The hadronic contribution to the running of the electroweak mixing - PowerPoint PPT Presentation

The hadronic contribution to the running of the electroweak mixing angle a. Helmholtz-Institut Mainz, Johannes Gutenberg-Universität Mainz c. John von Neumann-Institut für Computing, DESY Zeuthen d. Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University 37th International Symposium on Lattice Field Theory 武汉 , 17th June 2019 Marco Cè a,b Miguel Teseo San José Pérez a,b Antoine Gérardin c Harvey B. Meyer a,b Kohtaroh Miura a,d Konstantin Ottnad b Jonas Wilhelm b Hartmut Wittig a,b b. PRISMA + Cluster of Excellence and Institut für Kernphysik, Johannes Gutenberg-Universität Mainz

introduction – the electroweak mixing angle 𝑋 17/06/2019 The hadronic contribution to the running of the electroweak mixing angle Marco Cè (HIM, JGU Mainz) depends on the renormalization scheme and on the energy scale 𝑔 𝑎 𝑁 2 𝑁 2 𝑕 ′2 1 / 13 the electroweak mixing (Weinberg) angle 𝜄 W parametrizes the mixing between the SU(2) 𝑀 and U(1) 𝑍 sectors of the Standard Model. At tree level, sin 2 𝜄 W = 𝑕 2 + 𝑕 ′2 , where 𝑕 and 𝑕 ′ are the SU(2) 𝑀 and U(1) 𝑍 coupling respectively • it is a free parameter of the Standard Model • sin 2 𝜄 W = 1 − • 𝑎 vector coupling 𝑤 𝑔 = 𝑈 𝑔 − 2𝑅 𝑔 sin 2 𝜄 efg • weak charge of the proton 𝑅 𝑋 (𝑞) ∼ 1 − 4 sin 2 𝜄 W the precise numerical value of sin 2 𝜄 W

the running – a precision test of the SM [PDG 2018] 17/06/2019 The hadronic contribution to the running of the electroweak mixing angle Marco Cè (HIM, JGU Mainz) and global fjts to EW precision data physics Thomson limit [Erler, Ferro-Hernández 2017] theory: running, in the MS scheme [talk by J. Wilhelm, Had. Struct., Fri. 15:00] ⇒ non-perturbative QCD efgects [Becker et al. 2018] at low 𝑅 2 , upcoming experiments: at high 𝑅 2 2 / 13 • measurements at colliders 0.245 RGE Running Particle Threshold Measurements NuTeV NuTeV SLAC-E158 SLAC-E158 • MOLLER @ JLab 0.24 Qweak Qweak ) • P2 @ MESA, Mainz µ APV APV ( W eDIS eDIS 0.235 θ 2 sin LEP 1 LEP 1 Tevatron Tevatron LHC LHC SLC SLC 0.23 • sin 2 𝜄 0 = 0.238 68(5) in the 0.225 − − − − 4 3 2 1 2 3 4 10 10 10 10 1 10 10 10 10 µ [GeV] • the running at scales ≲ 𝛭 QCD is sin 2 𝜄 W (𝑅 2 ) = sin 2 𝜄 0 [1 + 𝛦 sin 2 𝜄 W (𝑅 2 )] afgected by non-perturbative QCD

the hadronic contribution 𝑘 𝛿 17/06/2019 The hadronic contribution to the running of the electroweak mixing angle Marco Cè (HIM, JGU Mainz) [next talk by M. T. San José Pérez] [Burger et al. 2015; Gülpers et al. 2015] 𝜈 , 𝑘 𝑎 𝑑𝛿 𝜈 𝑑, ̄ 4 𝑘 𝑈 3 ̄ 3 𝑑𝛿 𝜈 𝑑, of the e.m. current and the vector part of the 𝑎 current 𝛲 𝑎𝛿 [Jegerlehner 1986; 2011] 4π𝛽 𝛲 𝑎𝛿 3 / 13 𝜉 (0)⟩ the leading hadronic contribution to the running of sin 2 𝜄 W is 𝛦 had sin 2 𝜄 W (𝑅 2 ) = − 𝑆 (𝑅 2 ), 𝑆 (𝑅 2 ) = 𝛲 𝑎𝛿 (𝑅 2 ) − 𝛲 𝑎𝛿 (0), sin 2 𝜄 W proportional to the subtracted hadronic vacuum polarization (𝑅 𝜈 𝑅 𝜉 − 𝜀 𝜈𝜉 𝑅 2 )𝛲 𝑎𝛿 (𝑅 2 ) = 𝛲 𝑎𝛿 𝜈𝜉 (𝑅 2 ) = ∫ d 4 𝑦 𝑓 i𝑅𝑦 ⟨𝑘 𝑎 𝜈 (𝑦)𝑘 𝛿 𝜈 = 2 𝑣𝛿 𝜈 𝑣 − 1 𝑒𝛿 𝜈 𝑒 − 1 𝑡𝛿 𝜈 𝑡 + 2 3 ̄ 3 ̄ 3 ̄ 𝜈 = 1 𝑣𝛿 𝜈 𝑣 − 1 𝑒𝛿 𝜈 𝑒 − 1 𝑡𝛿 𝜈 𝑡 + 1 4 ̄ 4 ̄ 4 ̄ 𝜈 − sin 2 𝜄 W 𝑘 𝛿 𝜈 = 𝑘 𝑈 3 • can be extracted from phenomenology using dispersion relations • or can be computed ab initio on the lattice • similarly, the hadronic contribution to the running of 𝛽 QED is given by 𝛲 𝛿𝛿 𝑆 (𝑅 2 )

the time-momentum representation (TMR) method ∑ 17/06/2019 The hadronic contribution to the running of the electroweak mixing angle Marco Cè (HIM, JGU Mainz) ⇒ no loss of signal in the tail of the connected correlator case, the kernel has a shorter range 𝜈 [Gérardin, Harris, Meyer 2018] [Gérardin et al. 2019; talk by A. Gérardin, Had. Struct., Tue. 14:40] 𝜈 𝑙 (0)⟩, introduced for the HVP contribution to (𝑕 − 2) 𝜈 𝑙=1 ⟨𝑘 𝑎 3 ( [Bernecker, Meyer 2011; Francis et al. 2013] 𝛲 𝑎𝛿 ∞ 0 4 / 13 𝑅𝑦 0 𝐻 𝑎𝛿 (𝑦 0 ) = −1 0 − 4 𝑆 (𝑅 2 ) = ∫ d𝑦 0 𝐻 𝑎𝛿 (𝑦 0 )[𝑦 2 𝑅 2 sin 2 2 )], 3 ∫ d 3 𝑦 𝑙 (𝑦)𝑘 𝛿 ⇒ using correlators from 𝑂 f = 2 + 1 Mainz efgort in computing (𝑕 − 2) HVP • non-perturbatively 𝒫(𝑏) -improved vector currents • two discretizations: local-local and local-conserved • w.r.t. the (𝑕 − 2) HVP • expect 𝛦 had sin 2 𝜄 W to be more sensitive at cut-ofg efgects, especially at high 𝑅 2 • but much simpler large-distance systematic

lattice correlators 1 𝐷 𝑔 1 ,𝑔 2 𝜈𝜉 𝐸 𝑔 1 ,𝑔 2 𝜈𝜉 the 𝑎𝛿 correlator is given by 1 6√3 𝜈𝜉 𝐻 08 , 3𝐻 88 , 𝜈 , … Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 (𝑦)], (𝑦) = −⟨Tr{𝐸 −1 5 / 13 𝜈𝜉 (𝑦) = 6[𝐷 ℓ,ℓ 𝜈𝜉 (𝑦)], 𝐻 88 𝐻 08 1 2𝐷 ℓ,ℓ 2√3[𝐷 ℓ,ℓ 𝐻 33 with SU(3) 𝐺 notation, in the isospin-symmetric limit (light quark ℓ : either 𝑣 or 𝑒 ): 𝜈𝜉 (𝑦) = 1 𝜈𝜉 (𝑦), 𝜈𝜉 (𝑦) = 1 𝜈𝜉 (𝑦) + 2𝐷 𝑡,𝑡 𝜈𝜉 (𝑦) + 2𝐸 ℓ−𝑡,ℓ−𝑡 𝜈𝜉 (𝑦) − 𝐷 𝑡,𝑡 𝜈𝜉 (𝑦) + 𝐸 2ℓ+𝑡,ℓ−𝑡 where the connected and disconnected Wick’s contractions are 𝑔 1 (𝑦, 0)𝛿 𝜈 𝐸 −1 𝑔 2 (0, 𝑦)𝛿 𝜉 }⟩, (𝑦) = ⟨Tr{𝐸 −1 𝑔 1 (𝑦, 𝑦)𝛿 𝜈 } Tr{𝐸 −1 𝑔 2 (0, 0)𝛿 𝜉 }⟩, 𝐻 𝛿𝛿 = 𝐻 33 + 1 𝐻 𝑎𝛿 = ( 2 − sin 2 𝜄 W )(𝐻 𝛿𝛿 ) − where 𝐻 𝛿𝛿 is the e.m. current correlator, relevant for e.g. 𝛦 had 𝛽 QED (𝑅 2 ) , 𝑏 HVP

ensembles 5.4 480 200 4.1 64 128 D200* 4.4 465 285 3.1 48 128 N200* 440 E250 § 345 3.1 48 128 N203* 6.4 410 410 3.1 48 128 N202 4.4 420 4.2 192 from the CLS initiative 2.4 17/06/2019 The hadronic contribution to the running of the electroweak mixing angle Marco Cè (HIM, JGU Mainz) 4.2 475 260 3.2 64 192 J303 4.2 460 345 48 96 128 N302* 5.1 420 420 2.4 0.050 48 128 N300 4.1 490 130 6.2 420 2.1 0.064 5.8 460 280 2.8 32 96 H105* 5.0 440 355 2.8 32 96 H102 415 N101 415 2.8 0.086 32 96 H101 𝑛 𝜌 𝑀 𝑀 [ fm ] 𝑏 [ fm ] 𝑀/𝑏 𝑈 /𝑏 tree-level Lüscher-Weisz gauge action, non-perturbatively 𝒫(𝑏) -improved Wilson fermions, open BCs in time [Bruno et al. 2015, Bruno, Korzec, Schaefer 2017] 32 3.9 128 0.076 96 H200 5.3 460 285 3.7 48 128 N401* 4.3 440 350 48 2.4 32 128 4.1 280 460 5.8 C101* 96 6 / 13 48 4.1 220 470 4.6 S400 𝑛 𝜌 [ MeV ] 𝑛 𝐿 [ MeV ] * disconnected contribution available, § periodic BCs in time

preliminary results 0.008 26(21) E250: physical meson masses, 𝑏 = 0.064 26(74) fm l.c. l.l. 33 0.035 70(36) 0.035 29(36) 88 0.026 00(12) 0.025 59(12) 08 ⎧ 8 ⎪ ⎨ ⎪ ⎩ −0.002 484(39) −0.005 888(40) −0.010 329(41) Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 10 7 / 13 6 4 0 1 2 3 4 5 𝑢 0 𝑅 2 0.00 0.01 0.02 0.03 0.04 08 , conn. 0.05 0.06 0.07 0.08 𝛲 𝑆 (𝑅 2 ) 33 2 88 , conn. 0 𝑅 2 [ GeV 2 ] at 𝑅 2 = 1 GeV 2 𝑅 2 = 0.24 GeV 2 𝛦 had sin 2 𝜄 W (𝑅 2 ) = 𝑅 2 = 1 GeV 2 𝑅 2 = 4.22 GeV 2

preliminary results – including disconnected 08 33 0.030 11(11) 0.029 72(11) 88 0.025 40(5) 0.025 00(5) 08 0.003 93(6) 88 −0.000 32(7) −0.000 32(7) −0.001 09(29) 10 ⎧ ⎪ ⎨ ⎪ ⎩ −0.002 115(10) −0.005 427(14) −0.009 874(15) connected only! Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 l.l. l.c. 8 4 0.00 0.02 0.04 0.06 0.08 𝛲 𝑆 (𝑅 2 ) 33 88 , conn. 08 , conn. 0 1 2 3 5 08 , disc. 6 4 2 0 𝑢 0 𝑅 2 8 / 13 88 , disc. 𝛲 𝑆 (𝑅 2 ) 0.000 −0.001 −0.002 N200: 𝑁 𝜌 ≈ 285 MeV , 𝑏 = 0.064 26(74) fm 𝑅 2 [ GeV 2 ] at 𝑅 2 = 1 GeV 2 𝑅 2 = 0.24 GeV 2 𝛦 had sin 2 𝜄 W (𝑅 2 ) = 𝑅 2 = 1 GeV 2 𝑅 2 = 4.22 GeV 2

preliminary results – including disconnected −0.000 32(7) l.l. 33 0.030 11(11) 0.029 72(11) 88 0.025 40(5) 0.025 00(5) 08 0.003 93(6) 88 −0.000 32(7) 08 10 −0.001 09(29) ⎧ ⎪ ⎨ ⎪ ⎩ −0.002 138(13) −0.005 457(16) −0.009 905(17) Marco Cè (HIM, JGU Mainz) The hadronic contribution to the running of the electroweak mixing angle 17/06/2019 l.c. 8 / 13 8 4 0.00 0.02 0.04 0.06 0.08 𝛲 𝑆 (𝑅 2 ) 33 88 , conn. 08 , conn. 0 1 3 2 5 88 , disc. 6 4 2 0 𝑢 0 𝑅 2 08 , disc. 𝛲 𝑆 (𝑅 2 ) 0.000 −0.001 −0.002 N200: 𝑁 𝜌 ≈ 285 MeV , 𝑏 = 0.064 26(74) fm 𝑅 2 [ GeV 2 ] at 𝑅 2 = 1 GeV 2 𝑅 2 = 0.24 GeV 2 𝛦 had sin 2 𝜄 W (𝑅 2 ) = 𝑅 2 = 1 GeV 2 𝑅 2 = 4.22 GeV 2