Symmetries and vanishing couplings
in string-derived low-energy
effective field theory
Tatsuo Kobayashi
１．Introduction
2. Abelian Discrete Symmetries
3． Non-Abelian Discrete Symmetries
4． Anomalies
5. Explicit stringy computations
6. Summary
1. Introduction
Now, we have lots of 4D string models leading to
(semi-)realistic massless spectra such as
SU(3)xSU(2)xU(1) gauge groups,
three chiral genenations,
vector-like matter fields and lots of singlets
with and without chiral exotic fields,
e.g. in
heterotic models,
type II intersecting D-brane models,
type II magnetized D-brane models,
etc.
What about their 4D low-energy effective theories ?
Are they realistic ?
What about the quark/lepton masses and mixing angles ?
4D low-energy effective field theory
We have to control couplings in 4D LEEFT.
realization of quark/lepton mass and mixing
including the neutrino sector,
avoiding the fast proton decay,
stability of the LSP, suppressing FCNC, etc
Abelian and non-Abelian discrete symmetries
are useful in low-energy model building
to control them.
Abelian discrete symmetries
ZN symmetry
R-symmetric and non-R-symmetric
Flavor symmetry
R-parity, matter parity,
baryon triality, proton hexality
Non-Abelian discrete flavor symm.
Recently, in field-theoretical model building,
several types of discrete flavor symmetries have
been proposed with showing interesting results,
e.g. S3, D4, A4, S4, Q6, Δ(27),Δ(54), ......
Review: e.g
Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘10
⇒ large mixing angles
 2/3
1/ 3
0 


in the lepton sector
  1/ 6
1/ 3
1/ 2 
one Ansatz: tri-bimaximal   1 / 6 1 / 3  1 / 2 
String-derived 4D LEEFT
Can we derive these Abelian and non-Abelian
discrete symmetries from string theory ?
Which symmetires can appear in 4D LEEFT
derived from string theory ?
One can compute couplings of 4D LEEFT
in string theory.
(These are functions of moduli.)
Control on anomalies is one of stringy features.
What about anomalies of discrete symmetries ?
In this talk, we study heterotic orbifold models.
2. Abelian discrete symmetries
coupling selection rule
A string can be specified by
X (   ) its boundary condition.
X (  0)
Two strings can be connected
to become a string if their
boundary conditions fit each other.
coupling selection rule
symmetry
Heterotic orbifold models
S1/Z2 Orbifold
There are two singular points,
which are called fixed points.
X ~ X
X  e / 2 ~ ( X  e / 2)
Orbifolds
T2/Z3 Orbifold
There are three fixed points on Z3 orbifold
(0,0), (2/3,1/3), (1/3,2/3) su(3) root lattice
Orbifold = D-dim. Torus /twist
Torus = D-dim flat space/ lattice
Closed strings on orbifold
Untwisted and twisted strings
Twisted strings are associated with fixed points.
“Brane-world” terminology:
untwisted sector bulk modes
twisted sector
brane (localized) modes
Heterotic orbifold models
S1/Z2 Orbifold
X (   )   X (  0)
X (   )  e / 2  ( X (  0)  e / 2)
X (   )   X (  0)  n e, n  0, 1 (mod 2)
Heterotic orbifold models
S1/Z2 Orbifold
twisted string
X (   )   X (  0)  n e, n  0, 1 (mod 2)
X (   )  X (  0)
X (   )  (1) m X (  0)  n e,
m, n  0, 1 (mod 2)
untwisted string
Z2 x Z2 in Heterotic orbifold models
S1/Z2 Orbifold
X (   )  (1) X (  0)  n e,
m, n  0, 1 (mod 2)
m
two Z2’s
twisted string
 1 0 

,
 0  1
untwisted string
Z2 even for both Z2
1

0
0

 1
Closed strings on orbifold
Untwisted and twisted strings
Twisted strings (first twisted sector)
X (   )  X (  0)  n e1 , n  0, 1, 2 (mod 3)
  120 twist,

up to lattice   3me1  n(e1  e2 )
second twisted sector
X (   )   2 X (  0)  n e1 , n  0, 1, 2 (mod 3)
untwisted sector
X (   )  X (  0)
Z3 x Z3 in Heterotic orbifold models
T2/Z3 Orbifold
X (   )   X (  0)  n e,
m, n  0, 1 ,2
(mod 3)
m
two Z3’s
twisted string (first
 0 0 


 0  0 ,
 0 0 


twisted sector)
1 0

0 
0 0

0 

0 ,   exp( 2i / 3)
 2 
untwisted string
vanishing Z3 charges for both Z3
3. Non-Abelian discrete symmetries
Heterotic orbifold models
S1/Z2 Orbifold
String theory has two Z2’s.
In addition, the Z2 orbifold has the geometrical
symmetry, i.e. Z2 permutation.
X (   )  (1) m X (  0)  n e,
m, n  0, 1 (mod 2)
D4 Flavor Symmetry
Stringy symmetries require that Lagrangian has the
permutation symmetry between 1 and 2, and each
coupling is controlled by two Z2 symmetries.
Flavor symmeties: closed algebra S2 U(Z2xZ2)
0 1
,
 1  
1 0
D4 elements
 1,
 1 0 

 1  
 0  1
 1,
1 0 

 3  
 0  1
 i 2 ,
3
modes on two fixed points ⇒ doublet
untwisted (bulk) modes ⇒ singlet
Geometry of compact space
 origin of finite flavor symmetry
Abelian part (Z2xZ2) : coupling selection rule
S2 permutation : one coupling is the same as another.
T.K., Raby, Zhang, ‘05 T.K., Nilles, Ploger, Raby, Ratz, ‘07
Explicit Heterotic orbifold models
T.K. Raby, Zhang ’05, Buchmuller, Hamaguchi, Lebedev, Ratz, ’06
Lebedev, Nilles, Raby, Ramos-Sanchez, Ratz, Vaudrevange,Wingerter, ‘07
2D Z2 orbifold
1 generation in bulk
two generations on two fixed points
Heterotic orbifold models
T2/Z3 Orbifold
two Z3’s
0 
 0 0   1 0

 

 0  0 ,  0  0 ,   exp( 2i / 3)
 0 0   0 0 2 

 

Z3 orbifold has the S3 geometrical symmetry,
0

0
1

1
0
0
0

1 ,
0 
1

0
0

0
0
1
0

1
0 
Their closed algebra is Δ(54).
T.K., Nilles, Ploger, Raby, Ratz, ‘07
Heterotic orbifold models
T2/Z3 Orbifold
has Δ(54) symmetry.
localized modes on three fixed points
Δ(54) triplet
bulk modes
Δ(54) singlet
T.K., Nilles, Ploger, Raby, Ratz, ‘07
4. Discrete anomalies
4-1. Abelian discrete anomalies
Symmetry
violated
quantum effects
U(1)-G-G anomalies
anomaly free condition
 q T ( R)  0
2
ZN-G-G anomalies
anomaly free condition
 q T ( R)  0
2
(mod N )
Abelian discrete anomalies:
path integral
  '
Zn transformation
path integral measure
DD  J DD
J  exp[ A
1
A
N
1
32 2
d
i
32 2
d
4
x tr ( F

~
F )]
 q T ( R)
2
4
x tr ( F

~
F )  integer
ZN-G-G anomalies
anomaly free condition
 q T ( R)  0
2
(mod N )
Heterotic orbifold models
There are two types of Abelian discrete symmetries.
T2/Z3 Orbifold
X (   )   m X (  0)  n e,
m, n  0, 1 ,2
(mod 3)
two Z3’s
One is originated from twists,
the other is originated from shifts.
Both types of discrete anomalies  q T2 ( R)
are universal for different groups G.
Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, ‘08
Heterotic orbifold models
U(1)-G-G anomalies
 q T ( R)
2
are universal for different groups G.
4D Green-Schwarz mechanism
due to a single axion (dilaton),
which couples universally with gauge sectors.
ZN-G-G anomalies may also be cancelled
by 4D GS mechanism.
There is a certain relations between
U(1)-G-G and ZN-G-G anomalies,
anomalous U(1) generator is a linear combination
of anomalous ZN generators.
Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, ‘08
4-2. Non-Abelian discrete anomalies
Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, ‘08
Non-Abelian discrete group
finite elements
Each element generates an Abelian symmetry.
( gk ) Nk  1
We check ZN-G-G anomalies for each element.
 qk T2 ( R)  0 (mod Nk )
All elements are free from ZN-G-G anomalies.
The full symmetry G is anomaly-free.
Some ZN symmetries for elements gk are anomalous.
The remaining symmetry corresponds to
the closed algebra without such elements.
G  {g1 , g 2 ,, g M }
Non-Abelian discrete anomalies
matter fields
= multiplets under non-Abelian
discrete symmetry
Each element is represented by a matrix on the
multiplet.
det ( g k )  1
q
k
T2 ( R)  0 (mod N k )
Such a multiplet does not contribute to
ZN-G-G anomalies.
String models lead to certain combinations of
multiplets.
limited pattern of non-Abelian discrete
anomalies
Heterotic string on Z2 orbifold:
D4 Flavor Symmetry
Flavor symmeties: closed algebra S2 U(Z2xZ2)
modes on two fixed points ⇒ doublet
untwisted (bulk) modes ⇒ singlet
0 1
,
 1  
1 0
 1 0 

 1  
 0  1
1 0 

 3  
 0  1
The first Z2 is always anomaly-free, while the
others can be anomalous.
However, it is simple to arrange models such that
the full D4 remains.
e.g. left-handed and right-handed quarks/leptons
1+2
Such a pattern is realized in explicit models.
Heterotic models on Z3 orbifold
two Z3’s
 0 0 


 0  0 ,
 0 0 


1 0

0 
0 0

0 

0 ,   exp( 2i / 3)
 2 
Z3 orbifold has the S3 geometrical symmetry,
0

0
1

1
0
0
0

1 ,
0 
1

0
0

0
0
1
0

1
0 
Their closed algebra is Δ(54).
The full symmetry except Z2 is always anomaly-free.
That is, the Δ(27) is always anomaly-free.
Abe, et. al. work in progress
5. Explicit stringy computation
T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11
Explicit string computations tell something more.
Heterotic string on orbifold:
4D string, E8xE8 gauge part,
r-moving fermionic string,
6D string on orbifold, ghost
A mode has a definite “quantum number”
in each part.
Vertex operators
Boson
V-1  e

N Li
 X 
3
 X 
i
i
i 1
N Li

i
e
RNS
6D string
ghost
iqm H m
e
iP I X I
gauge
P I : quantum numbers under gauge groups (weights, roots)
qm : 10D Lorentz representa tion, (spinor, vector)
Fermion
V-1/2  e
 / 2
N Li
 X 
3
i
i 1
 X 
i
N Li

i
e
iqm( s ) H m
e
iP I X I
3-pt correlation function
Vertex operators:
V-1  e

N Li
 X 
3
i
i 1
V-1/2  e
 / 2
 X 
i
N Li
 X 
3
i
i 1
N Li
e
 X 
i
iqm H m
N Li
e
e
iP I X I
(s)
iqm
Hm
e
i
iP I X I
i
V-1V-1/2V-1/2
The correlation function vanishes unless
I
P
  0.
That is the momentum conservation in the string of
the gauge part, i.e. the gauge invariance.
String on 6D orbifold
The 6D part is quite non-trivial.
V-1V-1/2V-1/2

 DX X 
NL
 X 
NL
e  S 
For simplicity , we consider t he case with N L  0.
S


DX

X
e
 

NL
world-sheet instanton + quantum part
Classical solution
The world-sheet instanton, which
corresponds to string moving from a fixed point
to others.
symmetries of torus (sub)lattice
Suppose that only holomorphic instanton can appear.
X cl  ah( z )
a  difference of fixed points  torus (sub) lattice
Example: T2/Z3
different fixed points
a  SU(3) weight  adjoint lattice
Z3 symmetries (twist invariance)
twist invariance
+ H-momentum conservation
=> discrete R-symmetry
symmetries of torus (sub)lattice
Suppose that only holomorphic instanton can appear.
X cl  ah( z )
a  difference of fixed points  torus (sub)latti ce
Example: T2/Z3
three strings on the same fixed points
a  adjoint lattice
Z6 symmetries
enhanced symmetries
for certain couplings
Rule 4
Font, Ibanez, Nilles, Quevedo, ’88
T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11
T2/Z3 orbifold
Higher order couplings
S


DX

X
e
 

NL
The situation is the same.
Z3 symmetries among localized strings on different fixed points
Z6 symmetries among localized strings on a fixed point
Z6 enhanced symmetries only for the couplings
of the same fixed points.
=> Z3 twist invarince if the matter on the different fixed
points is included
Classical solution
T2/Z2
X cl  ah( z )
Similarly, the classical solution corresponding to
couplings on the same fixed points has enhanced
symmetires, e.g. Z4 and Z6, depending the torus
lattice, SO(5) torus and SU(3) torus.
T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11
String on 6D orbifold
The 6D part is quite non-trivial.
V-1V-1/2V-1/2

S


DX

X
e
 

NL
world-sheet instanton + quantum part
Only instantons with fine-valued action contribute.
That leads to certain conditions on combinations among
twists.
T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11
(anti-) Holomorphic instanton
Condition for fine-valued action
Vertex operators:
k : twist of the   th string



k
holomorphi c instanton appears :
1    1  k   0 for 0  k  1

anti - holomorphi c instanton appears :
1    k   0 for 0  k  1

Summary
We have studied discrete symmetries
and their anomalies.
Explicit stringy computations tell something
more.
String-derived massless spectra are (semi-)
realistic, but their 4D LEEFT are not so,
e.g. derivation of quark/lepton masses and
mixing angles are still challenging.
Applications of the above discrete symmetires
and explicit stringy computations would be
important.
5. Something else
T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11
Explicit stringy calculations would tell us something more.
Vertex operators:
V-1  e

i
NL
 X 
3
i
i 1
V-1/2  e
 / 2
 X 
i
i
NL
 X 
3
i
V0  e
 e
3
jv
iqm
H
m
X
j
e
i
NL
 e iqm H
jv
j 1
q1mv  (0,0,1,0,0),
iqm H
 X 
i
i 1

i
NL
qm2v  (0,0,0,1,0),
V-1V-1/2V-1/2V0 V0
m
e iP
(s)
H
X
j
e iqm
m
I
m
X
I
i
e iP
I
X
I
i
V
-1
qm3v  (0,0,0,0,1),
3-pt correlation function
Explicit stringy calculations
V-1V-1/2V-1/2
Vertex operators:
 X  e
 X   X
 F   DX X 
NL

 DX X
S
NL
cl
 NL 
   
r 0  r 
NL
NL
qu
cl
NL





s 0  s 
NL
 1 2 3
  X qu

NL
 e X 
 S cl
N L r
cl
X cl


 X qu   X qu  1 2 3
r
e
s
 S cl  S qu
 X 
cl
 1 2 3
N L s
Rule 5
T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11
NL
N L s
 NL  N L  N L 
 S cl
N L r


F      
e X cl 
 X cl


r  0  r  s  0  s  X cl




 X qu   X qu  1 2 3
r
X   X 
r
qu
qu
s
s

r s
No holomorphi c instanton : X cl  0
Rule 5:
No anti - holomorphi c instanton :  X cl  0
Only holomorphi c instanton
N L  s  NL  N L
Only anti - holomorphi c instanton
No instanton
NL  r  NL  N L
NL  N L
Z3 twist invariance
 NL 
F    
r 0  r 
NL
NL





s 0  s 
NL

 e X 
 S cl
N L r
cl
X cl
 X 

 X qu   X qu  1 2 3
r
s
X   X 
s
r
qu
qu

r s
N L  r  N L  s  0 mod 3
Z3 twist invariance
N L s
cl