Supersymmetry Relics in One-Flavor QCD from a New1/NExpansion

Abstract

We suggest a new large-N c limit for multiflavor QCD. Since fundamental and two-index antisymmetric representations are equivalent in SU(3), we have the option to define SUN c QCD keeping quarks in the latter. We can then define a new 1=N c expansion (at a fixed number of flavors N f ) that shares appealing properties with the topological (fixed N f =N c ) expansion while being more suitable for theoretical analysis. In particular, for N f 1, our large-N c limit gives a theory that we recently proved to be equivalent, in the bosonic sector, to N 1 supersymmetric gluodynamics. Using known properties of the latter, we derive several qualitative and semiquantitative predictions for N f 1 massless QCD that can be easily tested in lattice simulations. Finally, we comment on possible applications for pure SU(3) Yang-Mills theory and real QCD. DOI: 10.1103/PhysRevLett.91.191601 PACS numbers: 11.15.Pg, 11.30.Pb, 12.38.Aw, 12.38.Lg Very few techniques are available for analytical studies of nonperturbative properties of nonsupersymmetric gauge theories such as QCD. Among the most promising ones, large-N expansions play a special role, in particular, because of their conjectured connection to string theories.The simplest and the oldest 1=N c expansion in QCD is that suggested by 't Hooft c . The 't Hooft expansion led to a number of notable successes in such issues as the Zweig rule, the 0 mass formula (see below), and so on. Unfortunately, nobody succeeded in fully solving QCD even to the leading order in the 't Hooft expansion.In the range of questions where the quark loops are important, a better approximation is provided by the topological expansion (TE) [2] in which N f =N c is kept fixed in the large-N c limit, rather than N f . Then, in the leading order, TE preserves all planar diagrams, including quark loops. This is easily seen by slightly modifying [2] the 't Hooft double-line notation-adding a flavor line to the single color line for quarks. In the leading (planar) diagrams the quark loops are ''empty'' inside, since gluons do not attach to the flavor line. Needless to say, obtaining analytic results in TE is even harder than in the 't Hooft case.

QCD with one compact spatial dimension

“…Our interest in this problem was initiated from a quite different source: a discussion of the validity of the "orientifold QCD" programme of Armoni, Shifman, and Veneziano [4][5][6]. These authors proposed a large-N c equivalence between N = 1 super Yang-Mills and QCD with a single fermion flavor in the symmetric or antisymmetric tensor representation.…”

The ’t Hooft vertex revisited

“…An example to illustrate this procedure is the antisymmetric two indexed representation of SU(N). This representation has been extensively used in [22,23,24,25] for an alternative approach to the large N c limit. The basic N(N − 1)/2 fermion fields take the form…”

Phases ofN=∞QCD-like gauge theories onS3×S1and nonperturbative orbifold-orientifold equivalences

“…The parity argument is based on local order parameters on R 3,1 , in particular, our topologically nontrivial order parameters such as the imaginary parts of Wilson lines only emerge upon compactification. 19 Therefore, there is no clash between broken CPT and broken P for QCD-like theories formulated on R 2,1 × S 1 and the CPT and P theorems which are formulated on R 3,1 . However, it would be incorrect to make a general statement such as in vector-like gauge theories, CPT (or P) cannot be spontaneously broken.…”

“…The QCD-like theories described above, for each value of n f , are related to each other via a chain of orbifold and orientifold projections (see refs. [14][15][16][17][18][19][20] and references therein). The case where n f = 1 is particularly interesting, as it relates supersymmetric N = 1 SYM theory to nonsupersymmetric theories such as QCD(BF/AS/S).…”