compat=1.1.0 aainstitutetext: Department of Physics
Indian Institute of Science Education and Research Pune, India

# Amplitude’s positivity vs. subluminality: Causality and Unitarity
Constraints on dimension 6 & 8 Gluonic operators in the SMEFT

Diptimoy Ghosh a,2 Rajat Sharma a,3 Farman Ullah
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We derive the causality and unitarity constraints on dimension 6 and dimension
8 Gluon field strength operators in the Standard Model Effective Field Theory
(SMEFT). In the first part of the paper, we use the ‘amplitude analysis’ i.e.
dispersion relation for $2\rightarrow 2$ scattering in the forward limit, to
put bounds on the Wilson coefficients. We show that the dimension 6 operators
can exist only in the presence of certain dimension 8 operators. It is
interesting that the square of the dimension 6 Wilson coefficients can be
constrained in this case even at the tree level. In the second part of this
work, we successfully rederive all these bounds using the classical causality
argument that demands that the speed of fluctuations about any non-trivial
background should not exceed the speed of light. We also point out some
subtleties in the superluminality analysis regarding whether the low-frequency
phase velocity can always be used as the relevant quantity for Causality
violation: as an example, we show that, due to these subtleties, if a small
pion mass is added in the chiral Lagrangian, it is unclear if any strict
positivity bound can be derived on the dimension 8 Wilson coefficient.
Finally, we mention an interesting non-relativistic example where the
subluminality requirement produces a stronger bound than the ‘amplitude
analysis’.

##  1 Introduction

It is well known that dynamics at very high energy scales or short distances
are irrelevant to describe low energy or long distance phsyics, i.e. very
different energy scales are ‘decoupled’ from each other. For example, we don’t
need to know the fine details of nuclei to understand the properties of
electronic energy levels in atoms. It’s mainly on this idea that the framework
of Effective Field Theory (EFT) is built (see burgess ; Skiba:2010xn for a
review), in which one constructs the Lagrangian for the low energy (IR) theory
in terms of some physical cut-off energy scale ($\Lambda$). The Lagrangian for
the EFT, a priori, must contain all the possible operators consistent with the
symmetries of the theory e.g., Lorentz invariance and gauge invariance, and
the coefficients of these operators can have arbitrary values. However, it has
been shown in recent years that sacred principles like relativistic Causality
and Unitarity do impose non-trivial constraints on these coefficients and
carve out the allowed parameter space causality ; DSD2 ; derhamcausalEFT ;
snowmass . This is interesting also phenomenologically since it leads to
enhanced statistical power for experiments sensitive to these operators
because one can incorporate IR consistency bounds into the prior probability
distribution.

One of the first attempts in this direction due to PhysRevD.31.3027 ;
causality ; posscalar ; posgravity (which we refer to as the ‘amplitude
analysis’) exploits well-established fundamental principles like micro-
causality (leading to analyticity analyticity_Bros ; analyticity_Lehmann ) and
unitarity of the S-matrix to constrain the EFT parameter space. This involves
using dispersion relations for $2\rightarrow 2$ scattering amplitudes in the
forward limit. The ‘amplitude analysis’ has successfully given linear
positivity bounds on dimension 8 operators in a variety of theories but hasn’t
had much success with dimension 6 operators111See recent developments made to
constrain dim 6 operators using S-matrix Bootstrap methods dim6_Elias_miro .
containing 4 fields; although, one can derive certain sum rules 0907.5413 ;
1202.1532 ; 1807.06940 ; 2008.07551 ; 2010.04723 ; 2206.13524 ; dim6_amartya .
The reason for this lack of success is the fact that for such an operator, the
scattering amplitude grows as the Mandelstam variable $s$ at the tree level.
However, in this article, we show that it’s possible to constrain the square
of the coefficient of dimension 6 operators containing 3 gluon fields w.r.t
those of dim 8 operators. Such an operator appears in the SMEFT. Similar
positivity bounds for the electroweak gauge bosons were obtained in
constraintvectorboson . A lot of effort has been focused on constraining the
parameter space of SMEFT using various methods (see, for example,
Zhou_positivity_multifields ; constraintvectorboson ; Flavor_constraints ;
minimal_flovor_violation ; Trott_EFT ; consistentSMEFT and the references
therein), since it is and will be the main aim of present and future particle
physics experiments to measure these coefficients. Also, verifying whether the
experimentally measured coefficients satisfy our theoretical constraints
allows us to test fundamental properties of the UV theory such as locality and
Lorentz invariance up to very high energies through experimental signatures at
accessible scales consistentSMEFT .

The other method often employed to put constraints on the Wilson coefficients
is based on the classical causality argument causality . One demands that the
propagation of perturbations over any non-trivial background should respect
causality i.e. the speed of signal propagation should not be superluminal
causality ; snowmass ; derhamcausalEFT ; Goon . It is well known that if the
wavefront velocity i.e. infinite frequency limit of the phase velocity is
(sub)luminal then causality is preserved. Naively, one might conclude that it
is not possible to put constraints on the EFT coefficients since the EFT is
valid, by definition, in the low-frequency regime i.e. $\omega/\Lambda\ll 1$.
However, one can instead consider signal velocity (for a precise definition,
see BrillouinLeon1960Wpag ) which, for non-dispersive mediums (the case of our
interest), is equal to the group/phase velocity. Thus, the low frequency
group/phase velocity can be directly associated with causality obviating the
need to take the high-frequency limit. One can also use analyticity in the
form of the Kramers-Kronig relation Jackson:1998nia , which, for dissipative
backgrounds, demands that the phase velocity cannot decrease with increasing
frequency Jackson:1998nia ; snowmass . Therefore, the superluminal phase
velocity in the EFT can be associated with causality violation. However, in
general, it is not very clear how one can determine the dispersive properties
of the background medium, which is essential in the usefulness of the Kramers-
Kronig relation. We will discuss this in some more detail in section 4.

It is not always necessary that small superluminal low energy speed violates
causality as the observations detecting causality violation may turn out to be
unmeasurable within the valid regime of EFT Goon ; Drummond . Therefore for
generic EFTs, particularly gravitational ones, scattering phase shift or time
delay is perhaps a better probe to detect causality violations CEMZ ;
timedelay . However, for homogeneous backgrounds (as considered in this paper)
signals can be allowed to propagate over large distances, and in that case,
even the small superluminality can be detected within the EFT regime.
Therefore, using this method one can try to rederive or even hope to improve
the bounds on EFT coefficients obtained by the ‘amplitude analysis’.

We would like to stress that, a priori, it is unclear if the two methods
always provide the same constraints (or equivalently, whether using any one of
them is enough to maximally constrain the space of EFTs), as naively, they
don’t seem to be related at all. One is purely based on the classical
causality of wave propagation and another on scattering amplitudes which
relies on Unitarity, and Froissart bound in addition to micro-causality (the
two methods could be somewhat related since the classical causality analysis
secretly might also depend on analyticity in the form of Kramers-Kronig
relation snowmass and unitarity (for dissipative mediums) however, the
connection is unclear, as we discuss in section 4).

For the classical causality/superluminality analysis, we also point out some
subtleties that arise when mass-like terms exist in the dispersion relation.
We show, in the particular case of the chiral Lagrangian, that this may lead
to deviation from strict positivity. However, in the case of gluonic
operators, we demonstrate the mass-like terms can be removed by choosing
particular configurations of non-trivial background and polarization of
perturbation. This helps us derive constraints on the Wilson coefficients of
gluonic operators by demanding subluminal phase velocity as our measure for
causality. We show that the superluminality analysis for dim 6 and 8 gluonic
operators, in a nontrivial way, reproduces all bounds that we obtain from the
‘amplitude analysis’. This is the novel and main result of our work. Finally,
we mention a non-relativistic example following Baumann_2016 , where
superluminality gives stronger bounds than the amplitude analysis. From the
examples given in our work, one can gather that one should use both analyses
whenever possible in order to get the maximum amount of information on an IR
effective theory222For fermions, it is unclear how one can implement the
superluminality analysis. However, the ‘amplitude analysis’ can still be
carried out fermions_ampl ..

The rest of the paper is organized as follows: In section 2, we define all the
notations and conventions we have used throughout the paper to avoid any
confusion. In section 3, we derive positivity constraints on dim 6 and dim 8
gluonic operators in SMEFT using the ‘amplitude analysis’ with an overview of
the method first. In section 4, we first discuss a few subtleties in the
superluminality analysis, followed by a demonstration that all the bounds can
also be reproduced by the superluminality analysis. The summary of our results
is presented in section 5.

##  2 Notation and Conventions

We use $\eta_{\mu\nu}=(+,-,-,-)$ metric and work in Lorentz gauge.

Greek indices $\mu,\nu$, etc. run over four space-time coordinate labels
0,1,2,3 with $x^{0}$ being the time coordinate.

Repeated indices are summed over unless otherwise specified.

The complex conjugate and hermitian adjoint of a vector or a matrix A are
denoted by $A^{*}$ and $A^{{\dagger}}$.

The gluon field strength tensor is defined as
$\displaystyle{G^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A_{\mu}^{a}+g_{s}f^{abc}A^{b}_{\mu}A_{\nu}^{c}}$
and its dual tensor as
$\displaystyle{\widetilde{G}^{a}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}G^{a\;\rho\sigma}\;\;(\varepsilon_{0123}=+1)}$
where $A_{\mu}=t_{a}A_{\mu}^{a}$ expresses the gluon field; $t_{a}$ are the
generators of SU(3); $a$, $b$, $c$ = 1, 2, …, 8 are color indices; $g_{s}$ is
the coupling constant of the strong force; $f^{abc}$ and $d^{abc}$ are the
total anti-symmetric and symmetric structure constants of SU(3) respectively,
defined as

$\left[t^{a},t^{b}\right]=if^{abc}t^{c}\;;\;\quad\left\\{t^{a},t^{b}\right\\}=\frac{1}{3}\delta^{ab}+d^{abc}t^{c}$

For convenience we define
$\displaystyle{F^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A_{\mu}^{a}}$.

In scattering amplitudes, we consider $p_{1}$ and $p_{2}$ as incoming momenta
and $p_{3}$ and $p_{4}$ as outgoing, and define Mandelstam invariants:

$s=(p_{1}+p_{2})^{2},\quad t=(p_{1}-p_{3})^{2},\quad u=(p_{1}-p_{4})^{2}$
satisfying $s+t+u=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}$ where $m_{i}$ are
masses of the particles.

##  3 Positivity constraints from unitarity and analyticity

In this section, we derive constraints on dimension 6 and dimension 8 Gluonic
operators of the SMEFT using dispersion relations for $2\rightarrow 2$
scattering amplitude. Let us first review how the ‘amplitude analysis’ works
to put constraints on the Wilson coefficients of a general EFT.

###  3.1 Overview

In this section, we give a brief overview, following the discussion in the
seminal paper causality , of how the dispersion relations along with well-
established principles like micro-causality (leading to analyticity of
S-matrix), unitarity, and locality, can help in constraining the low-energy
EFT parameters which from the IR side can have arbitrary values, to begin
with. We slightly modify the discussion in accordance with our work but the
main underlying concepts are the same.

Let us consider $s\leftrightarrow u$ symmetric $2\rightarrow 2$ scattering
amplitude, $\mathcal{M}(s,t)$ for a process in which exchanged particles or
particles inside a loop are of the same mass $m$ (or massless as would be the
case in our work), in the forward limit i.e. initial and final states are
exactly same. We define
$\displaystyle{\mathcal{A}(s)=\mathcal{M}(s,t)|_{t\rightarrow 0}}$ (forward
limit) and take integral around contour as shown in fig. 1

Figure 1: Analytic structure of $\mathcal{A}(s)$ in complex s plane. The
contour is symmetric about $s=2m^{2}$ and go over branch cuts from $-\infty$
to 0 and $4m^{2}$ to $\infty$.

$\mathcal{I}=\oint_{C}\frac{ds}{2\pi i}\frac{\mathcal{A}(s)}{(s-2m^{2})^{3}}$
(1)

where $m$ is the mass of the exchanged particles (or regularized mass for
massless exchanged particles), and $\Lambda$ is the energy cut-off scale of
EFT in consideration. It is possible that the theory considered allows for the
loops leading to branch cuts starting from $-\infty$ to $0$ and $4m^{2}$ to
$+\infty$. That is why we probe the $s\rightarrow 2m^{2}$ limit instead of
$s\rightarrow 0$ (which would be okay in case the branch cut doesn’t go
through or extend up to 0).
Now we evaluate the integral (1); since
$\lim_{|s|\to\infty}\mathcal{A}(s)/|s|^{2}\to 0$ at infinity due to the
Froissart bound (more precisely, $\mathcal{A}(s)<s\;ln^{2}s$) froissart ; jim-
martin , the integral over the arc at infinity vanishes and we are left just
with the integral of discontinuity of $\mathcal{A}(s)$ across the branch cuts,

$\mathcal{I}=\frac{1}{2\pi
i}\int_{cuts}ds\frac{Disc\mathcal{A}(s)}{(s-2m^{2})^{3}}$

But the integral can also be evaluated in terms of the residues at the poles:
at $s=2m^{2}$ and $s=m^{2},\;3m^{2}$ (due to $s$-channel and $u$-channel in
exchange diagrams). Thus, we get

$\frac{1}{2}\mathcal{A}^{\prime\prime}(s=2m^{2})+\sum_{s^{*}=m^{2},3m^{2}}\frac{res\mathcal{A}(s=s^{*})}{(s^{*}-2m^{2})^{3}}=\frac{2}{2\pi
i}\int_{s>4m^{2}}ds\frac{Disc\mathcal{A}(s)}{(s-2m^{2})^{3}}$ (2)

In the above equation, we have mapped the integral over the negative branch
cut to the positive one using $s\leftrightarrow u$ crossing symmetry. Now,
$Disc\mathcal{A}(s)=2iIm\mathcal{A}(s)$ and from optical theorem (for which
initial and final states are required to be identical), we have
$Im\mathcal{A}(s)=\sqrt{s(s-4m^{2})}\sigma(s)$ where $\sigma(s)$ is the total
cross-section of the scattering,

$\displaystyle\frac{1}{2}\mathcal{A}^{\prime\prime}(s=2m^{2})+\sum_{s^{*}=m^{2},3m^{2}}\frac{res(\mathcal{A}(s=s^{*}))}{(s^{*}-2m^{2})^{3}}=\frac{2}{\pi}\int_{s>4m^{2}}ds\frac{\sqrt{s(s-4m^{2})}\sigma(s)}{(s-2m^{2})^{3}}$
(3)

For further analysis, we’ll take the $s\leftrightarrow u$ symmetric
$\mathcal{A}(s)$ to be of a particular form, which we’ll be encountering in
further sections.

We concern ourselves with operators only up to dimension 8 in the low-energy
EFT and also assume that it contains only 6 and 8 dimension operators in
addition to 4-dimensional terms. If the theory allows taking $t\rightarrow 0$
i.e. forward limit without causing any divergence problem (which will be the
case here) then we can write the $s\leftrightarrow u$ symmetric forward
scattering amplitude at tree level as

$\displaystyle\mathcal{A}(s)=\lambda+b\frac{m^{2}}{\Lambda^{2}}$
$\displaystyle+\frac{1}{\Lambda^{4}}(c_{1}s^{2}+c_{1}u^{2}+c_{3}m^{4})$ (4)
$\displaystyle+\frac{1}{\Lambda^{4}}\left(c_{4}\frac{s^{3}}{s-m^{2}}+c_{5}\frac{s^{2}m^{2}}{s-m^{2}}+c_{6}\frac{sm^{4}}{s-m^{2}}+c_{7}\frac{m^{6}}{s-m^{2}}\right)$
$\displaystyle+\frac{1}{\Lambda^{4}}\left(c_{4}\frac{u^{3}}{u-m^{2}}+c_{5}\frac{u^{2}m^{2}}{u-m^{2}}+c_{6}\frac{um^{4}}{u-m^{2}}+c_{7}\frac{m^{6}}{u-m^{2}}\right)$

Note: If we take the exchange particles to be massless, then it might not be
possible to take the forward limit, even with the regularized mass because we
also need to put $m\rightarrow 0$ at some stage, which is one of the main
problems in performing this analysis for EFTs of gravity. Then for the
massless case, either the t-channel shouldn’t exist or the numerator in the
t-channel exchange contribution should converge to 0 faster than t in
$t\rightarrow 0$ limit avoiding the t-channel pole problem which would be the
case for our EFT in consideration.
Putting (4) in (3) we get,

$\frac{2}{\Lambda^{4}}\left(c_{1}+c_{4}\right)=\frac{2}{\pi}\int_{s>4m^{2}}ds\frac{\sqrt{s(s-4m^{2})}\sigma(s)}{(s-2m^{2})^{3}}$
(5)

The integrand in the r.h.s of the above equation is positive as the cross-
section $\sigma(s)>0$, which then makes the r.h.s manifestly positive.
Therefore, the above equation shows that the coefficient of the term which
goes as $s^{2}$ upon taking $m\rightarrow 0$ limit of $\mathcal{A}(s)$ (l.h.s
of the above equation) is positive. The same result can be obtained by using a
different contour as shown in appendix C.

###  3.2 Relative bounds on dim 6 and dim 8 operators

It is clear from the previous section that we cannot put any constraint on the
contribution that grows slower than $s^{2}$. This is because for the integral
to vanish over the arc at infinity, we need minimum $n=3$ in
$\displaystyle{\oint_{c}\frac{\mathcal{A}(s)}{(s-2m^{2})^{n}}}$ but then there
is no contribution to the residue from terms growing less than $s^{2}$,
preventing us from constraining them. If we take $n=2$ and even ignore the
contribution from arcs at infinity, then we get the contribution from dim 6
operators towards the residue in $\text{eq}^{\text{n}}$(5). However, for $n=2$
the integrand on r.h.s of (5) takes form $\sigma(s)/s$ which makes the
integral have a non-definitive sign. Thus, the l.h.s also have a non-
definitive sign preventing us from constraining the Wilson coefficients using
positivity arguments.

This is usually the case for dimension 6 operators containing four fields as
their contribution to the tree level amplitude grows as $s$. However, if the
dim 6 operator also gives rise to terms containing only three fields then one
indeed gets an $s^{2}$ piece through exchange diagrams. The constraint in this
case, however, is on the square of the Wilson coefficient (and not on the
sign). Also, dimension 8 operators containing four fields give contributions
at the same order ($\displaystyle{s^{2}/\Lambda^{4}}$) via contact diagrams.
Thus, we produce relative bounds on dim 6 and dim 8 operators. Similar bounds
for the electroweak gauge bosons were obtained in constraintvectorboson .

Let us now apply the above analysis for dimensions 6 and 8 Gluonic operators
in the SMEFT. The independent Gluonic operators for SU(3) as mentioned in dim6
and dim8 ; dim_8_008 are given in table 1.

$X^{3}$ | $X^{4}$
---|---
$Q_{G^{3}}^{(1)}$ | $f^{abc}G_{\mu}^{a\nu}G_{\nu}^{b\rho}G_{\rho}^{c\mu}$ | $Q_{G^{4}}^{(1)}$ | $\left(G_{\mu\nu}^{a}G^{a\mu\nu}\right)\left(G_{\rho\sigma}^{b}G^{b\rho\sigma}\right)$
$Q_{G^{3}}^{(2)}$ | $f^{abc}\widetilde{G}_{\mu}^{a\nu}G_{\nu}^{b\rho}G_{\rho}^{c\mu}$ | $Q_{G^{4}}^{(2)}$ | $\left(G_{\mu\nu}^{a}\widetilde{G}^{a\mu\nu}\right)\left(G_{\rho\sigma}^{b}\widetilde{G}^{b\rho\sigma}\right)$
|  | $Q_{G^{4}}^{(3)}$ | $\left(G_{\mu\nu}^{a}G^{b\mu\nu}\right)\left(G_{\rho\sigma}^{a}G^{b\rho\sigma}\right)$
|  | $Q_{G^{4}}^{(4)}$ | $\left(G_{\mu\nu}^{a}\widetilde{G}^{b\mu\nu}\right)\left(G_{\rho\sigma}^{a}\widetilde{G}^{b\rho\sigma}\right)$
|  | $Q_{G^{4}}^{(5)}$ | $\left(G_{\mu\nu}^{a}G^{a\mu\nu}\right)\left(G_{\rho\sigma}^{b}\widetilde{G}^{b\rho\sigma}\right)$
|  | $Q_{G^{4}}^{(6)}$ | $\left(G_{\mu\nu}^{a}G^{b\mu\nu}\right)\left(G_{\rho\sigma}^{a}\widetilde{G}^{b\rho\sigma}\right)$
|  | $Q_{G^{4}}^{(7)}$ | $d^{abc}d^{dec}\left(G_{\mu\nu}^{a}G^{b\mu\nu}\right)\left(G_{\rho\sigma}^{d}G^{e\rho\sigma}\right)$
|  | $Q_{G^{4}}^{(8)}$ | $d^{abc}d^{dec}\left(G_{\mu\nu}^{a}\widetilde{G}^{b\mu\nu}\right)\left(G_{\rho\sigma}^{d}\widetilde{G}^{e\rho\sigma}\right)$
|  | $Q_{G^{4}}^{(9)}$ | $d^{abc}d^{dec}\left(G_{\mu\nu}^{a}G^{b\mu\nu}\right)\left(G_{\rho\sigma}^{d}\widetilde{G}^{e\rho\sigma}\right)$

Table 1: Dimension 6 and 8 gluonic operators in the SMEFT

The Lagrangian at dimensions 6 and 8 takes the following form:

$L^{(6)}=\frac{c_{6}}{\Lambda^{2}}Q_{G^{3}}^{(1)}+\frac{c_{6}^{\prime}}{\Lambda^{2}}Q_{G^{3}}^{(2)}\;;\;\;\;L^{(8)}=\frac{c_{8}^{(i)}}{\Lambda^{4}}Q_{G^{4}}^{(i)}$

where $c_{6}$, $c_{6}^{\prime}$ and $c_{8}^{(i)}$ are dimensionless Wilson
coefficients and $\Lambda$ is the UV cut-off for the EFT. The above-mentioned
dimension 6 operators have parts that contain three fields and three
derivatives, e.g.,
$\displaystyle{f^{abc}F_{\mu}^{a\nu}F_{\nu}^{b\rho}F_{\rho}^{c\mu}}$ in
$Q_{G^{3}}^{(1)}$. Also, dimension 8 operators have parts containing four
fields with four derivatives,
e.g.,$\left(F_{\mu\nu}^{a}F^{b\mu\nu}\right)\left(F_{\rho\sigma}^{a}F^{b\rho\sigma}\right)$
in $Q_{G^{4}}^{(3)}$. Both kinds of terms give $s^{2}/\Lambda^{4}$ growth in
the amplitude, the former via an exchange diagram and the latter via a contact
diagram. Therefore as discussed above, using the ‘amplitude analysis’ we can
constrain squares of $c_{6}$ and $c_{6}^{\prime}$ relative to $c_{8}^{(i)}$.
We calculate scattering amplitude for $gg\rightarrow gg$, written explicitly
as
$|p_{1},\epsilon_{1},a;p_{2},\epsilon_{2},b\rangle\rightarrow|p_{3},\epsilon_{3},d;p_{4},\epsilon_{4},e\rangle$
(where $p$’s are the momenta, $\epsilon$’s are polarizations and Latin indices
denote color of particles), at tree level getting contribution from Feynman
diagrams in fig. 2. To relate the imaginary part of amplitude to the cross-
section (optical theorem), one needs to take identical initial and final
states; therefore, we consider $a=d$ and $b=e$ for our calculations.

Below, we just present those terms that give $s^{2}$ contribution to
$\mathcal{A}(s)$333We took help of the FeynCalc Feyncalc1 ; Feyncalc2 ;
Feyncalc3 package to verify our calculation. , for reasons mentioned above.

$\displaystyle\mathcal{A}(s)$ $\displaystyle=\mathcal{M}(s,t)|_{t\rightarrow
0}$ (6)
$\displaystyle=\bigg{[}\Big{\\{}9f^{abc}f^{ab}_{\;\;\;c}\left(\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}^{\prime
2}-c_{6}^{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\right)+8\delta^{ab}\left(\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(1)}-c_{8}^{(2)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\right)+4(1+\delta^{ab})\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(3)}-c_{8}^{(4)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\right)$
$\displaystyle\hskip
14.22636pt+8d^{abc}d^{ab}_{\;\;\;c}\left(\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(7)}-c_{8}^{(8)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\right)\Big{\\}}\times\Big{\\{}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}|s\epsilon_{1}\cdot\epsilon_{2}^{*}-2\;\epsilon_{1}\cdot
p_{2}\epsilon_{2}^{*}\cdot
p_{1}|^{2}+|s\epsilon_{1}\cdot\epsilon_{2}-2\;\epsilon_{1}\cdot
p_{2}\epsilon_{2}\cdot
p_{1}|^{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{\\}}$
$\displaystyle\hskip
5.69046pt-2\Big{\\{}9f^{abc}f^{ab}_{\;\;\;c}\;\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}^{\prime
2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-8\delta^{ab}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(2)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-4(1+\delta^{ab})\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\;c_{8}^{(4)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-8\;d^{abc}d^{ab}_{\;\;\;c}\;\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(8)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{\\}}\times\Big{\\{}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}s^{2}|\epsilon_{1}|^{2}|\epsilon_{2}|^{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{\\}}$
$\displaystyle\hskip
5.69046pt+4\Big{\\{}9f^{abc}f^{ab}_{\;\;\;c}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}c_{6}^{\prime}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-4\delta^{ab}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(5)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-2(1+\delta^{ab})\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(6)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-4d^{abc}d^{ab}_{\;\;\;c}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(9)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{\\}}$
$\displaystyle\hskip
128.0374pt\times\varepsilon^{\mu\nu\rho\sigma}Re\Big{\\{}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\epsilon_{1\mu}^{*}\epsilon_{2\nu}p_{1\rho}p_{2\sigma})(s\epsilon_{1}\cdot\epsilon_{2}^{*}-2\;\epsilon_{1}\cdot
p_{2}\epsilon_{2}^{*}\cdot p_{1})$
$\displaystyle\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\hskip
176.407pt+(\epsilon_{1\mu}^{*}\epsilon_{2\nu}^{*}p_{1\rho}p_{2\sigma})(s\epsilon_{1}\cdot\epsilon_{2}-2\;\epsilon_{1}\cdot
p_{2}\epsilon_{2}\cdot
p_{1})\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{\\}}\bigg{]}$

It should be noted that $a$ and $b$ in the above expression are not
contracted, they are colors of incoming and outgoing particles whereas $c$
(color index) and Lorentz indices are contracted.

Figure 2: Exchange diagrams (first three) get contribution just from dim 6
operators ; contact diagram gets contribution from both dim 6 and dim 8 but
the dim 6 contribution is of order $s/\Lambda^{2}$ as explained in the text.

For the t-channel, we get $f^{adc}f^{bec}$ factor in the numerator which is
zero for our particular choice of colors, $a=d$ and $b=e$. Therefore, there is
no contribution from the t-channel and we are allowed to take the forward
limit without any divergence issues due to the t-channel pole. This isn’t
possible for massless theories like gravity where one gets $s^{2}/t$
contribution from the t-channel.

For further analysis we work in the COM frame and we consider general complex
polarizations transverse to the momentum,

$\displaystyle p_{1}=\\{E,0,0,E\\}\;,\;p_{2}=\\{E,0,0,-E\\}\;\text{(COM
frame)}$
$\displaystyle\epsilon_{1}=\epsilon_{3}=\\{0,\alpha,\beta,0\\}\;,\;\epsilon_{2}=\epsilon_{4}=\\{0,\gamma,\delta,0\\}$

Substituting momenta and polarizations in 6 we get,

$\displaystyle\mathcal{A}(s)=$ $\displaystyle
4\frac{s^{2}}{\Lambda^{4}}\Big{[}(2\delta^{ab}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(1)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+(1+\delta^{ab})\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(3)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+2d^{abc}d^{ab}_{\;\;\;c}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(7)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}|\alpha\gamma+\beta\delta|^{2}+|\alpha\gamma^{*}+\beta\delta^{*}|^{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})$
(7) $\displaystyle\hskip
11.38092pt+(2\delta^{ab}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(2)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+(1+\delta^{ab})\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(4)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+2d^{abc}d^{ab}_{\;\;\;c}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(8)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}|\alpha^{*}\delta-\beta^{*}\gamma|^{2}+|\alpha^{*}\delta^{*}-\beta^{*}\gamma^{*}|^{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})$
$\displaystyle\hskip
11.38092pt-(2\delta^{ab}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(5)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+(1+\delta^{ab})\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(6)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+2d^{abc}d^{ab}_{\;\;\;c}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(9)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})$
$\displaystyle\hskip 85.35826pt\times
Re\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\\{(\alpha^{*}\delta-\beta^{*}\gamma)(\alpha\gamma^{*}+\beta\delta^{*})+(\alpha^{*}\delta^{*}-\beta^{*}\gamma^{*})(\alpha\gamma+\beta\delta)\\}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{]}$
$\displaystyle-$ $\displaystyle
9\frac{s^{2}}{\Lambda^{4}}f^{abc}f^{ab}_{\;\;\;c}\Big{[}|\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\alpha\gamma+\beta\delta\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})-\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}^{\prime}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\alpha^{*}\delta^{*}-\beta^{*}\gamma^{*}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})|^{2}+|\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\alpha\gamma^{*}+\beta\delta^{*}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})-\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}^{\prime}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\alpha^{*}\delta-\beta^{*}\gamma\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})|^{2}\Big{]}$

where $Re(z)$ denotes the real part of z. From the above expression we can see
that contribution from $L^{(6)}$ is always negative which means that if we
don’t consider dimension 8 gluonic operators in our EFT then from dispersion
relation, we’ll get

$-9\frac{s^{2}}{\Lambda^{4}}f^{abc}f^{ab}_{\;\;\;c}\Big{[}|\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\alpha\gamma+\beta\delta\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})-\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}^{\prime}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\alpha^{*}\delta^{*}-\beta^{*}\gamma^{*}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})|^{2}+|\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\alpha\gamma^{*}+\beta\delta^{*}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})-\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}^{\prime}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\alpha^{*}\delta-\beta^{*}\gamma\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})|^{2}\Big{]}\geq
0$

which could be satisfied for arbitrary polarizations only when
$c_{6}=0=c_{6}^{\prime}$. It means that dim 6 gluonic operators cannot exist
without the presence of higher dimensional operators in the SMEFT. In appendix
B, we present another example of a dim 6 operator, involving scalar and
fermion, with a similar conclusion.
We get different expressions for $\mathcal{A}(s)$ (having $s^{2}$ dependence)
for different polarizations and colors leading to multiple constraints on
linear combinations of coefficients e.g., for $a=d=1,\;b=e=2$ and
polarizations specified below we get the following expressions:

$\sqrt{2}\epsilon_{1}$ | $\sqrt{2}\epsilon_{2}$ | $\mathcal{A}(s)/(s^{2}/\Lambda^{4})$
---|---|---
$\displaystyle{\\{0,1,i,0\\}}$ | $\displaystyle{\\{0,1,-i,0\\}}$ | $\displaystyle{(4c_{8}^{(4)}-9c_{6}^{\prime 2})+(4c_{8}^{(3)}-9c_{6}^{2})}$
$\displaystyle{\\{0,1,1,0\\}}$ | $\displaystyle{\\{0,1,-1,0\\}}$ | $\displaystyle{2(4c_{8}^{(4)}-9c_{6}^{\prime 2})}$
$\displaystyle{\\{0,1,1,0\\}}$ | $\displaystyle{\\{0,1,1,0\\}}$ | $\displaystyle{2(4c_{8}^{(3)}-9c_{6}^{2})}$
$\displaystyle{\\{0,\sqrt{2},0,0\\}}$ | $\displaystyle{\\{0,1,1,0\\}}$ | $\displaystyle{4(c_{8}^{(4)}+c_{8}^{(3)}-c_{8}^{(6)})-9(c_{6}-c_{6}^{\prime})^{2}}$

We can only constrain the $s\leftrightarrow u$ symmetric $2\rightarrow 2$
scattering amplitudes using the amplitude analysis. Therefore, we consider
real polarizations or particular combinations of helicity amplitudes, such
that the scattering amplitude is $s\leftrightarrow u$ symmetric, to constrain
the Wilson coefficients. In table 2, we list all the different constraints we
get on Wilson coefficients of dim 6 and dim 8 operators from considering
different colors and polarizations. Our constraints when considered together
are stronger than those presented in consistentSMEFT . They are also
consistent with the expressions obtained for Wilson coefficients in num_values
from different UV theories. The inclusion of the dimension 6 operators and the
resulting new constraints are the novel results of this section.

Colors | $\sqrt{2}\epsilon_{1}=\\{0,1,1,0\\}$ | $\sqrt{2}\epsilon_{1}=\\{0,1,1,0\\}$ |  $\sqrt{2}\epsilon_{1}=\left\\{0,1,-1,0\right\\}$ |  $\sqrt{2}\epsilon_{1}=\\{0,1,1,0\\}$
---|---|---|---|---
$\sqrt{2}\epsilon_{2}=\\{0,1,1,0\\}$ | $\sqrt{2}\epsilon_{2}=\\{0,1,-1,0\\}$ |  $\sqrt{2(A+B)}\epsilon_{2}=\left\\{0,\sqrt{A}+\sqrt{B},\sqrt{A}-\sqrt{B},0\right\\}$ |  $\sqrt{2(A+B)}\epsilon_{2}=\left\\{0,\sqrt{A}+\sqrt{B},\sqrt{B}-\sqrt{A},0\right\\}$
$a=1,b=8$ | $\displaystyle{c_{8}^{(3)}+\frac{2}{3}c_{8}^{(7)}}>0$ | $\displaystyle{c_{8}^{(4)}+\frac{2}{3}c_{8}^{(8)}}>0$ |  $\displaystyle{4\left(c_{8}^{(3)}+\frac{2}{3}c_{8}^{(7)}\right)\left({c_{8}^{(4)}+\frac{2}{3}c_{8}^{(8)}}\right)}>\left(c_{8}^{(6)}+\frac{2}{3}c_{8}^{(9)}\right)^{2}$
$a=1,b=2$ | $\displaystyle{c_{8}^{(3)}>\frac{9}{4}c_{6}^{2}}$ | $\displaystyle{c_{8}^{(4)}>\frac{9}{4}c_{6}^{\prime 2}}$ |  $\displaystyle{4\left(c_{8}^{(3)}-\frac{9}{4}c_{6}^{2}\right)\left(c_{8}^{(4)}-\frac{9}{4}c_{6}^{\prime 2}\right)>\left(c_{8}^{(6)}-\frac{9}{2}c_{6}c_{6}^{\prime}\right)^{2}}$
$a=1,b=4$ | $\displaystyle{c_{8}^{(3)}+\frac{1}{2}c_{8}^{(7)}>\frac{9}{16}c_{6}^{2}}$ | $\displaystyle{c_{8}^{(4)}+\frac{1}{2}c_{8}^{(8)}>\frac{9}{16}c_{6}^{\prime 2}}$ |  $\displaystyle{4\left(c_{8}^{(3)}+\frac{1}{2}c_{8}^{(7)}-\frac{9}{16}c_{6}^{2}\right)\left(c_{8}^{(4)}+\frac{1}{2}c_{8}^{(8)}-\frac{9}{16}c_{6}^{\prime 2}\right)>\left(c_{8}^{(6)}+\frac{1}{2}c_{8}^{(9)}-\frac{9}{8}c_{6}c_{6}^{\prime}\right)^{2}}$
$a=4,b=8$ | $\displaystyle{c_{8}^{(3)}+\frac{3}{2}c_{8}^{(7)}>\frac{27}{16}c_{6}^{2}}$ | $\displaystyle{c_{8}^{(4)}+\frac{3}{2}c_{8}^{(8)}>\frac{27}{16}c_{6}^{\prime 2}}$ |  $\displaystyle{4\left(c_{8}^{(3)}+\frac{3}{2}c_{8}^{(7)}-\frac{27}{16}c_{6}^{2}\right)\left(c_{8}^{(4)}+\frac{3}{2}c_{8}^{(8)}-\frac{27}{16}c_{6}^{\prime 2}\right)>\left(c_{8}^{(6)}+\frac{3}{2}c_{8}^{(9)}-\frac{27}{8}c_{6}c_{6}^{\prime}\right)^{2}}$
$a=b=1$ | $\displaystyle{c_{8}^{(1)}+c_{8}^{(3)}+\frac{1}{3}c_{8}^{(7)}}>0$ | $\displaystyle{c_{8}^{(2)}+c_{8}^{(4)}+\frac{1}{3}c_{8}^{(8)}}>0$ |  $\displaystyle{4\left(c_{8}^{(1)}+c_{8}^{(3)}+\frac{1}{3}c_{8}^{(7)}\right)\left(c_{8}^{(2)}+c_{8}^{(4)}+\frac{1}{3}c_{8}^{(8)}\right)}>\left(c_{8}^{(5)}+c_{8}^{(6)}+\frac{1}{3}c_{8}^{(9)}\right)^{2}$
$a=b=4$ | $\displaystyle{c_{8}^{(1)}+c_{8}^{(3)}+c_{8}^{(7)}}>0$ | $\displaystyle{c_{8}^{(2)}+c_{8}^{(4)}+c_{8}^{(8)}}>0$ |  $\displaystyle{4\left(c_{8}^{(1)}+c_{8}^{(3)}+c_{8}^{(7)}\right)\left(c_{8}^{(2)}+c_{8}^{(4)}+c_{8}^{(8)}\right)}>\left(c_{8}^{(5)}+c_{8}^{(6)}+c_{8}^{(9)}\right)^{2}$

Table 2: The table contains the constraints on dim 6 and dim 8 operators’
Wilson coefficients obtained using the amplitude analysis. $\epsilon_{1}$ and
$\epsilon_{2}$ denote the polarizations of particles 1 and 2 respectively. The
color of particle 1 is denoted by ‘$a$’ and that of particle 2 by ‘$b$’.
$A=4(2\delta^{ab}c_{8}^{(1)}+(1+\delta^{ab})c_{8}^{(3)}+2d^{abc}d^{ab}_{\;\;\;c}c_{8}^{(7)})-9f^{abc}f^{ab}_{\;\;\;c}c_{6}^{2}$;
$B=4(2\delta^{ab}c_{8}^{(2)}+(1+\delta^{ab})c_{8}^{(4)}+2d^{abc}d^{ab}_{\;\;\;c}c_{8}^{(8)})-9f^{abc}f^{ab}_{\;\;\;c}c_{6}^{\prime
2}$

For future convenience, we will refer to a particular constraint in table 2
using notation $C(i,j)$ where $i$ refers to the row, and $j$ to the column no.
of the table; for example, $C(2,1)$ refers to the constraint
$\displaystyle{c_{8}^{(3)}>\frac{9}{4}c_{6}^{2}}$.

The bounds in table 2 together significantly reduce the allowed parameter
space of the Wilson coefficients. Interestingly, from constraint C(2,1) and
C(2,2) we can directly see that for the existence of dim 6 operators we need
some specific dim 8 operators to be present. For example, for
$Q_{G^{3}}^{(1)}$ we need $Q_{G^{4}}^{(3)}$ to exist and for $Q_{G^{3}}^{(2)}$
we need $Q_{G^{4}}^{(4)}$. This aspect was appreciated in the case of the
electroweak bosons in constraintvectorboson . More bounds involving dim 8
operators can be obtained by relaxing the elastic forward scattering limit as
shown in Convex_geometry .

##  4 Superluminality

We saw in the previous section that assuming an EFT to have a UV-completion
that is Lorentz invariant and unitary one gets positivity constraints on the
Wilson coefficients using $2\rightarrow 2$ forward scattering amplitudes. It
is well known that one can also reproduce some of these constraints by
referring only to IR physics. It turns out that arbitrary values or signs of
these coefficients can lead to superluminal propagation of field fluctuations
over non-trivial backgrounds causality . Therefore, instead of working with an
S-matrix, we can work with classical wave propagation to derive interesting
bounds on the Wilson coefficients by demanding the EFT to be compatible with
causality. A priori, it is not absolutely clear whether this analysis would
give weaker or stronger bounds compared to the ‘amplitude analysis’.

In this section, we first look at the classical causality/subluminality
analysis more closely; we investigate how this analysis is modified for
massive fields. We then proceed to apply this analysis to the Gluonic
operators considered in sec. 3.2. Let us first briefly outline how
superluminality analysis can be used to put bounds on the Wilson coefficients
by demanding that signal velocity cannot be superluminal.

Consider the following Goldstone Lagrangian,

$\mathcal{L}=\frac{1}{2}(\partial\pi)^{2}+\frac{c_{3}}{\Lambda^{4}}(\partial\pi)^{4}$
(8)

Naively, it may seem that the 8-dimensional operator will not contribute to
the free propagation of perturbations as it is not a quadratic term. However,
this is only true for trivial backgrounds; for non-trivial backgrounds with
non-vanishing derivatives one can get terms quadratic in perturbations,

$\mathcal{L}\hskip
5.0pt{\supset}\frac{c_{3}}{\Lambda^{4}}(\partial\pi_{0})^{2}(\partial\pi)^{2}$
(9)

where $\pi_{0}$ is the background field. One has to be slightly careful in
choosing a background as it might happen that the background itself disallows
wave-like (propagating) solutions in which case it would not be possible to
run this analysis. Therefore, we usually choose a background whose scale of
variation is much larger than that of field fluctuations, allowing for wave-
like solutions.

It is also important to point out that given a general dispersion relation,
one cannot always directly demand phase or group velocity to be subluminal as
it is well known in the literature that they both can be superluminal while
remaining in perfect agreement with causality Fox ; superluminalvel ;
Recami_2009 . Instead, one usually demands the wavefront velocity be luminal.
However, as already mentioned in sec.1, one has the so-called Kramers-Kronig
relation G.M.Shore ; Jackson:1998nia

$n(\infty)=n(0)-\frac{2}{\pi}\int_{0}^{\infty}\frac{d\omega}{\omega}\textit{Im}\hskip
2.0ptn(\omega)$ (10)

where $n(\omega)$ denotes the refractive index of the medium. For dissipative
mediums, one has ${\rm Im}\hskip 1.0ptn(\omega)>0$ which implies that the high
frequency phase velocity is larger than the low frequency one. And since the
wavefront velocity - infinite frequency limit of phase velocity - dictates the
speed of information transfer, superluminal phase or group velocity can be
related to violation of causality. However, if the medium allows for
exhibiting gain (e.g., in the case of a Laser) then one can get ${\rm
Im}\hskip 1.0ptn(\omega)<0$ and in that case, the low-frequency phase velocity
cannot be a statement about causality.

The situation gets even more involved if we have a mass-like term in the
equation of motion. For example, take the massive Klein-Gordon equation

$\partial^{2}\phi+m^{2}\phi=0$ (11)

From the above $\text{eq}^{\text{n}}$ we get the following dispersion relation
and phase velocity,

$\displaystyle\omega^{2}=|\vec{k}|^{2}+m^{2}\qquad;\qquad
v_{p}^{2}=1+\frac{m^{2}}{|\vec{k}|^{2}}$ (12)

In this case phase velocity is not a good object to consider since
$v_{p}(\infty)$=1 (preserving causality) while being superluminal in the low
energy limit (EFT regime). The group velocity on the other hand is
(sub)luminal

$v_{g}=\frac{|\vec{k}|}{\sqrt{|\vec{k}|^{2}+m^{2}}}$ (13)

It was shown in Fox that it is the group velocity which is equal to the signal
velocity for Klein-Gordon modes with real mass.

###  4.1 Massive goldstone boson

Consider the following Lagrangian

$\mathcal{L}=\frac{1}{2}(\partial\pi)^{2}+\frac{c_{3}}{\Lambda^{4}}(\partial\pi)^{4}-\frac{1}{2}m^{2}\pi^{2}-J\pi$
(14)

where $J$ is what sources the background, with the following EOM

$\partial^{2}\pi\left(1+\frac{4c_{3}(\partial\pi)^{2}}{\Lambda^{4}}\right)+\frac{8c_{3}}{\Lambda^{4}}(\partial_{\nu}\pi)(\partial^{\mu}\partial^{\nu}\pi)(\partial_{\mu}\pi)+m^{2}\pi+J=0$
(15)

The linearised EOM for the fluctuations $\xi=\pi-\pi_{0}$ with
$\partial_{\mu}\pi_{0}=C_{\mu}$(constant) reads

$\displaystyle\partial^{2}\xi+\frac{8c_{3}}{\Lambda^{4}}C^{\mu}C^{\nu}\partial_{\mu}\partial_{\nu}\xi-\frac{4c_{3}C^{2}}{\Lambda^{4}}m^{2}\xi+m^{2}\xi=0$
(16)

where we have assumed that the background terms on the l.h.s are canceled by
the source $J$. Taking Fourier transform we get,

$\displaystyle
k_{\mu}k^{\mu}=-\frac{8c_{3}}{\Lambda^{4}}\left(C.k\right)^{2}-\frac{4c_{3}C^{2}}{\Lambda^{4}}m^{2}+m^{2}$
(17)

If the mass term is absent then subluminality demands $c_{3}>0$ as obtained by
causality . As mentioned in the above discussion, for a Klein-Gordon like
dispersion relation it is the group velocity that should be (sub)luminal since
the group velocity is the signal velocity. It might seem unclear whether this
holds true here (since the dispersion relation is more general here, and not
exactly Klein-Gordon like). However, with the choice of a purely space-like
background, $C_{\mu}=\left(0,0,0,C_{(3)}\right)$, the dispersion relation
takes the following form,

$\omega^{2}=ak^{2}+m^{\prime 2}$ (18)

where, $a=1-\frac{8c_{3}C^{2}_{(3)}}{\Lambda^{4}}$ and $m^{\prime
2}=m^{2}\left(1+\frac{4c_{3}C^{2}_{(3)}}{\Lambda^{4}}\right)$. This form is
very close to Klein-Gordon since the parameter $a$ is very close to unity;
therefore, one can perform the analysis assuming that the group velocity is
signal velocity.

We now consider the propagation of perturbations along the z-axis, then the
expression for group velocity from $\text{eq}^{\text{n}}$(17) reads,

$\displaystyle
v_{g}=\frac{d\omega}{dk}=\frac{k+\frac{8c_{3}C_{(3)}}{\Lambda^{4}}(C_{(0)}\omega-
C_{(3)}k)}{\sqrt{k^{2}+m^{2}-\frac{4c_{3}C^{2}m^{2}}{\Lambda^{4}}-\frac{8c_{3}}{\Lambda^{4}}(C_{(0)}\omega-
C_{(3)}k)^{2}}+\frac{8c_{3}C_{(0)}}{\Lambda^{4}}(C_{(0)}\omega-C_{(3)}k)}$
(19)

We choose the background to be varying only in the z-direction i.e.
$C_{\mu}=\left(0,0,0,C_{(3)}\right)$, then demanding the group velocity to be
subluminal gives

$\frac{m^{2}}{k^{2}}\left(1+\frac{4c_{3}C_{(3)}^{2}}{\Lambda^{4}}\right)+\frac{8c_{3}C_{(3)}^{2}}{\Lambda^{4}}>0$
(20)

Taking the limit $k\gg m$, we can drop the second term in the parenthesis and
get

$\frac{m^{2}}{k^{2}}+8c_{3}\varepsilon>0$ (21)

where, $0<\varepsilon=\frac{C_{(3)}^{2}}{\Lambda^{4}}\ll 1$. Finally, we get

$\displaystyle c_{3}>-\frac{m^{2}/k^{2}}{8\varepsilon}$ (22)

This is a ratio of two small positive quantities. Therefore, one does not get
a strict positivity bound in this case from the superlumianlity analysis,
unlike for the massless pions. However, the amplitude analysis still gives a
strict positivity bound on $c_{3}$ as it is unaffected by the mass term.

###  4.2 Gluon field strength operators

We now attempt to derive constraints on the Wilson coefficients of the Gluonic
operators considered in sec. 3.2, using the superluminality analysis. As we
will see below, here we will not need to take recourse to the Kramers-Kronig
relation because we will be able to choose backgrounds where the dispersion
relation takes the simplest form i.e it is non-dispersive:

$\displaystyle\omega=v|\vec{k}|$ (23)

where $v$ is a constant. In this case, phase and group velocities are the same
and are equal to the signal velocity. Due to the non-abelian nature of
Lagrangian, here the calculations are tedious and are given in Appendix A.
Here, we just state the main results highlighting the key assumptions that
went into the analysis. We’ll work with one higher-dimensional operator at a
time keeping the calculations and analysis easy to follow.

First, we’ll take dim 8 operator
$Q_{G^{4}}^{(1)}=\left(G_{\mu\nu}^{a}G^{a\mu\nu}\right)\left(G_{\rho\sigma}^{b}G^{b\rho\sigma}\right)$
in addition to the four dimensional term in the Lagrangian

$L=-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}+\frac{c_{8}^{(1)}}{\Lambda^{4}}\left(G_{\mu\nu}^{a}G^{a\mu\nu}\right)\left(G_{\rho\sigma}^{b}G^{b\rho\sigma}\right)$

giving equation of motion,

$\displaystyle-\partial_{\alpha}G^{f,\alpha\beta}+\frac{8c_{8}^{(1)}}{\Lambda^{4}}\left(2G_{\mu\nu}^{a}(\partial_{\alpha}G^{a,\mu\nu})G^{f,\alpha\beta}+G_{\mu\nu}^{a}G^{a,\mu\nu}\partial_{\alpha}G^{f,\alpha\beta}\right)$
(24) $\displaystyle=$ $\displaystyle-
g_{s}G^{a,\beta\nu}A_{\nu}^{h}f^{afh}+\frac{8c_{8}^{(1)}}{\Lambda^{4}}g_{s}f^{bfj}G^{a}_{\mu\nu}G^{a,\mu\nu}G^{b,\beta\sigma}A_{\sigma}^{j}$

We expand $A^{a,\mu}=A_{0}^{a,\mu}+h^{a,\mu}$ where we choose background
$A_{0}^{a,\mu}$ to be of a particular color ‘$a$’ with
$\partial^{\nu}A^{a,\mu}=constant$; such background also solves the equation
of motion (24). We look at the linearised equation of motion for the
perturbation $h^{a,\mu}$ of the same color ‘$a$’ as that of the background,

$\displaystyle-\partial_{\alpha}\partial^{\alpha}h^{a,\beta}+\frac{32c_{8}^{(1)}}{\Lambda^{4}}F^{a}_{0\mu\nu}F_{0}^{a\alpha\beta}\partial_{\alpha}\partial^{\mu}h^{a,\nu}=0$
(25)

where color index ‘$a$’ is not contracted, and we’ll drop the color index for
$F_{0}^{\mu\nu}$ terms which should be assumed to have color ‘$a$’. Above
$\text{eq}^{\text{n}}$ when expanded in terms of plane waves gives,

$\displaystyle
k_{\mu}k^{\mu}=-\frac{32c_{8}^{(1)}}{\Lambda^{4}}(F_{0\mu\nu}\epsilon^{\nu}k^{\mu})^{2}$
(26)

When a similar procedure is done considering all 6 and 8 dimensional
operators, we get the following equation (color index ‘$c$’ is contracted but
not ‘$a$’),

$\displaystyle k_{\mu}k^{\mu}=-\frac{32}{\Lambda^{4}}\Bigg{\\{}$
$\displaystyle\left(c_{8}^{(1)}+c_{8}^{(3)}+d^{aa}_{\hskip
8.53581ptc}d^{aac}c_{8}^{(7)}\right)\Big{(}F_{0\mu\nu}\epsilon^{\nu}k^{\mu}\Big{)}^{2}$
(27) $\displaystyle+$
$\displaystyle\left(c_{8}^{(2)}+c_{8}^{(4)}+d^{aa}_{\hskip
8.53581ptc}d^{aac}c_{8}^{(8)}\right)\left(\widetilde{F}_{0\mu\nu}\epsilon^{\nu}k^{\mu}\right)^{2}$
$\displaystyle+$ $\displaystyle\left(c_{8}^{(5)}+c_{8}^{(6)}+d^{aa}_{\hskip
8.53581ptc}d^{aac}c_{8}^{(9)}\right)\left(F_{0\mu\nu}\widetilde{F}_{0\alpha\beta}\epsilon^{\nu}\epsilon^{\beta}k^{\mu}k^{\alpha}\right)\Bigg{\\}}$

If we look at the perturbations of the same color as that of the background,
then we are effectively considering only the abelian part of our theory. Since
there is no analog of dim 6 operators $Q^{(1)}_{G^{3}}$ and $Q^{(2)}_{G^{3}}$
in the abelian gauge theory, there is no contribution from dim 6 operators
towards the wave propagation. However, we still see signs of the existence of
different gluon colors in the form of $d^{aa}_{\hskip 8.53581ptc}d^{aac}$
factors which give different values for different colors in consideration.

By choosing particular background and polarization for the perturbation, we
can get different bounds on Wilson coefficients $c_{8}^{(i)}$ using dispersion
relation (27). Consider the z-axis along the direction of propagation of
perturbation, perturbation of the same color as that of background let say 1)
with polarization, $\epsilon=\\{0,1,1,0\\}/\sqrt{2}$. Choose the background
such that only non-zero components of $F^{\mu\nu}$ are $F^{01}=-F^{10}$ and
$F^{02}=-F^{20}$ (we have dropped $0$ from $F_{0}^{\mu\nu}$ to avoid confusion
with time component), then we get

$\displaystyle k_{\mu}k^{\mu}=-\frac{16}{\Lambda^{4}}\Bigg{\\{}$
$\displaystyle\left(c_{8}^{(1)}+c_{8}^{(3)}+\frac{1}{3}c_{8}^{(7)}\right)\omega^{2}\Big{(}F_{01}+F_{02}\Big{)}^{2}$
(28) $\displaystyle+$
$\displaystyle\left(c_{8}^{(2)}+c_{8}^{(4)}+\frac{1}{3}c_{8}^{(8)}\right)\left(k^{3}\right)^{2}\Big{(}F_{02}-F_{01}\Big{)}^{2}$
$\displaystyle+$
$\displaystyle\left(c_{8}^{(5)}+c_{8}^{(6)}+\frac{1}{3}c_{8}^{(9)}\right)\omega
k^{3}\left(F_{02}+F_{01}\right)\left(F_{02}-F_{01}\right)\Bigg{\\}}$

Now, if we take $F_{01}=F_{02}$ then for perturbations to be causal we need

$\displaystyle c_{8}^{(1)}+c_{8}^{(3)}+\frac{1}{3}c_{8}^{(7)}\geq 0$ (29)

which is same as the constraint $C(5,1)$ obtained in sec 3.2. Similarly by
choosing different polarization and background, explicitly given in the table
3, we can reproduce $C(5,2)$ and $C(5,3)$. Also, if instead of color 1 we
choose color 4 for both background and perturbations then we get C(6,1),
C(6,2) and C(6,3).

Now to probe dim 6 operators using superluminality analysis, we need to
consider the background and perturbations of different colors. This makes the
analysis rather involved. We’ll first consider operator
$Q_{G^{3}}^{(1)}=f^{abc}G_{\mu}^{a\nu}G_{\nu}^{b\rho}G_{\rho}^{c\mu}$ for
which we have,

$\displaystyle
L=-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}+\frac{c_{6}}{\Lambda^{2}}f^{abc}G_{\mu}^{a\nu}G_{\nu}^{b\rho}G_{\rho}^{c\mu}$

giving equation of motion,

$\displaystyle-\partial_{\alpha}G^{f,\alpha\beta}+\frac{6c_{6}}{\Lambda^{2}}f^{fbc}\partial_{\alpha}\left(G^{c,\rho\alpha}G^{b,\beta\rho}\right)=-g_{s}G^{a,\beta\nu}A_{\nu}^{h}f^{afh}+\frac{6c_{6}}{\Lambda^{2}}g_{s}f^{abc}\left(f^{afh}G^{b,\nu\rho}G_{\hskip
1.0pt\rho}^{c,\beta}A_{\nu}^{h}\right)$ (30)

We again expand $A^{a,\mu}=A_{0}^{a,\mu}+h^{a,\mu}$ where we choose background
$A_{0}^{a,\mu}$ to be of particular color ‘$a$’ with
$\partial^{\nu}A^{a,\mu}=constant$, and look at the linearised equation of
motion for perturbation $h^{a,\mu}$ of different color ‘$f$’,

$\displaystyle-\partial_{\alpha}\left(H^{f,\alpha\beta}+g_{s}f^{faj}A_{0}^{a,\alpha}h^{j,\beta}+g_{s}f^{fia}h^{i,\alpha}A_{0}^{a,\beta}\right)$
(31)
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}f^{fba}\partial_{\alpha}\Big{(}F_{0}^{a,\rho\alpha}(H^{b,\beta}_{\hskip
11.38109pt\rho}+g_{s}f^{baj}A_{0}^{a,\beta}h^{j}_{\rho}+g_{s}f^{bia}h^{i,\beta}A^{a}_{0\rho})$
$\displaystyle\hskip 56.9055pt-F_{0}^{a,\beta\rho}(H_{\hskip
1.0pt\rho}^{b,\alpha}+g_{s}f^{baj}A^{a}_{0\rho}h^{j,\alpha}+g_{s}f^{bia}h^{i}_{\rho}A_{0}^{a,\alpha})\Big{)}$
$\displaystyle=-g_{s}\left(H^{d,\beta\nu}+g_{s}f^{daj}A_{0}^{a,\beta}h^{j,\nu}+g_{s}f^{dia}h^{i,\beta}A_{0}^{a,\nu}\right)A_{0\nu}^{a}f^{dfa}-g_{s}F_{0}^{a,\beta\nu}h_{\nu}^{h}f^{afh}$
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}g_{s}f^{daa}\left(f^{dfh}F_{0}^{a,\nu\rho}F_{0\rho}^{a,\beta}h_{\nu}^{h}\right)$
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}g_{s}f^{dba}\left(f^{dfa}(H^{b,\nu\rho}+g_{s}f^{baj}A_{0}^{a,\nu}h^{j,\rho}+g_{s}f^{bia}h^{i,\nu}A_{0}^{a,\rho})F_{0\rho}^{a,\beta}A_{0\nu}^{a}\right)$
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}g_{s}f^{dac}\left(f^{dfa}F_{0}^{a,\nu\rho}(H_{\hskip
1.0pt\rho}^{c,\beta}+g_{s}f^{caj}A^{a}_{0\rho}h^{j,\beta}+g_{s}f^{cia}h^{i}_{\rho}A_{0}^{a,\beta})A_{0\nu}^{a}\right)$

In the above equation and the following equations, ‘$f$’ and ‘$a$’ are not
contracted but are free color indices and we’ll again drop ‘$a$’ from
$F_{0}^{\mu\nu}$ terms which should always be assumed to have color ‘$a$’.

For further analysis, we take WKB approximation in which the scale of
variation of background($r$) is much larger than that of perturbations
($\omega^{-1}$). We also choose the background field to be arbitrarily small,
then at leading order in $r\omega$ and $A_{0}$ we get,

$\displaystyle-\partial_{\alpha}\partial^{\alpha}h^{f,\beta}+2g_{s}f^{fba}A_{0}^{a,\alpha}\partial_{\alpha}h^{b,\beta}-g_{s}f^{fba}A_{0\rho}\partial^{\beta}h^{b,\rho}$
(32)
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}f^{fba}\Big{(}F_{0\rho\alpha}\partial^{\alpha}\partial^{\beta}h^{b,\rho}+F_{0}^{\beta\rho}\partial_{\alpha}\partial^{\alpha}h_{\rho}^{b}\Big{)}=0$

To get the dispersion relation for perturbation of color ‘f’, we try to write
the differential equation just in terms of color ‘f’. So we assume a
particular solution for perturbation of some different color ‘$b$’ which
satisfies the EOM,

$\displaystyle-\partial^{\alpha}h^{b,\rho}+2g_{s}f^{bga}A_{0}^{a,\alpha}h^{g,\rho}-g_{s}f^{bga}A_{0\nu}\delta^{\alpha\rho}h^{g,\nu}+\frac{6c_{6}}{\Lambda^{2}}f^{bga}\left(F_{0}^{\sigma\alpha}\partial^{\rho}h^{g}_{\sigma}+F_{0}^{\rho\sigma}\partial^{\alpha}h_{\sigma}^{g}\right)=0$
(33)

The above particular solution for other colors ‘$b$’ modifies the
$\text{eq}^{\text{n}}$(32) to,

$\displaystyle-\partial^{\alpha}\partial_{\alpha}h^{f,\beta}=$ $\displaystyle
2g_{s}^{2}f^{fba}f^{gba}\left(2A_{0}^{\alpha}A_{0\alpha}h^{g,\beta}-\frac{3}{2}A_{0\nu}A_{0}^{\beta}h^{g,\nu}\right)$
(34)
$\displaystyle+g\frac{6c_{6}}{\Lambda^{2}}f^{fba}f^{gba}\left(5F_{0}^{\beta\rho}A_{0}^{\alpha}\partial_{\alpha}h^{g,\rho}-A_{0\nu}F_{0}^{\rho\beta}\partial^{\rho}h^{g,\nu}+5F_{0\rho\alpha}A_{0}^{\alpha}\partial^{\beta}h^{g,\rho}\right)$
$\displaystyle+36\frac{c_{6}^{2}}{\Lambda^{4}}f^{fba}f^{gba}\left(F_{0}^{\beta\rho}F_{0}^{\sigma\alpha}\partial_{\alpha}\partial_{\rho}h^{g}_{\sigma}+F^{\beta}_{0\rho}F_{0}^{\rho\sigma}\partial^{\alpha}\partial_{\alpha}h^{g}_{\sigma}+2F_{0}^{\rho\alpha}F_{0}^{\sigma\alpha}\partial^{\beta}\partial_{\rho}h^{g}_{\sigma}\right)$

Since we have assumed a particular type of solution for other colors, we want
to see how that affects the perturbation of color ‘$f$’, so we try to write
the differential equation just in terms of perturbation of color ‘$f$’ as
mentioned before. But it is not possible to replace the mass-like term (the
term inside parenthesis in the first line) in the above equation unless we
have an explicit solution for perturbations of all color $h^{g,\beta}$. So,
for now, we’ll assume that we can choose a particular background $A_{0}$ such
that the mass-like term vanishes. We’ll give below an explicit example of
background and polarization of the perturbation where this assumption is
satisfied.

In the second and third lines of the $\text{eq}^{\text{n}}$(34), when $g\neq
f$ we again substitute $h^{g}$ in terms of other colors using the solution
assumed $\text{eq}^{\text{n}}$(33). But this gives terms of higher order in
$A_{0}$ or $\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)$ which we can ignore
w.r.t the terms where $g=f$. Therefore only terms with $g=f$ survives at
$\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)$ and leading order in $A_{0}$,

$\displaystyle-\partial^{\alpha}\partial_{\alpha}h^{f,\beta}=$ $\displaystyle
g\frac{6c_{6}}{\Lambda^{2}}f^{fba}f^{fba}\left(5F_{0}^{\beta\rho}A_{0}^{\alpha}\partial_{\alpha}h^{f,\rho}-A_{0\nu}F_{0}^{\rho\beta}\partial^{\rho}h^{f,\nu}+5F_{0\rho\alpha}A_{0}^{\alpha}\partial^{\beta}h^{f,\rho}\right)$
(35) $\displaystyle+$ $\displaystyle
36\frac{c_{6}^{2}}{\Lambda^{4}}f^{fba}f^{fba}\left(F_{0}^{\beta\rho}F_{0}^{\sigma\alpha}\partial_{\alpha}\partial_{\rho}h^{f}_{\sigma}+F^{\beta}_{0\rho}F_{0}^{\rho\sigma}\partial^{\alpha}\partial_{\alpha}h^{f}_{\sigma}+2F_{0}^{\rho\alpha}F_{0}^{\sigma\alpha}\partial^{\beta}\partial_{\rho}h^{f}_{\sigma}\right)$

We expand the above equation in terms of waves with transverse polarization
and consider the spatial wave vector, $\vec{\kappa}$ to be complex in general.
We then get the following dispersion relation,

$\displaystyle\omega^{2}=|\vec{k}|^{2}+36\frac{c_{6}^{2}}{\Lambda^{4}}(f^{fba})^{2}\left(F_{0}^{\beta\rho}k_{\rho}\epsilon_{\beta}\right)^{2}$
(36)

where $\vec{k}$ denotes the real part of the spatial wave vector. After
considering all dim 6 and dim 8 operators we get the following dispersion
relation,

$\displaystyle\frac{k_{\mu}k^{\mu}}{4}=$
$\displaystyle\frac{9}{\Lambda^{4}}f^{afc}f^{af}_{\;\;\;c}\Big{[}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}F_{0}^{\mu\nu}k_{\mu}\epsilon_{\nu}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})-\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}^{\prime}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\widetilde{F}_{0}^{\alpha\beta}k_{\alpha}\epsilon_{\beta}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\Big{]}^{2}$
(37)
$\displaystyle-\frac{4}{\Lambda^{4}}\Big{[}(2\delta^{af}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(1)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+(1+\delta^{af})\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(3)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+2d^{afc}d^{af}_{\;\;\;c}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(7)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left(F_{0}^{\mu\nu}k_{\mu}\epsilon_{\nu}\right)^{2}$
$\displaystyle\hskip
19.91684pt+(2\delta^{af}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(2)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+(1+\delta^{af})\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(4)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+2d^{afc}d^{af}_{\;\;\;c}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(8)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left(\widetilde{F}_{0}^{\alpha\beta}k_{\alpha}\epsilon_{\beta}\right)^{2}$
$\displaystyle\hskip
19.91684pt-(2\delta^{af}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(5)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+(1+\delta^{af})\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(6)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+2d^{afc}d^{af}_{\;\;\;c}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(9)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left(F_{0}^{\mu\nu}k_{\mu}\epsilon_{\nu}\right)\left(\widetilde{F}_{0}^{\alpha\beta}k_{\alpha}\epsilon_{\beta}\right)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{]}$

where ‘$f$’ denotes the color of perturbation and ‘$a$’ of the background.
Note that the above dispersion relation is only valid, assuming that mass-like
term in the $\text{eq}^{\text{n}}$(34) vanish. We had a similar situation in
the previous section where we had a mass-like term dependent on the background
field, which refrained us from getting a constraint on the Wilson coefficients
of the theory. But in this case, since it depends on contracted four-vectors,
it is possible to make the mass-like term zero along with non-zero derivatives
of background by choosing an appropriate $A_{0}^{\mu}$.

Colors | $\sqrt{2}\epsilon=\\{0,1,1,0\\}$ | $\sqrt{2}\epsilon=\\{0,1,1,0\\}$
---|---|---
$A_{0\mu}=E\\{\sqrt{2}(x+y),x+y,-(x+y),0\\}$ | $A_{0\mu}=E\\{\sqrt{2}t,t,-t,0\\}$
$a=1,f=8$ | $\displaystyle{c_{8}^{(3)}+\frac{2}{3}c_{8}^{(7)}}>0$ | $\displaystyle{c_{8}^{(4)}+\frac{2}{3}c_{8}^{(8)}}>0$
$a=1,f=2$ | $\displaystyle{c_{8}^{(3)}>\frac{9}{4}c_{6}^{2}}$ | $\displaystyle{c_{8}^{(4)}>\frac{9}{4}c_{6}^{\prime 2}}$
$a=1,f=4$ | $\displaystyle{c_{8}^{(3)}+\frac{1}{2}c_{8}^{(7)}>\frac{9}{16}c_{6}^{2}}$ | $\displaystyle{c_{8}^{(4)}+\frac{1}{2}c_{8}^{(8)}>\frac{9}{16}c_{6}^{\prime 2}}$
$a=4,f=8$ | $\displaystyle{c_{8}^{(3)}+\frac{3}{2}c_{8}^{(7)}>\frac{27}{16}c_{6}^{2}}$ | $\displaystyle{c_{8}^{(4)}+\frac{3}{2}c_{8}^{(8)}>\frac{27}{16}c_{6}^{\prime 2}}$
$a=f=1$ | $\displaystyle{c_{8}^{(1)}+c_{8}^{(3)}+\frac{1}{3}c_{8}^{(7)}}>0$ | $\displaystyle{c_{8}^{(2)}+c_{8}^{(4)}+\frac{1}{3}c_{8}^{(8)}}>0$
$a=f=4$ | $\displaystyle{c_{8}^{(1)}+c_{8}^{(3)}+c_{8}^{(7)}}>0$ | $\displaystyle{c_{8}^{(2)}+c_{8}^{(4)}+c_{8}^{(8)}}>0$
Colors | $\sqrt{2}\epsilon=\left\\{0,1,-1,0\right\\}$ | $\sqrt{2}\epsilon=\\{0,1,1,0\\}$
$A_{0\mu}=E\\{\sqrt{2(D+B)}t,{(\sqrt{D}+\sqrt{B})}t,{(\sqrt{D}-\sqrt{B})}t,0\\}$ | $A_{0\mu}=E\\{\sqrt{2(D+B)}t,(\sqrt{D}+\sqrt{B})t,(\sqrt{B}-\sqrt{D})t,0\\}$
$a=1,f=8$ |  $\displaystyle{4\left(c_{8}^{(3)}+\frac{2}{3}c_{8}^{(7)}\right)\left({c_{8}^{(4)}+\frac{2}{3}c_{8}^{(8)}}\right)}>\left(c_{8}^{(6)}+\frac{2}{3}c_{8}^{(9)}\right)^{2}$
$a=1,f=2$ |  $\displaystyle{4\left(c_{8}^{(3)}-\frac{9}{4}c_{6}^{2}\right)\left(c_{8}^{(4)}-\frac{9}{4}c_{6}^{\prime 2}\right)>\left(c_{8}^{(6)}-\frac{9}{2}c_{6}c_{6}^{\prime}\right)^{2}}$
$a=1,f=4$ |  $\displaystyle{4\left(c_{8}^{(3)}+\frac{1}{2}c_{8}^{(7)}-\frac{9}{16}c_{6}^{2}\right)\left(c_{8}^{(4)}+\frac{1}{2}c_{8}^{(8)}-\frac{9}{16}c_{6}^{\prime 2}\right)>\left(c_{8}^{(6)}+\frac{1}{2}c_{8}^{(9)}-\frac{9}{8}c_{6}c_{6}^{\prime}\right)^{2}}$
$a=4,f=8$ |  $\displaystyle{4\left(c_{8}^{(3)}+\frac{3}{2}c_{8}^{(7)}-\frac{27}{16}c_{6}^{2}\right)\left(c_{8}^{(4)}+\frac{3}{2}c_{8}^{(8)}-\frac{27}{16}c_{6}^{\prime 2}\right)>\left(c_{8}^{(6)}+\frac{3}{2}c_{8}^{(9)}-\frac{27}{8}c_{6}c_{6}^{\prime}\right)^{2}}$
$a=f=1$ |  $\displaystyle{4\left(c_{8}^{(1)}+c_{8}^{(3)}+\frac{1}{3}c_{8}^{(7)}\right)\left(c_{8}^{(2)}+c_{8}^{(4)}+\frac{1}{3}c_{8}^{(8)}\right)}>\left(c_{8}^{(5)}+c_{8}^{(6)}+\frac{1}{3}c_{8}^{(9)}\right)^{2}$
$a=f=4$ |  $\displaystyle{4\left(c_{8}^{(1)}+c_{8}^{(3)}+c_{8}^{(7)}\right)\left(c_{8}^{(2)}+c_{8}^{(4)}+c_{8}^{(8)}\right)}>\left(c_{8}^{(5)}+c_{8}^{(6)}+c_{8}^{(9)}\right)^{2}$

Table 3: The table contains the constraints on dim 6 and dim 8 operators’
Wilson coefficients obtained using the superluminality analysis. $A_{0\mu}$
and $\epsilon$ represent the background field and polarization of the
perturbation, respectively. The color of the background is denoted by ‘$a$’
and that of perturbation by ‘$f$’.
$D=4(2\delta^{ab}c_{8}^{(1)}+(1+\delta^{ab})c_{8}^{(3)}+2d^{abc}d^{ab}_{\;\;\;c}c_{8}^{(7)})-9f^{abc}f^{ab}_{\;\;\;c}c_{6}^{2}$;
$B=4(2\delta^{ab}c_{8}^{(2)}+(1+\delta^{ab})c_{8}^{(4)}+2d^{abc}d^{ab}_{\;\;\;c}c_{8}^{(8)})-9f^{abc}f^{ab}_{\;\;\;c}c_{6}^{\prime
2}$

We now try to get different constraints on Wilson coefficients by considering
particular polarization and the background. Consider the perturbation with
polarization $\epsilon=\\{0,1,1,0\\}/\sqrt{2}$ and choose background of the
form $A_{0\mu}=E\\{\sqrt{2}t,t,-t,0\\}$ where E is some arbitrarily small
constant. Under this configuration, mass-like term in
$\text{eq}^{\text{n}}$(34) vanish and we get following non-zero components of
$F_{\mu\nu}$, $F_{01}=-F_{10}$ and $F_{02}=-F_{20}$ with $F_{01}=-F_{02}$,
which reduces the dispersion relation (37) to the following form,

$\displaystyle
k_{\mu}k^{\mu}=\frac{72}{\Lambda^{4}}f^{afc}f^{af}_{\;\;\;c}(c^{\prime}_{6})^{2}\left(k^{3}\right)^{2}\left(F_{02}\right)^{2}-\frac{32}{\Lambda^{4}}\left(c_{8}^{(4)}+2d^{afc}d^{af}_{\;\;\;c}c_{8}^{(8)}\right)\left(k^{3}\right)^{2}\left(F_{02}\right)^{2}$
(38)

and for the perturbation to be causal (dictated by the phase velocity) we
require,

$\displaystyle
9f^{afc}f^{af}_{\;\;\;c}(c^{\prime}_{6})^{2}-4\left(c_{8}^{(4)}+2d^{afc}d^{af}_{\;\;\;c}c_{8}^{(8)}\right)<0$
(39)

We can reproduce the $C(1,2),C(2,2),C(3,2)$ and $C(4,2)$ bounds of table 2
using the above relation by choosing different colors for perturbations and
background. The remaining bounds of table 2 can also be reproduced by choosing
different background and polarization configurations, details of which have
been relegated to the appendix A. In the table 3, we present all the bounds
obtained on dim 6 and 8 gluonic operators using the superluminality analysis
by considering different configurations of the background and perturbation.

###  4.3 A Non-relativistic Example: Stronger bound from Superluminality

Since we got similar bounds from both superluminality and amplitude analysis,
therefore, one might think that this is always the case. Here, following
Baumann_2016 we present a counter-example where superluminality gives a
stronger bound. Consider the following Lagrangian

$\displaystyle\mathcal{L}=\frac{1}{2}(\partial\pi)^{2}-\frac{4c_{3}}{3\Lambda^{2}}\dot{\pi}^{3}+\frac{2c_{4}}{3\Lambda^{4}}\dot{\pi}^{4}$
(40)

This Lagrangian emerges from the EFT of Inflation inflation Lagrangian in a
particular limit Baumann_2016 .
The $2\rightarrow 2$ forward scattering amplitude at tree level is given by

$\displaystyle\mathcal{A}(s)=\left(c_{4}-(2c_{3})^{2}\right)\frac{s^{2}}{\Lambda^{4}}$
(41)

Performing the ‘amplitude analysis’ gives the following bound

$\displaystyle c_{4}>(2c_{3})^{2}$ (42)

As a side remark, it is important to note that, due to subtleties related to
spontaneously broken Lorentz invariance, the derivation of the above bounds in
Baumann_2016 may not be completely rigorous, see Creminelli_2022 for a recent
discussion.
Now, let us check what superluminality gives for this Lagrangian. We derive
linearised EOM for the fluctuations, $\xi=\pi+\alpha t$, where $\alpha$ is a
small quantity.

$\displaystyle\ddot{\xi}+\frac{8c_{3}}{\Lambda^{2}}\alpha\ddot{\xi}+\frac{8c_{4}}{\Lambda^{4}}\alpha^{2}\ddot{\xi}-\partial_{i}^{2}\xi=0$
(43)

Up to $O(\alpha)$ we have

$\displaystyle\ddot{\xi}+\frac{8c_{3}}{\Lambda^{2}}\alpha\ddot{\xi}-\partial_{i}^{2}\xi=0$
(44)

then the phase velocity of the fluctuation is given by

$\displaystyle v_{phase}^{2}=1-\frac{8c_{3}\alpha}{\Lambda^{2}}$ (45)

Since $\alpha$ can have any sign, therefore, the only way to preserve
(sub)luminality is by taking $c_{3}=0$.
Now, up to $O(\alpha^{2})$ we have

$\displaystyle\ddot{\xi}+\frac{8c_{4}\alpha^{2}}{\Lambda^{2}}\ddot{\xi}-\partial_{i}^{2}\xi=0$
(46)

The phase velocity is given by

$\displaystyle v_{phase}^{2}=1-\frac{8c_{4}\alpha^{2}}{\Lambda^{4}}$ (47)

(Sub)luminality demands, $c_{4}\geq 0$. Therefore, superluminality in this
particular case gives a stronger bound than the ‘amplitude analysis’.

##  5 Summary

We have derived constraints on dim 6 and dim 8 gluon field strength operators
of SMEFT using both the amplitude and superluminality analysis. This
significantly reduces the parameter space of the Wilson coefficients.
Interestingly, these bounds imply that dim 6 operators can only exist in the
presence of certain dim 8 operators.
The amplitude analysis filters out terms growing as even power of s, $s^{2n}$
where $n\geq 1$ ($n=1$ in our case). It is because of this filtering property
that one is not able to put any bounds on the Wilson coefficients of dim 6
operators comprising four fields, e.g.
$c\phi^{2}\partial_{\mu}\phi\partial^{\mu}\phi$ as their contribution to the
tree level scattering amplitude (forward limit) grow as $s$. But for dim 6
operators comprising of three fields, like some terms in $Q_{G^{3}}^{(1)}$ and
$Q_{G^{3}}^{(2)}$, one indeed gets an $s^{2}$ dependence at tree level due to
exchange diagrams. It is this feature that allowed us to put constraints on
the square of the Wilson coefficients of dim 6 gluon field strength operators
in SMEFT. We obtained constraints on the magnitude of dim 6 operators’ Wilson
coefficients in terms of those of dim 8 operators. In appendix B, we have
given another example of a dim 6 operator (containing 3 fields) whose
magnitude can be constrained in terms of dim 8 operators.

In the context of superluminality analysis, we have mentioned the subtleties
involving the relation between low-frequency phase velocity and causality. We
showed, in the case of chiral Lagrangian, that it is unclear if one gets a
strict positivity bound from superluminality when the pion is considered to be
massive. The reason for this is that the superluminality analysis takes into
account the contribution from operators of all dimensions (unlike the
‘amplitude analysis’). As we have argued in our work, the contribution of the
dimension 4 operator (other than the kinetic term) to the dispersion relation
of the perturbation makes it unclear if one can use the phase velocity to
dictate the superluminality of the perturbation. However, in the case of the
gluon field strength operators, we managed to get rid of the mass-like terms
in the dispersion relation by choosing specific background and polarization of
the perturbation. This was possible because the mass-like term entirely
depended on four-vector contractions which could be made zero despite having a
non-zero field. We showed that interestingly and in a non-trivial way, the
superluminality analysis for gluonic operators in the SMEFT reproduces all the
bounds obtained from the ‘amplitude analysis’.

The above discussion might give the impression that the amplitude analysis
always gives similar or stronger bounds than the superluminality analysis.
However, this is not always true. In sec. 4.3, following Baumann_2016 , we
showed an example of a non-relativistic theory (motivated by the EFT of
inflation inflation ) that the superluminality gives stronger constraints than
the ‘amplitude analysis’ in this case. Thus, we conclude that it is not clear
which of the two analyses will give stronger bounds for a particular theory.
Hence, ideally, one should perform both analyses in order to obtain the
maximum amount of constraints on an IR effective theory.

## Acknowledgments

DG acknowledges support through the Ramanujan Fellowship and MATRICS Grant of
the Department of Science and Technology, Government of India. DG would also
like to acknowledge support from the ICTP through the Associates Programme
(2020-2025). DG thank Alex Azatov, Paolo Creminelli, and Amartya Harsh Singh
for fruitful discussions.

## Appendix

##  Appendix A Details of the subluminality analysis for gluons

  * •

For operator
$Q_{G^{4}}^{(1)}=\left(G_{\mu\nu}^{a}G^{a\mu\nu}\right)\left(G_{\rho\sigma}^{b}G^{b\rho\sigma}\right)$
we have,

$L=-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}+\frac{c_{8}^{(1)}}{\Lambda^{4}}\left(G_{\mu\nu}^{a}G^{a\mu\nu}\right)\left(G_{\rho\sigma}^{b}G^{b\rho\sigma}\right)$

Applying Euler’s Lagrange equation,

$\partial_{\alpha}\frac{\partial
L}{\partial\left(\partial_{\alpha}A^{f}_{\beta}\right)}-\frac{\partial
L}{\partial A_{\beta}^{f}}=0$

for the above Lagrangian, we get EOM as,

$\displaystyle-\partial_{\alpha}G^{f,\alpha\beta}+\frac{8c_{8}^{(1)}}{\Lambda^{4}}\left(2G_{\mu\nu}^{a}(\partial_{\alpha}G^{a,\mu\nu})G^{f,\alpha\beta}+G_{\mu\nu}^{a}G^{a,\mu\nu}\partial_{\alpha}G^{f,\alpha\beta}\right)$
(48) $\displaystyle=$ $\displaystyle-
g_{s}G^{a,\beta\nu}A_{\nu}^{h}f^{afh}+\frac{8c_{8}^{(1)}}{\Lambda^{4}}g_{s}f^{bfj}G^{a}_{\mu\nu}G^{a,\mu\nu}G^{b,\beta\sigma}A_{\sigma}^{j}$

We expand $A^{a,\mu}=A_{0}^{a,\mu}+h^{a,\mu}$ where we choose background
$A_{0}^{a,\mu}$ of particular color ‘$a$’ having
$\partial^{\nu}A^{a,\mu}=constant$, and look at the linearised equation of
motion for perturbation $h^{a,\mu}$ also of same color ‘$a$’, then we get

$\displaystyle-\partial_{\alpha}H^{f,\alpha\beta}+\frac{8c_{8}^{(1)}}{\Lambda^{4}}\Big{(}$
$\displaystyle
2G_{0\mu\nu}^{a}\big{(}\partial_{\alpha}H^{a,\mu\nu}+g_{s}f^{a1c}A_{0}^{1,\mu}h^{c,\nu}+g_{s}f^{ac1}A_{0}^{1,\nu}h^{c,\mu}\big{)}G_{0}^{f,\alpha\beta}$
(49)
$\displaystyle+G_{0\mu\nu}^{a}G_{0}^{a,\mu\nu}\partial_{\alpha}H^{f,\alpha\beta}\Big{)}=0$

where f=a,
$\displaystyle{G^{a}_{0\mu\nu}=\partial_{\mu}A^{a}_{0\nu}-\partial_{\nu}A_{0\mu}^{a}+g_{s}f^{abc}A^{b}_{0\mu}A_{0\nu}^{c}}$
and $H^{a}_{\mu\nu}=\partial_{\mu}h^{a}_{\nu}-\partial_{\nu}h_{\mu}^{a}$
since $A^{a}_{0}$ is non zero only for particular ‘$a$’,
$\displaystyle{G^{a}_{0\mu\nu}=F^{a}_{0\mu\nu}=\partial_{\mu}A^{a}_{0\nu}-\partial_{\nu}A_{0\mu}^{a}}$
and is also non-zero only for color index= ‘$a$’. From now we’ll stop writing
color index for background for convenience as it is fixed to be ‘$a$’.

$\displaystyle-\partial_{\alpha}H^{a,\alpha\beta}+\frac{8c_{8}^{(1)}}{\Lambda^{4}}\Big{(}2F_{0\mu\nu}\partial_{\alpha}H^{a,\mu\nu}F_{0}^{\alpha\beta}+F_{0\mu\nu}F_{0}^{\mu\nu}\partial_{\alpha}H^{a,\alpha\beta}\Big{)}=0$
(50)

Since we are working in Lorentz gauge, $\partial_{\alpha}h^{\alpha}=0$ , then
writing $H^{a,\mu\nu}$ explicitly

$\displaystyle-\partial_{\alpha}\partial^{\alpha}h^{a,\beta}\left(1-\frac{8c_{8}^{(1)}}{\Lambda^{4}}F_{0\mu\nu}F_{0}^{\mu\nu}\right)+\frac{32c_{8}^{(1)}}{\Lambda^{4}}F_{0\mu\nu}F_{0}^{\alpha\beta}\partial_{\alpha}\partial^{\mu}h^{a,\nu}=0$
(51)

Rescaling
$\displaystyle{h^{a,\mu}\rightarrow\frac{h^{a,\mu}}{\left(1-\frac{8c_{8}^{(1)}}{\Lambda^{4}}F_{0\mu\nu}F_{0}^{\mu\nu}\right)}}$
and considering terms only up to
$\displaystyle{\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)}$,

$\displaystyle-\partial_{\alpha}\partial^{\alpha}h^{a,\beta}+\frac{32c_{8}^{(1)}}{\Lambda^{4}}F_{0\mu\nu}F_{0}^{\alpha\beta}\partial_{\alpha}\partial^{\mu}h^{a,\nu}=0$
(52)

Taking the Fourier transform and multiplying the $\text{eq}^{\text{n}}$ by
normalized polarization of perturbation: $\epsilon_{\beta}$,

$\displaystyle
k^{2}\cdot\epsilon_{\beta}\tilde{h}^{a,\beta}=\frac{32c_{8}^{(1)}}{\Lambda^{4}}F_{0\mu\nu}F_{0}^{\alpha\beta}\epsilon_{\beta}k_{\alpha}k^{\mu}\tilde{h}^{a,\nu}$
(53)

Also, we can write $\tilde{h}^{a,\nu}=-\epsilon^{\nu}\hskip
2.0pt\tilde{h}^{a,\rho}\epsilon_{\rho}$ considering polarization to be
transverse and therefore having components only in spatial direction
($\Rightarrow\epsilon_{\nu}\epsilon^{\nu}=-1$)

$\displaystyle
k^{2}=-\frac{32c_{8}^{(1)}}{\Lambda^{4}}(F_{0\mu\nu}\epsilon^{\nu}k^{\mu})^{2}$
(54)

Doing the above calculation for other operators gives
$\text{eq}^{\text{n}}$(27).

  * •

For operator
$Q_{G^{3}}^{(1)}=f^{abc}G_{\mu}^{a\nu}G_{\nu}^{b\rho}G_{\rho}^{c\mu}$,

$\displaystyle
L=-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}+\frac{c_{6}}{\Lambda^{2}}f^{abc}G_{\mu}^{a\nu}G_{\nu}^{b\rho}G_{\rho}^{c\mu}$

we get EOM as,

$\displaystyle-\partial_{\alpha}G^{f,\alpha\beta}+\frac{6c_{6}}{\Lambda^{2}}f^{fbc}\partial_{\alpha}\left(G^{c,\rho\alpha}G^{b,\beta}_{\hskip
11.38109pt\rho}\right)=-g_{s}G^{a,\beta\nu}A_{\nu}^{h}f^{afh}+\frac{6c_{6}}{\Lambda^{2}}g_{s}f^{abc}\left(f^{afh}G^{b,\nu\rho}G_{\hskip
1.0pt\rho}^{c,\beta}A_{\nu}^{h}\right)$ (55)

We again expand $A^{a,\mu}=A_{0}^{a,\mu}+h^{a,\mu}$ where we choose background
$A_{0}^{a,\mu}$ of particular color ‘$a$’ with
$\partial^{\nu}A^{a,\mu}=constant$, and look at the linearised equation of
motion for perturbation $h^{a,\mu}$ of color f.
Then EOM takes the following form for color f,

$\displaystyle-\partial_{\alpha}\left(H^{f,\alpha\beta}+g_{s}f^{faj}A_{0}^{a,\alpha}h^{j,\beta}+g_{s}f^{fia}h^{i,\alpha}A_{0}^{a,\beta}\right)$
(56)
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}f^{fba}\partial_{\alpha}\Big{(}F_{0}^{a,\rho\alpha}(H^{b,\beta}_{\hskip
11.38109pt\rho}+g_{s}f^{baj}A_{0}^{a,\beta}h^{j}_{\rho}+g_{s}f^{bia}h^{i,\beta}A^{a}_{0\rho})$
$\displaystyle\hskip 56.9055pt-F_{0}^{a,\beta\rho}(H_{\hskip
1.0pt\rho}^{b,\alpha}+g_{s}f^{baj}A^{a}_{0\rho}h^{j,\alpha}+g_{s}f^{bia}h^{i}_{\rho}A_{0}^{a,\alpha})\Big{)}$
$\displaystyle=-g_{s}\left(H^{d,\beta\nu}+g_{s}f^{daj}A_{0}^{a,\beta}h^{j,\nu}+g_{s}f^{dia}h^{i,\beta}A_{0}^{a,\nu}\right)A_{0\nu}^{a}f^{dfa}-g_{s}F_{0}^{a,\beta\nu}h_{\nu}^{h}f^{afh}$
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}g_{s}f^{daa}\left(f^{dfh}F_{0}^{a,\nu\rho}F_{0\rho}^{a,\beta}h_{\nu}^{h}\right)$
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}g_{s}f^{dba}\left(f^{dfa}(H^{b,\nu\rho}+g_{s}f^{baj}A_{0}^{a,\nu}h^{j,\rho}+g_{s}f^{bia}h^{i,\nu}A_{0}^{a,\rho})F_{0\rho}^{a,\beta}A_{0\nu}^{a}\right)$
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}g_{s}f^{dac}\left(f^{dfa}F_{0}^{a,\nu\rho}(H_{\hskip
1.0pt\rho}^{c,\beta}+g_{s}f^{caj}A^{a}_{0\rho}h^{j,\beta}+g_{s}f^{cia}h^{i}_{\rho}A_{0}^{a,\beta})A_{0\nu}^{a}\right)$

For this, we work in WKB approximation where the scale of variation of
background(r) is much larger than that of perturbations ($\omega^{-1}$), then
at order $g_{s}$, $g_{s}^{2}$, 1/$\Lambda^{2}$ and leading order of r$\omega$
in each of them,

$\displaystyle-\partial_{\alpha}\left(H^{f,\alpha\beta}+g_{s}f^{faj}A_{0}^{a,\alpha}h^{j,\beta}+g_{s}f^{fia}h^{i,\alpha}A_{0}^{a,\beta}\right)$
(57)
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}f^{fba}\partial_{\alpha}\Big{(}F_{0}^{a,\rho\alpha}(H^{b,\beta}_{\hskip
11.38109pt\rho}+g_{s}f^{baj}A_{0}^{a,\beta}h^{j}_{\rho}+g_{s}f^{bia}h^{i,\beta}A^{a}_{0\rho})$
$\displaystyle\hskip 56.9055pt-F_{0}^{a,\beta\rho}(H_{\hskip
1.0pt\rho}^{b,\alpha}+g_{s}f^{baj}A^{a}_{0\rho}h^{j,\alpha}+g_{s}f^{bia}h^{i}_{\rho}A_{0}^{a,\alpha})\Big{)}$
$\displaystyle=-g_{s}\left(H^{d,\beta\nu}+g_{s}f^{daj}A_{0}^{a,\beta}h^{j,\nu}+g_{s}f^{dia}h^{i,\beta}A_{0}^{a,\nu}\right)A_{0\nu}^{a}f^{dfa}$
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}g_{s}f^{dba}f^{dfa}H^{b,\nu\rho}F_{0\rho}^{a,\beta}A_{0\nu}^{a}+\frac{6c_{6}}{\Lambda^{2}}g_{s}f^{dac}f^{dfa}F_{0}^{a,\nu\rho}H_{\hskip
1.0pt\rho}^{c,\beta}A_{0\nu}^{a}$

In above $\text{eq}^{\text{n}}$ since we don’t know the relative order of
$g_{s}$ and 1/$\Lambda^{2}$, we ignore only those terms which are definitely
of less order than $g_{s}$, $g_{s}^{2}$, 1/$\Lambda^{2}$ like
$g_{s}^{2}/\Lambda^{2}$. Also, in the above equation and all further equations
‘$f$’ and ‘$a$’ are not contracted but are considered particular color indices
and we’ll also drop ‘$a$’ from background terms.

Lorentz gauge, $\partial_{\alpha}h^{\alpha}=0$ implies

$\displaystyle-\partial_{\alpha}\partial_{\alpha}h^{f,\beta}-2g_{s}f^{faj}A_{0}^{a,\alpha}\partial_{\alpha}h^{j,\beta}+g_{s}f^{faj}A_{0\nu}\partial^{\beta}h^{j,\nu}$
(58)
$\displaystyle+g_{s}^{2}f^{dfa}f^{daj}\left(A_{0}^{\beta}h^{j,\nu}-h^{j,\beta}A_{0}^{\nu}\right)A_{0}^{\nu}$
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}f^{fba}\Big{(}F_{0\rho\alpha}\partial^{\alpha}\partial^{\beta}h^{b,\rho}+F_{0}^{\beta\rho}\partial_{\alpha}\partial^{\alpha}h_{\rho}^{b}\Big{)}$
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}g_{s}f^{fba}\Big{(}f^{baj}A_{0}^{\beta}F_{0}^{\rho\alpha}\partial_{\alpha}h_{\rho}^{j}-f^{baj}F_{0}^{\rho\alpha}A_{0\rho}\partial_{\alpha}h^{j,\beta}$
$\displaystyle\hskip
71.13188pt+f^{baj}A_{0}^{\alpha}F_{0}^{\beta\rho}\partial_{\alpha}h_{\rho}^{f}+f^{baj}H^{j,\nu\rho}F_{0\hskip
2.0pt\rho}^{\beta}A_{0\nu}+f^{bac}F_{0}^{\nu\rho}H_{\rho}^{c,\hskip
1.0pt\beta}A_{0\nu}\Big{)}=0$

We can also choose the amplitude (max$|A_{0}(x^{\mu})|$) of the background to
be arbitrary small without affecting other quantities, which would make the
terms of order $A_{0}^{2}$ less relevant in comparison to terms of lower
order,

$\displaystyle-\partial_{\alpha}\partial_{\alpha}h^{f,\beta}+2g_{s}f^{fba}A_{0}^{a,\alpha}\partial_{\alpha}h^{b,\beta}-g_{s}f^{fba}A_{0\rho}\partial^{\beta}h^{b,\rho}$
(59)
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}f^{fba}\Big{(}F_{0\rho\alpha}\partial^{\alpha}\partial^{\beta}h^{b,\rho}+F_{0}^{\beta\rho}\partial_{\alpha}\partial^{\alpha}h_{\rho}^{b}\Big{)}=0$

Now for also some other color ‘$b$’ we’ll have similar wave equation,

$\displaystyle-\partial^{\alpha}\partial_{\alpha}h^{b,\rho}+2g_{s}f^{bga}A_{0}^{a,\alpha}\partial_{\alpha}h^{g,\rho}-g_{s}f^{bga}A_{0\sigma}\partial^{\rho}h^{g,\sigma}$
(60)
$\displaystyle+\frac{6c_{6}}{\Lambda^{2}}f^{bga}\left(F_{0}^{\sigma\alpha}\partial_{\alpha}\partial^{\rho}h^{g}_{\sigma}+F_{0}^{\rho\sigma}\partial^{\alpha}\partial_{\alpha}h_{\sigma}^{g}\right)=0$

We consider the following solution of (60),

$\displaystyle-\partial^{\alpha}h^{b,\rho}+2g_{s}f^{bga}A_{0}^{a,\alpha}h^{g,\rho}-g_{s}f^{bga}A_{0\nu}\delta^{\alpha\rho}h^{g,\nu}+\frac{6c_{6}}{\Lambda^{2}}f^{bga}\left(F_{0}^{\sigma\alpha}\partial^{\rho}h^{g}_{\sigma}+F_{0}^{\rho\sigma}\partial^{\alpha}h_{\sigma}^{g}\right)=0$
(61)

and substitute in (59),

$\displaystyle-\partial^{\alpha}\partial_{\alpha}h^{f,\beta}=$ $\displaystyle
2g_{s}^{2}f^{fba}f^{gba}\left(2A_{0}^{\alpha}A_{0\alpha}h^{g,\beta}-\frac{3}{2}A_{0\nu}A_{0}^{\beta}h^{g,\nu}\right)$
(62)
$\displaystyle+g\frac{6c_{6}}{\Lambda^{2}}f^{fba}f^{gba}\left(5F_{0}^{\beta\rho}A_{0}^{\alpha}\partial_{\alpha}h^{g,\rho}-A_{0\nu}F_{0}^{\rho\beta}\partial^{\rho}h^{g,\nu}+5F_{0\rho\alpha}A_{0}^{\alpha}\partial^{\beta}h^{g,\rho}\right)$
$\displaystyle+36\frac{c_{6}^{2}}{\Lambda^{4}}f^{fba}f^{gba}\left(F_{0}^{\beta\rho}F_{0}^{\sigma\alpha}\partial_{\alpha}\partial_{\rho}h^{g}_{\sigma}+F^{\beta}_{0\rho}F_{0}^{\rho\sigma}\partial^{\alpha}\partial_{\alpha}h^{g}_{\sigma}+2F_{0}^{\rho\alpha}F_{0}^{\sigma\alpha}\partial^{\beta}\partial_{\rho}h^{g}_{\sigma}\right)$

Now we try to get the differential equation just in terms of perturbation of
color ‘$f$’ assuming that we can choose a particular background $A_{0}$ such
that mass term vanishes. Then in the above equation in second and third terms
of r.h.s, we write $h^{g}$ in terms of other colors. When $g\neq f$ we get
terms of higher order in $A_{0}$ or
$\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)$ from (60), therefore only
$g=f$ survives at $\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)$ and leading
order in $A_{0}$.

$\displaystyle-\partial^{\alpha}\partial_{\alpha}h^{f,\beta}=$ $\displaystyle
g\frac{6c_{6}}{\Lambda^{2}}f^{fba}f^{fba}\left(5F_{0}^{\beta\rho}A_{0}^{\alpha}\partial_{\alpha}h^{f,\rho}-A_{0\nu}F_{0}^{\rho\beta}\partial^{\rho}h^{f,\nu}+5F_{0\rho\alpha}A_{0}^{\alpha}\partial^{\beta}h^{f,\rho}\right)$
$\displaystyle+$ $\displaystyle
36\frac{c_{6}^{2}}{\Lambda^{4}}f^{fba}f^{fba}\left(F_{0}^{\beta\rho}F_{0}^{\sigma\alpha}\partial_{\alpha}\partial_{\rho}h^{f}_{\sigma}+F^{\beta}_{0\rho}F_{0}^{\rho\sigma}\partial^{\alpha}\partial_{\alpha}h^{f}_{\sigma}+2F_{0}^{\rho\alpha}F_{0}^{\sigma\alpha}\partial^{\beta}\partial_{\rho}h^{f}_{\sigma}\right)$

Taking the Fourier transform and multiplying by normalized polarization of
perturbation color ‘$f$’ $\epsilon_{\beta}$,

$\displaystyle k^{\mu}k_{\mu}(\epsilon_{\beta}\widetilde{h}^{f,\beta})=$
$\displaystyle
ig\frac{6c_{6}}{\Lambda^{2}}f^{fba}f^{fba}\epsilon_{\beta}\left(5F_{0}^{\beta\rho}A_{0}^{\alpha}k_{\alpha}\widetilde{h}^{f,\rho}-A_{0\nu}F_{0}^{\rho\beta}k^{\rho}\widetilde{h}^{f,\nu}+5F_{0\rho\alpha}A_{0}^{\alpha}k^{\beta}\widetilde{h}^{f,\rho}\right)$
(63) $\displaystyle-$ $\displaystyle
36\frac{c_{6}^{2}}{\Lambda^{4}}f^{fba}f^{fba}\epsilon_{\beta}\left(F_{0}^{\beta\rho}F_{0}^{\sigma\alpha}k_{\alpha}k_{\rho}\widetilde{h}^{f}_{\sigma}+F^{\beta}_{0\rho}F_{0}^{\rho\sigma}k^{\alpha}k_{\alpha}\widetilde{h}^{f}_{\sigma}+2F_{0}^{\rho\alpha}F_{0}^{\sigma\alpha}k^{\beta}k_{\rho}\widetilde{h}^{f}_{\sigma}\right)$

We consider polarization to be transverse and since we can write
$\tilde{h}^{f,\nu}=-\epsilon^{\nu}\hskip
2.0pt\tilde{h}^{f,\rho}\epsilon_{\rho}$, we get

$\displaystyle
k^{\mu}k_{\mu}\left(1-36\frac{c_{6}^{2}}{\Lambda^{4}}(f^{fba})^{2}F^{\beta}_{0\rho}F_{0}^{\rho\sigma}\epsilon_{\beta}\epsilon_{\sigma}\right)=-$
$\displaystyle
ig\frac{6c_{6}}{\Lambda^{2}}(f^{fba})^{2}\epsilon_{\beta}\left(5F_{0}^{\beta\rho}A_{0}^{\alpha}k_{\alpha}\epsilon^{\rho}-A_{0\nu}F_{0}^{\rho\beta}k^{\rho}\epsilon^{\nu}\right)$
(64) $\displaystyle+$ $\displaystyle
36\frac{c_{6}^{2}}{\Lambda^{4}}(f^{fba})^{2}\left(F_{0}^{\beta\rho}F_{0}^{\sigma\alpha}k_{\alpha}k_{\rho}\epsilon_{\beta}\epsilon_{\sigma}\right)$

Considering $k^{\mu}$ to be complex in general then for the real part of
dispersion relation at leading order we get,

$\displaystyle
k^{\mu}k_{\mu}=36\frac{c_{6}^{2}}{\Lambda^{4}}(f^{fba})^{2}\left(F_{0}^{\beta\rho}k_{\rho}\epsilon_{\beta}\right)^{2}$
(65)

After considering all dim 6 and dim 8 operators we’ll get following dispersion
relation,

$\displaystyle\frac{k_{\mu}k^{\mu}}{4}=$
$\displaystyle\frac{9}{\Lambda^{4}}f^{afc}f^{af}_{\;\;\;c}\Big{[}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}F_{0}^{\mu\nu}k_{\mu}\epsilon_{\nu}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})-\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{6}^{\prime}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\widetilde{F}_{0}^{\alpha\beta}k_{\alpha}\epsilon_{\beta}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\Big{]}^{2}$
(66)
$\displaystyle-\frac{4}{\Lambda^{4}}\Big{[}(2\delta^{af}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(1)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+(1+\delta^{af})\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(3)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+2d^{afc}d^{af}_{\;\;\;c}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(7)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left(F_{0}^{\mu\nu}k_{\mu}\epsilon_{\nu}\right)^{2}$
$\displaystyle\hskip
19.91684pt+(2\delta^{af}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(2)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+(1+\delta^{af})\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(4)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+2d^{afc}d^{af}_{\;\;\;c}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(8)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left(\widetilde{F}_{0}^{\alpha\beta}k_{\alpha}\epsilon_{\beta}\right)^{2}$
$\displaystyle\hskip
19.91684pt-(2\delta^{af}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(5)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+(1+\delta^{af})\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(6)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+2d^{afc}d^{af}_{\;\;\;c}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{8}^{(9)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left(F_{0}^{\mu\nu}k_{\mu}\epsilon_{\nu}\right)\left(\widetilde{F}_{0}^{\alpha\beta}k_{\alpha}\epsilon_{\beta}\right)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Big{]}$

where ‘$f$’ denotes the color of perturbation and ‘$a$’ of the background;
given the mass term in (62) vanish.
Consider the perturbation with polarization $\epsilon=\\{0,1,1,0\\}/\sqrt{2}$
and choose background of the form $A_{0\mu}=E\\{\sqrt{2}(x+y),x+y,-(x+y),0\\}$
where E is some arbitrary small constant. Under this configuration, we have
$A_{0\mu}A_{0}^{\mu}=0$ and $\epsilon^{\mu}A_{0\mu}=0$ i.e. mass-like term in
$\text{eq}^{\text{n}}$(62) vanish. For the chosen background, we get the
following non-zero components of $F_{\mu\nu}$, $F_{01}=-F_{10}$,
$F_{02}=-F_{20}$ and $F_{12}=-F_{21}$ with $F_{01}=F_{02}=F_{12}/\sqrt{2}$,
which reduces the dispersion relation (66) to the following form,

$\displaystyle
k_{\mu}k^{\mu}=\frac{72}{\Lambda^{4}}f^{afc}f^{af}_{\;\;\;c}(c_{6})^{2}\left(\omega
F_{01}\right)^{2}-\frac{32}{\Lambda^{4}}\left(c_{8}^{(3)}+2d^{afc}d^{af}_{\;\;\;c}c_{8}^{(7)}\right)\left(\omega
F_{01}\right)^{2}$ (67)

then by demanding perturbation to be causal, we get

$\displaystyle
9f^{afc}f^{af}_{\;\;\;c}(c_{6})^{2}-4\left(c_{8}^{(3)}+2d^{afc}d^{af}_{\;\;\;c}c_{8}^{(7)}\right)<0$
(68)

Choosing different combinations of colors for perturbation and background
leads to constraints C(1,1), C(2,1), C(3,1), and C(4,1). Now, if we choose the
background,
$A_{0\mu}=E\\{\sqrt{2(D+B)}t,(\sqrt{D}+\sqrt{B})t,(\sqrt{B}-\sqrt{D})t,0\\}$
where,
$D=4(2\delta^{ab}c_{8}^{(1)}+(1+\delta^{ab})c_{8}^{(3)}+2d^{abc}d^{ab}_{\;\;\;c}c_{8}^{(7)})-9f^{abc}f^{ab}_{\;\;\;c}c_{6}^{2}$;
$B=4(2\delta^{ab}c_{8}^{(2)}+(1+\delta^{ab})c_{8}^{(4)}+2d^{abc}d^{ab}_{\;\;\;c}c_{8}^{(8)})-9f^{abc}f^{ab}_{\;\;\;c}c_{6}^{\prime
2}$
with perturbation of the same polarization as before, then the subluminality
condition gives the following constraint,

$\displaystyle-2\sqrt{DB}<4(2\delta^{ab}c_{8}^{(5)}+(1+\delta^{ab})c_{8}^{(6)}+2d^{abc}d^{ab}_{\;\;\;c}c_{8}^{(9)})-18f^{abc}f^{ab}_{\;\;\;c}c_{6}c_{6}^{\prime}$
(69)

Similarly by choosing the background to be
$A_{0\mu}=E\\{\sqrt{2(D+B)}t,(\sqrt{D}+\sqrt{B})t,(\sqrt{D}-\sqrt{B})t,0\\}$
along with the polarization $\epsilon=\\{0,1,-1,0\\}/\sqrt{2}$, we get the
following constraint

$\displaystyle
2\sqrt{DB}>4(2\delta^{ab}c_{8}^{(5)}+(1+\delta^{ab})c_{8}^{(6)}+2d^{abc}d^{ab}_{\;\;\;c}c_{8}^{(9)})-18f^{abc}f^{ab}_{\;\;\;c}c_{6}c_{6}^{\prime}$
(70)

Combining the above two constraints we can reproduce C(1,3), C(2,3), C(3,3)
and C(4,3).

##  Appendix B An example with Scalar and Fermion

In this appendix, we consider another example of operators of dimensions 6 and
8 which, when subjected to the ‘amplitude analysis’, get relative bounds on
their Wilson coefficients. Consider the following Lagrangian,

$L^{(6)}=\frac{c_{6}}{\Lambda^{2}}\phi\partial_{\mu}\bar{\Psi}\partial^{\mu}\Psi\;;\;\;\;L^{(8)}=i\frac{c_{8}}{\Lambda^{4}}(\partial^{\mu}\partial_{\nu}\phi\partial_{\mu}\bar{\Psi}\gamma^{\nu}\Psi\phi-\partial^{\mu}\partial_{\nu}\phi\bar{\Psi}\gamma^{\nu}\partial_{\mu}\Psi\phi)$

where $\Psi$ represents a fermionic field and $\phi$ is a real scalar field.
The second term in $L^{(8)}$ has to be present for it to be hermitian. Note
that, the operator $L^{(6)}$ (with $\phi$ identified as the Higgs doublet, and
the normal derivatives replaced by appropriate covariant derivatives) can be
written in terms of a linear combination of SMEFT operators of the Warsaw
basis up to total derivatives using the EOM (see $\text{eq}^{\text{n}}(6.4)$
of dim6 ).

We calculate $2\rightarrow 2$ scattering amplitude with two scalars and
fermions of positive helicities, $\mathcal{M}(\phi f_{2}^{+}\rightarrow\phi
f_{4}^{+})$, at tree level up to
$\displaystyle{\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)}$. Since the
dimension 6 operator considered has 3 fields, we expect to get a contribution
scaling like $c_{6}^{2}s^{2}$ in the amplitude $\mathcal{M}(s,t)$.

Figure 3: The first two exchange diagrams represent $s$ and $u$-channel
contributions, and get contribution from $L^{(6)}$. The third contact diagram
gets contribution from $L^{(8)}$.

The tree level amplitude gets contribution from the Feynman diagrams in figure
3 and is given by:

$\displaystyle\mathcal{M}(\phi f_{2}^{+}\phi f_{4}^{+})=$
$\displaystyle-\frac{c_{6}^{2}}{\Lambda^{4}}\left\\{\overline{u}_{+}(p_{4})(\cancel{p}_{1}+\cancel{p}_{2})v_{-}(p_{2})\frac{s}{4}+\overline{u}_{+}(p_{4})(\cancel{p}_{2}-\cancel{p}_{3})v_{-}(p_{2})\frac{u}{4}\right\\}$
(71)
$\displaystyle+\frac{c_{8}}{\Lambda^{4}}\Big{\\{}-\overline{u}_{+}(p_{4})\cancel{p}_{1}v_{-}(p_{2})\frac{u}{2}+\overline{u}_{+}(p_{4})\cancel{p}_{1}v_{-}(p_{2})\frac{s}{2}$
$\displaystyle\hskip
36.98866pt+\overline{u}_{+}(p_{4})\cancel{p}_{3}v_{-}(p_{2})\frac{s}{2}-\overline{u}_{+}(p_{4})\cancel{p}_{3}v_{-}(p_{2})\frac{u}{2}\Big{\\}}$

which using spinor helicity formalism (for detailed introduction check scalar
) can be written as:

$\mathcal{M}(\phi f_{2}^{+}\phi
f_{4}^{+})=\frac{c_{8}}{\Lambda^{4}}\Big{\\{}\frac{-u}{2}[41]\langle
12\rangle+\frac{s}{2}[41]\langle 12\rangle+\frac{s}{2}[43]\langle
32\rangle-\frac{u}{2}[43]\langle
32\rangle\Big{\\}}-\frac{c_{6}^{2}}{\Lambda^{4}}\Big{\\{}[41]\langle
12\rangle\frac{s}{4}-[43]\langle 32\rangle\frac{u}{4}\Big{\\}}$

We now take the forward limit to get $\displaystyle{\mathcal{A}(s)=}$
$\displaystyle{\mathcal{M}(s,t)|_{t\rightarrow
0}=\frac{s^{2}}{\Lambda^{4}}\left(2c_{8}-\frac{c_{6}^{2}}{2}\right)}$. We
don’t have to worry about t-channel pole divergence since the t-channel
doesn’t exist for the process considered. From positivity condition discussed
in sec. 3.1, we get

$\displaystyle{4c_{8}>c_{6}^{2}}$

This puts an upper bound on the magnitude of $c_{6}$ in terms of $c_{8}$
similar to what we obtained for the gluonic operators. It also implies that
the 6-dimensional operator that we have considered in this example cannot
exist on its own, it needs some other operator which gives a positive
contribution proportional to $s^{2}$ in $\mathcal{A}(s)$ to survive.
One might not have expected to get $s^{2}$ dependence from exchange diagrams
as there are only two derivatives present in $L^{(6)}$ (unlike the gluonic
case which has three derivatives). However, the fermion propagator has $1/p$
dependence instead of the $1/p^{2}$ dependence for gluons, and more
importantly, spinors $\overline{u}$ and $v$ have implicit momentum factors.
These momentum factors, in our case, manifest themselves in the form of
Mandelstam variables once we take forward limit, leading to $s^{2}$ dependence
of exchange diagrams.

##  Appendix C The arc variable

Figure 4: Contour $C$ in s-complex plane where $s$ represents some energy
scale such that $\Lambda_{\text{qcd}}<s\ll\Lambda$.

In sec 3, we derived the constraints by calculating the residue at
$\lim_{m^{2}\to 0}(s\sim m^{2})\to 0$. However, since QCD is confined at low
energies it would be preferable to employ a method that circumvents the need
to calculate the residue at $s\sim 0$.

To do this, one can define the arc variable 2011.00037

$a(s)\equiv\int_{\cap_{s}}\frac{ds^{\prime}}{\pi
i}\frac{\mathcal{M}(s^{\prime})}{s^{\prime 3}}$ (72)

where $\cap_{s}$ represents a counterclockwise semicircular path as shown in
figure 4. Also, the Cauchy theorem implies that the integral over the contour
$C=\cap_{s}+\cap_{\infty}+\cap_{l_{1}}+\cap_{l_{2}}$ vanishes. Moreover, due
to the Froissart bound, the integral over the arc at infinity i.e.
$\cap_{\infty}$ vanishes. Therefore,

$a(s)=-\left[\int_{l_{1}}\frac{ds^{\prime}}{\pi
i}\frac{\mathcal{M}(s^{\prime})}{s^{\prime
3}}+\int_{l_{2}}\frac{ds^{\prime}}{\pi
i}\frac{\mathcal{M}(s^{\prime})}{s^{\prime 3}}\right]$ (73)

Using crossing symmetry and real analyticity,
$\mathcal{M}(s+i\epsilon)=\mathcal{M}^{*}(-s+i\epsilon)$, we can relate the
amplitude over $l_{2}$ to the amplitude over $l_{1}$,

$\displaystyle a(s)$ $\displaystyle=\int_{s}^{\infty}\frac{ds^{\prime}}{\pi
i}\frac{\mathcal{M}(s^{\prime})}{s^{\prime
3}}+\int_{-\infty}^{-s}\frac{ds^{\prime}}{\pi
i}\frac{\mathcal{M}(s^{\prime})}{s^{\prime 3}}$ (74)
$\displaystyle=\int_{s}^{\infty}\frac{ds^{\prime}}{\pi
i}\frac{\mathcal{M}(s^{\prime})}{s^{\prime
3}}-\int_{s}^{\infty}\frac{ds^{\prime}}{\pi
i}\frac{\mathcal{M}^{*}(s^{\prime})}{s^{\prime
3}}=\frac{2}{\pi}\int_{s}^{\infty}ds^{\prime}\frac{\text{Im}\mathcal{M}(s^{\prime})}{s^{\prime
3}}$ (75)

The optical theorem relates the imaginary part of amplitude to the cross-
section, $\text{Im}\mathcal{M}(s^{\prime})=s^{\prime}\sigma(s^{\prime})$,

$a(s)=\frac{2}{\pi}\int_{s}^{\infty}ds^{\prime}\frac{\sigma(s^{\prime})}{s^{\prime
2}}>0$ (76)

We can systematically compute the arc variable, $a(s)$, as an expansion in $s$
using $\text{eq}^{n}$(72) withing the validity of the EFT regime. For
amplitude of the form, $\mathcal{M}(s)=\sum_{n=0}c_{2n}s^{2n}$, which is the
case for gluon-gluon scattering, the arc variable is given by the Wilson
coefficient, $a(s)=c_{2}>0$ i.e. the coefficient of $s^{2}$ in the amplitude
is always positive.

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