OPE for null Wilson loops and open spin chains

Maximal helicity-violating scattering amplitudes in N=4$N=4$ supersymmetric Yang–Mills theory are dual to Wilson loops on closed null polygons. We perform their operator product expansion analysis in two-dimensional kinematics in the soft-collinear approximation which corresponds to the case when some light-cone distances vanish. We construct the expansion in terms of multi-particle “heavy”–light operators, where the “heavy” fields are identified with the Wilson lines defining the OPE channel and the light fields emerge from the curvature of the contour. The correlation function of these define the remainder function. We study the dilatation operator for these operators at one-loop order and find that it corresponds to a non-compact open spin chain. This provides an alternative view on elementary excitations propagating on the GKP string at weak coupling, which now correspond to particles traveling along an open spin chain. The factorized structure of the Wilson loop in the soft limit allows one to represent the two-loop correction to the octagon Wilson loop as a convolution formula and find the corresponding remainder function.     

One-loop photon–photon scattering in a thermal,deconfining SU(2) Yang–Mills plasma

For a deconfining thermal SU(2) Yang–Mills plasma we discuss the role of (anti)calorons in introducing non-thermal behavior effectively described in terms of Planck’s quantum of action ?$?$. This non-thermality cancels exactly between the ground-state estimate and its free quasiparticle excitations. Kinematic constraints in 4-vertex scattering and the counting of radial loop variables versus the number of independent constraints on them are re-visited. Next, we consider thermal 2→2$2\to 2$ one-loop scattering of the modes remaining massless upon the (anti)caloron induced adjoint Higgs mechanism (thermal ground state after spatial coarse graining). Starting with stringent analytical arguments, we are able to exclude the contribution to photon–photon scattering from diagrams containing at least one three-vertex and, in a next step, a vast majority of all possible configurations involving two four-vertices. By numerical analysis we show that the remaining contribution of the overall S channel is severely suppressed compared those of the T and U channels, meaning that the creation of a pair of massive vector modes by a pair of photons and vice versa practically does not occur in the Yang–Mills plasma. For the T and U channels the domain of loop integration represents less than 10⁻⁷$10^{-7}$ times the volume of the unconstrained integration region. The thus introduced photon–photon correlation should affect the Cosmic Microwave Background’s polarization at low redshift. An adaption of the here-developed methods to the analysis of irreducible bubble diagrams could prove the conjecture of hep-th/0609033 on the termination of the loop expansion of thermodynamical quantities at a finite irreducible order.     

Composite representation invariants and unoriented topological string amplitudes

Sinha and Vafa [1] had conjectured that the SO   Chern–Simons gauge theory on S³$S^3$ must be dual to the closed A  -model topological string on the orientifold of a resolved conifold. Though the Chern–Simons free energy could be rewritten in terms of the topological string amplitudes providing evidence for the conjecture, we needed a novel idea in the context of Wilson loop observables to extract cross-cap c=0,1,2$c=0,1,2$ topological amplitudes. Recent paper of Marino [2] based on the work of Morton and Ryder [3] has clearly shown that the composite representation placed on the knots and links plays a crucial role to rewrite the topological string cross-cap c=0$c=0$ amplitude. This enables extracting the unoriented cross-cap c=2$c=2$ topological amplitude. In this paper, we have explicitly worked out the composite invariants for some framed knots and links carrying composite representations in U(N)$U(N)$ Chern–Simons theory. We have verified generalised Rudolph's theorem, which relates composite invariants to the invariants in SO(N)$\mathit{SO}(N)$ Chern–Simons theory, and also verified Marino's conjectures on the integrality properties of the topological string amplitudes. For some framed knots and links, we have tabulated the BPS integer invariants for cross-cap c=0$c=0$ and c=2$c=2$ giving the open-string topological amplitude on the orientifold of the resolved conifold.     

Two-point functions of quenched lattice QCD in Numerical Stochastic Perturbation Theory. (II) The gluon propagator in Landau gauge

This is the second of two papers devoted to the perturbative computation of the ghost and gluon propagators in SU(3)$\mathit{SU}(3)$ Lattice Gauge Theory. Such a computation should enable a comparison with results from lattice simulations in order to reveal the genuinely non-perturbative content of the latter. The gluon propagator is computed by means of Numerical Stochastic Perturbation Theory: results range from two up to four loops, depending on the different lattice sizes. The non-logarithmic constants for one, two and three loops are extrapolated to the lattice spacing a→0$a\to 0$ continuum and infinite volume V→∞$V\to \infty$ limits.     

Simulating the all-order hopping expansion II: Wilson fermions

We investigate the extension of the Prokof'ev–Svistunov worm algorithm to Wilson lattice fermions in an external scalar field. We effectively simulate by Monte Carlo the graphs contributing to the hopping expansion of the two-point function on a finite lattice to arbitrary order. Tests are conducted for a constant background field i.e. free fermions at some mass. For the method introduced here this is expected to be a representative case. Its advantage is that we know the exact answers and can thus make stringent tests on the numerics. The approach is formulated in both two and three space–time dimensions. In D=2$D=2$ Wilson fermions enjoy special positivity properties and the simulation is similarly efficient as in the Ising model. In D=3$D=3$ the method also works at sufficiently large mass, but there is a hard sign problem in the present formulation hindering us to take the continuum limit.     

Some properties of the Alday–Maldacena minimum

The Alday–Maldacena solution, relevant to the n=4$n=4$ gluon amplitude in N=4$N=4$ SYM at strong coupling, was recently identified as a minimum of the regularized action in the moduli space of solutions of the AdS₅${\mathit{AdS}}_5$σ  -model equations of motion. Analogous solutions of the Nambu–Goto equations for the n=4$n=4$ case are presented and shown to form (modulo the reparametrization group) an equally large but different moduli space, with the Alday–Maldacena solution at the intersection of the σ  -model and Nambu–Goto moduli spaces. We comment upon the possible form of the regularized action for n=5$n=5$. A function of moduli parameters za$z_a$ is written, whose minimum reproduces the BDDK one-loop five-gluon amplitude. This function may thus be considered as some kind of Legendre transform of the BDDK formula and has its own value independently of the Alday–Maldacena approach.     

Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

We study the properties of the conformal blocks of the conformal field theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate fields, the conformal blocks obey second order differential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero–Sutherland Hamiltonian with non-trivial braiding properties. A generalized duality property relates the two types of second order degenerate fields. By studying this duality we found that the excited states of the Calogero–Sutherland Hamiltonian are characterized by two partitions, or in the case of _WAk−1${\mathrm{WA}}_{k-1}$ theories by k   partitions. By extending the conformal field theories under consideration by a u(1)$u(1)$ field, we find that we can put in correspondence the states in the Hilbert state of the extended CFT with the excited non-polynomial eigenstates of the Calogero–Sutherland Hamiltonian. When the action of the Calogero–Sutherland integrals of motion is translated on the Hilbert space, they become identical to the integrals of motion recently discovered by Alba, Fateev, Litvinov and Tarnopolsky in Liouville theory in the context of the AGT conjecture. Upon bosonization, these integrals of motion can be expressed as a sum of two, or in general k, bosonic Calogero–Sutherland Hamiltonian coupled by an interaction term with a triangular structure. For special values of the coupling constant, the conformal blocks can be expressed in terms of Jack polynomials with pairing properties, and they give electron wave functions for special Fractional Quantum Hall states.     

A complete calculation for direct detection of Wino dark matter

In the anomaly-mediated supersymmetry (SUSY) breaking scenario, neutral gaugino of SUL₍₂₎$\mathit{SU}{(2)}_L$ multiplet, Wino, can be the lightest SUSY particle and become a candidate for dark matter. We calculated scattering cross section of Wino dark matter with nucleon, which is responsible for direct detection of the dark matter, on the assumption that the SUSY particles and the heavier Higgs bosons have masses of the order of the gravitino mass in the SUSY standard model. In such a case, the Wino–nucleon coupling is generated by loop processes. We have included two-loop contribution to Wino–gluon interaction in the calculation, since it is one of the leading contributions to the Wino–nucleon coupling. It was found that the spin-independent scattering cross section with proton is ^{10−(46–48)} cm²${10}^{-(46\text{–}48)}{\text{cm}}^2$. While it is almost independent of the Wino mass, the result is quite sensitive to the Higgs boson mass due to the accidental cancellation.     

Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey–Wilson polynomials

Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey–Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey–Wilson polynomials in terms of a degree ?   (?=1,2,…$?=1,2,\dots$) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree ??1$??1$ and thus not constrained by any generalisation of Bochner's theorem.     

Minimal affinizations as projective objects

We prove that the specialization to q=1$q=1$ of a Kirillov–Reshetikhin module for an untwisted quantum affine algebra of classical type is projective in a suitable category. This yields a uniform character formula for the Kirillov–Reshetikhin modules. We conjecture that these results hold for specializations of minimal affinization with some restriction on the corresponding highest weight. We discuss the connection with the conjecture of Nakai and Nakanishi on q$q$-characters of minimal affinizations. We establish this conjecture in some special cases. This also leads us to conjecture an alternating sum formula for Jacobi–Trudi determinants.