# Infrared divergences in $e^{+}e^{-}\rightarrow 2\;\text{jets}$ in the light
front coherent state formalism

Deepesh Bhamre<EMAIL_ADDRESS>Department of Physics,
University of Mumbai, Santacruz (East), Mumbai-400098, India  Anuradha Misra
Formerly Department of Physics, University of Mumbai, Santacruz (East),
Mumbai-400098, India<EMAIL_ADDRESS>Centre for Excellence in
Theoretical and Computational Sciences (CETACS), University of Mumbai,
Santacruz (East), Mumbai-400098, India

###### Abstract

We study infrared (IR) divergences in light front quantum chromodynamics using
a coherent state basis in light front time-ordered Hamiltonian perturbation
theory. In computation of the S-matrix elements in Hamiltonian formalism, the
IR divergences appear in the form of vanishing energy denominators. We
consider the process $e^{+}e^{-}\rightarrow 2\;\text{jets}$ at
$\mathcal{O}(g^{2})$ in strong coupling, construct the coherent state
representing the outgoing particles, and explicitly show that the ‘true’ IR
divergences cancel to this order when the matrix elements are calculated
between coherent states instead of Fock states.

Keywords

Infrared divergences, Coherent State Formalism, Light Front QCD

##  I Introduction

Infrared (IR) divergences in quantum electrodynamics (QED) and quantum
chromodynamics (QCD) are known to cancel for infrared safe observables when
contributions of real emissions are added to virtual corrections, which
essentially amounts to summing over physically indistinguishable states in the
cross section calculations. This real+virtual cancellation takes place at the
level of cross section and is well established as a successful approach to the
IR problem.

An alternative approach, proposed originally for QED by Kulish and Faddeev
(referred to as KF approach henceforth), is based on the method of asymptotic
dynamics Kulish and Faddeev (1970) and addresses the issue of cancellation of
IR divergences at the amplitude level. The KF approach is based on Hamiltonian
perturbation theory wherein the S-matrix elements are obtained by calculating
the matrix elements between coherent states, which are the asymptotic states
obtained by applying the asymptotic evolution operator to the Fock states. The
rationale behind the use of coherent states in place of Fock states, in the
calculation of S-matrix elements, is that for theories containing long-range
interactions (as also for theories like QCD where the initial and final states
are bound states), the underlying assumption in the Lehmann, Symanzik and
Zimmermann formalism - that the asymptotic states are eigenstates of free
particle Hamiltonian - is no longer valid. The method of asymptotic dynamics
is based on the observation that the interaction Hamiltonian does not approach
zero at the asymptotic limits ($t\rightarrow\pm\infty$), and hence the correct
asymptotic states need to be obtained by applying the “asymptotic evolution
operator” to the Fock states. The asymptotic evolution operator contains the
asymptotic Hamiltonian which is obtained by taking the limiting value of
interaction Hamiltonian at infinite times. It was shown by Kulish and Faddeev
that the IR divergences in QED cancel at the amplitude level itself when the
S-matrix elements are calculated between coherent states constructed in this
manner, instead of the Fock states. The coherent state method was subsequently
discussed in the context of QCD also (see Refs. Catani and Ciafaloni (1985);
Catani  _et al._ (1986); Catani and Ciafaloni (1987); Ciafaloni (1985, 1989);
Del Duca  _et al._ (1989); Forde and Signer (2004); Hannesdottir and Schwartz
(2020a, b)).

A coherent state approach, in the context of light front QED (LFQED), was
developed by one of us in Ref. Misra (1994) and was shown to lead to
cancellation of IR divergences in the one-loop vertex correction in QED.
Application of the coherent state basis to the discretized light cone
quantization (DLCQ) method of bound state calculations was discussed in Misra
(1996) and preliminary study of coherent states in light front QCD (LFQCD) was
performed in Refs. Misra (2000) and Misra (2005). IR divergences in fermion
self-energy correction in LFQED were discussed in Refs. More and Misra (2012,
2013, 2014) and it was shown that the IR divergences in fermion self-energy
correction cancel to all orders at amplitude level in LFQED.

In this work, we focus on the construction of coherent states and cancellation
of IR divergences in coherent state basis in LFQCD. In this preliminary study,
we consider the process $e^{+}e^{-}\rightarrow 2\;\text{jets}$ to
$\mathcal{O}(g^{2})$ in strong coupling and show by explicit calculations the
cancellation of ‘true’ IR divergences. In the coherent state approach, a
transition amplitude receives, in addition to the conventional Fock state
contributions that are IR-divergent, contributions from higher-order terms of
the coherent state. These too are IR-divergent by construction since the phase
space is limited to the region where only those particles that are either soft
or are collinear to external particles are emitted/absorbed. These two IR-
divergent contributions are expected to cancel between themselves leading to
an IR-finite transition amplitude. A somewhat similar study has been performed
by Forde and Signer in equal-time QCD in Ref. Forde and Signer (2004), wherein
the authors have also discussed the advantages and practical use of an
amplitude approach as compared to the cross section approach.

A brief introduction to coherent states in massless QFTs and of recent
developments is provided in Ref. Agarwal  _et al._ (2023). Unless an explicit
distinction between the ‘soft’ and ‘collinear’ divergences is made, the
generic term ‘infrared divergences’ will be used to denote both in this paper.

In Sec.II, we review the derivation of LFQCD Hamiltonian presented in Ref.
Brodsky  _et al._ (1998). In Sec.III, a brief account of IR divergences
appearing in LFQCD is given and the coherent state formalism is described.
Further, the coherent state, to the order necessary for our calculation, is
constructed in this section. The asymptotic region of phase space which leads
to IR divergences and its relevance in the construction of coherent states is
also described. In Sec.IV, we show the cancellation of IR divergences at
$\mathcal{O}(g^{2})$ in the coherent state formalism for the process
$e^{+}e^{-}\rightarrow 2\;\text{jets}$, and conclude in Sec.V. Appendix A
contains a detailed calculation of one of the terms of the amplitude,
primarily to illustrate the technique of light cone time-ordered perturbation
theory being used in our work.

##  II Light Front QCD Hamiltonian

In this section, we review the derivation of LFQCD Hamiltonian following Ref.
Brodsky  _et al._ (1998). The standard QCD Lagrangian (without the gauge-
fixing terms) is given by

$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}_{a}F_{\mu\nu}^{a}+\frac{1}{2}[\bar{\Psi}_{c}(i\gamma^{\mu}D_{\mu}^{cc^{\prime}}-m\delta_{cc^{\prime}})\Psi_{c^{\prime}}+H.c.],$
(1)

where $F^{\mu\nu}_{a}$ is the field tensor given by

$F^{\mu\nu}_{a}=\partial^{\mu}A^{\nu}_{a}-\partial^{\nu}A^{\mu}_{a}-gf^{abd}A^{\mu}_{b}A^{\nu}_{d}.$
(2)

The fermionic (quark) fields in Eq.(1) carry color indices
$c,c^{\prime}=1,2,3$. $A^{\mu}_{a}$ in Eq.(2) are eight real-valued color
vector potentials with $a=1,2,...,8$ being the gluon index. The Hermitian,
traceless 3 x 3 matrix

$A^{\mu}=\frac{1}{2}\begin{bmatrix}\frac{1}{\sqrt{3}}A^{\mu}_{8}+A^{\mu}_{3}&A^{\mu}_{1}-iA^{\mu}_{2}&A^{\mu}_{4}-iA^{\mu}_{5}\\\
A^{\mu}_{1}+iA^{\mu}_{2}&\frac{1}{\sqrt{3}}A^{\mu}_{8}-A^{\mu}_{3}&A^{\mu}_{6}-iA^{\mu}_{7}\\\
A^{\mu}_{4}+iA^{\mu}_{5}&A^{\mu}_{6}+iA^{\mu}_{7}&-\frac{2}{\sqrt{3}}A^{\mu}_{8}\\\
\end{bmatrix}\\\ $ (3)

belongs to the $SU(3)_{color}$ gauge group. $A^{\mu}$ is constructed using the
definition

$(A^{\mu}(x))_{cc^{\prime}}\equiv T^{a}_{cc^{\prime}}A^{\mu}_{a}(x),$ (4)

where the color matrices $T^{a}$ are related to Gell-Mann matrices
$\lambda^{a}$ through

$T^{a}=\frac{1}{2}\lambda^{a}.$ (5)

The color matrices obey the relations

$[T^{a},T^{b}]_{cc^{\prime}}=if^{abd}T^{d}_{cc^{\prime}};\;\;Tr(T^{a}T^{b})=\frac{1}{2}\delta^{ab}.$
(6)

The covariant derivative appearing in Eq.(1) is given by

$D^{\mu}_{cc^{\prime}}=\partial^{\mu}\delta_{cc^{\prime}}+igT^{a}_{cc^{\prime}}A^{\mu}_{a}.$
(7)

The Hamiltonian can be obtained from the energy-momentum stress tensor
$T^{\mu\nu}$. The light cone Hamiltonian is

$\begin{split}P^{-}=P_{+}=&\int d^{2}{\bf{x}}_{\perp}dx^{-}T^{+-}\\\ =&\int
d^{2}{\bf{x}}_{\bf\perp}dx^{-}\bigg{(}F^{+\kappa}_{a}F_{\kappa+}^{a}+\frac{1}{4}F^{\kappa\lambda}_{a}F_{\kappa\lambda}^{a}+\frac{1}{2}[i\bar{\Psi}\gamma^{+}T^{a}D^{a}_{+}\Psi+H.c.]\bigg{)}\end{split}$
(8)

Using light cone gauge $A^{+}_{a}=0$, it is seen that the Euler-Lagrange
equations

$-\partial^{+}\partial_{-}A^{-}_{a}-\partial^{+}\partial_{i}A^{i}_{\perp
a}=gJ^{+}_{a}$ (9)

and

$2i\partial_{-}\Psi_{-}=(m\beta-i\alpha^{i}_{\perp}T^{a}D^{a}_{\perp
i})\Psi_{+}$ (10)

are constraint equations. Here $\alpha^{k}=\gamma^{0}\gamma^{k}$,
$\beta=\gamma^{0}$, and $\Psi_{\pm}=\Lambda_{\pm}\Psi$ are spinor projections
obtained using the operators $\Lambda_{\pm}=\frac{1}{2}(1\pm\alpha^{3})$. The
dependent degrees of freedom $A^{-}_{a}$ and $\Psi_{-}$ are eliminated by
inverting Eqs.(9) and (10) respectively. The free solutions, in terms of the
independent degrees of freedom, are

$\tilde{A}^{\mu}_{a}=\bigg{(}0,-\frac{1}{\partial^{+}}\partial_{i}A^{i}_{a},A_{\perp}^{a}\bigg{)}$
(11)

and

$\begin{split}\tilde{\Psi}=&\tilde{\Psi}_{+}+\tilde{\Psi}_{-}\\\
=&\tilde{\Psi}_{+}+\frac{(m\beta-i\alpha^{i}\partial_{i})}{2i\partial_{-}}\tilde{\Psi}_{+}.\end{split}$
(12)

Thus, the Hamiltonian in Eq.(8), now rewritten in terms of the free solutions
and thereby only the independent degrees of freedom in the theory, is given by

$\begin{split}P^{-}=\int
d^{2}{\bf{x}}_{\perp}dx^{-}\bigg{[}&\frac{1}{2}\bar{\Psi}\gamma^{+}\frac{m^{2}+(i\nabla_{\perp})^{2}}{(i\partial^{+})}\Psi+\frac{1}{2}A^{\mu}_{a}(i\nabla_{\perp})^{2}A_{\mu}^{a}\\\
&+gJ^{\mu}_{a}A_{\mu}^{a}+\frac{g^{2}}{4}B^{\mu\nu}_{a}B_{\mu\nu}^{a}+\frac{g^{2}}{2}J^{+}_{a}\frac{1}{(i\partial^{+})^{2}}J^{+}_{a}\\\
&+\frac{g^{2}}{2}(\bar{\Psi}\gamma^{\mu}T^{a}A_{\mu}^{a})\frac{\gamma^{+}}{(i\partial^{+})}(\gamma^{\nu}T^{b}A_{\nu}^{b}\Psi)\bigg{]},\end{split}$
(13)

where we have discarded the tilde notation i.e.
$\tilde{A}^{\mu}_{a}\rightarrow A^{\mu}_{a}$ and
$\tilde{\Psi}_{+}\rightarrow\Psi_{+}$ from Eq.(13) onwards. In Eq.(13),
$J^{\mu}_{a}=\bar{\Psi}_{c}\gamma^{\mu}T^{a}_{cc^{\prime}}\Psi_{c^{\prime}}+f^{abd}F^{\mu\alpha}_{b}A_{\alpha}^{d}$
is the color Maxwell current and $B^{\mu\nu}_{a}\equiv
f^{abc}A^{\mu}_{b}A^{\nu}_{c}$. The first two terms in Eq.(13) form the free
part of the Hamiltonian. Rest of the terms contain the strong coupling
constant ‘$g$’ and are the interaction terms of the theory. Using the
expressions for $J^{\mu}_{a}$ and $B^{\mu\nu}_{a}$, the interaction part of
the Hamiltonian is expanded to obtain the interaction Hamiltonian,

$H_{\text{int}}=V_{1}+V_{2}+V_{3}+W_{1}+W_{2}+W_{3}+W_{4},$ (14)

where

$\displaystyle V_{1}$ $\displaystyle\equiv\int
d^{2}{\bf{x}}_{\perp}dx^{-}g\bar{\Psi}\gamma^{\mu}T^{a}\Psi A^{a}_{\mu}$ (15)
$\displaystyle V_{2}$ $\displaystyle\equiv\int
d^{2}{\bf{x}}_{\perp}dx^{-}gf^{abd}(\partial^{\mu}A^{\nu}_{b})A_{\nu}^{d}A_{\mu}^{a}$
(16) $\displaystyle V_{3}$ $\displaystyle\equiv\int
d^{2}{\bf{x}}_{\perp}dx^{-}\frac{g^{2}}{4}f^{abd}f^{aef}A^{\mu}_{b}A^{\nu}_{d}A_{\mu}^{e}A_{\nu}^{f}$
(17) $\displaystyle W_{1}$ $\displaystyle\equiv\int
d^{2}{\bf{x}}_{\perp}dx^{-}\frac{g^{2}}{2}(\bar{\Psi}\gamma^{+}T^{a}\Psi)\frac{1}{(i\partial^{+})^{2}}(\bar{\Psi}\gamma^{+}T^{a}\Psi)$
(18) $\displaystyle W_{2}$ $\displaystyle\equiv\int
d^{2}{\bf{x}}_{\perp}dx^{-}\frac{g^{2}}{2}\bigg{[}(\bar{\Psi}\gamma^{+}T^{a}\Psi)\frac{1}{(i\partial^{+})^{2}}f^{abd}(\partial^{+}A^{\mu}_{b})A_{\mu}^{d}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+f^{abd}(\partial^{+}A^{\mu}_{b})A_{\mu}^{d}\frac{1}{(i\partial^{+})^{2}}(\bar{\Psi}\gamma^{+}T^{a}\Psi)\bigg{]}$
(19) $\displaystyle W_{3}$ $\displaystyle\equiv\int
d^{2}{\bf{x}}_{\perp}dx^{-}\frac{g^{2}}{2}f^{abd}f^{aef}(\partial^{+}A^{\mu}_{b})A_{\mu}^{d}\frac{1}{(i\partial^{+})^{2}}(\partial^{+}A^{\nu}_{e})A_{\nu}^{f}$
(20) $\displaystyle W_{4}$ $\displaystyle\equiv\int
d^{2}{\bf{x}}_{\perp}dx^{-}\frac{g^{2}}{2}(\bar{\Psi}\gamma^{\mu}T^{a}A_{\mu}^{a})\frac{\gamma^{+}}{(i\partial^{+})}(\gamma^{\nu}T^{b}A_{\nu}^{b}\Psi).$
(21)

In addition to the usual interactions in equal-time QCD, which are denoted by
$V$ in Eq.(14), the interaction Hamiltonian also contains nonlocal
interactions denoted here by $W$. The presence of the $W$ type i.e.,
instantaneous interaction terms is a result of eliminating the dependent
degrees of freedom from the theory. Each of the interaction terms above is
represented by a vertex as shown in Fig.1. Propagators with small vertical
lines in Figs.1(d) through 1(g) denote nonlocal (instantaneous) interactions.

(a) $V_{1}$ (b) $V_{2}$

(c) $V_{3}$ (d) $W_{1}$

(e) $W_{2}$ (f) $W_{3}$

(g) $W_{4}$

Figure 1: Vertices in LFQCD

The momentum-space expansion of the fields appearing in the Hamiltonian is

$\begin{split}&\Psi_{cf}(x)=\sum_{\lambda}\int[dp](b_{p\lambda cf}u_{p\lambda
cf}e^{-ip{\cdot}x}+d^{\dagger}_{p\lambda cf}v_{p\lambda cf}e^{ip{\cdot}x})\\\
&A^{a}_{\mu}(x)=\sum_{\lambda}\int[dp](a_{p\lambda a}\epsilon_{p\lambda
a}e^{-ip{\cdot}x}+a^{\dagger}_{p\lambda a}\epsilon^{*}_{p\lambda
a}e^{ip{\cdot}x}),\end{split}$ (22)

where $\lambda$ is the spin (polarization) of the fermion (boson) and the
color index is given by $c\ (a)$ for the fermion (boson). The fermion also
carries the index $f$ for flavor quantum number. The light front integral
measure is

$\int[dp]\equiv\int_{-\infty}^{\infty}\frac{d^{2}{\bf{p}}_{\perp}}{(2\pi)^{3/2}}\int_{0}^{\infty}\frac{dp^{+}}{\sqrt{2p^{+}}}$
(23)

and the commutation (anti-commutation) relations for bosons (fermions) are

$[a_{p\lambda
a},a^{\dagger}_{p^{\prime}{\lambda}^{\prime}a^{\prime}}]=\delta(p^{+}-p^{\prime+})\delta^{2}({\bf{p}}_{\perp}-{\bf{p^{\prime}}}_{\perp})\delta_{\lambda}^{\lambda^{{}^{\prime}}}\delta_{a}^{a^{\prime}},$
(24) $\\{b_{p\lambda
cf},b^{\dagger}_{p^{\prime}{\lambda}^{\prime}c^{\prime}f^{\prime}}\\}=\\{d_{p\lambda
cf},d^{\dagger}_{p^{\prime}{\lambda}^{\prime}c^{\prime}f^{\prime}}\\}=\delta(p^{+}-p^{\prime+})\delta^{2}({\bf{p}}_{\perp}-{\bf{p^{\prime}}}_{\perp})\delta_{\lambda}^{\lambda^{{}^{\prime}}}\delta_{c}^{c^{\prime}}\delta_{f}^{f^{\prime}},$
(25)

respectively, with the rest of the commutators (anticommutators) being zero.

##  III IR divergences in LFQCD and the coherent state formalism

Figure 2: IR divergence due to emission of a soft or collinear gluon

It is well known that IR divergences occur when massless propagators go on
shell. In addition to the ‘soft’ divergences, gauge theories with a vertex
involving all massless particles contain another type of divergence called
‘collinear’ divergence. For example, consider a process in which a final-state
massless quark is accompanied by a gluon, as shown in Fig.2. The quark
propagator has momentum $p+k$ and can become on shell in the following
situations:
(i) $k^{\mu}\rightarrow 0\implies(p+k)^{2}\rightarrow p^{2}=0$ leading to soft
divergence.
(ii) $k^{\mu}\parallel(p+k)^{\mu}\implies
k^{\mu}=\lambda(p+k)^{\mu}\implies(p+k)^{2}\rightarrow(1+\lambda)^{2}p^{2}=0$
leading to collinear divergence.

In the light front formalism, the condition
${\bf{k}}=(k^{+},{\bf{k}}_{\perp})\rightarrow\bf{0}$ could either lead to a
‘spurious’ or a ‘true’ IR divergence. The dispersion relation in light front
coordinates is given by $k^{-}=\frac{{\bf{k}}_{\perp}^{2}}{2k^{+}}$ for
massless particles. Due to this form of the mass-shell condition,
$k^{+}\rightarrow 0$, ${\bf{k}}_{\perp}\rightarrow 0$ does not necessarily
lead to soft divergences. If $k^{+}\rightarrow 0$ faster than
${\bf{k}}_{\perp}^{2}$, then $k^{-}\rightarrow\infty$. This type of divergence
is a spurious IR divergence. It is a manifestation of UV divergence of the
equal-time theory and hence will not be considered in this paper. The true
soft divergence of light front theory is when all the components of $k^{\mu}$
(including $k^{-}$) approach $0$. In the time-ordered Hamiltonian perturbation
theory, which is applied in Sec.IV, the IR divergences occur when the energy
transferred at a vertex is zero, and thus in turn are manifested in the form
of vanishing energy denominators in the perturbation series.

###  III.1 Coherent state formalism - an introduction

Conventionally, Fock representation is used for describing external states in
a scattering process. Chung had suggested that this was an inappropriate
choice because of which IR divergences appear in QED, and an alternate
representation was proposed in which the transition matrix elements were shown
to be free of IR divergences to all orders Chung (1965). In this
representation, an external electron is superposed with an infinite number of
soft photons. Kulish and Faddeev observed that the choice of asymptotic states
is associated with the form of asymptotic Hamiltonian used Kulish and Faddeev
(1970). We give below a brief description of the method of asymptotic
dynamics, which is used to construct the asymptotic states. The states thus
obtained, when employed in the calculation of S-matrix elements, lead to IR-
finite transition amplitudes.

The time-evolution operator in light front field theory is given by

$U(x^{+},x^{+}_{0})=T^{+}\,\text{exp}\bigg{(}-i\int_{x^{+}_{0}}^{x^{+}}dx^{\prime+}\,H_{\text{int}}(x^{\prime+})\bigg{)},$
(26)

where $H_{\text{int}}$ is the interaction part of the Hamiltonian and $T^{+}$
stands for the light front time-ordered product. The Møller operators are
defined using the time-evolution operator as

$\begin{split}\Omega_{\pm}\equiv\ &U(0,\mp\infty)\\\ =\
&T^{+}\,\text{exp}\bigg{(}-i\int_{\mp\infty}^{0}dx^{\prime+}\,H_{\text{int}}(x^{\prime+})\bigg{)},\end{split}$
(27)

and the S-matrix element is given by

$\begin{split}\matrixelement{f}{S}{i}=&\matrixelement{f}{\Omega^{\dagger}_{-}\Omega_{+}}{i}\\\
=&\matrixelement{f}{T^{+}\,\text{exp}\bigg{(}-i\int_{-\infty}^{\infty}dx^{\prime+}\,H_{\text{int}}(x^{\prime+})\bigg{)}}{i}.\end{split}$
(28)

The evolution of asymptotic states is conventionally presumed to be governed
by the free Hamiltonian $H_{0}$ as

$\ket{s(t)}=e^{-ix^{+}H_{0}}\ket{s(0)}.$ (29)

However, this assumption is incorrect for theories involving long-range
interactions and for theories where asymptotic states are bound states. To
deal with the presence of nonvanishing interactions in the large-time limit,
this assumption is modified in the coherent state formalism. The evolution of
asymptotic states is now considered to be governed by

$\ket{s(t)}=e^{-ix^{+}H_{A(\Delta)}}\ket{s(0)},$ (30)

where $H_{A(\Delta)}$ is the asymptotic Hamiltonian which is obtained by
taking the $|x^{+}|\rightarrow\infty$ limit of the full interaction
Hamiltonian, $H_{\text{int}}$. Thus, $H_{A(\Delta)}$ incorporates the long-
range interactions which are responsible for IR divergences. These long-range
interactions describe emission/absorption of soft gauge bosons and the
splitting of a massless particle into collinear massless particles. The
parameter $\Delta$ appearing in $H_{A(\Delta)}$ is a cutoff for the separation
of long-range interactions from $H_{\text{int}}$. The asymptotic Møller
operators are defined analogously,

$\begin{split}\Omega_{A(\Delta)\pm}\equiv\ &U_{A(\Delta)}(0,\mp\infty)\\\ =\
&T^{+}\,\text{exp}\bigg{(}-i\int_{\mp\infty}^{0}dx^{\prime+}\,H_{\Delta}(x^{\prime+})\bigg{)}.\end{split}$
(31)

Now onwards, $H_{A(\Delta)}$ will be replaced by $H_{\Delta}$ for brevity.

A coherent state can now be defined as the state formed by operating the
corresponding asymptotic Møller operator on a Fock state. Thus, initial
(incoming) and final (outgoing) coherent states are given by

$\begin{split}&\ket{s:\text{coh}}\equiv\Omega_{A(\Delta)+}^{\dagger}\ket{s}\\\
&\bra{s:\text{coh}}\equiv\bra{s}\Omega_{A(\Delta)-},\end{split}$ (32)

respectively.

###  III.2 Construction of $\ket{q_{p_{3}}\bar{q}_{p_{4}}:\text{{coh}}}$

The transition amplitude for the process $e^{+}e^{-}\rightarrow 2\;jets$, in
Fock basis, is given by

$T_{fi}=\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{T}{e^{-}_{p_{1}}e^{+}_{p_{2}}}$
(33)

and can be easily shown to contain IR divergences. Here, the subscript in
$p_{i}$ denotes, in addition to the momentum of an external fermion, the other
quantum numbers like color and spin also. In Sec.IV, we show that if a
coherent state basis is used instead of the Fock basis, the IR divergences are
eliminated and the modified expression for the amplitude, given by

$T_{fi}=\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}{T}{e^{-}_{p_{1}}e^{+}_{p_{2}}},$
(34)

is IR finite. Note that since we restrict ourselves to IR divergences in QCD,
the initial state is taken to be a Fock state i.e., the lowest-order term in
the series expansion (in electromagnetic coupling) of the coherent state
$\ket{e^{-}_{p_{1}}e^{+}_{p_{2}}:\text{coh}}$. The outgoing coherent state
$\ket{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}$, to be used in Sec.IV for
calculating Eq.(34), is constructed to $\order{g^{2}}$, which is the required
order in the following discussion. Using Eq.(31) and Eq.(32), the outgoing
coherent state $\ket{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}$ is given by

$\begin{split}\ket{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}=&\Omega^{\dagger}_{A(\Delta)-}\ket{q_{p3}\bar{q}_{p4}}\\\
=&T^{+}\,\text{exp}\bigg{(}i\int_{\infty}^{0}dx^{\prime+}\,H_{\Delta}(x^{\prime+})\bigg{)}\ket{q_{p_{3}}\bar{q}_{p_{4}}}\\\
=&T^{+}\,\text{exp}\bigg{(}-i\int_{0}^{\infty}dx^{\prime+}\,H_{\Delta}(x^{\prime+})\bigg{)}\ket{q_{p_{3}}\bar{q}_{p_{4}}}.\end{split}$
(35)

The $|x^{+}|\rightarrow\infty$ limit of the interaction Hamiltonian
$H_{\text{int}}$ in Eq.(14) leads to the asymptotic Hamiltonian
$H_{\Delta}(x^{\prime+})$ in Eq.(35). The interaction terms $V_{1}$ and
$W_{1}$, given by Eqs.(15) and (18) respectively, are the only ones that
contribute to $\ket{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}$ in the calculation
of
$\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}{T}{e^{-}_{p1}e^{+}_{p2}}$
to $\mathcal{O}(g^{2})$. Hence, the coherent state up to $\mathcal{O}(g^{2})$,
considering only the terms that contribute to the process under consideration,
is

$\begin{split}\ket{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}=&\bigg{(}1+(-i)\int_{0}^{\infty}dx_{1}^{+}\,V_{1}(x_{1}^{+})\Theta_{\Delta}+(-i)\int_{0}^{\infty}dx_{1}^{+}\,W_{1}(x_{1}^{+})\Theta_{\Delta}\bigg{)}\ket{q_{p_{3}}\bar{q}_{p_{4}}}\\\
\equiv&\ket{q_{p_{3}}\bar{q}_{p_{4}}}+\Omega^{\dagger}_{\Delta(V_{1})}\ket{q_{p_{3}}\bar{q}_{p_{4}}}+\Omega^{\dagger}_{\Delta(W_{1})}\ket{q_{p_{3}}\bar{q}_{p_{4}}},\end{split}$
(36)

where $\Theta_{\Delta}$ is a product of Heaviside step functions and ensures
that only those terms in the potential are included that survive in the limit
$|x^{+}|\rightarrow\infty$, as explained below.
Using Eq.(15) and Eq.(22), the second term of Eq.(36) becomes

$\begin{split}&\Omega^{\dagger}_{\Delta(V_{1})}\ket{q_{p_{3}}\bar{q}_{p_{4}}}\\\
=&(-i)\int_{0}^{\infty}dx_{1}^{+}\int
d^{2}{\bf{x}}_{1\perp}dx^{-}_{1}g\bar{\Psi}\gamma^{\mu}T^{a}\Psi
A^{a}_{\mu}\Theta_{\Delta}\ket{q_{p_{3}}\bar{q}_{p_{4}}}\\\
=&(-i)g\int_{0}^{\infty}dx_{1}^{+}\int
d^{2}{\bf{x}}_{1\perp}dx^{-}_{1}\int\prod_{i}[dk_{i}]\Theta_{\Delta}(b^{\dagger}_{k_{1}}\bar{u}_{k_{1}}e^{ik_{1}{\cdot}x_{1}}+d_{k_{1}}\bar{v}_{k_{1}}e^{-ik_{1}{\cdot}x_{1}})\\\
&\gamma^{\mu}T^{a}(b_{k_{2}}u_{k_{2}}e^{-ik_{2}{\cdot}x_{1}}+d^{\dagger}_{k_{2}}v_{k_{2}}e^{ik_{2}{\cdot}x_{1}})(a_{k_{3}}\epsilon^{a}_{\mu
k_{3}}e^{-ik_{3}{\cdot}x_{1}}+a^{\dagger}_{k_{3}}\epsilon^{*a}_{\mu
k_{3}}e^{ik_{3}{\cdot}x_{1}})\ket{q_{p_{3}}\bar{q}_{p_{4}}},\end{split}$ (37)

where the symbol $\prod$ stands for product over momenta $k_{i}$ and summation
over the spin and color quantum numbers. The presence of the oscillating
factors $\int dx^{+}e^{i\sum_{i}k^{-}_{i}x^{+}}$ in Eq.(37) implies that only
those terms survive in the $|x^{+}|\rightarrow\infty$ limit, for which the
condition $\sum_{i}k^{-}_{i}\rightarrow 0$ is satisfied. Thus, in the limit
$|x^{+}|\rightarrow\infty$,

$\begin{split}&\Omega^{\dagger}_{\Delta(V_{1})}\ket{q_{p_{3}}\bar{q}_{p_{4}}}\\\
=&(-i)g\int_{0}^{\infty}dx_{1}^{+}\int
d^{2}{\bf{x}}_{1\perp}dx^{-}_{1}\int\prod_{i}[dk_{i}]\Theta_{\Delta}(\bar{u}_{k_{1}}\gamma^{\mu}T^{a}u_{k_{2}}\epsilon^{*a}_{\mu
k_{3}}b^{\dagger}_{k_{1}}b_{k_{2}}a^{\dagger}_{k_{3}}\\\
&e^{i(k_{1}-k_{2}+k_{3}){\cdot}x_{1}}+\bar{v}_{k_{1}}\gamma^{\mu}T^{a}v_{k_{2}}\epsilon^{*a}_{\mu
k_{3}}d_{k_{1}}d^{\dagger}_{k_{2}}a^{\dagger}_{k_{3}}e^{-i(k_{1}-k_{2}-k_{3}){\cdot}x_{1}})b^{\dagger}_{p_{3}}d^{\dagger}_{p_{4}}\ket{0}.\end{split}$
(38)

The rest of the terms in Eq.(37) are either zero (viz.
$...a_{k_{3}}\ket{q_{p_{3}}\bar{q}_{p_{4}}}=0$) or eliminated by
$\Theta_{\Delta}$ function because $\sum_{i}k^{-}_{i}$ can never approach zero
for those terms. Performing space and time integrations and using

$\begin{split}&b^{\dagger}_{k_{1}}b_{k_{2}}a^{\dagger}_{k_{3}}b^{\dagger}_{p_{3}}d^{\dagger}_{p_{4}}\ket{0}=\delta({k_{2}}-p_{3})a^{\dagger}_{k_{3}}b^{\dagger}_{k_{1}}d^{\dagger}_{p_{4}}\ket{0}\\\
&d_{k_{1}}d^{\dagger}_{k_{2}}a^{\dagger}_{k_{3}}b^{\dagger}_{p_{3}}d^{\dagger}_{p_{4}}\ket{0}=-\delta({k_{1}}-p_{4})a^{\dagger}_{k_{3}}b^{\dagger}_{p_{3}}d^{\dagger}_{k_{2}}\ket{0},\end{split}$
(39)

where
$\delta(k-p)\equiv\delta^{2}({\bf{k}}_{\perp}-{\bf{p}}_{\perp})\delta(k^{+}-p^{+})\delta_{spins}\delta_{colors}\delta_{flavors}$,
to integrate over $k_{1}$ and $k_{2}$, one obtains

$\begin{split}\Omega^{\dagger}_{\Delta(V_{1})}\ket{q_{p_{3}}\bar{q}_{p_{4}}}=-g\int[dk]\Theta_{\Delta}&\Bigg{[}\frac{\bar{u}_{(p_{3}-k)}\not{\epsilon}^{*a}_{k}T^{a}u_{p_{3}}a^{\dagger}_{k}b^{\dagger}_{(p_{3}-k)}d^{\dagger}_{p_{4}}\ket{0}}{\sqrt{2p^{+}_{3}}\sqrt{2(p^{+}_{3}-k^{+})}(p^{-}_{3}-k^{-}-(p_{3}-k)^{-})}\\\
&+\frac{\bar{v}_{p_{4}}\not{\epsilon}^{*a}_{k}T^{a}v_{(p_{4}-k)}a^{\dagger}_{k}b^{\dagger}_{p_{3}}d^{\dagger}_{(p_{4}-k)}\ket{0}}{\sqrt{2p^{+}_{4}}\sqrt{2(p^{+}_{4}-k^{+})}(p^{-}_{4}-k^{-}-(p_{4}-k)^{-})}\Bigg{]}.\end{split}$
(40)

A similar procedure leads to

$\begin{split}&\Omega^{\dagger}_{\Delta(W_{1})}\ket{q_{p_{3}}\bar{q}_{p_{4}}}\\\
=&g^{2}\int[dk]\Theta_{\Delta}\Bigg{[}\frac{1}{(2\pi)^{3/2}\sqrt{2p^{+}_{3}}\sqrt{2p^{+}_{4}}\sqrt{2(p^{+}_{3}+p^{+}_{4}-k^{+})}}\\\
&\frac{\bar{u}_{k}\gamma^{+}T^{a}u_{p_{3}}\bar{v}_{p_{4}}\gamma^{+}T^{a}v_{(p_{3}+p_{4}-k)}b^{\dagger}_{k}d^{\dagger}_{(p_{3}+p_{4}-k)}\ket{0}}{(p^{+}_{3}-k^{+})^{2}(p^{-}_{3}+p^{-}_{4}-k^{-}-(p_{3}+p_{4}-k)^{-})}\Bigg{]}\end{split}$
(41)

for the third term in Eq.(36). Note that
$\Omega^{\dagger}_{\Delta(V_{1})}\ket{q_{p_{3}}\bar{q}_{p_{4}}}$ is an
$\mathcal{O}(g)$ and
$\Omega^{\dagger}_{\Delta(W_{1})}\ket{q_{p_{3}}\bar{q}_{p_{4}}}$ is an
$\mathcal{O}(g^{2})$ contribution to the outgoing coherent state.

###  III.3 The asymptotic region

In Sec.III.2, the function $\Theta_{\Delta}$ is introduced to ensure that only
those terms appear in the definition of coherent state for which the condition
$\sum_{i}k^{-}_{i}<\Delta$ is satisfied. This leads to the vanishing energy
denominators in Eqs.(40) and (41). Thus, $\Theta_{\Delta}$ restricts the
region of phase space to only those values where the energy transferred at a
vertex tends to zero in the coherent state. In this section, we illustrate
this idea of the region of phase space to be included in asymptotic states
through the example of a $qqg$ vertex. Consider the quark-quark-gluon vertex
shown in Fig.3.

Figure 3: The qqg vertex

Without loss of generality, the momentum of incoming quark can be written as

$p^{\mu}=\bigg{(}p^{+},\frac{p^{2}_{x}}{2p^{+}},p_{x},0\bigg{)}.$ (42)

The momentum of outgoing gluon is parametrized as

$k^{\mu}=\bigg{(}\alpha p^{+},\frac{(\alpha+\beta)^{2}p^{2}_{x}}{2\alpha
p^{+}},(\alpha+\beta)p_{x},0\bigg{)},$ (43)

where $0\leq\alpha\leq 1$ and $\beta$ can be either positive or negative.
Applying 3-momentum conservation, the momentum of outgoing quark becomes

$(p-k)^{\mu}=\bigg{(}(1-\alpha)p^{+},\frac{(1-\alpha-\beta)^{2}p^{2}_{x}}{2(1-\alpha)p^{+}},(1-\alpha-\beta)p_{x},0\bigg{)}.$
(44)

In the Hamiltonian formulation, IR divergences occur when the light front
energy transferred at the vertex tends to zero. Hence, for IR divergences to
occur in this example, the criterion is $|p^{-}-k^{-}-(p-k)^{-}|\rightarrow 0$
which can be implemented through the Heaviside step function

$\Theta_{\Delta}=\Theta(\Delta-|p^{-}-k^{-}-(p-k)^{-}|).$ (45)

In terms of the parameters $\alpha$ and $\beta$,

$\begin{split}\Theta_{\Delta}(\alpha,\beta)=&\Theta\bigg{(}\Delta-\frac{p^{2}_{x}}{2p^{+}}\bigg{|}1-\frac{(\alpha+\beta)^{2}}{\alpha}-\frac{(1-\alpha-\beta)^{2}}{(1-\alpha)}\bigg{|}\bigg{)}\\\
=&\Theta\bigg{(}\Delta-\frac{p^{2}_{x}}{2p^{+}}\frac{\beta^{2}}{\alpha(1-\alpha)}\bigg{)}.\end{split}$
(46)

Thus, infrared divergences arise due to following possibilities:

Case 1: Purely (hard) collinear gluon
$\beta\rightarrow 0,\ \alpha\neq 0$
$\implies\Theta_{\Delta}=\theta(\alpha-\Delta)\theta(\Delta-\beta)$.

Case 2: Soft + Soft-collinear gluon
$\beta\rightarrow 0,\ \alpha\rightarrow 0$ such that $k^{-}\rightarrow 0$
$\implies\Theta_{\Delta}=\theta(\Delta-\alpha)[\theta(\sqrt{\alpha\Delta}-\beta)-\theta(-\sqrt{\alpha\Delta}-\beta)]$.
In addition to these two cases of ‘true’ IR divergences, there also exists
spurious IR divergence in light front theories which occurs when the
parameters $\beta\rightarrow 0,\ \alpha\rightarrow 0$, but the gluon energy
component $k^{-}\rightarrow\infty$. The spurious IR divergence is a
manifestation of ultraviolet divergence of the equal-time theory.

The constraint $\Theta_{\Delta}=\Theta(\Delta-|p^{-}-k^{-}-(p-k)^{-}|)$
includes both the soft and collinear regions and hence in this formalism, both
soft and collinear divergences can be shown to cancel at the amplitude level
in inclusive cross sections like in the process under consideration.

##  IV Cancellation of IR divergences to $\mathcal{O}(g^{2})$

The transition amplitude for the process $e^{+}e^{-}\rightarrow
2\;\text{jets}$ in coherent state basis is given by

$T_{fi}=\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}{T}{e^{-}_{p_{1}}e^{+}_{p_{2}}}$
(47)

Employing the expansion for the outgoing coherent state
$\ket{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}$ upto $\mathcal{O}(g^{2})$ given by
Eq.(36)

$\ket{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}=\ket{q_{p_{3}}\bar{q}_{p_{4}}}+\Omega^{\dagger}_{\Delta(V_{1})}\ket{q_{p_{3}}\bar{q}_{p_{4}}}+\Omega^{\dagger}_{\Delta(W_{1})}\ket{q_{p_{3}}\bar{q}_{p_{4}}},$

the transition amplitude to $\mathcal{O}(g^{2})$ becomes

$\begin{split}T_{fi}=&\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}{T}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
=&\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{T}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
&+\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{\Omega_{\Delta(V_{1})}T}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
&+\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{\Omega_{\Delta(W_{1})}T}{e^{-}_{p_{1}}e^{+}_{p_{2}}}.\end{split}$
(48)

The initial state $\ket{e^{+}e^{-}}$ is considered as a Fock state, and not a
coherent state, since only QCD IR divergences are being dealt with here. Thus,
the time-ordered perturbation series for the transition amplitude is

$\begin{split}T_{fi}=&\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}{T}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
=&\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{V+V\frac{1}{p^{-}_{i}-H_{0}}V+V\frac{1}{p^{-}_{i}-H_{0}}V\frac{1}{p^{-}_{i}-H_{0}}V+...}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
&+\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{\Omega_{\Delta(V_{1})}V+V\frac{1}{p^{-}_{i}-H_{0}}V+V\frac{1}{p^{-}_{i}-H_{0}}V\frac{1}{p^{-}_{i}-H_{0}}V+...}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
&+\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{\Omega_{\Delta(W_{1})}V+V\frac{1}{p^{-}_{i}-H_{0}}V+V\frac{1}{p^{-}_{i}-H_{0}}V\frac{1}{p^{-}_{i}-H_{0}}V+...}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
&+...,\end{split}$ (49)

where $V$ denotes a generic interaction potential, $p^{-}_{i}$ is the total
energy of the initial state and $H_{0}$ is the free light front Hamiltonian.
Note that at $\mathcal{O}(g^{0})$, Eq.(49) leads to the tree-level QED diagram
with an outgoing $\ket{q\bar{q}}$ Fock state and hence contains no QCD IR
divergences. Terms of odd order in $g$ are absent in Eq.(49). Also, the
initial state $\ket{e^{-}e^{+}}$ connects to a $\ket{q\bar{q}}$ state only
through a photon. Hence, the lowest order at which QCD IR divergences occur is
$\mathcal{O}(g^{2})$, in which Eq.(49) reduces to

$\begin{split}T_{fi}=&\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{V_{1}\frac{1}{p^{-}_{i}-H_{0}}V_{1}\frac{1}{p^{-}_{i}-H_{0}}V_{em}\frac{1}{p^{-}_{i}-H_{0}}V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
&+\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{W_{1}\frac{1}{p^{-}_{i}-H_{0}}V_{em}\frac{1}{p^{-}_{i}-H_{0}}V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
&+\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{\Omega_{\Delta(V_{1})}V_{1}\frac{1}{p^{-}_{i}-H_{0}}V_{em}\frac{1}{p^{-}_{i}-H_{0}}V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
&+\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{\Omega_{\Delta(W_{1})}V_{em}\frac{1}{p^{-}_{i}-H_{0}}V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}},\end{split}$
(50)

where $V_{em}$ is the electromagnetic interaction. The first two terms in
Eq.(50) would be the contributions obtained if one used the Fock basis instead
of coherent state basis. The additional two terms are present because coherent
state basis is used to calculate the transition amplitude. In the rest of this
section, both these contributions are calculated. It is shown that each
contribution contains IR-divergent expressions and that the IR divergences
cancel between them.

###  IV.1 $\mathcal{O}(g^{0})$ coherent state contribution to
$\mathcal{O}(g^{2})$ transition amplitude

The first two terms of Eq.(50) represent the contribution of
$\mathcal{O}(g^{0})$ term of $\ket{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}$ to
$T_{fi}$ ($\mathcal{O}(g^{2})$) in the coherent state basis. These terms are
also the ones that would give the $\mathcal{O}(g^{2})$ transition amplitude in
the Fock basis. The expressions for these, which are obtained by inserting
complete set of states between operators and evaluating the resulting inner
products, are given below. To illustrate the technique, one of the terms of
Eq.(50) is explicitly calculated in Appendix A. The diagrams corresponding to
the Fock state contribution are shown in Fig.4 and Fig.5. We define

$\begin{split}T_{1}\equiv&\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{V_{1}\frac{1}{p^{-}_{i}-H_{0}}V_{1}\frac{1}{p^{-}_{i}-H_{0}}V_{em}\frac{1}{p^{-}_{i}-H_{0}}V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
=&\ T_{1a}+T_{1b}+T_{1c}+T_{1d},\end{split}$ (51)

where

$\begin{split}T_{1a}=&\frac{e^{2}g^{2}}{(2\pi)^{6}}\int\frac{d^{2}{\bf{k}}_{\perp}dk^{+}}{\prod_{i}\sqrt{2p^{+}_{i}}}\Bigg{[}\frac{\bar{u}_{p_{3}}\not{\epsilon}^{a}_{k}u_{(p_{3}-k)}\bar{u}_{(p_{3}-k)}\not{\epsilon}^{*a}_{k}u_{p_{3}}\bar{u}_{p_{3}}}{2(p^{+}_{1}+p^{+}_{2})2k^{+}2p^{+}_{3}2(p^{+}_{3}-k^{+})}\\\
&\frac{\not{\epsilon}_{(p_{1}+p_{2})}v_{p_{4}}\bar{v}_{p_{2}}\not{\epsilon}^{*}_{(p_{1}+p_{2})}u_{p_{1}}C_{F}\delta_{c_{3}c_{4}}}{(p^{-}_{3}-k^{-}-(p_{3}-k)^{-})(p^{-}_{1}+p^{-}_{2}-p^{-}_{3}-p^{-}_{4})(p^{-}_{1}+p^{-}_{2}-(p_{1}+p_{2})^{-})}\Bigg{]}\end{split}$
(52)
$\begin{split}T_{1b}=&\frac{e^{2}g^{2}}{(2\pi)^{6}}\int\frac{d^{2}{\bf{k}}_{\perp}dk^{+}}{\prod_{i}\sqrt{2p^{+}_{i}}}\Bigg{[}\frac{\bar{u}_{p_{3}}\not{\epsilon}_{(p_{1}+p_{2})}v_{p_{4}}\bar{v}_{p_{4}}\not{\epsilon}^{*a}_{k}v_{(p_{4}-k)}\bar{v}_{(p_{4}-k)}\not{\epsilon}^{a}_{k}}{2(p^{+}_{1}+p^{+}_{2})2k^{+}2p^{+}_{4}2(p^{+}_{4}-k^{+})}\\\
&\frac{v_{p_{4}}\bar{v}_{p_{2}}\not{\epsilon}^{*}_{(p_{1}+p_{2})}u_{p_{1}}C_{F}\delta_{c_{3}c_{4}}}{(p^{-}_{4}-k^{-}-(p_{4}-k)^{-})(p^{-}_{1}+p^{-}_{2}-p^{-}_{3}-p^{-}_{4})(p^{-}_{1}+p^{-}_{2}-(p_{1}+p_{2})^{-})}\Bigg{]}\end{split}$
(53)
$\begin{split}T_{1c}=&-\frac{e^{2}g^{2}}{(2\pi)^{6}}\int\frac{d^{2}{\bf{k}}_{\perp}dk^{+}}{\prod_{i}\sqrt{2p^{+}_{i}}}\Bigg{[}\frac{\bar{u}_{p_{3}}\not{\epsilon}^{*a}_{k}u_{(p_{3}+k)}\bar{u}_{(p_{3}+k)}\not{\epsilon}_{(p_{1}+p_{2})}v_{(p_{4}-k)}\bar{v}_{(p_{4}-k)}}{2(p^{+}_{1}+p^{+}_{2})2k^{+}2(p^{+}_{3}+k^{+})2(p^{+}_{4}-k^{+})}\\\
&\frac{\not{\epsilon}^{a}_{k}v_{p_{4}}\bar{v}_{p_{2}}\not{\epsilon}^{*}_{(p_{1}+p_{2})}u_{p_{1}}C_{F}\delta_{c_{3}c_{4}}}{(p^{-}_{4}-k^{-}-(p_{4}-k)^{-})(p^{-}_{1}+p^{-}_{2}-(p_{3}+k)^{-}-(p_{4}-k)^{-})(p^{-}_{1}+p^{-}_{2}-(p_{1}+p_{2})^{-})}\Bigg{]}\end{split}$
(54)
$\begin{split}T_{1d}=&-\frac{e^{2}g^{2}}{(2\pi)^{6}}\int\frac{d^{2}{\bf{k}}_{\perp}dk^{+}}{\prod_{i}\sqrt{2p^{+}_{i}}}\Bigg{[}\frac{\bar{u}_{p_{3}}\not{\epsilon}^{a}_{k}u_{(p_{3}-k)}\bar{u}_{(p_{3}-k)}\not{\epsilon}_{(p_{1}+p_{2})}v_{(p_{4}+k)}\bar{v}_{(p_{4}+k)}}{2(p^{+}_{1}+p^{+}_{2})2k^{+}2(p^{+}_{3}-k^{+})2(p^{+}_{4}+k^{+})}\\\
&\frac{\not{\epsilon}^{*a}_{k}v_{p_{4}}\bar{v}_{p_{2}}\not{\epsilon}^{*}_{(p_{1}+p_{2})}u_{p_{1}}C_{F}\delta_{c_{3}c_{4}}}{(p^{-}_{3}-k^{-}-(p_{3}-k)^{-})(p^{-}_{1}+p^{-}_{2}-(p_{3}-k)^{-}-(p_{4}+k)^{-})(p^{-}_{1}+p^{-}_{2}-(p_{1}+p_{2})^{-})}\Bigg{]}.\end{split}$
(55)

$T_{1a},T_{1b},T_{1c}$ and $T_{1d}$ correspond to the diagrams in
Figs.4(a)-4(d), respectively. The expressions for $T_{1a}$ through $T_{1d}$
i.e., Eqs.(52)-(55) are IR-divergent in the following limits:
$T_{1a}$ : $(p^{-}_{3}-k^{-}-(p_{3}-k)^{-})\rightarrow 0$
$T_{1b}$ : $(p^{-}_{4}-k^{-}-(p_{4}-k)^{-})\rightarrow 0$
$T_{1c}$ : $(p^{-}_{4}-k^{-}-(p_{4}-k)^{-})\rightarrow 0$;
$(p^{-}_{1}+p^{-}_{2}-(p_{3}+k)^{-}-(p_{4}-k)^{-})\rightarrow 0$
$T_{1d}$ : $(p^{-}_{3}-k^{-}-(p_{3}-k)^{-})\rightarrow 0$;
$(p^{-}_{1}+p^{-}_{2}-(p_{4}+k)^{-}-(p_{3}-k)^{-})\rightarrow 0$.

(a) $T_{1a}$ (b) $T_{1b}$

(c) $T_{1c}$ (d) $T_{1d}$

Figure 4: Regular diagrams corresponding to outgoing Fock state

Next, we define

$\begin{split}T_{2}\equiv&\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{W_{1}\frac{1}{p^{-}_{i}-H_{0}}V_{em}\frac{1}{p^{-}_{i}-H_{0}}V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
=&-\frac{e^{2}g^{2}}{(2\pi)^{6}}\int\frac{d^{2}{\bf{k}}_{\perp}dk^{+}}{\prod_{i}\sqrt{2p^{+}_{i}}}\Bigg{[}\frac{\bar{u}_{p_{3}}\gamma^{+}u_{k}\bar{u}_{k}\not{\epsilon}_{(p_{1}+p_{2})}v_{(p_{3}+p_{4}-k)}\bar{v}_{(p_{3}+p_{4}-k)}}{2(p^{+}_{1}+p^{+}_{2})2k^{+}2(p^{+}_{3}+p^{+}_{4}-k^{+})(p^{+}_{3}-k^{+})^{2}}\\\
&\frac{\gamma^{+}v_{p_{4}}\bar{v}_{p_{2}}\not{\epsilon}^{*}_{(p_{1}+p_{2})}u_{p_{1}}C_{F}\delta_{c_{3}c_{4}}}{(p^{-}_{3}+p^{-}_{4}-k^{-}-(p_{3}+p_{4}-k)^{-})(p^{-}_{1}+p^{-}_{2}-(p_{1}+p_{2})^{-})}\Bigg{]}\end{split}$
(56)

which is the amplitude of the diagram in Fig.5. $T_{2}$ given in Eq.(56) is
IR-divergent when $(p^{-}_{3}+p^{-}_{4}-k^{-}-(p_{3}+p_{4}-k)^{-})\rightarrow
0$. Note that $T_{1}$ and $T_{2}$ calculated in this section, and hence Figs.4
and 5, would have been the only contributions to the transition amplitude if
Fock basis was used, thus leading to an IR-divergent transition amplitude in
the Fock basis. Additional contributions arising due to the use of coherent
state basis are calculated below.

Figure 5: Instantaneous diagram corresponding to outgoing Fock state

###  IV.2 $\mathcal{O}(g)$ and $\mathcal{O}(g^{2})$ coherent state
contributions to $\mathcal{O}(g^{2})$ transition amplitude

In coherent state basis, in addition to the contribution calculated in
Sec.IV.1, there are additional contributions to the transition amplitude
arising from $\mathcal{O}(g)$ and $\mathcal{O}(g^{2})$ terms in the expansion
of coherent state. The $\mathcal{O}(g)$ term of
$\ket{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}$ is given by Eq.(40). Inserting
this in the third term of Eq.(50) leads to the following contribution to
$T_{fi}$:

$\begin{split}T_{3}\equiv&\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{\Omega_{\Delta(V_{1})}V_{1}\frac{1}{p^{-}_{i}-H_{0}}V_{em}\frac{1}{p^{-}_{i}-H_{0}}V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
=&\ T_{3a}+T_{3b}+T_{3c}+T_{3d},\end{split}$ (57)

where

$\begin{split}T_{3a}=&-\frac{e^{2}g^{2}}{(2\pi)^{6}}\int\frac{d^{2}{\bf{k}}_{\perp}dk^{+}\Theta_{\Delta}}{\prod_{i}\sqrt{2p^{+}_{i}}}\Bigg{[}\frac{\bar{u}_{p_{3}}\not{\epsilon}^{a}_{k}u_{(p_{3}-k)}\bar{u}_{(p_{3}-k)}\not{\epsilon}^{*a}_{k}u_{p_{3}}\bar{u}_{p_{3}}}{2(p^{+}_{1}+p^{+}_{2})2k^{+}2p^{+}_{3}2(p^{+}_{3}-k^{+})}\\\
&\frac{\not{\epsilon}_{(p_{1}+p_{2})}v_{p_{4}}\bar{v}_{p_{2}}\not{\epsilon}^{*}_{(p_{1}+p_{2})}u_{p_{1}}C_{F}\delta_{c_{3}c_{4}}}{(p^{-}_{3}-k^{-}-(p_{3}-k)^{-})(p^{-}_{1}+p^{-}_{2}-p^{-}_{3}-p^{-}_{4})(p^{-}_{1}+p^{-}_{2}-(p_{1}+p_{2})^{-})}\Bigg{]}\end{split}$
(58)
$\begin{split}T_{3b}=&-\frac{e^{2}g^{2}}{(2\pi)^{6}}\int\frac{d^{2}{\bf{k}}_{\perp}dk^{+}\Theta_{\Delta}}{\prod_{i}\sqrt{2p^{+}_{i}}}\Bigg{[}\frac{\bar{u}_{p_{3}}\not{\epsilon}_{(p_{1}+p_{2})}v_{p_{4}}\bar{v}_{p_{4}}\not{\epsilon}^{*a}_{k}v_{(p_{4}-k)}\bar{v}_{(p_{4}-k)}\not{\epsilon}^{a}_{k}}{2(p^{+}_{1}+p^{+}_{2})2k^{+}2p^{+}_{4}2(p^{+}_{4}-k^{+})}\\\
&\frac{v_{p_{4}}\bar{v}_{p_{2}}\not{\epsilon}^{*}_{(p_{1}+p_{2})}u_{p_{1}}C_{F}\delta_{c_{3}c_{4}}}{(p^{-}_{4}-k^{-}-(p_{4}-k)^{-})(p^{-}_{1}+p^{-}_{2}-p^{-}_{3}-p^{-}_{4})(p^{-}_{1}+p^{-}_{2}-(p_{1}+p_{2})^{-})}\Bigg{]}\end{split}$
(59)
$\begin{split}T_{3c}=&\frac{e^{2}g^{2}}{(2\pi)^{6}}\int\frac{d^{2}{\bf{k}}_{\perp}dk^{+}\Theta_{\Delta}}{\prod_{i}\sqrt{2p^{+}_{i}}}\Bigg{[}\frac{\bar{u}_{p_{3}}\not{\epsilon}^{*a}_{k}u_{(p_{3}+k)}\bar{u}_{(p_{3}+k)}\not{\epsilon}_{(p_{1}+p_{2})}v_{(p_{4}-k)}\bar{v}_{(p_{4}-k)}}{2(p^{+}_{1}+p^{+}_{2})2k^{+}2(p^{+}_{3}+k^{+})2(p^{+}_{4}-k^{+})}\\\
&\frac{\not{\epsilon}^{a}_{k}v_{p_{4}}\bar{v}_{p_{2}}\not{\epsilon}^{*}_{(p_{1}+p_{2})}u_{p_{1}}C_{F}\delta_{c_{3}c_{4}}}{(p^{-}_{4}-k^{-}-(p_{4}-k)^{-})(p^{-}_{1}+p^{-}_{2}-(p_{3}+k)^{-}-(p_{4}-k)^{-})(p^{-}_{1}+p^{-}_{2}-(p_{1}+p_{2})^{-})}\Bigg{]}\end{split}$
(60)
$\begin{split}T_{3d}=&\frac{e^{2}g^{2}}{(2\pi)^{6}}\int\frac{d^{2}{\bf{k}}_{\perp}dk^{+}\Theta_{\Delta}}{\prod_{i}\sqrt{2p^{+}_{i}}}\Bigg{[}\frac{\bar{u}_{p_{3}}\not{\epsilon}^{a}_{k}u_{(p_{3}-k)}\bar{u}_{(p_{3}-k)}\not{\epsilon}_{(p_{1}+p_{2})}v_{(p_{4}+k)}\bar{v}_{(p_{4}+k)}}{2(p^{+}_{1}+p^{+}_{2})2k^{+}2(p^{+}_{3}-k^{+})2(p^{+}_{4}+k^{+})}\\\
&\frac{\not{\epsilon}^{*a}_{k}v_{p_{4}}\bar{v}_{p_{2}}\not{\epsilon}^{*}_{(p_{1}+p_{2})}u_{p_{1}}C_{F}\delta_{c_{3}c_{4}}}{(p^{-}_{3}-k^{-}-(p_{3}-k)^{-})(p^{-}_{1}+p^{-}_{2}-(p_{3}-k)^{-}-(p_{4}+k)^{-})(p^{-}_{1}+p^{-}_{2}-(p_{1}+p_{2})^{-})}\Bigg{]}.\end{split}$
(61)

The $\Theta_{\Delta}$ function that appears in the expressions for $T_{3a}$,
$T_{3b}$, $T_{3c}$ and $T_{3d}$ given by Eqs.(58)-(61) restricts the region of
phase space and hence the integration region to only those values for which
the following conditions are satisfied.
$T_{3a}$ : $(p^{-}_{3}-k^{-}-(p_{3}-k)^{-})\rightarrow 0$
$T_{3b}$ : $(p^{-}_{4}-k^{-}-(p_{4}-k)^{-})\rightarrow 0$
$T_{3c}$ : $(p^{-}_{4}-k^{-}-(p_{4}-k)^{-})\rightarrow 0$;
$(p^{-}_{1}+p^{-}_{2}-(p_{3}+k)^{-}-(p_{4}-k)^{-}\rightarrow 0$
$T_{3d}$ : $(p^{-}_{3}-k^{-}-(p_{3}-k)^{-})\rightarrow 0$;
$(p^{-}_{1}+p^{-}_{2}-(p_{4}+k)^{-}-(p_{3}-k)^{-})\rightarrow 0$.
The diagrams corresponding to $T_{3}$ are represented schematically by Fig.6,
in which the dashed lines represent emission/absorption of a soft/collinear
gluon. These particles, lying in the restricted phase space through the
$\Theta_{\Delta}$ function, are indistinguishable from the final-state
particles.

The last term of Eq.(50) involves the $\mathcal{O}(g^{2})$ contribution to the
coherent state $\ket{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}$. The expression for
this term is

$\begin{split}T_{4}\equiv&\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{\Omega_{\Delta(W_{1})}V_{em}\frac{1}{p^{-}_{i}-H_{0}}V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}}\\\
=&\frac{e^{2}g^{2}}{(2\pi)^{6}}\int\frac{d^{2}{\bf{k}}_{\perp}dk^{+}\Theta_{\Delta}}{\prod_{i}\sqrt{2p^{+}_{i}}}\Bigg{[}\frac{\bar{u}_{p_{3}}\gamma^{+}u_{k}\bar{u}_{k}\not{\epsilon}_{(p_{1}+p_{2})}v_{(p_{3}+p_{4}-k)}\bar{v}_{(p_{3}+p_{4}-k)}}{2(p^{+}_{1}+p^{+}_{2})2k^{+}2(p^{+}_{3}+p^{+}_{4}-k^{+})(p^{+}_{3}-k^{+})^{2}}\\\
&\frac{\gamma^{+}v_{p_{4}}\bar{v}_{p_{2}}\not{\epsilon}^{*}_{(p_{1}+p_{2})}u_{p_{1}}C_{F}\delta_{c_{3}c_{4}}}{(p^{-}_{3}+p^{-}_{4}-k^{-}-(p_{3}+p_{4}-k)^{-})(p^{-}_{1}+p^{-}_{2}-(p_{1}+p_{2})^{-})}\Bigg{]}.\end{split}$
(62)

$\Theta_{\Delta}$ restricts the region of integration to only those values
where $(p^{-}_{3}+p^{-}_{4}-k^{-}-(p_{3}+p_{4}-k)^{-})\rightarrow 0$.

$T_{fi}$ is thus the sum total of Eqs.(52)-(56) and (58)-(62). The regions of
phase space in which $T_{3a}$ to $T_{3d}$ and $T_{4}$ are nonvanishing are the
same which lead to IR divergences in $T_{1a}$ to $T_{1d}$ and $T_{2}$
respectively. The expressions being equal and opposite, the IR divergences
cancel and the total amplitude
$T_{fi}=\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}{T}{e^{-}_{p_{1}}e^{+}_{p_{2}}}$
is IR-finite.

(a) $T_{3a}$ (b) $T_{3b}$

(c) $T_{3c}$ (d) $T_{3d}$

Figure 6: Regular diagrams corresponding to outgoing coherent state

##  V Summary and Outlook

We show that in LFQCD the total amplitude $T_{fi}$ for the process
$e^{+}e^{-}\rightarrow 2\;\text{jets}$ to $\mathcal{O}(g^{2})$ is free of IR
divergences. After reviewing the derivation of LFQCD Hamiltonian, we take a
brief look at IR divergences appearing in LFQCD. The coherent state formalism
is then described and the outgoing coherent state required for calculating the
$e^{+}e^{-}\rightarrow 2\;\text{jets}$ amplitude is constructed. Noting that
vanishing energy denominators in the amplitude in Fock basis lead to
soft/collinear divergences, we show that these divergences get eliminated when
higher order coherent state contributions are added to the amplitude. The
cancellation of IR divergences takes place through $\Theta_{\Delta}$ function,
which restricts the integration to only that region of phase space which leads
to IR divergences. Hence, to $\mathcal{O}(g^{2})$, we show that the total
amplitude
$T_{fi}=\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}:\text{coh}}{T}{e^{-}_{p_{1}}e^{+}_{p_{2}}}$
is free of soft and collinear divergences and thus IR-finite.

Parton showering algorithms that are currently used for studying high-energy
scattering phenomena employ the conventional technique of cancellation of IR
divergences at the cross-section level. Recently, construction of all-order
amplitude-level parton-shower algorithms is being attempted Ángeles Martínez
_et al._ (2016). The coherent state approach is expected to play an important
role in such algorithms by providing a way to deal with IR divergences at the
amplitude level itself. Keeping this aspect in view, the preliminary study
carried out in this work needs to be taken further. Our followup work will
deal with cancellation of IR divergences, in the coherent state formalism, in
multijet events which are now being regularly observed in the currently
operating colliders. Higher-order calculations will also be carried out in the
near future with the aim of developing an all-order proof of cancellation of
IR divergences in pQCD.

## Acknowledgments

D.B. would like to acknowledge the financial support provided by University
Grants Commission (UGC), India. A.M. would like to thank ICTP for their kind
hospitality and support under the Senior Associate program.

##  Appendix A Amplitude $T_{3a}$

The details of the calculation for the expression of amplitude $T_{3a}$ are
presented below. The amplitude $T_{3a}$ corresponds to the diagram of
Fig.6(a). $T_{3}$ is given by Eq.(57). Inserting complete sets of states
between operators and writing only those states of the complete set which give
a nonvanishing contribution, the amplitude becomes

$\begin{split}T_{3}=&\int
d^{3}kd^{3}k^{\prime}\prod_{i=1}^{2}d^{3}q_{i}\prod_{i=1}^{2}d^{3}q^{\prime}_{i}\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{\Omega_{\Delta(V_{1})}V_{1}}{q_{q_{1}}\bar{q}_{q_{2}}}\matrixelement{q_{q_{1}}\bar{q}_{q_{2}}}{\frac{1}{p^{-}_{1}+p^{-}_{2}-H_{0}}}{q_{q^{\prime}_{1}}\bar{q}_{q^{\prime}_{2}}}\\\
&\matrixelement{q_{q^{\prime}_{1}}\bar{q}_{q^{\prime}_{2}}}{V_{em}}{\gamma_{k}}\matrixelement{\gamma_{k}}{\frac{1}{p^{-}_{1}+p^{-}_{2}-H_{0}}}{\gamma_{k^{\prime}}}\matrixelement{\gamma_{k^{\prime}}}{V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}},\end{split}$
(63)

where $d^{3}p\equiv d^{2}{\bf{p}}_{\perp}dp^{+}$.
Using the identities

$\begin{split}&\innerproduct{\gamma_{k}}{\gamma_{k^{\prime}}}=\delta(k-k^{\prime})\\\
&\innerproduct{q_{q_{1}}\bar{q}_{q_{2}}}{q_{q^{\prime}_{1}}\bar{q}_{q^{\prime}_{2}}}=\delta(q_{1}-q^{\prime}_{1})\delta(q_{2}-q^{\prime}_{2})\end{split}$
(64)

to eliminate the momenta $k^{\prime}$, $q^{\prime}_{1}$ and $q^{\prime}_{2}$
(and other corresponding quantum numbers) from Eq.(63) reduces the amplitude
to

$T_{3}=\int
d^{3}kd^{3}q_{1}d^{3}q_{2}\frac{\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{\Omega_{\Delta(V_{1})}V_{1}}{q_{q_{1}}\bar{q}_{q_{2}}}\matrixelement{q_{q_{1}}\bar{q}_{q_{2}}}{V_{em}}{\gamma_{k}}\matrixelement{\gamma_{k}}{V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}}}{({p^{-}_{1}+p^{-}_{2}-q^{-}_{1}-q^{-}_{2}})({p^{-}_{1}+p^{-}_{2}-k^{-}})}.$
(65)

Consider the inner product
$\matrixelement{\gamma_{k}}{V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}}$. Substituting
for $V_{em}=e\int d^{2}{\bf{x}}_{\perp}dx^{-}\bar{\Psi}\gamma^{\mu}\Psi
A_{\mu}$, we get

$\begin{split}\matrixelement{\gamma_{k}}{V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}}=&e\int
d^{2}{\bf{x}}_{\perp}dx^{-}\int\prod_{i}[dk_{i}]\bra{0}a_{k}(b^{\dagger}_{k_{1}}\bar{u}_{k_{1}}e^{ik_{1}{\cdot}x}+d_{k_{1}}\bar{v}_{k_{1}}e^{-ik_{1}{\cdot}x})\\\
&\gamma^{\mu}(b_{k_{2}}u_{k_{2}}e^{-ik_{2}{\cdot}x}+d^{\dagger}_{k_{2}}v_{k_{2}}e^{ik_{2}{\cdot}x})(a_{k_{3}}\epsilon^{a}_{\mu
k_{3}}e^{-ik_{3}{\cdot}x}\\\ &+a^{\dagger}_{k_{3}}\epsilon^{*a}_{\mu
k_{3}}e^{ik_{3}{\cdot}x})b^{\dagger}_{p_{1}}d^{\dagger}_{p_{2}}\ket{0}.\end{split}$
(66)

The only nonvanishing string of creation/annihilation operators gives
$\matrixelement{0}{a_{k}d_{k_{1}}b_{k_{2}}a^{\dagger}_{k_{3}}b^{\dagger}_{p_{1}}d^{\dagger}_{p_{2}}}{0}=\delta(k_{3}-k)\delta(k_{2}-p_{1})\delta(k_{1}-p_{2})$.
Integrating over momenta $k_{i}$ in Eq.(66) using these delta functions, we
obtain

$\matrixelement{\gamma_{k}}{V_{em}}{e^{-}_{p_{1}}e^{+}_{p_{2}}}=\frac{e\bar{v}_{p_{2}}\gamma^{\mu}u_{p_{1}}\epsilon^{*}_{\mu
k}\delta(p_{1}+p_{2}-k)}{(2\pi)^{3/2}\sqrt{2p^{+}_{1}}\sqrt{2p^{+}_{2}}\sqrt{2k^{+}}}.$
(67)

Similarly,

$\matrixelement{q_{q_{1}}\bar{q}_{q_{2}}}{V_{em}}{\gamma_{k}}=\frac{e\bar{u}_{q_{1}}\gamma^{\mu}v_{q_{2}}\epsilon_{\mu
k}\delta(q_{1}+q_{2}-k)}{(2\pi)^{3/2}\sqrt{2q^{+}_{1}}\sqrt{2q^{+}_{2}}\sqrt{2k^{+}}}.$
(68)

The (Hermitian conjugate of) outgoing coherent state
$\Omega^{\dagger}_{\Delta(V_{1})}\ket{q_{p_{3}}\bar{q}_{p_{4}}}$ given by
Eq.(40) is substituted in
$\matrixelement{q_{p_{3}}\bar{q}_{p_{4}}}{\Omega_{\Delta(V_{1})}{V_{1}}}{q_{q_{1}}\bar{q}_{q_{2}}}$.
This leads to four nonvanishing terms, out of which the one containing the
string
$d_{p_{4}}b_{(p_{3}-k)}a_{k}b^{\dagger}_{n_{1}}d^{\dagger}_{n_{2}}a^{\dagger}_{n_{3}}b^{\dagger}_{q_{1}}d^{\dagger}_{q_{2}}$,
where $n_{i}$ are momenta labels for fields appearing in the interaction
$V_{1}$, eventually gives the amplitude $T_{3a}$. The other three terms give
the three remaining amplitudes $T_{3b-3d}$. $T_{3a}$ is now obtained by
performing momentum integrations using delta functions, and is given by
Eq.(58).

## References

  * Kulish and Faddeev (1970) P. P. Kulish and L. D. Faddeev, Theor. Math. Phys. 4, 745 (1970).
  * Catani and Ciafaloni (1985) S. Catani and M. Ciafaloni, Nucl. Phys. B 249, 301 (1985).
  * Catani  _et al._ (1986) S. Catani, M. Ciafaloni, and G. Marchesini, Nucl. Phys. B 264, 588 (1986).
  * Catani and Ciafaloni (1987) S. Catani and M. Ciafaloni, Nucl. Phys. B 289, 535 (1987).
  * Ciafaloni (1985) M. Ciafaloni, Phys. Lett. B 150, 379 (1985).
  * Ciafaloni (1989) M. Ciafaloni, Adv. Ser. Direct. High Energy Phys. 5, 491 (1989).
  * Del Duca  _et al._ (1989) V. Del Duca, L. Magnea, and G. F. Sterman, Nucl. Phys. B 324, 391 (1989).
  * Forde and Signer (2004) D. A. Forde and A. Signer, Nucl. Phys. B 684, 125 (2004), arXiv:hep-ph/0311059 .
  * Hannesdottir and Schwartz (2020a) H. Hannesdottir and M. D. Schwartz, “A finite $s$-matrix,” (2020a), arXiv:1906.03271 [hep-th] .
  * Hannesdottir and Schwartz (2020b) H. Hannesdottir and M. D. Schwartz, Phys. Rev. D 101, 105001 (2020b).
  * Misra (1994) A. Misra, Phys. Rev. D 50, 4088 (1994).
  * Misra (1996) A. Misra, Phys. Rev. D 53, 5874 (1996).
  * Misra (2000) A. Misra, Phys. Rev. D 62, 125017 (2000).
  * Misra (2005) A. Misra, Few Body Syst. 36, 201 (2005).
  * More and Misra (2012) J. D. More and A. Misra, Phys. Rev. D 86, 065037 (2012).
  * More and Misra (2013) J. D. More and A. Misra, Phys. Rev. D 87, 085035 (2013), arXiv:1302.3522 [hep-th] .
  * More and Misra (2014) J. D. More and A. Misra, Phys. Rev. D 89, 105021 (2014).
  * Agarwal  _et al._ (2023) N. Agarwal, L. Magnea, C. Signorile-Signorile, and A. Tripathi, Phys. Rept. 994, 1 (2023), arXiv:2112.07099 [hep-ph] .
  * Brodsky  _et al._ (1998) S. J. Brodsky, H.-C. Pauli, and S. S. Pinsky, Phys. Rept. 301, 299 (1998), arXiv:hep-ph/9705477 .
  * Chung (1965) V. Chung, Phys. Rev. 140, B1110 (1965).
  * Ángeles Martínez  _et al._ (2016) R. Ángeles Martínez, J. R. Forshaw, and M. H. Seymour, Phys. Rev. Lett. 116, 212003 (2016), arXiv:1602.00623 [hep-ph] .


akultät für Physik und Astronomie, Institut für Theoretische Physik II, Ruhr-
Universität Bochum, 44780 Bochum, Germany

# Spectral reconstruction of Euclidean correlator moments in lattice QCD

John Bulava
for the Baryon Scattering (BaSc) Collaboration FF

###### Abstract

A novel application of lattice QCD spectral reconstruction is presented, in
which euclidean correlation function data in a fixed time range are used to
infer values outside the range, enabling a model-independent investigation of
the asymptotic large-time behavior. Moments of the correlator are also
determined, and reconstructed correlation matrices between different moments
are included in a variational optimization similar to the standard Generalized
Eigenvalue Problem (GEVP). These ideas are illustrated using a single-nucleon
correlation function determined on an $N=2+1$ ensemble of gauge configurations
at $m_{\pi}=200{\rm MeV}$.

In a finite spatial volume, the spectrum of QCD is discrete. The energy gaps
suggest a strategy for the study of low-lying states in lattice QCD: take the
large-time limit of Euclidean correlation functions to suppress unwanted
excited state contamination. Solutions of a Generalized Eigenvalue Problem [1]
can be used to construct correlators asymptotically dominated by a single
state. However, (nearly) all correlation functions suffer from an exponential
degradation of the signal-to-noise ratio with increasing Euclidean time [2,
3]. Unfortunately, in practice the most precise data at early times is
discarded to fit the large-time region to a few-state ansatz. Choosing the
lower bound for such fits is often delicate: the statistical error can be
decreased at the expense of increased systematic error due to excited state
contamination. It is possible that this interplay is (at least partly)
responsible for the long-standing difficulty in reproducing the experimental
value for the nucleon axial charge [4, 5].

A model-independent alternative to discarding the early-time data is presented
here. It is based on the spectral reconstruction approach proposed by Backus
and Gilbert [6] and first used in lattice QCD by Hansen, Lupo, and Tantalo
[7]. Ref. [8] is similar in spirit to this work and is demonstrably of
comparable effectiveness [9]. Other approaches to spectral reconstruction in
lattice QCD are reviewed in Ref. [10]. The general problem of spectral
reconstruction employs Euclidean two-point correlator data

$\displaystyle C(t)=\int{\rm d}\omega\,\rho(\omega)\,{\rm e}^{-\omega
t},\qquad\rho(\omega)=\sum_{n}A_{n}\,\delta(\omega-E_{n})$ (1)

known for integer values of $t/a$ in the range $t\in[t_{\rm min},t_{\rm max}]$
with statistical errors. The direct determination of $\rho(\omega)$ from the
correlator data is an ill-posed problem, which can be ameliorated by instead
seeking smeared spectral densities of the form $\tilde{\rho}[f]=\int{\rm
d}\omega f(\omega)\,\rho(\omega)$ with a particular smearing kernel
$f(\omega)$ specified _a priori_. Previous applications of Ref. [7] use
$f(\omega)$ to approximate the Dirac-$\delta$ distribution [11], Heaviside
step function [12], or implement the principle-value prescription [13]. This
work however determines correlator moments, defined as

$\displaystyle D(\alpha,\tau)=\int{\rm
d}\omega\,\rho(\omega)\,\omega^{\alpha}\,{\rm e}^{-\omega\tau}.$ (2)

The different symbol emphasizes that $\tau$ need not be constrained to the
values of $t$ provided by the data, suggesting that the temporal resolution
may be increased with non-integer $\tau/a$ and the asymptotic limit probed
with $\tau>t_{\rm max}$. Furthermore, taking the (rational) power $\alpha$
different from zero suppresses or enhances excited state contamination.
Correlators with different $\alpha$ may be viewed as employing different
interpolating operators, which can accordingly form a correlation matrix for
use in a Generalized eigenvalue problem (GEVP).

The reconstruction approach employed here is detailed in Ref. [7], with
further discussion of the reliable estimation of statistical and systematic
errors in Refs. [11, 14]. Consider an estimator for $D(\alpha,\tau)$ from Eq.
2 which is a linear combination of all input correlator timeslices

$\displaystyle\hat{D}(\alpha,\tau)=\sum_{t=t_{\rm min}}^{t_{\rm
max}}g_{t}(\alpha,\tau)\,C(t)\equiv\int{\rm
d}\omega\,\hat{f}(\omega)\,\rho(\omega).$ (3)

Evidently $\hat{D}(\alpha,\tau)$ is a smeared spectral density smeared with
the kernel $\hat{f}(\omega)=\sum_{t}g_{t}(\alpha,\tau)\,{\rm e}^{-\omega t}$,
in contrast to $D(\alpha,\tau)$ which is smeared with
$f(\omega)=\omega^{\alpha}\,{\rm e}^{-\omega\tau}$. Any difference between
$\hat{f}(\omega)$ and $f(\omega)$ is a systematic error in the estimator
$\hat{D}(\alpha,\tau)$. The systematic and statistical errors are quantified
by

$\displaystyle A[g]=\int_{\omega_{0}}^{\infty}{\rm
d}\omega\,\left\\{f(\omega)-\hat{f}(\omega)\right\\}^{2},\qquad
B[g]=\sum_{tt^{\prime}}g_{t}g_{t^{\prime}}\,{\rm
Cov}\left\\{C(t),C(t^{\prime})\right\\},$ (4)

respectively. The coefficients $\\{g_{t}\\}$ are chosen to balance these two
considerations by minimizing $G_{\lambda}[g]=(1-\lambda)A[g]+\lambda B[g]$ for
a particular $\lambda\in[0,1]$. Small $\lambda$ results in a reconstructed
kernel $\hat{f}(\omega)$ close to the desired one $f(\omega)$, but at the
expense of large statistical error. Large $\lambda$ gives a statistically
precise result, but with a large systematic error due to the difference
between $\hat{f}(\omega)$ and $f(\omega)$. As is customary in lattice QCD data
analysis, an ideal $\lambda$ is sought in the statistics-limited regime
wherein the systematic error is reliably smaller than the statistical error.

To test these ideas, a single-nucleon correlator from an $N_{\rm f}=2+1$
ensemble of gauge configurations with $m_{\pi}=200\,{\rm MeV}$ is employed.
This correlator is computed over the time range $[t_{\rm min},t_{\rm
max}]=[2a,25a]$ and was used to determine the nucleon mass $m_{\rm N}$ in Ref.
[15]. Further details of the interpolating operators and measurement procedure
can be found there. For the reconstruction, the lower bound of the integration
in Eq. 4 is fixed at $a\omega_{0}=0.3$, and the optimal $\lambda$ chosen as in
Ref. [14] by demanding that the variation of the estimator among a set of
reconstructions which impose different constraints is three times smaller than
the statistical error. As a first test of the procedure, a reconstruction is
preformed using timeslices $[t_{\rm min},t_{\rm max}]=[2a,20a]$ to infer
$t=21a,\dots,25a$. The results of this test are shown in the left panel of
Fig. 1, which demonstrates that the standard estimator for the correlator and
$D(0,\tau)$ agree within $1.5\sigma$.

Figure 1: Two tests of the correlator reconstruction procedure. Left: using
correlator data from times $t/a=[2,20]$ to infer $D(0,t)$ at later times and
compare with direct measurements of $C(t)$. The error on the relative
difference between these two estimators is computed using the bootstrap
procedure. Right: using times $t/a=[2,25]$ to infer the effective mass using
Eq. 5 with $(\alpha,\beta)=(1,0)$. The naive forward-difference definition is
also shown for comparison. The horizontal band is the result of a two-state
fit (from Ref. [15]) over the horizontal range shown.

Consider now the family of effective masses

$\displaystyle m_{\rm
eff}(\alpha,\beta|\tau)=\left\\{\frac{D(\alpha,\tau)}{D(\beta,\tau)}\right\\}^{1/(\alpha-\beta)}$
(5)

for which $\lim_{\tau\rightarrow\infty}m_{\rm eff}(\alpha,\beta|\tau)=m_{\rm
N}\times\left\\{1+{\rm O}({\rm e}^{-2m_{\pi}\tau})\right\\}$. Due to the
finite spatial volume, the lowest non-interacting $p$-wave state has an energy
close to $m_{\rm N}+2m_{\pi}$. The case $(\alpha,\beta)=(1,0)$ coincides with
the standard forward-difference effective mass up to $O(a^{2})$ while
different $(\alpha,\beta)$ have varying excited state contamination. A second
test of the reconstruction procedure is shown in the right panel of Fig. 1,
which compares the forward difference effective mass with
$(\alpha,\beta)=(1,0)$. The reconstructed effective mass is considerably more
precise at later times, and asymptotically agrees with the value obtained from
a two-state fit. It should be emphasised, however, that the spectral
reconstruction approach does not impose a model for the time dependence.

Figure 2:  Different definitions of the effective mass by varying $\alpha$ and
$\beta$ in Eq. 5. In this manner the excited state contamination can be
suppressed or enhanced relative to the conventional definition, which employs
$(\alpha,\beta)=(1,0)$.

Fig. 2 shows various $(\alpha,\beta)$. A significant enhancement or reduction
of the excited state contamination is evident similar to a variation in the
level of quark field smearing, but here achieved for a single correlator. This
naturally suggests employing different moments in a correlation matrix
$A_{ij}(t)=D(\alpha_{i}+\alpha_{j},t)$. Rather than the standard GEVP, which
employs a second metric timeslice $t_{0}$, the variation optimization is
applied directly to $B_{ij}(t)=D(\alpha_{i}+\alpha_{j}+1,t)$, using $A(t)$ as
a metric. This ‘equal-time’ GEVP is therefore

$\displaystyle B(t)v_{n}(t)=\lambda_{n}(t)A(t)v_{n}(t),$ (6)

where the eigenvalues $\lambda_{n}(t)$ approach directly the states of
interest for large $t$. The results from a two-dimensional equal-time GEVP are
shown in Fig. 3, which show a further reduction in excited state contamination
compared to the input ‘diagonal’ effective masses. Furthermore, a rough
estimate of the first excited state is provided with an energy near $m_{\rm
N}+2m_{\pi}$. Interactions likely shift the multi-hadron excited states
significantly from these naive expectations, however.

Figure 3:  The eigenvalue $\lambda_{n}(t)$ from a two-by-two GEVP as in Eq. 6
with $\\{\alpha_{1},\alpha_{2}\\}=\\{-0.5,2\\}$. The lower horizontal band is
the two-state fit from Ref. [15], as in Fig. 1. The right panel is zoomed out
to show $\lambda_{1}(t)$, and the upper horizontal band is $m_{\rm
N}+2m_{\pi}$.

In conclusion, the reconstruction of correlator moments presented here
provides model-independent determinations of the correlator at arbitrary
Euclidean time separations. Unlike few-state fits, these estimates employ the
entire range of correlator data, including those at precise early times.
However, no miracle has been achieved: the asymptotic behaviour shown in Figs.
1, 2, and 3 is consistent with a two-state fit and has a similar statistical
precision. Nonetheless, it serves as a valuable model-independent confirmation
of the two-state ansatz over the limited time range of input correlator data.
In addition to the investigation of large-time asymptotics detailed here, the
model-independent interpolation of correlator data may be useful in the
comparison of lattice QCD vector-vector correlators and smeared experimental
$R$-ratio data. It is likely that this approach can also be extended to treat
the simultaneous reconstruction of multiple correlation functions in analogy
to the ratios constructed to determine energy differences, as well as three-
point correlation functions.

###### Acknowledgements.

I thank my co-authors of Ref. [15] and my colleagues in the Baryon Scattering
Collaboration (BaSc) for sharing the single-nucleon correlator used here.

## References

  * [1] C. Michael and I. Teasdale, Extracting Glueball Masses From Lattice QCD, Nucl. Phys. B215 (1983) 433.
  * [2] G. Parisi, The Strategy for Computing the Hadronic Mass Spectrum, Phys. Rept. 103 (1984) 203–211.
  * [3] G. P. Lepage, The Analysis of Algorithms for Lattice Field Theory, in Theoretical Advanced Study Institute in Elementary Particle Physics, 6, 1989.
  * [4] R. Gupta, S. Park, M. Hoferichter, E. Mereghetti, B. Yoon, and T. Bhattacharya, Pion–Nucleon Sigma Term from Lattice QCD, Phys. Rev. Lett. 127 (2021), no. 24 242002, [arXiv:2105.12095].
  * [5] L. Barca, G. Bali, and S. Collins, Toward N to N$\pi$ matrix elements from lattice QCD, Phys. Rev. D 107 (2023), no. 5 L051505, [arXiv:2211.12278].
  * [6] G. Backus and F. Gilbert, The resolving power of gross earth data, Geophysical Journal International 16 (1968), no. 2 169–205.
  * [7] M. Hansen, A. Lupo, and N. Tantalo, Extraction of spectral densities from lattice correlators, Phys. Rev. D99 (2019), no. 9 094508, [arXiv:1903.06476].
  * [8] G. Bailas, S. Hashimoto, and T. Ishikawa, Reconstruction of smeared spectral function from Euclidean correlation functions, PTEP 2020 (2020), no. 4 043B07, [arXiv:2001.11779].
  * [9] A. Barone, S. Hashimoto, A. Jüttner, T. Kaneko, and R. Kellermann, Inclusive semi-leptonic $B_{(s)}$ mesons decay at the physical $b$ quark mass, in 39th International Symposium on Lattice Field Theory, 11, 2022\.  arXiv:2211.15623.
  * [10] A. Rothkopf, Inverse problems, real-time dynamics and lattice simulations, EPJ Web Conf. 274 (2022) 01004, [arXiv:2211.10680].
  * [11] J. Bulava, M. T. Hansen, M. W. Hansen, A. Patella, and N. Tantalo, Inclusive rates from smeared spectral densities in the two-dimensional O(3) non-linear $\sigma$-model, JHEP 07 (2022) 034, [arXiv:2111.12774].
  * [12] P. Gambino, S. Hashimoto, S. Mächler, M. Panero, F. Sanfilippo, S. Simula, A. Smecca, and N. Tantalo, Lattice QCD study of inclusive semileptonic decays of heavy mesons, JHEP 07 (2022) 083, [arXiv:2203.11762].
  * [13] R. Frezzotti, N. Tantalo, G. Gagliardi, F. Sanfilippo, S. Simula, and V. Lubicz, Spectral-function determination of complex electroweak amplitudes with lattice QCD, Phys. Rev. D 108 (2023), no. 7 074510, [arXiv:2306.07228].
  * [14] J. Bulava, The spectral reconstruction of inclusive rates, PoS LATTICE2022 (2023) 231, [arXiv:2301.04072].
  * [15] J. Bulava, A. D. Hanlon, B. Hörz, C. Morningstar, A. Nicholson, F. Romero-López, S. Skinner, P. Vranas, and A. Walker-Loud, Elastic nucleon-pion scattering at $m_{\pi}=200\,{\rm MeV}$ from lattice QCD, Nucl. Phys. B 987 (2023) 116105, [arXiv:2208.03867].