# Phase ambiguity of the measure for continuum Majorana fermions

## Abstract

Integrating over a continuum Majorana fermion formally yields a functional pfaffian. We show that the phase of this pfaffian is ambiguous, as it depends on the choice of basis. This ambiguity is naturally resolved within a non-perturbative lattice definition, allowing us to discuss the relation between the phase of the lattice pfaffian and the effective θ angle of the theory. We also resolve an apparent paradox regarding the induced θ angle when a theory of N Dirac fermions in a real representation of the gauge group is re-expressed in terms of 2N Majorana fermions. We discuss how all this is reflected in chiral perturbation theory.

- Authors:

- Publication Date:

- Research Org.:
- San Francisco State Univ., CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); Israel Science Foundation (ISF)

- OSTI Identifier:
- 1550589

- Alternate Identifier(s):
- OSTI ID: 1594606

- Report Number(s):
- arXiv:1904.08600v3

Journal ID: ISSN 2470-0010; PRVDAQ; 034507

- Grant/Contract Number:
- FG03-92ER40711; SC0013682; 491/17

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 100 Journal Issue: 3; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Effective field theory; Lattice field theory; Path integrals

### Citation Formats

```
Golterman, Maarten, and Shamir, Yigal. Phase ambiguity of the measure for continuum Majorana fermions. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.100.034507.
```

```
Golterman, Maarten, & Shamir, Yigal. Phase ambiguity of the measure for continuum Majorana fermions. United States. doi:10.1103/PhysRevD.100.034507.
```

```
Golterman, Maarten, and Shamir, Yigal. Fri .
"Phase ambiguity of the measure for continuum Majorana fermions". United States. doi:10.1103/PhysRevD.100.034507.
```

```
@article{osti_1550589,
```

title = {Phase ambiguity of the measure for continuum Majorana fermions},

author = {Golterman, Maarten and Shamir, Yigal},

abstractNote = {Integrating over a continuum Majorana fermion formally yields a functional pfaffian. We show that the phase of this pfaffian is ambiguous, as it depends on the choice of basis. This ambiguity is naturally resolved within a non-perturbative lattice definition, allowing us to discuss the relation between the phase of the lattice pfaffian and the effective θ angle of the theory. We also resolve an apparent paradox regarding the induced θ angle when a theory of N Dirac fermions in a real representation of the gauge group is re-expressed in terms of 2N Majorana fermions. We discuss how all this is reflected in chiral perturbation theory.},

doi = {10.1103/PhysRevD.100.034507},

journal = {Physical Review D},

number = 3,

volume = 100,

place = {United States},

year = {2019},

month = {8}

}

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DOI: 10.1103/PhysRevD.100.034507

DOI: 10.1103/PhysRevD.100.034507

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