Anomalies of
discrete symmetries
Takeshi Araki
（威） （荒木）
・T.A, Prog. Theor. Phys. 117 (2007).
・T.A, T.Kobayashi, J.Kubo, S.R.Sanchez, M.Ratz,
and P.K.S.Vaudrevange, arXiv:0805.0207 (hep­th).
Who is Takeshi Araki?
Discrete flavor symmetries
Recently, a lot of models with a flavor symmetry based on
( S3 ; A4 ; DN ; QN ¢ ¢ ¢ )
a finite group have been proposed ,
because it is possible....
・to reduce the number of free
parameters in the Yukawa sector.
・to realize hierarchical structure
of fermion masses.
・to realize bi­maximal structure
of MNS matrix.
VCKM
small
mixing
quark
VS
bi­large
mixing
lepton
VM N S
Motivation
There are too many models!
Most of the models are meaningless because a flavor
symmetry is only one ( if there exist it ). But we have no criterion to distinguish them.
I thought it is important to consider a new criterion.
Anomalies of discrete symmetries.
That is, I want to consider a thing such as gauge anomalies.
A[U(1)Y ]3 =
Xh
(YLf )3
f
A[SU(2)L ]2 ¡U (1)Y =
A[SU (3)C ]2 ¡U (1)Y =
X
¡
(YRf )3
i
=0
YLf `2 (rf ) = 0
A[SU(2)L ]2 ¡D =
YLf `3 (rf ) = 0
A[SU(3)C ]2 ¡D =
f
X
f
A[U(1)Y ]2 ¡D =
？
Motivation
To this end, there are two problems.
1, How do I define anomalies
for discrete symmetries?
2, Why are discrete anomalies forbidden?
I would like to talk about these topics in this presentation.
Fujikawa's method
[ Fujikawa; P RL42(1979) ]
à ! ei®(x)°5 Ã
Let us consider a continuous chiral rotation, ,
in Euclidean space­time.
½Z
¾
Z
Z=
¹
D ÃDÃ
DA¹ exp
d4 x
L(Ã; A)
0
¹
¹
DÃDÃ
! J ¡1 DÃDÃ
½
¾¡2
Z
h
i
J ¡1 = det d4 x 'yn (x) ei®(x)°5 'm (x)
(
' exp ¡2
1 Z
X
n
L = L + ®(x)@¹ j5¹
ei®(x)°5 ' 1 + i®(x)°5
d4 x 'yn;i (x) [i®(x)°5 ]ij 'n;j (x)
)
½ Z
¾
®(x)
¹º½¾
= exp ¡i d4 x
Tr
[²
F¹º (A)F½¾ (A)]
16¼2
@¹ j5¹ =
=0
Hence the current does not conserve at the quantum level.
Discrete anomalies
I want to consider the same calculation for discrete ones.
à ! ei®° 5 Ã
Let us consider a discrete chiral rotation, , in Euclidean space­time. For instance ® = (2¼=N )q :
Z=
Z
¹
DÃDÃ
DA¹ exp
½Z
d4 x
L(Ã; A)
¹
¹
DÃDÃ
! J ¡1 DÃDÃ
½
¾¡2
Z
£
¤
J ¡1 = det d4 x 'yn (x) ei®° 5 'm (x)
0
¾
L =L
？
In this case, we can not use an infinitesimal transformation
because is a discrete parameter.
®
Discrete anomalies
Definition
D'n (x) = ¸n 'n (x)
Ãi (x) =
We can expand the Jacobian as follows.
½
¾
Z
J
¡1
=
det
4
d x
'yn (x)
Cnm =
Z
Z
£
e
i®° 5
¤
'm (x)
£
d4 x 'yn (x) e
¤
i®° 5
¡2
2
~pm = C~nm
= C~np C
Zn
¡2
´ fdet Cnm g
'm (x) = ±nm +
d4 x 'yn;i (x) [i®°5 ]2ij 'm;j (x)
Z
Z
= d4 x d4 y 'yn;i (x) [i®°5 ]ij ±jk ± 4 (x ¡ y) [i®°5 ]kl 'm;l (y)
Z
Z
= d4 x d4 y 'yn;i (x) [i®°5 ]ij 'p;j (x)'yp;k (y) [i®°5 ]kl 'm;l (y)
1
X
Z
Z
X
an 'n;i (x)
n
'n;i (x)'yn;j (y) = ±ij ± 4 (x ¡ y)
d4 x 'yn;i (x)'m;i (x) = ±nm
d4 x 'yn;i (x) [i®°5 ]ij 'm;j (x)
1
[i®°5 ]2ij 'm;j (x)
2!
Z
1
+ d4 x 'yn;i (x) [i®°5 ]3ij 'm;j (x) + ¢ ¢ ¢
3!
+
d4 x 'yn;i (x)
1 2
1 ~3
= ±nm + C~nm + C~nm
+ C
+ ¢¢¢
2!
3! nm
Then we use to obtain
det A = exp fTr ln Ag
J ¡1
=
=
=
det
½Z
(
¡
d4 x 'yn;i (x) e
exp ¡2
1 Z
X
n
½ Z
exp ¡i d4 x
¢
i®° 5
¾¡2
' (x)
ij m;j
)
d4 x 'yn;i (x) [i®°5 ]ij 'n;j (x)
¾
®
Tr [²¹º½¾ F¹º F½¾ ] :
2
16¼
Remaining calculation
is the same as that of
conventional one.
Anomalies of non­abelian discrete symmetries
Let us consider non­abelian discrete flavor transformations.
ÃL;® ! U®¯ ÃL;¯ ´ (eiX )®¯ ÃL;¯
( ®; ¯ : generations )
iY
ÃR;® ! V®¯ ÃR;¯ ´ (e )®¯ ÃR;¯
UUy = V V y = 1
These are unitary transformations (　).
det(U ) = det(eiX ) = ei´
2¼
´; » /
N
det(V ) = det(eiY ) = ei»
Let us calculate the Jacobian for these transformations.
½
¾
Z
J ¡1 = exp
´i
32¼2
d4 x Tr [²¹º½¾F¹º (L)F½¾ (L)]
½
¾
Z
»i
4
¹º½¾
£ exp ¡
d
x
Tr
[²
F¹º (R)F½¾ (R)]
2
32¼
»
´
Here, and correspond to the abelian parts of and .
V
U
Therefore, we only have to take
into account its abelian parts.
Stringy originated discrete symmetries
We want to consider the situation that discrete flavor anomalies are forbidden. But there is no reason....
We assumed that discrete flavor symmetries originate from string theory.
・In fact, we can derive non­abelian discrete flavor symmetries
[ T:Kobayashi; et al: N P B768(2007) ]
from heterotic orbifold models.
・Such discrete symmetries reflect geometrical symmetries of internal space.
・The geometrical operations are embedded into the gauge group.
・There might be some relation between discrete anomalies
and that of anomalous U(1) gauge symmetries.
We expected that we might get some constraint for discrete
anomalies from anomaly freedom of anomalous U(1) if there
exist such a relation.
1
[ T:Kobayashi; et al: N P B768(2007) ]
S /Z 2 Divide a 1­dimensional space by a 1­dimensional lattice.
a
b
identify a with b
1 S torus
a=b
Identify the points which are related by a reflection.
1
S /Z 2 orbifold
Z2
The points of both ends are invariant
fixed point
under ­ twist. Z2
There are brane matter fields living on these fixed points.
1
[ T:Kobayashi; et al: N P B768(2007) ]
S /Z 2 Let us consider two states as multiplet.
µ
¶
jx >
jx >
jy >
jy >
Then, allowed n­point couplings must be invariant under
・rotational part ( twist )　i=1»n
µ i
¶
µ
¶µ i
¶
µ i
¶
jx >
¡1 0
jx >
jx >
!
= ¡1
;
jy i >
0 ¡1
jy i >
jyi >
・translational part ( lattice )
µ i
¶
µ
¶µ i
¶
µ i
¶
jx >
1 0
jx >
jx >
!
=
¾
;
3
jy i >
0 ¡1
jy i >
jy i >
・permutation between two fixed points
µ i
¶
µ
¶µ i
¶
µ i
¶
jx >
0 1
jx >
jx >
!
=
¾
;
1
jy i >
1 0
jy i >
jy i >
these transformations.
1
[ T:Kobayashi; et al: N P B768(2007) ]
S /Z 2 n­point couplings must be invariant under the combinations of such transformations, too.
The product of such generators lead to the eight elements
§1; §¾1 ; §i¾2 ; §¾3 :
D4 symmetry
D4
That is, n­point couplings must be invariant under symmetry,
and the multiplets are doublets of .
D4
Building blocks
The symmetries of 6­dimensional orbifolds can be obtained by
combining symmetries of the following blocks. [ T:Kobayashi; et al: N P B768(2007) ]
Z 6­II orbifold models
Let us calculate anomalies of discrete flavor symmetries in heterotic orbifold models. We mainly focus on the orbifold models.
Z6 ¡ II
(Note that, we also considered different orbifold models, so that
our results are more generally valid.)
We pay attention to only abelian (transrational) parts of non­
abelian discrete flavor symmetries ( ).
D4 £ Zn s
In the orbifold, there are three translational symmetries.
Z6 ¡ II
Z2f lavor
0
£ Z2f lavor £ Z3f lavor
We define anomaly coefficients as
AZnf lavor ¡G¡G
1 X (f ) (f )
=
qn `(r ) :
n (f )
r
Z 6­II orbifold models
・KRZ model [ Kobayashi; Raby; Zhang; N P B704(2005) ]
G = SU (4) ­ SU (2)L ­ SU (2)R ­ SO(10) ­ SU (2)
0
Z3
is anomalous.
But it can be canceled by the GS mechanism.
Anomaly free
Discrete version of the GS mechanism
µ
¶
2¼
®=
N
Consider a discrete transformation.
© ! e¡i®q N ©
Anomaly = ¡iAG [®W a (G)Wa (G)]F
© : chiral supermultiplet
We can cancel this anomaly by the gauge kinetic term,
kG [SW a (G)Wa (G)]F
AG
S !S +i
®
kG
Cancel
kG [SW a (G)Wa (G)]F + iAG [®W a (G)Wa (G)]F
if we correspondingly shift the dilaton supermultiplet.
( Note that the Kahler potential is invariant. )
For instance, the anomaly cancellation conditions for the SM gauge groups are given by
A3
A2
A1
AG
=
=
=
:
k3
k2
k1
12
( It is difficult to build realistic models with higher levels in string theory. )
Z 6­II orbifold models
・BHLR model [ Buchmuller; Hamaguchi; Lebedev; Ratz; N P B785(2006) ]
G = SU (3) ­ SU (2) ­ SU (4) ­ SU (2)
0
is anomalous.
But it can be canceled by the GS mechanism.
Z3
Anomaly free
Empirical relations
By conducting many calculations for various models, we can obtain the following empirical relations.
AZ flavor ¡G¡G
2
AZ flavor ¡G¡G
3
nanom
= 2
mod 1
2
nanom
= 3
mod 1
3
anom
n
nanom
Here, and are the orbifold parameters and we found
3
2
these parameters are related to anomalous U(1) . V : shift
anom
anom
anom
tanom
=
k
V
+
n
W
+
n
W3
2
2
3
U (1)
W : wilson lines
¸ : lattice vector
Anticipation
・There might be some relation between discrete flavor
anomalies and U(1) gauge anomalies.
・Anomaly freedom might be guaranteed by GS mechanism
of anomalous U(1).
D 4 flavor symmetry
D
has eight elements which can be written as products of the
4
g
two generators and .
h
GD4 = fe; g; h; gh; hg; ghg; hgh; ghghg:
0
f lavor
Z2
; Z2f lavor
g
corresponds to translatiomal part ( ).
corresponds to parmutational part ( non­abelian part ).
h
・If this flavor symmetry originate from heterotic
D4
orbifold compactifications,
・If the empirical relations really exist,
g
Anomalies corresponding to are forbidden!!
h
( Anomalies corresponding to are future work. )