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ONE-LOOP NEUTRINO MASS IN SU(5)*
Ilja Doršner
University of Split
Brda 2016
Selected topics in flavor and collider physics October 21st, 2016
*I. Doršner, S. Fajfer, N. Košnik, work in progress.
• ONE-LOOP NEUTRINO MASS WITH LEPTOQUARKS
• ONE-LOOP NEUTRINO MASS MECHANISM IN SU(5)
• CONCLUSIONS
OUTLINE
ONE-LOOP NEUTRINO MASS MECHANISM
STANDARD MODEL TWO SCALAR LEPTOQUARK MULTIPLETS
+
◎C.-K. Chua, X.-G. He, W.-Y. P. Hwang, Phys. Lett. B479 (2000).
◎
SCALAR LEPTOQUARKS 1
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
LEPTOQUARK (LQ) MULTIPLETS:
◎I. Doršner, S. Fajfer, A. Greljo, J.F. Kamenik, N. Košnik, Phys. Rept. 641 (2016).
◎
1
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
SCALAR LEPTOQUARKS
LQ NOMENCLATURE :
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
�mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
◎W. Büchmuller et al., Phys. Lett. B 191, 442 (1987).
◎
ν MASS LQs:
SCALAR LEPTOQUARKS VS. ν MASS
+ ( ⋁ )
1
yij5i5j10
yij5i5j15
yij10i5j45
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
1
yij5i5j10
yij5i5j15
yij10i5j45
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
1
yij5i5j10
yij5i5j15
yij10i5j45
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
1
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
�mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
p DECAY LQs: ( )
SCALAR LEPTOQUARKS VS. p DECAY 1
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
1
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
�mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
ν MASS VS. p DECAY 1
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
�mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
A(NOTHER) WORD ABOUT NOMENCLATURE 1
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
1
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
≡ Yukawa coupling matrix
i, j (= 1, 2, 3) are flavor indices
a, b (= 1, 2) are SU(2) indices
U ≡ Pontecorvo-Maki-Nakagawa-Sakata unitary mixing matrix
1
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
1
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
1
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
≡ Yukawa coupling matrix
i, j (= 1, 2, 3) are flavor indices
a, b (= 1, 2) are SU(2) indices
A(NOTHER) WORD ABOUT NOMENCLATURE
ONE-LOOP NEUTRINO MASS
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
ONE-LOOP NEUTRINO MASS
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
ONE-LOOP NEUTRINO MASS
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
ONE-LOOP NEUTRINO MASS
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
1
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
ONE-LOOP NEUTRINO MASS
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
ONE-LOOP NEUTRINO MASS
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
IMPORTANT ISSUES
WHAT HAPPENED WITH THE LQ DIQUARK COUPLINGS?
LQ MASSES ARE FREE PARAMETERS… 1
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
ARE ALL A PRIORI UNKNOWN MATRICES…
RECENT DEVELOPMENTS
◎K. Cheung, T. Nomura, H. Okada, arXiv:1610.02322.
◎
“A testable radiative neutrino mass model without additional symmetries and explanation for the b → sℓ+ℓ− anomaly”
ONE-LOOP ν MASSES IN SU(5)
2
SCALAR LQs SU(5)
S3 ⌘ (3,3, 1/3) 45
R2 ⌘ (3,2, 7/6) 45˜R2 ⌘ (3,2, 1/6) 10,15˜S1 ⌘ (3,1, 4/3) 45
S1 ⌘ (3,1, 1/3) 5,45¯S1 ⌘ (3,1,�2/3) 10
d e
u ⌫
u e
s µ
c µ
Q = �1
3
Q = +2
3
Q = +5
3
li = e, µ
qi = (u, d), (c, s)
yi
yue yde ycµ ysµ
yd⌫
yu⌫
|yde| 0.34⇣ mLQ
1TeV
⌘
|yue| 0.36⇣ mLQ
1TeV
⌘
ONE-LOOP ν MASSES IN SU(5)
1
10
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
1
10
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
1
10
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
UNIFICATION OF GAUGE
COUPLINGS
UNIFICATION
TYPE II SEE-SAW ν MASSES
THE STANDARD MODEL SYMMETRY
BREAKING
CHARGED LEPTON &
DOWN-TYPE QUARK MASSES
◎ ⦿ ◆
SCALAR REPRESENTATIONS IN SU(5):
THE STANDARD
MODEL SYMMETRY BREAKING
◉
1
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
◉H. Georgi, S. L. Glashow, Phys. Rev. Lett. 32 (1974). ◎H. Murayama, T. Yanagida, Mod. Phys. Lett. A7 (1992). ⦿I. Doršner, P. Fileviez Perez, Nucl. Phys. B723 (2005). ◆H. Georgi, C. Jarlskog., Phys. Lett. B86 (1979).
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
CHARGED FERMION MASSES
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
ONE-LOOP ν MASSES IN SU(5)
1
10
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
1
10
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
SCALAR REPRESENTATIONS IN SU(5):
1
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
+4/33
(S3)
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
�51045
COMPRISE THE STANDARD MODEL FERMIONS
ONE-LOOP ν MASSES IN SU(5)
1
10
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
1
10
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
1
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
A POSSIBLE SU(5) SET-UP:
1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
+4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1, S1
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
�51045
1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
+4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
�51045
ONE-LOOP NEUTRINO MASS
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
1
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
d ⌫
d e
u ⌫
u e
s µ
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
10i5j5 & 10i5j45
5i5j15
�51545
�
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
10i5j5 & 10i5j45
5i5j15
�51545
�
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
10i5j5 & 10i5j45
5i5j15
�51545
�
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
≡ dimensionful parameter
ONE-LOOP ν MASSES IN SU(5)
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
�51045
�
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
+4/33
(S3)
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
�51045
p DECAY
1
(S3)
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
�51045
�
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
⌫ ⌫
(S1/33 , ˜R�1/3⇤
2 )
b b
(S1/3⇤3 , ˜R�1/3
2 )
hHi
Figure 5: One-loop diagram of neutrino mass generation. Scalar LQ fields that mix due tothe couplings with the SM Higgs boson H are bracketed. Flavor indices are omitted.
in Ref. [95] also addresses the nature of neutrino mass that is of Majorana typethrough an inverse seesaw mechanism.
2.2. Low-energy scenarios of neutrino massLQs have been used to address the question of neutrino masses and associ-
ated mixing parameters within extensions of the SM that do not necessarily aimor lead to unification [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].Namely, it has been stipulated that the neutrino masses are of the one-loop ortwo-loop origin, where some of the fields that participate in the loops are scalarLQs. This approach is used to naturally explain or justify observed smallnessof neutrino masses.
Let us outline the main features of this approach. We require the presenceof ˜R2. More specifically, we need a term proportional to yRL
2¯dR⌫L ˜R�1/3
2 , wherewe omit flavor indices. Note that ˜R�1/3
2 couples neutrino to the right-chiraldown-type quark. The other necessary ingredient is the presence of either S1
or S3. Both of these LQs couple the leptonic doublet with the quark doubletrepresentation. They thus couple neutrino to the left-chiral down-type quark.The required couplings are either yLL
1¯dCL⌫LS1 or yLL
3¯dCL⌫LS
1/33 . At this point
it is sufficient to introduce mixing between ˜R2 and either S1 or S3 through theHiggs boson H of the SM to generate neutrino mass(es) at the loop level. Theparticles in the loop are leptoquarks and the down-type quarks. The schematicdepiction of this approach can be written as follows:
yRL2
¯dR⌫L ˜R�1/32
% ˜R2HS1 ! yLL1
¯dCL⌫LS1
& ˜R2HS3 ! yLL3
¯dCL⌫LS1/33
(23)
The one-loop diagram of neutrino mass generation that corresponds to the mix-ing between ˜R2 and S3 is given in Fig. 5. The loop is closed through a massinsertion for the down-type quarks. To have a loop with the up-type quarksone would need to start with R2/3
2 leptoquark. It is also possible to generateneutrino masses through the loops that involve leptoquarks of vector nature.This has been done in Ref. [110]. We discuss several explicit realizations of thisapproach in what follows.
24
p DECAY
◎
◎I. Doršner, S. Fajfer, N. Košnik, Phys. Rev. D 86, 015013 (2012).
1
(S3)
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
�51045
�
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
2
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
LEPTOQUARK MULTIPLET COULD BE LIGHT IF NEEDED…
1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
�51045
p DECAY
1
10
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
1
10
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
1
S3, R2, S1, S1, S1
yij5i5j10
yij5i5j15
yij10i5j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
Q
1/3
−2/3
4/3
3
SCALAR LQs SU(5)
S3 ⌘ (3,3, 1/3) 45
R2 ⌘ (3,2, 7/6) 45
R2 ⌘ (3,2, 1/6) 10,15
S1 ⌘ (3,1, 4/3) 45
S1 ⌘ (3,1, 1/3) 5,45
S1 ⌘ (3,1,�2/3) 10
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
1
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
1
R�1/3 ⇤2
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
3
SCALAR LQs SU(5)
S3 ⌘ (3,3, 1/3) 45
R2 ⌘ (3,2, 7/6) 45
R2 ⌘ (3,2, 1/6) 10,15
S1 ⌘ (3,1, 4/3) 45
S1 ⌘ (3,1, 1/3) 5,45
S1 ⌘ (3,1,�2/3) 10
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
1
R2/3 ⇤2
R�1/3 ⇤2
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
1
R2/3 ⇤2
R�1/3 ⇤2
S�2/33
S1/33
S4/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
1
R2/3 ⇤2
R�1/3 ⇤2
S�2/33
S1/33
S4/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
1
R2/3 ⇤2
R�1/3 ⇤2
S�2/33
S1/33
S4/33
S1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
1
R2/3 ⇤2
R5/32
R2/3 ⇤2
R�1/3 ⇤2
S�2/33
S1/33
S4/33
S1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
1
R2/3 ⇤2
R5/32
R2/3 ⇤2
R�1/3 ⇤2
S�2/33
S1/33
S4/33
S1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
5/3
p DECAY
3
SCALAR LQs SU(5)
S3 ⌘ (3,3, 1/3) 45
R2 ⌘ (3,2, 7/6) 45
R2 ⌘ (3,2, 1/6) 10,15
S1 ⌘ (3,1, 4/3) 45
S1 ⌘ (3,1, 1/3) 5,45
S1 ⌘ (3,1,�2/3) 10
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
1
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
1
R�1/3 ⇤2
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
3
SCALAR LQs SU(5)
S3 ⌘ (3,3, 1/3) 45
R2 ⌘ (3,2, 7/6) 45
R2 ⌘ (3,2, 1/6) 10,15
S1 ⌘ (3,1, 4/3) 45
S1 ⌘ (3,1, 1/3) 5,45
S1 ⌘ (3,1,�2/3) 10
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
[ ]
3
SCALAR LQs SU(5)
S3 ⌘ (3,3, 1/3) 45
R2 ⌘ (3,2, 7/6) 45
R2 ⌘ (3,2, 1/6) 10,15
S1 ⌘ (3,1, 4/3) 45
S1 ⌘ (3,1, 1/3) 5,45
S1 ⌘ (3,1,�2/3) 10
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
1
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
1
R�1/3 ⇤2
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
3
SCALAR LQs SU(5)
S3 ⌘ (3,3, 1/3) 45
R2 ⌘ (3,2, 7/6) 45
R2 ⌘ (3,2, 1/6) 10,15
S1 ⌘ (3,1, 4/3) 45
S1 ⌘ (3,1, 1/3) 5,45
S1 ⌘ (3,1,�2/3) 10
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
*
0 0 0 0
0 0 0 0
4 × 4
p DECAY
3
SCALAR LQs SU(5)
S3 ⌘ (3,3, 1/3) 45
R2 ⌘ (3,2, 7/6) 45
R2 ⌘ (3,2, 1/6) 10,15
S1 ⌘ (3,1, 4/3) 45
S1 ⌘ (3,1, 1/3) 5,45
S1 ⌘ (3,1,�2/3) 10
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
1
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
1
R�1/3 ⇤2
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
3
SCALAR LQs SU(5)
S3 ⌘ (3,3, 1/3) 45
R2 ⌘ (3,2, 7/6) 45
R2 ⌘ (3,2, 1/6) 10,15
S1 ⌘ (3,1, 4/3) 45
S1 ⌘ (3,1, 1/3) 5,45
S1 ⌘ (3,1,�2/3) 10
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
[ ]
3
SCALAR LQs SU(5)
S3 ⌘ (3,3, 1/3) 45
R2 ⌘ (3,2, 7/6) 45
R2 ⌘ (3,2, 1/6) 10,15
S1 ⌘ (3,1, 4/3) 45
S1 ⌘ (3,1, 1/3) 5,45
S1 ⌘ (3,1,�2/3) 10
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
1
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
1
R�1/3 ⇤2
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
3
SCALAR LQs SU(5)
S3 ⌘ (3,3, 1/3) 45
R2 ⌘ (3,2, 7/6) 45
R2 ⌘ (3,2, 1/6) 10,15
S1 ⌘ (3,1, 4/3) 45
S1 ⌘ (3,1, 1/3) 5,45
S1 ⌘ (3,1,�2/3) 10
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
(3,1, 1/3)
(3,1, 4/3)
(3,1,�2/3)
SU(3)⇥ SU(2)⇥ U(1)
*
0 0 0 0
0 0 0 0
ONE-LOOP ν MASSES
1
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
1
R�1/3 ⇤2
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
[ ] 1
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
1
R�1/3 ⇤2
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
* m11 m12
m12 m22
1
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
1
R�1/3 ⇤2
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
cθ sθ
−sθ cθ
1
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
5i5j10
1
R�1/3 ⇤2
S1/33
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
(S1)
(R2)
(S3, S1)
(S3)
(R2, S1, S1)
10i5j5 & 10i5j45
5i5j15
�51545
→
m1 0
0 m2
→
2 2
2 2
m11 m12
m12 m22
2 2
2 2
2
2
ONE-LOOP ν MASSES 1
(m⌫)ij =3s✓c✓16⇡2
X
k=d,s,b
mk[B0(0,m2k,m
21)�B0(0,m
2k,m
22)]{yiky0jk + yjky
0ik}
B0(0,m2k,m
21)�B0(0,m
2k,m
22) =
m22[lnm
22 � lnm2
k]
m22 �m2
k
� m21[lnm
21 � lnm2
k]
m21 �m2
k
R2/3 ⇤2
R5/32
R2/3 ⇤2
R�1/3 ⇤2
S�2/33
S1/33
S4/33
S1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
1
(m⌫)ij =3s✓c✓16⇡2
X
k=d,s,b
mk[B0(0,m2k,m
21)�B0(0,m
2k,m
22)]{yiky0jk + yjky
0ik}
B0(0,m2k,m
21)�B0(0,m
2k,m
22) =
m22[lnm
22 � lnm2
k]
m22 �m2
k
� m21[lnm
21 � lnm2
k]
m21 �m2
k
R2/3 ⇤2
R5/32
R2/3 ⇤2
R�1/3 ⇤2
S�2/33
S1/33
S4/33
S1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
B0 – Passarino-Veltman function
ONE-LOOP ν MASSES
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
1
y0ij10i5j45
(m⌫)ij =3s✓c✓16⇡2
X
k=d,s,b
mk[B0(0,m2k,m
21)�B0(0,m
2k,m
22)]{yiky0jk + yjky
0ik}
B0(0,m2k,m
21)�B0(0,m
2k,m
22) =
m22[lnm
22 � lnm2
k]
m22 �m2
k
� m21[lnm
21 � lnm2
k]
m21 �m2
k
R2/3 ⇤2
R5/32
R2/3 ⇤2
R�1/3 ⇤2
S�2/33
S1/33
S4/33
S1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
1
(m⌫)ij =3s✓c✓16⇡2
X
k=d,s,b
mk[B0(0,m2k,m
21)�B0(0,m
2k,m
22)]{yiky0jk + yjky
0ik}
B0(0,m2k,m
21)�B0(0,m
2k,m
22) =
m22[lnm
22 � lnm2
k]
m22 �m2
k
� m21[lnm
21 � lnm2
k]
m21 �m2
k
R2/3 ⇤2
R5/32
R2/3 ⇤2
R�1/3 ⇤2
S�2/33
S1/33
S4/33
S1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
ONE-LOOP ν MASSES 1
(m⌫)ij =3s✓c✓16⇡2
X
k=d,s,b
mk[B0(0,m2k,m
21)�B0(0,m
2k,m
22)]{yiky0jk + yjky
0ik}
B0(0,m2k,m
21)�B0(0,m
2k,m
22) =
m22[lnm
22 � lnm2
k]
m22 �m2
k
� m21[lnm
21 � lnm2
k]
m21 �m2
k
R2/3 ⇤2
R5/32
R2/3 ⇤2
R�1/3 ⇤2
S�2/33
S1/33
S4/33
S1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
yij dCL ieL jS
4/33
(S3)
1
y0 ⇠ (MTe �Md)/v45
y0ij10i5j45
(m⌫)ij =3s✓c✓16⇡2
X
k=d,s,b
mk[B0(0,m2k,m
21)�B0(0,m
2k,m
22)]{yiky0jk + yjky
0ik}
B0(0,m2k,m
21)�B0(0,m
2k,m
22) =
m22[lnm
22 � lnm2
k]
m22 �m2
k
� m21[lnm
21 � lnm2
k]
m21 �m2
k
R2/3 ⇤2
R5/32
R2/3 ⇤2
R�1/3 ⇤2
S�2/33
S1/33
S4/33
S1
10i & 5i (i = 1, 2, 3)
126 � (5, 10, 15, ,45 )
yij16i16j126
�yij uCL i⌫L jS
�2/33
2�1/2yij uCL ieL jS
1/33
2�1/2yij dCL i⌫L jS
1/33
1
(S1)
(R2)
(S3, S1)
S3, R2, S1, S1, S1
yij10i5j5
yij10i10j5
yij5i5j10
yij5i5j15
yij10i5j45
yij10i10j45
5
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
Me – charged lepton mass matrix Md – down-type quark mass matrix
A LIST OF BENEFITS
p DECAY CONSTRAINTS CAN BE ACCOMMODATED
LQ MASSES COULD BE CONSTRAINED THROUGH THE GAUGE COUPLING UNIFICATION…
RELEVANT YUKAWA COUPLING MATRICES ARE RELATED TO FERMION MASSES AND/OR POSSESS ADDITIONAL SYMMETRY. THIS NOT ONLY REDUCES THE TOTAL NUMBER OF PARAMETERS BUT HELPS RELATE LEPTOQUARK DECAY PATTERNS TO NEUTRINO MASSES…
◎
◎P. Fileviez Perez, T. Han, Gui-Yu Huang, T. Li, K. Wang, Phys. Rev. D 78, 071301, (2008).
CONCLUSIONS
SU(5) CAN ACCOMMODATE WITH EASE THE ONE-LOOP NEUTRINO MASS MECHANISM THAT IS BASED ON THE LEPTOQUARK MULTIPLET MIXING. THE USE OF SU(5) CAN INCREASE PREDICTIVITY OF THE SET-UP. THIS COULD ESPECIALLY BE REFLECTED IN THE DECAY PATTERNS OF THE RELEVANT LEPTOQUARK MULTIPLETS.
THANK YOU
dorsner@fesb.hr
100 105 108 1011 1014 101710
20
30
40
50
60
ΜGeV
Α i#
Α1#1
Α2#1
Α3#1
(GeV)
1
m = 10 GeVLQ2
1
10
15
45
H ⌘ (1,2,�1/2)
yRL2 , yLL
1 , yLL3
yRL2
L � �yRL2 ij d
iRR
a2✏
abLj,bL
L � �yRL2 ij d
iRe
jLR
2/32 + (yRL
2 U)ij diR⌫
jLR
�1/32
S3 ⌘
R2 ⌘
R2 ⌘
S1 ⌘
S1 ⌘
S1 ⌘
mDdLdR
SU(3)⇥ SU(2)⇥ U(1)
(3,3, 1/3)
(3,2, 7/6)
(3,2, 1/6)
◎
◎H. Murayama, T. Yanagida, Mod. Phys. Lett. A7 (1992).
STANDARD MODEL + ( 2 × )
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