle) and excited quarks (lower)

95% CL mass limit of weak dijet resonances (TeV)
---
$\int L\/dt$ |  pp Collider $\sqrt{s}$ (TeV)
(fb${}^{-1})$ | 13 | 14 | 27 | 75 | 100 | 150 | 300 | 500
| W′ SSM $\rightarrow q\overline{q}^{\prime}$
101 | 3.1 | 3.3 | 5.2 | 9.7 | 11.3 | 14.0 | 19.2 | —
102 | 4.0 | 4.3 | 7.2 | 15.2 | 18.4 | 24.0 | 36.2 | 47.2
$1.4\times 10^{2}$ | 4.2 | 4.4 | 7.5 | 16.0 | 19.5 | 25.6 | 39.1 | 51.2
$2.0\times 10^{2}$ | 4.3 | 4.6 | 7.8 | 16.8 | 20.7 | 27.3 | 42.2 | 56.3
103 | 4.9 | 5.2 | 9.0 | 20.6 | 25.7 | 34.9 | 57.1 | 79.8
$2.5\times 10^{3}$ | 5.2 | 5.6 | 9.7 | 22.7 | 28.5 | 39.1 | 65.7 | 94.0
$3.0\times 10^{3}$ | 5.3 | 5.6 | 9.9 | 23.1 | 29.1 | 40.0 | 67.4 | 96.9
104 | 5.7 | 6.0 | 10.8 | 25.7 | 32.6 | 45.4 | 78.5 | 115.4
$3.0\times 10^{4}$ | 6.0 | 6.4 | 11.5 | 27.9 | 35.7 | 50.1 | 88.3 | 132.1
105 | 6.4 | 6.8 | 12.3 | 30.3 | 38.9 | 55.1 | 98.6 | 150.0
| Z′ SSM $\rightarrow q\overline{q}$
101 | 2.5 | 2.6 | 4.0 | 7.2 | 8.4 | 10.2 | — | —
102 | 3.5 | 3.7 | 6.0 | 12.0 | 14.4 | 18.4 | 27.0 | 35.0
$1.4\times 10^{2}$ | 3.6 | 3.8 | 6.3 | 12.8 | 15.4 | 19.8 | 29.1 | 38.5
$2.0\times 10^{2}$ | 3.8 | 4.0 | 6.6 | 13.6 | 16.5 | 21.3 | 31.8 | 42.2
103 | 4.4 | 4.7 | 8.0 | 17.4 | 21.4 | 28.5 | 45.0 | 61.7
$2.5\times 10^{3}$ | 4.8 | 5.1 | 8.8 | 19.6 | 24.3 | 32.8 | 53.2 | 74.2
$3.0\times 10^{3}$ | 4.9 | 5.2 | 8.9 | 20.0 | 24.9 | 33.6 | 54.9 | 76.8
104 | 5.3 | 5.7 | 9.9 | 22.8 | 28.7 | 39.3 | 65.9 | 94.7
$3.0\times 10^{4}$ | 5.7 | 6.1 | 10.8 | 25.4 | 32.1 | 44.4 | 76.1 | 111.4
105 | 6.2 | 6.6 | 11.7 | 28.1 | 35.7 | 49.9 | 87.3 | 129.9
| RS graviton $\rightarrow q\overline{q}$,gg
101 | 2.1 | 2.2 | 3.6 | 7.1 | 8.8 | 11.6 | 18.0 | —
102 | 2.9 | 3.1 | 5.0 | 10.4 | 12.8 | 16.8 | 27.1 | 38.8
$1.4\times 10^{2}$ | 3.0 | 3.2 | 5.2 | 10.9 | 13.5 | 17.8 | 28.5 | 41.0
$2.0\times 10^{2}$ | 3.1 | 3.3 | 5.5 | 11.5 | 14.2 | 18.9 | 29.9 | 43.3
103 | 3.7 | 4.0 | 6.7 | 14.5 | 18.0 | 24.0 | 39.5 | 56.0
$2.5\times 10^{3}$ | 4.1 | 4.3 | 7.4 | 16.3 | 20.3 | 27.4 | 45.0 | 65.1
$3.0\times 10^{3}$ | 4.2 | 4.4 | 7.5 | 16.7 | 20.7 | 28.1 | 46.3 | 66.9
104 | 4.6 | 4.9 | 8.5 | 19.2 | 24.0 | 32.8 | 55.2 | 79.8
$3.0\times 10^{4}$ | 5.0 | 5.4 | 9.3 | 21.6 | 27.1 | 37.3 | 63.6 | 93.3
105 | 5.5 | 5.8 | 10.3 | 24.1 | 30.6 | 42.4 | 73.4 | 108.8
Table 4:  Mass limit sensitivity for weakly produced dijet resonances at
proton-proton colliders. As a function of collision energy (columns) and
integrated luminosity (rows) the largest resonance mass for which a 95% CL
exclusion can be expected is listed for three strongly produced models: heavy
bosons $W^{\prime}$ (upper), $Z^{\prime}$ (middle) and Randall-Sundrum
graviton (lower)

Acknowledgements

Work supported by the Fermi National Accelerator Laboratory, managed and
operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359
with the U.S. Department of Energy. The U.S. Government retains and the
publisher, by accepting the article for publication, acknowledges that the
U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide
license to publish or reproduce the published form of this manuscript, or
allow others to do so, for U.S. Government purposes. This work was also
supported by the Turkish Energy, Nuclear and Mineral Research Agency.

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