Corrections to the SU(3)×SU(3){\bf SU(3)\times SU(3)} Gell-Mann-Oakes-Renner relation and chiral couplings L8rL^r_8 and H2rH^r_2

Next to leading order corrections to the $SU(3) \times SU(3)$ Gell-Mann-Oakes-Renner relation (GMOR) are obtained using weighted QCD Finite Energy Sum Rules (FESR) involving the pseudoscalar current correlator. Two types of integration kernels in the FESR are used to suppress the contribution of the kaon radial excitations to the hadronic spectral function, one with local and the other with global constraints. The result for the pseudoscalar current correlator at zero momentum is $\psi_5(0) = (2.8 \pm 0.3) \times 10^{-3} GeV^{4}$, leading to the chiral corrections to GMOR: $\delta_K = (55 \pm 5)$. The resulting uncertainties are mostly due to variations in the upper limit of integration in the FESR, within the stability regions, and to a much lesser extent due to the uncertainties in the strong coupling and the strange quark mass. Higher order quark mass corrections, vacuum condensates, and the hadronic resonance sector play a negligible role in this determination. These results confirm an independent determination from chiral perturbation theory giving also very large corrections, i.e. roughly an order of magnitude larger than the corresponding corrections in chiral $SU(2) \times SU(2)$. Combining these results with our previous determination of the corrections to GMOR in chiral $SU(2) \times SU(2)$, $\delta_\pi$, we are able to determine two low energy constants of chiral perturbation theory, i.e. $L^r_8 = (1.0 \pm 0.3) \times 10^{-3}$, and $H^r_2 = - (4.7 \pm 0.6) \times 10^{-3}$, both at the scale of the $\rho$-meson mass.Comment: Revised version with minor correction

Chiral corrections to the SU(2)×SU(2)SU(2)\times SU(2) Gell-Mann-Oakes-Renner relation

The next to leading order chiral corrections to the $SU(2)\times SU(2)$ Gell-Mann-Oakes-Renner (GMOR) relation are obtained using the pseudoscalar correlator to five-loop order in perturbative QCD, together with new finite energy sum rules (FESR) incorporating polynomial, Legendre type, integration kernels. The purpose of these kernels is to suppress hadronic contributions in the region where they are least known. This reduces considerably the systematic uncertainties arising from the lack of direct experimental information on the hadronic resonance spectral function. Three different methods are used to compute the FESR contour integral in the complex energy (squared) s-plane, i.e. Fixed Order Perturbation Theory, Contour Improved Perturbation Theory, and a fixed renormalization scale scheme. We obtain for the corrections to the GMOR relation, $\delta_\pi$, the value $\delta_\pi = (6.2, \pm 1.6)$. This result is substantially more accurate than previous determinations based on QCD sum rules; it is also more reliable as it is basically free of systematic uncertainties. It implies a light quark condensate $\simeq \equiv |_{2\,\mathrm{GeV}} = (- 267 \pm 5 MeV)^3$. As a byproduct, the chiral perturbation theory (unphysical) low energy constant $H^r_2$ is predicted to be $H^r_2 (\nu_\chi = M_\rho) = - (5.1 \pm 1.8)\times 10^{-3}$, or $H^r_2 (\nu_\chi = M_\eta) = - (5.7 \pm 2.0)\times 10^{-3}$.Comment: A comment about the value of the strong coupling has been added at the end of Section 4. No change in results or conslusion

The scalar gluonium correlator: large-beta_0 and beyond

The investigation of the scalar gluonium correlator is interesting because it carries the quantum numbers of the vacuum and the relevant hadronic current is related to the anomalous trace of the QCD energy-momentum tensor in the chiral limit. After reviewing the purely perturbative corrections known up to next-next-to-leading order, the behaviour of the correlator is studied to all orders by means of the large-beta_0 approximation. Similar to the QCD Adler function, the large-order behaviour is governed by the leading ultraviolet renormalon pole. The structure of infrared renormalon poles, being related to the operator product expansion are also discussed, as well as a low-energy theorem for the correlator that provides a relation to the renormalisation group invariant gluon condensate, and the vacuum matrix element of the trace of the QCD energy-momentum tensor.Comment: 14 pages, references added, discussion of IR renormalon pole at u=3 extended, similar version to appear in JHE

Generalised ladders and single-valued polylogarithms

We introduce and solve an infinite class of loop integrals which generalises the well-known ladder series. The integrals are described in terms of single-valued polylogarithmic functions which satisfy certain differential equations. The combination of the differential equations and single-valued behaviour allow us to explicitly construct the polylogarithms recursively. For this class of integrals the symbol may be read off from the integrand in a particularly simple way. We give an explicit formula for the simplest generalisation of the ladder series. We also relate the generalised ladder integrals to a class of vacuum diagrams which includes both the wheels and the zigzags.Comment: 27 pages, 7 figure

Symbols of One-Loop Integrals From Mixed Tate Motives

We use a result on mixed Tate motives due to Goncharov (arXiv:alg-geom/9601021) to show that the symbol of an arbitrary one-loop 2m-gon integral in 2m dimensions may be read off directly from its Feynman parameterization. The algorithm proceeds via recursion in m seeded by the well-known box integrals in four dimensions. As a simple application of this method we write down the symbol of a three-mass hexagon integral in six dimensions.Comment: 13 pages, v2: minor typos correcte

The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM

We provide an analytic formula for the (rescaled) one-loop scalar hexagon integral $\tilde\Phi_6$ with all external legs massless, in terms of classical polylogarithms. We show that this integral is closely connected to two integrals appearing in one- and two-loop amplitudes in planar $\\mathcal{N}=4$ super-Yang-Mills theory, $\Omega^{(1)}$ and $\Omega^{(2)}$. The derivative of $\Omega^{(2)}$ with respect to one of the conformal invariants yields $\tilde\Phi_6$, while another first-order differential operator applied to $\tilde\Phi_6$ yields $\Omega^{(1)}$. We also introduce some kinematic variables that rationalize the arguments of the polylogarithms, making it easy to verify the latter differential equation. We also give a further example of a six-dimensional integral relevant for amplitudes in $\\mathcal{N}=4$ super-Yang-Mills.Comment: 18 pages, 2 figure

The last of the simple remainders

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New differential equations for on-shell loop integrals

We present a novel type of differential equations for on-shell loop integrals. The equations are second-order and importantly, they reduce the loop level by one, so that they can be solved iteratively in the loop order. We present several infinite series of integrals satisfying such iterative differential equations. The differential operators we use are best written using momentum twistor space. The use of the latter was advocated in recent papers discussing loop integrals in N=4 super Yang-Mills. One of our motivations is to provide a tool for deriving analytical results for scattering amplitudes in this theory. We show that the integrals needed for planar MHV amplitudes up to two loops can be thought of as deriving from a single master topology. The master integral satisfies our differential equations, and so do most of the reduced integrals. A consequence of the differential equations is that the integrals we discuss are not arbitrarily complicated transcendental functions. For two specific two-loop integrals we give the full analytic solution. The simplicity of the integrals appearing in the scattering amplitudes in planar N=4 super Yang-Mills is strongly suggestive of a relation to the conjectured underlying integrability of the theory. We expect these differential equations to be relevant for all planar MHV and non-MHV amplitudes. We also discuss possible extensions of our method to more general classes of integrals.Comment: 39 pages, 8 figures; v2: typos corrected, definition of harmonic polylogarithms adde

Lattice QCD determination of m_b, f_B and f_Bs with twisted mass Wilson fermions

We present a lattice QCD determination of the b quark mass and of the B and B_s decay constants, performed with N_f=2 twisted mass Wilson fermions, by simulating at four values of the lattice spacing. In order to study the b quark on the lattice, two methods are adopted in the present work, respectively based on suitable ratios with exactly known static limit and on the interpolation between relativistic data, evaluated in the charm mass region, and the static point, obtained by simulating the HQET on the lattice. The two methods provide results in good agreement. For the b quark mass in the MSbar scheme and for the decay constants we obtain m_b(m_b)=4.29(14) GeV, f_B=195(12) MeV, f_Bs=232(10) MeV and f_Bs/f_B=1.19(5). As a byproduct of the analysis we also obtain the results for the f_D and f_Ds decay constants: f_D=212(8) MeV, f_Ds=248(6) MeV and f_Ds/f_D=1.17(5).Comment: 23 pages, 10 figures, 2 tables. Added appendix showing the agreement of the data for the ratios with the HQE prediction. Matching JHEP published versio