The perturbative CFT optical theorem and high-energy string scattering in AdS at one loop

We derive an optical theorem for perturbative CFTs which computes the double discontinuity of conformal correlators from the single discontinuities of lower order correlators, in analogy with the optical theorem for flat space scattering amplitudes. The theorem takes a purely multiplicative form in the CFT impact parameter representation used to describe high-energy scattering in the dual AdS theory. We use this result to study four-point correlation functions that are dominated in the Regge limit by the exchange of the graviton Regge trajectory (Pomeron) in the dual theory. At one-loop the scattering is dominated by double Pomeron exchange and receives contributions from tidal excitations of the scattering states which are efficiently described by an AdS vertex function, in close analogy with the known Regge limit result for one-loop string scattering in flat space at finite string tension. We compare the flat space limit of the conformal correlator to the flat space results and thus derive constraints on the one-loop vertex function for type IIB strings in AdS and also on general spinning tree level type IIB amplitudes in AdS.

More on topological vertex formalism for 5-brane webs with O5-plane

We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of [1]. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO(N) gauge theories and the pure G2 gauge theory. The obtained results agree with the results known in the literature. We also compute the partition function of the pure SU(3) gauge theory with the Chern-Simons level 9. At the end we rewrite the O-vertex in a form of a vertex operator.

The full vertex functions constrained by the longitudinal and transverse Ward–Takahashi identities in massless QED3 theory

In this paper, we find apart from the Ward–Takahashi (WT) identity, the identity between gamma matrices can also constrain the vertex functions in low-dimensional gauge theories. In (1 + 1) dimensions, the identity between gamma matrices gives the identity between vector and axial-vector vertex functions while in (2 + 1) dimensions it leads to the identity between vector and tensor vertex functions. Then, we derive the expressions of the full scalar, vector and tensor vertex functions in (2 + 1) dimensions Quantum Electrodynamics (QED3) by using the longitudinal and transverse WT identities for vector and tensor currents. Furthermore, we find that in the chiral limit with zero fermion masses, the contribution of Wilson line in full vector vertex function is eliminated and the full vector vertex function is strictly expressed in terms of the fermion propagators when using the identity between vector and tensor vertex functions to further constraint the vertex functions.

The constraint equation of the vertex function of quantum anomaly and Dyson–Schwinger equations in massless QED2 theory

In this paper, we calculate the quantum anomaly for the longitudinal and the transverse Ward–Takahashi (WT) identities for vector and axial-vector currents in QED2 theory by means of the point-splitting method. It is found that the longitudinal WT identity for vector current and transverse WT identity for axial-vector current have no anomaly while the longitudinal WT identity for axial-vector current and the transverse WT identity for vector current have anomaly in QED2 theory. Moreover, we study the four WT identities in massless QED2 theory and get the result that the four WT identities together give the constraint equation of the vertex function of quantum anomaly. At last, we discuss the Dyson–Schwinger equations in massless QED2 theory. It is found that the vertex function of the quantum anomaly has corrections for the fermion propagator and Schwinger model.

Zumkeller Labeling of Complete Graphs

Let G (V, E) be a graph with vertex set V and edge set E. The process of assigning natural numbers to the vertices of G such that the product of the numbers of adjacent vertices of G is a Zumkeller number on the edges of G is known as Zumkeller labeling of G. This can be achieved by defining an appropriate vertex function of G. In this article, we show the existence of this labeling to complete graph and fan graph.

Why is the mission impossible? Decoupling the mirror Ginsparg–Wilson fermions in the lattice models for two-dimensional Abelian chiral gauge theories

It is known that the four-dimensional Abelian chiral gauge theories of an anomaly-free set of Wely fermions can be formulated on the lattice preserving the exact gauge invariance and the required locality property in the framework of the Ginsparg–Wilson relation. This holds true in two dimensions. However, in the related formulation including the mirror Ginsparg–Wilson fermions, and therefore having a simpler fermion path-integral measure, it has been argued that the mirror fermions do not decouple: in the 345 model with Dirac– and Majorana–Yukawa couplings to the XY-spin field, the two-point vertex function of the (external) gauge field in the mirror sector shows a singular non-local behavior in the paramagnetic strong-coupling phase. We re-examine why the attempt seems to be a “Mission: Impossible” in the 345 model. We point out that the effective operators to break the fermion number symmetries (‘t Hooft operators plus others) in the mirror sector do not have sufficiently strong couplings even in the limit of large Majorana–Yukawa couplings. We also observe that the type of Majorana–Yukawa term considered is singular in the large limit due to the nature of the chiral projection of the Ginsparg–Wilson fermions, but a slight modification without such a singularity is allowed by virtue of their very nature. We then consider a simpler four-flavor axial gauge model, the $1^4(-1)^4$ model, in which the U(1)$_A$ gauge and Spin(6)(SU(4)) global symmetries prohibit the bilinear terms but allow the quartic terms to break all the other continuous mirror fermion symmetries. We formulate the model so that it is well behaved and simplified in the strong-coupling limit of the quartic operators. Through Monte Carlo simulations in the weak gauge-coupling limit, we show numerical evidence that the two-point vertex function of the gauge field in the mirror sector shows regular local behavior, and we argue that all you need is to kill the continuous mirror fermion symmetries with would-be gauge anomalies non-matched, as originally claimed by Eichten and Preskill. Finally, by gauging a U(1) subgroup of the U(1)$_A$$\times$ Spin(6)(SU(4)) of the previous model, we formulate the $2 1 (-1)^3$ chiral gauge model, and argue that the induced fermion measure term satisfies the required locality property and provides a solution to the reconstruction theorem formulated by Lüscher. This gives us “A New Hope” for the mission to be accomplished.

Pseudo-scalar mesons at finite temperatures from a Dyson-Schwinger-Bethe-Salpeter approach

The truncated Dyson-Schwinger–Bethe-Salpeter equations are employed at non-zero temperature. The truncations refer to a rainbow-ladder approximation augmented with an interaction kernel which facilitates a special temperature dependence. At low temperatures, T → 0, we recover a quark propagator from the Dyson-Schwinger (gap) equation smoothly interpolating to the T = 0 results. Utilizing that quark propagator we evaluate the Bethe-Salpeter vertex function in the pseudo-scalar qq̅ channel for the lowest boson Matsubara frequencies and find a competition of qq̅ bound states and quasi-free two-quark states at T = O (100 MeV).