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Continuing our previous study cite{Du:2011se} of permutation sum of color ordered tree amplitudes of gluons, in this note, we prove the large-$z$ behavior of their cyclic sum and the combination of cyclic and permutation sums under BCFW deformation. Unlike the permutation sum, the study of cyclic sum and the combination of cyclic and permutation sums is much more difficult. By using the generalized Bern-Carrasco-Johansson (BCJ) relation, we have proved the boundary behavior of cyclic sum with nonadjacent BCFW deformation. The proof of cyclic sum with adjacent BCFW deformation is a little bit simpler, where only Kleiss-Kuijf (KK) relations are needed. Finally we have presented a new observation for partial-ordered permutation sum and applied it to prove the boundary behavior of combination sum with cyclic and permutation.
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We review recent developments in QCD sum rule applications to semileptonic B->pi and D->pi transitions.
There is growing evidence that on-shell gluon scattering amplitudes in planar N=4 SYM theory are equivalent to Wilson loops evaluated over contours consisting of straight, light-like segments defined by the momenta of the external gluons. This equivalence was first suggested at strong coupling using the AdS/CFT correspondence and has since been verified at weak coupling to one loop in perturbation theory. Here we perform an explicit two-loop calculation of the Wilson loop dual to the four-gluon scattering amplitude and demonstrate that the relation holds beyond one loop. We also propose an anomalous conformal Ward identity which uniquely fixes the form of the finite part (up to an additive constant) of the Wilson loop dual to four- and five-gluon amplitudes, in complete agreement with the BDS conjecture for the multi-gluon MHV amplitudes.
Recently, Bjerrum-Bohr, Damgaard, Feng and Sondergaard derived a set of new interesting quadratic identities of the Yang-Mills tree scattering amplitudes. Here we comment that these quadratic identities of YM amplitudes actually follow directly from the KLT relation for graviton-dilaton-axion scattering amplitudes (in 4 dimensional spacetime). This clarifies their physical origin and also provides a simpler version of the new identities. We also comment that the recently discovered Bern-Carrasco-Johansson identities of YM helicity amplitudes can be verified by using (repeatedly) the Schouten identity. We also point out additional quadratic identities that can be written down from the KLT relations.
We present an improved calculation on the pionic twist-3 distribution amplitudes $phi^{pi}_{p}$ and $phi^{pi}_{sigma}$, which are studied within the QCD sum rules. By adding all the uncertainties in quadrature, it is found that $<xi^2_p>=0.248^{+0.076}_{-0.052}$, $<xi^4_p>=0.262^{+0.080}_{-0.055}$, $<xi^2_sigma>=0.102^{+0.035}_{-0.025}$ and $<xi^4_sigma>=0.094^{+0.028}_{-0.020}$. Furthermore, with the help of these moments, we construct a model for the twist-3 wave functions $psi^{pi}_{p,sigma}(x,mathbf{k}_bot)$, which have better end-point behavior and are helpful for perturbative QCD approach. The obtained twist-3 distribution amplitudes are adopted to calculate the $Btopi$ transition form factor $f^+_{Bpi}$ within the QCD light-cone sum rules up to next-to-leading order. By suitable choice of the parameters, we obtain a consistent $f^+_{Bpi}$ with those obtained in the literature.
Subspace codes, especially cyclic constant subspace codes, are of great use in random network coding. Subspace codes can be constructed by subspaces and subspace polynomials. In particular, many researchers are keen to find special subspaces and subspace polynomials to construct subspace codes with the size and the minimum distance as large as possible. In [14], Roth, Raviv and Tamo constructed several subspace codes using Sidon spaces, and it is proved that subspace codes constructed by Sidon spaces has the largest size and minimum distance. In [12], Niu, Yue and Wu extended some results of [14] and obtained several new subspace codes. In this paper, we first provide a sufficient condition for the sum of Sidon spaces is again a Sidon space. Based on this result, we obtain new cyclic constant subspace codes through the sum of two and three Sidon spaces. Our results generalize the results in [14] and [12].
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