Quark masses are determined indirectly from lattice QCD and experimental hadron spectra.
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Peer-reviewed literature demonstrates that quark masses are determined indirectly by combining lattice QCD simulations with experimental hadron spectrum data and hadron masses.
We determine the light baryon spectrum on ensembles generated by the Coordinated Lattice Simulations (CLS) effort, employing N$_{f}$ = 2 + 1 flavours of non-perturbatively improved Wilson fermions. The hadron masses are interpolated and extrapolated within the quark mass plane, utilizing three distinct trajectories, two of which intersect close to the physical quark mass point and the third one approaching the SU(3) chiral limit. The results are extrapolated to the continuum limit, utilizing six different lattice spacings ranging from a ≈ 0.10 fm down to below 0.04 fm. The light pion mass varies from M$_{π}$ ≈ 429 MeV down to 127 MeV. In general, the spatial extent is kept larger than four times the inverse pion mass and larger than 2.3 fm, with additional small and large volume ensembles to investigate finite size effects. We determine the Wilson flow scales $ \sqrt{t_{0,\textrm{ph}}}={0.1449}_{(9)}^{(7)}\textrm{fm} $ [1] and $ {t}_0^{\ast}\approx {t}_{0,\textrm{ph}} $ [2] from the octet cascade (Ξ baryon). Determining the light baryon spectrum in the continuum limit, we find the nucleon mass $ {m}_N={941.7}_{(7.6)}^{(6.5)} $ MeV and the other stable baryon masses to agree with their experimental values within sub-percent level uncertainties. Moreover, we determine SU(3) and SU(2) chiral perturbation theory low energy constants, including the octet and the Ω baryon sigma terms σ$_{πN}$ = 43.9(4.7) MeV, $ {\sigma}_{\pi \Lambda}={28.2}_{(5.4)}^{(4.3)} $ MeV, $ {\sigma}_{\pi \S
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Olynyk, Hadron Mass Calculations in QCD , Phys. Rev. D 27 (1983) 227 [ INSPIRE ]. P. de Forcrand et al., Exploring Hadron Masses in Lattice QCD With Light Quarks and an Improved Fermion Action , Phys. Lett. B 200 (1988) 143 [ INSPIRE ]. ADS Google Scholar A pe collaboration, β = 6 . 0 quenched Wilson fermions , Phys. Lett. B 258 (1991) 195 [ INSPIRE ]. M. Guagnelli et al., The Quenched mass spectrum in lattice QCD on a 1-gigaflops computer , Nucl. Phys. B 378 (1992) 616 [ INSPIRE ]. ADS Google Scholar M.-P. Lombardo, G. Parisi and A. Vladikas, Lattice QCD spectroscopy with an improved Wilson fermion action , Nucl. Phys. B 395 (1993) 388 [ hep-lat/9206023 ] [ INSPIRE ].
UKQCD collaboration, Quenched light hadron mass spectrum and decay constants: The effects of O ( a ) improvement at β = 6 . 2, Nucl. Phys. B 407 (1993) 331 [ hep-lat/9307009 ] [ INSPIRE ]. F. Butler et al., Hadron masses from the valence approximation to lattice QCD , Nucl. Phys. B 430 (1994) 179 [ hep-lat/9405003 ] [ INSPIRE ]. QCD-TARO collaboration, Quenched Wilson hadron spectroscopy on a 32 3 × 48 lattice at β = 6 . 3, Nucl. Phys. B Proc. Suppl. 34 (1994) 338 [ INSPIRE ]. T. Bhattacharya, R. Gupta, G. Kilcup and S.R. Sharpe, Hadron spectrum with Wilson fermions , Phys. Rev. D 53 (1996) 6486 [ hep-lat/9512021 ] [ INSPIRE ].
CP-PACS collaboration, Light hadron spectrum and quark masses from quenched lattice QCD , Phys. Rev. D 67 (2003) 034503 [ hep-lat/0206009 ] [ INSPIRE ]. K.D. Born et al., Hadron Properties in Lattice QCD With Dynamical Fermions , Phys. Rev. D 40 (1989) 1653 [ INSPIRE ]. MT( c ) collaboration, The Hadron spectrum in QCD with dynamical staggered fermions , Nucl. Phys. B 389 (1993) 445 [ INSPIRE ]. K.M. Bitar et al., Hadron spectrum in QCD at 6 /g 2 = 5 . 6, Phys. Rev. D 42 (1990) 3794 [ INSPIRE ]. K.M. Bitar et al., Hadron spectrum in QCD with valence Wilson fermions and dynamical staggered fermions at 6 /g 2 = 5 . 6, Phys. Rev. D 46 (1992) 2169 [ hep-lat/9204008 ] [ INSPIRE ]. M.
PACS-CS collaboration, SU(2) and SU(3) chiral perturbation theory analyses on baryon masses in 2 + 1 flavor lattice QCD , Phys. Rev. D 80 (2009) 054502 [ arXiv:0905.0962 ] [ INSPIRE ]. W. Bietenholz et al., Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks , Phys. Rev. D 84 (2011) 054509 [ arXiv:1102.5300 ] [ INSPIRE ]. S.R. Beane et al., High Statistics Analysis using Anisotropic Clover Lattices: (IV) Volume Dependence of Light Hadron Masses , Phys. Rev. D 84 (2011) 014507 [ arXiv:1104.4101 ] [ INSPIRE ]. QCDSF and UKQCD collaborations, Isospin breaking in octet baryon mass splittings , Phys. Rev.
Borsányi et al., Ab initio calculation of the neutron-proton mass difference , Science 347 (2015) 1452 [ arXiv:1406.4088 ] [ INSPIRE ]. ADS Google Scholar R. Horsley et al., Isospin splittings of meson and baryon masses from three-flavor lattice QCD + QED , J. Phys. G 43 (2016) 10LT02 [ arXiv:1508.06401 ] [ INSPIRE ]. Google Scholar ALPHA collaboration, Effects of Heavy Sea Quarks at Low Energies , Phys. Rev. Lett. 114 (2015) 102001 [ arXiv:1410.8374 ] [ INSPIRE ]. ALPHA collaboration, How perturbative are heavy sea quarks? , Nucl. Phys. B 943 (2019) 114612 [ arXiv:1809.03383 ] [ INSPIRE ]. N. Husung, P. Marquard and R.
D 103 (2021) 094508 [ arXiv:2101.00689 ] [ INSPIRE ]. D. Mohler, S. Schaefer and J. Simeth, CLS 2 + 1 flavor simulations at physical light- and strange-quark masses , EPJ Web Conf. 175 (2018) 02010 [ arXiv:1712.04884 ] [ INSPIRE ]. Google Scholar G.S. Bali et al., Hyperon couplings from N f = 2 + 1 Lattice QCD , PoS LATTICE2019 (2019) 099 [ arXiv:1907.13454 ] [ INSPIRE ]. RQCD collaboration, Nucleon axial structure from lattice QCD , JHEP 05 (2020) 126 [ arXiv:1911.13150 ] [ INSPIRE ]. D. Krause, JUWELS: Modular Tier-0/1 Supercomputer at the Jülich Supercomputing Centre , JLSRF 5 (2019) A135 [ INSPIRE ]. D. Krause and P.
Lattice computation of the strange quark mass in QCD
We present a determination of the strange quark mass using lattice QCD. Particular focus is put on the definition and renormalization of the mass. The latter is done non-perturbatively, using a recursive finite-size scaling technique. The hadronic regime of QCD, where the kaon mass is used as input of the calculation, is connected with the perturbative regime, where the strange quark mass can be translated into the MSbar scheme. A summary plot of the present lattice computations using dynamical (sea) quarks is included.
Published as: Acta Phys.Polon. B36 (2005) 3377-3388
arXiv categories: hep-ph hep-lat
Particular focus is put on the definition and renormalization of the mass. The latter is done non-perturbatively, using a recursive finite-size scaling technique. The hadronic regime of QCD, where the kaon mass is used as input of the calculation, is connected with the perturbative regime, where the strange quark mass can be translated into the MS ¯ ¯ MS \overline{{\rm MS}} scheme. A summary plot of the present lattice computations using dynamical (sea) quarks is included.
1 2 superscript subscript 𝑔 0 2 Tr subscript 𝐹 𝜇 𝜈 subscript 𝐹 𝜇 𝜈 subscript 𝑞 ¯ 𝑞 subscript 𝛾 𝜇 subscript 𝜇 subscript 𝐴 𝜇 subscript 𝑚 𝑞 𝑞 \displaystyle-\frac{1}{2g_{0}^{2}}{\rm Tr}\{F_{\mu\nu}F_{\mu\nu}\}+\sum_{q}\bar{q}\,(\gamma_{\mu}(\partial_{\mu}+A_{\mu})+m_{q})\,q\,. (1) The bare gauge coupling g 0 subscript 𝑔 0 g_{0} and bare quark masses m q subscript 𝑚 𝑞 m_{q} are the free parameters of QCD. In order to make predictions, we need after a suitable regularization, e.g. on a Euclidean space–time lattice, to fix the free parameters through a set of N f + 1 subscript 𝑁 f 1 N_{\rm f}+1 physical quantities from experiment.
( 1 ), extract from the high-energy regime of QCD the so-called renormalization group invariant (RGI) parameters associated with the running renormalized gauge coupling and masses. In addition a number of other predictions can be made for phenomenologically relevant quantities. A crucial point becomes immediately clear: we need a tool to relate the hadronic, low-energy regime of QCD with the high-energy ( > 2 absent 2 >2 – 10 GeV 10 GeV 10\,{\rm GeV} ) regime, where perturbation theory in the gauge coupling applies. In the following we will demonstrate that such a tool is provided by the Euclidean space–time lattice. Figure 1: The hadron spectrum from quenched lattice QCD.
The predictive power of lattice QCD can be seen in Fig. 1 , which is taken from [ 1 ] . The data points represent various hadron masses m h subscript 𝑚 ℎ m_{h} computed by the CP-PACS Collaboration [ 2 ] . The input used in the lattice simulations are m ρ subscript 𝑚 𝜌 m_{\rho} , m π subscript 𝑚 𝜋 m_{\pi} and m ϕ subscript 𝑚 italic-ϕ m_{\phi} , which fix the lattice spacing a 𝑎 a , the mass of the degenerate up and down quarks, and the strange quark mass (a chiral extrapolation is needed for the mass of the up and down quarks). The hadron spectrum is extrapolated to the continuum limit a → 0 → 𝑎 0 a\to 0 .
Given that quark polarization effects have been neglected (the so-called quenched approximation), the agreement between data points and experimental values marked by the horizontal lines is remarkable. In this article we will be concerned with the computation of the strange quark mass from lattice QCD. In the limit of vanishing quark masses (chiral limit) QCD possesses a large chiral symmetry. This symmetry is spontaneously broken and the spectrum contains eight pseudo-scalar Goldstone bosons, which correspond to the observed eight lightest hadrons ( π 𝜋 \pi ’s, K 𝐾 K ’s, η 𝜂 \eta ). The latter are not exactly massless due to the non-zero quark masses.
Quark masses are the explicit symmetry-breaking parameters [ 3 , 4 ] and they are treated as perturbations of the chiral limit in the framework of chiral perturbation theory [ 5 ] . At lowest order the quark mass ratios m u / m d = 0.56 subscript 𝑚 𝑢 subscript 𝑚 𝑑 0.56 m_{u}/m_{d}=0.56 and m s / m d = 20.1 subscript 𝑚 𝑠 subscript 𝑚 𝑑 20.1 m_{s}/m_{d}=20.1 can be determined from the pion and kaon masses [ 6 , 7 ] . Chiral perturbation theory, though, cannot determine the absolute scale of the quark masses; lattice computations are required for this. The strange quark mass can be determined almost directly through lattice computations; few assumptions are needed, which we will explain below.
For the up and down quark masses, more difficult chiral extrapolations are needed. The value of the strange quark mass is required as an input for phenomenological predictions of the Standard Model, such as the CP-violating ratio ϵ ′ / ϵ superscript italic-ϵ ′ italic-ϵ \epsilon^{\prime}/\epsilon . Also for physics beyond the Standard Model the quark masses are very important parameters. 2 The strange quark mass from lattice QCD In the following we will use Wilson’s formulation of lattice QCD, including Symanzik’s O( a 𝑎 a ) improvement. A review of principles of lattice computations can be found in Ref. [ 8 ] .
(13) The subscript “QCD” means that an estimate of the electromagnetic effects has been subtracted from the experimental numbers [ 29 ] in order to obtain a pure QCD kaon mass, as we have on the lattice. The lattice spacing could be set by computing on the lattice the kaon decay constant, a number a F K 𝑎 subscript 𝐹 𝐾 aF_{K} , and dividing it by the experimental value of F K subscript 𝐹 𝐾 F_{K} , which can be obtained from the decay rate of K + ⟶ μ + ν μ ⟶ superscript 𝐾 superscript 𝜇 subscript 𝜈 𝜇 K^{+}\longrightarrow\mu^{+}\nu_{\mu} .
We also mention the most recent compilation of the strange quark mass from QCD sum rules [ 41 ] , which quotes m ¯ s ( 2 GeV ) = 99 ( 28 ) MeV subscript ¯ 𝑚 𝑠 2 GeV 99 28 MeV \overline{m}_{s}(2\,{\rm GeV})=99(28)\,{\rm MeV} , in agreement with the range of the lattice results; see also [ 42 , 43 , 44 ] . In conclusion, Fig. 4 demonstrates that non-perturbative
parameters. The hadron mass calculation in lattice QCD can therefore gives also light quark masses(up, down … uncertainty on the pdf’s. Lepton-lepton, lepton-hadron and hadron-hadron interactions probe com- plementary aspects … in lepton-hadron scattering and of lepton pair pro- duction cross sections in hadron-hadron collisions
Everything we examined (3)
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