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# Non-perturbative renormalisation with interpolating momentum schemes

N. Garron  School of Mathematics, Computer Science and Engineering, Liverpool
Hope University, Hope Park, Liverpool L16 9JD, UK  Theoretical Physics
Division, Department of Mathematical Sciences, University of Liverpool,
Liverpool L69 3BX, UK  C. Cahill  Theoretical Physics Division, Department of
Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK  M.
Gorbahn  Theoretical Physics Division, Department of Mathematical Sciences,
University of Liverpool, Liverpool L69 3BX, UK
J. A. Gracey  Theoretical Physics Division, Department of Mathematical
Sciences, University of Liverpool, Liverpool L69 3BX, UK  P. E. L. Rakow
Theoretical Physics Division, Department of Mathematical Sciences, University
of Liverpool, Liverpool L69 3BX, UK

## Abstract

Hadronic matrix elements evaluated on the lattice can be converted to a
continuum scheme such as $\overline{\text{MS}}$ using intermediate non-
perturbative renormalisation schemes. Discretisation effects on the lattice
and convergence of the continuum perturbation theory are both scheme dependent
and we explore this dependence in the framework of the Rome-Southampton method
for generalised kinematics. In particular, we implement several non-
exceptional interpolating momentum schemes, where the momentum transfer is not
restricted to the symmetric point defined in RI/SMOM. Using flavour non-
singlet quark bilinears, we compute the renormalisation factors of the quark
mass and wave function for $N_{f}=3$ flavours of dynamical quarks. We
investigate the perturbative and non-perturbative scale-dependencies. Our
numerical results are obtained from lattice simulations performed with Domain-
Wall fermions, based on ensembles generated by RBC-UKQCD collaborations; we
use two different lattice spacings $1/a\sim 1.79$ and $2.38$ GeV. We also give
the numerical values for the relevant anomalous dimensions and matching
coefficients at next-to-next-to-leading order.

LTH 1290

##  1 Introduction

One major goal of lattice QCD is to determine the hadronic matrix elements of
hadronic operators, with particular emphasis on flavour non-singlet quark
bilinear operators, such as the local scalar, vector, axial vector or
pseudoscalar operators. To produce physically meaningful results we need
renormalisation factors to relate the bare lattice results to quantities in
standard renormalisation schemes, such as the modified minimal subtraction
scheme, $\overline{\text{MS}}$. This conversion can be done in lattice
perturbation theory but it is well known that it results in large systematic
error because the perturbative series converges poorly. Instead, in the last
couple of decades, more efficient methods have been developed, the most
popular ones being the Schrödinger Functional [1, 2] and Rome-Southampton [3]
frameworks. In this work we follow the latter. In both cases, the strategy
comprises two distinct steps: First the renormalisation factors are obtained
non-perturbatively from lattice simulations in a given intermediate scheme,
hence the name Non-Perturbative Renormalisation (NPR). Then these factors are
converted to $\overline{\text{MS}}$ providing us with a bridge between low and
high energies. This second step is done perturbatively, but in the framework
of the Rome-Southampton method, it can be done in standard continuum
perturbation theory so that we can take advantage of the multi-loop
computation available in the literature. In principle the method is relatively
simple: one makes lattice measurements of Green’s functions with bilinear
operators inserted into quark $2$-point functions (for example), and compares
the lattice results with continuum perturbative results. This method can also
be applied to more complicated Green functions, higher twist bilinears, four-
quark operators, etc. It is worth noting that although traditionally we work
in massless limit, a massive momentum scheme has been developed in [4].

In early lattice studies the operator insertion was at zero momentum transfer,
see for example [3, 5] for the early direction of this approach. This choice
of kinematics corresponds to an exceptional momentum configuration and is
subject to potential infrared issues. The origin of the problem and potential
solutions were already discussed in detail in [6]. These infrared problems can
become even more severe when the chiral limit is taken, and had lead to
significant discrepancies in the case of neutral meson mixing [7, 8]. With the
improvement in lattice analyses as well as the use of dynamical fermions, the
refinement of the non-perturbative structure that is of phenomenological
interest became suspect due to the emergence of the latent infrared issues.

Therefore in [9, 10, 11, 12], an extension to the zero momentum insertion for
this class of Green’s functions was developed. In particular the Green’s
functions were computed with a non-zero momentum flowing through the operator
and external quark legs at what is now termed the fully Symmetric MOMentum
(SMOM) subtraction point***To ease the notation we drop the ‘RI’ in RI/SMOM..
This circumvents any potential ambiguity in assessing infrared effects and in
particular in the chiral limit. In either scenario of zero or non-zero
operator momentum flow lattice measurements were assisted by matching to the
continuum limit. By this we mean that one can compute the same Green’s
function for any of the above operators in the continuum field theory at
several loop orders in the $\overline{\text{MS}}$ scheme. In the zero momentum
case this has been carried out to three loops, but for the non-zero momentum
case this has only been carried out to two loops, [9, 10, 11] through there
has been progress towards the next loop order in [13, 14]. Such continuum
matching allows one to accurately tune to the ultraviolet before extrapolating
to the non-perturbative or low energy region where measurements are made and
errors estimated.

The effect of using a momentum configuration which is infrared secure can be
seen for example in [15, 16, 17, 8, 18, 19] where the renormalisation factors
for the various flavour non-singlet quark bilinear or four-quark operators
were calculated for both momentum configurations. A quantity which can be used
as a benchmark is the ratio of the vector and axial vector operators. On
symmetry grounds this has to be unity for Domain-Wall fermions and thus
provides a stringent test of determining which momentum configuration was more
reliable. It is evident from the analysis of, for example, [15, 18, 19] that
the SMOM ratio is unity over virtually the full momentum range and well into
the deep infrared in marked contrast to the zero momentum case. While this
augurs well for significant improvement in lattice measurements and errors,
one question naturally arises which will form the main focus of our article.

Now that it is established that the SMOM non-exceptional momentum
configuration gives a clear improvement, it is only one of many possible
configurations that can be studied. Indeed, for example in [9, 10], a more
general off-shell case was developed which is termed IMOM to denote the
interpolating momentum subtraction configuration. In the SMOM definition the
momentum squared of the three external momenta were all equal. By contrast in
the IMOM case the magnitude of the operator momentum flow is allowed to vary
and this is governed by an additional kinematic parameter $\omega$. For
example the SMOM case corresponds to $\omega$ $=$ $1$. Varying $\omega$
between $0$ and $4$, which are the respective values giving infrared or
collinear singularities, allows one to search for a value which can improve
the matching to the continuum, the non-perturbative behaviour or maybe even
reduce the lattice artefacts. This therefore leads to our investigation to see
if there is an optimal value for $\omega$, possibly different from one, which
minimises errors and improves benchmarks such as that studied in [15, 18, 19].
This would be the appropriate next stage after the success of SMOM lattice
studies especially in the chiral limit. For instance the ratio of vector and
axial vector renormalisation constants can be examined in the chiral limit to
gauge pion mass effects. While $\omega$ will be our main parameter the overall
mass scale $\mu$ at which measurements are made will also be important.
Therefore to have a thorough investigation we have performed a lattice
computation of various operators on several different lattices for half
integer values of $\omega$ between $0$ and $4$ for various values of $\mu$.
This has allowed us to identify an optimal area of the
$(\omega,\mu^{2})$-plane where there is an effective plateau upon which
measurements are reliably accurate. Moreover the picture appears to be
consistent for the operators we consider, though the actual locations are not
necessarily the same point for each case. It suggests that there should be a
similar qualitative behaviour for operators with covariant derivatives, which
we do not consider here.

Although the article primarily concentrates on the lattice computations and
analysis we were indebted throughout to the various two loop
$\overline{\text{MS}}$ continuum results accumulated over the last decade, [9,
10, 11, 18, 20, 12], to facilitate the continuum matching. Moreover the
measurement of quark bilinear currents in the IMOM configuration will give
crucial direction to $3$-point quark Green’s functions such as those relevant
to kaon mixing for example, [7, 8]. In that scenario the effective $4$-point
functions can be mimicked by products of the Green’s functions we consider
here. It is particularly important in this case, as it has been argued that
the choice of kinematics was responsible for the discrepancy observed in the
literature.

The paper is organised as follows: In Section 2 we outline the kinematics
setup, explaining in detail how our incoming and outgoing momenta are chosen
to lead to a given value of $\omega$ for a given value of $\mu$. In Section 3
we explain how we obtain the $Z$-factors, and discuss our choice of
projectors. In Section 4, we present our results for the $Z-$factors and their
scale-evolutions. We conclude in Section 5. Appendix A shows some perturbative
results, such as the values of the conversion factors for matching from IMOM
to $\overline{\text{MS}}$, computed at Next-to-Next-To-Leading Order (NNLO)
More details about the lattice simulations can be found in Appendix B.

##  2 Kinematics

We consider the interaction in the diagram below, where the incoming momentum
is defined as $p_{1}$, the outgoing momentum as $p_{2}$, and the momentum
transfer as $q$.

Figure 1: We consider an incoming momentum $p_{1}$, outgoing momentum $p_{2}$
and a momentum transfer $q=p_{1}-p_{2}$.

The method is based on the Rome-Southampton regularisation independent
momentum scheme: the fermions are off-shell, we work in the chiral limit and
the renormalisation scale is given by the choice of external momenta. In the
original $\mathrm{RI/MOM}$ scheme (or $\mathrm{RI{}^{\prime}/MOM}$) [3], the
incoming and outgoing momenta are equal, $p_{1}=p_{2}$, so that there is no
momentum transfer, $q=0$. The renormalisation scale $\mu$ is defined by
$\mu^{2}=p_{1}^{2}=p_{2}^{2}$. In contrast, in the $\mathrm{SMOM}$ schemes,
the incoming and outgoing momenta are different: they are chosen such that
$q^{2}=p_{1}^{2}=p_{2}^{2}$; naturally the renormalisation scale is given by
$\mu=\sqrt{q^{2}}$. The advantage is to avoid exceptional kinematics which
behave badly in the infrared regime [9].

The above schemes can be generalised further. We retain the condition
$p_{1}^{2}=p_{2}^{2}=\mu^{2}$ but drop the requirement $q^{2}=\mu^{2}$. We
define the variable $\omega=q^{2}/\mu^{2}$ to parameterise the kinematics.
$\omega=0$ corresponds to the original RI/MOM and RI’/MOM schemes, $\omega=1$
corresponds to the SMOM scheme. The possible values of $\omega$ range between
0 and 4. Let us denote by $\alpha$ the angle between the incoming and outgoing
momenta. We choose our frame such that the momenta take the following form
(the time component is represented by the last entry):

$\displaystyle p_{1}$ $\displaystyle=$ $\displaystyle\mu\left[1,0,0,0\right],$
(2.1) $\displaystyle p_{2}$ $\displaystyle=$
$\displaystyle\mu\left[\cos\alpha,\sin\alpha,0,0\right].$ (2.2)

This gives

$q=p_{1}-p_{2}=\mu\left[1-\cos\alpha,-\sin\alpha,0,0\right]\;,$ (2.3)

and

$\displaystyle q^{2}$ $\displaystyle=$
$\displaystyle\mu^{2}\left((1-\cos\alpha)^{2}+\sin^{2}\alpha)\right)\;$ (2.4)
$\displaystyle=$ $\displaystyle 2\mu^{2}\left(1-\cos\alpha\right)\;$ (2.5)
$\displaystyle\equiv$ $\displaystyle\omega\mu^{2}\;,$ (2.6)

where we have defined $\omega=2(1-\cos\alpha)$. Clearly the parameter $\omega$
can take any real value between 0 and 4, and the value of $\omega$ is directly
related to the value of $\alpha$ as shown in Fig 2.

Figure 2: The parameter $\omega$ is dependent upon the value of the angle
$\alpha$ between the incoming and outgoing momenta.

We can check that the extreme cases $\omega=0$ and $\omega=4$ correspond to
$p_{1}=p_{2}$ and $p_{1}=-p_{2}$, respectively, and therefore these situations
suffer from infrared and collinear singularities. We also note that $\omega=0$
is the choice made in the original (exceptional) kinematics of a standard
RI/MOM scheme while $\omega=1$ corresponds to the SMOM kinematic of [9]. This
has been known for a while, see for example [9, 10]. Different choices of
$\omega$ are referred to as interpolating momentum schemes [11, 20, 21].
However, to our knowledge, there is no numerical study or practical
implementation of such kinematics.

Let us now turn to the lattice implementation. If $L$ is the physical spatial
extent of the lattice (we do not consider the temporal extent here for
simplicity), we have

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{L}\left[l,0,0,0\right]\;,$ (2.7) $\displaystyle
p_{2}$ $\displaystyle=$ $\displaystyle\frac{2\pi}{L}\left[m,n,0,0\right]\;,$
(2.8)

where $l,m$ and $n$ are dimensionless (they are the Fourier modes if they are
chosen to be integers). In order to obtain any desired value of $\mu$ and
$\omega$ we take advantage of (partially) twisted boundary conditions, which
allow $l,m$ and $n$ to be any real numbers. For any pair $(\mu,\omega)$, the
numbers $(l,m,n)$ can be obtained from the kinematic constraints
$p_{1}^{2}=p_{2}^{2}=\mu^{2}$ and $q^{2}=\omega\mu^{2}$ by solving

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle 2\pi l/L\;,$ (2.9)
$\displaystyle l^{2}$ $\displaystyle=$ $\displaystyle m^{2}+n^{2}\;,$ (2.10)
$\displaystyle\omega l^{2}$ $\displaystyle=$ $\displaystyle(l-m)^{2}+n^{2}\;.$
(2.11)

In order to show a concrete example, let us anticipate and consider one of the
ensembles used in this work, with spatial extent of $L/a=32$ and inverse
lattice spacing of $a^{-1}=2.383(9)$ GeV. In that case, we can see that the
following choice:

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{L}\left[2.1372,0,0,0\right],$ (2.12) $\displaystyle
p_{2}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{L}\left[1.60291,1.41363,0,0\right],$ (2.13)

corresponds to $\mu=1$ GeV and $\omega=0.5$ with an uncertainty at the
permille level.

It is worth noting another choice of kinematics given by †††Here we assume for
simplicity that T=L, but in general the time component should be rescaled if
this is not the case.

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{L}\left[m,m,m,m\right],$ (2.14) $\displaystyle
p_{2}$ $\displaystyle=$ $\displaystyle\frac{2\pi}{L}\left[\pm m,\pm m,\pm
m,\pm m\right].$ (2.15)

Depending on the number of minus signs in $p_{2}$, we can obtain any integer
value of $\omega$ between $0$ and $4$. With this choice we also expect the
quark wave renormalisation factor so to have similar lattice artefacts,
whereas in the setup mentioned above they will have different discretisation
effects (except in some special cases). In that case, one can expect the
discretisation effects to be $\omega$-independent. We note that the same
procedure with fewer than four components can lead to non-integer value of
$\omega$: if the last component is fixed to zero, we can achieve the values
$\omega=0,4/3,8/3,4$.

In this work we only consider the setup given in Eqs. (2.1)-(2.2) as we also
want to investigate arbitrary non-integer values of $\omega$. We chose to vary
$\omega$ between 0.5 and 4 with a step of 1/2. In addition, we can use the
$\omega$ dependence of the discretisation effects to obtain an additional
handle to estimate the systematic uncertainties.

##  3 Implementation

###  3.1 Lattice implementation

For the bilinears, omitting flavour and spin-colour indices for clarity (we
only consider flavour non-singlet operators) we define
$O_{\Gamma}=\bar{\psi}\Gamma\psi$, with
$\Gamma=\mathbb{1},\gamma_{\mu},\gamma_{\mu}\gamma_{5},\gamma_{5}$ (S,V,A,P).
We sketch here the computation of the propagators and of the bilinears needed
to renormalise the quark mass and the quark wave function. We drop the volume
factors and set $a=1$ for simplicity.

Following [22], the momentum source propagators are computed by first solving

$\sum_{x}D(y,x)\tilde{G}_{x}(p)=e^{ip.y}\mathbb{1}\;,$ (3.16)

for a given momentum $p$, on Landau-gauged fixed configurations. In the
previous equation, $D$ denotes a generic Dirac matrix in spin-colour space,
regardless of the lattice discretisation. Then we obtain the momentum source
propagators by multiplying by the corresponding phase factor

$G_{x}(p)=\tilde{G}_{x}(p)e^{-ip.x}=\sum_{z}D^{-1}(x,z)e^{ip.(z-x)}.$ (3.17)

So within our conventions, $G_{x}(p)$ is an incoming propagator (with respect
to $x$) with momentum $p$. After gauge average, the outgoing one is then given
by $G_{x}(-p)=\gamma_{5}G_{x}(p)^{\dagger}\gamma_{5}$. The advantage of the
method proposed in [22] compared to a traditional Fourier transom is that
there sum over $z$ in Eq.(3.17) significantly reduces the statistical
uncertainties. For the bilinears, we consider the diagram shown in Fig 1 and
compute ($\langle...\rangle$ denotes the gauge average)

$\displaystyle V_{\Gamma}(p_{2},p_{1})$ $\displaystyle=$
$\displaystyle\langle\psi(p_{2})O_{\Gamma}\bar{\psi}(p_{1})\rangle\;,$ (3.18)
$\displaystyle=$ $\displaystyle\sum_{x}\langle G_{x}(-p_{2})\Gamma
G_{x}(p_{1})\rangle\;.$ (3.19)

Defining $G(p)=\sum_{x}G_{x}(p)$, we then amputate the external legs

$\Pi_{\Gamma}=\langle G^{-1}(-p_{2})\rangle V_{\Gamma}(p_{2},p_{1})\langle
G^{-1}(p_{1})\rangle.$ (3.20)

This quantity $\Pi_{\Gamma}$ is obviously a matrix in spin-colour space, the
renormalisation factors $Z_{\Gamma}$ are then defined by projecting it to the
corresponding tree level value. A possible choice (which defines a so-called
$\gamma_{\mu}$-scheme) is to use the same $\Gamma$ matrix as in the vertex

$\Lambda_{\Gamma}=\frac{1}{F}{\cal{P}}[\Pi_{\Gamma}]=\frac{1}{F}\text{Tr}[\Gamma\Pi_{\Gamma}],$
(3.21)

where the trace is taken in colour space and $F$ is the corresponding tree
level value.

###  3.2 Z-factors

Let us now specify our conventions for the renormalisation factors. We denote
the renormalised quantities with an index $R$, whereas the bare quantities do
not carry extra indices. The quark wave function renormalisation factor
$Z_{q}$ is defined by $\psi_{R}=\sqrt{Z_{q}}\psi$, and $G_{R}=Z_{q}G$
represents the renormalised fermion propagator. For the bilinears defined
above, the renormalised quantities are denoted by
$O_{\Gamma}^{R}=Z_{\Gamma}O_{\Gamma}$. Note that for the corresponding Green’s
functions, we have as usual
$\Pi_{\Gamma}^{R}=Z_{\Gamma}/Z_{q}\times\Pi_{\Gamma}$.

The renormalisation factors for the bilinears are then obtained by imposing

$\lim_{m\to
0}\frac{Z_{\Gamma}(\mu,\omega)}{Z_{q}(\mu)}\times\Lambda_{\Gamma}|_{(\mu^{2}=p_{1}^{2}=p_{2}^{2},(p_{1}-p_{2})^{2}=\omega\mu^{2})}=1,$
(3.22)

where $m$ is the (bare) quark mass (before taking the chiral limit, we work
with degenerate quark masses for simplicity). Note that we could take also the
chiral limit by sending the renormalised mass to zero, here we assume that
there is no additive mass renormalisation ‡‡‡With Domain-Wall fermions, a
small additive renormalisation of the quark mass is also necessary. For
simplicity we also assume the residual mass has already been included; here
the quark mass renormalises multiplicatively.. There are several possible ways
to define the quark field renormalisation constant $Z_{q}$. For example in
$\mathrm{RI{}^{\prime}/MOM}$, one imposes a condition directly on the
fermionic propagator

$Z_{q}(\mu)=-i\lim_{m\to
0}\left[\frac{1}{12p^{2}}\text{Tr}\left[\not{p}G^{-1}(p)\right]\right]_{\mu^{2}=p^{2}}\;.$
(3.23)

We now turn to the quark mass; The choice of the field renormalisation fixes
the renormalisation factor $Z_{m}$, which relates the bare and renormalised
masses as $m_{R}=Z_{m}m$. To be specific, we write in Fourier space the
inverse bare fermion propagator

$G^{-1}(p)=i\not{p}+m+\Sigma=i\not{p}(1+\Sigma^{V})+m(1+\Sigma^{S})\;.$ (3.24)

and similarly, having a generic MOM scheme in mind, we write for the
renormalised propagator,

$G_{R}^{-1}(p)=i\not{p}+m_{R}+\Sigma_{R}=i\not{p}(1+\Sigma_{R}^{V})+m_{R}(1+\Sigma_{R}^{S})\;.$
(3.25)

Since $G_{R}=Z_{q}G$, we have

$\displaystyle(1+\Sigma_{R}^{V})$ $\displaystyle=$ $\displaystyle
Z_{q}^{-1}(1+\Sigma^{V})\;,$ (3.26) $\displaystyle(1+\Sigma_{R}^{S})$
$\displaystyle=$ $\displaystyle Z_{q}^{-1}Z_{m}^{-1}(1+\Sigma^{S})\;.$ (3.27)

Traditionally in a MOM scheme, one defines the renormalisation factors $Z_{m}$
and $Z_{q}$ by imposing that the renormalised parts are finite in the chiral
limit at a certain renormalisation point. For example the renormalisation
factor $Z_{m}$ can then be extracted from

$Z_{m}(\mu)=\frac{1}{Z_{q}(\mu)}\lim_{m\to
0}\left[\frac{1}{12m}\text{Tr}\left[G^{-1}(p)\right]\right]_{\mu^{2}=p^{2}}\;,$
(3.28)

which is equivalent to

$\lim_{m\to
0}\left[\frac{1}{12m_{R}}\text{Tr}\left[G_{R}^{-1}(p)\right]\right]_{\mu^{2}=p^{2}}=1\;.$
(3.29)

In [9], it was suggested to replace this condition by

$\lim_{m\to
0}\left\\{\frac{1}{12m_{R}}\left[\text{Tr}\left[G_{R}^{-1}(p)\right]_{\mu^{2}=p^{2}}-\frac{i}{2}\text{Tr}\left[q_{\mu}\Pi^{\mu}_{A,R}\gamma_{5}\right]_{\mu^{2}=p_{1}^{2}=p_{2}^{2}=(p_{1}-p_{2})^{2}}\right]\right\\}=1\;.$
(3.30)

The motivation for this definition of $Z_{m}$ is that it leads to
$Z_{m}=1/Z_{P}$ if $Z_{P}$ is defined from Eq.(3.21) at the same kinematic
point. This follows from requiring that the Ward-Takahashi identities (WTI)
given below are satisfied both for the bare and renormalised quantities (for
degenerate quark masses)

$\displaystyle q_{\mu}\Pi_{V^{\mu}}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle-i(G^{-1}(p_{2})-G^{-1}(p_{1}))\;,$ (3.31) $\displaystyle
q_{\mu}\Pi_{A^{\mu}}(p_{1},p_{2})$ $\displaystyle=$ $\displaystyle
2im\Pi_{P}(p_{1},p_{1})-i\left(\gamma_{5}G^{-1}(p_{2})+G^{-1}(p_{1})\gamma_{5}\right)\;,$
(3.32)

in the case of degenerate quark masses. Imposing this condition with the SMOM
kinematics leads to maintaining the WTI on the renormalised quantities. In
particular the procedure leads to $Z_{A}=Z_{V}$.

###  3.3 Our definitions

We deviate from the choice made in [9] for several reasons. Firstly, some
derivations in [9] rely on the SMOM kinematics, which we want to generalise
here. Secondly, rather than imposing the renormalisation condition directly on
the quark propagators, we take advantage of the good chiral properties of the
Domain-Wall discretisation and extract the renormalisation factors from the
bilinears. Thirdly, in our framework, we use the local vector and axial vector
current which are related to the conserved (or partially conserved) ones by
the finite “renormalisation” factors $Z_{V}$ and $Z_{A}$ respectively. In this
paper we are using Domain Wall fermions, so we have $Z_{A}=Z_{V}$ up to small
artefacts. However the SMOM scheme can also be used for other lattice
fermions, with $Z_{A}\neq Z_{V}$ so we do not want to restrict ourselves to
the case $Z_{A}=Z_{V}$.In this paper we are using Domain Wall fermions, so we
have $Z_{A}=Z_{V}$ up to small artefacts. However the SMOM scheme can also be
used for other lattice fermions, with $Z_{A}\neq Z_{V}$ so we do not want to
restrict ourselves to the case $Z_{A}=Z_{V}$. Our values of $Z_{A}=Z_{V}$ are
known from previous studies on the same lattice, and are given in Appendix B
for convenience. Additionally, within our framework we define $Z_{m}$ directly
as

$Z_{m}=1/Z_{S}\;\;$ (3.33)

and we also define

$\displaystyle\frac{Z_{V}}{Z_{q}(\mu,\omega)}\lim_{m\to
0}\left[\Lambda_{V}\right]_{\text{IMOM}}=1\;,$ (3.34)
$\displaystyle\frac{Z_{S}(\mu,\omega)}{Z_{q}(\mu,\omega)}\lim_{m\to
0}\left[\Lambda_{S}\right]_{\text{IMOM}}=1\;.$ (3.35)

where the shorthand IMOM stands for
$\mu^{2}=p_{1}^{2}=p_{2}^{2},(p_{1}-p_{2})^{2}=\omega\mu^{2}$. (These
equations are used for the quark wave function and the mass renormalisation
factors.) More explicitly, we define

$\displaystyle Z_{q}(\mu,\omega)$ $\displaystyle=$ $\displaystyle
Z_{V}\lim_{m\to 0}\left[\Lambda_{V}\right]_{\text{IMOM}}\;,$ (3.36)
$\displaystyle Z_{m}(\mu,\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{Z_{V}}\lim_{m\to
0}\left[\frac{\Lambda_{S}}{\Lambda_{V}}\right]_{\text{IMOM}}\;,$ (3.37)

where $\Lambda_{V,S}$ are the bare (and local) vertices defined above. Note
that both $Z_{m}$ and $Z_{q}$ acquire an $\omega$-dependence through these
definitions. This dependence is discussed in section 4. The main advantage of
this approach is that the Green’s functions of the bilinears have a much safer
infrared behaviour than the quark propagators.

In order to completely fix the definition of $Z_{q}$ and $Z_{m}$ in Eqs.
(3.36) and (3.37) we need to specify our the choice of projector ${\cal
P}_{\mu}$ for the vector Green’s function,

$\Lambda_{V}=\frac{1}{F}{\cal P}_{\mu}[\Pi_{V^{\mu}}]\;.$ (3.38)

We implement the so-called $\gamma_{\mu}$ and $\not{q}$-projectors (the trace
is taken over both Dirac and colour indices)

$\displaystyle\Lambda_{V}^{(\gamma_{\mu})}$ $\displaystyle=$
$\displaystyle\frac{1}{48}\text{Tr}[\gamma_{\mu}\Pi_{V^{\mu}}]\;,$ (3.39)
$\displaystyle\Lambda_{V}^{(\not{q})}$ $\displaystyle=$
$\displaystyle\frac{q^{\mu}}{12q^{2}}\text{Tr}[\not{q}\Pi_{V^{\mu}}\,]\;,$
(3.40)

where obviously $q=p_{1}-p_{2}$ is the momentum transfer of the IMOM kinematic
defined above, $\mu^{2}=p_{1}^{2}=p_{2}^{2},(p_{1}-p_{2})^{2}=\omega\mu^{2}$.
Plugging these definitions in Eqs (3.36) and (3.37), we see that we have
defined two kinds of IMOM schemes similarly to was done in the SMOM case. In
order to keep track of what projector was used, we introduce the respective
notations $Z_{m}^{(\gamma_{\mu})}(\mu,\omega)$ and
$Z_{m}^{(\not{q})}(\mu,\omega)$ for the renormalisation factors.

The renormalisation factors for the other bilinears are defined exactly in the
same way as Eqs 3.34,3.35. This, together with a lattice formulation which
preserves chiral symmetry, leads to $Z_{V}=Z_{A}$ and $Z_{S}=Z_{P}$, apart
from the physical effects of spontaneous chiral symmetry breaking, which may
become important at low $p^{2}$ and $q^{2}$. This is discussed in Section 4.
Naturally for a lattice formulation which break explicitly chiral symmetry,
one can follow [9] with the modification $q^{2}=\omega\mu^{2}$.

###  3.4 Nomenclature

To define unambiguously a momentum scheme for a composite operator one needs
to specify the kinematics (the choice of momenta) and the choice of
projectors, not only for the Green’s function corresponding to the operator in
question, but also for the quark wave function.§§§ Strictly speaking we also
need to specify the gauge, but it is standard to assume that the computation
is performed in the Landau gauge.  Historically, when the Rome-Southampton
method was introduced in [3], the kinematics was the exceptional case,
$p_{1}=p_{2}$ and $Z_{q}$ was defined through a $\gamma_{\mu}$-projector, i.e.
Eq. (3.39). Although the authors of [3] mention explicitly that other choices
were possible, by convention “RI/MOM scheme” refers to this specific choice of
kinematics and wave function. Similarly, it also became standard to call
“RI′/MOM scheme” a scheme in which the kinematics is still the exceptional
case but the quark wave function is renormalised through a
$\not{q}$-projector, ie Eq. (3.40). (In [3] the definition of the projector
differs from the one given here, but both definitions are equivalent up to
lattice artefacts).

We turn now to the SMOM schemes. By definition, the “S” of “SMOM” refers to
the situation where $p_{1}^{2}=p_{2}^{2}=(p_{1}-p_{2})^{2}$, ie $\omega=1$, or
“Symmetric” [9]. For the quark wave function, the convention in [9] was that
“RI/SMOM” refer to a $\not{q}$-projector and “RI/SMOM${}_{\gamma_{\mu}}$” to a
$\gamma_{\mu}$-projector. Strictly speaking there is also a subtlety in the
definition of $Z_{m}$, see section 3.2.

Here, following the conventions adopted by RBC-UKQCD (see for example [8]), we
use the notations $Z^{(X)}(\mu,\omega)$, where $X\in(\gamma_{\mu},\not{q})$
indicates the choice of the projector for the wave function. Obviously the
choice of kinematics is explicitly given by $\omega$. When needed, we also
refer to these schemes as IMOM${}^{(\gamma_{\mu})}$ and IMOM(／q).

Name | Kinematics | Projector for $Z_{q}$
---|---|---
RI/MOM | $p_{1}=p_{2}$ | $\gamma_{\mu}$
RI′/MOM | $p_{1}=p_{2}$ | $\not{q}$
RI/SMOM${}_{\gamma_{\mu}}$ | $p_{1}^{2}=p_{2}^{2}=(p_{1}-p_{2})^{2}$ | $\gamma_{\mu}$
RI/SMOM | $p_{1}^{2}=p_{2}^{2}=(p_{1}-p_{2})^{2}$ | $\not{q}$
IMOM${}^{(\gamma_{\mu})}$ | $p_{1}^{2}=p_{2}^{2}=\mu^{2},\;(p_{1}-p_{2})^{2}=\omega\mu^{2}$ | $\gamma_{\mu}$
IMOM(／q) | $p_{1}^{2}=p_{2}^{2}=\mu^{2},\;(p_{1}-p_{2})^{2}=\omega\mu^{2}$ | $\not{q}$

###  3.5 Non-perturbative running

To compute the non-perturbative scale evolution of $Z_{m}$: we define
$\Sigma_{m}$ as

$\Sigma^{(X)}_{m}(a,\mu,\mu_{0},\omega,\omega_{0})=\lim_{m\rightarrow
0}\frac{Z^{(X)}_{m}(a,\mu,\omega)}{Z^{(X)}_{m}(a,\mu_{0},\omega_{0})}\;,$
(3.41)

where $X$ can either be $\gamma_{\mu}$ or $\not{q}$. We can then take the
continuum limit $a^{2}\rightarrow 0$ and define

$\sigma^{(X)}_{m}(\mu,\mu_{0},\omega,\omega_{0})=\lim_{a^{2}\rightarrow
0}\Sigma^{(X)}_{m}(a,\mu,\mu_{0},\omega,\omega_{0})\;.$ (3.42)

Similarly, for the quark wave function we define $\sigma_{q}$, by just
substituting $Z_{q}$ for $Z_{m}$ in the above definitions. Since in practice
we have only two lattice spacings, we add a systematic error as an estimate of
the residual discretisation effects. This error is obtained by adding half the
difference beween the extrapolated value and the value on the the finest
lattice spacing (see Appendix D). We now have for each value of
$\mu,\mu_{0},\omega,\omega_{0}$ a central value of $\sigma_{m,q}$, some
statistical error and a systematic error from the continuum extrapolation In
the following we will always sum the systematic and statistical uncertainties
in quadrature except when stated otherwise. We found that the chiral
extrapolations are well under control and that an associated systematic error
would be negligible compared to the other sources of error. The main
motivation for looking at these quantities rather than the $Z$-factors
themselves is the existence of the continuum limit. We can also compare the
non-perturbative running obtained from our lattice simulation with the
perturbative prediction.

##  4 Results

To determine the relationship between lattice results and
$\overline{\mathrm{MS}}$ results we need to compare lattice and perturbation
theory in a kinematic region where both perturbation theory and lattice
results are reliable. In this region the perturbative and lattice results
should run in the same way, so the ratio should be constant. We will see
deviations from constancy in regions where perturbation theory converges
slowly (for example at small $\mu$ or small $q^{2}=\omega\mu^{2}$. We will
also see non-constancy if lattice artefacts are important, which is likely to
happen at large $\mu^{2}$ and large $\omega\mu^{2}$. We also note that our
procedure to estimate the residual discretisation errors could potentially
underestimate them in the large $\mu^{2}$ and large $\omega\mu^{2}$ regions.

In the following sections we will first keep $\omega$ constant and compare the
$\mu$ dependence of the lattice results and the continuum. An advantage of
IMOM is that we have two parameters, $\mu^{2}$ and $\omega$, so we can better
test that we are working in a region where it is sensible to trust both
perturbative and lattice results. Accordingly, we next will vary both $\mu$
and $\omega$ and study if a plateau is emerging where we can reliably
determine $Z$.

###  4.1 Quark mass renormalisation factor

We now apply this scheme to determine $Z$ for the mass operator and quark
wave-function. The running of the quark mass is shown in Fig 3 for the
$\gamma_{\mu}$-scheme and Fig. 4 for the $q\\!\\!\\!/$-scheme. In these plots
we show both the non non-perturbative scale evolution
$\sigma^{(\gamma_{\mu})}_{m}(\mu,\mu_{0},\omega,\omega_{0})$ and the
perturbative prediction
$u^{(\gamma_{\mu})}_{m}(\mu,\mu_{0},\omega,\omega_{0})$. In order to study the
$\mu$-evolution for fixed values of $\omega$, we set
$\omega=\omega_{0}=0.5,1.0,1.5,\ldots,4.0$ and let $\mu$ vary between 1 and 4
GeV. We find a good agreement for intermediate values of $\mu$ and $\omega$,
where both perturbation theory and lattice artefacts are expected to be under
good control. In fact perturbation theory works surprisingly well even for
small values of $\mu$, where significant discrepancies with the lattice
results only emerge at $\mu\simeq 1$GeV. Out of the two schemes, perturbation
theory and lattice results agree best in the $\gamma_{\mu}$-scheme. The onset
of lattice artefacts for large values of $\mu$ and $\omega$ becomes only
relevant for $q^{2}\gtrsim 25\mathrm{GeV}^{2}$. This becomes particularly
visible for large values of $\omega=4$, where perturbation theory also becomes
less reliable. The discrepancy of lattice results and perturbation theory
occurs already at values of $q^{2}\gtrsim 10\mathrm{GeV}^{2}$ in the
$q\\!\\!\\!/$-scheme.

We show our results for the scale evolution of the quark mass in the
$(\mu,\omega)$ plane in Fig. 5 for the $\gamma_{\mu}$-scheme and in Fig. 6 we
show the same quantity for the $\not{q}$-scheme. Here we allow
$\omega\neq\omega_{0}$ and show the ratio of the non-perturbative running
divided by the perturbative prediction as a function of $\mu$ and $\omega$ at
LO, NLO and NNLO, while fixing $\omega_{0}=2$ and $\mu_{0}=2.5$ GeV. The
discrepancy from one can then be compared with the combined lattice and
perturbative uncertainties that have been added in quadrature.

Figure 3: Comparison of the non-perturbative and perturbative running for
$Z_{m}^{(\gamma_{\mu})}$. Note that for $\omega=1$ the perturbative running is
known at N3LO. Figure 4: Same as Fig. 3 for $Z_{m}^{(\not{q})}$.  Figure 5:
Ratio of the non-perturbative running over the perturbative prediction for
$Z_{m}^{(\gamma_{\mu})}$. The central value is shown on the left and the error
on the right. This ratio is exactly one (by definition) for the point
($\mu=2.5$ GeV, $\omega=2$).  Figure 6: Same as Fig. 5 for
$Z_{m}^{(\not{q})}$.

As expected we observe that increasing the order of the perturbative expansion
improves the agreement with the non-perturbative evolution. The corner of the
planes are affected by larger systematic errors, in particular where
$\omega=4$ and/or $\mu>3.5$ GeV where the discretisation effects become more
sizeable. However our data agree with NNLO at a few percent level for a large
part of the $(\mu,\omega)$plane: approximately for $1.5\mbox{ GeV}\leq\mu\leq
3.5\mbox{ GeV}$ and $1\leq\omega\leq 3$.

For completeness, we study the next order in perturbation theory for the case
$\omega=1$, as the three loop matching has been computed in [13, 14] and the
$\overline{\text{MS}}$ four-loop anomalous dimension in can be found in [23].
We take $\mu_{0}=2$ GeV and $\mu=2.5$ GeV and compute the running in different
schemes, see Table 1. In Table 2, we show the contributions of the higher
orders. Although the corrections to the leading order contributions are rather
small, we see that the convergence in $\overline{\text{MS}}$ seems to be much
better than in the MOM schemes, in the sense the $(i+1)$-th correction is much
smaller than the $i$-th order. However, the important observations here are
that the leading order gives by far the main contribution and that the
perturbative series seems to converge nicely to the non-perturbative values
¶¶¶The attentive reader would have noticed that in Table 1 The NLO for
$\gamma_{\mu}$ and $q\\!\\!\\!/$ are identical. This seems to be nothing but a
numerical accident: adding more significant figures lead to 0.941112 for
$\gamma_{\mu}$ and 0.944132 for $q\\!\\!\\!/$..

Scheme | LO | NLO | NNLO | NNNLO | NP
---|---|---|---|---|---
$\overline{\text{MS}}$ | 0.9537 | 0.9456 | 0.9441 | 0.9439 |
$\overline{\text{MS}}\leftarrow\gamma_{\mu}$ | 0.9537 | 0.9350 | 0.9389 | 0.9426 |
$\overline{\text{MS}}\leftarrow q\\!\\!\\!/$ | 0.9537 | 0.9451 | 0.9462 | 0.9475 |
$\gamma_{\mu}$ | 0.9537 | 0.9411 | 0.9357 | 0.9318 | 0.9307(62)
$q\\!\\!\\!/$ | 0.9537 | 0.9441 | 0.9415 | 0.9400 | 0.9436(46)
Table 1: Running between 2 and 2.5 GeV for the quark mass. First we perform the computation directly in $\overline{\text{MS}}$ then and in the SMOM ($\omega=1$) schemes $\gamma_{\mu}$ and $q\\!\\!\\!/$. In this case the running is known at NNNLO. The lattice results are denoted by NP (non-perturbative). We also show the running computed first non-perturbatively then converted to $\overline{\text{MS}}$, we denote them by $\overline{\text{MS}}\leftarrow\gamma_{\mu}$ and $\overline{\text{MS}}\leftarrow q\\!\\!\\!/$, respectively.  Scheme | NLO-LO | NNLO-NLO | NNNLO-NNLO
---|---|---|---
$\overline{\text{MS}}$ | -0.0081 | -0.0015 | -0.0002
$\gamma_{\mu}$ | -0.0126 | -0.0054 | -0.0040
$q\\!\\!\\!/$ | -0.0096 | -0.0026 | -0.0017
Table 2:  Study of the convergence of the perturbative series for running of
the quark mass between 2 and 2.5 GeV in $\overline{\text{MS}}$,
SMOM-$\gamma_{\mu}$ and $q\\!\\!\\!/$.

Finally for completeness, in Tables 3 and 7 we give our results for the non-
perturbative running of the quark mass, with $\mu_{0}=2$ GeV and
$\omega=\omega_{0}=0.5,1.0,\ldots,4.0$. In Appendix C, we show more numerical
results, where we vary our parameters $\omega,\omega_{0},\mu$ and $\mu_{0}$ in
various ways. Our conclusion remains that the non-perturbative results agree
very well with the perturbative ones as long as we stay from the corner of the
$(\omega,\mu)$ plane.

$\omega/\mu=$ | $1.0$ | $1.5$ | $2.5$ | $3.0$ | $3.5$ | $4.0$
---|---|---|---|---|---|---
$0.5$ | $1.219(123)$ | $1.131(21)$ | $0.922(8)$ | $0.866(13)$ | $0.827(18)$ | $0.793(26)$
$1.0$ | $1.343(64)$ | $1.092(16)$ | $0.931(6)$ | $0.882(11)$ | $0.846(17)$ | $0.816(24)$
$1.5$ | $1.306(52)$ | $1.110(12)$ | $0.937(5)$ | $0.894(11)$ | $0.861(17)$ | $0.831(25)$
$2.0$ | $1.214(49)$ | $1.099(11)$ | $0.944(5)$ | $0.904(10)$ | $0.878(14)$ | $0.860(17)$
$2.5$ | $1.176(37)$ | $1.085(9)$ | $0.949(5)$ | $0.916(9)$ | $0.898(10)$ | $0.893(7)$
$3.0$ | $1.180(39)$ | $1.070(7)$ | $0.952(6)$ | $0.925(8)$ | $0.911(8)$ | $0.910(4)$
$3.5$ | $1.169(27)$ | $1.065(6)$ | $0.959(5)$ | $0.937(6)$ | $0.928(4)$ | $0.935(5)$
$4.0$ | $1.142(27)$ | $1.049(5)$ | $0.970(2)$ | $0.968(10)$ | $1.004(36)$ | $1.144(116)$
Table 3: Non-perturbative running for the quark mass in the $\gamma_{\mu}$ scheme.  $\omega/\mu=$ | $1.0$ | $1.5$ | $2.5$ | $3.0$ | $3.5$ | $4.0$
---|---|---|---|---|---|---
$0.5$ | $1.182(128)$ | $1.110(21)$ | $0.933(8)$ | $0.886(9)$ | $0.860(8)$ | $0.846(8)$
$1.0$ | $1.302(73)$ | $1.074(18)$ | $0.944(5)$ | $0.907(5)$ | $0.888(4)$ | $0.885(11)$
$1.5$ | $1.266(57)$ | $1.082(10)$ | $0.951(3)$ | $0.922(3)$ | $0.912(3)$ | $0.924(17)$
$2.0$ | $1.157(49)$ | $1.074(11)$ | $0.959(4)$ | $0.936(5)$ | $0.933(4)$ | $0.958(22)$
$2.5$ | $1.127(43)$ | $1.060(10)$ | $0.966(5)$ | $0.950(5)$ | $0.950(3)$ | $0.969(16)$
$3.0$ | $1.126(45)$ | $1.047(8)$ | $0.969(5)$ | $0.958(6)$ | $0.962(3)$ | $0.985(15)$
$3.5$ | $1.114(32)$ | $1.042(7)$ | $0.976(4)$ | $0.970(4)$ | $0.982(7)$ | $1.018(28)$
$4.0$ | $1.080(30)$ | $1.023(7)$ | $0.989(6)$ | $1.009(25)$ | $1.081(71)$ | $1.274(182)$
Table 4: Non-perturbative running for the quark mass in the $q\\!\\!\\!/$
scheme.

###  4.2 Quark wave function renormalisation factor

We also study the convergence for the quark wave function, $Z_{q}^{(X)}$, see
Table 5 and Table 6. Here the situation is very different from the quark mass
mainly as there is no contribution at leading order (in the Landau gauge). In
the $q\\!\\!\\!/$-scheme, the non-perturbative results differ significantly
from the perturbative predictions; however one should keep in mind that the
running is very small in magnitude. It is also interesting to have a close
look at the $q\\!\\!\\!/$ scheme where there the perturbative prediction is
known at N3LO. As we can see the perturbative series to converge very poorly
in the sense that the relative difference decrease very slowly as we increase
the order. The difference between the non-perturbative result and the N3LO,
namely $\sim 1.0195-1.0113\sim 0.0082$, could be explained by higher order in
the perturbative series.

Scheme | LO | NLO | NNLO | NNNLO | NP
---|---|---|---|---|---
$\overline{\text{MS}}$ | 1.0 | 1.0048 | 1.0062 | 1.0064 |
$\overline{\text{MS}}\leftarrow\gamma_{\mu}$ | 1.0 | 1.0069 | 1.0078 | N.A. |
$\overline{\text{MS}}\leftarrow q\\!\\!\\!/$ | 1.0 | 1.0195 | 1.0175 | 1.0146 |
$\gamma_{\mu}$ | 1.0 | 1.0017 | 1.0020 | N.A | 1.0037(20)
$q\\!\\!\\!/$ | 1.0 | 1.0048 | 1.0081 | 1.0113 | 1.0195(25)
Table 5: Running between 2 and 2.5 GeV for the quark wave function in $\overline{\text{MS}}$ and in the SMOM schemes $\gamma_{\mu}$($\omega=1$) and $q\\!\\!\\!/$. In this case the running is known at NNNLO.  Scheme | NLO-LO | NNLO-NLO | NNNLO-NNLO
---|---|---|---
$\overline{\text{MS}}$ | 0.0048 | 0.0013 | 0.0003
$\gamma_{\mu}$ | 0.0017 | 0.0003 |
$q\\!\\!\\!/$ | 0.0048 | 0.0033 | 0.0032
Table 6: Study of the convergence of the perturbative series for running of
the quark wave function between 2 and 2.5 GeV in $\overline{\text{MS}}$,
SMOM-$\gamma_{\mu}$ and $q\\!\\!\\!/$.

Keeping these facts in mind, we show our results in Fig. 7 in 8 for
$X=\gamma_{\mu}$ and $X=\not{q}$ respectively. Once again we divide the non-
perturbative running by the perturbative prediction, we fix $\omega_{0}=2$ and
$\mu_{0}=2.5$ GeV and let $\omega$ and $\mu$ vary.

Figure 7: Same as Fig. 5 for $Z_{q}^{(\gamma_{\mu})}$ Figure 8: Same as Fig. 5
for $Z_{q}^{(\not{q})}$. Figure 9: Same as Fig. 3 for $Z_{q}^{(\gamma_{\mu})}$
Figure 10: Same as Fig. 3 for $Z_{q}^{(\not{q})}$.

###  4.3 Study of the $\omega$-(in)dependence of $\sigma_{q}$

Due to the vector WTI, Eq 3.31, we might expect $Z_{q}^{(q\\!\\!\\!/)}$ to be
$\omega$-independent up to lattice artefacts. This is illustrated in Fig. 11
where we show $\sigma_{m}^{(q\\!\\!\\!/)}(\mu,\mu_{0},\omega,\omega_{0})$ as a
function of $\omega=\omega_{0}$ for different values of $\mu$ and $\mu_{0}$.
We can see that this property is rather well satisfied as long as the scales
remain moderate, say $\mu,\mu_{0}\leq 3$ GeV. However it is worth noting that
this invariance is only true in the continuum, as can be seen in Fig. 12. At
finite lattice spacing, within our small statistical errors, the lattice
artefacts are visible and $Z_{q}^{q\\!\\!\\!/}$ clearly show a $\omega$
dependence. These figures also suggest that the lattice artefacts are well
under control, with two lattice spacings, as long we do not go too high in
energy. Note that in these plots, the systematic errors due to the continuum
extrapolations are not included. The non perturbative-scale evolution of
$Z_{q}$, $\sigma_{q}(\mu,\mu_{0})$ with $\mu_{0}=2$ GeV and
$\omega=\omega_{0}$ for various values of $\mu$ and $\omega$ can be found in
Tables 7 and 8.

Also it is interesting to notice in Fig 12 how the discretisation effects
depend on $\omega$. Clearly for $\omega=2.0$ and $2.5$ the lattice artefacts
are much smaller than for $\omega=1.0$. Of course this effect is quantity-
dependent but it this could be useful in future computations.

|
---|---
|
|
Figure 11:  $\sigma_{q}^{(q\\!\\!\\!/)}(\mu,\mu_{0},\omega,\omega_{0})$ for $\omega_{0}=\omega$, statistical error only.  |
---|---
Figure 12: Example of continuum extrapolations for $\sigma_{q}^{(q\\!\\!\\!/)}(\mu,\mu_{0},\omega,\omega_{0})$.  $\omega/\mu=$ | $1.0$ | $1.5$ | $2.5$ | $3.0$ | $3.5$ | $4.0$
---|---|---|---|---|---|---
$0.5$ | $0.972(8)$ | $0.993(4)$ | $1.008(4)$ | $1.014(8)$ | $1.023(14)$ | $1.040(26)$
$1.0$ | $0.976(8)$ | $0.994(3)$ | $1.004(2)$ | $1.007(5)$ | $1.012(8)$ | $1.021(15)$
$1.5$ | $0.978(4)$ | $0.998(2)$ | $1.003(1)$ | $1.005(2)$ | $1.006(3)$ | $1.004(3)$
$2.0$ | $0.990(7)$ | $0.998(2)$ | $1.003(0)$ | $1.005(1)$ | $1.007(1)$ | $1.008(1)$
$2.5$ | $0.987(5)$ | $0.997(2)$ | $1.001(1)$ | $1.002(2)$ | $1.002(3)$ | $1.003(4)$
$3.0$ | $0.985(4)$ | $0.999(2)$ | $1.000(2)$ | $0.998(4)$ | $0.993(9)$ | $0.978(18)$
$3.5$ | $0.989(5)$ | $1.001(2)$ | $0.997(2)$ | $0.993(6)$ | $0.982(13)$ | $0.959(27)$
$4.0$ | $0.990(5)$ | $0.999(1)$ | $0.994(3)$ | $0.983(8)$ | $0.957(22)$ | $0.887(60)$
Table 7: Non-perturbative running for the quark wave function in the $\gamma_{\mu}$ scheme.  $\omega/\mu=$ | $1.0$ | $1.5$ | $2.5$ | $3.0$ | $3.5$ | $4.0$
---|---|---|---|---|---|---
$0.5$ | $0.935(27)$ | $0.974(7)$ | $1.018(7)$ | $1.035(16)$ | $1.059(31)$ | $1.096(56)$
$1.0$ | $0.951(19)$ | $0.978(6)$ | $1.017(6)$ | $1.035(14)$ | $1.058(28)$ | $1.096(51)$
$1.5$ | $0.950(15)$ | $0.973(4)$ | $1.017(4)$ | $1.037(11)$ | $1.064(25)$ | $1.106(49)$
$2.0$ | $0.942(15)$ | $0.976(3)$ | $1.020(3)$ | $1.040(8)$ | $1.069(20)$ | $1.118(45)$
$2.5$ | $0.942(14)$ | $0.974(3)$ | $1.019(1)$ | $1.039(3)$ | $1.060(9)$ | $1.087(20)$
$3.0$ | $0.937(12)$ | $0.978(4)$ | $1.017(2)$ | $1.033(2)$ | $1.048(2)$ | $1.060(2)$
$3.5$ | $0.943(10)$ | $0.979(3)$ | $1.014(2)$ | $1.027(3)$ | $1.039(3)$ | $1.050(2)$
$4.0$ | $0.940(10)$ | $0.975(4)$ | $1.013(3)$ | $1.027(8)$ | $1.046(18)$ | $1.084(40)$
Table 8: Non-perturbative running for the quark wave function in $q\\!\\!\\!/$
scheme. Here we expect the results to be $\omega$-independent.

###  4.4 Study of the chiral symmetry breaking effects

One of the original arguments to motivate the RI/SMOM schemes is a drastic
reduction in the effects of spontaneous chiral symmetry. Even though these
effects are physical, they can prevent a clean determination of the
renormalisation factors because they are absent from the perturbative
calculations. This is particularly true for quantities like $\Pi_{P}$ and
$\Sigma^{S}$ where the presence of pseudo-Goldstone poles can completely
dominate the signal. In practice, the vertex functions from which we want to
extract the Z-factors are ’polluted’ by negative powers of the quark mass or
the momentum scale. Of course these chiral symmetry breaking effects are non-
perturbative and disappear for high-momentum. However, in practice we want to
keep the Rome-Southampton windows as open as possible, so it is always
desirable to reduce these infrared contamination or at least keeping them
under control.

If chiral symmetry is exactly realised then we should find that $Z_{S}=S_{P}$
and $Z_{V}=Z_{A}$. Therefore, in order to study these chiral symmetry breaking
affects as a function of $\omega$, we study the deviation of $Z_{V}$ from
$Z_{A}$ and $Z_{S}$ from $Z_{P}$. We start by the analysis of the ratio
$Z_{V}/Z_{A}$, see Fig. 13 for $\omega=1$ using the the
${\gamma_{\mu}}$-projector. This ratio collapses quickly to one as the energy
scale increases: at $\mu=2$ GeV, the deviation from one is at most at the per-
mille level. We also find the quark mass dependence to be linear within our
statistical error - as expected - therefore we extrapolate to the chiral limit
using a straight line. Our results for the $q\\!\\!\\!/$-projector are very
similar.

|
---|---
Figure 13: $Z_{V}/Z_{A}$ for the ${\gamma_{\mu}}$-projector as a function of
$\mu$ for $\omega=1$. On the left panel we show the results at finite quark
masses as a function of the energy scale. As an example of chiral
extrapolations, on the right panel we show our results for $\mu=1$ GeV,
$\omega=1$. We perform a linear extrapolation (solid line), a quadratic
extrapolation is shown as a dash line for illustration.

In Fig. 13 we show our results in the chiral limit for the different values of
$\omega$ and find that $Z_{V}/Z_{A}$ is always very close to one. At our
lowest energy point $\mu=1$ GeV, we can observe a deviation from one at the
order of a percent. Our results seem to indicate a trend that $Z_{V}/Z_{A}$
increases toward one when $\omega$ increases between 0.5 and 2
($Z_{V}/Z_{A}\sim 0.98$ at $\omega=1$ and $Z_{V}/Z_{A}\sim 0.99$ at
$\omega=2$), but this could well be a statistical effect. All together we find
that $Z_{V}/Z_{A}=1$ to a very good approximation for all values of $\omega$.
This true for both values of the lattice spacing and both projectors
$X\in(\gamma_{\mu},q\\!\\!\\!/)$.

|
---|---
|
|
Figure 14: $Z_{V}/Z_{A}$ for the ${\gamma_{\mu}}$ projector as a function of
$\omega$ for various values of $\mu$.

We also study $(\Lambda_{S}-\Lambda_{P})/\Lambda_{V}$, which is proportional
to $Z_{V}(1/Z_{S}-1/Z_{P})$. Our results are shown in 15. Here it is clear
that the IR effects due to chiral symmetry breaking are affected by the value
of $\omega$. We can observe that at $\mu=1.5$ GeV,
$(\Lambda_{S}-\Lambda_{P})/\Lambda_{V}\sim 0.1$. The same quantity reduces to
$\sim 0.03$ for $\omega=2.0$. It will be interesting to perform a similar
study on four-quark operators, especially those where the IR contamination can
be important and a source of disagreement (see for example the section on BSM
kaon mixing in FLAG [24] and [8]).

|
---|---
|
|
Figure 15: $(\Lambda_{S}-\Lambda_{P})/\Lambda_{V}$ for the ${\gamma_{\mu}}$
projector as a function of $\omega$ for various values of $\mu$. Figure 16:
$Z_{S}$ and $Z_{P}$ at finite quark masses versus the renormalisation scale
for $\omega=1$.

##  5 Conclusions and outlook

As a proof of concept we have implemented several IMOM schemes, defined via
two different projectors, and for various kinematics. In this framework we
have determined the renormalisation factors and non-perturbative scale
evolution functions of the quark mass and of the quark wave function. To
compare our lattice results with NNLO results in continuum perturbation theory
we also present the numerical form of the perturbative scheme conversion
factors for these schemes in these general kinematics.

We find that the non-pertubative and perturbative results agree very well as
long as we stay from the corner of the $\omega,\mu$ plane, with one exception,
namely $Z_{q}^{(q\\!\\!\\!/)}$. Where, we argued that the reason for this
relatively bad agreement is the poor convergence of the perturbative
expansion. Clearly having several values of $\omega$ help to have a better
handle on the systematic errors coming from the NPR procedure. As an
application, we have seen a example where the discretisation effects depend
significantly on $\omega$ (see $\sigma_{q}^{(q\\!\\!\\!/)}$ where the
discretisation effects for $\omega=2,2.5$ are much smaller than for
$\omega=1$) as well as the perturbative convergence. In this proof of concept
study only two lattice spacings have been used. Adding a finer lattice could
potentially allow us to test the agreement of the perturbative and non-
perturbative window even further.

Generally speaking, it is well-known that the SMOM kinematics ($\omega=1$)
leads to much cleaner determinations of the renormalisation factors than the
$\omega=0$ case. In this work, we have shown that increasing the value of
$\omega$ leads to smaller contributions of the pseudo-Goldstone poles
contamination in $\Lambda_{P}-\Lambda_{S}$. It will be interesting to extend
this study to the case of four-quark operators where the infrared
contaminations due to chiral symmetry breaking are significantly more
sizeable.

##  6 Acknowledgements

This work was supported by the Consolidated Grant ST/T000988/1 and the work of
JAG by a DFG Mercator Fellowship. The quark propagators were computed on the
DiRAC Blue Gene Q Shared Petaflop system at the University of Edinburgh,
operated by the Edinburgh Parallel Computing Centre on behalf of the STFC
DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS
National E-infrastructure capital grant ST/K000411/1, STFC capital grant
ST/H008845/1, and STFC DiRAC Operations grants ST/K005804/1 and ST/K005790/1.
DiRAC is part of the National E-Infrastructure.

We warmly thank our colleagues of the RBC and UKQCD collaborations. We are
particularly indebted to Peter Boyle, Andreas Jüttner, J Tobias Tsang for many
interesting discussions. We also thank Peter Boyle for his help with the UKQCD
hadron software. We wish to thank Holger Perlt for his early contribution in
this area. N.G. thanks his collaborators from the California Lattice (CalLat)
collaboration and in particular those working on NPR: David Brantley, Henry
Monge-Camacho, Amy Nicholson and André Walker-Loud.

##  Appendix A Perturbation Theory

In order to make contact to phenomenology, results obtained from lattice
simulations need to be converted to a scheme that is used in continuum
perturbation theory such as $\overline{\text{MS}}$. The non-perturbative
renormalisation schemes used in this work serve as an intermediate scheme that
can be defined on the Lattice and in the continuum. This enables us to compare
the non-perturbative with the perturbative change of the renormalisation
scheme parameters $\mu$ and $\omega$. This appendix provides the ingredients
for the perturbative scheme changes at NNLO.

The dependence of the strong coupling constant on the dimensional
regularisation scale $\nu$ in the $\overline{\text{MS}}$ scheme is governed by
the $\beta$-function

$\nu^{2}\frac{d}{d\nu^{2}}\left(\frac{\alpha_{s}}{4\pi}\right)\equiv-\sum_{i=0}^{\infty}\beta_{i}\left(\frac{\alpha_{s}}{4\pi}\right)^{i+2}\;.$
(A.43)

Although it is known at five-loop [25, 26, 27], we will only use the first
three coefficients in the expansion to be consistent with our NNLO analysis of
the running. In the following we will need the first two coefficients

$\displaystyle\beta_{0}$ $\displaystyle=11-\frac{2}{3}N_{f}\;,$ (A.44)
$\displaystyle\beta_{1}$ $\displaystyle=102-\frac{38}{3}N_{f}\;,$ (A.45)

where we fixed the number of colours $N_{c}=3$ in this and the following
expressions. Using $\alpha_{S}(M_{Z})=0.1179(10)$ and $M_{Z}=91.1876(21)$ GeV
[28] we find the required values of the strong coupling constant for $N_{f}=3$
flavours given in Table 9.

$\nu$ /GeV | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0
---|---|---|---|---|---|---|---
$\alpha_{S}(\nu)$ | 0.4698 | 0.3467 | 0.2950 | 0.2652 | 0.2452 | 0.2307 | 0.2196
Table 9: Values of the strong running coupling as a function of the
dimensional regularisation scheme $\nu$. They are obtained in the
$\overline{\text{MS}}$ scheme in the three-flavour theory.

To arrive at three active flavours, we decoupled the bottom and charm quark at
$\nu_{b}=5$ GeV and $\nu_{c}=1.5$ GeV respectively using central values
$m_{b}(m_{b})=4.13$ GeV and $m_{c}(m_{c})=1.27$ GeV for the quark masses in
the $\overline{\text{MS}}$ scheme[28]. The relevant threshold effects and
running of the strong coupling constant are implemented using RunDec[29].

The fields and masses in the $\mathrm{IMOM}^{(X)}$ schemes are related to the
$\mathrm{\overline{MS}}$ scheme as
$\psi_{\overline{\text{MS}}}(\nu)=(Z_{q}^{\overline{\text{MS}}}/Z_{q}^{(X)})\psi_{X}(\mu,\omega)$
and
$m^{\mathrm{\overline{MS}}}(\nu)=(Z_{m}^{\overline{\text{MS}}}/Z_{m}^{(X)})m_{X}(\mu,\omega)$,
where $X\in(\gamma_{\mu},\not{q})$. The continuum perturbation theory
expansion of the conversion factors

$\frac{Z_{i}^{\overline{\text{MS}}}(\epsilon,\nu)}{Z_{i}^{(X)}(\epsilon,\mu,\omega)}\overset{\epsilon\to
0}{=}C_{i}^{(X)}(\nu,\mu,\omega)\,,\qquad i\in\\{q,m\\}\;,$ (A.46)

are known up to two loops [10, 20] for the IMOM scheme and up to three loops
[13, 14] in the SMOM limit, i.e. where $\omega=1$. Writing

$C_{i}^{(X)}=1+\frac{\alpha_{s}}{4\pi}C_{i}^{(X,1)}+\frac{\alpha_{s}^{2}}{(4\pi)^{2}}C_{i}^{(X,2)}$
(A.47)

and setting $\mu=\nu$ we find the numerical conversion factors as a function
of $\omega$ given in Table. 10.

$\omega$ | $C_{m}^{(\not{q},1)}$ | $C_{m}^{(\not{q},2)}$ | $C_{m}^{(\gamma,1)}$ | $C_{m}^{(\gamma,2)}$
---|---|---|---|---
$0.5$ | $-2.422$ | $-64.756+5.988N_{f}$ | $-3.248$ | $-89.07+7.571N_{f}$
$1.0$ | $-0.646$ | $-22.608+4.014N_{f}$ | $-1.979$ | $-55.032+6.162N_{f}$
$1.5$ | $0.778$ | $10.344+2.432N_{f}$ | $-0.964$ | $-28.916+5.035N_{f}$
$2.0$ | $1.994$ | $38.567+1.08N_{f}$ | $-0.098$ | $-6.829+4.072N_{f}$
$2.5$ | $3.071$ | $63.81-0.115N_{f}$ | $0.667$ | $12.741+3.222N_{f}$
$3.0$ | $4.042$ | $86.961-1.195N_{f}$ | $1.358$ | $30.56+2.454N_{f}$
$3.5$ | $4.933$ | $108.544-2.184N_{f}$ | $1.99$ | $47.076+1.752N_{f}$
$4.0$ | $5.757$ | $128.894-3.1N_{f}$ | $2.575$ | $62.576+1.102N_{f}$
$\omega$ | $C_{q}^{(\not{q},1)}$ | $C_{q}^{(\not{q},2)}$ | $C_{q}^{(\gamma,1)}$ | $C_{q}^{(\gamma,2)}$
$0.5$ | $0$ | $-25.464+2.333N_{f}$ | $0.825$ | $1.53+0.75N_{f}$
$1.0$ | $0$ | $-25.464+2.333N_{f}$ | $1.333$ | $9.599+0.185N_{f}$
$1.5$ | $0$ | $-25.464+2.333N_{f}$ | $1.742$ | $15.476-0.269N_{f}$
$2.0$ | $0$ | $-25.464+2.333N_{f}$ | $2.093$ | $20.137-0.658N_{f}$
$2.5$ | $0$ | $-25.464+2.333N_{f}$ | $2.403$ | $24.001-1.004N_{f}$
$3.0$ | $0$ | $-25.464+2.333N_{f}$ | $2.684$ | $27.292-1.316N_{f}$
$3.5$ | $0$ | $-25.464+2.333N_{f}$ | $2.942$ | $30.147-1.603N_{f}$
$4.0$ | $0$ | $-25.464+2.333N_{f}$ | $3.182$ | $32.66-1.869N_{f}$
Table 10:  The perturbative expansion coefficients of the conversion factors
$C_{m}^{(\gamma)}$, $C_{m}^{(\not{q})}$, $C_{q}^{(\gamma)}$ and
$C_{q}^{(\not{q})}$ as a function of $\omega$. Note that $C_{q}^{(\not{q})}$
is independent of $\omega$.

The scheme transformation can then be written as a product

$U_{i}^{(X)}(\mu_{1},\mu_{0},\omega_{1},\omega_{0})=C_{i}^{(X)}(\nu_{1},\mu_{1},\omega_{1})U_{i}^{\overline{\text{MS}}}(\nu_{1},\nu_{0})C_{i}^{(X)^{-1}}(\nu_{0},\mu_{0},\omega_{0})\bigg{|}_{\begin{subarray}{l}\mu_{0}=\nu_{0}\\\
\mu_{1}=\nu_{1}\end{subarray}}$ (A.48)

of the conversion factor, its inverse, and the and the $\overline{\text{MS}}$
evolution kernel $U_{i}^{\overline{\text{MS}}}(\nu_{1},\nu_{0})$. The
evolution Kernel fulfils the renormalisation group equation

$\nu^{2}\frac{d}{d\nu^{2}}U_{i}^{\overline{\text{MS}}}(\nu,\nu_{0})=-\gamma_{i}^{\overline{\text{MS}}}U_{i}^{\overline{\text{MS}}}(\nu,\nu_{0})$
(A.49)

and we expand the anomalous dimensions

$\gamma_{i}^{(X)}\equiv\sum_{k=0}^{\infty}\gamma_{i}^{(X,k)}\left(\frac{\alpha_{s}}{4\pi}\right)^{k+1}\;,$
(A.50)

where $X$ denotes the renormalisation scheme. Hence we can transform for
example a light quark masses renormalised in given scheme $X$ at different
kinematic points $(\mu_{1},\omega_{1})$ and $(\mu_{0},\omega_{0})$ via

$m^{(X)}(\mu_{1},\omega_{1})=U_{m}^{(X)}(\mu_{1},\mu_{0},\omega_{1},\omega_{0})m^{(X)}(\mu_{0},\omega_{0})\,,$
(A.51)

where the logarithms $\log\mu_{1}/\mu_{0}$ are summed using renormalisation
group improved perturbation theory. Explicitly, we find at NNLO

$\begin{split}U_{i}^{(X)}(\mu_{1},\mu_{0},\omega_{1},\omega_{0})=&\left(1+\frac{\alpha_{s}(\mu_{1})}{4\pi}J^{(X,1)}_{i}(\omega_{1})+\frac{\alpha_{s}^{2}(\mu_{1})}{(4\pi)^{2}}J^{(X,2)}_{i}(\omega_{1})\right)\left(\frac{\alpha_{s}(\mu_{0})}{\alpha_{s}(\mu_{1})}\right)^{\gamma^{(0)}_{i}/\beta_{0}}\times\\\
&\qquad\left(1-\frac{\alpha_{s}(\mu_{0})}{4\pi}J^{(X,1)}_{i}(\omega_{0})+\frac{\alpha_{s}^{2}(\mu_{0})}{(4\pi)^{2}}\left[(J^{(X,1)}_{i}(\omega_{0}))^{2}-J^{(X,2)}_{i}(\omega_{0})\right]\right)\,,\end{split}$

where the $J_{X}^{(1)}$ and $J_{X}^{(2)}$ are given by

$\begin{split}J^{(X,1)}_{i}=&\frac{\gamma_{i}^{(X,1)}}{\beta_{0}}-\frac{\beta_{1}\gamma_{i}^{(0)}}{\beta_{0}^{2}}\;,\\\
J^{(X,2)}_{i}=&\frac{1}{2}\left({J^{(X,1)}}^{2}+\frac{\gamma_{i}^{(X,2)}}{\beta_{0}}+\frac{\beta_{1}^{2}\gamma_{i}^{(0)}}{\beta_{0}^{3}}-\frac{\beta_{1}\gamma_{i}^{(X,1)}}{\beta_{0}^{2}}-\frac{\beta_{2}\gamma_{i}^{(0)}}{\beta_{0}^{2}}\right)\;.\end{split}$
(A.52)

The leading order anomalous dimensions

$\begin{split}\gamma_{q}^{(0)}&=0\;,\\\ \gamma_{m}^{(0)}&=4\;\end{split}$
(A.53)

are scheme independent. The expressions for the NLO and NNLO anomalous
dimensions are given in Table 11 for different values of $\omega$ as a
function of the number of flavours $N_{f}$.

$\omega$ | $\gamma_{m}^{(\not{q},1)}$ | $\gamma_{m}^{(\not{q},2)}$
---|---|---
$0.5$ | $93.981-3.837N_{f}$ | $2985.27-398.857N_{f}+6.256N_{f}^{2}$
$1.0$ | $74.434-2.653N_{f}$ | $1816.8-273.08N_{f}+3.623N_{f}^{2}$
$1.5$ | $58.78-1.704N_{f}$ | $948.781-176.459N_{f}+1.515N_{f}^{2}$
$2.0$ | $45.395-0.893N_{f}$ | $240.849-95.917N_{f}-0.288N_{f}^{2}$
$2.5$ | $33.558-0.175N_{f}$ | $-364.298-25.958N_{f}-1.882N_{f}^{2}$
$3.0$ | $22.868+0.473N_{f}$ | $-896.718+36.367N_{f}-3.322N_{f}^{2}$
$3.5$ | $13.075+1.066N_{f}$ | $-1374.46+92.858N_{f}-4.641N_{f}^{2}$
$4.0$ | $4.006+1.616N_{f}$ | $-1809.31+144.71N_{f}-5.862N_{f}^{2}$
$\omega$ | $\gamma_{m}^{(\gamma,1)}$ | $\gamma_{m}^{(\gamma,2)}$
$0.5$ | $103.057-4.387N_{f}$ | $3655.83-479.684N_{f}+8.367N_{f}^{2}$
$1.0$ | $89.101-3.541N_{f}$ | $2704.63-382.794N_{f}+6.487N_{f}^{2}$
$1.5$ | $77.943-2.865N_{f}$ | $1993.76-308.337N_{f}+4.984N_{f}^{2}$
$2.0$ | $68.413-2.288N_{f}$ | $1409.35-246.123N_{f}+3.701N_{f}^{2}$
$2.5$ | $59.993-1.777N_{f}$ | $905.536-191.915N_{f}+2.567N_{f}^{2}$
$3.0$ | $52.396-1.317N_{f}$ | $458.454-143.458N_{f}+1.544N_{f}^{2}$
$3.5$ | $45.441-0.895N_{f}$ | $53.89-99.383N_{f}+0.607N_{f}^{2}$
$4.0$ | $39.005-0.505N_{f}$ | $-317.4-58.784N_{f}-0.26N_{f}^{2}$
$\omega$ | $\gamma_{q}^{(\gamma,1)}$ | $\gamma_{q}^{(\gamma,2)}$
$0.5$ | $13.257-0.783N_{f}$ | $417.983-65.569N_{f}+1.741N_{f}^{2}$
$1.0$ | $7.667-0.444N_{f}$ | $200.702-36.683N_{f}+0.988N_{f}^{2}$
$1.5$ | $3.171-0.172N_{f}$ | $43.556-14.519N_{f}+0.382N_{f}^{2}$
$2.0$ | $-0.684+0.062N_{f}$ | $-79.969+3.809N_{f}-0.137N_{f}^{2}$
$2.5$ | $-4.102+0.269N_{f}$ | $-181.296+19.56N_{f}-0.597N_{f}^{2}$
$3.0$ | $-7.195+0.456N_{f}$ | $-266.635+33.428N_{f}-1.014N_{f}^{2}$
$3.5$ | $-10.033+0.628N_{f}$ | $-339.81+45.844N_{f}-1.396N_{f}^{2}$
$4.0$ | $-12.666+0.788N_{f}$ | $-403.372+57.097N_{f}-1.751N_{f}^{2}$
$\omega$ | $\gamma_{q}^{(\not{q},1)}$ | $\gamma_{q}^{(\not{q},2)}$
$0\dots 4$ | $22.333-1.333N_{f}$ | $1088.54-146.397N_{f}+3.852N_{f}^{2}$
Table 11:  The NLO and NNLO coefficients of anomalous dimensions
$\gamma_{m}^{(\not{q})}$, $\gamma_{m}^{(\gamma)}$, $\gamma_{q}^{(\gamma)}$,
$\gamma_{q}^{(\not{q})}$ as a function of $\omega$. Note that
$\gamma_{q}^{(\not{q})}$ is independent of $\omega$.

##  Appendix B Simulation Details

Our numerical work is based on RBC-UKQCD data, the lattice details can be
found in [30]. We compute the propagators using Landau gauge-fixed 2+1,
Domain-Wall (Shamir [31])/Iwasaki lattices. The values of the parameters can
be found in [7].

In a nutshell, we use two lattice spacings (we refer to them as $24^{3}$ and
$32^{3}$):

$\displaystyle a^{-1}$ $\displaystyle=$ $\displaystyle 1.785(5)\ \text{GeV}\ \
\ (24^{3})\,,$ (B.54) $\displaystyle a^{-1}$ $\displaystyle=$ $\displaystyle
2.383(9)\ \text{GeV}\ \ \ (32^{3}\,,$ (B.55)

for each lattice spacing we have three different sea quark masses, $am=0.005,\
0.010,\ 0.020$ for the $24^{3}\times 64\times 16$ lattice and $am=0.004,\
0.006,\ 0.008$ for the $32^{3}\times 64\times 16$ lattice.

We take the chiral limit on each lattice spacing using the values

$\displaystyle am_{res}$ $\displaystyle=$ $\displaystyle 0.003152(43)\ \ \ \
(24^{3}),$ (B.56) $\displaystyle am_{res}$ $\displaystyle=$ $\displaystyle
0.0006664(76)\ \ \ (32^{3}).$ (B.57)

Our values for $Z_{V}$ are [17]

$\displaystyle Z_{V}=0.71651(46)\ \ \ \ (24^{3}),$ (B.58) $\displaystyle
Z_{V}=0.74475(12)\ \ \ \ (32^{2}).$ (B.59)

##  Appendix C Extensive study for $\sigma_{m}$

In this appendix we provide our numerical results for running of the quark
mass. These results are given for
$\sigma^{(X)}_{m}(\mu,\mu_{0},\omega,\omega_{0})$ as defined in Eq. (3.41),
such that

$Z^{(X)}_{m}(\mu,\omega)=\sigma^{(X)}_{m}(\mu,\mu_{0},\omega,\omega_{0})Z^{(X)}_{m}(\mu_{0},\omega_{0})\;.$
(C.60)

###  C.1 Study of the $\omega$-dependence for fixed energy scales

In order to study the $\omega$-dependence, we first fix $\mu$ and $\mu_{0}$ to
some reasonable values, where we expect a rather good control over both the
perturbative errors and the lattice artefacts. We choose
$(\mu_{0},\mu)=(2.5\;\text{GeV},1.5\;\text{GeV})$. We then vary $\omega$, but
first we only consider the ‘diagonal’ case, ie $\omega_{0}=\omega$. We show
our results in Tables 12 and 13. The middle column shows the running itself
obtained from the lattice, while the other three columns on the right show the
ratio of the non-perberturbative running over the perturbative prediction at
Leading Order (LO), Next-to-Leading Order (NLO) and Next-to-Next-to-Leading
Order (NNLO). The errors quoted there combine an estimate of the
discretisaiton errors and the statistical one.

$(\mu_{0},\mu)=(2.5,1.5)$ | $\sigma_{m}^{(\gamma_{\mu})}$ | $\sigma_{m}/LO$ | $\sigma_{m}/NLO$ | $\sigma_{m}/NNLO$
---|---|---|---|---
$\omega=\omega_{0}\downarrow$ |  |  |  |
$0.5$ | $1.229(27)$ | $1.091(24)$ | $1.035(23)$ | $1.006(22)$
$1.0$ | $1.174(18)$ | $1.042(16)$ | $0.999(16)$ | $0.984(15)$
$1.5$ | $1.184(17)$ | $1.051(15)$ | $1.016(15)$ | $1.009(15)$
$2.0$ | $1.162(16)$ | $1.032(14)$ | $1.005(14)$ | $1.004(14)$
$2.5$ | $1.143(14)$ | $1.014(12)$ | $0.994(12)$ | $0.998(12)$
$3.0$ | $1.124(12)$ | $0.998(11)$ | $0.982(11)$ | $0.990(11)$
$3.5$ | $1.110(11)$ | $0.985(9)$ | $0.975(9)$ | $0.986(9)$
$4.0$ | $1.080(5)$ | $0.959(5)$ | $0.952(5)$ | $0.967(5)$
Table 12: Running for the quark mass, $\gamma_{\mu}$ scheme for $(\mu_{0},\mu)=(2.5\text{GeV},1.5\text{GeV})$. $(\mu_{0},\mu)=(2.5,1.5)$ | $\sigma_{m}^{(\not{q})}$ | $\sigma_{m}/LO$ | $\sigma_{m}/NLO$ | $\sigma_{m}/NNLO$
---|---|---|---|---
$\omega=\omega_{0}\downarrow$ |  |  |  |
$0.5$ | $1.194(23)$ | $1.059(21)$ | $1.012(20)$ | $0.992(20)$
$1.0$ | $1.138(18)$ | $1.010(16)$ | $0.980(16)$ | $0.995(16)$
$1.5$ | $1.138(13)$ | $1.010(11)$ | $0.991(11)$ | $0.985(11)$
$2.0$ | $1.117(13)$ | $0.992(12)$ | $0.981(12)$ | $1.000(12)$
$2.5$ | $1.096(13)$ | $0.973(11)$ | $0.970(11)$ | $0.971(11)$
$3.0$ | $1.080(13)$ | $0.959(12)$ | $0.962(12)$ | $0.984(12)$
$3.5$ | $1.068(11)$ | $0.948(9)$ | $0.956(10)$ | $0.963(10)$
$4.0$ | $1.033(12)$ | $0.917(10)$ | $0.929(10)$ | $0.952(11)$
Table 13: Same as Table 12.

We observe the non-perturbative running agrees extremely well with NNLO
predictions for all values of $\omega\leq 3$. Looking at the perturbative
convergence and stability, our data seem to favour the region $\omega\sim 2$.

###  C.2 Study of the $\omega$-dependence for larger energy ranges

Here we fix $\mu=2$ GeV and let $\mu_{0}$ vary over the full range. Again we
only consider the case $\omega_{0}=\omega$. The rows $\mu_{0}=2$ give
trivially one, but we leave the results in order to guide the eyes. We give
our results in Table 14.

| $\sigma_{m}$ | $\sigma_{m}/LO$ | $\sigma_{m}/NLO$ | $\sigma_{m}/NNLO$
---|---|---|---|---
$\mu_{0}\downarrow$ (GeV) |  |  |  |
$\bm{\omega=0.5}$ |  |  |  |
$1.0$ | $0.826(83)$ | $1.015(102)$ | $1.145(115)$ | $1.246(126)$
$1.5$ | $0.884(17)$ | $0.949(19)$ | $0.982(19)$ | $1.001(20)$
$2.0$ | $1$ | $1$ | $1$ | $1$
$2.5$ | $1.084(10)$ | $1.034(9)$ | $1.014(9)$ | $1.005(9)$
$3.0$ | $1.153(17)$ | $1.062(15)$ | $1.029(15)$ | $1.014(15)$
$3.5$ | $1.206(24)$ | $1.081(22)$ | $1.038(21)$ | $1.019(21)$
$4.0$ | $1.253(35)$ | $1.099(31)$ | $1.048(30)$ | $1.027(29)$
$\bm{\omega=1.0}$ |  |  |  |
$1.0$ | $0.740(42)$ | $0.910(51)$ | $0.999(56)$ | $1.045(59)$
$1.5$ | $0.916(13)$ | $0.984(14)$ | $1.010(15)$ | $1.021(15)$
$2.0$ | $1$ | $1$ | $1$ | $1$
$2.5$ | $1.074(7)$ | $1.025(7)$ | $1.009(7)$ | $1.004(6)$
$3.0$ | $1.133(13)$ | $1.044(12)$ | $1.018(12)$ | $1.010(12)$
$3.5$ | $1.179(21)$ | $1.057(19)$ | $1.024(18)$ | $1.014(18)$
$4.0$ | $1.220(31)$ | $1.070(27)$ | $1.030(26)$ | $1.018(26)$
$\bm{\omega=1.5}$ |  |  |  |
$1.0$ | $0.764(35)$ | $0.939(43)$ | $1.010(46)$ | $1.032(47)$
$1.5$ | $0.901(10)$ | $0.968(11)$ | $0.989(11)$ | $0.994(11)$
$2.0$ | $1$ | $1$ | $1$ | $1$
$2.5$ | $1.067(6)$ | $1.017(6)$ | $1.005(6)$ | $1.003(6)$
$3.0$ | $1.118(13)$ | $1.030(12)$ | $1.010(11)$ | $1.006(11)$
$3.5$ | $1.159(20)$ | $1.039(18)$ | $1.013(18)$ | $1.008(18)$
$4.0$ | $1.196(31)$ | $1.049(27)$ | $1.018(26)$ | $1.012(26)$
$\bm{\omega=2.0}$ |  |  |  |
$1.0$ | $0.825(34)$ | $1.014(42)$ | $1.074(45)$ | $1.078(45)$
$1.5$ | $0.910(10)$ | $0.978(11)$ | $0.994(11)$ | $0.995(11)$
$2.0$ | $1$ | $1$ | $1$ | $1$
$2.5$ | $1.059(6)$ | $1.010(5)$ | $1.001(5)$ | $1.000(5)$
$3.0$ | $1.106(12)$ | $1.019(11)$ | $1.003(11)$ | $1.002(11)$
$3.5$ | $1.137(17)$ | $1.020(15)$ | $0.999(15)$ | $0.998(15)$
$4.0$ | $1.160(21)$ | $1.018(18)$ | $0.994(18)$ | $0.993(18)$
Table 14: Running for the quark mass, $X=\gamma_{\mu},\mu=2\;\text{GeV}$. We
only consider $\omega=\omega_{0}$ and let $\mu_{0}$ vary between 1 and 4 GeV.

###  C.3 Study of the running in the non-degenerate $\omega$ case.

Here we fix again both energy scales $(\mu_{0},\mu)=(2.5\;\text{
GeV},1.5\;\text{ GeV})$, and we allow $\omega\neq\omega_{0}$. Our results are
shown in Tables 15 16, for $X=\gamma_{\mu}$ and in Tables 17 and 18 for
$X=q\\!\\!\\!/$.

$(\mu_{0},\mu)=(2.5,1.5)$ | $\sigma_{m}$ | $\sigma_{m}/LO$ | $\sigma_{m}/NLO$ | $\sigma_{m}/NNLO$
---|---|---|---|---
$\omega\downarrow$ |  |  |  |
$\bm{\omega_{0}=0.5}$ |  |  |  |
$0.5$ | $1.229(27)$ | $1.091(24)$ | $1.035(23)$ | $1.006(22)$
$1.0$ | $1.308(29)$ | $1.161(26)$ | $1.067(23)$ | $1.023(23)$
$1.5$ | $1.373(30)$ | $1.219(27)$ | $1.094(24)$ | $1.038(23)$
$2.0$ | $1.425(32)$ | $1.265(28)$ | $1.113(25)$ | $1.048(24)$
$2.5$ | $1.473(34)$ | $1.307(30)$ | $1.131(26)$ | $1.057(24)$
$3.0$ | $1.515(35)$ | $1.345(31)$ | $1.145(27)$ | $1.065(25)$
$3.5$ | $1.551(36)$ | $1.377(32)$ | $1.156(27)$ | $1.069(25)$
$4.0$ | $1.568(33)$ | $1.391(29)$ | $1.154(24)$ | $1.062(22)$
$\bm{\omega_{0}=1.0}$ |  |  |  |
$0.5$ | $1.103(18)$ | $0.979(16)$ | $0.969(16)$ | $0.967(16)$
$1.0$ | $1.174(18)$ | $1.042(16)$ | $0.999(16)$ | $0.984(15)$
$1.5$ | $1.232(19)$ | $1.094(17)$ | $1.024(16)$ | $0.998(15)$
$2.0$ | $1.279(20)$ | $1.136(18)$ | $1.042(16)$ | $1.008(16)$
$2.5$ | $1.322(21)$ | $1.174(19)$ | $1.059(17)$ | $1.017(16)$
$3.0$ | $1.360(22)$ | $1.208(20)$ | $1.073(18)$ | $1.024(17)$
$3.5$ | $1.392(22)$ | $1.236(20)$ | $1.083(17)$ | $1.028(17)$
$4.0$ | $1.407(22)$ | $1.249(20)$ | $1.080(17)$ | $1.021(16)$
$\bm{\omega_{0}=1.5}$ |  |  |  |
$0.5$ | $1.059(17)$ | $0.940(15)$ | $0.961(15)$ | $0.977(16)$
$1.0$ | $1.128(16)$ | $1.001(14)$ | $0.992(14)$ | $0.994(14)$
$1.5$ | $1.184(17)$ | $1.051(15)$ | $1.016(15)$ | $1.009(15)$
$2.0$ | $1.229(18)$ | $1.091(16)$ | $1.035(15)$ | $1.019(15)$
$2.5$ | $1.271(19)$ | $1.128(17)$ | $1.051(16)$ | $1.028(16)$
$3.0$ | $1.307(20)$ | $1.161(18)$ | $1.065(17)$ | $1.035(16)$
$3.5$ | $1.338(20)$ | $1.188(18)$ | $1.075(16)$ | $1.040(16)$
$4.0$ | $1.353(13)$ | $1.201(12)$ | $1.073(10)$ | $1.033(10)$
$\bm{\omega_{0}=2.0}$ |  |  |  |
$0.5$ | $1.002(15)$ | $0.890(13)$ | $0.934(14)$ | $0.963(14)$
$1.0$ | $1.067(14)$ | $0.947(13)$ | $0.964(13)$ | $0.980(13)$
$1.5$ | $1.119(15)$ | $0.993(14)$ | $0.987(13)$ | $0.994(14)$
$2.0$ | $1.162(16)$ | $1.032(14)$ | $1.005(14)$ | $1.004(14)$
$2.5$ | $1.201(17)$ | $1.066(15)$ | $1.021(14)$ | $1.013(14)$
$3.0$ | $1.236(18)$ | $1.097(16)$ | $1.034(15)$ | $1.020(15)$
$3.5$ | $1.265(18)$ | $1.123(16)$ | $1.044(15)$ | $1.024(15)$
$4.0$ | $1.279(12)$ | $1.136(11)$ | $1.043(10)$ | $1.018(10)$
Table 15: Running for the quark mass, $\gamma_{\mu}$-projector for
$(\mu_{0},\mu)=(2.5\text{GeV},1.5\text{GeV})$.

$(\mu_{0},\mu)=(2.5,1.5)$ | $\sigma_{m}$ | $\sigma_{m}/LO$ | $\sigma_{m}/NLO$ | $\sigma_{m}/NLO$
---|---|---|---|---
$\omega\downarrow$ |  |  |  |
$\bm{\omega_{0}=2.5}$ |  |  |  |
$0.5$ | $0.952(12)$ | $0.845(10)$ | $0.907(11)$ | $0.947(12)$
$1.0$ | $1.013(11)$ | $0.899(10)$ | $0.937(10)$ | $0.964(10)$
$1.5$ | $1.064(12)$ | $0.944(11)$ | $0.960(11)$ | $0.978(11)$
$2.0$ | $1.105(13)$ | $0.981(11)$ | $0.978(11)$ | $0.988(11)$
$2.5$ | $1.143(14)$ | $1.014(12)$ | $0.994(12)$ | $0.998(12)$
$3.0$ | $1.176(15)$ | $1.044(13)$ | $1.007(13)$ | $1.005(13)$
$3.5$ | $1.203(14)$ | $1.068(13)$ | $1.016(12)$ | $1.008(12)$
$4.0$ | $1.217(9)$ | $1.080(8)$ | $1.015(8)$ | $1.003(8)$
$\bm{\omega_{0}=3.0}$ |  |  |  |
$0.5$ | $0.911(11)$ | $0.809(10)$ | $0.887(10)$ | $0.935(11)$
$1.0$ | $0.970(10)$ | $0.861(9)$ | $0.915(9)$ | $0.952(10)$
$1.5$ | $1.018(11)$ | $0.903(9)$ | $0.938(10)$ | $0.965(10)$
$2.0$ | $1.057(11)$ | $0.938(10)$ | $0.955(10)$ | $0.975(10)$
$2.5$ | $1.092(12)$ | $0.970(10)$ | $0.970(10)$ | $0.983(10)$
$3.0$ | $1.124(12)$ | $0.998(11)$ | $0.982(11)$ | $0.990(11)$
$3.5$ | $1.150(12)$ | $1.021(11)$ | $0.992(11)$ | $0.995(11)$
$4.0$ | $1.163(7)$ | $1.032(6)$ | $0.990(6)$ | $0.988(6)$
$\bm{\omega_{0}=3.5}$ |  |  |  |
$0.5$ | $0.879(10)$ | $0.780(9)$ | $0.872(10)$ | $0.927(11)$
$1.0$ | $0.935(9)$ | $0.830(8)$ | $0.899(9)$ | $0.943(9)$
$1.5$ | $0.982(9)$ | $0.871(8)$ | $0.922(9)$ | $0.957(9)$
$2.0$ | $1.019(10)$ | $0.905(8)$ | $0.938(9)$ | $0.966(9)$
$2.5$ | $1.054(10)$ | $0.935(9)$ | $0.953(9)$ | $0.975(9)$
$3.0$ | $1.084(11)$ | $0.962(10)$ | $0.965(10)$ | $0.982(10)$
$3.5$ | $1.110(11)$ | $0.985(9)$ | $0.975(9)$ | $0.986(9)$
$4.0$ | $1.122(5)$ | $0.996(4)$ | $0.973(4)$ | $0.980(4)$
$\bm{\omega_{0}=4.0}$ |  |  |  |
$0.5$ | $0.846(10)$ | $0.751(9)$ | $0.853(10)$ | $0.915(11)$
$1.0$ | $0.901(9)$ | $0.799(8)$ | $0.880(9)$ | $0.931(9)$
$1.5$ | $0.945(9)$ | $0.839(8)$ | $0.902(9)$ | $0.944(9)$
$2.0$ | $0.982(10)$ | $0.871(9)$ | $0.918(9)$ | $0.954(9)$
$2.5$ | $1.014(10)$ | $0.900(9)$ | $0.932(9)$ | $0.962(10)$
$3.0$ | $1.044(11)$ | $0.927(10)$ | $0.945(10)$ | $0.969(10)$
$3.5$ | $1.068(10)$ | $0.948(9)$ | $0.954(9)$ | $0.973(10)$
$4.0$ | $1.080(5)$ | $0.959(5)$ | $0.952(5)$ | $0.967(5)$

Table 16: Running for the quark mass, $\gamma_{\mu}$-projector for
$(\mu_{0},\mu)=(2.5\text{GeV},1.5\text{GeV})$ (cont.).

.

$(\mu_{0},\mu)=(2.5,1.5)$ | $\sigma_{m}$ | $\sigma_{m}/LO$ | $\sigma_{m}/NLO$ | $\sigma_{m}/NNLO$
---|---|---|---|---
$\omega\downarrow$ |  |  |  |
$\bm{\omega_{0}=0.5}$ |  |  |  |
$0.5$ | $1.194(23)$ | $1.059(21)$ | $1.012(20)$ | $0.992(20)$
$1.0$ | $1.294(26)$ | $1.148(23)$ | $1.052(21)$ | $0.986(20)$
$1.5$ | $1.378(29)$ | $1.223(26)$ | $1.085(23)$ | $1.047(22)$
$2.0$ | $1.447(33)$ | $1.284(30)$ | $1.109(26)$ | $1.032(24)$
$2.5$ | $1.513(38)$ | $1.343(34)$ | $1.133(29)$ | $1.081(27)$
$3.0$ | $1.574(42)$ | $1.397(37)$ | $1.155(30)$ | $1.069(28)$
$3.5$ | $1.631(41)$ | $1.448(36)$ | $1.175(29)$ | $1.112(28)$
$4.0$ | $1.661(40)$ | $1.474(35)$ | $1.177(28)$ | $1.084(26)$
$\bm{\omega_{0}=1.0}$ |  |  |  |
$0.5$ | $1.050(17)$ | $0.932(15)$ | $0.943(15)$ | $1.002(16)$
$1.0$ | $1.138(18)$ | $1.010(16)$ | $0.980(16)$ | $0.995(16)$
$1.5$ | $1.212(20)$ | $1.076(17)$ | $1.010(16)$ | $1.057(17)$
$2.0$ | $1.274(22)$ | $1.130(20)$ | $1.033(18)$ | $1.043(18)$
$2.5$ | $1.332(26)$ | $1.182(23)$ | $1.056(21)$ | $1.092(22)$
$3.0$ | $1.386(29)$ | $1.230(26)$ | $1.076(22)$ | $1.081(22)$
$3.5$ | $1.436(28)$ | $1.275(24)$ | $1.095(21)$ | $1.123(22)$
$4.0$ | $1.461(33)$ | $1.297(29)$ | $1.096(24)$ | $1.095(24)$
$\bm{\omega_{0}=1.5}$ |  |  |  |
$0.5$ | $0.985(11)$ | $0.874(10)$ | $0.924(10)$ | $0.933(10)$
$1.0$ | $1.068(10)$ | $0.948(8)$ | $0.960(9)$ | $0.927(8)$
$1.5$ | $1.138(13)$ | $1.010(11)$ | $0.991(11)$ | $0.985(11)$
$2.0$ | $1.196(16)$ | $1.061(15)$ | $1.013(14)$ | $0.972(13)$
$2.5$ | $1.250(21)$ | $1.110(19)$ | $1.035(17)$ | $1.018(17)$
$3.0$ | $1.301(24)$ | $1.155(21)$ | $1.055(19)$ | $1.007(19)$
$3.5$ | $1.348(21)$ | $1.197(19)$ | $1.074(17)$ | $1.047(17)$
$4.0$ | $1.372(19)$ | $1.218(17)$ | $1.075(15)$ | $1.020(14)$
$\bm{\omega_{0}=2.0}$ |  |  |  |
$0.5$ | $0.921(9)$ | $0.818(8)$ | $0.896(9)$ | $0.961(9)$
$1.0$ | $0.998(9)$ | $0.886(8)$ | $0.930(8)$ | $0.954(8)$
$1.5$ | $1.064(10)$ | $0.944(9)$ | $0.960(9)$ | $1.014(10)$
$2.0$ | $1.117(13)$ | $0.992(12)$ | $0.981(12)$ | $1.000(12)$
$2.5$ | $1.168(17)$ | $1.037(15)$ | $1.002(15)$ | $1.047(16)$
$3.0$ | $1.216(20)$ | $1.079(18)$ | $1.022(17)$ | $1.036(17)$
$3.5$ | $1.260(18)$ | $1.118(16)$ | $1.040(15)$ | $1.077(15)$
$4.0$ | $1.283(19)$ | $1.139(17)$ | $1.041(16)$ | $1.051(16)$

Table 17: Same as Table 16 for the $q\\!\\!\\!/$-projector. Again we have
$(\mu_{0},\mu)=(2.5\text{GeV},1.5\text{GeV})$.

.

$(\mu_{0},\mu)=(2.5,1.5)$ | $\sigma_{m}$ | $\sigma_{m}/LO$ | $\sigma_{m}/NLO$ | $\sigma_{m}/NNLO$
---|---|---|---|---
$\omega\downarrow$ |  |  |  |
$\bm{\omega_{0}=2.5}$ |  |  |  |
$0.5$ | $0.862(7)$ | $0.765(6)$ | $0.864(7)$ | $0.889(7)$
$1.0$ | $0.935(7)$ | $0.830(6)$ | $0.899(7)$ | $0.884(7)$
$1.5$ | $0.997(8)$ | $0.885(7)$ | $0.927(7)$ | $0.939(7)$
$2.0$ | $1.048(10)$ | $0.930(9)$ | $0.949(9)$ | $0.928(9)$
$2.5$ | $1.096(13)$ | $0.973(11)$ | $0.970(11)$ | $0.971(11)$
$3.0$ | $1.141(15)$ | $1.013(13)$ | $0.989(13)$ | $0.961(13)$
$3.5$ | $1.181(12)$ | $1.048(11)$ | $1.005(11)$ | $0.998(10)$
$4.0$ | $1.203(21)$ | $1.068(19)$ | $1.007(18)$ | $0.974(17)$
$\bm{\omega_{0}=3.0}$ |  |  |  |
$0.5$ | $0.818(6)$ | $0.726(5)$ | $0.842(6)$ | $0.912(7)$
$1.0$ | $0.887(7)$ | $0.787(6)$ | $0.875(7)$ | $0.906(7)$
$1.5$ | $0.945(7)$ | $0.839(6)$ | $0.903(6)$ | $0.962(7)$
$2.0$ | $0.993(8)$ | $0.881(7)$ | $0.923(8)$ | $0.949(8)$
$2.5$ | $1.038(11)$ | $0.921(10)$ | $0.943(10)$ | $0.994(11)$
$3.0$ | $1.080(13)$ | $0.959(12)$ | $0.962(12)$ | $0.984(12)$
$3.5$ | $1.119(11)$ | $0.994(10)$ | $0.978(10)$ | $1.022(10)$
$4.0$ | $1.139(20)$ | $1.011(18)$ | $0.979(17)$ | $0.996(17)$
$\bm{\omega_{0}=3.5}$ |  |  |  |
$0.5$ | $0.780(6)$ | $0.692(5)$ | $0.822(6)$ | $0.859(7)$
$1.0$ | $0.845(6)$ | $0.750(5)$ | $0.855(6)$ | $0.853(6)$
$1.5$ | $0.901(6)$ | $0.800(5)$ | $0.882(6)$ | $0.906(6)$
$2.0$ | $0.947(8)$ | $0.840(7)$ | $0.902(7)$ | $0.894(7)$
$2.5$ | $0.990(11)$ | $0.879(10)$ | $0.921(10)$ | $0.936(10)$
$3.0$ | $1.030(13)$ | $0.914(12)$ | $0.939(12)$ | $0.926(12)$
$3.5$ | $1.068(11)$ | $0.948(9)$ | $0.956(10)$ | $0.963(10)$
$4.0$ | $1.086(17)$ | $0.964(15)$ | $0.957(15)$ | $0.939(14)$
$\bm{\omega_{0}=4.0}$ |  |  |  |
$0.5$ | $0.741(8)$ | $0.658(7)$ | $0.799(9)$ | $0.871(9)$
$1.0$ | $0.804(8)$ | $0.713(7)$ | $0.830(8)$ | $0.865(8)$
$1.5$ | $0.856(9)$ | $0.760(8)$ | $0.856(9)$ | $0.919(10)$
$2.0$ | $0.900(12)$ | $0.799(10)$ | $0.876(11)$ | $0.907(12)$
$2.5$ | $0.941(15)$ | $0.835(13)$ | $0.894(14)$ | $0.949(15)$
$3.0$ | $0.979(17)$ | $0.869(15)$ | $0.912(16)$ | $0.940(16)$
$3.5$ | $1.014(15)$ | $0.900(13)$ | $0.928(13)$ | $0.977(14)$
$4.0$ | $1.033(12)$ | $0.917(10)$ | $0.929(10)$ | $0.952(11)$

Table 18: Running for the $q\\!\\!\\!/$-projector (cont).

.

##  Appendix D Continuum extrapolation

As an example of continuum extrapolations, we show
$\Sigma_{m}^{(\gamma_{\mu})}(\mu,\mu_{0},\omega,\omega_{0})$ in Figs. 17 and
18. The magenta error bar is a systematic error, obtained by adding half the
difference beween the extrapolated value,
$\sigma_{m}^{(\gamma_{\mu})}(\mu,\mu_{0},\omega,\omega_{0})$, and the value
obtained on the finest lattice.

|
---|---
|
Figure 17: Continuum extrapolations for $\Sigma_{m}$. We choose $(X,\mu,\mu_{0})=(\gamma_{\mu},2\mbox{ GeV},3\mbox{ GeV})$ and show our results for the different values of $\omega=\omega_{0}$ from 0.5 (top left corner) to 2.0 (bottom right corner). We included an estimation of the error due to the continuum extrapolation (error bar in magenta, see text).  |
---|---
|
Figure 18: Same as Fig. 17 for $\omega=\omega_{0}=2.5,3.0,\ldots,4.0$.

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