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# Positivity bounds on Minimal Flavor Violation

Quentin Bonnefoy a,b Emanuele Gendy a,c Christophe Grojean
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron (DESY), D-22607 Hamburg, GermanyInstitute of
Theoretical Physics, Universität Hamburg, 22761 Hamburg, GermanyInstitut für
Physik, Humboldt-Universität zu Berlin, D-12489 Berlin, Germany

###### Abstract

From general analyticity and unitarity requirements on the UV theory,
positivity bounds on the Wilson coefficients of the dimension-8 operators
composed of 4 fermions and two derivatives appearing in the Standard Model
Effective Field Theory have been derived recently. We explore the fate of
these bounds in the context of models endowed with a Minimal Flavor Violation
(MFV) structure, models in which the flavor structure of higher dimensional
operators is inherited from the one already contained in the Yukawa sector of
the Standard Model Lagrangian. Our goal is to check whether the general
positivity bounds translate onto bounds on the Yukawa coefficients and/or on
elements of the CKM matrix. MFV fixes the coefficients of dimension-8
operators up to some multiplicative flavor-blind factors and we find that, in
the most generic setup, the freedom left by those unspecified coefficients is
enough as not to constrain the parameters of the renormalizable Yukawa sector.
On the contrary, the latter shape the allowed region for the former. Requiring
said overall coefficients to take natural $\order{1}$ values could give rise
to bounds on the Yukawa couplings. Remarkably, at leading order in an
expansion in powers of the Yukawa matrices, no bounds on the CKM entries can
be retrieved.

††preprint: DESY 20-201

##  1 Introduction

Effective Field Theories (EFTs) are one of the most powerful and general ways
to describe a physical system, when one does not know or care about its
dynamics below some length scale $\Lambda^{-1}$. In fact, unless one is
addressing the final theory of everything, this is always the case. The
procedure behind EFTs consists in writing down the most general, non-
renormalizable Lagrangian as a series of gauge invariant operators built out
of the relevant degrees of freedom, and suppressed by appropriate inverse
powers of $\Lambda$. Then, as long as one is working at masses and exchanged
momenta $m,|\vec{p}|\ll\Lambda$, such Lagrangian can be employed to make
predictions by retaining the relevant terms in the expansion. The Standard
Model (SM) itself has to be considered as an effective description of Nature,
and indeed it corresponds to the leading order, low energy approximation of
the more complete Standard Model Effective Field Theory (SMEFT). The latter is
defined as the EFT where the SM degrees of freedom are employed to build all
the operators invariant under the SM gauge group, and takes the general form:

$\displaystyle\mathcal{L}_{SMEFT}=\mathcal{L}^{(4)}_{SM}+\sum_{n\geq
5}\frac{c_{n}}{\Lambda^{n-4}}\mathcal{O}^{(n)}\ .$ (1)

Here $\mathcal{L}^{(4)}_{SM}$ is the dimension-4, renormalizable part of the
Lagrangian, $\Lambda$ is a UV scale until which the SMEFT is valid and above
which its predictivity is lost, and $\mathcal{O}^{(n)}$ are gauge-invariant,
dimension-$n$ operators. The $c_{n}$ are coefficients which, after taking into
account $\hbar$ dimensions, and possible selection rules (for instance baryon
number conservation), are to be taken in principle to be of $\order{1}$.

However, this is not the end of it. Indeed, not all the apparently healthy
EFTs are actually consistent. Instead, requiring that their UV completion
respects unitarity and analyticity, two properties that we wish upon any
Quantum Field Theory (QFT), imposes bounds on some of the Wilson coefficients
entering the EFT Lagrangian irobstruct . In particular, the simplest bounds
arise from positivity requirements on the $s^{2}$ coefficient of the IR $2\to
2$ amplitude, with $s$ the Mandelstam variable $s=(p_{1}+p_{2})^{2}$. This
property has had a variety of applications, from pion physics pions1 ; pions2
to Quantum Gravity qg  and to the derivation of the $a$-theorem atheorem , see
also softness  and references therein. Recently, efforts have been made to
constrain the coefficients in the SMEFT expansion sumrules ; Gu:2020thj ,
particularly regarding vector boson scattering SMEFTconsistency ;
vectorbosonscattering ; aQGC  (see also Refs. cones ;
electronpositronscattering ; Yamashita:2020gtt ; dimension6bounds ;
Bellazzini:2020cot ; Tolley:2020gtv ; Gu:2020ldn ; Trott:2020ebl  for recent
developments).

Finally, Ref. flavorconstr  obtained results about the coefficients of
dimension-8 operators composed of 4 fermions and two derivatives appearing in
the SMEFT. There, interesting bounds are derived, for instance for the flavor
violating coefficients, whose magnitude turns out to be bounded from above by
products of flavor conserving ones. Then, it is natural to check whether these
bounds are compatible with the Minimal Flavor Violation (MFV) ansatz MFV ;
MFVreview . The latter is one of the simplest methods of constraining the
flavor structure of any higher dimensional operator in the SM effective theory
containing fermions, in a way that does not clash with the stringent
experimental bounds on flavor violation. It does this by stating that all the
relevant building blocks are already contained in the Yukawa sector of the SM.
Thus, the flavor structure of any operator involving fermions is fixed by the
dimension-4 ones, up to some overall multiplicative factors. This applies in
particular to four-fermions dimension-8 operators, whose coefficients are
constrained by positivity bounds. Consequently, assuming MFV, one can ask
whether the positivity bounds yield constraints on the parameters of the
dimension-4 Lagrangian. For instance, flavor-violating dimension-8 couplings
are proportional to entries of the CKM matrix in MFV, and positivity
constraints will involve both dimension-4 and dimension-8 coefficients.
Therefore, our initial goal consists in checking whether one can extract from
the EFT consistency some bounds on the Yukawa and CKM parameters of the SM, to
be then compared with experimental value.

It is a known fact that, whatever the values of the SM parameters, there exist
MFV-compatible UV-completions of the SMEFT four-Fermi operators, an example
being a heavy vector coupled in a flavor-blind way. So the positivity bounds
cannot be as powerful enough as to make some of values of the SM parameters
inconsistent. Nonetheless, the positivity bounds will restrict the allowed
region for the unspecified EFT coefficients that the MFV assumption already
reduced down to flavor-blind global factors. And, the specific values of the
renormalizable couplings influence the shape of the allowed region of the
flavor-blind MFV overall factors. Remarkably, while this applies to fermion
masses, the structure of the positivity bounds is such that the CKM matrix
elements completely disappear from the most stringent bounds at leading order.

To go further, we consider the expectation that the allowed region for the EFT
coefficients should enable the flavor-bling MFV factors to be order one.
Otherwise, MFV would be cornered by the EFT consistency to unnatural
realizations (or specific ones, like the aforementioned case of a flavor-blind
heavy vector), which would question its use in the first place. Consequently,
we study whether the assumption of order one EFT coefficients now yields
interesting constraints on fermion masses and CKM elements, when implemented
in the bounds involving dimension-4 and dimension-8 coefficients. We study
this case analytically when all the coefficients are degenerate and equal to
one, and numerically when they vary independently in a neighborhood of unity
and find indeed a bound on the largest fermion mass. In the present case,
those bounds are ineffective, phenomenologically speaking as they are by far
satisfied by experimental values, and theoretically speaking since the values
that violate them lie in a region where the MFV expansion breaks down,
preventing us from obtaining any relevant bound on fermion masses.
Nevertheless, they show how in some cases restrictions on higher dimensional
operators can get reflected on renormalizable parameters.

The paper is organized as follows. In Section 2, we provide a brief review of
MFV, and express the coefficients of dimension 8, 4-fermions operators in
Section 2.2, accordingly. In Section 3, we turn to study the bounds, to remove
their dependence on external states, and to find the allowed regions for the
global factors that were left undetermined under the MFV assumption. Section 4
is devoted to final comments and conclusions. Several appendices complete this
paper. Appendix A describes the flavor structures that arise when resumming
the MFV expansion with large Yukawas, and Appendix B shows that a flavor
structure introduced in Section 2.2 is redundant when there are only two
flavors. Then, Appendix C discusses the impact on the positivity bounds of the
SMEFT operators that we neglected. The general bounds in the three flavor case
are presented in Appendix D and finally, Appendix E displays some coefficient
redefinitions that we use throughout the paper to simplify the expression of
the positivity bounds.

##  2 Minimal Flavor Violation

Minimal Flavor Violation (MFV) MFV ; MFVreview  is an ansatz constraining the
flavor structure of higher dimensional operators in the Standard Model
Effective Field Theory (SMEFT). It comes from noticing that, if we do not
consider the Yukawa operators, the Lagrangian of the Standard Model enjoys a
$U(3)^{5}$ global symmetry, that acts on the quark and lepton flavor space.
Its non-abelian subgroup can be split as:

$\displaystyle SU(3)^{3}_{q}$ $\displaystyle=SU(3)_{Q}\otimes SU(3)_{u}\otimes
SU(3)_{d}$ $\displaystyle SU(3)^{2}_{l}$ $\displaystyle=SU(3)_{L}\otimes
SU(3)_{e},$ (2)

which means that the flavor vector $i\in\\{Q,u,d,L,e\\}$111In this work we
omit $L$ and $R$ chirality subscripts, and indicate with lowercase $u$, $d$
and $e$ the right-handed up, down quark and electron respectively, and with
uppercase $Q$ and $L$ the quark and lepton doublets. transforms as a
fundamental of $SU(3)_{i}$. Then the Yukawa sector of the Lagrangian

$\displaystyle\mathcal{L}_{\text{Yukawa}}=\bar{Q}Y_{d}dH+\bar{Q}Y_{u}u\tilde{H}+\bar{L}Y_{e}eH+\text{
h.c.}$ (3)

can be made formally invariant under this group if we promote the Yukawa
matrices, $Y_{u,d,e}$, to spurion fields transforming as in Table 1.

|  $SU(3)_{Q}$ |  $SU(3)_{u}$ |  $SU(3)_{d}$ |  $SU(3)_{L}$ |  $SU(3)_{e}$
---|---|---|---|---|---
$Y_{u}$ |  $\mathbf{3}$ |  $\mathbf{\bar{3}}$ |  $\mathbf{1}$ |  $\mathbf{1}$ |  $\mathbf{1}$
$Y_{d}$ |  $\mathbf{3}$ |  $\mathbf{1}$ |  $\mathbf{\bar{3}}$ |  $\mathbf{1}$ |  $\mathbf{1}$
$Y_{e}$ |  $\mathbf{1}$ |  $\mathbf{1}$ |  $\mathbf{1}$ |  $\mathbf{3}$ |  $\mathbf{\bar{3}}$
Table 1: Transformation properties of the Yukawa matrices treated as spurions
under the MFV assumption.

Minimal Flavor Violation is the requirement that any higher dimensional
operator has to be built out of $Y$ matrices and Standard Model fields, and
must be formally invariant under the flavor group, taking into account the
transformation properties in Table 1. Notice that the spurions transform under
the $U(1)$ abelian factors of $U(3)^{5}$, too. We do not treat those
explicitly, but they turn out to be useful, e.g., to get rid of terms like
$\sim\big{(}Y_{u}\big{)}^{m}\big{(}Y_{u}^{\dagger}\big{)}^{n}\left(\bar{Q}Q\right)^{p}$
with $m\neq n$.

After building the Lagrangian, we can set the spurion fields to their vacuum
values, namely the physical Yukawas. The latter are defined up to the freedom
of changing the fermion fields basis, i.e., they are defined up to
$U(3)^{3}_{q}\otimes U(3)^{2}_{l}$ transformations. In the following, we
choose a basis where they read:

$\displaystyle Y_{u}$ $\displaystyle=\lambda_{u}$ $\displaystyle Y_{d}$
$\displaystyle=V_{\textit{CKM}}\lambda_{d}$ $\displaystyle Y_{e}$
$\displaystyle=\lambda_{e}\ ,$ (4)

where the $\lambda$’s are diagonal matrices containing the diagonal Yukawa
couplings, e.g., $\lambda_{u}=\text{diag}(y_{u},y_{c},y_{t})$, and
$V_{\textit{CKM}}$ is the CKM matrix. Another basis that we will occasionally
mention is the following,

$\displaystyle Y_{u}$ $\displaystyle=V_{\textit{CKM}}^{\dagger}\lambda_{u}$
$\displaystyle Y_{d}$ $\displaystyle=\lambda_{d}$ $\displaystyle Y_{e}$
$\displaystyle=\lambda_{e}\ ,$ (5)

related to the previous one via a $U(3)_{Q}$ transformation. As we will make
clear in Section 3.1, our discussion cannot and does not depend on the
particular choice of basis. The MFV framework is relevant and particularly
convenient from a theoretical point of view, since it drastically reduces the
number of free parameters entering the Lagrangian at each mass-dimension
MFVandU2 . In addition, its phenomenological value resides in the fact that,
in an unconstrained setting, the coefficients of operators contributing to
Flavor Changing Neutral Currents (FCNC) would be naturally of $\order{1}$.
This is in contradiction with experiments, at least up to New Physics scales
of $\sim\order{10^{3}\,\text{TeV}}$. MFV is an efficient way, albeit perhaps
drastic, to justify such behaviour MFVreview .

###  2.1 Dimension 8 independent fermionic operators

As shown in Ref. irobstruct , requiring unitarity and analyticity to be
properties of the theory up to the UV imposes bounds on some of the
coefficients of the Effective Field Theory expansion. What we wish to see is
to what degree these bounds are compatible with the MFV assumptions. Thus, the
most interesting approach consists perhaps on focussing on 4-fermions
operators. However, the lowest order, dimension 6 operators, giving no $s^{2}$
contribution to the forward amplitude, are unaffected by these bounds (see
however Ref. dimension6bounds  for recent efforts towards constraining such
operators). Thus, as anticipated, we will focus on how these requirements
affect the coefficients of operators formed by 4 fermionic fields and two
derivatives. Bounds for those coefficients have been obtained in Ref.
flavorconstr , whence we will take part of the terminology and conventions
adopted in the rest of the paper. In particular, operators formed with fields
of one kind only, i.e., those of the schematic form
$\mathcal{O}\sim\partial^{2}(\bar{\psi}_{m}\Gamma\psi_{n})(\bar{\psi}_{p}\Gamma\psi_{q})$,
$\psi\in\\{u,d,Q\\}$, where $\Gamma$ is some combination of Dirac and SM gauge
matrices and we only made flavor indices explicit, will be dubbed self-
quartic. We will refer to those formed with two kinds of fields,
$\mathcal{O}\sim\partial^{2}(\bar{\psi}_{m}\Gamma\psi_{n})(\bar{\chi}_{p}\Gamma\chi_{q})$,
$\psi,\chi\in\\{u,d,Q\\}$ and $\psi\neq\chi$, as cross-quartic. The way
Lorentz indices are contracted is not shown here. We restrict for this
discussion to the quark sector only. The extension of our methods to include
leptons can then be readily found222This is true provided one does not include
right-handed neutrinos in the discussion. Then, a generalization of MFV
accounting for the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing matrix is
needed MLFV .. The list of independent operators we are interested in, then,
can be read off Table 2. Operators of the form
$\mathcal{O}=\partial_{\mu}(\bar{\psi}_{m}\gamma_{\nu}\psi_{n})\partial^{\mu}(\bar{\chi}_{p}\gamma^{\nu}\chi_{q})$,
$\psi\neq\chi$ are also present, and are independent from the ones listed in
Table 2. However, since they do not contribute to the forward amplitude, there
are no bounds on their Wilson coefficients flavorconstr .

Type | Content | Operator | Symmetry
---|---|---|---
self-quartic | (4-u) | $\mathcal{O}_{1}[u]=c^{u,1}_{mnpq}\partial_{\mu}(\bar{u}_{m}\gamma_{\nu}u_{n})\partial^{\mu}(\bar{u}_{p}\gamma^{\nu}u_{q})$ |
$\mathcal{O}_{3}[u]=c^{u,3}_{mnpq}\partial_{\mu}(\bar{u}_{m}T^{a}\gamma_{\nu}u_{n})\partial^{\mu}(\bar{u}_{p}T^{a}\gamma^{\nu}u_{q})$ |  $c_{mnpq}=c_{pqmn}$ $c_{mnpq}=c^{*}_{nmqp}$
(4-Q) | $\mathcal{O}_{1}[Q]=c^{Q,1}_{mnpq}\partial_{\mu}(\bar{Q}_{m}\gamma_{\nu}Q_{n})\partial^{\mu}(\bar{Q}_{p}\gamma^{\nu}Q_{q})$
$\mathcal{O}_{2}[Q]=c^{Q,2}_{mnpq}\partial_{\mu}(\bar{Q}_{m}\tau^{I}\gamma_{\nu}Q_{n})\partial^{\mu}(\bar{Q}_{p}\tau^{I}\gamma^{\nu}Q_{q})$
$\mathcal{O}_{3}[Q]=c^{Q,3}_{mnpq}\partial_{\mu}(\bar{Q}_{m}T^{a}\gamma_{\nu}Q_{n})\partial^{\mu}(\bar{Q}_{p}T^{a}\gamma^{\nu}Q_{q})$
$\mathcal{O}_{4}[Q]=c^{Q,4}_{mnpq}\partial_{\mu}(\bar{Q}_{m}T^{a}\tau^{I}\gamma_{\nu}Q_{n})\partial^{\mu}(\bar{Q}_{p}T^{a}\tau^{I}\gamma^{\nu}Q_{q})$
(4-d) | $\mathcal{O}_{1}[d]=c^{d,1}_{mnpq}\partial_{\mu}(\bar{d}_{m}\gamma_{\nu}d_{n})\partial^{\mu}(\bar{d}_{p}\gamma^{\nu}d_{q})$
$\mathcal{O}_{3}[d]=c^{d,3}_{mnpq}\partial_{\mu}(\bar{d}_{m}T^{a}\gamma_{\nu}d_{n})\partial^{\mu}(\bar{d}_{p}T^{a}\gamma^{\nu}d_{q})$ |
cross-quartic | (2-u)(2-Q) | $\mathcal{O}_{K1}[u,Q]=-a^{uQ,1}_{mnpq}\left(\bar{u}_{m}\gamma_{\mu}\partial_{\nu}u_{q}\right)\left(\bar{Q}_{n}\gamma^{\nu}\partial^{\mu}Q_{p}\right)$ |  $a^{\psi\chi}_{mnpq}=a^{\chi\psi}_{nmqp}$ $a_{mnpq}=a^{*}_{qpnm}$
$\mathcal{O}_{K3}[u,Q]=-a^{uQ,3}_{mnpq}\left(\bar{u}_{m}T^{a}\gamma_{\mu}\partial_{\nu}u_{q}\right)\left(\bar{Q}_{n}T^{a}\gamma^{\nu}\partial^{\mu}Q_{p}\right)$
(2-d)(2-Q) | $\mathcal{O}_{K1}[d,Q]=-a^{dQ,1}_{mnpq}\left(\bar{d}_{m}\gamma_{\mu}\partial_{\nu}d_{q}\right)\left(\bar{Q}_{n}\gamma^{\nu}\partial^{\mu}Q_{p}\right)$
$\mathcal{O}_{K3}[d,Q]=-a^{dQ,3}_{mnpq}\left(\bar{d}_{m}T^{a}\gamma_{\mu}\partial_{\nu}d_{q}\right)\left(\bar{Q}_{n}T^{a}\gamma^{\nu}\partial^{\mu}Q_{p}\right)$
(2-d)(2-u) | $\mathcal{O}_{K1}[d,u]=-a^{du,1}_{mnpq}\left(\bar{d}_{m}\gamma_{\mu}\partial_{\nu}d_{q}\right)\left(\bar{u}_{n}\gamma^{\nu}\partial^{\mu}u_{p}\right)$
$\mathcal{O}_{K3}[d,u]=-a^{du,3}_{mnpq}\left(\bar{d}_{m}T^{a}\gamma_{\mu}\partial_{\nu}d_{q}\right)\left(\bar{u}_{n}T^{a}\gamma^{\nu}\partial^{\mu}u_{p}\right)$
Table 2: List of independent self-quartic and cross-quartic operators. $T^{a}$
are the $SU(3)_{C}$ QCD generators and $\tau^{I}=\frac{\sigma^{I}}{2}$ are the
$SU(2)_{L}$ EW generators.

Therefore, restricting to the operators listed in Table 2, we can see that
there are $2+4+2=8$ independent self-quartic tensors $c_{mnpq}$. As stated
already in Ref. flavorconstr , imposing the symmetry requirements
$c_{mnpq}=c_{pqmn}$ and $c_{mnpq}=c^{*}_{nmqp}$ leaves
$\frac{1}{2}N_{f}^{2}(N_{f}^{2}+1)$ independent real entries in each tensor.
Indeed, the first condition is a symmetry condition on the complex
$N_{f}^{2}\times N_{f}^{2}$ matrix $c_{mnpq}$ whose rows are indexed by
$(m,n)$ and columns by $(p,q)$, so that it leaves
$2\times\frac{1}{2}N_{f}^{2}(N_{f}^{2}+1)$ unconstrained real entries. The
second condition further halves them. On the other hand, there are $2+2+2=6$
independent cross-quartic structure of operators. Each $a_{mnpq}$ tensor has
only to obey the hermiticity condition $a_{mnpq}=a^{*}_{qpnm}$, thus each of
them contains $N_{f}^{4}$ independent real entries. Since
$a^{\psi\chi}_{mnpq}=a^{\chi\psi}_{nmqp}$, fixing one $a^{\psi\chi}_{mnpq}$
tensor automatically fixes the one with $\psi\leftrightarrow\chi$. Overall, we
will deal with $6+8=14$ independent types of operators, and
$2N_{f}^{2}(5N_{f}^{2}+2)$ independent operators overall.

###  2.2 MFV ansatz for dimension 8 operators

We now wish to enforce the MFV assumption on the list of four-fermion
dimension-8 operators. This means that all EFT coefficients in Table 2 must be
written in terms of Yukawa spurions and flavor-blind EFT coefficients.

Since our goal is to study possible bounds on the entries of the Yukawa
matrices, then to be compared with phenomenological values, one must in
principle depart from the latter and treat the fermion masses and the entries
of the CKM matrix as generic. However, this would prevent us from performing a
proper power counting, as all powers of those matrices could have in principle
the same magnitude. In contrast, phenomenological studies of MFV MFV ;
MFVreview  rely on the measured values of the fermion masses or of the CKM
elements to define a consistent expansion. Large Yukawas demand further care,
but can also be treated consistently Feldmann:2008ja ; Kagan:2009bn . We
follow the same approach, giving up on full generality and on constraints on
the smallest Yukawas; we stick to cases where there exists a Yukawa much
larger than the others, so that we can fix all remaining ones to zero at first
order in a consistent MFV expansion. This assumption is realized in particular
by the phenomenological values of the Yukawas. We will consider two simplified
scenarios, with respectively 2 and 3 flavors. Then, the largest Yukawa,
$y_{c}$ in the former case and $y_{t}$ in the latter, is the only one non-
vanishing at leading order. We keep them as free parameters in all the
expressions below - see however the discussion at the end of this section for
the case of $y_{t}$. In keeping only the first relevant order in this
expansion, we will see that, at least in the proper realizations of
$N_{f}=2,~{}3$, there is always a choice of basis in flavor space such that
the CKM matrix $V_{\textit{CKM}}$ makes no appearance in the computations, and
no hope of putting any bound on its entries can be retained. This is due to
the fact that only the up-Yukawa matrix $Y_{u}$ will enter our expressions,
and we can always pick a basis where $V_{\textit{CKM}}$ is placed exclusively
in $Y_{d}$, the down-Yukawa matrix. This basis is nothing but the one of Eq.
(4), since in our approximations and when restricted to quark-type Yukawas, it
becomes, for $N_{f}=3$,

$Y_{u}=\text{diag}(0,0,y_{t}),\quad Y_{d}=\text{diag}(0,0,0).$ (6)

The expansion that we use depends on the size of the largest Yukawa. By
assumption, we neglect any term where $Y_{d}$ appears, but an expansion in the
up-Yukawa matrix $Y_{u}$ demands that the entries of the matrix are $<1$, to
ensure a consistent, non-divergent expansion. While this works for $N_{f}=2$
due to the smallness of the charm quark Yukawa, this does not hold for the
top, so that the expansion has to be resummed when $N_{f}=3$. We start by
discussing the naive expansion, and discuss at the end of this section how it
should be modified to account for the top-Yukawa resummation.

Numerically, for $N_{f}=3$, and since the Yukawa matrices will always appear
in pairs, these approximations amount to neglecting terms of order
$\order{\left({y_{c}}/{y_{t}}\right)^{2}}\sim\order{\left({y_{b}}/{y_{t}}\right)^{2}}\sim\order{10^{-3}}$
at most, when setting the Yukawas to their real values. This gives a measure
of how much we can let $y_{t}$ vary without spoiling our approximation. In
addition, focusing only on quarks is justified, too, at this level. Indeed,
since $\left({y_{\tau}}/{y_{t}}\right)^{2}\sim 10^{-4}$, the only bilinears
formed by leptons that would be added to this order are of the form
$\bar{L}_{m}\Gamma L_{m}$ or $\bar{e}_{m}\Gamma e_{m}$. Thus, they only
contribute trivially to the flavor tensor structure, and bounds for the
operators built with them can be retrieved, e.g., looking at the ones built
with $d$ quark fields. All this is somehow weaker for $N_{f}=2$. There, the
biggest contributions we neglected have an approximate size of
$\left({y_{s}}/{y_{c}}\right)^{2}\sim\left({y_{\mu}}/{y_{c}}\right)^{2}\sim
10^{-2}$.

Let us now ask what the MFV ansatz implies for operators containing 4 right-
handed up-type quarks, when we work at order $\order{Y_{u}^{2}Y_{d}^{0}}$ in
the expansion. There are two possible operators containing 4 up-quark fields:

$\displaystyle\mathcal{O}_{1}[u]$
$\displaystyle=c^{u,1}_{mnpq}\partial_{\mu}(\bar{u}_{m}\gamma_{\nu}u_{n})\partial^{\mu}(\bar{u}_{p}\gamma^{\nu}u_{q})$
$\displaystyle\mathcal{O}_{3}[u]$
$\displaystyle=c^{u,3}_{mnpq}\partial_{\mu}(\bar{u}_{m}T^{a}\gamma_{\nu}u_{n})\partial^{\mu}(\bar{u}_{p}T^{a}\gamma^{\nu}u_{q}),$

where only flavor indices are shown. To obtain the MFV expansion of the
$c^{u}$ coefficients, it is useful to define two objects333Obviously, after
the spurions freeze to their expectation values, $M=\tilde{M}$ in our choice
of basis.:

$\displaystyle{M}\equiv Y_{u}Y_{u}^{\dagger}\ ,$ (7)
$\displaystyle\tilde{M}\equiv Y_{u}^{\dagger}Y_{u}\ .$ (8)

Let us study what happens for the physical case $N_{f}=3$ (requiring instead
$N_{f}=2$ can only impose additional constraints that can always be enforced
at a later moment). The product of quark bilinears
$\sim\bar{u}_{m}u_{n}\bar{u}_{p}u_{q}$ is a
$(\mathbf{\bar{3}}\otimes\mathbf{3})\otimes(\mathbf{\bar{3}}\otimes\mathbf{3})$
of $SU(3)_{u}$, and can be decomposed as
$\mathbf{1}_{1}\oplus\mathbf{1}_{2}\oplus\mathbf{8}_{1}\oplus\mathbf{8}_{2}\oplus\mathbf{27}$,
since the $\mathbf{10}$ and the $\mathbf{\overline{10}}$ vanish because of the
exchange symmetry. Then, at $\order{Y_{u}^{0}Y_{d}^{0}}$,
$c_{mnpq}^{u,i}=\rho^{u,i}_{1}\delta_{mn}\delta_{pq}+\rho^{u,i}_{3}\delta_{mq}\delta_{pn}$.
Because of the $SU(3)_{Q}$ index carried by $Y_{u}$, there is no invariant we
can build with just one copy of it. However, the contraction $\tilde{M}$
defined earlier is a singlet of $SU(3)_{Q}$ and contains a
$\mathbf{1}\oplus\mathbf{8}$ of $SU(3)_{u}$. Its trace can be reabsorbed
through a redefinition of the $\rho^{u,i}_{1}$ and $\rho^{u,i}_{3}$
coefficients, while with its traceless part we can build two further
structures, so that:

$\displaystyle c^{u,i}_{mnpq}$
$\displaystyle=\rho^{u,i}_{1}(\delta_{mn}\delta_{pq})+\rho^{u,i}_{2}(\tilde{M}_{mn}\delta_{pq}+\delta_{mn}\tilde{M}_{pq})+\rho^{u,i}_{3}(\delta_{mq}\delta_{pn})+$
$\displaystyle+\rho^{u,i}_{4}(\tilde{M}_{mq}\delta_{pn}+\delta_{mq}\tilde{M}_{pn}),$
(9)

where all of the $\rho$ coefficients are unconstrained, and can be taken of
$\order{1}$. In Table 3, we list the shape that the MFV ansatz forces on the
Wilson coefficients respectively of the self-quartic and cross-quartic kinds
of operators previously listed. These can be obtained in a similar manner as
what was just shown.

Type | Content | Operator
---|---|---
self-quartic | (4-u) | $c^{u,i}_{mnpq}=\rho^{u,i}_{1}(\delta_{mn}\delta_{pq})+\rho^{u,i}_{2}(\tilde{M}_{mn}\delta_{pq}+\delta_{mn}\tilde{M}_{pq})+\rho^{u,i}_{3}(\delta_{mq}\delta_{pn})+\rho^{u,i}_{4}(\tilde{M}_{mq}\delta_{pn}+\delta_{mq}\tilde{M}_{pn})$
i=1,3
(4-Q) | $c^{Q,i}_{mnpq}=\rho^{Q,i}_{1}(\delta_{mn}\delta_{pq})+\rho^{Q,i}_{2}(M_{mn}\delta_{pq}+\delta_{mn}M_{pq})+\rho^{Q,i}_{3}(\delta_{mq}\delta_{pn})+\rho^{Q,i}_{4}(M_{mq}\delta_{pn}+\delta_{mq}M_{pn})$
i=1,2,3,4
(4-d) | $c^{d,i}_{mnpq}=\rho^{d,i}_{1}(\delta_{mn}\delta_{pq})+\rho^{d,i}_{3}(\delta_{mq}\delta_{pn})$
i=1,3
cross-quartic | (2-u)(2-Q) | $a^{uQ,i}_{mnpq}=\rho^{uQ,i}_{1}(\delta_{mq}\delta_{np})+\rho^{uQ,i}_{2}(\tilde{M}_{mq}\delta_{np})+\rho^{uQ,i}_{3}(\delta_{mq}M_{np})+\rho^{uQ,i}_{4}((Y_{u})_{nq}(Y^{\dagger}_{u})_{mp})$
i=1,3
(2-d)(2-Q) | $a^{dQ,i}_{mnpq}=\rho^{dQ,i}_{1}(\delta_{mq}\delta_{np})+\rho^{dQ,i}_{2}(\delta_{mq}M_{np})$
i=1,3
(2-d)(2-u) | $a^{du,i}_{mnpq}=\rho^{du,i}_{1}(\delta_{mq}\delta_{np})+\rho^{du,i}_{2}(\delta_{mq}\tilde{M}_{np})$
i=1,3

Table 3: $\mathcal{O}(Y_{u}^{2}Y_{d}^{0})$-MVF expansion of the self-quartic
and cross-quartic operators

The above tables define the EFT coefficients on which we will soon apply
positivity bounds. The coefficients can be seen to respect all the symmetry
properties required by Table 2. However, the number of independent
coefficients is drastically reduced, as now the only free parameters are
flavor-blind overall coefficients, i.e., the objects we named $\rho_{A}^{i}$,
and their number is independent on $N_{f}$. In particular, as already stated,
the number of independent real coefficients in a unconstrained setting is
$2N_{f}^{2}(5N_{f}^{2}+2)$ (176 for $N_{f}=2$ and 846 for $N_{f}=3$), while
after imposing MFV we are left with 44 independent real coefficients
$\rho_{A}^{i}$ at the $\mathcal{O}(Y_{u}^{2}Y_{d}^{0})$ order in the MFV
expansion, independently on the number of flavor (actually, for $N_{f}=2$, it
turns that 6 coefficients are redundant).

We should nevertheless pause to comment on the $N_{f}=3$ case. We kept $y_{t}$
as a generic parameter in our analysis, but are eventually interested in its
phenomenological value, which is $\sim\order{1}$. Therefore, the truncation to
$\order{Y_{u}^{2}Y_{d}^{0}}$ we perform is not in principle justified for the
physical $N_{f}=3$ case: higher-order terms such as

$\displaystyle\left(Y_{u}Y^{\dagger}_{u}\right)^{n}_{ij}\sim
y_{t}^{2n}\delta_{3i}\delta_{3j}\ .$ (10)

should be properly resummed. Interestingly, the resummation does not bring new
flavor violation beyond the one contained in the first non-trivial contraction
$Y_{u}Y_{u}^{\dagger}$. This is more transparent in the basis of (5), where

$\displaystyle\left(Y_{u}Y^{\dagger}_{u}\right)^{n}_{ij}\sim
y_{t}^{2n}\left(V_{\textit{CKM}}^{*}\right)_{3i}\left(V_{\textit{CKM}}\right)_{3j}\
.$ (11)

This means that the flavor violation structure can be obtained from our naive
expansion in Table 3, up to a redefinition of the parameters to account for
the resummation. Namely, that means that one should remove the explicit
$y_{t}$ dependence by fixing $y_{t}=1$ and turn the EFT coefficients into
$\order{1}$ arbitrary functions of $y_{t}$,
$\rho_{i}^{x,j}\rightarrow\rho(y_{t})_{i}^{x,j}$. A proof of such behaviour is
given in Appendix A. When we freeze the spurions to their background values,
functions of $y_{t}$ become simple numbers, which means that any explicit
$y_{t}$-dependence simply disappears from the expansion.

##  3 Analysis of the bounds

At this point, we have all the machinery we need to pursue our goal, namely to
confront the bounds obtained in Ref. flavorconstr  with the MFV hypothesis
dictating the expansion of the various dimension-8 operators in power of the
Yukawa, as listed in Table 3. First of all, the positivity constraints of Ref.
flavorconstr  depend not only on the Wilson coefficients $c_{mnpq}$ and
$a_{mnpq}$, but also on some arbitrary external states, dubbed $\alpha$ and
$\beta$ and consisting in generic complex vectors of unit norm in flavor
space. Thus, we need to disentangle the former from the latter first, to
obtain expressions that depend on the operator coefficients only. Secondly,
many of the inequalities contain linear combinations of coefficients coming
from distinct independent operators. To simplify the computations, we define
new coefficients via suitable linear transformations. We will carry this
latter simplification first, and then proceed to show how we removed the
dependencies on the $\alpha$’s and $\beta$’s.

Let us see, for example, how this works in the case of operators containing 4
up fields. The bounds on them are obtained flavorconstr  by scattering the
following states:

$\displaystyle\ket{\psi_{1}}$ $\displaystyle=\alpha_{mi}\ket{\bar{u}_{mi}},$
$\displaystyle\ket{\psi_{2}}$ $\displaystyle=\beta_{mi}\ket{u_{mi}},$
$\displaystyle\ket{\psi_{3}}$
$\displaystyle=\beta^{*}_{mi}\ket{\bar{u}_{mi}},$
$\displaystyle\ket{\psi_{4}}$ $\displaystyle=\alpha^{*}_{mi}\ket{u_{mi}},$
(12)

where $m$ and $i$ are flavor and gauge indices respectively. The amplitude
then reads:

$\displaystyle\mathcal{A}=4s^{2}\left[\left(c^{u,1}_{mnpq}-\frac{1}{6}c^{u,3}_{mnpq}\right)\alpha_{mi}^{*}\beta_{ni}\beta^{*}_{pj}\alpha_{qj}+\frac{1}{2}c^{u,3}_{mnpq}\alpha_{mi}^{*}\beta_{nj}\beta^{*}_{pj}\alpha_{qi}\right].$
(13)

Marginalizing over the gauge indices444We factor out from now on the gauge
dependence from the $\alpha$’s and $\beta$’s, so that they only contain flavor
indices. This yields conservative bounds, albeit not necessarily the strongest
ones., two different bounds are obtained:

$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}\left(c_{mnpq}^{u,1}+\frac{1}{3}c_{mnpq}^{u,3}\right)$
$\displaystyle>0\,,$
$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}c_{mnpq}^{u,3}$
$\displaystyle>0\ .$ (14)

Here, as mentioned, $\alpha_{n}$ and $\beta_{n}$ are arbitrary $N_{f}$
components complex vectors of unit norm. They parametrize the external states,
since the bounds are obtained by constraining the $s^{2}$ coefficient of a
$2\to 2$ scattering of generic superpositions of flavor eigenstates.
Therefore, the inequalities in Eq. (14) have to be fulfilled for all
$\alpha$’s, $\beta$’s.

We perform a linear transformation on Eq. (14) by defining:

$\displaystyle\xi^{u,1}_{k}\equiv\rho^{u,1}_{k}+\frac{1}{3}\rho^{u,3}_{k}\quad\textrm{and}\quad\xi^{u,3}_{k}\equiv\rho^{u,3}_{k}\quad\textrm{for}\quad
k=1,2,3,4,$ (15)

so that, defining $c(\xi)^{u,i}_{mnpq}$ as in the first line of Table 3 but
with $\rho\to\xi$, i.e.,

$\displaystyle c(\xi)^{u,i}_{mnpq}$
$\displaystyle=\xi^{u,i}_{1}(\delta_{mn}\delta_{pq})+\xi^{u,i}_{2}(\tilde{M}_{mn}\delta_{pq}+\delta_{mn}\tilde{M}_{pq})+\xi^{u,i}_{3}(\delta_{mq}\delta_{pn})$
$\displaystyle+\xi^{u,i}_{4}(\tilde{M}_{mq}\delta_{pn}+\delta_{mq}\tilde{M}_{pn})$
$\displaystyle i$ $\displaystyle=1,3,$

the bounds become simply

$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}c(\xi)_{mnpq}^{u,i}$
$\displaystyle>0,$ $\displaystyle i$ $\displaystyle={1,3}.$ (16)

Since all the bounds are expressed as inequalities on linear combinations of
the flavor structure tensors as in Eq. (14), it is always possible to perform
a linear redefinition such as Eq. (15) to bring them to a form analogous to
Eq. (16). From now on, we will do this on all operators, and show both bounds
and flavor tensors as functions of $\xi$’s. Their explicit dependence on the
original $\rho$ coefficients is shown in Appendix E. In conclusion, the bounds
we have to study are all of the form: 0rCl”lCl α_mα^*_qβ_nβ^*_p
c(ξ)_mnpq^X,i&¿0, X=u, Q, d,
α_mα^*_qβ_nβ^*_p a(ξ)^X,i_mnpq¿0, X=uQ, dQ, du. We first prove that it is
always possible to find some $\xi$’s such that these constraints can be
satisfied for any $\alpha$, $\beta$. Indeed, for the self-quartic operators,
one can notice that, looking at the pattern in which the flavor indices are
summed, if we choose $\xi_{1}^{i}=\xi_{2}^{i}=0$, then the bounds can be
expressed as

$\displaystyle\xi_{3}^{i}|\alpha|^{2}|\beta|^{2}+\xi_{4}^{i}\left(\alpha_{m}A_{mq}\alpha_{q}^{*}|\beta|^{2}+\beta^{*}_{p}A_{pn}\beta_{n}|\alpha|^{2}\right)>0,$
(17)

where $A=\tilde{M}$, $A=M$ and $A=0$ for the (4-u), (4-Q) and (4-d) cases
respectively. In the former two cases, being the product of an invertible
matrix and its hermitian conjugate, $A$ is (semi-)positive definite. Thus
$\xi_{1}^{i}=\xi_{2}^{i}=0$, $\xi_{3}^{i}>0$ and $\xi_{4}^{i}>0$ is, in this
setting, an allowed region in the parameter space, and fulfills the bounds
$\forall\alpha,\beta$. In the (4-d) case, $A=0$, and
$\\{\xi_{1}^{i}=0,\xi_{3}^{i}>0\\}$ is an always allowed region of the
parameter space. Similar conclusions can be drawn for the cross-quartic
operators. As a consequence, there exists at least one region that is a
solution of Eqs. (3) and (3), with the coefficients expressed as per Table 3.

However, we wish to study the anatomy of the bounds when all the $\rho$
coefficients of the MVF expansions are of $\mathcal{O}(1)$, which is the
natural realization of the MFV ansatz. By fixing the coefficients, the bounds
become functions of SM parameters alone. We then verify whether they are
strict enough as to impose constraints on the parameters of the dimension-4
Lagrangian. In the two flavor case, that means $y_{c}$, the charm-quark Yukawa
coupling. Naively, that also means $y_{t}$ when $N_{f}=3$, however, as we
discussed previously, $y_{t}$ should be absorbed in the $\rho$ (equivalently,
in the $\xi$) coefficients. In that case, we can only check whether all the
EFT coefficients can be consistently $\mathcal{O}(1)$.

###  3.1 Flavor violation and CKM-(in)dependence of the positivity bounds

Before going any further, some clarifications are in order. All along the
discussion we made, it looks like there is no place for any flavor violation
at all. Indeed, in our approximation, the only matrix involved in building the
flavor invariants is $Y_{u}$, which we chose to be diagonal, and the only
physical parameter shaping the allowed region is $y_{t}$, while there is no
sign of the CKM matrix. Obviously, our discussion cannot depend on the
specific basis that we pick. In this section we show that this is the case and
that the CKM matrix only enters the bounds at subleading order with respect to
our approximations. Imagine we chose, instead of Eq. (4), the basis (5). Then,
we could have diagonalized $Y_{u}$, and consequently $M$ and $\tilde{M}$, at a
later time, by exploiting the redundancy contained in the definition of
expressions like Eq. (3) or (3). Indeed, we know that any square matrix can be
decomposed as

$\displaystyle Y_{u}=U\Sigma\tilde{U}^{\dagger},$ (18)

where $U$ and $\tilde{U}$ are unitary matrices and $\Sigma$ is diagonal. Then:

$\displaystyle M$
$\displaystyle=Y_{u}Y_{u}^{\dagger}=U\Sigma\Sigma^{*}U^{\dagger}$ (19)
$\displaystyle\tilde{M}$
$\displaystyle=Y_{u}^{\dagger}Y_{u}=\tilde{U}\Sigma^{*}\Sigma\tilde{U}^{\dagger}.$
(20)

Thus, $M$ and $\tilde{M}$ are diagonalized by $U$ and $\tilde{U}$
respectively. Then, in Eq. (3) or (3), we could have rotated555More precisely,
we can multiply by the identity $\mathbbm{1}_{N_{f}}=UU^{\dagger}$ so that
$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}c_{mnpq}^{u,i}=$
$\displaystyle=\alpha_{m}\left(U_{mm^{\prime}}U^{\dagger}_{m^{\prime}m^{\prime\prime}}\right)\left(U_{q^{\prime\prime}q^{\prime}}U^{\dagger}_{q^{\prime}q}\right)\alpha^{*}_{q}\left(U_{n^{\prime\prime}n^{\prime}}U^{\dagger}_{n^{\prime}n}\right)\beta_{n}\beta^{*}_{p}\left(U_{pp^{\prime}}U^{\dagger}_{p^{\prime}p^{\prime\prime}}\right)c_{m^{\prime\prime}n^{\prime\prime}p^{\prime\prime}q^{\prime\prime}}\equiv$
$\displaystyle\equiv\left(\tilde{\alpha}_{m^{\prime}}U^{\dagger}_{m^{\prime}m^{\prime\prime}}\right)\left(U_{q^{\prime\prime}q^{\prime}}\tilde{\alpha}^{*}_{q^{\prime}}\right)\left(U_{n^{\prime\prime}n^{\prime}}\tilde{\beta}_{n^{\prime}}\right)\left(\tilde{\beta}^{*}_{p^{\prime}}U^{\dagger}_{p^{\prime}p^{\prime\prime}}\right)c_{m^{\prime\prime}n^{\prime\prime}p^{\prime\prime}q^{\prime\prime}}$
both $\alpha$ and $\beta$ (and their hermitian conjugates) using $U$ or
$\tilde{U}$. This does not modify the space that $\alpha$ and $\beta$ span,
since unitary matrices conserve norms. Therefore we can explore the $\alpha$’s
and $\beta$’s space with the diagonalized version of $M$ and $\tilde{M}$. In
particular, $\tilde{M}=\lambda_{u}^{2}$ is diagonal to begin with, while we
can rotate $M\to\Sigma^{*}\Sigma=\lambda_{u}^{2}$ using
$U=V_{\textit{CKM}}^{\dagger}$, $\tilde{U}=\mathbbm{1}_{N_{f}}$. These are the
same matrices we got when we started with the basis in Eq. (4) in the first
place. The freedom to absorb unitary matrices in the generic vectors $\alpha$
and $\beta$ arises from specific properties of the positivity bounds. First,
those bounds are obtained in Ref. flavorconstr  in a high-energy limit where
all SM fermions are considered massless. In this limit, the mass terms
disappear and they do not single out anymore the preferred basis that
diagonalizes them. In addition, only dimension-8 operators are constrained by
the bounds, so that the dimension-8 EFT coefficients are the only spurions
that break the flavor symmetry and enter the bound. Thus, the flavor symmetry
can be used to absorb irrelevant parameters, here in the sense of not entering
the positivity bounds, among the ones that form the dimension-8 EFT
coefficients. In our case, the CKM matrix is precisely such an irrelevant
parameter. Notice that this statement derives from the use of the full flavor
group. Consequently, it does not hold if we only scatter a subset of the
flavor states (said differently, if we imposed some conditions on
$\alpha,\beta$). Indeed, the restriction of the flavor group to those states
may not be sufficient to remove all the CKM dependence from the bounds. We
will see an example of this in Section 3.3.2, when a two-flavor scenario is
embedded in $N_{f}=3$. There, by scattering the two first flavors only, we
obtain a subset of the $N_{f}=3$ bounds which depends on the entries of the
CKM matrix. Nevertheless, the full $N_{f}=3$ bounds are more stringent and do
not depend on the latter.

One can then ask at which order in the expansion in powers of Yukawa matrices
a CKM contribution would appear so that it could not be rotated away. From
what we just saw, this has to be a combination containing both up- and down-
Yukawa matrices. If we define $N_{mn}\equiv{(Y_{d}Y_{d}^{\dagger})}_{mn}$,
i.e., the analogous of $M$ for the down Yukawa matrix, we notice that this,
too, contains a $\mathbf{8}$ of $SU(3)_{q}$. Thus, if we expand a bit further,
we can add for example to the second line of Table 3 a term like
$\sim\tilde{\rho}_{1}(N_{mn}\delta_{pq}+\delta_{mn}N_{pq})+\tilde{\rho}_{2}(n\leftrightarrow
q)$. With this example we can see that the freedom left by the redundancy in
the definition of the external states is larger than the symmetry of the
Lagrangian alone. Indeed, by exploiting the flavor $U(3)^{3}$, we can
diagonalize either $N$ or $M$, but not both. However, from the point of view
of the bounds, adding only the aforementioned terms corresponds to a shift
$M\to M+N$ in the second line of Table 3. This matrix can then be
diagonalized, since it is hermitian, too. Nonetheless, its eigenvalues depend
now explicitly on the entries of the CKM matrix, that consequently enter the
bounds in any case at this level in the expansion.

Conversely, the fact that the entries of $V_{\textit{CKM}}$ are relevant only
at such a subleading order means that the bounds are not really sensitive to
their values, and even relatively large modifications for them do not affect
much the structure of the bounds.

###  3.2 Disentangling the external states.

As already stated666We focus here on the $c_{mnpq}$ as a generalization to the
$a_{mnpq}$ is straightforward. For the sake of simplicity, and since this
analysis applies everywhere, we here drop any superscript on $c_{mnpq}$.,
positivity conditions like Eq. (16) have to be fulfilled for every value of
$\alpha$ and $\beta$, since they simply label arbitrary in-states. However, to
obtain bounds that are purely expression of the EFT coefficients, one has to
disentangle the latter from the external states. This section is devoted to
show how this can be done in the case under consideration. We can start by
removing the dependence of the bounds on either $\alpha$ or $\beta$. Suppose
we fix $\beta$ and define $C(\beta)_{mq}=c_{mnpq}\beta_{n}\beta^{*}_{p}$.
Notice that this matrix is hermitian, and thus diagonalizable, and even if it
were not, its antihermitian part would drop out of expressions like Eq. (16).
Then the positivity requirement (16) takes the form
$C(\beta)_{mq}\alpha_{m}\alpha^{*}_{q}>0$, and this inequality has to be
satisfied for any unit $N_{f}$-vector $\alpha$. This is equivalent to asking
that the matrix $C(\beta)$ is positive definite, i.e., that its real
eigenvalues $r(\beta)_{I}$, $I=1,\ldots,N_{f}$, are all positive. Thus, we can
trade

$\displaystyle\begin{cases}c_{mnpq}\beta_{n}\beta^{*}_{p}\alpha_{m}\alpha^{*}_{q}>0\\\
\forall\alpha,\beta\ \textrm{with
}\norm{\alpha}=\norm{\beta}=1\end{cases}\iff\begin{cases}r(\beta)_{I}>0\quad
I=1,\ldots,N_{f}\\\ \forall\beta\ \textrm{with }\norm{\beta}=1\end{cases}\ .$
(22)

The conditions on the r.h.s of Eq. (22) are necessary and sufficient. They are
necessary since, if we find a negative eigenvalue for some
$\beta=\hat{\beta}$, we can pick $\alpha=\hat{\alpha}$ to be an eigenvector
associated to that eigenvalue and the quartic expression on the l.h.s of Eq.
(22) evaluated at $\hat{\alpha},~{}\hat{\beta}$ would be negative. They are
sufficient because, if there is some value of $\alpha$, and $\beta$, say
$\hat{\alpha},~{}\hat{\beta}$, in which the l.h.s of Eq. (22) is negative, or
in other words $C(\hat{\beta})_{mq}\hat{\alpha}_{m}\hat{\alpha}^{*}_{q}<0$,
then $C(\hat{\beta})$ has to have at least one negative eigenvalue. Indeed,
one can decompose $\hat{\alpha}$ on the basis $\\{v^{I}_{m}\\}$ of
eigenvectors of $C(\hat{\beta})$, i.e., write
$\hat{\alpha}_{m}=\hat{\alpha}_{I}v^{I}_{m}$, and obtain
$C(\hat{\beta})_{mq}\hat{\alpha}_{m}\hat{\alpha}^{*}_{q}=\sum_{I}r(\beta)_{I}\absolutevalue{\hat{\alpha}_{I}}^{2}\norm{v^{I}}^{2}<0$.

Another way to phrase the conditions in the r.h.s of Eq. (22) is by noticing
that expressions like Eq. (16) can be viewed as quadratic homogeneous
polynomials in the complex components of $\alpha$, with zero linear term.
Then, requiring that the polynomial is greater than zero reduces to asking the
multidimensional parabola to point upwards in any direction parametrized by
$\alpha$. Were this not the case, we could find an eigendirection with
negative hessian eigenvalue, along which we would end up in the negative
region.

Since $\alpha$ and $\alpha^{*}$ always appear in pairs, and so do $\beta$ and
$\beta^{*}$, we can remove a total phase from each of them. Moreover, they are
of fixed unit norm. Then, they contain $2N_{f}-2$ free real parameters each,
and the l.h.s of Eq. (22) depends on $4N_{f}-4$ parameters. We trade it for
the $N_{f}$ conditions on the eigenvalues, each condition depending only on
the $2N_{f}-2$ real parameters contained in $\beta$. This rapidly turns out to
be inconvenient for large values of $N_{f}$, but it works well for
$N_{f}=2,~{}3$. In particular, for $N_{f}=2$, the two eigenvalues are positive
if and only if the trace and the determinant of $C(\beta)_{mq}$ are positive.
Notice that the discrepancy in the counting between the two requirements lies
only in the number of free parameters we have to marginalize over. Instead,
having shown that the r.h.s. and the l.h.s. of Eq. (22) are equivalent, they
impose the same conditions on the $c_{mnpq}$ after all the $\alpha$’s and
$\beta$’s are removed.

###  3.3 A benchmark case: (4-Q) operators

We start by studying the positivity bounds (3) for the (4-Q) operators, and
work it out step by step, the procedure for the other cases being very
similar. Under the MFV assumption, the coefficients of the self-quartic (4-Q)
operators take the form:

$\displaystyle c(\xi)^{Q,i}_{mnpq}$
$\displaystyle=\xi^{Q,i}_{1}(\delta_{mn}\delta_{pq})+\xi^{Q,i}_{2}(M_{mn}\delta_{pq}+\delta_{mn}M_{pq})+\xi^{Q,i}_{3}(\delta_{mq}\delta_{pn})+$
$\displaystyle+\xi^{Q,i}_{4}(M_{mq}\delta_{pn}+\delta_{mq}M_{pn}),$
$\displaystyle i$ $\displaystyle={1,2,3,4}.$

In order to see what are the consequences imposed by Eq. (3) on the
coefficients $\xi^{Q,i}_{A}$, we will first consider for simplicity a 2-flavor
setting which can be implemented in two slightly different ways: it will
successively describe a theory of two generations only (Section 3.3.1), and
the restriction of a 3-flavor setting to the lightest two flavors (Section
3.3.2). Then, we proceed to study $N_{f}=3$.

####  3.3.1 Positivity bounds on true $N_{f}=2$ ansatz

First of all, if we reduce the symmetry group to be $SU(2)$, one can verify
that $(M_{mq}\delta_{pn}+\delta_{mq}M_{pn})$ is not an independent structure,
and its coefficient $\rho_{4}$ can be reabsorbed through a redefinition of the
remaining three. This can be seen both by counting the allowed singlets in the
tensor product, or in a more direct way, as shown in Appendix B. This way, we
can remove 4 $\xi_{4}^{Q,i}$ coefficients. However, since they provides just
an innocuous redundancy, we will keep them at first and set them to zero at a
later moment, highlighting the physical consequences of the two choices. Now,
we need to parametrize the generic complex unit vector
$\beta\in\mathbb{C}^{2}$. A possible parametrization is:

$\displaystyle\beta$ $\displaystyle=\begin{pmatrix}xe^{i\theta_{x}}\\\
ye^{i\theta_{y}}\\\ \end{pmatrix}\equiv
e^{i\theta_{y}}\begin{pmatrix}xe^{i\tilde{\theta}_{x}}\\\ y\\\ \end{pmatrix}$
$\displaystyle\textrm{with}\quad x^{2}+y^{2}$ $\displaystyle=1,$ (23)

where all the parameters are real positive and
$\tilde{\theta}_{x}=\theta_{x}-\theta_{y}$. As mentioned, we can remove the
total phase and set $\theta_{y}=0$. In the flavor basis (4), the up-Yukawa
matrix is simply

$\displaystyle Y_{u}$ $\displaystyle=\begin{pmatrix}y_{u}&0\\\
0&y_{c}\end{pmatrix}\sim y_{c}\begin{pmatrix}0&0\\\ 0&1\end{pmatrix},$ (24)

$y_{u}$, $y_{c}$ being the up and charm Yukawa respectively, and in the last
equality, we specifically assumed a mass hierarchy and kept only the leading
term. Although we also set $y_{c}$ to zero in the $N_{f}=3$ case, we keep it
here since it corresponds to the largest mass of this two-flavor theory. Now,
as anticipated, we can translate the positivity condition (3) as two
conditions on the eigenvalues of
$C(\beta)_{mq}=c_{mnpq}\beta_{n}\beta^{*}_{p}$, or, equivalently, on its
determinant and trace.

The assumed mass hierarchy, $y_{c}\gg y_{u}\sim 0$, ensures that the trace and
the determinant of $C(\beta)$ depend only on $x$ and not on $y$ nor on
$\tilde{\theta}_{x}$. They are given by:

$\displaystyle\Tr[C(\beta)]$
$\displaystyle=2x^{2}\left(\xi^{Q,i}_{2}+\xi^{Q,i}_{4}\right)y_{c}^{2}+\xi^{Q,i}_{4}y_{c}^{2}+\xi^{Q,i}_{1}+2\xi^{Q,i}_{3}$
(25) $\displaystyle\det[C(\beta)]$
$\displaystyle=x^{4}\left(\xi^{Q,i}_{2}+\xi^{Q,i}_{4}\right)^{2}y_{c}^{4}-x^{2}y_{c}^{2}\left(\xi^{Q,i}_{2}+\xi^{Q,i}_{4}\right)\left(y_{c}^{2}\left(\xi^{Q,i}_{2}-\xi^{Q,i}_{4}\right)-2\xi^{Q,i}_{3}\right)+$
$\displaystyle\quad+\left(\xi^{Q,i}_{4}y_{c}^{2}+\xi^{Q,i}_{3}\right)\left(\xi^{Q,i}_{1}+\xi^{Q,i}_{3}\right).$
(26)

The trace is a linear function of $x^{2}$ which varies within the interval
$[0,1]$. Thus, it is positive for any relevant value of $x$ if and only if its
values at the boundaries of the interval are also positive. The determinant,
on the other hand, is a quadratic polynomial in $x^{2}$. One can verify that
such parabola is positive in $[0,1]$ if and only if777This is jusfied like
this: assuming it is positive in 0 and 1: if it has negative discriminant, it
is positive in the whole interval. Otherwise, if the discriminant is positive,
and if the parabola opens downwards, i.e., $a<0$, it is also positive within
the interval. If $\Delta>0$ and $a>0$, we then need to make sure that the
minimum falls outside $[0,1]$. This is done by requiring
$x_{min}^{2}-x_{min}>0\longrightarrow b(b+2a)>0$.:

  * •

it is positive at the boundaries;

  * •

one of the following conditions is met:

$\displaystyle\Delta<0\text{ or }a<0\text{ or }b(b+2a)>0,$

where we parametrized $\det[C(\beta)]\equiv ax^{4}+bx^{2}+c$.

Putting everything together, and after some simplifications, we get the full
set of conditions:

$\displaystyle\text{(4-Q) ($N_{f}$=2): }\begin{cases}\bullet\
\xi^{Q,i}_{4}y_{c}^{2}+\xi^{Q,i}_{3}>0,\\\ \bullet\
2y_{c}^{2}\left(\xi^{Q,i}_{2}+\xi^{Q,i}_{4}\right)+\xi^{Q,i}_{1}+\xi^{Q,i}_{3}>0,\\\
\bullet\ \xi^{Q,i}_{1}+\xi^{Q,i}_{3}>0,\\\ \bullet\
y_{c}^{4}(\xi^{Q,i}_{4}-\xi^{Q,i}_{2})\left(\xi^{Q,i}_{2}+3\xi^{Q,i}_{4}\right)+8\xi^{Q,i}_{3}\xi^{Q,i}_{4}y_{c}^{2}+4\left(\xi^{Q,i}_{3}\right)^{2}>0\quad\textrm{or}\\\
\quad\left(-4y_{c}^{2}\left(\xi^{Q,i}_{1}\xi^{Q,i}_{4}+\xi^{Q,i}_{2}\xi^{Q,i}_{3}\right)+y_{c}^{4}\left(\xi^{Q,i}_{2}-\xi^{Q,i}_{4}\right)^{2}-4\xi^{Q,i}_{1}\xi^{Q,i}_{3}\right)<0.\end{cases}$
(27)

The allowed region specified by these bounds is shown in Fig. 1 as a function
of the unique relevant parameter $y_{c}$.

One can notice in particular that the natural MFV benchmark point
$\xi_{1,2,3}=1$ is compatible with the positivity bound (3) if and only if
$y_{c}^{2}<2\left(1+\sqrt{2}\right)$. However, as mentioned already, a
consistent MFV expansion in $Y_{u}$ requires $y_{c}<1$. So for any consistent
MVF expansion, the positivity bounds are easily satisfied.

Bounds on (4-Q) operators coefficients for $N_{f}=2$

Figure 1: Plot showing the allowed region (in color) obtained for the (4-Q)
operators restricted to $N_{f}=2$ with generic $\xi$ values, as $y_{c}$
changes. Every region associated to a larger $y_{c}$ value is contained in the
previous ones: for instance, blue and dark orange regions are allowed for any
$y_{c}$ roughly smaller than 4, but forbidden for larger values of $y_{c}$.
The redundant $\xi_{4}$ has been set to 0. In this case $\xi_{3}>0$, and using
that the bounds are invariant under a full rescaling, we have set $\xi_{3}=1$,
and plot the remaining two independent coefficients. As explained in the text,
values of $y_{c}>1$ are unphysical and are only plotted for visual reasons.
The black point represents the natural MFV benchmark point $\xi_{1,2,3}=1$.
The red line contours the region corresponding to the threshold value of
$y_{c}=\sqrt{2(1+\sqrt{2})}$: for bigger values of $y_{c}$, the natural
benchmark point does not belong to the allowed region any more. The region
$\xi_{1}<-1$ is excluded for any value of $y_{c}$. For the physical value
$y_{c}\approx 10^{-2}$, almost all points $(\xi_{1}\geq-1,\xi_{2})$ are
allowed.

####  3.3.2 $N_{f}=2$ revisited: projected $N_{f}=3$ onto $N_{f}=2$

Another approach one could follow to describe the $N_{f}=2$ case, is to take
the full $N_{f}=3$ setting, and to restrict all flavor indices to be
$\\{1,2\\}$. In our flavor basis (4), this turns out to be trivial, and it
leads to the same result as depicted in Fig. 1. However, in the present case,
one can modify the bounds using a $U(3)^{3}$ transformation: although the full
$N_{f}=3$ bounds (to be discussed in the next section) are basis-independent,
what we identify with the first two flavors is a basis dependent statement,
and so are the bounds derived using the restricted $N_{f}=2$ approach of this
section. As an example, we rotate to the basis of Eq. (5), and only then
perform the projection. Consequently, the restricted version of $M$ is now:

$\displaystyle M_{ij}$
$\displaystyle=\left(Y_{u}Y^{\dagger}_{u}\right)_{ij}\sim(V_{\textit{CKM}})_{3i}(V_{\textit{CKM}}^{*})_{3j}\
,$ $\displaystyle i,j$ $\displaystyle=1,2\ ,$

while $\tilde{M}=0$. In comparison with the usual value of $M_{ij}\sim
y_{t}^{2}(V_{\textit{CKM}})_{3i}(V_{\textit{CKM}}^{*})_{3j}$, we fixed
$y_{t}=1$ to account for the $y_{t}$-resummation, as explained at the end of
Section 2.2. At this point, we are left only with a $U(2)^{3}$ symmetry, part
of which, $U(2)_{Q}$ for the present (4-Q) case, can be used to diagonalize
$M$. This can always be done since $M$ is still hermitian. Here we parametrize
$V_{\textit{CKM}}$ through the Wolfenstein parametrization up to order
$\mathcal{O}(\lambda^{5})$, where $\lambda=\sin(\theta_{c})$, $\theta_{c}$
being the Cabibbo angle:

$\displaystyle
V_{\textit{CKM}}\approx\left(\begin{array}[]{ccc}1-\frac{1}{2}\lambda^{2}-\frac{1}{8}\lambda^{4}&\
\lambda&\ A\lambda^{3}(\rho-i\eta)\\\
-\lambda+\frac{1}{2}A^{2}\lambda^{5}[1-2(\rho+i\eta)]&\
1-\frac{1}{2}\lambda^{2}-\frac{1}{8}\lambda^{4}\left(1+4A^{2}\right)&\
A\lambda^{2}\\\ A\lambda^{3}[1-(1-\frac{1}{2}\lambda^{2})(\rho+i\eta)]&\
-A\lambda^{2}+\frac{1}{2}A\lambda^{4}[1-2(\rho+i\eta)]&\
1-\frac{1}{2}A^{2}\lambda^{4}\\\ \end{array}\right).$

Diagonalizing the 2$\times 2$ matrix $M$, one can see that its only non-zero
eigenvalue is:

$\sigma\equiv A^{2}\lambda^{4},$ (28)

which again we require to be $<1$ to ensure a consistent expansion of the
unitary $V_{CKM}$ matrix. One can then easily map the positivity bounds on the
$\xi_{1,2}$ parameter space using the results of the previous section by
substituting $y_{c}^{2}$ by $\sigma$. Note that the values of $\xi_{1,2}$
compatible with the positivity bounds now depend on $\sigma$, which itself
depends explicitly on the CKM entries, contrary to the general property
presented in Section 3.1. Indeed, when going back to how the bounds where
found in the first place, we see that the setting studied in this section
corresponds to a $2\to 2$ scattering where the initial and final states are
restricted to the first two flavors. However, fixing them breaks the flavor
symmetry down to $U(2)^{3}$. The latter is then too small to absorb all the
CKM parameters, which consistently enter the bounds. This is different in the
full $N_{f}=3$ case, as we now discuss.

####  3.3.3 $N_{f}=3$

Now we wish to tackle the $N_{f}=3$ setup. We can approximate
$M=\left(Y_{u}Y^{\dagger}_{u}\right)_{ij}\sim\delta_{3i}\delta_{3j}$,
$i,j=1,2,3$, again after fixing $y_{t}=1$. The only non-zero eigenvalue of
this matrix is obviously $1$. Barring a total irrelevant phase, we can
parametrize the complex unit vector $\beta\in\mathbb{C}^{3}$ as

$\displaystyle\beta$ $\displaystyle=\begin{pmatrix}xe^{i\theta_{x}}\\\
ye^{i\theta_{y}}\\\ z\\\ \end{pmatrix},\quad\textrm{with}\quad
x^{2}+y^{2}+z^{2}=1.$ (29)

As before, the positivity bounds (3) mapped onto the $\xi^{Q,i}_{1,2,3,4}$
space will be obtained by requiring that the eigenvalues of the matrix
$C(\beta)$ are all positive. For simplicity, let us first compute these
eigenvalues for the natural benchmark point with all $\xi=1$. Again, because
of the mass hierarchy, $y_{t}\gg y_{c},y_{u}$, the characteristic polynomial
depends only on $z$ and not on $x,y,\theta_{x},\theta_{y}$. It factorizes
nicely:

$\displaystyle p(t)=$
$\displaystyle-\left(t-z^{2}-1\right)\left[t^{2}-4t\left(z^{2}+1\right)+4\left(1+z^{2}+z^{4}\right)\right],$
(30)

so its first eigenvalue is simply $t_{1}(z)=z^{2}+1$ and it is always
positive. To avoid unpleasant radicals, we can evaluate the sum and product of
the remaining two eigenvalues. This is equivalent to taking the trace and the
determinant of $C(\beta)$ and subtract and factor out $t_{1}$ respectively:

$\displaystyle t_{2}(z)+t_{3}(z)=4\left(z^{2}+1\right),$ (31) $\displaystyle
t_{2}(z)t_{3}(z)=4\left(z^{4}+z^{2}+1\right),$ (32)

which both remain positive for any value of $z$. And to conclude that the
benchmark point $\xi^{Q,i}_{1,2,3,4}=1$ is fully consistent with the
positivity bounds (3).

We can then extend our analysis to generic values of the $\xi$ coefficients,
as we did for $N_{f}=2$. The explicit computations are shown in Appendix D,
while we report here only the resulting expression:

$\displaystyle\text{(4-Q) ($N_{f}$=3): }\begin{cases}\bullet\
\xi^{Q,i}_{3}>0,\\\ \bullet\ \xi^{Q,i}_{1}+\xi^{Q,i}_{3}>0,\\\ \bullet\
\xi^{Q,i}_{3}+\xi^{Q,i}_{4}>0,\\\ \bullet\
\xi^{Q,i}_{1}+2\xi^{Q,i}_{2}+2\xi^{Q,i}_{3}+3\xi^{Q,i}_{4}>0,\\\ \bullet\
\xi^{Q,i}_{1}+2\xi^{Q,i}_{2}+\xi^{Q,i}_{3}+2\xi^{Q,i}_{4}>0,\\\ \bullet\
\left((\xi^{Q,i}_{2}-\xi^{Q,i}_{4})^{2}-4\xi^{Q,i}_{3}\xi^{Q,i}_{2}\right)<4\xi^{Q,i}_{1}\left(\xi^{Q,i}_{4}+\xi^{Q,i}_{3}\right)\\\
\qquad\qquad\text{ or
}\left(\xi^{Q,i}_{4}-\xi^{Q,i}_{2}+2\xi^{Q,i}_{3}\right)\left(\xi^{Q,i}_{2}+3\xi^{Q,i}_{4}+2\xi^{Q,i}_{3}\right)>0.\end{cases}$
(33)

Those constraints are illustrated in Fig. 2.

Bounds on (4-Q) operators coefficients for $N_{f}=3$

Figure 2: Plot showing in yellow the allowed region obtained for the (4-Q)
(or, equivalently, (4-u)) operators with generic $\xi$ values. Using the
scaling invariance of the bounds and since $\xi_{3}>0$, we have set
$\xi_{3}=1$, and plot the remaining three independent coefficients. The red
dot indicates the natural MFV benchmark point, $\xi_{1,2,3,4}=1$, that can be
seen being inside the allowed region.

###  3.4 (4-u), (4-d) self-quartic and cross-quartic operators

Here we continue the discussion for the remaining operators. We will see that,
in most cases, we have already done most of the work that was needed, and the
bounds for the coefficients of these operators can simply be obtained by
taking appropriate limits of the ones in Eqs. (27) and (33), or by carefully
looking at the order in which the indices are summed.

We start by going through the remaining self-quartic operators, and then
address the cross-quartic ones.

##### (4-u) operators

As we have seen, the bounds we have found until now for the coefficients of
the (4-Q) operators have turned out to depend exclusively on the eigenvalues
of the matrix $M$. This is the case for the (4-u) operators, too, with the
exchange $M\to\tilde{M}$. However, the eigenvalues of these matrices coincide.
This is clearly a basis-independent statement, but it can be easily seen in
the basis in Eq. (5), where they coincide. Thus, the resulting bounds are the
same for the (4-u) operators as for the (4-Q) ones, with the simple
replacement $\xi^{Q,i}_{A}\to\xi^{u,i}_{A}$ in Eqs. (27) and (33). Similarly
to what we did for the $(4-Q)$ case, we can, in the $N_{f}=2$ case, exploit
the redundancy of $(\tilde{M}_{mq}\delta_{pn}+\delta_{mq}\tilde{M}_{pn})$ to
remove two $\xi_{4}^{u,i}$ coefficients.

An exception where the simple exchange $\xi^{Q,i}_{A}\to\xi^{u,i}_{A}$ does
not work is the case studied in Section 3.3.2. Indeed, in this setting the two
matrices differ, and in particular $\tilde{M}=0$. The bounds are then
retrieved in this case by sending $y_{c}\to 0$ in Eq. (27). This gives simply

$\displaystyle\text{(4-u) ($N_{f}$=2 \text{revisited}):
}\begin{cases}\xi^{u,i}_{3}>0,\\\ \xi^{u,i}_{1}+\xi^{u,i}_{3}>0.\\\
\end{cases}$ (34)

##### (4-d) operators

As anticipated, the bounds here can be obtained by applying a formal limit to
the ones we have already. Indeed, we see that by 888This is just a trick to
get to the result, so in particular one does not need to worry about spoiling
the Yukawa hierarchy that led to the approximation at the beginning of Section
3.4. sending $\tilde{M}\to 0$ or $M\to 0$ in the first two lines of Table 3
respectively, we retrieve the tensor structure associated to the (4-d)
operators. Then, the $N_{f}=2$ and $N_{f}=3$ cases produce the same bounds,
the ones we already saw in Eq. (34):

$\displaystyle\text{(4-d): }\begin{cases}\xi^{d,i}_{3}>0,\\\
\xi^{d,i}_{1}+\xi^{d,i}_{3}>0.\end{cases}$ (35)

Figure 3 shows a plot of the corresponding allowed region. Again, we can note
that the natural MFV benchmark point, $\xi^{d,i}_{1,3}=1$, is compatible with
the positivity constraints.

Bounds on (4-d) operators coefficients for $N_{f}=2$

Figure 3: Plot showing in yellow the allowed parameter space for
$\xi^{d,i}_{1}$ and $\xi^{d,i}_{3}$, $i=1,2$. The black dot indicates the
natural MFV benchmark point, $\xi_{1,3}=1$, that can be seen being inside the
allowed region.

##### Cross-quartic operators

All the bounds on the cross-quartic operators give in fact much less
information than the ones on the self-quartic ones, in our MFV setting. This
is due to the fact that, looking at the index disposition in $a_{mnpq}$ for
the (2-d)(2-Q) and (2-d)(2-u) operators, we see that the objects in the l.h.s
of Eq. (3) are essentially linear combinations of products of the norms of
$\alpha$ and $\beta$, meaning terms of the form
$\alpha_{m}A_{mq}\alpha_{q}^{*}\beta_{n}B_{np}\beta_{p}^{*}$, where one
between $A_{mn}$ and $B_{mn}$ is a $\delta_{mn}$, while the other is either a
$\delta$, or $M$, or $\tilde{M}$. Then, since all three of these matrices are
(semi-)positive definite, taking all $\xi=1$ means that Eq. (3) turns into a
sum of positive terms and the bounds are trivially satisfied.

The (2-u)(2-Q) case, however, has an additional term
$\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}(Y_{u})_{nq}(Y^{\dagger}_{u})_{mp}$.
This is nothing but the modulus squared of $\beta_{n}Y_{u,nq}\alpha^{*}_{q}$,
which is then also positive. Moreover, we can rest assured that, as long as we
pick only positive values for the $\xi$ coefficients, the bounds will be
fulfilled, so we can definitely find acceptable $\order{1}$ values for them,
independently on the renormalizable Lagrangian parameters. This kills any hope
of bounding them through these operators. To get a full picture, one can here,
too, allow for generic $\xi$ values, whose shape is dictated by the Yukawa
couplings. Results as well as specific plots are shown in Appendix D.

Thus, in the end we have proven that the natural MFV benchmark point where all
the flavor-blind factors of the MFV expansion are chosen to be one, the
positivity constraints of Ref. flavorconstr  are all satisfied.

Moreover, since the constraints themselves are unchanged if multiplied by a
positive factor, what we have showed is actually that any configurations where
the $\xi$ coefficients are degenerate and positive are compatible with the
positivity requirements. Because of this scaling invariance, and of the
particular shape of the constraints, when considering generic values for the
$\xi$ coefficients, we can always rescale one of them to be $1$ or $-1$, once
per every type of operators. Taking this into account, we see that, for
physical values of the parameters, at least in the $N_{f}=3$ case, the
parameter space spanned by the $\xi$ coefficients is at least cut by a factor
of two. Since there are a total of 14 independent types of operators under
consideration, the overall allowed region is at least $2^{14}$ times smaller
than the one with no positivity restriction999This has obviously to be
understood as the result of a limit, meaning that if the space of parameters
is restricted to a box of volume $\mathcal{V}$, then the allowed region has a
volume $\sim 2^{-14}\mathcal{V}$, where $\mathcal{V}$ is eventually sent to
infinity..

##  4 Discussion and Conclusions

In Ref. flavorconstr , using arguments that rely on the analyticity and
unitarity of the theory in the UV, the authors obtained positivity constraints
on the coefficients of dimension-8 operators with 4 fermions. Starting from
that result, we showed that Minimal Flavor Violation, perhaps the simplest way
to generalize the Standard Model flavor structure to higher dimensional
operators, can be made consistent with those positivity constraints. To show
this, we have first identified the bounds on the (flavor-blind) parameters
that control the MFV expansion of the EFT coefficients. Such bounds are
obtained after we disentangle the physical parameters from other parameters
describing initial and final states of the $2\to 2$ scattering processes. We
have shown how this can be done in the specific case where the scattered
states contain non-trivial flavor structure. This allowed us to find bounds on
the various coefficients that parametrize the dimension-8 operators under the
MFV assumption. In the space spanned by these coefficients, the positivity
constraints become in general non-linear. This is a general feature that the
linear nature of the positivity bounds exhibited in the simplest cases (e.g.
that of a single scalar or a single flavor of an uncharged Weyl fermion) is
gradually lost when the number of degrees of freedom describing the scattered
initial and final states is increased, for instance by considering non-trivial
internal quantum numbers or several flavors, thereby increasing the number of
dimension-8 operators to be studied.

As a result, the natural benchmark point, where all the flavor-blind
parameters that enter the MFV expansion of EFT operators are degenerate and
equal to unity, has proven to trivially satisfy the aforementioned positivity
constraints. This is true independently of the parameters of the
renormalizable SM Lagrangian, namely the fermion masses and the entries of the
CKM matrix. More generally, every setting where the flavor-blind parameters
are degenerate is seen to be compatible with the positivity conditions. Still,
the positivity constraints are such as to reduce the full parameter space by a
factor of $2^{14}$.

Remarkably, MFV is not restraining as to turn the positivity constraints on
the dimension-8 operators into restrictions on the physical input parameters
of the SM defined by the dimension-4 operators.

An immediate consequence is that, for flavor models which are less restricting
than MFV, such as the so called $U(2)^{5}$ model U(2)5 ; MFVandU2 , and which
reduce to MFV for some values of the parameters, there exist at least an
allowed region of the parameter space where the free coefficients are
compatible with the bounds.

It would also be interesting to derive (maybe) tighter constraints following
the approach of Ref. electronpositronscattering  and considering the
scattering of states that are no longer SM gauge eigenstates. For instance,
after suitable redefinitions accounting for the different operator basis used
there, the bounds used in our work can be mapped into the ones in Eqs.
(7)–(10) of Ref. electronpositronscattering , while Eqs. (11)–(12) are missing
from our analysis101010The precise signs of the bounds differ between Ref.
electronpositronscattering  and Ref. flavorconstr , which is likely to be due
to different conventions. We chose conventions so that the signs are those
given in Ref. flavorconstr . The fact that some bounds are missing in our
analysis and in Ref. flavorconstr  is anyway independent of these sign
conventions.. However, in Ref. electronpositronscattering  only one flavor
family is taken into account, whereas, due to the non-linearity of the
additional bounds obtained, the $N_{f}\neq 1$ case cannot be straightforwardly
tackled using, e.g., the approach we outlined in Section 3. Therefore, we
sticked for our analysis to the bounds of flavorconstr , and left the study of
the most general scatterings to future work.

Finally, we point out that the analysis carried out in this paper immediately
extends to the positivity constraints obtained in Ref. dimension6bounds  on
the dimension-6 4-Fermi operators. Those bounds are derived under the
assumption of an improved behavior of the scattering amplitude in the deep UV
at energies above the EFT cutoff scale, behavior which is actually not
encountered in simple UV-completions of the dimension-6 4-Fermi operators
obtained by integrating out massive vectors. Nevertheless, taking those bounds
at face value and upon some linear redefinitions, they are of the same shape
as those in Eqs. (3) and (3), with $c_{mnpq}$ and $a_{mnpq}$ substituted by
suitable combinations of their dimension-6 counterparts. This implies that the
techniques we have exposed in the present paper also allow to derive
constraints on the dimension-6 coefficients in a MVF setup, similar to, e.g.,
Eq. (33). We leave the explicit presentation of the bounds in the dimension-6
case, as well as an analysis of their phenomenological impact, for future
work.

## Acknowledgments

We acknowledge support by the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy EXC 2121 “Quantum Universe” - 390833306. We thank M.
Montull for early discussions on positivity bounds and for comments on the
draft of this paper, and J. Gu with insightful discussions on the
phenomenological relevance of the positivity constraints. We also thanks J.
Louis for stimulating conversations in the early stages of this work.

##  Appendix A Flavor-violation and large Yukawas

Our MFV expansion needs a little more justification, in particular concerning
the resummation of the top-Yukawa $y_{t}$. Indeed, while $Y_{d}$ has
eigenvalues $\ll 1$, and thus allows for the approximation we explained in
Section 2.2, this is not true for $Y_{u}$, whose biggest eigenvalue is
$y_{t}\sim 1$. Consequently there is in principle no clear expansion in powers
of $Y_{u}$ as long as we keep all the coefficients in the operator expansion
of $\order{1}$. However, in Ref. MFVreview , for example, it is stated that
any $\order{1}$ term in the (4-Q) case has to be of the form
$\left(Y_{u}Y^{\dagger}_{u}\right)^{n}$. In the basis of Eq. (5), this reads:

$\left(Y_{u}Y^{\dagger}_{u}\right)^{n}_{ij}\sim
y_{t}^{2n}(V_{\textit{CKM}}^{*})_{3i}(V_{\textit{CKM}})_{3j}\ ,$ (36)

while the structure for the (4-u) case similarly reads
$\left(Y^{\dagger}_{u}Y_{u}\right)^{n}_{ij}\sim
y_{t}^{2n}\delta_{3i}\delta_{3j}$. Consequently, considering more Yukawa
matrices does not change the flavor structure of the couplings, but simply
demands to resum the powers of $y_{t}$. We dwell a bit on the details of this
conclusion below.

###  A.1 Group theory argument

The restriction to Eq. (36) can be justified like this: in $SU(3)$, the
invariant tensors are $\varepsilon_{abc}$, $\varepsilon^{abc}$ and
$\delta^{a}_{b}$. Imagine we want to build a contribution to the (4-Q) case
using $n$ powers of $Y_{u}$ and $m$ powers of $Y_{u}^{\dagger}$. Then, since
$\bar{Q}Q\bar{Q}Q$ is a singlet under $SU(3)_{u}$, we need to contract all of
the $SU(3)_{u}$ indices of the various $Y_{u}$ and $Y_{u}^{\dagger}$ using
$\varepsilon$’s or $\delta$’s. If we contract the indices of three $Y_{u}$’s
using an $\varepsilon$, then for this product not to vanish, the $SU(3)_{q}$
indices of those matrices need to be fully antisymmetrized as well. However,
this is but a singlet under $SU(3)_{Q}\otimes SU(3)_{u}$, and contributes as a
redefinition of the coefficient of one operator of order
$(Y_{u})^{n-3}(Y_{u}^{\dagger})^{m}$. Similarly for $Y_{U}^{\dagger}$. Then,
we can only consistently use $\delta^{a}_{b}$ to contract the $SU(3)_{u}$
indices to hope to build a new structure. This proves that the building block
is actually $Y_{u}Y_{u}^{\dagger}$ which is a
$\bar{\mathbf{3}}\otimes\mathbf{3}$ of $SU(3)_{Q}$, and we have to take $n=m$.
Then, we want eventually to contract the remaining indices with those from
$\bar{Q}Q\bar{Q}Q$. We want to prove that this, too, can be done exclusively
with $\delta$’s. Indeed, suppose we want to employ $\varepsilon_{abc}$.
Similarly to what happened before, if we use it to contract three upstairs
indices coming from three copies of $Y_{u}Y_{u}^{\dagger}$, symmetry imposes
that the downstairs indices are antisymmetrized, too, giving rise to an
uninteresting singlet. If we contract it with two upstairs indices from two
$Y_{u}Y_{u}^{\dagger}$ and one from a $Q$, then the two downstairs indices
form the two $Y_{u}Y_{u}^{\dagger}$ need to be antisymmetrized with an
$\varepsilon^{abc}$. However the product of two epsilon is but a sum of
products of deltas. Similarly, if we contract $\varepsilon_{abc}$ with the two
fundamental indices of the two $Q$, then the antifundamental ones of the two
$\bar{Q}$ need to be contracted with an $\varepsilon^{abc}$, again giving
products of $\delta$’s. Thus a series in
$\left(Y^{\dagger}_{u}Y_{u}\right)^{n}$ includes all allowed contractions.
Similar reasoning holds for the (4-u) case, with the exchange of $SU(3)_{Q}$
and $SU(3)_{u}$ indices, and $Y_{u}Y_{u}^{\dagger}\to Y_{u}^{\dagger}Y_{u}$.
For the (4-d) operators, there are simply no possible non-trivial insertions
of $Y_{u}$. (2-d)(2-Q) and (2-d)(2-u) are similar to the former cases, with
the exception that there are now only one index in the fundamental and one in
the antifundamental of $SU(3)_{Q}$ in the structure coming from the spinors.

Finally, (2-Q)(2-u) requires a bit of attention. Again, contracting three
$Y_{u}$’s or three $Y_{u}^{\dagger}$’s with an epsilon tensor eventually
produces a $SU(3)_{Q}\otimes SU(3)_{u}$ singlet. However, the spinor structure
provides an index in the fundamental and one in the antifundamental for both
$SU(3)_{u}$ and $SU(3)_{Q}$. So we can contract two $Y_{u}$ or two
$Y_{u}^{\dagger}$ with one index coming from the spinors. We cannot use an
upstairs and a downstairs epsilon from the same group as that would reduce to
sum of products of deltas. To sum up, we can use two epsilon tensors, one for
each group, each with one index contracted to one coming from the spinor
structure. Let us take the first one to be a $SU(3)_{u}$ $\varepsilon^{abc}$.
Its remaining two indices can only be contracted with two downstairs
$SU(3)_{u}$ indices from two $Y_{u}$’s. Then, the $SU(3)_{Q}$ fundamental
indices that these two matrices carry have to be antisymmetrized. This has to
be done with the only remaining possible epsilon tensor, giving a structure
as:

$\displaystyle(\bar{u}u)_{\bar{u}_{1}u_{1}}(\bar{Q}Q)_{\bar{q}_{1}q_{1}}(Y_{u})_{q_{2}\bar{u}_{2}}(Y_{u})_{q_{3}\bar{u}_{3}}\varepsilon_{\bar{u}_{1}\bar{u}_{2}\bar{u}_{3}}\varepsilon_{q_{1}q_{2}q_{3}}(Y_{u}^{\dagger})_{u_{1}\bar{q}_{1}}$
(37)

The other case, i.e., picking a $SU(3)_{u}$ $\varepsilon_{abc}$, gives the
hermitian conjugate of (37). However, this term is subleading, as it can be
immediately seen by plugging the leading contribution $(Y_{u})_{q\bar{u}}\sim
y_{t}\delta_{3\bar{u}}\delta_{3q}$. Moreover, it is not $U(1)_{u}$ invariant.
In conclusion, only $\delta$ factors can be used to contract indices
consistently. This forces the operators to be of the form already contained in
the (2-u)(2-Q) line of Table 3, times an arbitrary number of
$Y^{\dagger}_{u}Y_{u}$ of $Y_{u}Y_{u}^{\dagger}$, suitably contracted in. The
latter can anyhow be reabsorbed in a redefinition of the overall $\rho$
coefficients, as we show in the next section.

###  A.2 Non-linear realization

One can also phrase the argument in favor of the single flavor-violating
structure in Eq. (36) in a non-linear language. Indeed, when $y_{t}$ is ${\cal
O}(1)$, the necessary resummation of the expansion in powers of $y_{t}$ means
that the flavor group is non-linearly realized Feldmann:2008ja ; Kagan:2009bn
. In the basis (5) where the up-type Yukawa is diagonal, the EFT is an
expansion around the vev

$\langle Y_{u}\rangle=\left(\begin{matrix}0&0&0\\\ 0&0&0\\\
0&0&y_{t}\end{matrix}\right)\ ,$ (38)

breaking $U(3)_{Q}\times U(3)_{u}$ down to $U(2)_{Q}\times U(2)_{u}\times
U(1)_{3}$. Following Ref. Kagan:2009bn , the building blocks for the EFT are
found as follows. We first identify the Goldstone modes,

$Y_{u}=e^{i\hat{\rho}_{Q}}\left(\begin{matrix}\phi_{u}&0\\\
0&y_{t}\end{matrix}\right)e^{-i\hat{\rho}_{u}}\ ,\text{ with
}\hat{\rho}_{i}=\left(\begin{matrix}0&\rho_{i}\\\
\rho_{i}^{\dagger}&\theta_{i}\end{matrix}\right)\ ,$ (39)

with $\rho_{i}$ a complex 2-vector and $\theta_{Q}=-\theta_{u}\equiv\theta$,
with $\theta$ a real field. The fields transform as

$e^{i\hat{\rho}_{i}}\rightarrow
V_{i}e^{i\hat{\rho}_{i}}U^{\dagger}_{i}(\hat{\rho}_{i},V_{i})\
,\quad\left(\begin{matrix}\phi_{u}&0\\\ 0&y_{t}\end{matrix}\right)\rightarrow
U_{Q}(\hat{\rho}_{Q},V_{Q})\left(\begin{matrix}\phi_{u}&0\\\
0&y_{t}\end{matrix}\right)U^{\dagger}_{u}(\hat{\rho}_{u},V_{u})\ ,$ (40)

where $U_{i}$ are functions of $V_{i}$ and of the Goldstones that belong to
$U(2)_{i}\times U(1)_{3}$:

$U_{i}=\left(\begin{matrix}U_{i}^{2\times 2}&0\\\
0&e^{i\phi_{3}}\end{matrix}\right)\ .$ (41)

We can also dress the down-type Yukawa matrix as well as some of the quark
fields to obtain fields that transform as linear representations of
$U(2)_{Q}\times U(2)_{u}\times U(1)_{3}$ under the full flavor group:

$\tilde{Y}_{d}\equiv e^{-i\hat{\rho}_{Q}}Y_{d}\ ,\quad\tilde{Q}\equiv
e^{-i\hat{\rho}_{Q}}Q\ ,\quad\tilde{u}\equiv e^{-i\hat{\rho}_{u}}u\ .$ (42)

The new fields can be split as follows:

$\tilde{Y}_{d}=\left(\begin{matrix}\phi_{d}\\\
\phi^{\prime\dagger}_{d}\end{matrix}\right)\
,\quad\tilde{Q}=\left(\begin{matrix}\tilde{Q}^{(2)}\\\
\tilde{t}_{L}\end{matrix}\right)\
,\quad\tilde{u}=\left(\begin{matrix}\tilde{u}^{(2)}\\\
\tilde{t}_{R}\end{matrix}\right)\ ,$ (43)

where $\phi_{d}$ is a $2\times 3$ matrix, $\phi^{\prime}_{d}$ a 3-vector,
$\tilde{Q}^{(2)}$ a doublet of $U(2)_{Q}$, $\tilde{t}_{L}$ a singlet, and
similarly for $\tilde{u}$. The components transform as

$\displaystyle\phi_{d}\rightarrow U_{Q}^{2\times 2}\phi_{d}V_{d}^{\dagger}\
,\quad\phi^{\prime}_{d}\rightarrow e^{-i\phi_{3}}V_{d}\phi^{\prime}_{d}\
,\quad\tilde{Q}^{(2)}\rightarrow U_{Q}^{2\times 2}\tilde{Q}^{(2)}\ ,$ (44)
$\displaystyle\tilde{t}_{L}\rightarrow e^{i\phi_{3}}\tilde{t}_{L}\
,\quad\tilde{u}^{(2)}\rightarrow U_{u}^{2\times 2}\tilde{u}^{(2)}\
,\quad\tilde{t}_{R}\rightarrow e^{i\phi_{3}}\tilde{t}_{R}\ .$ (45)

The fields above, together with $d_{R},\phi_{u},y_{t}$ and the invariance
under $U(2)_{Q}\times U(2)_{u}\times U(1)_{3}\times U(3)_{d}(\times
U(3)_{L}\times U(3)_{e})$, are the building blocks for the EFT. One should in
principle also use the covariant derivatives obtained from the Maurer-Cartan
form

$e^{-i\hat{\rho}_{Q}}\partial_{\mu}e^{i\hat{\rho}_{Q}}$ (46)

but they are identically zero when we freeze the Yukawa spurions to their
background values. Summing up, the different fields and their representations
are:

| $U(2)_{Q}$ | $U(2)_{u}$ | $U(1)_{3}$ | $U(3)_{d}$
---|---|---|---|---
Fermions
$\tilde{Q}^{(2)}$ | $\mathbf{2}$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$
$\tilde{t}_{L}$ | $\mathbf{1}$ | $\mathbf{1}$ | $+1$ | $\mathbf{1}$
$\tilde{u}^{(2)}$ | $\mathbf{1}$ | $\mathbf{2}$ | $0$ | $\mathbf{1}$
$\tilde{t}_{R}$ | $\mathbf{1}$ | $\mathbf{1}$ | $+1$ | $\mathbf{1}$
$\tilde{d}$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $\mathbf{3}$
Spurions
$\phi_{u}$ | $\mathbf{2}$ | $\mathbf{\bar{2}}$ | $0$ | $\mathbf{1}$
$y_{t}$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $\mathbf{1}$
$\phi_{d}$ | $\mathbf{2}$ | $\mathbf{1}$ | 0 | $\mathbf{\bar{3}}$
$\phi^{\prime}_{d}$ | $\mathbf{1}$ | $\mathbf{1}$ | -1 | $\mathbf{3}$

The background values of the spurions are obtained from
$Y_{u}=\text{diag}(y_{u},y_{c},y_{t})$,
$Y_{d}=V_{\textit{CKM}}\text{diag}(y_{d},y_{s},y_{b})$. We see that all
spurions but $y_{t}$ are small, so that there exists an expansion in terms of
small Yukawas and CKM elements. Every EFT term constructed from
$\phi_{u},\phi_{d},\phi^{\prime}_{d}$ up to a given order is then completed by
multiplying it by an arbitrary function of $y_{t}$. In particular, the
approximation we have been discussing in this appendix is the one where all
$y$s but $y_{t}$ are zero. At this order, the only fermion bilinears that can
enter the dimension-8 coefficients in Table 2 in a flavor-invariant way are

$\overline{\tilde{Q}^{(2)}}\gamma^{\mu}\tilde{Q}^{(2)}\
,\quad\overline{\tilde{t}_{L}}\gamma^{\mu}\tilde{t}_{L}\
,\quad\overline{\tilde{u}^{(2)}}\gamma^{\mu}\tilde{u}^{(2)}\
,\quad\overline{\tilde{t}_{R}}\gamma^{\mu}\tilde{t}_{R}\
,\quad\overline{\tilde{d}}\gamma^{\mu}\tilde{d}\ .$ (47)

When we freeze the spurions to their background values, the Goldstone fields
$\hat{\rho}$ are zero in the basis where $Y_{u}$ is diagonal, one can simply
remove the tildes in the expression above and rename
$\tilde{t}_{L}=Q_{3},\tilde{t}_{R}=u_{3}$. In the basis where
$Y_{u}=V_{\textit{CKM}}^{\dagger}\text{diag}(y_{u},y_{c},y_{t})$ and
$Y_{d}=\text{diag}(y_{d},y_{s},y_{b})$, Eq. (47) becomes

$\displaystyle\left[\delta_{ij}-\left(V_{\textit{CKM}}^{*}\right)_{3i}\left(V_{\textit{CKM}}\right)_{3j}\right]\overline{Q_{i}}\gamma^{\mu}Q_{j}\
,$
$\displaystyle\quad\left(V_{\textit{CKM}}^{*}\right)_{3i}\left(V_{\textit{CKM}}\right)_{3j}\overline{Q_{i}}\gamma^{\mu}Q_{j}\
,$ (48) $\displaystyle\overline{u^{(2)}}\gamma^{\mu}u^{(2)}\
,\quad\overline{u_{3}}$ $\displaystyle\gamma^{\mu}u_{3}\
,\quad\overline{d}\gamma^{\mu}d\ .$

This is consistent with Eq. (36): the two terms in the first line can be
combined to reconstruct $\delta_{ij}\overline{Q_{i}}\gamma^{\mu}Q_{j}$ and
$\left(V_{\textit{CKM}}^{*}\right)_{3i}\left(V_{\textit{CKM}}\right)_{3j}\overline{Q_{i}}\gamma^{\mu}Q_{j}$,
which are the flavor structures that are obtained from
$\left(Y_{u}Y^{\dagger}_{u}\right)^{n}$. The $u$-quark terms in the second
line can be combined to reconstruct
$\delta_{ij}\overline{u_{i}}\gamma^{\mu}u_{j}$ and
$\delta_{3i}\delta_{3j}\overline{u_{i}}\gamma^{\mu}u_{j}$, which are the
flavor structures that are obtained from
$\left(Y^{\dagger}_{u}Y_{u}\right)^{n}$. The $d$-quark terms are flavor
diagonal, as they should at order $Y_{d}^{0}$.

##  Appendix B Redundancy of the $\rho_{4}$ structure in $N_{f}=2$

Here we provide a proof showing that, in $SU(2)$,
$\bar{u}_{m}u_{n}\bar{u}_{p}u_{q}(\tilde{M}_{mq}\delta_{pn}+\delta_{mq}\tilde{M}_{pn})$
is redundant with respect to the other structures contained in the first line
of Table 3, and the corresponding coefficient, which we called $\rho_{4}$, can
be reabsorbed in the definitions of the remaining three. An analogous
discussion can be done for
$\bar{u}_{m}u_{n}\bar{u}_{p}u_{q}(\tilde{M}_{mq}\delta_{pn}+\delta_{mq}\tilde{M}_{pn})$
and the second line of Table 3. First of all, define
$X_{mn}\equiv\bar{u}_{m}u_{n}$. $SU(2)$ does not distinguish between
fundamental and antifundamental indices, and all summations need to be
performed with $\varepsilon_{ab}$, the only invariant tensor. Then

$\displaystyle
X_{ij}X_{kl}\varepsilon_{jk}\varepsilon_{lm}\varepsilon_{in}\tilde{M}_{nm}$
$\displaystyle=\left(X_{\\{ij\\}}+\frac{1}{2}X_{ab}\varepsilon_{ab}\varepsilon_{ij}\right)\left(X_{\\{kl\\}}+\frac{1}{2}X_{cd}\varepsilon_{cd}\varepsilon_{kl}\right)\varepsilon_{jk}\varepsilon_{lm}\varepsilon_{in}\tilde{M}_{nm}=$
$\displaystyle=X_{\\{ij\\}}X_{\\{kl\\}}\varepsilon_{jk}\varepsilon_{lm}\varepsilon_{in}\tilde{M}_{nm}+X_{ab}\varepsilon_{ab}\varepsilon_{nk}\varepsilon_{lm}X_{\\{kl\\}}\tilde{M}_{nm}+$
$\displaystyle\quad-\frac{1}{4}\varepsilon_{ab}X_{ab}\varepsilon_{cd}X_{cd}\varepsilon_{nm}\tilde{M}_{nm}.$
(49)

The second piece can be absorbed by a shift in $\rho_{1}$, while the third one
with a shift on $\rho_{2}$. The first piece, instead, splits as:

$\displaystyle
X_{\\{ij\\}}X_{\\{kl\\}}\varepsilon_{jk}\varepsilon_{lm}\varepsilon_{in}\tilde{M}_{nm}$
$\displaystyle=X_{\\{ij\\}}X_{\\{kl\\}}\varepsilon_{jk}\varepsilon_{lm}\varepsilon_{in}\left(\tilde{M}_{\\{nm\\}}+\frac{1}{2}\varepsilon_{nm}\varepsilon_{ef}\tilde{M}_{ef}\right)=$
$\displaystyle=X_{\\{ij\\}}X_{\\{kl\\}}\tilde{M}_{\\{nm\\}}\varepsilon_{jk}\varepsilon_{lm}\varepsilon_{in}+\frac{1}{2}X_{\\{ij\\}}X_{\\{kl\\}}\varepsilon_{jk}\varepsilon_{il}\varepsilon_{ef}M_{ef}.$
(50)

Here, the second term is reabsorbed through a shift on $\rho_{3}$, while the
first one vanishes:

$\displaystyle
X_{\\{ij\\}}X_{\\{kl\\}}M_{\\{nm\\}}\varepsilon_{jk}\varepsilon_{lm}\varepsilon_{in}$
$\displaystyle\overset{j\leftrightarrow
k}{=}-X_{\\{ij\\}}X_{\\{kl\\}}M_{\\{nm\\}}\varepsilon_{kj}\varepsilon_{lm}\varepsilon_{in}\overset{(ij)\leftrightarrow(kl)}{=}$
$\displaystyle=-X_{\\{ij\\}}X_{\\{kl\\}}M_{\\{nm\\}}\varepsilon_{il}\varepsilon_{jm}\varepsilon_{kn}\overset{m\leftrightarrow
n}{=}$
$\displaystyle=-X_{\\{ij\\}}X_{\\{kl\\}}M_{\\{nm\\}}\varepsilon_{il}\varepsilon_{jn}\varepsilon_{km}\overset{i\leftrightarrow
j}{=}$
$\displaystyle=-X_{\\{ij\\}}X_{\\{kl\\}}M_{\\{nm\\}}\varepsilon_{jl}\varepsilon_{in}\varepsilon_{km}\overset{l\leftrightarrow
k}{=}$
$\displaystyle=-X_{\\{ij\\}}X_{\\{kl\\}}M_{\\{nm\\}}\varepsilon_{jk}\varepsilon_{lm}\varepsilon_{in}.$
(51)

In conclusion, the examined term provides no new structure and can be set to
0.

##  Appendix C Contribution of other SMEFT operators

One comment should also be made about the contributions of other SMEFT
operators to the positivity bounds. Indeed, in the usual version of the latter
that we consider in this paper, they are associated to the $s^{2}$ coefficient
of a $2\to 2$ forward amplitude. By dimensional analysis only, one sees that
such a $s^{2}$ growth can be obtained from a dimension-8 four-fermions contact
term, as we considered in the main text, but also from the product of two
coefficients, both of dimension-6, or one of dimension-5 and the other of
dimension-7, or one of dimension-8 and at least one of dimension-4.

Let us discuss first the case of dimension-6 operators. We mentioned already
that dimension-6 four-fermions operators do not enter the positivity bounds,
due to their softer UV behaviour when compared to dimension-8 operators.
Nevertheless, there are operators at dimension-6 which, when combined
together, contribute at tree level to the four-fermions amplitudes with a UV
behaviour similar to that of dimension-8 four-fermions operators. A simple
example is the following: consider the dimension-6 operator that couples the
photon to the up-type right-handed quark current,

${\cal L}\supset\frac{c}{\Lambda^{2}}\partial_{\mu}J_{\nu}F^{\mu\nu}\ ,$ (52)

where $J_{\mu}=\overline{u}_{m}\gamma_{\mu}u_{m}$. This coupling generates a
four-fermion amplitude whose s-channel component is depicted below and reads

|  $\displaystyle{\cal A}=$ $\displaystyle\
i\frac{c^{2}}{\Lambda^{4}}\overline{v_{2}}\gamma_{\nu}u_{1}P^{2}\left(\frac{\eta^{\nu\sigma}}{P^{2}}-(1-\xi)\frac{P^{\nu}P^{\sigma}}{(P^{2})^{2}}\right)P^{2}\overline{u_{3}}\gamma_{\sigma}v_{4}$
(53) $\displaystyle\approx$ $\displaystyle\
i\frac{c^{2}}{\Lambda^{4}}P^{2}\overline{v_{2}}\gamma_{\mu}u_{1}\overline{u_{3}}\gamma^{\mu}v_{4}\
,$ (53)
---|---

where $u_{i},v_{i}$ are (anti)particle polarizations and we used the fact that
the particles are effectively massless at high energies, so that
$\not{p}u(p),\not{p}v(p)\approx 0$. We would have obtained the same result
with the four fermion interaction
$\frac{c^{2}}{\Lambda^{4}}\partial_{\mu}J_{\nu}\partial^{\mu}J^{\nu}$, which
is of the kind subject to positivity bounds. Thus, $c^{2}$ should be added to
the combination of dimension-8 coefficients that are constrained to be
positive, modifying the bound. However, Eq. (52) is a redundant operator that
can consistently be ignored in the first place. Indeed, enforcing the photon
equation of motion derived from Eq. (52),

$\partial^{\mu}F_{\mu\nu}-\frac{c}{\Lambda^{2}}\Box J_{\nu}=0\ ,$ (54)

we find that
$\frac{c}{\Lambda^{2}}\partial_{\mu}J_{\nu}F^{\mu\nu}=\frac{c^{2}}{\Lambda^{4}}\partial_{\mu}J_{\nu}\partial^{\mu}J^{\nu}$,
consistently with our previous analysis. Thus, we can set $c=0$ at no cost.
Using the Warsaw basis for the dimension-6 SMEFT Warsaw , and considering the
high-energy phase where all particles are massless and the electroweak
symmetry unbroken, it is straightforward to check that there are no
dimension-6 contributions to the four-fermion positivity bounds at tree level.
This differs from the case of $2\to 2$ gauge bosons scattering, where non-
redundant trilinear couplings enter the bounds at tree-level and strenghthen
the bounds on dimension-8 operators vectorbosonscattering ; aQGC ;
Yamashita:2020gtt .

Similarly, contributions that could spoil our bounds come from combinations of
dimension-5 and dimension-7, or of dimension-4 and dimension-8 terms. Since in
the SMEFT the only dimension-5 term is the Weinberg operator $HHL_{i}L_{j}$,
which does not contain quarks, the former possibility is irrelevant. Instead,
combining one dimension-4 and one dimension-8 operator could give rise to a
diagram with the same shape as the one in Eq. (53) or an analogous one with an
intermediate Higgs boson. The former could arise from combining a gauge
interaction and a dimension-8 operator of the schematic form
$\mathcal{O}\sim\partial^{3}X\bar{\psi}\psi$, with $X$ any of the SM field
strengths, and the latter from combining a Yukawa interaction and an operator
$\mathcal{O}\sim\partial^{4}H\bar{\psi}\psi$. However, in both cases the
dimension-8 operator is proportional to some equations of motion dimension8_1
; dimension8_2 , and can thus be reabsorbed by redefining the coefficients of
lower-dimensional operators, as just shown for the dimension-6 ones.

Those conclusions can be reached in a number of ways, including Hilbert series
techniques and on-shell methods. In conclusion, it is useful to have in mind
that specifying the basis of operators is necessary when writing positivity
bounds.

##  Appendix D Bounds for generic $\xi$ values

###  D.1 Self-quartic

We report here the bounds that we get when we allow for generic values of the
$\rho$ coefficients in the $c_{mnqp}$ and $a_{mnpq}$ tensors. The procedure is
exactly the same as in the (4-Q) case with $N_{f}=2$, which was shown in
Section 3.3.1. We keep explicit factors of $y_{t}^{2}$ to make it easier to
identify the contributions of contracted Yukawa matrices, but we remind that
they should be put to $1$ for consistency of the expansion.

Let us see for example how the (4-Q) case changes. Again, we impose the first
eigenvalue of the matrix $C(\beta)$, as well as the sum and product of the
remaining two, to be positive:

$\displaystyle t_{1}(z)$
$\displaystyle=\xi^{Q,i}_{3}+\xi^{Q,i}_{3}y_{t}^{2}z^{2},$ (55) $\displaystyle
t_{2}(z)+t_{3}(z)$
$\displaystyle=\xi^{Q,i}_{1}+2\xi^{Q,i}_{3}+\xi^{Q,i}_{4}y_{t}^{2}+2y_{t}^{2}z^{2}(\xi^{Q,i}_{2}+\xi^{Q,i}_{4}),$
(56) $\displaystyle t_{2}(z)t_{3}(z)$
$\displaystyle=(\xi^{Q,i}_{1}+\xi^{Q,i}_{3})(\xi^{Q,i}_{3}+\xi^{Q,i}_{4}y_{t}^{2})$
$\displaystyle\hphantom{=}+z^{2}y_{t}^{2}(\xi^{Q,i}_{2}+\xi^{Q,i}_{4})(y_{t}^{2}(\xi^{Q,i}_{4}-\xi^{Q,i}_{2})+2\xi^{Q,i}_{3})$
$\displaystyle\hphantom{=}+z^{4}y_{t}^{4}(\xi^{Q,i}_{2}+\xi^{Q,i}_{4})^{2}.$
(57)

The first two are linear objects in $z^{2}$, which varies within the interval
$[0,1]$. Thus, they are positive $\forall z$ if and only if their value at the
boundaries of the interval is also positive. However, the third expression is
a quadratic polynomial in $z^{2}$, which we can parametrize as
$az^{4}+bx^{2}+c$. Then, as already seen, we require this polynomial to be
positive at the boundaries and to satisfy $\Delta<0\lor a<0\lor b(b+2a)>0$.
Putting everything together, and after some simplifications, we get the full
set of conditions:

$\displaystyle\text{(4-Q): }\begin{cases}\bullet\ \xi^{Q,i}_{3}>0,\\\ \bullet\
\xi^{Q,i}_{1}+\xi^{Q,i}_{3}>0,\\\ \bullet\
\xi^{Q,i}_{3}+\xi^{Q,i}_{4}y_{t}^{2}>0,\\\ \bullet\
\xi^{Q,i}_{1}+2\xi^{Q,i}_{2}y_{t}^{2}+2\xi^{Q,i}_{3}+3\xi^{Q,i}_{4}y_{t}^{2}>0,\\\
\bullet\
\xi^{Q,i}_{1}+2\xi^{Q,i}_{2}y_{t}^{2}+\xi^{Q,i}_{3}+2\xi^{Q,i}_{4}y_{t}^{2}>0,\\\
\bullet\
y_{t}^{2}\left(y_{t}^{2}(\xi^{Q,i}_{2}-\xi^{Q,i}_{4})^{2}-4\xi^{Q,i}_{3}\xi^{Q,i}_{2}\right)<4\xi^{Q,i}_{1}\left(\xi^{Q,i}_{4}y_{t}^{2}+\xi^{Q,i}_{3}\right)\\\
\qquad\qquad\textrm{or}\quad\left(y_{t}^{2}(\xi^{Q,i}_{4}-\xi^{Q,i}_{2})+2\xi^{Q,i}_{3}\right)\left(y_{t}^{2}(\xi^{Q,i}_{2}+3\xi^{Q,i}_{4})+2\xi^{Q,i}_{3}\right)>0.\end{cases}$
(58)

Again, since within our basis choice $M$ and $\tilde{M}$ are the same,
identical results apply to the $\xi^{u,i}_{k}$, $k=1,2,3,4$.

###  D.2 Cross-quartic operators

The $C(\beta)$ matrix turns out to have two equal eigenvalues, and the
conditions are simply:

$\displaystyle\text{(2-u)(2-Q): }\begin{cases}\bullet\ \xi^{uQ,i}_{1}>0,\\\
\bullet\ \xi^{uQ,i}_{1}+\xi^{uQ,i}_{2}y_{t}^{2}>0,\\\ \bullet\
\xi^{uQ,i}_{1}+\xi^{uQ,i}_{3}y_{t}^{2}>0,\\\ \bullet\
\xi^{uQ,i}_{1}+(\xi^{uQ,i}_{2}+\xi^{uQ,i}_{3}+\xi^{uQ,i}_{4})y_{t}^{2}>0.\end{cases}$
(59)

A plot of the results is reported in Fig. 4.

Bounds on (2-u)(2-Q) operators coefficients for $N_{f}=3$

Figure 4: Plots showing in yellow the allowed region obtained for the
(2-u)(2-Q) operators, with generic $\xi$ values. Here $\xi_{1}>0$, so we have
rescaled it to 1, and we show the allowed region for the remaining three.
$y_{t}$ is set to 1. The red dot indicates the naturak MFV benchmark point,
$\xi_{1,2,3,4}=1$, which can be seen being inside the allowed region.

Here we also report the bounds for the (2-d)(2-Q) and the (2-d)(2-u)
operators, which can both be computed directly or obtained from the previous
ones with appropriate limits. Again, since $M$ and $\tilde{M}$ coincide, so do
the bounds:

$\displaystyle\text{(2-d)(2-Q): }\begin{cases}\bullet\ \xi^{dQ,i}_{1}>0,\\\
\bullet\ \xi^{dQ,i}_{1}+\xi^{dQ,i}_{2}y_{t}^{2}>0.\end{cases}$ (60)
$\displaystyle\text{(2-d)(2-u): }\begin{cases}\bullet\ \xi^{du,i}_{1}>0,\\\
\bullet\ \xi^{du,i}_{1}+\xi^{du,i}_{2}y_{t}^{2}>0.\end{cases}$ (61)

##  Appendix E $\rho$ dependence of $\xi$ coefficients

##### Self-quartic

For the operators built out of (4-Q) fields, the coefficient looks like:

$\displaystyle c^{Q,i}_{mnpq}$
$\displaystyle=\rho^{Q,i}_{1}(\delta_{mn}\delta_{pq})+\rho^{Q,i}_{2}(M_{mn}\delta_{pq}+\delta_{mn}M_{pq})+\rho^{Q,i}_{3}(\delta_{mq}\delta_{pn})+$
$\displaystyle+\rho^{Q,i}_{4}(M_{mq}\delta_{pn}+\delta_{mq}M_{pn})$ (62)

The conditions coming from Eq. (12) of Ref. flavorconstr  are:

$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}\left(c_{mnpq}^{Q,1}+\frac{1}{4}c_{mnpq}^{Q,2}+\frac{1}{3}c_{mnpq}^{Q,3}+\frac{1}{12}c_{mnpq}^{Q,4}\right)$
$\displaystyle>0,$
$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}\left(c_{mnpq}^{Q,2}+\frac{1}{3}c_{mnpq}^{Q,4}\right)$
$\displaystyle>0,$
$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}\left(c_{mnpq}^{Q,3}+\frac{1}{4}c_{mnpq}^{Q,4}\right)$
$\displaystyle>0,$
$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}c_{mnpq}^{Q,4}$
$\displaystyle>0.$ (63)

Then, we can define the linearly transformed coefficients $\xi(\rho)$:

$\displaystyle\begin{cases}\bullet\
\xi^{Q,1}_{k}&\equiv\rho^{Q,1}_{k}+\frac{1}{4}\rho^{Q,2}_{k}+\frac{1}{3}\rho^{Q,3}_{k}+\frac{1}{12}\rho^{Q,4}_{k},\\\
\bullet\ \xi^{Q,2}_{k}&\equiv\rho^{Q,2}_{k}+\frac{1}{3}\rho^{Q,4}_{k},\\\
\bullet\ \xi^{Q,3}_{k}&\equiv\rho^{Q,3}_{k}+\frac{1}{4}\rho^{Q,4}_{k},\\\
\bullet\ \xi^{Q,4}_{k}&\equiv\rho^{Q,4}_{k},\end{cases}$ $\displaystyle k$
$\displaystyle=1,2,3,4,$ (64)

to turn the bounds into

$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}c(\xi)_{mnpq}^{Q,i}$
$\displaystyle>0$ $\displaystyle i$ $\displaystyle=1,2,3,4,$ (65)

i.e., in the form of Eq. (3). For the (4-d) operators, the bounds are the same
as in the (4-u) case:

$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}\left(c_{mnpq}^{d,1}+\frac{1}{3}c_{mnpq}^{d,3}\right)$
$\displaystyle>0,$
$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}c_{mnpq}^{u,3}$
$\displaystyle>0.$ (66)

Thus, with a redefinition identical to (15), we recast them into the desired
form in Eq. (3).

##### Cross-quartic

The bounds on the (2-u)(2-Q) operators are:

$\displaystyle\alpha_{m}\alpha^{*}_{q}\beta_{n}\beta^{*}_{p}\left(a^{uQ,1}_{mnpq}+\frac{1\pm
3}{12}a^{uQ,3}_{mnpq}\right)>0.$ (67)

Upon defining:

$\displaystyle\begin{cases}\xi^{uQ,1}_{k}&\equiv\rho^{uQ,1}_{k}+\frac{1}{3}\rho^{uQ,3}_{k},\\\
\xi^{uQ,3}_{k}&\equiv\rho^{uQ,1}_{k}-\frac{1}{6}\rho^{uQ,3}_{k},\end{cases}$
$\displaystyle k$ $\displaystyle=1,2,3,4,$ (68)

they are recast as in Eq. (3). The shape of the bounds on the remaining cross
quartic operators is the same, therefore identical redefinitions

$\displaystyle\begin{cases}\xi^{dQ,1}_{k}&\equiv\rho^{dQ,1}_{k}+\frac{1}{3}\rho^{dQ,3}_{k},\\\
\xi^{dQ,3}_{k}&\equiv\rho^{dQ,1}_{k}-\frac{1}{6}\rho^{dQ,3}_{k},\end{cases}$
$\displaystyle k$ $\displaystyle=1,2,$ (69)
$\displaystyle\begin{cases}\xi^{du,1}_{k}&\equiv\rho^{du,1}_{k}+\frac{1}{3}\rho^{du,3}_{k},\\\
\xi^{du,3}_{k}&\equiv\rho^{du,1}_{k}-\frac{1}{6}\rho^{du,3}_{k},\end{cases}$
$\displaystyle k$ $\displaystyle=1,2,$ (70)

are needed to recast the bounds in the form of Eq. (3).

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