A little large N group theory

We discuss the group theory relevant to the ground-state baryons in large Nc QCD. For very large representation, the group generators become classical variables. We find the form of the classical generators for the completely symmetric N index representation of SU(m) as N → ∞ and derive an integral formula for the matrix elements of an arbitrary polynomial in the group generators between low-spin baryon states in the large N limit. ∗Research supported in part by the National Science Foundation under Grant #PHY-9218167. The idea of replacing the SU(3) gauge symmetry of QCD with an SU(N) symmetry and studying the N → ∞ limit, as articulated by ’t Hooft in [1], has led to an important qualitative understanding of some of the properties of QCD, such as Zweig’s rule and the narrowness of resonances. Witten [2] later provided a conceptual framework to include baryons in the theory. In Witten’s picture, a baryon is described by a Hartree-Fock equation with each quark moving in the mean potential generated by all the other quarks in the baryon. Using this description for the baryons, [3, 4, 5] (see also [6]) one can show that to leading order in N , the low-spin baryons have a spin-flavor symmetry which we will denote as SU(m), e.g. for an Nf -flavor theory m = 2Nf . In this letter, we study the group theory associated with the large representations of this SU(m). In doing so, we develop an elegant integral formula for matrix elements of SU(m) generators between low-spin baryon states. This formula provides an insight into the nature of the large N enhancement of these matrix elements. An N quark baryon lives in the completely antisymmetric representation of the color SU(N). An s-wave ground state, to satisfy fermi statistics, must then be completely symmetric under the SU(m) symmetry. Therefore, the representation of SU(m) relevant to large N baryons is the completely symmetric combination of N defining representations. We will examine the matrix elements of the group generators in this representation, which we will denote by T α β , to leading order in N , and will discover that many properties of the generators can be obtained rather simply in this limit. Our primary tool will be the fact that for large N , the commutator of two group generators is lower order in N than the product. The product is order N, while the commutator is order N (times the structure constants, which depend on m but not on N). Thus to leading order in N , the generators can be simultaneously diagonalized—they become essentially classical variables [6] [3] [7]. Explicitly calculating the traces in the representation space of products of generators and taking the N → ∞ limit leads to the following form for the generators: T α β |u〉 = T α β (u) |u〉 T α β (u) = N (