重访低能定理与无质量QCD中的OPE
Low-energy theorem revisited and OPE in massless QCD
AI总结：
本文利用维度正规化重访无质量QCD中的NSVZ型低能定理，揭示其要求积分三点关联函数存在无限加法重整化，并将该抵消项与OPE发散接触项相联系，证明其非微扰有限性。
摘要：
我们通过维度正规化实现文献[1]中导出的SU($N$) QCD（含$N_f$个无质量夸克）的NSVZ型低能定理（LET），并对其重新考察。该LET将左侧的$n$点关联函数与右侧在零动量处额外插入Tr$F^2$的$n+1$点关联函数联系起来。首先，我们证明，对于左侧某算符$O$的$2$点关联函数，该LET意味着右侧的积分$3$点关联函数在微扰论中除乘法重整化外，一般还需要一个无限的加法重整化。其次，我们将上述完全由LET确定的抵消项与Tr$F^2$和$O$的OPE在动量表示中某一系数里相应的发散接触项联系起来，从而借助LET将文献[2]中最初针对$O$=Tr$F^2$出现的独立论证推广到任意算符$O$。第三，我们通过直接计算验证，文献[3]中首先计算到微扰论$g^4$阶、并在文献[2]中计算到所有阶的后一发散接触项，确实与LET所蕴含的结果一致。第四，我们在坐标表示和动量表示中分别计算上述OPE系数的发散接触项，并讨论它们之间的关系。第五，我们证明，在QCD的渐近自由相中，LET中前述抵消项虽然在微扰论中逐阶发散，但在对所有微扰阶重求和后实际上是非微扰有限的。最后，我们简要回顾该LET在维度正规化的规范不变框架下对大-$N$ QCD中微扰和非微扰重整化的意义。该LET在含$N_f$个无质量夸克的SU($N$) QCD共形窗口内部及上方的意义将在后续论文中给出。
英文摘要：
We revisit a low-energy theorem (LET) of NSVZ type in SU($N$) QCD with $N_f$ massless quarks derived in [1] by implementing it in dimensional regularization. The LET relates $n$-point correlators in the lhs to $n+1$-point correlators with the extra insertion of Tr$F^2$ at zero momentum in the rhs. First, we demonstrate that, for $2$-point correlators of an operator $O$ in the lhs, the LET implies that, in general, the integrated $3$-point correlator in the rhs needs in perturbation theory an infinite additive renormalization in addition to the multiplicative one. Second, we relate the above counterterm -- that is completely fixed by the LET -- to a corresponding divergent contact term in a certain coefficient of the OPE of Tr$F^2$ with $O$ in the momentum representation, thus extending by means of the LET to any operator $O$ an independent argument that first appeared for $O$=Tr$F^2$ in [2]. Third, we verify by direct computation that the latter divergent contact term first computed in [3] to order $g^4$ in perturbation theory and to all orders in [2] actually agrees with the one implied by the LET. Fourth, we evaluate the divergent contact terms for the above OPE coefficient both in the coordinate and momentum representation and discuss their relation. Fifth, we demonstrate that in the asymptotically free phase of QCD the aforementioned counterterm in the LET -- though divergent order by order in perturbation theory -- is actually finite nonperturbatively after resummation to all perturbative orders. Finally, we briefly recall the implications of the LET in the gauge-invariant framework of dimensional regularization for the perturbative and nonperturbative renormalization in large-$N$ QCD. The implications of the LET inside and above the conformal window of SU($N$) QCD with $N_f$ massless quarks will appear in a forthcoming paper.
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