AdS/CFT Correspondence and QCD with Quarks in Fundamental Representations

The most straightforward use of AdS/CFT correspondence gives versions of QCD where quarks are in adjoint representations. Using an asymmetric orbifold approach we obtain nonsupersymmetric QCD with four quark flavors in fundamental representations of color.


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The interplay between gauge field theories and string theory is one of the most fertile in high-energy theory. Indeed string theory had its beginnings as an attempted theory of strong interactions [1,2] to be replaced by the gauge field theory of quantum chromodynamics (QCD) [3,4]. Nevertheless the interconnection between these theories constantly portrays them not as competitors but most recently as dual descriptions. The notion that strings describe some large N colors limit of QCD was suggested already by 't Hooft [5] in 1974. An excellent review is provided by the 1987 book by Polyakov [6]. A large step forward was taken with the identification by Maldacena [7] of the AdS/CFT correspondence For example, compactifying a superstring on an AdS 5 × S 5 manifold there is a duality of descriptions of either a type IIB superstring in ten spacetime dimensions or an N = 4 supersymmetric SU (N ) gauge field theory in four spacetime dimensions.
This correspondence provides a powerful tool to investigate gauge field theory. Independently of whether the superstring can provide a correct theory of quantum gravity the AdS/CFT correspondence can suggest interesting models for non-gravitational physics. Beyond being merely a tool, it can suggest directions to extend the standard model and even additional particles that may be lying in the TeV regime awaiting discovery [8]. As a tool, it may be used to study QCD and very interesting results have been obtained by Polchinski and Strassler [9][10][11] about the relationship of QCD to string theory from the AdS/CFT correspondence. This is of special interest in view of the above history because the string theory was originally displaced by QCD thirty years ago.
This recent work on string QCD sheds light on how QCD describes hard scattering by pointlike constituents [12] whereas the string gave only infinitely soft scattering. The resolution come from the importance of the curved geometry of the fifth dimension in the AdS 5 manifold.
Here we make an observation about the group theory of orbifolding The reconciliation of these two differences with QCD can be arranged simultaneously by the following asymmetric orbifold procedure.
The fermions which survive the orbifolding are those which are invariant under a product of a Z p transformation and a SU (N ) p gauge transformation. These fermions are in the bifundamental representations of SU (N ) p : The complex scalars which survive are also those which are invariant under a product of a Z p transformation and a SU (N ) p gauge transformation but now follows from the real and v i = (α a 1 , α a 2 , α a 3 ). Here a 1 = (A 2 + A 3 ), a 2 = (A 3 + A 1 ) and a 3 = (A 1 + A 2 ) although we note that the subscript orderings of A µ , a i are arbitrary and we can therefore choose Both Eq.(1) and Eq.(2) can be conveniently displayed in quiver or moose diagrams as we shall illustrate for p = 3 in Figure 1.
For simplicity we choose the lowest possible value of p (p = 3) which works.
Note that for p = 2 the only choice for A µ is A µ = (1, 1, 1, 1) and consequently a i = (0, 0, 0) which means the quiver diagram for scalars, from Eq.(2), is disconnected into two separate SU (N )s and no progress toward QCD is possible. Choosing p = 3 is, however, sufficient. In this case the unique choice to attain N = 0 is A µ = (1, 1, 2, 2) and therefore a i = (0, 0, 1). The scalar quiver is shown in Figure 1(a). Each SU (N ) has two adjoints and two (N + N * ) fundamentals of scalars.
If we now spontaneously break the SU (N ) p symmetry in a way which respects the Z p symmetries of the quiver diagram, and identify the QCD gauge group as the diagonal subgroup of the three SU (N ) groups then the quarks will again be in adjoints just as in the N = 4 case but without the supersymmetry.
Instead we choose an asymmetrical symmetry breaking where two of the SU (N )s, say the two lower nodes in Figure 1 The dynamical studies in [9,10] presumably carry over to the present case since they depend only on the AdS geometry. Using the present AdS/CFT version of string QCD will, however, help arrange the correct color group theory factors to appear in the calculations.