Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Proton decay matrix elements on lattice Jun-Sik Yoo 1 1 Department of Physics and Astronomy Stony Brook University 2019 Lattice Workshop for US-Japan Intensity Frontier Incubation, BNL, March 25-27, 2019 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Introduction Figure 1: Proton decay image from (HYPER-K, ) → Π + ¯ p − ℓ 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Introduction Figure 2: Energy scale of search, Zoltan Ligeti 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Baryon asymmetry Nonzero net baryon number n B − ¯ n B ∼ 10 − 10 n γ Sakharov’s conditions ♣ At least one B violating process ♣ C- and CP-violation ♣ interactions outside of thermal equilibrium 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference GUT, SUSY-GUT GUT Symmetry group to be G ⊃ SU (3) C ⊗ SU (2) L ⊗ U (1) Y ♣ Gauge problem ♣ Charge quantization problem ♣ Coupling unification ♣ Baryon asymmetry SUSY-GUT ♣ Superpartners to particles ♣ Better unification at higher scale 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference GUT,SUSY-GUT (a) d=4 operator (b) d=5 operator (c) d=6 operator Figure 3: Possible BV operators in (SUSY-)GUT 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference GUT,SUSY-GUT (a) ∼ Λ GUT (b) ∼ Λ SUSY (c) ∼ Λ EW Figure 4: Proton decay operator at different scales Model parameters come into Wilson coefficients (a) Y qq , Y ql , Y ud , Y ue (b) M H C (c) m ˜ l , m ˜ q , triangle loop integrals, ... 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Effective operators Figure 5: Four-fermion effective operators Effective operator : O ΓΓ ′ = ( qq ) Γ ( q ℓ ) Γ ′ , ( XY ) Γ = ( X T C P Γ Y ) C := (Charge Conjugation Matrix) ℓ |O ΓΓ ′ | p � SM = C ΓΓ ¯ � Π¯ ℓ | p � GUT ∼ C ΓΓ � Π¯ v ℓ � Π | ( qq ) Γ P Γ ′ q | p � , where C ΓΓ ′ is a wilson coefficient, Π is a meson, and p is a proton. 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Decay rate The decay rate Γ is calculated from the hadronic matrix element, � Π( p ′ ) | O ΓΓ ′ ( q ) | N ( p , s ) � � ( q 2 ) − i / q � W ΓΓ ′ W ΓΓ ′ ( q 2 ) = ¯ u N ( p , s ) (1) v ℓ P Γ ′ 0 1 m N v ℓ P Γ ′ W ΓΓ ′ ( q 2 ) u N ( p , s ) + O ( m l / m N ) ¯ = ¯ v ℓ u N ( p , s ) 0 where Π a meson, N a nucleon, and W 0 , 1 decay form factor(AOKI et al., 2000). Then the decay rate is 2 � � = ( m 2 p − m 2 Π ) 2 � � p → Π + ¯ p → Π + ¯ � C I W I � � � � Γ ℓ ℓ . (2) � � 0 32 π m 3 � � p � � I 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Experimental bound Figure 6: Current proton decay bound in SK, (ABE et al., 2018) 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Lattice QCD Formulate a strongly interacting theory on a finite, discrete euclidean spacetime → Lattice QCD Numerically compute observables via importance sampling �O� = 1 � D [Φ] e − S E [Φ] O [Φ] Z = 1 � N k =1 O (Φ k ) N ◦ fully nonperturbative Figure 7: 3D lattice predictions from first principle ◦ fully gauge invariant 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Matrix elements 3pt-function (Meson)-(Decay Operator)-(Proton) Figure 8: 3pt function, Y.Aoki C 3 pt ( t , t ′ ) = � p ′ · � x ′ � 0 | J Π ( x ′ ) O ( x ) ¯ e − i � q · � x e i � J N ( x 0 ) | 0 � � x ,� x ′ = C 2 pt Π ( t ′ − t , � Tr[ PC 2 pt p ′ ) ( t , � p )] p p ′ ) |O| N ( � √ Z Π × � Π( � p ) � ¯ u N ( � p ) � Z p 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Matrix elements √ Z Π C 3 pt ( t , t ′ ) Define the ratio R 3 ( t , t ′ ) = � Z p . C 2 pt p ′ )Tr[ PC 2 pt Π ( t ′ − t ,� ( t ,� p )] p As t → ∞ , R 3 ( t , t ′ ) → � Π( p ′ ) | O ΓΓ ′ ( q ) | N ( p , s ) � , giving decay form factors W 0 , 1 ( q 2 ) 0 ( q 2 ) − iq 4 Tr[ R 3 P L P 4 ] = W Γ L W Γ L 1 ( q 2 ) . m N Tr[ R 3 P L iP 4 γ j ] = q j W Γ L 1 ( q 2 ) m N Momentum transfer is chosen to be q 2 ∼ 0 : � q + � p = � p ′ 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Matrix elements 2pt correlation function Ex) Kaon Interpolating operator for Kaon : J K + ( x ) = ¯ s ( x ) γ 5 u ( x ). C 2 pt � e i � p · � x � 0 | J K ( t , � x ) J † K (0 ,� K ( t , � p ) = 0) | 0 � � x Z K p ) e − E ( � p ) t = 2 E ( � where √ Z K = � 0 | J K | K � 1 Identity matrix used is: 1 = � p | K ; � p � p ) � K ; � p | + . . . � 2 E ( � 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Matrix elements 2pt correlation function Ex) Kaon [ 1 13] 0.750 [ 2 13] [ 3 13] [ 4 13] 0.725 log (C2pt) vs. t 0.700 log(C(t)/C(t+1)) 0.675 0.650 0.625 0.600 0.575 2 4 6 8 10 12 t/a Figure 9: Kaon 2pt function with momentum [0 1 1] in log plot 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Lattice settings RBC/UKQCD generated N f = 2 + 1 dynamic Domain wall Fermion, gauge action Iwasaki-DSDR Lattice size 24 3 × 64( L ∼ 4 . 8 fm ), L 5 = 24, β = 1 . 633, m ℓ a = 0 . 00107 , m h a = 0 . 0850 , m res = 0 . 00228 a − 1 = 1 . 0 GeV, m π a = 139, m K a = 505, m π L ∼ 3 . 4 Deflated CG with 2000 Eigenvectors (basis 1000) Generated 32+1 AMA samples on 102 gauge configurations with 3 source-sink separation, i.e., t sep ∈ { 8 , 9 , 10 } To meet the kinematic condition, chose the most suitable two sets of � p for each meson. 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Matrix elements < K 0 |( us ) L u L | p > 0.075 0.070 0.065 W 0 a 2 0.060 0.055 2 = 2.355 0.050 p = [0, 1, 1] 0.045 0 1 2 3 4 5 6 t / a Figure 10: decay form factor W LL 0 ( p → K 0 e + ) at t sep = 8 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Matrix elements + |(ud) L d R |p > < + |(ud) L d L |p > < < K + |(ds) L u R |p > < K + |(ds) L u L |p > < K + |(ud) L s R |p > < K + |(ud) L s L |p > < K + |(us) L d R |p > < K + |(us) L d L |p > < K 0 |(us) L u R |p > < K 0 |(us) L u L |p > 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 W 0 a 2 Figure 11: Decay matrix elements w/ different src-sink separation { 8,9,10 } 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Matrix elements Figure 12: Decay matrix elements w/ different src-sink separation { 8,9,10 } 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
Introduction Effective operators Matrix elements on lattice Excited States Future projects Reference Matrix elements Bare value, but multiplicative renormalization only − → ratio can be compared with renormalized values � � W ΓΓ ′ ( Channel ) � � W norm 0 = (3) � � 0 W ΓΓ ′ ( � K + | ( ds ) Γ u Γ ′ | p � ) � � 0 � � 2019 Lattice Workshop for US-Japan Intensit JS Yoo Proton Decay / 34
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