###### Abstract

We develop a general expression for weighted cross sections in leptonic
annihilation to hadrons based on time-ordered perturbation theory (TOPT). The
analytic behavior of the resulting integrals over spatial momenta can be
analyzed in the language of Landau equations and infrared (IR) power counting.
For any infrared-safe weight, the cancellation of infrared divergences is
implemented locally at the integrand level, and in principle can be evaluated
numerically in four dimensions. We go on to show that it is possible to
eliminate unphysical singularities that appear in time-ordered perturbation
theory for arbitrary amplitudes. This is done by reorganizing TOPT into an
equivalent form that combines classes of time orderings into a “partially
time-ordered perturbation theory”. Applying the formalism to leptonic
annihilation, we show how to derive diagrammatic expressions with only
physical unitarity cuts.

YITP-SB-2023-28

Local infrared safety in
time-ordered perturbation theory

George Sterman and Aniruddha Venkata

C.N. Yang Institute for Theoretical Physics and Department of Physics and
Astronomy

Stony Brook University, Stony Brook, NY 11794-3840 USA

###### Contents

  1. 1 Introduction
  2. 2 Time-ordered perturbation theory and leptonic annihilation
    1. 2.1 Cross sections in TOPT
    2. 2.2 Unphysical cuts in TOPT
  3. 3 Local Infrared safety
    1. 3.1 Reorganized cross sections
    2. 3.2 Analysis of pinch surfaces
    3. 3.3 Logarithmic singularities and cancellation
    4. 3.4 All-order example: energy in a “cone”
    5. 3.5 Low-order example
  4. 4 Posets in TOPT
    1. 4.1 Definitions
    2. 4.2 Examples in vacuum polarization diagrams
  5. 5 PTOPT: TOPT without unphysical singularities
    1. 5.1 Time integrals of extremal vertices; the covering set
    2. 5.2 Integrals of induced extrema
  6. 6 Vacuum polarization graphs, leptonic annihilation, and weighted cross sections
  7. 7 Summary
  8. A Embedded Extrema
  9. B An explicit algorithm to identify $\gamma^{k+1}$

##  1 Introduction

In the light of prospects for increasingly high-statistics data from the Large
Hadron Collider and proposed facilities, the need for precision in
perturbative calculations of collider cross sections is widely recognized [1,
2, 3]. A recurring theme in these discussions has been the possibility of
carrying out calculations directly in four dimensions [4, 5, 6]. For physical
observables like cross sections, this involves canceling infrared (IR)
singularities at the level of integrands, resulting in expressions amenable to
numerical calculation [7, 8, 9]. The potential for such an approach is
implicit in formal proofs of the infrared safety of jet and related weighted
cross sections [10, 11], based on light-cone ordered perturbation theory, and
in proofs of factorization based on time-ordered perturbation theory (TOPT)
[12, 13, 14, 15]. In each case, cancellations of infrared divergences locally
in momentum space become manifest after integration over loop energies or
light-cone variables. The same fundamental result was found in Refs. [7, 8,
9], which carry out energy integrals using the technique of Loop Tree Duality
[4, 16]. In this paper, inspired by the results of [7, 8, 9], and by the
general analyses of perturbation theory in Refs. [17, 18], we return to time-
ordered perturbation theory, to provide a complementary perspective on these
important results.

We will reconsider infrared (IR) safe cross sections for hadronic states
produced in lepton pair annihilation from this point of view. In this case,
unitarity implies that cross sections can be computed as sums over the final
states that correspond to cuts of vacuum polarization diagrams. The
contribution of any fixed cut, of course, includes infrared divergences, which
cancel in the sum over cuts. The result of our analysis is a manifestly power-
counting finite expression for such a cross section, given as a sum over time-
ordered, cut vacuum polarization diagrams. All terms coming from a given
diagram to this sum are evaluated as one three-dimensional integration per
loop of the original uncut diagram, without change of variables as we sum over
the cuts (states) of the diagram.

In the course of our discussion in TOPT, we will encounter apparent,
unphysical contributions to the cross section, associated with the vanishing
of energy denominators corresponding to cuts that divide the diagram into more
than two parts. We’ll see how these unphysical contributions cancel in the sum
over time orders. This observation opens the door to a reformulation of TOPT,
in which unphysical energy denominators are altogether absent. Such a form, in
fact, has already been derived in Ref. [18], from a distinct but related point
of view. We will present an alternative derivation, based on what we will call
partially time-ordered perturbation theory (PTOPT). These results, like those
in Ref. [18], are quite general, and provide a reformulation for arbitrary
amplitudes, and to light-cone- as well as time-ordered perturbation theory.

In Sec. 2 we will review some properties of time-ordered perturbation theory,
and we will observe that within a single time-ordered diagram, many
contributions to the imaginary part are unphysical, involving the disconnected
production of finite-energy particles out of the vacuum. We will show,
however, that upon summing over time orders, these contributions cancel among
themselves at the amplitude level. In Sec. 3 we construct a locally IR finite
formula for weighted cross sections in TOPT for leptonic annihilation to
hadrons. Each such expression is closely related to the imaginary part of the
vacuum polarization amplitude for the current that couples to quantum
chromodynamics in the Standard Model. We will derive the analogs of Landau
equations for this class of cross sections, along with power counting at the
“pinch surfaces” where these equations are satisfied.

In subsequent sections, we show how to reorganize TOPT to eliminate these
unphysical singularities manifestly, by developing partially time-ordered
perturbation theory, with results similar to those of Ref. [18]. These partial
sums over time orders can be carried out algorithmically, and in effect reduce
the number of terms necessary to compute amplitudes and cross sections. To
this end, in Sec. 4, we will recall a mathematical structure that underlies
each time order: the partially ordered set (poset), and show by examples how
partial ordering can be used to eliminate unphysical cuts.

Section 5 is dedicated to combining the full TOPT contributions associated
with each poset, implementing the constraints associated with the poset. In
doing so, we will sum over all time orders consistent with a poset. Finally,
in Sec. 6, we will return to weighted cross sections in leptonic annihilation,
and adapt our formulas for the weighted cross section case. We conclude with a
short summary and comments on future directions.

##  2 Time-ordered perturbation theory and leptonic annihilation

Time-ordered perturbation theory provides a systematic method for integrating
all loop energies in an arbitrary diagram [14, 15]. For the treatment of
electroweak annihilation, the diagrams of interest are two-point correlators
of electroweak currents. More generally, we can consider any amplitude
$A\left(q^{\prime}_{j},q_{i}\right)$, where $q_{i}$ represents a set of
incoming momenta, and $q_{j}^{\prime}$ a set of outgoing momenta, any or all
of which may be off-shell. In time-ordered perturbation theory, the expression
for such an amplitude is of the form

$A\left(q^{\prime}_{j},q_{i}\right)\ =\sum_{G\in{G_{A}}}\;\int d{\cal
L}_{G}\;\sum_{\tau_{G}}\;\mathbb{N}_{\tau_{G}}\;\prod_{i=1}^{N_{G}}\frac{1}{2\omega_{i}}\prod_{s=1}^{V_{G}-1}\frac{i}{E_{s}-\sum_{j\in
s}\omega_{j}+i\epsilon},$ (2.1)

where $G_{A}$ represents the set of graphs in Lorentz-invariant perturbation
theory that contribute to the amplitude and $\tau_{G}$ represents the time
orders of the $V_{G}$ vertices of each graph $G$, with $N_{G}$ lines. Each
ordering, $\tau_{G}$ of the vertices specifies a set of states, labeled
$s=1\dots V_{G}-1$, whose total energy is the sum of the on-shell energies of
all the particles in that state, defined to flow from earlier to later times.
The denominators in Eq. (2.1) are “energy deficits”, for each state $s$, the
difference between the sum of the on-shell energies
$\omega_{i}=\sqrt{\vec{p}_{i}^{2}+m_{i}^{2}}$ for lines of mass $m_{i}$ in
state $s$ and the external energy, $E_{s}$, that has flowed into the diagram
before state $s$. The external net energy of each state depends on the
specific ordering $\tau_{G}$.

In Eq. (2.1), $\int d{\cal L}_{G}$ represents the spatial loop momentum space
of the graph $G$, which is the same for every time order. Each line momentum
$\vec{p}_{j}$ is a linear function of loop and external momenta, which in this
way determine the energies $\omega_{j}$. The factor $\mathbb{N}_{\tau_{G}}$
represents the perturbative numerator, consisting of overall constants and
spin-dependent momentum factors, with the energies of every line evaluated on-
shell, that is with energies $\omega_{i}$ defined as above. The signs of these
energies are always positive for flow from the earlier to later vertex
connected by the line in question. In this way, the numerator factor depends
on the time order.

###  2.1 Cross sections in TOPT

We consider annihilation cross sections for leptons to hadrons, at lowest
order in electroweak couplings. In this case, we can represent our cross
sections as

$\displaystyle\sigma(Q)\ $ $\displaystyle\equiv$ $\displaystyle\
\sum_{N}\,(2\pi)^{4}\delta^{4}\left(q-P_{N}\right)\,\big{|}\langle
0\big{|}\,{\cal J}(0)\,|N\rangle|^{2}\,,$ (2.2)

where the $\cal J$ represents an electroweak current, implicitly contracted
with tree-level leptonic tensors and vector boson propagators, which we
suppress, and where $q^{2}=Q^{2}$. The latter will play no role in these
arguments, and we will consistently absorb them into the currents. As a total
cross section, $\sigma(Q)$ is related by the optical theorem to a forward-
scattering amplitude, here a two-point correlation function of the currents
${\cal J}$,

$\displaystyle\sigma(Q)\ $ $\displaystyle=$ $\displaystyle\ {\rm
Im}\,\Pi(Q)\,,$ $\displaystyle\Pi(Q)\ $ $\displaystyle=$ $\displaystyle\
i\,\int d^{4}x\,e^{-iq\cdot x}\langle 0|T[{\cal J}(0){\cal J}(x)]|0\rangle\,.$
(2.3)

In this paper, we will use time-ordered perturbation theory to derive
expressions for sets of weighted cross sections that generalize Eq. (2.2),

$\displaystyle\Sigma[f,Q]\ $ $\displaystyle\equiv$ $\displaystyle\
\sum_{N}\,(2\pi)^{4}\delta^{4}\left(q-P_{N}\right)\,f(N)\,\langle 0|\,{\cal
J}(0)\,|N\rangle|^{2}\,,$ (2.4)

with $f(N)$ a function of the kinematic variables that describe state $N$, and
with currents normalized as for the total cross section, Eq. (2.2). Our goal
will be to derive a set of expressions that exhibit manifestly the
cancellation of infrared divergences in such quantities [10, 11, 19]. First,
however, we will discuss a bit more how the general relations (2.2) and (2.3)
are realized in TOPT.

In the notation of Eq. (2.1), for the two-point correlation function of Eq.
(2.3), a single momentum $q$ flows into and out of the diagram, and we may
choose to work in its rest frame, $q^{\mu}=(Q,\vec{0})$. In this case, our
vacuum polarization diagram $\Pi(Q)$ is given by

$\displaystyle\Pi(Q)\ $ $\displaystyle=$ $\displaystyle\sum_{G\in
G_{\Pi}}\;\int d{\cal
L}_{G}\;\prod_{i=1}^{N_{G}}\frac{1}{2\omega_{i}}\;\sum_{\tau_{G}}\;\mathbb{N}_{\tau_{G}}\;\pi_{\tau_{G}}(Q,{\cal
L}_{G})\,,$ $\displaystyle\pi_{\tau_{G}}(Q,{\cal L}_{G})\ $ $\displaystyle=$
$\displaystyle\ \prod_{s=1}^{V_{G}-1}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}+i\epsilon}\,,$ (2.5)

where the sum is now over the set of diagrams $G_{\Pi}$ that contribute to the
two-point correlation function that mediates the leptonic annihilation
process. We have suppressed vector indices of the currents associated with the
decay of the mediating vector bosons. In this expression, the state-dependent
factor $\lambda_{s}$ enforces the condition that the net external energy in
state $s$ is positive only when state $s$ occurs after momentum $q$ flows into
the diagram, here and below at vertex $i$, and before the same energy flows
out, here and below at vertex $o$,

$\displaystyle\lambda_{s}\ $ $\displaystyle=$ $\displaystyle\ 1\,,\quad
o>s\geq i\,,$ $\displaystyle\lambda_{s}\ $ $\displaystyle=$ $\displaystyle\
-1\,,\quad i>s\geq o$ (2.6) $\displaystyle=$ $\displaystyle\ 0\,,\quad{\rm
otherwise}\,.$

In Eq. (2.5), the set of states $s$ depends implicitly on the time order,
$\tau_{G}$. Finally, we notice that energy denominators are negative semi-
definite for massless particles and negative definite for massive particles
except when $o>s\geq i$.

Each covariant diagram with $V_{G}$ vertices provides $V_{G}!$ terms, the sum
over $\tau_{G}$ in Eq. (2.5), each with $V_{G}-1$ denominators. The vanishing
of a subset of these denominators results in a branch cut in the function
$\Pi(Q)$. Each such branch cut corresponds to an on-shell intermediate state
that is at or above threshold. These are the physical, or unitarity cuts of
the underlying diagram. In time-ordered perturbation theory, such states
separate a vacuum polarization diagram $G$ into a connected amplitude and
complex-conjugate amplitude, where external momentum flows into the amplitude
and out of the complex conjugate. Clearly, such cuts are possible only in
orderings for which the vertex $i$ is earlier than vertex $o$. By unitarity in
the form of the optical theorem, the sum of these singularities gives the
total cross section for leptonic annihilation or equivalently the decay width
of the relevant off-shell electroweak boson. We shall see below, however, that
even when $o>i$ in the uncut diagram, many cuts of $\pi_{\tau_{G}}$ in Eq.
(2.5) are actually unphysical, corresponding to the production of particles
out of the vacuum. One of the aims of our discussion is to reorganize the sum
of time orders to show how these unphysical singularities always cancel. In
Secs. 4 and 5, we will show how to reorganize the time-ordered series,
reducing the number of terms and eliminating unphysical singularities without
having to rely on this cancellation.

Setting aside for a moment the cancellation of unphysical cuts, let us write
the imaginary part of the amplitude $\Pi(Q)$, Eq. (2.5), as a sum over its
intermediate states, and generalize the resulting sum over final states to
more general weighted cross sections. To do so, we will need first to write
the TOPT expression for $\Pi(Q)$ as a sum over fixed states, given by cuts $C$
of an arbitrary diagram. Here each $C$ is a particular state $s$ in Eq. (2.5)
that is set on shell. Summing over $C$, we know from the optical theorem that
we reconstruct the imaginary part of the forward scattering graph in Eq.
(2.5),

$\displaystyle{\rm Im}\,\Pi(Q)\ $ $\displaystyle=$ $\displaystyle\
\sum_{G}\,\sum_{C}\,\sum_{\tau_{L[G/C]}}\,\sum_{\tau_{R[G/C]}}\,\pi^{(C)}_{\tau_{L[G/C]}\,\cup\,\tau_{R[G/C]}}(Q)$
$\displaystyle\pi^{(C)}_{\tau_{L[G/C]}\,\cup\,\tau_{R[G/C]}}(Q)\ $
$\displaystyle=$ $\displaystyle\ \int d{\cal
L}_{G}\;\mathbb{N}_{\tau_{G}}\;\prod_{i=1}^{N}\frac{1}{2\omega_{i}}\;\prod_{s=C+1}^{V_{G}-1}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}-i\epsilon}$ $\displaystyle\hskip 14.22636pt\times\
(2\pi)\,\delta\left(Q-\sum_{j\in
C}\omega_{j}\right)\prod_{s=1}^{C-1}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}+i\epsilon}\,,$

where $\tau_{L[G/C]}$ represents the time orders of the graph $G$ on the left
of the cut, $C$, and $\tau_{R[G/C]}$ represents the time orders of the graph
$G$ on the right of cut $C$. The union of these orders,
$\tau_{L[G/C]}\cup\tau_{R[G/C]}$, uniquely specifies an ordering, $\tau_{G}$
of the full diagram, with all vertices in the conjugate after all vertices in
the amplitude111In Eq. (LABEL:eq:forwardscattering-cuts) and below, factors
associated with later times in the uncut diagram are written to the left,
following time ordering conventions. In all figures, however, we will choose
our final states to the right of initial states. Thus, $\tau_{L[G/C]}$ refers
to time orders on the left of the cut diagram.. For a given underlying
diagram, $G$, orderings $\tau_{L[G/C]}$ and $\tau_{R[G/C]}$ specify
diagrammatic contributions to the amplitude and complex conjugate amplitude,
respectively, to produce final state $C$ by the action of current ${\cal J}$.
As above, the function $\mathbb{N}_{\tau_{G}}$ absorbs overall factors
associated with vertices, and the overall factor of $i$ in Eq. (2.3), all of
which we do not exhibit. We note, however, that each vertex is associated with
a factor of $i$, and that when all denominators are real and nonzero, the
diagram is real. Note that to satisfy the delta function we must have
$\lambda_{C}=+1$, so that the external momentum must flow into the diagram in
the amplitude ($s<C$) and out from the complex conjugate ($s>C$). Then
$\lambda_{s}=0$ or $1$, and not $-1$, in both the amplitude and complex
conjugate.

A complete discussion of the proof of Eq. (LABEL:eq:forwardscattering-cuts),
including symmetry factors, is given in Ref. [14], but here we observe that it
involves the relationship between time orders of the full diagram and of
amplitudes separated by its cuts,

$\displaystyle\sum_{\tau_{G}}\sum_{C}\ =\
\sum_{C}\sum_{\tau_{L[G/C]}}\sum_{\tau_{R[G/C]}}\,,$ (2.8)

which the summations satisfy by definition. A subtle feature of the sums is
that the diagrams $L[G/C]$ and $R[G/C]$ specified by these orderings need not
be connected on the left and right of the cut. As a result, Eq.
(LABEL:eq:forwardscattering-cuts) includes unphysical terms that are not found
on the right-hand side of Eq. (2.2), which defines the cross section in terms
of connected amplitudes. In the next sub-section, we show how such
contributions occur, and that they always cancel in the sum over time orders,
as they must by the optical theorem. Notice further that many of the final
states $C$ on both sides of this relation do not contribute to the imaginary
part, because they correspond to states for which $\lambda_{s}$ equals $0$ or
$-1$ in Eq. (2.6). The full cross section, of course, involves a leptonic
tensor, and depends on the specific electroweak currents at the vertices we
have labeled $i$ and $o$. As noted above, for simplicity, we have suppressed
these familiar factors, which play no direct role in our discussion, and refer
to expressions like Eq. (LABEL:eq:forwardscattering-cuts) as cross sections.

We now consider a weighted lepton annihilation cross section where the sum
over final states $C$ is weighted by a set of functions $f_{C}(p_{i})$, which
depend only on particle momenta appearing in state $C$. The relevant TOPT
energy factors, denominators, and energy delta function for this weighted
cross section can be organized as

$\displaystyle\Sigma[f,Q]\ $ $\displaystyle=$ $\displaystyle\
\sum_{G}\sum_{C}\sum_{\tau_{L[G/C]}}\sum_{\tau_{R[G/C]}}\,\int d{\cal
L}_{G}\;\sum_{\tau_{G}}\;\mathbb{N}_{\tau_{G}}\;\prod_{i=1}^{N_{G}}\frac{1}{2\omega_{i}}\,\sigma^{(C)}_{\tau_{L[G/C]}\cup\tau_{R[G/C]}}[f,{\cal
L}_{G}]\,,$ $\displaystyle\sigma^{(C)}_{\tau_{G}}[f,{\cal L}_{G},Q]$
$\displaystyle=$
$\displaystyle\prod_{s=C+1}^{V_{G}-1}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}-i\epsilon}\ f_{C}(\vec{q}_{1}\ldots\vec{q}_{k_{C}})$ (2.9)
$\displaystyle\hskip 14.22636pt\times\ (2\pi)\,\delta\left(Q-\sum_{j\in
C}\omega_{j}\right)\prod_{s=1}^{C-1}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}+i\epsilon}\,,$

where we understand that $\tau_{G}=\tau_{L[G/C]}\cup\tau_{R[G/C]}$. Here
$f_{C}(\\{\vec{q}_{i}\\})$ is a weight function, depending in general on the
spatial momenta, $\\{\vec{q}_{i}\\}=\\{\vec{q}_{1}\ldots\vec{q}_{k_{C}}\\}$
and masses of particles in the state $C$. This expression differs from the
corresponding factor in the imaginary part of the forward scattering graph
only by the weight function $f_{C}$, and setting $f_{C}=1$, gives back Eq.
(LABEL:eq:forwardscattering-cuts). At this point, we observe that the sum over
time orders in Eq. (2.9) can be thought of as generated from connected
diagrams only, so that the sum does not include the disconnected diagrams
encountered in Eq. (LABEL:eq:forwardscattering-cuts). Because, as we observed
above and will show in the next section, these contributions cancel among
themselves, the sums are nevertheless effectively identical.

In what follows, we will study infrared safe weight functions, for which

$\displaystyle
f_{C}(\vec{q}_{1},\ldots\vec{q}_{i}\dots\vec{q}_{j-1},\xi\vec{q}_{i},\vec{q}_{j+1},\dots\vec{q}_{k_{C}})\
$ $\displaystyle=$ $\displaystyle
f_{C/j}(\vec{q}_{1},\ldots(1+\xi)\vec{q}_{i},\dots\vec{q}_{j-1},\vec{q}_{j+1},\dots\vec{q}_{k_{C}})\,,$

for any real $\xi>-1$, where $C/j$ denotes a state with $k_{C}-1$ particles.
Such weights are unchanged by the emission of zero-momentum particles
($\xi=0$) or by the emission or recombination of massless collinear particles.
This is a familiar condition, of course, which ensures the cancellation of
soft and collinear (collectively, infrared) singularities. Requirements on the
manner in which the weight functions approach the equalities of Eq.
(LABEL:eq:irs) are discussed in Refs. [11, 19].

Our goal below is to exhibit an expression for such a weighted cross section
in which the cancellations implied by Eq. (LABEL:eq:irs) can be made explicit,
so that all cancellations take place in four dimensions, without infrared
regularization. First, however, we resolve the apparent difference between the
sums over time-ordered diagrams in the unitarity condition, Eq.
(LABEL:eq:forwardscattering-cuts), and in the corresponding expression for
weighted cross sections, Eq. (2.9).

###  2.2 Unphysical cuts in TOPT

Let us now turn our attention to unphysical cuts in time-ordered perturbation
theory. By unphysical cuts, we refer to states that contribute to the forward-
scattering diagram but which, when they go on-shell (that is, when the
corresponding denominator vanishes), do not separate the diagram into
connected diagrams to the right and left. Such cuts of the diagram do not
appear in the sum over final states in the optical theorem, Eq. (2.3). What we
will show is that the sum over all time-ordered diagrams that share any
specific unphysical cut vanishes. This result appears to be the analog in TOPT
of the cancellation of so-called spurious singularities in loop-tree duality
[9].

To establish some notation, we say the vertices are in the set
$V=\\{i,b_{1},b_{2},b_{3},\ldots b_{n},o\\}$, and $i$, $o$ are the vertices
where the external momentum $q$ flows in and out respectively. We take
$(b_{P_{1}},b_{P_{2}},\ldots,b_{P_{n+2}})$ to represent an arbitrary time
order labeled by the permutation $P$. The way in which unphysicsl cuts appear
in TOPT is illustrated by the low-order example in Fig. 1. In TOPT,
intermediate states may include particles created from the vacuum, with no
available external energy. If such a state precedes the vertex at which
momentum flows into the diagram, as in state $C_{1}$ of the figure, the
corresponding energy denominator is negative semi-definite, vanishing only if
all particles are massless, and then only on the set of zero measure where all
particles carry zero spatial momentum. All denominators in states such as this
are of the form $\frac{i}{-\sum_{j}\omega_{j}+i\epsilon}$. We shall refer to
such states as “vacuum states”.

State $C_{2}$ in Fig. 1 describes the amplitude for the same production of
particles from the vacuum, which, however, appears mixed with particles that
share the energy flowing into the diagram. Denominators like those of state
$C_{2}$ can be of the form $\frac{i}{Q-\sum_{j}\omega_{j}+i\epsilon}$. The
resulting mixed denominators can vanish in this configuration and can
contribute to the imaginary part of the specific time-ordered diagram. We
refer to these states as “pseudo-physical”.

Figure 1: An example of a graph with unphysical cuts of two types. Here, no
external momentum is ordered earlier than state $C_{1}$, while momentum $Q$
enters prior to state $C_{2}$. The state $C_{1}$ is a vacuum cut and the state
$C_{2}$ is a pseudo-physical cut.

We now prove that the cross section evaluated on a pseudo-physical cut is
identically zero. To do so, we start with the expression for the denominators,
$s$, of the amplitude $a_{L[G/C]}\left(Q,{\cal L}_{L[G/C]}\right)$, $C\geq s$
for any final state $C$ of diagram $G$, Eq. (LABEL:eq:forwardscattering-cuts)
summed over its time orders,

$\displaystyle 2\pi\delta\left(Q-\sum_{j\in
C}{\omega}_{j}\right)\,a_{L[G/C]}\left(Q,{\cal L}_{L[G/C]}\right)$
$\displaystyle=$ $\displaystyle\ 2\pi\delta\left(Q-\sum_{j\in
C}{\omega}_{j}\right)$ (2.11) $\displaystyle\times\
\sum_{\tau_{L[G/C]}}\prod_{s=1}^{C}\frac{i}{Q\lambda_{s}-\sum\limits_{j\in
s}\omega_{j}+i\epsilon}\,,$

where we exhibit as well the energy-conserving delta function of final state
$C$. (Note that $C$ denotes both the final state and the integer label on the
vertex that immediately precedes the final state.) At this point, diagram
$a_{L}$ may or may not be connected. What we will show is that if it is not
connected, the sum over its time orderings vanishes when state $C$ is on-
shell.

A useful technique in our analysis, here and below, will be a representation
of the denominators in terms of integrals over times,

$2\pi\delta\left(Q-\sum_{j\in
C}\omega_{j}\right)\,a_{L[G/C]}\,=\,\sum_{P(L[G/C])}\prod_{\alpha_{P}=1}^{C}\int_{-\infty}^{t_{\alpha_{P}+1}}dt_{\alpha_{P}}\
e^{-i\left(\delta_{\alpha_{P},i}Q\;+\;\eta^{(\alpha_{P})}_{j}(\omega_{j}-i\epsilon)\right)\,t_{\alpha}}\,.$
(2.12)

Here, we have written the sum over time orders as the sum over all ordered
permutations, $P(L[G/C])$, of the set of $C$ time variables $\\{t_{\alpha}\\}$
associated with the vertices $\alpha$ of diagram $L[G/C]$. For each order,
$P$, we define $\eta^{(\alpha_{P})}_{j}$ as the incidence matrix of the vertex
$\alpha_{P}$, defined in a given time order by

$\displaystyle\eta^{(\alpha_{P})}_{j}\ $ $\displaystyle=$ $\displaystyle
1\,,\quad{\rm line}\ j\ {\rm enters}\ {\rm vertex}\ \alpha_{P}\,,$
$\displaystyle\eta^{(\alpha_{P})}_{j}\ $ $\displaystyle=$ $\displaystyle-1\,,\
{\rm line}\ j\ {\rm leaves}\ {\rm vertex}\ \alpha_{P}\,,$
$\displaystyle\eta^{(\alpha_{P})}_{j}\ $ $\displaystyle=$ $\displaystyle
0\,,\quad{\rm otherwise}\,.$ (2.13)

All lines are emitted and then absorbed, except for lines that appear in
(final) state $C$. As a result, we have in Eq. (2.12), for any state $s\leq
C$,

$\displaystyle\sum_{\alpha_{P}\leq s}\eta^{(\alpha_{P})}_{j}\ $
$\displaystyle=$ $\displaystyle\ 0\,,\ j\,\hbox to0.0pt{/\hss}{\in}\,s\,,$
$\displaystyle\sum_{\alpha_{P}\leq s}\eta^{(\alpha_{P})}_{j}\ $
$\displaystyle=$ $\displaystyle\ -1\,,\ j\in s\,.$ (2.14)

The term $\delta_{\alpha_{P},i}Q$ in Eq. (2.12) contributes the energy flowing
into the diagram at vertex $i$, which must always be in the diagram specified
by the time order $\tau_{L[G/C]}$, for the final state $C$ to be on-shell.
Using this result and Eq. (2.14), we readily see that each $\alpha_{P}$ time
integral produces the TOPT denominator in Eq. (2.11) for the state immediately
following it, times a phase whose exponent is proportional to the following
time, $t_{\alpha_{P}+1}$. We define $t_{C+1}=\infty$, for the final integral
in each order, which produces the delta function that enforces the condition
that the energy flowing in, $Q$, equals the energy flowing out ($\sum_{j\in
C}\omega_{j}$).

For every ordering $P(L)$ in Eq. (2.12), the energy-conservation delta
function is generated by the final time integral, which is unbounded on both
sides: $\int_{-\infty}^{\infty}dt_{C}$. The vertex associated with this
integral depends, of course, on the specific time order. At this point, it
will be useful to rewrite (2.12) by introducing an auxiliary maximum time,
$t_{max}$ that is the same for all orders. We do this by using the following
identity for each order $P(L)$,

$\displaystyle\int_{-\infty}^{\infty}dt_{C}\,e^{-i\left(\delta_{C,i}Q\;+\;\eta^{(C)}_{j}(\omega_{j}-i\epsilon)\right)\,t_{C}}\prod_{\alpha=1}^{C-1}\int_{-\infty}^{t_{\alpha+1}}dt_{\alpha}\,e^{-i\left(\delta_{\alpha,i}Q\;+\;\eta^{(\alpha)}_{j}(\omega_{j}-i\epsilon)\right)\,t_{\alpha}}$
(2.15) $\displaystyle=\ \left[-i(Q-\sum\limits_{j\in
C}\tilde{\omega}_{j})\right]\,\int_{-\infty}^{\infty}dt_{max}\,\int_{-\infty}^{t_{max}}dt_{C}\,e^{-i\left(\delta_{C,i}Q\;+\;\eta^{(C)}_{j}(\omega_{j}-i\epsilon)\right)\,t_{C}}$
$\displaystyle\times\
\prod_{\alpha=1}^{C-1}\int_{-\infty}^{t_{i+1}}dt_{\alpha}\,e^{-i\left(\delta_{\alpha_{P},i}Q\;+\;\eta^{(\alpha_{P})}_{j}(\omega_{j}-i\epsilon)\right)\,t_{\alpha}}\,.$

On the right-hand side of this equation, the integral over $t_{C}$ produces a
denominator that cancels the overall factor of $-i(Q-\sum\limits_{j\in
C}\tilde{\omega}_{j})$ in Eq. (2.15), while the new, $t_{max}$, integral
reproduces the energy conservation delta function, of the same argument. We
can implement this identity for all time the orders, $P$ in Eq. (2.12).

In the case when the amplitude breaks up into two (or more) disconnected
pieces, $G_{1}$ and $G_{2}$, we may write it as the product of two factors,
with $n_{1}$ and $n_{2}$ vertices each, $n_{1}+n_{2}=C$, with a sum over time
orders, or permutations $P^{(1)}$ and $P^{(2)}$, of the independent
disconnected vertices, whose times can be integrated independently to
$t_{max}$ in Eq. (2.15). For definiteness, we assume that vertex $i$, at which
external energy flows in, attaches to the subdiagram $G_{1}$, with $n_{1}$
vertices. For such a diagram, we then have in place of Eq. (2.12), the
equivalent form

$\displaystyle a_{L[G/C]}\left(Q,{\cal
L}_{L[G/C]}\right)2\pi\delta\left(Q-\sum_{j\in C}{\omega}_{j}\right)$
$\displaystyle=$ $\displaystyle\ [-i(Q-\sum_{j\in
C}{\omega}_{j})]\int_{-\infty}^{\infty}dt_{max}$ (2.16) $\displaystyle\times\
\left(\sum_{P^{(1)}}\prod_{\alpha=1}^{n_{1}}\int_{-\infty}^{t_{\alpha+1}}dt_{\alpha}e^{-i(\delta_{P^{(1)}_{\alpha},i}Q+\eta^{(P^{(1)}_{\alpha})}_{j}(\omega_{j}-i\epsilon))t_{\alpha}}\right)$
$\displaystyle\times\
\left(\sum_{P^{(2)}}\prod_{\alpha=1}^{n_{2}}\int_{-\infty}^{\tilde{t}_{\alpha+1}}d\tilde{t}_{\alpha}e^{-i(\eta^{(P^{(2)}_{\alpha})}_{j}(\omega_{j}-i\epsilon))\tilde{t}_{\alpha}}\right),$

where now we have defined $t_{n_{1}+1}=\tilde{t}_{n_{2}+1}=t_{max}$. Note that
because of this independence, the number of terms in the sums of permutations
is $n_{1}!n_{2}!$, down from $C!=(n_{1}+n_{2})!$.

The connected subdiagrams each possess the properties of the sums over
vertices in Eq. (2.14). As a result, the final time integral $t_{max}$
inherits the same energy-conservation phase, and the integrals in (2.16) can
be done explicitly, giving for each choice of $P^{(1)}$ and $P^{(2)}$, the
factor

$\displaystyle-i\left(Q-\sum_{j\in
C}{\omega}_{j}\right)2\pi\delta\left(Q-\sum_{j\in
C}{\omega}_{j}\right)\times\left(\prod\limits_{s_{2}=1}^{n_{2}}\frac{i}{-\sum\limits_{j\in
s_{2}}{\omega}_{j}+i\epsilon}\right)\left(\prod\limits_{s_{1}=1}^{n_{1}}\frac{i}{Q\lambda_{s,1}-\sum\limits_{j\in
s_{1}}{\omega}_{j}+i\epsilon}\right)\,,$ (2.17)

where $s_{1}$ and $s_{2}$ label states within the two disconnected factors,
and where $\lambda_{s,1}$ is defined by analogy to Eq. (2.6), this time for
the diagram with $n_{1}$ vertices, through which the external energy $Q$
flows. Recall that in the case of an amplitude, $\lambda_{s}=1$ or $0$ only.
Reorganized in this fashion, the time integrals in Eq. (2.16) and the
denominators that appear in Eq. (2.17) lack a denominator that cancels the
overall factor $Q-\sum_{j\in C}\omega_{j}$. For fixed time orders of the full
diagram, such a denominator is always present, but it cancels in the sum over
time orders, and the integrand vanishes almost everywhere in phase space
because energy is conserved.

In summary, we have found that in every term, Eq. (2.17) that results from the
sum over the relative time orders of the two disconnected diagrams while
keeping their internal time orders fixed, the energy conservation delta
function is multiplied by its own argument. It is easy to check that when the
argument of the delta function vanishes, the cut denominators are generically
finite (except in a region with vanishing measure) while the quantity
$\left(Q-\sum_{j\in C}\tilde{\omega}_{j}\right)$ multiplies the delta
function, forcing the integral over the phase space to zero.

##  3 Local Infrared safety

Our interest here is in IR safe weighted cross sections, inclusive cross
sections in which the IR singularities of their exclusive channels cancel
among themselves. In this section, we construct a general expression that
implements this cancellation locally in momentum space. In principle, this can
eliminate the need for infrared regularization.

###  3.1 Reorganized cross sections

When integrated over loop momenta as it stands, the arbitrary weighted cross
section Eq. (2.9) is a sum of infrared divergent terms in general, which,
however, cancel in the sum over final states $C$ for each time ordering
$\tau_{G}$. To make this cancellation manifest, we use the distribution
identity,

$\displaystyle 2\pi\delta(x)=\frac{i}{x+i\epsilon}-\frac{i}{x-i\epsilon}$
(3.1)

to rewrite the TOPT expression for a general weighted cross section,
$\Sigma[f,Q]$, Eq. (2.9). Recalling the identity for sums over states in Eq.
(2.8), we express $\Sigma[f,Q]$ as

$\displaystyle\Sigma[f,Q]$ $\displaystyle=$ $\displaystyle\sum_{G}\
\sum_{\tau_{G}}\,\int d{\cal
L}_{G}\;\mathbb{N}_{\tau_{G}}\;\prod_{i=1}^{N_{G}}\frac{1}{2\omega_{i}}\;\sum_{C}\,\sigma^{(C)}_{\tau_{L[G/C]}\cup\tau_{R[G/C]}}[f,{\cal
L}_{G}]\,.$ (3.2)

In this expression, we sum over all cuts of the vacuum diagram $G$ at fixed
time-order $\tau_{G}$ and have used the independence of
$\mathbb{N}_{\tau_{G}}$ from the choice of $C$. As noted above, the full set
of cuts of any diagram will include in general unphysical cuts, that is, cuts
that include disconnected subdiagrams to the left and/or right of the cut. In
the previous section we have shown, however, that all such terms cancel once
the sum over time orders is carried out. In fact, in Sec. 6 we will show that
we can re-express the cross section in a diagrammatic form in which all such
cuts are absent. For now, however, we continue with this expression, in the
knowledge that further cancellations will occur in a result that we will show
is already infrared finite.

We now apply the identity, Eq. (3.1) to the integrand of (3.2), using (2.9) to
get,

$\displaystyle\sum_{C=1}^{n}\;\sigma^{(C)}_{\tau_{L[G/C]}\cup\tau_{R[G/C]}}[f,{\cal
L}_{G},Q]\ $ $\displaystyle=$ $\displaystyle\sum_{C}\
\prod_{s=C+1}^{n+1}\frac{i}{Q\lambda_{s}-\sum_{j\in s}\omega_{j}-i\epsilon}\
f_{C}(\vec{q}_{1}\ldots\vec{q}_{k_{C}})$ (3.3)
$\displaystyle\times\,\left(\frac{i}{Q\lambda_{s}-\sum_{j\in
C}\omega_{j}+i\epsilon}\,-\,\frac{i}{Q\lambda_{s}-\sum_{j\in
C}\omega_{j}-i\epsilon}\right)$ $\displaystyle\times\
\prod_{s=1}^{C-1}\frac{i}{Q\lambda_{s}-\sum_{j\in s}\omega_{j}+i\epsilon}\,.$

Then, simply collecting terms with a fixed denominator structure yields,

$\displaystyle\sum_{C}\;\sigma^{(C)}_{\tau_{L[G/C]}\cup\tau_{R[G/C]}}[f,{\cal
L}_{G},Q]\ $ $\displaystyle=$ $\displaystyle\
\left(\prod_{s=1}^{n+1}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}+i\epsilon}f_{n+1}\right.$ (3.4) $\displaystyle\left.+\
\sum_{C=1}^{n}\prod_{s=C+1}^{n+1}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}-i\epsilon}(f_{C}-f_{C+1})\prod_{s=1}^{C}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}+i\epsilon}\right.$ $\displaystyle\left.-\
\prod_{s=1}^{n+1}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}-i\epsilon}f_{1}\right)\,.$

In this equality, we have suppressed the arguments of the functions $f_{C}$,
which are the momenta of individual particle lines, and hence linear
combinations of the spatial loop momenta of diagram $G$. We notice that if all
weight functions $f_{C}$ were indeed equal to unity, we would get back the
imaginary part of the forward scattering graph. The first and last terms in
this expression have the analytic structure of the vacuum polarization diagram
and its complex conjugate and hence are individually IR finite [10]. Although
Eq. (3.4) is a simple reorganization of a standard expression, we will show
that it provides a locally finite expression for the set of leptonic
annihilation cross sections under consideration so long as the weight function
satisfies the usual criteria for infrared safety in Eq. (LABEL:eq:irs). We are
not aware of such an expression in the previous literature.

In Eq. (3.4), the sum of terms labeled $C=1\dots n$ are somewhat unusual,
having no energy-conserving delta function. Rather, they are given entirely by
products of denominators with opposite $i\epsilon$ prescriptions. We will
refer to the product of denominators with $+i\epsilon$ as the “generalized
amplitude”, and the product with $-i\epsilon$ as the “generalized conjugate
amplitude”. In the following subsection, we will study how infrared
singularities can arise in TOPT generally, and in the product of generalized
amplitudes and complex conjugates, and go on to verify that the sum in Eq.
(3.4) is infrared safe locally in momentum space without the need for infrared
regularization.

###  3.2 Analysis of pinch surfaces

The individual terms in Eq. (3.4) have the same infrared singularities that
are found in cross sections for fixed final states, which have explicit
energy-conservation delta functions. In that case, singularities can be
identified from solutions to the Landau equations for the amplitude and
complex conjugate for each point in final-state phase space. These solutions
are satisfied on subspaces of momentum space sometimes referred to as pinch
surfaces [15].

The analysis leading to pinch surfaces in amplitudes can be applied to the
spatial integrals in Eq. (3.2) for each choice of order, $\tau_{G}$, and cut,
$C$ in the form of Eq. (3.4), to derive the Landau equations. We begin by
combining state denominators via a Feynman parametrization, chosen so that the
imaginary parts all add with the same sign at every point in parameter space.
This requires that we factor out a $(-1)$ for each denominator in the
generalized conjugate amplitude, with a $-i\epsilon$,

$\displaystyle\int d{\cal
L}_{G}\;\frac{\mathbb{N}_{\tau_{G}}}{\prod_{i=1}^{N}2\omega_{i}}\left(f_{C}-f_{C+1}\right)\;\prod_{s=C+1}^{n+1}\frac{i}{Q-\sum_{j\in
s}\omega_{j}-i\epsilon}\ \prod_{s=1}^{C}\frac{i}{Q-\sum_{j\in
s}\omega_{j}+i\epsilon}$ $\displaystyle=\ \int\frac{d{\cal
L}_{G}}{\prod_{i=1}^{N}2\omega_{i}}\;\frac{\mathbb{N}_{\tau_{G}}\,\left(f_{C}-f_{C+1}\right)\,(-1)^{n-C}\,[d\alpha_{s}]_{n+1}}{\left(\sum_{s=1}^{C}\alpha_{s}\left(Q-\sum_{j\in
s}\omega_{j}\right)-\sum_{s=C+1}^{n+1}\alpha_{s}\left(Q-\sum_{j\in
s}\omega_{j}\right)+i\epsilon\right)^{n+1}}\,,$ (3.5)

where, as usual,

$\displaystyle[d\alpha_{s}]_{n+1}\ =\ n!\,\int_{0}^{1}d\alpha_{n+1}\dots
d\alpha_{1}\,\delta\left(1-\sum_{s=1}^{n+1}\alpha_{s}\right)\,.$ (3.6)

We would like to show that for any infrared safe weight function $f$, Eq.
(3.5) is finite without infrared regularization. To do so, we must identify
the origin and strength of IR singularities in these TOPT expressions.

As in covariant perturbation theory, infrared singularities in Eq. (3.5) can
arise whenever the loop integrals are pinched between coalescing
singularities. In this case, of course, we have three integrals per loop
remaining. Singularities arise from two sources, the product of particle
energies $\omega_{i}$, and from the full parameterized denominator. We note
that each energy factor, $\omega_{i}$ depends quadratically on the spatial
loop momenta of particle $i$, and always produces a pinch at the point
$\omega_{i}=0$, for line $i$ massless.

The identification of pinches from sets of on-shell denominators with lines of
nonzero energy in Eq. (3.5) requires a time-ordered perturbation theory
version of Landau equations, which follow the same pattern as for integrals in
covariant perturbation theory. In the parameterized form, an off-shell
denominator $D_{s}\neq 0$ must have $\alpha_{s}=0$. Derivatives with respect
to each loop momentum component of denominators with $D_{s}=0$ must vanish,

$\displaystyle\frac{\partial}{\partial
l^{\mu}}\,\left[\sum_{s=1}^{C}\alpha_{s}\left(\sum_{i\in
s}\omega_{i}\right)-\sum_{s=C+1}^{n+1}\alpha_{s}\left(\sum_{j\in
s}\omega_{j}\right)\right]\ =\ 0\,.$ (3.7)

Because the derivative of the energy $\omega_{i}$ of a line with respect to
its momentum gives its velocity,
$\vec{\beta}_{i}=\partial\omega_{i}/\partial\vec{p}_{i}$, the Landau equations
are given as linear sums in velocities. For an arbitrary loop momentum $l$, we
can thus write

$\displaystyle\sum_{s=1}^{C}\alpha_{s}\left(\sum_{i\in
s}\eta_{l,i}\,\vec{\beta}_{i}\right)-\sum_{s=C+1}^{n+1}\alpha_{s}\left(\sum_{j\in
s}\eta_{l,j}\,\vec{\beta}_{j}\right)\ =\ 0\,,$ (3.8)

with the $\eta_{l,i,}=\pm 1,0$ incidence matrices,

$\displaystyle\eta_{l.i}\ $ $\displaystyle=$ $\displaystyle\ \frac{\partial
p_{i}^{\mu}}{\partial l^{\mu}}\quad{\rm any}\ \mu\,.$ (3.9)

To be specific, we define the momenta $p_{i}^{\mu}$ to be in the direction of
energy flow, so that $p_{i}^{0}=\omega(\vec{p}_{i})\geq 0$. Note that for the
amplitude this is the direction toward the final state, $C$.

The equations (3.8) can be satisfied for any loop that appears in a subset of
denominators $\\{t_{i}\\}$, $i=1\dots k$, with $t_{1}\leq t_{C}$ and
$t_{k}\geq C+1$, because such terms include denominators with both
“$i\epsilon$” prescriptions. A loop whose on-shell states appear only with
$+i\epsilon$ or only with $-i\epsilon$ cannot give a solution to Eq. (3.8),
unless all lines that carry this loop have collinear momenta. We can think of
such a loop as internal to a jet of collinear moving particles. These
solutions can be given a physical interpretation in the sense of Coleman and
Norton [15, 20], by identifying the Feynman parameters $\alpha_{s}$ as times
so that their products with velocities are translations. The Landau equation
for any loop then describes a sum of translations relative to the origin, in
the direction of the velocities of lines carried by that loop.

For loops that are internal to a given jet, their contributions to Eq. (3.8)
cancel within each state, because $\eta_{li}=-\eta_{li^{\prime}}$ if the line
flows forward along line $i$ within the jet and back along line $i^{\prime}$.
For loops that extend over states with $s<C$ to states with $s>C$, Eq. (3.8)
requires that the contribution of states in these two categories cancel for
the two jets individually. Thus, for the jet along which the loop flows
forward, we have

$\displaystyle\sum_{s=1}^{C}\alpha_{s}-\sum_{s=C+1}^{n+1}\alpha_{s}\ =\ 0\,,$
(3.10)

which has the interpretation that the jet flows forward away from the origin
through a sequence of states up to state $C$, and then backward to the origin
for the same amount of time in the same direction. Thus, for such terms, the
only momentum configurations that produce pinch singularities are those in
which on-shell states differ only by the rearrangement of collinear momenta
and the emission or absorption of lines with zero momentum. These states are
characterized by “jets” of fixed total momentum, accompanied by arbitrary
numbers of soft lines. The succession of on-shell states with different
numbers of jets always results in at least one internal loop of the amplitude
or complex conjugate that flows between two jets. For such a loop, the two jet
velocities in Eq. (3.8) appear times parameters with only positive or only
negative signs, which is inconsistent with a pinch. The only exception is for
loops that carry zero momentum between different jets. Such “soft” loops are
unconstrained by the Landau equations, in both TOPT and covariant perturbation
theory, although as noted above, there is a pinch whenever any line carries
exactly zero momentum [21].

We now turn to the role of off-shell states. Let us refer to the full set of
states $\\{s_{i}\\}$ for time order $\tau_{G}$ of vacuum polarization diagram
$G$ as ${\cal S}=\\{s_{1}\dots s_{n+1}\\}$. At a given pinch surface, states
in ${\cal S}$ are either on-shell, with $Q-\sum_{j\in s_{i}}\omega_{j}=0$, or
off-shell, $Q-\sum_{j\in s_{i}}\omega_{j}\neq 0$. These simple considerations
limit how off-shell states can appear at a pinch surface. First of all, states
adjacent to the external vertices ($i$ and $o$ above) at which momenta flow
into and out of the forward-scattering diagram, can always be off-shell. These
states correspond to the “hard part” of the scattering cross section. For an
arbitrary pinch surface, $\zeta$ there is thus an “earliest” state,
$s^{[\zeta]}_{\rm min}\geq s_{1}$, at which the relevant set of jets first
appears, and a “latest” state, $s^{[\zeta]}_{\max}\leq s_{n+1}$, in the
complex conjugate amplitude, where they last appear.

We next examine when a subset of states can be off-shell at pinch surface
$\zeta$. Let us denote such an ordered subset as
$\Gamma^{[\zeta]}=\\{\sigma^{[\zeta]}_{i}\\}\subset{\cal S}$,
$\sigma^{[\zeta]}_{i}=\sigma^{[\zeta]}_{\rm min}\dots\leq\sigma^{[\zeta]}_{\rm
max}$, and assume that all of these states are off-shell, that is,
$Q-\sum_{j\in\sigma^{[\zeta]}_{i}}\omega_{j}\neq 0$, and consecutive. There
may be a number of these sets; first, let us consider the case with only a
single such set, $\Gamma^{[\zeta]}$. For any such set of off-shell states,
$\Gamma^{[\zeta]}$, we must have two sets of on-shell states: on-shell states
$\\{s^{[\zeta]}\\}_{\Gamma<}$ with $s^{[\zeta]}_{\rm min}\leq
s^{[\zeta]}<\sigma^{[\zeta]}_{\rm min}$ that are before the off-shell states,
and similarly another set of on-shell states, $\\{s^{[\zeta]}\\}_{\Gamma>}$,
all of whose elements are between $\sigma^{[\zeta]}_{\rm max}$ and
$s^{[\zeta]}_{\rm max}$.

Because states are ordered, the off-shell states of $\Gamma^{[\zeta]}$ may be
either entirely in the generalized amplitude or in the generalized conjugate
amplitude, or may extend between them both. In all cases, however, one of the
two sets of states, $\\{s^{[\zeta]}\\}_{\Gamma<,}$ and
$\\{s^{[\zeta]}\\}_{\Gamma_{>}}$ will have only $+i\epsilon$ or only
$-i\epsilon$ in its denominators. We use these considerations to show that at
the pinch surface, $\zeta$, no sets of lines that appear in the states of
$\Gamma^{[\zeta]}$ can carry finite momentum between any pair of distinct jets
of pinch surface $\zeta$.

To see why, consider the case that $\Gamma^{[\zeta]}$ is in the amplitude (all
$+i\epsilon$ in its denominators). We suppose the two jets in question have
velocities $\vec{\beta}_{a}$ and $\vec{\beta}_{b}$, in different directions.
Let us suppose that some set of lines carries finite momenta and connects
these two jets. We will then be able to find at least one loop consisting of
lines within the two jets in all the on-shell states of
$\\{s^{[\zeta]}\\}_{\Gamma<}$, closing the loop with the lines of finite
momentum that appear in the states of $\Gamma^{[\zeta]}$ and lines that appear
only in the off-shell states, $s\leq s^{[\zeta]}_{\rm min}$. The Landau
equation for this loop would have no contributions from the states in
$\Gamma^{[\zeta]}$ or from states with $s\leq s^{[\zeta]}_{\rm min}$, because
the Feynman parameters of off-shell states are all zero. They do of course get
contributions from the on-shell states, $\\{s^{[\zeta]}\\}_{\Gamma<}$. We can
always define the loop in question to flow through one line of jet $a$ in the
direction of the $a$-jet energy flow for each on-shell state. The velocity of
all such lines is the same as the jet velocity, $\vec{\beta}_{a}$. The loop
then flows back, against the direction of the $b$-jet, for which the
corresponding particle velocities are all $\vec{\beta}_{b}$. The resulting
Landau equations, Eq. (3.8), are then

$\displaystyle\sum_{s\in\\{s^{[\zeta]}\\}_{\Gamma<}}\alpha_{s}\,\left(\vec{\beta}_{a}-\vec{\beta}_{b}\right)\
=\ 0\,.$ (3.11)

Because $\vec{\beta}_{a}$ and $\vec{\beta}_{b}$ are in different directions,
there can be no solution to Eq. (3.11) for nonzero parameters $\alpha_{s}$. As
a result, $\zeta$ is not a pinch surface after all. The generalization of this
result to sets $\Gamma^{[\zeta]}$ in the complex conjugate amplitude or
between the amplitude and conjugate is immediate. Several sequences of off-
shell states also follow the same pattern.

The only remaining possibility for an off-shell $\Gamma^{[\zeta]}$ is one for
which some set of off-shell states is associated with one or more internal
loop momenta of a jet, which are not in the direction of the jet. Such a loop
will also take a set of consecutive states $\sigma^{[\zeta]}$ off-shell. We
will therefore need to consider this possibility in our discussion of local
finiteness.

###  3.3 Logarithmic singularities and cancellation

To identify which pinch surfaces result in actual infrared singularities in
individual terms, we must recall the power counting analysis of covariant
perturbation theory [22, 23]. Rather than repeating the details of this
analysis, we can rely on the basic result. For a pinch surface of a cut vacuum
polarization diagram, we identify the space of “normal” variables, which
parameterize the space perpendicular to the pinch surface, and assign a
dimensionless scaling variable (conventionally denoted by $\lambda$) so that
each normal variable vanishes linearly as the scaling variable vanishes. The
fundamental starting point of this analysis, when applied to leptonic
annihilation, which shows that singularities are at worst logarithmic, can be
summarized as

$\displaystyle 2L_{J}+4L_{S}+p_{\rm num}-N_{J}-2N_{S}\ \geq\ 0\,,$ (3.12)

where $L_{J}$ and $N_{J}$ are the total number of loops and lines in jet
subdiagrams, respectively, while $L_{S}$ and $N_{S}$ are the soft loops and
lines, and $p_{\rm num}$ is the scaling dimension of numerator factors at the
pinch surface in question. We will see an example below.

We can apply Eq. (3.12) to TOPT expressions like Eq. (3.5) in a
straightforward fashion. First, using the graphical Euler identity in the form

$\displaystyle L_{J}+L_{S}\ =\ N_{J}+N_{S}-V+1\,,$ (3.13)

we see immediately that Eq. (3.12) may be written equivalently as

$\displaystyle L_{J}+3L_{S}+p_{\rm num}-N_{S}-(V-1)\ \geq\ 0\,,$ (3.14)

where here $V$ is the total number of vertices in the on-shell TOPT diagram
found by contracting all states that are off-shell at this pinch surface. This
is also the number of states that are pinched on-shell. In this expression, of
course, $L_{J}$ remains the total number of internal loops of all jets, and
$L_{S}$ is the number of loops that carry zero momentum in all three remaining
components. For leptonic annihilation [22], the natural normal variables are
all three remaining components of the $L_{S}$ soft loops, $\vec{l}_{\rm
soft}\sim\lambda$, and for each jet loop momentum $\vec{l}_{\rm jet}$,
$l_{\perp,\,{\rm jet}}^{2}\sim\lambda$. The dependence of all denominators is
then linear in $\lambda$. Because the number of denominators is $V-1$, the
inequality of Eq. (3.14) ensures that divergences in TOPT are at worst
logarithmic, just as in covariant perturbation theory.

Notice that it is necessary to saturate the inequality (3.14) for a specific
time order and pinch surface to contribute to an infrared divergence. This
eliminates infrared divergences for time orders where a vertex that connects
on-shell jet lines and/or soft lines appears between off-shell denominators,
simply because choosing such an order sacrifices at least one on-shell state
for an off-shell state. For this reason, in identifying pinch surfaces, we may
capture their leading infrared behavior by their reduced diagrams, found by
shrinking any loop momenta internal to the jets but not in the jet direction
to points. The resulting time-ordered reduced diagrams automatically have the
maximum number of on-shell denominators. The logarithmic nature of infrared
divergences applies to each term in the original expression of the weighted
cross section, Eq. (2.9) as well as to its linear combinations in Eq. (3.4)
and (3.5) for an arbitrary weight function, $f_{C}$.

In summary, a general weighted cross section encounters at worst logarithmic
singularities in individual terms in the sum found by combining Eqs. (3.2) and
(3.4),

$\displaystyle\Sigma[f,Q]$ $\displaystyle=$ $\displaystyle\sum_{G}\
\sum_{\tau_{G}}\,\int d{\cal
L}_{G}\;\mathbb{N}_{\tau_{G}}\;\prod_{i=1}^{N_{G}}\frac{1}{2\omega_{i}}$
(3.15) $\displaystyle\
\times\left(\prod_{s=1}^{n+1}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}+i\epsilon}f_{n+1}\right.$ $\displaystyle\left.+\
\sum_{C=1}^{n}\prod_{s=C+1}^{n+1}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}-i\epsilon}(f_{C}-f_{C+1})\prod_{s=1}^{C}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}+i\epsilon}\right.$ $\displaystyle\left.-\
\prod_{s=1}^{n+1}\frac{i}{Q\lambda_{s}-\sum_{j\in
s}\omega_{j}-i\epsilon}f_{1}\right)\,.$

In this form, we can use the properties of infrared safe weight functions
$f_{C}$, Eq. (LABEL:eq:irs) to show that this expression is locally finite in
loop momentum space if the sum over $C$ is carried out before integration.

We first observe that the first and third terms in parentheses are infrared
finite because their denominators can produce no pinches, just as for the
total cross section. For the remaining terms in the sum, which involve
denominators with both $i\epsilon$ signs, let us start by considering a
“leading” pinch surface, at which every state in Eq. (3.15) is pinched on-
shell. Each pinch surface (leading or not) is at worst logarithmically
divergent in the integral over normal variables. Differences $f_{C}-f_{C+1}$
then need only vanish as any power of the normal variables for the ${\cal
L}_{G}$ integrals to be finite. This is precisely the condition for IR safety
found in Refs. [11, 19]. For any infrared safe weight function, by Eq.
(LABEL:eq:irs), the value of any $f_{C}$ is the same for every state that is
pinched on shell. This ensures that the difference $f_{C}-f_{C+1}$ in Eq.
(3.15) vanishes as a power of the normal variables. The integral in Eq. (3.15)
is then finite for all leading pinch surfaces. To extend this result to pinch
surfaces with intermediate off-shell states, we simply note that for an off-
shell state $\sigma^{[\zeta]}$, the corresponding denominator
$D_{\sigma^{[\zeta]}}$ is real, and the terms proportional to
$f_{\sigma^{[\zeta]}}$ cancel between conjugate terms in (3.15). The sum of
all terms in Eq. (3.15) is thus finite at an arbitrary pinch surface. This is
what we set out to demonstrate.

It is worth noting that a special class of event weights are those that count
jet final states. These weights take the value unity when the state satisfies
some criterion, and zero elsewhere. For IR safe jet algorithms, the weights
will be identical for all states associated with a given pinch surface [22].
For such states, the differences $f_{C}-f_{C+1}$ are exactly zero for the full
range of momentum space where the weight function is unity for both states, a
range that includes all pinch surfaces.

###  3.4 All-order example: energy in a “cone”

It is useful to give an explicit example that illustrates the local finiteness
of Eq. (3.4) for a specific infrared safe cross section at all orders. To this
end, we begin by introducing an abbreviated notation for energy denominators,

$\displaystyle D_{s}\ =\ E_{s}-\sum_{j\in s}\omega_{j}\,,$ (3.16)

where in our case, $E_{s}=Q\lambda_{s}$, with $\lambda=\pm 1,0$. In these
terms, a contribution to the integrand for a general weighted cross section,
Eq. (3.4) becomes

$\displaystyle\sum_{C=1}^{n+1}\,\sigma^{(C)}_{\tau_{G}}[f,{\cal L}_{G},Q]$
$\displaystyle=$ $\displaystyle\
\left(\prod_{s=1}^{n+1}\frac{i}{D_{s}+i\epsilon}f_{n+1}+\
\sum_{C=1}^{n}\prod_{s=C+1}^{n+1}\frac{i}{D_{s}-i\epsilon}(f_{C}-f_{C+1})\prod_{s=1}^{C}\frac{i}{D_{s}+i\epsilon}\right.$
(3.17) $\displaystyle\left.-\
\prod_{s=1}^{n+1}\frac{i}{D_{s}-i\epsilon}f_{1}\right)\,.$

To illustrate how Eq. (3.17) provides a locally finite sum, we consider the
example of a “cone” weight, defined as the energy flowing into a cone fixed in
space,

$\displaystyle f_{s}^{(\Omega)}\ =\ \sum_{i\in_{s}\Omega}\omega_{i}\,.$ (3.18)

Here, $i\in_{s}\Omega$ means that particle $i$ flows into angular region
$\Omega$ in state $s$. For simplicity, we consider angular region $\Omega$ to
be fixed, so that this quantity is not a jet cross section in the usual sense.
Clearly, this weight for state $s$ is related to the denominator $D_{s}$ by

$\displaystyle f_{s}^{(\Omega)}\ =\ Q\,-\,\sum_{i\hbox
to0.0pt{/\hss}\in_{s}\Omega}\omega_{i}\,-\,D_{s}\,,$ (3.19)

where $\sum_{i\hbox to0.0pt{/\hss}\in_{s}\Omega}$ sums the on-shell energies
of all the particles of state $s$ that are outside cone $\Omega$. This leads
to a relation between weights in consecutive final states,

$\displaystyle f_{C}^{(\Omega)}-f_{C+1}^{(\Omega)}\ $ $\displaystyle=$
$\displaystyle\ D_{C+1}\,-\,D_{C}\,+\,\sum_{i\hbox
to0.0pt{/\hss}\in_{C+1}\Omega}\omega_{i}\ -\ \sum_{i\hbox
to0.0pt{/\hss}\in_{C}\Omega}\omega_{i}$ (3.20) $\displaystyle\equiv$
$\displaystyle D_{C+1}\,-\,D_{C}\,+\,\delta^{(\Omega)}\omega_{C}\,,$

where in the first relation we use (3.19), and in the second we define the
quantity $\delta^{(\Omega)}\omega_{C}$. As we see from its definition,
$\delta^{(\Omega)}\omega_{C}$ counts the energy of particles that are radiated
outside the cone in the transition from state $C$ to state $C+1$, or that was
outside the cone in state $C$, but whose energy is absorbed into the cone in
the transition. We then have, from the general relation, Eq. (3.17),

$\displaystyle\sum_{C=1}^{n+1}\,\sigma^{(C)}_{\tau_{G}}[f^{(\Omega)},{\cal
L}_{G},Q]$ $\displaystyle=$ $\displaystyle\
\left(\prod_{s=1}^{n+1}\frac{i}{D_{s}+i\epsilon}\left(Q\,-\,\sum_{i\hbox
to0.0pt{/\hss}\in_{n+1}\Omega}\omega_{i}\,-\,D_{n+1}\right)\right.$ (3.21)
$\displaystyle\left.+\
\sum_{C=1}^{n}\prod_{s=C+1}^{n+1}\frac{i}{D_{s}-i\epsilon}\left(D_{C+1}\,-\,D_{C}\,+\,\delta^{(\Omega)}\omega_{C}\right)\prod_{s=1}^{C}\frac{i}{D_{s}+i\epsilon}\right.$
$\displaystyle\left.-\
\prod_{s=1}^{n+1}\frac{i}{D_{s}-i\epsilon}\left(Q\,-\,\sum_{i\hbox
to0.0pt{/\hss}\in_{1}\Omega}\omega_{i}\,-\,D_{1}\right)\right)\,.$

On the right, all terms $D_{i}$ cancel, using the distribution identity,

$\displaystyle
D_{i}\,\left(\frac{i}{D_{i}-i\epsilon}\,-\,\frac{i}{D_{i}+i\epsilon}\right)\
=\ D_{i}\,2\pi\delta\left(D_{i}\right)\ =\ 0\,.$ (3.22)

We thus have for the cone cross section an expression that depends on energies
flowing into the cone in the first and last state, and on the energy transfer
variables, $\delta\omega_{C}$,

$\displaystyle\sum_{C=1}^{n+1}\,\sigma^{(C)}_{\tau_{G}}[f^{(\Omega)},{\cal
L}_{G},Q]$ $\displaystyle=$ $\displaystyle\
\prod_{s=1}^{n+1}\frac{i}{D_{s}+i\epsilon}\left(Q-\sum_{i\hbox
to0.0pt{/\hss}\in_{n+1}\Omega}\omega_{i}\right)\ -\
\prod_{s=1}^{n+1}\frac{i}{D_{s}-i\epsilon}\left(Q-\sum_{j\hbox
to0.0pt{/\hss}\in_{1}\Omega}\omega_{j}\right)$ (3.23) $\displaystyle\hskip
14.22636pt+\
\sum_{C=1}^{n}\prod_{s=C+1}^{n+1}\frac{i}{D_{s}-i\epsilon}\,\delta^{(\Omega)}\omega_{C}\,\prod_{s=1}^{C}\frac{i}{D_{s}+i\epsilon}\,.$

For this expression, we recall that at an arbitrary pinch surface, all finite
energy is carried in jets of exactly collinear particles. For any such
configuration, if state $C$ and state $C+1$ are both on-shell,
$\delta^{(\Omega)}\omega_{C}=0$ because consecutive states have exactly the
same energy flow. That is, the quantities $\delta^{(\Omega)}\omega_{C}$ vanish
for every such final state $C$. The contributions of these terms are
integrable.

For nonzero values of any set of $\delta^{(\Omega)}\omega_{C}$ in Eq. (3.23),
following the general argument of the previous subsection, we consider a set
of states $\Gamma^{[\zeta]}$, that are off-shell at an arbitrary pinch surface
$\zeta$, due to loop momenta circulating in the subdiagram of a jet or between
soft lines. As above, on-shell states appear both before and after
$\Gamma^{[\zeta]}$. Since all jets must appear with the same total cone energy
in each on-shell state at the pinch surface, all nonzero contributions to
$\delta^{(\Omega)}\omega_{C}$ must cancel as we sum over off-shell states
$\Gamma^{[\zeta]}$. Thus, the product of off-shell denominators associated
with states $\Gamma^{[\zeta]}$ is multiplied by a sum of energy transfers
$\delta^{(\Omega)}\omega_{C}$ that cancel at the pinch surface.

It is worth noting that in the special case where $\Omega$ grows to the entire
two-sphere, there is no radiation outside the cone and all the
$\delta^{(\Omega)}\omega_{C}$ are zero, and we find

$\displaystyle\sum_{C=1}^{n+1}\,\sigma^{(C)}_{\tau_{G}}[f^{(S_{2})}]$
$\displaystyle=$ $\displaystyle\
Q\,\left[\prod_{s=1}^{n+1}\frac{i}{D_{s}+i\epsilon}\ -\
\prod_{s=1}^{n+1}\frac{i}{D_{s}-i\epsilon}\right]\,.$ (3.24)

This, as expected, is the energy times the imaginary part of the uncut TOPT
diagram, in our notation, the total cross section.

###  3.5 Low-order example

To illustrate the formalism further with an explicit example, we consider the
two next-to-lowest-order vacuum polarization diagrams in Fig. 2.

(1) (2)

Figure 2: Two loop vacuum polarization diagrams: (1) vector exchange, (2)
fermion self-energy.

For simplicity, we take the trace on the external vector current indices, and
we do not consider color in our examples. From the discussion above, the
contributions of these diagrams to a general weighted cross section are given
by

$\displaystyle\Sigma_{i}[f,Q]\ $ $\displaystyle=$ $\displaystyle\
ig^{2}\,\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\,\frac{d^{3}\vec{k}^{\prime}}{(2\pi)^{3}}\frac{\mathbb{N}_{i}}{2^{4}\omega_{k}\,\omega_{q-k}\,\omega_{k^{\prime}}\,\omega_{k-k^{\prime}}}\,\left(\delta_{i1}\,\frac{1}{2\omega_{q-k^{\prime}}}+\delta_{i2}\,\frac{1}{2\omega_{k}}\right)$
$\displaystyle\times\left[\frac{f_{iA}}{D^{+}_{iC}\,D^{+}_{iB}\,D^{+}_{iA}}\
-\ \frac{f_{iC}}{D^{-}_{iC}\,D^{-}_{iB}\,D^{-}_{iA}}\ +\
\frac{f_{iB}-f_{iA}}{D^{-}_{iC}\,D^{-}_{iB}\,D^{+}_{iA}}\ +\
\frac{f_{iC}-f_{iB}}{D^{-}_{iC}\,D^{+}_{iB}\,D^{+}_{iA}}\right]\,,$

where $i=1$ refers to the gluon exchange diagram and $i=2$ to the fermion self
energy diagram. 222The case where the two-particle weights $f_{A}$ and $f_{C}$
and the three-particle state weight $f_{B}$ are defined to give unity for a
two-jet final state and zero otherwise, is of interest. An example is the
original cone jet cross section of Ref. [24] at this order. Using the symmetry
between the two-particle states, $A$ and $C$, the expression, Eq.
(LABEL:eq:sigma_i) is readily seen to reduce to the total cross section minus
the three-jet cross section, as noted early on in Ref. [25], due to the exact
cancellation of three- and two-particle momentum configurations in the two-jet
region. At this order, the resulting expression needs no infrared
regularization.

In terms of the momentum assignments shown in the figure, energy denominators
in Eq. (LABEL:eq:sigma_i) are given by

$\displaystyle D^{\pm}_{1A}\ $ $\displaystyle=$ $\displaystyle\ D^{\pm}_{2A}\
=\ Q-2\omega_{k}\pm i\epsilon$ $\displaystyle D^{\pm}_{1B}\ $ $\displaystyle=$
$\displaystyle\ D^{\pm}_{1B}\ =\
Q-\omega_{k}-\omega_{k^{\prime}}-\omega_{k-k^{\prime}}\pm i\epsilon\,,$
$\displaystyle D^{\pm}_{1C}\ $ $\displaystyle=$ $\displaystyle\
Q-2\omega_{k^{\prime}}\pm i\epsilon\,,$ $\displaystyle D^{\pm}_{2C}\ $
$\displaystyle=$ $\displaystyle\ Q-2\omega_{k}\pm i\epsilon\,.$ (3.26)

The numerators simplify significantly in TOPT, where we must evaluate line
momenta on-shell, $k^{2}=k^{\prime}{}^{2}=0$, and the $\mathbb{N}_{i}$ in Eq.
(LABEL:eq:sigma_i) become

$\displaystyle\mathbb{N}_{1}\ $ $\displaystyle=$ $\displaystyle\
-32\left(k^{\prime}\cdot(k-q)\right)\left(k\cdot(k^{\prime}-q)\right)\,,$
$\displaystyle\mathbb{N}_{2}\ $ $\displaystyle=$ $\displaystyle\ -32\,k\cdot
k^{\prime}\,q\cdot k\,.$ (3.27)

To exhibit the pinch surfaces of these diagrams, we can choose spherical
coordinates, with $\vec{k}$ in the $\hat{z}$ ($\theta_{k}=0$) direction, and
the polar angle, $\theta^{\prime}$ of $\vec{k}^{\prime}$ measured relative to
$\hat{z}$. It is now natural to use the notation $k=|\vec{k}|=\omega_{k}$, and
$k^{\prime}=|\vec{k}^{\prime}|=\omega_{k}^{\prime}$, but to leave
$\omega_{k-k^{\prime}}$ to represent $|\vec{k}-\vec{k}^{\prime}|$, which is
given by

$\displaystyle\omega_{k-k^{\prime}}\ $ $\displaystyle=$
$\displaystyle\sqrt{k^{2}+k^{\prime}{}^{2}-2kk^{\prime}\cos\theta^{\prime}}$
(3.28) $\displaystyle=$
$\displaystyle\sqrt{(k-k^{\prime})^{2}+2kk^{\prime}(1-\cos\theta^{\prime})}\,.$

We assume that our weight functions can be expressed in terms of these
variables only.

In Eq. (LABEL:eq:sigma_i), the first and second terms in square brackets are
free of collinear pinches altogether, simply because their energy denominators
enter with the same sign for the $i\epsilon$s. We denote the remaining,
potentially singular, terms as $\hat{\Sigma}_{1}$ and $\hat{\Sigma}_{2}$. For
$\hat{\Sigma}_{1}$, corresponding to gluon exchange, Fig. 2(1), we have

$\displaystyle\hat{\Sigma}_{1}[f,Q]\ $ $\displaystyle=$ $\displaystyle\
i\,\frac{g^{2}}{8\pi^{4}}\,\int_{0}^{\infty}\,dk\,dk^{\prime}\,\int_{-1}^{1}d\cos\theta^{\prime}\;\frac{k^{\prime}\cdot(q-k)\,k\cdot(q-k^{\prime})}{\omega_{k-k^{\prime}}}$
$\displaystyle\times\left[\frac{f_{1B}(k,k^{\prime})-f_{1A}(k)}{(Q-2k^{\prime}-i\epsilon)\,(Q-k-k^{\prime}-\omega_{k-k^{\prime}}-i\epsilon)\,(Q-2k+i\epsilon)}\right.$
$\displaystyle\left.+\
\frac{f_{1C}(k^{\prime})-f_{1B}(k,k^{\prime})}{(Q-2k^{\prime}-i\epsilon)\,(Q-k-k^{\prime}-\omega_{k-k^{\prime}}+i\epsilon)\,(Q-2k+i\epsilon)}\right]\,,$

where we have indicated the momentum dependence of the weight functions. For
non-zero numerators, this integral has both collinear and soft pinches. There
is a collinear pinch at $k=Q/2,\,\theta^{\prime}=0$, with $|\vec{k}|\
>|\vec{k}^{\prime}|$, between the second and third denominators of the first
term in square brackets. Similarly, a pinch appears at
$k^{\prime}=Q/2,\,\theta^{\prime}=0$ with $|\vec{k}^{\prime}|\ >|\vec{k}|$
between the first and second denominators of the second term in square
brackets. These limits correspond to the gluon parallel to the quark (when
$k=Q/2$) or the antiquark (when $k^{\prime}=Q/2$). The soft pinch, at which
all three denominators vanish, appears in both terms when $k$ and $k^{\prime}$
both approach $Q/2$ with $\theta^{\prime}=0$.

All of these pinches lead to logarithmically divergent power counting,
corresponding to two denominators vanishing while two variables ($k$ or
$k^{\prime}$ and $\theta^{\prime}$) approach the endpoints of their
integration regions. In all three cases, of course, the differences of the
weight functions in the numerators vanish, so long as they depend only on
overall energy flow. A simple example would be a weight function like the
thrust, for which $f_{A}=1$, while $f_{B}\sim
1-2k\cdot(k-k^{\prime})/Q^{2}\sim 1-2kk^{\prime}(1-\cos\theta^{\prime})/Q^{2}$
near $\theta^{\prime}=0$.

For $\hat{\Sigma}_{2}$, since the states $A$ and $C$ are identical for diagram
2, we can write

$\displaystyle\hat{\Sigma}_{2}[f,Q]\ $ $\displaystyle=$ $\displaystyle\
i\,\frac{g^{2}}{8\pi^{4}}\,\int_{0}^{\infty}\,dk\,dk^{\prime}\,\int_{-1}^{1}d\cos\theta^{\prime}\;\frac{kk^{\prime}{}^{2}Q(1-\cos\theta^{\prime})}{\omega_{k-k^{\prime}}}$
(3.30)
$\displaystyle\times\left[\frac{f_{2B}(k,k^{\prime})-f_{2A}(k)}{(Q-2k-i\epsilon)\,(Q-k-k^{\prime}-\omega_{k-k^{\prime}}-i\epsilon)\,(Q-2k+i\epsilon)}\right.$
$\displaystyle\left.+\
\frac{f_{2A}(k)-f_{2B}(k,k^{\prime})}{(Q-2k-i\epsilon)\,(Q-k-k^{\prime}-\omega_{k-k^{\prime}}+i\epsilon)\,(Q-2k+i\epsilon)}\right]\,.$
$\displaystyle=$ $\displaystyle\
i\,\frac{g^{2}}{8\pi^{4}}\,\int_{0}^{\infty}\,dk\,dk^{\prime}\,\int_{-1}^{1}d\cos\theta^{\prime}\;\frac{kk^{\prime}{}^{2}Q(1-\cos\theta^{\prime})}{\omega_{k-k^{\prime}}}$
$\displaystyle\times\left[\frac{f_{2B}(k,k^{\prime})-f_{2A}(k)}{(\omega_{k-k^{\prime}}-(k-k^{\prime}))^{2}}(2\pi
i)\delta\left(Q-k-k^{\prime}-\omega_{k-k^{\prime}}\right)\right]\,.$

In the second relation, we have used the delta function to evaluate the
squared denominator, $Q-2k$. This denominator is negative semi-definite and
vanishes only at the integration endpoint, $\cos\theta^{\prime}\to 1$, where
$k^{\prime}$ is collinear to $k$. Although it has an apparent double pole at
$2k=Q$, the explicit numerator factor of $1-\cos\theta^{\prime}$ reduces this
to a single pole, and logarithmic power counting. This relative suppression is
a contribution “$+1$” to the term $p_{\rm num}$ to infrared power counting in
Eq. (3.12), applied to the pinch surface where the self-energy consists of two
collinear lines. Here again, in Eq. (3.30), the difference of weight functions
in the numerator vanishes for any infrared safe weight and renders the
integral finite. This confirms that in the original form of the integration,
the two- and three-particle singularities cancel. For this example, the
suppression appears locally in loop momentum space, with the standard form of
the self energy subdiagram. At higher orders, however, the suppression
requires in general integration over the internal loop momentum of the uncut
self-energies ($k^{\prime}$ here), to realize their contributions to the
suppression factor $p_{\rm num}$ in Eq. (3.12). For diagrams with more than a
single self-energy on cut lines, we believe it will be natural to use
alternative integrands for self-energies and their counterterms, which
eliminate higher-order poles for the single-particle final state, as described
for amplitudes in Refs. [5, 6].

##  4 Posets in TOPT

We now return to the question of pseudo-physical cuts in TOPT. We recall that
a pseudo-physical cut of a time-ordered graph disconnects the diagram into
more than two connected parts. In Sec 2.2, we saw that the cross section
evaluated on any pseudo-physical cut vanishes upon summing over time orders
for the fixed cut. In this section, we develop a poset formalism in order to
show how pseudo-physical cuts can be avoided entirely. We note that posets
have also been useful in discussions of eikonal exponentiation [26] and
coordinate-space amplitudes [27].

The treatment that follows has much in common with the recent discussion of
“flow-oriented” [17] and “cross-free” representations of perturbation theory
[18]. Here, we work directly from TOPT to find a number of related results,
which, we believe, will provide intuition for applications, one of which we
discuss in Sec. 6.

In this section, we will review some standard poset terminology, introduce our
method, and show how it works in representative examples involving vacuum
polarization diagrams. A general discussion will be provided in Sec. 5.

###  4.1 Definitions

The method we will use to reorganize TOPT is based on the construction of
partially ordered sets (posets) on the vertices of the diagrams. To do so, we
impose a binary relationship among the vertices, which partitions the set of
time orders into distinct posets. A poset is a set together with a binary
relationship. For any TOPT diagram, our binary relation can be defined from
the incidence matrix introduced in the integral for an arbitrary amplitude in
Eq. (2.12) and defined in Eq. (2.13). We denote the incidence matrix by
$\eta^{(b)}_{j}$, where the superscript $(b)$ represents a vertex and the
subscript $j$ represents a line. Entry $\eta^{(b)}_{j}$ is $+1$ if the line
$j$ enters vertex $b$, $-1$ if the line $j$ exits vertex $b$ and, zero
otherwise. Consider a TOPT diagram, given in the notation of Eq. (2.12). If
there is a line joining vertices $b_{i},b_{j}$, and $t_{b_{i}}\geq t_{b_{j}}$,
for each line $k$ connecting both $b_{i},b_{j}$,
$\eta^{(b_{i})}_{k}=1=-\eta^{(b_{j})}_{k}$. On the other hand, if
$t_{b_{j}}\geq t_{b_{i}}$ for each line $k$ between $b_{i},b_{j}$,
$\eta^{(b_{j})}_{k}=1=-\eta^{(b_{i})}_{k}$.

We begin by defining an ordering between the vertices $\geq$,
$V=\\{b_{1},b_{2},b_{3}\ldots,b_{n}\\}$ of a Feynman graph $G$. It satisfies:

  * •

If $b_{i}$ and $b_{j}$ are connected directly by one or more lines, then
either $b_{i}\geq b_{j}$ or $b_{j}\geq b_{i}$, i.e., $b_{i}$ and $b_{j}$ are
ordered by $\geq$. The ordering $\geq$, abstracted from time ordering is
transitive, $a\geq b,\ b\geq c\ \rightarrow a\geq c$. It follows from the
transitivity of the binary that if two vertices $b_{i}$ and $b_{j}$ are
related by a sequence of increasing times, they are ordered by $\geq$.

  * •

If $b_{i}$ and $b_{j}$ are not connected by a line or a sequence of vertices
that are increasing, then $b_{i}\not\geq b_{j}$ and $b_{j}\not\geq b_{i}$,
i.e., $b_{i}$ and $b_{j}$ are not ordered by $\geq$. It will be useful to
introduce the notation $b_{i}\sim b_{j}$, to mean $b_{i}\not\geq b_{j}$ and
$b_{j}\not\geq b_{i}$. 333We shall assume that our TOPT diagrams do not
include lowest-order tadpole subdiagrams, in which a single line emerges and
is absorbed at a single vertex. Such lines, which are removed by normal
ordering, cannot be assigned a poset (or time) order.

We will say that $b_{i}>b_{j}$ if $b_{i}\geq b_{j}$ and $b_{i}\neq b_{j}$
i.e., we require that $b_{i},b_{j}$ are distinct in addition to being related.

Having introduced the binary relation, we are ready to define the posets. A
poset $D$ is the ordered pair $D=(V,\geq)$, where the set $V$ is the set of
vertices of a Feynman graph, taken together with the binary relationship
$\geq$.

It is clear that multiple time orders are compatible with a given poset, and
in each time order, we can uniquely identify an underlying poset. Therefore,
posets partition the set of all time orders.

Our posets have the same information as the incidence matrix in the TOPT
amplitudes, Eqs. (2.12), (2.14). It is clear that it is possible to read off a
poset, given an incidence matrix. Also notice that since the integrand in Eq.
(2.12) only depends on the incidence matrix, this is also fixed by the poset.
Therefore, posets are in one-to-one correspondence with incidence matrices.

Let us make a few more definitions that relate to posets in order to develop
this language more fully. An idea that we will find use for in what follows is
that of minimal/maximal elements. An element $b_{i}$ is said to be a minimal
element of a poset $D$ if $\forall b_{j}\in D,b_{j}\geq b_{i}$ or $b_{j}\sim
b_{i}$. A sequence of reducing vertices, starting from any vertex, always ends
at a minimal element.

An element $b_{i}$ is said to be a maximal element of a poset $D$ if $\forall
b_{j}\in D,b_{i}\geq b_{j}$ or $b_{i}\sim b_{j}$. A sequence of increasing
vertices, starting from any vertex, always ends at a maximal element.

We will also find it useful to distinguish how two elements are related to
each other. In particular if $a<b$, then either there exists a single line
between the vertices $a$ and $b$ or there exists a sequence of increasing,
connected vertices that starts at $a$ and ends at $b$. It will be useful to
reserve the symbol $\prec$ for the first scenario when $a$ and $b$ are
directly connected through a line. It is therefore natural to define a
“covering relation”. For elements $x,y\in V$, we say that $y$ covers $x$ if
$y>x$ and there is no $z\in V$ such that $y>z>x$. We will denote it by $x\prec
y$. In a given graph, if $x\prec y$, then $x,y$ are vertices that are directly
connected by one or more lines, while $y\geq x$ includes the possibility that
$x,y$ are vertices that are connected by a sequence of lines.

Having defined our posets, we will now go on to show how to use them to
eliminate pseudo-physical cuts in some explicit examples of vacuum
polarization diagrams. A generalization to all processes at arbitrary loops
will be given subsequently in Sec. 5.

###  4.2 Examples in vacuum polarization diagrams

Let us now look at some low order examples that will show how the time
integrals over minimal and maximal vertices can be carried out explicitly
within a given poset. For definiteness, our examples will be drawn from the
vacuum polarization diagrams of the lepton annihilation processes. They
illustrate features of quite general application and open the way to an
analogous development for general amplitudes and Green functions in the next
section.

For vacuum polarization diagrams, we label vertices by permutations of the
list, $V=(i,b_{1},b_{2},\dots,b_{n},o)$, where the external momentum flows
into the diagram at vertex $i$, and out at vertex $o$. We can then divide all
posets into four classes.

  * •

Posets with $o>i$ for which the external energy arrives at vertex $i$ before
it leaves at vertex $o$.

  * •

Posets with $i>o$ for which the external energy leaves at vertex $o$ before it
arrives at vertex $i$

  * •

Posets with $o\sim i$, which split into two sub-classes by the imposition of
an additional relationship in a time order: $t_{o}>t_{i}$ and $t_{i}>t_{o}$.
At the level of the poset, there is no definite order among the vertices
$i(o)$, where the external energy enters(leaves). However, imposing the
additional relationship $t_{o}>t_{i}$ restricts us to time orders within the
poset where the external energy arrives at vertex $i$ before it leaves at
vertex $o$.

As we will see, there are unphysical cuts that one would like to eliminate in
all four classes of posets. In what follows we will focus on time orders that
satisfy $t_{o}>t_{i}$. These are all time orders in posets with $o>i$, and in
posets with $o\sim i$, with time orders $t_{o}>t_{i}$. These time orders carry
at least one physical cut but contain both physical and unphysical cuts in
general. There are no denominator singularities in time orders with
$t_{i}>t_{o}$, and the integrand is completely real (and negative semi-
definite).

Using the notation introduced in Eq. (2.5), we consider here diagrammatic
integrands, including numerator factors, written as sums over time orders
$\tau_{G}$ of vacuum polarization integrands for an arbitrary diagram, $G$,

$\displaystyle\pi_{G}[Q,{\cal L}_{G}]\ $ $\displaystyle=$
$\displaystyle\sum_{\tau_{G}}\;\mathbb{N}_{\tau_{G}}\;\pi_{\tau_{G}}(Q,{\cal
L}_{G})\,.$ (4.1)

We observe that although the notation $\mathbb{N}_{\tau_{G}}$ suggests that
the numerator factor depends on the time order, it is the same for all time
orders within a poset $D$. The numerator factor of a graph in covariant
perturbation theory is derived from the Feynman rules and the numerator factor
in TOPT is obtained by replacing the energy of a line $j$ by the on-shell
value of the energy $\pm\omega_{j}$, where the sign is determined by the
direction of the line alone. Therefore, the numerator factor is the same for
all time orders within a given poset. We will henceforth use the notation
$\mathbb{N}_{D}$ to emphasize the dependence of the numerator factor.

Given that posets partition time orders into non-overlapping subsets, we may
write $\pi_{G}$ in Eq. (4.1) as

$\displaystyle\pi_{G}(Q,{\cal L}_{G})\ $ $\displaystyle=$
$\displaystyle\sum_{\tau_{G}}\;\mathbb{N}_{D}\;\pi_{\tau_{G}}(Q,{\cal L}_{G})$
(4.2) $\displaystyle=$
$\displaystyle\sum_{D}\;\mathbb{N}_{D}\;\sum_{\tau_{D}}\pi_{\tau_{D}}(Q,{\cal
L}_{G})\,,$

where the second equality follows from the fact that the sum of all time
orders $\tau_{G}$ of graph $G$ is the sum over posets, $D$, and within each
poset the sum over time orders contained in $D$,

$\sum_{\tau_{G}}=\sum_{D}\sum_{\tau_{D}}\;.$ (4.3)

Our interest is in obtaining an expression for $\pi_{D}(Q,{\cal L}_{G})$ that
is free of pseudo-physical cuts. We first notice that if a poset $D$ has a
single minimum $i$ (where the external momentum flows in), and a single
maximum $o$ (where the external momentum flows out), every time order we
induce on the poset exclusively carries physical cuts. To see this, we argue
that in an arbitrary time order, and on any cut $C$ of that time order, we may
start at vertex $v$ that lies to the left of $C$, and follow a sequence of
vertices that are less than $v$ (in the poset ordering) to reach the unique
minimum $i$. Therefore every vertex on the left of the cut $C$ is ordered with
respect to $i$. A similar argument shows that every vertex to the right of
such a cut is also ordered with respect to the unique maximum $o$. There may
or may not be more than one time order for such a poset, depending on the
incidence matrix, but every cut of such a time-ordered diagram will be
physical.

If, on the other hand, there is more than one extremum in a poset, the poset
is guaranteed to include time orders that differ by exchanging the relative
positions of the minima or maxima among themselves. Our strategy in handling
posets without a unique minimum or a unique maximum element is to reduce the
problem to one where there exists a unique extremum. This will involve
combining time orders that exchange extrema and other unordered pairs of
vertices.

As a warm-up consider an example with a unique maximum but two minima, the
three loop graph in Fig. 3. Although the figure shows a particular time order
of its six vertices, $i$, $o$, and $b_{1}\dots b_{4}$, it represents only the
poset structure. We recall that in TOPT every line carries energy forward in
time (to the right in a TOPT diagram). This information provides us with an
incidence matrix, and hence a poset. In this case, the relevant binary
relations are

$\displaystyle\hskip 28.45274pti<b_{3}<b_{4}<o\,,$ (4.4) $\displaystyle\hskip
28.45274ptb_{1}<b_{2}<b_{4}\,,$ $\displaystyle\hskip 28.45274ptb_{1}<b_{3}\,,$
$\displaystyle\hskip 28.45274ptb_{1}\sim i\,,$ $\displaystyle\hskip
28.45274ptb_{2}\sim i,b_{3}\,.$

Within this poset, there are two minima, $b_{1}$ and $i$. Vertex $b_{1}$ is
covered by both $b_{2}$ and $b_{3}$. There are five time orders. Two time
orders have $t_{b_{2}}>t_{b_{3}}$, which we label by permutation as

$\displaystyle t_{b_{1}}<t_{i}\quad(1i324o)\,,$ (4.5) $\displaystyle
t_{i}<t_{b_{1}}\quad(i1324o)\,.$

The other three time orders, which have $t_{b_{3}}>t_{b_{2}}$, are given by

$\displaystyle t_{i}<t_{b_{1}}<t_{b_{2}}<t_{b_{3}}\quad(i1234o)\,,$ (4.6)
$\displaystyle t_{b_{1}}<t_{i}<t_{b_{2}}<t_{b_{3}}\quad(1i234o)\,,$
$\displaystyle t_{b_{1}}<t_{b_{2}}<t_{i}<t_{b_{3}}\quad(12i34o)\,.$

In TOPT, after time integration, each of these orderings gives one or more
pseudo-physical cuts. For example, the contribution of $(i1324o)$ can be
written as

$\displaystyle\pi_{(i1324o)}\ $ $\displaystyle=$ $\displaystyle\
e^{iQt_{o}}\;\int_{-\infty}^{t_{o}}dt_{b_{4}}e^{-i(\omega_{ch}-\omega_{d}+i\epsilon)t_{b_{4}}}\int_{-\infty}^{t_{b_{4}}}dt_{b_{2}}e^{-i(\omega_{ef}-\omega_{h}+i\epsilon)t_{b_{2}}}\int_{-\infty}^{t_{b_{2}}}dt_{b_{3}}e^{-i(\omega_{ag}-\omega_{f}+i\epsilon)t_{b_{3}}}$
$\displaystyle\times\
\int_{-\infty}^{t_{b_{3}}}dt_{b_{1}}e^{-i(-\omega_{efg}+i\epsilon)t_{b_{1}}}\int_{-\infty}^{t_{b_{1}}}dt_{i}e^{-i(Q-\omega_{ab}+i\epsilon)t_{i}}$
$\displaystyle=$ $\displaystyle\
\frac{i}{Q-\omega_{bd}+i\epsilon}\,\frac{i}{Q-\omega_{bch}+i\epsilon}\,\frac{i}{Q-\omega_{bcef}+i\epsilon}\frac{i}{Q-\omega_{abefg}+i\epsilon}\,\frac{i}{Q-\omega_{ab}+i\epsilon}\,,$

where for compactness we use the notation

$\displaystyle\omega_{ab\dots c}\ =\
\omega_{a}+\omega_{b}+\dots+\omega_{c}\,.$ (4.8)

We have also set the outgoing energy $Q^{\prime}$ to equal $Q$, to suppress
the phase that provides the energy conserving delta function after the
integral over $t_{o}$ in this case. In this representative TOPT term, the
denominator $Q-\omega_{abefg}$ corresponds to a pseudo-physical final state
because the amplitude with this final state consists of two disconnected
parts, one that includes vertex $i$ and provides particles $a$ and $b$ in the
final state, and one in which particles $e$, $f$, and $g$ emerge from the
vacuum into the final state. We shall see, however, that such singularities
are absent in an evaluation of integrals based on the poset. In other words,
pseudo-physical denominators all cancel in the sum over the time orderings
that make up the poset. This will be the case separately for the combinations
of time orders in Eq. (4.5) and (4.6).

The case of the two orders in Eq. (4.5) is particularly simple. Adding the two
orders together, we find that times $t_{b_{1}}$ and $t_{i}$ integrate
independently to $t_{b_{3}}$, and the pseudo-physical cut disappears in TOPT,
even though each of these orderings gives one or more pseudo-physical cuts,

$\displaystyle\pi_{(i1324o)}+\pi_{(1i324o)}$ $\displaystyle=$ $\displaystyle
e^{iQt_{o}}\;\int_{-\infty}^{t_{o}}dt_{b_{4}}e^{-i(\omega_{ch}-\omega_{d}+i\epsilon)t_{b_{4}}}\int_{-\infty}^{t_{b_{4}}}dt_{b_{2}}e^{-i(\omega_{ef}-\omega_{h}+i\epsilon)t_{b_{2}}}$
$\displaystyle\times\
\int_{-\infty}^{t_{b_{2}}}dt_{b_{3}}e^{-i(\omega_{ag}-\omega_{f}+i\epsilon)t_{b_{3}}}$
$\displaystyle\times\
\int_{-\infty}^{t_{b_{3}}}dt_{b_{1}}e^{-i(-\omega_{efg}+i\epsilon)t_{b_{1}}}\int_{-\infty}^{t_{b_{3}}}dt_{i}e^{-i(Q-\omega_{ab}+i\epsilon)t_{i}}$
$\displaystyle=\
\left(\frac{i}{Q-\omega_{bd}+i\epsilon}\,\frac{i}{Q-\omega_{bch}+i\epsilon}\,\frac{i}{Q-\omega_{bcef}+i\epsilon}\,\frac{i}{Q-\omega_{ab}+i\epsilon}\right)\,\frac{i}{-\omega_{efg}}\,,$

where the rightmost fraction, which is negative semi-definite for massless
lines, is the result of the integral
$\int\limits_{-\infty}^{t_{b_{3}}}dt_{1}e^{-i(-\omega_{efg}+i\epsilon)t_{b_{1}}}$.
The remainder of the integral gives the factor in parentheses, consisting of
four denominators, each of which provides a unitarity cut, corresponding to a
physical final state.

Figure 3: A three loop example of a graph with two minima. Vertices $1$ and
$i$ are both minima of this graph, and our procedure eliminates the minimum
$1$, through the process of successive integration. This graph is to be
understood as a poset graph rather than a time-ordered graph.

A representation of the diagram after the $t_{b_{1}}$ integral is given in
Fig. 4a, which includes a modified vertex, which absorbs the fraction
$\frac{i}{-\omega_{efg}}$. This vertex, the denominator combined with coupling
constants, is real and is effectively local in time (in this case at
$t_{b_{3}}$). At this stage, all remaining vertices are ordered between $i$
and $o$ in the poset. We have thus achieved what we set out to do, reduce the
diagram in Fig. 3 to one with a unique maximum and unique minimum. The cuts of
the modified diagram in Fig. 4a do not disconnect the graph into more than two
connected subdiagrams, and are therefore unitarity cuts, that is, physical
cuts of the forward scattering graph.

We can now turn to the other possibility, $t_{b_{3}}>t_{b_{2}}$, the component
of this poset given by the orders in Eq. (4.6). Rather than reproducing the
four-dimensional time integral as in the previous case, we shall simply show
the integrals over minimum vertices. Examining the ranges of integrations
possible in Eq. (4.6), we find that we can carry out the $t_{1}$ integral from
$-\infty$ to $t_{2}$ independently of the value of $t_{i}$. This gives the
$t_{2}$-dependent factor

$\int\limits_{-\infty}^{t_{2}}dt_{1}\;e^{-i(-\omega_{efg}+i\epsilon)t_{1}}=\frac{i}{-\omega_{efg}}\;e^{-i(-\omega_{efg}+i\epsilon)t_{2}}.$
(4.10)

The composite graph obtained after integrating out vertex $1$ is shown in Fig.
4b. In this case, the vertex $2$ is still a minimum of the poset, and we would
like to carry out the $t_{2}$ integral as well. We can do this because in the
combination of time orders in Eq. (4.6), the upper limit of the $t_{2}$
integral is $t_{3}$, independent of $t_{i}$. We thus find,

$\int\limits_{-\infty}^{t_{3}}dt_{2}\;e^{-i(-\omega_{efg}+i\epsilon)t_{2}}e^{-i(\omega_{ef}-\omega_{h}+i\epsilon)t_{2}}=\frac{i}{-\omega_{gh}}\;e^{-i(-\omega_{gh}+i\epsilon)t_{3}}.$
(4.11)

A graphical representation of the diagram that results after this step is
shown in Fig. 4c.

(a)  (b)  (c)

Figure 4:  LABEL:sub@ex1-int-a The poset ordered graph in Fig. 3. after
integrating out the minimum vertex $1$, with the choice $2\geq 3$. Here the
black dot represents the composite, modified vertex $3$. LABEL:sub@ex1-int-b
Integrating the time of vertex $1$, making the choice $3\geq 2$. The composite
vertex $2$ is represented with a black dot. $2$ is still a minimum of the new
poset. LABEL:sub@ex1-int-c The time of the composite vertex $2$, which was a
new minimum in the choice $3\geq 2$, has been integrated. The new composite
vertex $3$ is represented with a black dot. Here, all vertices lie between
$i,o$.

The complete result for the time orders in Eq. (4.6), analogous to Eq. (4.2)
for the time orders of Eq. (4.5), is now easily found to be

$\displaystyle\pi_{(i1234o)}+\pi_{(1i234o)}+\pi_{(12i34o)}$ $\displaystyle=$
$\displaystyle\left(\frac{i}{Q-\omega_{bd}+i\epsilon}\,\frac{i}{Q-\omega_{cbh}+i\epsilon}\,\frac{i}{Q-\omega_{ab}+i\epsilon}\,\right)\frac{i}{-\omega_{gh}}\,\frac{i}{-\omega_{efg}}\,.$

Together, the two negative semi-definite denominators combine with couplings
to form the composite vertex 3 in Fig. 4c. Again, the remaining denominators
are either physical (in parentheses) or negative semi-definite. The full
expression for this poset ($D$), $\pi_{D}$, is given by the sum of this result
with Eq. (4.2).

Figure 5: A two loop example of a graph with two minima and two maxima.
Vertices $1$ and $i$ are both minima of this graph, and Vertices $2$ and $o$
are both maxima of this graph.

For the next example, consider the two loop graph in Fig. 5. In this example,
at the level of the poset, there exists no unique minimum or unique maximum.
Our process of eliminating the additional extrema starts with eliminating the
vertex we have labeled $1$. The minimum vertex, $1$ is connected to mutually
un-ordered vertices $2$ and $o$. To integrate out the minimum $1$, we will
order these vertices as before. The two choices in ordering are $t_{o}>t_{2}$
and $t_{2}>t_{o}$, and as before, we must sum over both choices. Making the
choice $t_{o}>t_{2}$ and integrating $t_{1}$, we obtain a unique minimum $i$
and a unique maximum $o$, giving

$\int\limits_{-\infty}^{t_{u}}dt_{1}\;e^{-i(-\omega_{dec}+i\epsilon)t_{1}}=\frac{i}{-\omega_{dec}}\;e^{-i(-\omega_{dec}+i\epsilon)t_{u}},$
(4.13)

where the upper limit $t_{u}$ is $t_{2}$ if $t_{o}>t_{2}$ and $t_{o}$ if
$t_{2}>t_{o}$. Having made the choice $t_{o}>t_{2}$, the rest of the graph is
fully ordered and we may use TOPT for the remaining denominators,

$\pi_{D}^{(1)}=\left(\frac{i}{Q-\omega_{ac}+i\epsilon}\,\frac{i}{Q-\omega_{ab}+i\epsilon}\right)\,\frac{i}{-\omega_{ced}}.$
(4.14)

The ordered graph that yields these denominators is shown in Fig. 6a.

The choice of ordering $t_{2}>t_{o}$, leaves us with a single maximum vertex
$2$, which is not $o$. We would therefore like to integrate the maximum time,
$t_{2}$, from $t_{o}$ to $\infty$,

$\int\limits^{\infty}_{t_{o}}dt_{2}\;e^{-i(\omega_{bed}-i\epsilon)t_{2}}=\frac{i}{-\omega_{bed}}\;e^{-i(\omega_{bed}-i\epsilon)t_{o}}.$
(4.15)

This procedure is represented diagrammatically in Figs. 6b, 6c. What remains
is the integral of $t_{i}$ in the range $-\infty$ to $t_{o}$ which is easily
carried out. The contribution of the order $t_{2}>t_{o}$ to the poset
denominator is

$\pi_{D}^{(2)}=\left(\frac{i}{Q-\omega_{ab}+i\epsilon}\right)\,\frac{i}{-\omega_{bed}}\,\frac{i}{-\omega_{ced}}\,.$
(4.16)

As before, the full expression for the integrand from this poset is the sum of
the two choices in orderings included in the poset,
$\pi_{D}=\pi_{D}^{(1)}+\pi_{D}^{(2)}$.

(a)  (b)  (c)

Figure 6:  LABEL:sub@ex2-int-a The poset ordered graph in Fig. 5. after
integrating the time of the minimum vertex, $1$, with the choice $o\geq 2$.
Here the black dot represents the composite, modified vertex $2$.
LABEL:sub@ex2-int-b Integrating $t_{1}$, making the choice $2\geq o$. The
composite vertex $o$ is represented with a black dot. Vertex $2$ is the new
maximum of the poset. However, since the maximum is not $o$, we continue by
integrating the time for vertex $2$. LABEL:sub@ex2-int-c The composite vertex
$o$, after $t_{2}$ has been integrated. The new composite vertex $o$ is
represented with a black dot.  Figure 7: A four loop example of a graph with
two minima and two maxima. Vertices $1$ and $i$ are both minima of this graph,
and vertices $2$ and $o$ are both maxima of this graph.

As a final example, consider the poset graph in Fig. 7, which has both a
minimum (vertex 1) and a maximum (vertex 2). This example captures our
procedure at four loops. In this poset, time $t_{1}$ is bounded from above by
$t_{2}$ if $t_{2}<t_{3}$, and by $t_{3}$ if $t_{3}<t_{2}$. In either case,

$\int\limits_{-\infty}^{t_{u}}dt_{1}\;e^{-i(-\omega_{ijk}+i\epsilon)t_{1}}=\frac{i}{-\omega_{ijk}}\;e^{-i(-\omega_{ijk}+i\epsilon)t_{u}},$
(4.17)

where the upper limit $t_{u}$ is $t_{2}$ if $t_{3}>t_{2}$ and $t_{3}$ if
$t_{2}>t_{3}$. Making the first choice, $t_{3}>t_{2}$, confines every
remaining vertex to lie between $i\text{ and }o$. A graphical representation
of this reduced graph is in Fig 8a. It is now fully ordered and as above, we
can reconstruct the remaining integrals from the rules of TOPT, to find.

$\pi_{D}^{(1)}=\left(\frac{i}{Q-\omega_{af}}\,\frac{i}{Q-\omega_{kae}}\,\frac{i}{Q-\omega_{adjk}}\,\frac{i}{Q-\omega_{ghad}}\,\frac{i}{Q-\omega_{gcd}}\,\frac{i}{Q-\omega_{ab}}\right)\frac{i}{-\omega_{ijk}}\,,$
(4.18)

where we have suppressed the $+i\epsilon$ terms present in each physical
denominator.

Making the other choice, $t_{2}>t_{3}$, and integrating the minimum time,
$t_{1}$ from $-\infty$ to $t_{3}$ leaves $2$ as a maximum element. The reduced
graph we obtain at this stage is shown in Fig. 8b. We can now integrate over
$t_{2}$,

$\int\limits^{\infty}_{t_{3}}dt_{2}\;e^{-i(\omega_{igh}-i\epsilon)t_{2}}=\frac{i}{-\omega_{igh}}\;e^{-i(\omega_{igh}-i\epsilon)t_{3}}.$
(4.19)

The reduced graph we obtain is shown in Fig. 8c. This is a fully ordered
diagram, for which every vertex lies between $i$ and $o$. We can proceed using
TOPT rules for the new diagram, giving the final result

$\pi_{D}^{(2)}=\left(\frac{i}{Q-\omega_{af}}\,\frac{i}{Q-\omega_{kae}}\,\frac{i}{Q-\omega_{ghad}}\,\frac{i}{Q-\omega_{gcd}}\,\frac{i}{Q-\omega_{ab}}\right)\,\frac{i}{-\omega_{igh}}\,\frac{i}{-\omega_{ijk}}\,,$
(4.20)

where again we suppress $+i\epsilon$ terms in all the denominators. The full
poset integrand is once again $\pi_{D}=\pi_{D}^{(1)}+\pi_{D}^{(2)}$.

Let us summarize the process of integrating out of extrema that we have seen
so far in these examples. We first identify a minimum vertex, $v$ ( $v\neq
i$), and order the vertices that cover $v$. We then integrate out the minimum,
at the cost of adding the energy carried by $v$ into the next vertex, say
$v+1$. If there were more than one minimum, we will sequentially repeat these
steps for all minima. We continue this process until each vertex that remains
is greater than $i$. We use an identical procedure with maxima, integrating
them to generate negative definite denominators. After the elimination of both
minima and maxima, the remaining vertices are all ordered relative to $i$ and
$o$ and therefore carry only physical cuts of the graph.

We have seen in these examples that the resulting ordered diagrams include
composite vertices with multiple lines emanating from or flowing into them. At
each stage, the graph with composite vertices yields a new time-ordered graph,
with fewer pseudo-physical cuts but the same physical cuts as in the original
graph. The new graph thus obtained has fewer vertices and a modified incidence
matrix. The composite vertex denotes a factorized expression for long time
processes that were initiated in the vacuum and do not affect the cuts of the
short distance process. The composite vertices may be interpreted intuitively
as effective vertices in the short distance function, where all the long
distance behavior has been absorbed into factorized, negative semi-definite,
vacuum denominators. We next turn to a general implementation of these
methods.

(a)  (b)  (c)

Figure 8: LABEL:sub@ex3-int-a The poset ordered graph in Fig 7 after
integrating the time of the minimum vertex $1$, with the choice $3\geq 2$.
Here the black dot represents the composite, modified vertex $2$.
LABEL:sub@ex3-int-b Integrating vertex $1$, making the choice $2\geq 3$. The
composite vertex $3$ is represented with a black dot. Vertex $2$ is a new
maximum of the poset. However, since the maximum is not $o$, we continue
integrating out vertex $2$. LABEL:sub@ex3-int-c The vertex $2$, which was the
new maximum in the choice $2\geq 3$, after $t_{2}$ has been integrated. The
new composite vertex $3$ is represented with a black dot.

##  5 PTOPT: TOPT without unphysical singularities

In this section, we describe a generalization to arbitrary order of the
examples of the previous section, based on graphs’ poset structures. We will
term this approach, “partially time-ordered perturbation theory” (PTOPT). We
will construct an algorithm that enables us to re-express the sum over time-
ordered diagrams of any graph into a sum with fewer terms, all of whose
denominators are either negative semi-definite, or physical. In this
construction, all pseudo-physical cuts will be eliminated. We will present our
discussion for an arbitrary amplitude with any numbers of incoming and
outgoing momenta. Our result, given in Eq. (5.37) below, is essentially
equivalent to that given Ref. [18]. In the following section, we show how
PTOPT can be applied to cross sections written as sums over cuts, as in the
weighted cross sections of Sec. 2, to provide expressions that involve only
physical singularities, and which separate universal and process-dependent
dynamics.

In this construction, it will be useful to turn our attention to integral
representation Eq. (2.12) for the time-ordered denominators of functions like
$\pi_{\tau_{G}}$ in Eq. (4.1), written as sums over posets. Compared to Eq.
(2.12), however, we associated with every vertex $\alpha$ an external energy
$E_{\alpha}$, defined to flow into that vertex. In a multiloop diagram, with a
fixed number of external lines, most $E_{\alpha}$ are zero. For an outgoing
physical momentum flowing out of vertex $\alpha$, $E_{\alpha}$ is negative.
The resulting functions are then not limited to vacuum polarizations and may
represent any scattering configuration, with arbitrary numbers of incoming and
outgoing lines. To emphasize the generality of these considerations, we denote
our functions as $F_{G}(\\{E_{\alpha}\\},{\cal L}_{G})$ for graph $G$, and
$F_{D}(\\{E_{\alpha}\\},{\cal L}_{G})$ for poset $D$. When we return to the
analysis of vacuum polarizations, we revert to the notation of $\pi_{G}$ or
$\pi_{D}$.

In these terms, our expression for the integral associated with an arbitrary
graph $G$ with $n$ vertices at fixed spatial loop momenta is

$\displaystyle
2\pi\delta\left(\sum_{\alpha}E_{\alpha}\right)\,F_{G}\left(\\{E_{\alpha}\\},{\cal
L}_{G}\right)$ $\displaystyle=$
$\displaystyle\sum_{\tau}\mathbb{N}_{\tau}\,\prod_{\alpha=1}^{n}\int_{-\infty}^{t_{\alpha+1}}dt_{\alpha}e^{-i(E_{\alpha}+\eta^{(\tau_{\alpha})}_{j}(\omega_{j}-i\epsilon))t_{\alpha}}\,$
$\displaystyle=$ $\displaystyle
2\pi\left(\sum_{\alpha}E_{\alpha}\right)\,\sum_{\tau}\
\mathbb{N}_{\tau}F_{\tau}\left(\\{E_{\alpha}\\},{\cal L}_{G}\right)$
$\displaystyle=$ $\displaystyle
2\pi\delta\left(\sum_{\alpha}E_{\alpha}\right)\,\sum_{D}\sum_{\tau_{D}}\mathbb{N}_{D}\,F_{\tau_{D}}\left(\\{E_{\alpha}\\},{\cal
L}_{G}\right)$ $\displaystyle=$ $\displaystyle
2\pi\delta\left(\sum_{\alpha}E_{\alpha}\right)\,\sum_{D}\mathbb{N}_{D}\,F_{D}\left(\\{E_{\alpha}\\},{\cal
L}_{G}\right)\,,$

where we define, $t_{n+1}=\infty$ in the first equality. The final, $t_{n}$,
integral gives the delta function that enforces the energy conservation in
$G$. Here, $\eta^{(\tau_{\alpha})}_{j}$ is the incidence matrix of the vertex
$b_{\tau_{\alpha}}$ in the time order labeled by $\tau$. The second equality
defines $F_{\tau}$ as the coefficient of the energy-conserving delta function
from the full integral associated with time order $\tau$. In the third, we
reorganize the sum over time orders into a sum over posets, $D$, as in Eq.
(4.2), recalling that every time order is associated with a single poset.
Finally, the fourth equality defines $F_{D}$ as the complete contribution to
$F_{G}$ from all the time orders within poset $D$. The functions $F_{D}$ will
be the subject of the following discussion.

###  5.1 Time integrals of extremal vertices; the covering set

We consider the time integrals allowed within a given partial order for an
arbitrary graph. Again, in this section, we will not restrict to forward
scattering graphs that one encounters in leptonic annihilation. Rather, we
treat graphs with a generic flow of external momenta. We will label this
general partially-ordered diagram, or poset diagram, as $D$, defined on a set
of vertices $V$ with ordering $\geq$. In the following, we will use $D$ to
refer both to the poset and to the ordered diagram. Note that we will only
encounter vertices that are local in time, whose time integrals we will carry
out. After our first time integral, however, the composite vertices we
encounter generally will not be local in space.

The object of interest is the function $F_{D}$ for a fixed poset $D=(V,\geq)$,
which as we have noted before, is fixed by an incidence matrix
$\eta^{(\alpha)}_{j}$. (For compactness of notation, we do not label
$\eta^{(\alpha)}_{j}$ by $D$.) We represent schematically the multidimensional
region of vertex times restricted to the orders associated with a specific
poset $D$ as

$2\pi\delta\left(\sum_{\alpha=1}^{n}E_{\alpha}\right)\,F_{D}\left(\\{E_{\alpha}\\},{\cal
L}_{G}\right)=\prod_{\alpha=1}^{|V|}\int_{X_{D}}dt_{\alpha}e^{-i(E_{\alpha}+\eta^{(\alpha)}_{j}(\omega_{j}-i\epsilon))t_{\alpha}}\,,$
(5.2)

where we have formally carried out the sum over time orders consistent with
poset $D$, corresponding to the integral of times over a region labeled by
$X_{D}$. We let $|V|$ represent the number of vertices. The region $X_{D}$ is
defined by

$X_{D}=\\{(t_{1},t_{2}\ldots t_{|V|})|b_{k}\geq b_{j}\implies t_{k}\geq
t_{j}\\}\,.$ (5.3)

The union of all possible regions $X_{D}$ is the full set of time integrals in
Eq. (5).

For a given poset $D$, we can identify a unique set $M^{1}_{D}$ of embedded
extrema (minima and maxima) of $D$. We define an embedded vertex $v\in
M^{1}_{D}$, as one whose removal leaves behind a connected diagram consisting
of vertices $V/v$. Every connected diagram has a non-empty set of embedded
vertices (see Appendix A) . The time integrals of minima in $X_{D}$ extend
independently to negative infinity, and those of maxima to positive infinity.
We can always begin the integration process for $X_{D}$ by picking a subset of
embedded extremal vertices that form an anti-chain (mutually incomparable
elements). We will do so for a subset of the extremal embedded vertices,
chosen to have the largest number of such vertices that form an anti-chain.
Such a subset may include both maxima and minima, because the minima may not
be ordered with respect to all maxima, and vice-versa.

Let us call this set the first extremal antichain, $A^{1}$, defined to
satisfy,

$\displaystyle A^{1}\ $ $\displaystyle=$ $\displaystyle\ \\{x\in M^{1}|x,y\in
A^{1}\implies x\sim y\\}$ (5.4) $\displaystyle=$
$\displaystyle\left\\{b_{1,1},\dots,b_{1,r_{1}}\,;\,d_{1,1},\dots,d_{1,s_{1}}\right\\}\
\equiv\ \left\\{e_{1,i}\right\\}\,,$

where we denote by $r_{1}$ the number of minimal elements, $b_{1,i}$, in
$A^{1}$ and by $s_{1}$ the number of maximal elements, $d_{1,i}$, and where
$e_{i,j}$ labels these vertices collectively, with $j=1\dots r_{1}+s_{1}$.
$A^{1}$ represents a set of vertices whose times can be integrated
simultaneously and independently of each other.444This choice for $A^{1}$ is
not unique, and we need not adhere to $A^{1}$ being the largest possible
antichain in any one step. Clearly, each of these integrals must extend from a
fixed upper (lower) limit to infinity (negative infinity). We now turn to the
determination of these limits.

Having identified the set $A^{1}$, we identify a companion set, $C^{1}$, of
vertices that cover the minimal vertices in $A^{1}$ or are covered by the
maximal vertices in $A^{1}$, that is, vertices that are connected to one or
more elements of $A^{1}$ by single line(s) with no intervening vertices,

$\displaystyle C^{1}$ $\displaystyle=$ $\displaystyle\\{x\in V|\exists v\in
A^{1}:(v\prec x)\,{\rm or}\,(x\prec v)\\}$ (5.5) $\displaystyle=$
$\displaystyle\bigcup\limits_{i=1}^{r_{1}}C_{b_{1,i}}\bigcup\limits_{j=1}^{s_{1}}C_{d_{1,j}},$

where we have defined the covering set for each element in $A^{1}$ by

$\displaystyle C_{b_{1,i}}$ $\displaystyle=$ $\displaystyle\\{x\in
V|b_{1,i}\prec x\\},$ $\displaystyle C_{d_{1,i}}$ $\displaystyle=$
$\displaystyle\\{x\in V|x\prec d_{1,i}\\}.$ (5.6)

By construction, vertices in $C^{1}$ are connected to extrema by lines that
either emerge directly from minima or flow directly into maxima, as denoted by
the covering relation $\prec$, defined in Sec. 4.1. If there are several such
vertices for any given extremal vertex, they form an anti-chain, that is, they
are not mutually ordered within the poset $D$. We note as well that a single
element of $C^{1}$ can cover more than a single extremum. We will refer to the
elements of $C^{1}$ as the first covering set for the first extremal
antichain, $A^{1}$. We will use the covering set $C^{1}$ to organize the time
integrals for the extremal vertices in $A^{1}$. This will enable us to begin a
recursive analysis over induced poset diagrams and to do all subsequent time
integrals in the same way as the first.

In the time integral over $X_{D}$, corresponding to the partially ordered
diagram $D$, Eq. (5.2), we must include all choices within poset $D$ of the
earliest covering vertices for all minimum elements of $A^{1}$ and of the
latest covered vertex for all maximum elements. Region $X_{D}$, and thus our
integral, is given by a sum over these choices, which we will denote as
$\gamma^{1}\subset C^{1}$. It is worth pointing out that it is possible to
construct the full list of compatible choices $\gamma^{1}$ of next-to-extremal
covering vertices systematically, by summing over available choices of
covering (covered) vertices for each minimum (maximum) in turn. Because next-
to-extremal vertices may cover more than one extremum, the number of resulting
sets is not a simple product of the number of covering vertices for the
extrema. For a concrete algorithm to generate all consistent $\gamma^{1}$, see
Appendix B.

Given a choice of next-to-extremal vertices, the integration range of each
extremal vertex is -$\infty$ to the time of its earliest covering vertex for a
minimum, and from the time of its latest covered vertex to $\infty$ for each
maximum. We shall describe these choices collectively as sets of next-to-
extremal covering vertices, and label them among the elements of $C^{1}$ as
$x^{(e_{1,i})}_{a}$, where index $e_{1,i}$ identifies the extremal vertex
connected to the vertex $x_{a}^{(e_{1,i})}$, while index $a$ identifies the
choice of covering (covered) vertex in set $C_{e_{1,i}}$ that covers (is
covered by) the minimal (maximal) vertex $e_{1,i}$.

In terms of the next-to-extremal covering elements, $\gamma^{1}$ is given by

$\displaystyle\gamma^{1}\ =\ \left\\{\cup_{e_{i}=1\dots
r_{1}+s_{1}}\;x_{a}^{(e_{1,i})}\right\\}\,.$ (5.7)

Whenever two extrema, say $e_{1,i}$ and $e_{1,i^{\prime}}$, have the same
next-to-extremal covering vertex, we have
$x^{(e_{1,i})}_{a}=x^{(e_{1,i^{\prime}})}_{a^{\prime}}$. The number of
elements in $\gamma^{1}$ is therefore less than or equal to the number of
elements in $C^{1}$. In this notation, once we choose $\gamma^{1}$, the
integrals over the times of all extremal vertices in set $A^{1}$, minima and
maxima, can be done explicitly and independently.

To relate the integration region for each $\gamma^{1}$ to the original
integration region $X_{D}$ for poset $D$ in Eq. (5.3), we proceed as follows.
As noted above, the next-to-extremal vertices for any $e_{1,i}$ form an
antichain (mutually incomparable vertices) and therefore need additional
ordering at the poset level to specify the ranges of the $e_{1,i}$ time
integrals. This additional ordering we impose is fixed by our choice in
$x_{a}^{(e_{1,i})}$. For each choice of $\gamma^{1}$, we construct a partial
order on the antichains in $C^{1}$, by defining a new binary relation, denoted
$\geq_{\gamma^{1}}$ over the set $V$ by

$\displaystyle\hskip 42.67912pt{\rm If}\ a\geq b,\ {\rm then}\
a\geq_{\gamma^{1}}b\,,$ $\displaystyle\hskip 42.67912pt\forall
y\in{C_{b_{1,i}}}:y\geq_{\gamma^{1}}x_{a}^{(b_{1,i})}>_{\gamma^{1}}b_{1,i}\,,$
$\displaystyle\hskip 42.67912pt\forall
y\in{C_{d_{1,i}}}:d_{1,i}>_{\gamma^{1}}x_{a}^{(d_{1,i})}\geq_{\gamma^{1}}y\,.$
(5.8)

Here, the subscript on $\geq_{\gamma^{1}}$ reflects the ordering that remains
after we identify each extremal vertex $e_{1,i}$ with its corresponding vertex
$x_{a}^{(e_{1,i})}\in C_{e_{1,i}}$ in the covering set $\gamma^{1}$.

The relations, Eq. (5.8) that define the ordering $\geq_{\gamma^{1}}$ identify
new posets, whose mutually-disjoint integration regions allow us to integrate
the times $t_{e_{1,i}}$ explicitly. Again, although we formally have
constructed $\geq_{\gamma^{1}}$ as a “stronger” binary relationship than the
original ordering $\geq$ of the poset, it is the natural ordering inherited
from $\geq$ when every extremum $e_{1,i}$ is identified with the specific
covering element $x_{a}^{(e_{1,i})}$, that is, contracting the vertex
$e_{1,i}$ onto the corresponding $x_{a}^{(e_{1,i})}$ in $\gamma^{1}$. We,
therefore, think of a new, lower-order graph with all $e_{1,i}$ removed from
the set of vertices. Every line that previously emanated from a minimum
$b_{1,i}$, and was not absorbed at vertex $x^{(b_{1,i})}_{a}$ now emanates
from $x^{(b_{1,i})}_{a}$, and therefore $x^{(b_{1,i})}_{a}$ is “less than”
every other vertex of $C^{1}$ that was connected to $b_{1,i}$, since after the
contraction of $b_{1,i}$ onto $x^{(b_{1,i})}_{a}$ there is at least one line
starting from $x^{(b_{1i})}_{a}$ and ending on any such vertex. Maxima are
contracted on their nearest covered vertices in an exactly analogous fashion.

The combination of the new, contracted vertex and the induced order defines a
new poset,

$\displaystyle D\left[\gamma^{1}\right]\ =\
\left\\{V\left[\gamma^{1}\right]\,,\,\geq_{\gamma^{1}}\right\\}\,,\quad\quad
V\left[\gamma^{1}\right]\ =\ V\setminus\cup_{i}e_{1,i}\,,$ (5.9)

with $\geq_{\gamma^{1}}$ defined by Eq. (5.8). The integration region for post
$D[\gamma^{1}]$, analogous to the original region, Eq. (5.3), is given by

$X_{D\left[\gamma^{1}\right]}=\\{(t_{1},t_{2}\ldots\hat{t}_{e_{1,1}}\ldots\hat{t}_{e_{1,r_{1}+s_{1}}}\dots
t_{|V|})|b_{k}\geq_{\gamma^{1}}b_{j}\implies t_{k}\geq t_{j}\\}\,,$ (5.10)

where $\hat{t}_{e_{1,i}}$ means that $t_{e_{1,i}}$ is excluded from the list.
Again, the notation $X_{D\left[\gamma^{1}\right]}$ identifies the subregion of
$X_{D}$ where the vertices in $\gamma^{1}$ are the full set of next-to-
extremal covering vertices.

It is clear that any time order in region $X_{D}$ has a unique set
$\gamma^{1}$, and that distinct choices of $\gamma^{1}$ correspond to
different time orders. Therefore, when we sum over all choices, of
$\gamma^{1}$, we exhaust all time orders within $X_{D}$. Explicitly, we may
represent the full poset integration region, $X_{D}$ as

$\displaystyle X_{D}\ \ $ $\displaystyle=$ $\displaystyle\
\bigcup\limits_{\gamma^{1}}X_{D\left[\gamma^{1}\right]}\times\bigcup_{\begin{subarray}{c}t_{b_{i,1}}\\\
i=1\end{subarray}}^{r_{1}}(-\infty,t_{x_{a}^{(b_{1,i})}})\times\bigcup_{\begin{subarray}{c}t_{d_{i,1}}\\\
i=1\end{subarray}}^{s_{1}}(t_{x_{a}^{(d_{1,i})}},\infty)\,,$ (5.11)

where again, $\gamma^{1}$ is specified, as in Eq. (5.7), as the choice of
next-to-extremal elements $x_{a}^{(e_{1,i})}$.

In summary, for a given choice of $\gamma^{1}$, we can carry out the
$t_{e_{1,i}}$ integrals up to (or down to) the times
$t_{x_{a}^{(e_{1,i})}}\in\gamma^{1}$. We can thus rewrite the full integral
for $F_{D}$, Eq. (5.2) using Eq. (5.11) as

$\displaystyle
2\pi\delta\left(\sum_{\alpha=1}^{n}E_{\alpha}\right)\,F_{D}\left(\\{E_{\alpha}\\},{\cal
L}_{G}\right)\ $ $\displaystyle=$ $\displaystyle\
\sum_{\gamma^{1}}\;\prod_{\alpha=1}^{|V[\gamma^{1}]|}\int_{X_{D}[\gamma^{1}]}dt_{\alpha}e^{-i(E_{\alpha}+\eta^{(\alpha)}_{j}(\omega_{j}-i\epsilon))t_{\alpha}}$
(5.12) $\displaystyle\times\
\prod_{i=1}^{r_{1}}\int_{-\infty}^{t[x^{(b_{1,i})}_{a}]}dt_{1,i}\,e^{-i\left(E_{i}+\sum_{j}\eta^{(b_{1,i})}_{j}(\omega_{j}-i\epsilon)\right)t_{1,i}}$
$\displaystyle\times\
\prod_{i=1}^{s_{1}}\int^{\infty}_{t[x^{(d_{1,i})}_{a}]}dt_{1,i}\,e^{-i\left(E_{i}+\sum_{j}\eta^{(d_{1,i})}_{j}(\omega_{j}-i\epsilon)\right)t_{1,i}}\,.$

Here we have labeled the times of the extremal vertices, over which we are
integrating, with the subscripts of the corresponding vertices $e_{1,i}$
themselves, and the times of the next-to-extremal covering vertices
$x^{(e_{1,i})}_{a}$ in a hopefully obvious notation. We note that each of the
limits in the integrals over extremal vertices are times that appear in the
remaining integration measure of $X_{D[\gamma^{1}]}$.

In Eq. (5.12), the time integrals for minima all take the form

$\displaystyle\int_{-\infty}^{t[x^{(b_{1,i})}_{a}]}dt_{1,i}\,e^{-i\left(E_{b_{1i}}+\sum_{j}\eta^{(b_{1,i})}_{j}(\omega_{j}-i\epsilon)\right)t_{1,i}}\
=\ \
\frac{i}{\Delta_{b_{1,i}}[\gamma^{1}]}\,e^{-i\left(E_{b_{1i}}+\sum_{j}\eta^{(b_{1,i})}_{j}(\omega_{j}-i\epsilon)\right)t[x^{(b_{1,i})}_{a}]}$
(5.13)

where we define

$\displaystyle\Delta_{b_{1,i}}[\gamma^{1}]\ =\
E_{b_{1i}}-\sum_{j}\left|\eta^{(b_{1,i})}_{j}\right|\omega_{j}+i\epsilon\,\,.$
(5.14)

We note that every nonzero $\eta_{j}^{(b_{1,i})}=-1$, since they all
correspond to minimum vertices, which only emit particles. Such a denominator
has the standard form of an energy deficit in the channel of vertex $b_{1,i}$,
where external energy $E_{b_{1,i}}$ flows in. We will refer to the set of
lines, $j$ for which $\eta^{(b_{1,i})}_{j}=-1$, all of whose on-shell energies
flow out of vertex $b_{1,i}$, as the “cut” of diagram $D$ that separates it
into two diagrams, one consisting of vertex $b_{1,i}$ and one with the
remaining vertices, $V\setminus b_{1,i}$. Since all $b_{1,i}$ are embedded
vertices, all the diagrams $V\setminus b_{1,i}$ are connected.

Similarly, for all the time integrals for maxima in Eq. (5.12) we have

$\displaystyle\int^{\infty}_{t[x^{(d_{1,i})}_{a}]}dt_{1,i}\,e^{-i\left(E_{d_{1i}}+\sum_{j}\eta^{(d_{1,i})}_{j}(\omega_{j}-i\epsilon)\right)t_{1,i}}$
$\displaystyle=$
$\displaystyle\frac{i}{\Delta_{d_{1,i}}[\gamma^{1}]}\,e^{-i\left(E_{d_{1i}}+\sum_{j}\eta^{(d_{1,i})}_{j}(\omega_{j}-i\epsilon)\right)t[x^{(d_{1,i})}_{a}]}\,.$

In this case we define

$\displaystyle\Delta_{d_{1,i}}[\gamma^{1}]$ $\displaystyle=$ $\displaystyle\
-\left(E_{d_{1i}}+\sum_{j}\eta^{(d_{1,i})}_{j}\omega_{j}-i\epsilon\right)$
(5.16) $\displaystyle=$ $\displaystyle\
(-E_{d_{1i}})-\sum_{j}\eta^{(d_{1,i})}_{j}\omega_{j}+i\epsilon\,,$

where in the second equality we use that $\eta^{(d_{1,i})}$ is either zero or
$1$ for maximum vertices. When $-E_{d_{1,i}}$ is a positive energy flowing out
of vertex $d_{1,i}$, we again have a standard energy deficit form in the
channel of vertex $d_{1,i}$. As for minimal vertices, will refer to the set of
lines, $j$ for which $\eta^{(d_{1,i})}_{j}=1$, as the “cut” of diagram $D$
that separates vertex $d_{1,j}$ from a connected diagram with vertices
$V\setminus d_{1,j}$.

Substituting the integrals in Eqs. (5.13) to (5.16) into the expression for
$F_{D}$, Eq. (5.12), we now have

$\displaystyle
2\pi\delta\left(\sum_{\alpha=1}^{n}E_{\alpha}\right)\,F_{D}\left(\\{E_{\alpha}\\},{\cal
L}_{G}\right)$ $\displaystyle=$
$\displaystyle\sum_{\gamma^{1}}\;\prod_{i=1}^{r_{1}}\frac{i}{\Delta_{b_{1,i}}[\gamma^{1}]}\;\prod_{i=1}^{s_{1}}\frac{i}{\Delta_{d_{1,i}}[\gamma^{1}]}$
$\displaystyle\times\
\prod_{\alpha=1}^{|V[\gamma^{1}]|}\int_{X_{D}[\gamma^{1}]}dt_{\alpha}e^{-i(E_{\alpha}+\eta^{(\alpha)}_{j}(\omega_{j}-i\epsilon))t_{\alpha}}$
$\displaystyle\times\
\prod_{i=1}^{r_{1}}e^{-i\left(E_{b_{1,i}}+\sum_{j}\eta^{(b_{1,i})}_{j}(\omega_{j}-i\epsilon)\right)t[x^{(b_{1,i})}_{a}]}\prod_{i=1}^{s_{1}}e^{-i\left(E_{d_{1,i}}+\sum_{j}\eta^{(d_{1,i})}_{j}(\omega_{j}-i\epsilon)\right)t[x^{(d_{1,i})}_{a}]}\,.$

In this expression, every next-to-extremal vertex $x^{(e_{1,i})}_{a}$ appears
in the product over the vertices ($\alpha$) of diagram $D[\gamma^{1}]$. The
integral may thus be written in a more compact form by combining these
contributions in the phases, using the definitions

$\displaystyle E[\gamma^{1}]_{\alpha}\ $ $\displaystyle=$ $\displaystyle\
E_{\alpha}+\sum_{i=1}^{r_{1}}E_{b_{1,i}}\,\delta_{\alpha,x^{(b_{1,i})}_{a}}+\sum_{i=1}^{s_{1}}E_{d_{1,i}}\,\delta_{\alpha,x^{(d_{1,i})}_{a}}$
$\displaystyle\eta[\gamma^{1}]^{(\alpha)}_{a}\ $ $\displaystyle=$
$\displaystyle\
\eta_{j}^{(\alpha)}+\sum_{i=1}^{r_{1}}\eta^{(b_{1,i})}_{j}\delta_{\alpha,x^{(b_{1,i})}_{a}}+\sum_{i=1}^{s_{1}}\eta^{(d_{1,i})}_{j}\delta_{\alpha,x^{(d_{1,i})}_{a}}\,.$
(5.18)

An important feature of the new incidence matrix
$\eta[\gamma^{1}]_{j}^{(\alpha)}$ is that every line $j$ for which it takes a
non-zero value is connected to vertex $\alpha$ in $D[\gamma^{1}]$, either
directly or through an embedded extremal vertex whose time has been integrated
up to the time of vertex $\alpha$.

We can now write the full poset integral as

$\displaystyle
2\pi\delta\left(\sum_{\alpha=1}^{n}E_{\alpha}\right)\,F_{D}\left(\\{E_{\alpha}\\},{\cal
L}_{G}\right)$ $\displaystyle=$ $\displaystyle\
\sum_{\gamma^{1}}\;\prod_{i=1}^{r_{1}}\frac{i}{\Delta_{b_{1,i}}[\gamma^{1}]}\;\prod_{j=1}^{s_{1}}\frac{i}{\Delta_{d_{1,j}}[\gamma^{1}]}$
(5.19) $\displaystyle\times\
\prod_{\alpha=1}^{|V[\gamma^{1}]|}\int_{X_{D}[\gamma^{1}]}dt_{\alpha}e^{-i(E[\gamma^{1}]_{\alpha}+\eta[\gamma^{1}]^{(\alpha)}_{j}(\omega_{j}-i\epsilon))t_{\alpha}}\,.$

In this form, the original integral for poset $D$ is expressed as a sum over
$\gamma^{1}$ (which is constructed explicitly in App. B) of products of energy
deficit denominators multiplied by an integral of the original form, Eq.
(5.2), this time for the poset diagram $D[\gamma^{1}]$, with a reduced number
of time integrals remaining. The energy deficit denominators correspond to the
cuts of $D$ associated with each of the embedded extremal vertices in the set
$A^{1}$. Because these vertices make up an antichain, these cuts have no lines
in common. Finally, we recall from Eq. (5.9) that the original list of
vertices, $V$ is given by the union of $V[\gamma^{1}]$ with the extremal
vertices,

$\displaystyle
V[\gamma^{1}]\prod_{i=1}^{r_{1}}\cup\,b_{1,i}\,\prod_{j=1}^{s_{1}}\cup\,d_{1,j}\
=\ V\,.$ (5.20)

We will encounter generalizations of all of these features below.

Equation (5.19) already exhibits the basic structure of our results. Let us
summarize its essential features. The new, induced, poset diagram
$D[\gamma^{1}]$, defined by Eqs. (5.8) and (5.9), is a connected diagram with
a set of vertices $V[\gamma^{1}]$, all of which are local in time. The number
of vertices has decreased by identifying pairs of embedded extremal and next-
to-extremal vertices, with the resulting “composite” vertices inheriting the
next-to-extremal times $t[x_{j}^{(e_{1,i})}]$. The extremal vertices of
$D[\gamma^{1}]$ can be labeled $\\{b_{2,i},d_{2,i}\\}$.

For these induced, embedded minimal vertices, the new incidence matrix defined
in Eq. (5.18), $\eta[\gamma^{1}]^{(b_{2,i})}_{j}$ takes only the values
$0,-1$, with only outgoing lines, although now these lines may have been
emitted originally by the first round of vertices, $\\{e_{1,i}\\}$, over whose
times we have already integrated. These lines thus emerge from a subdiagram of
the original poset $D$ that is connected. Similar considerations apply to
maximal extrema in $D[\gamma^{1}]$, for which
$\eta[\gamma^{1}]^{(d_{2,i})}_{j}=0,+1$. In the new diagram, by analogy to
Eqs. (5.5) and (5.6) we can identify the set of next-to-extremal vertices,
$C^{2}=\\{C_{e_{2,i}}\\}$, which will be labeled $x^{(e_{2,i})}_{a}$.

Examples of this process are given above in the integrations from Fig. 3 to
Fig. 4a and to Fig. 4b, which correspond to different choices of $\gamma^{1}$
in this case. The explicit denominators of Eq. (5.19) provide the results of
the time integrals of the chosen antichain $A^{1}$ of embedded extremal
vertices. By construction, they correspond to non-overlapping cuts, each
separating the original diagram into two connected parts. As such, they are
all “physical” denominators, in the sense we have identified above. The poset
$D[\gamma^{1}]$ and the integral over the region $X_{D[\gamma^{1}]}$ in Eq.
(5.19) have all the properties of the original poset $D$ and integral over
region $X_{D}$ of Eq. (5.2) that we used in the forgoing analysis. In the
following subsection, we use this recursive structure to derive a general
expression for $F_{D}$ in which all unphysical denominators are eliminated.

###  5.2 Integrals of induced extrema

The relation, Eq. (5.19) clearly carries fewer time integrals, and the
remaining time integrals are in the same form that we encountered in Eq.
(5.2). It is therefore straightforward to extend the reasoning recursively in
order to carry out all the time integrals.

Let us assume that we are given the result after $k-1$ iterations of the
procedure outlined above. Each step consists of integration over the induced
embedded extrema of an induced poset. We assume that the result takes the form

$\displaystyle
2\pi\delta\left(\sum_{\alpha=1}^{n}E_{\alpha}\right)\,F_{D}\left(\\{E_{\alpha}\\},{\cal
L}_{G}\right)$ $\displaystyle=$ $\displaystyle\
\sum_{\gamma^{k-1}}\;\prod_{l=1}^{k-1}\prod_{i=1}^{r_{l}}\frac{i}{\Delta_{b_{l,i}}[\gamma^{l}]}\;\prod_{j=1}^{s_{l}}\frac{i}{\Delta_{d_{l,j}}[\gamma^{l}]}$
(5.21) $\displaystyle\times\
\prod_{\alpha=1}^{|V[\gamma^{k-1}]|}\int_{X_{D}[\gamma^{k-1}]}dt_{\alpha}e^{-i(E[\gamma^{k-1}]_{\alpha}+\eta[\gamma^{k-1}]^{(\alpha)}_{j}(\omega_{j}-i\epsilon))t_{\alpha}}\,.$

This integral has a set of properties that hold in the initial case, Eq.
(5.19), corresponding to $k=2$, and which will remain true recursively.

(1) Poset structure. $D[\gamma^{k-1}]$ is a poset with integration region
$X_{D[\gamma^{k-1}]}$, where $\gamma^{k-1}$ is specified by a set of vertices
from the original diagram $D$, and all vertices in $V_{D[\gamma^{k-1}]}$ are
local in time. That is, each remaining vertex $\alpha$ is associated with a
time integral over $t_{\alpha}$. As for any such diagram, among the vertices
of $D[\gamma^{k-1}]$, there is a non-empty set $M^{k}$ of embedded extremal
vertices, $\\{b_{k,i},d_{k,i}\\}$, each with a corresponding covering set,
$C^{k}=\cup_{i}C^{k}_{e_{k,i}}$, of potential next-to-extremal vertices,
$C^{k}_{e_{k,i}}=\\{x^{(e_{k,i})}_{a}\\}$.

(2) Inductive functional dependence. In Eq. (5.21), the explicit denominators
and phases are defined by

$\displaystyle\Delta_{b_{l,i}}[\gamma^{l}]\ =\
E_{b_{l,i}}[\gamma^{l-1}]-\sum_{j}\left|\eta[\gamma^{l-1}]^{(b_{l,i})}_{j}\right|\omega_{j}+i\epsilon\,\,,$
(5.22)

and

$\displaystyle\Delta_{d_{l,i}}[\gamma^{l}]$ $\displaystyle=$ $\displaystyle\
\left(-E_{d_{l,i}}[\gamma^{l-1}]\right)-\sum_{j}\eta[\gamma^{l-1}]^{(d_{l,i})}_{j}\omega_{j}+i\epsilon\,,$
(5.23)

in terms of the inductive relations

$\displaystyle E_{\alpha}[\gamma^{l}]$ $\displaystyle=$ $\displaystyle\
E_{\alpha}[\gamma^{l-1}]+\sum_{i=1}^{r_{l}}E_{b_{l,i}}[\gamma^{l-1}]\,\delta_{\alpha,x^{(b_{l,i})}_{a}}+\sum_{i=1}^{s_{l}}E_{d_{k,i}}[\gamma^{l-1}]\,\delta_{\alpha,x^{(d_{l,i})}_{a}}\,,$
$\displaystyle\eta[\gamma^{l}]^{(\alpha)}_{j}\ $ $\displaystyle=$
$\displaystyle\
\eta[\gamma^{l-1}]_{j}^{(\alpha)}+\sum_{i=1}^{r_{l}}\eta[\gamma^{l-1}]^{(b_{l,i})}_{j}\delta_{\alpha,x^{(b_{l,i})}_{a}}+\sum_{i=1}^{s_{l}}\eta[\gamma^{l-1}]^{(d_{l,i})}_{j}\delta_{\alpha,x^{(d_{l,i})}_{a}}.$

The expressions, Eqs. (5.22), (5.23) and (LABEL:eq:E-eta-gamma-k) are direct
generalizations of Eqs. (5.13), (LABEL:eq:max-integrals) and (5.18), with
$\gamma^{1}$ replaced by $\gamma^{l}$, and $e_{1,i}$ by $e_{l,i}$.

(3) Denominators, cuts, and merged sets. The explicit denominators,
$\Delta_{e_{l,i}}$ ($e=b,d$) in Eq. (5.21), with $l\leq k-1$ all correspond to
cuts of the original diagram, $D$, each of which separates $D$ into exactly
two connected components. One of these components is associated with an
embedded extremal vertex, $e_{l,i}\in A^{l-1}$, of $D[\gamma^{l-1}]$, and with
a connected set of vertices, $\lambda[e_{l,i}]$ of the original diagram. We
will refer to $\lambda[e_{l,i}]$ as the “merged set” of vertices for
$e_{l,i}$, corresponding to one or more series of time integrals that
terminate at $t_{e_{l,i}}$. It may be a minimum or maximum. In either case,
its merged set is given by

$\displaystyle\lambda[e_{1,i}]\ $ $\displaystyle=$ $\displaystyle e_{1,i}\,,$
$\displaystyle\lambda[e_{l,i}]\ $ $\displaystyle=$ $\displaystyle\
\left\\{e_{l,i}\,\prod_{e_{l^{\prime},j}}\cup\,\lambda[e_{l^{\prime},j}]\,\bigg{|}\,x^{(e_{l^{\prime},j})}_{a^{\prime}}=e_{l,i}\,,l^{\prime}\leq
l-1\right\\}\,,$ (5.25)

where the range of index $i$ depends on the set $\gamma^{l-1}$. That is, the
merged set of extremal vertex $e_{l,i}$ is found by merging all the merged
sets of extremal vertices $e_{l^{\prime},i}$, $l^{\prime}\leq l-1$ for which
the vertex $e_{l,i}$ is the nearest vertex in its covering set. For a minimal
extremal vertex in $D[\gamma^{l-1}]$, all lines that cross the cut
$\Delta_{b_{l,i}}[\gamma^{l}]$ emerge from the minimum $b_{l,i}$ and are
absorbed in $D[\gamma^{l-1}]\setminus b_{l,i}$. For a maximal extremal vertex,
lines that cross the cut $\Delta_{d_{l,i}}[\gamma^{l}]$ emerge from
$D[\gamma^{l-1}]\setminus d_{l,i}$ and are absorbed at $d_{l,i}$. In both
cases, in terms of the original poset diagram $D$, the lines of each cut
connect $\lambda[e_{l,i}]$ with $D\setminus\lambda[e_{l,i}]$, both of which
are connected.

(4) Partial nesting. Any set of vertices picked from different
$\lambda[b_{k,i}]$ form an antichain, and similarly for maximal sets
$\lambda[d_{k,i}]$. Together, they satisfy the following properties, which may
be described as partial nesting [18],

$\displaystyle\lambda[e_{l_{1},i}]\subset\lambda[e_{l_{2},i^{\prime}}]\quad{\rm
or}\quad\lambda[e_{l_{1},i}]\cap\lambda[e_{l_{2},i^{\prime}}]\ $
$\displaystyle=$ $\displaystyle\ 0\,,l_{1}\ <l_{2}\,,$
$\displaystyle\lambda[e_{l,i_{1}}]\cap\lambda[e_{l,i_{2}}]\ $ $\displaystyle=$
$\displaystyle\ 0\,,i_{1}\ \neq i_{2}\,,$ $\displaystyle
V[\gamma^{k}]\prod_{i}\,\cup\,\lambda[b_{k,i}]\,\prod_{l}\,\cup\,\lambda[d_{k,l}]\
$ $\displaystyle=$ $\displaystyle\ V\,.$ (5.26)

Taken together, these conditions imply that the sets of vertices
$\lambda[e_{l,i}]$ are either nested, according to index $l$, or disjoint, and
that, together with $V[\gamma^{k}]$, they include all the vertices of the
original diagram $D$. This implies that the cuts represented by the
denominators $\Delta_{e_{l,i}}$ do not cross, since this would lead to a non-
nested relationship between at least two successive sets $\lambda[e_{l,i}]$
and $\lambda[e_{l+1,j}]$. Comparing to Eq. (5.20) above for the first set of
integrals, we observe that indeed $\lambda[b_{1,i}]\ =\ b_{1,i}$ and
$\lambda[d_{1,i}]\ =\ d_{1,i}$, as in Eq. (5.25).

The specific manner in which the posets and the sets $\gamma^{k-1}$ and
$\lambda[e_{k,i}]$ appear will emerge from the following analysis, where we
describe how to carry out the $k$th iteration, the next set of time integrals
in Eq. (5.21). We will see that all of the features (1) – (4) of