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Short Range Operator Contributions to 0 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,

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SLIDE 1

Short Range Operator Contributions to 0νββ decay from LQCD

Henry Monge-Camacho1,2

1 College of William & and Mary 2LBNL

36th International Symposium on Lattice Field Theory July 24, 2018

Henry Monge-Camacho (W&M, LBNL) July 24, 2018 1 / 14

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SLIDE 2

Motivation

ν ? ¯ ν

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

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SLIDE 3

Motivation

0νββ Half life

1 T1/2 = G(Qββ, Z)|M|2ηββ

ηββ Light neutrino Heavy neutrino

1 me

Uelml mN Ueh/mh 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

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SLIDE 4

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

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SLIDE 5

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

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SLIDE 6

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

O++

1+ = (¯

qLτ+γµqL)(¯ qRτ+γµqR) O++

2± = (¯

qRτ+qL) (¯ qRτ+qL) + (¯ qLτ+qR)(¯ qLτ+qR) O++

3± = (¯

qLτ+qL)(¯ qLτ+qL) + (¯ qRτ+qR)(¯ qRτ+qR) O++

4± = (¯

qLτ+γµqL ∓ ¯ qRτ+γµqR)(¯ qLτ+qR − ¯ qRτ+qL) O++

5± = (¯

qLτ+γµqL ± ¯ qRτ+γµqR)(¯ qLτ+qR + ¯ qRτ+qL)

Henry Monge-Camacho (W&M, LBNL) July 24, 2018 4 / 14

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SLIDE 7

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

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SLIDE 8

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´ t0 ti

¯ dγ5u

Henry Monge-Camacho (W&M, LBNL) July 24, 2018 5 / 14

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SLIDE 9

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´ t0 tf ti

¯

dγ5u ¯ dγ5u

Henry Monge-Camacho (W&M, LBNL) July 24, 2018 5 / 14

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SLIDE 10

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 t0 tf ti

¯

dγ5u |¯ u Γ 1 d ¯ u Γ 2 d| ¯ dγ5u

Henry Monge-Camacho (W&M, LBNL) July 24, 2018 5 / 14

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SLIDE 11

π− → π+

  • A. Nicholson et al. (2018). In: arXiv: 1805.02634 [nucl-th]

Ri(t) = a4 π| O++

i+ |π

(a2Z π

0 )3

+ Re.s.(t)

Henry Monge-Camacho (W&M, LBNL) July 24, 2018 6 / 14

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SLIDE 12

π− → π+ Results

These matrix elements become inputs for nucleon potentials. For example: V nn→pp

i

(|q|) = −Oi g2

A

4F 2

π

τ+

1 τ+ 2

σ1 · qσ2 · q (|q|2 + m2

π)2

Henry Monge-Camacho (W&M, LBNL) July 24, 2018 7 / 14

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SLIDE 13

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]

1.5 2.0 2.5 3.0 3.5 4.0 4.5 µ [GeV] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (ZA/ZV − 1)×103

a15m310 : SMOMγµ a12m310 : SMOMγµ a09m310 : SMOMγµ a15m310 : SMOM/

q

a12m310 : SMOM/

q

a09m310 : SMOM/

q

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)

  • 1C. 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

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SLIDE 14

Renormalization Constants Running

Renormalization Group ⇒ cont. running Σ(µ1, µ2) = Z(µ1)Z(µ2)−1 In the Lattice: Σ(µ1, µ2, a) = Σ(µ1, µ2)cont + ∆a2 Fit assuming smooth µ dependence to obtain Σ(µ1, µ2)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

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SLIDE 15

Four-quark Feynman-Hellman Method: π− → π+

Analog of method implemented for baryons and bilinear currents 2 ∂λEλ = n| Hλ |n Sλ = λ

  • d4x ¯

ψΓ 1ψ ¯ ψΓ 2ψ

∂λEλ

For a meson effective mass: ∂meff ∂λ

  • λ=0

= −∂λC(t + τ) + ∂λC(t − τ) − 2cosh(meff τ)∂λC(t) 2τC(t)sinh(meff τ) For long enough t ∂meff

∂λ

  • λ=0

≈ J00

2E 2

∂λC(t)

Matrix element is pulled down with ∂λ N(t) =

  • d4x
  • Ω|TO(t)J (x)O†(0)|Ω
  • 2C. 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

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SLIDE 16

Lattice Implementation:

Brute force calculation on small Lattice:

π+

  • d4x
  • Ω|TO(t)J (x)O†(0)|Ω
  • =

y0 ∈ V

π−

δ(y0 − y)Γ1 δ(y0 − y)Γ2 t

Hubbard-Stratanovich Transformation: e−λ2

d4x(ψΓψ)2 = α ∞

  • −∞

dσe−

  • d4x{ σ2

4 +λiσ(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

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SLIDE 17

Lattice Implementation:

Four-quark is recovered after σ integration:

=

  • dσ

Numerical implementation:

π+

  • d4x
  • Ω|TO(t)J (x)O†(0)|Ω
  • =

π−

σΓ2

  • σΓ1
  • t

Henry Monge-Camacho (W&M, LBNL) July 24, 2018 12 / 14

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SLIDE 18

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

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SLIDE 19

LBNL: Chia Cheng Chang, Andr´ e Walker-Loud, W&M and LLNL: David Brantley BNL: Enrico Rinaldi FZJ: Evan Berkowitz JLab: B´ alint J´

  • 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