Lattice Parton Collaboration ($\rm{\bf LPC}$)

# Distribution Amplitudes of $K^{*}$ and $\phi$ at Physical Pion Mass from
Lattice QCD

Jun Hua  INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology,
Key Laboratory for Particle Astrophysics and Cosmology (MOE), School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Min-Huan Chu  INPAC, Shanghai Key Laboratory for Particle Physics and
Cosmology, Key Laboratory for Particle Astrophysics and Cosmology (MOE),
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China  Shanghai Key Laboratory for Particle Physics and Cosmology, Key
Laboratory for Particle Astrophysics and Cosmology (MOE), Tsung-Dao Lee
Institute, Shanghai Jiao Tong University, Shanghai 200240, China  Peng Sun
Corresponding author<EMAIL_ADDRESS>Department of Physics and Institute of
Theoretical Physics, Nanjing Normal University, Nanjing, Jiangsu, 210023,
China  Wei Wang  Corresponding author<EMAIL_ADDRESS>INPAC, Shanghai
Key Laboratory for Particle Physics and Cosmology, Key Laboratory for Particle
Astrophysics and Cosmology (MOE), School of Physics and Astronomy, Shanghai
Jiao Tong University, Shanghai 200240, China  Ji Xu  INPAC, Shanghai Key
Laboratory for Particle Physics and Cosmology, Key Laboratory for Particle
Astrophysics and Cosmology (MOE), School of Physics and Astronomy, Shanghai
Jiao Tong University, Shanghai 200240, China  School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China  Yi-Bo
Yang  CAS Key Laboratory of Theoretical Physics, Institute of Theoretical
Physics, Chinese Academy of Sciences, Beijing 100190, China  School of
Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced
Study, UCAS, Hangzhou 310024, China  International Centre for Theoretical
Physics Asia-Pacific, Beijing/Hangzhou, China  Jian-Hui Zhang  Center of
Advanced Quantum Studies, Department of Physics, Beijing Normal University,
Beijing 100875, China  Qi-An Zhang  Shanghai Key Laboratory for Particle
Physics and Cosmology, Key Laboratory for Particle Astrophysics and Cosmology
(MOE), Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai
200240, China

###### Abstract

We present the first lattice QCD calculation of the distribution amplitudes of
longitudinally and transversely polarized vector mesons $K^{*}$ and $\phi$
using large momentum effective theory. We use the clover fermion action on
three ensembles with 2+1+1 flavors of highly improved staggered quarks (HISQ)
action, generated by MILC collaboration, at physical pion mass and {0.06,
0.09, 0.12} fm lattice spacings, and choose three different hadron momenta
$P_{z}=\\{1.29,1.72,2.15\\}$ GeV. The resulting lattice matrix elements are
nonperturbatively renormalized in a hybrid scheme proposed recently. An
extrapolation to the continuum and infinite momentum limit is carried out. We
find that while the longitudinal distribution amplitudes tend to be close to
the asymptotic form, the transverse ones deviate rather significantly from the
asymptotic form. Our final results provide crucial ab initio theory inputs for
analyzing pertinent exclusive processes.

Introduction.–Searching for new physics beyond the standard model (SM) is a
primary goal of particle physics nowadays. A unique possibility of doing so is
to investigate flavor-changing neutral current processes which are highly
suppressed in the SM. Some prominent examples of such processes include $B\to
K^{*}\ell^{+}\ell^{-}$ and $B_{s}\to\phi\ell^{+}\ell^{-}$ decays. Recent
experimental analyses by Belle and LHCb collaborations Wei:2009zv ;
Aaij:2015esa ; Aaij:2015oid ; Aaij:2017vbb ; Aaij:2020nrf  have revealed
notable tensions between the SM predictions of such processes and data, and
attracted quite considerable theoretical interests (see Refs.
Capdevila:2017bsm ; Buttazzo:2017ixm ; Cerri:2018ypt  and many references
therein). Various new physics interpretations have been proposed to resolve
such tensions, but to firmly establish their existence requires an accurate
and reliable theoretical understanding of the dynamics of weak decays.

In the low recoil region (high $q^{2}$), the $B\to K^{*}$ and $B_{s}\to\phi$
form factors can be directly calculated on the lattice (see for instance Refs.
Horgan:2013hoa ; Horgan:2013pva ), however these decays at large recoil are
also of experimental interests, and for instance the $P^{\prime}_{5}$ anomaly
has attracted many theoretical and experimental attentions Descotes-
Genon:2012isb ; LHCb:2015svh . In the latter kinematics region, decay
amplitudes are split into short-distance hard kernels and long-distance
universal inputs. The universal inputs that enter include the light-cone
distribution amplitudes (LCDAs) of the vector mesons $K^{*},\phi$ which, to
the leading-twist accuracy, specify the longitudinal momentum distribution
amongst the valence quark and antiquark in the meson. While the hard
scattering kernel is perturbatively calculable, the LCDAs can only be
extracted from nonperturbative methods or from fits to relevant data. A
reliable knowledge of LCDAs is essential in making predictions on physical
observables, and in particular the transition form factors at large recoil can
be typically affected by ${\cal O}(10\%)$ by the non-asymptotic terms of LCDAs
in light-cone sum rules approach Ball:1997rj ; Ball:2004rg . To date most of
the available analyses have made use of estimates based on QCD sum rules
Ball:2007zt  or Dyson-Schwinger equation Gao:2014bca , but a first-principle
description of LCDAs for the vector $(K^{*},\phi)$ meson is still missing.

Lattice QCD provides an ideal ab initio tool to access nonperturbative
quantities in strong interaction. Though some lowest moments of the $\rho$
LCDA have been studied in Ref. Braun:2016wnx , a direct calculation of the
entire distribution has not been feasible until the proposal of large momentum
effective theory (LaMET) Ji:2013dva ; Ji:2014gla  recently. This is realized
by simulating on the lattice appropriately chosen equal-time correlations, and
then convert them to the latter through a perturbative matching. Since LaMET
was proposed, a lot of progress has been achieved in calculating various
parton distribution functions Ji:2020ect ; Cichy:2018mum  (and many references
therein) as well as distribution amplitudes for light pseudoscalar mesons
Zhang:2017bzy ; Chen:2017gck ; Zhang:2020gaj . Other variants have also been
explored in Refs. Ma:2014jla ; Ma:2017pxb ; Radyushkin:2017cyf .

In this Letter, we present the first lattice calculation of LCDAs for vector
mesons $K^{*},\phi$ in LaMET with the clover fermion action, on three
ensembles with 2+1+1 flavors of highly improved staggered quarks (HISQ) action
Follana:2006rc , generated by MILC collaboration Bazavov:2012xda , at physical
pion mass and 0.06, 0.09 and 0.12 fm lattice spacings. To improve the signal-
to-noise ratio of simulation, we take the smearing transformation of
hyperubi(HYP) fat link Hasenfratz:2001hp , the other simulation setup is given
in Table I. A hybrid renormalization scheme Ji:2020brr  is used to renormalize
bare quantities, after which an extrapolation is taken to the continuum limit
as well as to the infinite momentum limit based on data at three hadron
momenta, $P_{z}=\\{1.29,1.72,2.15\\}$ GeV. It should be noticed that, a
momentum boost close to or larger than the inverse lattice spacing may
introduce uncontrolled discretization effects, while the dispersion relation
is satisfied with the ${\cal O}(a^{2})$ corrections as shown in the
supplemental material supplementary . In the calculation we neglect the strong
decays of $K^{*},\phi$ due to their narrow decay widths. The finite width
corrections should be solved with a proper finite-volume analysis, which is
beyond the scope of this work. Our final results indicate that, while the
longitudinal LCDAs are close to the asymptotic form, the transverse ones
deviate considerably from the asymptotic form.

Table 1: Information on the simulation setup. The light and strange quark mass(both valence and sea quark) of the clover action are tuned such that $m_{\pi}$=140 MeV and $m_{\eta_{s}}$=670 MeV. Ensemble |  $a$(fm) | $L^{3}\times T$ | $c_{\mathrm{SW}}$ | $m_{u/d}$ | $m_{s}$
---|---|---|---|---|---
a12m130 | 0.12 | 48$\times$ 64 | 1.05088 | -0.0785 | -0.0191
a09m130 | 0.09 | 64$\times$ 96 | 1.04239 | -0.0580 | -0.0174
a06m130 | 0.06 | 96$\times$192 | 1.03493 | -0.0439 | -0.0191

LCDAs from LaMET.–The leading-twist LCDAs for longitudinally and transversely
polarized vector mesons, $\Phi_{V,L}$ and $\Phi_{V,T}$, are defined as follows
Ali:1993vd :

$\displaystyle\int d\xi^{-}e^{-ixp^{+}\xi^{-}}\langle
0|\bar{\psi}_{1}(0)n\\!\\!\\!/_{+}U(0,\xi^{-})\psi_{2}(\xi^{-})|V\rangle$
$\displaystyle=f_{V}n_{+}\cdot\epsilon\Phi_{V,L}(x),$ (1) $\displaystyle\int
d\xi^{-}e^{-ixp^{+}\xi^{-}}\langle
0|\bar{\psi}_{1}(0)\sigma^{+{\mu_{\perp}}}U(0,\xi^{-})\psi_{2}(\xi^{-})|V\rangle$
$\displaystyle=f_{V}^{T}[\epsilon^{+}p^{\mu_{\perp}}-\epsilon^{{\mu_{\perp}}}p^{+}]\Phi_{V,T}(x),$
(2)

where $U(0,\xi^{-})=P{\rm exp}\big{[}ig_{s}\int_{\xi_{-}}^{0}ds{n}_{+}\cdot
A(s{n}_{+})\big{]}$ is the gauge-link defined along the minus lightcone
direction, $\epsilon$ is the polarization vector of the vector meson, and
${n}_{+}$ is the unit vector along the plus lightcone direction. $f_{V}$ and
$f_{V}^{T}$ are the decay constants defined by the local vector and tensor
current, respectively. Here for $K^{*}$, $\psi_{1}$ denotes the strange quark
field and $\psi_{2}$ is the light $u/d$ quark. For the $\phi$ meson, both
$\psi_{1,2}$ are strange quark fields.

According to LaMET, the above LCDAs can be obtained by first calculating the
following bare equal-time correlations on the lattice

$\displaystyle\langle
0|\bar{\psi}_{1}(0)\gamma^{t}U(0,z\hat{z})\psi_{2}(z\hat{z})|V\rangle$
$\displaystyle=H_{V,L}(z)\epsilon^{t}f_{V},$ (3) $\displaystyle\langle
0|\bar{\psi}_{1}(0)\sigma_{\nu\rho}U(0,z\hat{z})\psi_{2}(z\hat{z})|V\rangle$
$\displaystyle=H_{V,T}(z)f_{V}^{T}[{\epsilon}_{\nu}p_{\rho}-{\epsilon}_{\rho}p_{\nu}],$

where the Lorentz indices in the second line are chosen as
$\\{\nu,\rho\\}={z,y}$, and the gauge-link $U(0,z\hat{z})$ is along the $z$
direction. The quantities $H_{V,\\{L,T\\}}(z)$ can be renormalized
nonperturbatively in an appropriate scheme Ji:2020brr ; Ishikawa:2016znu ;
Chen:2016fxx ; Green:2017xeu ; Stewart:2017tvs ; Alexandrou:2017huk ;
Braun:2018brg . Here we choose the hybrid scheme Ji:2020brr  proposed recently
which has the advantage that the renormalization factor does not introduce
extra nonperturbative effects at large $z$ which distort the IR property of
the bare correlations. This scheme works as follows: At $|z|\leq z_{S}$ where
$z_{S}$ is within the region where the leading-twist approximation is valid,
we can choose the RI/MOM scheme Stewart:2017tvs  to avoid certain
discretization effects (alternative choices include, e.g., the ratio
Radyushkin:2017cyf  scheme), while for $|z|>z_{S}$ one applies the gauge-link
mass subtraction scheme

$\displaystyle
H_{V}^{R}(z,a,P_{z})=\frac{H_{V}(z,a,P_{z})}{Z(z,a)}\theta(z_{S}-|z|)$
$\displaystyle\hskip 10.00002pt+H_{V}(z,a,P_{z})e^{-\delta
m(\tilde{\mu})z}Z_{\rm hybrid}(z_{S},a)\theta(|z|-z_{S}),$ (4)

where the superscript $R$ denotes the renormalized quantity, $\tilde{\mu}$
denotes the intrinsic scale dependence of the gauge-link including both UV and
IR. We have chosen $Z(z,a)$ as the RI/MOM renormalization factor computed from

$\displaystyle Z(z,a)$ $\displaystyle=\frac{1}{12}{\rm
Tr}\Big{[}\big{<}S(p)\big{>}^{-1}\times\big{<}S(p|z)\big{>}\gamma_{z}\gamma_{5}$
$\displaystyle\times\prod_{n}U_{z}(n\hat{z})S(p|0)\big{<}S(p)\big{>}^{-1}\gamma_{z}\gamma_{5}\Big{]}_{p^{2}=-\mu_{R}^{2},\atop
p_{z}=0}.$ (5)

The $Z_{\rm hybrid}$ denotes the endpoint renormalization constant which can
be determined by imposing a continuity condition at $z=z_{\rm S}$,

$\displaystyle Z_{\rm hybrid}(z_{\rm S},a)=e^{\delta m(\tilde{\mu})z_{\rm
S}}/{Z(z_{\rm S},a)}\,.$ (6)

The mass counterterm $\delta m(\tilde{\mu})$ can be extracted from the RI/MOM
renormalization factor Ji:2020brr . The $z_{S}$ are chosen as $0.24fm$ and
$0.36fm$ within perturbative region, and their difference is treated as a
systematic error.

By Fourier transforming $H^{R}_{V,\\{L,T\\}}$ to momentum space, we then
obtain the quasi-DAs

$\displaystyle\tilde{\Phi}_{V,\\{L,T\\}}(y,P_{z})$ $\displaystyle=\int
dze^{-iyP_{z}z}H^{R}_{V,\\{L,T\\}}(z,P_{z}),$ (7)

where the continuum limit has been taken. It can be factorized into the LCDAs
through the factorization theorem Ji:2015qla :

$\displaystyle\tilde{\Phi}_{V,\\{L,T\\}}(y,P_{z},\mu_{R})$
$\displaystyle=\int_{0}^{1}dx\,C_{V,\\{L,T\\}}(x,y,P_{z},\mu_{R},\mu)\Phi_{V,\\{L,T\\}}(x,\mu),$
(8)

where the matching kernel $C_{V,\\{L,T\\}}$ was derived first in the
transverse momentum cutoff scheme in Ref. Xu:2018mpf  and then in the RI/MOM
scheme in Ref. Liu:2018tox . The $\mu$ and $\mu_{R}$ reflect the generic
renormalization scale dependence of LCDAs and quasi-DAs. The matching formula
and more details of the hybrid scheme can be found in the supplemental
material supplementary .

Numerical setup.–On the lattice, one directly calculates the two-point
correlation function defined as:

$\displaystyle C_{2}^{m}(z,\vec{P},t)$ $\displaystyle=\int
d^{3}ye^{-i\vec{P}\cdot\vec{y}}$ $\displaystyle\times\langle
0|\bar{\psi}_{1}(\vec{y},t)\Gamma_{1}U(\vec{y},\vec{y}+z\hat{z})\psi_{2}(\vec{y}+z\hat{z},t)$
$\displaystyle\times\bar{\psi}_{2}(0,0)\Gamma_{2}\psi_{1}(0,0)|0\rangle,$ (9)

where the longitudinal polarization case ($m=L$) has $\Gamma_{1}=\gamma_{t}$,
and $\Gamma_{2}=\gamma_{z}$, and the transverse polarization case ($m=T$) has
$\Gamma_{1}=\sigma_{zy}$, and $\Gamma_{2}=\gamma_{x}/\gamma_{y}$. Then the
quasi-DAs can be extracted from the following parameterization:

$\displaystyle\frac{C^{m}_{2}(z,\vec{P},t)}{C^{m}_{2}(z=0,\vec{P},t)}=\frac{H_{V,m}^{b}(z)(1+c_{m}(z)e^{-\Delta
Et})}{(1+c_{m}(0)e^{-\Delta Et})},$ (10)

where $c_{m}(z)$ and $\Delta E$ are free parameters accounting for the excited
state contaminations, and $H_{V,m}^{b}(z)$ is the bare matrix elements for
quasi-DA. When $t$ is large enough, the excited state contaminations
parameterized by $c_{m}(z)$ and $\Delta E$ are suppressed exponentially, and
the ratio defined in Eq. (10) approaches the ground state matrix element
$H_{V,m}^{b}(z)$. Based on the comparison between the joint two-state fit and
constant fit shown in the supplemental material supplementary , we choose to
use the constant fit in the range of $t\geq 0.54~{}$fm to provide a
conservative error estimate in the following calculation.

The numerical simulation is based on three ensembles with 2+1+1 flavors of
HISQ Follana:2006rc  at physical pion mass with 0.06, 0.09 and 0.12 fm lattice
spacings. The momentum smeared grid source Yang:2015zja  with the source
positions $(x_{0}+j_{x}L/2,y_{0}+j_{y}L/2,z_{0}+j_{z}L/2)$ are used in the
calculation, where $(x_{0},y_{0},z_{0})$ is a random position and
$j_{x,y,z}=0/1$. It allows us to obtain the even momenta in unit of $2\pi/L$
with $\sim 8$ times of the statistics. We also repeat the calculation at 8, 6,
4 time slices and fold the data in the normal and reversed time directions,
which is equivalent to having $570\times 8\times 8\times 2$, $730\times
8\times 6\times 2$ and $970\times 8\times 4\times 2$ measurements at three
ensembles at $a=$0.06, 0.09 and 0.12 fm, respectively. We have further
reversed the $\hat{z}$ direction in Eq. (9) to double the statistics. Based on
the numerical results, we confirmed that the dispersion relation can be
satisfied for all the cases up to the ${\cal O}(a^{2}p^{4})$ correction, and
the continuum extrapolation in the coordinate space or momentum space provides
consistent results supplementary .

Results.–After renormalization in the hybrid scheme, we perform a phase
rotation $e^{izP_{z}/2}$ to the renormalized correlation, so that the
imaginary part directly reflects the flavor asymmetry between the strange and
up/down quarks. Taking the transversely-polarized $K^{*}$ as an example, we
show in Fig. 1 the real (upper panel) and imaginary part (lower panel) of the
renormalized quasi-DA matrix elements $e^{izP_{z}/2}H_{K^{*},T}(z)$ with the
momentum $P_{z}=2\pi/L\times 10=2.15$ GeV. As shown in the upper panel, the
matrix elements at different lattice spacings are consistent with each other,
indicating that linear divergences arising from the gauge-link have been
canceled up to the current numerical uncertainty. In the lower panel, we find
a positive imaginary part at all the lattice spacings, which corresponds to a
non-zero asymmetry with the peak at $x<1/2$. This is consistent with
expectations that lighter quarks carry less momentum of the parent meson.

Figure 1: The two-point correlation function for the transversely-polarized
$K^{*}$ in coordinate space. We make a phase rotation by multiplying a factor
$e^{izP_{z}/2}$ with $P_{z}=2.15$GeV.

As one can see from Fig. 1, the uncertainty of lattice data grows rapidly with
the spatial separation of the nonlocal operator. Thus, to have a reasonable
control of uncertainties in the final result we need to truncate the
correlation at certain point. The missing long-range information can be
supplemented by a physics-based extrapolation proposed in Ref. Ji:2020brr ,
which removes unphysical oscillations in a naive truncated Fourier transform
with the price of altering the endpoint distribution (at $x\sim 0$ or $1$),
which cannot be reliably predicted by LaMET anyway due to increasingly
important higher-twist contributions. Following Ref. Ji:2020brr , we adopt the
following extrapolation form:

$\displaystyle H_{V,\\{L,T\\}}(z,P_{z})$
$\displaystyle=\Big{[}\frac{c_{1}}{(-i\lambda)^{a}}+e^{i\lambda}\frac{c_{2}}{(i\lambda)^{b}}\Big{]}e^{-\lambda/\lambda_{0}},$
(11)

where the exponential term accounts for the finite correlation length for a
hadron at finite momentum, and the two algebraic terms account for a power law
behavior of the momentum distribution at $x$ close to 0 and 1, respectively.
$\lambda=zP_{z}$, and the parameters $c_{1,2},a,b,\lambda_{0}$ are determined
by a fitting to the lattice data in the region where it exhibits an
exponential decay behavior. To account for systematics from such an
extrapolation, we have done two different extrapolations, one including the
exponential term and the other not, and taken their difference as an estimate
of systematics. This can be attributed as a source of the uncertainty from the
inverse problem, and a more systematic strategy to handle the inverse problem
of the Fourier transform is available in Ref. Karpie:2018zaz . The detailed
comparison of two extrapolations can be found in the supplemental material
supplementary .

Figure 2: Quasi-DA and LCDA extracted from it for the transversely-polarized
$K^{*}$ using data at $a=0.09~{}$fm, $P_{z}=2.15$ GeV.

After renormalization and extrapolation, we can Fourier transform to momentum
space and apply the corresponding matching. In Fig. 2, we show as an example
the comparison of the quasi-DA and extracted LCDA for the transversely
polarized $K^{*}$. The results correspond to the case with $P_{z}=2.15$ GeV,
and $a=0.09~{}$fm. One notices that there is a non-vanishing tail for quasi-DA
(yellow curve) in the unphysical region ($x>1$ or $x<0$), but it becomes much
better for the LCDA (blue curve) after the perturbative matching is applied.

Figure 3: The continuum limit of the LCDA for the transversely-polarized
$K^{*}$, extrapolated from three different lattice spacings.

We have performed a simple extrapolation to the continuum limit using the
results at three different lattice spacings and the following formula

$\displaystyle\psi(a)=\psi(a\to 0)+c_{1}a+\mathcal{O}(a^{2}),$ (12)

with $\mathcal{O}(a)$ correction being due to the mixed action effect from the
clover valence fermion on HISQ sea. As an example, we show the extrapolated
results for the transversely polarized $K^{*}$ in Fig. 3 for three different
momenta, $P_{z}=\\{1.29,1.72,2.15\\}$ GeV. From this figure, one can see that
the asymmetry slightly increases with $P_{z}$. Defining the asymmetry as
$c_{asy}=\int^{1/2}_{0}dx\phi(x)/\int^{1}_{1/2}dx\phi(x)$, we find $c_{asy}$
is 1.090(15), 1.176(07), 1.227(08) for the three momenta. Since the strange
quark is heavier than the up/down quark, a slight preference of $x<1/2$ to
$x>1/2$ is expectable. It suggests that a large $P_{z}$ extrapolation is
essential to suppress the power corrections and reproduce this correct
preference behavior. Such a behavior is also observed in the study of Kaon
LCDAs in Ref. Zhang:2020gaj .

Figure 4: LCDAs for the longitudinally-polarized $K^{*}$(upper panel) and
transversely-polarized $K^{*}$(lower panel). The results are extrapolated to
the continuous limit ($a\to 0$) and the infinite momentum limit
($P_{z}\to\infty$). Regions with $x<0.1,x>0.9$ are shaded, as systematic
errors in these regions are difficult to estimate.

Figure 5: Similar to Fig. 4, but for the $\phi$ vector meson.

After matching from Quasi-DA to LCDAs with $\mu=2$ GeV, our final results for
LCDAs of the $K^{*}$ and $\phi$ are given in Fig. 4 and 5, respectively, where
the upper and lower panels correspond to the longitudinal and transverse
polarization cases. In these figures, we have made a $P_{z}\to\infty$
extrapolation using the following simple form:

$\displaystyle\psi(P_{z})=\psi(P_{z}\to\infty)+\frac{c_{2}}{P^{2}_{z}}+\mathcal{O}\Big{(}\frac{1}{P^{4}_{z}}\Big{)}.$
(13)

We have chosen two renormalization scale (1.82GeV and 3.04GeV) and treated
their difference an estimate of systematic error from matching. It is worth
emphasizing that the endpoint regions are difficult to access in LaMET. The
endpoint can be roughly estimated from the largest attainable $\lambda$
(conjugate variable of $x$ in the Fourier transform) as $1/\lambda_{\rm max}$.
In the present calculation, we have $\lambda_{\rm max}\approx 14$
(specifically $z^{P_{z}=1.29GeV}_{max}\approx 2.1fm$,
$z^{P_{z}=1.72GeV}_{max}\approx 1.6fm$, $z^{P_{z}=2.15GeV}_{max}\approx
1.3fm$), thus we take a conservative estimate of the predictable region as
$[0.1,0.9]$. Beyond this region, we plot a shaded area with systematic errors
difficult to estimate. As a comparison, we also show in Fig. 4 the asymptotic
form $6x(1-x)$, the model results from earlier QCD sum rule calculations
Ball:2007zt  and the Dyson-Schwinger equations(DSE) results Gao:2014bca  in
Fig. 5. Our results indicate that while the longitudinal LCDAs tend to be
close to the asymptotic form, the transverse LCDAs have relatively large
deviations from the asymptotic form. These behaviors might have important
implications on the study of semileptonic $B\to K^{*}\ell^{+}\ell^{-}$ decay
towards the search for new physics, and can be explored in the future.

Summary.–We have presented the first lattice QCD calculation of LCDAs of
longitudinally and transversely polarized vector mesons $K^{*},\phi$ using
LaMET. We did not consider the $\rho$ meson due to its large width which will
introduce sizable systematic errors. The continuum and infinite momentum
limits are taken based on calculations at physical light and strange quark
mass with three lattice spacings and momenta. Our final results are then
compared to the asymptotic form and QCD sum rule results. While the
longitudinal LCDAs tend to be close to the asymptotic form, the transverse
ones have relatively large deviations from the asymptotic form. Our final
results provide crucial ab initio theory inputs for analyzing pertinent
exclusive processes.

Acknowledgement.–We thank Xiangdong Ji, Liuming Liu, Maximilian Schlemmer, and
Andreas Schäfer for valuable discussions. We thank the MILC collaboration for
providing us their HISQ gauge configurations. The LQCD calculations were
performed using the Chroma software suite Edwards:2004sx  and QUDA
Clark:2009wm ; Babich:2011np ; Clark:2016rdz  through HIP programming model
Bi:2020wpt . The numerical calculation is supported by Strategic Priority
Research Program of Chinese Academy of Sciences, Grant No. XDC01040100. The
setup for numerical simulations was conducted on the $\pi$ 2.0 cluster
supported by the Center for High Performance Computing at Shanghai Jiao Tong
University, HPC Cluster of ITP-CAS, and Jiangsu Key Lab for NSLSCS. JH, WW,
and JX are supported in part by Natural Science Foundation of China under
grant No. 11735010, 11911530088, U2032102, 11653003, by Natural Science
Foundation of Shanghai under grant No. 15DZ2272100. JH is supported by NSFC
under grant 11947215. PS is supported by Natural Science Foundation of China
under grant No. 11975127 as well as Jiangsu Specially Appointed Professor
Program. YBY is also supported by the Strategic Priority Research Program of
Chinese Academy of Sciences, Grant No. XDB34030303. JHZ is supported in part
by National Natural Science Foundation of China under Grant No. 11975051, and
by the Fundamental Research Funds for the Central Universities.

## References

  * (1) J. T. Wei et al. [Belle], Phys. Rev. Lett. 103, 171801 (2009) doi:10.1103/PhysRevLett.103.171801 [arXiv:0904.0770 [hep-ex]].
  * (2) R. Aaij et al. [LHCb], JHEP 09, 179 (2015) doi:10.1007/JHEP09(2015)179 [arXiv:1506.08777 [hep-ex]].
  * (3) R. Aaij et al. [LHCb], JHEP 02, 104 (2016) doi:10.1007/JHEP02(2016)104 [arXiv:1512.04442 [hep-ex]].
  * (4) R. Aaij et al. [LHCb], JHEP 08, 055 (2017) doi:10.1007/JHEP08(2017)055 [arXiv:1705.05802 [hep-ex]].
  * (5) R. Aaij et al. [LHCb], Phys. Rev. Lett. 125, no.1, 011802 (2020) doi:10.1103/PhysRevLett.125.011802 [arXiv:2003.04831 [hep-ex]].
  * (6) B. Capdevila, A. Crivellin, S. Descotes-Genon, J. Matias and J. Virto, JHEP 01, 093 (2018) doi:10.1007/JHEP01(2018)093 [arXiv:1704.05340 [hep-ph]].
  * (7) D. Buttazzo, A. Greljo, G. Isidori and D. Marzocca, JHEP 11, 044 (2017) doi:10.1007/JHEP11(2017)044 [arXiv:1706.07808 [hep-ph]].
  * (8) A. Cerri, V. V. Gligorov, S. Malvezzi, J. Martin Camalich, J. Zupan, S. Akar, J. Alimena, B. C. Allanach, W. Altmannshofer and L. Anderlini, et al. CERN Yellow Rep. Monogr. 7, 867-1158 (2019) doi:10.23731/CYRM-2019-007.867 [arXiv:1812.07638 [hep-ph]].
  * (9) R. R. Horgan, Z. Liu, S. Meinel and M. Wingate, Phys. Rev. D 89, no.9, 094501 (2014) doi:10.1103/PhysRevD.89.094501 [arXiv:1310.3722 [hep-lat]].
  * (10) R. R. Horgan, Z. Liu, S. Meinel and M. Wingate, Phys. Rev. Lett. 112, 212003 (2014) doi:10.1103/PhysRevLett.112.212003 [arXiv:1310.3887 [hep-ph]].
  * (11) S. Descotes-Genon, J. Matias, M. Ramon and J. Virto, JHEP 01, 048 (2013) doi:10.1007/JHEP01(2013)048 [arXiv:1207.2753 [hep-ph]].
  * (12) R. Aaij et al. [LHCb], JHEP 02, 104 (2016) doi:10.1007/JHEP02(2016)104 [arXiv:1512.04442 [hep-ex]].
  * (13) P. Ball and V. M. Braun, Phys. Rev. D 55, 5561-5576 (1997) doi:10.1103/PhysRevD.55.5561 [arXiv:hep-ph/9701238 [hep-ph]].
  * (14) P. Ball and R. Zwicky, Phys. Rev. D 71, 014029 (2005) doi:10.1103/PhysRevD.71.014029 [arXiv:hep-ph/0412079 [hep-ph]].
  * (15) P. Ball, V. M. Braun and A. Lenz, JHEP 08, 090 (2007) doi:10.1088/1126-6708/2007/08/090 [arXiv:0707.1201 [hep-ph]].
  * (16) F. Gao, L. Chang, Y. X. Liu, C. D. Roberts and S. M. Schmidt, Phys. Rev. D 90, no.1, 014011 (2014) doi:10.1103/PhysRevD.90.014011 [arXiv:1405.0289 [nucl-th]].
  * (17) V. M. Braun, P. C. Bruns, S. Collins, J. A. Gracey, M. Gruber, M. Göckeler, F. Hutzler, P. Pérez-Rubio, A. Schäfer and W. Söldner, et al. JHEP 04, 082 (2017) doi:10.1007/JHEP04(2017)082 [arXiv:1612.02955 [hep-lat]].
  * (18) X. Ji, Phys. Rev. Lett. 110, 262002 (2013) doi:10.1103/PhysRevLett.110.262002 [arXiv:1305.1539 [hep-ph]].
  * (19) X. Ji, Sci. China Phys. Mech. Astron. 57, 1407-1412 (2014) doi:10.1007/s11433-014-5492-3 [arXiv:1404.6680 [hep-ph]].
  * (20) X. Ji, Y. Liu, Y. S. Liu, J. H. Zhang and Y. Zhao, [arXiv:2004.03543 [hep-ph]].
  * (21) K. Cichy and M. Constantinou, Adv. High Energy Phys. 2019, 3036904 (2019) doi:10.1155/2019/3036904 [arXiv:1811.07248 [hep-lat]].
  * (22) J. H. Zhang, J. W. Chen, X. Ji, L. Jin and H. W. Lin, Phys. Rev. D 95, no.9, 094514 (2017) doi:10.1103/PhysRevD.95.094514 [arXiv:1702.00008 [hep-lat]].
  * (23) J. H. Zhang et al. [LP3], Nucl. Phys. B 939, 429-446 (2019) doi:10.1016/j.nuclphysb.2018.12.020 [arXiv:1712.10025 [hep-ph]].
  * (24) R. Zhang, C. Honkala, H. W. Lin and J. W. Chen, Phys. Rev. D 102, no.9, 094519 (2020) doi:10.1103/PhysRevD.102.094519 [arXiv:2005.13955 [hep-lat]].
  * (25) Y. Q. Ma and J. W. Qiu, Phys. Rev. D 98, no.7, 074021 (2018) doi:10.1103/PhysRevD.98.074021 [arXiv:1404.6860 [hep-ph]].
  * (26) Y. Q. Ma and J. W. Qiu, Phys. Rev. Lett. 120, no.2, 022003 (2018) doi:10.1103/PhysRevLett.120.022003 [arXiv:1709.03018 [hep-ph]].
  * (27) A. V. Radyushkin, Phys. Rev. D 96, no.3, 034025 (2017) doi:10.1103/PhysRevD.96.034025 [arXiv:1705.01488 [hep-ph]].
  * (28) E. Follana et al. [HPQCD and UKQCD], Phys. Rev. D 75, 054502 (2007) doi:10.1103/PhysRevD.75.054502 [arXiv:hep-lat/0610092 [hep-lat]].
  * (29) A. Bazavov et al. [MILC], Phys. Rev. D 87, no.5, 054505 (2013) doi:10.1103/PhysRevD.87.054505 [arXiv:1212.4768 [hep-lat]].
  * (30) A. Hasenfratz and F. Knechtli, Phys. Rev. D 64, 034504 (2001) doi:10.1103/PhysRevD.64.034504 [arXiv:hep-lat/0103029 [hep-lat]].
  * (31) X. Ji, Y. Liu, A. Schäfer, W. Wang, Y. B. Yang, J. H. Zhang and Y. Zhao, Nucl. Phys. B 964, 115311 (2021) doi:10.1016/j.nuclphysb.2021.115311 [arXiv:2008.03886 [hep-ph]].
  * (32) See the supplementary material of this work for a collection of perturbative matching kernel, and more detailed results, which includes Ref. Regge:1959mz .
  * (33) T. Regge, Nuovo Cim. 14, 951 (1959). doi:10.1007/BF02728177
  * (34) A. Ali, V. M. Braun and H. Simma, Z. Phys. C 63, 437-454 (1994) doi:10.1007/BF01580324 [arXiv:hep-ph/9401277 [hep-ph]].
  * (35) T. Ishikawa, Y. Q. Ma, J. W. Qiu and S. Yoshida, [arXiv:1609.02018 [hep-lat]].
  * (36) J. W. Chen, X. Ji and J. H. Zhang, Nucl. Phys. B 915, 1-9 (2017) doi:10.1016/j.nuclphysb.2016.12.004 [arXiv:1609.08102 [hep-ph]].
  * (37) J. Green, K. Jansen and F. Steffens, Phys. Rev. Lett. 121, no.2, 022004 (2018) doi:10.1103/PhysRevLett.121.022004 [arXiv:1707.07152 [hep-lat]].
  * (38) I. W. Stewart and Y. Zhao, Phys. Rev. D 97, no.5, 054512 (2018) doi:10.1103/PhysRevD.97.054512 [arXiv:1709.04933 [hep-ph]].
  * (39) C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, H. Panagopoulos and F. Steffens, Nucl. Phys. B 923, 394-415 (2017) doi:10.1016/j.nuclphysb.2017.08.012 [arXiv:1706.00265 [hep-lat]].
  * (40) V. M. Braun, A. Vladimirov and J. H. Zhang, Phys. Rev. D 99, no.1, 014013 (2019) doi:10.1103/PhysRevD.99.014013 [arXiv:1810.00048 [hep-ph]].
  * (41) X. Ji, A. Schäfer, X. Xiong and J. H. Zhang, Phys. Rev. D 92, 014039 (2015) doi:10.1103/PhysRevD.92.014039 [arXiv:1506.00248 [hep-ph]].
  * (42) J. Xu, Q. A. Zhang and S. Zhao, Phys. Rev. D 97, no.11, 114026 (2018) doi:10.1103/PhysRevD.97.114026 [arXiv:1804.01042 [hep-ph]].
  * (43) Y. S. Liu, W. Wang, J. Xu, Q. A. Zhang, S. Zhao and Y. Zhao, Phys. Rev. D 99, no.9, 094036 (2019) doi:10.1103/PhysRevD.99.094036 [arXiv:1810.10879 [hep-ph]].
  * (44) Y. B. Yang, A. Alexandru, T. Draper, M. Gong and K. F. Liu, Phys. Rev. D 93, no.3, 034503 (2016) doi:10.1103/PhysRevD.93.034503 [arXiv:1509.04616 [hep-lat]].
  * (45) J. Karpie, K. Orginos and S. Zafeiropoulos, JHEP 11, 178 (2018) doi:10.1007/JHEP11(2018)178 [arXiv:1807.10933 [hep-lat]].
  * (46) R. G. Edwards et al. [SciDAC, LHPC and UKQCD], Nucl. Phys. B Proc. Suppl. 140, 832 (2005) doi:10.1016/j.nuclphysbps.2004.11.254 [arXiv:hep-lat/0409003 [hep-lat]].
  * (47) M. A. Clark, R. Babich, K. Barros, R. C. Brower and C. Rebbi, Comput. Phys. Commun. 181, 1517-1528 (2010) doi:10.1016/j.cpc.2010.05.002 [arXiv:0911.3191 [hep-lat]].
  * (48) R. Babich, M. A. Clark, B. Joo, G. Shi, R. C. Brower and S. Gottlieb, doi:10.1145/2063384.2063478 [arXiv:1109.2935 [hep-lat]].
  * (49) M. A. Clark, B. Joó, A. Strelchenko, M. Cheng, A. Gambhir and R. Brower, [arXiv:1612.07873 [hep-lat]].
  * (50) Y. J. Bi, Y. Xiao, W. Y. Guo, M. Gong, P. Sun, S. Xu and Y. B. Yang, PoS LATTICE2019, 286 (2020) doi:10.22323/1.363.0286 [arXiv:2001.05706 [hep-lat]].


#  Relation between polar and running masses of heavy quarks using the
principle of maximum conformality

Daniel Salinas-Arizmendi<EMAIL_ADDRESS>Ivan Schmidt
<EMAIL_ADDRESS>Departamento de Física, Universidad Técnica Federico Santa
María, y Centro Científico-Tecnológico de Valparaíso, Casilla 110-V,
Valparaíso, Chile

(August 28, 2024)

###### Abstract

The relation between the polar and running heavy quark masses to order
$\mathcal{O}\left(\alpha_{s}^{4}\right)$ in perturbative quantum
chromodynamics (pQCD) can be obtained with the help of the Principle of
Maximum Conformity (PMC), which is a formalism that provides a rigorous method
for eliminating renormalization scale and scheme ambiguities for observables
in pQCD. Using PMC, an optimal renormalization scale for the heavy quark mass
ratio is determined, independent of the renormalization scale and scheme.
Precise values are then obtained for the PMC masses of the heavy quarks
$M_{b}^{\text{PMC}}=4.842$ GeV, $M_{t}^{\text{PMC}}=172.239$ GeV, and
$\overline{m}_{t}^{\text{PMC}}=162.56$ GeV.

###### pacs:

14.60.St, 11.30.Hv, 12.60.-i

##  I Introduction

The mass of quarks plays a significant role in the phenomenology of high-
energy physics. For example, the b quark mass is essential for determining the
decay of the B meson and is the dominant channel in the decay of the Higgs
boson into a pair of quarks, whereas the top quark is used indirectly in the
determination of the mass of the Higgs boson.

In lowest order perturbation theory, there is no need to choose a
renormalization scheme when considering corrections to quantum quark masses,
which are defined differently according to the renormalization scheme. One of
the most widely used renormalization schemes is the modified minimal
subtraction scheme, $\overline{\text{MS}}$, which corresponds to a specific
choice of the finite parts of the counterterms. Then the renormalized mass and
the coupling constant depend explicitly on the renormalization scale, through
the renormalization group equations (RGE), that is of the order of the scale
of the process. Another widely used renormalization scheme is the on-shell
subtraction scheme, where the renormalized mass is defined to all orders as
the position of the pole in the particle propagator.

High-order QCD corrections to the quark mass ratio have been calculated up to
$\mathcal{O}\left(\alpha_{s}^{3}\right)$ (gray1990three, ; melnikov2000three,
) and $\mathcal{O}\left(\alpha_{s}^{4}\right)$ (marquard2015quark, ;
Steinhauser2016xkf, ; kataev2020multiloop, ), arguing that the renormalization
scale should be set to the typical momentum scale of the process, in order to
eliminate large logarithms; this estimated scale is then varied over an
arbitrary range in order to determine its uncertainty. Nevertheless, this
conventional procedure gives scheme-dependent predictions and thus violates
the fundamental principle of renormalization group invariance. The mass
relationship in the resulting non-compliant pQCD series presents divergences
in the renormalization procedure, through the choice of scale and scheme,
introducing uncertainties; in the particular case of the Polar Mass, the
contributions grow factorially in higher orders, and the contribution of the
renormalon as $n!\left(\beta_{0}\alpha_{s}\right)^{n}$, preventing
convergence.

The Principle of Maximum Conformity (PMC) (brodsky2012scale, ;
brodsky2012setting, ; brodsky2012eliminating, ; mojaza2013systematic, ;
brodsky2014systematic, ; wu2013renormalization, ; singlescale1, ), underlying
the BLM method (brodsky1983elimination, ), presents the best alternative to
solve the problem exposed in the previous paragraph, providing a systematic
way to remove renormalization scheme and scale ambiguities. The PMC determines
the renormalization scale by absorbing all the conformal terms that govern the
running coupling behavior, and therefore the resulting pQCD series matches the
conformal series.

The starting point of the PMC configuration is to introduce the
$\delta$-$\mathcal{R}$enomalization Scheme (mojaza2013systematic, ;
brodsky2014systematic, ), which consists in a generalization of the
conventional scheme used in dimensional regularization
(bollini1972dimensional, ; veltman1972regularization, ), in which a constant
$-\delta$ is subtracted in addition to the standard $\ln 4\pi-\gamma_{E}$
subtraction of the $\overline{\text{MS}}$ scheme. The subtraction $\delta$
defines an infinite set of renormalization schemes, in which the physical
results are independent of $\delta$, therefore scheme independent. Since the
PMC predictions do not depend on the choice of renormalization scheme, the PMC
scaling satisfies the RGE invariance principles (brodsky2012self, ).

  * •

In short, the PMC procedure consists in first choosing a renormalization
scheme, $\mathcal{R}_{\delta}$ for example, $\alpha_{s}^{\mathcal{R}}$ and a
renormalization scale. Then the pQCD observable is chosen. Next identifying
the $\\{\beta_{i}\\}$ terms, using the active quark flavor number $N_{f}$
series coefficients, shifting the scale of the strong coupling $\alpha_{s}$ at
each orden to the PMC scale to eliminate all the no-conformal terms. Finally,
expressing the observable pQCD using a conformal series, independent of the
initial renormalization scale and scheme to a fixed order.

In the present work, the on-shell renormalization scheme (and the
$\overline{\text{MS}}$ scheme, as appropriate) is initially chosen as the
starting point, to be able to determine the mass ratio between the two schemes
as a means to obtain a PMC quark mass ratio after applying the procedure,
where the new determination is independent of the choice of the initial
scheme. The relationship between the Polar-$\overline{\text{MS}}$ masses for
the quarks (Steinhauser2016xkf, ; marquard2016ms, ; kataev2020multiloop, ) is
given by

$H_{q}\left(\mu\right)=\frac{M\left(\mu\right)}{\overline{m}}=1+\sum_{n\geq
1}h_{n}\left(\frac{\alpha_{s}}{\pi}\right)^{n},$ (1)

where $M$ and $\overline{m}$ are the polar quark mass and
$\overline{\text{MS}}$ quark mass, respectively, $\alpha_{s}$ is de strong
running coupling constant, and $h_{n}$ are the perturbative coefficients,

$h_{n}=h_{n,0}+h_{n,1}N_{f}+h_{n,2}N_{f}^{2}+\cdots+h_{n,n-1}N_{f}^{n-1},$ (2)

where $N_{f}$ is the active flavor quarks number. The $N_{f}$ terms come from
the UV-divergent diagrams of the process, which dynamically depend on the
virtuality of the fundamental quark and gluon subprocesses. We will show an
optimal way to set the scales at each order of perturbation theory and with a
single PMC scale, in contrast to the conventional scaling formalism, therefore
producing a relationship which converges to the final value. In the same way,
the inverse relationship (Steinhauser2016xkf, ; marquard2016ms, ;
kataev2020multiloop, ) is

$\overline{H}_{q}\left(\mu\right)=\frac{\overline{m}\left(\mu\right)}{M}=1+\sum_{n\geq
1}\overline{h}_{n}\left(\frac{\alpha_{s}}{\pi}\right)^{n}$ (3)

with $\overline{h}_{n}$ are the perturbative coefficients, similar to (2),
will be useful for the determination of the mass in the modified minimum
subtraction scheme.

##  II Determination of the polar mass and running relationship for heavy
quarks in PMC setting

In this section we present an improved analysis of the relationship between
the on-shell and running quark masses by the use of the PMC. However, before
applying the PMC formalism, it will be necessary to obtain the dependence of
the mass relation on the initial scale, from the above mass relation with
coefficients $N_{f}$ on a fixed scale. This can be achieved by using the scale
dependence of the strong coupling constant up to four-loop level
(mojaza2013systematic, ; brodsky2014systematic, ), i.e.

$\displaystyle a_{s}(Q^{*})$ $\displaystyle=$ $\displaystyle
a_{s}(Q)-\beta_{0}\ln\left(\frac{Q^{*2}}{Q^{2}}\right)a_{s}^{2}(Q)$
$\displaystyle+\left[\beta_{0}^{2}\ln^{2}\left(\frac{Q^{*2}}{Q^{2}}\right)-\beta_{1}\ln\left(\frac{Q^{*2}}{Q^{2}}\right)\right]a_{s}^{3}(Q)$
$\displaystyle+\left[-\beta_{3}^{3}\ln^{2}\left(\frac{Q^{*2}}{Q^{2}}\right)+\frac{5}{2}\beta_{0}\beta_{1}\ln^{2}\left(\frac{Q^{*2}}{Q^{2}}\right)\right.$
$\displaystyle\left.-\beta_{2}\ln\left(\frac{Q^{*2}}{Q^{2}}\right)\right]a_{s}^{4}(Q)+\mathcal{O}\left(a_{s}^{5}\right)$

where $a_{s}=\alpha_{s}/\pi$, and $Q^{*}$ and $Q$ are two arbitrary
renormalization scales.

Choosing as the initial renormalization scale the mass of the heavy quark
involved, the expression for the polar-running mass in Eq. (1), dependening on
the active quark flavor number $N_{f},$ can be written as

$\displaystyle H_{q}$ $\displaystyle=$ $\displaystyle
1+h_{1,0}a_{s}\left(\mu_{r}^{init}\right)+\left(h_{2,0}+h_{2,1}N_{f}\right)a_{s}^{2}\left(\mu_{r}^{init}\right)$
$\displaystyle+\left(h_{3,0}+h_{3,1}N_{f}+h_{3,2}N_{f}^{2}\right)a_{s}^{3}\left(\mu_{r}^{init}\right)+\left(\right.h_{4,0}$
$\displaystyle\left.+h_{4,1}N_{f}+h_{4,2}N_{f}^{2}+h_{4,3}N_{f}^{3}\right)a_{s}^{4}\left(\mu_{r}^{init}\right)+\mathcal{O}\left(a_{s}^{5}\right).$

The coefficients $h_{i,j}$, at four loop level, are given in Ref
(Steinhauser2016xkf, ):

$\begin{array}[]{lll}h_{1,0}=1.333,&&h_{2,0}=14.485,\\\
h_{2,1}=-1,041,&&h_{3,0}=217.903,\\\ h_{3,1}=-27.961,&&h_{3,2}=0,653,\\\
h_{4,0}=4357.4\pm 1.64,&&h_{4,1}=-\left(834.548\pm 0.04\right),\\\
h_{4,2}=-45.431,&&h_{4,3}=-0.678.\end{array}$ (6)

The scheme-$\mathcal{R}_{\delta}$ (mojaza2013systematic, ;
brodsky2014systematic, ) exposes the special degeneracy of the coefficients in
different perturbative orders of the series, ensuring the correspondence
between the $N_{f}$-terms and the $\left\\{\beta_{i}\right\\}$-terms, through
Eq. (II), order by order in the series. Then from Eq. (II), the general form
of the observable can be derived

$\displaystyle H_{q}$ $\displaystyle=$ $\displaystyle
1+r_{1,0}a_{s}\left(\mu_{r}^{init}\right)+\left(r_{2,0}+\beta_{0}r_{2,1}\right)a_{s}^{2}\left(\mu_{r}^{init}\right)$
$\displaystyle+\left(r_{3,0}+\beta_{1}r_{2,1}+2\beta_{0}r_{3,1}+\beta_{0}^{2}r_{3,2}\right)a_{s}^{3}\left(\mu_{r}^{init}\right)$
$\displaystyle+\left(r_{4,0}+\beta_{2}r_{2,1}+2\beta_{1}r_{3,1}+\frac{5}{2}\beta_{0}\beta_{1}r_{3,2}\right.$
$\displaystyle\left.+3\beta_{0}r_{4,1}+3\beta_{0}^{2}r_{4,2}+\beta_{0}^{3}r_{4,3}\right)a_{s}^{4}\left(\mu_{r}^{init}\right)$

Here, for later convenience, we have transformed the series $N_{f}$ into the
required series $\left\\{\beta_{i}\right\\}$. The expressions for $\beta_{0},\
\beta_{1}$ and $\beta_{2}$ of the series above, have been determined in
(baikov2017five, ). The $r_{i,0}$ with $i=(1,\ldots,4)$ are scale-invariant
conformal coefficients, and the $r_{i,j}$ with $1\leq j<i\leq 4$ are non-
conforming coefficients which must be absorbed in the coupling constant.

By applying PMC, we absorb all the non-conforming coefficients that control
the operating behavior of the coupling constant into the coupling constant.
The non-conformal terms are removed, and the result of the pQCD series is
transformed into a conformal mass ratio:

$\displaystyle H_{q}^{\text{PMC}}$ $\displaystyle=$ $\displaystyle
1+r_{1,0}a_{s}\left(Q_{1}\right)+r_{2,0}a_{s}^{2}\left(Q_{2}\right)$
$\displaystyle+r_{3,0}a_{s}^{3}\left(Q_{3}\right)+r_{4,0}a_{s}^{3}\left(Q_{4}\right)$
$\displaystyle=$ $\displaystyle
1-0.424307\alpha_{s}\left(Q_{1}\right)+0.182631\alpha_{s}^{2}\left(Q_{2}\right)$
$\displaystyle-3.39816\alpha_{s}^{3}\left(Q_{3}\right)-1.60253\alpha_{s}^{4}\left(Q_{4}\right),$

where the conformal coefficients are independent of the renormalization scale
$\mu_{r}$, the resulting series becomes scheme independent, and the ambiguity
of the conventional renormalization scheme is removed. Here $Q_{i}\
\left(i=1,\ldots,4\right)$ are the PMC scales, for the mass relation on shell-
running mass PMC.

The PMC procedure simultaneously determines the process scale at each order
(mojaza2013systematic, ). From the mass relation, the PMC scales $Q_{i}$ are
determined up to N3LO order by the following formula

$\ln\frac{Q_{k}^{2}}{Q^{2}}=\frac{R_{k,1}+\Delta_{k}^{\left(1\right)}\left(a_{s}\right)R_{k,2}+\Delta_{k}^{\left(2\right)}\left(a_{s}\right)R_{k,3}}{1+\Delta_{k}^{\left(1\right)}\left(a_{s}\right)R_{k,1}+\left(\Delta_{k}^{\left(1\right)}\left(a_{s}\right)\right)^{2}\left(R_{k,2}-R_{k,1}\right)+\Delta_{k}^{\left(2\right)}\left(a_{s}\right)R_{k,1}^{2}},$
(9)

where

$\displaystyle R_{k,j}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{j}\frac{r_{k+j,j}}{r_{k,0}},$ (10)
$\displaystyle\Delta_{k}^{\left(1\right)}\left(a_{s}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2!}\left[\frac{\partial\beta}{\partial
a_{s}}+\left(k+p-2\right)\frac{\beta}{a_{s}}\right]$ (11)
$\displaystyle\Delta_{k}^{\left(2\right)}\left(a_{s}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{3!}\left[\beta\frac{\partial^{2}\beta}{\partial
a_{s}^{2}}+\left(\frac{\partial\beta}{\partial a_{s}}\right)^{2}\right.$
$\displaystyle+3\left(k+p-2\right)\frac{\beta}{a_{s}}\frac{\partial\beta}{\partial
a_{s}}$
$\displaystyle\left.+\left(k+p-2\right)\left(k+p-3\right)\frac{\beta^{2}}{a_{s}^{2}}\right],$

and $\beta$ comes from the RG equation of the $\beta$-function.

PMC scales are perturbative in nature, therefore, we need to know more loop
terms to have more accurate predictions, so $Q_{4}$ is unknown since we
require the terms $\\{\beta_{i}\\}$ in the order $a_{s}^{5}$. However, $Q_{4}$
can be set as the last determined scale $Q_{3}$, which guarantees the scheme
independence of our prediction due to the commensurate scale relations among
(CSR, ).

The inverse relationship, comparing running mass with pole mass,
$\overline{H}_{q}$, conventionally determined to order
$\mathcal{O}\left(a_{s}^{4}\right)$, for a renormalization scale
$\mu_{r}^{init}=M$, is

$\displaystyle\overline{H}_{q}$ $\displaystyle=$ $\displaystyle
1+\bar{h}_{1,0}a_{s}\left(\mu_{r}^{init}\right)+\left(\bar{h}_{2,0}+\bar{h}_{2,1}N_{f}\right)a_{s}^{2}\left(\mu_{r}^{init}\right)$
$\displaystyle+\left(\bar{h}_{3,0}+\bar{h}_{3,1}N_{f}+\bar{h}_{3,2}N_{f}^{2}\right)a_{s}^{3}\left(\mu_{r}^{init}\right)$
$\displaystyle+\left(\bar{h}_{4,0}+\bar{h}_{4,1}N_{f}+\bar{h}_{4,2}N_{f}^{2}+\bar{h}_{4,3}N_{f}^{3}\right)a_{s}^{4}\left(\mu_{r}^{init}\right),$

where the coefficients are set out in (Steinhauser2016xkf, ):

$\begin{array}[]{lll}\bar{h}_{1,0}=-1.333,&&\bar{h}_{2,0}=-15.374,\\\
\bar{h}_{2,1}=1.041,&&\bar{h}_{3,0}=-226.283,\\\
\bar{h}_{3,1}=28.229,&&\bar{h}_{3,2}=-0.653,\\\ \bar{h}_{4,0}=-4455.25\pm
1.64,&&\bar{h}_{4,1}=845.941\pm 0.04,\\\
\bar{h}_{4,2}=-45.517,&&\bar{h}_{4,3}=0.678.\end{array}$ (14)

Performing the PMC adjustment, the following conformal series is acquired

$\displaystyle\overline{H}_{q}^{\text{PMC}}$ $\displaystyle=$ $\displaystyle
1+\overline{r}_{1,0}a_{s}\left(\overline{Q}_{1}\right)+\overline{r}_{2,0}a_{s}^{2}\left(\overline{Q}_{2}\right)$
$\displaystyle+\overline{r}_{3,0}a_{s}^{3}\left(\overline{Q}_{3}\right)+\overline{r}_{4,0}a_{s}^{3}\left(\overline{Q}_{4}\right)$
$\displaystyle=$ $\displaystyle
1+0.424307\alpha_{s}\left(\overline{Q}_{1}\right)-0.272706\alpha_{s}^{2}\left(\overline{Q}_{2}\right)$
$\displaystyle+3.27051\alpha_{s}^{3}\left(\overline{Q}_{3}\right)+1.84591\alpha_{s}^{4}\left(\overline{Q}_{4}\right)$

The $\overline{Q}_{i}\ \left(i=1,\ldots,4\right)$, are the PMC scales at each
order of the perturbation, determined by equations (9), (10), (11) and (II),
using the conformal coefficients $\overline{r}_{i,0}$ and non-conformal
coefficients $\overline{r}_{i,j}$ of the inverse relationship.

The results presented in this section are $H_{q}^{\text{PMC}}$ and
$\overline{H}_{q}^{\text{PMC}}$, which initially come from the series
determined in the On-Shell and $\overline{\text{MS}}$ schemes. After following
the PMC formalism, they become scheme independent. The initial schemes are
used as a way to determine the generalized relationship for polar and running
mass, and using this we can determine the mass relations of the heavy quarks
$M_{q}^{\text{PMC}}$ and their corresponding mass differences $\Delta
M_{q}^{\text{PMC}}$ in the PMC setting.

##  III Numerical results and discussions

For numerical computations, we will use the central value for the running
masses of $c$, $b$ y $t$-quarks, namely
$\overline{m}_{c}\left(\overline{m}_{c}^{2}\right)=1.27\pm 0.02$ GeV,
$\alpha_{s}\left(\overline{m}_{c}\right)=0.38\pm 0.03$,
$\overline{m}_{b}\left(\overline{m}_{b}^{2}\right)=4.18_{-0.03}^{+0.04}$ GeV,
$\alpha_{s}\left(\overline{m}_{b}\right)=0.233\pm 0.008$, and from cross-
section measurements
$\overline{m}_{t}\left(\overline{m}_{t}^{2}\right)=162.5_{-1.5}^{+2.1}$ GeV,
$\alpha_{s}\left(\overline{m}_{t}\right)=0.1083$ (particle2022review, ) as
input parameters. Also $\alpha_{s}\left(M_{Z}\right)=0.1181$.

###  III.1 Charm quark mass relations

The determination of the charm quark mass plays an essential role in the
estimation of the CKM parameters. The relationship of its on-shell and running
mass with QCD corrections $\left(\alpha_{s}\right)$ have been obtained at
three-loops (hoang2006charm, ) and four-loops (kataev2020multiloop, ). The
results $H_{c}$, are presented in Table 1, where $H_{c}^{\left(i\right)}\
$represents the approximate relation to each perturbative order with $i=$LO,
NLO, etc., where we have fixed $\mu_{r}=\mu^{init}=2$ GeV for the conventional
scale, and in the PMC scaling we take $\mu^{init}=2$ GeV for the initial
scale.

For the relationship $H_{c}$, we need to introduce four PMC scales. As
indicated in the previous section, Eq. (9), since there are not enough $\beta$
terms that can determine its optimal value from the last scale, we set
$Q_{4}^{\left(c\right)}$ equal the last determined scale of the process.

The PMC multi-scales are

|  | $H_{c}^{\left(1\right)}$ | $H_{c}^{\left(2\right)}$ | $H_{c}^{\left(3\right)}$ | $H_{c}^{\left(4\right)}$ |  | $\overline{H}_{c}^{\left(1\right)}$ | $\overline{H}_{c}^{\left(2\right)}$ | $\overline{H}_{c}^{\left(3\right)}$ | $\overline{H}_{c}^{\left(4\right)}$
---|---|---|---|---|---|---|---|---|---|---
Conv. |  | $1.16124$ | $1.31224$ | $1.51842$ | $1.88291$ |  | $0.949974$ | $0.934186$ | $0.927642$ | $0.924158$
PMCs |  | $1.10938$ | $1.09126$ | $1.14729$ | $1.15544$ |  | $0.947859$ | $0.950617$ | $0.944311$ | $0.943946$
PMC |  | $1.22555$ | $1.12481$ | $1.7668$ | $1.97739$ |  | $0.945018$ | $0.949683$ | $0.941924$ | $0.941442$
Table 1: Numerical results for the c-quark $H_{c}$ mass ratio and inverse
$\overline{H}_{c}$ ratio, under conventional (Conv.) scaling correction, PMCs
and PMC, respectively, corrected up to four-loop. $Q=2$ GeV (and $Q=M_{z}$
(particle2022review, )).

$\begin{array}[]{c}Q_{1}^{\left(c\right)}=1.136\ \text{GeV,
}Q_{2}^{\left(c\right)}=0.950\ \text{GeV},\\\ Q_{3}^{\left(c\right)}=1.00\
\text{GeV.}\end{array}$ (16)

which are smaller than $\mu^{init}$, and different, showing that they are
controlled by a different series $\left\\{\beta_{i}\right\\}$ at each
perturbative order. The QCD perturbative corrections to the pole-running
charm-quark mass ratio are

$H_{c}^{\text{PMC}}=1+0.2255-0.1007+0.6420+0.2106.$ (17)

An alternative approach is to use the PMC single scale setting (PMCs), since
it helps to reduce the dependency on a residual scale when this becomes
considerable. The idea is to set a single effective scale. The details of the
procedure are illustrated in Ref. (singlescale1, ). In our case, the single
scale is determined from the following perturbative relation, fixed to order
next-to-next-to-leading log (N2LL) as

$\ln\left(\frac{Q_{s}^{\left(c\right)}}{Q}\right)^{2}=-1.171+12.56a_{s}(Q)+104.8a_{s}^{2}(Q),$
(18)

where the coefficients are determined according to the formulas shown in
Appendix A.

Figure 1: Single-scale PMC values for $H_{c}$, determined up to NNLL. $Q=2$
GeV.

The PMC single-scale values are shown in Figure 1, up to orden N2LL
$Q_{s}^{(c)}=1.1134$ GeV. The scales at different orders are
$Q_{s}^{\left(c\right),\text{LL}}<Q_{s}^{\left(c\right),\text{NLL}}<Q_{s}^{\left(c\right),\text{N}^{2}\text{LL}}$.

The best result for the PMC Polar mass of the charm quarks comes from
performing the renormalization scale adjustment using PMCs, since it presents
a faster convergence of the observable. The values in each approximation are:

$\begin{array}[]{l}\left.M_{c}^{\text{PMC}}\right|_{\text{NLO}}=1.4089\
\text{GeV}\\\ \left.M_{c}^{\text{PMC}}\right|_{\text{N}^{2}\text{LO}}=1.3859\
\text{GeV}\\\ \left.M_{c}^{\text{PMC}}\right|_{\text{N}^{3}\text{LO}}=1.4571\
\text{GeV}\\\ \left.M_{c}^{\text{PMC}}\right|_{\text{N}^{4}\text{LO}}=1.4674\
\text{GeV}\end{array}$ (19)

The perturbative PMC Pole mass $M_{c}^{\text{PMC}}$ contributes to the central
value in (particle2022review, ), and is comparable to the results shown in
(kataev2020multiloop, ; marquard2016ms, ). To quantify the theoretical
prediction, we will estimate the error by defining it

$E\left(M_{q}\right)=\frac{\lvert M_{q}^{\text{data}}-M_{q}\rvert}{1\
\text{GeV}},$ (20)

which compares the numerical result obtained with the value extracted from the
experimental or phenomenological data. So for the c-quark, the conventional
fit $E(M_{q}^{\text{Conv.}})\simeq 0.7$, and our result
$E(M_{q}^{\text{PMCs}})\simeq 0.2$.

The values of the inverse relationship, Eq.(3), of the charm quark
$\overline{H}_{c}$ are presented in Table 1 from PMC and PMCs, compared to the
conventional scale setting. When applying the PMC procedure, the values of the
masses shown in table 2 are obtained. Finally, the difference between the PMC
polar mass and the running $\Delta M_{c}=M_{c}-\overline{m}_{c}$ charm quark,
approaches a better result, with a difference of $\Delta
M_{c}^{\text{PMC}}\sim 100$ MeV.

|  |  $M_{c}$ (GeV) |  |  $\overline{m}_{c}$ (GeV) |  |  $\left|\Delta M_{c}\right|$ (GeV)
---|---|---|---|---|---|---
Conv. |  | $2.39129$ |  | $1.54334$ |  | $0.847947$
PMCs |  | $1.46741$ |  | $1.57639$ |  | $-0.108979$
PMC |  | $2.51128$ |  | $1.57221$ |  | $0.939072$
Table 2: Values for the polar mass, running mass and the difference between
them for the charm quark, using the conventional setting (Conv.), PMCs, and
PMC.

###  III.2 Bottom quark mass relations

The $b$ quark plays a significant role in modern particle physics, and the
precise knowledge of the corresponding mass parameter is necessary for
accurate theoretical predictions, e.g. a precise bottom quark mass is required
for B-meson decay calculations, which are often proportional to the fifth
power of quarks mass (marquard2015quark, ).

|  | $H_{b}^{\left(1\right)}$ | $H_{b}^{\left(2\right)}$ | $H_{b}^{\left(3\right)}$ | $H_{b}^{\left(4\right)}$ |  | $\overline{H}_{b}^{\left(1\right)}$ | $\overline{H}_{b}^{\left(2\right)}$ | $\overline{H}_{b}^{\left(3\right)}$ | $\overline{H}_{b}^{\left(4\right)}$
---|---|---|---|---|---|---|---|---|---|---
Conv. |  | $1.09462$ | $1.14138$ | $1.17515$ | $1.20652$ |  | $0.949889$ | $0.935519$ | $0.930128$ | $0.927575$
PMCs |  | $1.0922$ | $1.07932$ | $1.11288$ | $1.11699$ |  | $0.947965$ | $0.950712$ | $0.944444$ | $0.944082$
PMC |  | $1.11205$ | $1.08069$ | $1.14816$ | $1.1586$ |  | $0.945343$ | $0.949918$ | $0.94233$ | $0.941862$
Table 3: Numerical results for the b-quark $H_{b}$ mass ratio and inverse
$\overline{H}_{b}$ ratio, under conventional (Conv.) scaling correction, PMCs
and PMC, respectively, corrected up to four-loop. $Q=2$ GeV (and $Q=M_{z}$).

Perturbative bottom-quark mass relations with QCD corrections are presented at
three-loop (chetyrkin2000relation, ) and four-loop, in Refs.
(melnikov2000three, ; Steinhauser2016xkf, ; kataev2020multiloop, ). After
using the PMC formalism for the $b$ quark, it gives the following numerical
series

$H_{b}^{\text{PMC}}=1+0.4684-1.131+0.2820+0.04365$ (21)

in contrast to the displayed value $\mathcal{O}\left(\alpha_{s}^{3}\right)$
(particle2022review, ) and $\mathcal{O}\left(\alpha_{s}^{4}\right)$ in
(marquard2015quark, ; kataev2020multiloop, ), where the PMC multi-scale
adjustment delivers the following scales

$\begin{array}[]{c}Q_{1}^{\left(b\right)}=1.129\ \text{GeV,
}Q_{2}^{\left(b\right)}=0.5682\ \text{GeV},\\\ Q_{3}^{\left(b\right)}=1.006\
\text{GeV,}\end{array}$ (22)

considering that $Q_{4}^{(b)}$ is set equal to the last determined scale in
the main order. Optionally, in the single-scale fit (PMCs), for
$Q_{s}^{(b),\text{NNLLO}}$ the mass ratio $H_{b}$ is presented in Table 3,
where an initial scaling choice of $2$ GeV is taken for the conventional
scaling calculation, compared to the multi-scale Eq. (22) and the single-scale
fit, with values presented in Figure 2, given by

$\ln\left(\frac{Q_{s}^{(b)}}{Q^{2}}\right)^{2}=-1.1714+12.38a_{s}(Q)+96.38a_{s}^{2}(Q),$
(23)

to order N2LL.

Our result for the PMC polar mass of the bottom-quark, presented in different
orders, is

$\begin{array}[]{l}\left.M_{b}^{\text{PMC}}\right|_{\text{NLO}}=4.6484\
\text{GeV}\\\ \left.M_{b}^{\text{PMC}}\right|_{\text{N}^{2}\text{LO}}=4.5173\
\text{GeV}\\\ \left.M_{b}^{\text{PMC}}\right|_{\text{N}^{3}\text{LO}}=4.7993\
\text{GeV}\\\ \left.M_{b}^{\text{PMC}}\right|_{\text{N}^{4}\text{LO}}=4.8429\
\text{GeV}\end{array}$ (24)

These results show that the QCD correction contributes $14.5\%$ to the total
value. The value obtained for PMC Polar Mass, $M_{b}^{\text{PMC}}=4.84$ GeV,
considering QCD four-loop corrections, with
$E\left(M_{b}^{\text{PMC}}\right)\simeq 0.08$, is much closer to the central
value PDG($2022$) $M_{b}=4.78\pm 0.06$ GeV (particle2022review, ), than the
value obtained by conventional scale setting $M_{b}^{\text{Conv.}}$,
$E\left(M_{b}^{\text{Conv.}}\right)\simeq 0.26$, which contains the ambiguity
of the renormalization scheme. In turn, the value determined by the second
PMCs approach is closely bound to the central value. The numerical correction
to order $\mathcal{O}\left(\alpha_{s}^{4}\right)$ for the inverse relation
$\overline{H}_{b}^{\text{PMC}}$ is presented in Table 3 .

In Table 4, the results of the PMC mass of the $b$ quark are shown in both
relationships. The difference in PMC mass for the $b$ quarks is narrowed by
$0.34$ GeV.

|  |  $M_{b}$ (GeV) |  |  $\overline{m}_{b}$ (GeV) |  |  $\left|\Delta M_{b}\right|$ (GeV)
---|---|---|---|---|---|---
Conv. |  | $5.04326$ |  | $4.43381$ |  | $0.609449$
PMCs |  | $4.66903$ |  | $4.51271$ |  | $0.317034$
PMC |  | $4.84294$ |  | $4.5021$ |  | $0.340842$
Table 4: Obtained values for the polar mass, running mass and the difference
between them for the b-quark, using the conventional setting (Conv.), PMCs,
and PMC. Figure 2: Single-scale PMC value for $H_{b}$, determined up to NNLL.
$Q=2$ GeV.

###  III.3 Top quark mass relations

The top quark’s mass is an essential theoretical parameter, which has been
extracted from Tevatron and LHC experimental data in Ref.(particle2022review,
; hoang2020top, ). The renormalized mass of the top quark, in the polar-
running mass relationship with QCD corrections up to
$\mathcal{O}\left(\alpha_{s}^{3}\right)$ and
$\mathcal{O}\left(\alpha_{s}^{4}\right)$ have been determined in
(chetyrkin2000relation, ) and (marquard2015quark, ; marquard2016ms, ;
Steinhauser2016xkf, ; kataev2020multiloop, ) respectively, and with mixed QCD-
EW corrections $\left(\alpha\alpha_{s}\right)$ in (jegerlehner2003alpha, ;
kataev2022notes, ). The polar mass is sensitive to small momenta, implying a
rather large perturbative correction; thus truncating the series in an
inevitably finite order becomes scale-scheme dependent. The Principle of
Maximum Conformality considerably improves the mass ratios for the top quark,
obtaining results independent of the renormalization scheme. The PMC mass
ratio (and inverse ratio) at different orders of the Perturbative Theory (PT)
are shown in Table 5, for conventional scaling, PMCs and PMCs, taking an
initial scale of $80$ GeV. The value of the ratio under the PMC formalism, up
to $\mathcal{O}\left(\alpha_{s}^{4}\right)$, is

$H_{t}^{\text{PMC}}=1+\left(49.4-4.90+5.39+0.359\right)\times 10^{-3},$ (25)

This is a more convergent result than Eq. (II) for the conventional case. The
effective scales found are

$\begin{array}[]{c}Q_{1}^{\left(t\right)}=44.86\ \text{GeV,
}Q_{2}^{\left(t\right)}=16.24\ \text{GeV},\\\ Q_{3}^{\left(t\right)}=40.25\
\text{GeV.}\end{array}$ (26)

These values are independent of the initial renormalization scale up to some
residual dependency from the truncated $\beta$ function, which is less than
the quoted precision. Table 5 presents the results for PMCs, where the unique
scale is shown in Figure 3, up to order NNLL. The result for the PMC polar
mass of the top quark, for different orders of the perturbation, is presented
below:

$\begin{array}[]{l}\left.M_{t}^{\text{PMC}}\right|_{\text{NLO}}=172.10\
\text{GeV}\\\ \left.M_{t}^{\text{PMC}}\right|_{\text{N}^{2}\text{LO}}=171.30\
\text{GeV}\\\ \left.M_{t}^{\text{PMC}}\right|_{\text{N}^{3}\text{LO}}=172.18\
\text{GeV}\\\ \left.M_{t}^{\text{PMC}}\right|_{\text{N}^{4}\text{LO}}=172.24\
\text{GeV}\end{array}$ (27)

The PMC polar mass for the top quark presents a significant improvement
compared to the conventional determination. The current central value is
$172.5$ GeV (particle2022review, ) and contributes to the value determined in
the (wang2018precise, ) process of top-quark pair production at the hadron
colliders. The result for PMC/running mass of the top quark obtained is
illustrated in Table 6, which also presents an improvement. The top quark PMC
mass correction can be used as a good fit parameter in the oblique corrections
of $\Delta\rho$ (bochkarev1995scheme, ) and determined in the PMC
configuration in (yu2021new, ). On the other hand, the difference between the
top PMC masses for the two initial schemes has a value of $\Delta
M_{t}^{\text{PMC}}=9.7$ GeV.

Figure 3: Single-scale PMC value for $H_{t}$, determined up to NNLL. $Q=80$ GeV. |  | $H_{t}^{\left(1\right)}$ | $H_{t}^{\left(2\right)}$ | $H_{t}^{\left(3\right)}$ | $H_{t}^{\left(4\right)}$ |  | $\overline{H}_{t}^{\left(1\right)}$ | $\overline{H}_{t}^{\left(2\right)}$ | $\overline{H}_{t}^{\left(3\right)}$ | $\overline{H}_{t}^{\left(4\right)}$
---|---|---|---|---|---|---|---|---|---|---
Conv. |  | $1.04595$ | $1.05574$ | $1.05876$ | $1.05995$ |  | $0.949889$ | $0.93699$ | $0.932718$ | $0.930977$
PMCs |  | $1.04782$ | $1.04436$ | $1.04904$ | $1.04933$ |  | $0.948079$ | $0.950814$ | $0.944588$ | $0.944229$
PMC |  | $1.0494$ | $1.0445$ | $1.04988$ | $1.05024$ |  | $0.945765$ | $0.950205$ | $0.942825$ | $0.942374$
Table 5: Numerical results for the top quark $H_{t}$ mass ratio and inverse $\overline{H}_{t}$ ratio, under conventional (Conv.) scaling correction, PMCs and PMC, respectively, corrected up to four-loop. $Q=80$ GeV (and $Q=M_{z}$). |  |  $M_{t}$ (GeV) |  |  $\overline{m}_{t}$ (GeV) |  |  $\left|\Delta M_{t}\right|$ (GeV)
---|---|---|---|---|---|---
Conv. |  | $173.831$ |  | $160.593$ |  | $13.2376$
PMCs |  | $172.091$ |  | $162.86$ |  | $9.21144$
PMC |  | $172.239$ |  | $162.56$ |  | $9.67998$
Table 6: Obtained values for the polar mass, running mass and the difference
between them for the top quark, using the conventional setting (Conv.), PMCs,
and PMC.

##  IV Summary

The conventional determination of the mass ratio for heavy quarks assigns an
arbitrary range and systematic error to their pQCD corrections in fixed order,
breaking the renormalization group’s invariance. The PMC formalism presents a
solution to the previously mentioned problem, with respect to the pole and
running masses relationship, through a systematic adjustment of the
renormalization scale of the process. The PMC formalism results are
independent on the initial renormalization scale. The PMC scale of the process
is determined from the absorption process of the non-conformal terms, which
presents a significant theoretical advantage, since they are determined and
not chosen arbitrarily. A second improvement is that at each order of the
correction, a different scale adjustment is presented, which helps the series
to convergence; in cases where the series converges slowly, it is helpful to
introduce a single universal PMC scale, which improves the determination of
the observable. In this work this situation is present in the determination of
the mass ratio for the quark charm, where the prediction made with single-
scale adjustment is much better than in the other case.

The results achieved in this work are $M_{b}^{\text{PMC}}=4.842$ GeV,
$M_{t}^{\text{PMC}}=172.239$ GeV, and $\overline{m}_{t}^{\text{ PMC}}=162.56$
GeV, improving the accuracy that can be obtained in some tests of the standard
model and the sensitivity of new phenomena, because the formalism eliminates
the ambiguities of the scheme and the renormalization scale. The polar mass
PMCs can be utilized in the threshold phenomena (penin1999top, ; hoang2000top,
), and a better correction can influence the predictions for the cross-section
of quark production.

Acknowledgement: This work was supported by Project PIIC No. 026/2019 DPP-USM,
and by ANID PIA/APOYO AFB180002 (Chile).

##  Appendix A Coefficients for the PMC single-scale setting up to NNLO

The single universal renormalization scale PMC is

$\ln\frac{Q_{s}^{2}}{Q^{2}}=T_{0}+T_{1}a_{s}(Q)+T_{2}a_{s}^{2}(Q)+\mathcal{O}\left(a_{s}^{3}\right),$
(28)

where the $T_{i}$ are process-dependent coefficients. We can get a NNLL
$Q_{s}$, whose three coefficients are

$\displaystyle T_{0}=$ $\displaystyle-\frac{r_{2,1}}{r_{1,0}},$ (29)
$\displaystyle T_{1}=$
$\displaystyle\frac{(p+1)\left(r_{2,0}r_{2,1}-r_{1,0}r_{3,1}\right)}{p\
r_{1,0}^{2}}+\frac{(p+1)\left(r_{2,1}^{2}-r_{1,0}r_{3,2}\right)}{2r_{1,0}^{2}}\beta_{0},$
$\displaystyle T_{2}=$
$\displaystyle\frac{(p+1)^{2}\left(r_{1,0}r_{2,0}r_{3,1}-r_{2,0}^{2}r_{2,1}\right)+p(p+2)\left(r_{1,0}r_{2,1}r_{3,0}-r_{1,0}^{2}r_{4,1}\right)}{p^{2}r_{1,0}^{3}}+\frac{(p+2)\left(r_{2,1}^{2}-r_{1,0}r_{3,2}\right)}{2r_{1,0}^{2}}\beta_{1}$
$\displaystyle-\frac{p(p+1)r_{2,0}r_{2,1}^{2}+(p+1)^{2}\left(r_{2,0}r_{2,1}^{2}-2r_{1,0}r_{2,1}r_{3,1}-r_{1,0}r_{2,0}r_{3,2}\right)+(p+1)(p+2)r_{1,0}^{2}r_{4,2}}{2p\
r_{1,0}^{3}}\beta_{0}$
$\displaystyle+\frac{(p+1)(p+2)\left(r_{1,0}r_{2,1}r_{3,2}-r_{1,0}^{2}r_{4,3}\right)+(p+1)(1+2p)\left(r_{1,0}r_{2,1}r_{3,2}-r_{2,1}^{3}\right)}{6r_{1,0}^{3}}\beta_{0}^{2}.$

## References

  * (1) N. Gray, D. J. Broadhurst, W. Grafe and K. Schilcher, “Three Loop Relation of Quark (Modified) Ms and Pole Masses,” Z. Phys. C 48, 673-680 (1990) doi:10.1007/BF01614703
  * (2) K. Melnikov and T. v. Ritbergen, “The Three loop relation between the MS-bar and the pole quark masses,” Phys. Lett. B 482, 99-108 (2000) doi:10.1016/S0370-2693(00)00507-4 [arXiv:hep-ph/9912391 [hep-ph]].
  * (3) P. Marquard, A. V. Smirnov, V. A. Smirnov and M. Steinhauser, “Quark Mass Relations to Four-Loop Order in Perturbative QCD,” Phys. Rev. Lett. 114, no.14, 142002 (2015) doi:10.1103/PhysRevLett.114.142002 [arXiv:1502.01030 [hep-ph]].
  * (4) P. Marquard, A. V. Smirnov, V. A. Smirnov, M. Steinhauser and D. Wellmann, “$\overline{\rm MS}$-on-shell quark mass relation up to four loops in QCD and a general SU$(N)$ gauge group,” Phys. Rev. D 94, no.7, 074025 (2016) doi:10.1103/PhysRevD.94.074025 [arXiv:1606.06754 [hep-ph]].
  * (5) M. Steinhauser, “Relation between the pole and $\overline{MS}$ quark mass in QCD,” PoS DIS2016, 148 (2016) doi:10.22323/1.265.0148
  * (6) A. L. Kataev and V. S. Molokoedov, “Multiloop contributions to the on-shell-$\overline{{\mathrm{MS}}}$ heavy quark mass relation in QCD and the asymptotic structure of the corresponding series: the updated consideration,” Eur. Phys. J. C 80, no.12, 1160 (2020) doi:10.1140/epjc/s10052-020-08673-6 [arXiv:1807.05406 [hep-ph]].
  * (7) S. J. Brodsky, G. P. Lepage and P. B. Mackenzie, “On the Elimination of Scale Ambiguities in Perturbative Quantum Chromodynamics,” Phys. Rev. D 28, 228 (1983) doi:10.1103/PhysRevD.28.228
  * (8) S. J. Brodsky and X. G. Wu, “Scale Setting Using the Extended Renormalization Group and the Principle of Maximum Conformality: the QCD Coupling Constant at Four Loops,” Phys. Rev. D 85, 034038 (2012) [erratum: Phys. Rev. D 86, 079903 (2012)] doi:10.1103/PhysRevD.85.034038 [arXiv:1111.6175 [hep-ph]].
  * (9) S. J. Brodsky and L. Di Giustino, Phys. Rev. D 86, 085026 (2012) doi:10.1103/PhysRevD.86.085026 [arXiv:1107.0338 [hep-ph]].
  * (10) S. J. Brodsky and X. G. Wu, “Eliminating the Renormalization Scale Ambiguity for Top-Pair Production Using the Principle of Maximum Conformality,” Phys. Rev. Lett. 109, 042002 (2012) doi:10.1103/PhysRevLett.109.042002 [arXiv:1203.5312 [hep-ph]].
  * (11) M. Mojaza, S. J. Brodsky and X. G. Wu, “Systematic All-Orders Method to Eliminate Renormalization-Scale and Scheme Ambiguities in Perturbative QCD,” Phys. Rev. Lett. 110, 192001 (2013) doi:10.1103/PhysRevLett.110.192001 [arXiv:1212.0049 [hep-ph]].
  * (12) S. J. Brodsky, M. Mojaza and X. G. Wu, “Systematic Scale-Setting to All Orders: The Principle of Maximum Conformality and Commensurate Scale Relations,” Phys. Rev. D 89, 014027 (2014) doi:10.1103/PhysRevD.89.014027 [arXiv:1304.4631 [hep-ph]].
  * (13) X. G. Wu, S. J. Brodsky and M. Mojaza, “The Renormalization Scale-Setting Problem in QCD,” Prog. Part. Nucl. Phys. 72, 44-98 (2013) doi:10.1016/j.ppnp.2013.06.001 [arXiv:1302.0599 [hep-ph]].
  * (14) H. J. Lu and S. J. Brodsky, “Commensurate scale relations in quantum chromodynamics,” Nucl. Phys. B Proc. Suppl. 39BC, 309-311 (1995) doi:10.1016/0920-5632(95)00093-O
  * (15) J. M. Shen, X. G. Wu, B. L. Du and S. J. Brodsky, “Novel All-Orders Single-Scale Approach to QCD Renormalization Scale-Setting,” Phys. Rev. D 95, no.9, 094006 (2017) doi:10.1103/PhysRevD.95.094006 [arXiv:1701.08245 [hep-ph]].
  * (16) S. J. Brodsky and X. G. Wu, “Self-Consistency Requirements of the Renormalization Group for Setting the Renormalization Scale,” Phys. Rev. D 86, 054018 (2012) doi:10.1103/PhysRevD.86.054018 [arXiv:1208.0700 [hep-ph]].
  * (17) P. A. Baikov, K. G. Chetyrkin and J. H. Kühn, “Five-Loop Running of the QCD coupling constant,” Phys. Rev. Lett. 118, no.8, 082002 (2017) doi:10.1103/PhysRevLett.118.082002 [arXiv:1606.08659 [hep-ph]].
  * (18) R. L. Workman [Particle Data Group], “Review of Particle Physics,” PTEP 2022, 083C01 (2022) doi:10.1093/ptep/ptac097
  * (19) A. H. Hoang, “What is the Top Quark Mass?,” Ann. Rev. Nucl. Part. Sci. 70, 225-255 (2020) doi:10.1146/annurev-nucl-101918-023530 [arXiv:2004.12915 [hep-ph]].
  * (20) A. H. Hoang and A. V. Manohar, “Charm quark mass from inclusive semileptonic B decays,” Phys. Lett. B 633, 526-532 (2006) doi:10.1016/j.physletb.2005.12.020 [arXiv:hep-ph/0509195 [hep-ph]].
  * (21) K. G. Chetyrkin and M. Steinhauser, “The Relation between the MS-bar and the on-shell quark mass at order $alpha_{s}^{3}$,” Nucl. Phys. B 573, 617-651 (2000) doi:10.1016/S0550-3213(99)00784-1 [arXiv:hep-ph/9911434 [hep-ph]].
  * (22) F. Jegerlehner and M. Y. Kalmykov, “$O(\alpha\alpha_{s})$ relation between pole- and MS-bar mass of the t quark,” Acta Phys. Polon. B 34, 5335-5344 (2003) [arXiv:hep-ph/0310361 [hep-ph]].
  * (23) A. L. Kataev and V. S. Molokoedov, “Notes on Interplay between the QCD and EW Perturbative Corrections to the Pole-Running-to-Top-Quark Mass Ratio,” JETP Lett. 115, no.12, 704-712 (2022) doi:10.1134/S0021364022600902 [arXiv:2201.12073 [hep-ph]].
  * (24) A. I. Bochkarev and R. S. Willey, “On the scheme dependence of the electroweak radiative corrections,” Phys. Rev. D 51, 2049-2052 (1995) doi:10.1103/PhysRevD.51.R2049 [arXiv:hep-ph/9407261 [hep-ph]].
  * (25) Q. Yu, H. Zhou, J. Yan, X. D. Huang and X. G. Wu, “A new analysis of the pQCD contributions to the electroweak parameter $\rho$ using the single-scale approach of principle of maximum conformality,” Phys. Lett. B 820, 136574 (2021) doi:10.1016/j.physletb.2021.136574 [arXiv:2105.07230 [hep-ph]].
  * (26) S. Q. Wang, X. G. Wu, Z. G. Si and S. J. Brodsky, “A precise determination of the top-quark pole mass,” Eur. Phys. J. C 78, no.3, 237 (2018) doi:10.1140/epjc/s10052-018-5688-1 [arXiv:1703.03583 [hep-ph]].
  * (27) A. A. Penin and A. A. Pivovarov, Nucl. Phys. B 550, 375-396 (1999) doi:10.1016/S0550-3213(99)00205-9 [arXiv:hep-ph/9810496 [hep-ph]].
  * (28) A. H. Hoang, M. Beneke, K. Melnikov, T. Nagano, A. Ota, A. A. Penin, A. A. Pivovarov, A. Signer, V. A. Smirnov and Y. Sumino, et al. “Top - anti-top pair production close to threshold: Synopsis of recent NNLO results,” Eur. Phys. J. direct 2, no.1, 3 (2000) doi:10.1007/s1010500c0003 [arXiv:hep-ph/0001286 [hep-ph]].
  * (29) C. G. Bollini and J. J. Giambiagi, “Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter,” Nuovo Cim. B 12, 20-26 (1972) doi:10.1007/BF02895558
  * (30) G. ’t Hooft and M. J. G. Veltman, “Regularization and Renormalization of Gauge Fields,” Nucl. Phys. B 44, 189-213 (1972) doi:10.1016/0550-3213(72)90279-9