Light meson decay constants beyond the quenched approximatio

We calculate the effects of including dynamical fermion loops in the lattice QCD estimates of meson decay constants, by extrapolating the results from negative flavour numbers after a suitable matching of the pion and rho mass. For moderately light quarks,

hep-lat/9510048 26 Oct 1995

We calculate the effects of including dynamical fermion loops in the lattice QCD estimates of meson decay constants, by extrapolating the results from negative flavour numbers after a suitable matching of the pion and rho mass. For moderately light quarks,

The computation of meson decay constants with lattice QCD simulations provides for light mesons a crucial test of the lattice approach and for heavy mesons the essential information which allows to extract the values of the still poorly known weak mixing angles. While the estimates within the quenched approximation where internal fermion loops are neglected have reached a considerable degree of accuracy, those based on full QCD simulations are still a ected by large errors which at present make the evaluation of the impact of the sea quark contribution di cult to extract. In both cases, the extraction of the physical values of the decay constants requires the evaluation of operator renormalization constants which are known in most cases only perturbatively and the extrapolations of the nal result to the continuum and, for light mesons, to the chiral limit. A recent analysis in the quenched approximation suggests 1] that the quenched results do not extrapolate to the correct experimental values, but lie somewhat below them. In this letter we present a study of meson decay constants in the unquenched case with a method which allows to reach more precise estimates and a rst evidence for sea quark e ects. The method, already discussed in refs. 2] and 3], extrapolates the unquenched results from a theory with a negative number of avours to the physical case. The elds appearing in such a theory, called bermions, obey Bose statistics and are governed by the square of the hermitian operator Q, de ned as 5 times the lattice Euclidean Dirac operator: X y S U;]= Sg U]+ (x)Q2 (x) (1) where Sg is the standard Wilson action for the gauge sector and (x) is the bermion eld. Bermion elds have various indices which are omitted to shorten the notation: spin, colour and bermion{ avour. A theory with a positive number nb of bermion avours corresponds to a negative number nf= 2nb of fermion avours. For the lattice Dirac operator we follow the standard Wilson formulation: 3 1 (x) 1 X U (x)(1 Q](x)= 2K 5 ) (x+ ) 2 5=0 3 X (2)+ 1 5 U y(x )(1+ ) (x ) 2=0 where K is the usual Wilson hopping parameter related to the bare mass. Such a theory exists in the Euclidean space only. The integration over bermion elds leads to a negative power of the det

erminant of the Dirac operator and therefore gauge con gurations leading to small eigenvalues are sampled in the Monte 2x

We calculate the effects of including dynamical fermion loops in the lattice QCD estimates of meson decay constants, by extrapolating the results from negative flavour numbers after a suitable matching of the pion and rho mass. For moderately light quarks,

Carlo more frequently than in the fermion case where they are suppressed. In the limit of very light quark masses one can expect di culties1 . An example is given by the mass mS of the avour singlet pseudoscalar. According to the Witten{Veneziano expression 4] m2= m2+(nf=Nc ) 2, its mass for negative avour numbers is lighter S NS than the pion mass mNS and may lead to severe problems when approaching the chiral limit. We have monitored such a mass, indeed smaller than the pion mass, and in the results that we discuss we have made a safe choice of bare parameters. Detailed results on the pseudoscalar singlet will be presented elsewhere 5]. Changing the avour content of the theory does change its lattice cuto and, at xed value of the Wilson hopping parameter K, the correction to the quark mass. The extrapolation from negative avour numbers may be complicated by these variations. In order to minimize such an e ect we have extrapolated theories after matching the values of two independent physical quantities, chosen to be the pion and the rho mass. The results that we present are obtained at two di erent values of the ratio R2 of the pion over rho mass squared, 0.7 and 0.5, and of the rho mass, 0.71 and 0.66, respectively. The simulations were performed on a 25 Giga op machine of the APE series. The update procedure was for the gauge sector a Cabibbo{Marinari pseudo{heat 6] bath followed by three overrelaxation sweeps and for the bermion sector a heat bath followed by a number from three to seven overrelaxation sweeps 7]. The pseudoscalar and vector decay constants are de ned in the continuum by

h0j( 1 h0j( 1

5 2

)contj (p)i= f p; m2 2)cont j (p; )i= f

(3)

where 1 and 2 are avour indices and is the polarization vector. The operators chosen to extract the rho and pion decay constants are the standard lattice local operators:

P5 (~; t)= i 1(~; t) 5 2(~; t); x x x Vk (~; t)= 1(~; t) k 2(~; t) x x x (k= 1; 2; 3); (4) A0(~; t)= 1(~; t) 0 5 2(~; t); x x x and the the decay constants were extracted from the behaviour at large times of the space integrated correlations given byThis problem is not present if the bermions are subject to an external background gauge eld like in the case of the Schrodinger functional. We thank M. Luscher for this remark.1

We calculate the effects of including dynamical fermion loops in the lattice QCD estimates of meson decay constants, by extrapolating the results from negative flavour numbers after a suitable matching of the pion and rho mass. For moderately light quarks,

G55(t)= G05(t)= G (t)=using the following formulae:q q

X

X

x~

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