# Understanding some lines from 't Hooft's paper on large-N QCD in 1+1d

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... 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.