Non-perturbative quark mass renormalisation and running in \(N_{\scriptstyle \mathrm{f}}=3\) QCD

Eur. Phys. J. C (2018) 78:387 https://doi.org/10.1140/epjc/s10052-018-5870-5 Regular Article - Theoretical Physics Non-perturbative quark mass renormalisation and running in Nf = 3 QCD ALPHA Collaboration I. Campos1, P. Fritzsch2,a , C. Pena3 , D. Preti4 , A. Ramos5 , A. Vladikas6 1 Instituto de Física de Cantabria IFCA-CSIC, Avda. de los Castros s/n, 39005 Santander, Spain 2 Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland 3 Instituto de Física Teórica UAM-CSIC & Dpto. de Física Teórica, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain 4 INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Turin, Italy 5 School of Mathematics, Trinity College Dublin, Dublin 2, Ireland 6 INFN, Sezione di Tor Vergata c/o Dipartimento di Fisica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy Received: 21 February 2018 / Accepted: 4 May 2018 / Published online: 18 May 2018 © The Author(s) 2018 Abstract We determine from first principles the quark mass anomalous dimension in Nf = 3 QCD between the electroweak and hadronic scales. This allows for a fully non-perturbative connection of the perturbative and nonperturbative regimes of the Standard Model in the hadronic sector. The computation is carried out to high accuracy, employing massless O(a)-improved Wilson quarks and finite-size scaling techniques. We also provide the matching factors required in the renormalisation of light quark masses from lattice computations with O(a)-improved Wilson fermions and a tree-level Symanzik improved gauge action. The total uncertainty due to renormalisation and running in the determination of light quark masses in the SM is thus reduced to about 1%. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . 1 2 Strategy . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 Quark running and RGI masses . . . . . . . . . 2 2.2 Step scaling functions . . . . . . . . . . . . . . 3 2.3 Renormalisation schemes . . . . . . . . . . . . 3 2.4 Determination of RGI quark masses . . . . . . 6 3 Running in the high-energy region . . . . . . . . . . 6 3.1 Determination of Z P and P . . . . . . . . . . 6 3.2 Determination of the anomalous dimension . . 7 3.3 Connection to RGI masses . . . . . . . . . . . 9 4 Running in the low-energy region . . . . . . . . . . 9 5 Hadronic matching and total renormalisation factor . 11 a e-mail: 6 Conclusions . . . . . . . . . . . . . . . . . . . . . 13 Appendix A: Systematic uncertainties in the determination of step scaling functions . . . . . . . . . . . . . 14 A.1 Tuning of the critical mass . . . . . . . . . . . 14 A.2 Tuning of the gauge coupling . . . . . . . . . . 15 A.3 Perturbative values of boundary improvement coefficients . . . . . . . . . . . . . . . . . . . 15 Appendix B: Simulation details . . . . . . . . . . . . . 17 Appendix C: Tables . . . . . . . . . . . . . . . . . . . 17 References . . . . . . . . . . . . . . . . . . . . . . . . 22 1 Introduction In the paradigm provided by the Standard Model (SM) of Particle Physics, quark masses are fundamental constants of Nature. More specifically, Quantum Chromodynamics (QCD), the part of the SM that describes the fundamental strong interaction, is uniquely defined by the values of the quark masses and the strong coupling constant. Apart from this intrinsic interest, precise knowledge of the values of quark masses is crucial for the advancement of frontier research in particle physics – one good illustration being the fact that the values of the bottom and charm quark masses are major sources of uncertainty in several important Higgs branching fractions, e.g., (H → bb̄) and (H → cc̄) [1– 5]. Quark masses are couplings in the QCD Lagrangian, and have to be treated within a consistent definition of the renormalised theory. A meaningful determination can only be achieved by computing physical observables as a function of quark masses, and matching the result to the experimental 123 387 Page 2 of 23 values. A non-perturbative treatment of QCD is mandatory to avoid the presence of unquantified systematic uncertainties in such a computation: the asymptotic nature of the perturbative series, and the strongly coupled nature of the interaction at typical hadronic energy scales, implies the presence of an irreducible uncertainty in any determination that does not treat long-distance strong interaction effects from first principles. Lattice QCD (LQCD) is therefore the best-suited framework for a high-precision determination of quark masses. Indeed, following the onset of the precision era in LQCD, the uncertainties on the values of both light and heavy quark masses have dramatically decreased in recent years [6–22]. The natural observables employed in a LQCD computation of quark masses are hadronic quantities, considered at energy scales around or below 1 GeV. This requires, in particular, to work out the renormalisation non-perturbatively. Then, in order to make contact with the electroweak scale, where the masses are used to compute the QCD contribution to high-energy observables, the masses have to be run with the Renormalisation Group (RG) across more than two orders of magnitude in energy. While high-order perturbative estimates of the anomalous dimension of quark masses in various renormalisation schemes exist [23–25], a non-perturbative determination is mandatory to match the current percentlevel precision of the relevant hadronic observables. In this work we present a high-precision determination of the anomalous dimension of quark masses in QCD with three light quark flavours, as well as of the renormalisation constants required to match bare quark masses.1 This is a companion project of the recent high-precision determination of the β function and the QCD parameter in Nf = 3 QCD by the ALPHA Collaboration [29–31]. We will employ the Schrödinger Functional [32,33] as an intermediate renormalisation scheme that allows to make contact between the hadronic scheme used in the computation of bare quark masses and the perturbative schemes used at high energies, and employ well-established finite-size recursion techniques [34–45] to compute the RG running nonperturbatively. Our main result is a high-precision determination of the mass anomalous dimension between the electroweak scale and hadronic scales at around 200 MeV, where contact with hadronic observables obtained from simulations by the CLS effort [46] can be achieved. The paper is structured as follows. In Sect. 2 we describe our strategy, which (similar to the determination of QCD ) involves using two different definitions of the renormalised coupling at energies above and below an energy scale around 2 GeV. Sections 3 and 4 deal with the determination of the anomalous dimension above and below that scale, respectively. Section 5 discusses the determination of the renormal1 Preliminary results have appeared as conference proceedings in [26– 28]. 123 Eur. Phys. J. C (2018) 78:387 (...truncated)