Meson-Baryon Scattering in QCD_2 for any Coupling

Extending earlier work on strong-coupling meson-baryon scattering in QCD_2, we derive the effective meson-baryon action for any value of the coupling constant, in the large-N_c limit. Colour degrees of freedom play an important role, and we show that meson-baryon scattering can be formulated as a relativistic potential problem. We distinguish two cases that are non-trivial for unequal quark masses, and present the resulting equations for meson-baryon scattering amplitudes.


Introduction
The problem of deriving resonances from QCD is clearly of importance, both as a fundamental test of QCD and of specific non-perturbative methods. Experimentally, this field continues to bring surprises, including enhancements nearpp thresholds [1,2,3], an exotic K + N resonance [4,5,6], and two new mesons containing c ands quarks [7,8]. These discoveries demonstrate that non-perturbative QCD is not yet fully 'solved' [9], and underline the interest in developing new non-perturbative methods. The dynamics governing the formation of resonances in QCD is still imperfectly understood, especially the relation between the spectrum and chiral symmetry breaking ( χ SB), possible 'exotic' non-qq mesons and non-qqq baryons, and combinations of light and heavy quarks. QCD in one space and one time dimension, QCD 2 , is an interesting laboratory for studying many of these issues. There is no spontaneous χ SB in two dimensions, but a trace of the phenomenon does exist, in the sense that m P S ∝ √ e c m q , where m P S is the mass of the lightest pseudoscalar meson, e c is the gauge coupling and m q is the quark mass. This formula for m P S is analogous to the case in four dimensions, with e c replacing Λ QCD .
Motivated by this similarity and the tractability of QCD 2 , we investigated in [10] the problem of resonances in the meson-baryon channel in this model. This question is still relevant both as a crude model for the real world in four dimensions and also as a testing ground for non-perturbative methods that might have relevance there and elsewhere.
In [10] we worked in the strong-coupling limit, e c /m q ≫ 1, namely in the limit where the QCD 2 coupling is much larger than the quark mass. This would correspond in QCD 4 to quark masses being small compared to the QCD scale, m q /Λ QCD ≪ 1, the limit where explicit χ SB is small. Moreover, we worked in the approximation that the baryons are heavy compared to the mesons, which is justified by the large-N c limit in QCD. In this double limit, we found no resonances in meson-baryon scattering, only elastic scattering. This result suggest that quark masses are essential complications in QCD, at least in two dimensions.
In the present work we relax one of our two previous assumptions, that of strong coupling, while retaining the assumption that baryons are heavy. This discussion of general coupling may equivalently be regarded as introduc-ing finite quark masses. We show that this generalization necessitates the re-introduction of colour degrees of freedom, which were decoupled in the previous strong-coupling limit. The scattering problem may be formulated as a relativistic potential problem that differs essentially from the previous strong-coupling, low-mass case. We derive the corresponding new equations for meson-baryon scattering amplitudes, valid for any value of the QCD 2 coupling.
In Section 2 we discuss the general formulation for QCD 2 , in bosonized form, which is necessary for obtaining the meson-baryon scattering amplitude for an arbitrary value of the coupling. We use this formulation in Section 3 to derive the effective meson-baryon action, for arbitrary coupling in the heavy-baryon limit. We show that in Section 4 that, for light mesons and heavy baryons, the scattering problem can be transformed to problem with relativistic potentials, and we evaluate that potential in two cases that are non-trivial for unequal quark masses. Section 5 contains our derivation of the equations for scattering amplitudes, and Section 6 discusses the prospects for future work.

General Formulation
We start from the discussion of [11], which presented an effective action for bosonized QCD 2 with N f flavours and N c colours: where the traces are over both colour and flavour. The second term is the result of integrating out the gauge fields, which was made possible by the quadratic dependence on the gauge fields in two dimensions in the light-cone gauge, which we chose. Finally, S 0 [u] is the Wess-Zumino-Witten (WZW) action, representing free fermions.
The above action is exact for any value of the gauge coupling e c . The matrices u represent the bosonic version of the quark bilinears, symbolically Ψ ⊗Ψ, with both colour and flavour indices. The symmetry groups are SU(N c ) and U(N f ), for colour and flavour, respectively, including the U(1) of baryon number, and where m q is the quark mass,m the normal-ordering scale, and C ≡ 1 2 e γ ≈ 0.891. For a detailed discussion, see [12]. For the time being, we take all quarks to have the same mass.
When using the product scheme for bosonization and taking the strong-coupling limit e c /m q → ∞, one can eliminate the second term in (1), after first choosing an appropriate normal-ordering scalẽ m, as described in [12]. The colour degrees of freedom h are completely eliminated in this limit, and one gets an effective action in terms of flavour degrees of freedom only: Note that here the traces are over flavour only.
As shown in [10], this action results in elastic scattering only, and is therefore unsuitable for modelling QCD 4 realistically. Scattering is a soluble problem in this approximation, with the transmission and reflection coefficients T and R given in [10] for all meson energies.
When dealing with finite coupling, we should not use the product scheme, but rather the 'u-scheme', where the original matrices u ∈ U(N F × N c ). The two schemes are equivalent only when the quark masses are zero, or equivalently in the strong-coupling limit. One important aspect of the difference, at finite mass, is that the 'u-scheme' admits multi-soliton solutions [11] that do not exist in the product scheme. Each of the solitons in these solutions carries both colour and flavour, yet the total multi-soliton solutions are colour-neutral. These individual coloured solitons were interpreted in [11] as constituent quarks. When the quark masses are equal, the solutions can be obtained analytically, and are given by the sine-Gordon profile. For unequal quark masses, the solutions were obtained numerically.

Extension to Arbitrary Coupling
We first consider the second term of (1), for arbitrary coupling: with details to be found in [12]. As already mentioned, this term represents the interactions, as it arises from integrating out the gauge potentials. However, we will see that, for the physical situation we discuss, this term does not contribute to meson-baryon scattering for any coupling. As a result, the latter is described by the effective actionS eff [u], whereas in the strong coupling limit it is described byS eff [g].
In order to describe meson-baryon scattering, we follow the four-dimensional example [13], and take u to be of the form corresponding to a classical soliton Φ c representing one of the baryons described in [11], and a small fluctuation δΦ around it, representing the meson. The resulting action is then expanded to second order in δΦ, yielding a linear equation of motion for δΦ in the soliton background. The latter serves as an external potential in which the meson is propagating.
We start by evaluating and obtain the equations of motion for the meson field by varying with respect to δΦ. The variation of (7) with respect to δΦ is proportional to To compute its variation with respect to δΦ, we need only the second term M 2 of M, as the first term M 1 is independent of δΦ.
We take for the soliton a diagonal ansatz, following [11]: The part of M that contributes to the effective action is its colour projection (8). We note that Tr f &c M 2 = 0, and thus (14) The mesons δΦ have to be diagonal in colour, so We recall that the flavour structure of the mesons is independent of their colour indices, and restrict our attention to mesons that have no U(1) flavour part. In this way, we may be sure that classical solutions lead to stable particles, since their non-vanishing flavour quantum numbers put them in a different sector from the vacuum. We then have as advertized earlier, and the effective meson-baryon action is with u depending on δΦ for fixed Φ c as in (9).

Evaluation of the Potential
The equation of motion for δΦ is obtained from (17), by first varying with respect to u and then varying u with respect to δΦ. To first order in δΦ, we find The resulting equation of motion is then where m is the diagonal mass matrix: m = δ ij m j with (possibly different) entries m j corresponding to flavours j. We note that there is the possibility of an overall scale ambiguity in m, since, when the masses are different, there is a question which normal-ordering scale to use, as discussed in the Appendix of [11]. The resulting equation of motion for δΦ is where µ ≡ m √ 8π.
As discussed before, both Φ c and δΦ are diagonal in colour. Moreover, Φ c is diagonal in flavour too. So, taking the ααjj ′ matrix element of the equation of motion (20), we find Examining the classical solutions for the quark solitons inside the baryons as in [11], we see that, for a given colour index α, there is only one flavour for which Φ c is non-zero. We can now distinguish three cases.
• The first is when an index α and indices j and j ′ are chosen in such a way that both (Φ c ) αj and (Φ c ) αj ′ are zero. In such a case, where(Φ c ) αj = 0 and (Φ c ) αj ′ = 0.
Thus δΦ αjj ′ is a free field with squared mass given by the average of m 2 j and m 2 j ′ in this case, which we do not discuss further.
• The second case is that of j = j ′ , with α such that (Φ c ) αj is a quark soliton inside the baryon. In this case, This case is analogous to the case in Section 2 of [10], with the difference that here the potential is determined by Φ c which, for the case of unequal masses, is not necessarily of the sine-Gordon type.
• The third case is when j is different from j ′ , now with one of the Φ c being a soliton and the other vanishing. Taking (Φ c ) αj to be the soliton, we obtain where j ′ = j and (Φ c ) αj ′ = 0 . This case is analogous to Section 4 of [10], but with the same difference as in (23), i.e., that for unequal masses the soliton is not of the sine-Gordon type.

Evaluation of Meson-Baryon Scattering
The equations which determine the static solution (Φ c ) αj were derived in [11]. For completeness, we include them here too. First one defines where the (χ c ) αj are canonical fields, whose equations of motion are Note the extra factor 2 in front of the mass term, as compared with Eq. (22) of [11], due to an error in this reference.
The baryon number § associated with any given flavour l is given by Combining the last two equations, we find for the total baryon number.
We now continue in a similar manner to [10], starting with the first nontrivial case (23) identified above. As the soliton solutions are such that there is a unique correspondence between the colour index α and the flavour index j, we suppress α in what follows. Putting we find and the equation for u j (x) is We define the potential V j for this scattering process via and find (33) § In our normalization, a single quark carries one unit of baryon number.
In our normalization the outgoing wave has coefficient 1, which is more convenient for numerical calculations, and the wave for x → −∞ is now in this case.
In the second non-trivial case (24), we put To eliminate the first derivative term in u, we substitute This results in We note that the last three lines vanish when all the quark masses are equal, as then the soliton is a sine-Gordon one. Thus, the scattering would then be only elastic, as found in [10].
The potential of the scattering is defined here via so that Taking we get and the wave for in this case.

Discussion
We have shown that meson-baryon scattering in QCD 2 in the large-N c limit is non-trivial for non-zero quark masses, and is described by two distinct effective potentials when the quark masses are unequal. These effective potentials are not of the sine-Gordon type found in previous cases, and we expect the scattering amplitudes also to be non-trivial. Their calculation will require numerical analysis, that we postpone for a future occasion.
Clearly QCD 2 is not a complete laboratory for studying non-perturbative physics in QCD 4 . However, it is already quite a rich system, and full QCD can only be richer still. We have already pointed out the existence of constituent quark solitons in QCD 2 [11], and this numerical analysis may cast light on their importance in scattering, where the additive quark model has long been an intriguing approximation.
It is intriguing that the introduction of unequal quark masses is an essential complication. We recall that the light-cone wave functions for mesons containing unequal quark masses are expected not to be symmetric, resulting in a net colour field that underlies this effect. One task of numerical analysis will be to see what new physics this produces in meson-baryon scattering and possibly in resonant states. We note that two of the recent new puzzles in non-perturbative QCD concern systems with unequal quark masses [4,5,7,8]. It would be hubristic to suggest that the continuation of our QCD 2 studies will cast light on these puzzles, but they will provide some extra motivation for our future work in this direction.