Massive Modes in Magnetized Brane Models

Progress of Theoretical Physics, Nov 2012

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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Massive Modes in Magnetized Brane Models

Yuta Hamada 0 Tatsuo Kobayashi 0 0 Department of Physics, Kyoto University , 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. - 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)


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Yuta Hamada, Tatsuo Kobayashi. Massive Modes in Magnetized Brane Models, Progress of Theoretical Physics, 2012, pp. 903-923, 128/5, DOI: 10.1143/PTP.128.903