New spectral representation and evaluation of f π and the quark condensate $$\left\langle {\bar qq} \right\rangle $$ in the terms of string tension

Abstract

New spectral representations for f π and chiral condensate are derived in QCD and used for calculations in the large-N c limit. Both quantities are expressed in this limit through string tension σ and gluon correlation length T g without fitting parameters. As a result, one obtains $$\left\langle {\bar qq} \right\rangle = - N_c \sigma ^2 T_g a_1 $$ , $$f_\pi = \sqrt {N_c } \sigma T_g a_2 $$ , with a 1=0.0823, a 2=0.30. Taking σ=0.18 GeV2 and T g=1 GeV−1, as known from analytic and lattice calculations, this yields $$\left\langle {\bar qq} \right\rangle $$ (μ=2 GeV)=−(0.225 GeV)3, f π=0.094 GeV, which is close to the standard values.