# Tadpoles and vacuum bubbles in light-front quantization

S.S. Chabysheva  Department of Physics, University of Idaho, Moscow ID 83844
USA  J.R. Hiller  Department of Physics, University of Idaho, Moscow ID 83844
USA  Department of Physics and Astronomy, University of Minnesota-Duluth,
Duluth, Minnesota 55812 USA

###### Abstract

We develop a method by which vacuum transitions may be included in light-front
calculations. This allows tadpole contributions which are important for
symmetry-breaking effects and yet are missing from standard light-front
calculations. These transitions also dictate a nontrivial vacuum and
contributions from vacuum bubbles to physical states. In nonperturbative
calculations these separate classes of contributions (tadpoles and bubbles)
cannot be filtered; instead, we regulate the bubbles and subtract the vacuum
energy from the eigenenergy of physical states. The key is replacement of
momentum-conserving delta functions with model functions of finite width; the
width becomes the regulator and is removed after subtractions. The approach is
illustrated in free scalar theory, quenched scalar Yukawa theory, and in a
limited Fock-space truncation of $\phi^{4}$ theory.

##  I Introduction

Recent calculations of the critical coupling in two-dimensional $\phi^{4}$
theory [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] have shown that there is a
discrepancy in the nonperturbative equivalence of equal-time and light-front
quantization. Although this discrepancy can be explained with a computed shift
in the renormalized mass [14, 5, 6, 8], this explanation is a correction to
the light-front calculation, rather than a direct calculation. In addition,
calculations with coordinates that interpolate between equal-time and light-
front quantizations [15, 16], indicate that the light-front limit should
obtain the same critical coupling as obtained in equal time quantization [17].
The key is the inclusion of tadpole contributions, which on the light front
requires zero modes, modes with zero longitudinal momentum, to represent
transitions to and from the vacuum, as illustrated in Fig. 1(a).111These
figures were drawn with JaxoDraw [18]. In a nonperturbative calculation, where
one cannot pick and choose classes of diagrams, the presence of vacuum
transitions necessarily imports (divergent) vacuum bubbles, of a sort shown in
Fig. 1(b), as well as tadpoles.222This is a separate question from
perturbative equivalence, which has been generally established. For recent
discussions, see [19, 20, 21].

|
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(a) | (b)
Figure 1:  Tadpole graph (a) and vacuum bubble (b) in $\phi^{4}$ theory. Note
the momentum-conserving transitions to and from the vacuum that imply light-
front zero-mode contributions.

Zero-mode contributions to physical states and the corresponding nontriviality
of the light-front vacuum [22, 23, 19, 20] are of broader interest than just
the critical coupling in $\phi^{4}$ theory. They enter into any discussion of
symmetry breaking, such as the Higgs mechanism, and of vacuum condensates
[24]. More recently they have been identified as possible contributions to
higher-twist distribution functions [25]. For these reasons, we explore a
possible method for inclusion, within the context of two-dimensional scalar
theories; extension to three and four dimensions should be
straightforward.333For alternative methods, see [26] and [27].

Contributions such as tadpoles and vacuum bubbles that involve transitions to
and from the vacuum must rely on terms normally excluded from light-front
Hamiltonians. These are terms with only creation operators or only
annihilation operators. With light-front longitudinal momenta constrained to
be nonnegative, momentum conservation requires that the operators create or
annihilate zero momentum. On this basis they are always dropped. However,
depending on the zero-momentum behavior of the Fock-space wave functions,
matrix elements of such terms need not be zero.

For example, consider light-front quantization [28, 29, 30, 31, 32, 33, 34] of
a two-dimensional scalar theory. We define light-front coordinates [28] and
momenta as $x^{\pm}=t\pm z$ and $p^{\pm}=E\pm p_{z}$, with $x^{+}$ chosen as
the light-front time. The mass-shell condition for the total two-momentum
$(P^{-},P^{+})$ is then $M^{2}=P^{+}P^{-}$, which gives the fundamental bound-
state eigenproblem as

${\cal P}^{-}|\psi(P^{+})\rangle=\frac{M^{2}}{P^{+}}|\psi(P^{+})\rangle,$ (1)

where ${\cal P}^{-}$ is the light-front Hamiltonian. A typical Fock-state wave
function for an eigenstate of this Hamiltonian satisfies an equation of the
following form:

$\sum_{i}^{n}\frac{\mu^{2}}{p_{i}^{+}}\psi_{n}(p_{1}^{+},\ldots,p_{n}^{+})+\frac{1}{\sqrt{p_{1}^{+}p_{2}^{+}p_{3}^{+}\ldots}}\times\mbox{contributions
from other Fock sectors}=\frac{M^{2}}{P^{+}}\psi_{n},$ (2)

with $n$ the number of constituents. The symmetry for bosons then requires
that the small momentum behavior of this wave function is

$\psi_{n}\sim\frac{1}{\sqrt{\prod_{i}^{n}p_{i}^{+}}\sum_{i}^{n}\frac{1}{p_{i}^{+}}}.$
(3)

The matrix element of a vacuum transition in $\phi^{4}$ theory reduces to

$\displaystyle\langle\psi_{n+4}|\int\frac{\prod_{i}^{4}dp_{i}^{+}}{\sqrt{\prod_{i}^{4}p_{i}^{+}}}\delta(\sum_{i}^{4}p_{i}^{+})\prod_{i}^{4}a^{\dagger}(p_{i}^{+})|\psi_{n}\rangle$
(5)
$\displaystyle\sim\int\frac{\prod_{i}^{n+4}dp_{i}^{+}}{\prod_{i}^{n+4}p_{i}^{+}}\frac{\delta(\sum_{i}^{4}p_{n+i}^{+})}{\left(\sum_{i}^{n+4}\frac{1}{p_{i}^{+}}\right)\left(\sum_{i}^{n}\frac{1}{p_{i}^{+}}\right)}\delta(P^{+}-\sum_{i}^{n}p_{i}^{+}).$

With $Q\equiv\sum_{i}^{4}p_{n+i}^{+}$, $p_{n+i}^{+}=x_{i}Q$, and
$\prod_{i=n+1}^{n+4}dp_{i}^{+}=Q^{3}dQ\prod_{i}^{4}dx_{i}\delta(1-\sum_{i}^{4}x_{i})$,
this becomes

$\displaystyle\langle\psi_{n+4}|\int\frac{\prod_{i}^{4}dp_{i}^{+}}{\sqrt{\prod_{i}^{4}p_{i}^{+}}}\delta(\sum_{i}^{4}p_{i}^{+})\prod_{i}^{4}a^{\dagger}(p_{i}^{+})|\psi_{n}\rangle$
$\displaystyle\sim\int
dQ\delta(Q)\int\frac{\prod_{i}^{4}dx_{i}\delta(1-\sum_{i}^{4}x_{i})}{\left(\prod_{i}^{4}x_{i}\right)\left(\sum_{i}^{4}\frac{1}{x_{i}}\right)}\int\frac{\prod_{i}^{n}dp_{i}^{+}\delta(P^{+}-\sum_{i}^{n}p_{i}^{+})}{\left(\prod_{i}^{n}p_{i}^{+}\right)\left(\sum_{i}^{n}\frac{1}{p_{i}^{+}}\right)},$

which is finite and nonzero. Thus, such vacuum-transition terms cannot be
ignored automatically.

Vacuum transitions also generate vacuum bubbles which make contributions
proportional to the size $L$ of the spatial dimension as expressed through
$4\pi\delta(0)=\int dx^{-}\equiv L$. A nonperturbative calculation requires a
cutoff, to regulate this infinity, and a subtraction of the vacuum energy from
any eigenenergy of a physical state. We regulate by replacing delta functions
of momentum with model functions $\delta_{\epsilon}$ that have a width
parameter $\epsilon$ and take the limit of $\epsilon\rightarrow 0$ at the end
of a calculation.444This is equivalent to the regulator used in [19] for the
calculation of a time-ordered product. The parameter $R$ in their Eq. (22),
introduced as the radius of a circle approaching infinity, is essentially
$1/\epsilon$. However, the circle at infinity is something that arises in
diagrammatic calculations rather than nonperturbative eigenvalue problems. For
the (nontrivial) vacuum state, we compute a finite energy density, with the
model parameter $\epsilon$ related to the spatial volume $L$ in a model-
dependent way: $4\pi\delta_{\epsilon}(0)=L$.

For any finite width $\epsilon$, there will be additional modes present in any
calculation. We call these ephemeral modes, since they are not zero modes but
instead disappear in the limit of zero width. The remaining imprint is
essentially a zero-mode contribution, but obtained as a limit. Contributions
to massive states, beyond the vacuum-energy shift and tadpoles, are generally
negligible for weak coupling; however, for strong coupling, the Fock-state
momentum wave functions can be become broad enough that they overlap with
ephemeral modes. Depending on the zero-momentum behavior of these wave
functions, there can be additional contributions from vacuum transitions.

Such contributions cannot be readily captured with the DLCQ formalism [35],
and DLCQ calculations with constrained zero modes [36, 37, 38] are incomplete.
For good resolution of the ephemeral modes, the DLCQ resolution $K$ must
satisfy $1/K\ll\epsilon/P^{+}$. Also, the integrals that must be represented
by the rectangular DLCQ grid are highly singular, for which the grid is ill
suited. Calculations would be best undertaken with a basis function expansion,
for which matrix elements can be computed once and for all with an adaptive
Monte Carlo integration, such as is available in the VEGAS package [39].

Even with antiperiodic boundary conditions, the DLCQ approach cannot neglect
zero modes. With such boundary conditions one can avoid the constraint
equation, but the approximation to the integral operators in ${\cal P}^{-}$ is
the midpoint rule with an error no better than the $1/K^{2}$ for periodic
boundary conditions, where the integrals are approximated by the trapezoidal
rule.555Without a solution to the constraint equation, the error in DLCQ with
periodic boundary conditions is of order $1/K$, unless the endpoints (the zero
modes) make no contribution to the integrals. The trouble is that the
coefficient of the $1/K^{2}$ correction is small only if the integrand is
slowly varying. If instead there is rapid variation, such as can happen near
zero momentum, the approximation becomes quite poor except at very high
resolution. In other words, if zero-mode contributions are important,
antiperiodic boundary conditions do not provide an approximation any better
than periodic boundary conditions.

To explore the inclusion of vacuum transitions, we first consider $\phi^{4}$
theory in more detail; in Sec. II we consider the leading tadpole and vacuum-
bubble contributions. Next, in Sec. III, we develop an analytic solution for a
free scalar as a generalized coherent state of ephemeral modes. The vacuum
bubble contributions replicate the one-loop calculation emphasized by Collins
[22]. We also consider the solution for a shifted scalar with nonzero vacuum
expectation value. This is done in the continuum, without interpolation from
equal-time quantization and without discretization. Finally, we consider
quenched scalar Yukawa theory in lowest-order Fock truncation in Sec. IV, to
see the subtraction of the vacuum energy of the neutral scalar in the charge-
zero sector from the dressed scalar energy in the charge-one sector. Numerical
calculations are postponed to future work.

##  II Lowest-order $\phi^{4}$ theory

The Lagrangian for two-dimensional $\phi^{4}$ theory is

${\cal
L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}\mu^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4},$
(7)

where $\mu$ is the mass of the boson and $\lambda$ is the coupling constant.
The light-front Hamiltonian density is

${\cal H}=\frac{1}{2}\mu^{2}\phi^{2}+\frac{\lambda}{4!}\phi^{4}.$ (8)

The mode expansion for the field is666For convenience we drop the $+$
superscript and will from here on write light-front momenta such as $p^{+}$ as
just $p$.

$\phi(x^{+}=0,x^{-})=\int\frac{dp}{\sqrt{4\pi
p}}\left\\{a(p)e^{-ipx^{-}/2}+a^{\dagger}(p)e^{ipx^{-}/2}\right\\}.$ (9)

The nonzero commutation relation is

$[a(p),a^{\dagger}(p^{\prime})]=\delta(p-p^{\prime}).$ (10)

The light-front Hamiltonian is ${\cal P}^{-}={\cal P}^{-}_{0}+{\cal
P}^{-}_{\rm int}$, with

$\displaystyle{\cal P}^{-}_{0}$ $\displaystyle=$ $\displaystyle\int
dp\frac{\mu^{2}}{p}a^{\dagger}(p)a(p)+\frac{\mu^{2}}{2}\int\frac{dp_{1}dp_{2}}{\sqrt{p_{1}p_{2}}}\delta_{\epsilon}(p_{1}+p_{2})\left[a(p_{1})a(p_{2})+a^{\dagger}(p_{1})a^{\dagger}(p_{2})\right],$
(11) $\displaystyle{\cal P}^{-}_{\rm int}$ $\displaystyle=$
$\displaystyle{\cal P}^{-}_{04}+{\cal P}^{-}_{40}+{\cal P}^{-}_{22}+{\cal
P}^{-}_{13}+{\cal P}^{-}_{31},$ (12)

where

$\displaystyle{\cal P}^{-}_{04}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{24}\int\frac{dp_{1}dp_{2}dp_{3}dp_{4}}{4\pi\sqrt{p_{1}p_{2}p_{3}p_{4}}}\delta_{\epsilon}(\sum_{i}^{4}p_{i})a(p_{1})a(p_{2})a(p_{3})a(p_{4}),$
(13) $\displaystyle{\cal P}^{-}_{40}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{24}\int\frac{dp_{1}dp_{2}dp_{3}dp_{4}}{4\pi\sqrt{p_{1}p_{2}p_{3}p_{4}}}\delta_{\epsilon}(\sum_{i}^{4}p_{i})a^{\dagger}(p_{1})a^{\dagger}(p_{2})a^{\dagger}(p_{3})a^{\dagger}(p_{4}),$
(14) $\displaystyle{\cal P}^{-}_{22}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{4}\int\frac{dp_{1}dp_{2}}{4\pi\sqrt{p_{1}p_{2}}}\int\frac{dp^{\prime}_{1}dp^{\prime}_{2}}{\sqrt{p^{\prime}_{1}p^{\prime}_{2}}}\delta(p_{1}+p_{2}-p^{\prime}_{1}-p^{\prime}_{2})$
$\displaystyle\rule{144.54pt}{0.0pt}\times
a^{\dagger}(p_{1})a^{\dagger}(p_{2})a(p^{\prime}_{1})a(p^{\prime}_{2}),$
$\displaystyle{\cal P}^{-}_{13}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{6}\int\frac{dp_{1}dp_{2}dp_{3}}{4\pi\sqrt{p_{1}p_{2}p_{3}(p_{1}+p_{2}+p_{3})}}a^{\dagger}(p_{1}+p_{2}+p_{3})a(p_{1})a(p_{2})a(p_{3}),$
(16) $\displaystyle{\cal P}^{-}_{31}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{6}\int\frac{dp_{1}dp_{2}dp_{3}}{4\pi\sqrt{p_{1}p_{2}p_{3}(p_{1}+p_{2}+p_{3})}}a^{\dagger}(p_{1})a^{\dagger}(p_{2})a^{\dagger}(p_{3})a(p_{1}+p_{2}+p_{3}).$
(17)

The subscripts indicate the number of creation and annihilation operators in
each term.

To isolate the contribution from the tadpole and vacuum bubble in Fig. 1, we
consider only two terms in the Fock-state expansion of the eigenstate

$|\psi(P)\rangle=\psi_{1}a^{\dagger}(P)|0\rangle+\cdots+\int\prod_{i}^{5}dp_{i}\delta(P-\sum_{i}^{5}p_{i})\psi_{5}(p_{1},\ldots,p_{5})\frac{1}{\sqrt{5!}}\prod_{i}^{5}a^{\dagger}(p_{i})|0\rangle+\cdots,$
(18)

in order to represent the five constituents in the intermediate states, and we
keep only the first term of ${\cal P}^{-}_{0}$ and the first three terms of
${\cal P}^{-}_{\rm int}$, as the only terms that connect the two Fock sectors.
We then consider the eigenvalue problem

$({\cal P}^{-}_{0}+{\cal P}^{-}_{\rm
int})|\psi(P)\rangle=\left(\frac{M^{2}}{P}+P_{\rm
vac}^{-}\right)|\psi(P)\rangle,$ (19)

where we include the shift of vacuum energy $P_{\rm vac}^{-}$, to be obtained
from solving the corresponding vacuum eigenvalue problem

$({\cal P}^{-}_{0}+{\cal P}^{-}_{\rm int})|{\rm vac}\rangle=P_{\rm
vac}^{-}|{\rm vac}\rangle,$ (20)

with $|{\rm vac}\rangle$ the lowest eigenstate. Projection of the eigenvalue
problem for the lowest massive state onto Fock sectors yields a system of
equations for the Fock-state wave functions

$\displaystyle\frac{\mu^{2}}{P}\psi_{1}$ $\displaystyle+$
$\displaystyle\frac{\lambda}{\sqrt{24}}\int\frac{\prod_{i}^{4}dp_{i}}{4\pi\sqrt{\prod_{i}^{4}p_{i}}}\delta_{\epsilon}(\sum_{i}^{4}p_{i})\psi_{5}(p_{1},\ldots,p_{5})=\left(\frac{M^{2}}{P}+P_{\rm
vac}^{-}\right)\psi_{1},$ (21)
$\displaystyle\left(\sum_{i}^{5}\frac{\mu^{2}}{p_{i}}\right)\psi_{5}$
$\displaystyle+$
$\displaystyle\frac{\lambda}{24}\frac{1}{5}\left[\frac{\delta_{\epsilon}(\sum_{i}^{4}p_{i})}{4\pi\sqrt{\prod_{i}^{4}p_{i}}}+(p_{5}\leftrightarrow
p_{1},p_{2},p_{3},p_{4})\right]\psi_{1}$ $\displaystyle+$ $\displaystyle
20\frac{\lambda}{4}\int\frac{dp^{\prime}_{1}dp^{\prime}_{2}}{4\pi\sqrt{p_{1}p_{2}p^{\prime}_{1}p^{\prime}_{2}}}\delta(p_{1}+p_{2}-p^{\prime}_{1}-p^{\prime}_{2})\psi_{5}(p^{\prime}_{1},p^{\prime}_{2},p_{3},p_{4},p_{5})=\frac{M^{2}}{P}\psi_{5},$

where we have invoked a sector-dependent energy shift, with no $P_{\rm
vac}^{-}$ in the top Fock sector.777In the top sector, there are, of course,
no vacuum corrections from higher Fock sectors.

The second equation can be solved iteratively with respect to the self-
coupling of the five-constituent Fock state in the third term of (II); this
corresponds to a diagrammatic expansion. The leading term in the expansion
generates the vacuum bubble in Fig. 1(b) that contributes $P_{\rm vac}^{-}$ in
(21). The second term, where the self interaction acts once, produces the
tadpole in Fig. 1(a). Both are written explicitly in (23) and (24) below.
Subtraction of $P_{\rm vac}^{-}$ from both sides of (21) eliminates the
divergent bubble.

From (21) and (II), the contributions take the forms

${\rm
bubble}\rightarrow\int\frac{\prod_{i}^{5}dp_{i}}{\prod_{i}^{4}p_{i}}\delta(P-\sum_{i}^{5}p_{i})\frac{\delta_{\epsilon}(\sum_{i}^{4}p_{i})^{2}}{\frac{M^{2}}{P}-\sum_{i}^{5}\frac{\mu^{2}}{p_{i}}}\sim-\int\delta_{\epsilon}(Q)^{2}\frac{dQ}{\mu^{2}}\int\frac{\prod_{i}^{4}dx_{i}}{\prod_{i}^{4}x_{i}}\delta(1-\sum_{i}^{4}x_{i}),$
(23) ${\rm
tadpole}\rightarrow\int\frac{\prod_{i}^{5}dp_{i}\delta_{\epsilon}(\sum_{i}^{4}p_{i})}{\sqrt{\prod_{i}^{4}p_{i}}}\frac{\delta(P-\sum_{i}^{5}p_{i})}{\frac{M^{2}}{P}-\sum_{i}^{5}\frac{\mu^{2}}{p_{i}}}\int\frac{dp^{\prime}_{1}dp^{\prime}_{2}}{\sqrt{p_{4}p_{5}p^{\prime}_{1}p^{\prime}_{2}}}\frac{\delta(p_{4}+p_{5}-p^{\prime}_{1}-p^{\prime}_{2})}{\frac{M^{2}}{P}-\sum_{i}^{3}\frac{\mu^{2}}{p_{i}}-\frac{\mu^{2}}{p^{\prime}_{1}}-\frac{\mu^{2}}{p^{\prime}_{2}}}\frac{\delta_{\epsilon}(\sum_{i}^{3}p_{i}+p^{\prime}_{1})}{\sqrt{\prod_{i}^{3}p_{i}p^{\prime}_{1}}}$
(24)

The expression for the bubble diverges as $\epsilon\rightarrow 0$ and is
proportional to $\delta(0)=L/4\pi$; however, the same expression is obtained
for the nontrivial vacuum energy $P_{\rm vac}^{-}$ and is subtracted.

The expression for the tadpole contribution can be simplified by noting that
$\delta(P-\sum_{i}^{5}p_{i})$ reduces to $\delta(P-p_{5})$, which can be used
to do the $p_{5}$ integral and
$\delta(p_{4}+p_{5}-p^{\prime}_{1}-p^{\prime}_{2})$ becomes
$\delta(p_{4}+P-p^{\prime}_{1}-p^{\prime}_{2})$, which can be used to do the
$p^{\prime}_{2}$ integral. Finally,
$\delta_{\epsilon}(\sum_{i}^{3}p_{i}+p^{\prime}_{1})$ can be written
$\delta_{\epsilon}(p_{4}-p^{\prime}_{1})$ and used to do the $p^{\prime}_{1}$
integral. These leave

${\rm
tadpole}\sim\int\frac{\prod_{i}^{4}dp_{i}}{\prod_{i}^{4}p_{i}}\frac{1}{p_{4}P}\frac{\delta_{\epsilon}(\sum_{i}^{4}p_{i})}{\left[\frac{M^{2}}{P}-\sum_{i}^{4}\frac{\mu^{2}}{p_{i}}-\frac{\mu^{2}}{P}\right]^{2}}\sim\frac{1}{P}\int\delta_{\epsilon}(Q)dQ\int\frac{\prod_{i}^{4}dx_{i}}{(\prod_{i}^{4}x_{i})x_{4}\left(\sum_{i}^{4}\frac{\mu^{2}}{x_{i}}\right)^{2}},$
(25)

which is finite and inversely proportional to $P$, the mark of a light-front
self-energy correction.

Of course, in a nonperturbative calculation, these contributions cannot be
separated. However, with the bubbles regulated, one can solve the
eigenproblems for the vacuum and the massive states and then carry out the
necessary $P_{\rm vac}^{-}$ subtraction prior to taking the width parameter
$\epsilon$ to zero.

##  III Free scalar

###  III.1 Free vacuum

The free vacuum $|{\rm vac}\rangle$ is an eigenstate of the free scalar
Hamiltonian ${\cal P}^{-}_{0}$ in (11):

${\cal P}^{-}_{0}|{\rm vac}\rangle=P_{\rm vac}^{-}|{\rm vac}\rangle.$ (26)

We will show that the vacuum is a generalized coherent state,

$|{\rm vac}\rangle=\sqrt{Z}e^{A^{\dagger}}|0\rangle,$ (27)

where

$A^{\dagger}=\int_{0}^{\infty}\frac{dp_{1}dp_{2}}{\sqrt{p_{1}p_{2}}}\frac{f(p_{1},p_{2})}{\frac{1}{p_{1}}+\frac{1}{p_{2}}}a^{\dagger}(p_{1})a^{\dagger}(p_{2}).$
(28)

For such a state, we have

$a(p)|{\rm
vac}\rangle=2\int\frac{dp^{\prime}}{\sqrt{pp^{\prime}}}\frac{f(p,p^{\prime})}{\frac{1}{p}+\frac{1}{p^{\prime}}}a^{\dagger}(p^{\prime})|{\rm
vac}\rangle$ (29)

and

$\displaystyle a(p_{1})a(p_{2})|{\rm vac}\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{p_{1}p_{2}}}\frac{f(p_{1},p_{2})}{\frac{1}{p_{1}}+\frac{1}{p_{2}}}|{\rm
vac}\rangle$
$\displaystyle+4\int\frac{dp^{\prime}_{1}dp^{\prime}_{2}}{\sqrt{p_{1}p_{2}p^{\prime}_{1}p^{\prime}_{2}}}\frac{f(p_{1},p^{\prime}_{1})}{\frac{1}{p_{1}}+\frac{1}{p^{\prime}_{1}}}\frac{f(p_{2},p^{\prime}_{2})}{\frac{1}{p_{2}}+\frac{1}{p^{\prime}_{2}}}a^{\dagger}(p^{\prime}_{1})a^{\dagger}(p^{\prime}_{2})|{\rm
vac}\rangle.$

With these we can apply ${\cal P}^{-}_{0}$ to obtain

$\displaystyle{\cal P}^{-}_{0}|{\rm vac}\rangle$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}}{2}\int\frac{dp_{1}dp_{2}}{\sqrt{p_{1}p_{2}}}\delta_{\epsilon}(p_{1}+p_{2})\left[a^{\dagger}(p_{1})a^{\dagger}(p_{2})+\frac{2}{\sqrt{p_{1}p_{2}}}\frac{f(p_{1},p_{2})}{\frac{1}{p_{1}}+\frac{1}{p_{2}}}\right.$
$\displaystyle\hskip
72.26999pt\left.+4\int\frac{dp^{\prime}_{1}dp^{\prime}_{2}}{\sqrt{p_{1}p_{2}p^{\prime}_{1}p^{\prime}_{2}}}\frac{f(p_{1},p^{\prime}_{1})}{\frac{1}{p_{1}}+\frac{1}{p^{\prime}_{1}}}\frac{f(p_{2},p^{\prime}_{2})}{\frac{1}{p_{2}}+\frac{1}{p^{\prime}_{2}}}a^{\dagger}(p^{\prime}_{1})a^{\dagger}(p^{\prime}_{2})\right]|{\rm
vac}\rangle$ $\displaystyle+\int dp\frac{\mu^{2}}{p}a^{\dagger}(p)\int
dp^{\prime}\frac{2}{\sqrt{pp^{\prime}}}\frac{f(p,p^{\prime})}{\frac{1}{p}+\frac{1}{p^{\prime}}}a^{\dagger}(p^{\prime})|{\rm
vac}\rangle.$

The solution to ${\cal P}^{-}_{0}|{\rm vac}\rangle=P_{\rm vac}^{-}|{\rm
vac}\rangle$ is then possible if

$P_{\rm
vac}^{-}=\frac{\mu^{2}}{2}\int\frac{dp_{1}dp_{2}}{\sqrt{p_{1}p_{2}}}\delta_{\epsilon}(p_{1}+p_{2})\frac{2}{\sqrt{p_{1}p_{2}}}\frac{f(p_{1},p_{2})}{\frac{1}{p_{1}}+\frac{1}{p_{2}}}$
(32)

and the symmetrized coefficients of $a^{\dagger}(p_{1})a^{\dagger}(p_{2})$ sum
to zero:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}}{2}\frac{\delta_{\epsilon}(p_{1}+p_{2})}{\sqrt{p_{1}p_{2}}}+2\mu^{2}\int\frac{dp^{\prime}_{1}dp^{\prime}_{2}}{p^{\prime}_{1}p^{\prime}_{2}\sqrt{p_{1}p_{2}}}\delta_{\epsilon}(p^{\prime}_{1}+p^{\prime}_{2})\frac{f(p_{1},p^{\prime}_{1})}{\frac{1}{p_{1}}+\frac{1}{p^{\prime}_{1}}}\frac{f(p_{2},p^{\prime}_{2})}{\frac{1}{p_{2}}+\frac{1}{p^{\prime}_{2}}}$
$\displaystyle+\frac{1}{2}\left[\frac{\mu^{2}}{p_{1}}+\frac{\mu^{2}}{p_{2}}\right]\frac{2}{\sqrt{p_{1}p_{2}}}\frac{f(p_{1},p_{2})}{\frac{1}{p_{1}}+\frac{1}{p_{2}}}.$

In the second term of (III.1), we can compute

$\displaystyle\int\frac{dp^{\prime}_{1}dp^{\prime}_{2}}{p^{\prime}_{1}p^{\prime}_{2}}\delta_{\epsilon}(p^{\prime}_{1}+p^{\prime}_{2})\frac{f(p_{1},p^{\prime}_{1})}{\frac{1}{p_{1}}+\frac{1}{p^{\prime}_{1}}}\frac{f(p_{2},p^{\prime}_{2})}{\frac{1}{p_{2}}+\frac{1}{p^{\prime}_{2}}}$
$\displaystyle=p_{1}p_{2}\int QdQ\delta_{\epsilon}(Q)\int
dx\frac{f(p_{1},xQ)f(p_{2},(1-x)Q)}{(p_{1}+xQ)(p_{2}+(1-x)Q)}=f(p_{1})f(p_{2})\int
QdQ\delta_{\epsilon}(Q)=0.$

The sum of coefficients in (III.1) is then zero if

$f(p_{1},p_{2})=-\frac{1}{2}\delta_{\epsilon}(p_{1}+p_{2}).$ (35)

This determines the vacuum state.

With this solution for the coherent-state wave function, the energy of the
vacuum is

$\displaystyle P_{\rm vac}^{-}$ $\displaystyle=$
$\displaystyle-\frac{\mu^{2}}{2}\int\frac{dp_{1}dp_{2}}{p_{1}p_{2}}\frac{\delta_{\epsilon}(p_{1}+p_{2})^{2}}{\frac{1}{p_{1}}+\frac{1}{p_{2}}}=-\frac{\mu^{2}}{2}\int\frac{QdQdx}{Q^{2}x(1-x)}\frac{\delta_{\epsilon}(Q)^{2}}{\frac{1}{Q}\frac{1}{x(1-x)}}$
(36) $\displaystyle=$ $\displaystyle-\frac{\mu^{2}}{2}\int
dQ\delta_{\epsilon}(Q)^{2}=-\frac{\mu^{2}}{2}\delta_{\epsilon}(0)\int_{0}^{\infty}dQ\delta_{\epsilon}(Q)=-\frac{\mu^{2}}{2}\frac{L}{4\pi}\frac{1}{2}=-\frac{\mu^{2}L}{16\pi}.$

Here $L$ is the (infinite) volume of light-front space; however,
$\delta_{\epsilon}$ at finite $\epsilon$ regulates $P_{\rm vac}^{-}$ when it
is embedded in a nonperturbative calculation.

Figure 2:  One-loop self-energy graph as a simple vacuum bubble which
contributes to the vacuum state of a free scalar. The dashed line indicates
the intermediate state of two ephemeral modes.

This result is proportional to the one-loop vacuum bubble computed by Collins
[22]. The equivalent perturbative calculation, corresponding to the loop in
Fig. 2, is

$2\frac{\mu^{2}}{2}\int\frac{dq_{1}dq_{2}}{\sqrt{q_{1}q_{2}}}\delta_{\epsilon}(q_{1}+q_{2})\frac{1}{\frac{M^{2}}{P}-\frac{\mu^{2}}{q_{1}}-\frac{\mu^{2}}{q_{2}}}\frac{\mu^{2}}{2}\frac{\delta_{\epsilon}(q_{1}+q_{2})}{\sqrt{q_{1}q_{2}}}=2\frac{\mu^{4}}{4}\int\frac{dq_{1}dq_{2}}{q_{1}q_{2}}\frac{\delta_{\epsilon}(q_{1}+q_{2})^{2}}{-\frac{\mu^{2}}{q_{1}}-\frac{\mu^{2}}{q_{2}}},$
(37)

which matches (36). The leading factor of 2 comes from the two possible
contractions of the double scalar creation and annihilation operators.

Any massive state in the free theory has a Fock-state wave function that is a
product of delta functions of the individual particle momenta $p_{i}$. This
part of the state will not mix with the ephemeral modes, provided the width
parameter $\epsilon$ is chosen such that it is much less than all the $p_{i}$.
For example, the single-particle state with mass $\mu$ and momentum $P$ is
just $a^{\dagger}(P)|{\rm vac}\rangle$. It is an eigenstate of ${\cal
P}^{-}_{0}-P_{\rm vac}^{-}$ with eigenvalue $\mu^{2}/P$. Weak couplings that
broaden the momentum-space wave functions only slightly will produce
effectively unmixed contributions, but calculations with strong couplings
require more care.

###  III.2 Shifted scalar

Next we consider the shifted free scalar where $\phi\rightarrow\phi+v$. The
new Lagrangian is

${\cal L}={\cal L}_{v=0}-\mu^{2}v\phi-\frac{1}{2}\mu^{2}v^{2},$ (38)

and the Hamiltonian is ${\cal P}^{-}={\cal P}^{-}_{0}+{\cal P}^{-}_{\rm int}$
with the interaction part

${\cal P}^{-}_{\rm int}=\int
dx^{-}\mu^{2}v\phi=\sqrt{4\pi}\mu^{2}v\int\frac{dp}{\sqrt{p}}\delta_{\epsilon}(p)[a(p)+a^{\dagger}(p)]+\frac{1}{2}\mu^{2}v^{2}L.$
(39)

The constant term represents the shift in the energy of the vacuum and is
therefore proportional to the spatial size $L$.

The vacuum state $|{\rm vac}\rangle_{v}$ is now an eigenstate of ${\cal
P}^{-}$

$\left({\cal P}^{-}_{0}+{\cal P}^{-}_{\rm int}\right)|{\rm
vac}\rangle_{v}=P_{\rm vac}^{-}|{\rm vac}\rangle_{v}.$ (40)

It can be constructed from the free vacuum as $e^{B}|{\rm vac}\rangle$ with

$B\equiv v\int dp\sqrt{4\pi p}\,\delta_{\epsilon}(p)[a^{\dagger}(p)-a(p)].$
(41)

This works because

$e^{B}\phi(x^{-})e^{-B}=\phi(x^{-})+v$ (42)

and

$e^{B}{\cal P}^{-}_{0}e^{-B}={\cal P}^{-}_{0}+{\cal P}^{-}_{\rm int}.$ (43)

This then permits

$\left({\cal P}^{-}_{0}+{\cal P}^{-}_{\rm int}\right)|{\rm
vac}\rangle_{v}=e^{B}{\cal P}^{-}_{0}e^{-B}e^{B}|{\rm vac}\rangle=e^{B}{\cal
P}^{-}_{0}|{\rm vac}\rangle=P_{\rm vac}^{-}e^{B}|{\rm vac}\rangle=P_{\rm
vac}^{-}|{\rm vac}\rangle_{v}.$ (44)

Thus, in both the free and shifted cases, the vacuum is a generalized coherent
state of ephemeral modes.

The state is also correctly normalized, because

${}_{v}\langle{\rm vac}|{\rm vac}\rangle_{v}=\langle{\rm
vac}|e^{B^{\dagger}}e^{B}|{\rm vac}\rangle=\langle{\rm vac}|e^{-B}e^{B}|{\rm
vac}\rangle=\langle{\rm vac}|{\rm vac}\rangle=1.$ (45)

The vacuum expectation value of the field can also be computed:

${}_{v}\langle{\rm vac}|\phi(x^{-})|{\rm vac}\rangle_{v}$ $\displaystyle=$
$\displaystyle\langle{\rm vac}|e^{B^{\dagger}}\phi(x^{-})e^{B}|{\rm
vac}\rangle=\langle{\rm vac}|e^{-B}\phi(x^{-})e^{B}|{\rm vac}\rangle$ (46)
$\displaystyle=$ $\displaystyle\langle{\rm vac}|(\phi(x^{-})-v)|{\rm
vac}\rangle=-v.$

This restores the shift.

##  IV Quenched scalar Yukawa theory

In order to look at a case with more structure, we consider scalar Yukawa
theory [40], for which the Lagrangian is

${\cal
L}=|\partial_{\mu}\chi|^{2}-m^{2}|\chi|^{2}+\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}\mu^{2}\phi^{2}-g\phi|\chi|^{2},$
(47)

where $\chi$ is a complex scalar field with mass $m$ and $\phi$ is a real
scalar field with mass $\mu$. The two fields are coupled by a Yukawa term with
strength $g$. In two dimensions, the light-front Hamiltonian density is

${\cal H}=m^{2}|\chi|^{2}+\frac{1}{2}\mu^{2}\phi^{2}+g\phi|\chi|^{2}.$ (48)

The mode expansions for the fields are (9) for $\phi$ and

$\chi=\int\frac{dp}{\sqrt{4\pi
p}}\left[c_{+}(p)e^{-ipx^{-}/2}+c_{-}^{\dagger}(p)e^{ipx^{-}/2}\right].$ (49)

The nonzero commutation relations of the creation and annihilation operators
are (10) and

$[c_{\pm}(p),c^{\dagger}_{\pm}(p^{\prime})]=\delta(p-p^{\prime}).$ (50)

In terms of these operators, the quenched light-front Hamiltonian ${\cal
P}^{-}=\int dx^{-}{\cal H}={\cal P}^{-}_{0}+{\cal P}^{-}_{\rm int}$ is
specified by

$\displaystyle{\cal P}^{-}_{0}$ $\displaystyle=$ $\displaystyle\int
dp\frac{m^{2}}{p}\left[c_{+}^{\dagger}(p)c_{+}(p)+c_{-}^{\dagger}(p)c_{-}(p)\right]+\int
dq\frac{\mu^{2}}{q}a^{\dagger}(q)a(q)$
$\displaystyle+\frac{\mu^{2}}{2}\int\frac{dq_{1}dq_{2}}{\sqrt{q_{1}q_{2}}}\delta_{\epsilon}(q_{1}+q_{2})\left[a(q_{1})a(q_{2})+a^{\dagger}(q_{1})a^{\dagger}(q_{2})\right],$

and

${\cal P}^{-}_{\rm int}=g\int\frac{dpdq}{\sqrt{4\pi
pq(p+q)}}\left\\{\left[c_{+}^{\dagger}(p+q)c_{+}(p)+c_{-}^{\dagger}(p+q)c_{-}(p)\right]a(q)+{\rm
h.c.}\right\\}.$ (52)

Pair creation and annihilation terms are suppressed for the complex scalar;
without this quenching, the theory is unstable [41]. This also suppresses
ephemeral modes for the complex scalar, which would need to appear in pairs to
conserve charge, leaving only those of the neutral scalar. The vacuum in the
charge-zero sector is that of the free scalar, as given in the previous
section; this provides the value of $P_{\rm vac}^{-}$ for subtraction in the
charge-one sector.

We seek eigenstates of ${\cal P}^{-}$, for which the two-dimensional light-
front mass eigenvalue problem is

${\cal P}^{-}|\psi(P)\rangle=\left(\frac{M^{2}}{P}+P_{\rm
vac}^{-}\right)|\psi(P)\rangle.$ (53)

We limit this to the charge-one sector. This sector is characterized as a
single complex scalar dressed by a cloud of neutrals. For the present purposes
we will consider only a severe Fock-space truncation that keeps no more than
two neutrals. The Fock-state expansion for the eigenstate is then

$\displaystyle|\psi(P)\rangle$ $\displaystyle=$
$\displaystyle\psi_{0}c_{+}^{\dagger}(P)|0\rangle+\int
dqdp\,\delta(P-q-p)\psi_{1}(q)a^{\dagger}(q)c_{+}^{\dagger}(p)|0\rangle$
$\displaystyle+\int
dq_{1}dq_{2}dp\,\delta(P-q_{1}-q_{2}-p)\psi_{2}(q_{1},q_{2})\frac{1}{\sqrt{2}}a^{\dagger}(q_{1})a^{\dagger}(q_{2})c_{+}^{\dagger}(p)|0\rangle.$

The normalization condition
$\langle\psi(P^{\prime})|\psi(P)\rangle=\delta(P^{\prime}-P)$ becomes

$1=|\psi_{0}|^{2}+\int dq|\psi_{1}|^{2}+\int dq_{1}dq_{2}|\psi_{2}|^{2}.$ (55)

To construct the eigenvalue problem for the wave functions, we act with ${\cal
P}^{-}_{0}$ and ${\cal P}^{-}_{\rm int}$ on the eigenstate and then project
onto the three Fock sectors included in the truncation. Terms that generate
higher Fock sectors are dropped. For ${\cal P}^{-}_{0}$ we have

$\displaystyle{\cal
P}^{-}_{0}|\psi(P)\rangle=\frac{m^{2}}{P}\psi_{0}c_{+}^{\dagger}(P)|0\rangle+\int
dqdp\,\delta(P-q-p)\left(\frac{\mu^{2}}{q}+\frac{m^{2}}{p}\right)\psi_{1}(q)a^{\dagger}(q)c_{+}^{\dagger}(p)|0\rangle$
$\displaystyle+\int
dq_{1}dq_{2}dp\,\delta(P-q_{1}-q_{2}-p)\left(\frac{\mu^{2}}{q_{1}}+\frac{\mu^{2}}{q_{2}}+\frac{m^{2}}{p}\right)\psi_{2}(q_{1},q_{2})\frac{1}{\sqrt{2}}a^{\dagger}(q_{1})a^{\dagger}(q_{2})c_{+}^{\dagger}(p)|0\rangle$
$\displaystyle+\frac{\mu^{2}}{2}\int\frac{dq_{1}dq_{2}}{\sqrt{q_{1}q_{2}}}\delta_{\epsilon}(q_{1}+q_{2})\psi_{0}a^{\dagger}(q_{1})a^{\dagger}(q_{2})c_{+}^{\dagger}(P)|0\rangle+\frac{\mu^{2}}{2}\int\frac{dq_{1}dq_{2}}{\sqrt{q_{1}q_{2}}}\delta_{\epsilon}(q_{1}+q_{2})\psi_{2}(q_{1},q_{2})c_{+}^{\dagger}(P)|0\rangle.$

The last two terms violate momentum conservation but only by amounts of order
$\epsilon$, the width of $\delta_{\epsilon}$. For ${\cal P}^{-}_{\rm int}$ we
find

$\displaystyle{\cal P}^{-}_{\rm
int}|\psi(P)\rangle=\frac{g}{\sqrt{4\pi}}\int\frac{dqdp}{\sqrt{qp(p+q)}}\delta(P-p-q)\left[\psi_{1}(q)c_{+}^{\dagger}(P)+\psi_{0}a^{\dagger}(q)c_{+}^{\dagger}(p)\right]|0\rangle$
$\displaystyle+\sqrt{2}\frac{g}{\sqrt{4\pi}}\int\frac{dq_{1}dq_{2}dp}{\sqrt{q_{2}p(p+q_{2})}}\delta(P-p-
q_{1}-q_{2})\left[\psi_{2}(q_{1},q_{2})a^{\dagger}(q_{1})c_{+}^{\dagger}(p+q_{2})\right.$
$\displaystyle\hskip
238.49121pt\left.+\psi_{1}(q_{1})a^{\dagger}(q_{1})a^{\dagger}(q_{2})c_{+}^{\dagger}(p)\right]|0\rangle.$

Projection of ${\cal P}^{-}|\psi(P)\rangle=\left(\frac{M^{2}}{P}+P_{\rm
vac}^{-}\right)|\psi(P)\rangle$ onto each of the three Fock sectors yields the
following three equations:

$\frac{m^{2}}{P}\psi_{0}+\frac{g}{\sqrt{4\pi}}\int_{0}^{P}\frac{dq\,\psi_{1}(q)}{\sqrt{qP(P-q)}}+\frac{\mu^{2}}{\sqrt{2}}\int\frac{dq_{1}dq_{2}}{\sqrt{q_{1}q_{2}}}\delta_{\epsilon}(q_{1}+q_{2})\psi_{2}(q_{1},q_{2})=\left(\frac{M^{2}}{P}+P_{\rm
vac}^{-}\right)\psi_{0},$ (58)
$\left(\frac{\mu^{2}}{q}+\frac{m^{2}}{P-q}\right)\psi_{1}(q)+\frac{g}{\sqrt{4\pi}}\frac{\psi_{0}}{\sqrt{qP(P-q)}}+\sqrt{2}\frac{g}{\sqrt{4\pi}}\int_{0}^{P-q}\frac{dq^{\prime}\,\psi_{2}(q,q^{\prime})}{\sqrt{q^{\prime}(P-q)(P-q-q^{\prime})}}=\frac{M^{2}}{P}\psi_{1}(q),$
(59)

and

$\displaystyle\left(\frac{\mu^{2}}{q_{1}}+\frac{\mu^{2}}{q_{2}}+\frac{m^{2}}{P-q_{1}-q_{2}}\right)\psi_{2}(q_{1},q_{2})+\frac{\mu^{2}}{\sqrt{2}}\delta_{\epsilon}(q_{1}+q_{2})\frac{\psi_{0}}{\sqrt{q_{1}q_{2}}}$
$\displaystyle+\frac{1}{\sqrt{2}}\frac{g}{\sqrt{4\pi}}\left[\frac{\psi_{1}(q_{1})}{\sqrt{q_{2}(P-q_{1})(P-q_{1}-q_{2})}}+\frac{\psi_{1}(q_{2})}{\sqrt{q_{1}(P-q_{2})(P-q_{1}-q_{2})}}\right]=\frac{M^{2}}{P}\psi_{2}(q_{1},q_{2}).$

The vacuum energy $P_{\rm vac}^{-}$ appears only in the first equation,
because the Fock-space truncation prevents any such correction in all but the
lowest Fock sector.

We build a matrix representation for these equations by introducing basis-
function expansions

$\psi_{1}(q)=\frac{1}{\sqrt{P}}\sum_{n}a_{n}f_{n}(q),\;\;\psi_{2}(q_{1},q_{2})=\frac{1}{P}\sum_{nj}b_{nj}g_{nj}(q_{1},q_{2}),$
(61)

with $p=P-q$ and $p=P-q_{1}-q_{2}$, respectively, $\tilde{m}\equiv m/\mu$, and

$\displaystyle f_{-1}(q)$ $\displaystyle=$
$\displaystyle\frac{C_{-1}}{\sqrt{q(P-q)}}\frac{P\delta_{\epsilon}(q)}{\frac{1}{q}+\frac{\tilde{m}^{2}}{P-q}}=C_{-1}\sqrt{qP}\delta_{\epsilon}(q),$
(62) $\displaystyle f_{n}(q)$ $\displaystyle=$
$\displaystyle\frac{C_{n}}{\sqrt{q(P-q)}}\frac{(q/P)^{n}}{\frac{1}{q}+\frac{\tilde{m}^{2}}{P-q}},\;n\geq
0,$ (63)
$g_{-10}(q_{1},q_{2})=\frac{D_{-10}\sqrt{P}}{\sqrt{q_{1}q_{2}(P-q_{1}-q_{2})}}\frac{P\delta_{\epsilon}(q_{1}=q_{2})}{\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{\tilde{m}^{2}}{P-q_{1}-q_{2}}}=D_{-10}P\sqrt{q_{1}q_{2}}\frac{\delta_{\epsilon}(q_{1}+q_{2})}{q_{1}+q_{2}},$
(64)
$g_{nj}(q_{1},q_{2})=\frac{D_{nj}\sqrt{P}}{\sqrt{q_{1}q_{2}(P-q_{1}-q_{2})}}\frac{(q_{1}^{j}q_{2}^{(n-j)}+q_{1}^{(n-j)}q_{2}^{j})/P^{n}}{\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{\tilde{m}^{2}}{P-q_{1}-q_{2}}},\;n\geq
0,j=0,\ldots,n/2.$ (65)

These have the desired small-momentum behavior shown in (3). The negative
index $n=-1$ is reserved for the ephemeral-mode contributions. The
normalization condition (55) reduces to

$1=|\psi_{0}|^{2}+\sum_{nm}B_{nm}^{(1)}a_{n}a_{m}+\sum_{nj,ml}B_{nj,ml}^{(2)}b_{nj}b_{ml},$
(66)

where the overlaps between the nonorthogonal basis functions are the symmetric
matrices

$B_{nm}^{(1)}=\frac{1}{P}\int
f_{n}(q)f_{m}(q)dq,\;\;B_{nj,ml}^{(2)}=\frac{1}{P^{2}}\int
g_{nj}(q_{1},q_{2})g_{ml}(q_{1},q_{2})dq_{1}dq_{2}.$ (67)

The normalization coefficients $C_{n}$ and $D_{nj}$ are fixed by requiring
$B_{nn}^{(1)}=1$ and $B_{nj,nj}^{(2)}=1$, and one can show that the $n=-1$
basis functions are orthogonal to the others, making $B_{-1,n}^{(1)}=0$ and
$B_{-10,nj}^{(2)}=0$ for $n\geq 0$.

The system of equations (58-IV) becomes, with $\lambda\equiv
g/\sqrt{4\pi}\mu^{2}$ and $\tilde{M}\equiv M/\mu$,

$\tilde{m}^{2}\psi_{0}-\frac{P}{\mu^{2}}P_{\rm
vac}^{-}\psi_{0}+\lambda\sum_{n}V_{n}^{(0)}a_{n}+\sum_{nj}U_{nj}^{(0)}b_{nj}=\tilde{M}^{2}\psi_{0},$
(68) $\sum_{m}T_{nm}^{(1)}a_{m}+\lambda
V_{n}^{(0)}\psi_{0}+\lambda\sum_{ml}V_{n,ml}^{(1)}b_{ml}=\tilde{M}^{2}\sum_{m}B_{nm}^{(1)}a_{m},$
(69)
$\sum_{ml}T_{nj,ml}^{(2)}b_{ml}+U_{nj}^{(0)}\psi_{0}+\lambda\sum_{m}V_{m,nj}^{(1)}a_{m}=\tilde{M}^{2}\sum_{m}B_{nj,ml}^{(2)}b_{ml},$
(70)

The various matrix elements are defined by

$T_{nm}^{(1)}=\int
dqf_{n}(q)\left(\frac{1}{q}+\frac{\tilde{m}^{2}}{P-q}\right)f_{m}(q),$ (71)
$T_{nj,ml}^{(2)}=\frac{1}{P}\int
dq_{1}dq_{2}g_{nj}(q_{1},q_{2})\left(\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{\tilde{m}^{2}}{P-q_{1}-q_{2}}\right)g_{ml}(q_{1},q_{2}),$
(72)
$U_{nj}^{(0)}=\frac{1}{\sqrt{2}}\int\frac{dq_{1}dq_{2}}{\sqrt{q_{1}q_{2}}}\delta_{\epsilon}(q_{1}+q_{2})g_{nj}(q_{1},q_{2}),$
(73)
$V_{n}^{(0)}=\sqrt{P}\int\frac{dq\,f_{n}(q)}{\sqrt{qP(P-q}},\;\;V_{n,ml}^{(1)}=\sqrt{\frac{2}{P}}\int\frac{dqdq^{\prime}\,f_{n}(q)g_{ml}(q,q^{\prime})}{\sqrt{q^{\prime}(P-q)(P-q-q^{\prime})}}.$
(74)

Details of matrix element computations are left to Appendix A. These include
the definition of a factor $\beta\equiv
1/\left[\int_{0}^{\infty}qdq\delta_{\epsilon}(q)^{2}\right]$ which enters the
normalization for ephemeral modes.888The value of $\beta$ depends on the model
used for $\delta_{\epsilon}$; it is not zero because the integral is over only
half the real line. Physical results are independent of $\beta$. With these
matrix elements, the system of equations can be written as

$\tilde{m}^{2}\psi_{0}-\frac{P}{\mu^{2}}P_{\rm
vac}^{-}\psi_{0}+\lambda\frac{\sqrt{\beta}}{2}a_{-1}+\lambda\sum_{n\geq
0}V_{n}^{(0)}a_{n}-\frac{P\sqrt{12\beta}}{\mu^{2}}b_{-10}+\frac{D_{00}}{2\sqrt{2}}b_{00}=\tilde{M}^{2}\psi_{0}$
(75) $n=-1:\;\;-\frac{2P}{\mu^{2}}\beta P_{\rm
vac}^{-}a_{-1}+\frac{1}{2}\sqrt{\beta}C_{0}a_{0}+\frac{1}{2}\sqrt{\beta}\lambda\psi_{0}+\lambda\sqrt{12}g_{-10}=\tilde{M}^{2}a_{-1}$
(76) $n=0:\;\;\frac{1}{2}\sqrt{\beta}C_{0}a_{-1}+\sum_{m\geq
0}T_{0m}^{(1)}a_{m}+\lambda V_{0}^{(0)}\psi_{0}+\lambda\sum_{ml\geq
0}V_{0,ml}^{(1)}b_{ml}=\tilde{M}^{2}\sum_{m\geq 0}B_{0m}^{(1)}a_{m},$ (77)
$n>0:\;\;\sum_{m\geq 0}T_{nm}^{(1)}a_{m}+\lambda
V_{n}^{(0)}\psi_{0}+\lambda\sum_{ml\geq
0}V_{n,ml}^{(1)}b_{ml}=\tilde{M}^{2}\sum_{m\geq 0}B_{nm}^{(1)}a_{m},$ (78)
$n=-1:\;\;-\frac{12P}{\mu^{2}}\beta P_{\rm
vac}^{-}b_{-10}+\frac{1}{2}D_{-10}D_{00}b_{00}-\frac{\sqrt{12}P}{\mu^{2}}\sqrt{\beta}P_{\rm
vac}^{-}\psi_{0}+\lambda\sqrt{12}a_{-1}=\tilde{M}^{2}b_{-10},$ (79)
$n=0:\;\;\frac{1}{2}D_{-10}D_{00}b_{-10}+\sum_{ml\geq
0}T_{00,ml}^{(2)}b_{ml}+\frac{D_{00}}{2\sqrt{2}}\psi_{0}+\lambda\sum_{m\geq
0}V_{m,00}^{(1)}a_{m}=\tilde{M}^{2}\sum_{ml\geq 0}B_{00,ml}^{(2)}b_{ml},$ (80)
$n>0:\;\;\sum_{ml\geq 0}T_{nj,ml}^{(2)}b_{ml}+\lambda\sum_{m\geq
0}V_{m,nj}^{(1)}a_{m}=\tilde{M}^{2}\sum_{ml\geq 0}B_{nj,ml}^{(2)}b_{ml}.$ (81)

Cancellation of the infinite vacuum energy $P_{\rm vac}^{-}$ in (75), (76),
and (79) is achieved if $a_{-1}=0$ and $b_{-10}=-\psi_{0}/\sqrt{12\beta}$.
These values correspond to the structure of the vacuum; in other words, as a
part of solving the dressed particle state, we have reconstituted the vacuum
as the foundation of the physical eigenstate and thereby cancelled the
(infinite) vacuum energy. The projection onto $f_{-1}$, which is equation
(76), is no longer needed or used. The factor $\beta$ disappears in the
eigenstate by cancelling in the product $b_{-10}f_{-10}\propto b_{-10}D_{-10}$
with $D_{-10}=\sqrt{6\beta}$.

This leaves a finite matrix problem with finite corrections due to nonzero
matrix elements of vacuum transitions. In particular, there is a finite matrix
element ($D_{00}/2\sqrt{2}$) coupling the three-particle sector ($b_{00}$) to
the one-particle sector ($\psi_{0}$) between (75) and (80). The two extra
particles are ephemeral modes.

In this severe Fock-space truncation, the matrix elements are simple enough to
invoke $\epsilon\rightarrow 0$ explicitly. A more general calculation would
require a model for $\delta_{\epsilon}$ and extrapolation of the limit
$\epsilon\rightarrow 0$ numerically, in addition to consideration of several
$\delta_{\epsilon}$ models to confirm model independence.

##  V Summary

We have developed a formalism by which vacuum transitions can be included in
light-front calculations and have argued that they must be included to have
full equivalence with equal-time quantization and to be consistent with the
perturbative equivalence of the two quantizations. The latter equivalence
follows, at least on a formal level, as a choice of coordinates for evaluation
of Feynman diagrams, with proper care as emphasized in [19, 20]. In that
context, contributions such as nonzero vacuum bubbles and tadpoles are
recovered. These have been missing from nonperturbative calculations due to
the neglect of vacuum transitions in light-front Hamiltonians. The inclusion
of such transitions means that the light-front vacuum is not trivial and
instead can be characterized as a generalized coherent state of ephemeral
modes, even for a free theory.

Our approach is based on the realization that vacuum transition matrix
elements are nonzero with respect to Fock-state wave functions with the
correct small-momentum behavior. These matrix elements lead to tadpole
contributions as well as disconnected vacuum bubbles. The vacuum bubbles are
regulated by the introduction of a finite width $\epsilon$ in momentum-
conserving delta functions, so that a bubble’s proportionality to $\delta(0)$
is replaced by $\delta_{\epsilon}(0)=L/4\pi$, where $L$ is the light-front
spatial volume. The width $\epsilon$ is taken to zero (and $L$ to infinity)
after the (infinite) vacuum energy is subtracted. The modes with momentum of
order $\epsilon$ that are removed in this limit are the ephemeral modes. They
represent the accumulation of contributions at zero momentum.

The use of proper basis functions is critical. A standard DLCQ approximation
[35, 29] cannot capture these effects, partly because the zero-mode
contributions form sets of measure zero and partly because the DLCQ grid
provides a poor approximation to integral operators with modes of order
$\epsilon\ll P^{+}$, for either periodic or antiperiodic boundary conditions.

We have considered several applications of these ideas. The most basic was to
show that the vacuum bubbles and tadpoles expected in $\phi^{4}$ theory are in
fact reproduced. We next considered the free scalar case in detail,
constructing the vacuum state as a generalized coherent state of ephemeral
modes and extending this to include the shifted scalar, with recovery of the
correct vacuum expectation value. The shifted case can, of course, be handled
in DLCQ by inclusion of the constraint equation for the spatial average of the
field [36, 37, 38]. Here, however, we have an exact analytic solution with no
discretization. Also, the analytic solution contains the one-loop vacuum
bubble discussed by Collins [22] as a prime example of light-front vacuum
structure in perturbation theory.

To illustrate how the approach functions in an interacting theory, we
considered the charge-one sector of quenched scalar Yukawa theory. There we
have shown how the vacuum subtraction can be implemented and how strong
coupling can result in residual effects from ephemeral modes, which in the
limit translate to zero-mode effects.

This work was done in two dimensions. The extension to three and four
dimensions should be straightforward. The transverse momenta have the full
range of $-\infty$ to $\infty$ and therefore can be balanced without being
individually zero. The coherent state for the free scalar vacuum would be
built from an operator such as

$A^{\dagger}=\int\frac{dp_{1}^{+}dp_{2}^{+}d\vec{p}_{\perp}}{\sqrt{p_{1}^{+}p_{2}^{+}}}\frac{f(p_{1}^{+},p_{2}^{+},\vec{p}_{\perp})}{\frac{1}{p_{1}^{+}}+\frac{1}{p_{2}^{+}}}\delta_{\epsilon}(p_{1}^{+}+p_{2}^{+})a^{\dagger}(p_{1}^{+},\vec{p}_{\perp})a^{\dagger}(p_{2}^{+},-\vec{p}_{\perp})$
(82)

that creates two ephemeral modes with opposite transverse momenta.

The ideal demonstration that our approach is useful would be to compute the
critical coupling in $\phi^{4}$ theory. The tadpole contributions that were
absent previously [5] would now be included. Such a calculation is a natural
next step.

###### Acknowledgements.

This work was supported in part by the Minnesota Supercomputing Institute and
the Research Computing and Data Services at the University of Idaho through
grants of computing time.

##  Appendix A Matrix elements for scalar Yukawa theory

We compute the matrix elements needed to resolve the system of Fock-space
equations for scalar Yukawa theory. The matrices are defined in (67) through
(74). With the definition of the model-dependent999For any $\delta_{\epsilon}$
model that scales properly with $\epsilon$, $\beta$ is independent of
$\epsilon$. factor $\beta$

$\frac{1}{\beta}=\int_{0}^{\infty}dq\,q\delta_{\epsilon}(q)^{2},$ (83)

the basis function overlaps (67) are, for $n,m\geq 0$,

$\displaystyle B_{-1-1}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{(C_{-1})^{2}}{P}\int
dq\,qP\delta_{\epsilon}(q)^{2}=\frac{(C_{-1})^{2}}{\beta},$ (84)
$\displaystyle B_{-1n}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{C_{-1}C_{n}}{P}\int
dq\frac{\sqrt{qP}\delta_{\epsilon}(q)}{\sqrt{q(P-q)}}\frac{(q/P)^{n}}{\frac{1}{q}+\frac{\tilde{m}^{2}}{q}}=\frac{C_{-1}C_{n}}{P}\int
dq\,q(q/P)^{n}\delta_{\epsilon}(q)\rightarrow 0,$ (85) $\displaystyle
B_{nm}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{C_{n}C_{m}}{P}\int_{0}^{P}\frac{dq}{q(P-q)}\frac{(q/P)^{n+m}}{\left(\frac{1}{q}+\frac{\tilde{m}^{2}}{q}\right)^{2}}=C_{n}C_{m}\int_{0}^{1}\frac{x^{n+m+1}(1-x)dx}{(1-x+\tilde{m}^{2}x)^{2}},$
(86) $\displaystyle B_{-10,-10}^{(2)}$ $\displaystyle=$
$\displaystyle(D_{-10})^{2}\int
dq_{1}dq_{2}\,q_{1}q_{2}\frac{\delta_{\epsilon}(q_{1}+q_{2})^{2}}{(q_{1}+q_{2})^{2}}=(D_{-10})^{2}\int
dx\,x(1-x)\int QdQ\delta_{\epsilon}(Q)^{2}=\frac{(D_{-10})^{2}}{6\beta},$
$\displaystyle B_{-10,nj}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{D_{-10}D_{nj}}{P}\int
dq_{1}dq_{2}\frac{\delta_{\epsilon}(q_{1}+q_{2})}{q_{1}+q_{2}}\frac{q_{1}q_{2}}{q_{1}+q_{2}}\frac{q_{1}^{j}q_{2}^{n-j}+q_{1}^{n-j}q_{2}^{j}}{P^{n}}\rightarrow
0,$ (88) $\displaystyle B_{nj,ml}^{(2)}$ $\displaystyle=$ $\displaystyle
D_{nj}D_{ml}\int_{0}^{1}dx_{1}\int_{0}^{1-x_{1}}dx_{2}\frac{(x_{1}^{j}x_{2}^{n-j}+x_{1}^{n-j}x_{2}^{j})(x_{1}^{l}x_{2}^{m-l}+x_{1}^{m-l}x_{2}^{j})}{x_{1}x_{2}(1-x_{1}-x_{2})\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{\tilde{m}^{2}}{1-x_{1}-x_{2}}\right)^{2}}.$
(89)

The normalization conditions, that diagonal elements of $B^{(1)}$ and
$B^{(2)}$ be unity, yield $C_{-1}=\sqrt{\beta}$ and $D_{-10}=\sqrt{6\beta}$.

The matrix elements for kinetic energy are, with $P_{\rm vac}^{-}$ defined in
(36),

$\displaystyle T_{-1-1}^{(1)}$ $\displaystyle=$ $\displaystyle(C_{-1})^{2}\int
dq\,qP\delta_{\epsilon}(q)^{2}\left[\frac{1}{q}+\frac{\tilde{m}^{2}}{q}\right]=(C_{-1})^{2}P\int\delta_{\epsilon}(q)^{2}=-(C_{-1})^{2}\frac{2P}{\mu^{2}}P_{\rm
vac}^{-},$ (90) $\displaystyle T_{-1n}^{(1)}$ $\displaystyle=$ $\displaystyle
C_{-1}C_{n}\int
dq\delta_{\epsilon}(q)(q/P)^{n}=\frac{1}{2}C_{-1}C_{n}\delta_{n0},$ (91)
$\displaystyle T_{nm}^{(1)}$ $\displaystyle=$ $\displaystyle
C_{n}C_{m}\int\frac{dx\,x^{n+m}}{1-x+\tilde{m}^{2}x},$ (92) $\displaystyle
T_{-10,-10}^{(2)}$ $\displaystyle=$ $\displaystyle\frac{(D_{-10})^{2}}{P}\int
dq_{1}dq_{2}P^{2}q_{1}q_{2}\frac{\delta_{\epsilon}(q_{1}+q_{2})^{2}}{(q_{1}+q_{2})^{2}}\left(\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{\tilde{m}^{2}}{P-q_{1}-q_{2}}\right)=-(D_{-10})^{2}\frac{2P}{\mu^{2}}P_{\rm
vac}^{-},$ $\displaystyle T_{-10,nj}^{(2)}$ $\displaystyle=$ $\displaystyle
D_{-10}D_{nj}\int
dq_{1}dq_{2}\frac{\delta_{\epsilon}(q_{1}+q_{2})}{q_{1}+q_{2}}\frac{q_{1}^{j}q_{2}^{n-j}+q_{1}^{n-j}q_{2}^{j}}{P^{n}}=\frac{1}{2}D_{-10}D_{00}\delta_{n0},$
(94) $\displaystyle T_{nj,ml}^{(2)}$ $\displaystyle=$ $\displaystyle
2D_{nj}D_{ml}\int
dx_{1}dx_{2}\frac{x_{1}^{j}x_{2}^{n-j}(x_{1}^{l}x_{2}^{m-l}+x_{1}^{m-l}x_{2}^{l})}{(x_{1}+x_{2})(1-x_{1}-x_{2})+\tilde{m}^{2}x_{1}x_{2}}.$
(95)

In $T_{-1n}^{(1)}$ and $T_{-10,nj}^{(2)}$ we have used $\int
dq\,\delta_{\epsilon}(q)=\frac{1}{2}$, which follows from integrating only
over positive $q$.

The potential terms have the following matrix elements:

$\displaystyle U_{-10}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{D_{-10}}{\sqrt{2}}\int\frac{dq_{1}dq_{2}}{\sqrt{q_{1}q_{2}}}\delta_{\epsilon}(q_{1}+q_{2})P\sqrt{q_{1}q_{2}}\frac{\delta_{\epsilon}(q_{1}+q_{2})}{q_{1}+q_{2}}=-\frac{D_{-10}}{\sqrt{2}}\frac{2P}{\mu^{2}}P_{\rm
vac}^{-},$ (96) $\displaystyle U_{nj}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\int\frac{dq_{1}dq_{2}}{\sqrt{q_{1}q_{2}}}\delta_{\epsilon}(q_{1}+q_{2})\frac{D_{nj}\sqrt{P}}{\sqrt{q_{1}q_{2}(P-q_{1}-q_{2})}}\frac{(q_{1}^{j}q_{2}^{n-j}+q^{n-j}q_{2}^{j})/P^{n}}{\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{\tilde{m}^{2}}{P-q_{1}-q_{2}}}=\frac{D_{00}}{2\sqrt{2}}\delta_{n0},$
$\displaystyle V_{-1}^{(0)}$ $\displaystyle=$ $\displaystyle
C_{-1}\sqrt{P}\int\frac{dq}{\sqrt{qP(P-q)}}\sqrt{qP}\delta_{\epsilon}(q)=\frac{1}{2}C_{-1},$
(98) $\displaystyle V_{n}^{(0)}$ $\displaystyle=$ $\displaystyle
C_{n}\int\frac{dx\,x^{n}}{1-x+\tilde{m}^{2}x},$ (99) $\displaystyle
V_{-1,-10}^{(1)}$ $\displaystyle=$ $\displaystyle\sqrt{2}C_{-1}D_{-10}\int
dq\,q\delta_{\epsilon}(q)^{2}=\frac{\sqrt{2}}{\beta}C_{-1}D_{-10}=\sqrt{12},$
(100) $\displaystyle V_{n,-10}^{(1)}$ $\displaystyle=$
$\displaystyle\sqrt{2}C_{n}D_{-10}\int
dqdq^{\prime}\left(\frac{q}{P}\right)^{n+1}\frac{\delta_{\epsilon}(q+q^{\prime})}{q+q^{\prime}}=\sqrt{2}C_{n}D_{-10}\int
dq\left(\frac{q}{P}\right)^{n+1}\delta_{\epsilon}(q)\rightarrow 0,$
$\displaystyle V_{-1,nj}^{(1)}$ $\displaystyle=$
$\displaystyle\sqrt{2}C_{-1}D_{nj}\int\frac{dqdq^{\prime}\delta_{\epsilon}(q)}{q^{\prime}(P-q^{\prime})}q\frac{q^{j}q^{\prime
n-j}+q^{n-j}q^{\prime j}}{P^{n}}\rightarrow 0,$ (102) $\displaystyle
V_{n,ml}^{(1)}$ $\displaystyle=$
$\displaystyle\sqrt{2}C_{n}D_{ml}\int\frac{dx_{1}dx_{2}x_{1}^{n+1}}{1-x_{1}+\tilde{m}^{2}x_{1}}\frac{x_{1}^{l}x_{2}^{m-l}+x_{1}^{m-l}x_{2}^{l}}{(x_{1}+x_{2})(1-x_{1}-x_{2})+\tilde{m}^{2}x_{1}x_{2}}.$
(103)

In $V_{n,-10}^{(1)}$ we have used a representation of the Dirac delta function

$\delta(q)=\int dq^{\prime}\frac{\delta(q+q^{\prime})}{q+q^{\prime}}$ (104)

which follows from

$\int dqf(q)dq^{\prime}\frac{\delta(q+q^{\prime})}{q+q^{\prime}}=\int
QdxdQf(xQ)\frac{\delta(Q)}{Q}=\frac{1}{2}f(0).$ (105)

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# Fine-tuning wav2vec2 for speaker recognition

###### Abstract

This paper explores applying the wav2vec2 framework to speaker recognition
instead of speech recognition. We study the effectiveness of the pre-trained
weights on the speaker recognition task, and how to pool the wav2vec2 output
sequence into a fixed-length speaker embedding. To adapt the framework to
speaker recognition, we propose a single-utterance classification variant with
cross-entropy or additive angular softmax loss, and an utterance-pair
classification variant with BCE loss. Our best performing variant achieves a
1.88% EER on the extended voxceleb1 test set compared to 1.69% EER with an
ECAPA-TDNN baseline. Code is available at github.com/nikvaessen/w2v2-speaker.

Index Terms—  speaker recognition, wav2vec2, transfer learning

##  1 Introduction

In the field of natural language processing (NLP) it has become standard to
fine-tune self-supervised pre-trained models, such as BERT [1], XLNet [2], and
T5 [3], on a wide variety of NLP tasks. Recently, this framework of pre-
training and fine-tuning has also been successfully used in automatic speech
recognition with wav2vec2 [4]. The aim of this study is to explore the
feasibility of fine-tuning the wav2vec2 pre-trained network on a different
task than speech recognition, namely _speaker_ recognition.

The BERT and wav2vec2 network have commonalities in their design. Both have
stacks of transformer layers and they use self-supervised, contrastive pre-
training with masked input. However, they differ in three major aspects: 1)
the input tokens to the encoder in wav2vec2 are raw audio processed by a CNN
instead of WordPiece embeddings, 2) wav2vec2 uses relative positional
embeddings computed by a CNN instead of sinusoidal positional embeddings, 3)
there is no class token and equivalent next-sentence prediction task in the
pre-training procedure of wav2vec2.

The class token in BERT is used when fine-tuning on sentence-pair
classification tasks, e.g., entailment, and for single-sentence classification
tasks, e.g., sentiment analysis. It is not used for single sentence tagging
tasks, e.g., named entity recognition. In this respect, speech recognition is
analogous to sentence tagging where each output token can represent a phone or
letter. Hence, speech recognition does not require a class token. In contrast,
speaker recognition corresponds to either sentence-pair classification, or
single-sentence classification, and might therefore benefit from a class token
which summarizes the whole input sequence.

Our work focuses on the following questions. First and foremost, we want to
find out whether the pre-trained wav2vec2 weights are an effective
initialization for speaker recognition. We further want to explore if wav2vec2
can be adapted to speaker recognition without a class token and next-sentence
prediction task. One solution is to pool the variable-length sequence of
wav2vec2 embeddings into a fixed-size speaker embedding. This raises the
questions if pooling is an effective replacement for a class token, and if so,
which pooling method is most suitable.

##  2 Related work

Wav2vec2 has already been applied to a variety of speech-related tasks. In [5]
the network is used for speaker recognition and language identification in
both a single and multi-task learning setting. The authors of [6] show good
performance for language identification with 25 languages but modify wav2vec2
to use log-mel spectogram input instead of raw waveforms. In [7] the (frozen)
wav2vec2 embeddings are input to a learnable downstream model for carrying out
emotion recognition. Meanwhile [8] manages to fine-tune the wav2vec2 model
itself on emotion recognition with CTC loss by using emotion-labeled phonetic
units. The LeBenchmark [9] uses the wav2vec2 models as a baseline and
encapsulates speech recognition, spoken language understanding, emotion
recognition and speech translation in a single benchmark.

##  3 Methodology

###  3.1 The wav2vec2 architecture

The wav2vec2 framework [4] applies the concept of self-supervised pre-training
with transformers to automatic speech recognition. In Fig. 1 we show a general
overview of the network architecture during fine-tuning. The next subsections
summarize each component.

Fig. 1: Overview of the Wav2vec2 architecture. Shapes are specified for a
batch of $B$ audio samples with a length of 3 seconds.

####  3.1.1 Feature extraction

The first step is to encode a raw audio waveform (normalized to zero mean and
unit variance) into learned representations with a discrete time unit. The
feature extractor consists of 7 consecutive 1-dimensional convolutions with
512 channels and respective kernel sizes of (10, 3, 3, 3, 3, 2, 2) and stride
(5, 2, 2, 2, 2, 2, 2). The output of the first convolutional layer is group
normalized [10] such that each of the 512 channel sequences has zero mean and
unit variance before GELU activation [11] is applied. The other convolutional
layers do not have any normalization layers and their output is directly
activated with GELU. The output of the feature extractor is an encoded vector
sequence with dimensionality 512. Each vector has a receptive field of 20 ms
which is similar to the window sizes in spectral-based representations.

####  3.1.2 Projection, SpecAugment & positional embedding

After the feature extraction each encoded vector representation in the
sequence is independently normalized to zero mean and unit variance and
projected into 768 dimensions by a single, shared fully-connected layer called
the feature projector. On all projections dropout is applied (but no
activation). Then, masking is applied over the whole sequence analogous to
SpecAugment [12]; 0 or more random sets of consecutive vectors (masking in
time domain) as well as 0 or more random sets of consecutive channels (masking
in “frequency” domain) have their values blanked to 0. This masked projected
sequence is then convolved by a single layer with a kernel size of 128, a
stride of 1, padding of 64 and 16 groups followed by GELU activation in order
to create a relative positional embedding for each projected representation.
This relative positional embedding is summed with the original input of the
convolution, which changes the receptive field from 20 ms to 2.5 s. As a final
step each vector is independently normalized with LayerNorm and dropout is
applied again.

####  3.1.3 Transformer

The masked and projected sequence with both local and positional information
is fed through an encoder with 12 consecutive transformer layers. Each
transformer layer consists of a residual 12-headed self-attention module and a
residual 2-layer feed forward network with respectively 3072 and 768 units.
LayerDrop [13, 14] is applied such that each transformer layer is potentially
skipped. The final output sequence, with each representation potentially
having both local and global information due to self-attention, is used in a
downstream task.

###  3.2 Three wav2vec2 variants for speaker recognition

The original wav2vec2 framework fine-tunes on speech recognition by
independently labeling each wav2vec2 output embedding with a shared fully-
connected layer, and optimizes with CTC loss [15]. We propose two adaptions to
this design for the speaker recognition task, inspired by BERTs [1] single-
sentence and sentence-pair classification setup.

####  3.2.1 Speaker recognition as single-utterance classification

The current paradigm in speaker recognition with deep neural networks is to
train models with a classification-based approach [16, 17]. To mimic this
architectural paradigm two modifications to wav2vec2 are made. First, the
sequence of wav2vec2 embeddings is reduced to a single embedding during
training (see subsection 3.3). Secondly, we add a fully connected layer which
uses the pooled embedding to classify each speaker in the training data with
cross-entropy (CE) or angular additive softmax (AAM) [18, 19] loss. A trial is
evaluated with the cosine similarity between two pooled embeddings. We refer
to these variants as w2v2-ce and w2v2-aam.

####  3.2.2 Speaker recognition as utterance-pair classification

The second approach directly computes a similarity score. The two audio
segments of a speaker recognition trial are first processed independently up
to the encoder part of the network. Then, the two sequences of input tokens
are concatenated, accompanied by special tokens at the embedding level: A
start (all $+1$), separator (all $-1$), and end (all $-1$) token. The first
wav2vec2 embedding in the output sequence (corresponding to the start token)
is used as input to a logistic regression with one dense layer. The singular
output is the logit for the BCE loss and the score for evaluating a trial.
During training a batch consists of 8 speakers, 4 utterances per speaker, and
16 same and different speaker-pairs. We refer to this third variant as
w2v2-bce.

###  3.3 Pooling methods

We propose several pooling methods to reduce the variable-length sequence of
wav2vec2 embeddings to a fixed-size speaker embedding. We first consider the
standard statistical pooling methods mean, max, mean&std and quantile. They
aggregate each dimension over the time axis. The mean&std variant doubles the
embedding dimensionality while quantile pooling expands each dimension five-
fold with quantiles (0, 0.25, 0.5, 0.75, 1). We also assess taking the first,
middle or last embedding of the sequence as a “pooling” strategy as well as
randomly selecting the index. Lastly we consider inserting a “start” token
(all values $+1$) before the input sequence of the encoder and then selecting
the first output token as the speaker embedding. This is termed first&cls.
Unlike BERT our “start” token token does not have a meaningful prior.

##  4 Experiments

###  4.1 Data

All experiments are conducted on the VoxCeleb datasets [20, 21], which consist
of interviews of celebrities extracted from YouTube. The VoxCeleb2 dev set,
which contains ~1.1 M audio files over 6 k speakers, is used for training. A
validation set is created based on ~2 % of the dev set data which includes all
speakers but does not overlap in recordings. For this validation set a random
trial set of 5 k same and 5 k different pairs is generated, from which we
compute the validation EER. This is used to select the best checkpoint during
a training run as well as to tune hyperparameters. For evaluation we use the
cleaned original (vox1-o, 40 speakers, ~37 k trials), extended (vox1-e, 1251
speakers, ~580 k trials) and hard (vox1-h, 1190 speakers, ~550 k trials, equal
nationality and sex) test sets from voxceleb 1[21]. There is no speaker
overlap between the voxceleb1 test sets and the voxceleb2 dev set. The
experiments with the wav2vec2 network use the pre-trained weights111See
https://huggingface.co/facebook/wav2vec2-base on Librispeech [22] released on
HuggingFace [23] by Fairseq [24].

###  4.2 Computational budget and fair comparison

We compare the performances of models under similar computational budgets.
Each network is trained with a batch size of 3.2M audio samples [4] by
randomly sampling 3 seconds from 66 different audio files. No data
augmentation techniques are used. We train for 100k iterations, approximately
6 epochs, with Adam [25] and a OneCycle learning rate schedule [26]. The
maximum learning rate (LR) of the cycle is found by 1) performing an LR range
test [27] with 5k iterations and 2) tuning on a 7-sized grid centered around
the LR with the steepest slope and bounded by the LRs where the loss
respectively started and stopped decreasing. Models are evaluated with a
cosine score between speaker embeddings lacking any further post-processing,
except for the wav2vec2-bce variant which computes scores directly.

###  4.3 Baseline systems

We train two popular baselines models for speaker recognition, x-vector [16,
28] and ECAPA-TDNN [17, 28], and compare them to the three wav2vec2 adaptions
w2v2-ce, w2v2-aam and w2v2-bce. All five models have the computation budget
described in subsection 4.2. The X-vector and ECAPA-TDNN networks use
40-dimensional filterbanks as input. The w2v2-ce and w2v2-aam variants use
mean&std pooling which was chosen as it is also used in the x-vector network
architecture. The w2v2-aam variant and ECAPA-TDNN use the AAM softmax loss
with a scale of $30$ and a margin of $0.2$ in this and further experiments.
The feature encoder part of the wav2vec2 architecture is frozen for the whole
training procedure which corresponds to [4].

###  4.4 Variation in pooling

Next we explore the different poolings methods proposed in Section 3.3. This
was carried out for the single-utterance classification variants: w2v2-ce and
w2v2-aam. We use the same training settings as in the baseline comparison and
only vary the pooling method. Note therefore that the learning rate was tuned
to the “mean&std” setup.

Table 1: EER performance of standard filterbank-based speaker recognition
networks as well as the fine-tuned wav2vec2 variations. We ran $N=4$ runs to
compute the standard deviations. Bold indicates best performance.

|  | EER performance (mean, std in %)
---|---|---
NETWORK |  | vox1-o |  | vox1-e |  | vox1-h
x-vector [16] |  | 5.22 | 0.12 |  | 5.60 | 0.05 |  | 8.75 | 0.05
ECAPA-TDNN [17] |  | 1.61 | 0.03 |  | 1.69 | 0.03 |  | 3.10 | 0.05
w2v2-ce |  | 2.25 | 0.20 |  | 2.58 | 0.10 |  | 4.91 | 0.13
w2v2-aam |  | 1.91 | 0.12 |  | 2.22 | 0.04 |  | 4.33 | 0.08
w2v2-bce |  | 7.28 | 0.22 |  | 7.19 | 0.22 |  | 11.34 | 0.83

###  4.5 Ablation study

We perform several ablations on the best-performing wav2vec2 variant for
speaker recognition. These ablations are trained with 100k iterations except
in the experiments for the batch size. The first set of ablations study the
effect of not freezing the feature extractor in the fine-tuning procedure as
well as randomly initialising the whole network and thus not using any pre-
trained weights. The second set of ablations explore the relative importance
of the regularisation techniques in the network architecture. We first disable
only LayerDrop, then sequentially disable dropout and the masking of certain
frames as well. The third set simply studies the effect of increasing (with
50k iterations) or decreasing (with 200k iterations) the chosen batch size by
a factor of 2. The last ablations involve the learning rate schedule. We first
test two constant learning rates: $3\times 10^{-6}$ is the LR where the loss
started increasing in the LR range test and $10^{-5}$ is the LR with the
steepest decrease. The second schedule exponentially decays the LR from
$10^{-5}$ to $3\times 10^{-6}$. The final schedule is the tri-stage learning
rate schedule similar to [4] which includes a warm-up phase linearly
increasing the LR from $10^{-7}$ to $10^{-5}$ in 10k iterations, a constant
phase of 40k iterations, and an exponentially decreasing stage from $10^{-5}$
to $10^{-7}$ for the remaining iterations.

##  5 Results

###  5.1 Baseline comparison

The LR range test and grid-tuning approach found the following learning rates:
$10^{-4}$ for x-vector, $10^{-3}$ for ECAPA-TDNN, $9\times 10^{-5}$ for
w2v2-ce, $5\times 10^{-5}$ for w2v2-aam, and $3\times 10^{-5}$ for w2v2-bce.
Table 1 shows four runs with these learning rates. We see that ECAPA-TDNN
performs best on all test sets. The w2v2-aam network is the best performing
wav2vec2 variant. Both w2v2-ce and w2v2-aam manage to improve on the x-vector
architecture. Modeling speaker recognition as utterance-pair classification
(w2v2-bce) performed worst.

###  5.2 Pooling methods

Table 2: The performance of different pooling strategies for the single utterance classification architectures on the extended voxceleb1 test set. Each method was trained with $N=3$ random seeds. The evaluation of random pooling was repeated four times for each run.  |  | EER on vox1-e (mean, std in %)
---|---|---
pooling |  | w2v2-ce |  | w2v2-aam
max |  | 4.79 | 0.55 |  | 2.27 | 0.04
quantile |  | 2.75 | 0.17 |  | 2.21 | 0.03
mean |  | 2.69 | 0.06 |  | 2.11 | 0.05
mean&std |  | 2.60 | 0.08 |  | 2.18 | 0.03
first |  | 2.61 | 0.10 |  | 2.15 | 0.05
first&cls |  | 2.52 | 0.11 |  | 2.06 | 0.03
middle |  | 2.55 | 0.07 |  | 2.18 | 0.03
last |  | 2.58 | 0.07 |  | 2.18 | 0.03
random ($N=1$) |  | 2.56 | 0.002 |  | 2.40 | 0.002
random ($N=3$) |  | 2.70 | 0.13 |  | 2.37 | 0.03

Table 2 compares the 9 different pooling strategies with the w2v2-ce and
w2v2-aam networks. We observe that first&cls pooling performs best for both
networks. The difference between first&cls pooling and the other pooling
methods is more pronounced for w2v2-aam than for w2v2-ce. The low inter-model
variance of random pooling shows that each wav2vec2 embedding is a stand-alone
speaker embedding. Not shown, using random pooling in the evaluation for
networks which were trained with first&cls pooling degrades the EER with
$0.2\%$ points. Moreover, creating ensembles out of different embeddings in
the output sequence did not improve performance. This suggests that the
transformer layer processes each embedding in the sequence similarly.

###  5.3 Ablation study

Table 3 shows the results of the ablation study. We see that freezing the
feature extractor leads to worse performance but is more stable across runs
with different seeds. We also note a large degradation in performance when
initializing with random weights instead of the pre-trained weights. The
regularisation settings for fine-tuning on speech recognition are also
beneficial for fine-tuning on speaker recognition. Increasing the batch size
beyond 3.2M audio samples does not increase performance, although it does
decrease the variance slightly. Using a learning rate schedule with a warm-up
phase, such as the tri-stage or OneCycle schedule, is critical for stable and
good performance.

Table 3: EER performance under varying ablated configurations for the w2v2-aam
network variant. Each configuration was run with $N=3$ random seeds. One run
with exponential decay diverged and we therefore show $N=2$ results for that
row.

|  | EER on vox-e
---|---|---
ablation |  | mean(%) |  | std (%)
default w2v2-aam first&cls setup |  | 2.06 |  | 0.03
unfrozen feature extractor |  | 1.88 |  | 0.08
& no pre-trained weights |  | 5.08 |  | 0.08
no layerdrop ($p=0$) |  | 2.45 |  | 0.05
& no dropout ($p=0$) |  | 2.39 |  | 0.04
&& no time masking ($p=0$) |  | 2.39 |  | 0.09
batch size 32 |  | 2.12 |  | 0.10
batch size 128 |  | 2.05 |  | 0.01
constant (1e-5) |  | 50.0 |  | 0.00
constant (3e-6) |  | 3.71 |  | 0.08
exponential decay ($N=2$) |  | 2.42 |  | 0.07
tri-stage |  | 1.97 |  | 0.04

##  6 Conclusion and future work

We have shown that the wav2vec2 framework can be successfully adapted to the
speaker recognition task and that the pre-trained weights used for fine-tuning
on speech recognition are also useful for fine-tuning on speaker recognition.
Good results with first&cls pooling indicate that including a class token in
the pre-training procedure is promising future work. Modeling speaker
recognition as a paired-utterance classification problem did not perform well.
Future work in that direction might involve limiting the attention mechanism
to the opposite utterance to simplify the learning task.

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