Kaon semileptonic vector form factor with Twisted Mass fermions
We investigate the vector form factor relevant for the semileptonic decay using maximally twisted-mass fermions with 4 dynamical flavours (). Our simulations feature pion masses ranging from MeV to approximately MeV and lattice spacing values as small as fm. Our main result for the vector form factor at zero 4-momentum transfer is where the uncertainty is both statistical and systematic. By combining our result with the experimental value of we obtain , which satisfies the unitarity constraint of the Standard Model at the permille level.
00 \journalnameNuclear Physics B Proceedings Supplement \runauth \jidnuphbp \jnltitlelogoNuclear Physics B Proceedings Supplement
1 Introduction and simulation details
Meson semileptonic decays are very interesting phenomena because the measure of their rates combined with lattice QCD calculations allows us to extract the elements of the CKM matrix. Specifically, in this contribution we present our calculation of the vector form factor of the kaon semileptonic decay at zero 4-momentum transfer, which allows us to extract the value of .
This is possible thanks to the relation between the vector current responsible for the decay and two form factors:
where . The definition of the scalar form factor is
which implies the relation . By calculating on the lattice and using the experimental result of we can then extract the value of .
We used the ensembles produced by the ETM Collaboration with using the Twisted Mass action Frezzotti:2003zc (); Frezzotti:2003ni (), which include in the sea, beside the contribution of two degenerate light quarks, the strange and charm quarks. Our simulation features pion masses ranging from MeV to approximately MeV and three values of the lattice spacing, the smallest being approximately fm. Valence quarks were simulated using the Osterwalder-Seiler action Osterwalder:1977pc (), while the gauge fields were implemented using the Iwasaki action Iwasaki:1985we (). For each lattice spacing we used three values of the bare strange quark mass to allow for a smooth interpolation of our data to the physical value , which we determined in our paper Carrasco:2014cwa (). Different values of the spatial momenta were simulated using Twisted Boundary conditions Bedaque:2004kc (); deDivitiis:2004kq (), allowing us to cover both the spacelike and timelike region of the 4-momentum transfer. For further details about the simulation the reader should see ref. Carrasco:2014cwa ().
We studied a combination of three-points correlation functions in order to extract the form factors and as functions of the 4-momentum transfer , light quark mass and the lattice spacing . We then performed a chiral and continuum extrapolation in order to obtain the physical value of .
Our result is where the uncertainty is both statistical and systematic.This allows us to extract the value of the CKM matrix element , which is compatible with the unitarity constraint of the Standard Model at the permille level.
2 Extraction of the form factors
Our data consists of three point correlation functions connecting moving pions and kaons through a vector current inserted at a time distance from the source and from the sink. The behaviour of these correlation functions for large allows us to extract the matrix elements of the vector current by studying the following quantity
The matrix elements and can be extracted from the plateaux as
Thus, we obtain the form factors through the relations
and subsequentely calculate from eq.(2).
An example of the extraction of the matrix elements can be seen in fig.(1). Meson masses were calculated by isolating the ground state of two points correlation functions of pseudoscalar mesons at rest.
3 Analysis of the form factors
The first step in our analysis was to study the dependence of and on the 4-momentum transfer in order to interpolate our data to . This was done using the expansion Hill:2006bq () (up to ) and the condition is imposed as a constraint. An example can be seen in fig.(2). We also tried to fit the dependence using other fit ansatz (e.g. polynomial expression in ), obtaining nearly identical results.
Thus, after interpolating our data to the physical value of the strange quark mass using a quadratic spline procedure, we performed the chiral and continuum extrapolation using the following SU(2) ChPT prediction at NLO Flynn:2008tg ():
where , is the lattice spacing and the parameters , and are determined by our fit.
In order to estimate the systematic uncertainty induced by the chiral extrapolation we also fitted our data using the SU(3) ChPT ansatz beyond the NLO:
where , and are determined in our fit. The full expression for can be found in Gasser:1984gg (); Gasser:1984ux (). In eq.(7) the Ademollo Gatto theorem Ademollo:1964sr () is satisfied in the continuum limit, i.e. in the SU(3) limit and the deviations from this value are quadratic in . In fig.(3) we show the chiral and continuum extrapolation of our data, using eqs. (6) and (7). It can be seen that the results at the physical point are compatible within the uncertainties.
Thus we combined the two results obtaining
where indicates the statistical uncertainty which includes the one induced by the fitting procedure and the error induced by the numerical inputs needed for the analysis, namely the values of the light quark mass , the lattice spacing and the SU(2) ChPT low energy constants and , which were determined in Carrasco:2014cwa (). The part of the uncertainty is the one induced by the difference in the results corresponding to the two chiral extrapolations we performed. It should be noticed that there are two lattice points calculated at the same lattice spacing and light quark mass but different volumes. They turn out to be well compatible within the uncertainties, allowing us to state that finite size effects can be safely neglected in our analysis.
4 An outlook on a possible extension
As a possible extension of our analysis we performed a multi-combined fit of the , and dependencies of the form factors in order to predict them not only at , but on the entire region accessible to experiments, i.e from to
We opted for the same strategy used in Lubicz:2010bv (), i.e. we performed a global fit using functional forms of the form factors derived by expanding in powers of the NLO SU(3) ChPT predictions for the form factors Gasser:1984ux (); Gasser:1984gg (). As a constraint we included in the analysis the Callan-Treiman theorem Callan:1966hu (), which relates in the SU(2) chiral limit the scalar form factor calculated at the unphysical to the ratio of the decay constants . A preliminary result for the form factors is presented in fig.(4).
The authors would like to thank Francesco Sanfilippo for the useful discussions about the work here presented.
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