大动量有效理论与短程算符展开：对比与互补
Large-Momentum Effective Theory vs. Short-Distance Operator Expansion: Contrast and Complementarity
摘要：
大动量有效理论（LaMET）和短程算符乘积展开（SD-OPE）是从大动量强子的坐标空间关联函数中提取部分子分布函数的两种不同方法，但它们在无限动量极限下是等价的。LaMET在$Λ_{\rm QCD}/[x(1-x)P^z]$中实现动量空间扩展，以直接计算Bjorken $x\in [x_{\rm min}\sim Λ_{\rm QCD}/P^z, x_{\rm max}\sim 1-x_{\min}]$中间区域中的PDF $f(x)$。SD-OPE在小欧几里德距离$z$处应用微扰QCD，以提取与PDF的傅里叶变换相对应的前导-扭转相关的范围$[0,λ_{\rm max}]$，$h(λ=zP^z)$。类似于量子力学的不确定性原理，不完全的领先扭转相关不能轻易地转换为动量空间的局部分布，解决"逆问题“的方法基本上涉及对$λ_{\rm max}$之外的缺失信息进行建模。另一方面，短距离相关性，以及预期的终点渐近性，可用于在现象学上拟合LaMET互补区域中的PDF：$x\in [0,x_{\rm min}]$和$[x_{\rm max}, 1]$。我们使用最近的结果从ANL/BNL合作的π价夸克分布来证明这一点。
英文摘要：
Although equivalent in the infinite-momentum limit, large-momentum effective theory (LaMET) and short-distance operator product expansion (SD-OPE) are two different approaches to extract parton distribution functions (PDFs) from coordinate-space correlation functions in large-momentum hadrons. LaMET implements a momentum-space expansion in $Λ_{\rm QCD}/[x(1-x)P^z]$ to directly calculate PDFs $f(x)$ in a middle region of Bjorken $x\in [x_{\rm min}\sim Λ_{\rm QCD}/P^z, x_{\rm max}\sim 1-x_{\min}]$. SD-OPE applies perturbative QCD at small Euclidean distances $z$ to extract a range $[0,λ_{\rm max}]$ of leading-twist correlations, $h(λ=zP^z)$, corresponding to the Fourier transformation of PDFs. Similar to the quantum mechanical uncertainty principle, an incomplete leading-twist correlation cannot be readily converted to a momentum-space local distribution, and the methods to solve the ``inverse problem'' involve essentially modelling of the missing information beyond $λ_{\rm max}$. On the other hand, short-distance correlations, along with the expected end-point asymptotics, can be used to phenomenologically fit the PDFs in the LaMET-complementary regions: $x\in [0,x_{\rm min}]$ and $[x_{\rm max}, 1]$. We use the recent results of the pion valence quark distribution from the ANL/BNL collaboration to demonstrate this point.
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