多重手征对称性的Gell-Mann-Oakes-Renner关系
Gell-Mann-Oakes-Renner relation for multiple chiral symmetries
摘要：
作为考虑重叠费米子的手征微扰理论的第一步，我们调查是否有任何歧义的π介子质量的表达式中产生的多重手征对称性。问题是，在计算Ginsparg Wilson手征对称性的守恒流时，不同的手征对称性表达式会导致不同的守恒流。这意味着所有狄拉克算符的金斯帕-威尔逊表达式（包括重叠算符）的π介子和π介子衰变常数的定义是模糊的。本文利用重整化群映射方法研究了连续Ginsparg-Wilson“Dirac-operator的局部手征对称性变换。“我们发现，这自然会导致守恒电流的表达式不同于标准表达式，但与选择哪种Ginsparg-Wilson对称性无关。我们恢复了大量狄拉克算子，传播子和手征凝聚的标准表达式。在此基础上，我们继续计算映射理论中的π介子质量作为夸克质量的函数，并发现了$F_π$和$m_π$的唯一表达式，恢复了通常的Gell-Mann-Oakes-Renner关系，暴露了用其修正值代替手征凝聚。我们假设这个论证可以直接应用到格理论。
英文摘要：
As a first step towards considering a chiral perturbation theory for overlap fermions, we investigate whether there are any ambiguities in the expression for the pion mass resulting from multiple chiral symmetries. The concern is that, calculating the conserved current for Ginsparg Wilson chiral symmetries in the usual way, different expressions of the chiral symmetries lead to different currents. This implies an ambiguity in the definition of the pion and pion decay constant for all Ginsparg-Wilson expressions of the Dirac operator, including the overlap operator. We use a renormalisation group mapping procedure to consider local chiral symmetry transformations for a continuum Ginsparg-Wilson "Dirac-operator." We find that this naturally leads to an expression for the conserved current that differs from the standard expression by cut-off artefacts, but is independent of which of the Ginsparg-Wilson symmetries is chosen. We recover the standard expressions for the massive Dirac operator, propagator, and chiral condensate. With this in place, we proceed to calculate the pion mass in the mapped theory as a function of the quark mass, and discover a unique expression for $F_π$ and $m_π$, recovering the usual Gell-Mann-Oakes-Renner relation, baring the substitution of the chiral condensate with its modified value. We hypothesise that the argument can be carried directly over to the lattice theory.
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