aainstitutetext: Maryland Center for Fundamental Physics, Department of
Physics, University of Maryland, College Park, MD 20742, USAbbinstitutetext:
Università degli Studi and INFN Roma Tre, Via della Vasca Navale 84, I-00146,
Romeccinstitutetext: Department of Physics, University of California at Santa
Barbara (UCSB), 5105 Broida Hall, Santa Barbara, CA 93106, USAddinstitutetext:
Mitchell Institute for Fundamental Physics and Astronomy, Department of
Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA

#  Energy-peak based method to measure top quark mass via $B$-hadron decay
lengths

Kaustubh Agashe a Sagar Airen b Roberto Franceschini c Joseph Incandela d
Doojin Kim a and Deepak Sathyan<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We develop a method for the determination of the top quark mass using the
distribution of the decay length of the $B$-hadrons originating from its
decay. This technique is based on our earlier observation regarding the
location of the peak of the $b$ quark energy distribution. Such “energy-peak”
methods enjoy a greater degree of model-independence with respect to the
kinematics of top quark production compared to earlier proposals. The CMS
experiment has implemented the energy-peak method using associated $b$-jet
energy as an approximation for $b$ quark energy. The new method uses
$B$-hadron decay lengths, which are related to $b$ quark energies by
convolution. The advantage of the new decay length method is that it can be
applied in a way that evades jet-energy scale (JES) uncertainties. Indeed, CMS
has measured the top quark mass using $B$-hadron decay lengths, but they did
not incorporate the energy-peak method. Therefore, mismodeling of top quark
transverse momentum remains a large uncertainty in their result. We
demonstrate that, using energy-peak methods, this systematic uncertainty can
become negligible. We show that with the current LHC data set, a sub-GeV
statistical uncertainty on the top quark mass can be attained with this
method. To achieve a comparable systematic uncertainty as is true for many
methods based on exclusive or semi-inclusive observables using hadrons, we
find that the quark-hadron transition needs to be described significantly
better than is the case with current fragmentation functions and hadronization
models.

††preprint:  UMD-PP-022-10 MI-HEE-790

##  1 Motivation for a new method

A precise measurement of the top quark mass is well-motivated for various
reasons: see, for example, Section 1.2 of Ref. Agashe:2013hma . Ideally, we
should aim for a method that is least sensitive to unknowns, e.g. the modeling
of the kinematics of the production mechanism of top quarks at the LHC and
other aspects of top quark physics which may affect the extraction of the top
quark mass from the data. In this paper we refer to such a method as “model-
independent.”

In this work we will assume that the top quark decays as in the SM into a
bottom quark and a $W$ boson at leading order (neglecting CKM mixing):

$\displaystyle t$ $\displaystyle\rightarrow$ $\displaystyle bW\,$ (1)

We are looking for a strategy that does not depend on knowledge of the fine
details of the top quark’s production mechanism. Indeed, even if one assumes
the SM to be the correct theory of Nature (in light of the absence of
compelling evidence for new physics thus far!), there are still uncertainties
in SM production itself, such as higher-order QCD effects or uncertainties in
PDFs that have not yet been calculated. Therefore, it is very useful to have a
mass measurement method that is model-independent.

Furthermore, the goal of devising production-model-independent measurement
methods of the top quark mass is motivated by the existence of models in which
there is a potentially significant new physics contribution to a $t\bar{t}$
final state. Examples in the past included stealth stops in SUSY (i.e.,
decaying to top quark plus neutralino/gravitino and being almost degenerate
with that combination: see, for example, Section 1.6.1.2 of Ref.
Agashe:2013hma  for more details) or those proposed to explain the “anomaly”
in the top quark forward-backward asymmetry seen at the Tevatron (see, for
example, Section 1.4.3 of Ref. Agashe:2013hma  for more details). Even though
these particular proposals may be ruled out today, one cannot completely rule
out the possibility of new physics in top quark samples.

However, most of the current methods for the measurement of the top quark mass
assume the SM matrix element in production. For instance, this is true of all
of the methods used in the latest Tevatron-LHC combination for top mass
ATLAS:2014wva , which reported a central value of 173 GeV with an uncertainty
of $\sim 0.8$ GeV.111This feature is obvious for the “matrix-element” method.
On the other hand, any method that is based on a full reconstruction of the
top quark decay, with either leptonic or hadronic $W$ boson decays, may appear
to be model-independent but faces ambiguities that rely upon Monte Carlo to
resolve. For example, combinatorial issues, such as determining which $b$-jet
of two in $t\bar{t}$ events does one combine with a given reconstructed $W$
boson or in the case of leptonic $W$ boson decay, an additional discrete
ambiguity in the longitudinal neutrino momentum. To address these issues, one
relies on the simulation of these ambiguities, using the SM matrix element.
Other methods, currently too imprecise to be included in the Tevatron-LHC
combination, have tried to maximize the degree of model independence that one
can attain in mass measurement. Examples include methods based on kinematic
endpoints Chatrchyan:2013boa  and the location of the $b$-jet energy peak
CMS:2015jwa , both of which have been implemented by the CMS collaboration,
each with a central value of (approximately) 174 GeV with an uncertainty of
$\sim 2$ GeV, which is consistent with the result from the combination. To
reduce the uncertainty, besides using more data than in these two early
attempts, it is compulsory to add corrections for final state QCD radiation
off the top quark to take into account off-shell top quark production. The
inclusion of radiative corrections is by itself a source of model-dependence
of these two methods, but at a quantitatively smaller (and better understood)
level of theory uncertainty compared to methods that are model-dependent from
the very start. As things stand, it would then appear that the top quark mass
is known model-independently with an uncertainty of 2 GeV, as opposed to 0.8
GeV (or less, see e.g. Ref. CMS:2022kcl ) upon assuming pure SM production and
combining TeVatron and LHC results, i.e., there seems to be room for
improvement in the model-independent arena.

Irrespective of the above considerations of model-independent production, new
methods for measurement of the top quark mass are valuable, especially if they
are simpler and can be complementary to existing ones by involving different
systematics.

Finally, once tested on the top quark, such new methods can be more
confidently used to measure the masses of any new particles that may be
discovered, either at the LHC or future colliders. This is especially the case
for new particles decaying semi-invisibly, i.e., into visible/SM and invisible
particles, where a full reconstruction of their decay chains on an event-by-
event basis is not possible.

With the above motives in mind, we have developed a new method for top quark
measurement that aims to be as model-independent as possible. It is based on
the general idea of an “energy-peak” as follows. We consider the energy
distribution of a massless particle in the laboratory frame arising from the
two-body decay of a heavy particle produced unpolarized, whose boost
distribution is arbitrary. Remarkably, the location of the peak in this energy
distribution is identical to its single-valued energy in the rest frame of the
parent, which depends on the parent’s mass and that of the other decay
product. It follows that if one knows the mass of the latter, the mass of the
parent can be inferred from a measurement of the energy peak. It’s then clear
that this energy-peak idea can furnish a measurement of the top quark mass via
the energy of the bottom quark from its decay (given the precisely known value
of the $W$ boson mass), which, based on the “parent-boost-invariance” alluded
to above, is largely insensitive to details of the production mechanism of the
top quark. As mentioned above, most other methods assume pure SM production
which itself has uncertainties and certainly does not take into account the
possibility of any Beyond Standard Model (BSM) contributions.

The original proposal along this line was to simply use the $b$-jet (i.e.,
inclusive) energy as a very good approximation to the bottom quark energy.
This method has been successfully implemented by the CMS Collaboration
CMS:2015jwa . However, this method is afflicted by the jet energy scale (JES)
uncertainty. This drawback can be circumvented by using instead the decay
length of a $B$-hadron (i.e., more exclusive) contained in the $b$-jet as a
proxy for the bottom quark energy. In particular, one can make use of the fact
that the $B$-hadron decay lengths are related to $b$ quark energies by a
double convolution of the B hadron decay exponential and the b-quark
fragmentation function. On the flip side, the $B$-hadron decay length method
is sensitive to hadronization and fragmentation dynamics and so complementary
to the $b$-jet energy method is assured. The CMS collaboration also performed
a top quark mass measurement using $B$-hadron decay length CMSlifetime , where
they assumed SM production.

In this paper, our new proposal is to then appropriately dovetail these two
methods to obtain a “best of both worlds” determination of the top quark mass
that is based on a measurement of the $B$-hadron decay length, but improved by
the energy-peak concept as outlined above to obtain a result that is both free
of JES uncertainty and largely independent of the top quark production model.

The remainder of the paper is structured as follows. In Section 2, we review
the invariant energy peak result and describe how it is used to measure the
top quark mass using $b$-jet energy spectrum. Next, we describe how the same
idea can be put to use to measure the top mass using $B$-hadron decay lengths
and discuss the associated systematic uncertainties. In Section 3, we
thoroughly discuss how to implement the new method to measure the top quark
mass. In Section 4, we discuss the robustness of our method and present our
detailed expectations for the statistical and various systematic
uncertainties. Finally, in Section 5 we summarize our results and note
possibilities for future improvements.

##  2 Basic idea

We will now briefly summarize the invariant energy-peak idea and how it can be
used to measure the top quark mass. First, we show how to obtain a model-
independent measurement of the top quark mass by measuring the energy peak of
the $b$ quark energy distribution in the laboratory frame. Then we describe
how to measure the energy-peak of the $b$ quarks in two ways: (1) by using the
$b$-jet energy-peak as our proxy for the $b$ quark energy-peak and (2) by
using the $B$-hadron decay length distribution as our proxy. We then discuss
the associated systematic uncertainties and experimental effects of each
method.

###  2.1 $b$ energy-peak

A new method to measure the top quark mass was proposed in Ref. Agashe:2012bn
using only the directly observed laboratory-frame $b$-jet energy distribution.
The starting point is that in the rest-frame of the top quark, the bottom
quark is “monochromatic” in energy, with a value that is a simple function of
the masses:

$\displaystyle E_{b}^{\rm rest}$ $\displaystyle=$
$\displaystyle\frac{m_{t}^{2}-m_{W}^{2}+m_{b}^{2}}{2\;m_{t}}\,.$ (2)

However, the top quark is boosted in the laboratory frame, and the magnitude
of this boost and its direction with respect to that of the bottom quark in
the top quark’s rest-frame varies event-by-event. Thus, we obtain a
distribution of the observed bottom quark energy. Now let’s assume only that
top quarks are produced (i) unpolarized, (ii) with a typical222Further details
on the conditions of the boost distribution are given in Ref. Agashe:2012bn .
boost distribution, and (iii) they undergo the above two-body decay without
final state radiation (FSR). In particular, there is no hard gluon emission
off the bottom quark. In this case, it was shown Agashe:2012bn  that the
location of the peak in the bottom quark energy distribution is given by

$\displaystyle E_{b}^{\rm lab,\>peak}$ $\displaystyle=$
$\displaystyle\frac{m_{t}^{2}-m_{W}^{2}+m_{b}^{2}}{2\;m_{t}}.$ (3)

This value is identical to the bottom quark energy in the rest-frame of the
top quark: see Eq. (2).333This is just a specific case of a general result for
two-body decays which was shown in Ref. Agashe:2012bn . It was then applied to
the measurement of masses of hypothetical new particles in Ref. Agashe:2013eba
. Thus, assuming $m_{W}$ from an independent measurement, we get a simple
measurement of the top quark mass from the bottom quark “energy-peak”. The
energy-peak method is quasi-model-independent, i.e., it does not depend on
many details of the production mechanism of the top quarks, such as the
presence of initial state radiation (ISR), whether the top quark is produced
singly or in pairs via SM production or via a new physics effect. It is also
not affected by uncertainties in parton distribution functions (PDFs).

Of course, the bottom quark hadronizes, resulting in a $b$-jet, which is what
is experimentally detected. However, being a largely inclusive quantity,
$b$-jet energy is a good approximation to the original bottom quark energy,
i.e., heuristically speaking, the probability for hadronization is unity.

###  2.2 $B$-hadron decay length

We expect that the features in the energy distribution of the bottom quark
(such as the peak mentioned above) can be “translated” into features in the
decay lifetime/length of the $B$-hadron contained inside the $b$-jet. Let us
now explore this carefully. For simplicity, let’s assume here that the bottom
quark can only decay to one kind of $B$-hadron. (This is for illustration only
and is not done in the actual analysis.) First, take the case of fixed energy
of bottom quark, denoted simply by $E_{b}$, i.e., dropping the superscript
“lab” henceforth in this section. Even in this case, it is clear that we
obtain a distribution of $B$-hadron energies (denoted by $E_{B}$) originating
from it. Namely, the probability for the $B$-hadron energy to be between the
values $E_{B}$ and $E_{B}+dE_{B}$ is given by the fragmentation function,
denoted here by $D\left(E_{B};E_{b}\right)$, where we are keeping track of a
possible $E_{b}$ dependence. We normalize $D$ such that

$\displaystyle\int dE_{B}\;D\left(E_{B};E_{b}\right)=1\;\hbox{for any
(fixed)}\;E_{b}\,.$ (4)

Thus, as shorthand, denoting the probability distribution functions or pdf’s
for the energies $E_{b}$ and $E_{B}$ by $f$ and $F$, respectively, we have the
“convolution” relation:

$\displaystyle F\left(E_{B}\right)$ $\displaystyle=$ $\displaystyle\int
dE_{b}\;f\left(E_{b}\right)D\left(E_{B};E_{b}\right)\,,$ (5)

where $\int dE_{b}\;f\left(E_{b}\right)=1$ (as is true for $F$ and other pdf’s
discussed below). Note that from the discussion above we have:

$\displaystyle\hbox{maximum of}\;f\left(E_{b}\right)$ $\displaystyle=$
$\displaystyle f\left(E_{b}^{\rm rest}\right),\;{\rm with}~{}E_{b}^{\rm
rest}~{}\hbox{given by Eq.~{}(\ref{Ebrest})}\,$ (6)

While for fixed $E_{B}$, given the probabilistic nature of the decay process,
we will obtain a distribution of decay times of the bottom quark, whose mean
value is given by:

$\displaystyle\tau_{B}^{\rm lab}$ $\displaystyle=$
$\displaystyle\gamma_{B}{\tau}_{B}^{\rm rest}$ (7) $\displaystyle=$
$\displaystyle\frac{E_{B}}{m_{B}}\tau_{B}^{\rm rest}\,,$

where $\tau_{B}^{\rm rest}$ is the proper lifetime of the
$B$-hadron.444$\tau_{B}^{\rm rest}$ is in the $B$-hadron’s rest frame, not
that of the top quark! Just for clarity, we re-introduce the superscript “lab”
on the LHS here. It follows that in the laboratory frame (for a fixed
$B$-hadron energy) the mean decay length for a collection of $b$ quarks with
fixed energy is given by:

$\displaystyle\lambda_{B}$ $\displaystyle=v_{B}\tau^{\rm lab}_{B}=$
$\displaystyle\beta_{B}c\cdot\gamma_{B}\tau^{\rm rest}_{B}$ (8)
$\displaystyle=$
$\displaystyle\sqrt{1-\left(\frac{m_{B}}{E_{B}}\right)^{2}}c\cdot\frac{E_{B}}{m_{B}}\tau^{\rm
rest}_{B}\,$

For relativistic $b$ quarks, $E_{B}\gg m_{B}$, and the above equation becomes

$\displaystyle\lambda_{B}$ $\displaystyle\approx$ $\displaystyle
c\frac{E_{B}}{m_{B}}\tau^{\rm
rest}_{B}\left[1+\mathcal{O}\left(\left(\frac{m_{B}}{E_{B}}\right)^{2}\right)\right]\,$
(9)

In other words, the probability for the $B$-hadron to have a decay length
between the values of $L_{B}$ and $L_{B}+dL_{B}$ is proportional to
$\exp\left(-L_{B}/\lambda_{B}\right)dL_{B}$. Since $E_{B}$ varies event-by-
event, as per Eq. (5), it’s clear that $\lambda_{B}$ itself has probability
given by
$g\left(\lambda_{B}\right)d\lambda_{B}~{}=~{}F\left(E_{B}\right)dE_{B}$. Thus:

$\displaystyle g\left(\lambda_{B}\right)$ $\displaystyle=$
$\displaystyle\frac{F\left(E_{B}\right)}{\frac{d\lambda_{B}}{dE_{B}}}$ (10)
$\displaystyle\approx$ $\displaystyle
F\left(E_{B}\right)\frac{m_{B}}{c\tau^{\rm rest}_{B}}\,$ (11)

where in the second line, we have made the relativistic approximation in Eq.
(9) to simplify the equations in the discussion below. We will retain the full
form of Eq. (8) in the calculations presented in later sections. We will have
another convolution (via the decay exponential) to get the
$L_{B}$-distribution from that of $\lambda_{B}^{\rm lab}$.

Combining all of the above ingredients, we finally have an (albeit somewhat
schematic) expression for the probability distribution $G\left(L_{B}\right)$
for the quantity that we actually measure in this method:

$\displaystyle G\left(L_{B}\right)$ $\displaystyle=$ $\displaystyle\int
d\lambda_{B}\Big{[}\frac{g\left(\lambda_{B}\right)}{\lambda_{B}}\Big{]}\exp\left(-\frac{L_{B}}{\lambda_{B}}\right)$
(12) $\displaystyle\approx$ $\displaystyle\int
dE_{B}\frac{F\left(E_{B}\right)}{E_{B}}\frac{m_{B}}{c\tau_{B}^{\rm
rest}}\exp\left(-\frac{L_{B}m_{B}}{c\tau_{B}^{\rm rest}E_{B}}\right)$ (13)
$\displaystyle=$ $\displaystyle\int dE_{B}\int
dE_{b}\;f\left(E_{b}\right)D\left(E_{B};E_{b}\right)\frac{m_{B}}{c\tau_{B}^{\rm
rest}E_{B}}\exp\left(-\frac{L_{B}m_{B}}{c\tau_{B}^{\rm rest}E_{B}}\right)$
(14)

where in the first line the factor of $1/\lambda_{B}$ is from proper
normalization of the decay exponential; the second line invokes the
relativistic approximation and in the third line, we have related the
distribution of $E_{B}$ to that of $E_{b}$ as per Eq. (5). For the sake of
clarity, we summarize relevant variables and pdf’s here:

$\displaystyle G\left(L_{B}\right)$ $\displaystyle\rightarrow$
$\displaystyle\hbox{pdf of decay length of $B$-hadron},\;L_{B}$ $\displaystyle
f\left(E_{b}\right)$ $\displaystyle\rightarrow$ $\displaystyle\hbox{pdf of
energy of bottom quark},\;E_{b}$ $\displaystyle\hbox{location of peak
of}\;f\left(E_{b}\right)$ $\displaystyle\rightarrow$
$\displaystyle\frac{m_{t}^{2}-M_{W}^{2}+m_{b}^{2}}{2\;m_{t}}$ $\displaystyle
D\left(E_{B};E_{b}\right)$ $\displaystyle\rightarrow$ bottom quark
fragmentation function $\displaystyle\tau_{B}^{\rm rest}$
$\displaystyle\rightarrow$ $\displaystyle\hbox{mean decay lifetime of
$B$-hadron in its rest frame}\,.$

Thus, we can hope to deduce the $f\left(E_{b}\right)$ distribution and hence
the location of its peak, by twice “de-convolving” the observed decay length:
once to remove the exponential decay law and a second time to remove the
fragmentation. Equivalently, the simple and robust predictions for properties
of the distribution of $f\left(E_{b}\right)$ (for example, the location of the
peak in terms of masses and a few other features) can then “materialize” in
the distribution of $L_{B}$ via Eq. (14). A concrete strategy for achieving
this is discussed in Section 3.3.

The CDF Abulencia:2006rz  and CMS CMSlifetime  Collaborations have determined
the top quark mass based on measurements of the decay length using one of the
original proposals in Ref. Hill:2005zy . In particular, the measured decay
length in the transverse plane, $L^{\rm mean}_{B,\;xy}$, is used and they
relate the top quark mass to the mean of the distribution of this quantity. It
is not possible to make a simple and robust (i.e., model-independent)
statement about the distribution of $L_{B,\;xy}$ as was just presented for
$L_{B}$ above.555Note that the energy-peak idea in Ref. Agashe:2012bn  was not
known when these analyses were performed! In these analyses it is assumed that
the top quark is produced via the SM matrix element and simulations were then
performed to obtain the $L^{\rm mean}_{B,\;xy}$ as a fully numerical
“function” of the top quark mass. Because of this, the analyses are maximally
exposed to uncertainties in the modeling of all aspects of top quark
production and decay, and in the transition from quarks to hadrons. On the
contrary, with the energy-peak method, we expect to track the top quark mass
down to the peak of the $\lambda_{B}$ distribution, which can be predicted
without simulation. The exact relation between this peak and the top quark
mass does however still require complete knowledge of quark-hadron transition
dynamics. The uncertainties in this knowledge will result in uncertainties on
the top mass that are in common with the $L^{\rm mean}_{B,\;xy}$ method and
will be discussed in detail below.

###  2.3 Systematics and experimental effects

The two methods described in the previous two subsections have very different
systematics. The $b$-jet energy-peak method was applied by the CMS
Collaboration to measure the top quark mass CMS:2015jwa  where it was found
that the jet energy scale was a major source of uncertainty. There are of
course many other experimental effects resulting from the selection criteria,
background estimation, etc., which were found to be under control.

The new method using $B$-hadron decay lengths involves some very different
experimental effects and systematics. As mentioned earlier $B$-hadron decay
lengths have been used by CDF and CMS to measure the top quark mass. Because
our proposal makes use of decay lengths it should be sensitive to the same
experimental details like secondary vertex reconstruction, tracker resolution,
and so on. These systematics were under control in the previous implementation
using decay lengths, and we expect the same for our new method. The biggest
sources of uncertainty in the previous work Abulencia:2006rz ; CMSlifetime
were due to the sensitivity to the hadronization model and top quark
production modeling. As discussed in the previous section, the new method we
propose should have a negligible top quark production modeling uncertainty
while the uncertainty associated with hadronization remains. Clearly, the new
proposal described in Section 2.2 is not plagued by JES uncertainty.
Therefore, the two methods inspired by the invariant energy-peak result
complement each other well because of the different systematics.

Now, we will elaborate further on how the model-independent nature of the two
methods shields them from the uncertainties associated with production
modeling. We will consider the following three small and independent,
dimensionless parameters. First, $\delta_{\rm prod}$ will denote the relative
uncertainty in our knowledge of the unpolarized part of the cross-section for
top quark production. This includes effects such as limited precision in the
calculation of QCD production (due to PDF uncertainties or absence of
corrections at higher-order in $\alpha_{s}$ etc.) as well as the potential
contribution of unknown BSM physics. Obviously, focusing on the two-body decay
of the top quark, one can see that $\delta_{\rm prod}$ will not change the
energy peak of Eq. (3).

Next, suppose that $f_{\rm pol}$ is the fraction of the total production
cross-section that corresponds to polarized top quarks. If the top quark is
polarized, then, the proof for energy-peak of Eq. (3) does not strictly hold.
In particular, the peak position can be $O(1)$ different if the top quarks
are, say, purely left or right-handed. Combining the polarized and unpolarized
parts of the production can therefore result in a shift of the energy peak
from Eq. (3) by $O\left(f_{\rm pol}\right)$. Of course, the new position can
be calculated given the matrix element in production, i.e., the boost
distribution of top quarks and degree of top quark polarization.

We will focus on pair-produced top quarks so that in the SM, $f_{\rm pol}$
originates from SM $Z$ boson exchange and is thus negligible, namely,
$\sim\left(\alpha_{\rm EW}^{2}/\alpha_{s}^{2}\right)\times\hbox{the ratio of
gluon to}\;q\bar{q}$ PDFs. Again, the shift in the energy peak resulting from
this $Z$ exchange is in principle known. Thus, any relevant $f_{\rm pol}$
actually comes from new physics and hence is (a priori) unknown and
constitutes an error to the prediction of the energy-peak location.

Finally, consider the contribution from hard gluon radiation off a bottom
quark, which is part of an effective three-body decay of the top
quark.666Clearly, the energy associated with soft radiation off the bottom
quark is included in the $b$-jet energy so that this case is still effectively
a two-body decay. For such a decay, even in the rest frame of the top, we get
a non-trivial, “delocalized” (i.e., not the $\delta$-function of a strictly
two-body decay) energy spectrum for the bottom quark. This distribution can be
calculated assuming the SM matrix element in the top quark decay, which is, of
course, independent of the top quark production mechanism. Once again the
proof of the energy-peak proxy for a single-valued $b$-jet energy in the top
quark rest frame does not apply and the new peak position will depend on the
top quark boost distribution which can be $O(1)$ different than in Eq. (3). Of
course, if we know the matrix element in production, then the location of the
new energy peak is calculable. Let us denote by $\epsilon_{\rm FSR}$ the size
of this FSR contribution relative to the two-body decay of the top quark. It
is suppressed by (perturbative) $\sim\alpha_{s}/\pi$ ($\sim 5\%$) and perhaps
even further if we impose a jet-veto (for the extra radiation). Upon adding
this to the leading-order (two-body), the energy peak might thus shift from
Eq. (3) by $O\left(\epsilon_{\rm FSR}\right)$.

Combining all the above three effects, we get the following schematic result
for the observed lab-frame energy peak

$\displaystyle E_{b}^{\rm lab,mode}$ $\displaystyle=$
$\displaystyle\frac{m_{t}^{2}-m_{W}^{2}+m_{b}^{2}}{2\;m_{t}}\left\\{1+\epsilon_{\rm
FSR}\Big{[}a_{QCD}+O\left(\delta_{\rm prod}\right)\Big{]}+O\left(f_{\rm
pol}\right)\right\\}\,,$ (16)

where we assume that the schematic terms inside the curly brackets have
unknown O(1) coefficients.777The exact coefficients are computed in Ref.
Agashe:2016bok . Thus, the relative uncertainty in the top quark mass using
the energy-peak method is:

$\displaystyle{\left.\frac{\delta m_{t}}{m_{t}}\right|_{\rm energy-peak}}$
$\displaystyle=$ $\displaystyle\Big{[}\mathcal{O}\left(\epsilon_{\rm
FSR}\delta_{\rm prod}\right)+\mathcal{O}\left(f_{\rm pol}\right)\Big{]}.$ (17)

On the other hand, the relative error in most of the other methods, where it
is crucial to assume the SM matrix element in production to produce templates
for the observables of interest can be written as:

$\displaystyle{\left.\frac{\delta m_{t}}{m_{t}}\right|_{\rm SM-templates}}$
$\displaystyle=$ $\displaystyle\mathcal{O}\left(\delta_{\rm prod}\right)\,.$
(18)

Therefore, assuming top quarks are produced unpolarized, i.e.,
$\mathcal{O}(f_{\rm pol})\approx 0$, both the $b$-jet energy-peak method and
the $B$-hadron decay length method via an energy-peak approach feature a rough
“double”-suppression as far as uncertainties in the production mechanism are
concerned whereas there is only a single level of suppression for matrix
element-based methods.

##  3 General description of the analysis

###  3.1 Signal and background definition

We take our baseline selection criteria for the $t\bar{t}$ sample along the
lines of what has already been used by the CMS and ATLAS Collaborations
Chatrchyan:2012ea ; ATLAScuts ; namely opposite-flavor dilepton final states,

$pp\to e^{+}\;\mu^{-}\;\textrm{ or }\;e^{-}\mu^{+}+{\rm jets},$ (19)

where the observed collection of jets contains two $b$-tagged jets. This
channel has the best chances to reduce possible issues with our method, as
some of the requirements for the separation of $t\bar{t}$ events from the
background are not necessary. For instance, neither a $Z$-mass veto for the
invariant mass of dileptons (which is needed for same-flavor dileptons) nor a
missing transverse energy (denoted by MET or $E_{T}^{\rm miss}$) cut are
required. The absence of these requirements makes the selection more inclusive
than that in other channels, and hence the chance to preserve the energy peak
in Eq. (3) is maximized. Despite such a light set of event selection criteria,
the choice of the opposite-flavor leptonic final state gives a high level of
signal purity. Indeed, the resulting $S/B$ observed at the LHC for $\sqrt{s}=$
7 TeV is $\sim 20$, as seen in Table 1 of Ref. Chatrchyan:2012ea .

In addition to the ultra-clean opposite-flavor, fully-leptonic channel, we
also consider same-flavor fully leptonic and general semi-leptonic final
states

$\displaystyle pp$ $\displaystyle\to$ $\displaystyle e^{+}\;e^{-}\;\textrm{ or
}\;\mu^{-}\mu^{+}+{\rm jets}\,,$ (20) $\displaystyle pp$ $\displaystyle\to$
$\displaystyle\ell^{\pm}+{\rm jets}\,,$ (21)

where $\ell=e,\mu$. These channels need additional event selection criteria to
reach a good level of $S/B$ CMS:2022kcl . These channels bring additional
events that increase overall statistics, and so a balance between purity and
efficiency needs to be taken into consideration to achieve an optimal result.
We address this aspect of the analysis below.

In any case, some background contamination is expected and needs to be dealt
with to achieve the best precision in the top mass extraction. Obviously, the
$b$-jet energy distribution from background processes is not correlated with
the top quark mass. One strategy to deal with $b$-jets from the background is
to subtract the expected shape using a Monte Carlo template. We will work
under the assumption that this subtraction can be successfully administered
based upon the limited effect of residual background uncertainty achieved in
recent analyses, e.g., Ref. CMS:2022kcl .

Some remarks are in order about the backgrounds to the processes we are
focusing on here. The dominant background turns out to be from single top
quark production processes,

$d\bar{u}\to W^{*}\to bt,\;qb\to q^{\prime}t\;,\;\textrm{and}\;gb\to Wt,$

all of which contain an on-shell top quark (or anti-quark). Thus, one might
naively expect the energy peak from this background to be the same as our
signal in Eq. (19) from pair production. This turns out not to be the case
because some of the requirements for the production mechanism model-
independence are not satisfied by single top production. First of all, the
energy of the $b$-jet (out of a total of up to two in the event) coming from
top decay is of course correlated with the top quark mass, but since the
production of a single top quark always involves a $W$ boson, the top quark is
polarized. Thus, we expect a shift in the energy peak with respect to Eq. (3).
The shift is calculable and will be fully removed when the background
templates match their actual shape in the data. Still, as such removal may be
imperfect, it is important to keep in mind any possible adverse effects of
having $b$-jets that do not originate from QCD pair production of top quarks.
Secondly, and maybe more importantly, the properties of the other $b$ jet
emerging in single top production are not at all related to the top quark
mass. Therefore, the inclusion of such $b$-jets in the analysis would result
in a significant bias in the extracted top quark mass beyond the inherent
sources listed in Eq. (16).

Next, we point out that our peculiar choice of observable for the
determination of the top quark mass is associated with an energy peak that is
rather broad, as shown in Figure 1. To reliably extract the peak position from
the data, it is thus important to minimize any potential distortion of the
distribution in the vicinity of the peak. As we will delineate in the next
section, our event selection scheme is designed to not only sufficiently
suppress the backgrounds but also accommodate this requirement. Indeed, a
simple analytic fitting function for the energy distribution has been proposed
in Ref. Agashe:2012bn :

$\displaystyle f^{\rm fit}\left(E_{b};E^{\rm rest}_{b},w\right)$
$\displaystyle=$
$\displaystyle\frac{1}{N}\exp\Big{[}-w\left(\frac{E_{b}}{E_{b}^{\rm
rest}}+\frac{E_{b}^{\rm rest}}{E_{b}}\right)\Big{]}\,,$ (22)

where $w$ and $E_{b}^{\rm rest}$ are the fitting parameters and $N$ is the
overall normalization factor. The $w$ parameter describes the width of the
distribution, while $E_{b}^{\rm rest}$ takes care of the peak position, and
its best-fit value is equated to the rest-frame energy value in Eq. (2).

Figure 1: Energy spectrum of the bottom quark.

This was inspired by properties of the energy distribution, in addition to the
peak position, that can be proven from first principles888For example, we can
show that the energy distribution must be a function of the combination
$E_{b}^{\rm lab}/E_{b}^{\rm rest}$. and checked against a parton-level
inclusive calculation, (see LHS of Fig. 1 of Ref. Agashe:2012bn ).999Note that
the CMS implementation used a different fitting function, essentially a
Gaussian in logarithm of $E_{b}$ (instead of the above form). The form adopted
by CMS still has the expected properties reported in Ref. Agashe:2012bn .
Finally, this fitting procedure was shown to be robust against realistic
effects such as the application of selection criteria and smearing of measured
quantities due to detector effects as implemented in Delphes de-
Favereau:2013dk , (see RHS of Fig. 1 of Ref. Agashe:2012bn ). For these
reasons we will define our peak-energy proxy and evaluate the goodness of our
top mass extraction based upon an analysis that uses a template from the
family in Eq. (22), or possible extensions of this family based on arguments
presented in Ref. Agashe:2012bn .

###  3.2 Event generation and selection criteria for $B$-Hadron decay length

For the final states in Eqs. (19)-(21) we consider the processes

$\displaystyle pp$ $\displaystyle\rightarrow$ $\displaystyle t\bar{t}\
(t\rightarrow W^{+}b,\ W^{+}\rightarrow\nu_{l}l^{+}),\ (\bar{t}\rightarrow
W^{-}\bar{b},\ W^{-}\rightarrow\bar{\nu_{l}}l^{-})\,,$ (23) $\displaystyle pp$
$\displaystyle\rightarrow$ $\displaystyle t\bar{t}\ (t\rightarrow W^{+}b,\
W^{+}\rightarrow ff^{{}^{\prime}}),(\bar{t}\rightarrow W^{-}\bar{b},\
W^{-}\rightarrow\bar{\nu_{l}}l^{-})+c.c.\,.$ (24)

We use MadGraph5_aMC@NLO Alwall:2014hca  as the parton-level Monte Carlo event
generator for proton-proton collisions to generate $t\bar{t}$ events,
specifying their decay chains to be either semi-leptonic or dileptonic with
the center-of-mass energy being $\sqrt{s}=14\text{ TeV}$. First, we generate
fully inclusive events, and then we impose event selection criteria similar to
what the CMS and ATLAS Collaborations use to eliminate background events
CMS:2016iru . After some further optimization for semi-leptonic events,
described below, we require either one electron with $p_{T}>25\text{ GeV}$ and
$|\eta|<2.4$ or one muon with $p_{T}>25\text{ GeV}$ and $|\eta|<2.1$. We also
require at least 4 jets with $p_{T}>25\text{ GeV}$ and $|\eta|<2.5$.101010A
residual JES dependence associated with event selection criteria that makes
use of calorimeter-based jet energies was avoided via the use of track-based
jets in Ref. CDF:2009qxf . For same-flavor dilepton events, as a baseline, we
require exactly two leptons with $p_{T}>25\text{ GeV}$ and $|\eta|<2.4$, at
least 2 jets with $p_{T}>25\text{ GeV}$ and $|\eta|<2.5$, $E_{T}^{\rm
miss}>40\text{ GeV}$, and for the invariant mass of the two leptons,
$M_{\ell\ell}$, we require $|M_{\ell\ell}-m_{Z}|>15\text{ GeV}$ to remove
backgrounds containing a real $Z$ boson and $M_{\ell\ell}>20\text{ GeV}$ to
remove backgrounds from photon conversions and QCD resonances. In Section 4 we
show the effect of variations of these optimal selection criteria.111111The
proposed cuts we show are similar to those used by experiments to eliminate
background events. We expect our minimal changes not to affect background
estimates on identifying $t\bar{t}$ events. For comparison, the CMS selection
for a determination of $m_{t}$ using a $B$-hadron decay length measurement are
reported in Table 1, together with a summary of our selection criteria. Our
selection is identical for the $\eta$ ranges but differs on the $p_{T}$
requirements. As detailed later, our analysis prefers to have equal $p_{T}$
thresholds for all final states, hence we softened the jet $p_{T}$ and
hardened the lepton $p_{T}$ requirements to a medium value of 25 GeV, which
should be attainable for experiments at the LHC and HL-LHC.

|  | Ref. CMS:2016iru  | Optimal choice for our analysis
---|---|---|---
$\ell+{\rm jets}$ | $e$ |  $p_{T}>30$ GeV, $\eta<2.4$ |  $p_{T}>25$ GeV, $\eta<2.4$
$\mu$ |  $p_{T}>26$ GeV, $\eta<2.1$ |  $p_{T}>25$ GeV, $\eta<2.1$
$j$ |  $N_{j}\geq 4$, $p_{T}>30$ GeV, $\eta<2.5$ |  $N_{j}\geq 4$, $p_{T}>25$ GeV, $\eta<2.5$
$2\ell+{\rm jets}$ | $e,\mu$ |  $p_{T}>20$ GeV, $\eta<2.4$ |  $p_{T}>25$ GeV, $\eta<2.4$
SF |  $M_{\ell\ell}>20$ GeV, $|M_{\ell\ell}-m_{Z}|>15$ GeV |  $M_{\ell\ell}>20$ GeV, $|M_{\ell\ell}-m_{Z}|>15$ GeV
OF | - | -
$j$ |  $p_{T}>30$ GeV, $\eta<2.5$ |  $p_{T}>25$ GeV, $\eta<2.5$
|  $E_{T}^{\rm miss}>40$ GeV |  $E_{T}^{\rm miss}>40$ GeV

Table 1: Baseline selection of the events used in our analysis and an
optimized choice that we use to minimize bias of the measured top quark mass.

The events that have passed our selection cuts are fed into Pythia 8.2
Sjostrand:2014zea  for showering and hadronization. The $B$-hadron decay
lengths are then fitted to the double convolution function to extract the $b$
quark energy peak as per Eq. (LABEL:ansatz_LB).

###  3.3 Extracting energy peak from $B$-hadron decay lengths

For a realistic discussion, we need to take into account some crucial
differences between $b$-jets and $B$-hadrons. The most obvious this is the
fact that a bottom quark can hadronize into one of the many $B$ hadron
species, with different masses and mean rest-frame lifetimes. Therefore,
constructing a fitting function for $B$-hadron energy or length spectrum is
unavoidably more involved than what we sketched in Eq. (14). The probability
of decay into a particular $B$ hadron species needs to be taken into account,
thus we introduce fractions $f_{i}$ for the production yield of each hadron
$B_{i}$, whose mass, energy, and lifetime need to be correctly tracked in our
formula. This leads to the following more appropriate fitting function for the
spectra:

$\displaystyle G^{\rm fit}\left(L_{B};E^{\rm rest}_{b},w,\nu\right)$
$\displaystyle=$ $\displaystyle\int_{E_{B,\rm
min}}^{E_{b}}dE_{B}\int_{E_{b,\rm min}}^{E_{b,\rm
max}}dE_{b}\frac{1}{N(w)}\exp\bigg{[}-w\left(\frac{E_{b}}{E_{b}^{\rm
rest}}+\frac{E_{b}^{\rm rest}}{E_{b}}\right)^{\nu}\bigg{]}\times$
$\displaystyle\sum_{i}D_{i}\left(\frac{E_{B_{i}}}{E_{b}};E_{b}\right)\frac{f_{i}m_{B_{i}}}{c\tau_{B_{i}}^{\rm
rest}\sqrt{E_{B}^{2}-m_{B_{i}}^{2}}}\exp\left(-\frac{L_{B}m_{B_{i}}}{c\tau_{B_{i}}^{\rm
rest}\sqrt{E_{B}^{2}-m_{B_{i}}^{2}}}\right),$

where the functions, variables, and parameters in this expression play the
following roles:

$\displaystyle G^{\rm fit}\left(L_{B};E^{\rm rest}_{b},w\right)$
$\displaystyle\rightarrow$ fitting function for the decay length $(L_{B})$
distribution $\displaystyle\hbox{best-fit value of parameter}\;E^{\rm
rest}_{b}$ $\displaystyle\rightarrow$
$\displaystyle\frac{m_{t}^{2}-M_{W}^{2}+m_{b}^{2}}{2\;m_{t}}$
$\displaystyle\tau_{B}^{\rm rest}$ $\displaystyle\rightarrow$ mean decay
lifetime of $B$-hadron in its rest frame $\displaystyle\hbox{parameter}\;w$
$\displaystyle\rightarrow$ width of fitting function
$\displaystyle\hbox{parameter}\;\nu$ $\displaystyle\rightarrow$ tails weight
of fitting function $\displaystyle i$ $\displaystyle\rightarrow$
$\displaystyle B\text{-hadron species}$ $\displaystyle
D_{i}\left(\frac{E_{B_{i}}}{E_{b}};E_{b}\right)$ $\displaystyle\rightarrow$
$\displaystyle\hbox{bottom quark fragmentation function for species }i$
$\displaystyle f_{i}$ $\displaystyle\rightarrow$ $\displaystyle\text{relative
proportion of species }i$ $\displaystyle N(w)$ $\displaystyle\rightarrow$
normalization factor

Concerning hadronization, we note that the relative fractions $f_{i}$, in
principle, depend on the kinematics of the bottom quark, see e.g. LHCb:2014ofc
; HFLAV:2019otj ; LHCb:2013vfg ; ATLAS:2015esn  for the $p_{T}$ and $\eta$
dependence of $\Lambda_{b}$ and $B_{s}$ production, and therefore on the
production mechanism, including a subtle dependence on the color environment,
which raises some doubt about the universal applicability of results from
$e^{+}e^{-}$ machines to hadron machines at the level of precision that our
method may require.

In order to disentangle the role of uncertainty in $D_{i}$, discussed in
detail below in Sec. 4.3, from other uncertainties in the method, we largely
eliminate it here by computing our templates using exactly the same $D_{i}$ as
in the generation of data with Pythia. In particular, for all $b$ jets in our
simulation sample, we extract the energies of the $b$ quarks before showering
and the energies of the subsequently formed $B$-hadron to obtain the
distribution of $D_{i}(\frac{E_{B_{i}}}{E_{b}};E_{b})$ for each hadron species
$B_{i}$ in Pythia.

We stress that by measuring this fragmentation function from Pythia itself and
plugging it into our double integral, we essentially remove the unknowns due
to fragmentation in our procedure. Therefore, the results we will obtain in
this way can be considered to be the best achievable results for negligible
uncertainties originating from fragmentation. We will elaborate further below
about the possible impact of uncertainties in fragmentation and hadronization.
We remark that our $D_{i}$ functions do have a dependency on $E_{b}$. This
dependence arises in part from the fact that some showering physics is
encapsulated in these functions. We verified that the $E_{b}$ dependence is
quite weak and so in the following we will drop it to keep our exercise
computationally manageable. In the following, we calculate the sensitivity of
the top mass determination to knowledge about the relative fractions and
fragmentation functions in order to estimate the uncertainties in the final
result that should be associated with them. Results on these matters are
presented in Section 4.

To compute the templates of Eq. (LABEL:ansatz_LB), we need to fix some
parameters. Concerning the integrals, we set the lower and upper limits of the
$E_{b}$ integration range to 40 GeV and 450 GeV, respectively. For the energy
$E_{B}$ we impose a minimum energy requirement, $E_{B}>7$ GeV and a
fragmentation boundary constraint $E_{B}<E_{b}$. These range choices were
chosen to minimize both error and bias in our result. Variations of these
ranges have been tested and are discussed in Section 4.

For our top quark mass extraction, we will consider only a limited range of
$L_{B}$, to maximize the stability of our results and the precision
attainable. We found that the calculation of the double convolution becomes
numerically challenging when evaluated for very large decay lengths. As there
is only 5% of the total data at $L_{B}>20$ mm, we use the range
$L_{B}\in[0,20]$ mm. For our measurement, there is no lower limit on the range
of $B$-hadron decay lengths, but in practice, experiments have a minimum
threshold of $\mathcal{O}(1)$ mm on decay length. We find that our results
change negligibly when we increase the lower limit of the $B$-hadron decay
length to 1 mm. As this threshold is experiment-dependent, we quote results
for a fitting range starting at 0 mm in the following.

The exponent $\nu$ in Eq. (LABEL:ansatz_LB) is in principle a parameter of our
template that would need to be floated in the fit. This is a new parameter
compared to our original ansatz of Ref. Agashe:2012bn , which corresponds to
$\nu=1$. Variations of $\nu$ mostly change the shape of the $b$ quark energy
spectrum in the tail regions, which have a greater importance for this
application than in our earlier work. Its value participates in the definition
of our template and we find $\nu=0.3$ as optimal for minimizing overall bias
in our result.

| best
---|---
$\nu$ | 0.3
$E_{b}$ range | [40,450] GeV
$E_{B}$ | $7{\,\rm GeV}<E_{B}<E_{b}$
$L_{B}$ | [0,20] mm
Table 2:  Summary of the parameters that we fixed to compute our template Eq.
(LABEL:ansatz_LB).

Having set these hyper-parameters, as summarized in Table 2, all that remains
to be done is to fit the decay length data to the fitting function Eq.
(LABEL:ansatz_LB), and obtain $E_{b}^{\text{rest}}$ and its uncertainty from a
$\chi^{2}$ analysis. Then $m_{\text{top}}$ is obtained from
$E_{b}^{\text{rest}}$.

Ideally, we want to have a precise measurement, that stems from a narrow
$\chi^{2}$, with no bias from our template fits. In other words, the minimum
of the $\chi^{2}$ should coincide with the input top quark mass. We estimate
the bias by doing pseudo-experiments for various combinations of the hyper-
parameters and we pick the values which lead to the smallest bias. These
optimal values are collected in Table 2. The stability of the estimated bias
and details on the whole procedure are given in Section 4 together with our
main result.

##  4 Results

###  4.1 Measurement of the top quark mass

The measurement of the top quark mass starts from the observed decay length
spectrum, inclusive of all species of $B$-hadrons that are produced at the LHC
in $t\bar{t}$ events. A prediction from this observable from Pythia 8.2 is
shown in Figure 2 for $m_{t}=173.0\textrm{ GeV}$. Using our template Eq.
(LABEL:ansatz_LB) we can fit the best value for $m_{t}$ with a simple
$\chi^{2}$ minimization and find $m_{t}=172.50\pm 0.35$ on this particular
spectrum. An example $\chi^{2}$ shape is displayed in Figure 3 for
illustration.

The result shown in Figure 2 is just one instance of a representative input
mass; $m_{t\text{,~{}input}}=173.0$ GeV. In the following, we present the
results obtained for various other masses and we quote the performance of our
method by taking the average of the uncertainties obtained. In particular, we
apply our template fitting procedure to top quark masses over the range
$170\text{ GeV}\leq m_{t\text{,~{}input}}\leq 176\text{ GeV}$ with the hyper-
parameters fixed to the values in Table 2 to obtain the following expression
for the uncertainty, which is discussed in detail below:

$\delta m_{t}^{(E_{B},~{}peak)}=\frac{0.5~{}\rm
GeV}{\sqrt{\mathcal{L}/100\,{\rm fb}^{-1}}}({\rm stat}.)\oplus 0.5~{}{\rm
GeV}\cdot\left(\frac{0.1\%}{\frac{\delta{D_{i}}}{D_{i}}}\right)\;\oplus
0.3~{}{\rm GeV}\cdot\left(\frac{5\%}{\frac{\delta{f_{i}}}{f_{i}}}\right)\;.$
(26)

Figure 2: Pythia 8.2 decay length spectrum for input $m_{t}=173$ GeV,
normalized for $\sim 230$/fb at LHC 14 TeV summing fully leptonic and
semileptonic $t\bar{t}$ decays (blue). Result of the fit not corrected for the
bias (orange).

Figure 3: $\chi^{2}$ profile in the vicinity of the best-fit parameters in the
space $(E^{\rm rest}_{b}$, $w)$ (left) and for fixed $w$ as a function of
$E_{b}^{\rm rest}$ (right).

Given the need to compute templates, our method incurs a risk of returning a
value of the top quark mass that has a small uncertainty, due to a sharp
$\chi^{2}$ profile, but is somewhat shifted with respect to the input value.
As the event generators currently available in QCD are not fully up to the
task of dealing with the $10^{-3}$ precision that a top quark mass measurement
currently demands, we take the position that a robust method for the
measurement of the top quark mass should be designed with the goal of a small
bias as well as a small error bar. In fact, lacking a reliable event generator
for the precision we need, this approach has the potential to avoid a large
bias when applied to real-world data, which will certainly differ from the
Monte Carlo data on which the bias and the error bar have been optimized. When
we set all the parameters to their optimum values, we obtain an average bias
in the $m_{t}$ measurement that is $-0.18\pm 0.12$ GeV, which is below the
expected systematic uncertainty. It is also not very sensitive to the choice
of hyper-parameters, as detailed below Section 4.2, thus we consider our
method to have negligible bias.

With an integrated luminosity of $100~{}{\rm fb}^{-1}$, we find that at 14 TeV
LHC the expected statistical uncertainty for a $\chi^{2}$ fit of our
templates, assuming a total production cross-section for $t\bar{t}$ at 985 pb
Czakon:2013goa , is $500$ MeV. Thus we expect this measurement to be already
competitive with regard to the statistical uncertainty over other historically
more frequently pursued methods in LHC Run 3. We further expect that the
statistical uncertainty will go down to about 100 MeV at HL-LHC for 3000
$\rm{fb}^{-1}$. Hence, a thorough discussion of the systematic uncertainties
associated with this method is now in order.

###  4.2 Uncertainty from template definition

Figure 4: Bias as a function of the limits on the $b$-quark energy range in
the $E_{b}$ integral of Eq. (LABEL:ansatz_LB). Subplots are titled by the
common $p_{T}$ cut on leptons and jets used for the selection of events.

Figure 5: Bias as a function of the lower and higher limit on the b-quark
energy range on which we perform the $E_{b}$ integral in Eq.
(LABEL:ansatz_LB). The lower (higher) limit is fixed at 450 GeV (40 GeV) in
the right (left) plot.

Figure 6: The variation of the bias as a function of the cut on minimum
B-hadron energy. The lower and the higher limit on b-quark energy is fixed at
40 GeV and 450 GeV respectively, as in Table 2.

The definition of our template Eq. (LABEL:ansatz_LB) requires that some
parameters be fixed and a phase-space region be specified in which we want to
carry out the measurement as summarized for our analysis in Tables 1 and 2. As
noted above, the goal for these choices was a minimization of both the bias
and the error in our measurement. Variations of these choices need to be
explored to demonstrate the stability of the procedure. It is important to
verify that it does not hinge on a too particular or peculiar a choice of the
phase-space region or a too narrow definition of the template parameters. In
fact, a mismatch between the phase-space used to compute the templates and the
phase-space imposed on the data through event selection criteria may be
reflected in a mismatch between the extracted top quark mass and its true
value.

We quantify this possible mismatch with a study of the impact of variations of
the $p_{T}$ cut on each particle and of the $E_{b}$ range in our template
definition. For this study, we fix the choice of the length fitting range
$L_{B}\in[0,20]\text{ mm}$ as per our optimal measurement strategy. We find
that using unequal $p_{T}$ cuts for different particles tends to introduce
larger biases, hence in the following, we consider a common value for the
$p_{T}$ cut of both leptons and jets. We vary the common $p_{T}$ cut on jets
and leptons in the event selection criteria and the limits on bottom quark
energy and we obtain the bias (in GeV) on the top quark mass shown in Figure
4. We found that the choice of the $L_{B}$ fitting range is not very important
as long as the tail of the spectrum is excluded. All in all, an analysis can
be carried out with a common $p_{T}$ threshold at 25 GeV in the energy range
$E_{b}\in[40,450]$ GeV. As shown broadly in Figure 4 and detailed in Figure 6,
a variation of the highest energy considered in the template would have a very
mild effect on the bias. Variations of the lowest energy considered in the
template may have more impact, however, the bias is around $\frac{80{\rm
MeV}}{\left(\frac{\delta E}{E}/10^{-2}\right)}$, which is reasonable to
neglect in view of the expected $\delta E/E$ performance for $b$-jets in Run3
and HL-LHC experiments. Variations of the minimum $B$-hadron energy can lead
to bias of the extracted $m_{t}$ by around 160 MeV for a 10% error on $E_{B}$.
Experimental analysis will have to select $B$-hadron decay modes for which
such precision on the measurement of $E_{B}$ can be attained.

###  4.3 Hadronization and fragmentation uncertainties

In our analysis, we use multiple $B$-hadron species, which differ in mass,
mean decay length, and production rate. The central values that we used for
these quantities are given in Table 3. To estimate the sensitivity of the
result to these inputs in the ansatz, we vary them one by one and see how the
prediction for the extracted top quark mass shifts on a fixed data set.
Clearly, future measurements will be needed to better fix these quantities,
especially if they can be measured directly at the LHC in phase-space regions
similar to $b$ quarks from $t\bar{t}$ production.

Hadron | Mass (MeV) ParticleDataGroup:2020ssz  | Lifetime ($10^{-12}$ s) HFLAV:2019otj  | Fraction
---|---|---|---
$B^{\pm}$ | 5279.34 $\pm$ 0.12 | 1.638 $\pm$ 0.004 | 42.9 %
$B^{0}$ | 5279.65 $\pm$ 0.12 | 1.519 $\pm$ 0.004 | 42.9 %
$B_{s}^{0}$ | 5366.88 $\pm$ 0.14 | 1.516 $\pm$ 0.006 | 9.5 %
$\Lambda_{b}^{0}$ | 5619.69 $\pm$ 0.17 | 1.471 $\pm$ 0.009 | 3.6 %
Table 3: Properties of the four most prominent species of $B$ hadrons from $b$-quark hadronization. Production fractions are taken from Pythia 8.2 Monash tune default. Parameter | Sensitivity
---|---
$m_{B_{i}}$ | $\simeq 1$
$\tau_{B_{i}}^{\text{rest}}$ | $\lesssim 1$
$f_{i}$ | $\simeq 0.04$
Table 4: Sensitivity of the top quark mass measurement to the properties of B
hadron species involved. The sensitivity that we quote is the maximum
sensitivity across the hadron species.

Table 4 summarizes how the extracted top quark mass changes upon variations of
$B$ hadrons properties. We quote a sensitivity

$\Delta_{\xi_{B_{j}}}=\frac{{\delta m_{t}\over
m_{t}}}{{\delta\xi_{B_{j}}\over\xi_{B_{j}}}}$ (27)

computed as the ratio of the relative variation on the extracted $m_{t}$ over
the variation of the parameter $\xi_{B_{j}}$ for each $B$ hadron property
$\xi_{B_{j}}$ listed in Table 3. The shift in the extracted mass tends to be
bigger when properties of $B^{\pm}$ or $B^{0}$ hadrons are varied because
these species have a higher production fraction in $b$ quark fragmentation. In
the right column, we quote the maximum shift in $m_{t}$ that we get among the
four hadron species considered. The result is that for masses and lifetimes we
need to know the physical values with the same or better precision that we
want to extract the top quark mass. The present knowledge of these quantities
seems more than adequate to warrant the resulting uncertainty on the top quark
mass to be negligible.

Production fractions tend to impact the extracted top quark mass to a much
milder degree than masses and lifetimes, hence relatively large uncertainties
up to 5% or so can be tolerated and still give rise to sub-leading
uncertainties in the extracted top quark mass. We ascribe this result to the
fact that the properties of different $B$-hadron species are rather similar.
The universality of the presently measured values of $f_{i}$ is questionable
when different production environments and different kinematic regimes are
compared, so we cannot really use the uncertainty of present measurements at
LEP and the TeVatron for the $f_{i}$ as a guideline for the impact of $f_{i}$
uncertainty on the extraction of $m_{t}$. In any case, it seems that if the
LHC will attain even just the same $\mathcal{O}(10\%)$ TeVatron precision on
the measurement of $f_{i}$, this uncertainty will have an impact on the top
mass extraction of order 100 MeV at most. It should be noted that the
fractions $f_{i}$ have been proven to be functions of the transverse momentum
of the hadron (see e.g. Section 4.1.3 of HFLAV:2019otj  for a recent summary).
However, this effect should be of even more sub-leading importance for the
$f_{i}$ values, hence it has not been assessed in detail.

The fragmentation function $D_{i}$ for each hadron species is necessary for
our template calculation, hence any uncertainty on the $D_{i}$ induces an
uncertainty on the extracted $m_{t}$. There are many possible reasons for
uncertainty on $D_{i}$, starting from the experimental data that are used to
measure it, the theoretical quantities and methods involved in the extraction
of $D_{i}$ from the data, and more practical details such as the possible
sensitivity to kinematics and color environment between the experiments in
which the measurement is carried out and those in which the function is used.
(For example, the use of $e^{+}e^{-}$ data to extract $D_{i}$ to be used at a
hadron collider.)

It is not possible for us to list all possible sources of uncertainty on
$D_{i}$, which depend crucially on the method used to obtain such functions.
In order to obtain a meaningful and useful result we aim at characterizing the
shape of $D_{i}$ using the first few Mellin moments of the $D_{i}$ function.
This kind of characterization is analogous to the characterization of
probability density functions through their mean, standard deviation, skewness
and kurtosis. Indeed, we define the Mellin moments of $D(z)$ such as $\langle
z\rangle$ is the first moment, and $\int z^{i}D(z)\,dz$ is the $i$-th
moment.121212We recall that, in principle, from the infinite series of Mellin
moments, one would be able to faithfully describe the $D_{i}$ function in
Mellin space, but we limit the calculation to the first few coefficients, as
the extraction of high Mellin moments is usually challenging from both
experimental and theoretical points of view.

Armed with this characterization of $D_{i}$ in terms of Mellin moments, we
compute the sensitivity of the extracted $m_{t}$ to the first three the Mellin
moments. In this way any possible change in $D_{i}$ due to future, improved
theoretical tools used in their extraction, or better data, or any correction
factor which may be used to account for kinematic or color effects can be cast
as a shift of the Mellin moments and translated into a shift in the top quark
mass.

To probe the dependence of our result on the Mellin moments of the
fragmentation functions $D_{i}$, we perform two analyses. In each analysis, we
compare two or more physical setups, each of which is internally consistent
and physically distinguishable from the others and which, in particular,
yields a different fragmentation function. In the first analysis, we compare
two different tunes of Pythia 8.2, namely the default tune of Monash (Tune:pp
14) and a tune by ATLAS derived from Monash (Tune:pp 21). These two tunes give
different $D_{i}$ functions shown in Figure 7. The difference in the Mellin
moments of these two tunes is given in Table 5 and results in a change in the
extracted top quark mass well above a few GeV. Such a large difference is the
result of the current poor knowledge of the dynamics of hadronization when
compared with the ambitious goals of the top quark mass extraction. In order
to do a more tailored job in our assessment of the sensitivity to $D_{i}$, we
produce a new $D_{i}$ whose value at each $z$ is given by the Monash tune
weighted at 90% and the ATLAS tune weighted at 10%. The weighted average of
$D_{i}$ functions that we obtain from the two tunes results in a difference in
the extracted top quark mass of 1.7 GeV, thus it is better suited to mimic a
situation where two future determinations of $D_{i}$, improved over the
current ones, will be compared in a top quark mass extraction. The resulting
change in the extracted $m_{t}$ is used to compute a sensitivity ratio as in
Eq. (27), from which we observe that a knowledge of the (Mellin moments of)
$D_{i}$ with precision about 3 times better than the desired precision on
$m_{t}$ is needed if one does not want to see the $m_{t}$ measurement spoiled
by poor knowledge of $D_{i}$.

This conclusion is corroborated by the second analysis in which we exploited
the fact that in our procedure we have extracted $D_{i}$ function directly
from Monte Carlo truth, namely, we have extracted a set of $D_{i}$ functions
for each hadron species and for each $m_{t}$ for which we have generated Monte
Carlo data. Therefore we can mimic a mismatch between the $D_{i}$ functions
appropriate for the data and a slightly incorrect one used in the computation
of templates by simply trying the extraction of $m_{t}$ from one data sample
using templates Eq. (LABEL:ansatz_LB) based on $D_{i}$ that correspond to a
different value of $m_{t}$. The result of several exercises of this type is
reported in Figure 8 where we show the extracted $m_{t}$ for several input
values and several choices of the $D_{i}$. The choices of the $D_{i}$ are
labeled by $m_{t}$ values along the horizontal axes. In this case, we choose
to parameterize the effect by looking at the Mellin moments of the $D_{i}$
functions. We summarize these findings and quote our result as a sensitivity
reported in Table 6. As for the first analysis, we find that a knowledge of
the (Mellin moments of) $D_{i}$ with precision about 3 times better than the
desired precision on $m_{t}$ is needed if one does not want to see the $m_{t}$
measurement spoiled by a poor knowledge of $D_{i}$.

Other analyses are possible to asses the sensitivity to $D_{i}$. For instance,
one could try to use $D_{i}$ computed from experimental inputs using
calculations to some order in perturbation theory, e.g. following recent
efforts to improve the knowledge of fragmentation from first principles
calculations Fickinger:2016rfd ; Maltoni:2022bpy ; Czakon:2021ohs ;
Czakon:2022pyz  which contain scale parameters, such as the fragmentation and
renormalization scales. As customary in pQCD studies the change on $D_{i}$
that follows from the variation of these scales can be taken as a source of
uncertainty in the results, in our case on the templates Eq. (LABEL:ansatz_LB)
and the $m_{t}$ extraction to which they give rise. We do not attempt this
kind of analysis here, as it has more to do with the work of extraction of
$D_{i}$. Furthermore, effects on $D_{i}$ from this type of theory uncertainty
can be captured in any case by our statement on the Mellin moments described
above.

It is important to remark that the present knowledge of $D_{i}$, be it taken
as a phenomenological parameter tuned over data, e.g one of the Pythia tunes
we have compared, or from first principle computations applied on data, is far
from the necessary precision to not spoil a determination of $m_{t}$ that aims
at a sub-GeV precision. As far as we know this situation is common to other
methods that explicitly use hadrons produced in $t\bar{t}$ events, see e.g.
Corcella:2017rpt ; Agashe:2016bok ; CMSlifetime ; ATLAS:2022jbw ;
ATLAS:2019ezb . Therefore we urge for improvements on the determination of
fragmentation functions both on the experimental and theoretical side.

Mellin Moment | $\delta\langle z^{n}\rangle/\langle z^{n}\rangle$ | $\delta m_{t}(10\%\text{ reweighting})$ | Sensitivity
---|---|---|---
$\langle z\rangle$ | 2.8 % |  | 3.5
$\langle z^{2}\rangle$ | 5.2 % |  1.7 GeV | 2.5
$\langle z^{3}\rangle$ | 7.2 % |  | 1.4
Table 5:  For each of the first three Mellin moments of the $D_{i}$ we report: the difference between the default Pythia tune (Tune:pp 14) and the ATLAS tune (Tune:pp 21); the effect on the extracted $m_{t}$ stemming from a 10% contamination of the ATLAS tune into the Monash tune; the sensitivity of the extracted $m_{t}$ to each Mellin moment.  Mellin Moment | $\delta\langle z^{n}\rangle/\langle z^{n}\rangle$ | $\delta m_{t}^{{(171\to 176)}}$ | Sensitivity
---|---|---|---
$\langle z\rangle$ | 0.53 % |  | 3.8
$\langle z^{2}\rangle$ | 0.91 % |  3.5 GeV | 2.2
$\langle z^{3}\rangle$ | 1.23 % |  | 1.6
Table 6:  For each of the first three Mellin Moments of the fragmentation
function we report: their change due to varying the $m_{t}$ value that labels
the $D_{i}$ extracted from the Monte Carlo truth from 171 GeV to 176 GeV; the
change on the extracted $m_{t}$ due to using the $D_{i}$ extracted from the
Monte Carlo truth for $m_{t}=176$ GeV on the data sample for $m_{t}=171$ GeV;
the sensitivity of the extracted $m_{t}$ to each Mellin moment. Figure 7:
Fragmentation function obtained from Pythia 8 using a ATLAS tune (Tune:pp 21)
and the default tune (Tune:pp 14) for $m_{t}=171$ GeV. Figure 8: Effect on the
extracted $m_{t}$ from the change of the fragmentation function as
parameterized by changing $m_{t}$ in the data used to the MC truth on which
the fragmentation functions is measured. The $m_{t}$ used to measure the
fragmentation is on the horizontal axis; the measurement is shown as a black
line for each subplot corresponding to a correct $m_{t}$ used to generate
data. The blue line is shown as a reference, as it corresponds to an unbiased
measurement.

###  4.4 Uncertainties due to top quark production modeling

As mentioned earlier, CDF and CMS have already implemented a method to measure
top quark mass using $B$-hadron decay length Abulencia:2006rz ; CMSlifetime .
More specifically, they use the mean transverse decay length of $B$-hadrons in
top quark events. Like our method, their results are not plagued by jet energy
scale uncertainty. The largest uncertainty in the latest CMS analysis is due
to modeling of the top quark $p_{T}$ distribution CMSlifetime , which shows
how the measurement is in general sensitive to the production mechanism of the
top quarks. Our method, if applied on fully inclusive data samples and with a
perfect ansatz for the shape of the $b$ quark energy distribution, would be
independent of top quark production mechanism, hence we expect our results to
have little sensitivity to top quark $p_{T}$. In fact, some degree of
sensitivity is unavoidable because we use only part of the data and our ansatz
is unavoidably not perfect. Indeed we tune hyper-parameters by doing numerical
experiments to minimize the bias of the method and we do that assuming
Standard Model production. This procedure and the event selection cuts could
lead to a non-zero sensitivity to top quark $p_{T}$ and to other aspects of
the production mechanism of the top quark in general. In the following, we
describe how we probed this effect and we concluded that our method is indeed
practically insensitive to top quark $p_{T}$.

In order to quantify the sensitivity of our method to changes in the top quark
$p_{T}$ distribution we perform the following exercise, which also gives us
the opportunity to compare our method with the $L_{xy}$ methods of CMS in this
particular respect. To carry out extraction of $m_{t}$ from the $L_{xy}$
spectrum we generate the $L_{xy}$ distribution with the same event selection
criteria as we use earlier for our method. In order to learn the response of
the $L_{xy}$ distribution to changes of $m_{t}$ we generate event samples for
7 different values of $m_{t}$ from 170 GeV to 176 GeV and we fit each bin
count of the $L_{xy}$ distribution using a quadratic polynomial as a function
of $m_{t}$. Having learned how each bin count of the $L_{xy}$ distribution
depends on $m_{t}$ we generate a sample for a test value of $m_{t}$, that we
call $m_{t,test}$ and we obtain a top quark mass measurement from a $\chi^{2}$
minimization. As a check of the goodness of this procedure, we checked that
this method returns $m_{t,~{}true}$ free of any bias.

For a fixed luminosity we find that the statistical error obtained on the
$m_{t}$ measurement from the $L_{xy}$ spectrum is better than the result of
our method in Eq. (26). However, the transverse decay length method is highly
sensitive to top-quark $p_{T}$ compared to our method. We quantify the
sensitivity of the transverse length method and of our method by performing
the top mass extraction on $L_{xy}$ and $L_{xyz}$ spectra, respectively,
obtained by reweighting each event of the $m_{t,test}$ sample according to the
top quark $p_{T}$ in that event. The new weights are obtained using

$\text{w}_{\text{new}}=\text{w}_{\text{old}}\cdot\left[1+\alpha\cdot\Theta(p_{T}<400)\cdot(p_{T}-200)\right],$
(28)

where $\text{w}_{\text{new}}$ and $\text{w}_{\text{old}}$ are the new and the
old weights, $\Theta(c)=1$ when $c$ is true and $\Theta(c)=0$ otherwise, and
$\alpha$ adjusts the amount of re-weighting. We stop re-weighting at a $p_{T}$
threshold of 400 GeV, as there are fewer data at such large $p_{T}$, hence the
impact on the measurement is modest, and the entailed larger statistical
uncertainty makes it less meaningful to manipulate the distribution at such
large $p_{T}$. Figure 9 shows how the two methods compare. It is clear that
even for a very soft re-weighting, the transverse decay length prediction
shifts appreciably, whereas predictions using the full decay length fitted
over our templates Eq. (LABEL:ansatz_LB) do not move nearly as much. For
concreteness one can look at the reweighting for $\alpha=10^{-4}$, which
roughly corresponds to the theoretical uncertainty in top $p_{T}$ spectrum
shown in Ref. CMS:2021vhb , this is roughly moving the average top quark
$p_{T}$ by around 0.5%. At this value of $\alpha$, the prediction using
$L_{xy}$ method shifts by $\approx 600$ MeV, whereas for $L_{xyz}$ the shift
is only $\approx 50$ MeV.

Figure 9: Bottom plot is the result of $L_{xy}$ method when reweighting the
top quark $p_{T}$ distribution. The green line/data points represent a $\sim
0.5\%$ change in the average top quark $p_{T}$. The top plot shows the results
of our method with an inset included to show how small the shift is for the
same reweighting of top quark $p_{T}$ distribution.

##  5 Conclusions

The LHC and its High-Lumi upgrade face the formidable task of improving our
understanding of the Standard Model, which is already very well established.
Nevertheless, there are corners of the theory that are in urgent need of
further study. The existence of multiple proposals for a Higgs “factory”, or
an even more versatile factory that can also produce top quarks and
electroweak bosons is a clear indication of the need to sharpen our knowledge
of the Standard Model with higher precision and potentially see discrepancies
resulting from new physics.

The top quark mass is a very special parameter in this respect because it
involves a number of perturbative and non-perturbative issues on the theory
side, as well as a variety of experimental challenges resulting from the
relatively complex decay modes of top quarks. At present, there are
measurements that claim sub-GeV precision for the top quark mass, but further
scrutiny and caution are needed because of the myriad delicate physical
phenomena that affect how the process is manifested in the data, how the
complex data are analyzed by the experimental teams, and finally how this is
all translated into a theoretical interpretation. In view of this, and the
long period ahead of us before a lepton collider will be ready to study top
quarks, there is an urgent need to develop new methods to determine the top
quark mass at the LHC. These methods can also be applied to advantage at
future colliders. The hope is that, by getting a number of high-precision
measurements, with different machines, experiments, and methods with different
and complementary sources of systematic uncertainty, a coherent picture will
emerge that provides greater clarity and a detailed understanding of the
delicate phenomena alluded to above in order to have a similarly clear and
robust measurement of $m_{t}$. For this program to take place, it is necessary
to diversify the types of observables and the theory approaches used to
extract $m_{t}$.

With this in mind, we proposed a new observable that is essentially the
transposition of the energy peak proposal of Ref. Agashe:2012bn  from the
energy domain into the decay length domain. The shift from an energy
observable to a measured length has as most dramatic effect the elimination of
much of the jet energy scale uncertainty from the list of experimental
uncertainties for the measurement.

The flip side of this proposal is the increase of the importance played by
theoretical uncertainties tied to the description of the formation of long-
distance objects, i.e. the mesons and hadrons whose decays we observe in
detectors. This phenomenon tracks back to the great mystery of confinement
dynamics in QCD. The difficulty and depth of this issue have made it such that
it has been only possible to present phenomenological descriptions for the
transition of quarks into hadrons, even after many decades of perturbative QCD
being the established theory of strong interactions. While a deeper
understanding of hadronization may be attained in the future, we have
investigated how much the present ignorance of the dynamics of hadron
formation would impact the extraction of the top quark mass using a method
based on hadron decay length measurements.

To carry out our study we used the Monte Carlo event generator Pythia 8.2 and
we explored a number of aspects of the description of the parton shower and
hadronization models of this type of event generator. The convenience of using
an event generator allowed us to mimic the same process for the extraction of
$m_{t}$ that the experiments would carry out on real data. Following a
standard procedure, we extract $m_{t}$ by minimizing a $\chi^{2}$ obtained by
comparison of template length spectra that we compute with Pythia 8.2 for
several values of $m_{t}$ and separate spectrum, for a simulated sample that
is also based Pythia 8.2, which plays the role of the real data.

The observable that we use to extract $m_{t}$ is the three-dimensional
$B$-hadron decay length, $L_{xyz}$, and we carry out the $\chi^{2}$ analysis
using a carefully crafted template that is given in Eq. (LABEL:ansatz_LB).
This template returns the distribution of the $L_{xyz}$ observables for a
given $m_{t}$, exploiting the fact that the average distance traveled by a
$B$-hadron in the laboratory has a one-to-one correspondence with the energy
of the $b$-quark that originated the $b$-flavored hadron. The key for this
link to exist is that the distribution of the energy of the $b$-quark is, in
turn, related to $m_{t}$ by the invariance of the peak position discussed in
Ref. Agashe:2012bn . With this chain of links between observables in mind, we
have formulated Eq. (LABEL:ansatz_LB) as a candidate template for the
measurement of $m_{t}$. We have verified that a measurement of $m_{t}$ can be
carried out with this template without introducing a significant bias. We have
also verified that a number of “hyper-parameters” of our template can be set
to the values collected in Table 2 that are optimal for the stability of our
result, maintain negligible bias, and insure a competitive uncertainty in the
measurement of $m_{t}$ when $\mathcal{O}(100/{\rm fb})$ or more of LHC data
are used in the analysis.

Having established that a robust and precise measurement of $m_{t}$ can be
carried out with templates of the form Eq. (LABEL:ansatz_LB) on the $L_{xyz}$
measurements, we then stress-tested our method with the aim of assessing the
sensitivity of our result to the many theoretical and experimental quantities
that are used to compute our templates. We have tested in particular how well
the dynamics of parton shower and hadronization that determines the
fragmentation functions $D_{i}$ needs to be known in order not to spoil the
precision of this method. With two different assessments, summarized in Tables
5 and 6, we showed that fragmentation functions known at the sub-percent level
are needed. Indeed fragmentation dynamics has emerged as the most important
challenge to attain an uncertainty in $m_{t}$ at the 500 MeV level. As a
potential second challenging source of uncertainty, we have identified the
hadronization fractions, which account for the abundance of each hadron
species produced in the phase-space covered by the experiment. The expected
theoretical uncertainty from the hadronization fractions, the fragmentation
function, and collected statistics at the LHC are given in Eq. (26), which
formulates our final predictions for the uncertainty on $m_{t}$ in terms of
the levels of uncertainty in the hadronization functions and fractions.

It is no accident that we have not included in our summary of the most
prominent uncertainties a possible source of uncertainty from the mis-modeling
of top quark production. In fact, we tested explicitly how much our extracted
top quark mass would change if the transverse boost of the top quark in the
data were to differ from that predicted by the event generator prediction and
we found that even for a mis-modeling larger than the current NNLO theory
error band, the uncertainty on $m_{t}$ would still be negligible compared to
the LHC target of few hundred MeV. This is not at all true of the method that
comes closest to our proposal, which is the CMS measurement CMSlifetime  that
makes use of $L_{xy}$, the decay length projected onto the plane transverse to
the beam axis. As in our method, the choice of decay length is motivated by
the elimination of jet energy scale uncertainties but in the CMS case,
templates based on any invariance of the underlying $b$-quark energy are not
employed. As a consequence, the CMS result is very sensitive to the top quark
production kinematics. Indeed, the uncertainty related to top quark transverse
boost was found to be the largest theoretical uncertainty, exceeding 1 GeV in
Ref. CMSlifetime . The striking reduction of sensitivity to top quark
transverse momentum in our method is seen in Figure 9, highlighting the
benefit of decay length distribution templates rooted in energy-peak
invariance.

We stress that our method may be inferior to other proposals when one looks
solely at the statistical part of the uncertainty. Indeed we found that our
implementation of the CMS method based on $L_{xy}$ gives a smaller statistical
uncertainty than our methodHowever, given the extremely large sample of top
quarks that has already been recorded by the LHC experiments and the even
larger sample that will be collected at the High-Lumi LHC, statistical
uncertainty will not be an issue and essentially all of the LHC measurements
will ultimately be limited by systematic uncertainties.

Further study is necessary to transform our proposal into a real measurement
of $m_{t}$. For sure, a thorough study of the measurement uncertainties for
the lengths would be necessary. Indeed each $B$-hadron species is detected in
different decay modes, whose detection may be more or less amenable to the
measurement of the three-dimensional decay length. On a related topic, one can
imagine that each species of $B$-hadrons being measured with different
accuracy, the measured length spectra for each hadron species can be
translated into a “particle-level” length spectra, in which the effects of
detector mismeasurement have been removed. This procedure is applied routinely
in SM studies at the LHC, especially for kinematic properties of SM reactions,
and goes by the name of detector unfolding. Given the different role played by
each hadron species and potentially different degrees of understanding of the
fragmentation function $D_{i}$ and the hadronization fractions $f_{i}$ for
each species, it might be useful to explore how to carry out this type of
measurement on unfolded data for each hadron species.

Another branch of development that is worth mentioning is a possible reduction
of complexity in the computation of templates Eq. (LABEL:ansatz_LB). This type
of ansatz requires a double integral, one of which is over $E_{B}$, and in
part serves the purpose of convolving the exponential decay law with the
result of the integral over $E_{b}$. As the exponential decay law is a piece
of extremely well-established physics, one could potentially consider it a
part of the detector smearing so that the measurement of a length can be
transformed into a measurement of average decay length, or, if one wishes, a
measurement of $E_{B}$, through the relation $\lambda_{B_{i}}=\langle
L_{i}\rangle=\gamma\beta\cdot c\tau_{i,0}$ where $\gamma=E_{B_{i}}/m_{B_{i}}$
and $\gamma=1/\sqrt{1-\beta^{2}}$.

While it was not an issue for our work to generate adequately many sets of
precise templates from the double integral Eq. (LABEL:ansatz_LB) it might
become a more important issue in an actual measurement with collider data. As
a matter of fact, the $E_{B}$ integral is rendered more complicated by the
presence of the exponential, which plays the detrimental role of “diluting”
the information and forces us to perform a more complicated integral to undo
its effect for each and every template that we compute. Therefore, one can
envisage being more economical and undoing the exponential part of the
integral once and for all in the unfolding. Such an observation may enable a
quicker path to more thorough explorations of showering and hadronization
effects or changes in top quark production kinematics for which it may be
necessary to generate numerous dedicated sets of templates. Amusingly, the
unfolding of the exponential convolution from $E_{B}$ to $L_{B}$ space could
be attempted using semi-analytical techniques for the inversion of Laplace
transforms Stehfest_1970 ; CohenLaplace .

## Acknowledgements

K. A. and J. I. thank O. W. (Wally) Greenberg for organizing the “50 Years of
Quarks and Color” symposium at the University of Maryland, where K. A. and J.
I. first discussed the basic idea in this paper. The authors acknowledge
Snowmass 2021, as it motivated them to make significant progress in this work.
J. I. acknowledges the support of DOE grant DE-SC0011702 and the Joe and Pat
Yzurdiaga endowed chair in experimental science. The work of K. A., S. A. and
D. S. is supported by NSF Grant No. PHY-2210361 and by the Maryland Center for
Fundamental Physics. The work of D. K. is supported by the DOE Grant No. DE-
SC0010813.

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Parameterized Verification under TSO with Data Types

Uppsala University, Sweden Indian Institute of Technology Bombay
UC Berkeley, USA

Parosh Abdulla1 Mohamad Faouzi Atig	1 Florian Furbach1  Adwait Godbole  3 Yacoub G. Hendi1 Shankaranarayanan Krishna	2 Stephan Spengler1
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