Flavor structure from magnetic fluxes and non-Abelian Wilson lines

0.26 MB·English

by  Hiroyuki Abe

WU-HEP-09-05
KUNS-2249
YITP-09-122
0
1
0
2 Flavor structure from magnetic fluxes and

n
a non-Abelian Wilson lines
J

2
1

] Hiroyuki Abe1,∗, Kang-Sin Choi2,†, Tatsuo Kobayashi2,‡
h
-t and Hiroshi Ohki2,3,§
p
e
h
[ 1Department of Physics, Waseda University, Tokyo 169-8555, Japan

1 2
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
v
8 3Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
8
7
1
.
1
0 Abstract
0
1
We study the flavor structure of 4D effective theories, which are derived from
:
v
extra dimensional theories with magnetic fluxes and non-Abelian Wilson lines. We
i
X studyzero-modewavefunctions andcomputeYukawacouplingsaswellasfour-point
r couplings. In our models, we also discuss non-Abelian discrete flavor symmetries
a
such as D , ∆(27) and ∆(54).
4
∗email: [email protected]
†email: [email protected]
‡email: [email protected]
§email: [email protected]
1 Introduction
Recently, extra dimensional field theory, in particular string-derived one, plays an im-
portant role in particle physics and cosmology. When we start with extra dimensional
field theory, it is one of most important issues how to realize a chiral spectrum in four
dimensional (4D) effective field theory. The magnetic flux background is one of interest-
ing ways to realize a 4D chiral theory. Indeed, several field-theoretical models and string
models, i.e. magnetized D-brane models, have been studied [1, 2, 3, 4, 5, 6, 7, 8, 9].
Furthermore, magnetized D-brane models are T-duals of intersecting D-brane models.
Within the latter type of model building, a number of interesting models have been con-
structed [4, 5, 6, 10, 11, 12].1.
Wavefunction profiles of zero-modes are quasi-localized on the torus with magnetic
flux background. The number of zero-modes is determined by the size of magnetic flux.
Since we know zero-mode profile explicitly, we can compute concretely 3-point and higher
order couplings of 4D effective field theory by overlap integrals of zero-mode profiles [7,
14, 15, 16, 17]. That is an important aspect of magnetized extra dimensional models.
Moreover, such a 4D effective field theory can have Abelian and non-Abelian discrete
flavor symmetries, which are originated from localization behavior of zero-modes in extra
dimensions [18].2
In addition to magnetic fluxes, we can introduce constant gaugebackgrounds and non-
trivial twisted boundary conditions aswell asorbifoldboundaryconditions [7, 23, 24, 25].3
That leads richer structure in model building such as zero-mode spectra and zero-mode
profiles.
Non-AbelianWilsonlines, i.e. theso-calledtoronbackgrounds[29], arealsointeresting
backgrounds [30, 7, 31, 32, 33]. They can break gauge groups with reducing their ranks.
For a certain case with magnetic fluxes and non-Abelian Wilson lines, zero-mode profiles
have been given [7]. Our purpose of this paper is to study more about models with
magnetic fluxes and non-Abelian Wilson lines. We analyze zero-mode profiles in generic
case and compute 3-point couplings. Furthermore, we study flavor symmetries. We also
study the orbifold compactification.
This paper is organized as follows. In section 2, we give a brief review on the extra
dimensional models with magnetic fluxes and non-Abelian Wilson lines. In section 3,
we study zero-mode wavefunctions on the torus compactification with magnetic fluxes
and non-Abelian Wilson lines. We compute Yukawa couplings in section 4 and study
flavor symmetries of our models in section 5. In section 6, we also study the orbifold
compactification. Section 7 is devoted to conclusion and discussion. In Appendix A we
show useful calculations, which are relevant to Yukawa couplings and in Appendix B we
compute four-point couplings as an example of higher order couplings.
1 See for a review [13] and references therein.
2
Similar non-Abeliandiscrete flavorsymmetries havebeen derivedwithin the frameworkofheterotic
orbifold models [19, 20, 21]. Analysis on their anomalies are also important [22].
3 Geometrical backgrounds other than tori and toroidal orbifolds have also been studied [26, 27, 28].
2
2 Non-Abelian Wilson lines
2.1 Higher dimensional super Yang-Mills theory
Our starting point is N = 1 U(N) super Yang-Mills theory in D = 4 + 2n dimensions
with n = 1 or 3. Its Lagrangian is written by
1 i
L = − Tr FMNF + Tr λ¯ΓMD λ , (1)
4g2 MN 2g2 M
(cid:0) (cid:1) (cid:0) (cid:1)
where M,N = 0,··· ,(D −1). Here, λ denotes gaugino fields, ΓM is the gamma matrix
for D dimensions and the covariant derivative D is given as
M
D λ = ∂ λ−i[A ,λ], (2)
M M M
where A is the vector field.
M
Here, we consider the torus(T2)n as 2n-dimensional extra dimensions anddenote their
coordinates by y (m = 4,··· ,2n+ 3). We use the orthogonal coordinates and choose
m
the torus metric such that y is identified by y +1 on the torus. The gaugino fields λ
m m
and the vector fields A corresponding to the compact directions are decomposed as
m
λ(x,y) = χ (x)⊗ψ (y), (3)
n n
n
X
A (x,y) = ϕ (x)⊗φ (y), (4)
m n,m n,m
n
X
where x denotes the coordinates of four-dimensional uncompact space R3,1. Here, we
are interested only in zero-modes, ψ (y) and φ (y). Thus, we omit the mode index
0 0,m
corresponding to n = 0 and write them as ψ(y) and φ (y).
m
2.2 Non-Abelian Wilson lines
Here, we consider T2 of (T2)n, whose coordinates are denoted as (y ,y ). As a U(N)
4 5
gauge background, we introduce the following form of (Abelian) magnetic flux,
f 1 0
F = 2π a Na , (5)
45 0 0
(cid:18) (cid:19)
where 1 denotes (N ×N ) identity matrix. For example, we use the following gauge,
Na a a
A = −F y , A = 0, (6)
4 45 5 5
for the U(N ) part. Then, their boundary conditions can be written as
a
∂ χ 1 0
A (y +1,y ) = A (y ,y )+ m 4 Na , χ = 0,
m 4 5 m 4 5 0 0 4
(cid:18) (cid:19)
∂ χ 1 0
A (y ,y +1) = A (y ,y )+ m 5 Na , χ = −2πf y . (7)
m 4 5 m 4 5 0 0 5 a 4
(cid:18) (cid:19)
3
This background breaks the gauge group U(N) to U(N )×U(N−N ). The zero mode
a a
ψ(y) corresponding to the gaugino is also decomposed as
A B
ψ = , (8)
C D
(cid:18) (cid:19)
depending on their U(N ) × U(N − N ) charges. That is, A and D correspond to the
a a
gaugino fields of unbroken symmetries, U(N ) and U(N − N ), respectively, while B
a a
and C correspond to bi-fundamental representations, (N ,N −N ) and (N ,N −N ),
a a a a
respectively.
Since only the U(1) part of U(N ) has the non-trivial background, its charge q is
a
relevant, thatis, A,B,C andD havechargesq = 0,1,−1and0, respectively. Forexample,
the zero-mode of B elements satisfies the following equation,
Γ˜m(∂ −iA )B(y) = 0, (9)
m m
for m = 4,5, where A denotes the U(N ) gauge background (6). Also, the zero-mode of
m a
C elements satisfies Γ˜m(∂ +iA )C(y) = 0, while the zero-modes of A and D elements
m m
satisfy Γ˜m∂ A(y) = 0 and Γ˜m∂ D(y) = 0. Here, Γ˜m corresponds to the gamma matrix
m m
for the two-dimensional torus T2, e.g.
0 1 0 −i
Γ˜4 = , Γ˜5 = , (10)
1 0 i 0
(cid:18) (cid:19) (cid:18) (cid:19)
and ψ(y) is the two component spinor,
ψ
ψ = + , (11)
ψ
−
(cid:18) (cid:19)
that is, A,B,C and D also have two components, A ,B ,C and D .
± ± ± ±
In particular, we are interested in matter fields. Thus, let us concentrate on B and C
fields. Because of (7), the spinor field, e.g. B, satisfies the following boundary conditions,
B(y +1,y ) = eiχ4B(y ,y ), (12)
4 5 4 5
B(y ,y +1) = eiχ5B(y ,y ). (13)
4 5 4 5
Here, we write these boundary conditions as
B(y +1,y ) = Ω (y ,y )B(y ,y ), (14)
4 5 4 4 5 4 5
B(y ,y +1) = Ω (y ,y )B(y ,y ). (15)
4 5 5 4 5 4 5
The above case corresponds to Ω (y ,y ) = eiχ4 and Ω (y ,y ) = eiχ5. Then, the consis-
4 4 5 5 4 5
tency forthecontractible loop, i.e. (y ,y ) → (y +1,y ) → (y +1,y +1) → (y ,y +1) →
4 5 4 5 4 5 4 5
(y ,y ) requires
4 5
Ω−1(y ,y +1)Ω−1(y +1,y +1)Ω (y +1,y )Ω (y ,y ) B(y ,y ) = B(y ,y ). (16)
5 4 5 4 4 5 5 4 5 4 4 5 4 5 4 5
(cid:0) (cid:1)
4
The left hand side reduces to e−2πifaψ(y ,y ) in the above background. This condition
4 5
for B leads to the quantization condition of the magnetic flux f . That is, the magnetic
a
flux f should be quantized such that f = integer. The consistency condition for the C
a a
fields also leads to the same quantization condition, i.e. f = integer.
a
When we introduce a non-trivial background for the SU(N ) part of U(N ), the situ-
a a
ation changes. That modifies the boundary conditions on, for example, B,
B(y +1,y ) = Ω (y ,y )B(y ,y ) = eiχ4ω B(y ,y ), (17)
4 5 4 4 5 4 5 4 4 5
B(y ,y +1) = Ω (y ,y )B(y ,y ) = eiχ5ω B(y ,y ), (18)
4 5 5 4 5 4 5 5 4 5
where ω are constant elements of SU(N ). Then, the consistency condition (16) reduces
m a
to
ω−1ω−1ω ω e−2πifa = 1 . (19)
5 4 5 4 Na
If ω and ω commute each other, that would require again e−2πifa = 1. Thus, it is
4 5
interesting that ω and ω do not commute each other, that is, non-Abelian Wilson lines.
4 5
In particular, we consider the case that ω−1ω−1ω ω corresponds to the center of SU(N ),
5 4 5 4 a
that is,
ω−1ω−1ω ω = e2πiMa/Na1 , (20)
5 4 5 4 Na
where M is an integer. In this case, the consistency condition (19) requires that the
a
magnetic flux should satisfy f = M /N (mod 1).
a a a
Wedenote P = g.c.d.(M ,N ), m = M /P andn = N /P .4 A solutionof Eq. (20)
a a a a a a a a a
is given as
ω = Pˆ , ω = Qˆ−ma, (21)
4 a 5 a
where
0 1 0 0 1 0 0 0
Pa Pa
0 0 1 0 0 ρ1 0 0
Pˆ =  Pa , Qˆ = ρ(na−1)/2 Pa , (22)
a ··· a ···
1 0 0 0  0 0 0 ρna−11 
 Pa   Pa
   
with ρ ≡ e2πi/na.
These non-Abelian Wilson lines break the gauge group U(N ) further. The following
a
condition on the U(N ) gauge field,
a
A = w A ω−1 = w A ω−1, (23)
µ 4 µ 4 5 µ 5
is required. Then, the gauge group U(N ) breaks to U(P ).
a a
4 Here, g.c.d. denotes the greatest common divisor.
5
3 Matter fields
Here, we consider the following form of U(N) magnetic fluxes,
f 1 0
1 N1
F = 2π ... . (24)
45  
0 f 1
 n Nn 
 
This form of magnetic fluxes breaks U(N) to U(N ) for f = integer. Furthermore, the
i i i
gauge group is broken to U(P ) when we choose f = M /N with P = g.c.d.(M ,N )
i i Q i i i i i i
andnon-AbelianWilsonlinessuchthattheysatisfytheconsistencyconditionlikeEq.(16).
Q
Now, let us focus on the (N +N )×(N +N ) block in U(N), which has the magnetic
a b a b
flux,
ma1
F = 2π na Na . (25)
mb1
(cid:18) nb Nb(cid:19)
We use the same gauge as Eq. (6), i.e.
ma1
A = −2π na Na y , A = 0. (26)
4 mb1 5 5
(cid:18) nb Nb(cid:19)
Similarly to Eq. (7), we denote their boundary conditions as
∂ χa1 0
A (y +1,y ) = A (y ,y )+ m 4 Na ,
m 4 5 m 4 5 0 ∂ χb1
(cid:18) m 4 Nb(cid:19)
∂ χa1 0
A (y ,y +1) = A (y ,y )+ m 5 Na , (27)
m 4 5 m 4 5 0 ∂ χb1
(cid:18) m 5 Nb(cid:19)
where
m m
χa = 0, χa = −2π ay , χb = 0, χb = −2π by . (28)
4 5 n 4 4 5 n 4
a b
We decompose the gaugino fields of this block in a way similar to Eq. (8). That is,
A and D correspond to adjoint matter fields of U(N ) and U(N ), respectively, while B
a b
and C correspond to bi-fundamental representations, (N ,N ) and (N ,N ), respectively.
a b a b
Among them, we concentrate on the field B, which satisfies the boundary conditions,
B(y +1,y ) = ΩaB(y ,y )(Ωb)† = ei(χa4−χb4)ωaB(y ,y )(ωb)†,
4 5 4 4 5 4 4 4 5 4
B(y ,y +1) = ΩaB(y ,y )(Ωb)† = ei(χa5−χb5)ωaB(y ,y )(ωb)†. (29)
4 5 5 4 5 5 5 4 5 5
Here, ωa,b are non-Abelian Wilson lines, which are given as Eqs. (21) and (22). Then,
4,5
the gauge symmetries are broken to U(P ) and U(P ). We study zero-mode profiles of B
a b
fields in what follows.
6
3.1 Integer magnetic fluxes
Before considering the models with fractional magnetic fluxes and non-Abelian Wilson
lines, it would be convenient to review briefly the models with integer magnetic fluxes
and no Wilson lines, that is,
n = n = 1, wa,b = wa,b = 1. (30)
a b 4 5
Then, the boundary condition (29) reduces to
B (y +1,y ) = B (y ,y ),
pq 4 5 p,q 4 5
B (y ,y +1) = e−2πimy4B (y ,y ), (31)
pq 4 5 p,q 4 5
where m = m −m . Suppose that m > 0. Then, the B component for each B element
a b + p,q
has m independent solutions for the zero-mode Dirac equation (9) with above boundary
condition (31). These solutions are given by
Θj(y4,y5) = e−mπ(l+mj )2+2πim(l+mj )y4−πmy52−2πm(l+mj )y5
l
X
j
= e−πmy52 ϑ m (mz,mτ), (32)
0
(cid:20) (cid:21)
j
wherez = y +iy , j = 0,1,··· ,m−1andτ = i. Here, ϑ m (mz,mτ) denotestheJacobi
4 5 0
(cid:20) (cid:21)
theta function. On the other hand, the B component has no normalizable zero-modes.
−
Similarly, the C fields have the same solutions as B , but the C has no normalizable
− + +
zero-modes.
When m < 0, the B and C fields have the |m| independent solutions with the same
− +
wavefunctions as above except replacing m by |m|. However, the B and C fields have
+ −
no normalizable zero-modes.
3.2 Fractional magnetic fluxes
Here, we study zero-mode profiles in the models with fractional magnetic fluxes and non-
Abelian Wilson lines.
3.2.1 n = n
a b
First, let us study the magnetic flux (25) for n = n = n . In this case, the non-Abelian
a b
Wilson lines break the gauge group U(N )×U(N ) to U(P )×U(P ), where P = N /n
a b a b a a
and P = N /n. Following this breaking pattern, we decompose the fields B as
b b
B B ···
00 01
B B ···
B =  10 11 . (33)
···
B B ··· B 
n−1,0 n−1,1 n−1,n−1
 
 
7
EachofB componentsis(P ×P )matrix-valuedfields, whichcorrespondtobi-fundamental
pq a b
(P ,P¯ ) fields under U(P )×U(P ). The boundary condition (29) due to the non-Abelian
a b a b
Wilson lines is written as
B (y +1,y ) = B (y ,y ),
pq 4 5 p+1,q+1 4 5
Bpq(y4,y5 +1) = ρ−(map−mbq)e−2πnimy4Bp,q(y4,y5), (34)
where m is used asm = m −m , andB = B = B . That leads to the boundary
a b p+n,q p,q+n p.q
condition,
B (y +n,y ) = B (y ,y ),
pq 4 5 pq 4 5
B (y ,y +n) = e−2πimy4B (y ,y ). (35)
pq 4 5 pq 4 5
Suppose that mn > 0. Then, similarly to section 3.1, the B component for B has
+ p,q
nm independent solutions for the zero-mode Dirac equation (9) with the above condition
(35). These solutions are given by
Θj(y4,y5) = e−nmπ(l+njm)2+2πim(l+njm)y4−πnmy52−2πm(l+njm)y5
l
X
j
= e−πnmy52 ϑ nm (mz,nmτ), (36)
0
(cid:20) (cid:21)
where j = 0,1,··· ,nm − 1 and τ = i. On the other hand, the B component has no
−
normalizable zero-modes. One finds that these solutions satisfy the boundary conditions,
Θj(y4 +1,y5) = e2πnijΘj(y4,y5),
Θj(y4,y5+1) = e−2πnimy4Θj+m(y4,y5). (37)
Thus, the zero-mode solutions with the boundary conditions (34) due to non-Abelian
Wilson lines can be written in terms of Θj of Eq. (36) as
n−1
Bpjq(y4,y5) = cjpq e2πi(map−mbq)nrΘj+mr, (38)
r=0
X
where j = 0,1,...,m−1. Here, cj is a constant normalization, which can be determined
pq
by the boundary conditions. Note that the boundary condition (34) relates B and
p,q
B . Thus, there are the series
p+1,q+1
B → B → ··· → B = B . (39)
p,q p+1,q+1 p+n,q+n p,q
The periodicity of the series is equal to n, and there are n independent series. Each of the
series has m independent solutions (38) and the total number of independent zero-modes
is equal to mn.
We have concentrated on the B fields. Similarly, when mn > 0, the C fields have
+ −
the same solutions as B . However, the B and C have no normalizable zero-modes
+ − +
8
for mn > 0. On the other hand, when mn < 0 the B and C have normalizable zero-
− +
modes with the same wavefunctions as the above, while B and C have normalizable
+ −
zero-modes.
We have considered the zero-mode profiles of fermionic fields. If 4D N=1 supersym-
metry is preserved, the scalar mode has the same zero-mode profiles as its fermionic
superpartner.
3.2.2 n 6= n
a b
Next, we study the model with n 6= n . In this case, the non-Abelian Wilson lines break
a b
the gauge group U(N )×U(N ) to U(P )×U(P ), where P = N /n and P = N /n .
a b a b a a a b b b
Similarly to the previous subsection, we decompose the fields B as
B B ··· B
00 01 0,nb−1
B B ···
B =  10 11 . (40)
···
B B ··· B 
 na−1,0 na−1,1 na−1,nb−1
 
Each of B components is (P ×P ) matrix-valued fields. The boundary condition (29)
pq a b
due to the non-Abelian Wilson lines is written as
B (y +1,y ) = B (y ,y ),
pq 4 5 p+1,q+1 4 5
Bpq(y4,y5 +1) = e−2πi(mnaa−mnbb)y4e−2πi(mnaap−mnabq)Bp,q(y4,y5), (41)
where B = B = B . This boundarycondition relates B andB . Then,
p+na,q p,q+nb p,q p,q p+1,q+1
similarly to (39), there are the following series
B → B → ··· → B = B . (42)
p,q p+1,q+1 p+Qab,q+Qab p,q
Here, the periodicity of the series is obtained by Q ≡ l.c.m.(n ,n ),5 and the number
ab a b
of independent series is equal to k ≡ g.c.d.(n ,n ). Obviously, there is the relation,
ab a b
Q = nanb. The above boundary condition (41) leads to the boundary condition,
ab kab
B (y +Q ,y ) = B (y ,y ),
pq 4 ab 5 pq 4 5
Bpq(y4,y5 +Qab) = e−k2aπbiIaby4Bp,q(y4,y5). (43)
Here we have defined ’the intersection number’ I ≡ n m − n m analogous to inter-
ab b a a b
secting brane models. There are S = nanbI independent zero-mode solutions, which
ab k2 ab
ab
satisfy the boundary condition (43). Those functions are obtained as
Θj(y4,y5) = e−πSab(n+Sjab)2+2πQiaSbab(n+Sjab)y4−πQS2aabby52−2πQSaabb(n+Sjab)y5
n
X
= e−πQS2aabby52ϑ Sjab ((Sab/Qab)z,Sabτ), (44)
0
(cid:20) (cid:21)
5 Here, l.c.m. denotes the least common multiple.
9
where τ = i. These wavefunctions satisfy the following boundary conditions,
Θj(y4 +1,y5) = e2πinkaanbbjΘj(y4,y5),
Θj(y4,y5 +1) = e2πi(mnaa−mnbb)y4Θj−kIaabb(y4,y5). (45)
Thus, the zero-mode wavefunctions, which satisfy the boundary conditions (41), are ob-
tained as
Bpjq(y4,y5) = cjpqQab−1e2πi(mnaap−mnbbq)rΘj+kIaabbr(y4,y5), (46)
r=0
X
where j = 0,1,··· , Iab −1. Hence, the number of the independent zero-modes is equal
kab
to M = Sab = Iab in each of the series (42), and there are the k independent series.
ab Qab kab ab
Thus, the total number of zero-modes is equal to M k = I .
ab ab ab
As an illustrating example, we consider the model with n = 2,n = 4 and m =
a b a
m = 3. Then, we have k = g.c.d.(n ,n ) = 2 6= 1, Q = 4, S = 12 and I = 6. We
b ab a b ab ab ab
decompose the bi-fundamental fields B with the 2×4 matrix entries as
B B B B
B = 00 01 02 03 . (47)
B B B B
10 11 12 13
(cid:18) (cid:19)
From the wave function formula in Eq. (46), one obtains the three independent solutions
labeled by j = 0,1,2 for each component of B and these are represented by linear
pq
combinations of Θi in Eq. (44). For example, the B and B fields are
00 01
Bj = Θj +Θj+3 +Θj+6 +Θj+9,
00
Bj = Θj +e−3π2iΘj+3 +e−3πiΘj+6 +e−92πiΘj+9. (48)
01
Obviously, the y -direction boundary condition can connect some of B components
4 p,q
follows
B → B → B → B → B , (49)
00 11 02 13 00
B → B → B → B → B . (50)
01 12 03 10 01
Since there are k = 2 independent series, there are I = 6 zero mode solutions in this
ab ab
background.
3.3 Another representation of solutions
Intheprevioussection, wehavepresented solutionsintermsoftheΘj functions. However,
by using the properties of the theta function, one can represent the wave functions (38)
and (46) as a single theta function as
j m m
Bpjq(y4,y5) = Cpj,qe−πI˜aby52 ×ϑ M0ab I˜abz + nap− nbq ,I˜abτ , (51)
(cid:20) (cid:21)(cid:18) (cid:18) a b (cid:19) (cid:19)
10