In 't Hooft's paper "A two-dimensional model for mesons", the author shows that two-dimensional (1+1) QCD in the large-N limit interestingly gives a theory of mesons. 't Hooft calculates the "mesonic spectrum" of the theory in a way that I don't precisely understand.

Let me first say what I do follow.

  • p.1-2: The Lagrangian is declared, light-cone gauge chosen. In this gauge, the Faddeev-Popov ghosts decouple. We choose lightcone coordinates. Since we are working in 1+1 dimensions, there are no transverse dimensions.
  • pp.3: Feynman rules are read off the Lagrangian. Here the double-line notation for gluon propagators is used, which is relevant for taking the large-N limit. The dressed propagator (i.e. propagator with self-energy) is given formally.
  • pp.4: In the large-N limit, only planar diagrams with gluon-loops contribute to the self-energy. With a bit of thinking, this leads to a simple "bootstrap" equation (Fig. 3 in paper, Eq. 10 in paper).
  • pp.5: With an arbitrary infrared-cutoff, the self-energy is calculated and given by Eq. 14 in the paper. Interestingly, in the 'physical' limit where the IR regulator is taken to zero, $\lambda\rightarrow 0$, the quark propagator pole shifts towards $k^2\rightarrow \infty$, which suggests that there are no physical quark states (confining!...?)

Now onto what I do not understand.

  • pp.5: The author introduces an arbitrary diagrammatic blob $\psi (p,r)$, out of which come two quarks: one with mass $m_1$ and momentum $p$, the other with mass $m_2$ and momentum $r-p$. Presumably when re-summed this should be the creation of a meson (nontrivial two-quark) state. The author states that this blob $\psi(p,r)$ satisfies an inhomogeneous Bethe-Salpeter equation (Fig. 4 in paper), but that only the homogeneous part matters for the spectrum of two-particle states.

I do not precisely understand the statements in bold (I know what a Bethe-Salpeter equation is by the way). Firstly, is the inhomogeneous Bethe-Salpeter equation the following? I'm partly shooting in the dark here...

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Secondly, why exactly does this blob (on the left) dictate the spectrum of two-particle states? Specifically, how can I know this before doing any calculations, i.e. at that point in the paper, where it is declared? I would expect the following color-singlet correlator to govern mesonic states:

$$\int d^4x \, e^{ikx} \, \langle \bar \psi (x) \psi (x) \bar \psi (0) \psi (0)\rangle$$

where here $\psi$ represents the quark-fields (not the $\psi(p,r)$ blob from before), with all indices omitted.


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