Mesons in Relativistic Mock Meson Model Dae

We derive formulas for the decay constants fP and fV of pseudoscalar and vector mesons in the relativistic mock meson model. Using these formulas, we obtain fP and fV of Bs, Bd, Ds, and Dd mesons as functions of the mock meson parameter β. Then by using the values of β which are obtained by the variational method in the relativistic quark model, we obtain the decay constants fP and fV of the heavy mesons, and the corresponding ratios fV /fP . The results are compared with other calculations and existing experimental results. 1: e-mail: [email protected] 2: e-mail: [email protected] We study the decay constants of heavy pseudoscalar and vector mesons in the relativistic mock meson model of Godfrey, Isgur, and Capstick [1, 2, 3], in which the heavy meson state composed of a light quark q and a heavy antiquark Q̄ is represented as |M(K) >= ∫ dpΦ(p)χss̄ φcc̄ |q( mq m K+ p, s) Q̄( mQ̄ m K − p, s̄) >, (1) where K is the mock meson momentum, m ≡ mq+mQ̄, and Φ(p), χss̄, and φcc̄ are momentum, spin, and color wave functions respectively. We take the momentum wave function Φ(p) as a Gaussian wave function Φ(p) = 1 ( √ πβ)3/2 e /2β . (2) The decay constant of the pseudoscalar and vector mesons, fP and fV respectively, are defined by < 0| Q̄γγ5q |MP (K) >= fPK, < 0| Q̄γq |MV (K, ε) >= fVmV ε. (3) The meson state in (1) is written explicitly in the meson rest frame (where pq = −pQ̄) as |MP (0) > = √ 2mP ∫ dpq (2π)3/2 √ 2Eq 2EQ̄ Φ(pq) 1 √ Nc × 1 √ 2 [a†↑(pq, c)b † ↓(pQ̄, c̄)− a†↓(pq, c)b†↑(pQ̄, c̄)] |0 >, (4) where the arrow indicates a state with spin up (down) along a fixed axis and c is the colour index which is summed. Whereas we wrote the pseudoscalar meson state in (4), we can also write the vector meson state in the same way with the spin combinations for the vector states, which are given by (↑↑), 1/ √ 2 (↑↓ + ↓↑) and (↓↓). In (4) we adopted the normalization of the creation and annihilation operators given by {a(p, s), a(p, s′)} = (2π) 2E δss′δ(p−p), and then the meson state in (4) is normalized by < MP (0)|MP (0) >= 2mP δ(0), and also in the same way for the vector meson states. 2 Since we are concerned with the matrix elements in the left hand sides of (3) with the meson states in (4), it is convenient to represent the meson states by ΨP ≡ − < 0| qQ̄ |MP (0) >, ΨV ≡ − < 0| qQ̄ |MV (0) >, (5) with which the formulas in (3) are written as Tr( γγ5ΨP ) = fPmP , T r( γ ΨV ) = fVmV ε . (6) If both two quarks inside the meson are static, the spinor combinations of u(0)v̄(0) for the pseudoscalar and vector meson states are given respectively as [4, 5] P (0, 0) = − 1 √ 2 1 + γ 2 γ, V (0, 0, ε) = 1 √ 2 1 + γ 2 6 ε, (7) where the polarization vectors of the vector meson are given by ε± = (1/ √ 2)(0, 1,±i, 0) and εμ3 = (0, 0, 0, 1). However, since the quarks inside the mock meson represented by (4) are not static, we boost the spinors by using the formulas u(k) = 6 k +m