aainstitutetext: FZU - Institute of Physics of the Czech Academy of Sciences &
CEICO
Na Slovance 2, 182 21 Prague 8, Czech Republicbbinstitutetext: Institute of
Mathematical Sciences, Homi Bhabha National Institute (HBNI)
IV Cross Road, C. I. T. Campus, Taramani, Chennai 600 113, Tamil Nadu,
Indiaccinstitutetext: Instituto de Física Teórica, Universidade Estadual
Paulista (IFT-UNESP)
R. Dr. Bento Teobaldo Ferraz 271, São Paulo 01140-070, Brazil

# Renormalisation in $\text{T}\overline{\text{T}}$-Deformed (Non-)Integrable
Theories

Subhroneel Chakrabarti b Arkajyoti Manna c and Madhusudhan Raman
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

That the exact quantum S-matrix of $\text{T}\overline{\text{T}}$-deformed
field theories is known has interesting consequences for their perturbative
renormalisation. Recent investigations into the interplay between
renormalisation and integrable deformations have focused on those cases where
the (undeformed) seed theories are integrable. In this paper, we study the
perturbative renormalisation of non-integrable
$\text{T}\overline{\text{T}}$-deformed field theories, focusing on a theory of
scalars with a quartic interaction. We show that in this theory, the $4$-point
$1$-loop S-matrix exhibits features similar to its integrable cousins. We also
carry out a similar analysis for the case of an integrable quantum field
theory comprising a pair of chiral bosons. Our discussion of this latter
theory supplies the first explicit example of a perturbative field theory
computation in the formalism due to Sen.

##  1 Introduction and Methodology

Solvable irrelevant deformations of two-dimensional quantum field theories
such as the $\text{T}\overline{\text{T}}$ deformation (Zamolodchikov:2004ce, ;
Cavaglia:2016oda, ; Smirnov:2016lqw, ) have several novel and interesting
features, that have resulted in the widespread attention they have received
over the last few years. (We refer the reader to Jiang:2019hxb  for a
comprehensive review of recent developments in this field.) The central
property of interest in this paper is the following rather remarkable result:
the exact quantum S-matrix $S_{\lambda}$ of the
$\text{T}\overline{\text{T}}$-deformed theory is related to the quantum
S-matrix $S_{0}$ of the undeformed seed theory as

$S_{\lambda}=S_{\text{CDD}}\,S_{0}\ .$ (1)

Here, $S_{\text{CDD}}$ is a pure phase, known as the Castillejo-Dalitz-Dyson
(CDD) factor, and it captures the effect of $\text{T}\overline{\text{T}}$
deformations on the S-matrix of the seed theory.

Being an exact quantum statement, (LABEL:smatrix) has a profound consequences.
For instance, it implies that if we start with the classical
$\text{T}\overline{\text{T}}$-deformed Lagrangian and renormalise it following
the usual techniques, the S-matrix obtained from the renormalised Lagrangian
must be consistent with (LABEL:smatrix) order-by-order in perturbation theory.
Said differently, knowledge of the exact deformed S-matrix imposes strong
constraints on the renormalisation of the classical deformed theory. This is a
remarkable result, especially considering the irrelevant nature of
$\text{T}\overline{\text{T}}$ deformation.

Following this line of reasoning, it was shown in (Rosenhaus:2019utc, ) that a
$\text{T}\overline{\text{T}}$-deformed free scalar, with or without mass, can
be uniquely renormalised in such a way that amplitudes from renormalised
perturbation theory match (LABEL:smatrix). A similar success story for the
case of $\text{T}\overline{\text{T}}$-deformed free massive fermions was
reported in (Dey:2021jyl, ). In this paper, we extend these studies in two
important ways.

### Chiral and Integrable Theories

First, we study the perturbative renormalisation of a
$\text{T}\overline{\text{T}}$-deformed theory of left- and right-chiral bosons
whose deformed classical Lagrangian was determined and studied in
(Chakrabarti:2020dhv, ). This, like the previously studied cases in the
literature, is another example of an integrable quantum field theory. However,
owing to the fact that these theories can only be defined in Lorentzian
signature, it occupies a rather unusual position in the bestiary of two
dimensional quantum field theories.

Recall that the $\text{T}\overline{\text{T}}$ deformation is only rigorously
defined in Euclidean signature. The easiest way to see this is to observe that
the workhorse of this deformation — the Zamolodchikov factorisation formula —
is derived using Euclidean operator product expansions. In deriving
(LABEL:smatrix) it was implicitly assumed that there is no obstruction to
analytically continuing the (un)deformed Euclidean theory to the corresponding
Lorentzian theory. While this is ordinarily true for most quantum field
theories, in the case of chiral bosons this assumption fails. In
(Chakrabarti:2020dhv, ) it was _assumed_ that the classical deformed
Lagrangian can be determined using the Lorentzian version of the
$\text{T}\overline{\text{T}}$ operator. This requires substantiation. We use
(LABEL:smatrix) as the _definition_ of the $\text{T}\overline{\text{T}}$
deformation of a Lorentzian theory. It is then a non-trivial check to see if
the classical deformed Lagrangian obtained in (Chakrabarti:2020dhv, ) can also
be renormalised such that it matches with the CDD deformation of the S-matrix.
This is one of the goals of this paper.

This example is also interesting in its own right. A truly Lorentz covariant
polynomial action for chiral bosons with chirality imposed off-shell as well
as on-shell was only discovered recently (Sen:2015nph, ; Sen:2019qit, ). Such
chiral boson actions have several unusual features, such as inclusion of an
additional free field with wrong-signed kinetic term which completely
decouples from the dynamics, but not at the level of action. A perturbative
loop-level computation of S-matrix for field theories with these unusual
actions, to the best of our knowledge, has not been done so far.111While it
was explained in (Sen:2015nph, ) how one might go about determining the
Feynman rules for such actions, no explicit perturbative calculations with
this action were performed. This computation thus should also provide a new
result in the interacting quantum theories of chiral bosons.

### Non-Integrable Theories

Second, we consider the deformations of interacting non-integrable quantum
field theories and their renormalisation. In particular, we consider the case
of a massive scalar with a quartic interaction as our seed theory and consider
the perturbative renormalisation of its $\text{T}\overline{\text{T}}$
deformation.

There are two principal novelties in our choice of system. First, the S-matrix
of the undeformed theory is no longer just the identity matrix, since the seed
theory is interacting. Second, $4$-particle scattering does not fix the
structure of all higher-point scattering since the seed theory is not
integrable. In fact, as one might suspect, for these cases renormalisability
should run into serious problems given the irrelevant nature of the
deformation and the absence of constraints imposed by quantum integrability on
the S-matrix. Nonetheless, we show by an explicit $1$-loop computation that
due to low dimensionality, the $2\rightarrow 2$ scattering matrix at $1$-loop
uniquely fixes the renormalisation.

As observed by (Rosenhaus:2019utc, ) and (Dey:2021jyl, ), the classical
deformed Lagrangian (for massive free bosons or fermions) has only two mass
scales, viz. mass $m$ and $\text{T}\overline{\text{T}}$ coupling $\lambda$.
However, the quantum renormalised Lagrangian which reproduces the expected
S-matrix turns out to have multiple scales. Schematically, consider two
interaction terms in the classical deformed Lagrangian

$L_{\text{class.}}\subset\dots+\lambda\mathcal{O}_{1}+\lambda\mathcal{O}_{2}+\dots$
(2)

In the renormalised theory, these two terms become

$L_{\text{quant.}}\subset\dots+\alpha\mathcal{O}_{1}+\beta\mathcal{O}_{2}+\dots$
(3)

where

$\displaystyle\alpha$
$\displaystyle=\lambda+\lambda^{2}f_{1}(\lambda,m,\Lambda)\ ,$ (4)
$\displaystyle\beta$
$\displaystyle=\lambda+\lambda^{2}f_{2}(\lambda,m,\Lambda)\ ,$

with $\Lambda$ being the cut-off scale. This is certainly a novel feature: the
quantum theory sees the emergence of different couplings, all of which flow to
the same classical value, giving the impression that the classical theory has
a single coupling.

The mass term plays an important role here. Without the mass term, we only see
a renormalisation of the $\text{T}\overline{\text{T}}$ coupling $\lambda$, as
expected in any quantum field theory. The presence of the mass term allows for
the possibility of a new dimensionless parameter $m\lambda^{2}$ which is
crucial for the emergence of the new coupling. Similarly, if one has
additional scales in the undeformed action, like a coupling $\lambda_{1}$ for
a quartic interaction, then one can form additional dimensionless parameters.
New couplings emerging due to quantum effects ought to reflect this fact. This
is precisely what we find by explicit computation.

The paper is organised as follows. We first discuss the (integrable) case of
chiral bosons in Section 2. After this, we move on to the (non-integrable)
case of $\phi^{4}$ theory in Section 3. We conclude in Section 4 with some
comments and outline some immediate open questions. A summary of various
notations and conventions used in this paper can be found in Appendix A. The
Feynman rules and their derivation for the theory of chiral bosons in Sen’s
formalism are collected in Appendix B. Many of the computations in the text
are rather straightforward, but involve a number of integrals. To avoid
clutter in the main text, we have collected all of the integrals used in
Appendix C.

##  2 Renormalisation of Interacting Chiral Bosons

###  2.1 Free Chiral Bosons

The action for a free chiral boson in Sen’s formalism Sen:2015nph ;
Sen:2019qit  is 222Other known approaches to chiral bosons are
(Floreanini:1987as, ; Sonnenschein:1988ug, ; Pasti:1996vs, ). For advantages
of Sen’s formalism over this other approaches we refer to the original papers
(Sen:2015nph, ; Sen:2019qit, ) (also see (Andriolo:2020ykk, ;
Chakrabarti:2020dhv, )). Some recent works which have used Sen’s formalism to
great effect for self-dual field strengths in higher dimensions are
Andriolo:2020ykk ; Andriolo:2021gen ; Gustavsson:2020ugb .

$S=\int\left[\frac{1}{2}\text{d}\phi\wedge\star\text{d}\phi-2A\wedge\text{d}\phi\right]\
.$ (5)

The field $A_{\mu}$ is self-dual 1-form, i.e. it satisfies the condition
$A^{\mu}=\epsilon^{\mu\nu}A_{\nu}$ and thereby has a single independent
component $A_{1}=-A_{0}=a$. The case of an anti-self-dual $1$-form is arrived
at by making the replacement $A\rightarrow-B$ in the action, and anti-self-
duality implies $B_{0}=B_{1}=b$. In the interest of brevity, we will only
discuss the the case of self-dual 1-forms, and the reader should note that,
_mutatis mutandis_ , what we say here goes through for the case of anti-self-
dual 1-forms as well.

It is natural to expect that chiral theories cannot support mass terms. This
expectation is correct: a term of the form $A^{2}$ is identically zero, since
$A^{2}=\epsilon^{\mu\nu}A_{\mu}A_{\nu}=0$ by antisymmetry. The same, of
course, is true of anti-self-dual gauge fields.

The equations of motion are

$\displaystyle\delta_{\phi}S=0$
$\displaystyle\Rightarrow\quad\text{d}\left(\star\text{d}\phi+\text{d}\phi+2A\right)=0\
,$ (6) $\displaystyle\delta_{A}S=0$
$\displaystyle\Rightarrow\quad\left(\star\text{d}\phi-\text{d}\phi\right)=0\
.$

The free self-dual form $(\star\text{d}\phi+\text{d}\phi+2A)$ was shown in
Sen:2019qit  to decouple from the physical dynamics via a Hamiltonian analysis
of the constrained system. The physical field in this model is $A$,
corresponding to a self-dual $1$-form (i.e. a chiral boson) which satisfies a
free equation of motion $\text{d}A=0$.

Any classical solutions to this equation of motion can be parametrised via a
mode expansion

$A_{\mu}(x)=\int\frac{\text{d}k}{2\pi}\frac{1}{2\omega}\left(\varepsilon_{\mu}\,\alpha(k)\,e^{ik\cdot
x}+\text{c.c.}\right)\ ,$ (7)

for some choice of coefficients $\alpha(k)$ provided that

$k_{\mu}\varepsilon^{\mu}=0\ .$ (8)

This condition is a consequence of self-duality and the equations of motion.
The choice $\varepsilon_{\mu}\propto k_{\mu}$ solves the above constraint,
since $k^{2}=0$ for massless particles. The $\varepsilon_{\mu}$ are similar to
polarisation vectors encountered when studying gauge fields, and this analogy
is occasionally profitable; for example, when considering scattering
processes, we must associate factors of $\varepsilon_{\mu}$ to external lines.

Integrating by parts and working in momentum space via the two-dimensional
Fourier transform, we have

$S=\frac{1}{2}\int\frac{\mathrm{d}^{2}k}{(2\pi)^{2}}\,\left[\tilde{\phi}(k)\,k^{2}\,\tilde{\phi}(-k)-4i\tilde{A}^{\mu}(k)\,k_{\mu}\,\tilde{\phi}(-k)\right]\
,$ (9)

This is quadratic in the auxiliary field $\tilde{\phi}$ and on completing the
squares (in a procedure that is fully analogous to (Sen:2015nph, , Sec. 7)
which discusses the case of the self-dual 5-form in Type IIB supergravity) we
can derive an expression for the propagator of the self-dual gauge field.
Define

$\tilde{\varphi}(k)=\tilde{\phi}(k)-2i\tilde{A}^{\mu}(k)\frac{k_{\mu}}{k^{2}}\
,$ (10)

in terms of which the above action can be written as

$S=\frac{1}{2}\int\frac{\text{d}^{2}k}{(2\pi)^{2}}\left[\tilde{\varphi}(k)\,k^{2}\,\tilde{\varphi}(-k)-4\tilde{A}^{\mu}(k)\,\frac{k_{\mu}k^{\nu}}{k^{2}}\,\tilde{A}_{\nu}(-k)\right]\
.$ (11)

The term quadratic in $\tilde{\varphi}$ decouples and we are left with the
action for physical fields

$S=\int\frac{\text{d}^{2}k}{(2\pi)^{2}}\left[\tilde{A}^{\mu}(k)\,\left(-2\frac{k_{\mu}k^{\nu}}{k^{2}}\right)\,\tilde{A}_{\nu}(-k)\right]\
,$ (12)

from which we can read off the free propagator of the self-dual gauge field.
As pointed out in Sen:2015nph , up to a numerical factor this is precisely the
kinetic term for self-dual $1$-forms as discussed in Alvarez-Gaume:1983ihn .

It is useful to keep in mind that the field $A_{\mu}$ is self-dual off-shell;
to make self-duality manifest, we rewrite the above expression as

$S=\int\frac{\text{d}^{2}k}{(2\pi)^{2}}\left[\left(\frac{\tilde{A}^{\mu}(k)+\epsilon^{\mu\rho}\tilde{A}_{\rho}(k)}{2}\right)\left(-2\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\left(\frac{\tilde{A}^{\nu}(-k)+\epsilon^{\nu\sigma}\tilde{A}_{\sigma}(-k)}{2}\right)\right]\
,$ (13)

and on introducing the chiral projection operator

$\mathcal{P}_{\pm}^{\mu\nu}=\frac{1}{2}\left(\eta^{\mu\nu}\pm\epsilon^{\mu\nu}\right)\
,$ (14)

we can write the above expression as

$S=\int\frac{\text{d}^{2}k}{(2\pi)^{2}}\frac{1}{2}\left[\left(\mathcal{P}_{+}^{\mu\rho}\tilde{A}_{\rho}(k)\right)\left(-4\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\left(\mathcal{P}_{+}^{\nu\sigma}\tilde{A}_{\sigma}(-k)\right)\right]\
.$ (15)

The reader will note that this is the analogue of (Sen:2015nph, , eq. (7.6))
for the case of self-dual $1$-form fields, in what is hopefully more
transparent notation. We have also included a factor of $1/2$ in order to
remain consistent with the standard normalisations for field bilinears. We
will see that the presence of the projectors in the above expression is
crucial to ensure chiral propagation. This is easiest seen in lightcone
coordinates.

In terms of the self-dual $1$-forms, the $2$-point functions relevant to this
system are (as usual) given by $i$ times the inverse of the kinetic term.
Componently, this means:

$\left\langle\tilde{A}_{-}(k)\tilde{A}_{-}(-k)\right\rangle=\frac{-i}{4}\frac{k_{-}^{2}}{k^{2}}\
,$ (16)

and all other correlators identically vanish due to the projectors. A similar
expression is true for anti-self-dual gauge fields. Although elementary, this
brief exercise tells us that we really are working with chiral fields.

###  2.2 Relation to Free Scalars

We have seen that Sen’s formalism successfully reproduces the $2$-point
functions of free chiral theories. It is useful, now, to consider the question
of interactions. In particular, we’d like to explore the extent to which
interacting chiral theories are related to the more familiar subject of
interacting non-chiral theories.

####  2.2.1 Interacting Chiral Fields

Consider a field of the form

$S_{\mu}=A_{\mu}+B_{\mu}\ ,$ (17)

where $A_{\mu}$ and $B_{\mu}$ are self- and anti-self-dual $1$-form fields of
the kind we encountered in the previous section. Then, in terms of the
projectors we introduced in the previous section, we have

$\displaystyle\mathcal{P}^{\mu\nu}_{+}S_{\nu}=A_{\nu}\ ,$ (18)
$\displaystyle\mathcal{P}^{\mu\nu}_{-}S_{\nu}=B_{\nu}\ .$

However, the field $S_{\mu}$ itself has no definite chirality. Let us further
define the dual of $S_{\mu}$ to be $T^{\mu}=\epsilon^{\mu\nu}S_{\nu}$; it is
easy to see that

$T^{\mu}=A^{\mu}-B^{\mu}\ ,$ (19)

on using (anti-)self-duality. The pair of fields $(A,B)$ are chiral, so we
will refer to the pair of fields $(S,T)$ as non-chiral.

Now, consider a theory of free left- and right-chiral fields. In momentum
space, their kinetic terms would ordinarily be written in terms of the chiral
pair as

$\int\frac{\mathrm{d}^{2}k}{(2\pi)^{2}}\left[\tilde{A}^{\mu}(k)\left(-2\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\tilde{A}^{\nu}(-k)+\tilde{B}^{\mu}(k)\left(-2\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\tilde{B}^{\nu}(-k)\right]\
,$ (20)

but in terms of the non-chiral pair, we can rewrite this as

$\int\frac{\mathrm{d}^{2}k}{(2\pi)^{2}}\left[\tilde{S}^{\mu}(k)\left(-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\tilde{S}^{\nu}(-k)+\tilde{T}^{\mu}(k)\left(-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\tilde{T}^{\nu}(-k)\right]\
.$ (21)

At this stage, all we have is a simple rewriting. Observe, however, that in
terms of the fields $S$ and $T$ any interaction that couples left- and right-
chiral fields can be written as

$A\cdot B=\eta^{\mu\nu}A_{\mu}B_{\nu}=\frac{1}{2}S^{2}=-\frac{1}{2}T^{2}\ ,$
(22)

and more generally, for $g\geq 2$

$(A\cdot B)^{g}=\frac{1}{2^{g}}(S\cdot S)^{g}\ .$ (23)

We could have chosen to write this in terms of the field $T$ as well — this
choice is equally acceptable. We conclude from this discussion that at the
level of the action, we can write any theory of (equal numbers of) interacting
left- and right-chiral bosons as a theory of self-interacting scalars.

The relevance of this to our problem is as follows: from eq. 23 the leading
interaction term introduced by the $\text{T}\overline{\text{T}}$ flow we
studied in Chakrabarti:2020dhv  can be written as

$\frac{\lambda}{4}(A\cdot B)^{2}=\frac{\lambda}{16}(S\cdot S)^{2}\ .$ (24)

Further, in order to match normalisations for the kinetic term we must rescale
$S\rightarrow S/\sqrt{2}$. So the final Lagrangian in momentum space is

$\int\frac{\mathrm{d}^{2}k}{(2\pi)^{2}}\left[\frac{1}{2}\tilde{S}^{\mu}(k)\left(-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\tilde{S}^{\nu}(-k)+\frac{\lambda}{64}(\tilde{S}_{\mu}\tilde{S}^{\mu})^{2}\right]\
.$ (25)

At this point, one can derive the Feynman rules for this theory and do a
standard textbook field theory computation to check agreement with the CDD
factor. It turns out, however, that there is a much more direct way to
establish a match with the CDD factor for this case, which we will now
describe.

####  2.2.2 Ordinary Scalars Rewritten

The action in (LABEL:SLagrangian) is actually a familiar theory written in an
unfamiliar way, and in this section we make this correspondence explicit. It
is important to remember that by definition in (LABEL:SDef) the constituent
field (say) $A$ is more akin to a “field strength”. That is, in the more
standard formulations of chiral bosons, we would say that on-shell, $A$ is a
conserved current. Of course, this perspective sits harmoniously with Sen’s
formalism, since the equation of motion for $A$ is $\mathrm{d}A=0$, which can
be thought of as a conservation equation.

The $\text{T}\overline{\text{T}}$-deformed massless (ordinary) free boson
(say, $\varphi$) action is only a functional of the “field-strength” like
variable $V_{\mu}=\partial_{\mu}\varphi$. One can just as easily consider the
scattering of $V_{\mu}$ directly instead of considering the scattering of
$\varphi$. Of course, the Feynman rules need to be adjusted accordingly, but
the S-matrix, thanks to LSZ reduction, is impervious to any field
redefinition.

The Lagrangian for $\text{T}\overline{\text{T}}$-deformed massless ordinary
boson Cavaglia:2016oda  up to first order in
$\text{T}\overline{\text{T}}$-coupling is

$\displaystyle L$
$\displaystyle=-\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-\frac{\lambda}{4}(\partial_{\mu}\varphi\partial^{\mu}\varphi)^{2}\
,$ (26)
$\displaystyle=-\frac{1}{2}V_{\mu}V^{\mu}-\frac{\lambda}{4}(V_{\mu}V^{\mu})^{2}\
.$

The momentum space Feynman propagator for $V_{\mu}V_{\nu}$, as derived from
the $\varphi\,\varphi$ propagator, is

$\langle V_{\mu}(-k)V_{\nu}(k)\rangle=-i\frac{k_{\mu}k_{\nu}}{k^{2}}\;.$ (27)

This propagator is exactly the same as that of $\tilde{S}_{\mu}$ fields
defined in previous section. Furthermore, the vertex factors as derived from
(LABEL:Lagr_V) matches exactly with the vertex factor derived from
(LABEL:SLagrangian), provided we redefine the
$\text{T}\overline{\text{T}}$-coupling for chiral bosons as
$\lambda\rightarrow\tilde{\lambda}=\lambda/16$.

Therefore, to the leading order in $\text{T}\overline{\text{T}}$-coupling, the
deformed chiral bosons and deformed ordinary boson lead to a QFT with
identical degrees of freedom and identical Feynman rules.333There is a
redefinition of coupling constant, but since the coupling constant is
dimensionful and there are no natural mass scales in the undeformed theory,
this simply corresponds to a shift in choice of some reference scale, which
suggests simply that there is no canonical choice of coupling. The physics is
insensitive to this choice of scale. Therefore, following the result already
obtained for massless ordinary bosons in (Rosenhaus:2019utc, ), it
automatically follows that up to first order, the $4$-point S-matrix of the
$\text{T}\overline{\text{T}}$-deformed chiral boson theory agrees with the CDD
factor. Of course, the CDD factor does not fix the real part of the
$O(\lambda)$ contribution to S-matrix, which just like the case of free
bosons, lead to an expected renormalisation of the
$\text{T}\overline{\text{T}}$ coupling.

At this point, it is perhaps tempting to conclude that the two theories will
continue to be identical to all orders in the coupling. This, however, is not
true. Lorentz invariance and the definition of the field variables guarantee
that all interactions for chiral bosons will always be of the form
$\sim(\tilde{S}_{\mu}\tilde{S}^{\mu})^{n}$. However, the actual structure of
the stress tensor for chiral bosons is very different from the stress tensor
for free bosons. In fact, as explicitly established in (Chakrabarti:2020dhv,
), the closed form action is starkly different from that of the case of
ordinary massless bosons. So at higher orders, the interaction terms will
start to differ in two cases and one would need to resort to Feynman
diagrammatics to compute the S-matrix.

Also, in (Chakrabarti:2020dhv, ) it was shown that one can also find closed-
form $\text{T}\overline{\text{T}}$-deformed actions for arbitrary (but non-
zero) number of left- and right-chiral bosons. In particular, for the case
where there is a chiral asymmetry (i.e. unequal numbers of left- and right-
movers), one cannot re-write the action in terms of variables like $S_{\mu}$
and would have to work with the $A_{\mu}$ and the $B_{\mu}$ fields. We will
not look at these cases in this paper, but for completeness and in the hope
that they can be useful for other researchers looking to do perturbative
computations in Sen’s formalism, we give the Feynman rules in terms of $A$ and
$B$ fields in Appendix B.

The match with the CDD prediction vindicates the assumption made in
(Chakrabarti:2020dhv, ) regarding the viability of
$\text{T}\overline{\text{T}}$ deformation of a purely Lorentzian theory. In
fact, this perhaps suggests that the definition of
$\text{T}\overline{\text{T}}$ deformation in terms of the exact quantum
S-matrix, viz. (LABEL:smatrix), has wider applicability. The results of this
section are perhaps unsurprising given that the undeformed theory was also an
integrable quantum field theory without any mass scale. However, as pointed
out in (Rosenhaus:2019utc, ), the CDD factor derivation holds for any
undeformed theory, integrable or not. For non-integrable QFTs, the undeformed
S-matrix is no longer just identity matrix and there will be particle
production. However, that we are in low dimensions and that the deformed
S-matrix is still given by a CDD phase multiplying the undeformed S-matrix
together suggests that there should still be severe constraints on the
renormalisation of such theories. This is what we turn to next, where we
consider the theory of a massive scalar with quartic interaction as our
undeformed seed theory.

##  3 Renormalisation of $\phi^{4}$ Theory

Consider the Lagrangian for an interacting scalar field theory with a generic
interaction:

$\displaystyle\mathcal{L}=\frac{1}{2}\left(\partial_{\mu}\phi\right)^{2}+V(\phi)\,.$
(28)

The $\text{T}\overline{\text{T}}$-deformed Lagrangian for this seed theory is
Bonelli:2018kik

$\displaystyle\tilde{\mathcal{L}}=-\frac{1}{2\lambda}\frac{1-2\lambda
V}{1-\lambda V}+\frac{1}{2\lambda}\sqrt{\frac{(1-2\lambda V)^{2}}{(1-\lambda
V)^{2}}+2\lambda\frac{\left(\partial_{\mu}\phi\right)^{2}+2V}{1-\lambda
V}}\,,$ (29)

where $\lambda$ is the $\text{T}\overline{\text{T}}$ coupling. In this
section, we consider massive scalar theory with a bare $\phi^{4}$ potential:
$V=\frac{1}{2}m^{2}\phi^{2}+\lambda_{1}\phi^{4}$. Here we assume that the mass
$m$ is physical and therefore does not require any renormalisation. The
relevant Lagrangian for $2\rightarrow 2$ scattering can be obtained by
expanding (LABEL:ttbardeformed) upto quadratic order in both of the couplings

$\displaystyle\mathcal{\tilde{L}}=\frac{1}{2}\left(\partial_{\mu}\phi\right)^{2}$
$\displaystyle+\frac{m^{2}\phi^{2}}{2}+\lambda_{1}\phi^{4}+\frac{1}{4}\lambda
m^{4}\phi^{4}-\frac{\lambda}{4}\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}$
(30)
$\displaystyle+\lambda\lambda_{1}m^{2}\phi^{6}+\lambda^{2}\left(\frac{1}{4}\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{3}+\frac{m^{2}}{8}\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}\phi^{2}+\frac{m^{6}}{8}\phi^{6}\right)\,.$
(31)

It is useful to note that the mass dimensions of the
$\text{T}\overline{\text{T}}$ and bare $\phi^{4}$ couplings are $-2$ and $2$
respectively.

In order to obtain the Feynman rules for the
$\text{T}\overline{\text{T}}$-deformed theory, we have to rewrite the
Lagrangian in (LABEL:barelagrangian) in Lorentzian signature. This can be done
by substituting
$(m^{2},\lambda_{1},\lambda)\rightarrow(-m^{2},-\lambda_{1},-\lambda)$ and we
obtain

$\displaystyle\tilde{\mathcal{L}}_{\text{L}}=-\frac{1}{2}\left(\partial_{\mu}\phi\right)^{2}-$
$\displaystyle\frac{m^{2}\phi^{2}}{2}-\lambda_{1}\phi^{4}-\frac{1}{4}\lambda
m^{4}\phi^{4}+\frac{\lambda}{4}\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}$
(32)
$\displaystyle-\lambda\lambda_{1}m^{2}\phi^{6}+\lambda^{2}\left(\frac{1}{4}\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{3}-\frac{m^{2}}{8}\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}\phi^{2}-\frac{m^{6}}{8}\phi^{6}\right)\,.$
(33)

From this Lagrangian, the Feynman rules can be read off:

$\displaystyle\text{propagator}:$
$\displaystyle\quad\frac{-i}{p^{2}-m^{2}}\,,$ (34)
$\displaystyle\text{4-scalar vertex}:$
$\displaystyle\quad-i\Big{(}\lambda_{1}+\frac{\lambda
m^{4}}{4}\Big{)}+\frac{i\lambda}{4\cdot 3}\Big{[}(p_{1}\cdot p_{2})(p_{3}\cdot
p_{4})+(2\leftrightarrow 3)+(2\leftrightarrow 4)\Big{]}\,.$ (35)

Parametrizing the components of external momenta in terms of individual
rapidity $\theta_{i}$

$\displaystyle p^{0}_{i}=m\cosh\theta_{i}\,,\quad
p^{1}_{i}=-m\sinh\theta_{i}\,,$ (36)

we express the Mandelstam variables, in $(-,+)$ signature for all ingoing
external momenta as follows

$\displaystyle s=(p_{1}+p_{2})^{2}=2m^{2}(1-\cosh\theta_{12})\,,$ (37)
$\displaystyle u=(p_{1}+p_{3})^{2}=2m^{2}(1-\cosh\theta_{13})\,,$ (38)
$\displaystyle t=(p_{1}+p_{4})^{2}=2m^{2}(1-\cosh\theta_{14})\,.$ (39)

Here we define relative rapidities as $\theta_{ij}:=\theta_{i}-\theta_{j}$.
The parametrization (LABEL:thetaparametrization) also implies that various
momentum scalar products can be written as

$\displaystyle p_{1}\cdot p_{2}=-m^{2}\cosh\theta_{12}\,,\quad p_{1}\cdot
p_{4}=-m^{2}\cosh\theta_{14}\,,\quad p_{1}\cdot
p_{3}=-m^{2}\cosh\theta_{13}\,.$ (40)

All other momentum scalar products can be expressed in terms of the above
scalar products.

###  3.1 $2\rightarrow 2$ scattering

In the case of four particle scattering in $(1+1)$ dimensions, one can always
set the spatial component of the momentum vector as

$\displaystyle p_{1}=-p_{3}\,,\quad p_{2}=-p_{4}\,,$ (41)

leading to the temporal component ($p^{0}\equiv E$) to be related as
$E_{1}=\pm E_{3}$ and $E_{2}=\pm E_{4}$. By setting $E_{1}=E_{3}$ and using
$E_{1}^{2}=m^{2}+p_{1}^{2}$, any one of the three Mandelstam variables (say
$u$) can be set to zero

$\displaystyle u=(p_{1}+p_{3})^{2}=2m^{2}+2(-E_{1}E_{3}+p_{1}p_{3})=0\,.$ (42)

In the following analysis, we set $u=0$ (or $\theta_{13}=0$). Then the onshell
condition $s+t=4m^{2}$ for $u=0$, relates the two other relative rapidities as
$\theta_{14}=(i\pi-\theta_{12})$. Therefore the Mandelstam variables become
function of a single relative rapidity factor:

$\displaystyle s=2m^{2}(1-\cosh\theta)\,,\quad u=0\,,\quad
t=2m^{2}(1+\cosh\theta)\,,$ (43)

where we rewrite $\theta_{12}\equiv\theta$ to avoid clutter. At tree level the
4-scalar vertex for the integrable theory can now can be expressed as

$\displaystyle V_{\phi\phi\phi\phi}=-i\left(\lambda_{1}+\frac{\lambda
m^{4}}{4}\right)+\frac{i\lambda m^{4}}{12}\Big{[}2\cosh^{2}\theta+1\Big{]}\,.$
(44)

###  3.2 Tree amplitude

The tree level amplitude (Figure 1) now can be expressed as function of
rapidity

$\displaystyle\mathcal{A}^{\text{tree}}_{4}(\theta)$
$\displaystyle=-i4!\left[\left(\lambda_{1}+\frac{\lambda
m^{4}}{4}\right)-\frac{\lambda}{12}m^{4}(2\cosh^{2}\theta+1)\right]$
$\displaystyle=-24i\lambda_{1}+4i\lambda m^{4}\sinh^{2}\theta\,.$ (45)

This matches with result of (Rosenhaus:2019utc, ) by using a simple
substitution:

$\displaystyle\frac{1}{4}\lambda
m^{4}\longrightarrow\lambda_{1}+\frac{1}{4}\lambda m^{4}\,.$ (46)

Figure 1: Contact scattering diagram for massive $\phi^{4}$ theory at tree
level

###  3.3 From the CDD factor

We can quickly check that the tree level amplitude computed in the previous
section is precisely what one would expect based on the CDD factor. Recall,
The quantum S-matrix is related to the scattering amplitude via the following
relation (Rosenhaus:2019utc, ; Dey:2021jyl, )

$\displaystyle S(\theta)=1+\frac{T}{4m^{2}\sinh\theta}\,.$ (47)

At linear order in $\text{T}\overline{\text{T}}$ coupling this gives

$\displaystyle S_{\lambda}=S_{0}\left(1+i\lambda m^{2}\sinh\theta\right)\,.$
(48)

Therefore the amplitude for the $\text{T}\overline{\text{T}}$-deformed theory
$T_{\lambda}$ is related to the amplitude of the undeformed seed theory
$T_{0}$ as

$\displaystyle T_{\lambda}=T_{0}+i\lambda m^{2}T_{0}\sinh\theta+4i\lambda
m^{4}\sinh^{2}\theta\,.$ (49)

If the bare $\phi^{4}$ coupling is set to zero then $T_{0}=0$, which
corresponds to a seed theory describing free propagation and we recover the
tree amplitude computed in (Rosenhaus:2019utc, ) for the
$\text{T}\overline{\text{T}}$-deformed theory. With bare $\phi^{4}$ coupling
turned on, the amplitude for the undeformed seed theory is
$T_{0}=-24i\lambda_{1}$. In this case, the tree amplitude for the
$\text{T}\overline{\text{T}}$-deformed theory is

$\displaystyle\mathcal{A}^{\text{tree}}_{4}(\theta)\equiv
T_{\lambda}=-24i\lambda_{1}+4i\lambda m^{4}\sinh^{2}\theta\,,$ (50)

which precisely matches the result obtained by explicit computation in
(LABEL:4-pt-tree).

###  3.4 1-loop amplitude

Since the bare Lagrangian in (LABEL:barelagrangian) has sextic terms
proportional to $\lambda^{2}$ and $\lambda\lambda_{1}$, there exists two
classes of diagrams that contribute to the second order S-matrix for
$2\rightarrow 2$ scattering: first, the 1-loop bubble diagram given in Figure
2 and second, we have to evaluate the tadpole diagram in Figure 3.

Figure 2: 1-loop bubble diagram for massive $\phi^{4}$ theory.

We begin with the 1-loop bubble diagram in $s$-channel (see Figure 2). This is
given by the following integral

$\displaystyle\mathcal{A}^{1\text{-loop}}_{s}[1,2,3,4]=\int\frac{\mathrm{d}^{2}k}{(2\pi)^{2}}\frac{N_{s}}{(k^{2}-m^{2})[(k+p_{1}+p_{2})^{2}-m^{2}]}\,,$
(51)

where the numerator is

$\displaystyle N_{s}=\left[i\chi+\frac{i\lambda}{12}\Big{\\{}(p_{1}\cdot
p_{2})(k\cdot\overline{k+P})+(p_{1}\cdot
k)(p_{2}\cdot\overline{k+P})+(p_{1}\cdot\overline{k+P})(p_{2}\cdot
k)\Big{\\}}\right]$ (52)
$\displaystyle\times\left[i\chi+\frac{i\lambda}{12}\Big{\\{}(p_{3}\cdot
p_{4})(k\cdot\overline{k+P})+(p_{3}\cdot
k)(p_{4}\cdot\overline{k+P})+(p_{3}\cdot\overline{k+P})(p_{4}\cdot
k)\Big{\\}}\right]\,.$ (53)

Here $\chi:=\left(\lambda_{1}+\frac{\lambda m^{4}}{4}\right)$ and
$P:=(p_{1}+p_{2})$. Note that the derivative terms are adding up as opposed to
tree amplitude (where all momenta are ingoing) because one of the two loop
momenta will always be outgoing at the vertex.

We rewrite the integrand by classifying the contributions from diagrams
involving $(a)$ no derivative, $(b)$ single derivative, and $(c)$ both
derivative interactions at the vertices

$\displaystyle
N_{s}=(i\chi)^{2}+(i\chi)\left(\frac{i\lambda}{12}\right)\mathcal{M}^{(b)}+\left(\frac{i\lambda}{12}\right)^{2}\mathcal{M}^{(c)}\,.$
(54)

The $s$-channel contribution from diagrams involving no derivatives is

$\displaystyle\mathcal{A}^{(a)}_{s}=-72\left(\lambda_{1}+\frac{\lambda
m^{4}}{4}\right)^{2}\frac{(\pi+i\theta)}{\pi m^{2}\sinh\theta_{12}}\,.$ (55)

The contribution from the $u$\- and $t$-channel can be found by simply setting
$\theta=0$ and $\theta=(i\pi-\theta)$ respectively. We combine those results
to obtain the total contribution without derivative interactions at the
vertices:

$\displaystyle\mathcal{A}^{(a)}=\frac{72}{\pi
m^{2}}\left(\lambda_{1}+\frac{\lambda
m^{4}}{4}\right)^{2}\left(i-\frac{\pi}{\sinh\theta}\right)\,.$ (56)

Similarly, the contribution from diagrams with single derivative interaction
is

$\displaystyle\mathcal{A}^{(b)}$
$\displaystyle=-\frac{12im^{2}}{\pi}\lambda\left(\lambda_{1}+\frac{\lambda
m^{4}}{4}\right)\left[2\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+i\pi\left(2\sinh\theta+\frac{3}{\sinh\theta}\right)\right]\,,$
(57)

and diagrams with two derivative interactions is

$\displaystyle\mathcal{A}^{(c)}=\frac{i\lambda^{2}m^{6}}{12\pi}$
$\displaystyle\left[i\pi\frac{6(2+\cosh(2\theta))^{2}}{\sinh\theta}+8+57\frac{\Lambda^{2}}{m^{2}}+90\log\left(\frac{m^{2}}{\Lambda^{2}}\right)\right.$
(58)
$\displaystyle\qquad\qquad\qquad\qquad\left.+8\cosh(2\theta)\left(-2+3\frac{\Lambda^{2}}{m^{2}}+3\log\frac{m^{2}}{\Lambda^{2}}\right)\right]\,.$

Adding all three contributions, we obtain the amplitude from the $1$-loop
bubble diagram as

$\displaystyle\mathcal{A}_{\text{bubble}}=\frac{1}{12\pi m^{2}}$
$\displaystyle\left[i\lambda^{2}m^{6}\left\\{57\Lambda^{2}+8\cosh(2\theta)\big{(}3\Lambda^{2}+3m^{2}\log\left(\frac{m^{2}}{\Lambda^{2}}\right)-2m^{2}\big{)}+62m^{2}\right.\right.$
(59)
$\displaystyle\qquad\qquad\left.+18m^{2}\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+24i\pi
m^{2}\sinh^{3}\theta\right\\}$
$\displaystyle+144\lambda_{1}\Big{\\{}\lambda_{1}(6i-6\pi\text{csch}\theta)$
$\displaystyle\qquad\qquad\left.+\lambda
m^{4}\big{(}2\pi\sinh\theta-2i\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+3i\big{)}\Big{\\}}\right]\,.$

This matches with the result given in (Rosenhaus:2019utc, ) for 1-loop bubble
diagram when we set $\lambda_{1}=0$.

###  3.5 Tadpole contribution

Figure 3: 1-loop tadpole diagram for massive $\phi^{4}$ theory.

The relevant terms in the $\text{T}\overline{\text{T}}$-deformed Lagrangian
(in Lorentzian signature) for massive $\phi^{4}$ theory that give rise to
tadpoles are

$\displaystyle\tilde{\mathcal{L}}_{\lambda^{2},\lambda\lambda_{1}}=-\lambda\lambda_{1}m^{2}\phi^{6}+\lambda^{2}\left(\frac{1}{4}\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{3}-\frac{m^{2}}{8}\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}\phi^{2}-\frac{m^{6}}{8}\phi^{6}\right)\,.$
(60)

In order to find the tadpole contribution, we first write down the effective
Lagrangian where we Wick contract any two of the external scalars. During the
Wick contraction, it is important to keep track of how many ways one can
perform this operation. For terms with $\phi^{n}$ polynomial structure, this
number is simply ${}^{n}C_{2}$. For terms with derivative, these numbers are
the following

$\displaystyle\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{3}\longrightarrow
9\,,\quad\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}\longrightarrow
4\,.$ (61)

After Wick contraction, we obtain the following Lagrangian for the tadpole
diagram

$\displaystyle\tilde{\mathcal{L}}_{\text{tadpole}}$
$\displaystyle=\left(\tfrac{9}{4}\lambda^{2}\langle\partial_{\mu}\phi\partial^{\mu}\phi\rangle-\tfrac{\lambda^{2}m^{2}}{8}\langle\phi^{2}\rangle\right)\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}-\tfrac{\lambda^{2}m^{2}}{2}\phi^{2}\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)\langle\partial_{\mu}\phi\partial^{\mu}\phi\rangle$
(62)
$\displaystyle\quad-15\left(\lambda\lambda_{1}m^{2}+\tfrac{\lambda^{2}m^{6}}{8}\right)\phi^{4}\langle\phi^{2}\rangle\,.$

The Feynman rule for the vertex can be easily derived from the above
Lagrangian

$\displaystyle V_{\text{tadpole}}=\frac{i\kappa_{1}}{3}\Big{[}(p_{1}\cdot
p_{2})(p_{3}\cdot p_{4})$ $\displaystyle+(2\leftrightarrow
3)+(2\leftrightarrow 4)\Big{]}$ (63)
$\displaystyle+i\kappa_{2}\left\\{(1;234)+(2;34)+(p_{3}\cdot
p_{4})\right\\}-i\kappa_{3}\,,$ (64)

where the coefficients $\kappa_{i}$’s are

$\displaystyle\kappa_{1}$
$\displaystyle:=\left(\tfrac{9}{4}\lambda^{2}\langle\partial_{\mu}\phi\partial^{\mu}\phi\rangle-\tfrac{\lambda^{2}m^{2}}{8}\langle\phi^{2}\rangle\right)\,,$
(65) $\displaystyle\kappa_{2}$
$\displaystyle:=\tfrac{\lambda^{2}m^{2}}{2}\langle\partial_{\mu}\phi\partial^{\mu}\phi\rangle\,,$
(66) $\displaystyle\kappa_{3}$
$\displaystyle:=15\left(\lambda\lambda_{1}m^{2}+\tfrac{\lambda^{2}m^{6}}{8}\right)\langle\phi^{2}\rangle\,,$
(67)

and additionally we have used the following notations to denote various
momentum scalar products appearing in the vertex

$\displaystyle(1;234)$ $\displaystyle:=(p_{1}\cdot p_{2})+(p_{1}\cdot
p_{3})+(p_{1}\cdot p_{4})\,,$ (68) $\displaystyle(2;34)$
$\displaystyle:=(p_{2}\cdot p_{3})(p_{2}\cdot p_{4})\,.$ (69)

We set $u=0$ for $2\rightarrow 2$ scattering in two dimensions and rewrite the
Feynman rule in terms of relative rapidity $\theta$

$\displaystyle
V_{\text{tadpole}}=\frac{i\kappa_{1}}{3}m^{4}(2\cosh^{2}\theta+1)-2im^{2}\kappa_{2}-i\kappa_{3}\,.$
(70)

Making use of this Feynman rule, we obtain the contribution to the amplitude
from the tadpole diagram as

$\displaystyle\mathcal{A}_{\text{tadpole}}=4!i\lambda^{2}m^{4}$
$\displaystyle\left[\frac{(1+2\cosh^{2}\theta)}{3}\left(\tfrac{9}{4}\langle\partial_{\mu}\phi\partial^{\mu}\phi\rangle-\tfrac{m^{2}}{8}\langle\phi^{2}\rangle\right)\right.$
(71)
$\displaystyle\qquad\qquad-\tfrac{1}{6}\langle\partial_{\mu}\phi\partial^{\mu}\phi\rangle-15\big{(}\tfrac{\lambda_{1}}{\lambda
m^{2}}+\tfrac{m^{2}}{8}\big{)}\langle\phi^{2}\rangle\bigg{]}\,.$

Note that, the combinatoric factor for the diagrams with contact interaction
$\phi^{2}\partial_{\mu}\phi\partial^{\mu}\phi$ is 4 as opposed to $4!$ of the
other two classes of interactions. Using the expressions for the following two
point correlators

$\displaystyle\langle\phi^{2}\rangle=-\frac{1}{4\pi}\log\left(\frac{m^{2}}{\Lambda^{2}}\right)\,,\quad\langle\partial_{\mu}\phi\partial^{\mu}\phi\rangle=-\frac{1}{4\pi}\left[m^{2}\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+\Lambda^{2}\right]\,,$
(72)

we obtain the tadpole amplitude as

$\displaystyle\mathcal{A}_{\text{tadpole}}$
$\displaystyle=\frac{i\lambda^{2}m^{6}}{4\pi}\big{[}15-17\cosh(2\theta)\big{]}\log\left(\frac{m^{2}}{\Lambda^{2}}\right)$
(73)
$\displaystyle\qquad-\frac{i\lambda^{2}\Lambda^{2}m^{4}}{2\pi}\big{[}9\cosh(2\theta)+16\big{]}+\frac{90i\lambda\lambda_{1}m^{2}}{\pi}\log\left(\frac{m^{2}}{\Lambda^{2}}\right)\,.$

###  3.6 renormalisation

The amplitude at $\mathcal{O}(\lambda^{2},\lambda\lambda_{1},\lambda_{1}^{2})$
is a sum of the 1-loop bubble and the tadpole contributions:

$\displaystyle\mathcal{A}^{(2)}=\frac{1}{12\pi}$
$\displaystyle\left[-im^{6}\lambda^{2}\left\\{39\frac{\Lambda^{2}}{m^{2}}+\cosh
2\theta\left(30\frac{\Lambda^{2}}{m^{2}}+27\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+16\right)\right.\right.$
(74)
$\displaystyle\qquad\qquad\qquad\left.{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-24i\pi\sinh^{3}\theta}-63\log\left(\frac{m^{2}}{\Lambda^{2}}\right)-62\right\\}$
$\displaystyle\qquad+72\lambda\lambda_{1}\left\\{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}4\pi
m^{2}\sinh\theta}+11im^{2}\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+i6m^{2}\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}864\frac{\lambda_{1}^{2}}{m^{2}}(\pi\text{csch}\theta-i)}\right]\,.$

The S-matrices of the $\text{T}\overline{\text{T}}$-deformed and undeformed
seed theory are related to each other by the following relation upto quadratic
order in $\text{T}\overline{\text{T}}$ coupling $\lambda$

$\displaystyle S_{\lambda}=S_{0}\left(1+i\lambda
m^{2}\sinh\theta-\frac{1}{2}\lambda^{2}m^{4}\sinh^{2}\theta\right)\,.$ (75)

Repeating the analysis of cddanalysis and using (LABEL:smatrix-amplitude), we
find the relation between the scattering amplitudes of the two theories

$\displaystyle T_{\lambda}=T_{0}+4i\lambda
m^{4}\sinh^{2}\theta-2\lambda^{2}m^{6}\sinh^{3}\theta+iT_{0}\lambda
m^{2}\sinh\theta+\mathcal{O}(\lambda^{2}\lambda_{1},\ldots)\,.$ (76)

The amplitude of the undeformed massive $\phi^{4}$ theory upto quadratic order
in bare $\phi^{4}$ coupling is

$\displaystyle T_{0}=-24i\lambda_{1}+\frac{72\lambda_{1}^{2}}{\pi
m^{2}}(i-\pi\text{csch}\theta)\,.$ (77)

Therefore, the amplitude of the $\text{T}\overline{\text{T}}$-deformed theory
at $\mathcal{O}(\lambda^{2},\lambda\lambda_{1},\lambda_{1}^{2})$ is given by

$\displaystyle
T_{\lambda}^{(2)}=-2\lambda^{2}m^{6}\sinh^{3}\theta+24\lambda\lambda_{1}m^{2}\sinh\theta+\frac{72\lambda_{1}^{2}}{\pi
m^{2}}(i-\pi\text{csch}\theta)\,.$ (78)

Now let us first consider the terms with $\text{T}\overline{\text{T}}$
coupling $\lambda$ which are purely real and exactly matches with the real
part of the amplitude in (LABEL:1-loopresult). However, the imaginary part of
the amplitude can not be fixed by (LABEL:2ndordercddfactor) and should cancel
on adding counterterms to the bare Lagrangian. Let us collect all the
imaginary pieces in (LABEL:1-loopresult)

$\displaystyle\text{Im}\,\mathcal{A}^{(2)}$
$\displaystyle=\frac{-i}{12\pi}\left[m^{6}\lambda^{2}\left\\{69\frac{\Lambda^{2}}{m^{2}}-36\log\left(\frac{m^{2}}{\Lambda^{2}}\right)-46\right.\right.$
(79)
$\displaystyle\qquad\qquad\qquad\qquad\left.+\sinh^{2}\theta\left(60\frac{\Lambda^{2}}{m^{2}}+54\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+32\right)\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-72\lambda\lambda_{1}m^{2}\left\\{11\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+6\right\\}\right]\,.$

We assume that the both the bare $\phi^{4}$ and $\text{T}\overline{\text{T}}$
coupling are classical couplings and that both of them get renormalised.
Therefore we add counter terms at
$\mathcal{O}(\lambda^{2},\lambda\lambda_{1})$ to the bare Lagrangian in
(LABEL:barelagrangian), we have

$\displaystyle\mathcal{\tilde{L}}_{\text{renorm.}}=\frac{1}{2}\left(\partial_{\mu}\phi\right)^{2}$
$\displaystyle+\frac{m^{2}\phi^{2}}{2}+\left(\lambda_{1}+\frac{\lambda
m^{4}}{4}\right)\phi^{4}-\left(\frac{\lambda}{4}-\gamma\lambda^{2}\right)\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}+(\alpha\lambda^{2}+\beta\lambda\lambda_{1})\phi^{4}$
(80)
$\displaystyle+\lambda\lambda_{1}m^{2}\phi^{6}+\lambda^{2}\left(\frac{1}{4}\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{3}+\frac{m^{2}}{8}\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}\phi^{2}+\frac{m^{6}}{8}\phi^{6}\right)\,.$
(81)

We do not consider any counterterm at $\mathcal{O}(\lambda_{1}^{2})$ as the
this is completely fixed by the last term of (LABEL:2ndordercddfactor), since
the CDD factor does not involve bare $\phi^{4}$ coupling. We only need to
consider the tree level contribution to amplitude due to these counter terms
as they are already at $\mathcal{O}(\lambda^{2},\lambda\lambda_{1})$. The
amplitude due to counterterms is

$\displaystyle\mathcal{A}_{\text{count.}}$
$\displaystyle=24i\left[-(\alpha\lambda^{2}+\beta\lambda\lambda_{1})-\frac{\gamma}{3}\lambda^{2}m^{4}(2\sinh^{2}\theta+3)\right]\,,$
(82) $\displaystyle=-24i\left[(\alpha+\gamma
m^{4})\lambda^{2}+\beta\lambda\lambda_{1}+\frac{2}{3}\gamma\lambda^{2}m^{4}\sinh^{2}\theta\right]\,.$
(83)

The additional sign is due to change in metric signature. In order to get rid
of the imaginary part of the amplitude, we set

$\displaystyle\gamma$
$\displaystyle=-\frac{m^{2}}{32\pi}\left[10\frac{\Lambda^{2}}{m^{2}}+9\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+\frac{16}{3}\right]\,,$
(84) $\displaystyle\beta$
$\displaystyle=\frac{m^{2}}{4\pi}\left[11\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+6\right]\,,$
(85)

and $\alpha$ gets automatically fixed to be

$\displaystyle\alpha=\frac{m^{6}}{96\pi}\left[\frac{94}{3}+7\frac{\Lambda^{2}}{m^{2}}+39\log\left(\frac{m^{2}}{\Lambda^{2}}\right)\right]\,.$
(86)

This analysis shows that the renormalised $\text{T}\overline{\text{T}}$
Lagrangian involves two “new” couplings associated to $\phi^{4}$ and
$\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}$ interactions:

$\displaystyle\chi_{\text{renorm.}}$
$\displaystyle=\chi_{\text{bare}}+\sum_{n\geq 2}a_{n}\lambda^{n}+\sum_{n\geq
1}b_{n}(\lambda\lambda_{1})^{n}\,,$ (87)
$\displaystyle\omega_{\text{renorm.}}$ $\displaystyle=\sum_{n\geq
1}c_{n}\lambda^{n}\,.$ (88)

Here
$\chi_{\text{bare}}=\left(\frac{\lambda_{1}}{m^{4}}+\frac{\lambda}{4}\right)$
and $c_{1}=1$. The renormalised Lagrangian can be recast as

$\displaystyle\mathcal{\tilde{L}}_{\text{renorm.}}=\frac{1}{2}\left(\partial_{\mu}\phi\right)^{2}+\frac{1}{2}m^{2}\phi^{2}+\frac{1}{4}\chi_{\text{renorm.}}m^{4}\phi^{4}-\frac{1}{4}\omega_{\text{renorm.}}\left(\partial_{\mu}\phi\right)^{2}+\cdots\,,$
(89)

where at 1-loop, the $2\rightarrow 2$ scattering matrix fully fixes the
coefficients in the series expansion of renormalised couplings as

$\displaystyle a_{2}$
$\displaystyle=\frac{m^{2}}{24\pi}\left[\frac{94}{3}+7\frac{\Lambda^{2}}{m^{2}}+39\log\left(\frac{m^{2}}{\Lambda^{2}}\right)\right]\,,$
(90) $\displaystyle b_{1}$ $\displaystyle=\frac{1}{\pi
m^{2}}\left[11\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+6\right]\,,$ (91)
$\displaystyle c_{2}$
$\displaystyle=\frac{m^{2}}{8\pi}\left[10\frac{\Lambda^{2}}{m^{2}}+9\log\left(\frac{m^{2}}{\Lambda^{2}}\right)+\frac{16}{3}\right]\,.$
(92)

The results in this section show that the bare $\phi^{4}$ interaction
introduces a new family of dimensionless couplings $(\lambda\lambda_{1})^{n}$,
which starts appearing from the first order in perturbation theory with
coefficients $b_{n}$, comprised within $\chi_{\text{renorm}}$. This feature
supports our claim in the introduction that additional scales (here
$\lambda_{1}$) in the seed action can be used to form new dimensionless
parameters along with the $\text{T}\overline{\text{T}}$ parameter $\lambda$,
which then contributes to the emergence of novel couplings in the quantum
mechanically corrected theory.

A rudimentary, but non-trivial consistency check of our calculation is we
should reproduce the answer obtained in Rosenhaus:2019utc  if we set the bare
quartic coupling $\lambda_{1}=0$. This can be checked rather
straightforwardly.444Since Rosenhaus:2019utc  gave their result without the
finite pieces, one can only confirm agreement with their calculations without
the finite pieces. In our computation we have kept all the finite pieces
arising in our renormalisation scheme. As indicated in Rosenhaus:2019utc , the
knowledge of the finite pieces are important in the calculation of correlation
functions with renormalised Lagrangian. While the expectations from dimension
counting are vindicated by this explicit computation, we note that the
situation becomes more interesting for higher-point scattering amplitudes. A
systematic approach to analyse the higher point scattering will most
definitely shed light on this intriguing interplay of renormalisation and the
knowledge of exact quantum S-matrix.

##  4 Discussion

The central theme of this paper was the study of constraints imposed on the
renormalisation of $\text{T}\overline{\text{T}}$-deformed theories coming from
knowledge of the exact deformed S-matrix. We extended previous investigations
into this theme in two directions. Both these directions involved the study of
the relationship between deformed and undeformed S-matrices.

The first deformed theory we looked at was an integrable field theory
describing a pair of (left- and right-)chiral bosons. The deformation of this
theory shared similar qualitative features as that of an ordinary massless
(free) boson (Rosenhaus:2019utc, ). The strong constraints of integrability
along with a lack of any mass scale in the bare Lagrangian inevitably leads to
only an expected renormalisation of the $\text{T}\overline{\text{T}}$ coupling
$\lambda$. In this case too, the S-matrix takes exactly the form dictated by
the CDD factor. This computation contains some novelties, in that the seed
theory has some unusual features. While it has been very clear from the outset
that there is a meaningful way to extract Feynman rules for these kinds of
theories (viz. self-dual $p$-forms), and is already implicit in Sen:2015nph ,
to the best of our knowledge this is the first time an explicit derivation of
Feynman rules are presented starting from an action. We expect these technical
results to be useful for studies of (anti)-self-dual fields and their
interactions in higher dimensions as well.

The second example we tackled involved an exploration of the constraints
imposed by the CDD factor on renormalisability of a non-integrable quantum
field theory. The undeformed theory we considered was a massive $\phi^{4}$
theory, and therefore contained _two_ different mass scales in the bare
Lagrangian.555Recall that in two dimensions the $\phi^{4}$ coupling is
dimensionful. We studied the $4$-point scattering up to $1$-loop, for which
the computation was somewhat cumbersome. Nevertheless, the end result of the
computation was worthwhile: we found that the $1$-loop S-matrix for
$2\rightarrow 2$ scattering matches precisely with CDD factor. The quantum
action has new couplings that flow to the $\text{T}\overline{\text{T}}$
coupling at tree level, echoing similar observations made for integrable field
theories in (Rosenhaus:2019utc, ; Dey:2021jyl, ). In fact, we find an
additional contribution to one of the emergent couplings (compared to the case
of the free massive boson) which can be traced to the fact that there is an
additional mass scale in the undeformed theory, viz. the $\phi^{4}$ coupling.
This is consistent with the intuition that the existence of mass scales in the
undeformed theory is crucial for the emergence of new couplings in the quantum
action.

There are several questions that follow naturally from the results of this
paper. Firstly, unlike the integrable quantum field theories, non-integrable
QFTs do not have their $n$-point S-matrices completely fixed once the
$4$-point S-matrix is known. Since there is particle production, we will have
non-trivial higher-point S-matrices as well. The next S-matrix that should be
evaluated for the $\phi^{4}$ theory considered here is the $6$-point
scattering. However, given the number of interaction vertices and the addition
of two external legs, the diagrammatics becomes cumbersome. Even if one
perseveres and works out the $3\rightarrow 3$ amplitude, going to higher
points would be prohibitively difficult. We entertain the hope that it may be
possible to deploy recent advances in the computation of scattering
amplitudes. In particular, the perturbiner method (Rosly:1996vr, ;
Mafra:2015gia, ; Mizera:2018jbh, ; Gomez:2020vat, ) guarantees a Berends-
Giele-like recursion relation for _any_ quantum field theory and has recently
been extended to $1$-loop level as well Lee:2022aiu ; Upcoming:2022 . A
successful adaptation of the perturbiner method to evaluate the perturbative
S-matrices in these theories will hopefully reveal structures the CDD factor
imposes on renormalisability of non-integrable quantum field theories.

A second follow-up question involves the supersymmetric version of
$\text{T}\overline{\text{T}}$ deformations. There are analogous deformations
which are known to preserve both $\mathcal{N}=1$ and $\mathcal{N}=2$ SUSY
Baggio:2018rpv ; Chang:2018dge ; Chang:2019kiu ; Ferko:2019oyv ; Jiang:2019hux
; Cribiori:2019xzp ; Ebert:2020tuy . One would naturally expect an analogue of
CDD factor to exist for S-matrices of these deformed theories as well. The
existence of integrability and supersymmetry should impose very strong
constraints on the quantum dynamics and it is possible that nonrenormalisation
theorems can be systematically proved for these theories. Even if the
undeformed theory is non-integrable, given the indications we have uncovered
in this paper, one can reasonably expect some definite structure to exist
dictated by an intriguing interplay of supersymmetry, integrability, and
renormalisation.

We leave the exploration of these problems for the future.

###### Acknowledgements.

We would like to thank Divyanshu Gupta for collaboration at an earlier stage
of this project. SC would like to thank Renann Lipinski Jusinskas for numerous
discussions and suggestions as well as for sharing the preliminary results of
Upcoming:2022 . Research of SC has been supported by the Czech Science
Foundation - GAČR, project 19-06342Y. MR is supported by Grant No. 21/02253-0
and 19/21281-4, São Paulo Research Foundation (FAPESP).

##  Appendix A Conventions

We collect various conventions we have used throughout the paper in this
appendix.

We work in flat spacetime with signature $(-,+)$, and for discussing self-
duality we will need the Levi-Civita symbol $\epsilon^{\mu\nu}$ with
conventions $\epsilon^{01}=-\epsilon^{10}=1$.

Two-dimensional Fourier transforms are defined as

$\displaystyle f(x)$
$\displaystyle=\int\frac{\text{d}^{2}k}{(2\pi)^{2}}\,e^{ikx}\,\tilde{f}(k)\ ,$
(93) $\displaystyle\tilde{f}(k)$
$\displaystyle=\int\mathrm{d}^{2}x\,e^{-ikx}\,f(x)\ .$

Since we have just a single independent component in the self-dual form field
$A_{\mu}$, it is useful to work in lightcone coordinates. Let us define the
following lightcone coordinates and derivatives:

$X^{\pm}=\frac{X^{1}\pm
X^{0}}{\sqrt{2}}\quad\text{and}\quad\partial_{\pm}=\frac{1}{\sqrt{2}}\left(\partial_{1}\pm\partial_{0}\right)\
,$ (94)

and inversely,

$X^{0/1}=\frac{1}{\sqrt{2}}\left(X^{+}\mp
X^{-}\right)\quad\text{and}\quad\partial_{0/1}=\frac{1}{\sqrt{2}}\left(\partial_{+}\mp\partial_{-}\right)\
.$ (95)

In these coordinates, the metric is $\eta_{-+}=\eta_{+-}=1$ such that
$\eta=\eta^{-1}$. Correspondingly, the Levi-Civita symbol is now
$\epsilon_{-+}=-\epsilon_{+-}=-\epsilon^{-+}=\epsilon^{+-}=1$. Now, the self-
dual gauge field with components $A_{\mu}=(-a,a)$ in Cartesian coordinates is
such that its index can be raised and lowered using the metric, but it also
obeys a self-duality condition $A^{\mu}=\epsilon^{\mu\nu}A_{\nu}$. Demanding
that these two conditions are compatible yields that

$A_{+}=A^{-}=0\quad\text{and}\quad A_{-}=A^{+}\neq 0\ .$ (96)

We fix the non-zero components by computing an inner product, say $X\cdot A$,
in two coordinate systems and demanding that they agree. This allows us to
conclude that $A_{-}=A^{+}=\sqrt{2}a$. Similarly, for an anti-self-dual field
with components $B_{\mu}=(b,b)$ in Cartesian coordinates will have
$B_{+}=B^{-}=-\sqrt{2}b$.

As for the projectors, in lightcone coordinates we have that

$\left(\mathcal{P}_{+}\right)^{+-}=\left(\mathcal{P}_{+}\right)_{-+}=\left(\mathcal{P}_{-}\right)^{-+}=\left(\mathcal{P}_{-}\right)_{+-}=1\
,$ (97)

and the other possibilities identically zero. This is good: it tells us that
only $A_{-}$ and $B_{+}$ propagate, which makes sense because they are chiral.

Since we’re working in lightcone coordinates, for massless particles we will
have $p^{2}=2p_{+}p_{-}=0$ so either $p_{+/-}=0$. For the case of
$2\rightarrow 2$ scattering, we make the choice that particles labelled $1,3$
will have $p_{+}=0$ and for particles labelled $2,4$ we have $p_{-}=0$. This
ensures that $s=(p_{1}+p_{2})^{2}=2p_{1}\cdot
p_{2}=2\left(p_{1}\right)_{-}\left(p_{2}\right)_{+}=-2p_{1}\cdot p_{4}=-t$,
and that $u=(p_{1}+p_{3})^{2}=0$. These conventions make sense because
$A_{\mu}$ has non-zero component $A_{-}$ and that should have non-zero
momentum, and similarly with $B_{\mu}$ and $B_{+}$.

##  Appendix B Feynman Rules for Chiral Bosons

Here we give the explicit Feynman rules for the chiral boson theory in terms
of the original field variables following Sen’s formalism. We derive the
Feynman rules in such a way that it can be easily adapted for case of higher
self-dual $p$-form field strengths as well.

The action for a pair of left- and right-chiral bosons interacting via linear
deformation in the $\text{T}\overline{\text{T}}$ coupling is given by

$\displaystyle\mathcal{L}=\frac{1}{2}\Big{(}\partial_{\mu}\phi\partial^{\mu}\phi+\partial_{\mu}\tilde{\phi}\partial^{\mu}\tilde{\phi}\Big{)}-2\epsilon^{\mu\nu}\Big{(}A_{\mu}\partial_{\nu}\phi-
B_{\mu}\partial_{\nu}\tilde{\phi}\Big{)}+\frac{\lambda}{4}\Big{(}A_{\mu}B^{\mu}\Big{)}^{2}\,,$
(98)

where $\phi,\tilde{\phi}$ are the auxiliary scalars needed to describe chiral
1-form fields in Sen’s formalism.

Since the derivation of the propagators (in momentum space) have already been
discussed in the main text, we simply quote them here for completeness

$\displaystyle\langle A_{\mu}(-k)A_{\nu}(k)\rangle$
$\displaystyle=-\frac{i}{4}\frac{k_{\mu}k_{\nu}}{k^{2}}\,;$ (99)
$\displaystyle\langle B_{\mu}(-k)B_{\nu}(k)\rangle$
$\displaystyle=-\frac{i}{4}\frac{k_{\mu}k_{\nu}}{k^{2}}\;.$ (100)

Note that it is implicit that the propagators are always sandwiched between
appropriate projectors as pointed out in (Sen:2015nph, ). In particular, note
that there are no contractions between $A_{\mu}$ and $B_{\nu}$.

The vertex factor can be read of from the action as usual. There is only one
additional subtlety: the vertex must keep the (anti-)self-duality of each leg
manifest. Recall that in Sen’s formalism, the (anti-)self-duality holds off-
shell as well, so this is a crucial and necessary step. Ensuring the correct
duality property of each leg in a vertex is easily taken care of by inserting
appropriate factors of the projectors. For the specific interaction term under
consideration the vertex factor is

$\displaystyle\widehat{V}_{A_{\mu}B_{\nu}A_{\alpha}B_{\beta}}=V^{(4)}_{A_{\gamma}B_{\delta}A_{\rho}B_{\sigma}}\mathcal{P}^{\rho}{}_{-,\alpha}\mathcal{P}^{\sigma}{}_{+,\beta}\mathcal{P}^{\gamma}{}_{-,\mu}\mathcal{P}^{\delta}{}_{+,\nu}\
,$ (101)

where

$\displaystyle
V^{(4)}_{A_{\mu}B_{\nu}A_{\rho}B_{\sigma}}=\frac{i\lambda}{2}\big{(}\eta_{\mu\nu}\eta_{\rho\sigma}+\eta_{\mu\sigma}\eta_{\rho\nu}\big{)}\,,$
(102)

is the vertex factor that one would have written if $A$ and $B$ were ordinary
vector fields. This vertex factor ensures that only chiral or anti-chiral
particles are created or annihilated at the vertices, even off-shell.

Finally, since the field variables here are like field strengths, each
external leg carries an additional factor of $\sim ik_{\mu}$ (with appropriate
signs depending on the momentum flowing in or out as per a chosen convention).
For most cases, we actually know the action in terms of potential as well the
field strength. This indicates that once the wave function normalization is
fixed for the potential, that automatically fixes the normalisation for the
wave function of the field strength.

As an example, for ordinary scalars, if one adopts the normalisation that
$\langle 0|\varphi(x)|k\rangle=e^{ikx}$, then for the field strength
$V_{\mu}=\partial_{\mu}\varphi$ we automatically know the wave function
normalisation. In Sen’s formalism there is no such way to fix the numerical
factor for the external legs uniquely since the mapping between the self-dual
field strength and its potential is not uniquely known. This is however not a
problem, since with any choice of normalisation, the LSZ reduction guarantees
that the S-matrix is independent of any such wavefunction renormalisation.

This Feynman rules are consistent with the ones prescribed in Alvarez-
Gaume:1983ihn  (albeit there it was given for self-dual $5$-forms in $10$
dimensions). The difference is now they can be systematically derived from an
action following the procedure outlined in Sen:2015nph  and we have given an
explicit derivation for a concrete example to supplement the general
procedure. Of course, the end result of a computation does not depend on the
choice of field variables and for theories with equal number of chiral and
anti-chiral fields it might be simpler to work with the fields $S_{\mu}$
defined in the text. However, one can easily imagine theories which break
chiral symmetry or where only a self-dual field strength couples to other
matter fields, for those cases it is unavoidable that one works with the
Feynman rules that can be derived in the same spirit as it has been done in
this appendix.

##  Appendix C Useful Integrals

In this appendix, we collect all the integrals used in 1loopphi4 of 1-loop
amplitude for massive $\phi^{4}$ theory following Rosenhaus:2019utc . We begin
with the simplest integral that does not have any tensor structure in the
integrand

$\displaystyle
L:=\int\frac{\mathrm{d}^{2}k}{(2\pi)^{2}}\frac{1}{k^{2}-m^{2}}\frac{1}{(k+P)^{2}-m^{2}}=\frac{i}{4\pi}\frac{(i\pi-\theta)}{m^{2}\sinh\theta}\,,$
(103)

where $P:=p_{1}+p_{2}$ and $\theta:=\theta_{1}-\theta_{2}$, relative rapidity
between particles with momentum $p_{1}$ and $p_{2}$. This result has been used
in (LABEL:noderivative) giving the contribution of no derivative terms in
1-loop amplitude. Next we consider the following symmetric second rank 1-loop
integral

$\displaystyle
L_{\mu\nu}:=\frac{1}{4}\int\frac{\mathrm{d}^{2}k}{(2\pi)^{2}}\frac{k_{\mu}}{k^{2}-m^{2}}\frac{P_{\nu}-k_{\nu}}{(k+P)^{2}-m^{2}}\,.$
(104)

These class of integrals are required in the evaluation of the single
derivative terms appearing in the integrand of massive $\phi^{4}$ 1-loop
amplitude. In lightcone coordinates, the components of this tensor integral
are given by

$\displaystyle L_{++}$
$\displaystyle=-\frac{i}{16\pi}e^{(\theta_{1}+\theta_{2})}\left(1-\frac{(i\pi-\theta)}{\sinh\theta}\right)\,,$
(105) $\displaystyle L_{--}$
$\displaystyle=-\frac{i}{16\pi}e^{-(\theta_{1}+\theta_{2})}\left(1-\frac{(i\pi-\theta)}{\sinh\theta}\right)\,,$
(106) $\displaystyle L_{-+}$
$\displaystyle=-\frac{i}{16\pi}\left(\log\left(\frac{m^{2}}{\Lambda^{2}}\right)-(i\pi-\theta)\coth\theta\right)\,.$
(107)

Finally, we consider a class of rank-4 tensor integrals, appearing as double
derivative terms in the 1-loop integrand of (LABEL:phi4integrand)

$\displaystyle
L_{\mu\nu\rho\sigma}:=\frac{1}{16}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{k_{\mu}k_{\nu}}{k^{2}-m^{2}}\frac{\left(P_{\rho}-k_{\rho}\right)\left(P_{\sigma}-k_{\sigma}\right)}{(k+P)^{2}-m^{2}}\,.$
(108)

By definition, this tensor integral is symmetric within the first two and last
two indices and also between the pairs. Again, in lightcone coordinate, the
different components of this tensor integral has been evaluated in
Rosenhaus:2019utc  and we quote the results below for completeness

$\displaystyle L_{++++}$
$\displaystyle=-\frac{im^{2}}{64\pi}e^{2(\theta_{1}+\theta_{2})}\left(\frac{4}{3}+\frac{\cosh\theta}{3}-\frac{(i\pi\,-\theta)}{\sinh\theta}\right)$
(109) $\displaystyle L_{-+++}$
$\displaystyle=\frac{im^{2}}{192\pi}e^{(\theta_{1}+\theta_{2})}\Big{(}1-2\cosh\theta+3(i\pi-\theta)\frac{\cosh\theta}{\sinh\theta}\Big{)}$
(110) $\displaystyle L_{--++}$
$\displaystyle=\frac{im^{2}}{64\pi}\left(\frac{\Lambda^{2}}{m^{2}}-2\cosh\theta\log\frac{m^{2}}{\Lambda^{2}}+(i\pi-\theta)\frac{\cosh(2\theta)}{\sinh\theta}\right)$
(111) $\displaystyle L_{+-+-}$
$\displaystyle=\frac{im^{2}}{64\pi}\left(\frac{\Lambda^{2}}{m^{2}}-\frac{1}{3}\cosh\theta+2\log\frac{m^{2}}{\Lambda^{2}}+\frac{(i\pi-\theta)}{\sinh\theta}\right)\,.$
(112)

All other components of the rank-4 tensor integral are related to the above
integrals. For instance:

$\displaystyle L_{----}=e^{-4(\theta_{1}+\theta_{2})}L_{++++}\,,\quad
L_{+---}=e^{-2(\theta_{1}+\theta_{2})}L_{-+++}\,.$ (113)

Symmetry considerations also constrains some of the integrals as follows

$\displaystyle L_{++--}=L_{--++}\,,\quad L_{+-++}=L_{-+++}\,,$ (114)

and so on. These results have been used to determine the contributions coming
from single derivative terms ($\mathcal{A}^{(b)}$ in (LABEL:singlederivative))
and double derivative terms ($\mathcal{A}^{(c)}$ in (LABEL:doublederivative))
in massive $\phi^{4}$ 1-loop amplitude.

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