The Hilbert series is computed for the lepton flavor invariants of the
Standard Model with three generations including the right-handed neutrino
sector needed to generate light neutrino masses via the see-saw mechanism. We
also compute the Hilbert series of the quark flavor invariants for the case of
four generations.Comment: 6 page

A Butti

A Hanany

Amihay Hanany

Aneesh V. Manohar

B Feng

EE Jenkins

Elizabeth E. Jenkins

F Knop

Giuseppe Torri

J Gray

S Benvenuti

English

arXiv

Springer - Publisher Connector

J
H
E
P03(2011)096
Published for SISSA by Springer
Received: January 18, 2011
Accepted: March 4, 2011
Published: March 21, 2011
Hilbert series for flavor invariants of the
Standard Model1
Amihay Hanany,a Elizabeth E. Jenkins,b Aneesh V. Manoharb and Giuseppe Torria
aTheoretical Physics Group, The Blackett Laboratory, Imperial College London,
Prince Consort Road, London, SW7 2AZ, U.K.
bDepartment of Physics, University of California at San Diego,
La Jolla, CA 92093, U.S.A.
E-mail: a.hanany@imperial.ac.uk, ejenkins@ucsd.edu, amanohar@ucsd.edu,
giuseppe.torri08@imperial.ac.uk
Abstract: The Hilbert series is computed for the lepton flavor invariants of the Standard
Model with three generations including the right-handed neutrino sector needed to generate
light neutrino masses via the see-saw mechanism. We also compute the Hilbert series of
the quark flavor invariants for the case of four generations.
Keywords: Phenomenological Models
ArXiv ePrint: 1010.3161
1Imperial/TP/10/AH/06.
Open Access doi:10.1007/JHEP03(2011)096
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Contents
1 Introduction 1
2 Lepton invariants 3
3 Quark invariants for four generations 4
4 Conclusions 5
1 Introduction
A recent study of the lepton and quark flavor invariants was performed in the Standard
Model effective theory using methods from Invariant Theory [1, 2]. All flavor invariants
and the structure of their non-trivial relations (syzygies) were found in the case of physical
interest, namely the Standard Model effective theory with three generations of fermions,
including a dimension-five Majorana neutrino mass operator for the light neutrinos.
The number of invariants of each degree in the fermions mass matrices is given com-
pactly in terms of the Hilbert series for the invariants, which provides the necessary infor-
mation to determine the number of independent invariants of each degree in the basic mass
matrices. The Hilbert series for the three-generation Standard Model effective theory with
a dimension-five Majorana mass term was determined in ref. [2]. The flavor invariants in
the low-energy Standard Model involve the quark mass matrices mU and mD, the charged
lepton mass matrix mE , and the light neutrino Majorana mass matrix m5. The see-saw
mechanism with singlet neutrinos provides an elegant way to generate the low-energy Ma-
jorana mass term using a renormalizable gauge theory. In the see-saw theory, the Majorana
mass term m5 is instead replaced by two other matrices, the Majorana mass matrix M of
the heavy singlet neutrinos, and the Dirac mass matrix coupling the singlet and doublet
neutrinos, mν . The matrix
m5 = m
T
νM
−1mν (1.1)
is the Majorana mass matrix for the light neutrinos generated by integrating out the
heavy singlet neutrinos. The flavor invariants in the see-saw model are considerably more
complicated than in the Standard Model effective theory, and in ref. [2], the Hilbert series
in the see-saw theory was obtained for two generations only. The classification of the
lepton invariants for the see-saw theory with three generations proved to be too difficult,
and only a partial classification was given. This three-generation see-saw theory is of great
physical interest since it is the leading candidate theory for a renormalizable Standard
Model extension that includes neutrino masses.
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The Hilbert series for the three-generation see-saw theory can be computed using new
methods developed recently in refs. [3–7] and references therein, especially the Molien-Weyl
formula, which reduces the problem of computing the Hilbert series to that of evaluating
a small number of complex integrals. This formula is a standard tool in Invariant Theory,
and, given the action of a symmetry group on some mathematical objects, it allows one to
compute quantities which are invariant with respect to the given symmetry.
The Molien-Weyl formula consists of two pieces: the measure of integration, which is
the Haar measure of the symmetry group, and an integrand given by the so called plethystic
exponential, which, for a function f(t) vanishing at the origin, is defined as:
PE[f(t)] = exp
(
∞∑
r=1
f(tr)
r
)
. (1.2)
The flavor invariants are symmetric functions of the masses of the leptons or of the quarks,
which transform under specific representations of the flavor groups. Therefore, in order to
count all the possible flavor invariants of a given theory, it is necessary to generate first all
the possible symmetric products of the building blocks of these theories, i.e. the masses of
the particles. This is precisely what the plethystic exponential does: it acts like a generator
for symmetrization.
Once the Molien-Weyl formula is constructed for a given theory, one can proceed by
integrating over the Haar measure, thus obtaining the Hilbert series H(t), which, for the
cases we are interested in, is always guaranteed to be a rational function. The coefficients
of the Taylor expansion of the Hilbert series give the number of flavor invariants of a
certain degree.
Another interesting property of the theories we are examining is that the set of flavor
invariants, together with the usual operations of sum and product have the structure of
a ring which is finitely generated. Thus, it is an interesting problem to determine which
are the generators of the ring and the relations among them. This is easily done by the
plethystic logarithm of the Hilbert series which is defined as:
PL[H(t)] =
∞∑
r=1
µ(r) logH(tr)
r
, (1.3)
where µ(r) is the Mo¨bius function, which is defined as follows:
µ(r) :=


0 r has repeated prime factors
1 r = 1
(−1)n r is a product of n distinct primes
(1.4)
The first positive terms of the Taylor expansion of this function correspond to the
generators of the ring of invariants and the first negative terms correspond to the relations
among them (syzygies).
In addition, we determine the quark invariants of the Standard Model for four gener-
ations of quarks as an added bonus using the same methodology.
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2 Lepton invariants
The Hilbert series of a general theory is defined to be
H(t) =
∞∑
r=0
crt
r, (2.1)
where cr is the number of invariants of degree r and c0 = 1. Thus, the Hilbert series
determines the number and degree of all flavor invariants. A general Hilbert series H(t)
is of the form H(t) = N(t)/D(t), where the numerator N(t) and the denominator D(t) of
the Hilbert series are polynomials in t. For the see-saw theory with three generations of
leptons, the Molien-Weyl formula can be written as:
H(t) =
1
(3!)2
3∏
k=1
∮
dzk
2piizk
3∏
l=1
∮
dwl
2piiwl
∏
i<k
(zi − zk)
2
∏
j<l
(wj − wl)
2 ×
PE
[
3∑
i,j
zi
zj
t2 +
3∑
i,j
(
zi
wj
+
wi
zj
)
t+
3∑
i≤j
(
wiwj +
1
wiwj
)
t
]
. (2.2)
The numerator of the Hilbert series computed with the formula above is
N(t) = 1+t4+5t6+9t8+22t10+61t12+126t14+273t16+552t18+1038t20
+1880t22+3293t24+5441t26+8712t28+13417t30+19867t32+28414t34+39351t36
+52604t38+68220t40+85783t42+104588t44+123852t46+142559t48+159328t50
+173201t52+183138t54+188232t56+188232t58+183138t60+173201t62+159328t64
+142559t66+123852t68+104588t70+85783t72+68220t74+52604t76+39351t78
+28414t80+19867t82+13417t84+8712t86+5441t88+3293t90+1880t92+1038t94
+552t96+273t98+126t100+61t102+22t104+9t106+5t108+t110+t114, (2.3)
which is of degree dN = 114 and palindromic, i.e. t
dNN(1/t) = N(t). The denominator is
D(t) =
(
1− t2
)3(
1− t4
)4(
1− t6
)4(
1−t8
)2(
1−t10
)2(
1− t12
)3(
1− t14
)2(
1− t16
)
, (2.4)
which is of degree dD = 162. The plethystic logarithm of this Hilbert series can be written
as:
PL[H(t)] = 3t2 + 5t4 + 9t6 + 10t8 + 19t10 + 40t12 + 66t14 + 92t16 + 70t18−O(t20) (2.5)
There are three mass matrices in the lepton sector of the see-saw model, mν , mE and M ,
and their complex conjugates mν
†, mE
† andM † =M∗. The Dirac mass matrix of the light
neutrinos mν and the mass matrix of the charged leptons mE are complex 3× 3 matrices,
whereas the Majorana mass matrix of the heavy singlet neutrinos is a complex symmetric
3 × 3 matrix. Thus, the dimension of the vector space on which the flavor symmetry
transformations act is dimV = 48, since the complex 3 × 3 matrices mν and mE each
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contribute 18 and the complex symmetric matrix M contributes 12. Knop’s theorem [8]
states that
dimV ≥ dD − dN ≥ p, (2.6)
where p is the number of parameters. The number of parameters in the three mass matrices
in the see-saw theory for three generations of leptons is p = 21, which consists of 9 masses,
6 angles and 6 phases. Thus, Knop’s theorem gives 48 ≥ 48 ≥ 21.
From the point of view of Invariant Theory, computing flavor invariants in the Standard
Model is equivalent to computing chiral gauge invariant operators of supersymmetric gauge
theories. In particular, the Hilbert series determined from (2.2) is the same as the one for
counting gauge invariant operators in an N = 1 supersymmetric theory with gauge group
U(3) × U(3), with one field transforming in the adjoint representation of the first group,
corresponding to mE, one bi-fundamental field and its complex conjugate, corresponding
to mν and m
†
ν , and one field transforming in the rank-2 symmetric representation of the
second gauge group and its complex conjugate, corresponding to M and M †. In addition,
the superpotential is set to 0.
From this perspective, the number of parameters of the see-saw theory corresponds
to the dimension of the vacuum moduli space of the gauge theory, which can be elegantly
determined with an argument based on the Higgs mechanism. In the Higgs mechanism, a
massless vector multiplet becomes massive by ‘absorbing’ degrees of freedom from a chiral
multiplet. The gauge theory we are examining contains 9 degrees of freedom coming from
the adjoint field, 18 coming from the bi-fundamentals and 12 coming from the symmetric
rank-2 tensors. The vector multiplets becoming massive take away 18 degrees of freedom (9
per gauge group), leaving a total of 21, which correspond to the dimension of the moduli
space of the gauge theory. Note that this counting correctly reproduces the number of
parameters of the see-saw theory.
3 Quark invariants for four generations
The invariants and relations for three generations of quarks were determined previously in
ref. [2]. Here we give the Hilbert series for four generations of quarks, which is consider-
ably more complicated than the three-generation case. The general counting of physical
parameters of the up- and down-quark mass matrices, mU and mD, respectively, is given
in refs. [1, 2]. In the case of four quark generations, the parameters consist of 8 masses, 6
angles and 3 phases.
The Hilbert series for four generations of quarks is
H(t) =
1+t10+3t12+2t14+4t16+4t18+2t20+3t22+3t24+2t26+4t28+4t30+2t32+3t34+t36+t46
(1− t2)
2
(1− t4)
3
(1− t6)
4
(1− t8)
6
(1− t10) (1− t12)
.
(3.1)
The degree of the numerator is dN = 46, whereas the degree of the denominator is
dD = 110. The plethystic logarithm of this Hilbert series can be written as:
PL[H(t)] = 2t2 + 3t4 + 4t6 + 6t8 + 2t10 + 4t12 + 2t14 + 4t16 + 4t18 + t20−O(t24) (3.2)
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There are two 4 × 4 matrices mU and mD and their complex conjugates, so dimV = 64.
From the denominator, one sees that there are p = 17 parameters, as expected. Knop’s
theorem gives 64 ≥ 64 ≥ 17.
The multi-graded Hilbert series, where tu and td count powers of mU and mD respec-
tively, is found to be H(tu, td) = N(tu, td)/D(tu, td), where
N(tu, td) =
(
1 + t2ut
2
d
) (
1− t2ut
2
d + t
4
ut
4
d
) (
1− t2ut
4
d − t
4
ut
2
d + t
4
ut
6
d + t
4
ut
8
d + t
6
ut
4
d + 2t
6
ut
6
d
+ t6ut
8
d + t
8
ut
4
d + t
8
ut
6
d − t
8
ut
10
d − t
10
u t
8
d + t
12
u t
12
d
)
,
D(tu, td) =
(
1− t2u
) (
1− t4u
) (
1− t6u
) (
1− t8u
) (
1− t2d
) (
1− t4d
) (
1− t6d
) (
1− t8d
) (
1− t2ut
2
d
)
(
1− t4ut
2
d
)2 (
1− t2ut
4
d
)2 (
1− t4ut
4
d
)2 (
1− t6ut
2
d
) (
1− t2ut
6
d
)
, (3.3)
which is considerably more complicated than the Hilbert series of the three-flavor case
found in ref. [2]. The multi-graded version of eqs. (2.3), (2.4) has also been computed,
but the result is too complicated to give here; the numerator has over 6000 terms. The
plethystic logarithm of this multi-graded Hilbert series can be written as:
PL[H(tu, td)] = t
2
d + t
2
u + t
4
d + t
2
dt
2
u + t
4
u + t
6
d + t
4
dt
2
u + t
2
dt
4
u + t
6
u + t
8
d + t
6
dt
2
u + 2t
4
dt
4
u
+ t2dt
6
u + t
8
u + t
6
dt
4
u + t
4
dt
6
u + t
8
dt
4
u + 2t
6
dt
6
u + t
4
dt
8
u + t
8
dt
6
u + t
6
dt
8
u + t
10
d t
6
u (3.4)
+ 2t8dt
8
u + t
6
dt
10
u + t
12
d t
6
u + t
10
d t
8
u + t
8
dt
10
u + t
6
dt
12
u + t
10
d t
10
u −O(t
12
u )O(t
12
d )
4 Conclusions
In summary, we have used the methods of refs. [3–7] to find the Hilbert series for the
lepton invariants of the three-generation see-saw theory. This problem had proven to be
computationally too difficult previously, and so only partial results were given in refs. [1, 2].
The same techniques also were used in this work to find the Hilbert series for the quark
invariants of the Standard Model in the case of four generations.
We would like to thank H. Fritzsch, W. Plessas and M. Shifman for organizing the
2010 Oberwo¨lz workshop on QCD and Strings, where this work was started. We would
also like to thank John Davey and Ennio Salvioni for their kind help.
Open Access. This article is distributed under the terms of the Creative Commons
Attribution Noncommercial License which permits any noncommercial use, distribution,
and reproduction in any medium, provided the original author(s) and source are credited.
References
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– 6 –


Hilbert series for flavor invariants of the Standard Model

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Hilbert Series for Flavor Invariants of the Standard Model

Hilbert Series for Flavor Invariants of the Standard Model

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