Massive Modes in Magnetized Brane Models
Yuta Hamada
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Tatsuo Kobayashi
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Department of Physics, Kyoto University
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Kyoto 606-8502, Japan
We study higher dimensional models with magnetic fluxes, which can be derived from superstring theory. We study mass spectrum and wavefunctions of massless and massive modes for spinor, scalar and vector fields. We compute the 3-point couplings and higher order couplings among massless modes and massive modes in 4D low-energy effective field theory. These couplings have non-trivial behaviors, because wavefunctions of massless and massive modes are non-trivial.
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products of 3-point couplings. These low-energy effective field theories can also lead
to Abelian and non-Abelian discrete flavor symmetries, e.g. D4 and (27) flavor
symmetries.15), 16),) These non-Abelian discrete flavor symmetries are important
to derive the realistic quark and lepton mass matrices (see e.g. 18) and references
therein).
In addition to massless modes, massive modes also have important effects in
4D low-energy effective field theory. For example, they may induce the fast proton
decay and flavor changing neutral currents (FCNCs) (see e.g. 19)). Our purpose in
this paper is to study massive modes in the extra dimensional models with magnetic
fluxes. We study their mass spectrum and wavefunctions explicitly. Then, we study
compute 3-point couplings and higher order couplings including these massive modes.
These couplings have non-trivial behaviors, because wavefunctions of massless and
massive modes are non-trivial.
This paper is organized as follows. In 2, we briefly review the fermion
zeromodes on T 2 with the magnetic flux. Then, we study mass spectrum and
wavefunctions of higher modes explicitly. These analyses are extended to those for zero-modes
and higher modes of scalar and vector fields. Its extension to T 6 is straightforward.
In 3, we compute couplings among these modes. In 3.1, we give a brief review
on computations of the 3-point couplings and higher order couplings among
zeromodes. Then, we extend them to the computations of the 3-point and higher order
couplings including higher modes in 3.2. In 3.3, we also consider the couplings
including massive modes due to only the Wilson line effect, but not magnetic fluxes.
In 4, we give comments on some phenomenological implications of our results.
Section 5 is devoted to conclusion and discussion. In Appendix A, we show some useful
properties of the Hermite function. In Appendix B, we briefly review the vector field
in extra dimensions. In Appendix C, we show useful properties of the products of
zero-mode wavefunctions.
Mass spectrum and wavefunctions of massive modes
We consider the (4 + d)-dimensions, and denote four-dimensional and
d-dimensional coordinates by x and ym with = 0, , 3 and m = 1, , d, respectively.
We study the spinor field (x, ym) and the vector field AM (x, ym) with M =
0, , (3 + d). We decompose these fields as follows,
Here we choose the internal wavefunctions n(ym) as eigenfunctions of the internal
Dirac operator as
where m denote the gamma matrices in the internal space. The eigenvalues of mn
become masses of the modes n(x) in 4D effective field theory. Similarly, n,M (ym)
correspond to eigenfunctions of the internal Laplace operators, as will be shown
explicitly later. The scalar field in the (4 + d)-dimensions is also decomposed in a
similar way.
2.1. T 2 with magnetic flux
This magnetic flux is derived, e.g., from the following vector potential,
Their boundary conditions can be written as
Az =
for m = 0.
2.1.1. Fermion zero-modes
Furthermore, we can introduce non-vanishing Wilson lines by using
where is complex and corresponds to the degree of freedom of the Wilson line. It
is convenient to use the following notation,
We use the gamma matrices on T 2 as
Then, the zero-mode equation is written as
Note that the effect of the Wilson line is the shift of the wavefunctions j,qm(z)
to j,qm(z + ). The zero-mode wavefunction can be written by a product of the
Gaussian function and the Jacobi -function, i.e.,
D =
for the spinor with U (1) charge q. The charge q and magnetic flux m should satisfy
that qm = integer. They also satisfy the following boundary conditions,
When qm > 0, only the zero-mode +,0 has a solution, but ,0 has no solution.
Then, the chiral spectrum for the zero-modes is realized and the number of
zeromodes is equal to qm. Their zero-mode wavefunctions are written explicitly as
(+j,qm) represents the anti-particle of +j,qm, and is obtained from Eq. (2.16) by
replacing j,qm(z + , ) with j,qm(z + , ). These zero-mode wavefunctions
satisfy the following orthonormal condition,
When qm < 0, there appear the zero-modes for ,0, but not for +,0. The
number of their zero-modes is equal to |qm|, and their wavefunctions are obtained
similarly. In the following discussion, we assume qm > 0.
2.1.2. Fermion massive modes
The 2D Laplace operator is defined as
and it satisfies the following algebraic relations,
= m2n
[D, D] =
a =
a =
(...truncated)