# Proton-Proton Fusion and Tritium $\beta $ Decay from Lattice Quantum Chromodynamics

## Abstract

The nuclear matrix element determining the $$pp\to d e^+ \nu$$ fusion cross section and the Gamow-Teller matrix element contributing to tritium $$\beta$$-decay are calculated with lattice Quantum Chromodynamics (QCD) for the first time. Using a new implementation of the background field method, these quantities are calculated at the SU(3)-flavor-symmetric value of the quark masses, corresponding to a pion mass of $$m_\pi$$ ~ 806 MeV. The Gamow-Teller matrix element in tritium is found to be 0.979(03)(10) at these quark masses, which is within $$2\sigma$$ of the experimental value. Assuming that the short-distance correlated two-nucleon contributions to the matrix element (meson-exchange currents) depend only mildly on the quark masses, as seen for the analogous magnetic interactions, the calculated $$pp\to d e^+ \nu$$ transition matrix element leads to a fusion cross section at the physical quark masses that is consistent with its currently accepted value. Moreover, the leading two-nucleon axial counterterm of pionless effective field theory is determined to be $$L_{1,A}=3.9(0.1)(1.0)(0.3)(0.9)\ {\rm fm}^3$$ at a renormalization scale set by the physical pion mass, also in agreement with the accepted phenomenological range. This work concretely demonstrates that weak transition amplitudes in few-nucleon systems can be studied directly from the fundamental quark and gluon degrees of freedom and opens the way for subsequent investigations of many important quantities in nuclear physics.

- Authors:

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF); Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- Contributing Org.:
- NPLQCD Collaboration

- OSTI Identifier:
- 1393535

- Alternate Identifier(s):
- OSTI ID: 1374773

- Report Number(s):
- JLAB-THY-16-2362; DOE/OR/23177-3961; arXiv:1610.04545

Journal ID: ISSN 0031-9007; PRLTAO; TRN: US1702835

- Grant/Contract Number:
- NSF PHY11-25915; PHY1206498; PHY15-15738; AC02-05CH11231; AC05-00OR22725; SC001347; SC00-10337; FG02-00ER41132; SC0010495; SC0011090; FG02-04ER41302; AC05-06OR23177

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review Letters

- Additional Journal Information:
- Journal Volume: 119; Journal Issue: 6; Journal ID: ISSN 0031-9007

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Savage, Martin J., Shanahan, Phiala E., Tiburzi, Brian C., Wagman, Michael L., Winter, Frank, Beane, Silas R., Chang, Emmanuel, Davoudi, Zohreh, Detmold, William, and Orginos, Kostas. Proton-Proton Fusion and Tritium β Decay from Lattice Quantum Chromodynamics. United States: N. p., 2017.
Web. https://doi.org/10.1103/PhysRevLett.119.062002.
```

```
Savage, Martin J., Shanahan, Phiala E., Tiburzi, Brian C., Wagman, Michael L., Winter, Frank, Beane, Silas R., Chang, Emmanuel, Davoudi, Zohreh, Detmold, William, & Orginos, Kostas. Proton-Proton Fusion and Tritium β Decay from Lattice Quantum Chromodynamics. United States. https://doi.org/10.1103/PhysRevLett.119.062002
```

```
Savage, Martin J., Shanahan, Phiala E., Tiburzi, Brian C., Wagman, Michael L., Winter, Frank, Beane, Silas R., Chang, Emmanuel, Davoudi, Zohreh, Detmold, William, and Orginos, Kostas. Thu .
"Proton-Proton Fusion and Tritium β Decay from Lattice Quantum Chromodynamics". United States. https://doi.org/10.1103/PhysRevLett.119.062002. https://www.osti.gov/servlets/purl/1393535.
```

```
@article{osti_1393535,
```

title = {Proton-Proton Fusion and Tritium β Decay from Lattice Quantum Chromodynamics},

author = {Savage, Martin J. and Shanahan, Phiala E. and Tiburzi, Brian C. and Wagman, Michael L. and Winter, Frank and Beane, Silas R. and Chang, Emmanuel and Davoudi, Zohreh and Detmold, William and Orginos, Kostas},

abstractNote = {The nuclear matrix element determining the $pp\to d e^+ \nu$ fusion cross section and the Gamow-Teller matrix element contributing to tritium $\beta$-decay are calculated with lattice Quantum Chromodynamics (QCD) for the first time. Using a new implementation of the background field method, these quantities are calculated at the SU(3)-flavor-symmetric value of the quark masses, corresponding to a pion mass of $m_\pi$ ~ 806 MeV. The Gamow-Teller matrix element in tritium is found to be 0.979(03)(10) at these quark masses, which is within $2\sigma$ of the experimental value. Assuming that the short-distance correlated two-nucleon contributions to the matrix element (meson-exchange currents) depend only mildly on the quark masses, as seen for the analogous magnetic interactions, the calculated $pp\to d e^+ \nu$ transition matrix element leads to a fusion cross section at the physical quark masses that is consistent with its currently accepted value. Moreover, the leading two-nucleon axial counterterm of pionless effective field theory is determined to be $L_{1,A}=3.9(0.1)(1.0)(0.3)(0.9)\ {\rm fm}^3$ at a renormalization scale set by the physical pion mass, also in agreement with the accepted phenomenological range. This work concretely demonstrates that weak transition amplitudes in few-nucleon systems can be studied directly from the fundamental quark and gluon degrees of freedom and opens the way for subsequent investigations of many important quantities in nuclear physics.},

doi = {10.1103/PhysRevLett.119.062002},

journal = {Physical Review Letters},

number = 6,

volume = 119,

place = {United States},

year = {2017},

month = {8}

}

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