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The chirally rotated Schrodinger functional: theoretical expectations and perturbative tests

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 Added by Mattia Dalla Brida
 Publication date 2016
  fields
and research's language is English




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The chirally rotated Schrodinger functional ($chi$SF) with massless Wilson-type fermions provides an alternative lattice regularization of the Schrodinger functional (SF), with different lattice symmetries and a common continuum limit expected from universality. The explicit breaking of flavour and parity symmetries needs to be repaired by tuning the bare fermion mass and the coefficient of a dimension 3 boundary counterterm. Once this is achieved one expects the mechanism of automatic O($a$) improvement to be operational in the $chi$SF, in contrast to the standard formulation of the SF. This is expected to significantly improve the attainable precision for step-scaling functions of some composite operators. Furthermore, the $chi$SF offers new strategies to determine finite renormalization constants which are traditionally obtained from chiral Ward identities. In this paper we consider a complete set of fermion bilinear operators, define corresponding correlation functions and explain the relation to their standard SF counterparts. We discuss renormalization and O($a$) improvement and then use this set-up to formulate the theoretical expectations which follow from universality. Expanding the correlation functions to one-loop order of perturbation theory we then perform a number of non-trivial checks. In the process we obtain the action counterterm coefficients to one-loop order and reproduce some known perturbative results for renormalization constants of fermion bilinears. By confirming the theoretical expectations, this perturbative study lends further support to the soundness of the $chi$SF framework and prepares the ground for non-perturbative applications.



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The chirally rotated Schrodinger functional ($chi$SF) renders the mechanism of automatic $O(a)$ improvement compatible with Schrodinger functional (SF) renormalization schemes. Here we define a family of renormalization schemes based on the $chi$SF for a complete basis of $Delta F = 2$ parity-odd four-fermion operators. We compute the corresponding scale-dependent renormalization constants to one-loop order in perturbation theory and obtain their NLO anomalous dimensions by matching to the $overline{textrm{MS}}$ scheme. Due to automatic $O(a)$ improvement, once the $chi$SF is renormalized and improved at the boundaries, the step scaling functions (SSF) of these operators approach their continuum limit with $O(a^{2})$ corrections without the need of operator improvement.
The use of chirally rotated boundary conditions provides a formulation of the Schroedinger functional that is compatible with automatic O(a) improvement of Wilson fermions up to O(a) boundary contributions. The elimination of bulk O(a) effects requires the non-perturbative tuning of the critical mass and one additional boundary counterterm. We present the results of such a tuning in a quenched setup for several values of the renormalized gauge coupling, from perturbative to non-perturbative regimes, and for a range of lattice spacings. We also check that the correct boundary conditions and symmetries are restored in the continuum limit.
In a previous paper [1], we have discussed the non-perturbative tuning of the chirally rotated Schroedinger functional (XSF). This tuning is required to eliminate bulk O(a) cutoff effects in physical correlation functions. Using our tuning results obtained in [1] we perform scaling and universality tests analyzing the residual O(a) cutoff effects of several step-scaling functions and we compute renormalization factors at the matching scale. As an example of possible application of the XSF we compute the renormalized strange quark mass using large volume data obtained from Wilson twisted mass fermions at maximal twist.
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Symanzik showed that quantum field theory can be formulated on a space with boundaries by including suitable surface interactions in the action to implement boundary conditions. We show that to all orders in perturbation theory all the divergences induced by these surface interactions can be absorbed by a renormalization of their coefficients.
133 - P. Perez-Rubio , S. Sint 2008
We consider the Schrodinger functional with staggered one-component fermions on a fine lattice of size $(L/a)^3 times (T/a)$ where $T/a$ must be an odd number. In order to reconstruct the four-component spinors, two different set-ups are proposed, corresponding to the coarse lattice having size $(L/2a)^3 times (T/2a)$, with $T = T pm a$. The continuum limit is then defined at fixed $T/L$. Both cases have previously been investigated in the pure gauge theory. Here we define fermionic correlation functions and study their approach to the continuum limit at tree-level of perturbation theory.
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