Quark disconnected diagrams in chiral perturbation theory

Abstract

Abstract:We show how quark-disconnected and quark-connected contributions to hadronic n-point functions can be written as independent correlators for which one can derive expressions in partially quenched chiral effective theory. As an example we apply the idea to the case of the hadronic vacuum polarisation. In particular, we consider the cases of the N f = 2 theory without and with a partially quenched strange quark and also the N f = 2 + 1 theory. In the latter two cases a parameter-free prediction for the disconnected contribution at NLO in the effective theory is given. Finally we show how twisted boundary conditions can then be used in lattice QCD to improve the q 2 resolution in the connected contributions even when flavour singlet operators are considered.

Isospin breaking and chiral symmetry restoration

“…However, when considering the low-energy representation for the partition function and the susceptibilities in terms of GB fields, it is not so simple to separate the connected and disconnected parts if we take the isospin limit from the very beginning. A possible approach to perform such separation is to work within the partially quenched ChPT framework, as discussed in [27] for the vacuum polarization. We will however work within the isospin-breaking scenario we are considering here, which is very useful for this purpose, as noted first in [17] for the susceptibilities and used also in [28] for the vacuum polarization.…”

“…It is also independent of B 0 , unlike the individual quark condensates, which have only physical meaning when multiplied by the appropriate quark masses, since the m i B 0 products give meson masses. In addition, the dependence with the EM LEC disappears in the difference rðTÞ À rð0Þ, which is the quantity directly related to the susceptibility through (27).…”

“…The result is that rðTÞ À rð0Þ can be expressed as the susceptibility term in (27) plus a linear combination of @ðh" ssi T À h" ssi 0 Þ=@m and @ðh" ssi T À h" ssi 0 Þ=@m s , to this chiral order and neglecting Oðe 4 Þ and Oðm u À m d Þ isospin-breaking corrections in the right-hand side. Since the strange quark condensate has a much weaker dependence on temperature than the light one (or equivalently, we can approximately neglect the thermal functions evaluated on kaon and eta masses) we expect the T behavior of rðTÞ À rð0Þ to be dominated by the light scalar susceptibility also in the SUð3Þ case and therefore the sum rule (27) should hold approximately. In this case we have to one loop rðTÞ SUð3Þ À rð0Þ SUð3Þ…”

“…The sum rule (27) has another interesting consequence, regarding lattice simulations. In the staggered fermion lattice formalism, the need to introduce four different copies (tastes) for every quark flavor leads to the so-called taste violation [8,19,29].…”

“…Obviously, the staggered case includes additional operators and the spectrum of states is more complicated. However, we can use the sum rule (27) to estimate roughly the expected differences between the lattice staggered condensate and the continuum one, considering the lightest states. For fine-enough lattices, one has a ¼ ðM 2 ;a À M 2 Þ=M 2 ' ca 2 for the lightest tastes of squared mass M 2 ;a [8], where from the two smallest lattices in [8] we get c ' 140 fm À2 .…”