By Takehisa Fujita, Makoto Hiramotoa, Hidenori Takahashi

The authors current a unified description of the spontaneous symmetry breaking and its linked bosons in fermion box thought. there is not any Goldstone boson within the fermion box idea types of Nambu-Jona-Lasinio, Thirring and QCD2 after the chiral symmetry is spontaneously damaged within the new vacuum. The disorder of the Goldstone theorem is clarified, and the 'massless boson' anticipated by way of the concept is digital and corresponds to only a unfastened massless fermion and antifermion pair.Further, the authors talk about the precise spectrum of the Thirring version via the Bethe ansatz ideas, and the analytical expressions of the entire actual observables let the authors to appreciate the essence of the spontaneous symmetry breaking extensive. additionally, the authors research the boson spectrum in QCD2, and convey that bosons constantly have a finite mass for SU(Nc) shades. the matter of the sunshine cone prescription in QCD2 is mentioned, and it truly is proven that the trivial mild cone vacuum is liable for the inaccurate prediction of the boson mass.

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The boson mass for the NJL model is plotted as the function of GΛ2 . It is measured by the cutoff Λ. Further, we should note that Kleinert and Van den Bossche [12] also found No Goldstone Boson in Fermion Field Theory 29 that the bosons after the symmetry breaking are all massive, which is just consistent with our claim. Their method and approach are quite different from the present calculation, but both of the calculations agree with each other that there is no massless boson in the NJL model.

Since the massless Thirring model cannot be bosonized properly, there is no massless excitation spectrum in the model, and this is consistent with the Bethe ansatz solutions that the massless Thirring model has a finite gap and then the continuum spectrum starts right above the gap. Bethe Ansatz Solutions in Thirring Model 47 Also, we should stress that the bosonization of the massless Thirring model has a subtlety, and one must be very careful for treating it. If one makes a small approximation or a subtle mistake in calculating the spectrum of the Hamiltonian, then one would easily obtain unphysical massless excitations from the massless Thirring model.

22) holds true as operator equations under these conditions. Together with the Coulomb interaction part, we can write down the Hamiltonian for the Schwinger model as H =∑ p 1 1 † Π (p)Π(p) + 2 2 2πp L 2 Φ† (p)Φ(p) + g2 † Φ (p)Φ(p) . 23) 2π This is just the free massive boson Hamiltonian. It should be important to note that the Schwinger model has the right zero mode in the Hamiltonian of the boson field. However, as we saw in section 5, there is no corresponding zero mode in the massless Thirring model, and this leads to the finite gap of the spectrum in the massless Thirring model, which is indeed consistent with the fact that there should exist no physical massless boson in two dimensions.