I. Doršner, Leptoquark Mass Limit in SU(5)

LEPTOQUARK MASS LIMIT IN SU(5)*
Ilja Doršner
University of Sarajevo, Bosnia and Herzegovina
BALKAN WORKSHOP 2013 — BW2013
Vrnjačka Banja, Serbia
April 28, 2013
I. Doršner, Phys. Rev. D 86:055009, 2012, 1206.5998;
I. Doršner, S. Fajfer and N. Košnik, Phys. Rev. D 86:015013, 2012, 1204.0674.
*

• MINIMAL UNIFICATION OF MATTER
THE GEORGI-GLASHOW SU(5) SCENARIO
• d = 6 PROTON DECAY OPERATORS
SCALAR CONTRIBUTIONS
• MINIMAL VIABLE SU(5) UNIFICATION
• p-DECAY PREDICTIONS
SCALAR CONTRIBUTIONS
OUTLINE

THE STANDARD MODEL COMPRISES 15 FERMIONS.
THE GEORGI-GLASHOW SU(5) MODEL*
*See talk by Borut Bajc.

SU(5) SCENARIO*
*H. Georgi and S.L. Glashow (1974).
LEPTONS
QUARKS
FIFTEEN FERMIONS OF THE STANDARD MODEL:

LEPTONS
QUARKS
SU(5) SCENARIO*

FERMION MASSES
(SCALAR REPRESENTATIONS IN THE MINIMAL SU(5))
&
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1

NOTATION
(VACUUM EXPECTATION VALUE)
MD = Y1v⇤
45
1
2
Y3v⇤
5
ME = 3Y T
1 v⇤
45
1
2
Y T
3 v⇤
5
(Y1)ij10i5j45⇤
(Y3)ij10i5j5⇤
h4515
1 i = h4525
2 i = h4535
3 i = v45/
p
2
E†
RDLMdiag
D Mdiag
E ET
L D⇤
R = 4Y1v45
h55
i = v5/
p
2
|v5|2
/2 + 12|v45|2
= v2
t ¯t
(g 2)µ
45 2 126
&

WHAT GOES WRONG WITH SU(5)?*

FERMION MASSES*
v = 246 GeV
Y 10
ij 10i10j45
Y 5
ij10i5j45⇤
Y 10
ij 10i10j5
Y 5
ij10i5j5⇤
MD = Y 5
v⇤
45⇤
MD =
1
2
Y 5
v⇤
5
ME = 3Y 5 T
v⇤
45
1 5 T ⇤
1
|v5|2
/2 + 12|v45|2
= v2
v = 246 GeV
Y 10
ij 10i10j45
Y 5
ij10i5j45⇤
Y 10
ij 10i10j5
Y 5
ij10i5j5⇤
MD = Y 5
v⇤
45⇤
MD =
1
2
Y 5
v⇤
5
Y 10
ij 10i10j45
Y 5
ij10i5j45⇤
Y 10
ij 10i10j5
Y 5
ij10i5j5⇤
MD = Y 5
v⇤
45⇤
MD =
1
2
Y 5
v⇤
5
ME = 3Y 5 T
v⇤
45
ME =
1
2
Y 5 T
v⇤
5
p 5 5 T
Y 5
ij10i5j45⇤
Y 10
ij 10i10j5
Y 5
ij10i5j5⇤
MD = Y 5
v⇤
45⇤
MD =
1
2
Y 5
v⇤
5
ME = 3Y 5 T
v⇤
45
ME =
1
2
Y 5 T
v⇤
5
MU = 2
p
2(Y 5
Y 5 T
)v45
p 10 10 T
45
ME =
1
2
Y 5 T
v⇤
5
MU = 2
p
2(Y 5
Y 5 T
)v45
MU =
p
2(Y 10
+ Y 10 T
)v5
10 ⇥ 10 = 5 45 : MU
10 ⇥ 5 = 5 45 : ME, MD
10+1
⇥ 10+1
= 5
+2
45
+2
: MD
10+1
⇥ 5
3
= 5 2
45 2
: MU
3 3 6 6
*See talk by Borut Bajc.

FERMION MASSES
v = 246 GeV
Y 10
ij 10i10j45
Y 5
ij10i5j45⇤
Y 10
ij 10i10j5
Y 5
ij10i5j5⇤
MD = Y 5
v⇤
45⇤
MD =
1
2
Y 5
v⇤
5
ME = 3Y 5 T
v⇤
45
1 5 T ⇤
1
|v5|2
/2 + 12|v45|2
= v2
v = 246 GeV
Y 10
ij 10i10j45
Y 5
ij10i5j45⇤
Y 10
ij 10i10j5
Y 5
ij10i5j5⇤
MD = Y 5
v⇤
45⇤
MD =
1
2
Y 5
v⇤
5
Y 10
ij 10i10j45
Y 5
ij10i5j45⇤
Y 10
ij 10i10j5
Y 5
ij10i5j5⇤
MD = Y 5
v⇤
45⇤
MD =
1
2
Y 5
v⇤
5
ME = 3Y 5 T
v⇤
45
ME =
1
2
Y 5 T
v⇤
5
p 5 5 T
Y 5
ij10i5j45⇤
Y 10
ij 10i10j5
Y 5
ij10i5j5⇤
MD = Y 5
v⇤
45⇤
MD =
1
2
Y 5
v⇤
5
ME = 3Y 5 T
v⇤
45
ME =
1
2
Y 5 T
v⇤
5
MU = 2
p
2(Y 5
Y 5 T
)v45
p 10 10 T
45
ME =
1
2
Y 5 T
v⇤
5
MU = 2
p
2(Y 5
Y 5 T
)v45
MU =
p
2(Y 10
+ Y 10 T
)v5
10 ⇥ 10 = 5 45 : MU
10 ⇥ 5 = 5 45 : ME, MD
10+1
⇥ 10+1
= 5
+2
45
+2
: MD
10+1
⇥ 5
3
= 5 2
45 2
: MU
3 3 6 6

NOTATION
(MASS MATRICES AND UNITARY TRANSFORMATIONS)
UP-TYPE QUARKS, DOWN-TYPE QUARKS AND CHARGED LEPTONS:
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
5 =
0
@
H
1
A
(p ! e+
⇡0
) ⇠
↵2
v4
5m4
3
8
(VUD)11(VUD)13m⌧ mb
2

IS UNIFICATION WRONG WITHIN SU(5)?*
1
↵ 1
1
(p ! e+
⇡0
) ⇠
↵2
v4
5m4
3
8
2
(p ! µ+
⇡0
) ⇠
↵2
v4
5m4
3
8
(VUD)11(VUD)12m⌧ ms
2
(p ! e+
⇡0
) ⇠
↵2
v4
5m4 (VUD)11[mu +
3
4
md +
1
4
m⌧ ]
2
3
2
mb(VUD)13
2
(p ! µ+
⇡0
) ⇠
↵2
v4
5m4 (VUD)11[mu +
3
4
md +
1
4
m⌧ ]
2
3
2
ms(VUD)12
2
(p ! e+
⇡0
) ⇠
↵2
(VUD)11[mu +
3
md] +
1
(V †
UDU⇤
2 Mdiag
E U†
2 )11
2
p!⇡+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
d
p!K+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
s
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
↵ 1
2
↵ 1
3
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
i=1,2,3
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵
1
p!⇡+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
d
p!K+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
s
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
↵ 1
2
↵ 1
3
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵

50
M 1012
GeV
24 = (⌃8, ⌃3, ⌃(3,2), ⌃(¯3,2), ⌃24)
✏abcuT
a iCub j
3
3 c
10i 5i , i = 1, 2, 3
24 5 15
16i , i = 1, 2, 3
210 10 126 126
120
⌃3 = (1, 3, 0)
a = (1, 3, 1)
b = (3, 2, 1/6)
ADDRESSING NEUTRINO MASSES ALSO ADDRESSES UNIFICATION
IN A SATISFACTORY MANNER!
NEUTRINO MASSES WITHIN SU(5)?*
¶I. Doršner and P. Fileviez Pérez, Nucl. Phys. B 723:53-76, 2005, hep-ph/0504276.
‡B. Bajc and G. Senjanović, JHEP 0708 014, 2007, hep-ph/0612029.
‡¶

*See talk by Andrea Romanino.
UNIFICATION IN SU(5)*
1
↵ 1
1
(p ! e+
⇡0
) ⇠
↵2
v4
5m4
3
8
2
(p ! µ+
⇡0
) ⇠
↵2
v4
5m4
3
8
2
(p ! e+
⇡0
) ⇠
↵2
v4
5m4 (VUD)11[mu +
3
4
md +
1
4
m⌧ ]
2
3
2
mb(VUD)13
2
(p ! µ+
⇡0
) ⇠
↵2
v4
5m4 (VUD)11[mu +
3
4
md +
1
4
m⌧ ]
2
3
2
ms(VUD)12
2
(p ! e+
⇡0
) ⇠
↵2
(VUD)11[mu +
3
md] +
1
(V †
UDU⇤
2 Mdiag
E U†
2 )11
2
p!⇡+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
d
p!K+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
s
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
↵ 1
2
↵ 1
3
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
i=1,2,3
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵
1
p!⇡+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
d
p!K+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
s
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
↵ 1
2
↵ 1
3
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵

1
↵ 1
1
(p ! e+
⇡0
) ⇠
↵2
v4
5m4
3
8
2
(p ! µ+
⇡0
) ⇠
↵2
v4
5m4
3
8
2
(p ! e+
⇡0
) ⇠
↵2
v4
5m4 (VUD)11[mu +
3
4
md +
1
4
m⌧ ]
2
3
2
mb(VUD)13
2
(p ! µ+
⇡0
) ⇠
↵2
v4
5m4 (VUD)11[mu +
3
4
md +
1
4
m⌧ ]
2
3
2
ms(VUD)12
2
(p ! e+
⇡0
) ⇠
↵2
(VUD)11[mu +
3
md] +
1
(V †
UDU⇤
2 Mdiag
E U†
2 )11
2
UNIFICATION IN SU(5)*
p!⇡+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
d
p!K+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
s
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
↵ 1
2
↵ 1
3
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
i=1,2,3
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵
1
p!⇡+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
d
p!K+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
s
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
↵ 1
2
↵ 1
3
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵
*See talk by Andrea Romanino.

NOTATION
(MASS MATRICES AND UNITARY TRANSFORMATIONS)
MAJORANA NEUTRINOS:
QUALITATIVE ASPECTS OF NEUTRINO PHYSICS ARE NOT
RELEVANT FOR DISCUSSION OF p-DECAY!

HOW PREDICTIVE IS SU(5) FOR p-DECAY?*

≡ Yukawa coupling(s) ≡ Leptoquark mass
*S. Weinberg, Phys. Rev. D 22:1694, 1980.
p-DECAY WIDTHS
(SCALAR CONTRIBUTIONS*)

≡ Yukawa coupling(s) ≡ Leptoquark mass
*S. Weinberg, Phys. Rev. D 22:1694, 1980.
p-DECAY WIDTHS
a6 ⇠
Y 2
m2
LQ
E = DC
D = EC
U = UC
U†
D = VCKM
N = I
E = I
D = I

EXPERIMENTAL INPUT
(PROTON DECAY)
5
PROCESS ⌧p (1033
years)
p ! K+
¯⌫ 4.0
p ! ⇡+
¯⌫ 0.025
p ! ⇡0
e+
13.0
j = 1, 2, 3 j = 1, 2
La ⌘ (1, 2, 1/2)a = (⌫a ea)T
eC
a ⌘ (1, 1, 1)a
Qa ⌘ (3, 2, 1/6)a = (ua da)T

IS AN ACCURATE LIMIT?
KEY QUESTION…

LEPTOQUARK IN SU(5)
(p-DECAY MEDIATING SCALAR LEPTOQUARK)
THERE IS ONLY ONE SET OF PROTON DECAY MEDIATING
SCALARS IN THE MINIMAL SU(5) SETUP!
1
↵ 1
1
5 =
0
@
H
1
A
(p ! e+
⇡0
) ⇠
↵2
v4
5m4
3
8
2
(p ! µ+
⇡0
) ⇠
↵2
v4
5m4
3
8
2
(p ! e+
⇡0
) ⇠
↵2
v4
5m4 (VUD)11[mu +
3
4
md +
1
4
m⌧ ]
2
3
2
mb(VUD)13
2
(p ! µ+
⇡0
) ⇠
↵2
(V ) [m +
3
m +
1
m ]
2
3
m (V )
2

SU(5) Y 1
ij10i1j10⇤
Y 5
ij5i5j10
(3, 1, 2/3)
⌘ Y 1
ijuC T
a i C⌫C
j
⇤
a 2 1/2
✏abcY 5
ijdC T
a i CdC
b j c
Y 5
= Y 5 T
OH(d↵, e ) = a(d↵, e ) uT
L C 1
d↵ uT
L C 1
e
OH(d↵, eC
) = a(d↵, eC
) uT
L C 1
d↵ eC†
L C 1
uC⇤
OH(dC
↵ , e ) = a(dC
↵ , e ) dC
↵
†
L C 1
uC⇤
uT
L C 1
e
OH(dC
↵ , eC
) = a(dC
↵ , eC
) dC
↵
†
L C 1
uC⇤
eC†
L C 1
uC⇤
OH(d↵, d , ⌫i) = a(d↵, d , ⌫i) uT
L C 1
d↵ dT
L C 1
⌫i
OH(d↵, dC
, ⌫i) = a(d↵, dC
, ⌫i) dC†
L C 1
uC⇤
dT
↵ L C 1
⌫i
OH(d↵, dC
, ⌫C
i ) = a(d↵, dC
, ⌫C
i ) uT
L C 1
d↵ ⌫C
i
†
L C 1
dC⇤
OH(dC
↵ , dC
, ⌫C
i ) = a(dC
↵ , dC
, ⌫C
i ) dC†
L C 1
uC⇤
⌫C
i
†
L C 1
dC
↵
⇤
i(= 1, 2, 3)
d = 6 PROTON DECAY OPERATORS
(SCALAR CONTRIBUTIONS)
(3, 1, 1/3) 2 1/2
✏abcY 5
ijuC T
a i CdC
b j
⇤
c
⌘
2✏abc[Y 10
ij + Y 10
ji ]dT
a iCub j c
2 1/2
Y 5
ijuT
a iCej
⇤
a Y 1
ijdC T
a i C⌫C
j a2[Y 10
ij + Y 10
ji ]eC T
i CuC
a j a
2 1/2
Y 5
ijdT
a iC⌫j
⇤
a
SU(5) ⇥ U(1) Y 10
ij 10+1
i 10+1
j 50 2
(3, 1, 1/3) 2
⌘
12 1/2
✏abc[Y 10
ij + Y 10
ji ]uT
a iCdb j c
3 1/2
[Y 10
ij + Y 10
ji ]⌫C T
i CdC
a j a
↵, (= 1, 2)
↵ + < 4
L(= (1 5)/2)
MU,D,E ! Mdiag
U,D,E

SU(5) Y 1
ij10i1j10⇤
Y 5
ij5i5j10
(3, 1, 2/3)
⌘ Y 1
ijuC T
a i C⌫C
j
⇤
a 2 1/2
✏abcY 5
ijdC T
a i CdC
b j c
Y 5
= Y 5 T
OH(d↵, e ) = a(d↵, e ) uT
L C 1
d↵ uT
L C 1
e
OH(d↵, eC
) = a(d↵, eC
) uT
L C 1
d↵ eC†
L C 1
uC⇤
OH(dC
↵ , e ) = a(dC
↵ , e ) dC
↵
†
L C 1
uC⇤
uT
L C 1
e
OH(dC
↵ , eC
) = a(dC
↵ , eC
) dC
↵
†
L C 1
uC⇤
eC†
L C 1
uC⇤
OH(d↵, d , ⌫i) = a(d↵, d , ⌫i) uT
L C 1
d↵ dT
L C 1
⌫i
OH(d↵, dC
, ⌫i) = a(d↵, dC
, ⌫i) dC†
L C 1
uC⇤
dT
↵ L C 1
⌫i
OH(d↵, dC
, ⌫C
i ) = a(d↵, dC
, ⌫C
i ) uT
L C 1
d↵ ⌫C
i
†
L C 1
dC⇤
OH(dC
↵ , dC
, ⌫C
i ) = a(dC
↵ , dC
, ⌫C
i ) dC†
L C 1
uC⇤
⌫C
i
†
L C 1
dC
↵
⇤
i(= 1, 2, 3)
*P. Nath and P.F. Pérez, Phys. Rept. 441 (2007) 191-317.
WE WILL TAKE NEUTRINOS TO BE MAJORANA
PARTICLES IN WHAT FOLLOWS.

SU(5) Y 1
ij10i1j10⇤
Y 5
ij5i5j10
(3, 1, 2/3)
⌘ Y 1
ijuC T
a i C⌫C
j
⇤
a 2 1/2
✏abcY 5
ijdC T
a i CdC
b j c
Y 5
= Y 5 T
OH(d↵, e ) = a(d↵, e ) uT
L C 1
d↵ uT
L C 1
e
OH(d↵, eC
) = a(d↵, eC
) uT
L C 1
d↵ eC†
L C 1
uC⇤
OH(dC
↵ , e ) = a(dC
↵ , e ) dC
↵
†
L C 1
uC⇤
uT
L C 1
e
OH(dC
↵ , eC
) = a(dC
↵ , eC
) dC
↵
†
L C 1
uC⇤
eC†
L C 1
uC⇤
OH(d↵, d , ⌫i) = a(d↵, d , ⌫i) uT
L C 1
d↵ dT
L C 1
⌫i
OH(d↵, dC
, ⌫i) = a(d↵, dC
, ⌫i) dC†
L C 1
uC⇤
dT
↵ L C 1
⌫i
OH(d↵, dC
, ⌫C
i ) = a(d↵, dC
, ⌫C
i ) uT
L C 1
d↵ ⌫C
i
†
L C 1
dC⇤
OH(dC
↵ , dC
, ⌫C
i ) = a(dC
↵ , dC
, ⌫C
i ) dC†
L C 1
uC⇤
⌫C
i
†
L C 1
dC
↵
⇤
i(= 1, 2, 3)
⌘ Y 1
ijuC T
a i C⌫C
j
⇤
a 2 1/2
✏abcY 5
ijdC T
a i CdC
b j c
Y 5
= Y 5 T
OH (d↵, e ) = a(d↵, e ) uT
L C 1
d↵ uT
L C 1
e
OH (d↵, eC
) = a(d↵, eC
) uT
L C 1
d↵ eC†
L C 1
uC⇤
OH (dC
↵ , e ) = a(dC
↵ , e ) dC
↵
†
L C 1
uC⇤
uT
L C 1
e
OH (dC
↵ , eC
) = a(dC
↵ , eC
) dC
↵
†
L C 1
uC⇤
eC†
L C 1
uC⇤
OH (d↵, d , ⌫i) = a(d↵, d , ⌫i) uT
L C 1
d↵ dT
L C 1
⌫i
OH (d↵, dC
, ⌫i) = a(d↵, dC
, ⌫i) dC†
L C 1
uC⇤
dT
↵ L C 1
⌫i
OH (d↵, dC
, ⌫C
i ) = a(d↵, dC
, ⌫C
i ) uT
L C 1
d↵ ⌫C
i
†
L C 1
dC⇤
OH (dC
↵ , dC
, ⌫C
i ) = a(dC
↵ , dC
, ⌫C
i ) dC†
L C 1
uC⇤
⌫C
i
†
L C 1
dC
↵
⇤
i(= 1, 2, 3)
OH(d↵, e ) = a(d↵, e ) uT
L C 1
d↵ uT
L C 1
e
OH(d↵, eC
) = a(d↵, eC
) uT
L C 1
d↵ eC†
L C 1
uC⇤
OH(dC
↵ , e ) = a(dC
↵ , e ) dC
↵
†
L C 1
uC⇤
uT
L C 1
e
OH(dC
↵ , eC
) = a(dC
↵ , eC
) dC
↵
†
L C 1
uC⇤
eC†
L C 1
uC⇤
OH(d↵, d , ⌫i) = a(d↵, d , ⌫i) uT
L C 1
d↵ dT
L C 1
⌫i
OH(d↵, dC
, ⌫i) = a(d↵, dC
, ⌫i) dC†
L C 1
uC⇤
dT
↵ L C 1
⌫i
OH(d↵, dC
, ⌫C
i ) = a(d↵, dC
, ⌫C
i ) uT
L C 1
d↵ ⌫C
i
†
L C 1
dC⇤
OH(dC
↵ , dC
, ⌫C
i ) = a(dC
↵ , dC
, ⌫C
i ) dC†
L C 1
uC⇤
⌫C
i
†
L C 1
dC
↵
⇤
i(= 1, 2, 3)
↵, (= 1, 2)
↵ + < 4
L(= (1 5)/2)

p-DECAY WIDTHS
(p ! ¯⌫i⇡+
) =
(m2
p m2
⇡+ )2
32⇡f2
⇡m3
p
|↵ a(d1, dC
1 , ⌫i) + a(d1, d1, ⌫i)|2
(1 + D + F)2
(3, 1, 1/3)
(3, 3, 1/3)
(3, 1, 4/3)
(3, 1, 2/3)
SU(5) Y 10
ij 10i10j50
(3, 1, 1/3) 12 1/2
✏abc[Y 10
ij + Y 10
ji ]dT
a iCub j c
⌧ ⇠ 1
m > 1.0 ⇥ 1012
✓
↵
0.0112 GeV3
◆1/2
GeV
(p ! ⇡+
¯⌫)
(p ! K+ ¯⌫)
= 9.0
1
⌧ ⌘
⌧ ⇠ 1
PARTIAL LIFETIME

d = 6 PROTON DECAY COEFFICIENTS
SU(5) ⇥ U(1) Y 1
ij10+1
i 1+5
j 10⇤ 6
Y 5
ij5i 5j 10+6
(3, 1, 2/3)+6
⌘ Y 1
ijdC T
a i CeC
j
⇤
a 2 1/2
✏abcY 5
ijuC T
a i CuC
b j c
Y 5
= Y 5 T
a(d↵, e ) =
p
2
m2 (UT
(Y 10
+ Y 10 T
)D)1↵ (UT
Y 5
E)1
a(d↵, eC
) =
4
m2 (UT
(Y 10
+ Y 10 T
)D)1↵ (E†
C(Y 10
+ Y 10 T
)†
U⇤
C) 1
a(dC
↵ , e ) =
1
2m2 (D†
CY 5 †
U⇤
C)↵1 (UT
Y 5
E)1
a(dC
↵ , eC
) =
p
2
m2 (D†
CY 5 †
U⇤
C)↵1 (E†
C(Y 10
+ Y 10 T
)†
U⇤
C) 1
a(d↵, d , ⌫i) =
p
2
m2 (UT
(Y 10
+ Y 10 T
)D)1↵ (DT
Y 5
N) i
a(d↵, dC
, ⌫i) =
1
2m2 (D†
CY 5 †
U⇤
C) 1 (DT
Y 5
N)↵i
a(d↵, dC
, ⌫C
i ) =
2
m2 (UT
(Y 10
+ Y 10 T
)D)1↵ (N†
CY 1 †
D⇤
C)i
a(dC
↵ , dC
, ⌫C
i ) =
1
p
2m2
(D†
CY 5 †
U⇤
C) 1 (N†
CY 1 †
D⇤
C)i↵

SU(5) ⇥ U(1) Y 1
ij10+1
i 1+5
j 10⇤ 6
Y 5
ij5i 5j 10+6
(3, 1, 2/3)+6
⌘ Y 1
ijdC T
a i CeC
j
⇤
a 2 1/2
✏abcY 5
ijuC T
a i CuC
b j c
Y 5
= Y 5 T
a(d↵, e ) =
p
2
m2 (UT
(Y 10
+ Y 10 T
)D)1↵ (UT
Y 5
E)1
a(d↵, eC
) =
4
m2 (UT
(Y 10
+ Y 10 T
)D)1↵ (E†
C(Y 10
+ Y 10 T
)†
U⇤
C) 1
a(dC
↵ , e ) =
1
2m2 (D†
CY 5 †
U⇤
C)↵1 (UT
Y 5
E)1
a(dC
↵ , eC
) =
p
2
m2 (D†
CY 5 †
U⇤
C)↵1 (E†
C(Y 10
+ Y 10 T
)†
U⇤
C) 1
a(d↵, d , ⌫i) =
p
2
m2 (UT
(Y 10
+ Y 10 T
)D)1↵ (DT
Y 5
N) i
a(d↵, dC
, ⌫i) =
1
2m2 (D†
CY 5 †
U⇤
C) 1 (DT
Y 5
N)↵i
a(d↵, dC
, ⌫C
i ) =
2
m2 (UT
(Y 10
+ Y 10 T
)D)1↵ (N†
CY 1 †
D⇤
C)i
a(dC
↵ , dC
, ⌫C
i ) =
1
p
2m2
(D†
CY 5 †
U⇤
C) 1 (N†
CY 1 †
D⇤
C)i↵
*R.N. Mohapatra, Phys. Rev. Lett. 43, 893 (1979).

1
E = DC
D = EC
U = UC
N = I U†
D = VCKM
m > 2.2 ⇥ 1011
✓
|↵|
0.0112 GeV3
◆1/2
GeV
m > 2.2 ⇥ 1011
GeV
E = DC
D = EC
U = UC
U†
D = VCKM
N = I
E = I
D = I
m > 2.2 ⇥ 1011
✓
|↵|
0.0112 GeV3
◆1/2
GeV
m > 2.2 ⇥ 1011
GeV
p!⇡+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
d
E = DC
D = EC
U = UC
U†
D = VCKM
N = I
E = I
D = I
m > 2.2 ⇥ 1011
✓
|↵|
0.0112 GeV3
◆1/2
GeV
m > 2.2 ⇥ 1011
GeV

MINIMAL SU(5) IS VERY PREDICTIVE BECAUSE IT IS NOT VIABLE!

MINIMAL VIABLE SU(5)
(CHARGED FERMION MASSES)
1
⇤ ⌘
✏↵ ⌘Yij 10↵
i 10j 5⌘
1
⇤ ⌘
✏↵ ⌘Yij 10↵
i 10j 5⌘
Yij 10↵
i 5j 5⇤
↵
Yij 10↵
i
24
⇤
5j 5⇤
↵
X
i
(DT
YDN)↵i(DT
YDN)⇤
i =
1
v2
5
((Mdiag
D )2
)↵
X
i
(DT
YU N)↵i(DT
YU N)⇤
i =
4
v2
5
(V T
UD(Mdiag
U )2
V ⇤
UD)↵
1
⇤ ⌘
✏↵ ⌘Yij 10↵
i 10j 5⌘
Yij 10↵
i 5j 5⇤
↵
Yij 10↵
i
24
⇤
5j 5⇤
↵
X
i
(DT
YDN)↵i(DT
YDN)⇤
i =
1
v2
5
((Mdiag
D )2
)↵
X
i
(DT
YU N)↵i(DT
YU N)⇤
i =
4
v2
5
(V T
UD(Mdiag
U )2
V ⇤
UD)↵
1
⇤ ⌘
✏↵ ⌘Yij 10↵
i 10j 5⌘
Yij 10↵
i 5j 5⇤
↵
Yij 10↵
i
24
⇤
5j 5⇤
↵
X
i
(DT
YDN)↵i(DT
YDN)⇤
i =
1
v2
5
((Mdiag
D )2
)↵
X
i
(DT
YU N)↵i(DT
YU N)⇤
i =
4
v2
5
(V T
UD(Mdiag
U )2
V ⇤
UD)↵
CUTOFF
Y 10
ij 10i10j45
Y 5
ij10i5j45⇤
Y 10
ij 10i10j5
Y 5
ij10i5j5⇤
MD = Y 5 T
v⇤
45⇤
MD =
1
2
Y 5 T
v⇤
5
ME = 3Y 5
v⇤
45
ME =
1
2
Y 5
v⇤
5
Yij 10i10j45
Y 5
ij10i5j45⇤
Y 10
ij 10i10j5
Y 5
ij10i5j5⇤
MD = Y 5 T
v⇤
45⇤
MD =
1
2
Y 5 T
v⇤
5
ME = 3Y 5
v⇤
45
ME =
1
2
Y 5
v⇤
5
MU = 2
p
2(Y 5
Y 5 T
)v
MU =
p
2(Y 10
+ Y 10 T
)v
10 ⇥ 10 = 5 45 : MU
10 ⇥ 5 = 5 45 : ME, M
10+1
⇥ 10+1
= 5
+2
45
+2

PREDICTIONS*
(MINIMAL VIABLE SU(5))
(3, 1, 2/3)
⌘ Y 1
ijuC T
a i C⌫C
j
⇤
a 2 1/2
✏abcY 5
ijdC T
a i CdC
b j c
Y 5
= Y 5 T
OH(d↵, e ) = a(d↵, e ) uT
L C 1
d↵ uT
L C 1
e
OH(d↵, eC
) = a(d↵, eC
) uT
L C 1
d↵ eC†
L C 1
uC⇤
OH(dC
↵ , e ) = a(dC
↵ , e ) dC
↵
†
L C 1
uC⇤
uT
L C 1
e
OH(dC
↵ , eC
) = a(dC
↵ , eC
) dC
↵
†
L C 1
uC⇤
eC†
L C 1
uC⇤
OH(d↵, d , ⌫i) = a(d↵, d , ⌫i) uT
L C 1
d↵ dT
L C 1
⌫i
OH(d↵, dC
, ⌫i) = a(d↵, dC
, ⌫i) dC†
L C 1
uC⇤
dT
↵ L C 1
⌫i
OH(d↵, dC
, ⌫C
i ) = a(d↵, dC
, ⌫C
i ) uT
L C 1
d↵ ⌫C
i
†
L C 1
dC⇤
OH(dC
↵ , dC
, ⌫C
i ) = a(dC
↵ , dC
, ⌫C
i ) dC†
L C 1
uC⇤
⌫C
i
†
L C 1
dC
↵
⇤
*I. Doršner, S. Fajfer and N. Košnik, Phys. Rev. D 86:015013, 2012, 1204.0674.

PREDICTIONS*
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
i=1,2,3
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵
U†
D ⌘ VUD
U = UCK0
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
i=1,2,3
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵
U†
D ⌘ VUD
U = UCK0
*I. Doršner, Phys. Rev. D 86:055009, 2012, 1206.5998.

PREDICTIONS
(MINIMAL VIABLE SU(5))UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
i=1,2,3
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵
U†
D ⌘ VUD
U = UCK0
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
i=1,2,3
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵
MU = MT
U
U†
D ⌘ VUD
U = UCK0
(K0)11 = ei
5 =
0
@
H
1
A
2

PREDICTIONS
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
i=1,2,3
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵
U†
D ⌘ VUD
U = UCK0
(K0)11 = ei
5 =
0
@
H
1
A
(p ! e+
⇡0
) ⇠
↵2
v4
5m4
3
8
2
↵ 1
1
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
i=1,2,3
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵
U†
D ⌘ VUD
U = UCK0
(K0)11 = ei
5 =
0
@
H
1
A
5
a(dj, dC
k , ⌫i) =
2
m2 v2
5
(VUDMdiag
D )1k(DT
MDN)ji
X
i=1,2,3
(DT
MDN)↵i(DT
MDN)⇤
i = (Mdiag 2
D )↵
MU = MT
U
U†
D ⌘ VUD
U = UCK0
(K0)11 = ei
5 =
0
@
H
1
A
(p ! e+
⇡0
) ⇠
↵2
v4
5m4
3
8
2

PREDICTIONS
p!⇡+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
d
p!K+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
s
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
a(dj, dk, ⌫i) =
2
m2 v2
5
(Mdiag
U K0VUD)1j(DT
MDN)ki
a(dj, dC
, ⌫i) =
2
(VUDMdiag
)1k(DT
MDN)ji

PREDICTIONS
3 2 1 0 1 2 3
1.2
1.4
1.6
1.8
2.0
2.2
Φ
m1011
GeV
p K Ν

PREDICTIONS
ut of the way we are ready to
proton decay mediating scalar
o in the next section.
AY LEPTOQUARK
scalar that contributes to pro-
imensional scalar representa-
number violating dimension-
es are [8]
L C−1
dj dT
k L C−1
νi,
C
k
†
L C−1
uC∗
dT
j L C−1
νi,
1, 2) (j + k < 4) represent
γ5)/2. Our notation is such
for the d (s) quark. The color
tensor in the SU(3) space is
i) operators contribute exclu-
with anti-neutrinos in the fi-
ents for the p → π+
¯ν (p →
i=1,2,3
D
Clearly, the lepton mixing matrix does not affect proton decay
signatures through scalar exchange. It is also clear that the
p → π+
¯ν decay rate is significantly suppressed compared to
the p → K+
¯ν one. The suppression factor, as inferred from
Eq. (11), is proportional to (md/ms)2
.
For the decay widths for p → π+
¯ν and p → K+
¯ν channels
we find
Γp→π+ ¯ν = Cπ+ A (m2
u + m2
d + 2mumd cos φ)m2
d,
Γp→K+ ¯ν ≈ CK+ A (m2
u + m2
d + 2mumd cos φ)m2
s,
where we neglect terms suppressed by either (md/ms)2
or
|(VUD)12|2
in the expression for Γp→K+ ¯ν . Here, A =
4|α|2
|(VUD)11|2
/v4
, eiφ
= (K0)11 and we introduce
CK+ =
(m2
p − m2
K+ )2
32πf2
πm3
p
1 +
mp
3mΛ
(D + 3F)
2
. (12)
After we insert all low-energy parameters we find
Γp→π+ ¯ν /Γp→K+ ¯ν = 10−2
. (13)
m > 2.2 ⇥ 1011
✓
|↵|
0.0112 GeV3
◆1/2
GeV
m > 2.2 ⇥ 1011
GeV
p!⇡+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
d
p!K+ ¯⌫ ⇠ (m2
u + m2
d + 2mumd cos )m2
s
UT
MU UC = Mdiag
U DT
MDDC = Mdiag
D ET
MEEC = Mdiag
E
↵ 1
1
↵ 1
2

CONCLUSIONS
Predictions of the minimal viable version of
SU(5) for the two-body p-decay modes induced
through scalar leptoquark exchange exhibit
minimal (one-phase only) model dependence for
p → K+ ν and p → π+ ν channels.
There exists an accurate limit on the mass of the
scalar leptoquark.
The ratio of p-decay widths for channels with π+
and K+ in the final state is phase independent and
predicts strong suppression of the former width
with respect to the latter one.

THANK YOU!
CONTACT E-MAIL:
ILJA.DORSNER@IJS.SI

I. Doršner, Leptoquark Mass Limit in SU(5)

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