Short Range Operator Contributions to 0 decay from LQCD Henry - PowerPoint PPT Presentation

Short Range Operator Contributions to 0 νββ decay from LQCD Henry Monge-Camacho 1 , 2 1 College of William & and Mary 2 LBNL 36th International Symposium on Lattice Field Theory July 24, 2018 Henry Monge-Camacho (W&M, LBNL) July 24, 2018 1 / 14

Motivation ν ν ¯ ? Henry Monge-Camacho (W&M, LBNL) July 24, 2018 2 / 14

Motivation η ββ 0 νββ Half life Light neutrino Heavy neutrino T 1 / 2 = G ( Q ββ , Z ) | M | 2 η ββ 1 � U el m l � U eh / m h 1 m N m e Experiments focused on 0 + → 0 + B. C. Tiburzi, M. L. Wagman, F. Winter, E. Chang, Z. Davoudi, W. Detmold, K. Orginos, M. J. Savage, and P. E. Shanahan (2017). In: Phys. Rev. D96.5, p. 054505. arXiv: 1702.02929 [hep-lat] Henry Monge-Camacho (W&M, LBNL) July 24, 2018 3 / 14

Contributing Diagrams G. Prezeau, M. Ramsey-Musolf, and P. Vogel (2003). In: Phys. Rev. D68, p. 034016. arXiv: hep-ph/0303205 [hep-ph] used Effective Field Theory to Integrate out heavy modes and obtain the contributing operators are found to be: Henry Monge-Camacho (W&M, LBNL) July 24, 2018 4 / 14

Contributing Diagrams G. Prezeau, M. Ramsey-Musolf, and P. Vogel (2003). In: Phys. Rev. D68, p. 034016. arXiv: hep-ph/0303205 [hep-ph] used Effective Field Theory to Integrate out heavy modes and obtain the contributing operators are found to be: Henry Monge-Camacho (W&M, LBNL) July 24, 2018 4 / 14

Contributing Diagrams G. Prezeau, M. Ramsey-Musolf, and P. Vogel (2003). In: Phys. Rev. D68, p. 034016. arXiv: hep-ph/0303205 [hep-ph] used Effective Field Theory to Integrate out heavy modes and obtain the contributing operatorsare found to be: Decay Operators q L τ + γ µ q L )( ¯ q R τ + γ µ q R ) O ++ 1 + = ( ¯ O ++ q R τ + q L ) ( ¯ q R τ + q L ) + ( ¯ q L τ + q R )( ¯ q L τ + q R ) 2 ± = ( ¯ O ++ q L τ + q L )( ¯ q L τ + q L ) + ( ¯ q R τ + q R )( ¯ q R τ + q R ) 3 ± = ( ¯ q L τ + γ µ q L ∓ ¯ q R τ + γ µ q R )( ¯ O ++ q L τ + q R − ¯ q R τ + q L ) 4 ± = ( ¯ q L τ + γ µ q L ± ¯ q R τ + γ µ q R )( ¯ O ++ q L τ + q R + ¯ q R τ + q L ) 5 ± = ( ¯ Henry Monge-Camacho (W&M, LBNL) July 24, 2018 4 / 14

π − → π + Matrix Element The operators contributing to π − → π + process are O ++ 1 + , O ++ 2 + O ++ 3 + and O ′ ++ 1 + , O ′ ++ 2 + (color mixed). The corresponding 3-point correlation functions are computed as follows: Henry Monge-Camacho (W&M, LBNL) July 24, 2018 5 / 14

π − → π + Matrix Element The operators contributing to π − → π + process are O ++ 1 + , O ++ 2 + O ++ 3 + and O ′ ++ 1 + , O ′ ++ 2 + (color mixed). The corresponding 3-point correlation functions are computed as follows: π ´ t 0 t i ¯ d γ 5 u � Henry Monge-Camacho (W&M, LBNL) July 24, 2018 5 / 14

π − → π + Matrix Element The operators contributing to π − → π + process are O ++ 1 + , O ++ 2 + O ++ 3 + and O ′ ++ 1 + , O ′ ++ 2 + (color mixed). The corresponding 3-point correlation functions are computed as follows: π ` π ´ t f t 0 t i � ¯ d γ 5 u ¯ d γ 5 u � Henry Monge-Camacho (W&M, LBNL) July 24, 2018 5 / 14

π − → π + Matrix Element The operators contributing to π − → π + process are O ++ 1 + , O ++ 2 + O ++ 3 + and O ′ ++ 1 + , O ′ ++ 2 + (color mixed). The corresponding 3-point correlation functions are computed as follows: π ` π ´ O t f t 0 t i � ¯ u Γ 1 d ¯ u Γ 2 d | d γ 5 u ¯ d γ 5 u � | ¯ Henry Monge-Camacho (W&M, LBNL) July 24, 2018 5 / 14

π − → π + A. Nicholson et al. (2018). In: arXiv: 1805.02634 [nucl-th] R i ( t ) = a 4 � π | O ++ i + | π � + R e . s . ( t ) ( a 2 Z π 0 ) 3 Henry Monge-Camacho (W&M, LBNL) July 24, 2018 6 / 14

π − → π + Results These matrix elements become inputs for nucleon potentials. For example: g 2 σ 1 · q σ 2 · q V nn → pp τ + 1 τ + A ( | q | ) = − O i ( | q | 2 + m 2 i 2 4 F 2 π ) 2 π Henry Monge-Camacho (W&M, LBNL) July 24, 2018 7 / 14

Non-perturbative Renormalization A. Nicholson et al. (2018). In: arXiv: 1805.02634 [nucl-th] C. C. Chang et al. (2018). In: Nature 558.7708, pp. 91–94. arXiv: 1805.12130 [hep-lat] a15m310 : SMOM γ µ a15m310 : SMOM / q 1 . 2 a12m310 : SMOM γ µ a12m310 : SMOM / q a09m310 : SMOM γ µ 1 . 0 a09m310 : SMOM / q ( Z A /Z V − 1) × 10 3 0 . 8 0 . 6 0 . 4 0 . 2 0 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 µ [GeV] Method RI-SMOM: 1 Three Lattice spacings:0.09,0.12,0.15fm Projectors: γ and / q show agreement after MS conversion Step scaling functions are used to handle reduced renormalization windows (0.15) 1 C. Sturm, Y. Aoki, N. H. Christ, T. Izubuchi, C. T. C. Sachrajda, and A. Soni (2009). In: Phys. Rev. D80, p. 014501. arXiv: 0901.2599 [hep-ph] Henry Monge-Camacho (W&M, LBNL) July 24, 2018 8 / 14

Renormalization Constants Running Renormalization Group ⇒ cont. running Σ ( µ 1 , µ 2 ) = Z ( µ 1 ) Z ( µ 2 ) − 1 In the Lattice: Σ ( µ 1 , µ 2 , a ) = Σ ( µ 1 , µ 2 ) cont + ∆ a 2 Fit assuming smooth µ dependence to obtain Σ ( µ 1 , µ 2 ) cont R. Arthur and P. A. Boyle (2011). In: Phys. Rev. D83, p. 114511. arXiv: 1006.0422 [hep-lat] Henry Monge-Camacho (W&M, LBNL) July 24, 2018 9 / 14

Four-quark Feynman-Hellman Method: π − → π + Analog of method implemented for baryons and bilinear currents 2 � d 4 x ¯ ψΓ 1 ψ ¯ ψΓ 2 ψ ∂ λ E λ = � n | H λ | n � S λ = λ ∂ λ E λ For a meson effective mass: = − ∂ λ C ( t + τ ) + ∂ λ C ( t − τ ) − 2 cosh ( m eff τ ) ∂ λ C ( t ) ∂ m eff � � � 2 τ C ( t ) sinh ( m eff τ ) ∂ λ � λ = 0 � For long enough t ∂ m eff ≈ J 00 � � 2 E 2 ∂ λ � 0 λ = 0 ∂ λ C ( t ) Matrix element is pulled down with ∂ λ � � � d 4 x Ω | T O ( t ) J ( x ) O † ( 0 ) | Ω N ( t ) = 2 C. Bouchard, C. C. Chang, T. Kurth, K. Orginos, and A. Walker-Loud (2017). In: Phys. Rev. D96.1, p. 014504. arXiv: 1612.06963 [hep-lat] Henry Monge-Camacho (W&M, LBNL) July 24, 2018 10 / 14

Lattice Implementation: Brute force calculation on small Lattice: δ ( y 0 − y ) Γ 1 � � π + π − Ω | T O ( t ) J ( x ) O † ( 0 ) | Ω � d 4 x = � y 0 ∈ V t 0 δ ( y 0 − y ) Γ 2 Hubbard-Stratanovich Transformation: � ∞ d 4 x ( ψΓψ ) 2 = α e − λ 2 � � d 4 x { σ 2 d σ e − 4 + λ i σ ( ψΓψ ) } − ∞ D. J. Gross and A. Neveu (1974). In: Phys. Rev. D10, p. 3235 R. L. Stratonovich (1957). In: Doklady Akad. Nauk S.S.S.R. 115, p. 1097,J. Hubbard (1959). In: Phys. Rev. Lett. 3, pp. 77–80 Henry Monge-Camacho (W&M, LBNL) July 24, 2018 11 / 14

Lattice Implementation: Four-quark is recovered after σ integration: = � d σ Numerical implementation: � σ Γ 1 � � π + π − d 4 x Ω | T O ( t ) J ( x ) O † ( 0 ) | Ω � = t 0 � σ Γ 2 Henry Monge-Camacho (W&M, LBNL) July 24, 2018 12 / 14

Conclusions and Future Work: Reproduce π − → π + calculation with the new method Implement calculation using the Hubbard-Stratanovich transformation Apply method to nn → pp calculation Henry Monge-Camacho (W&M, LBNL) July 24, 2018 13 / 14

LBNL: Chia Cheng Chang, Andr´ e Walker-Loud, W&M and LLNL: David Brantley BNL: Enrico Rinaldi FZJ: Evan Berkowitz JLab: B´ alint J´ oo Liverpool Univ.: Nicolas Garron LLNL: Pavlos Vranas,Arjun Gambhir NERSC: Thorsten Kurth UNC: Amy Nicholson nVidia: Kate Clark Funded by: Nuclear Theory for Double-Beta Decay and Fundamental Symmetries (DBD Collaboration, DOE) Henry Monge-Camacho (W&M, LBNL) July 24, 2018 14 / 14