Nonsupersymmetric gauge coupling unfication in [SU(6)]$^4\times$ Z$_4$ and proton stability

29 Pages·0.24 MB·English

by  Abdel Pérez-Lorenzana

Nonsupersymmetric gauge coupling unfication in
[SU(6)]4 Z and proton stability
4
×
9
9 A. P´erez-Lorenzana1, William A. Ponce1,2, and Arnulfo Zepeda1
9 ∗ †
1

n 1-Departamento de F´ısica,
a
J
Centro de Investigacio´n y de Estudios Avanzados del IPN.
4
1
Apdo. Postal 14-740, 07000, M´exico D. F., M´exico.

1
v
1 2-Departamento de F´ısica, Universidad de Antioquia
0
3
A.A. 1226, Medell´ın, Colombia.
1
0
9
9
h/ Abstract.We systematically study the three family extension of the Pati-Salam gauge
p
- group with an anomaly-free single irreducible representation which contains the known quarks
p
e
and leptons without mirror fermions. In the context of this model we implement the survival
h
:
v hypothesis, the modified horizontal survival hypothesis, and calculate the tree level masses for
i
X
the gauge boson and fermion fields. We also use the extended survival hypothesis in order to
r
a
calculate the mass scales using the renormalization group equation. Theinteracting Lagrangean
with all the known and predicted gauge interactions is explicitly displayed. Finally the stability
of the proton in this model is established.
PACS: 11.15.Ex,12.10.Dm
∗e-mail: abdel@fis.cinvestav.mx
†e-mail: wponce@fis.cinvestav,mx
1
1. Introduction
The renormalizability of the original Pati-Salam[1] model for unification of flavors and
forces rests on the existence of conjugate or mirror partners of ordinary fermions. Mirror
fermions are fermions with quantum numbers with respect to the Standard Model (SM)
gauge group SU(3) SU(2) U(1) identical to those of the known quarks and leptons,
C L Y
⊗ ⊗
except thattheyhaveoppositehandedness fromordinaryfermions. Theirexistence vitiate
the survival hypothesis [2] according to which chiral fermions that can pair off while
respecting a symmetry will do so, acquiring masses grater than or equal to the mass scale
of that symmetry.
Today we know how to cancel anomalies without introducing unwanted mirror fer-
mions. As a matter of fact, the three family extension of the Pati-Salam model without
mirror fermions was presented recently in the literature, with some aspects of the model
briefly analyzed in the original reference[3]. But a systematic analysis of this model
is still lacking. In what follows we do such analysis, paying special attention to the
implementation of the survival hypothesis [2] and of the modified horizontal survival
hypothesis [4]. (For a technical explanation of the terminology used in this article see
Appendix A.)
The model under consideration unifies non-gravitational forces with three families of
flavors, using the gauge group
G SU(6) SU(6) SU(6) SU(6) Z
L R CR CL 4
≡ ⊗ ⊗ ⊗ ×
where indicates a direct product, a semidirect one, and Z (1,P,P2,P3) is the four-
4
⊗ × ≡
element cyclic group acting upon [SU(6)]4 such that if (A,B,C,D) is a representation of
[SU(6)]4 with A a representation of the first factor, B of the second, C of the third,
and D of the fourth, then P(A,B,C,D)=(B,C,D,A) and then Z (A,B,C,D) (A,B,C,D)
4
≡ ⊕
2
(B,C,D,A) (C,D,A,B) (D,A,B,C). The electric charge operator in G is defined as[3]
⊕ ⊕
Q = T +T +[Y +Y ]/2, (1)
EM ZL ZR (B L)L (B L)R
− −
where (B L) stands for the local Abelian factor of (Baryon Lepton) hyper-
L(R) L(R)
− −
charge associated with the diagonal generators Y =Diag(1, 1, 1, 1,1, 1) of
(B−L)L(R) 3 3 3 − − L(R)
SU(6)
CL(CR)
The irreducible representation (irrep) of G which contains the known fermions is
ψ(144) = Z ψ(¯6,1,1,6) = ψ(¯6,1,1,6) ψ(1,1,6,¯6) ψ(1,6,¯6,1) ψ(6,¯6,1,1).
4
⊕ ⊕ ⊕
The model described by the structure [G,ψ(144)] is a grand unification model which
contains the three family SM gauge group, the three family left-right symmetric exten-
sion of the SM[5] [SU(3) SU(2) SU(2) U(1) ] and the three family chiral color
C L R (B L)
⊗ ⊗ ⊗ −
extension of the SM[6] [SU(3) SU(3) SU(2) U(1) ]. Finally, [G,ψ(144)] is the
CR CL L Y
⊗ ⊗ ⊗
chiral extension of the vector-color-like model described by [7,8]GV SU(6) SU(6)
L C
≡ ⊗ ⊗
SU(6) Z and ψV(108) = ψV(¯6,6,1) ψV(6,1,¯6) ψV(1,¯6,6), where SU(6) in GV is
R 3 C
× ⊕ ⊕
the diagonal subgroup of SU(6) SU(6) G, and ψV(108) ψ(144)
CR CL
⊗ ⊂ ⊂
That [G,ψ(144)]is freeof anomalies anddoes not contain mirror fermions follows from
its particle content. To see this we first show that there is a unique way to embed the
SM gauge group for three families in [G,ψ(144)][3] and then write the quantum numbers
for ψ(144) with respect to the subgroups of the SM which are [the notation designates
behavior under (SU(3) , SU(2) , U(1) )]:
C L Y
ψ(¯6,1,1,6) 3(3,2,1/3) 6(1,2, 1) 3(1,2,1)
∼ ⊕ − ⊕
ψ(1,6,¯6,1) 3(¯3,1, 4/3) 3(¯3,1,2/3) 6(1,1,2) 9(1,1,0) 3(1,1, 2)
∼ − ⊕ ⊕ ⊕ ⊕ −
ψ(6,¯6,1,1) 9(1,2,1) 9(1,2, 1)
∼ ⊕ −
ψ(1,1,6,¯6) (8+1,1,0) 2(3,1,4/3) 2(¯3,1, 4/3) (3,1, 2/3) (¯3,1,2/3) 5(1,1,0)
∼ ⊕ ⊕ − ⊕ − ⊕ ⊕ ⊕
2(1,1,2) 2(1,1, 2),
⊕ −
3
where the ordinary left-handed quarks correspond to 3(3,2,1/3) in ψ(¯6,1,1,6), the or-
dinary right-handed quarks correspond to 3(¯3,1, 4/3) 3(¯3,1,2/3) in ψ(1,6,¯6,1), the
− ⊕
known left-handed leptons are in three of the six (1,2, 1) of ψ(¯6,1,1,6), and the known
−
right-handed charged leptons are in three of the six (1,1,2) of ψ(1,6,¯6,1). The exotic
leptons in ψ(¯6,1,1,6) belong to the vectorlike representation 3(1,2, 1) 3(1,2,1) (vec-
− ⊕
torlike with respect to the SM quantum numbers) and the exotic leptons in ψ(1,6,¯6,1)
belong to the vectorlike representation 3(1,1,2) 3(1,1, 2) 9(1,1,0),where three lineal
⊕ − ⊕
combinations of the nine states with quantum numbers (1,1,0) could be identified as the
right-handed neutrinos.
ψ(6,¯6,1,1) is formed by 36 exotic spin 1/2 Weyl fermions (we call them nones because
they have zero lepton and baryon numbers), 9 with positive electric charges, 9 with
negative (the charge conjugates to the positive ones), and 18 are neutrals; all together
constitute a vectorlike representation with respect to the SM.
Also all the particles in ψ(1,1,6,¯6) form a vectorlike representation with respect to
the SM, where 5(1,1,0) 2(1,1,2) 2(1,1, 2) stands for nine exotic fermions, five with
⊕ ⊕ −
zero electric charge (nones), two with electric charge +1 and the other two with electric
charge 1 (spin 1/2 dileptons); 2(3,1,4/3) 2(¯3,1, 4/3) refers to two exotic spin 1/2
− ⊕ −
leptoquarks with electric charge 2/3; (3,1, 2/3) (¯3,1,2/3) refers to one exotic spin 1/2
− ⊕
leptoquark with electric charge 1/3, and the nine states in (8+1,1,0)=(8,1,0)+(1,1,0)
−
(quaits)+(quone) are the so-called dichromatic fermion multiplets [6] (also nones) which
belong to the (3,¯3) representation of the SU(3) SU(3) subgroup of SU(6)
CR CL CR
⊗ ⊗
SU(6) .
CL
Notice that contrary to the original Pati-Salam model, the G symmetry and the rep-
resentation content of ψ(144) forbid mass terms for fermion fields at the unification scale,
but according to the survival hypothesis[2] the vectorlike substructures pointed in this
section (all the exotic particles in the model) should get masses at scales above M , the
Z
4
known weak interaction mass scale.
2. The Model
The model under consideration contains 140 spin 1 gauge boson fields, 144 spin 1/2 Weyl
fermion fields, and a conveniently large number of spin 0 scalar fields. We use for them
the following notation:
2..1 The gauge bosons
For the gauge boson fields we define:
a)-For the 70 gauge fields of SU(6) and SU(6)
CL CR
D G1 G1 X∼ Y∼ Z∼
1 2 3 1 1 1
 
G2 D G2 X∼ Y∼ Z∼
1 2 3 2 2 2
 
A = 1  G31 G32 D3 X∼3 Y∼3 Z∼3  (2)
CL(CR)  
√2  X X X D P P0 
 1 2 3 4 1− 
 
 
 Y Y Y P+ D P+ 
 1 2 3 1 5 2 
 
 Z1 Z2 Z3 P∼0 P2− D6 
 CL(CR)
 
where
1 2
D = (Gδ) + B + B , δ = 1,2,3;
δCL(CR) δ CL(CR) s30 (B−L)L(R) s15 1YL(R)
3 1 1
D = B B B ;
4CL(CR) −s10 (B−L)L(R) − √30 1YL(R) − √2 2YL(R)
3 4
D = B B ;
5CL(CR) s10 (B−L)L(R) − √30 1YL(R)
3 1 1
D = B B + B ;
6CL(CR) −s10 (B−L)L(R) − √30 1YL(R) √2 2YL(R)
5
with (Gδ) , δ,η = 1,2,3 the gauge fields associated with SU(3) (G1 =
η CL(CR) CL(CR) 1CL(CR)
B /√2 +B /√6, G2 = B /√2+B /√6, G3 =
1gCL(CR) 2gCL(CR) 2CL(CR) − 1gCL(CR) 2gCL(CR) 3CL(CR)
2B /√6suchthat (Gδ) = 0, andB andB arethegauge
− 2gCL(CR) δ δ CL(CR) 1gCL(CR) 2gCL(CR)
fields associated with the dPiagonal generators of SU(3) ). B is the gauge
CL(CR) (B−L)L(R)
boson associated with the generator Y , and B and B are two gauge
(B−L)L(R) 1YL(R) 2YL(R)
bosons associated with the SU(6) diagonalgenerators Y = Diag(2,2,2, 1, 4,
CL(CR) 1L(R)
− −
1)/√15 and Y = Diag(0,0,0, 1,0,1) respectively. X ,Y and Z are spin 1 lepto-
2L(R) δ δ δ
− −
quark gauge bosons with electric charges 2/3,1/3 and 2/3 respectively, with δ = 1,2,3
− −
a color index. P ,κ = 1,2; and P0 are spin 1 dilepton gauge bosons with electric charges
κ±
as indicated.
b)-For the 70 gauge fields of SU(6) and SU(6)
L R
A B+ H 0 B+ H0 B+
1 1′ 1′ 2 2 3
 
B A B H 0 B H0
1′− 2 4− 3′ 5− 4
 0 
A = 1  H∼′1 B4+ A3 B6′+ H5′0 B7+  (3)
L(R) √2  B2− H∼′30 B6′− A4 B8− H6′0 
 
 0 0 
 H∼2 B5+ H∼′5 B8+ A5 B9′+ 
 0 0 
 B3− H∼4 B7− H∼′6 B9′− A6 L(R)
 
where the diagonal and the primed entries in Eq.(3) are related to the physical fields as
explained in the Appendix B.
6
2..2 The Fermionic content
For the spin 1/2 Weyl fields we use the following definitions:
d d d e e0c e
1 2 3 −11 12 −13
 
u u u n0 n+ n0
1 2 3 − 11 12 − 13
 
ψ(¯6,1,1,6) =  s1 s2 s3 −e−21 e022c e−23  ψα (4)
L  c c c n0 n+ n0  ≡ a
 1 2 3 21 22 − 23 
 
 b b b e e0c e 
 1 2 3 − −31 32 − −33 
 
 t1 t2 t3 n031 n+32 n033 L
 
where the rows(columns) represent color(flavor) degrees of freedom, (u,d,c,s,b,t) are the
quark fields with colors δ = 1,2,3 as indicated, (e ,n ), i,j = 1,2,3 are lepton Weyl
ij ij
fields with electric charge as indicated, the minus signs are phases chosen for convenience,
and the upper c symbol stands for charge conjugation.
dc uc sc cc bc tc
1 1 1 1 1 1
 
dc uc sc cc bc tc
2 2 2 2 2 2
 
 dc uc sc cc bc tc 
ψ(1,6,¯6,1) =  3 3 3 3 3 3  ψA (5)
 E+ N0c E+ N0c E+ N0c  ≡ ∆
 11 − 11 − 21 21 − 31 31 
 
 E0 N E0 N E0 N 
 12 1−2 22 2−2 32 3−2 
 
 E1+3 −N103c E2+3 −N203c −E3+3 N303c L
 
where the rows (columns) now represent flavor (color) degrees of freedom. The notation
we are using with the lepton fields in ψ(1,6,¯6,1) unrelated in principle to the lepton fields
in ψ(¯6,1,1,6) is consistent with the SM quantum numbers for ψ(¯6,1,1,6) ψ(1,6,¯6,1)
⊕
presented in the Introduction. The known leptons (ν ,e ,ν ,µ ,ν ,τ ) and the known
e − µ − τ −
quarks are linear combinations of the leptons and quarks in ψ(¯6,1,1,6) ψ(1,6,¯6,1), up
⊕
to mixing with exotics. Our notation is such that a,b,..; A,B,...; α,β,...; ∆,Ω,... stand
for SU(6) , SU(6) , SU(6) , and SU(6) tensor indices respectively.
L R CL CR
7
For the sake of completeness we also write:
g1 g1 g1 x y z
1 2 3 r r r
 
g2 g2 g2 x y z
1 2 3 y y y
 
ψ(1,1,6,¯6) ψ∆ =  g13 g23 g33 xb yb zb  (6)
≡ α  x∼r x∼y x∼b l10 l1+ l20 
 
 
 y∼r y∼y y∼b l1− l30 l2− 
 
 z∼r z∼y z∼b l40 l2+ l50 L
 
where gi, i,j = 1,2,3 are the (quaits)+(quone) spin 1/2 nones; x,y and z are the spin
j
1/2 leptoquarks with electric charges 2/3, 1/3 and 2/3 respectively, l ,j = 1,2 are spin
− j±
1/2 dilepton fields with electric charges as indicated, and l0,j = 1,...5 are five nones with
j
zero electric charge.
2..3 The Scalar Content
In order to spontaneously break the G symmetry down to SU(3) U(1) , and to im-
C EM
⊗
plement at the same time the survival hypothesis and the horizontal survival hypothesis,
we need to introduce the following rather complicated scalar sector:
First we introduce the scalar fields φ and φ with Vacuum Expectation Values (VeVs)
1 2
such that φ φ M, where
1 2
h i ∼ h i ∼
[a,b] [α,δ] [∆,Ω] [A,B]
φ = φ (900) = Z φ (15,1,1,15) = φ +φ +φ +φ
j j 4 j j[α,δ] j[∆,Ω] j[A,B] j[a,b]
j = 1,2, and [.,.] stands for the commutator of the indices inside the brackets. The VeVs
for φ , j = 1,2 are conveniently chosen in the following directions:
j
φ[a,b] = √3M for [a,b] = [4,1] = [2,3] = [5,6];[α,δ] = [5,6]
h 1[α,δ]i
φ[∆,Ω] = √3M for [A,B] = [4,1] = [2,3] = [5,6];[∆,Ω] = [5,6]
h 1[A,B]i
[A,B]
φ = M for [a,b] = [A,B] = [4,1] = [2,3] = [6,5]
h 1[a,b]i
8
[α,δ]
φ = 0;j = 1,2
h j[∆,Ω]i
φ[a,b] = √3M for [a,b] = [1,2] = [6,3] = [4,5];[α,δ] = [4,5]
h 2[α,δ]i
φ[∆,Ω] = √3M for [A,B] = [1,2] = [6,3] = [4,5];[∆,Ω] = [4,5]
h 2[A,B]i
[A,B]
φ = M for [a,b] = [A,B] = [2,1] = [6,3] = [4,5].
h 2[a,b]i
It is easy to show[9] that φ + φ with the VeVs as indicated breaks
1 2
h i h i
G SU(2) SU(2) SU(3) SU(3) U(1) U(1) ,
−→ L⊗ R⊗ CL⊗ CR⊗ (B−L)L⊗ (B−L)R
the chiral extension of the left-right symmetric extension of the SM.
Next we introduce
φ = φ (5184) = Z φ (1,1,(15+21),(15+21)) = φab +φαη +φ∆Ω +φAB
3 3 4 3 3,αη 3,∆Ω 3,AB 3,ab
with the following VEVs:
φαη = M δαδη; α,η,∆,Ω = 1,..,6,
h 3,∆Ωi C Ω ∆
[∆,Ω]
φ = M for [∆,Ω] = ∆Ω Ω∆ = [A,B] = [4,6],
h 3,[A,B]i R −
φab = φAB = 0.
h 3,αηi h 3,abi
It then follows that
φαη
SU(6) SU(6) h 3,∆Ωi SU(6) SU(6)V,
CR⊗ CL −→ (CL+CR) ≡ C
[∆,Ω]
and that the main effect of φ is to break SU(2) U(1) in an appropriate
h 3,[A,B]i R⊗ (B−L)R
way as we will shortly show.
Finally we introduce
φ = φ (2592) = φ (6,¯6,6,¯6)+φ (¯6,6,¯6,6) = φaΩ +φAα
4 4 4 4 4,Aα 4,aΩ
with the following VEVs: φAα = 0, and φaΩ = M for (a,A) = (6,6); (Ω,α) = (1,1)
h 4,aΩi h 4,Aαi Z
= (2,2) = (3,3) = (4,4) = (5,5) = (6,6); and also for (a,A)=(Ω,α)=(5,5). As we will show
in the next section the main effect of φ is to break SU(2) U(1) down to U(1) .
4 L Y EM
h i ⊗
9
3. Tree level masses
The scalar fields and their VEVs introduced in the previous section allow for the following
tree level masses:
3..1 Masses for gauge bosons
A tedious calculation[9] in the sector of the covariant derivative in the Lagrangian shows
the following results:
1. φ + φ produces:
1 2
h i h i
3
(M) = g2M2 18 X 2 +2 Y 2 + Z 2 +2 P0 2 (7)
L ( " | δCL| | δCL| | δCL| | CL|
δ=1(cid:18) (cid:19)
X
5 8 6
+ P 2 + P 2 + B2 +B2 +24 c B 2 + c H0 2
| 1CL| | 2CL| 3 1YL 2YL# " i| iL| ′i| iL| #
i=1 i=1
X X
+ 12 3A2 +A2 +A2 +3A2 +(L R) ,
(cid:18) 1HL 2HL 1AL 2AL(cid:19) −→ )
where g is the gauge coupling constant for the simple group G, and the coefficients c and
i
c are such that c = c = c /2 = c = c /2 = c /3 = c = c = 1 and c /3 = c /2 = c =
′i 1 2 3 4 5 6 7 8 ′1 ′2 ′3
c /2 = c /3 = c = 1. (The relationship between the unprimed fields in Eq. (7) and the
′4 ′5 ′6
primed ones in Eq. (3) is presented in Appendix B.)
As it is clear from the former equation, φ + φ breaks G down to the chiral
1 2
h i h i
extension of the left-right symmetric extension of the SM.
2. For φ we split the analysis.
3
h i
2a. φαη produces:
h 3,∆Ωi
(M ) = 12g2M2Tr A2 2A A +A2 (8)
L C C CL − CL CR CR
3h i
= 6g2M2 2 X X 2 + Y Y 2 + Z Z 2
C " δ=1(cid:18)| δCL − δCR| | δCL − δCR| | δCL − δCR| (cid:19)
X
2
+ 2 P P 2 +(B B )2 +(B B )2
iCL iCR igL igR iYL iYR
| − | − −
i=1(cid:18) (cid:19)
X
10