Proton decay matrix element on the lattice with physical pion mass Jun-Sik Yoo (Theory Center, KEK) RIKEN R-CCS Seminar, January 6, 2021 Host: Field Theory Research Team(R-CCS) Jun-Sik Yoo (KEK) Proton Decay 1 / 36 Overview 1 Introduction Baryon Asymmetry in the Universe Proton decay in GUT 2 Lattice simulation Introduction to Lattice QCD Lattice simulation at physical point 3 Matrix elements on lattice Bare Calculation Renormalization 4 Possible extensions 5 Conclusion Jun-Sik Yoo (KEK) Proton Decay 2 / 36 Baryon Asymmetry in the Universe(BAU) Figure: Matter-antimatter asymmetry [worldsciencefestival.com] Jun-Sik Yoo (KEK) Proton Decay 3 / 36 Baryon Asymmetry in the Universe(BAU) Three possible origins of BAU: • In the Early Universe BV heavy particle decay processes • During the EW phase transition 1st order EW phase transition 2nd order PT based on the Higgs mass • After the EWPT (post-sphaleron) EW scale scalar boson decay to 6q; 6¯q Jun-Sik Yoo (KEK) Proton Decay 4 / 36 Baryon Asymmetry in the Universe(BAU) Three possible origins of BAU: • In the Early Universe BV heavy particle decay processes • During the EW phase transition 1st order EW phase transition 2nd order PT based on the Higgs mass • After the EWPT (post-sphaleron) EW scale scalar boson decay to 6q; 6¯q Jun-Sik Yoo (KEK) Proton Decay 5 / 36 Proton decay (SUSY-)GUT provides the effective operator of lowest dimension 6. (a) ∼ ΛGUT (b) ∼ ΛSUSY (c) ∼ ΛEW Figure: Proton decay operator at different scales Model parameters come into Wilson coefficients Yqq; Yql ; Yud ; Yue MHC ml~; mq~; triangle loop integrals, ... whereas the model independent content remains in the effective operator. Each model provides the different list of operators and the lifetime estimates Jun-Sik Yoo (KEK) Proton Decay 6 / 36 Experimental measurement of proton lifetime • Super Kamiokande Water Cherenkov detector Fiducial Volume 22.5kt • Hyper Kamiokande Water Cherenkov detector Fiducial Volume 187kt • DUNE LArTPC Fiducial Volume 40kt (10kt x4) • JUNO Liquid Scintillator(LS) Fiducial Volume 5kt Jun-Sik Yoo (KEK) Proton Decay 7 / 36 Experimental measurement of proton lifetime Current proton decay bound in SK, [Abe, 2018] Jun-Sik Yoo (KEK) Proton Decay 8 / 36 Matrix element in the decay amplitude Proton decay amplitude ¯ ΓΓ ¯ ΓΓ C hΠ`jpiGUT ∼ C hΠ`jOΓΓ0 jpiSM = C v¯`hΠj(¯q q)ΓPΓ0 qjpi, ΓΓ0 where C is a wilson coefficient, Π is a meson, p is a proton and (XY )Γ = (XPΓY ). 0 C hπ j(¯u d)χuLjpi + C hπ j(¯u d)χdLjpi 0 C hK j(¯u s)χuLjpi + C hK j(¯u s)χdLjpi + C hK j(¯u d)χsLjpi Four-fermion effective theory + C hK j(d¯ s)χuLjpi C hη j(¯u d)χuLjpi where χ is a chirality L; R of the decay operator. Jun-Sik Yoo (KEK) Proton Decay 9 / 36 Decay rate The decay rate Γ is calculated from the hadronic matrix element, 0 hΠ(p0)`¯(q)jOΓΓ jN(p; s)i ΓΓ0 2 iq= ΓΓ0 2 =v ¯`PΓ0 W0 (q ) − W1 (q ) uN (p; s) (1.1) mN ΓΓ0 2 =v ¯`PΓ0 W0 (q )uN (p; s) + O(ml =mN )v ¯`uN (p; s) where Π a meson, N a nucleon, and W0;1 decay form factor[Aoki,1999]. Then the decay rate is 2 (m2 − m2 )2 ¯ p Π X I ¯ Γ p ! Π + ` = 3 CI W0 p ! Π + ` : 32πmp I Jun-Sik Yoo (KEK) Proton Decay 10 / 36 Lattice QCD Lattice QCD QCD action implemented on the discretized 4D Euclidean spacetime lattice quark fields defined on the lattice site, gluon fields on the link between them direct computation of path integral (Monte Carlo method) R −S DUD[ ; ¯]Oe E 1 P hOi = = Oi R DUD[ ; ¯]e−SE N i where Oi is a random variable distributed −S along the probability density function e E . R DUD[ ; ¯]e−SE Systematic and statistical error sources Expensive computational cost Discretization error Finite volume effect Jun-Sik Yoo (KEK) Proton Decay 11 / 36 Figure: E.Berkowitz Previous study Jun-Sik Yoo (KEK) Proton Decay 12 / 36 Problems in achieving physical point For a lot of simulation on the lattice, non-physical quark mass mud is adopted for latt computational cost reasons, resulting in mπ > 140 MeV. Inversion in gauge configuration production (RHMC) cost is highly dependent on the quark mass ! 5 7 2 3 2 L MeV fm MeV a mK Cost ∼ Ls C1 + C2 fm mπ a mK fm mπ Inverting Dslash with physical mass - high computational cost S = M−1b, where M is an implementation of Dirac op. D= = @= + m. Finite volume effect - systematic errors of O(mπL) The size of the hadron cloud around the hadron, does not fit into the lattice volume, and it shifts the spectrum. Noise controlled by quark mass - Nucleon and 3/2 pions −m t− 3mπ t [Lepage] the correlation function of nucleon has noise due to pions O(e N 2 ). Jun-Sik Yoo (KEK) Proton Decay 13 / 36 Correlation functions of interpolating operators We define the hadron interpolating operators, abc a T b c JN = (u Cγ5d )u ¯a a Jπ+ = d γ5u a a JK + =s ¯ γ5u ; where u; d; s are quark fields with up, down, and strange flavors. (i) X −i~k·(x~0−~x) X (i) mN + EN −E t ~0 ¯ N e Tr[P+h0jJN (x ; t)JN (~x; 0)j0i] = ZN e 2EN x~0;~x i (i) (i) −E t ~0 Z e Π X −i~p·(x −~x) 0 X Π e h0jJΠ(x~ ; t)J¯Π(~x; 0)j0i = ; 2E (i) x~0;~x i Π (Meson)-(Decay Operator)-(Proton) C 3pt (t; t0) X i(~p0·~x0−~q·~x) 0 = e h0jJΠ(x )O(x)J¯N (x0)j0i ~x;~x0 Jun-Sik Yoo (KEK) Proton Decay 14 / 36 Lattice settings Lattice paramters lattice size 243 × 64 × 24 lattice size 323 × 64 × 12 gauge action Iwasaki-DSDR gauge action Iwasaki-DSDR fermion DWF fermion DWF β 1.633 β 1.75 lattice cutoff a−1 = 1:0230(20)GeV[1] lattice cutoff a−1 = 1:3787(48)GeV[1] ml a 0.00107 ml a 0.0001 mha 0.0850 mha 0.0450 mπa 0:1387 mπa 0:1046 mK a 0.5051 mK a 0:3602 mres 0.0022824(70) mres 0.0018915(75) mπL 3.3 mπL 3.4 Deflated CG 2000+1000 Deflated CG 2000+250 AMA 32+2 AMA 32+1 Ncfg 129 Ncfg 75(out of 112) Jun-Sik Yoo (KEK) Proton Decay 15 / 36 Kinematic Choice Energy-momentum conservation 2 2 q = −ml ∼ 0 on-shell condition Discrete momenta insertion on the lattice π K 3.0 3.0 2.5 2.5 2.0 2.0 n n | | i i p p m m | 1.5 | 1.5 p p 1.0 1.0 012 002 0.5 111 0.5 001 011 011 0.0 001 0.0 111 0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20 q2a2 q2a2 Jun-Sik Yoo (KEK) Proton Decay 16 / 36 Kinematic Choice Energy-momentum conservation 2 2 q = −ml ∼ 0 on-shell condition Discrete momenta insertion on the lattice 2 2 Π n~Π n~N q (GeV ) 24c, 32c [0 0 0] 0.010, -0.012 [1 1 1] π [0 1 0] 0.113, 0.095 [0 0 2] [0 0 0] -0.116, -0.14 [0 0 0] -0.034, -0.042 [0 1 1] K [0 1 0] 0.058, 0.056 [0 0 1] [0 0 0] 0.075, 0.074 Table: Momenta choice for physical kinematics Jun-Sik Yoo (KEK) Proton Decay 17 / 36 Two states fit of two point functions Pion 243 × 64 lattice (a−1 = 1:0230 GeV) Jun-Sik Yoo (KEK) Proton Decay 18 / 36 Two states fit of two point functions Pion 323 × 64 lattice (a−1 = 1:3787 GeV) Jun-Sik Yoo (KEK) Proton Decay 19 / 36 Two states fit of two point functions Kaon 243 × 64 lattice (a−1 = 1:0230 GeV) Jun-Sik Yoo (KEK) Proton Decay 20 / 36 Two states fit of two point functions Kaon 323 × 64 lattice (a−1 = 1:3787 GeV) Jun-Sik Yoo (KEK) Proton Decay 21 / 36 Two states fit of two point functions Proton 243 × 64 lattice (a−1 = 1:0230 GeV) Jun-Sik Yoo (KEK) Proton Decay 22 / 36 Two states fit of two point functions Proton 323 × 64 lattice (a−1 = 1:3787 GeV) Jun-Sik Yoo (KEK) Proton Decay 23 / 36 Two states fit of two point functions Dispersion relation 0.25 physical pion physical kaon physical proton 1.10 32c 0.45 32c 32c 24c 24c 24c 0.20 1.05 0.40 ) 2 0.15 1.00 (GeV 0.35 2 E 0.95 0.10 0.30 0.90 0.05 0.25 0.85 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 ~p2(GeV2) ~p2(GeV2) ~p2(GeV2) Figure: Dispersion relation compared with experimental measurement Jun-Sik Yoo (KEK) Proton Decay 24 / 36 Two states fit of three point functions For linear combinations 0 0 ( ΓΓ ΓΓ (EN −Eπ )(mN +EN ) Tr [P+C3pt ] − q~ · Tr [P+i~γC3pt ] ~q2 Tr[PCX ] = 1 ΓΓ0 ~q2 Tr [P+γj C3pt ] Plateau method q Tr[PC (t0; t; t )] Z (p~0)Z (p~) ΓΓ0 0 X 0 Π N R (t ; t; t0) = 2pt ~0 0 2pt CΠ (p ; t − t)Tr[P+CN (p~; t − t0)] Two states fit q Z i Z j j j Π N i 0 j m + E (ij) X −EΠ(t −t)−EN (t−t0) N N Tr[PCX ] = e j WC 2EΠ i;j=0;1 i 2EN where 8 ~ Γ 2 (EN −Eπ )(mN +EN )−(k·~q) 2 < W0 (q )(mN + EN ) + W1(q )(P = P+) 2 mN WC (q ) = ~ 2 ikj (EN −Eπ )−iqj (mN +EN )±(~q×k)j 2 : −ikj W0(q ) + W1(q )(P = P+γj ) mN Jun-Sik Yoo (KEK) Proton Decay 25 / 36 Two states fit of three point functions 2 0 Figure: Bare W0(q ) form factor for the matrix element hK j(us)LuLjpi on the (top) 24c (bottom) 32c lattice.