## Abstract

We provide a systematic renormalization group formalism for the mass effects in the relation of the pole mass *m*
^{pole}_{
Q
}
and short-distance masses such as the \( \overline{\mathrm{MS}} \) mass \( {\overline{m}}_Q \) of a heavy quark *Q*, coming from virtual loop insertions of massive quarks lighter than *Q*. The formalism reflects the constraints from heavy quark symmetry and entails a combined matching and evolution procedure that allows to disentangle and successively integrate out the corrections coming from the lighter massive quarks and the momentum regions between them and to precisely control the large order asymptotic behavior. With the formalism we systematically sum logarithms of ratios of the lighter quark masses and *m*
_{
Q
}, relate the QCD corrections for different external heavy quarks to each other, predict the \( \mathcal{O}\left({\alpha}_s^4\right) \) virtual quark mass corrections in the pole-\( \overline{\mathrm{MS}} \) mass relation, calculate the pole mass differences for the top, bottom and charm quarks with a precision of around 20 MeV and analyze the decoupling of the lighter massive quark flavors at large orders. The summation of logarithms is most relevant for the top quark pole mass *m*
^{pole}_{
t
}
, where the hierarchy to the bottom and charm quarks is large. We determine the ambiguity of the pole mass for top, bottom and charm quarks in different scenarios with massive or massless bottom and charm quarks in a way consistent with heavy quark symmetry, and we find that it is 250 MeV. The ambiguity is larger than current projections for the precision of top quark mass measurements in the high-luminosity phase of the LHC.

## References

R. Tarrach,

*The pole mass in perturbative QCD*,*Nucl. Phys.***B 183**(1981) 384 [INSPIRE].N. Gray, D.J. Broadhurst, W. Grafe and K. Schilcher,

*Three loop relation of quark (modified) Ms and pole masses*,*Z. Phys.***C 48**(1990) 673 [INSPIRE].K.G. Chetyrkin and M. Steinhauser,

*Short distance mass of a heavy quark at order α*^{3}_{ s },*Phys. Rev. Lett.***83**(1999) 4001 [hep-ph/9907509] [INSPIRE].K.G. Chetyrkin and M. Steinhauser,

*The Relation between the MS-bar and the on-shell quark mass at order α*^{3}_{ s },*Nucl. Phys.***B 573**(2000) 617 [hep-ph/9911434] [INSPIRE].K. Melnikov and T.v. Ritbergen,

*The three loop relation between the MS-bar and the pole quark masses*,*Phys. Lett.***B 482**(2000) 99 [hep-ph/9912391] [INSPIRE].P. Marquard, L. Mihaila, J.H. Piclum and M. Steinhauser,

*Relation between the pole and the minimally subtracted mass in dimensional regularization and dimensional reduction to three-loop order*,*Nucl. Phys.***B 773**(2007) 1 [hep-ph/0702185] [INSPIRE].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**(2015) 142002 [arXiv:1502.01030] [INSPIRE].P. Marquard, A.V. Smirnov, V.A. Smirnov, M. Steinhauser and D. Wellmann, \( \overline{\mathrm{MS}} \)

*-on-shell quark mass relation up to four loops in QCD and a general*SU(*N*)*gauge group*,*Phys. Rev.***D 94**(2016) 074025 [arXiv:1606.06754] [INSPIRE].P.A. Baikov, K.G. Chetyrkin and J.H. Kühn,

*Five-loop running of the QCD coupling constant*,*Phys. Rev. Lett.***118**(2017) 082002 [arXiv:1606.08659] [INSPIRE].B.A. Kniehl, A.V. Kotikov, A.I. Onishchenko and O.L. Veretin,

*Strong-coupling constant with flavor thresholds at five loops in the anti-MS scheme*,*Phys. Rev. Lett.***97**(2006) 042001 [hep-ph/0607202] [INSPIRE].A.S. Kronfeld,

*The perturbative pole mass in QCD*,*Phys. Rev.***D 58**(1998) 051501 [hep-ph/9805215] [INSPIRE].I.I.Y. Bigi, M.A. Shifman, N.G. Uraltsev and A.I. Vainshtein,

*The pole mass of the heavy quark. Perturbation theory and beyond*,*Phys. Rev.***D 50**(1994) 2234 [hep-ph/9402360] [INSPIRE].M. Beneke and V.M. Braun,

*Heavy quark effective theory beyond perturbation theory: renormalons, the pole mass and the residual mass term*,*Nucl. Phys.***B 426**(1994) 301 [hep-ph/9402364] [INSPIRE].M. Beneke,

*Renormalons*,*Phys. Rept.***317**(1999) 1 [hep-ph/9807443] [INSPIRE].A. Czarnecki, K. Melnikov and N. Uraltsev,

*Non-abelian dipole radiation and the heavy quark expansion*,*Phys. Rev. Lett.***80**(1998) 3189 [hep-ph/9708372] [INSPIRE].M. Beneke,

*A quark mass definition adequate for threshold problems*,*Phys. Lett.***B 434**(1998) 115 [hep-ph/9804241] [INSPIRE].A.H. Hoang, Z. Ligeti and A.V. Manohar,

*B decay and the Upsilon mass*,*Phys. Rev. Lett.***82**(1999) 277 [hep-ph/9809423] [INSPIRE].A.H. Hoang, Z. Ligeti and A.V. Manohar,

*B decays in the upsilon expansion*,*Phys. Rev.***D 59**(1999) 074017 [hep-ph/9811239] [INSPIRE].A.H. Hoang,

*1S and MS-bar bottom quark masses from Upsilon sum rules*,*Phys. Rev.***D 61**(2000) 034005 [hep-ph/9905550] [INSPIRE].A. Pineda,

*Determination of the bottom quark mass from the*Y(1*S*)*system*,*JHEP***06**(2001) 022 [hep-ph/0105008] [INSPIRE].A. Jain, I. Scimemi and I.W. Stewart,

*Two-loop jet-function and jet-mass for top quarks*,*Phys. Rev.***D 77**(2008) 094008 [arXiv:0801.0743] [INSPIRE].S. Fleming, A.H. Hoang, S. Mantry and I.W. Stewart,

*Jets from massive unstable particles: top-mass determination*,*Phys. Rev.***D 77**(2008) 074010 [hep-ph/0703207] [INSPIRE].A.H. Hoang, A. Jain, I. Scimemi and I.W. Stewart,

*Infrared renormalization group flow for heavy quark masses*,*Phys. Rev. Lett.***101**(2008) 151602 [arXiv:0803.4214] [INSPIRE].A.H. Hoang et al.,

*The MSR mass and the*\( \mathcal{O}\left({\varLambda}_{\mathrm{QCD}}\right) \)*renormalon sum rule*, arXiv:1704.01580 [INSPIRE].A.H. Hoang et al.,

*Top-anti-top pair production close to threshold: synopsis of recent NNLO results*,*Eur. Phys. J. direct***2**(2000) 3 [hep-ph/0001286] [INSPIRE].Particle Data Group collaboration, C. Patrignani et al.,

*Review of Particle Physics*,*Chin. Phys.***C 40**(2016) 100001 [INSPIRE].C. Ayala, G. Cvetič and A. Pineda,

*The bottom quark mass from the*Y(1*S*)*system at NNNLO*,*JHEP***09**(2014) 045 [arXiv:1407.2128] [INSPIRE].M. Beneke, P. Marquard, P. Nason and M. Steinhauser,

*On the ultimate uncertainty of the top quark pole mass*, arXiv:1605.03609 [INSPIRE].J. Komijani,

*A discussion on leading renormalon in the pole mass*,*JHEP***08**(2017) 062 [arXiv:1701.00347] [INSPIRE].N. Isgur and M.B. Wise,

*Weak decays of heavy mesons in the static quark approximation*,*Phys. Lett.***B 232**(1989) 113 [INSPIRE].S. Bekavac, A. Grozin, D. Seidel and M. Steinhauser,

*Light quark mass effects in the on-shell renormalization constants*,*JHEP***10**(2007) 006 [arXiv:0708.1729] [INSPIRE].P. Ball, M. Beneke and V.M. Braun,

*Resummation of (beta0 α*_{ s }*)**n corrections in QCD: Techniques and applications to the tau hadronic width and the heavy quark pole mass*,*Nucl. Phys.***B 452**(1995) 563 [hep-ph/9502300] [INSPIRE].A.H. Hoang and A.V. Manohar,

*Charm effects in the MS-bar bottom quark mass from*Y*mesons*,*Phys. Lett.***B 483**(2000) 94 [hep-ph/9911461] [INSPIRE].A.H. Hoang,

*Bottom quark mass from*Y*mesons: charm mass effects*, hep-ph/0008102 [INSPIRE].A.L. Kataev and V.S. Molokoedov,

*On the flavour dependence of the*\( \mathcal{O}\left({\alpha}_s^4\right) \)*correction to the relation between running and pole heavy quark masses*,*Eur. Phys. J. Plus***131**(2016) 271 [arXiv:1511.06898] [INSPIRE].A.H. Hoang and A.V. Manohar,

*Charm quark mass from inclusive semileptonic B decays*,*Phys. Lett.***B 633**(2006) 526 [hep-ph/0509195] [INSPIRE].M. Melles,

*Massive fermionic corrections to the heavy quark potential through two loops*,*Phys. Rev.***D 58**(1998) 114004 [hep-ph/9805216] [INSPIRE].O. Buchmuller and H. Flacher,

*Fit to moment from*\( B\to {X}_cl\overline{\nu} \)*and B*→*X*_{ s }*γ decays using heavy quark expansions in the kinetic scheme*,*Phys. Rev.***D 73**(2006) 073008 [hep-ph/0507253] [INSPIRE].P. Gambino, K.J. Healey and S. Turczyk,

*Taming the higher power corrections in semileptonic B decays*,*Phys. Lett.***B 763**(2016) 60 [arXiv:1606.06174] [INSPIRE].M. Butenschoen et al.,

*Top quark mass calibration for Monte Carlo event generators*,*Phys. Rev. Lett.***117**(2016) 232001 [arXiv:1608.01318] [INSPIRE].A.H. Hoang, S. Mantry, A. Pathak and I.W. Stewart,

*Extracting a short distance top mass with light grooming*, arXiv:1708.02586.CDF, D0 collaboration, T.E.W. Group,

*Combination of CDF and D0 results on the mass of the top quark using up to*9*.*7*fb*^{−1}*at the Tevatron*, arXiv:1407.2682 [INSPIRE].D0 collaboration, V.M. Abazov et al.,

*Combination of D0 measurements of the top quark mass*,*Phys. Rev.***D 95**(2017) 112004 [arXiv:1703.06994] [INSPIRE].ATLAS collaboration,

*Measurement of the top quark mass in the*\( t\overline{t}\to \)*dilepton channel from*\( \sqrt{s}=8 \)*TeV ATLAS data*,*Phys. Lett.***B 761**(2016) 350 [arXiv:1606.02179] [INSPIRE].CMS collaboration,

*Measurement of the top quark mass using proton-proton data at*\( \sqrt{s}=7 \)*and*8*TeV*,*Phys. Rev.***D 93**(2016) 072004 [arXiv:1509.04044] [INSPIRE].CMS collaboration,

*Updates on projections of physics reach with the upgraded CMS detector for high luminosity LHC*, CMS-DP-2016-064 (2016).

###
**Open Access**

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

ArXiv ePrint: 1706.08526

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## About this article

### Cite this article

Hoang, A.H., Lepenik, C. & Preisser, M. On the light massive flavor dependence of the large order asymptotic behavior and the ambiguity of the pole mass.
*J. High Energ. Phys.* **2017**, 99 (2017). https://doi.org/10.1007/JHEP09(2017)099

Received:

Accepted:

Published:

DOI: https://doi.org/10.1007/JHEP09(2017)099

### Keywords

- Heavy Quark Physics
- Perturbative QCD
- Quark Masses and SM Parameters
- Renormalization Regularization and Renormalons