Two Remarks Concerning the Signs of the Light Quark Masses

Abstract

Partial conservation of the nine axial currents $A^β_μ$ for $β=0,1,...,8$ applied to the vacuum expectation values of the equal-time commutators of the nine axial charges $Q_A^α(x_0)$ for $α=0,1,...,8$ with the corresponding axial current divergences $\partial^μA_μ^β$ implies that the non-vanishing current quark masses $m_u, m_d, m_s$ of the light quarks $u, d, s$ have the same sign. Under the more realistic assumption of partial conservation of only the eight axial currents of $SU(3)\otimes SU(3)$, i.e. $α=1,...,8$ and $β=1,...,8$ in the above, inequalities for the quark masses follow. They are trivially fulfilled if the three light quark masses have the same sign and imply, for non-vanishing quark masses $m_u, m_d, m_s$, that at least two of these masses have the same sign as their sum $m_u + m_d + m_s$. If one of the three quark masses vanishes, one of the other two might also. If both do not, they have the same sign (the same, of course, as their sum). Our assumptions include the standard vector SU(3) symmetry of the vacuum.