#### Natural fermion hierarchies from random Yukawa couplings

HJE
Natural fermion hierarchies from random Yukawa couplings
Gero von Gersdor 0
0 Department of Physics, Pontif cia Universidade Catolica de Rio de Janeiro
1 es de Sa~o Vicente 225 , Rio de Janeiro , Brazil
The Standard Model of particle physics requires Yukawa matrices with eigenvalues that di er by orders of magnitude. We propose a novel way to explain this fact without any small or large parameters. The mechanism is based on the observation that products of matrices of random order-one numbers have hierarchical spectra. The same mechanism can easily account for the hierarchical structure of the quark mixing matrix.
Yukawa; Beyond Standard Model; Quark Masses and SM Parameters
1 Introduction 2 3 4
given a convincing UV description in terms of spontaneously broken U(1) symmetries [1]
or in terms of wave-function localization in extra dimensions [2{4]. If avor violating new
physics (NP) is present at the TeV scale, these models typically feature somehow suppressed
avor changing neutral currents (FCNC) (as opposed to a generic
avor structure), but
nevertheless are tightly constrained, in particular by data on KK mixing and
decays (see, for instance, ref. [5] for a review of avor bounds in extra dimensions.).
! e
In this short note we suggest a new mechanism that provides a natural explanation
of the fermion mass hierarchies of the SM. Instead of invoking an order parameter, the
mechanism is based on completely random couplings of order one. Due to the properties
of the probability distributions for the e ective Yukawa couplings extreme ratios of
eigenvalues become completely common, and mass hierarchies are hence the rule rather than
the exception. We will refer to this mechanism as the stochastic hierarchy mechanism.
The peculiar almost-diagonal nature of the Cabbibo-Kobayashi-Maskawa (CKM) matrix
can also easily explained with our approach, as the alignment/misalignment of the up and
down Yukawa matrices can be controlled by some judicious choice of the matrix products.
The \randomness" hypothesis has previously been considered in the context of
neutrino mixing [6] and is commonly referred to as neutrino anarchy.
Moreover, anarchic
perturbations to a hierarchical model were investigated in ref. [7], and in the context of
{ 1 {
supersymmetry it was found that some hierarchies can also arise stochastically [8].
Furthermore, in models giving rise to Yukawa couplings of the type eq. (1.1), the unknown
O(1) numbers multiplying the suppression factors are often considered to be stochastic as
well. However, the goal of the present manuscript is the generation of the large (charged)
fermion hierarchies \out of nothing", i.e., from purely O(1) random couplings, which to
the best of our knowledge has never been achieved in the literature.
2
that they are random O(1) numbers. We will assume that they are real and follow some
\base distribution" (or \prior") with mean zero and variance
. A simple and natural
2
choice is the uniform distribution with
1 < Yaib < 1 ;
h
max jyij= min jyij ;
(2.1)
(2.2)
(2.3)
(2.4)
where Nf denotes the number of families. Note that the correlations between di erent
matrix elements vanish. For higher moments this is no longer true and the matrix elements
are in fact statistically dependent. Interestingly, for Nf = 3 and at priors, the variance is
independent of N and equals the one of the at prior ( 2 = 1=3).
{ 2 {
which has variance 2 = 1=3, but many of our results below are valid for any other
sym
We would like to nd the probability distribution for the largest mass hierarchies in
metric prior.
the matrix Yab,
where yi2 are the three eigenvalues of Y Y T . In theory, the analytical calculation of these
distributions is straightforward: one substitutes one of the matrix elements by h and
marginalizes (integrates) over the remaining ones. Unfortunately, this calculation is obstructed by
the resulting complicated region of integration, and one has to result to numerical
simulations. However, some aspects of these distributions can be computed analytically and
allow for a rough understanding of the mechanism.
Rather than calculating the full distributions we will focus on their lowest moments.
For instance, for mean and (co)variance of Yab one easily nds
hYabi = 0 ;
hYabYcdi =
1
Nf
Nf
To get a feeling for the typical size of eigenvalues, we can compute the mean of tr Y Y T .
htr Y Y T i = Nf Nf
N , and from eq. (2.6) and (2.5) we get the estimate
Thus, at least one eigenvalue has to be of O(1), and the geometric mean of the other two
is suppressed. Ordering y1; y2
y3, one gets from eq. (2.7) the estimate
nant multiplication theorem which causes the integrals over (Yi)ab and (Yj )ab to factorize.
For Nf = 3 and the
at prior, the variance is given by (2=9)N and hence the
determinant tends to be very suppressed despite the fact that the matrix elements are typically
of O(1). At rst this seems at odds with the fact that the covariance matrix is diagonal.
However, correlations of higher moments do not vanish, and it is those that make these
y3 <
y2
y2
y1
{ 3 {
Interestingly, this ratio is actually independent of 2 and hence valid for any prior, even
though the size of the largest eigenvalue is no longer independent of N and can be both
supressed or enhanced, depending on wether 2 is smaller or larger than 1=3.1
Finally, we comment that there is typically also a hierarchy between the lighter two
eigenvalues. To show this, we would have to examine the distributions of other quantities
(such as the principal minors), but we will not go into this much detail here. Instead we
show in
gure 1 the simulated distributions of the Nf = 3 eigenvalues for the case N = 7,
from which the hierarchical spectrum y1
y2
y3 is quite evident. We also provide,
for comparison, the cases of a Gaussian prior and a log-uniform prior, showing excellent
prior-independence.2 We also nd empirically
in about 80% of the cases, roughly independent of N , which shows that up-like and
leptonlike spectra are more common than down-like ones.
1The behaviour of hierarchies of random matrices at large Nf (rather than N ) has been studied previously
in the context of neutrino masses [9].
eigenvalues depends on the variance, but the hierarchies do not.
2All priors are chosen to have the same variance. As already mentioned, the absolute scale of the
ob0.4
r
P0.2
0.0
Uniform
Gaussian
Log Uniform
-8
-6
-2
0
-4
Log10(yi)
HJEP09(217)4
1 and more suppressed entries the more one moves
away from the diagonal. The CKM matrix is the ratio of the left handed rotation matrices
VCKM = VuLVdyL ;
(2.10)
where VuL and VdL are the unitary matrices diagonalizing the combinations Y uY u y and
Y dY d y respectively. Therefore, in order to obtain an approximately diagonal CKM matrix,
the two left handed rotations have to be similar, or equivalently, the up and down Yukawa
couplings have to be roughly aligned. This can easily be achieved as follows. In addition to
the physical Yukawas being products of several matrices, we stipulate that some of these
matrices are the same (we will see in the next section how this can be achieved in a model):
Y d = Y
q
1
YNqq Y1
d
YNdd ;
Y u = Y
q
1
YNqq Y1
u
YNuu :
(2.11)
Note that the rst Nq factors are common, and it is this common factor that guarantees a
certain degree of (left-handed) alignment. Without this factor, there would be no alignment
and the CKM matrix becomes democratic. To see why the CKM matrix is hierarchical, it
is instructive to diagonalize the respective factors
Y d = UqY^ qU~qyU~dY^ dU y ;
d
Y u = UqY^ qU~qyU~uY^ dUuy :
Here, Y^ q;u;d are diagonal and hierarchical (according to the numbers Nq, Nu, and Nd).
It is important to stress that the unitary matrices U~q;u;d and Uq;u;d are not hierarchical.
Making a gauge-invariant change of basis with the matrices Uq;u;d, we obtain the structure
(Y 0d)ij
q d
y^i y^j ;
(Y 0u)ij
q u
y^i y^j ;
where we have not written O(1) numbers arising from the U~q;u;d. In this new basis, the
matrices take the familiar Frogatt-Nielsen (FN) form, eq. (1.1). As is well known, for
hierarchical y^q;u;d, the CKM matrix scales as [1]
(2.12)
(2.13)
(2.14)
(VCKM)ij
exp
log y^iq=y^jq
;
{ 4 {
sen0.4
D
ity0.3
l
i
abb0.2
o
rP0.1
0.0
Vut
Vcb
Vus
-5
-4
-1
0
-3
Log10HVCKML
-2
HJEP09(217)4
values are indicated as vertical lines.
terms of the hierarchies of the eigenvalues y^iq of the common factor.
(again up to O(1) numbers), showing that the CKM hierarchy is determined entirely in
We can choose the numbers Nu, Nd and Nq in order to maximize the probabilities to
achieve SM-like values for the masses and mixings.3 We have simulated the distributions
for VCKM, drawing from
at priors for all the matrices involved in eq. (2.11), for the case
Nq = 5, Nd = 0, and Nu = 5. In this simulation we have made the additional selection y3=y2
greater (smaller) than y2=y1 for the down (up) sector respectively, as occuring in the SM.
The e ciency of this cut is about 0.13, which is a bit lower than the value 0:8 0:2 = 0:16 to
be expected from the considerations around eq. (2.9), because the eigenvalues in the up and
down sector become mildly correlated. The resulting distributions are shown in
gure 2.
One can see clearly that the true CKM angles appear in the bulk of the distributions and
are hence at their natural values.
3
An explicit model
Motivated by the observations of the previous section we now move on to construct a model
with e ective Yukawa couplings given by products of matrices. Consider replacing the SM
(say up-type) Yukawa interaction by the Lagrangian
Lu =
X Qi(=p + Miq)Qi
q
(QiKi Qi+1 + h:c:)
Nq
i=1
Nu
i=1
+ X Ui(=p + Miu)Ui
(UiKiuUi+1 + h:c:)
(Q1H~ Y0uU1 + h:c:) ;
(3.1)
where QN+1 = qL and UN+1
uR are chiral elds, the remaining Ui and Qi are vector-like
quarks, and H is the SM Higgs eld. The masses Mi are hermitian and the mass mixings
3An additional global suppression factor can be introduced for the down sector, mimicking the e ect of
large tan
in supersymmetry.
{ 5 {
Ki arbitrary 3 3 matrices. Lagrangians of this kind are familiar from discretizations of
extra dimensions [
10
] and composite Higgs models with partial compositeness [
11
]. They
have recently been reconsidered in the so-called clockwork mechanism [12]. In contrast to
these constructions, here no small parameter will be needed.4 Let us brie y comment on
how one can achieve that the fermions Ui (Qi) only couple to Ui+1 (Qi+1). One possibility
is to promote the spurions Ki and Mi to physical elds that obtain vacuum expectation
values via some mechanism. in fact, for vanishing couplings a large chiral symmetry (with a
U(3)L
U(3)R at each site) emerges. The nearest neighbor interaction can then be achieved
by introducing physical elds Mi and Ki only in the bifundamentals of U(3)L;i
U(3)R;i
and U(3)L;i
U(3)R;i+1 respectively.
We can integrate out the Ui nad Qi, yielding the e ective Lagrangian
L0u = qL ZNqq =p qL + uR ZNuu =p uR
H~ qL(Y~1q
Y~Nqq )yY0u(Y~1u
Y~Nuu )uR + h:c: ;
with the recursively de ned matrices
and
Zk = Y~ky(1 + Z
k 1)Y~k ;
Z1 = Y~1yY~1 ;
Y~k = (Mk
Ky
~
Y
k 1 k 1
) 1K ;
k
Y~1 = M1 1K1 :
(3.2)
(3.3)
(3.4)
We stress that the e ective Lagrangian eq. (3.2) is exact as long as none of the (true) masses
of the heavy elds are at or below the electroweak scale. The expressions for N > 1 quickly
get quite complicated. However, the recursive de nitions are well suited for numerical
simulations, and in particular are much easier to handle than the full diagonalization of
the mass matrix.
The down-type Yukawa couplings arise in the same way, by introducing Nd vectorlike
down quarks Di and replacing the second and third line of eq. (3.1). In the lepton sector,
we introduce N` vectorlike doublets Li, Ne vectorlike charged singlets Ei, and N vectorlike
neutral singlets Ni. Since our mechanism implies that the largest Yukawa coupling is of
O(1), we introduce heavy Majorana masses for the elds NN +1
R, implementing the
sea-saw mechanism [14{16]. The six integer numbers Nq; Nu; Nd; Ne and N are the only
parameters that we treat non-stochastically, They can be viewed as the analogue of the
FN charges in our model. We have simulated this model, using at priors with
m < Maib < m ;
m < Kaib < m ;
1 < Yi0j < 1 ;
(3.5)
where m is a heavy mass scale. It is clear that the physical Yukawa couplings (being
dimensionless parameters) cannot depend on the mass m and hence we will work in units
of m = 1, and analogous expressions hold for the down-quark and charged lepton sectors.
4Our Lagrangian also resembles somewhat the model presented in ref. [13] (section 5) which also uses
vectorlike fermions. In that model a judicious choice of the proto-Yukawa couplings ensures that after
integrating out the heavy fermions some of the SM fermions only couple to higher powers of the Higgs eld,
creating the observed hierarchies. Here all elds have linear couplings to the Higgs, and the hierarchies
appear via products of matrices, as explained below.
{ 6 {
0.0
0
N=4
N=7
N=10
N=0
1
N=1
2
Log10Hh0L
3
4
5
HJEP09(217)4
the SM values hd, he, and hu respectively.
Note that the dependence on the variance of the prior then only enters in the quantity Y 0,
and hence does not get magni ed with powers of N as would be the case in the simple toy
model of the previous section.
physical eigenvalues, determined from
We will rst consider the hierarchies h de ned in eq. (2.3), where the yi are now the
det hY~ u y(Zq) 1Y~ u
yi2Zui = 0 ;
where Zq;u = 1 + Zq:u and Y~ u = (Y~1q
N
Y~Nqq )yY0uY~1u
Y~Nuu . We will use as benchmarks
the SM hierarchies
hd
8:7
In gure 3 we plot the probabilities for the occurence of hierarchies greater than h0 as
a function of h0. To a good approximation, the eigenvalue distributions only depend on
the sums (Nq + Nu, N` + Ne etc.), so for simplicity we report only the results for these
sums (called N in the following). From the curve N = 0, corresponding to the SM, one
can see that even though h > 10 occurs with probability of roughly 1=3, larger hierarchies
are very unlikely. This however changes drastically when a few vectorlike fermions are
introduced. As is evident from the curves, rather large hierachies quickly become the rule
rather than the exception. In table 1 we quote explicitely the probabilities for the SM
model benchmarks. We also plot in
gure 4 the distributions for the actual eigenvalues
in the case N = 7. They do not look very di erent from the distributions of the simple
product struture obtained in the previous section (see gure 1). The independence of the
exact form of the matrix product shows the robustness of our mechanism.
The CKM mixing angles are well reproduced from Nq
6{8 (see gure 5) while absence
of alignment between the charged leptons and neutrinos require N`
0{1. One can also
ask the question whether one can nd a choice of parameters that is compatible with SU(5)
quantum numbers of a potential grand uni ed theory. Taking into account both the masses
and mixings, we nd that
Nq = Nu = Ne = 6 ;
N` = Nd = 0
(3.8)
{ 7 {
1.2
ity1.0
s
en0.8
D
0.0
ity0.8
s
n
y
eD0.6
l
ib0.4
roP0.2
Vut
Vcb
Vus
{ 8 {
p(h > hd)
p(h > he)
p(h > hu)
3:9
2:0
9:1
Upper block: priors chosen according to eq. (3.5). Lower block: modi ed prior eq. (3.9) with q
in which all angles follow the same distribution (dashed-dotted line).
works rather well. Furthermore, in order to avoid a too-large hierarchy between the two
heaviest neutrinos, we choose N
. 2. The case N` = N
= 0 then corresponds to standard
anarchic see-saw neutrinos. Taking N 6
= 0 in addition creates a small hierarchy in the
Neutrino Yukawa couplings, but as long as N` = 0 we do not generate any alignment with
the charged lepton sector, and mixing angles will stay large.
One could be worried that a large number of new fermions will give rise to Landau
poles for the gauge couplings in the UV. Notice however that up to now nothing has been
said about the mass scale for the new fermions. It is fully consistent (and in fact natural)
that these masses are at or just below the Planck scale. In this case the model is consistent
with the absence of any additional New Physics below the Planck scale. Notice that such
a high mass scale automatially suppresses dangerous FCNCs which would otherwise be an
issue due to the presence of the vector like quarks and leptons. An alternative scenario
would be an additional UV completion of the model at a lower scale, in which case a
detailed assessment of avor violating e ects would be necessary.
Finally, we would like to point out two more ways to even further improve the
performance of the mechanism. Firstly, assume that there is a reason for the mass mixings Ki
to be systematically suppressed with respect to the masses Mi. We can incorporate this
assumption easily by modifying the priors for Ki as
HJEP09(217)4
(3.9)
(3.10)
(3.11)
with q a dimensionless number q
1. In this case, one has approximately Zk
0, and
Thus in this approximation of small q the physical Yukawa couplings simply scale as qN
and the hierarchies, being ratios of eigenvalues, become independent of q. The probabilities
from these distributions are also given in table 1. We nd that hierarchies are generated
even more e ciently, presumably because accidentally small vectorlike masses have a larger
e ect or occur more common than in the case q = 1 considered above.
A second possible modi cation is the assumption of some kind of minimal avor
violation meachanism, that xes all mixings to be proportional to, say, Y0,
q m < Kaib < q m ;
~
Y
k
Mk 1K :
k
Ki / Y0 :
h0N .
{ 9 {
For simplicity we may assume that the masses Mi are proportional to the identity. There
are always mild hierarchies h0 in the random eigenvalues of Y0 (confer the N = 0 curve in
gure 3), and since there is a basis in which everything is diagonal simultaneously, these
hierarchies are coherently ampli ed h
mechanism to future work.
We leave the construction of an explicit
In this note we have presented a new explanation for the large hierarchy of fermion masses
present in the SM. It is based on the observation that products of a few random matrices
typically feature strong hierarchies in their eigenvalue spectrum, even though the individual
entries are of order unity. Moreover, we have shown that the peculiar form of the CKM
matrix can easily be recovered within this paradigm, as up and down Yukawa couplings can
become naturally aligned if they include common factors as in eq. (2.11). We have presented
a model that generates an e ective Yukawa coupling as a product of several matrices. This
model is by no means unique, and we expect that any model with such a product structure
has similar eigenvalue distributions. Finally, even though we have focused on the case of
real random matrices for simplicity, CP violation can easily be accommodated by including
random complex phases.
Acknowledgments
I would like to thank Arman Esmaili for useful discussions and comments on the manuscript,
and the Conselho Nacional de Desenvolvimento Cient co e Tecnologico (CNPq) for
support under fellowship number 307536/2016-5.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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