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# Energy-momentum tensor in $\Phi^{4}$ theory at one loop

Brean Maynard
Department of Physics, University of Connecticut, Storrs, CT 06269, U.S.A.
May 2024

###### Abstract

The energy-momentum tensor form factors are studied in $\Phi^{4}$ theory to
one-loop order with particular focus on the $D$-term, a particle property
which has attracted a lot of attention in the recent literature. It is shown
that the free Klein-Gordon theory value of the $D$-term $D_{\rm free}=-1$ is
reduced to $D_{\rm one\\!-\\!loop}=-\frac{1}{3}$ even if the $\Phi^{4}$
interaction is infinitesimally small. A companion work in $\Phi^{3}$ theory
confirms this result which may indicate that it is independent of the type of
interaction as long as the scalar theory is renormalizable. Dispersion
relations are studied. Various definitions of mean square radii including the
mechanical radius are investigated. The findings contribute to a better
understanding of the EMT properties of particles and their interpretation.

##  1 Introduction

The energy momentum tensor (EMT) is an important operator in field theory
through which particles couple to gravity, and whose matrix elements define
fundamental particle properties like mass, spin and the less well-known
$D$-term. Introduced in the 1960s [1, 2], EMT form factors received little
attention until the advent of generalized parton distribution functions
accessible in hard-exclusive reactions [3, 4, 5, 6, 7, 8] which allow one to
infer the spin [4] and mass[9] decompositions of the nucleon, the property
known as the $D$-term [10], and mechanical properties [11]. For reviews on
generalized parton distribution functions and EMT form factors, see [12, 13,
14, 15, 16, 17, 18, 19, 20, 21].

In view of the complexity of hadron structure and QCD, it is interesting to
study the EMT properties of particles in simpler theories. The simplest case
constitute free theories where EMT matrix elements and other properties can be
exactly evaluated. For instance, the $D$-term of a free, non-interacting spin-
zero particle is $D_{\rm free}=-1$ [22], while that of a free fermion vanishes
[23]. The arguably “next-to-simplest” case are theories with an
infinitesimally small interaction which can be solved perturbatively to lowest
order.

The purpose of this work is to investigate how the EMT form factors are
affected if the theory of a scalar field $\Phi$ is introduced to a $\Phi^{4}$
type interaction. To be more precise, the investigation will be carried out in
the perturbative regime under the assumption that the coupling constant
$\lambda\ll 1$ is “infinitesimally small” such that a one-loop calculation can
be assumed to be justified and higher orders ${\cal O}(\lambda^{2})$ can be
safely neglected. The goal of this work is to determine the EMT form factors
to one-loop order in the $\Phi^{4}$ theory.

At the first glance, in the situation of an infinitesimally small coupling
constant in a scalar theory, one could naively expect the one-loop result
$D_{\rm one\\!-\\!loop}$ to be not much different from the free-scalar theory
result $D_{\rm free}=-1$. However, this is not the case. Rather, the $D$-term
is strongly affected when interactions are introduced in the theory of a
scalar field, and its value is reduced at one-loop order to $D_{\rm
one\\!-\\!loop}=-\frac{1}{3}$. This was suspected (but not proven) in Ref.
[22]. The technical reason for this strong shift of the value of the $D$-term
is due to the necessity to introduce an improvement term to render Greens
functions with EMT insertions finite [24], see also [25, 26, 27, 28, 29, 30].

Interestingly, the same finding was obtained in another renormalizable scalar
theory, namely in $\Phi^{3}$ theory in [31]. This may indicate that the
$D$-term is affected in the same way by infinitesimal interactions,
independently of the type of interaction as long as it is renormalizable.
These findings are in line with observations in literature that the $D$-term
is the particle property most sensitive to the variations of the dynamics in a
system [20, 21, 22, 23].

The outline is as follows. In Sec. 2 the EMT form factors for spin-zero
particles are presented, free-field theory results reviewed, and $\Phi^{4}$
theory is introduced. In Secs. 3 and 4 the one-loop calculation is carried out
in respectively dimensional and Pauli-Villars regularization. In Sec. 5 the
properties of the $D(t)$ form factor are discussed, and Sec. 6 presents a non-
trivial cross check of the calculation by means of dispersion relations. The
Sec. 7 discusses several mean square radii. In Sec. 8 the conclusions are
presented. The Appendices contain technical details.

##  2 EMT form factors of a spin-zero particle

In this section, we introduce the EMT form factors for a spin-zero particle,
review what is known from the free theory case and discuss expectations when
interactions are included.

###  2.1 EMT form factors

The matrix elements of the EMT operator of a spin-zero particle are described
in terms of 2 form factors as [1, 2]

$\langle\vec{p}^{\,\prime\,}|T^{\mu\nu}(x)|\vec{p}\rangle=e^{i\Delta\cdot
x}\biggl{(}\frac{P^{\mu}P^{\nu}\\!}{2}\,A(t)+\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{2}\,D(t)\biggr{)}$
(1)

where $T^{\mu\nu}(x)$ denotes the EMT operator at the space-time position
$x^{\mu}$, and the variables are defined as

$P^{\mu}=p^{\,\prime\,\mu}+p^{\mu}\,,\;\;\;\Delta^{\mu}=p^{\,\prime\,\mu}-p^{\mu}\,,\;\;\;t=\Delta^{2}.$
(2)

The covariant normalization of the one-particle states is provided by

$\langle\vec{p}^{\,\prime}|\vec{p}\,\rangle=2\,\omega_{p}\,(2\pi)^{3}\,\delta^{(3)}(\vec{p}-\vec{p}^{\,\prime})\,,\quad\omega_{p}=\sqrt{\vec{p}^{\,\,2}+m^{2}}\,.$
(3)

At zero-momentum transfer, the form factors satisfy [1, 2], see for instance
also Ref. [32],

$\lim\limits_{t\to 0}A(t)=A(0)=1\,,\quad\lim\limits_{t\to 0}D(t)=D(0)\equiv
D\,.$ (4)

The constraint for $A(0)=1$ arises because in the limit $\vec{p}\to 0$ and
$\vec{p}^{\,\prime}\to 0$, only the $00$-component contributes in Eq. (1), and
$H=\int d^{3}x\;T^{00}(x)$ is the Hamiltonian of the system with
$H\,|\vec{p}\rangle=m\,|\vec{p}\rangle$ when $\vec{p}\to 0$. It is important
to notice that no analog constraint exists for $D(0)$ which therefore implies
a novel global property of a particle which is on the same footing as other
fundamental particle properties related to matrix elements of operators
corresponding to conserved currents like mass, spin, electric charge or
magnetic moment [10].

Notice that many authors define the vector $P^{\mu}$ not as a sum of $p^{\mu}$
and $p^{\prime\mu}$ but their average. Due to invariance under space-time
translations, the position $x^{\mu}$ of the EMT operator can be arbitrarily
changed which affects only the overall phase $e^{i\Delta\cdot x}$ in Eq. (1).
The position $x^{\mu}$ can therefore be set to the origin without loss of
generality.

###  2.2 Free Klein-Gordon theory

The classical Lagrangian of the theory of a real spin-zero particle of mass
$m$ is given by

$\mathcal{L}_{\rm
free}=\frac{1}{2}\,(\partial^{\rho}\Phi)\,(\partial_{\rho}\Phi)-\frac{1}{2}m^{2}\Phi^{2}\,.$
(5)

By exploring translational invariance of the theory, the Noether theorem
yields the canonical EMT which is symmetric and given by the expression

$T^{\mu\nu}_{\rm
free}(x)=(\partial^{\mu}\Phi)\,(\partial^{\nu}\Phi)-g^{\mu\nu}\mathcal{L}_{\rm
free}\,.$ (6)

The Euler-Lagrange equation of the free theory in Eq. (5) is the Klein-Gorden
equation $(\square+m^{2})\Phi(x)=0$ with the free field solution given by

$\Phi(x)=\int\frac{d^{3}k}{2\omega_{k}(2\pi)^{3}}\biggl{(}\hat{a}_{k}e^{-ik\cdot
x}+\hat{a}^{\dagger}_{k}e^{ik\cdot x}\biggr{)}\,$ (7)

where
$[\hat{a}_{k},\hat{a}_{k^{\prime}}]=2\omega_{k}(2\pi)^{3}\delta^{3}({\vec{k}-\vec{k^{\prime}}})$
and $\omega_{k}$ is defined in Eq. (3). The one-particle states are given by
$\ket{\vec{k}}=\hat{a}^{\dagger}_{k}\ket{0}$, where $\ket{0}$ is the free
theory vacuum state. Evaluating the matrix elements of $T^{\mu\nu}$ for
$\lambda=0$ yields for the form factors

$A(t)_{\rm free}=1\,,\quad D(t)_{\rm free}=-1\,.$ (8)

These results were first obtained in Ref. [2], see also Ref. [22].

###  2.3 The $\Phi^{4}$ theory

We consider the theory of a real scalar particle of mass $m$ with a self-
interaction $\Phi^{4}$ and the coupling constant $\lambda$. The classical
Lagrangian of this theory is given by

$\mathcal{L}=\frac{1}{2}\,(\partial^{\rho}\Phi)\,(\partial_{\rho}\Phi)-\frac{1}{2}m^{2}\Phi^{2}-\frac{\lambda}{4!}\,\Phi^{4}\,.$
(9)

The EMT of this theory is given by the expression

$T^{\mu\nu}(x)=T^{\mu\nu}_{\rm can}(x)+\Theta^{\mu\nu}_{\rm imp}(x)\,,\quad
T^{\mu\nu}_{\rm
can}(x)=(\partial^{\mu}\Phi)\,(\partial^{\nu}\Phi)-g^{\mu\nu}\mathcal{L}$ (10)

where the first term $T^{\mu\nu}_{\rm can}(x)$ is the symmetric canonical EMT
which follows from the Noether theorem, while $\Theta^{\mu\nu}_{\rm imp}(x)$
constitutes the improvement term which is given in $N$ space-time dimensions
by [24]

$\Theta^{\mu\nu}_{\rm
imp}(x)=-h(\partial^{\mu}\partial^{\nu}-g^{\mu\nu}\square)\,\Phi^{2}(x)\,,\;\;\;\;\;h=\frac{1}{4}\biggl{(}\frac{N-2}{N-1}\biggr{)},$
(11)

with $h=\frac{1}{6}$ in $N=3+1$ space-time dimensions. The improvement term
(11) can be derived from coupling the scalar theory (9) to a classical
gravitational background field in a non-minimal way in terms of the effective
action

$S_{\rm grav}=\int
d^{4}x\;\sqrt{-g}\biggl{(}\frac{1}{2}\,g^{\mu\nu}(x)(\partial_{\mu}\Phi)(\partial_{\nu}\Phi)-V(\Phi)-\frac{1}{2}\,h\,R\,\Phi^{2}\biggr{)}$
(12)

where $-\frac{1}{2}\,h\,R\,\Phi^{2}$ is a non-minimal coupling term, $R$ is
the Riemann scalar, and $g$ denotes the determinant of the space-time
dependent metric $g^{\mu\nu}(x)$. In this work,
$V(\Phi)=\frac{1}{2}m^{2}\Phi^{2}+\frac{1}{4!}\lambda\Phi^{4}$, but the
improvement term is valid also for other scalar theories. The EMT operator in
Eqs. (10, 11) follows from varying the action (12) according to

$T^{\mu\nu}(x)=\frac{2}{\sqrt{-g}}\,\frac{\delta S_{\rm grav}}{\delta
g_{\mu\nu}(x)}\,,$ (13)

where it is understood that after the variation, the metric is set to the
metric of the flat space. Notice that the Riemann scalar $R$ vanishes in flat
space, but its variation with respect to the metric is nevertheless non-zero.

On the quantum level, the addition of the improvement term makes Greens
functions of renormalized fields with an EMT insertion finite [24]. To be more
precise, the $h$ quoted in Eq. (11) removes UV divergences up to three-loops
in dimensional regularization in $\Phi^{4}$ theory. The operator expression
for the improvement term remains the same to all orders, albeit the value of
$h$ needs to be changed beyond three loops [30].

The coupling constant $\lambda\ll 1$ can be thought to be infinitesimally
small. In the $\Phi^{4}$ theory, the cross section for the scattering of the
$\Phi$-particles is proportional to $\lambda^{2}$. Clearly, for an
infinitesimally small $\lambda\ll 1$ the cross section is negligibly small and
the $\Phi$-particles are practically free. In that case, one could naively
assume that the EMT form factors hardly change compared to the free theory
result in Eq. (8) [22, 38]. But this is not the case, as it was suspected
(without proof) in Ref. [22] and will be explicitly demonstrated in the
subsequent sections.

##  3 EMT in $\Phi^{4}$ theory at one-loop in dimensional regularization

To one-loop order under renormalized perturbation theory, the Lagrangian in
Eq. (9) becomes

$\mathcal{L}=\frac{1}{2}(\partial_{\rho}\Phi)(\partial^{\rho}\Phi)-\frac{1}{2}(m^{2}+\delta_{m})\Phi^{2}-\frac{\lambda\Phi^{4}}{4!}.$
(14)

The mass counter term is set by the on-shell renormalization conditions,

$\left.M^{2}\left(p^{2}\right)\right|_{p^{2}=m^{2}}=0\quad\text{ and
}\left.\quad\frac{d}{dp^{2}}M^{2}\left(p^{2}\right)\right|_{p^{2}=m^{2}}=0$
(15)

where $-iM^{2}(p^{2})$ is the sum of one-particle-irreducible (1PI)
insertions. With these conditions, $\delta_{m}$ is set to one-loop order by

$-iM^{2}(p^{2})=-i\mu^{4-d}\lambda\;\frac{1}{2}\;\int\frac{d^{d}k}{(2\pi)^{d}}\frac{i}{k^{2}-m^{2}}+i\left(p^{2}\delta_{Z}-\delta_{m}\right),$
(16)

where the wave function renormalization constant $\delta_{Z}=0$ at one loop.
In dimensional regularization, Eq. (16) implies

$\delta_{m}=-\lambda\frac{\mu^{4-d}}{2(4\pi)^{\frac{d}{2}}}\;\frac{\Gamma(1-\frac{d}{2})}{m^{2-d}}.$
(17)

The coupling constant counterterm, $\delta_{\lambda}=0$ at the order $\lambda$
and becomes non-zero only at ${\cal O}(\lambda^{2})$, which is not considered
in this work.

###  3.1 Canonical EMT in dimensional regularization

Let us begin with the matrix elements of the canonical EMT which are given by

$\displaystyle\bra{\vec{p}^{\>\prime}_{\rm out}}T^{\mu\nu}_{\rm
can}(0)\ket{\vec{p}_{\rm in}}$ $\displaystyle=$ $\displaystyle\bra{\Omega_{\rm
out}}\hat{a}_{p^{\prime}}(\partial^{\mu}\Phi\partial^{\nu}\Phi-g^{\mu\nu}\mathcal{L})\hat{a}^{\dagger}_{p})\ket{\Omega_{\rm
in}}$ (18) $\displaystyle=$
$\displaystyle\bra{\Omega}T\biggl{(}\hat{a}_{p^{\prime}}(t^{\prime})(\partial^{\mu}\Phi\partial^{\nu}\Phi-g^{\mu\nu}\mathcal{L})\hat{a}^{\dagger}_{p}(t)\,e^{-i\int\frac{\lambda}{4!}\Phi^{4}(y)d^{4}y}\biggr{)}\ket{\Omega}\;,$

where $\ket{\Omega}$ denotes the vacuum of the interacting theory. In the last
step the distinction between in- and out-vacuum states is irrelevant and the
limits $t\to-\infty$ and $t^{\prime}\to\infty$ are understood. Here $\Phi$ is
redefined to be the renormalized wave function, and we will only calculate the
connected, amputated contributions. Expanding to ${\cal O}(\lambda)$ yields

$\displaystyle\bra{\vec{p}^{\>\prime}_{\rm out}}T^{\mu\nu}_{\rm
can}(0)\ket{\vec{p}_{\rm in}}$ $\displaystyle=$
$\displaystyle\frac{P^{\mu}P^{\nu}}{2}-\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{2}+\lambda\biggl{[}f^{\mu\nu}(\Delta)+g^{\mu\nu}\biggl{(}g(\Delta)+m^{2}\,h(\Delta)+j(\Delta))\biggr{)}\biggr{]}+g^{\mu\nu}\,\delta_{m}\
$ (19)

Figure 1:  Feynman diagrams for the evaluation of matrix elements of
$T^{\mu\nu}_{{\rm can}(n)}$ in $\Phi^{4}$ theory. The squares represent
insertions of $T^{\mu\nu}_{{\rm can}(n)}$ at respectively $n=0$ order in
$\lambda$ (with $T^{\mu\nu}_{{\rm can}(0)}=T^{\mu\nu}_{\rm free}(x)$ as
defined in Eq. (6)) and $n=1$ order (where $T^{\mu\nu}_{{\rm
can}(1)}=g^{\mu\nu}\frac{\lambda}{4!}\Phi^{4}$ which can be traced back to the
operator $-g^{\mu\nu}{\cal L}$ in $T^{\mu\nu}_{\rm can}$ in Eq. (10)).

The first two terms of ${\cal O}(\lambda^{0})$ correspond to the tree-level
diagram in Fig. 1a. The terms $f^{\mu\nu}(\Delta)$, $g(\Delta)$, $h(\Delta)$
are due to insertions of respectively the operators
$\partial^{\mu}\Phi\partial^{\nu}\Phi$,
$\partial_{\rho}\Phi\partial^{\rho}\Phi$ and $\Phi^{2}$ from the zeroth-order
canonical EMT and correspond to the loop diagram in Fig. 1b where the vertex
originates from the expansion of
$e^{-i\int\frac{\lambda}{4!}\Phi^{4}(y)d^{4}y}$ to first order in $\lambda$.
The term $j(\Delta)$ originates from $\frac{\lambda}{4!}\Phi^{4}$ in the
canonical EMT in conjunction with the zeroth order of
$e^{-i\int\frac{\lambda}{4!}\Phi^{4}(y)d^{4}y}$ and corresponds to the diagram
in Fig. 1c. The expressions $f^{\mu\nu}(\Delta)$, $g(\Delta)$, $h(\Delta)$,
$j(\Delta)$ are logarithmically or quadratically divergent in $d=4$ space-time
dimensions, and need to be regularized (indicated by the subscript “reg”
below) and are given by, cf. App. A.1,

$\displaystyle f^{\mu\nu}(\Delta)\ $ $\displaystyle=$ $\displaystyle i$
$\displaystyle\int_{\rm
reg}\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{\mu}}{k^{2}-m^{2}+i\epsilon}\;\frac{(k-\Delta)^{\nu}}{(k-\Delta)^{2}-m^{2}+i\epsilon}\,,$
$\displaystyle g(\Delta)\ $ $\displaystyle=$ $\displaystyle-\frac{i}{2}$
$\displaystyle\int_{\rm
reg}\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{\rho}}{k^{2}-m^{2}+i\epsilon}\;\frac{(k-\Delta)^{\rho}}{(k-\Delta)^{2}-m^{2}+i\epsilon}\,,$
$\displaystyle h(\Delta)\ $ $\displaystyle=$ $\displaystyle\ \frac{i}{2}$
$\displaystyle\int_{\rm
reg}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{2}-m^{2}+i\epsilon}\;\frac{1}{(k-\Delta)^{2}-m^{2}+i\epsilon}\,,$
$\displaystyle j(\Delta)\ $ $\displaystyle=$ $\displaystyle\frac{i}{2}$
$\displaystyle\int_{\rm
reg}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{2}-m^{2}+i\epsilon}\,.$ (20)

In dimensional regularization, one replaces
$d^{4}k/(2\pi)^{4}\to\mu^{d-4}d^{d}k/(2\pi)^{d}$ and introduces the
renormalization scale $\mu$. After exploring the Feynman parameterization

$\frac{1}{U\,V}=\int_{0}^{1}dx\,\frac{1}{(x\,U+(1-x)\,V)^{2}}\,,$

and carrying out a Wick rotation which results in $ik^{0}\to k_{E}^{0}$,
integrating over Euclidean momenta yields, see App. A.2,

$\displaystyle f^{\mu\nu}(\Delta)$ $\displaystyle=$
$\displaystyle\frac{\mu^{4-d}}{2(4\pi)^{\frac{d}{2}}}\int_{0}^{1}dx\biggl{(}g^{\mu\nu}\frac{\Gamma(1-\frac{d}{2})}{\overline{m}(x,t)^{2-d}}+\Delta^{\mu}\Delta^{\nu}(1-x)x\frac{2\,\Gamma(2-\frac{d}{2})}{\overline{m}(x,t)^{4-d}}\biggr{)}$
$\displaystyle g(\Delta)$ $\displaystyle=$
$\displaystyle\frac{\mu^{4-d}}{2(4\pi)^{\frac{d}{2}}}\int_{0}^{1}dx\biggl{(}-\frac{\Gamma(1-\frac{d}{2})\,d}{\overline{m}(x,t)^{2-d}}-\Delta^{2}(1-x)x\frac{2\,\Gamma(2-\frac{d}{2})}{\overline{m}(x,t)^{4-d}}\biggr{)}$
$\displaystyle h(\Delta)$ $\displaystyle=$
$\displaystyle\frac{\mu^{4-d}}{2(4\pi)^{\frac{d}{2}}}\int_{0}^{1}dx\biggl{(}-\,\frac{\Gamma(2-\tfrac{d}{2})}{\overline{m}(x,t)^{4-d}}\biggr{)}$
$\displaystyle j(\Delta)$ $\displaystyle=$
$\displaystyle\frac{\mu^{4-d}}{2(4\pi)^{\frac{d}{2}}}\;\frac{\Gamma(1-\frac{d}{2})}{m^{2-d}}$
(21)

where

$\overline{m}(x,t)^{2}=m^{2}-x(1-x)\,t\,.$ (22)

Inserting these results in Eq. (18), setting the number of space-time
dimensions to $d=4-\epsilon$ and expanding in $\epsilon$ yields the following
contributions to the EMT form factors from $T^{\mu\nu}_{\rm can}(0)$

$A(t)_{\rm one\\!-\\!loop,\,can}=1\,,\quad D(t)_{\rm
one\\!-\\!loop,\,can}=-1+\frac{2\lambda}{(4\pi)^{2}}\,\int_{0}^{1}dx(1-x)x\biggl{(}\frac{2}{\epsilon}+\ln{\frac{\mu^{2}}{\overline{m}(x,t)^{2}}}\biggr{)}\,,$
(23)

where it is understood that $\epsilon>0$ is small but non-zero and ${\cal
O}(\epsilon)$ neglected. Several comments are in order.

First, the canonical EMT has the Lorentz decomposition in Eq. (1) and is
therefore evidently conserved. Second, the form factor $A(t)$ is not altered
at one-loop as compared to the free theory case. Third, it is important to
stress that the one-loop result for $A(t)$ complies with the general
requirement $A(0)=1$ in Eq. (4) which is a cross check for the consistency of
the calculation. Fourth, the form factor $D(t)$ obtained in the one-loop
calculation is divergent and depends on the renormalization scale $\mu$.

The latter observation reflects the known fact that in order to obtain a
finite result for the EMT matrix elements, it is necessary to include the
improvement term [24]. We expect that this will remedy both problems, namely
make $D(t)$ (i) finite and (ii) renormalization scale independent without
spoiling EMT conservation and the constraint $A(0)=1$. We will show this
explicitly in Sec. 3.2.

###  3.2 Including the improvement term in dimensional regularization

Evaluating the matrix elements of the improvement term introduced in Eqs. (10,
11) in $\Phi^{4}$ theory yields a tree-level contribution independent of
$\lambda$ and a loop contribution proportional to $\lambda$. The results are
given as

$\bra{\vec{p}^{\>\prime}_{\rm out}}\Theta^{\mu\nu}_{\rm
imp}(0)\ket{\vec{p}_{\rm
in}}=\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{2}\,4h\,\biggl{(}1+\lambda\,h(\Delta)\biggr{)}$
(24)

with the integral $h(\Delta)$ as defined in Eq. (20). The improvement term
generates the structure $(\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2})$
which preserves EMT conservation and contributes only to $D(t)$. Setting
$d=4-\epsilon$, expanding in $\epsilon$ and neglecting ${\cal
O}(\epsilon)$-terms, the contribution to $D(t)$ is

$D(t)_{\rm
one\\!-\\!loop,\,imp}=4h\biggl{[}1-\frac{\lambda}{2(4\pi)^{2}}\biggl{(}\frac{2}{\epsilon}+\ln{\frac{\mu^{2}}{\overline{m}(x,t)^{2}}}\biggr{)}\biggr{]}\,,$
(25)

Setting $N=4$ space-time directions in Eq. (11), which implies
$h=\frac{1}{6}$, and combining the results (23, 25) to determine the EMT form
factors of the total EMT operator $T^{\mu\nu}(x)=T^{\mu\nu}_{\rm
can}(x)+\Theta^{\mu\nu}_{\rm imp}(x)$ at one-loop order in $\Phi^{4}$ theory
yields

$A(t)_{\rm one\\!-\\!loop}=1\,,\quad D(t)_{\rm
one\\!-\\!loop}=-\frac{1}{3}+\frac{2\lambda}{(4\pi)^{2}}\int_{0}^{1}dx\biggl{(}\frac{1}{6}-(1-x)x\biggr{)}\log\biggr{(}1-\frac{t}{m^{2}}x(1-x)\biggl{)}\,.$
(26)

As expected, the addition of the improvement term has preserved the EMT
conservation and the constrained $A(0)=1$ (in fact, it left the free theory
result $A_{\rm free}(t)=1$ unaffected). But it has affected $D(t)$ in the way
it was expected, namely making it finite and renormalization scale independent
(we recall that the running of the coupling constant begins at two loops, and
$\lambda$ is renormalization scale independent at one-loop order to which we
work here).

Notice that throughout Sec. 3, we have notationally distinguished the number
of space-time dimensions of the physical space $N=4$ in Eq. (11) and
$d=4-\epsilon$ in the integrals requiring regularization. This careful
distinction was deliberate and necessary. In Eq. (11) the number of space
dimensions $N=4$ must be kept fixed and must not be continued to $4-\epsilon$
in dimensional regularization. This can be understood in two ways.

First, had we continued $N=4$ to $4-\epsilon$ also in $h$ in Eq. (11), then we
would still obtain a finite one-loop result for $D(t)_{\rm one\\!-\\!loop}$
but the $\mu$-dependence would not cancel out. However, the form factors of a
conserved external current must be renormalization scale independent. Second,
the improvement term in Eq. (11) was not constructed with dimensional
regularization in mind, but is regularization scheme independent. We will
therefore cross check our calculation in Sec. 4 using the Pauli-Villars
method, rederive the dimensional regularization result in Eq. (26), and
confirm in this way that $N=4$ must be kept fixed in Eq. (11) in dimensional
regularization.

##  4 Pauli Villars Regularization

In the Pauli-Villars method, fictional particles are introduced with masses
$\Lambda_{i}\gg m$ in order to regulate the divergent integrals. In an
effective low energy theory, the cutoffs $\Lambda_{i}$ have a physical meaning
and ensure that the result of the calculation applies only to low energies
below the cutoffs $\Lambda_{i}$. In a renormalizable theory it is possible to
take the limit $\Lambda_{i}\to\infty$ after the renormalization of the
parameters of the theory.

To implement the Pauli-Villars regularization, we follow Callan, Coleman &
Jackiw [24]. In $\Phi^{4}$ theory, it is necessary to introduce two fictional
fields $\Phi_{1}$ and $\Phi_{2}$ in the Lagrangian as follows

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigl{(}(\partial_{\mu}\Phi)^{2}-m^{2}\Phi\bigr{)}-\frac{1}{2}\bigl{(}(\partial_{\mu}\Phi_{1})^{2}-\Lambda_{1}^{2}\Phi_{1}^{2}\bigr{)}-\frac{1}{2}\bigl{(}(\partial_{\mu}\Phi_{2})^{2}-\Lambda_{2}^{2}\Phi_{2}^{2}\bigr{)}$
(27)
$\displaystyle-\frac{1}{2}\delta_{m}\bigl{(}\Phi+a_{1}\Phi_{1}+a_{2}\Phi_{2}\bigr{)}^{2}-\frac{\lambda}{4!}\bigl{(}\Phi+a_{1}\Phi_{1}+a_{2}\Phi_{2}\bigr{)}^{4}$

which makes the theory non-hermitian. The propagator of the unregularized
theory

$\displaystyle\Pi(k^{2})=\frac{1}{k^{2}-m^{2}+i\epsilon}$

is then modified to a propagator of the form

$\displaystyle\Pi_{\text{mod}}(k^{2})=\frac{1}{k^{2}-m^{2}+i\epsilon}-\frac{a_{1}^{2}}{k^{2}-\Lambda_{1}^{2}+i\epsilon}-\frac{a_{2}^{2}}{k^{2}-\Lambda_{2}^{2}+i\epsilon}\,,\quad
a_{1}^{2}=\frac{m^{2}-\Lambda_{2}^{2}}{\Lambda_{1}^{2}-\Lambda_{2}^{2}}\,,\quad
a_{2}^{2}=\frac{\Lambda_{1}^{2}-m^{2}}{\Lambda_{1}^{2}-\Lambda_{2}^{2}}\,.$

Using the same on-shell renormalization scheme specified in Eq. (15), the 1PI
insertion in (16) now becomes

$-iM^{2}(p^{2})=-i\lambda\;\frac{1}{2}\;\int\frac{d^{d}k}{(2\pi)^{d}}\Pi_{\text{mod}}(k^{2})-i\delta_{m}.$
(28)

Thus, the mass counterterm in Eq. (27) is of the form

$\delta_{m}=-\frac{\lambda}{2}\int\frac{d^{4}k_{E}}{(2\pi)^{4}}\biggl{(}\frac{1}{k_{E}^{2}+m^{2}}-\frac{m^{2}-\Lambda_{2}^{2}}{(\Lambda_{1}^{2}-\Lambda_{2}^{2})(k_{E}^{2}+\Lambda_{1}^{2})}-\frac{\Lambda_{1}^{2}-m^{2}}{(\Lambda_{1}^{2}-\Lambda_{2}^{2})(k_{E}^{2}+\Lambda_{2}^{2})}\biggr{)}.$
(29)

The canonical EMT and the improvement term in the Pauli-Villars regularization
are given by [24]

$\displaystyle T^{\mu\nu}_{\rm can,PV}$ $\displaystyle=$
$\displaystyle\partial^{\mu}\Phi\partial^{\nu}\Phi-\partial^{\mu}\Phi_{1}\partial^{\nu}\Phi_{1}-\partial^{\mu}\Phi_{2}\partial^{\nu}\Phi_{2}-g^{\mu\nu}\mathcal{L}$
$\displaystyle\Theta^{\mu\nu}_{\rm imp,PV}$ $\displaystyle=$
$\displaystyle-h\bigl{(}\partial^{\mu}\partial^{\nu}-g^{\mu\nu}\square\bigr{)}\bigl{(}\Phi^{2}-\Phi_{1}^{2}-\Phi_{2}^{2}\bigr{)}$
(30)

with $h$ as defined in Eq. (11).

To simplify our work, we note that

$\displaystyle-\frac{\lambda}{4!}\bra{\Omega}T\biggl{(}a_{p^{\prime}}\bigl{(}\Phi^{4}+6c_{1}^{2}\phi_{1}^{2}\Phi^{2}+6c_{2}^{2}\phi_{2}^{2}\Phi^{2}\bigr{)}a_{p}^{\dagger}\biggr{)}\ket{\Omega}$
$\displaystyle\quad=-\frac{\lambda}{2}\int\frac{d^{4}k_{E}}{(2\pi)^{4}}\biggl{(}\frac{1}{k_{E}^{2}+m^{2}}-\frac{m^{2}-\Lambda_{2}^{2}}{(\Lambda_{1}^{2}-\Lambda_{2}^{2})(k_{E}^{2}+\Lambda_{1}^{2})}-\frac{\Lambda_{1}^{2}-m^{2}}{(m_{1}^{2}-\Lambda_{2}^{2})(k_{E}^{2}+\Lambda_{2}^{2})}\biggr{)}.$
(31)

Hence, to order $\lambda$, we find

$\displaystyle-\frac{\lambda}{4!}\bra{\Omega}T\biggl{(}a_{p^{\prime}}(\Phi^{4}+6c_{1}^{2}\Phi_{1}^{2}\Phi^{2}+6c_{2}^{2}\Phi_{2}^{2}\Phi^{2})a_{p}^{\dagger}\biggr{)}\ket{\Omega}-\frac{1}{2}\delta_{m}\bra{\Omega}T\biggl{(}a_{p^{\prime}}(\Phi+a_{1}\Phi_{1}+a_{2}\Phi_{2})^{2}a_{p}^{\dagger}\biggr{)}\ket{\Omega}=0$

and our calculation reduces to evaluating the following expression for the
total EMT

$\displaystyle\bra{p^{\prime}}T^{\mu\nu}_{PV}\ket{p}=F^{\mu\nu}(\Delta)+G^{\mu\nu}(\Delta)+H^{\mu\nu}(\Delta)+I^{\mu\nu}(\Delta)$
(32)

where,

$\displaystyle
F^{\mu\nu}(\Delta)=\bra{p^{\prime}}(\partial^{\mu}\Phi\partial^{\nu}\Phi-\partial^{\mu}\Phi_{1}\partial^{\nu}\Phi_{1}-\partial^{\mu}\Phi_{2}\partial^{\nu}\Phi_{2})\ket{p}$
$\displaystyle
G^{\mu\nu}(\Delta)=-\frac{g^{\mu\nu}}{2}\bra{p^{\prime}}((\partial_{\rho}\Phi)^{2}-(\partial_{\rho}\Phi_{1})^{2}-(\partial_{\rho}\Phi_{2})^{2})\ket{p}$
$\displaystyle
H^{\mu\nu}(\Delta)=-\frac{g^{\mu\nu}}{2}\bra{p^{\prime}}(-m^{2}\Phi^{2}+\Lambda_{1}^{2}\Phi_{1}^{2}+\Lambda_{2}^{2}\Phi_{2}^{2})\ket{p}$
$\displaystyle
I^{\mu\nu}(\Delta)=-h\bra{p^{\prime}}(\partial^{\mu}\partial^{\nu}-g^{\mu\nu}\Box)(\Phi^{2}-\Phi_{1}^{2}-\Phi_{2}^{2})\ket{p}.$
(33)

We find, after letting $\Lambda_{1}\to\Lambda$ and $\Lambda_{2}\to\Lambda$
with $\Lambda^{2}\gg\overline{m}^{2}$, that

$\displaystyle
F^{\mu\nu}(\Delta)=\biggl{[}p^{\prime\mu}p^{\nu}+p^{\prime\nu}p^{\mu}+\int_{0}^{1}dx\biggl{(}\frac{\lambda
g^{\mu\nu}}{2(4\pi)^{2}}\overline{m}^{2}\log{\frac{\overline{m}^{2}}{\Lambda^{2}}}+\frac{\lambda\Delta^{\mu}\Delta^{\nu}x(1-x)}{(4\pi)^{2}}\log{\frac{\Lambda^{2}}{\overline{m}^{2}}}\biggr{)}\biggr{]}$
$\displaystyle G^{\mu\nu}(\Delta)=-g^{\mu\nu}\biggl{[}p^{\prime}\cdot
p+\int_{0}^{1}dx\biggl{(}\frac{\lambda}{(4\pi)^{2}}\overline{m}^{2}\log{\frac{\overline{m}^{2}}{\Lambda^{2}}}+\frac{\lambda\Delta^{2}x(1-x)}{2(4\pi)^{2}}\log{\frac{\Lambda^{2}}{\overline{m}^{2}}}\biggr{)}\biggr{]}$
$\displaystyle
H^{\mu\nu}(\Delta)=g^{\mu\nu}\biggl{[}m^{2}-\int_{0}^{1}dx\frac{\lambda}{2(4\pi)^{2}}m^{2}\log{\frac{\Lambda^{2}}{\overline{m}^{2}}}\biggr{]}$
$\displaystyle
I^{\mu\nu}(\Delta)=h\biggl{[}2-\frac{\lambda}{(4\pi)^{2}}\log{\frac{\Lambda^{2}}{\overline{m}^{2}}}\biggr{]}(\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2})$
(34)

with the details of these derivations given in App. B .

After combining the results Eq. (34), simplifying, and letting
$\Lambda\to\infty$, we arrive at the one-loop matrix elements for the total
EMT,

$\displaystyle\bra{p^{\prime}}T^{\mu\nu}_{\rm
PV}\ket{p}=\frac{P^{\mu}P^{\nu}}{2}$
$\displaystyle-\frac{1}{3}\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{2}$
$\displaystyle+(\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2})\frac{\lambda}{(4\pi)^{2}}\int_{0}^{1}dx\biggl{(}-(1-x)x+\frac{1}{6}\biggr{)}\log{(1-\frac{\Delta^{2}}{m^{2}}x(1-x))}.$
(35)

From Eq. (4) we read of the results for $A(t)_{\rm one\\!-\\!loop}$ and
$D(t)_{\rm one\\!-\\!loop}$ in complete agreement with (26).

##  5 Properties of $D(t)$

After computing the EMT form factors in the $\Phi^{4}$ theory in one-loop
order in dimensional regularization in Eq. (26) and double-checking the
results in Pauli-Villars regularization, we now turn to a discussion of their
physical properties. In the following, the “one-loop” subscript will be
dropped for notational simplicity.

The one-loop calculation shows that the EMT form factor $A(t)=1$ remains a
trivial constant, while $D(t)$ exhibits an non-trivial $t$-dependence which
makes it worth discussing. For convenience, let us quote again the result

$D(t)=-\frac{1}{3}+\frac{2\lambda}{(4\pi)^{2}}\int_{0}^{1}dx\biggl{(}\frac{1}{6}-(1-x)x\biggr{)}\log\biggr{(}1-\frac{t}{m^{2}}x(1-x)\biggl{)}\,.$
(36)

The value of the $D$-term, the form factor $D(t)$ at $t=0$ is
$\lambda$-independent and given by

$D=-\frac{1}{3}\,.$ (37)

The integral in Eq. (36) can be solved in closed form for $t\neq 0$. Below the
threshold for particle production $t<4m^{2}$, the form factor $D(t)$ is real.
Above $t>4m^{2}$, it develops an imaginary part. The expressions are given by

$\displaystyle{\rm Re\,}D(t)=$
$\displaystyle-\frac{1}{3}+\frac{\lambda}{24\pi^{2}}$
$\displaystyle\times\begin{cases}\displaystyle\biggl{[}\frac{2m^{2}}{t}-\frac{1}{6}-\frac{2m^{2}}{t}\sqrt{1-\frac{4m^{2}}{t}}\;{\rm
arsinh}\biggl{(}\frac{\sqrt{-t}}{2m}\biggr{)}\biggr{]}&\mbox{for}\quad\;t<0,\\\
{}\\\
\displaystyle\biggl{[}\frac{2m^{2}}{t}-\frac{1}{6}-\frac{2m^{2}}{t}\sqrt{\frac{4m^{2}}{t}-1}\;{\rm
arcsin}\biggl{(}\frac{\,\sqrt{t}}{2m}\,\biggr{)}\biggr{]}&\mbox{for}\quad
0<t<4m^{2},\\\ {}\\\
\displaystyle\biggl{[}\frac{2m^{2}}{t}-\frac{1}{6}-\frac{2m^{2}}{t}\sqrt{1-\frac{4m^{2}}{t}}\,{\rm
arcosh}\biggl{(}\frac{\sqrt{t}}{2m}\biggr{)}\biggr{]}&\mbox{for}\quad
t>4m^{2},\end{cases}$ $\displaystyle{\rm Im\,}D(t)=$
$\displaystyle\frac{\lambda}{24\pi}$
$\displaystyle\times\begin{cases}\displaystyle 0\,\hskip
190.63345pt&\mbox{for}\quad t\leq 4m^{2},\\\ {}\\\
\displaystyle\;\frac{m^{2}}{t}\sqrt{1-\frac{4m^{2}}{t}}&\mbox{for}\quad
t>4m^{2}.\end{cases}$ (38)

In the vicinity of $t=0$, the form factor is continuous and (infinitely)
differentiable, with a unique Taylor expansion

$D(t)=-\frac{1}{3}+\frac{\lambda}{1440\pi^{2}}\biggl{[}\frac{t}{m^{2}}+\frac{t^{2}}{7\,m^{4}}+\frac{t^{3}}{42\,m^{6}}+\frac{t^{4}}{231\,m^{8}}+\dots\biggr{]}\;,$
(39)

which is independent of whether one approaches the point $t=0$ from the region
of positive or negative $t$. The limits $t\to\pm\infty$ in the respective
terms in Eq. (38) exist and are given by

$D_{\inf}=\lim_{t\to-\infty}D(t)=\lim_{t\to\infty}{\rm
Re\,}D(t)=-\frac{1}{3}-\frac{\lambda}{144\pi^{2}},\quad\lim_{t\to\infty}{\rm
Im\,}D(t)=0\,.$ (40)

At the threshold $t=4m^{2}$, the real part of $D(t)$ is continuous but not
differentiable and assumes at the cusp the value

$D_{\rm cusp}={\rm Re}\,D(4m^{2})=-\frac{1}{3}+\frac{\lambda}{72\pi^{2}}\,.$
(41)

The imaginary part of $D(t)$ is non-zero only above the threshold $t=4m^{2}$
and is always non-negative. It exhibits a global maximum at $t=6m^{2}$ with
the value

${\rm Im}\,D_{\max}={\rm Im}\,D(6m^{2})=\frac{\lambda}{144\sqrt{3}\pi}\,.$
(42)

Figure 2:  The form factor $D(t)$ in $\Phi^{4}$ theory at one loop as function
of $t$ in units of $m^{2}$. In all figures, 1 unit on the $y$-axis corresponds
to $a=\lambda/(144\pi^{2})$. (a) The space-like region plotted as function of
$(-t)>0$. (b) ${\rm Re}\,D(t)$ in the time-like region. (c) The same as (a)
and (b) but showing a wide $t$-range. (d) ${\rm Im}\,D(t)$ which is non-zero
only above the threshold $t=4m^{2}$. The vertical lines in (a)–(c) are the
asymptotic values $D_{\inf}$ defined in Eq. (40). The dots in (b) and (c) mark
the cusp defined in Eq. (41). The square in (d) denotes the maximum of ${\rm
Im}\,D(t)$ in Eq. (42).

In Fig. 2 we show the real and imaginary parts of $D(t)$ as functions of $t$
in units of $m^{2}$ under the assumption that $\lambda$ is small and the one-
loop approximation is justified. For $\lambda\ll 1$, it is always ${\rm
Re}\,D(t)<0$ while ${\rm Im}\,D(t)\geq 0$. If we literally assume $\lambda$ to
be infinitesimally small, then $D(t)\simeq-\frac{1}{3}$ is a trivial constant.
In order to show a non-trivial $t$ dependence of $D(t)$, we choose the unit on
the $y$-axis in Fig. 2 to correspond to $a=\lambda/(144\pi^{2})$.

In Fig. 2a, we show the space-like region where $D(t)$ is real and plotted as
a function of $(-t)>0$ as it is customary for form factors measured in
scattering experiments. We see that $D(t)$ decreases monotonically from
$D=-\frac{1}{3}$ to the asymptotic value $D_{\inf}$ in Eq. (40), i.e. the
absolute value $|D(t)|$ increases which we shall comment on below. The Fig. 2b
shows ${\rm Re}\,D(t)$ in the time-like region around the threshold $t=4m^{2}$
above which $D(t)$ becomes complex. The dot at $t=4m^{2}$ marks the cusp
defined in Eq. (41). In Fig. 2c, we plot a wide $t$-range to illustrate how
${\rm Re}\,D(t)$ approaches logarithmically the asymptotic value $D_{\inf}$ in
Eq. (40). The Fig. 2d shows ${\rm Im}\,D(t)$, which is non-zero only above the
threshold and exhibits a maximum at $t=6m^{2}$, which is defined in Eq. (42)
and marked in Fig. 2d by a square. ${\rm Im}\,D(t)$ approaches asymptotically
zero for $t\to\infty$. Higher loops would generate further cusps and
thresholds in respectively ${\rm Re}\,D(t)$ and ${\rm Im}\,D(t)$ at
$t=4n^{2}m^{2}$ for integer $n>1$. However, the relative size of the cusps
(relative to $D=-\frac{1}{3}$) or thresholds would be proportional to
$\lambda^{n}$ and not visible on the scale of Fig. 2 for $\lambda\ll 1$.

The growth of $|D(t)|$ in $\Phi^{4}$-theory for $t\to-\infty$ is interesting.
Indeed, hadronic models [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69], lattice QCD
[70, 71, 72, 73, 74, 75, 76, 77, 78], perturbative QCD [79, 80, 81], data-
based determinations [82, 83, 84], and QCD sum rules [85, 86, 87, 88] yield a
monotonically decreasing $|D(t)|$ for increasing $(-t)$ which is a behavior
shared by all hadronic form factors.111Some form factors like the neutron
electric form factor $G_{E}^{n}(t)$ or the nucleon gravitomagnetic form factor
$B(t)$ vanish at $t=0$ and increase initially with $(-t)$. But eventually
their magnitudes also decrease like other hadronic form factors. This
unfamiliar behavior of $D(t)$, from a hadronic physics perspective, deserves
some comments.

The $\Phi^{4}$ interaction for $\lambda>0$ is repulsive [89] in contrast to
the strong interaction, which is attractive and forms bound states. This
raises the question whether the growth of $|D(t)|$ might be a feature of a
repulsive interaction. In fact, if we could continue the theory (9) to
$\lambda<0$, then the interaction would become attractive and our $|D(t)|$
would exhibit a “more hadronic behavior” and decrease for $(-t)\to\infty$. The
continuation to negative $\lambda$ would not affect our perturbative
calculation, but it would make the theory (9) unbound from below and the
vacuum unstable.

However, in this context it is interesting to note that the (complex)
$|\Phi|^{4}$ theory continued analytically to $\lambda<0$ with an additional
positive term $C\,|\Phi|^{6}$ in the Lagrangian is a well-defined (albeit
effective and non-renormalizable) theory. This theory admits non-perturbative
soliton solutions referred to as $Q$-balls [91] which exhibit EMT properties
familiar from nuclear and hadronic physics [92, 93]. In this theory it is
possible to take a certain (so-called $Q$-cloud) limit [94], where the
additional term $C\,|\Phi|^{6}$ in the Lagrangian becomes irrelevant and can
effectively be set to zero, i.e. one basically deals with a complex
$|\Phi|^{4}$ theory continued to negative $\lambda$. Remarkably, the
appropriately rescaled $Q$-cloud solutions encountered in this limit still
exhibit hadron-type EMT properties [90].

These considerations illustrate that the growth of $|D(t)|$ in
$\Phi^{4}$-theory for $t\to-\infty$ in our calculation is natural. (Notice
that $Q$-balls and $Q$-clouds exist only in complex scalar theories, while our
calculation was in real $\Phi^{4}$ theory. But our 1-loop calculation would
look basically the same in complex $\Phi^{4}$ theory.)

Of course, one may also pragmatically argue that in the weak coupling limit
needed to justify the perturbative treatment, $D(t)\simeq-\frac{1}{3}$ is
nearly constant as one would expect for a form factor of a particle with no
internal structure. It is also not surprizing that in $\Phi^{4}$ theory the
form factors (trivially for $A(t)$ and non-trivially for $D(t)$) approach
asymptotically a constant for $(-t)\to\infty$ and do not vanish like hadronic
form factors do. The latter reflects the fact that hadrons are composed
particles. It is unlikely for a hadron to stay intact after absorbing a large
momentum transfer. The situation is of course distinctly different in
$\Phi^{4}$ theory.

##  6 Dispersion relation for $D(t)$ in $\Phi^{4}$ theory

The form factor $D(t)$ satisfies a dispersion relation [82], see also [83],
which provides a powerful test for the calculation. In our case, the form
factor does not go to zero for $|t|\to\infty$. In this case, one can either
formulate an unsubtracted dispersion relation for $D(t)-D_{\inf}$ which does
go to zero for $|t|\to\infty$, or subtracted dispersion relation for
$D(t)-D(0)$ which does not go to zero for $|t|\to\infty$. Both variants are
given by

$\displaystyle D(t)-D_{\inf}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{4m^{2}}^{\infty}ds\;\frac{{\rm
Im}\,D(s)}{s-t}\quad\;\mbox{unsubtracted case for}\,\quad t<4m^{2}\,,$ (43)
$\displaystyle D(t)-D(0)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{4m^{2}}^{\infty}ds\;\frac{{\rm
Im}\,t\,D(s)}{s(s-t)}\quad\mbox{subtracted case for}\quad t<4m^{2}\,.$ (44)

Notice that ${\rm Im}\,D_{\inf}={\rm Im}\,D(0)=0$ was used to simplify the
expressions on the right-hand side. In either case, one must subtract the
tree-level term $D(0)$, which is of order $\lambda^{0}$. Only the loop
contribution of order $\lambda$ has non-trivial analytical properties. We
checked analytically and numerically that our results (36, 38) satisfy the
dispersion relations in Eq. (43, 44). This is an important cross check for the
calculation.

The form factor $A(t)=1$ satisfies the dispersion relation trivially. Also,
here it is necessary to formulate the dispersion relation for the difference
$A(t)-1$, which is the quantity that goes to zero for $|t|\to\infty$. The real
and imaginary parts of $A(t)-1$ vanish, and the unsubtracted dispersion
becomes zero equals zero.

##  7 Radii

One of the reasons for the interest in EMT form factors is related to their
attractive interpretations. Fourier transforms of the EMT form factors give
insights into the distribution of matter and internal forces inside an
extended particle. For particles whose radius is much larger than the Compton
wavelength, one can discuss 3D densities [11, 20]. For particles of any mass,
including massless particles, one can discuss 2D densities [39, 40]. It is
interesting to investigate aspects of this interpretation in the $\Phi^{4}$
theory, and study the mean square radii of several quantities.

Let us begin with what has been referred to as the mean square radius of the
matter distribution in some works [69]. This radius is defined in terms of the
derivative of the EMT form factor $A(t)$ at $t=0$ as follows

$\langle r^{2}_{\rm mat}\rangle=-6\,A^{\prime}(0)\,.$ (45)

Since we found $A(t)=1$, we see that the mean square radius of the matter
distribution in the $\Phi^{4}$ theory is

$\langle r^{2}_{\rm mat}\rangle=0\,.$ (46)

In the 2D light-front formulation, the slope of $A(t)$ at $t=0$ (with a
prefactor of minus $4$) defines the so-called light-front momentum mean square
radius [40], which is of course also zero.

Another interesting quantity is the mean square scalar radius associated with
the trace of the EMT. The form factor of this trace of the EMT can be
expressed in terms of $A(t)$ and $D(t)$ based on Eq. (1). The mean square
radius of the EMT trace operator [42] is defined as

$\langle r^{2}_{\rm tr}\rangle=\langle r^{2}_{\rm
mat}\rangle-\frac{3}{2m^{2}}\;(1+3D)\,.$ (47)

This quantity has been referred to in literature also as the mass radius [95]
or scalar radius [69]. With our results for the EMT form factors, we obtain

$\langle r^{2}_{\rm tr}\rangle=0\,.$ (48)

Finally, let us discuss the mechanical radius, whose definition does not
involve derivatives of form factors, but rather is given by [96, 20]

$\langle r^{2}_{\rm
mech}\rangle=\frac{6\,D(0)}{\int_{-\infty}^{0}dt\;D(t)}\;.$ (49)

In the $\Phi^{4}$ theory, the numerator is finite while the integral in the
denominator diverges such that one obtains

$\langle r^{2}_{\rm mech}\rangle=0\;.$ (50)

The mean square radii discussed in these examples vanish in the $\Phi^{4}$
theory at one-loop order, which shows that the described particle is point-
like.

##  8 Conclusions

In this work, a study of EMT form factors was presented in $\Phi^{4}$ theory
in the weak-coupling regime at one-loop. This study is of interest as the EMT
form factors of hadrons have attracted a lot of attention in the recent past
[20, 21]. In view of the complexity of QCD, EMT studies in simpler theories
can be very instructive. After free field theory, the next-to-simplest case is
to solve an interacting theory perturbatively to lowest non-trivial order.

In free spin-zero theory, the EMT form factors can be evaluated exactly with
$A(t)_{\rm free}=1$ and $D(t)_{\rm free}=-1$ [2]. This implies for the
$D$-term $D_{\rm free}=-1$ of a free spin-zero boson [22]. In the weak
coupling regime $\lambda\ll 1$, the cross sections in $\Phi^{4}$ theory is
proportional to $\lambda^{2}$, i.e. the $\Phi$ quanta are nearly free
particles. One could therefore expect the $D$-term in a weakly interacting
$\Phi^{4}$ theory to be very similar to the free theory. But this is not the
case.

In this work, we have shown that the inclusion of a $\Phi^{4}$ interaction
reduces the $D$-term from $D_{\rm free}=-1$ in free case to $D=-\frac{1}{3}$
no matter how small the coupling constant $\lambda$ is. This was speculated
without proof in Ref. [22] based on the necessity to introduce an improvement
in the EMT in scalar field theory which is needed to render EMT matrix
elements finite in the presence of quantum interactions [24]. This result is
supported by calculations in another renormalizable scalar theory, the
$\Phi^{3}$ theory [31], and in line with observations in the literature that
the $D$-term is the particle property which is most sensitive to the dynamics
in a system [20, 21, 22, 23].

The calculation was carried out at one-loop in dimensional regularization and
checked independently in Pauli-Villars regularization. The consistency of the
calculations was checked in several ways: the EMT is of course conserved also
at one-loop, the form factor $A(t)$ satisfies the constraint $A(0)=1$ imposed
by Lorentz symmetry, and both EMT form factors $A(t)$ and $D(t)$ are
renormalization scale independent.

Assuming that $\lambda\ll 1$ and ${\cal O}(\lambda^{2})$ can be safely
neglected, our results yield the “full results” for the EMT form factors in
$\Phi^{4}$ theory in the weak coupling regime. The form factor $A(t)$ is not
altered by quantum corrections and remains $A(t)=1$ also in the interacting
case. For $D(t)$ the situation is different: at $t=0$ it assumes the exact
value $D=-\frac{1}{3}$ which is a drastic change from $D_{\rm free}=-1$, while
for $t\neq 0$ it exhibits a non-trivial $t$-dependence.

In the space-like region $t<0$ the form factor $D(t)$ shows an unfamiliar
behavior from the point of view of hadronic physics with $|D(t)|$ slowly
growing as $(-t)$ becomes large, while the form factors of hadrons decrease.
This behavior can be understood considering that the quanta in $\Phi^{4}$
theory are elementary fields, not composed particles. The weak increase of
$|D(t)|$ which logarithmically approaches a constant for asymptotically large
$(-t)$ might be related to the fact that the interaction in $\Phi^{4}$ theory
is repulsive. This interesting point deserves further studies.

Exploring the field theoretical setting, we continued analytically our
diagrammatic calculation to time-like $t>0$. At the threshold for two-particle
production, $t=4m^{2}$ nothing unusual happens to $A(t)$ which remains equal
to unity also in the entire time-like region. The form factor $D(t)$ becomes
complex for $t>4m^{2}$. We explicitly determined the analytical expressions
for ${\rm Re}\,D(t)$ and ${\rm Im}\,D(t)$ and showed that they satisfy the
dispersion relation [82].

Our study was rounded up by an investigation of several mean square radii
introduced in recent literature. In particular, we investigated the mean
square radius of the matter distribution $\langle r^{2}_{\rm
mat}\rangle=-6\,A^{\prime}(0)$, the mean square radius of the EMT trace
$\langle r^{2}_{\rm tr}\rangle=-6\,A^{\prime}(0)-\frac{3}{2m^{2}}\;(1+3D)$,
and the mechanical radius $\langle r^{2}_{\rm
mech}\rangle=6\,D/\int_{-\infty}^{0}dt\;D(t)$. All these radii vanish in the
$\Phi^{4}$ theory. On the one hand, this is not unexpected since the particles
in the $\Phi^{4}$ theory are elementary and point-like. On the other hand, it
is reassuring to observe that the various mean square radii recently
introduced to describe EMT properties of hadrons yield zero for pointlike
particles and hence have a sound physical basis. In this sense, our study
contributes to a better understanding of EMT properties and their
interpretation.

We stress that the $\phi^{4}$ results are not representative for “all weakly
coupled theories” and “all elementary particles”. QED is a well-known
counterexample [97, 98, 99, 100, 101, 102, 103]. The long-range nature of the
Coulomb force [100, 101], based on the masslessness of the photon in gauge
theory, gives rise to very different EMT properties for charged particles
independently of their spin. For instance, in one-loop order in QED the EMT
form factors of the electron vanish for asymptotically large $(-t)$ and no
mean square radii can be defined [103].

While our results are sufficient in the weak coupling limit, it would be
nevertheless instructive to study EMT properties in $\Phi^{4}$ theory beyond
one-loop. The interesting feature is the running of the coupling constant
$\lambda\to\lambda(\mu)$ starting at two-loop order. Form factors of conserved
external currents are expected to be renormalization scale independent. It
would be interesting to see how this is realized in a two-loop calculation in
$\Phi^{4}$ theory. This question goes beyond the scope of this work and is
left to future studies.

Acknowledgments. Discussions with John Collins, Andrew Dotson, Gerald Dunne,
Luchang Jin, Barbara Pasquini, and Matthew Sievert played important roles to
motivate and carry out this work. The author is indebted to Andrew Dotson for
checking parts of the calculation and Peter Schweitzer for help with
manuscript preparation. This work was supported by National Science Foundation
under Award No. 2111490, and U.S. Department of Energy under the umbrella of
the Quark-Gluon Tomography (QGT) Topical Collaboration with Award No. DE-
SC0023646.

##  Appendix A Dimensional regularization

This Appendix contains for reference the details of the EMT calculation in
dimensional regularization.

###  A.1 The expressions for $f^{\mu\nu}(\Delta)$, $g(\Delta)$, $h(\Delta)$,
$j(\Delta)$

Let us first derive the expressions for $f^{\mu\nu}(\Delta)$, $g(\Delta)$,
$h(\Delta)$, $j(\Delta)$ introduced in Eq. (20) which emerge from respectively
evaluating matrix elements of the operators
$\partial^{\mu}\Phi\partial^{\nu}\Phi$,
$\partial_{\rho}\Phi\partial^{\rho}\Phi$, $\Phi^{2}$, $\Phi^{4}$ in Eq. (19).
We find

$\displaystyle\bra{\Omega}T(a_{p^{\prime}}\partial^{\mu}\Phi(x)\partial^{\nu}\Phi(x)a_{p}^{\dagger}e^{-i\int\frac{\lambda}{4!}\Phi^{4}d^{4}y})\ket{\Omega}\phantom{\frac{1}{1}}$
$\displaystyle=(p^{\prime\mu}p^{\nu}+p^{\prime\nu}p^{\mu})e^{i\Delta\cdot
x}-\frac{i\lambda}{4!}\int
d^{4}y\bra{0}T(a_{p^{\prime}}\partial^{\mu}\Phi(x)\partial^{\nu}\Phi(x)a_{p}^{\dagger}\Phi^{4}(y))\ket{0}\phantom{\frac{1}{1}}$
$\displaystyle=\biggl{(}p^{\prime\mu}p^{\nu}+p^{\prime\nu}p^{\mu}+i\lambda\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{\mu}}{k^{2}-m^{2}+i\epsilon}\frac{(k-\Delta)^{\nu}}{(k-\Delta)^{2}-m^{2}+i\epsilon}\biggr{)}e^{i\Delta\cdot
x}\phantom{\frac{1}{1}}$
$\displaystyle=(p^{\prime\mu}p^{\nu}+p^{\prime\nu}p^{\mu}+\lambda\,f^{\mu\nu}(\Delta))e^{i\Delta\cdot
x},\phantom{\frac{1}{1}}$ (51)
$\displaystyle\frac{g^{\rho\sigma}}{2}\bra{\Omega}T(a_{p^{\prime}}\partial_{\rho}\Phi(x)\partial_{\sigma}\Phi(x)a_{p}^{\dagger}e^{-i\int\frac{\lambda}{4!}\Phi^{4}d^{4}y})\ket{\Omega}\phantom{\frac{1}{1}}$
$\displaystyle=\frac{g^{\rho\sigma}}{2}[p^{\prime}_{\rho}p_{\sigma}e^{i\Delta\cdot
x}+p^{\prime}_{\sigma}p_{\rho}e^{i\Delta\cdot x}-\frac{i\lambda}{4!}\int
d^{4}y\bra{0}T(a_{p^{\prime}}\partial_{\rho}\Phi(x)\partial_{\sigma}\Phi(x)a_{p}^{\dagger}\Phi^{4}(y))\ket{0}]\phantom{\frac{1}{1}}$
$\displaystyle=\biggl{(}p^{\prime}\cdot
p+\frac{i\lambda}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{\rho}}{k^{2}-m^{2}+i\epsilon}\frac{(k-\Delta)^{\rho}}{(k-\Delta)^{2}-m^{2}+i\epsilon}\biggr{)}e^{i\Delta\cdot
x}\phantom{\frac{1}{1}}$ $\displaystyle=(p^{\prime}\cdot p-\lambda
g(\Delta))e^{i\Delta\cdot x}\phantom{\frac{1}{1}}$ (52)
$\displaystyle\frac{(m^{2}+\delta_{m})}{2}\bra{\Omega}T(a_{p^{\prime}}\Phi(x)\Phi(x)a_{p}^{\dagger}e^{-i\int\frac{\lambda}{4!}\Phi^{4}d^{4}y})\ket{\Omega}$
$\displaystyle=\frac{(m^{2}+\delta_{m})}{2}\biggl{(}2e^{i\Delta\cdot
x}-\frac{i\lambda}{4!}\int
d^{4}y\bra{0}T(a_{p^{\prime}}\Phi(x)\Phi(x)a_{p}^{\dagger}\Phi^{4}(y))\ket{0}\biggr{)}$
$\displaystyle=(m^{2}+\delta_{m})\biggl{(}1+\frac{i\lambda}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{2}-m^{2}+i\epsilon}\frac{1}{(k-\Delta)^{2}-m^{2}+i\epsilon}\biggr{)}e^{i\Delta\cdot
x}$ $\displaystyle=(m^{2}+\delta_{m})(1+\lambda h(\Delta))e^{i\Delta\cdot
x},\phantom{\frac{1}{1}}$ (53)
$\displaystyle\frac{\lambda}{4!}\bra{\Omega}T(a_{p^{\prime}}\Phi(x)\Phi(x)\Phi(x)\Phi(x)a_{p}^{\dagger}e^{-i\int\frac{\lambda}{4!}\Phi^{4}d^{4}y})\ket{\Omega}$
$\displaystyle=\frac{\lambda}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{ie^{i(p^{\prime}-p)\cdot
x}}{k^{2}-m^{2}+i\epsilon}=\lambda j(\Delta)e^{i\Delta\cdot
x}.\phantom{\frac{1}{1}}$ (54)

Combining Eqs. (A.1–A.1) and letting $x\to 0$, we obtain the result for the
canonical EMT quoted in Eq. (19).

###  A.2 Evaluation of canonical EMT in dimensional regularization

We shall make use of the following integrals in dimensional regularization in
Euclidean space-time where $k^{0}=ik^{0}_{E}$

$\displaystyle\int\frac{dk^{4}}{(2\pi)^{4}}\frac{1}{(k^{2}-\Sigma)^{n}}\xrightarrow{}(-1)^{n}i\int\frac{d^{4}k_{E}}{(2\pi)^{4}}\frac{1}{(k_{E}^{2}+\Sigma)^{n}}=(-1)^{n}i\frac{1}{(4\pi)^{d/2}}\frac{\Gamma\left(n-\frac{d}{2}\right)}{\Gamma(n)}\left(\frac{1}{\Sigma}\right)^{n-\frac{d}{2}},$
$\displaystyle\int\frac{dk^{4}}{(2\pi)^{4}}\frac{k^{2}}{(k^{2}-\Sigma)^{n}}\xrightarrow{}(-1)^{n+1}i\int\frac{d^{4}k_{E}}{(2\pi)^{4}}\frac{k_{E}^{2}}{(k_{E}^{2}+\Sigma)^{n}}=(-1)^{n+1}i\frac{1}{(4\pi)^{d/2}}\frac{d}{2}\frac{\Gamma\left(n-\frac{d}{2}-1\right)}{\Gamma(n)}\left(\frac{1}{\Sigma}\right)^{n-\frac{d}{2}-1}.$

After changing the measure of our integrals from 4 to the arbitrary dimension
$d$, the integrals are evaluated as follows

$\displaystyle f^{\mu\nu}(\Delta)$
$\displaystyle=i\mu^{4-d}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{k^{\mu}}{k^{2}-m^{2}+i\epsilon}\frac{(k-\Delta)^{\nu}}{(k-\Delta)^{2}-m^{2}+i\epsilon}$
$\displaystyle=i\mu^{4-d}\int\frac{d^{d}k}{(2\pi)^{d}}k^{\mu}(k-\Delta)^{\nu}\int_{0}^{1}dx\frac{1}{(k^{2}-m^{2}+i\epsilon+(\Delta^{2}-2k\cdot\Delta)x)^{2}}\quad$
$\displaystyle=i\mu^{4-d}\int\int_{0}^{1}\frac{d^{d}kdx}{(2\pi)^{d}}(k^{\mu}k^{\nu}-\Delta^{\mu}\Delta^{\nu}x+\Delta^{\mu}\Delta^{\nu}x^{2})\frac{1}{(k^{2}-\overline{m}^{2}(x,t)+i\epsilon)^{2}}$
$\displaystyle=-\mu^{4-d}\int_{0}^{1}dx\biggl{(}-g^{\mu\nu}\frac{\Gamma(1-\frac{d}{2})}{2\overline{m}(x,t)^{2-d}(4\pi)^{\frac{d}{2}}}-\Delta^{\mu}\Delta^{\nu}(1-x)x\frac{\Gamma(2-\frac{d}{2})}{\overline{m}(x,t)^{4-d}(4\pi)^{\frac{d}{2}}}\biggr{)}$
$\displaystyle g(\Delta)$
$\displaystyle=-i\frac{\mu^{4-d}}{2}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{k_{\mu}}{k^{2}-m^{2}+i\epsilon}\frac{(k-\Delta)^{\mu}}{(k-\Delta)^{2}-m^{2}+i\epsilon}$
$\displaystyle=-i\frac{\mu^{4-d}}{2}\int\frac{d^{d}k}{(2\pi)^{d}}\int_{0}^{1}dx\frac{(k+\Delta
x)_{\mu}(k+\Delta(x-1))^{\mu}}{(k^{2}-\overline{m}^{2}(x,t)+i\epsilon)^{2}}$
$\displaystyle=-i\frac{\mu^{4-d}}{2}\biggl{[}\int\frac{d^{d}k}{(2\pi)^{d}}\int_{0}^{1}dx\frac{k^{2}}{(k^{2}-\overline{m}^{2}(x,t)+i\epsilon)^{2}}-\int\frac{d^{d}k}{(2\pi)^{d}}\int_{0}^{1}dx\frac{\Delta^{2}(1-x)x}{(k^{2}-\overline{m}^{2}(x,t)+i\epsilon)^{2}}\biggr{]}$
$\displaystyle=\int_{0}^{1}dx\frac{\mu^{4-d}}{2}\biggl{(}-\frac{\Gamma(1-\frac{d}{2})d}{2\overline{m}(x,t)^{2-d}(4\pi)^{\frac{d}{2}}}-\Delta^{2}(1-x)x\frac{\Gamma(2-\frac{d}{2})}{\overline{m}(x,t)^{4-d}(4\pi)^{\frac{d}{2}}}\biggr{)}$
$\displaystyle h(\Delta)$
$\displaystyle=\frac{i\mu^{4-d}}{2}\int\frac{d^{d}k}{(2\pi)^{d}}\int_{0}^{1}dx\frac{1}{(k^{2}-\overline{m}^{2}(x,t)+i\epsilon)^{2}}=-\frac{\mu^{4-d}}{2(4\pi)^{\frac{d}{2}}}\int_{0}^{1}dx\overline{m}(x,t)^{d-4}\Gamma(2-\frac{d}{2})$
$\displaystyle j(\Delta)$
$\displaystyle=\frac{\mu^{4-d}}{2}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{i}{k^{2}-m^{2}+i\epsilon}=\frac{\mu^{4-d}}{2(4\pi)^{\frac{d}{2}}}\frac{\Gamma(1-\frac{d}{2})}{(m^{2})^{(1-\frac{d}{2})}}.$

Using $d=4-\epsilon$ and combining the terms gives

$\displaystyle\lambda\left(f^{\mu\nu}(\Delta)+g^{\mu\nu}\left(g(\Delta)+m^{2}h(\Delta)+j(\Delta)\right)\right)=$
$\displaystyle=\int_{0}^{1}dx\frac{\lambda\mu^{\epsilon}}{(2\pi)^{4-\epsilon}}(\Delta^{\mu}\Delta^{\nu}-\Delta^{2}g^{\mu\nu})(1-x)x\pi^{2-\frac{\epsilon}{2}}\frac{\Gamma(\frac{\epsilon}{2})}{\overline{m}(x,t)^{\epsilon}}+g^{\mu\nu}\frac{\lambda\mu^{\epsilon}}{2(4\pi)^{2-\frac{\epsilon}{2}}}\frac{\Gamma(\frac{\epsilon}{2}-1)}{m^{\epsilon-2}}$
$\displaystyle\xrightarrow[]{\epsilon\to
0}\frac{\lambda}{(4\pi)^{2}}\int_{0}^{1}dx(\Delta^{\mu}\Delta^{\nu}-\Delta^{2}g^{\mu\nu})(1-x)x\biggl{(}\frac{2}{\epsilon}+\ln{\frac{\mu^{2}}{\overline{m}^{2}(x,t)}}\biggr{)}+\frac{g^{\mu\nu}m^{2}\lambda}{2(4\pi)^{2}}\biggl{(}\frac{2}{\epsilon}-\gamma+\log{\frac{4\pi\mu^{2}}{m^{2}}}\biggr{)}\,.$

Thus, the canonical EMT is

$\displaystyle\bra{\Omega}T(a_{p^{\prime}}T_{\text{can}}^{\mu\nu}(0)a_{p}^{\dagger})\ket{\Omega}$
$\displaystyle=\frac{P^{\mu}P^{\nu}}{2}-\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{6}+\frac{\lambda}{(4\pi)^{2}}\int_{0}^{1}dx(\Delta^{\mu}\Delta^{\nu}-\Delta^{2}g^{\mu\nu})(1-x)x\biggl{(}\frac{2}{\epsilon}+\ln{\frac{\mu^{2}}{\overline{m}^{2}(x,t)}}\biggr{)}$
$\displaystyle\hskip
122.34692pt+\frac{\lambda}{(4\pi)^{2}}\;\frac{g^{\mu\nu}m^{2}}{2}\biggl{(}\frac{2}{\epsilon}-\gamma+\log{\frac{4\pi\mu^{2}}{m^{2}}}\biggr{)}+g^{\mu\nu}\delta_{m}$
$\displaystyle=\frac{P^{\mu}P^{\nu}}{2}-\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{2}+\frac{\lambda}{(4\pi)^{2}}\int_{0}^{1}dx(\Delta^{\mu}\Delta^{\nu}-\Delta^{2}g^{\mu\nu})(1-x)x\biggl{(}\frac{2}{\epsilon}+\ln{\frac{\mu^{2}}{\overline{m}^{2}(x,t)}}\biggr{)}.$
(55)

From this result we read off the contribution from the canonical EMT to the
EMT form factors in dimensional regularization quoted in Eq. (23).

###  A.3 Evaluation of the improvement term in dimensional regularization

The improvement term consists of two terms which we will calculate each term
individually

$\displaystyle-h\bra{\Omega}T(a_{p^{\prime}}(\partial^{\mu}\partial^{\nu}-g^{\mu\nu}\Box)\Phi^{2}a_{p}^{\dagger})\ket{\Omega}=-h\biggl{(}\bra{\Omega}T(a_{p^{\prime}}\partial^{\mu}\partial^{\nu}\Phi^{2}a_{p}^{\dagger})\ket{\Omega}-g^{\mu\nu}\bra{\Omega}T(a_{p^{\prime}}\Box\Phi^{2}a_{p}^{\dagger})\ket{\Omega}\biggr{)}.$

Note that $h=\frac{1}{4}\frac{(N-2)}{(N-1)}=\frac{1}{6}$, with $N=4$ being the
number of space-time dimensions. It is important to iterate that $h$ is just a
constant that we set to $N=4$ based on the number of dimensions we are working
in. When we use dimensional regularization, we only assume the dimension
$d=4-\epsilon$ of the integral itself varies rather than the dimension $N=4$
of the manifold (physical space) we are working on, see the discussion at the
end of Sec. 3.2.

For the first term we obtain

$\displaystyle-h\bra{\Omega}T(a_{p^{\prime}}\partial^{\mu}\partial^{\nu}\Phi(x)\Phi(x)a_{p}^{\dagger})\ket{\Omega}$
$\displaystyle=-h\partial^{\mu}\partial^{\nu}\biggl{[}2e^{i\Delta\cdot
x}-i\mu^{4-d}\lambda\int d^{4}ye^{i\Delta\cdot
y}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{ie^{-ik\cdot(x-y)}}{k^{2}-m^{2}+i\epsilon}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{ie^{-iq\cdot(y-x)}}{q^{2}-m^{2}+i\epsilon}\biggr{]}$
$\displaystyle=\biggl{[}2h\Delta^{\mu}\Delta^{\nu}+L^{\mu\nu}(\Delta)\biggr{]}e^{i\Delta\cdot
x},$

where

$\displaystyle L^{\mu\nu}(\Delta)$
$\displaystyle=hi\mu^{4-d}\lambda\int\frac{d^{d}k}{(2\pi)^{d}}\frac{\Delta^{\mu}\Delta^{\nu}}{k^{2}-m^{2}+i\epsilon}\frac{1}{(\Delta+k)^{2}-m^{2}+i\epsilon}$
$\displaystyle=hi\mu^{4-d}\lambda\int\frac{d^{d}k\Delta^{\mu}\Delta^{\nu}}{(2\pi)^{d}}\int_{0}^{1}dx\frac{1}{((k+\Delta
x)^{2}-\overline{m}^{2}(x,t)+i\epsilon)^{2}}$
$\displaystyle=h\mu^{4-d}i\lambda\int\frac{d^{d}k\Delta^{\mu}\Delta^{\nu}}{(2\pi)^{d}}\int_{0}^{1}dx\frac{1}{(k^{2}-\overline{m}^{2}(x,t)+i\epsilon)^{2}}$
$\displaystyle=-h\mu^{4-d}\lambda\int_{0}^{1}dx\frac{\Delta^{\mu}\Delta^{\nu}}{(4\pi)^{\frac{d}{2}}}\Gamma\bigl{(}2-\frac{d}{2}\bigr{)}\overline{m}(x,t)^{(d-4)}.$

The second term is just the trace of the first term multiplied by
$g^{\mu\nu}$, i.e.

$\displaystyle
hg^{\mu\nu}\bra{\Omega}T(a_{p^{\prime}}\Box\Phi^{2}a_{p}^{\dagger})\ket{\Omega}=-g^{\mu\nu}\biggl{[}2h\Delta^{2}+L^{\rho}_{\;\rho}(\Delta)\biggr{]}e^{i\Delta\cdot
x}$

Combining terms and setting $d=4-\epsilon$ gives the result

$\displaystyle-h\bra{\Omega}T(a_{p^{\prime}}(\partial^{\mu}\partial^{\nu}-g^{\mu\nu}\Box)\Phi^{2}a_{p}^{\dagger})\ket{\Omega}=2h(\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2})\biggl{[}1-\frac{\mu^{\epsilon}\lambda}{2(4\pi)^{\frac{\epsilon}{2}}}\,\Gamma\bigl{(}2-\frac{d}{2}\bigr{)}\int_{0}^{1}dx\,\overline{m}(x,t)^{-\epsilon}\biggr{]}\,.$

Expanding this expression in $\epsilon$ yields the result for the improvement
term contribution to the EMT quoted in Eq. (25).

###  A.4 Total EMT in dimensional regularization

Combining the results from App. A.2 and App. A.3 and expanding in $\epsilon$
it follows,

$\displaystyle\bra{\Omega}T(a_{p^{\prime}}(T^{\mu\nu}(0)_{\rm
can}+\Theta^{\mu\nu}_{\rm
imp}(0))a_{p}^{\dagger})\ket{\Omega}=\frac{P^{\mu}P^{\nu}}{2}-\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{2}+2h(\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2})+$
$\displaystyle\hskip
56.9055pt+\frac{\lambda}{(4\pi)^{2}}\int_{0}^{1}dx(\Delta^{\mu}\Delta^{\nu}-\Delta^{2}g^{\mu\nu})\biggl{[}(1-x)x\biggl{(}\frac{2}{\epsilon}+\log{\frac{\mu^{2}}{\overline{m}^{2}(x,t)}}\biggr{)}-\frac{1}{6}\biggl{(}\frac{2}{\epsilon}+\log{\frac{\mu^{2}}{\overline{m}^{2}(x,t)}}\biggr{)}\biggr{]}$
$\displaystyle=\frac{P^{\mu}P^{\nu}}{2}-\frac{1}{3}\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{2}+\frac{\lambda}{(4\pi)^{2}}\int_{0}^{1}dx(\Delta^{\mu}\Delta^{\nu}-\Delta^{2}g^{\mu\nu})\biggl{(}\frac{1}{6}-(1-x)x\biggr{)}\log{\frac{m^{2}\bigl{(}1-\frac{\Delta^{2}}{m^{2}}x(1-x)\bigr{)}}{\mu^{2}}}$
$\displaystyle=\frac{P^{\mu}P^{\nu}}{2}-\frac{1}{3}\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{2}+\frac{\lambda}{(4\pi)^{2}}\int_{0}^{1}dx(\Delta^{\mu}\Delta^{\nu}-\Delta^{2}g^{\mu\nu})\biggl{(}\frac{1}{6}-(1-x)x\biggr{)}\log{\biggl{(}1-\frac{\Delta^{2}}{m^{2}}x(1-x)\biggr{)}}.$

Here it should be noted that explicit dependence on $\mu$ drops out since
$\mu$ can be separated from the original logarithm and
$\int_{0}^{1}dx\bigl{(}\frac{1}{6}-x(1-x)\bigr{)}\log{\frac{m^{2}}{\mu^{2}}}=0$.
The vanishing of the integral $\int_{0}^{1}\bigl{(}\frac{1}{6}-x(1-x)\bigr{)}$
ensures also the finiteness of the trace which in turn ensures the finiteness
of the EMT [24].

##  Appendix B Pauli Villars Regularization

In this Appendix we present details of the calculation in the Pauli-Villars
regularization. It is convenient to define

$\displaystyle M_{1}^{2}(\Delta)=\Lambda_{1}^{2}-\Delta^{2}x(1-x),\quad
M_{2}^{2}(\Delta)=\Lambda_{2}^{2}-\Delta^{2}x(1-x),$

and for brevity below $\overline{m}=\overline{m}(x,t)$. The results for
$F^{\mu\nu}(\Delta)$, $G^{\mu\nu}(\Delta)$, $H^{\mu\nu}(\Delta)$,
$I^{\mu\nu}(\Delta)$ in Eq. (34) follow from

$\displaystyle\bra{\Omega}T(a_{p^{\prime}}(\partial^{\mu}\Phi\partial^{\nu}\Phi-\partial^{\mu}\Phi_{1}\partial^{\nu}\Phi_{1}-\partial^{\mu}\Phi_{2}\partial^{\nu}\Phi_{2})a_{p}^{\dagger})\ket{\Omega}\phantom{\frac{1}{1}}$
$\displaystyle=\biggl{[}p^{\prime\mu}p^{\nu}+p^{\prime\nu}p^{\mu}+i\lambda\int\frac{d^{4}k}{(2\pi)^{4}}\int_{0}^{1}dx\biggl{(}\frac{g^{\mu\nu}}{4}k^{2}-\Delta^{\mu}\Delta^{\nu}x+\Delta^{\mu}\Delta^{\nu}x^{2}\biggr{)}\times$
$\displaystyle\hskip
28.45274pt\biggl{(}\frac{1}{(k^{2}-\overline{m}^{2}(x,t)+i\epsilon)^{2}}-\frac{a_{1}^{2}}{(k^{2}-M_{1}^{2}(\Delta)+i\epsilon)^{2}}-\frac{a_{2}^{2}}{(k^{2}-M_{2}^{2}(\Delta)+i\epsilon)^{2}}\biggr{)}\biggr{]}$
$\displaystyle\hskip
56.9055pt\xrightarrow[]{\Lambda_{1}^{2},\Lambda_{2}^{2}\to\Lambda^{2}\gg\overline{m}^{2}}\biggl{[}p^{\prime\mu}p^{\nu}+p^{\prime\nu}p^{\mu}+\int_{0}^{1}dx\frac{\lambda}{(4\pi)^{2}}\biggl{(}\frac{g^{\mu\nu}}{2}\overline{m}^{2}\log{\frac{\overline{m}^{2}}{\Lambda^{2}}}+\frac{\Delta^{\mu}\Delta^{\nu}x(1-x)}{2}\log{\frac{\Lambda^{2}}{\overline{m}^{2}}}\biggr{)}\biggr{]}=F^{\mu\nu}(\Delta),$
$\displaystyle-\frac{g^{\mu\nu}}{2}\bra{\Omega}T(a_{p^{\prime}}(\partial^{\mu}\Phi\partial_{\mu}\Phi-\partial^{\mu}\Phi_{1}\partial_{\mu}\Phi_{1}-\partial^{\mu}\Phi_{2}\partial_{\mu}\Phi_{2})a_{p}^{\dagger})\ket{\Omega}=$
$\displaystyle=\biggl{[}g^{\mu\nu}p^{\prime}\cdot p-i\lambda
g^{\mu\nu}\int\frac{d^{4}k}{2(2\pi)^{4}}\int_{0}^{1}dx(k^{2}-\Delta^{2}x(1-x))\times$
$\displaystyle\hskip
28.45274pt\biggl{(}\frac{1}{(k^{2}-\overline{m}^{2}(x,t)+i\epsilon)^{2}}-\frac{a_{1}^{2}}{(k^{2}-M_{1}^{2}(\Delta)+i\epsilon)^{2}}-\frac{a_{2}^{2}}{(k^{2}-M_{2}^{2}(\Delta)+i\epsilon)^{2}}\biggr{)}\biggr{]}$
$\displaystyle\hskip
56.9055pt\xrightarrow[]{\Lambda_{1}^{2},\Lambda_{2}^{2}\to\Lambda^{2}\gg\overline{m}^{2}}-\biggl{[}g^{\mu\nu}p^{\prime}\cdot
p+\int_{0}^{1}dx\frac{\lambda}{(4\pi)^{2}}\biggl{(}g^{\mu\nu}\overline{m}^{2}\log{\frac{\overline{m}^{2}}{\Lambda^{2}}}+\frac{g^{\mu\nu}\Delta^{2}x(1-x)}{2}\log{\frac{\Lambda^{2}}{\overline{m}^{2}}}\biggr{)}\biggr{]}=G^{\mu\nu}(\Delta)\,,$
$\displaystyle\frac{g^{\mu\nu}}{2}\bra{\Omega}T(a_{p^{\prime}}((m^{2}+\delta_{m})\Phi\Phi-\Lambda_{1}^{2}\Phi_{1}\Phi_{1}-\Lambda_{2}^{2}\Phi_{2}\Phi_{2})a_{p}^{\dagger})\ket{\Omega}]$
$\displaystyle=g^{\mu\nu}\biggl{[}m^{2}+\delta_{m}-\frac{m^{2}}{2}i\lambda\int_{0}^{1}\frac{d^{4}k}{(4\pi)^{2}}\int
dx\biggl{(}\frac{1}{(k^{2}-\overline{m}^{2}(x,t)+i\epsilon)^{2}}-\frac{a_{1}^{2}}{(k^{2}-M_{1}^{2}(\Delta)+i\epsilon)^{2}}-\frac{a_{2}^{2}}{(k^{2}-M_{2}^{2}(\Delta)+i\epsilon)^{2}}\biggr{)}\biggr{]}$
$\displaystyle\hskip
56.9055pt\xrightarrow[]{\Lambda_{1}^{2},\Lambda_{2}^{2}\to\Lambda^{2}\gg\overline{m}^{2}}g^{\mu\nu}\biggl{(}m^{2}+\delta_{m}-\int_{0}^{1}dx\frac{\lambda}{2(4\pi)^{2}}m^{2}\log{\frac{\Lambda^{2}}{\overline{m}^{2}}}\biggr{)}=H^{\mu\nu}(\Delta)+g^{\mu\nu}\delta_{m},$
$\displaystyle\frac{\lambda}{4!}\bra{\Omega}T(a_{p^{\prime}}(\Phi+a_{1}\Phi_{1}+a_{2}\Phi_{2})^{4}a_{p}^{\dagger}e^{-i\int\frac{\lambda}{4!}\Phi^{4}d^{4}x})\ket{\Omega}=\frac{\lambda}{2(4\pi)^{2}}\biggl{(}\Lambda^{2}+m^{2}\log{\frac{\Lambda^{2}}{m^{2}}}\biggr{)}.$

Similarly, for the improvement term it follows

$\displaystyle-h\bra{\Omega}T(a_{p^{\prime}}(\partial^{\mu}\partial^{\nu}\Phi\Phi-\partial^{\mu}\partial^{\nu}\Phi_{1}\Phi_{1}-\partial^{\mu}\partial^{\nu}\Phi_{2}\Phi_{2})a_{p}^{\dagger})\ket{\Omega}+hg^{\mu\nu}\bra{\Omega}T(a_{p^{\prime}}(\Box\Phi^{2}-\Box\Phi_{1}^{2}-\Box\Phi_{2}^{2})a_{p}^{\dagger})\ket{\Omega}$
$\displaystyle=(\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2})\biggl{(}2h-h\frac{\lambda}{(4\pi)^{2}}\log{\frac{\Lambda^{2}}{\overline{m}^{2}}}\biggr{)}=I^{\mu\nu}(\Delta).$

Combining the results yields for the total EMT

$\displaystyle\bra{\vec{p^{\prime}}}(T^{\mu\nu}(0)_{\rm
can}+\Theta^{\mu\nu}(0)_{\rm
imp})\ket{\vec{p}}=\frac{P^{\mu}P^{\nu}}{2}-\frac{1}{3}\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{2}$
$\displaystyle\hskip
28.45274pt+\frac{\lambda}{(4\pi)^{2}}\int_{0}^{1}dx\biggl{(}\frac{g^{\mu\nu}}{2}\bar{m}^{2}\log{\frac{\bar{m}^{2}}{\Lambda^{2}}}+\Delta^{\mu}\Delta^{\nu}x(1-x)\log{\frac{\Lambda^{2}}{\bar{m}^{2}}}-g^{\mu\nu}\bar{m}^{2}\log{\frac{\bar{m}^{2}}{\Lambda^{2}}}-\frac{g^{\mu\nu}\Delta^{2}x(1-x)}{2}\log{\frac{\Lambda^{2}}{\bar{m}^{2}}}$
$\displaystyle\hskip
28.45274pt+g^{\mu\nu}\biggl{(}\delta_{m}-\frac{1}{2}m^{2}\log{\frac{\Lambda^{2}}{\bar{m}^{2}}}\biggr{)}+\frac{1}{2}\biggl{(}\Lambda^{2}+m^{2}\log{\frac{\Lambda^{2}}{m^{2}}}\biggr{)}-h(\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2})\log{\frac{\Lambda^{2}}{\bar{m}^{2}}}\biggr{)}$
$\displaystyle=\frac{P^{\mu}P^{\nu}}{2}-\frac{1}{3}\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{2}+(\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2})\frac{\lambda}{(4\pi)^{2}}\int_{0}^{1}dx\biggl{(}(1-x)x-\frac{1}{6}\biggr{)}\log{\frac{\Lambda^{2}(1-\frac{\Delta^{2}x(1-x)}{\Lambda^{2}})}{m^{2}(1-\frac{\Delta^{2}}{m^{2}}x(1-x))}}$
$\displaystyle\xrightarrow[]{\Lambda\to\infty}\frac{P^{\mu}P^{\nu}}{2}-\frac{1}{3}\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{2}+(\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2})\frac{\lambda}{(4\pi)^{2}}\int_{0}^{1}dx\biggl{(}-(1-x)x+\frac{1}{6}\biggr{)}\log{\biggl{(}1-\frac{\Delta^{2}}{m^{2}}x(1-x)\biggr{)}},$

where in the last term, before taking $\Lambda\to\infty$, we made use of the
fact that the term proportional to $\log[\frac{\Lambda^{2}}{m^{2}}]$ can be
separated from the main logarithm and drops out due to
$\int_{0}^{1}dx\bigl{(}\frac{1}{6}-x(1-x)\bigr{)}\log[\frac{\Lambda^{2}}{m^{2}}]=0$.
We recall that the vanishing of
$\int_{0}^{1}dx\bigl{(}\frac{1}{6}-x(1-x)\bigr{)}$ ensures the finiteness of
the EMT [24], see the discussion at the end of App. A.4.

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