By Yu B.J., Xu M.

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**Example text**

Further, the new term is the only mass scale in this Hamiltonian since the coupling constant cannot serve as the mass scale. In fact, it is even worse than the dimensionless coupling constant case, since the coupling constant in the NJL model is proportional to the inverse square of the mass dimension. Thus, we define the new fermion mass MN by MN = 2G B. 23) The Bogoliubov angle θn can be determined from the following equation cot 2θn = |ppn | . 24) In this case, the vacuum changes drastically since the original vacuum is trivial.

For the massless boson, there should be a continuum spectrum, and this continuum spectrum of the massless boson should be differentiated from the continuum spectrum arising from the many body nature of the system. This differentiation must have been an extremely difficult task without having some analytic expression of the spectrum. In fact, even if one finds a continuum spectrum which has, for example, the dispersion of E = c0 p2 as often discussed in solid state physics, one sees that the spectrum has nothing to do with the Goldstone boson.

15b) is for i = i0 . In this case, the energy of the one particle-one hole states E(i1p1h 0) given as, N = |ki0 | − ∑ |ki |. 15) can be found at the specific value of ni0 and then from this ni0 value on, we find continuous spectrum of the 1 p−1h states. 15) for the lowest 1 p − 1h state. 17c) for ni = −1, −2, · · · , −N0 . 18) Bethe Ansatz Solutions in Thirring Model 41 where [X ] denotes the smallest integer value which is larger than X . In this case, we can express the lowest 1p − 1h state energy analytically 1p−1h E0 = −Λ (N0 + 1) − 2ni0 2(N0 + 1) −1 g tan + N0 π π Therefore, the lowest excitation energy ∆E0 vacuum state becomes 1p−1h ∆E01p−1h ≡ E01p−1h − Evtrue = .