晶格QCD中矢量电流的动量减法归一化研究

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"在本文中，作者探讨了在晶格量子色动力学（Lattice QCD）中如何使用动量减法（momentum-subtraction schemes）对矢量电流进行归一化，以理解和量化中间动量减法带来的非扰动伪影的系统误差。他们使用了包含nf = 2 + 1 + 1种海夸克的高度改进的交错夸克行为（highly improved staggered quark action），并强调其结果可适用于其他类型的夸克行为。研究关注的焦点在于，通过精确的晶格矢量Ward-Takahashi同一体（lattice vector Ward-Takahashi identity）对保留电流的重归一化方案，以及如何处理不受保守扰动影响的非保守矢量电流的ZV（归一化因子）。作者对比了两种不同的归一化方案：零动量传递时的矢量形状因子方法和RI-SMOM（Regularization Independent - Subtraction at Momentum Operating Momentum）动量减法。他们发现，ZV的确定在两种方法下仅因离散效应（discretization effects）而略有差异，特别是对于RI-SMOM中的任何动量传递值。RI'-MOM方案虽然常见，但可能存在O（1%）的非扰动污染，这限制了其精度。为了解决这个问题，他们提出了一个新的方法，即从局部矢量电流守恒率与保守矢量电流顶点函数的比例中定义RI'-MOM的ZV，证明这种方法是晶格QCD计算中安全有效的。此外，他们还首次使用RI-SMOM方案研究了包括淬灭量子电动力学（QED）影响在内的矢量电流重归一化。这一研究为理解和改进Lattice QCD计算中的归一化问题提供了重要见解，有助于提高未来计算的准确性和可靠性。" 在Lattice QCD的计算中，归一化是一个关键步骤，因为非扰动效应可能会影响计算结果的精确度。文章指出，使用动量减法时，非扰动伪影可能导致系统误差。通过对比不同归一化策略，作者揭示了RI-SMOM方案的局限性，并提出了一种改进的方法，确保ZV的计算更加准确，减少了非扰动污染。同时，考虑QED影响的研究进一步扩展了对晶格QCD理解的深度，使得计算能够更全面地反映现实世界的物理现象。

We now discuss momentum-subtraction renormalization

schemes on the lattice and the extent to which they make

use of Eq. (8).

III. MOMENTUM-SUBTRACTION SCHEMES

USED ON THE LATTICE

Momentum-subtraction schemes are useful interme-

diate schemes between the lattice regularization and the

continuum

MS scheme in which it is now standard to

quote results for scheme-dependent quantities. If the same

momentum-subtraction scheme is implemented both in

lattice QCD and in continuum QCD then the continuum limit

of the lattice results will be in the continuum momentum-

subtraction scheme (and should be independent of lattice

action at that point). They can then be converted to the

scheme using continuum QCD perturbation theory.

A momentum-subtraction scheme imposes renormaliza-

tion conditions on matrix elements between (in the cases

we consider) external quark states so that the tree-level

result is obtained, i.e., Z

is defined by

i¼hp

tree

ð10Þ

for some operator O

ψΓψ, and hp

j and jp

i external

quark states with momenta p

and p

, typically taken to

have large magnitude. This calculation must of course be

done in a fixed gauge, and this is usually taken to be

Landau gauge, which can be straightforwardly imple-

mented in lattice QCD. Effects from the existence of

Gribov copies under the gauge fixing could arise in general;

here we show that there are no such effects for Z

determined using the Ward-Takahashi identity.

Here we will concentrate on the RI-SMOM scheme

[14,15]. This scheme uses a symmetric kinematic configu-

ration with only one scale so that p

¼ p

¼ q

¼ μ

(where q ¼ p

− p

). The wave-function renormalization

is defined (using continuum notation) by

¼ −

12p

Trð

−1

ðpÞÞ: ð11Þ

The vector current renormalization follows from requiring

12q

Trðq

ðp

qÞ¼1: ð12Þ

The traces here are over spin and color and normalizations

arechosensothatZ

¼ Z

¼ 1 at tree lev el. The equations

above are giv en for the continuum SMOM scheme. On the

lattice we must take care to define the appropriate discre-

tization for q

and q

in the various places that they appear.

Below we will see what form q

must take in combination

with the vertex function for the conserved current.

The RI-SMOM scheme was defined with the vector

Ward-Takahashi identity in mind [15]. This reference

shows how the identity defines the projectors needed for

the vector vertex function in the continuum [given in

Eq. (12)] so that Z

¼ 1 for the conserved current. Here

we repeat this exercise, but now on the lattice. Returning to

the Ward-Takahashi identity in Eq. (8) we can multiply

both sides by

q and take the trace while dividing by

(where

q is a discretization of q to be defined later). This

gives

−2i

sinðaq

=2ÞTrðΛ

μ;þ

qÞ

½−TrðS

−1

ðp

qÞþTrðS

−1

ðp

qÞ: ð13Þ

We can simplify the right-hand side assuming that the

inverse propagator takes the general form S

−1

ðpÞ¼

pΣ

ðp

ÞþΣ

ðp

Þ in the continuum (from relativistic

invariance). Then, for the SMOM kinematics,

Tr ðS

−1

ðp

qÞ − TrðS

−1

ðp

qÞ¼TrðS

−1

ðqÞ

qÞ: ð14Þ

On the lattice this formula could be broken by discretiza-

tion effects. We do not see noticeable effects of this kind

with the HISQ action, however, as we will discuss in

Sec. IV B.

Using Eq. (14) in Eq. (13) and multiplying by i then

gives, from the Ward-Takahashi identity

sinðaq

=2ÞTrðΛ

μ;þ

qÞ¼−

TrðS

−1

ðqÞ

qÞ: ð15Þ

From Eq. (11) we see that the right-hand side of this

expression is Z

in the RI-SMOM scheme. Comparing the

left-hand side to Eq. (12) we see that this is Z

cons

the RI-SMOM scheme where Z

cons

is the Z

factor for

the conserved current and the Ward-Takahashi identity

requires us to discretize q

as 2 sinðaq

=2Þ=a [

q is defined

in Eq. (21)]. Then, from Eq. (15), we expect that

cons

ðSMOMÞ¼1 on the lattice and no further renormal-

ization is needed to match to

MS. Notice that this works for

any value of q.

We will show by explicit calculation that

cons

ðSMOMÞ¼1 for the HISQ action in Sec. IV B.

This is not true configuration by configuration, however.

It does require an average over gluon fields.

Another popular momentum-subtraction scheme is

-MOM [12,13], abbreviated here to MOM. In this

scheme Z

is defined in the same way, by Eq. (11),but

is defined by a different projector for the vector vertex

function and the kinematic configuration for the MOM case

is p

¼ p

¼ p so that q ¼ 0. Instead of Eq. (12) we have,

in the MOM scheme,

Trðγ

ðpÞÞ ¼ 1: ð16Þ

RENORMALIZING VECTOR CURRENTS IN LATTICE QCD … PHYS. REV. D 100, 114513 (2019)

114513-5

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