The Oldest Unsolved Problem:
the mystery of three generations
C.S. Lam
McGill and UBC, Canada
arXiv:1002.4176 (phys. Letts. B)
arXiv:1003.0498(talk, summary)
Who Ordered That?
Millikan
Blackett
1928: Dirac Eq
1932: positron
趙忠堯 (1930)
1933: Nobel Prize (Dirac)
1936: Nobel Prize (Anderson)
Isidor Issac Rabi
(1898—1988)
1936: muon (Anderson)
e+
oldest mystery in particle physics
Carl Anderson
(1905—1991)
e-
the plot thickens
• now there are three generations of quarks
and leptons. What tells them apart?
• Maybe a new set of (generation, horizontal)
quantum numbers to tell them apart?
Heisenberg invented
isotopic spin in 1932
to distinguish the newly discovered
neutron from the proton
a horizontal symmetry?
• if so, must be (spontaneously) badly broken to
account for the mass differences and mixing
a horizontal symmetry?
• if so, must be (spontaneously) broken to account
for mass differences and mixing
• is there any trace of symmetry left?
yes for leptons, very little for quarks
a horizontal symmetry?
• if so, must be (spontaneously) broken to account
for mass differences and mixing
• is there any trace of symmetry left?
yes for leptons, very little for quarks
• what is the unbroken symmetry for leptons?
a horizontal symmetry?
• if so, must be (spontaneously) broken to account
for mass differences and mixing
• is there any trace of symmetry left?
yes for leptons, very little for quarks
• what is the unbroken symmetry for leptons?
• is there a direct experimental test to decide
whether there is a horizontal symmetry?
residual leptonic symmetry
(regularity in neutrino mixing)
1  2  3
2
1 
U
 1
6
 1

0  e

2 3  
 
2  3  
2
beta decay
tri-bimaximal mixing
quarks
how is this regularity related to the unbroken horizontal symmetry, and
what symmetry might that be?
two .. math slides
• horizontal symmetry, symmetry breaking, and mass
matrices (L, R complications)
• mass matrices, masses, mixings, `residual’ symmetry of
the mass matrices and regularity of the mixing matrix
?
math slide I
3
horizontal symmetry
H (eL , eR , L , N R ,  )  G.H
symmetry breaking
H (eL , eR , L , N R ,  )  G.H
a
a
LR, RR, mass matrices
eL M eeR   L M N R  N R M N N R  h.c.  ......
1
2
M e  M e M e†
LL mass matrices
effective LL Hamiltonian
M  M M N1MT
eL M e eL ,  L M L
H eff  H eff (eL , L ,  a ,  a )
math slide II
It is reversible if F is non-degenerate
 1 2 2 
1
G1  2 2 1 
3

 2 1 2 
LL mass matrices
 1 2 2 
1
G2   2 1 2 
3

 2 2 1
1 0 0
G3  0 0 1 
0 1 0


(special unitary) residual symmetry operators
M e  diag (m , m , m )
2
e
2
2
[ F , M e ]  0  F  diag
diagonalization
U M U  diag
T
1  
2 3
2
1 
U   1
6
 1

2 0  e

2 3  

2  3  

same diagonalization
G eigenvalues: +1, -1, -1
[G, M ]  0, G  1
2
 G1 , G2 , G3
two.. math slides
• horizontal symmetry, symmetry breaking, and mass
matrices (L, R complications)
• mass matrices, masses, mixings, `residual’ symmetry of
the mass matrices and regularity of the mixing matrix
• relation between regularity of mixing and unbroken
horizontal symmetry (2.5 criteria)
• the case of tri-bimaximal mixing
?
2.5 criteria
1. horizontal symmetry contains residual
symmetry of mass matrices G  F , G
2. vev must be determined to obey 1.
vev
H (eL , eR , L , N R ,  )  G.H
a
H eff  H eff (eL , L , , )
a
F, G
H eff  H eff (eL , L , 
a
F 
a
 
G 
a
 
a
a
a
a
a
,
a
)
2.5 criteria (summary)
1. horizontal symmetry contains residual
symmetry of mass matrices G  F , G
2. vev must be determined to obey 1.
Fa a  a
Ga  a   a
3. residual symmetry should determine the mixing
matrix (reversibility)
F must be non-degenerate
tri-bimaximal mixing & variations
2
1 
 1
U 
6
 1

0 

2 3   G1 , G2 , G3

2  3
2
F
1


  


2

 

A4  F , G2 
G  F , G
S3   F , G3 
S4  {F , G1}  F , G1 , G2 , G3 
FNG , G1, G2 , G3   S4
3  1
An experimental test of
horizontal symmetry
eLeR
qL qR
•
Fermion masses in the SM come from the coupling with a single Higgs
•
In horizontal symmetry models, they come from couplings with several
Higgs.
•
Therefore the coupling of any of these Higgs is no longer proportional to the
fermion mass. Instead, they are proportional to the Clebsch-Gordan
coefficients of the horizontal symmetry group.
•
When a Higgs is found, the deviation of its fermion-pair decay rates from the
SM predictions may indicate the presence of a horizontal symmetry.
eLeR
qL qR
illustration: `SM Higgs’ is a horizontal singlet
1. the SM Higgs does not decay to a fermion pair whose L and R
belong to different (horizontal) irreducible representations (IR)
2. its decay rate to every member of an IR is the same, instead of
proportional to the square mass of the member
3. if the top quark (or the tau) belongs to an irreducible triplet, then
the Higgs decay rate is 1/3 to 1/9 of the SM rate. This makes
the Higgs detection more difficult
4. if the top quark (or the tau) belongs to an irreducible doublet,
then the Higgs decay rate is 1/2 to 1/4 of the SM rate
5. if all decay rates are the same as the SM rates, then horizontal
symmetry either does not exist, or all fermions belong to a 1dimensional IR
6. otherwise we know which IR each fermion belongs to, which
greatly constrains possible horizontal groups and models.
Conclusion
• The regularity of neutrino mixing suggests the presence of a
horizontal symmetry A4 , S3 , S 4 or
for
, groups containing
S4
leptons, but we do not know theoretically why these groups
• No simple regularity seems to be present in quark mixing. That
makes the idea of horizontal symmetry somewhat of an enigma
• An experimental test for the presence of horizontal symmetry is
suggested: measure the SM Higgs decay rates to fermion pairs and
compared them with the SM rates.
• The experimental results should give much better insight into this
oldest unsolved puzzle in particle physics: `who order that?’


M     


  


The coupling of the SM Higgs to fermion pairs is of the form
In the presence of a horizontal symmetry,
the mass matrix is then
The SM Higgs is a singlet, with
For irreducible representations,
hence
. .Its coupling to the ith lepton pair is
In SM, the Higgs to ith fermion pair decay rate is proportional to
With horizontal symmetry, if i belongs to an irreducible triplet, the it is
proportional to
where R is the contribution from non-singlet Higgs. Its value is model
dependent, but for those models in which the vev can be chosen for M to
be diagonal, then
A similar formula holds if i belongs to an irreducible doublet.
1. Horizontal symmetry?
2. An experimental test.