11institutetext: Department of Physics and Astronomy, University of Sheffield,
Hounsfield Road, Sheffield S3 7RH, UK

# Transformation properties of the transverse mass under transverse Lorentz
boosts at hadron colliders

Daniel R. Tovey

(Received: date / Revised version: date)

###### Abstract

The transverse mass of semi-invisibly decaying particles, calculated from the
transverse momenta of their decay products, has been used in a broad range of
searches and measurements at hadron colliders, such as the LHC. This variable
is invariant by construction under Lorentz boosts purely in the longitudinal
(beam) direction, thereby minimising sensitivity to fluctuations in the
fractions of the proton momentum carried by the colliding partons. In this
paper we examine, by contrast, the properties of the transverse mass under
boosts with a component also in the transverse plane perpendicular to the beam
direction. We show that this variable is invariant under such boosts in cases
where the boost is purely transverse and (a) the momenta of the decay products
are confined to the transverse plane in the rest frame of the parent particle
and/or (b) the transverse momenta of the decay products are perpendicular to
the boost direction. We discuss the transformation properties of the
transverse mass in the case of combined transverse and longitudinal boosts and
identify the criteria under which the transverse mass in the laboratory frame
can equal the rest mass of the parent particle, irrespective of its value in
the rest frame of the parent.

###### pacs:

11.80.CrKinematical properties and 14.70.FmW bosons

##  1 Introduction

A key challenge at hadron colliders such as the Large Hadron Collider (LHC)
Evans:2008zzb  is the identification and measurement of particles producing
both visible and invisible decay products. Such semi-invisibly decaying
particles include the Standard Model $W$ boson, Higgs boson (e.g. when
decaying to $\ell^{+}\ell^{-}\nu\bar{\nu}$ via $ZZ^{*}$ or $W^{+}W^{-}$) and
$\tau$ lepton, and many possible new beyond Standard Model states such as
$W^{\prime}$ bosons Aaboud:2017efa ; Sirunyan:2018mpc ; Sirunyan:2018lbg ;
Tanabashi:2018oca:wprime , dark matter mediator particles Aaboud:2018zpr ;
Boveia:2018yeb  and Vector-Like Quarks Aaboud:2017zfn ; Tanabashi:2018oca:vlq
. Kinematic variables constructed from the decay products of such particles
which take values strongly correlated with their mass provide effective tools
for both identifying their presence in events and measuring their masses.
Variables defined using momentum components in the transverse plane make
allowance for the lack of knowledge of the momenta of invisible particles in
the beam direction.

A simple example of such a variable is just the net transverse momentum of the
visible decay product(s) of the parent particle. The distribution of this
variable displays a Jacobian peak in the parent rest frame at half the parent
mass for massless decay products. The transverse momenta of visible decay
products are invariant under Lorentz boosts of the parent rest frame along the
beam direction. Hence the unknown value of this boost in each event, arising
from the varying fractions of the proton momenta carried by the colliding
partons, does not affect the position or shape of the peak. However,
transverse momenta are not invariant under boosts in the transverse plane,
generated, for instance, by recoil of the parent against hadrons in the final
state of the hard process. For this reason the position and shape of the peak
can be strongly affected by the physics of the hard process generating the
hadronic recoil, and identification efficiencies and mass measurements using
visible transverse momenta can be sensitive to recoil modelling uncertainties.
Nevertheless, the use of transverse momenta of visible decay products provides
an important route to measuring the masses of semi-invisibly decaying
particles, and offers the benefit of insensitivity to potentially significant
systematic uncertainties arising from experimental measurement of the net
momenta of invisible decay products via the missing transverse energy
($E^{\mathrm{miss}}_{\mathrm{T}}$). The most recent measurement of the mass of
the $W$ boson, by the ATLAS collaboration at the LHC, relies primarily upon
this technique Aaboud:2017svj . There has also been recent interest in the use
of the energies of visible decay products with similar motivation
Agashe:2012bn ; Bianchini:2019iey .

An alternative to the direct use of transverse momenta is provided by the
transverse mass ($m_{\mathrm{T}}$) vanNeerven:1982mz ; Arnison:1983rp ;
Smith:1983aa , defined as the invariant mass of the visible and invisible
decay products calculated assuming that each has zero net momentum in the beam
direction. Distributions of $m_{\mathrm{T}}$ also display a Jacobian peak, at
the mass of the parent particle. Values of $m_{\mathrm{T}}$ are also invariant
under Lorentz boosts in the beam direction, as they depend purely upon the
transverse momenta of the decay products (including
$E^{\mathrm{miss}}_{\mathrm{T}}$ for invisible particles). The sensitivity of
the position and shape of the peak to transverse boosts of the parent is
furthermore reduced in comparison with the visible transverse momentum
distribution discussed above (see Figure 1 (top)), due to the similarity of
the definition of $m_{\mathrm{T}}$ to that of the invariant mass. However,
$m_{\mathrm{T}}$ is not completely invariant under purely transverse boosts
Smith:1983aa  or indeed under more general combined transverse and
longitudinal boosts (see Figure 1 (bottom)). In this paper we will examine in
detail the transformation properties of this variable in these cases. For
simplicity the paper will focus on the special case of a singly-produced
massive parent ($\delta$) decaying to two massless visible ($v$) and invisible
($\chi$) daughter particles.

Figure 1:  Top: Distributions of transverse mass values in the parent rest
frame and in the laboratory frame for a $W$ boson decaying to a charged lepton
and neutrino and recoiling against a hadronic system with transverse momentum
of at least 5 GeV. Also shown (red/light dashed) is the equivalent
distribution of charged lepton transverse momenta measured in the laboratory
frame. Events were simulated with PYTHIA v.8.240 Sjostrand:2014zea ,
specifying the $qg\rightarrow Wq$ process with ISR, FSR and hadronisation
disabled. The transverse mass has been normalised to the per-event mass of the
$W$ boson $m$ for each event to illustrate the form of the Jacobian peak.
Bottom: The distribution of the ratio of the transverse mass values measured
in the laboratory and $W$ boson frames plotted as a function of the angle
$\xi$ between the boost and the beam-line ($\hat{z}$ direction) defined by
$\cos\xi\equiv(\beta_{\mathrm{z}}/\beta)$.

The paper begins by defining the coordinate system and kinematic quantities to
be used (Section 2). There follows (Section 3) an analysis of the values of
$m_{\mathrm{T}}$ measured in the laboratory frame in the presence of a purely
transverse Lorentz boost of the semi-invisibly decaying parent particle
arising from its transverse recoil. Next, the values of $m_{\mathrm{T}}$
measured in the laboratory frame in the presence of a more general combined
transverse and longitudinal Lorentz boost are considered (Section 4). The
paper finishes with a discussion of these findings (Section 5) and conclusions
(Section 6).

##  2 Coordinate system and kinematic variables

To simplify the analysis we assume initially that the momenta of both $\delta$
and the system against which it recoils lie in the transverse plane of the
laboratory frame – in other words we assume that the complete final state
system of the hard process is at rest in the beam direction (defined as
$\hat{\mathrm{z}}$). We later relax this assumption and investigate the non-
trivial consequences that this has for $m_{\mathrm{T}}$. We assume that the
masses of the visible and invisible particles can be neglected both in the
definition of the transverse mass and in the analysis that follows.

In the rest frame of $\delta$ the visible and invisible decay products have
momenta of magnitude $p_{0}=m/2$. Here, $m$ is the per-event mass of $\delta$,
sampled from a Breit-Wigner distribution of mean $m_{\delta}$ and width
$\Gamma_{\delta}$. The particle $\delta$ recoils in the laboratory frame
against a system of net momentum $\mathbf{p}_{\mathrm{b}}$. We define the
decay angle relative to the beam direction ($\hat{\mathrm{z}}$), in the rest
frame of $\delta$, as $\theta$ such that $\theta=\pi/2$ corresponds to decays
confined to the transverse plane while $\theta=0$ or $\pi$ correspond to
decays parallel to the beam direction. We define the decay angle $\phi$ in the
transverse plane of the rest frame of $\delta$, relative to the transverse
momentum $\mathbf{p}_{\delta,\mathrm{T}}$ of $\delta$, and hence to the
Lorentz boost, such that $\phi=0$ corresponds to a visible (invisible)
daughter momentum parallel (anti-parallel) to
$\mathbf{p}_{\delta,\mathrm{T}}$. We define the positive $\hat{\mathrm{x}}$
direction as being parallel to the transverse momentum of $\delta$ and hence
the boost direction in the transverse plane, and the positive
$\hat{\mathrm{y}}$ direction in the transverse plane such that the visible
daughter has positive $\hat{\mathrm{y}}$ momentum component
$p_{v,\mathrm{y}}$.

The recoil of $\delta$ leads to a boost of its decay products with velocity
${\bm{\beta}}=\mathbf{p}_{\mathrm{b}}/{(p_{\mathrm{b}}^{2}+m^{2})}$ in the
laboratory frame. The relativistic boost factor is
$\gamma=1/\sqrt{1-\beta^{2}}=\sqrt{p_{\mathrm{b}}^{2}+m^{2}}/m$. We denote the
rest frame of $\delta$ as the unprimed frame, and the detector / laboratory
frame as the primed frame.

The invariant mass of the daughters, $v$ and $\chi$, is equal to the per-event
mass $m$ of $\delta$ and can be defined as follows:

$\displaystyle m_{v\chi}^{2}$ $\displaystyle=$
$\displaystyle(E_{v}+E_{\chi})^{2}-(\mathbf{p}_{v}+\mathbf{p}_{\chi})^{2}$ (1)
$\displaystyle=$ $\displaystyle
m_{v}^{2}+m_{\chi}^{2}+2\left(E_{v}E_{\chi}-p_{v,\mathrm{x}}p_{\chi,\mathrm{x}}-p_{v,\mathrm{y}}p_{\chi,\mathrm{y}}\right.$
(3) $\displaystyle\left.\hskip 113.81102pt-
p_{v,\mathrm{z}}p_{\chi,\mathrm{z}}\right)$ $\displaystyle=$ $\displaystyle
m_{v}^{2}+m_{\chi}^{2}+2\left(E_{v,\mathrm{T}}E_{\chi,\mathrm{T}}\cosh\Delta\eta-
p_{v,\mathrm{x}}p_{\chi,\mathrm{x}}\right.$ (5) $\displaystyle\left.\hskip
113.81102pt-p_{v,\mathrm{y}}p_{\chi,\mathrm{y}}\right),$

where $E_{i,\mathrm{T}}^{2}\equiv p_{i,\mathrm{T}}^{2}+m_{i}^{2}$ and
$p_{i,\mathrm{T}}^{2}\equiv p_{i,\mathrm{x}}^{2}+p_{i,\mathrm{y}}^{2}$ for
particle $i$. $\Delta\eta$ is the difference in rapidity for particles $v$ and
$\chi$. The rapidity is defined by

$\eta\equiv\frac{1}{2}\ln\left(\frac{E_{i}+p_{i,\mathrm{z}}}{E_{i}-p_{i,\mathrm{z}}}\right)$
(6)

for particle $i$, and reduces to $\eta=-\ln(\tan(\theta/2))$ for massless $v$
and $\chi$. Differences in rapidity $\Delta\eta$ are invariant under Lorentz
boosts in the longitudinal ($\hat{\mathrm{z}}$) direction, but are not
invariant under boosts in other directions.

As noted in Section 1, the transverse mass $m_{\mathrm{T}}$ of the $v\chi$
system is defined as the invariant mass under the assumption that components
of the momenta of $v$ and $\chi$ in the beam direction are zero. This is
equivalent to setting $\Delta\eta=0$ in Eqn. (5):

$\displaystyle m_{\mathrm{T}}^{2}$ $\displaystyle=$
$\displaystyle\left(E_{v,\mathrm{T}}+E_{\chi,\mathrm{T}}\right)^{2}-\left(\mathbf{p}_{v,\mathrm{T}}+\mathbf{p}_{\chi,\mathrm{T}}\right)^{2}$
(7) $\displaystyle=$ $\displaystyle
m_{v}^{2}+m_{\chi}^{2}+2\left(E_{v,\mathrm{T}}E_{\chi,\mathrm{T}}-p_{v,\mathrm{x}}p_{\chi,\mathrm{x}}-p_{v,\mathrm{y}}p_{\chi,\mathrm{y}}\right)$
(8) $\displaystyle=$ $\displaystyle
2\left(p_{v,\mathrm{T}}p_{\chi,\mathrm{T}}-p_{v,\mathrm{x}}p_{\chi,\mathrm{x}}-p_{v,\mathrm{y}}p_{\chi,\mathrm{y}}\right),$
(9)

where we have required $m_{v}=m_{\chi}=0$ in the last line. In general
$m_{\mathrm{T}}\leq m$, because $\cosh\Delta\eta\geq 1$ and hence
$m_{\mathrm{T}}\leq m_{v\chi}$, as can be seen from the identity:

$m_{\mathrm{T}}^{2}=m_{v\chi}^{2}+2E_{v,\mathrm{T}}E_{\chi,\mathrm{T}}(1-\cosh\Delta\eta).$
(10)

The transverse mass is invariant under arbitrary longitudinal boosts of the
laboratory transverse plane along the beam direction, because it is defined
exclusively in terms of momentum components measured in the transverse plane.

##  3 Transformation properties of $m_{\mathrm{T}}$ under purely transverse
Lorentz boosts

Before considering the more general case of a combined transverse and
longitudinal Lorentz boost of $\delta$ in the laboratory frame we consider
first the simpler case of a purely transverse boost, for which ${\bm{\beta}}$
lies in the transverse plane and hence ${\bm{\beta}}=(\beta_{\mathrm{x}},0,0)$
and $\beta=\beta_{\mathrm{x}}$. This case was also considered in detail in
Ref. Smith:1983aa .

###  3.1 Transverse mass in the laboratory frame

With the definitions from Section 2 we can obtain the momentum components of
$v$ and $\chi$ in the laboratory (primed) frame from a Lorentz transformation
of the unprimed quantities:
$\displaystyle p_{v,\mathrm{x}}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma(p_{v,\mathrm{x}}+\beta E_{v})$ (11) $\displaystyle=$
$\displaystyle p_{0}\gamma(\sin\theta\cos\phi+\beta),$ (12) $\displaystyle
p_{\chi,\mathrm{x}}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma(p_{\chi,\mathrm{x}}+\beta E_{\chi})$ $\displaystyle=$
$\displaystyle p_{0}\gamma(-\sin\theta\cos\phi+\beta),$ $\displaystyle
p_{v,\mathrm{y}}^{\prime}$ $\displaystyle=$ $\displaystyle p_{v,\mathrm{y}}$
(14) $\displaystyle=$ $\displaystyle p_{0}\sin\theta\sin\phi,$ (15)
$\displaystyle p_{\chi,\mathrm{y}}^{\prime}$ $\displaystyle=$ $\displaystyle
p_{\chi,\mathrm{y}}$ (16) $\displaystyle=$ $\displaystyle-
p_{0}\sin\theta\sin\phi.$ (17)

We want to calculate the value of the transverse mass in the laboratory frame
($m^{\prime}_{\mathrm{T}}$) and so we must use the primed momentum components.
Substituting with the above Lorentz transformations and performing some
algebra we obtain the following:

$\displaystyle{m^{\prime}_{\mathrm{T}}}^{2}$ $\displaystyle=$ $\displaystyle
2\left(p^{\prime}_{v,\mathrm{T}}p^{\prime}_{\chi,\mathrm{T}}-p^{\prime}_{v,\mathrm{x}}p^{\prime}_{\chi,\mathrm{x}}-p^{\prime}_{v,\mathrm{y}}p^{\prime}_{\chi,\mathrm{y}}\right),$
(18) $\displaystyle=$
$\displaystyle\frac{m^{2}}{2}\Biggl{(}\sin^{2}\theta+(1-\gamma^{2})(1-\sin^{2}\theta\cos^{2}\phi)\Biggr{.}$
(21)
$\displaystyle\Biggl{.}+\Bigl{|}\left((\gamma^{2}(1-\sin^{2}\theta\cos^{2}\phi)-(1-\sin^{2}\theta\sin^{2}\phi))^{2}\right.\Bigr{.}\Biggr{.}$
$\displaystyle\Biggl{.}\Bigl{.}\left.+4\gamma^{2}\sin^{4}\theta\cos^{2}\phi\sin^{2}\phi\right)^{1/2}\Bigr{|}\Biggr{)}.$

###  3.2 Special cases

We can now examine the values of $m^{\prime}_{\mathrm{T}}$ for several special
cases.

  * •

$\gamma=1$, any $\phi$, any $\theta$: In this case $\delta$ is at rest in the
laboratory transverse plane. $m^{\prime}_{\mathrm{T}}$ is then just given by
$m^{\prime}_{\mathrm{T}}=m\sin\theta$, independent of $\phi$.

  * •

$\theta=\pi/2$, any $\phi$, any $\gamma$: In this case $\delta$ decays in the
transverse plane. The rapidity difference for $v$ and $\chi$ vanishes and
hence $m^{\prime}_{\mathrm{T}}=m$, independent of $\phi$ and $\gamma$.
$m^{\prime}_{\mathrm{T}}$ thus measures $m$ independently of the magnitude or
direction of the recoil. This special case is the origin of an invariant
population of events in the Jacobian peak of the $m^{\prime}_{\mathrm{T}}$
distribution for purely transverse boosts.

  * •

$\phi=\pi/2$, any $\theta$, any $\gamma$: In this case $\delta$ decays in its
rest frame perpendicularly to its direction of motion. The factors of $\gamma$
cancel from Eqn. (LABEL:eqn:master) and we obtain
$m^{\prime}_{\mathrm{T}}=m\sin\theta\leq m$, independently of $\gamma$. Such
events with $\theta\neq\pi/2$ therefore give rise to an invariant tail of
events in the $m^{\prime}_{\mathrm{T}}$ distribution below the Jacobian peak
at $m^{\prime}_{\mathrm{T}}=m$, for purely transverse boosts.

  * •

$\gamma>1$, $\phi=0$ or $\pi$, and $\theta\neq\pi/2$: In this case
$m^{\prime}_{\mathrm{T}}$ is given by

$m^{\prime}_{\mathrm{T}}=m\sqrt{1-\gamma^{2}\cos^{2}\theta},$ (23)

for $\beta<\sin\theta$, or equivalently $\gamma<1/\cos\theta$. When
$\gamma\geq 1/\cos\theta$ the direction of the momentum of $\chi$ (or $v$ for
$\phi=\pi$) in the transverse plane changes sign and becomes parallel to that
of $v$ (or $\chi$). This causes $m^{\prime}_{\mathrm{T}}$ to become
identically zero when the decay products are massless.

  * •

$\gamma\rightarrow\infty$: In this case the limit of Eqn. (LABEL:eqn:master)
is given by:

$\displaystyle m^{\prime}_{\mathrm{T}}$ $\displaystyle=$
$\displaystyle\frac{m\sin\theta\sin\phi}{\sqrt{1-\sin^{2}\theta\cos^{2}\phi}}$
(24) $\displaystyle=$ $\displaystyle\frac{m\
p_{v,\mathrm{y}}}{\sqrt{p_{v,\mathrm{y}}^{2}+p_{v,\mathrm{z}}^{2}}}$ (25)
$\displaystyle=$ $\displaystyle m\sin\psi,$ (26)

where $\psi$ is the visible daughter decay angle relative to the beam
direction projected into the $\hat{\mathrm{y}}-\hat{\mathrm{z}}$ plane. Highly
boosted events therefore acquire values of $m^{\prime}_{\mathrm{T}}$ which are
always less than that measured in the parent’s rest frame.

These cases are illustrated in Figure 2 for various values of $\theta$ and
$\phi$. Excepting the above special cases, the general dependence of
$m^{\prime}_{\mathrm{T}}$ on $\gamma$ resembles that observed for $\theta=0.9$
and $\phi=0.2$ in the figure. In general the shape of the
$m^{\prime}_{\mathrm{T}}$ distribution is affected by the recoil of $\delta$,
and $m^{\prime}_{\mathrm{T}}\leq m_{\mathrm{T}}\leq m$ (as noted also in Refs.
Smith:1983aa ; Nachman:2016qyc ). The constraint that
$m^{\prime}_{\mathrm{T}}/m_{\mathrm{T}}\leq 1$ can be observed in Figure 1
(bottom) when $\xi=\pi/2$ and hence $\beta_{\mathrm{z}}=0$.

Figure 2: Values of $m^{\prime}_{\mathrm{T}}$ normalised to $m$ plotted as a
function of $\gamma$ for different values of $\theta$ and $\phi$. The value of
$m^{\prime}_{\mathrm{T}}/m$ at $\gamma=1$ for $\theta=0.9$ is
$\sin\theta=0.783$. The maximum value of $\gamma$ for $\theta=0.9$ and
$\phi=0$ for which $m^{\prime}_{\mathrm{T}}$ is non-zero is
$\gamma=1/\cos\theta=1.609$. The curve in this case has equation
$m^{\prime}_{\mathrm{T}}/m=\sqrt{1-0.386\gamma^{2}}$. The limiting value of
$m^{\prime}_{\mathrm{T}}/m$ for $\theta=0.9$ and $\phi=0.2$, given by Eqn.
(26), is 0.243.

##  4 Transformation properties of $m_{\mathrm{T}}$ under combined transverse
and longitudinal Lorentz boosts

We now consider the more general case of a combined transverse and
longitudinal Lorentz boost of $\delta$ in the laboratory frame. In this case,
and with the definition that the boost points in the $+\hat{\mathrm{x}}$
direction in the transverse plane, we have
${\bm{\beta}}=(\beta_{\mathrm{x}},0,\beta_{\mathrm{z}})$ and
$\beta=\sqrt{\beta_{\mathrm{x}}^{2}+\beta_{\mathrm{z}}^{2}}$. We define the
angle of the boost relative to the beam direction as $\xi$ such that
$\tan\xi\equiv(\beta_{\mathrm{x}}/\beta_{\mathrm{z}})$ or equivalently
$\cos\xi\equiv(\beta_{\mathrm{z}}/\beta)$.

###  4.1 Transverse mass in the laboratory frame

With the definitions from Section 2 we can again obtain the momentum
components of $v$ and $\chi$ in the laboratory (primed) frame from a Lorentz
transformation of the unprimed quantities, using now the expression for a
boost in an arbitrary non-axial direction:

$\displaystyle p_{v,\mathrm{x}}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma\beta_{\mathrm{x}}E_{v}+p_{v,\mathrm{x}}+(\gamma-1)\frac{\beta_{\mathrm{x}}}{\beta^{2}}(\beta_{\mathrm{x}}p_{v,\mathrm{x}}+\beta_{\mathrm{z}}p_{v,\mathrm{z}})$
(27) $\displaystyle=$ $\displaystyle
p_{0}\Bigl{(}\gamma\beta_{\mathrm{x}}+\sin\theta\cos\phi\Bigr{.}$ (29)
$\displaystyle\Bigl{.}\hskip
28.45274pt+(\gamma-1)\frac{\beta_{\mathrm{x}}}{\beta^{2}}(\beta_{\mathrm{x}}\sin\theta\cos\phi+\beta_{\mathrm{z}}\cos\theta)\Bigr{)}$
$\displaystyle p_{v,\mathrm{y}}^{\prime}$ $\displaystyle=$ $\displaystyle
p_{v,\mathrm{y}}$ (30) $\displaystyle=$ $\displaystyle
p_{0}\sin\theta\sin\phi,$ (31) $\displaystyle p_{\chi,\mathrm{x}}^{\prime}$
$\displaystyle=$
$\displaystyle\gamma\beta_{\mathrm{x}}E_{\chi}+p_{\chi,\mathrm{x}}+(\gamma-1)\frac{\beta_{\mathrm{x}}}{\beta^{2}}(\beta_{\mathrm{x}}p_{\chi,\mathrm{x}}+\beta_{\mathrm{z}}p_{\chi,\mathrm{z}})$
(32) $\displaystyle=$ $\displaystyle
p_{0}\Bigl{(}\gamma\beta_{\mathrm{x}}-\sin\theta\cos\phi\Bigr{.}$ (34)
$\displaystyle\Bigl{.}\hskip
28.45274pt-(\gamma-1)\frac{\beta_{\mathrm{x}}}{\beta^{2}}(\beta_{\mathrm{x}}\sin\theta\cos\phi+\beta_{\mathrm{z}}\cos\theta)\Bigr{)}$
$\displaystyle p_{\chi,\mathrm{y}}^{\prime}$ $\displaystyle=$ $\displaystyle
p_{\chi,\mathrm{y}}$ (35) $\displaystyle=$ $\displaystyle-
p_{0}\sin\theta\sin\phi.$ (36)

Figure 3: Values of $m^{\prime}_{\mathrm{T}}$ normalised to $m$ plotted as a
function of $\log_{10}\gamma$ and $\xi$ for $\theta=\pi/2$ and $\phi=0$ (top)
and $\theta=0.9$ and $\phi=0.2$ (bottom). In the top figure the boundary of
the dark blue region, in which $m^{\prime}_{\mathrm{T}}=0$, is given by Eqn.
(37). With the values $\theta$ and $\phi$ used in the bottom figure, the value
of $m_{\mathrm{T}}/m$ in the $\delta$ rest frame is equal to
$\sin\theta=0.783$.

Substituting these primed momenta into Eqn. (9) we can obtain an expression
for $m^{\prime}_{\mathrm{T}}$ similar to that in Eqn. (LABEL:eqn:master) for
purely transverse boosts. We do not provide its specific analytical form here,
however we have plotted its dependence upon $\gamma$ and $\xi$ for two
representative cases for $\theta$ and $\phi$ in Figure 3. It is immediately
noticeable in this figure that, in contrast to the case of purely transverse
boosts, $m^{\prime}_{\mathrm{T}}$ can exceed $m_{\mathrm{T}}$ (which is given
by $m\sin\theta=0.783$ $m$ in this case), as observed also in Figure 1
(bottom) for $\xi\neq\pi/2$, and indeed can saturate the bound
$m^{\prime}_{\mathrm{T}}\leq m$.

###  4.2 Special cases

We can now again examine the values of $m^{\prime}_{\mathrm{T}}$ for several
special cases.

  * •

$\theta=\pi/2$ and $\phi=\pi/2$, any $\gamma$: In this case $\Delta\eta=0$ in
the $\delta$ rest frame and is invariant under the boost, because the momenta
of $v$ and $\chi$ are perpendicular to the boost direction. Consequently
$m^{\prime}_{\mathrm{T}}=m_{\mathrm{T}}=m$.

  * •

$\theta=\pi/2$ and $\phi\neq\pi/2$, any $\gamma$: In this case $\Delta\eta=0$
in the $\delta$ rest frame and so $m_{\mathrm{T}}=m$. However, any boost with
both transverse and longitudinal components increases $|\Delta\eta|$ and hence
$m^{\prime}_{\mathrm{T}}<m_{\mathrm{T}}$.

  * •

$\phi=\pi/2$ and $\theta\neq\pi/2$, any $\gamma$: In this case $\Delta\eta\neq
0$ in the $\delta$ rest frame and any combined transverse and longitudinal
boost decreases $|\Delta\eta|$ and increases $m^{\prime}_{\mathrm{T}}$. For
this reason $m_{\mathrm{T}}<m^{\prime}_{\mathrm{T}}\leq m$. Application of
such a boost can therefore ‘regenerate’ the Jacobian peak of events at the
transverse mass end-point. We shall discuss this further below (Section 5).

  * •

$\gamma>1$, $\phi=0$ or $\phi=\pi$, any $\theta$: In this case the decay axis
lies in the plane of the beam direction and the boost. As is the case when the
boost is purely transverse and $\phi=0$ or $\phi=\pi$, the direction of the
momentum of $\chi$ for $\phi=0$ (or $v$ for $\phi=\pi$) in the transverse
plane can change sign and become parallel to that of $v$ (or $\chi$). At this
point the transverse mass becomes identically zero (remembering that $v$ and
$\chi$ are massless). The criterion can be found by setting $\phi=0$ and
solving the equation $p_{\chi,\mathrm{x}}^{\prime}=0$, or equivalently
$m^{\prime}_{\mathrm{T}}=0$, for $\gamma$ (and similarly for
$p_{v,\mathrm{x}}^{\prime}=0$ when $\phi=\pi$). The result is:

$\gamma=\frac{-(\tan\xi\sin\theta+\cos\theta)\pm\sec^{2}\xi}{\tan\xi(\tan\xi\cos\theta-\sin\theta)}$
(37)

When $\theta=\pi/2$ this reduces further to $\gamma=1\pm(\tan\xi+\cot\xi)$.
This curve is visible in Figure 3 (top) as the boundary of the dark blue
region where $m^{\prime}_{\mathrm{T}}/m=0$. The ‘horns’ in this figure lying
at $\xi=0$, $\pm\pi/2$, and $\pm\pi$ correspond to cases where
$\beta_{\mathrm{x}}=0$ or $\beta_{\mathrm{z}}=0$. In the former case the boost
is purely longitudinal and hence $m^{\prime}_{\mathrm{T}}=m_{\mathrm{T}}$
independently of the relativistic boost factor $\gamma$. In the latter case
the boost is purely transverse and $m^{\prime}_{\mathrm{T}}=m$ independently
of $\gamma$, as noted in the second bullet in Section 3.2.

  * •

$\gamma\rightarrow\infty$: In this limit $m^{\prime}_{\mathrm{T}}$ reduces to
a simple form:

$\displaystyle m^{\prime}_{\mathrm{T}}$ $\displaystyle=$
$\displaystyle\frac{mp_{v,\mathrm{y}}}{\sqrt{E_{v}^{2}-(\bm{\beta}\cdot\bm{p}_{v})^{2}}}$
(38) $\displaystyle=$
$\displaystyle\frac{m\sin\theta\sin\phi}{\sqrt{1-\left(\sin\xi\sin\theta\cos\phi+\cos\xi\cos\theta\right)^{2}}}.$
(39)

When $\xi=\pi/2$, and hence $\beta_{\mathrm{x}}=1$, Eqn. (39) reduces to Eqn.
(26), as expected. This function is shown in Figure 4 (top) for the case
$\theta=0.9$ and $\phi=0.2$. The peaks in $m^{\prime}_{\mathrm{T}}$ occur
when:
$\displaystyle\beta_{\mathrm{x}}$ $\displaystyle=$
$\displaystyle\frac{\sin\theta\cos\phi}{\sqrt{1-\sin^{2}\theta\sin^{2}\phi}}$
$\displaystyle=$
$\displaystyle\frac{p_{v,\mathrm{x}}}{\sqrt{p_{v,\mathrm{x}}^{2}+p_{v,\mathrm{z}}^{2}}},$
$\displaystyle\beta_{\mathrm{z}}$ $\displaystyle=$
$\displaystyle\frac{\cos\theta}{\sqrt{1-\sin^{2}\theta\sin^{2}\phi}}$ (41)
$\displaystyle=$
$\displaystyle\frac{p_{v,\mathrm{z}}}{\sqrt{p_{v,\mathrm{x}}^{2}+p_{v,\mathrm{z}}^{2}}},$
(42)

and hence $\tan\xi=\tan\theta\cos\phi$. At these points

$\displaystyle p_{v,\mathrm{x}}^{\prime}$ $\displaystyle=$ $\displaystyle
p_{0}\gamma\sin\theta\cos\phi\left(1+\frac{1}{\sqrt{1-\sin^{2}\theta\sin^{2}\phi}}\right)\gg
p_{v,\mathrm{y}}^{\prime},$ (43) $\displaystyle p_{v,\mathrm{z}}^{\prime}$
$\displaystyle=$ $\displaystyle
p_{0}\gamma\cos\theta\left(1+\frac{1}{\sqrt{1-\sin^{2}\theta\sin^{2}\phi}}\right),$
(44) $\displaystyle p_{\chi,\mathrm{x}}^{\prime}$ $\displaystyle=$
$\displaystyle-
p_{0}\gamma\sin\theta\cos\phi\left(1-\frac{1}{\sqrt{1-\sin^{2}\theta\sin^{2}\phi}}\right)\gg
p_{\chi,\mathrm{y}}^{\prime},$ (45) $\displaystyle
p_{\chi,\mathrm{z}}^{\prime}$ $\displaystyle=$ $\displaystyle-
p_{0}\gamma\cos\theta\left(1-\frac{1}{\sqrt{1-\sin^{2}\theta\sin^{2}\phi}}\right).$
(46)

Consequently
$\beta_{\mathrm{x}}/\beta_{\mathrm{z}}=p_{v,\mathrm{x}}^{\prime}/p_{v,\mathrm{z}}^{\prime}=p_{v,\mathrm{x}}/p_{v,\mathrm{z}}$,
and the boost is hence parallel to the momentum of $v$ or $\chi$ in the
$\hat{\mathrm{x}}-\hat{\mathrm{z}}$ plane. The transverse mass in the
laboratory frame $m^{\prime}_{\mathrm{T}}$ equals $m$ and hence saturates the
physical bound and can exceed the value of $m_{\mathrm{T}}$ in the $\delta$
rest frame. We shall discuss the reason for this behaviour in Section 5. The
minimum value of $m^{\prime}_{\mathrm{T}}$ for $\gamma\rightarrow\infty$
occurs when:
$\displaystyle\beta_{\mathrm{x}}$ $\displaystyle=$
$\displaystyle\frac{\cos\theta}{\sqrt{1-\sin^{2}\theta\sin^{2}\phi}}$
$\displaystyle=$
$\displaystyle\frac{p_{v,\mathrm{z}}}{\sqrt{p_{v,\mathrm{x}}^{2}+p_{v,\mathrm{z}}^{2}}},$
$\displaystyle\beta_{\mathrm{z}}$ $\displaystyle=$
$\displaystyle\frac{\sin\theta\cos\phi}{\sqrt{1-\sin^{2}\theta\sin^{2}\phi}}$
(48) $\displaystyle=$
$\displaystyle\frac{p_{v,\mathrm{x}}}{\sqrt{p_{v,\mathrm{x}}^{2}+p_{v,\mathrm{z}}^{2}}},$
(49)

and hence $\tan\xi=\cot\theta/\cos\phi$. At these points
$m^{\prime}_{\mathrm{T}}=\sin\theta\sin\phi$.

Figure 4: Top: Values of $m^{\prime}_{\mathrm{T}}$ normalised to $m$ plotted
as a function of $\xi$ for $\theta=0.9$ and $\phi=0.2$ for
$\gamma\rightarrow\infty$. The positions of the maxima and minima given by
Eqns. 42 and 49 are denoted by light grey lines, as is the minimum value of
$m^{\prime}_{\mathrm{T}}/m=\sin\theta\sin\phi=0.156$. Bottom: An enlargement
of the top panel around $\xi=0$ (i.e. $\beta_{\mathrm{x}}=0$) with curves for
several other large, but finite, values of $\gamma$ also shown. The two
limiting values of $m^{\prime}_{\mathrm{T}}/m$ are shown – namely
$m^{\prime}_{\mathrm{T}}/m=\sin\phi=0.199$ when $\gamma\rightarrow\infty$ and
$m^{\prime}_{\mathrm{T}}/m=\sin\theta=0.783$ when $\beta_{\mathrm{x}}=0$.

##  5 Discussion

It is clear from the analysis of the dependence of $m^{\prime}_{\mathrm{T}}$
on $\gamma$ and $\xi$ presented in Section 4 that a combined transverse and
longitudinal boost of $\delta$ can lead to transverse mass values which exceed
those measured in the rest frame of $\delta$, and indeed can saturate the
bound $m^{\prime}_{\mathrm{T}}\leq m$. This observation is in stark contrast
to the case of purely transverse boosts, where $m^{\prime}_{\mathrm{T}}\leq
m_{\mathrm{T}}<m$ when $\theta\neq\pi/2$, as noted in Section 3. In order to
understand this we can consider the special case where
$\gamma\rightarrow\infty$ discussed in Section 4.2. At the peaks in Figure 4
(top), where $m^{\prime}_{\mathrm{T}}=m$, the ratios of $\hat{\mathrm{x}}$ and
$\hat{\mathrm{z}}$ momentum components observed in the laboratory frame for
both $v$ and $\chi$ (Eqn. (46)) are equal:
$p_{v,\mathrm{x}}^{\prime}/p_{v,\mathrm{z}}^{\prime}=p_{\chi,\mathrm{x}}^{\prime}/p_{\chi,\mathrm{z}}^{\prime}=\tan\theta\cos\phi$.
As $\gamma\rightarrow\infty$ we can neglect the $\hat{\mathrm{y}}$ components
of the momenta of these particles and proceed to calculate $\theta^{\prime}$,
the value of $\theta$ observed in the laboratory frame:
$\tan\theta^{\prime}=p_{\mathrm{T}}^{\prime}/p_{\mathrm{z}}^{\prime}\simeq
p_{\mathrm{x}}^{\prime}/p_{\mathrm{z}}^{\prime}=\tan\theta\cos\phi$ for both
$v$ and $\chi$. The equivalence of $\theta^{\prime}$ for both $v$ and $\chi$
causes their rapidities $\eta$ to be identical and hence $\Delta\eta=0$ and
$m^{\prime}_{\mathrm{T}}=m$. In general this condition is satisfied along a
curve in the $\gamma-\xi$ plane for specific values of $\theta$ and $\phi$,
leading to a ‘ridge’ with $m^{\prime}_{\mathrm{T}}=m$ in Figure 3 (bottom) and
the peaks with $m^{\prime}_{\mathrm{T}}=m$ in Figure 4 (top).

When comparing Figure 3 (bottom) with Figure 4 (top) two points may be noted:

  * •

Two of the peaks in $m^{\prime}_{\mathrm{T}}$ expected at large values of
$\gamma$ are not present when $\gamma\rightarrow\infty$. These are the peaks
in Figure 3 (bottom) located near $\xi=0$ and $\xi=\pm\pi$, where
$\beta_{\mathrm{x}}\sim 0$, which are not visible in Figure 4 (top).

  * •

When $\xi$ is exactly zero, $\beta_{\mathrm{x}}=0$ and hence the boost is
purely longitudinal. We therefore expect
$m^{\prime}_{\mathrm{T}}=m_{\mathrm{T}}=m\sin\theta$. However, Eqn. (39)
indicates that when $\gamma\rightarrow\infty$ and $\beta_{\mathrm{x}}=0$,
$m^{\prime}_{\mathrm{T}}=m\sin\phi$. Since we are, in general, free to pick
events with arbitrary $\theta$ and $\phi$ these values are apparently in
conflict.

The answer to these two conundrums is illustrated in Figure 4 (bottom), which
shows $m^{\prime}_{\mathrm{T}}/m$ over a narrow range of $\xi$ around $\xi=0$,
both for $\gamma\rightarrow\infty$ and for several large, but finite, values
of $\gamma$. At finite $\gamma$, the peak in $m^{\prime}_{\mathrm{T}}$ near
$\xi=0$, obtained when $\Delta\eta=0$, narrows and moves towards $\xi=0$ as
$\gamma$ becomes larger, with the function maintaining a value
$m^{\prime}_{\mathrm{T}}=m\sin\theta$ when crossing the $\xi=0$ axis and
$m^{\prime}_{\mathrm{T}}=m$ at the peak. When $\gamma\rightarrow\infty$ this
peak becomes infinitely narrow and only in this unphysical case does the
‘underlying’ value $m^{\prime}_{\mathrm{T}}=m\sin\phi$ then take over.

One can consider whether this analysis can be used to reduce the dependence of
the shape of the experimentally measured transverse mass distribution on the
recoil of the parent particle under consideration. In particular, the
selection of a population of events with transverse mass values invariant
under the Lorentz boosts generated by this recoil would minimise the
systematic uncertainties in the measurement of the mass of the parent particle
arising from the modelling of the recoil. This would be very useful for
measurements of, for instance, the mass of the $W$ boson. In the presence of
purely transverse Lorentz boosts a requirement that the transverse momentum
components of the decay products in the lab frame parallel to the boost
direction be equal would ensure that $\phi=\pi/2$ and hence that events are
indeed invariant under these boosts, as discussed in bullet 3 of Section 3.2.
Unfortunately however, this strategy fails in the presence of a combined
transverse and longitudinal Lorentz boost, as discussed in Section 4. In this
case, in order to isolate events at the transverse mass end-point unaffected
by the boost one would need to measure the rapidity of both the visible and
invisible decay products of the parent. The latter is inaccessible
experimentally and hence this strategy cannot be applied in practice.

##  6 Conclusions

In this paper we have studied the transformation properties of the transverse
mass under Lorentz boosts with a component transverse to the beam direction.
When a purely transverse boost is applied, we observed that the transverse
mass is invariant when (a) the momenta of the decay products of the parent
particle are confined to the transverse plane and/or (b) the transverse
momenta of these decay products are perpendicular to the boost direction. In
cases where the boost is purely transverse, the transverse mass in the
laboratory frame cannot exceed that in the rest frame of the parent particle.
Turning to cases with a longitudinal component to the boost in addition to a
transverse component, we observed that by contrast the transverse mass in the
laboratory frame can exceed that in the parent’s rest frame, and indeed can
saturate the bound provided by the rest mass of the parent. In such cases the
boost is such that the difference in rapidity between the momenta of the decay
products in the laboratory frame vanishes and hence the transverse mass in
this frame equals the invariant mass.

## Acknowledgements

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement no 694202). DRT also acknowledges financial support from the
UK Science and Technology Facilities Council (STFC).

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