O(Nc) and USp(Nc) QCD from String Theory

Toshiya Imoto 1 2 ) Tadakatsu Sakai 1 2 Shigeki Sugimoto 0 1 0 Institute for the Physics and Mathematics of the Universe, The University of Tokyo , Kashiwa 277-8568, Japan 1 Subject Index: 101, 121 2 Department of Physics, Nagoya University , Nagoya 464-8602, Japan We propose a holographic dual of large Nc quantum chromodynamics (QCD) with the gauge groups O(Nc) and USp(Nc) and Nf flavors of massless quarks. This is constructed by adding O6-planes to an intersecting D4-D8 system in type IIA superstring theory. The holographic dual description is formulated in Witten's D4-brane background with D8-branes and O6-planes embedded in it as probes. The D4-brane background gives rise to a smooth interpolation of D8-D8 pairs and an O6-O6 pair. We show that the resultant brane configuration explains geometrically the flavor symmetry breaking patterns in O(Nc) and USp(Nc) QCD, which are caused by quark bilinear condensates. We next discuss that baryons can be realized as D4-D4 pairs wrapped on S4, which intersect with the O6-plane. By analyzing the tachyons on it, we reproduce the stability conditions of the baryons that are expected from the gauge theory viewpoint. The stable baryon configurations are classified systematically using K-theory. We also give a similar analysis of the flux tubes and again reproduce the results that are consistent with QCD. - 1, 2, , Nc is the color index of the gauge group, and i = 1, 2, , Nf is the flavor index. Since the fundamental representation for these gauge groups is vector-like, there is no distinction between left-handed and right-handed components of the quark fields, and hence, the flavor symmetry is U (Nf ) for both G = O(Nc) and G = U Sp(Nc) cases.) Note that Nf must be even for the G = U Sp(Nc) case to avoid the global anomaly.6) Like SU (Nc) QCD, these models are considered to develop non-vanishing condensates: Jab qai qbj = c Jij . (for G = U Sp(Nc)) Here, c is a non-vanishing constant and J is the U Sp invariant anti-symmetric tensor. This implies that the flavor symmetry U (Nf ) is spontaneously broken as (for G = O(Nc)) U (Nf ) U Sp(Nf ) . (for G = U Sp(Nc)) This phenomenon should result from strong coupling gauge dynamics, and hence, it is rather difficult to prove that this really occurs by making a full analysis of the gauge dynamics. A natural question to ask then is whether we can demonstrate this phenomenon from string theory, following the same line in Ref. 5). The purpose of this paper is to show that it is indeed possible to analyze O(Nc) and U Sp(Nc) QCD using string theory. To this end, we begin by constructing a brane configuration that realizes O(Nc) and U Sp(Nc) QCD with massless flavors, by adding O6-planes in the D4-D8 configuration presented in Ref. 5). Then, the D4-branes are replaced with the corresponding supergravity (SUGRA) background obtained in Ref. 7), which is a holographic dual of large Nc strongly coupled YangMills (YM) theory. As in Ref. 5), the quarks are incorporated into the model by embedding probe D8-branes into the D4-brane background. Here, we use the probe approximation and ignore the backreaction of the O6-plane and D8-branes, which can be justified when Nf Nc. It is found that a brane interpolation mechanism that is observed in Ref. 5) leads to a natural explanation of the symmetry breaking pattern (1.2) from the physics of intersecting D-branes and O-planes. We also study the stability of baryons and flux tubes in O(Nc) and U Sp(Nc) QCD using our holographic description. The properties of these objects are quite different from the cases with G = SU (Nc). As discussed in Ref. 8), the baryon number in O(Nc) QCD is Z2-valued; that means a single baryon is stable, while two baryons can decay. The flux tubes also behave in a similar way. On the other hand, the baryons and flux tubes are totally unstable in U Sp(Nc) QCD. In the holographic description of QCD, the baryons and flux tubes can be realized as D-branes wrapped on S4 in the D4-brane background. We show that the stability conditions of these ) Here, we include the anomalous U(1) part of the flavor symmetry, since the effect of the anomaly vanishes in the large Nc limit. The effect of the anomaly can be incorporated as studied in 5.8 of Ref. 5). D-branes are in agreement with what we expect in QCD. Furthermore, since the stable D-brane configurations are classified using K-theory,9) the baryons and flux tubes correspond to the elements of K-groups. We find that the relevant K-groups that are used to classify the stable baryons and flux tubes again reproduce the above results. This refines the classification via homotopy groups given in Ref. 8), in which there are some discrepancies for Nf 3. This paper is organized as follows. In 2, we present a brane configuration that defines massless QCD with the gauge group G = O(Nc) and G = U Sp(Nc). In 3, we formulate the holographic dual description and show that it nicely explains the flavor symmetry breaking and the stability of baryons and flux tubes in QCD. We end this paper with a summary and discussion in 4. In Appendices A and B, we summarize the properties of intersecting Dp-Op systems and K-groups that are used in this paper, respectively. 2. Brane configuration of O(Nc) and U Sp(Nc) QCD In this section, we construct a brane configuration of O(Nc) and U Sp(Nc) QCD with massless flavors, by generalizing the model given in Ref. 5), which is proposed as a holographic dual of U (Nc) QCD with massless flavors. For this purpose, let us first review some key results in Ref. 5) with an emphasis on how the gluon and quarks emerge. This model is composed of Nc D4-branes and Nf pairs of D8- and D8-branes: Here, the x4 direction is compactified on S1 of radius MKK1 . In this paper, we work in the MKK = 1 unit. The D4-branes are wrapped around this circle, while the D8-branes and D8-branes are located at the antipodal points x4 = /2 and x4 = /2, respectively. Following Ref. 7), we impose the anti-periodic boundary condition along the S1 parametrized by x4 on all the fermions in the system, while all the bosonic fields are kept periodic. Then, the gluinos as well as the scalar fields on the D4-brane, which belong to the adjoint representation of the SU (Nc) gauge symmetry, become massive. In addition, the left-handed and right-handed components of the quark fields (qL and qR, respectively) are created as the massless modes in the 4-8 strings (open strings stretching from the D4-branes to the D8branes) and 4-8 strings, respectively. Then, the D4-brane world-volume theory flows to four-dimensional U (Nc) QCD with Nf massless quarks at low energy.) Note that the gauge symmetries on the D8-branes and D8-branes correspond to the chiral symmetries U (Nf )L and U (Nf )R, respectively. To obtain O(Nc) and U Sp(Nc) QCD, we consider an orientifold defined on the basis of the Z2 action (x4, x8, x9) (x4, x8, x9) together with the world-sheet ) We regard the diagonal U(1) part of the U(Nc) g (...truncated)