A strategy for implementing non-perturbative renormalisation of heavy-light four-quark operators in the static approximation

We discuss the renormalisation properties of the complete set of $\Delta B = 2$ four-quark operators with the heavy quark treated in the static approximation. We elucidate the role of heavy quark symmetry and other symmetry transformations in constraining their mixing under renormalisation. By employing the Schroedinger functional, a set of non-perturbative renormalisation conditions can be defined in terms of suitable correlation functions. As a first step in a fully non-perturbative determination of the scale-dependent renormalisation factors, we evaluate these conditions in lattice perturbation theory at one loop. Thereby we verify the expected mixing patterns and determine the anomalous dimensions of the operators at NLO in the Schroedinger functional scheme. Finally, by employing twisted-mass QCD it is shown how finite subtractions arising from explicit chiral symmetry breaking can be avoided completely.Comment: 41 pages, 6 figure

The chromomagnetic operator on the lattice

We study matrix elements of the "chromomagnetic" operator on the lattice. This operator is contained in the strangeness-changing effective Hamiltonian which describes electroweak effects in the Standard Model and beyond. Having dimension 5, the chromomagnetic operator is characterized by a rich pattern of mixing with other operators of equal and lower dimensionality, including also non gauge invariant quantities; it is thus quite a challenge to extract from lattice simulations a clear signal for the hadronic matrix elements of this operator. We compute all relevant mixing coefficients to one loop in lattice perturbation theory; this necessitates calculating both 2-point (quark-antiquark) and 3-point (gluon-quark-antiquark) Green's functions at nonzero quark masses. We use the twisted mass lattice formulation, with Symanzik improved gluon action. For a comprehensive presentation of our results, along with detailed explanations and a more complete list of references, we refer to our forthcoming publication [1].Comment: 7 pages, 1 figure. Talk presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, German

Perturbative and non-perturbative renormalization results of the Chromomagnetic Operator on the Lattice

The Chromomagnetic operator (CMO) mixes with a large number of operators under renormalization. We identify which operators can mix with the CMO, at the quantum level. Even in dimensional regularization (DR), which has the simplest mixing pattern, the CMO mixes with a total of 9 other operators, forming a basis of dimension-five, Lorentz scalar operators with the same flavor content as the CMO. Among them, there are also gauge noninvariant operators; these are BRST invariant and vanish by the equations of motion, as required by renormalization theory. On the other hand using a lattice regularization further operators with $d \leq 5$ will mix; choosing the lattice action in a manner as to preserve certain discrete symmetries, a minimul set of 3 additional operators (all with $d<5$) will appear. In order to compute all relevant mixing coefficients, we calculate the quark-antiquark (2-pt) and the quark-antiquark-gluon (3-pt) Green's functions of the CMO at nonzero quark masses. These calculations were performed in the continuum (dimensional regularization) and on the lattice using the maximally twisted mass fermion action and the Symanzik improved gluon action. In parallel, non-perturbative measurements of the $K-\pi$ matrix element are being performed in simulations with 4 dynamical ($N_f = 2+1+1$) twisted mass fermions and the Iwasaki improved gluon action.Comment: 7 pages, 1 figure, 3 tables, LATTICE2014 proceeding

Kaon oscillations in the Standard Model and Beyond using Nf=2 dynamical quarks

We compute non-perturbatively the B-parameters of the complete basis of four-fermion operators needed to study the Kaon oscillations in the SM and in its supersymmetric extension. We perform numerical simulations with two dynamical maximally twisted sea quarks at three values of the lattice spacing on configurations generated by the ETMC. Unwanted operator mixings and O(a) discretization effects are removed by discretizing the valence quarks with a suitable Osterwalder-Seiler variant of the Twisted Mass action. Operators are renormalized non-perturbatively in the RI/MOM scheme. Our preliminary result for BK(RGI) is 0.73(3)(3).Comment: 7 pages, 3 figures, 1 table, proceedings of the XXVII Int'l Symposyum on Lattice Field Theory (LAT2009), July 26-31 2009, Peking University, Beijing (China

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

A determination of the average up-down, strange and charm quark masses from Nf=2+1+1N_f=2+1+1

We present a lattice QCD determination of the average up-down, strange and charm quark masses based on simulations performed by the European Twisted Mass Collaboration with $N_f = 2 + 1 + 1$ dynamical fermions. We simulated at three different values of the lattice spacing, the smallest being approximately $0.06fm$, and with pion masses as small as $210 \text{MeV}$. Our results are: $m_{ud}(2\text{GeV})=3.70(17)\text{MeV}$, $m_s(2\text{GeV})=99.2(3.9)\text{MeV}$, $m_c(m_c)=1.350(49)\text{GeV}$, $m_s/m_{ud}=26.64(30)$ and $m_c/m_s=11.65(12)$

Light quark masses and pseudoscalar decay constants from Nf=2 Lattice QCD with twisted mass fermions

We present the results of a lattice QCD calculation of the average up-down and strange quark masses and of the light meson pseudoscalar decay constants with Nf=2 dynamical fermions. The simulation is carried out at a single value of the lattice spacing with the twisted mass fermionic action at maximal twist, which guarantees automatic O(a)-improvement of the physical quantities. Quark masses are renormalized by implementing the non-perturbative RI-MOM renormalization procedure. Our results for the light quark masses are m_ud^{msbar}(2 GeV)= 3.85 +- 0.12 +- 0.40 MeV, m_s^{msbar}(2 GeV) = 105 +- 3 +- 9 MeV and m_s/m_ud = 27.3 +- 0.3 +- 1.2. We also obtain fK = 161.7 +- 1.2 +- 3.1 MeV and the ratio fK/fpi=1.227 +- 0.009 +- 0.024. From this ratio, by using the experimental determination of Gamma(K-> mu nu (gamma))/Gamma(pi -> mu nu (gamma)) and the average value of |Vud| from nuclear beta decays, we obtain |Vus|=0.2192(5)(45), in agreement with the determination from Kl3 decays and the unitarity constraint.Comment: 20 pages, 5 figure

K→πK \to \pi matrix elements of the chromomagnetic operator on the lattice

We present the results of the first lattice QCD calculation of the $K \to \pi$ matrix elements of the chromomagnetic operator $O_{CM} = g\, \bar s\, \sigma_{\mu\nu} G_{\mu\nu} d$, which appears in the effective Hamiltonian describing $\Delta S = 1$ transitions in and beyond the Standard Model. Having dimension 5, the chromomagnetic operator is characterized by a rich pattern of mixing with operators of equal and lower dimensionality. The multiplicative renormalization factor as well as the mixing coefficients with the operators of equal dimension have been computed at one loop in perturbation theory. The power divergent coefficients controlling the mixing with operators of lower dimension have been determined non-perturbatively, by imposing suitable subtraction conditions. The numerical simulations have been carried out using the gauge field configurations produced by the European Twisted Mass Collaboration with $N_f = 2+1+1$ dynamical quarks at three values of the lattice spacing. Our result for the B-parameter of the chromomagnetic operator at the physical pion and kaon point is $B_{CMO}^{K \pi} = 0.273 ~ (70)$, while in the SU(3) chiral limit we obtain $B_{CMO} = 0.072 ~ (22)$. Our findings are significantly smaller than the model-dependent estimate $B_{CMO} \sim 1 - 4$, currently used in phenomenological analyses, and improve the uncertainty on this important phenomenological quantity.Comment: 20 pages, 4 figures, 2 table. Refined SU(3) ChPT analysis with no changes in the final result. Version to appear in PR

K^0-\bar{K}^0 mixing in the Standard Model from Nf=2+1+1 Twisted Mass Lattice QCD

We present preliminary results at {\beta} = 1.95 (a = 0.077 fm) on the first unquenched N_f=2+1+1 lattice computation of the B_K parameter which controls the neutral kaon oscillations in the Standard Model. Using N_f=2+1+1 maximally twisted sea quarks and Osterwalder-Seiler valence quarks we achieve O(a) improvement and a continuum-like renormalization pattern for the four-fermion operator. Our results are extrapolated/interpolated to the physical light/strange quark mass but not yet to the continuum limit. The computation of the relevant renormalization constants is performed non perturbatively in the RI'-MOM scheme using dedicated simulations with N_f=4 degenerate sea quark flavours produced by the ETM collaboration. We get B_K^{RGI} (a = 0.077) = 0.747(18), which when compared to our previous unquenched N_f=2 determination and most of the existing results, suggests a rather weak B_K^{RGI} dependence on the number of dynamical flavours. We are at the moment analysing lattice data at two additional {\beta} values which will allow us to perform an extrapolation to the continuum limit.Comment: 7 pages, 8 figures, Proceedings of Lattice 2011, XXIX International Symposium on Lattice Field Theory, Squaw Valley, Lake Tahoe, Californi

Going chiral: overlap versus twisted mass fermions

We compare the behavior of overlap fermions, which are chirally invariant, and of Wilson twisted mass fermions at full twist in the approach to the chiral limit. Our quenched simulations reveal that with both formulations of lattice fermions pion masses of O(250 MeV) can be reached in practical applications. Our comparison is done at a fixed value of the lattice spacing a=0.123 fm. A number of quantities are measured such as hadron masses, pseudoscalar decay constants and quark masses obtained from Ward identities. We also determine the axial vector renormalization constants in the case of overlap fermions.Comment: 22 pages, 10 figure