# Order parameters in quasi-1D spin systems

Garry Goldstein1<EMAIL_ADDRESS>
###### Abstract

In this work we extend the notion of what is meant by a meanfield. For the
purposes of this work meanfields are approximately maps - through some self
consistency relation - of a complex, usually manybody, problem to a simpler
more readily solvable problem. This mapping can then be solved to represent
properties of the complex many body problem using some self consistency
relations and the solution of the simpler problem. Prototypical examples of
simpler meanfield problems (meanfield systems) are the single site and free
particle problems - which are exactly solvable. Here we propose a new class of
simple meanfield systems where the simple problem to be solved is a 1D spin
chain. These meanfields are particularly useful for studying quasi-1D models,
where there is a 3D system composed of weakly coupled 1D spin chains with the
coupling in the transverse direction weaker then in the 1D direction. We
illustrate this idea by considering meanfields for the Ising (of any coupling
sign) and ferromagnetic Heisenberg models with one direction coupled much more
strongly then the other directions (quasi-1D systems) which map at meanfield
level onto the 1D Ising and 1D ferromagnetic Heisenberg models. We also
consider more exotic models to illustrate other methods of solving 1D systems,
namely the ferromagnetic $N$-state Potts model. Magnetic phase transition
temperatures and are obtained for all three models, we see that they
significantly differ from the usual meanfield estimates. Indeed if the 1D
direction has coupling $\Gamma$ and the transverse directions have coupling
$J$ with $\lambda\sim\frac{\Gamma}{J}\gg 1$ then regular meanfield would
predict the transition temperature to be $k_{B}T_{c}\sim\Gamma$ for all three
models while 1D meanfield predicts temperatures of
$k_{B}T_{c}\sim\frac{\Gamma}{\log\left(\lambda\right)}$ for the Ising and
Potts models and $k_{B}T_{c}\sim\frac{\Gamma}{\sqrt{\lambda}}$ for the
ferromagnetic Heisenberg model. Cluster 1D (ladders etc.) meanfield extensions
are also proposed.

##  I Introduction

Meanfields have been very successfully and almost universally used as a first
step towards the solution (understanding) of many complex manybody problems
[1, 3, 4, 5, 2]. In a meanfield like solution of a complex problem we map a
difficult manybody problem into a simpler one (a simpler system which can then
be efficiently solved), solve the problem and solve for self consistency to
relate the parameters of the solution of the simpler problem to the more
complex many body problem to be solved or understood. The prototypical
meanfield - simple - systems are the single site problem or the single
particle problem. In the single particle case the complex system is mapped
onto a quadratic Hamiltonian, through say a Hartree-Fock [5] meanfield, in a
self consistent manner whereby properties of the complex model namely ground
state energy, symmetry breaking, correlation functions may be often reliably
computed to some accuracy from the Hartree-Fock problem [5]. In the single
site problem - often a spin model is mapped onto a single spin problem which
can be efficiently solved. Then self consistency is imposed through equating
the parameters of the single site problem to the effects of the couplings to
the neighboring spins and their magnetization [1]. In some cases, the single
site problem may be extended, as in Dynamical Meanfield Theory (DMFT), to a
single site and a bath whereby the frequency dependence (but not momentum
dependence) of the Green’s functions of the manybody system may be modeled
self consistently through the dissipitative effects of the bath [6].

Here we propose another important simple system which may be effciently
analyzed as a meanfield system - the 1D system. Indeed many 1D models are
solvable through transfer matrix calculations [1, 7], Density Matrix
Renormalization Group (DMRG) [8, 9, 10] methods , Jordan-Wigner fermionization
[3, 5], or more generally the Bethe Ansatz (BA) methods [11, 12, 7, 13], and
Exact Diagonalization (ED) techniques [14, 15, 16] to name a few. In this work
we consider 3D spin systems where the coupling in one direction is much
stronger then the coupling in the other two (quasi-1D systems). For these
systems mapping the system to a 1D meanfield system and solving for self
consistency is more efficient then regular meanfield (we verify this in part
in Appendix A). We illustrate this idea through the 3D (quasi-1D) Ising model
whose meanfield is the 1D Ising model (of either sign of coupling) in an
external field (which is solvable by transfer matrix techniques) and the 3D
(quasi-1D) ferromagnetic Heisenberg model whose meanfield is the 1D
ferromagnetic Heisenberg model in an external field (which is solvable by BA
techniques). We choose these two examples because of their simplicity clarity
and because in both 3D and 1D these models are prototypical examples of models
with magnetic phase transitions. We also illustrate the ideas through a more
exotic related example the $N$-state ferromagnetic Potts model. We show
significant differences between regular meanfield results and 1D meanfield
results. Indeed assuming two couplings $\Gamma\gg J$ for longitudinal and
transverse directions and $\lambda\sim\frac{\Gamma}{J}\gg 1$; then regular
meanfield predicts a transition temperature of $k_{B}T_{c}\sim\Gamma$ for all
three models while 1D improved meanfield shows that
$k_{B}T_{c}\sim\frac{\Gamma}{\log\left(\lambda\right)}$ for the Ising and
Potts models and $k_{B}T_{c}\sim\frac{\Gamma}{\sqrt{\lambda}}$ for the
ferromagnetic Heisenberg model. Here $k_{B}$ is the Boltzmann constant and
$T_{c}$ is the critical temperature for the magnetic phase transition. Cluster
1D meanfields are also proposed, which should systematically improve meanfield
accuracy [1].

##  II Ising model

Consider a 3D quasi-1D ferromagnetic Ising model in an external field with the
following Hamiltonian:

$\displaystyle H_{I}=$
$\displaystyle-\Gamma\sum_{i}\sigma^{z}\left(i\right)\sigma^{z}\left(i+\hat{z}\right)$
$\displaystyle-J\sum_{\left\langle i,j\right\rangle,j\neq
i\pm\hat{z}}\sigma^{z}\left(i\right)\sigma^{z}\left(j\right)+B\sum_{i}\sigma^{z}\left(i\right)$
(1)

at a temperature $\beta$ and $\left\langle i,j\right\rangle$ are nearest
neighbors, here we will be interested in $\Gamma\gg J$ so that the 1D chains
along the z-axis are more strongly coupled then the transverse plane.
Antiferromagnetic Ising models are equivalent to ferromagnetic Ising models on
bipartite lattices. so need not be considered Now consider the following
meanfield Hamiltonian at unit temperature:

$H_{MF}=\gamma\sum_{i}^{N}\sigma^{z}\left(i\right)\sigma^{z}\left(i+1\right)+h\sum_{i}\sigma^{z}\left(i\right)$
(2)

Where for self consistency:

$\gamma=\beta\Gamma,\;h=\beta J\mathcal{N}m-\beta B$ (3)

here $\mathcal{N}\sim 2\left(d-1\right)$ is the number of nearest neighbors in
the weakly coupled directions and $d=3$ is the dimension of the system. Then
we know that the magnetization $m$ in the thermodynamic limit for the
Hamiltonian in Eq. (2) at unit temperature is given by 1 [1]:

$m=\frac{\sinh\left(h\right)}{\left[\exp\left(-4\gamma\right)+\left(\sinh\left(h\right)\right)^{2}\right]^{1/2}}$
(4)

This is an exact result (using the transfer matrix formalism) [1, 7]. Now we
substitute the meanfield relations in Eq. (3) and obtain:

$m=\frac{\sinh\left(\beta J\mathcal{N}m-\beta
B\right)}{\left[\exp\left(-4\beta\Gamma\right)+\left(\sinh\left(\beta
J\mathcal{N}m-\beta B\right)\right)^{2}\right]^{1/2}}$ (5)

Setting $B=0$ and linearizing to find the phase transition temperature we have
that:

$\displaystyle m$ $\displaystyle=\exp\left(2\beta\Gamma\right)\beta
J\mathcal{N}m,$ $\displaystyle 1$
$\displaystyle=\exp\left(2\beta\Gamma\right)\beta J\mathcal{N}$ (6)

leads to magnetism. Taking the logarithm of both sides we see that:

$2\beta\Gamma=-\log\left(\beta J\mathcal{N}\right)$ (7)

Introducing

$\lambda=\frac{\Gamma}{J\mathcal{N}}\gg 1$ (8)

We get that:

$2\lambda=-\frac{\log\left(\beta J\mathcal{N}\right)}{\beta J\mathcal{N}}$ (9)

We now write:

$\beta_{0}J\mathcal{N}=\frac{1}{2\lambda}$ (10)

Then:

$-\frac{\log\left(\beta J\mathcal{N}\right)}{\beta
J\mathcal{N}}\cong\frac{\log\left(2\lambda\right)}{\beta J\mathcal{N}}$ (11)

This means that:

$\beta_{1}J\mathcal{N}=\frac{1}{2\lambda\log\left(2\lambda\right)}$ (12)

In general we can iterate the solution through the relation (though Eq. (12)
is often enough):

$\beta_{n+1}J\mathcal{N}=-\frac{1}{2\lambda\log\left(\beta_{n}J\mathcal{N}\right)}$
(13)

We note that the usual meanfield equations the phase transition temperature
can be found through the following relationship:

$2\beta\Gamma+\beta J\mathcal{N}=1$ (14)

Which are much worse as it predicts:

$\beta J\mathcal{N}=\frac{1}{2\lambda+1}$ (15)

which is significantly different, for large $\lambda$, then Eq. (12), see
Appendix A.

##  III $N$ state Potts model

We focus on the ferromagnetic Potts model. We write:

$\displaystyle H_{Pott}=$
$\displaystyle-\Gamma\sum_{i}\delta\left(n\left(i\right),n\left(i+\hat{z}\right)\right)$
$\displaystyle-J\sum_{\left\langle i,j\right\rangle,j\neq
i\pm\hat{z}}\delta\left(n\left(i\right),n\left(j\right)\right)+B\sum_{i}\delta\left(n\left(i\right),1\right)$
(16)

Where $n\left(i\right)$ is the state of the i’th unit, with
$n\left(i\right)=1,2...,N$. We now write the Hamiltonian of the meanfield
system:

$H_{Pott}=-\Gamma\sum_{i}\delta\left(n\left(i\right),n\left(i+1\right)\right)-\mathcal{B}\sum_{i}\delta\left(n\left(i\right),1\right)$
(17)

Then we write the transfer matrix [7, 1] of the system as:

$\displaystyle T$ $\displaystyle=\left|V\right\rangle\left\langle
V\right|+\left(\exp\left(\mathcal{B}\right)-1\right)\left[\left|V\right\rangle\left\langle
1\right|+\left|1\right\rangle\left\langle V\right|\right]$
$\displaystyle\quad+\left(\exp\left(\Gamma\right)-1\right)\sum_{i=2}^{N}\left|i\right\rangle\left\langle
i\right|+$
$\displaystyle\quad+\left(\exp\left(\Gamma+2\mathcal{B}\right)-2\exp\left(\mathcal{B}\right)+1\right)\left|1\right\rangle\left\langle
1\right|$ (18)

Where we introduce the un-normalized vector

$\left|V\right\rangle=\sum_{i=1}^{N}\left|i\right\rangle$ (19)

Now we look for the biggest eigenvalue of the transfer matrix [7, 1] using the
ansatz:

$\displaystyle E\left(a\left|1\right\rangle+b\left|V\right\rangle\right)$
$\displaystyle=T\left(a\left|1\right\rangle+b\left|V\right\rangle\right)$ (20)

Now we linearIze everything for small $\mathfrak{B}$ in order to find the
magnetic phase transition and write:

$\left(\begin{array}[]{cc}\left(\exp\left(\beta\Gamma\right)-\left(2\exp\left(\beta\Gamma\right)-1\right)\beta\mathcal{B}\right)&\left(\left(N-2\right)+2\exp\left(\beta\Gamma\right)\right)\beta\mathcal{B}\\\
\beta\mathcal{B}&\left(N-1+\exp\left(\beta\Gamma\right)+\beta\mathcal{B}\right)\end{array}\right)\left(\begin{array}[]{c}a\\\
b\end{array}\right)=E\left(\begin{array}[]{c}a\\\ b\end{array}\right)$ (21)

For a full calculation to all orders see Appendix B. Now we treat the matrix
proportional to $\beta\mathcal{B}$ as a perturbation and we write:

$\left(\begin{array}[]{cc}\exp\left(\beta\Gamma\right)&0\\\
0&\left(N-1+\exp\left(\beta\Gamma\right)\right)\end{array}\right)+\beta\mathcal{B}\left(\begin{array}[]{cc}\left(2\exp\left(\beta\Gamma\right)-1\right)&\left(\left(N-2\right)+2\exp\left(\beta\Gamma\right)\right)\\\
1&1\end{array}\right)$ (22)

Then

$E\cong\left(N-1+\exp\left(\beta\Gamma\right)\right)+\beta\mathcal{B}$ (23)
$b\cong 1$ (24)
$a\cong\frac{\beta\mathcal{B}\left(N+2\exp\left(\beta\Gamma\right)-2\right)}{N-1}$
(25)

and

$M\cong\frac{2a}{N}=\frac{2\beta\mathcal{B}\left(N+2\exp\left(\beta\Gamma\right)-2\right)}{\sqrt{N}\left(N-1\right)}$
(26)

is the magnetization. This then means that at meanfield:

$\displaystyle\mathcal{B}$ $\displaystyle=M\mathcal{N}J$
$\displaystyle\Rightarrow 1$ $\displaystyle=\frac{2\mathcal{\beta
N}J\left(\left(N-2\right)+2\exp\left(\beta\Gamma\right)\right)}{N\left(N-1\right)}$
(27)

Now we will assume that $\beta\mathcal{N}J\ll 1$ so that $\left(N-2\right)\ll
2\exp\left(\beta\Gamma\right)$ as such we obtain:

$\displaystyle 1$ $\displaystyle=\frac{4\mathcal{\beta
N}J\exp\left(\beta\Gamma\right)}{N\left(N-1\right)}$
$\displaystyle\beta\Gamma$
$\displaystyle=\log\left(\frac{N\left(N-1\right)}{4\beta\mathcal{N}J}\right)$
$\displaystyle\lambda$
$\displaystyle=\frac{\log\left(\frac{N\left(N-1\right)}{4\beta\mathcal{N}J}\right)}{\beta
J\mathcal{N}}\cong\frac{\log\left(N\left(N-1\right)/4\right)}{\beta
J\mathcal{N}}$ $\displaystyle\beta J\mathcal{N}$
$\displaystyle\cong\frac{\log\left(N\left(N-1\right)/4\right)}{\lambda}$ (28)

Iterating we get that:

$\displaystyle 1$
$\displaystyle=2\frac{\log\left(N\left(N-1\right)/4\right)}{\lambda}\frac{\left(\left(N-2\right)+2\exp\left(\beta\Gamma\right)\right)}{N\left(N-1\right)}$
$\displaystyle\beta\Gamma$
$\displaystyle\cong\frac{1}{2}\log\left(\frac{\lambda
N\left(N-1\right)}{2\log\left(N\left(N-1\right)/4\right)}-N+2\right)$ (29)

##  IV  Ferromagnetic Heisenberg model

We consider the anisotropic 3D (quasi-1D) ferromagnetic Heisenberg model with
the following Hamiltonian:

$\displaystyle H=$
$\displaystyle-\Gamma\sum_{i}\vec{\sigma}\left(i\right)\cdot\vec{\sigma}\left(i+\hat{z}\right)$
$\displaystyle-J\sum_{\left\langle i,j\right\rangle,j\neq
i\pm\hat{z}}\vec{\sigma}\left(i\right)\cdot\vec{\sigma}\left(j\right)+B\sum_{i}\sigma_{i}^{z}$
(30)

with $\mathcal{J}\gg J$. Now we consider the following meanfield Hamiltonian
(which happens to be the Heisenberg model in an external field and as such
solvable by BA techniques [12, 11]):

$H_{MF}=-\Gamma\sum_{i}\vec{\sigma}\left(i\right)\cdot\vec{\sigma}\left(i+1\right)-h\sum_{i}\sigma^{z}\left(i\right)$
(31)

with

$h=\mathcal{N}J\mathcal{M}-B$ (32)

Where $\mathcal{M}$ will be chosen self consistently that is:

$\mathcal{M}=\left\langle\sigma^{z}\left(i\right)\right\rangle_{H_{MF}}$ (33)

We will now solve Eqs. (31) and (33) and find the critical transition
temperature where $\mathcal{M}$ vanishes. This will be an extended meanfield
treatment of the Hamiltonian in Eq. (30). Indeed the magnetic susceptibility
$\chi$ can be used to determine phase boundaries. We have that for the
ferromagnetic Heisenberg model at temperature $T$ the susceptibility is given
by [11, 12]:

$\chi=\mathcal{J}^{-1}\left(\frac{1}{6}\left(\frac{\Gamma}{T}\right)^{2}+0.581\left(\frac{\Gamma}{T}\right)^{3/2}+0.68\left(\frac{\Gamma}{T}\right)\right)+...$
(34)

Now to determine the phase transition temperature between magnetic and non-
magnetic phase we use the relationship:

$\displaystyle\chi h$ $\displaystyle=\mathcal{M}$
$\displaystyle\chi\mathcal{N}J\mathcal{M}$ $\displaystyle=\mathcal{M}$
$\displaystyle\chi\mathcal{N}J$ $\displaystyle=1.$ (35)

Where we have set $B=0$ to find the phase transition. As such for the phase
transition between magnetic and non-magnetic we write:

$\frac{\mathcal{N}J}{\Gamma}\left(\frac{1}{6}\left(\frac{\Gamma}{T}\right)^{2}+0.581\left(\frac{\Gamma}{T}\right)^{3/2}+0.68\left(\frac{\Gamma}{T}\right)\right)=1$
(36)

Now we have that

$\displaystyle\frac{\mathcal{N}J}{\Gamma}\left(\frac{1}{6}\left(\frac{\Gamma}{T}\right)^{2}+0.581\left(\frac{\Gamma}{T}\right)^{3/2}+0.68\left(\frac{\Gamma}{T}\right)\right)$
$\displaystyle\cong\frac{1}{6\lambda}\left(\frac{\Gamma}{T}\right)^{2}$ (37)

This means that at first approximation for the critical temperature:

$T_{0}\cong\Gamma\left(\frac{1}{6\lambda}\right)^{1/2}$ (38)

Now we write:

$\displaystyle\frac{\mathcal{N}J}{\Gamma}\left(\frac{1}{6}\left(\frac{\Gamma}{T}\right)^{2}+0.581\left(\frac{\Gamma}{T}\right)^{3/2}+0.68\left(\frac{\Gamma}{T}\right)\right)$
$\displaystyle\cong\frac{\mathcal{N}J}{\Gamma}\left(\frac{1}{6}\left(\frac{\Gamma}{T_{0}}\right)^{2}+0.581\left(\frac{\Gamma}{T_{0}}\right)^{3/2}+0.68\left(\frac{\Gamma}{T_{0}}\right)\right)$
(39)

As such the second approximation for the critical temperature gives:

$T_{1}\cong\Gamma\sqrt{\frac{1}{6\lambda}\left[1-2.23\left(\frac{1}{\lambda}\right)^{1/4}-1.67\left(\frac{1}{\lambda}\right)^{1/2}\right]}$
(40)

We can continue to iteratively solve the problem for the critical temperature
more and more accurately using the formula (though Eq. (40) is often
sufficient):

$T_{n+1}\cong\Gamma\sqrt{\frac{1}{6\lambda}\left[1-\frac{1}{\lambda}\left[0.581\left(\frac{\Gamma}{T_{n}}\right)^{3/2}-0.68\left(\frac{\Gamma}{T_{n}}\right)\right]\right]}$
(41)

##  V Conclusions

In this work we have extended the notion of the meanfield. We have proposed a
new class of meanfield models where 1D systems are the meanfield problems to
be solved. This combined with self consistency relations may be used to
efficiently solve for properties of anisotropic 3D or quasi-1D systems - where
one direction is much more strongly coupled then the other. In this work we
have illustrated this idea with the Ising and ferromagnetic Heisenberg models
which are solvable by transfer matrix and Bethe Ansatz techniques respectively
and found transition temperatures as preliminary results. We also studied a
more exotic model - the Potts model. We found significantly different results
then regular meanfield. Indeed we found magnetic phase transition temperatures
of $k_{B}T_{c}\sim\frac{\Gamma}{\log\left(\lambda\right)}$ for the Ising and
Potts models and $k_{B}T_{c}\sim\frac{\Gamma}{\sqrt{\lambda}}$ for the
ferromagnetic Heisenberg model, where as regular meanfield predicts
$k_{B}T_{c}\sim\Gamma$ for all three models. In future works it would be of
interest to extend these results to many other quasi-1D systems
systematically. Furthermore it would be of interest to study cluster 1D
meanfields, where a cluster of 1D systems is chosen as the meanfield system to
be solved. The accuracy of the calculation is systematically improved with
cluster size in cluster 1D meanfields.

Acknowledgements: The author would like to thank Natan Andrei for useful
discussions.

##  Appendix A Onsager solution

We note that the Onsager relation for $d=2$ ($\mathcal{N}=2$) is that the
critical temperature is given by [1, 17]:

$\displaystyle\sinh\left(2\beta J\right)\sinh\left(2\beta\Gamma\right)$
$\displaystyle=1$ $\displaystyle\exp\left(2\beta\Gamma\right)\cdot\beta J$
$\displaystyle\cong 1$ (42)

This is the exact same relationship as in Eq. (6) for $\mathcal{N}=1$, which
means that:

$\beta J=\frac{1}{2\lambda\log\left(2\lambda\right)}+...$ (43)

This means the meanfield is accurate within a factor of two while the regular
meanfield is off by $\sim\log\left(2\lambda\right)$ for large $\lambda$
showing significant improvement of our approach over regular meanfield in the
limit of strong anisotropy or nearly 1D systems.

##  Appendix B $N$-state Potts model in an external Field at unit temperature

Now we look for the biggest eigenvalue using the ansatz in Eq. (20). This
means that:

$\left(\begin{array}[]{cc}\left(\exp\left(J+2B\right)-\exp\left(B\right)+1\right)&\left(\left(N-2\right)\left[\exp\left(B\right)-1\right]+\exp\left(J+2B\right)-\exp\left(J\right)\right)\\\
\left[\exp\left(B\right)-1\right]&\left(N-2+\exp\left(B\right)+\exp\left(J\right)\right)\end{array}\right)\left(\begin{array}[]{c}a\\\
b\end{array}\right)=E\left(\begin{array}[]{c}a\\\ b\end{array}\right)$ (44)

This means that

$\det\left[\begin{array}[]{cc}\left(\exp\left(J+2B\right)-\exp\left(B\right)+1-E\right)&\left(\left(N-2\right)\left[\exp\left(B\right)-1\right]+\exp\left(J+2B\right)-\exp\left(J\right)\right)\\\
\left[\exp\left(B\right)-1\right]&\left(N-2+\exp\left(B\right)+\exp\left(J\right)-E\right)\end{array}\right]=0$
(45)

As such:

$E^{2}-E\left[\exp\left(J+2B\right)+N-1+\exp\left(J\right)\right]+\exp\left(2J+2B\right)+\left[N-1\right]\left[\exp\left(J+2B\right)-\exp\left(2B\right)+\exp\left(B\right)\right]=0$
(46)

As such (since we want the biggest eigenvalue):

$E_{+}=\frac{\left[\exp\left(J+2B\right)+N+\exp\left(J\right)-1\right]+\sqrt{\Delta\left(J,B,N\right)}}{2}$
(47)

Where

$\Delta\left(J,B,N\right)=\left[\exp\left(J+2B\right)-N+1+\exp\left(J\right)\right]^{2}-4\left[\exp\left(2J+2B\right)+\left[N-1\right]\left[\exp\left(B\right)-\exp\left(2B\right)-\exp\left(J\right)\right]\right]$
(48)

Now we have that

$\displaystyle\frac{a}{b}$
$\displaystyle=\frac{E_{+}-N+2-\exp\left(B\right)-\exp\left(J\right)}{\exp\left(B\right)-1}$
$\displaystyle=\frac{\exp\left(J+2B\right)+\sqrt{\Delta\left(J,B,N\right)}}{2\left(\exp\left(B\right)-1\right)}-\frac{1}{2}$
(49)

Now we have that the normalization of the vector is

$\left\|a\left|1\right\rangle+b\left|V\right\rangle\right\|=Nb^{2}+a^{2}+2ab$
(50)

Therefore the magnetization is given by:

$M=\frac{a^{2}+2ab}{Nb^{2}+a^{2}+2ab}$ (51)

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Dirac Masses and Mixings in the (geo)SM(EFT) and Beyond

Jim Talberta and Michael Trotta,b
a Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
b Department of Physics (visitor), University of Notre Dame, Notre Dame, IN,
46556, USA
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
We report a set of exact formulae for computing Dirac masses, mixings, and CP-
violation parameter(s) from 3$\times$3 Yukawa matrices $Y$ valid when
$YY^{\dagger}\rightarrow U^{\dagger}\,YY^{\dagger}\,U$ under global
$U(3)_{Q_{L}}$ flavour symmetry transformations $U$. The results apply to the
Standard Model Effective Field Theory (SMEFT) and its ‘geometric’ realization
(geoSMEFT). We thereby complete, in the Dirac flavour sector, the catalogue of
geoSMEFT parameters derived at all orders in the $\sqrt{2\langle
H^{\dagger}H\rangle}/\Lambda$ expansion. The formalism is basis-independent,
and can be applied to models with decoupled ultraviolet flavour dynamics, as
well as to models whose infrared dynamics are not minimally flavour violating.
We highlight these points with explicit examples and, as a further
demonstration of the formalism’s utility, we derive expressions for the
renormalization group flow of quark masses, mixings, and CP-violation at all
mass dimension and perturbative loop orders in the (geo)SM(EFT) and beyond.

###### Contents

  1. 1 Introduction
  2. 2 The Yukawa Sector of the (geo)SM(EFT)
    1. 2.1 Extension to Fermionic Mixing
  3. 3 All-Orders Masses, Mixings, and CP-Violation
    1. 3.1 Approach with Flavour Invariants
    2. 3.2 Numerical Checks
    3. 3.3 Applicability to other BSM Scenarios
  4. 4 Renormalization Group Flow
    1. 4.1 Computation of Quark Sector RGE at One-Loop Order
  5. 5 Summary and Outlook
  6. A Formulae for RGE Computation
    1. A.1 All-Orders Product Functions
    2. A.2 One-Loop RGE for $I_{i}$

##  1 Introduction

The flavour sector of the Standard Model (SM) sources the bulk of its free
parameters while simultaneously providing some of its richest phenomenology.
These free parameters originate in the renormalizable Yukawa interactions
between left- and right-chiral fermionic fields and the SU(2)L Higgs boson
doublet ($H$),

$\mathcal{L}_{SM}^{Y}\supset
Y^{u}_{pr}\,\overline{Q}_{L,p}\,\tilde{H}\,u_{R,r}+Y^{d}_{pr}\,\overline{Q}_{L,p}\,H\,d_{R,r}+Y^{e}_{pr}\,\overline{L}_{L,p}\,H\,e_{R,r}\,+\text{h.c.}\,,$
(1)

whose couplings have non-trivial structure in the flavour indices $\\{p,r\\}$;
here $\tilde{H}_{j}=\epsilon_{jk}H^{\dagger k}$. In the SM $Y$ are generically
$3\times 3$ complex matrices in flavour space. Field re-definitions allow one
to reduce the number of parameters in this structure to only 13 free
parameters in the (physical) fermion mass-eigenstate basis: six quark masses,
three quark mixing angles, one CP-violating Dirac phase, and finally three
charged lepton masses. The quark mixing angles and CP phase appear in the
Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix, which encodes all of the
flavour-violation present in the SM’s charged-current interactions. The Yukawa
operators of (1) are the only terms which explicitly break the global
$\mathcal{G}_{F}\sim U(3)^{5}$ flavour symmetry otherwise present in the SM
[1, 2].

Extracting values for the SM’s mass and mixing parameters is critical to
understanding its precision phenomenology, and Beyond-the-SM (BSM) physics
that may depend on (or even explain) this distinct flavour structure. In
practice this extraction is typically achieved via numerical methods for
computing matrix eigenvalues (giving Dirac masses) via diagonalizing these
matrices with (bi-)unitary transformations. A natural question to ask is: can
one obtain analytic expressions for mass and mixing parameters given arbitrary
forms for the complex Yukawa couplings ? Developing such a formalism with only
two fermion generations is straightforward, see Section 2. However, for three
generations standard diagonalization techniques become intractable for generic
$Y^{u,d,e}$.

In this paper we employ flavour invariants to derive exact, compact formulae
for the computation of Dirac mass, mixing, and CP violation parameters for
fully generic $3\times 3$ Yukawa couplings in not just the SM, but also its
generalization to the SM effective field theory (SMEFT) [3, 4],

$\mathcal{L}_{SMEFT}=\mathcal{L}_{SM}+\sum_{i}\frac{C_{i}^{(d)}}{\Lambda^{d-4}}\mathcal{Q}_{i}^{(d)}\,.$
(2)

Here the sum runs over the complete basis of non-renormalizable operators
$\mathcal{Q}_{i}^{(d)}$ composed of SM fields and invariant under the SM gauge
group $\mathcal{G}_{SM}\equiv SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$, at a
given mass dimension $d>4$, with associated Wilson coefficients $C_{i}^{(d)}$.
Such operators are induced generically when new physics is integrated out at a
scale $\Lambda>\overline{v}_{T}\equiv\sqrt{2\langle H^{\dagger}H\rangle}$,
with $\overline{v}_{T}$ the vacuum expectation value (vev) of the SM Higgs
doublet.111For a comprehensive review of the SMEFT formalism, see [5]. In
particular, we will employ the geometric realization of the SMEFT, the
geoSMEFT [6], in our computation. The geoSMEFT represents an all-orders
reorganization of (2), such that interactions are described on a curved
manifold in scalar field space(s). The degree of curvature depends on the
ratio $\overline{v}_{T}/\Lambda$, with the ‘flat’ limit (where
$\overline{v}_{T}/\Lambda\rightarrow 0$) corresponding to $\mathcal{L}_{SM}$.
The geoSMEFT factorizes into simple operator forms multiplying field-space
connections, with the latter encoding SM theory parameters valid at all orders
in $\overline{v}_{T}/\Lambda$. Our results will therefore complete the
geoSMEFT expressions in the Dirac flavour sector. In addition we show that the
formulae can be used to make predictions in ultraviolet scenarios when (e.g.)
spontaneous flavour-symmetry breaking leads to special textures for SM Yukawa
matrices, and even in theories that introduce additional flavour violation
into the low-energy spectrum, as long as the global $U(3)_{Q_{L}}$
transformation properties of $YY^{\dagger}$ are respected.

Besides their obvious predictive utility, our formulae may also be of use in
high(er)-order global SMEFT fits to existing data, especially to CKM mixing
elements (see [7]), or in studying non-standard flavour violation effects in
the SMEFT (see e.g. [8]). Towards the latter end, we use our formulae to
rapidly derive their renormalization group flow at all mass dimension and loop
orders in the (geo)SM(EFT), including an explicit numerical calculation of
quark sector RGE at one-loop perturbative order in minimally flavour-violating
(MFV) [9] theories. Our results therefore constitute a generic formalism for
studying flavour in (B)SM matching cases of the SMEFT, and also open the door
for related studies in the leptonic sector, when non-zero neutrino masses are
properly accounted for.

The paper develops as follows: in Section 2 we review the Yukawa sector of the
geoSMEFT, while in Section 3 we present the unique flavour invariants we
employ, and use them to derive the final formulae. In addition, we discuss the
domain of applicability of these expressions with demonstrated examples. In
Section 4 we derive generic, analytic expressions for the renormalization
group flow of the flavour parameters. Finally, we conclude in Section 5,
providing an outlook for the extension of this formalism into the lepton
sector. Some useful formulae are presented in Appendix A.

##  2 The Yukawa Sector of the (geo)SM(EFT)

The geoSMEFT [6] represents a re-organization of the SMEFT operator product
expansion (OPE) in (2), such that

$\mathcal{L}_{\text{SMEFT}}=\sum_{i}G_{i}\left(I,A,\phi,...\right)\,f_{i}\,.$
(3)

Here $G_{i}$ are field-space connections that depend on the group indices
$I,A$ of all internal symmetry groups, and real scalar field coordinates
$\phi_{I}$ of the SU(2)L Higgs doublet, normalized as follows:222$\phi_{4}$ is
expanded around the vev via $\phi_{4}\rightarrow\phi_{4}+\overline{v}_{T}$.

$H\left(\phi_{I}\right)=\frac{1}{\sqrt{2}}\left[\begin{array}[]{c}\phi_{2}+i\phi_{1}\\\
\phi_{4}-i\phi_{3}\end{array}\right]\,\,\,\Longrightarrow\,\,\,\tilde{H}\left(\phi_{I}\right)=\frac{1}{\sqrt{2}}\left[\begin{array}[]{c}\phi_{4}+i\phi_{3}\\\
-\phi_{2}+i\phi_{1}\end{array}\right]\,.$ (4)

On the other hand, $f_{i}$ are operator forms composed of the Lorentz-index-
carrying building blocks of $\mathcal{L}_{\text{SMEFT}}$, and which are
(largely)333Except powers of $D^{\mu}H$, which get grouped with $f_{i}$. This
residual scalar coordinate-dependence in the composite operator forms
manifests in powers of $\partial_{\mu}h$, upon the Higgs acquiring its vev.
independent of $\phi$. That is, upon the Higgs obtaining its vev, the tower of
interactions composing $G_{i}$ reduces to a number and emissions of $h$ (the
propagating Higgs field), while $f_{i}$ remains a distinct operator of SM
fields and derivatives. Critically, by using Hilbert series techniques one can
show that, at least for two- and three-point functions, the number of $f_{i}$
saturates to a constant value at arbitrary mass-dimension. This then allows
the field-space connections $G_{i}$, and thereby important theory parameters
(e.g. gauge boson masses, gauge couplings, weak mixing angles, the Higgs mass,
etc.), to be defined at all orders in the SMEFT’s characteristic
$\overline{v}_{T}/\Lambda$ expansion. For a complete description of the
geoSMEFT see [6], which builds on prior work [10, 11, 12, 13].

In what follows we are only concerned with the Yukawa sector of the effective
theory, i.e. with non-renormalizable SMEFT operators of the form

$Q^{6+2n}_{\overset{\psi
H}{\scriptscriptstyle{pr}}}=\left(H^{\dagger}H\right)^{n+1}\left(\bar{\psi}_{L,p}\psi_{R,r}H\right)\,\,\,\,\,\text{with}\,\,\,\,\,n\geq
0\,,$ (5)

with $\psi_{L}\in\\{Q,L\\}_{L}$ and $\psi_{R}\in\\{u,d,e\\}_{R}$, and where
$H\rightarrow\tilde{H}$ in the second bracket when $\psi_{R}=u_{R}$. Hence the
classification of two- and three-point functions in terms of their field space
connections and composite operator forms, as already present in [6], is
sufficient for our purposes. From there we recall the two-point Yukawa
function $Y(\phi)\overline{\psi}_{1}\psi_{2}$, whose field space connection
$Y(\phi)$ is given by

$Y_{pr}^{\psi_{1}}\left(\phi_{I}\right)=\frac{\delta\mathcal{L}_{\text{SMEFT}}}{\delta(\overline{\psi}^{I}_{2,p}\psi_{1,r})}\bigg{|}_{\mathcal{L}(\alpha,\beta,...)\rightarrow
0}\,,$ (6)

for a generic fermion sector $\psi\in\\{u,d,e\\}$. Here $\alpha,\beta,...$
denote effective gauge couplings etc., such that the notation
$\mathcal{L}(\alpha,\beta,...)\rightarrow 0$ implies that all Lagrangian terms
and spin connections with non-trivial Lorentz indices are sent to zero. As a
demonstrative example, the Higgs potential is given simply by
$V(\phi)=-\mathcal{L}_{\text{SMEFT}}|_{\mathcal{L}(\alpha,\beta,...)\rightarrow
0}$.

Using (6) one quickly arrives at the explicit expressions for the fermionic
field space connections,

$\displaystyle Y_{pr}^{\psi}\left(\phi_{I}\right)$
$\displaystyle=-H\left(\phi_{I}\right)\left[Y_{\psi}\right]_{pr}^{\dagger}+H\left(\phi_{I}\right)\sum_{n=0}^{\infty}C^{(6+2n)}_{\overset{\psi
H}{\scriptscriptstyle{pr}}}\left(\frac{\phi^{2}}{2}\right)^{n}\,,$ (7)

where again $H\rightarrow\tilde{H}$ when $\psi=u$, from which one defines the
all-orders Yukawa couplings

$[\mathcal{Y}^{\psi}]_{rp}=\frac{\delta(Y^{\psi}_{pr})^{\dagger}}{\delta
h}\bigg{|}_{\phi_{i}\rightarrow
0}=\frac{\sqrt{h}^{44}}{\sqrt{2}}\left([Y^{\psi}]_{rp}-\sum_{n=3}^{\infty}\frac{2n-3}{2^{{n-2}}}\tilde{C}^{(2n),\star}_{\overset{\psi
H}{\scriptscriptstyle{pr}}}\right)\,.$ (8)

Here $\sqrt{h}^{44}$ is the 44 component of the matrix square-root of the
expectation value for the scalar field space metric, $\langle
h^{IJ}\rangle^{1/2}$; $h^{IJ}(\phi)$ is the all-orders connection for
derivative Higgs operators: $\mathcal{L}\supset
h_{IJ}(\phi)\left(D_{\mu}\phi\right)^{I}\left(D^{\mu}\phi\right)^{J}$. Also
note that the Wilson coefficients have been normalized in (8) such that
$\tilde{C}_{i}^{(2n)}\equiv
C_{i}^{(2n)}\,\overline{v}_{T}^{2n-4}/\Lambda^{2n-4}$. Finally, we recall that
the fermion mass matrices are given after the Higgs acquires it vev,

$\left[M_{\psi}\right]_{rp}=\langle(Y_{pr}^{\psi})^{\dagger}\rangle\,.$ (9)

The flavour structure of the theory can be analyzed in either phase, and so in
what follows we generally work with the Yukawa matrices in (8).

###  2.1 Extension to Fermionic Mixing

Our goal is to extend the definition of all-orders parameters in the geoSMEFT
to the fermion mass and mixing sector, i.e. those quantities defined in the
fermion mass-eigenstate basis, where the different generations of a given
fermion family (e.g. $u$, $c$, $t$) can be physically distinguished by their
distinct masses. As in the SM, this requires the diagonalization of (8), which
can be achieved through a bi-unitary transformation on both the SU(2)L doublet
and singlet fermion fields:

$[U_{\psi L}^{\dagger}]_{ir}\,[\mathcal{Y}^{\psi}]_{rp}\,[U_{\psi
R}]_{pj}\equiv[D_{\psi}]_{ij}=\text{diag}\left(y_{\psi 1},y_{\psi 2},y_{\psi
3}\right)\,.$ (10)

One can rotate away the dependence on the singlet transformations by
constructing the Hermitian combinations $\mathcal{Y}\mathcal{Y}^{\dagger}$ and
diagonalizing these objects through unitary field transformations, i.e.

$[U_{\psi
L}^{\dagger}]_{ir}\,[\mathcal{Y}^{\psi}\mathcal{Y}^{\psi,\dagger}]_{rp}\,[U_{\psi
L}]_{pj}=\text{diag}\left(y^{2}_{\psi 1},y^{2}_{\psi 2},y^{2}_{\psi
3}\right)\,.$ (11)

Dropping the $L$ subscript from the mixing matrices, one then writes the
physical CKM matrix as the overlap between up and down-quark fermion mixing:

$V_{CKM}\equiv
U^{\dagger}_{u}U_{d}\equiv\left(\begin{array}[]{ccc}V_{ud}&V_{us}&V_{ub}\\\
V_{cd}&V_{cs}&V_{cb}\\\ V_{td}&V_{ts}&V_{tb}\\\ \end{array}\right)\,.$ (12)

As mentioned in the introduction, $V_{CKM}$ can be generically parameterized
in terms of three real mixing angles $\theta_{ij}$ and a Dirac CP-violating
phase $\delta$, and we do so with the standard Particle Data Group (PDG) [16]
representation,

$V_{CKM}=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}\,e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}\,e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}\,e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}\,e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}\,e^{i\delta}&c_{23}c_{13}\\\
\end{array}\right)\,,$ (13)

where $\cos\theta_{ij}\equiv c_{ij}$, etc. Note that while the structural form
of this matrix is identical to its SM counterpart, the mixing angles encode
the fact that there is an infinite tower of higher-order operator forms
contributing to the Yukawa connection (and therefore flavour violation), as
shown in (8). For notational simplicity we do not differentiate these
‘effective’ mixing elements from those in the flat limit where $C\rightarrow
0$.

##  3 All-Orders Masses, Mixings, and CP-Violation

Our goal in this section is to derive expressions for the Yukawa eigenvalues
$y_{i}^{2}$ and CKM mixing parameters $\theta_{ij}$ and $\delta$, in terms of
$\mathcal{Y}_{rp}$ alone, such that these Lagrangian parameters are defined at
all orders in the $\overline{v}_{T}/\Lambda$ expansion characterizing the
geoSMEFT. To this end we first consider the two-generation case, where one can
expand the flavour indices of $\mathcal{Y}$, finding

$\mathcal{Y}=\frac{\sqrt{h}^{44}}{\sqrt{2}}\left[\left(\begin{array}[]{cc}Y_{11}&Y_{22}\\\
Y_{21}&Y_{22}\end{array}\right)-\sum_{n=3}^{\infty}\frac{2n-3}{2^{{n-2}}}\left(\begin{array}[]{cc}\tilde{C}^{(2n),\star}_{11}&\tilde{C}^{(2n),\star}_{21}\\\
\tilde{C}^{(2n),\star}_{12}&\tilde{C}^{(2n),\star}_{22}\end{array}\right)\right]\,.$
(14)

Such an expansion holds for both quark families $\mathcal{Y}_{u,d}$. To avoid
bi-unitary transformations, we next form the Hermitian product
$\mathcal{Y}\mathcal{Y}^{\dagger}$, and take its magnitude (anticipating the
fact that no CP-violation is present in two-generation theories):

$|\mathcal{Y}\mathcal{Y}^{\dagger}|\equiv\left(\begin{array}[]{cc}|\mathbb{y}_{11}|&|\mathbb{y}_{12}|\\\
|\mathbb{y}_{12}|&|\mathbb{y}_{22}|\end{array}\right)\,\,\,\,\,\Longrightarrow
U=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right)\,.$ (15)

The off-diagonal elements of $|\mathcal{Y}\mathcal{Y}^{\dagger}|$ are
equivalent due to Hermiticity, and one recognizes that this object is now a
symmetric, real matrix, which is trivially diagonalized by $U$, leading to
Yukawa eigenvalues and a mixing angle that can be expressed as

$y_{i,j}^{2}=\frac{1}{2}\left(\mathbb{y}_{11}+\mathbb{y}_{22}\mp\sqrt{\mathbb{y}_{11}^{2}+4\mathbb{y}_{12}^{2}-2\mathbb{y}_{11}\mathbb{y}_{22}+\mathbb{y}_{22}^{2}}\right)\,,\,\,\,\,\,\,\,\,\,\,\,t_{2\theta}=\frac{2\,|\mathbb{y}_{12}|}{\left(|\mathbb{y}_{22}|-|\mathbb{y}_{11}|\right)}\,.$
(16)

As $\theta$ lives in the first quadrant by definition, $\sin\theta\geq 0$,
$\cos\theta\geq 0$, and hence $\tan\theta\geq 0$. Also, while we have derived
(16) in the arbitrary weak-interaction basis such that it holds in both the up
and down-sectors, one can rotate without loss of generality to a flavour basis
where either $Y_{d}$ or $Y_{u}$ is diagonal, respectively leading to either
$U_{d}=\mathbb{1}$ or $U_{u}=\mathbb{1}$. In this case
$\theta\rightarrow\theta_{C}$, where $\theta_{C}$ is the renowned (physical)
Cabibbo angle [17] describing the dominant CKM mixing between first and second
quark generations. While (16) is valid at all-orders in the geoSMEFT expansion
in $\overline{v}_{T}/\Lambda$, there is clearly a basis dependence to the
expressions which is undesirable when comparing with experiment. Furthermore,
when three fermion generations are present, obtaining analogous expressions
for $y_{i}^{2}$, $\theta_{ij}$, and $\delta$ becomes intractable if approached
with standard diagonalization techniques as above, regardless of concerns over
the basis-dependence of the results.

###  3.1 Approach with Flavour Invariants

To circumvent these issues we will instead employ the (rephasing and weak-
basis) invariant theory developed in [18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31]. While we leave the details of the invariant theory to these
prior references, we recall that flavour invariants are objects that do not
change under field redefinitions corresponding to unitary transformations of
SM fermions under the global $\mathcal{G}_{F}\sim U(3)^{5}$ flavour symmetry.
Such transformations correspond to basis changes of the Yukawa
connections,444We focus only on the quark sector from this point forward.
Additional discussion regarding extensions to leptons will be given in Section
5.

$Y^{u}\,\longrightarrow\,U^{\dagger}_{Q_{L}}\,Y^{u}\,U_{u_{R}}\,,\,\,\,\,\,\,\,\,\,\,Y^{d}\,\longrightarrow\,U^{\dagger}_{Q_{L}}\,Y^{d}\,U_{d_{R}}\,,$
(17)

which, as argued above, are the fundamental objects encoding the mass and
mixing parameters of the theory. Polynomials of $Y^{\psi}$ define a group ring
$\mathbb{C}$ when the linear combinations of all possible products of the
generators $Y^{\psi}$ (with complex coefficients) are formed. The
$\mathcal{G}_{F}$-invariant ring $\mathbb{C}^{\mathcal{G}_{F}}$ is that set of
polynomials unchanged under the action $\mathcal{G}_{F}$ (cf. (17)), and it
can be shown that $\mathbb{C}^{\mathcal{G}_{F}}$ is finitely generated. A
central result of [19] is an explicit representation of said generators,
denoted $I_{i}$ (and henceforth referred to simply as ‘invariants’), for the
quark sector of the SM with both two and three fermion generations. Here we
extend this analysis to the geoSMEFT, and further use the invariants to
extract our desired formulae.

In the case of three fermion generations, Hilbert Series techniques lead one
to conclude that there are 11 polynomially independent invariants $I_{i}$
[19]. In what follows we will actually use the representation and notation
given in [20] to present $I_{i}$. Here, after defining the Hermitian
combination $YY^{\dagger}\equiv\mathbb{Y}$, a complete set of 3D quark-sector
invariants are reported as

$\displaystyle I_{1}$
$\displaystyle\equiv\text{tr}\left(\mathbb{Y}_{u}\right)\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hat{I}_{3}\equiv\text{tr}\left(\text{adj}\,\mathbb{Y}_{u}\right)\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hat{I}_{6}\equiv\text{tr}\left(\mathbb{Y}_{u}\,\text{adj}\,\mathbb{Y}_{u}\right)=3\,\text{det}\,\mathbb{Y}_{u}\,,$
$\displaystyle I_{2}$
$\displaystyle\equiv\text{tr}\left(\mathbb{Y}_{d}\right)\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hat{I}_{4}\equiv\text{tr}\left(\text{adj}\,\mathbb{Y}_{d}\right)\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hat{I}_{8}\equiv\text{tr}\left(\mathbb{Y}_{d}\,\text{adj}\,\mathbb{Y}_{d}\right)=3\,\text{det}\,\mathbb{Y}_{d}\,,$
(18)

for invariants ‘unmixed’ between up and down sectors, and

$\displaystyle\hat{I}_{5}$
$\displaystyle\equiv\text{tr}\left(\mathbb{Y}_{u}\,\mathbb{Y}_{d}\right),\,\,\,\,\,\,\,\,\hat{I}_{7}\equiv\text{tr}\left(\text{adj}\,\mathbb{Y}_{u}\,\mathbb{Y}_{d}\right),\,\,\,\,\,\,\,\,\,\hat{I}_{9}$
$\displaystyle\equiv\text{tr}\left(\mathbb{Y}_{u}\,\text{adj}\,\mathbb{Y}_{d}\right),\,\,\,\,\,\,\,\,\,\hat{I}_{10}\equiv\text{tr}\left(\text{adj}\,\mathbb{Y}_{u}\,\text{adj}\,\mathbb{Y}_{d}\right)\,,$
(19)

for real, mixed invariants. Note that it is natural to consider
$\mathbb{Y}_{u}$ and $\mathbb{Y}_{d}$ when considering flavour symmetry
violating effects as these two invariants are not simultaneously
diagonalizable. This is why in meson mixing, for example, flavour violation is
always proportional to such invariants. One notes that (18) contains all of
the information required to extract the Yukawa eigenvalues. On the other hand,
(19) know about the overlap between up and down sectors, and are therefore
functions of the mixing elements $U^{u,d}_{ij}$. As there are four invariants,
this is sufficient to extract the four independent elements of the (unitary)
CKM mixing matrix. The expressions in (18)-(19) are all CP-even and $\geq 0$.
Additionally there is the complex, CP-Odd invariant

$I_{11}^{-}=-\frac{3i}{8}\text{det}\left[\mathbb{Y}_{u},\mathbb{Y}_{d}\right]\,,$
(20)

which is proportional to the Jarlskog determinant [23]. This invariant will
therefore be necessary for completely determining the sign of the Dirac CP-
violating phase below.

Note that, in the above expressions, $\text{adj}\,\mathbb{Y}$ are the adjoint
matrices555These matrices are adjoint under the $SU(3)_{Q_{L}}$ quark flavour
subgroup of the global $U(3)^{5}$ SM flavour symmetry, i.e. $U(3)^{5}\supset
U(3)^{3}/U(1)_{B}\sim SU(3)_{Q_{L}}\times U(3)_{u_{R}}\times U(3)_{d_{R}}$,
with $U(1)_{B}$ global baryon number. satisfying
$\mathbb{Y}\,\text{adj}\,\mathbb{Y}=\text{det}\,\mathbb{Y}$, i.e.

$\displaystyle\text{adj}\,\mathbb{Y}$
$\displaystyle=\mathbb{Y}^{2}-\text{tr}\left[\mathbb{Y}\right]\mathbb{Y}+\frac{1}{2}\left(\text{tr}^{2}\left[\mathbb{Y}\right]-\text{tr}\left[\mathbb{Y}^{2}\right]\right)\mathbb{1}\,,$
(21)

which holds for matrices that are diagonalized via unitary field
transformations. Generalized expressions for adjoint matrices diagonalized by
bi-unitary transformations can also be found. Both are derived directly from
the Cayley-Hamilton identity for $3\times 3$ matrices [20].

#### Invariants

A core observation of this work is that the structure of the flavour
invariants is the same regardless of the mass dimension considered in the
(geo)SM(EFT). This can be seen clearly in (8), where one notes that the
addition of higher-order SMEFT operators simply results in an all-orders
reparameterization of each individual matrix element of the Hermitian objects
$\mathbb{Y}$, which compose the polynomial invariants. Hence we can freely
form (18)-(20) with explicit dependence on the geoSMEFT’s Hermitian Yukawa
coupling,

$\mathbb{Y}_{rp}=\frac{\mathbb{h}}{2}\,\left(Y_{ri}Y^{\star}_{pi}-\sum_{n^{\prime}}^{\infty}f(n^{\prime})Y_{ri}\tilde{C}_{ip}^{(2n^{\prime})}-\sum_{n}^{\infty}f(n)\tilde{C}_{ir}^{(2n),\star}Y^{\star}_{pi}+\sum_{n,n^{\prime}}^{\infty}f(n)f(n^{\prime})\tilde{C}^{(2n),\star}_{ir}\tilde{C}_{ip}^{(2n^{\prime})}\right),$
(22)

where $\mathbb{h}\equiv(\sqrt{h}^{44})(\sqrt{h}^{44})^{\star}$,
$f(n)=\frac{2n-3}{2^{n-2}}$, and analogously for $f(n^{\prime})$. We observe
that (22) represents the fundamental BSM object in this formalism.

#### Quark Masses

Proceeding to the extraction of the quark Yukawa eigenvalues $y^{2}$, we solve
the system of equations in (18), finding

$\displaystyle y^{2}_{i}$
$\displaystyle=\frac{\left(-2\right)^{1/3}}{3\,\psi_{u}}\left(I_{1}^{2}-3\,\hat{I}_{3}+\left(-2\right)^{-1/3}\,I_{1}\,\psi_{u}+\left(-2\right)^{-2/3}\,\psi_{u}^{2}\right)\,,$
(23) $\displaystyle y^{2}_{j,k}$
$\displaystyle=\frac{1}{12\psi_{u}}(\left(-2\right)^{4/3}\,I_{1}^{2}-3\cdot\left(-2\right)^{4/3}\,\hat{I}_{3}+4\,I_{1}\,\psi_{u}$
$\displaystyle\mp\psi_{u}\,\sqrt{24\left(I_{1}^{2}-3\,\hat{I}_{3}\right)+\frac{6\cdot\left(-2\right)^{5/3}\left(I_{1}^{2}-3\,\hat{I}_{3}\right)^{2}}{\psi_{u}^{2}}-3\cdot\left(-2\right)^{4/3}\psi_{u}^{2}}+\left(-2\right)^{2/3}\psi_{u}^{2})$
(24)

for the up-quark Yukawa eigenvalues, where the $\psi_{u}$ parameter is given
by

$\psi_{u}=\left(-2\,I_{1}^{3}+9\,I_{1}\hat{I}_{3}-9\,\hat{I}_{6}+3\,\sqrt{-3\,I_{1}^{2}\hat{I}_{3}^{2}+12\,\hat{I}_{3}^{3}+4\,I_{1}^{3}\hat{I}_{6}-18\,I_{1}\hat{I}_{3}\hat{I}_{6}+9\,\hat{I}_{6}^{2}}\right)^{1/3}\,.$
(25)

Recall that by definition the functions $y^{2}$ and $I_{1,3,6}$ are $\geq 0$,
and of course $y^{2}$ is also real. While certain individual expressions on
the RHS of these equations are imaginary, we have checked that the eigenvalues
are in fact real. Note also that the system of equations in (18) permits six
different solutions, corresponding to the six different possible mass
hierarchies for $y_{u,c,t}$. The unmixed mass invariants remain constant under
the transposition of any two flavour-eigenstate indices, e.g.
$I_{1}=I_{1}(i\leftrightarrow j)$. However, by definition, one recognizes the
up quark as the lightest generation, the top as the heaviest, and the charm as
the intermediate,

$y_{u}^{2}\equiv\text{min}\\{y_{i}^{2},y_{j}^{2},y_{k}^{2}\\}\,,\,\,\,\,\,\,\,\,\,\,y_{c}^{2}\equiv\text{mid}\\{y_{i}^{2},y_{j}^{2},y_{k}^{2}\\}\,\,\,\,\,\,\,\,\,\,\,y_{t}^{2}\equiv\text{max}\\{y_{i}^{2},y_{j}^{2},y_{k}^{2}\\}\,.$
(26)

Furthermore, the expressions for the down-quark mass-eigenvalues are analogous
to (23)-(24), replacing

$\\{I_{1},\hat{I}_{3},\hat{I}_{6}\\}\longrightarrow\\{I_{2},\hat{I}_{4},\hat{I}_{8}\\}$
(27)

which, when implemented, results also in the notation change
$\psi_{u}\longrightarrow\psi_{d}$ in the final equations. The beauty of
(23)-(24) is that their RHS can be calculated in any arbitrary flavour basis,
whilst the LHS are always the mass eigenvalues (up to the Higgs vev).

#### CKM Parameters

Given the Yukawa/mass eigenvalues at all orders, one can then move to the four
real CKM parameters, which can be obtained by solving the system of equations
implied by $\hat{I}_{5,7,9,10,11}$. We will solve (19) for $|V_{cd}|^{2}$,
$|V_{cs}|^{2}$, $|V_{td}|^{2}$, and $|V_{ts}|^{2}$, and then use the unitarity
constraints of the CKM matrix to uniquely determine its remaining matrix
elements in terms of the invariants $\hat{I}_{5,7,9,10,11}$. We then have all
of the information required to determine the three mixing angles $s_{ij}$,
which upon using the final $I^{-}_{11}$ invariant will give us the phase
$\delta$. Critically, we observe that (19) does not permit multiple solutions
for $|V_{ij}|^{2}$.

Proceeding along these lines, it is straightforward to derive the following
compact expressions for the mixing angles $s_{ij}$:

$\displaystyle s_{13}$
$\displaystyle=\left[\frac{-\hat{I}_{10}-y_{b}^{2}\,\left(\hat{I}_{7}-\Delta_{ds}^{+}\Delta_{uc}^{+}\Delta_{ut}^{+}\right)-y_{u}^{2}\left(\hat{I}_{9}+y_{b}^{2}\left(\hat{I}_{5}-y_{b}^{2}\Delta_{ct}^{+}\right)-y_{d}^{2}\,y_{s}^{2}\,\Delta_{ct}^{+}\right)}{\Delta_{bd}^{-}\Delta_{bs}^{-}\Delta_{cu}^{-}\Delta_{ut}^{-}}\right]^{1/2}\,,$
(28) $\displaystyle s_{23}$
$\displaystyle=\left[\frac{\Delta_{tu}^{-}\left(-\hat{I}_{10}+y_{c}^{2}\left(-\hat{I}_{9}+\left(y_{b}^{4}+y_{d}^{2}y_{s}^{2}\right)\Delta_{ut}^{+}\right)+y_{b}^{2}\left(-\hat{I}_{7}+y_{c}^{2}\left(-\hat{I}_{5}+\Delta_{ct}^{+}\Delta_{ds}^{+}\right)+y_{u}^{2}\Delta_{ct}^{+}\Delta_{ds}^{+}\right)\right)}{\Delta_{ct}^{-}\left(\hat{I}_{10}+y_{u}^{2}\,\hat{I}_{9}+y_{b}^{2}\left(\hat{I}_{7}+y_{u}^{2}\left(\hat{I}_{5}-2\Delta_{ct}^{+}\Delta_{ds}^{+}\right)\right)-\left(y_{u}^{4}+y_{c}^{2}y_{t}^{2}\right)\left(y_{b}^{4}+y_{d}^{2}y_{s}^{2}\right)\right)}\right]^{1/2}$
(29) $\displaystyle s_{12}$
$\displaystyle=\left[\frac{\Delta_{db}^{-}\left(\hat{I}_{10}+y_{s}^{2}\left(\hat{I}_{7}-y_{c}^{2}y_{t}^{2}\Delta_{db}^{+}\right)\right)+y_{u}^{2}\,\Delta_{bd}^{-}\left(-\hat{I}_{9}-y_{s}^{2}\,\hat{I}_{5}+\Delta_{sb}^{+}\Delta_{ct}^{+}\Delta_{ds}^{+}\right)+y_{u}^{4}y_{s}^{2}\left(y_{b}^{4}-y_{d}^{4}\right)}{\Delta_{ds}^{-}\left(\hat{I}_{10}+y_{u}^{2}\,\hat{I}_{9}+y_{b}^{2}\left(\hat{I}_{7}+y_{u}^{2}\left(\hat{I}_{5}-2\Delta_{ct}^{+}\Delta_{ds}^{+}\right)\right)-\left(y_{b}^{4}+y_{d}^{2}y_{s}^{2}\right)\left(y_{u}^{4}+y_{c}^{2}y_{t}^{2}\right)\right)}\right]^{1/2}\,,$
(30)

which are given in terms of the Yukawa eigenvalues $y_{i}^{2}$ and the
difference/sum parameters $\Delta^{\pm}_{ij}$ defined by

$\Delta^{\pm}_{ij}\equiv y_{i}^{2}\pm y_{j}^{2}.$ (31)

Given the expressions in (23)-(24) for $y_{i}^{2}$, one then has the desired
final expressions entirely in terms of the flavour-invariants ${I}_{i}$. Note
that, by definition, these angles lie in the first quadrant, $0\leq
s_{ij}\leq\frac{\pi}{2}$. Finally, one can use $I^{-}_{11}$, along with
(28)-(30), to derive that

$\displaystyle s_{\delta}$
$\displaystyle=\frac{4}{3}\,I_{11}^{-}\,\left[\,\Delta_{tc}^{-}\Delta_{tu}^{-}\Delta_{cu}^{-}\Delta_{bs}^{-}\Delta_{bd}^{-}\Delta_{sd}^{-}\,s_{12}s_{13}s_{23}\left(1-s_{23}^{2}\right)^{1/2}\left(1-s_{12}^{2}\right)^{1/2}\left(1-s_{13}^{2}\right)\right]^{-1}\,.$
(32)

As $I^{-}_{11}$ is not guaranteed to be $\geq 0$, one sees that (32) uniquely
pins down the sign of $\delta$. These expressions therefore complete the list
of quark flavour parameters in the (geo)SM(EFT); (23)-(32) and their
renormalization group flow (cf. (43)-(47) in Section 4) represent the core
results of this work.

###  3.2 Numerical Checks

In order to probe the reliability and accuracy of (23)-(32) we performed a
series of numerical tests. Specifically we have built an automated script
that666Note that we do this computation in the arbitrary flavour / weak-
eigenstate basis where all matrix elements are a priori non-zero and complex.
To obtain our numerical results we use Mathematica’s built-in Eigenvalues and
Eigenvectors functional calls, and at no point in this numerical procedure was
any information about the flavour invariants $I_{i}$ given to the extraction
code.

  1. 1.

assembles $\mathcal{Y}^{\psi}_{rp}$ by randomly drawing up a list of $n-1$,
3$\times$3 arrays of complex numbers and assigning these arrays to
$Y^{\psi}_{rp}$ and $\tilde{C}^{(2n)}_{\psi H}$, thereby forming the effective
Yukawa coupling in (8). The Hermitian combination
$\left[\mathcal{Y}\mathcal{Y}^{\dagger}\right]_{rp}$ is then constructed.

  2. 2.

computes the invariants using (18)-(19), the Yukawa eigenvalues from
(23)-(24), the mixing angles from (28)-(30), and finally the Dirac phase from
(32).

  3. 3.

simultaneously computes the eigenvalues of the Hermitian coupling obtained in
(1) numerically.

  4. 4.

simultaneously computes the diagonalizing matrices $U_{(u,d)}$ numerically and
extracts the four physical parameters of the CKM matrix. This is achieved by

    1. (a)

computing the eigenvectors of
$\left[\mathcal{Y}^{(u,d)}\mathcal{Y}^{(u,d),\dagger}\right]$. These are
normalized to unit vectors $\mathbb{v}_{i}$ and then the numerical matrices
are defined by
$U_{(u,d)}\equiv\left(\mathbb{v}_{1}^{(u,d),T},\mathbb{v}_{2}^{(u,d),T},\mathbb{v}_{3}^{(u,d),T}\right)$.

    2. (b)

computing the CKM matrix as $V_{CKM}=U_{u}^{\dagger}\cdot U_{d}$.

    3. (c)

uniquely extracting the $s_{13}$ mixing angle from $V_{13}$,
$s_{13}=|V_{13}|$.

    4. (d)

uniquely extracting the $s_{23}$ mixing angle from $V_{23}$,
$s_{23}=|V_{23}|/\sqrt{1-s_{13}^{2}}$.

    5. (e)

uniquely extracting the $s_{12}$ mixing angle from $V_{12}$,
$s_{12}=|V_{12}|/\sqrt{1-s_{13}^{2}}$.

    6. (f)

uniquely extracting the $s_{\delta}$ phase in a phase-convention-independent
manner from the Jarlskog invariant $J$.

  5. 5.

subtracts the values obtained in (4) from those in (2). Their differences are
added to a list, and the process is repeated, at various $n$.

The result is that, in no draw and with no parameter, and at arbitrary mass-
dimension $n$ and new physics scales $\Lambda$,777We have run scans up to
$n=10$ and allowed for new physics between 2-10 TeV. are the differences
between the values computed with (23)-(32) in step (2) and those computed with
the numerical procedure between steps (3)-(4) greater than the numerical
tolerance we allow for ($10^{10}$ in this case). As is clear, these checks
therefore validate the expressions in (23)-(32) in a highly non-trivial
manner.

###  3.3 Applicability to other BSM Scenarios

Given (23)-(32), it is interesting to note when they do (and do not) apply.
Indeed, the formalism is in general complete when (18)-(20) are sufficient to
extract all of the flavour parameters of the low-energy theory. As described
above, these are determined by analyzing the global flavour group ring
$\mathbb{C}^{\mathcal{G}_{F}}$, based on the transformation properties of
$Y^{\psi}$ in (17). Hence (23)-(32) hold in any theory where (17) represents
the generic transformation properties of the Dirac Yukawa/mass matrices under
$\mathcal{G}_{F}$, such that $Y^{\psi}Y^{\psi\,\dagger}\rightarrow
U^{\dagger}\,Y^{\psi}Y^{\psi\,\dagger}\,U$ under global flavour
transformations $U\in U(3)_{Q_{L}}$, including theories where $Y^{\psi}$ or
$M^{\psi}$ originate from ultraviolet dynamics [19]. Of course, if
$Y^{\psi}Y^{\psi\,\dagger}$ respects these transformation properties, but
there are additional sources of flavour violation in the infrared spectrum,
(18)-(20) will be incomplete as there are just enough invariants to uniquely
determine the 10 (13) flavour parameters of the quark (quark $+$ charged
lepton) Yukawa sector. More invariants would be required to extract the values
of parameters associated to the additional flavour violation.

For academic purposes we also consider the case when global flavour rotations
are not family-universal, i.e. when

$\\{\mathbb{Y}_{u},\mathbb{Y}_{d}\\}\rightarrow\\{\mathbb{Y}_{u}^{\prime}\,,\,\mathbb{Y}_{d}^{\prime}\\}=\\{U^{u\,\dagger}_{\chi}\,\mathbb{Y}_{u}\,U^{u}_{\chi},U^{d\,\dagger}_{\chi}\,\mathbb{Y}_{d}\,U^{d}_{\chi}\\}\,\,\,\,\,\text{with}\,\,\,\,U^{d}_{\chi}\neq
U^{u}_{\chi}\,.$ (33)

Such a situation may be conceivable in the broken electroweak phase, e.g. Here
we observe that (18) (which we denote $\hat{I}_{\text{mass}}$), and therefore
(23)-(24), still hold as basis-independent expressions, since
$\hat{I}_{\text{mass}}\rightarrow\hat{I}^{\prime}_{\text{mass}}=\hat{I}_{\text{mass}}$
due to the cyclic property of the Trace. However, (19)-(20) do not exhibit
this invariance generically. This is clear, for example, in $\hat{I}_{5}$:

$\hat{I}^{\prime}_{5}=\text{tr}\left(\mathbb{Y}_{u}^{\prime}\mathbb{Y}_{d}^{\prime}\right)=\text{tr}\left(U_{\chi}^{u\,\dagger}\,\mathbb{Y}_{u}\,U_{\chi}^{u}U_{\chi}^{d\,\dagger}\,\mathbb{Y}_{d}\,U_{\chi}^{d}\right)\neq\hat{I}_{5}\,.$
(34)

This implies that predictions for CKM mixing parameters in such theories will
not hold. We have checked that these statements are true by generating random
unitary transformations $U_{\chi}^{\psi}$ numerically and constructing (33),
where $y_{i}^{2}$, $\theta_{ij}$ and $\delta$ are also generated randomly. We
compare the values computed with (23)-(32) to those extracted numerically, and
indeed confirm that while the Yukawa/mass eigenvalues are in perfect agreement
with the randomly drawn inputs, the mixing parameters are not. On the other
hand, we now give two examples from the literature where our formulae apply
exactly.

#### An effective theory of flavour in the ultraviolet

Our formalism can be used to analytically or numerically compute the mass,
mixing, and CP-violating parameters predicted from typical high-energy EFTs of
flavour (e.g. Froggatt-Nielsen models [32]), where some ultraviolet dynamics
at a scale $\Lambda_{F}$ breaks a flavour symmetry $\mathcal{G}_{F}$, yielding
specifically textured infrared Yukawa couplings. As an example we consider the
quark sector of the non-Abelian ‘universal texture zero’ (UTZ) model of [33].
Here the high-energy Lagrangian is given as an OPE of non-renormalizable
interactions between SM fields and heavy scalars $\\{\theta_{i},\Sigma,S\\}$,

$\mathcal{L}_{\text{UTZ}}\supset\psi_{p}\left({1\over
M_{3,f}^{2}}\theta_{3}^{p}\theta_{3}^{r}+{1\over
M_{23,f}^{3}}\theta_{23}^{p}\theta_{23}^{r}\Sigma+{1\over
M_{123,f}^{3}}(\theta_{123}^{p}\theta_{23}^{r}+\theta_{23}^{p}\theta_{123}^{r})S\right)\psi_{r}^{c}H+\mathcal{O}(1/M^{4})+\text{...}$
(35)

where the labels indicate a specific family sector, $f\in\\{u,d,e,\nu\\}$, and
whose terms are invariant under the SM gauge symmetries and
$\mathcal{G}_{F}\simeq\Delta\left(27\right)$ due to the triplet (anti-triplet)
$\Delta\left(27\right)$ charge assignments of the SM multiplets
(flavons).888The Higgs field is a trivial flavour singlet, as are the BSM
$\Sigma$ and $S$ fields, with the latter helpful for shaping the couplings and
the former associated to Grand Unified symmetry breaking. See [33] for further
details of the model’s implementation, including its scalar potential. Upon
$\\{\theta\\}$ developing vevs in specific directions of flavour-space (and
with specific scales $\\{v\\}$), the UTZ Lagrangian in (35) populates Dirac
mass matrices of the form

$\mathcal{M}^{D}_{f}=\left(\begin{array}[]{ccc}0&a\,e^{i\gamma}&a\,e^{i\gamma}\\\
a\,e^{i\gamma}&\left(b\,e^{-i\gamma}+2a\,e^{-i\delta}\right)e^{i(\gamma+\delta)}&b\,e^{i\delta}\\\
a\,e^{i\gamma}&b\,e^{i\delta}&1-2a\,e^{i\gamma}+b\,e^{i\delta}\end{array}\right)_{f}\,,$
(36)

for all Dirac fermions ($u$, $d$, …), where the free parameters
$\\{a,b,\gamma,\delta\\}_{f}$ are associated to ultraviolet dynamics, and
where we have removed two unphysical phases according to the discussion in
[34]. This object is then the only information required to form the invariants
in (18)-(19), and therefore also the mass and mixing formulae in (23)-(32),
which yield the physical flavour parameters. We focus on the quark sector of
the model, and input the best-fit values for the unconstrained ultraviolet
inputs $\\{a,b,\gamma,\delta\\}_{u,d}$ from [33] and compute, finding

$\frac{m_{u}}{m_{t}}=7.16\cdot
10^{-6}\,,\,\,\,\,\,\,\,\,\,\,\frac{m_{c}}{m_{t}}=0.0027\,,\,\,\,\,\,\,\,\,\,\,\frac{m_{d}}{m_{b}}=0.00090\,,\,\,\,\,\,\,\,\,\,\,\frac{m_{s}}{m_{b}}=0.020\,,$
(37)

for the mass eigenvalues (note that the Higgs vev is already encoded in the
numerical values for the ultraviolet parameters) and

$s_{12}=0.226,\,\,\,\,\,\,s_{23}=0.0191,\,\,\,\,\,s_{13}=0.0042,\,\,\,\,\,s_{\delta}=0.5609,$
(38)

for the mixing angles and Dirac phase, in perfect agreement with the numerical
values extracted in [33], up to $\mathcal{O}(1/M_{f}^{4})$. Extending the
analysis to higher-order operators in (35) is straightforward and no more
complicated.

#### On models with light leptoquarks

As examples of theories with non-minimal flavour violation in the infrared, we
consider models incorporating leptoquarks, exotic scalars that couple SM
SU(2)L doublet quark and lepton fields, e.g.

$\mathcal{L}\supset
y_{pr}^{LL}\,\overline{Q}_{L,p}^{C}\,\Theta_{3}\,L_{L,r}+z_{pr}^{LL}\,\overline{Q}_{L,p}^{C}\,\Theta_{3}^{\dagger}\,Q_{L,r}+\text{h.c.}\,,$
(39)

where we have given illustrative operators for a scalar triplet leptoquark
$\Theta_{3}$ which transforms as
$\Theta_{3}\sim\left({\bf{\bar{3}}},{\bf{3}},1/3\right)$ under
$\mathcal{G}_{SM}$, and where all gauge/internal index contractions are
implied.999For a somewhat comprehensive review of leptoquark models and basic
phenomenology, see [35]. Here the couplings $y^{LL}$ and $z^{LL}$ are also
$3\times 3$ matrices in flavour space, and are in general not simultaneously
diagonalized along with $Y^{\psi}$. There will therefore be additional
(physical) mixing parameters associated to the leptoquark couplings that
(23)-(32) cannot account for. Regardless, as discussed above, (23)-(32) can
still predict the Dirac parameters $y_{i}^{2}$, $\theta_{ij}$, and $\delta$.

To demonstrate this we focus on the toy leptoquark models developed in [36,
37]. Here the generic representation for the Yukawa matrices in the quark
sector can be written as

$Y_{u}^{\prime\prime}=P^{\dagger}\,\Lambda_{d}\,V_{CKM}^{\dagger}\cdot\left(\begin{array}[]{ccc}y_{u}&0&0\\\
0&y_{c}&0\\\
0&0&y_{t}\end{array}\right)\cdot\Lambda_{U}^{\dagger}\,P,\,\,\,\,\,\,\,\,Y_{d}^{\prime\prime}=P^{\dagger}\,\Lambda_{d}\,\cdot\left(\begin{array}[]{ccc}y_{d}&0&0\\\
0&y_{s}&0\\\ 0&0&y_{b}\end{array}\right)\cdot\Lambda_{D}^{\dagger}\,P\,,$ (40)

where the double prime ($\prime\prime$) notation simply indicates that, as is
clear, these matrices are in a special flavour basis that is neither the
traditional weak-eigenstate nor fermion mass-eigenstate basis. That is,
information about the structure of $y^{LL}$ in this model is already encoded
in (40) ($z^{LL}$ was not considered), through the introduction of the
rotations $\Lambda_{d,U,D}$ ($P$ is associated to the representation theory of
various flavour groups — see [37] for details).

Hence (40) are ideal candidates to test our conclusions from above, and to do
so we consider one of the explicit models in [37], based on a $D_{15}$
dihedral flavour symmetry. Without presenting the details of its ultraviolet
Lagrangian, we simply note that this model yields predictions for special
Yukawa textures,

$Y^{\prime\prime}_{u}=v_{u}\,\left(\begin{array}[]{ccc}a_{u}/v_{u}&0&0\\\
0&b_{u}\cos\frac{\pi}{15}&-c_{u}\sin\frac{\pi}{15}\\\
0&b_{u}\sin\frac{\pi}{15}&c_{u}\cos\frac{\pi}{15}\end{array}\right)\,,\,\,\,\,\,\,\,Y^{\prime\prime}_{d}=v_{d}\,\left(\begin{array}[]{ccc}a_{d}/v_{d}&0&0\\\
0&b_{d}&0\\\ 0&0&c_{d}\end{array}\right)\,,$ (41)

where $v_{u,d}$ are vevs associated to the breaking of $D_{15}$, and
$\\{a,b,c\\}_{u,d}$ are free ultraviolet ($D_{15}$-symmetric) Lagrangian
parameters that can be directly related to the mass eigenvalues of the theory:
$a_{u}^{2}\propto y_{t}^{2}$, $b_{u}^{2}\propto y_{u}^{2}$, $c_{u}^{2}\propto
y_{c}^{2}$, and analogously for the down sector. Plugging (41) into (23)-(32),
we obtain predictions for the Yukawa eigenvalues and CKM mixing parameters
that are equivalent to those from [37] (at particular numerical scan points)
when appropriate mass hierarchies are realized, as expected.

As further confirmation, we have done a more exhaustive scan of (40) without
any information associated to the special $D_{15}$ encoded — $y_{i}^{2}$,
$\theta_{ij}$, and $\delta$ are drawn randomly, as is the lone parametric
degree of freedom in $\Lambda_{d}$ (we have assumed a special texture for the
leptoquark coupling $y^{LL}$ in the fermion mass-eigenstate basis). We again
find exact agreement between the predictions of (23)-(32) and those values
extracted numerically in the scans.

##  4 Renormalization Group Flow

The formulae presented in Section 3 have a number of foreseeable applications
and benefits. As a particularly powerful example, we now use (23)-(32) to
perform an exact computation of the renormalization group flow (RGE) of Dirac
flavour parameters in terms of invariants, thereby allowing these parameters
to be computed at any scale in which the formulae in Section 3 hold.

We begin from the top, and find the $\psi_{u}$ parameters’ running is given by

$\displaystyle\psi_{u}^{2}\,\dot{\psi}_{u}$
$\displaystyle=-2I_{1}^{2}\dot{I}_{1}+3(\dot{I}_{1}\hat{I}_{3}+I_{1}\dot{\hat{I}}_{3})-3\dot{\hat{I}}_{6}$
$\displaystyle+\frac{3\,\left(36\hat{I}_{3}^{2}\dot{\hat{I}}_{3}+18\hat{I}_{6}\dot{\hat{I}}_{6}+12I_{1}^{2}\dot{I}_{1}\hat{I}_{6}+4I_{1}^{3}\dot{\hat{I}}_{6}-6I_{1}\dot{I}_{1}\hat{I}_{3}^{2}-6I_{1}^{2}\hat{I}_{3}\dot{\hat{I}}_{3}-18\left(\dot{I}_{1}\hat{I}_{3}\hat{I}_{6}+I_{1}\dot{\hat{I}}_{3}\hat{I}_{6}+I_{1}\hat{I}_{3}\dot{\hat{I}}_{6}\right)\right)}{2\,\left(\psi_{u}^{3}+2I_{1}^{3}-9I_{1}\hat{I}_{3}+9\hat{I}_{6}\right)}\,,$
(42)

and analogously for $\psi_{d}$ using (27). One notes that (42) is given
entirely in terms of the RGE of the flavour invariants themselves. Here
$\dot{x}\equiv\mu\,dx/d\mu$. Using (42), one can also derive the RGE for the
Yukawa eigenvalues, finding

$\displaystyle 3\,\psi_{u}\,\dot{(y_{i}^{2})}$
$\displaystyle=-3\,\dot{\psi}_{u}\,y_{i}^{2}+(-2)^{1/3}\left(2I_{1}\dot{I}_{1}-3\dot{\hat{I}}_{3}+(-2)^{-1/2}\left(\dot{I}_{1}\psi_{u}+I_{1}\dot{\psi}_{u}\right)+(-2)^{-2/3}2\psi_{u}\dot{\psi}_{u}\right)\,,$
$\displaystyle 12\,\psi_{u}\,\dot{(y_{j,k}^{2})}$
$\displaystyle=-12\,\dot{\psi}_{u}\,y_{j,k}^{2}+2(-2)^{4/3}I_{1}\dot{I}_{1}-3(-2)^{4/3}\dot{\hat{I}}_{3}+4(\dot{I}_{1}\psi_{u}+I_{1}\dot{\psi}_{u})+2(-2)^{2/3}\psi_{u}\dot{\psi}_{u}$
$\displaystyle\mp\frac{\dot{\psi}_{u}}{\psi_{u}}\,\left(24\,\psi_{u}^{2}\left(I_{1}^{2}-3\hat{I}_{3}\right)+6\left(-2\right)^{5/3}\left(I_{1}^{2}-3\,\hat{I}_{3}\right)^{2}-3\left(-2\right)^{4/3}\psi_{u}^{4}\right)^{1/2}$
$\displaystyle\mp\psi_{u}^{2}\frac{48\,I_{1}\dot{I}_{1}-72\,\dot{\hat{I}}_{3}-6(-2)^{4/3}\psi_{u}\dot{\psi}_{u}+12(-2)^{5/3}\left(\frac{(I_{1}^{2}-3\hat{I}_{3})(2I_{1}\dot{I}_{1}-3\dot{\hat{I}}_{3})}{\psi_{u}^{2}}-\frac{\dot{\psi}_{u}}{\psi_{u}^{3}}\left(I_{1}^{2}-3\hat{I}_{3}\right)^{2}\right)}{\left(96\,\psi_{u}^{2}\left(I_{1}^{2}-3\,\hat{I}_{3}\right)+24\left(-2\right)^{5/3}\left(I_{1}^{2}-3\,\hat{I}_{3}\right)^{2}-12\left(-2\right)^{4/3}\psi_{u}^{4}\right)^{1/2}}$
(43)

as well as the RGE for the mixing angles $s_{ij}$,

$\displaystyle\dot{s}_{13}$
$\displaystyle=\frac{1}{2}\left[-\frac{\mathcal{P}_{13}}{s_{13}\,\Delta_{bd}^{-}\Delta_{bs}^{-}\Delta_{cu}^{-}\Delta_{ut}^{-}}-s_{13}\sum_{(ij)\,\in\,\mathbb{s}_{1}}\frac{\dot{\Delta}^{-}_{ij}}{\Delta^{-}_{ij}}\right]\,,$
(44) $\displaystyle\dot{s}_{23}$
$\displaystyle=\frac{1}{2}\left[\frac{\left(\Delta^{-}_{tu}\mathcal{P}^{a}_{23}-s_{23}^{2}\,\Delta^{-}_{ct}\,\mathcal{P}^{b}_{23}\right)}{s_{23}\,\mathcal{D}_{23}}+s_{23}\left(\frac{\dot{\Delta}^{-}_{tu}}{\Delta^{-}_{tu}}-\frac{\dot{\Delta}^{-}_{ct}}{\Delta^{-}_{ct}}\right)\right]\,,$
(45) $\displaystyle\dot{s}_{12}$
$\displaystyle=\frac{1}{2}\left[\frac{\mathcal{P}^{a}_{12}}{s_{12}\mathcal{D}_{12}}-s_{12}\left(\frac{\dot{\Delta}^{-}_{ds}}{\Delta^{-}_{ds}}+\Delta^{-}_{ds}\,\frac{\mathcal{P}^{b}_{12}}{\mathcal{D}_{12}}\right)\right]\,\,,$
(46)

where, in the equation for $\dot{s}_{23}$, the index set $\mathbb{s}_{1}$ is
$\mathbb{s}_{1}=\\{(bs),(bd),(cu),(ut)\\}$, and the denominator functions
$\mathcal{D}_{ij}$ are given in (28)-(30), defining
$s_{ij}^{2}\equiv\mathcal{N}_{ij}/\mathcal{D}_{ij}$. The remaining product
functions $\mathcal{P}_{ij}$ appearing in (44)-(46) are tedious but
algebraically simple expressions, and we give them in Appendix A for
completeness. Finally, one can derive a compact expression for the RGE of the
Dirac phase $\delta$,

$\dot{s}_{\delta}=s_{\delta}\left[\frac{\dot{I}^{-}_{11}}{I_{11}^{-}}-\sum_{(ij)\,\in\,\mathbb{s}_{2}}\frac{\dot{\Delta}^{-}_{ij}}{\Delta^{-}_{ij}}-\dot{s}_{12}\frac{(1-2\,s_{12}^{2})}{s_{12}c_{12}^{2}}-\dot{s}_{23}\frac{(1-2\,s_{23}^{2})}{s_{23}c_{23}^{2}}-\dot{s}_{13}\frac{(1-3\,s_{13}^{2})}{s_{13}c_{13}^{2}}\right]\,,$
(47)

which depends on (46). Here the index set $\mathbb{s}_{2}$ is over the mass-
differences appearing on the RHS of (32),
$\mathbb{s}_{2}=\\{(tc),(tu),(cu),(bs),(bd),(sd)\\}$.

Figure 1: TOP LEFT: Running of the (normalized) unmixed invariants relevant
for up-quark Yukawa extraction at one-loop order. TOP RIGHT: The same for the
actual up-quark Yukawa eigenavlues (squared). Note that we have rescaled the
up and charm eigenvalues by $10^{9}$ and $10^{4}$ respectively, hence the
starred ($\star$) notation. BOTTOM LEFT: Running of the (normalized) unmixed
invariants relevant for down-quark Yukawa extraction at one-loop order. BOTTOM
RIGHT: The same for the actual down-quark Yukawa eigenavlues (squared). We
have again rescaled the eigenvalues for the purpose of plotting, multiplying
the bottom, strange, and down eigenvalues by $10^{3}$, $10^{5}$ and and
$10^{7}$ respectively.

Figure 2: LEFT: Running of the (normalized) mixed invariants relevant for
extracting the CKM mixing elements, at one-loop order. Note that $\bar{I}_{7}$
has been multiplied by 1.05 in order to visually distinguish it from
$\bar{I}_{5}$. RIGHT: Running of the mixing angles and Dirac CP-violating
phase of the CKM matrix, again at one-loop order

The RGE derived in (42)-(47) are again exact, and valid to all orders in the
$\overline{v}_{T}/\Lambda$ geoSMEFT expansion, and to all orders in the
perturbative loop expansion (which is implicitly embedded in $\dot{I}_{i}$).
They therefore open the door to any number of novel phenomenological
applications that depend on the radiative structure of flavour parameters in a
(geo)SMEFT context. Furthermore, one sees that, as must be the case given
(23)-(32), the running of all flavour parameters is given entirely by the
running of the flavour invariants $I$.

###  4.1 Computation of Quark Sector RGE at One-Loop Order

Since the RGE for each invariant was derived explicitly in [20] in the quark
sector (assuming MFV particle spectrum), it is now also straightforward to
compose the system of coupled differential equations, and obtain exact
(numerical) solutions for the running of quark mass eigenvalues, mixing
angles, and CP-violating phase at arbitrary mass-dimension orders in the
(geo)SM(EFT). We have done so using SM couplings, and show the results in
Figure 1 for the unmixed invariants and associated Yukawa eigenvalues. Note
that we have set the boundary conditions at $\mu=100$ GeV for these RGE using
the values reported for the mass and mixing parameters in the 2020 PDG
databook [16], and have neglected the electroweak correction (as in [20]) and
the running of the strong coupling for this illustration. For the invariants
we have normalized all of the functions to their values at $\mu=100$ GeV, i.e.
$\bar{I}_{i}=\hat{I}_{i}/\hat{I}_{i}(100)$ and similarly for the un-hatted
invariants in (18)-(19). Then, in Figure 2 we show the running of the mixed
variants relevant to extracting information about the CKM mixing matrix (left
panel), and then the CKM mixing elements themselves (right panel). Note that
while we have used the one-loop RGE in the above example calculation, two-loop
expressions for $\dot{I}_{i}$ are also provided in [20], and hence performing
the analogous computation of the flavour parameter evolution is no more
difficult.

##  5 Summary and Outlook

We have reported a set of compact, exact formulae for computing Dirac
Yukawa/mass eigenvalues and CKM mixing parameters directly from $3\times 3$
Yukawa couplings in arbitrary flavour bases. These formulae hold at all orders
in the $\overline{v}_{T}/\Lambda$ expansion characterizing the geoSMEFT, and
therefore complete the list of geometric expressions originally given in [6]
in the Dirac flavour sector. In addition to demonstrating their utility in
(e.g.) BSM model-building, we have also used our expressions to derive
analytic RGE for Dirac masses and mixings at all orders in
$\overline{v}_{T}/\Lambda$, and all perturbative loop orders. The practical
implementation of these latter formulae requires knowledge of the RGE of the
11 flavour invariants that exclusively compose them, which has been given at
one- and two-loop perturbative orders for quark flavour invariants in MFV
theories [20]. We use the results from [20] to perform an explicit one-loop
numerical computation within our formalism.

The results presented in this paper open the door to any number of interesting
physics applications, from aiding global SMEFT fits to experimental data in
the flavour sector (cf. [7]), to studying non-standard RGE-induced flavour
violation effects in the SMEFT (cf. [8]), to permitting exact predictions in
flavoured model building (cf. Section 3.3). In addition to these foreseeable
applications, we also note that an extension of our formalism to the lepton
sector would be desirable for describing (e.g.) neutrino mass and mixing to
all orders in $\overline{v}_{T}/\Lambda$ with the geoSMEFT. Of course, it is
clear that describing the mass sector of the charged leptons at all orders is
already achieved through (18), by replacing
$\mathbb{Y}_{u,d}\rightarrow\mathbb{Y}_{e}$. Furthermore, by $(a)$ adding to
the SM a neutrino Yukawa operator of the form $\mathcal{L}\supset
Y_{N}\,L\tilde{H}^{\dagger}\overline{N}$, with $N$ a singlet neutrino
transforming under $\mathcal{G}_{SM}$ as
$N\sim\left({\bf{1}},{\bf{1}},0\right)$, and $(b)$ forbidding lepton-number
violation, one can also use our formalism to describe leptonic mixing, for
renormalizable interactions only. To obtain an all-orders description in this
case one must first develop a geo$\nu$SMEFT. On the other hand, a formal
treatment of Majorana neutrinos is in principle achievable with the geoSMEFT
alone, if an all-orders field connection can be formalized for a tower of
operators whose lowest-dimension contribution is the dim-5 Weinberg operator
[38]. Then leptonic mixing will be described by six parameters, accounting for
the two additional Majorana phases. A set of 15 polynomially independent
invariants, as required in this case, has already been given in [19]. Upon
writing down the hypothetical ‘Weinberg connection,’ one must then find an
amenable basis for deriving relations analogous to (18)-(20). Furthermore, to
our knowledge, the complete set of RGE for the leptonic flavour invariants has
not yet been derived, as they have in [20] for the quark sector. We leave
these interesting extensions to future work.

## Acknowledgements

JT and MT are supported by the Villum Fund, project number 00010102. We thank
Andreas Helset for helpful insight early in the project, and JT acknowledges
interesting discussions with Fady Bishara, Tyler Corbett, Christophe Grojean,
Wolfgang G. Hollik, and Ayan Paul on related topics.

##  Appendix A Formulae for RGE Computation

In this appendix we report some of the functions employed in Section 4 for
completeness.

###  A.1 All-Orders Product Functions

In the derivation of the all-orders expressions for the mixing angles’ RGE,
cf. (44)-(46), product functions appear. They are given by

$\displaystyle\mathcal{P}_{13}$
$\displaystyle=y_{u}^{2}\left(\dot{\hat{I}}_{9}+\dot{(y_{b}^{2})}\left(\hat{I}_{5}-y_{b}^{2}\Delta_{ct}^{+}\right)+y_{b}^{2}\left(\dot{\hat{I}}_{5}-\dot{(y_{b}^{2})}\,\Delta^{+}_{ct}-y_{b}^{2}\dot{\Delta}^{+}_{ct}\right)-\dot{(y_{d}^{2})}\,y_{s}^{2}\Delta^{+}_{ct}-y_{d}^{2}\,\dot{(y_{s}^{2})}\,\Delta^{+}_{ct}-y_{d}^{2}y_{s}^{2}\dot{\Delta}^{+}_{ct}\right)$
$\displaystyle+\dot{\hat{I}}_{10}+\dot{(y_{b}^{2})}\left(\hat{I}_{7}-\Delta^{+}_{ds}\Delta^{+}_{uc}\Delta^{+}_{ut}\right)+\dot{(y_{u}^{2})}\left(\hat{I}_{9}+y_{b}^{2}\left(\hat{I}_{5}-y_{b}^{2}\Delta^{+}_{ct}\right)-y_{d}^{2}y_{s}^{2}\Delta^{+}_{ct}\right)$
$\displaystyle+y_{b}^{2}\left(\dot{\hat{I}}_{7}-\dot{\Delta}^{+}_{ds}\Delta^{+}_{uc}\Delta^{+}_{ut}-\Delta^{+}_{ds}\dot{\Delta}^{+}_{uc}\Delta^{+}_{ut}-\Delta^{+}_{ds}\Delta^{+}_{uc}\dot{\Delta}^{+}_{ut}\right)\,,$

for the (13) angle,

$\displaystyle\mathcal{P}_{23}^{a}$
$\displaystyle=y_{b}^{2}\left(\dot{(y_{c}^{2})}\left(\Delta_{ct}^{+}\Delta_{ds}^{+}-\hat{I}_{5}\right)+y_{c}^{2}\left(\dot{\Delta}_{ct}^{+}\Delta_{ds}^{+}+\Delta_{ct}^{+}\dot{\Delta}_{ds}^{+}-\dot{\hat{I}}_{5}\right)+\dot{(y_{u}^{2})}\Delta_{ct}^{+}\Delta_{ds}^{+}+y_{u}^{2}\dot{\Delta}_{ct}^{+}\Delta_{ds}^{+}+y_{u}^{2}\Delta_{ct}^{+}\dot{\Delta}_{ds}^{+}-\dot{\hat{I}}_{7}\right)$
$\displaystyle-\dot{\hat{I}}_{10}+\dot{(y_{c}^{2})}\left(\Delta_{ut}^{+}\left(y_{b}^{4}+y_{d}^{2}y_{s}^{2}\right)-\hat{I}_{9}\right)+\dot{(y_{b}^{2})}\left(y_{c}^{2}\left(\Delta_{ct}^{+}\Delta_{ds}^{+}-\hat{I}_{5}\right)+y_{u}^{2}\Delta_{ct}^{+}\Delta_{ds}^{+}-\hat{I}_{7}\right)$
$\displaystyle+y_{c}^{2}\left(\dot{\Delta}_{ut}^{+}\left(y_{b}^{4}+y_{d}^{2}y_{s}^{2}\right)+\Delta_{ut}^{+}\left(\dot{(y_{b}^{4})}+\dot{(y_{d}^{2})}y_{s}^{2}+y_{d}^{2}\dot{(y_{s}^{2})}\right)-\dot{\hat{I}}_{9}\right)\,,$
$\displaystyle\mathcal{P}_{23}^{b}$
$\displaystyle=\dot{\hat{I}}_{10}+\dot{(y_{u}^{2})}\hat{I}_{9}+y_{u}^{2}\dot{\hat{I}}_{9}-\left(\dot{(y_{u}^{4})}+\dot{(y_{c}^{2})}y_{t}^{2}+\dot{(y_{t}^{2})}y_{c}^{2}\right)\left(y_{b}^{4}+y_{d}^{2}y_{s}^{2}\right)-\left(\dot{(y_{b}^{4})}+\dot{(y_{d}^{2})}y_{s}^{2}+\dot{(y_{s}^{2})}y_{d}^{2}\right)\left(y_{u}^{4}+y_{c}^{2}y_{t}^{2}\right)$
$\displaystyle+\dot{(y_{b}^{2})}\left(\hat{I}_{7}+y_{u}^{2}\left(\hat{I}_{5}-2\Delta_{ct}^{+}\Delta_{ds}^{+}\right)\right)+y_{b}^{2}\left(\dot{\hat{I}}_{7}+\dot{(y_{u}^{2})}\left(\hat{I}_{5}-2\Delta_{ct}^{+}\Delta_{ds}^{+}\right)+y_{u}^{2}\left(\dot{\hat{I}}_{5}-2\dot{\Delta}_{ct}^{+}\Delta_{ds}^{+}-2\Delta_{ct}^{+}\dot{\Delta}_{ds}^{+}\right)\right)\,,$

for the (23) angle, and finally

$\displaystyle\mathcal{P}_{12}^{a}$
$\displaystyle=\dot{\Delta}_{db}^{-}\left(\hat{I}_{10}+y_{s}^{2}\left(\hat{I}_{7}-y_{c}^{2}y_{t}^{2}\Delta_{db}^{+}\right)\right)+\dot{(y_{u}^{2})}\Delta_{bd}^{-}\left(\Delta_{sb}^{+}\Delta_{ct}^{+}\Delta_{ds}^{+}-y_{s}^{2}\hat{I}_{5}-\hat{I}_{9}\right)$
$\displaystyle+\Delta_{db}^{-}\left(\dot{\hat{I}}_{10}+\dot{(y_{s}^{2})}\left(\hat{I}_{7}-y_{c}^{2}y_{t}^{2}\Delta_{db}^{+}\right)+y_{s}^{2}\left(\dot{\hat{I}}_{7}-\dot{(y_{c}^{2})}y_{t}^{2}\Delta_{db}^{+}-y_{c}^{2}\dot{(y_{t}^{2})}\Delta_{db}^{+}-y_{c}^{2}y_{t}^{2}\dot{\Delta}_{db}^{+}\right)\right)$
$\displaystyle+y_{u}^{2}\Delta_{bd}^{-}\left(-\dot{\hat{I}}_{9}-\dot{(y_{s}^{2})}\hat{I}_{5}-y_{s}^{2}\dot{\hat{I}}_{5}+\dot{\Delta}_{sb}^{+}\Delta_{ct}^{+}\Delta_{ds}^{+}+\Delta_{sb}^{+}\dot{\Delta}_{ct}^{+}\Delta_{ds}^{+}+\Delta_{sb}^{+}\Delta_{ct}^{+}\dot{\Delta}_{ds}^{+}\right)$
$\displaystyle+\dot{(y_{u}^{4})}y_{s}^{2}\left(y_{b}^{4}-y_{d}^{4}\right)+y_{u}^{4}\dot{(y_{s}^{2})}\left(y_{b}^{4}-y_{d}^{4}\right)+y_{u}^{4}y_{s}^{2}\left(\dot{(y_{b}^{4})}-\dot{(y_{d}^{4})}\right)+y_{u}^{2}\dot{\Delta}_{bd}^{-}\left(\Delta_{sb}^{+}\Delta_{ct}^{+}\Delta_{ds}^{+}-y_{s}^{2}\hat{I}_{5}-\hat{I}_{9}\right)\,,$
$\displaystyle\mathcal{P}_{12}^{b}$
$\displaystyle=-\left(\dot{(y_{b}^{4})}+\dot{(y_{d}^{2})}y_{s}^{2}+\dot{(y_{s}^{2})}y_{d}^{2}\right)\left(y_{u}^{4}+y_{c}^{2}y_{t}^{2}\right)-\left(\dot{(y_{u}^{4})}+\dot{(y_{c}^{2})}y_{t}^{2}+\dot{(y_{t}^{2})}y_{c}^{2}\right)\left(y_{b}^{4}+y_{s}^{2}y_{d}^{2}\right)$
$\displaystyle+y_{b}^{2}\left(\dot{\hat{I}}_{7}+\dot{(y_{u}^{2})}\left(\hat{I}_{5}-2\Delta_{ct}^{+}\Delta_{ds}^{+}\right)+y_{u}^{2}\left(\dot{\hat{I}}_{5}-2\dot{\Delta}_{ct}^{+}\Delta_{ds}^{+}-2\dot{\Delta}_{ds}^{+}\Delta_{ct}^{+}\right)\right)$
$\displaystyle+\dot{\hat{I}}_{10}+\dot{(y_{u}^{2})}\hat{I}_{9}+y_{u}^{2}\dot{\hat{I}}_{9}+\dot{(y_{b}^{2})}\left(\hat{I}_{7}+y_{u}^{2}\left(\hat{I}_{5}-2\Delta_{ct}^{+}\Delta_{ds}^{+}\right)\right)$

for the (12) angle. While somewhat tedious, these expressions amount to simple
polynomials in the Yukawa eigenvalues and flavour invariants, and it is clear
that every individual term comes with only a single derivative operator. Note
also that we have been careful to indicate when the derivatives are of the
_squared_ $\dot{\left(y_{i}^{2}\right)}$ or _quartic_
$\dot{\left(y_{i}^{4}\right)}=2y_{i}^{2}\dot{\left(y_{i}^{2}\right)}$ Yukawa
eigenvalues, and it is of course clear that
$\dot{\Delta}^{\pm}_{ij}=\dot{(y_{i}^{2})}\pm\dot{(y_{j}^{2})}$.

###  A.2 One-Loop RGE for $I_{i}$

The RGE equations for $\hat{I}_{i}$ were derived, for MFV theories, in [20] at
one- and two-loop perturbative order. We have used the one-loop expressions in
the sample numerical calculation performed in Section 4.1, and so we report
the results from [20]:

$\displaystyle\mu\frac{dI_{1}}{d\mu}$ $\displaystyle\simeq
2a_{0}I_{1}+2a_{1}\left(\frac{2}{3}I_{1}^{2}-2\hat{I}_{3}\right)+2a_{2}\left(\hat{I}_{5}-\frac{I_{1}I_{2}}{3}\right)\,,$
$\displaystyle\mu\frac{dI_{2}}{d\mu}$ $\displaystyle\simeq
2b_{0}I_{2}+2b_{1}\left(\frac{2}{3}I_{2}^{2}-2\hat{I}_{4}\right)+2b_{2}\left(\hat{I}_{5}-\frac{I_{1}I_{2}}{3}\right)\,,$
$\displaystyle\mu\frac{d\hat{I}_{3}}{d\mu}$ $\displaystyle\simeq
4a_{0}\hat{I}_{3}-2a_{1}\left(3\hat{I}_{6}-\frac{I_{1}\hat{I}_{3}}{3}\right)-2a_{2}\left(\hat{I}_{7}-\frac{I_{2}\hat{I}_{3}}{3}\right)\,,$
$\displaystyle\mu\frac{d\hat{I}_{4}}{d\mu}$ $\displaystyle\simeq
4b_{0}\hat{I}_{4}-2b_{1}\left(3\hat{I}_{8}-\frac{I_{2}\hat{I}_{4}}{3}\right)-2b_{2}\left(\hat{I}_{9}-\frac{I_{1}\hat{I}_{4}}{3}\right)\,,$
$\displaystyle\mu\frac{d\hat{I}_{5}}{d\mu}$
$\displaystyle\simeq\left(2a_{0}+2b_{0}\right)\hat{I}_{5}+\left(2a_{1}+2b_{2}\right)\left(\hat{I}_{7}-I_{2}\hat{I}_{3}+\frac{2I_{1}\hat{I}_{5}}{3}\right)+\left(2a_{2}+2b_{1}\right)\left(\hat{I}_{9}-I_{1}\hat{I}_{4}+\frac{2I_{2}\hat{I}_{5}}{3}\right)\,,$
$\displaystyle\mu\frac{d\hat{I}_{6}}{d\mu}$ $\displaystyle\simeq
6a_{0}\hat{I}_{6}\,,$ $\displaystyle\mu\frac{d\hat{I}_{7}}{d\mu}$
$\displaystyle\simeq\left(4a_{0}+2b_{0}\right)\hat{I}_{7}+\left(2a_{1}-2b_{2}\right)\left(\frac{I_{1}\hat{I}_{7}}{3}-I_{2}\hat{I}_{6}\right)-\left(2a_{2}-2b_{1}\right)\left(\hat{I}_{10}-\hat{I}_{3}\hat{I}_{4}+2\frac{I_{2}\hat{I}_{7}}{3}\right)\,,$
$\displaystyle\mu\frac{d\hat{I}_{8}}{d\mu}$ $\displaystyle\simeq
6b_{0}\hat{I}_{8}\,,$ $\displaystyle\mu\frac{d\hat{I}_{9}}{d\mu}$
$\displaystyle\simeq\left(4b_{0}+2a_{0}\right)\hat{I}_{9}+\left(2b_{1}-2a_{2}\right)\left(\frac{I_{2}\hat{I}_{9}}{3}-I_{1}\hat{I}_{8}\right)-\left(2b_{2}-2a_{1}\right)\left(\hat{I}_{10}-\hat{I}_{3}\hat{I}_{4}+2\frac{I_{1}\hat{I}_{9}}{3}\right)\,,$
$\displaystyle\mu\frac{d\hat{I}_{10}}{d\mu}$
$\displaystyle\simeq\left(4a_{0}+4b_{0}\right)\hat{I}_{10}+\left(2a_{1}+2b_{2}\right)\left(\frac{I_{1}\hat{I}_{10}}{3}-\hat{I}_{4}\hat{I}_{6}\right)+\left(2a_{2}+2b_{1}\right)\left(\frac{I_{2}\hat{I}_{10}}{3}-\hat{I}_{3}\hat{I}_{8}\right)\,,$
$\displaystyle\mu\frac{dI^{-}_{11}}{d\mu}$
$\displaystyle\simeq\left(6a_{0}+6b_{0}+2a_{1}I_{1}+2b_{1}I_{2}\right)I_{11}^{-}\,,$
(48)

where the $a$ and $b$ parameters are defined as

$\displaystyle a_{0}$
$\displaystyle=\frac{3}{8\pi^{2}}\left(I_{1}+I_{2}+\frac{I_{1}-I_{2}}{2n_{g}}\right)-2\frac{\alpha_{s}}{\pi},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a_{1}=\frac{3}{16\pi^{2}}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a_{2}=-\frac{3}{16\pi^{2}}\,,$
$\displaystyle b_{0}$
$\displaystyle=\frac{3}{8\pi^{2}}\left(I_{1}+I_{2}+\frac{I_{2}-I_{1}}{2n_{g}}\right)-2\frac{\alpha_{s}}{\pi},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b_{1}=\frac{3}{16\pi^{2}}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b_{2}=-\frac{3}{16\pi^{2}}\,,$
(49)

which depend on the strong coupling constant $\alpha_{s}$ and the number of
generations $n_{g}$. Note that the expressions in (48) - (49) neglect
subdominant electroweak corrections.

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