# Dirac spectrum and chiral condensate for QCD at fixed $\theta $ angle

## Abstract

We analyze the mass dependence of the chiral condensate for QCD at nonzero θ angle and find that in general the discontinuity of the chiral condensate is not on the support of the Dirac spectrum. To understand this behavior we decompose the spectral density and the chiral condensate into contributions from the zero modes, the quenched part, and a remainder which is sensitive to the fermion determinant and is referred to as the dynamical part. We obtain general formulas for the contributions of the zero modes. Expressions for the quenched part, valid for an arbitrary number of flavors, and for the dynamical part, valid for one and two flavors, are derived in the microscopic domain of QCD. We find that at nonzero θ angle the quenched and dynamical parts of the Dirac spectral density are strongly oscillating with an amplitude that increases exponentially with the volume V and a period of order of 1/V. The quenched part of the chiral condensate becomes exponentially large at θ ≠ 0 , but this divergence is canceled by the contribution from the zero modes. The oscillatory behavior of the dynamical part of the density is essential for moving the discontinuity of the chiralmore »

- Authors:

- Publication Date:

- Research Org.:
- State Univ. of New York (SUNY), Stony Brook, NY (United States)

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

- OSTI Identifier:
- 1509927

- Alternate Identifier(s):
- OSTI ID: 1610130

- Grant/Contract Number:
- FG02-88ER40388

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 99 Journal Issue: 7; 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; astronomy & astrophysics; physics; chiral perturbation theory; nonperturbative effects in field theory; quantum chromodynamics; spontaneous symmetry breaking

### Citation Formats

```
Kieburg, M., Verbaarschot, J. J. M., and Wettig, T. Dirac spectrum and chiral condensate for QCD at fixed θ angle. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.99.074515.
```

```
Kieburg, M., Verbaarschot, J. J. M., & Wettig, T. Dirac spectrum and chiral condensate for QCD at fixed θ angle. United States. https://doi.org/10.1103/PhysRevD.99.074515
```

```
Kieburg, M., Verbaarschot, J. J. M., and Wettig, T. Tue .
"Dirac spectrum and chiral condensate for QCD at fixed θ angle". United States. https://doi.org/10.1103/PhysRevD.99.074515.
```

```
@article{osti_1509927,
```

title = {Dirac spectrum and chiral condensate for QCD at fixed θ angle},

author = {Kieburg, M. and Verbaarschot, J. J. M. and Wettig, T.},

abstractNote = {We analyze the mass dependence of the chiral condensate for QCD at nonzero θ angle and find that in general the discontinuity of the chiral condensate is not on the support of the Dirac spectrum. To understand this behavior we decompose the spectral density and the chiral condensate into contributions from the zero modes, the quenched part, and a remainder which is sensitive to the fermion determinant and is referred to as the dynamical part. We obtain general formulas for the contributions of the zero modes. Expressions for the quenched part, valid for an arbitrary number of flavors, and for the dynamical part, valid for one and two flavors, are derived in the microscopic domain of QCD. We find that at nonzero θ angle the quenched and dynamical parts of the Dirac spectral density are strongly oscillating with an amplitude that increases exponentially with the volume V and a period of order of 1/V. The quenched part of the chiral condensate becomes exponentially large at θ ≠ 0 , but this divergence is canceled by the contribution from the zero modes. The oscillatory behavior of the dynamical part of the density is essential for moving the discontinuity of the chiral condensate away from the support of the Dirac spectrum. As important by-products of this work we obtain analytical expressions for the microscopic spectral density of the Dirac operator at nonzero θ angle for both one- and two-flavor QCD with nonzero quark masses.},

doi = {10.1103/PhysRevD.99.074515},

journal = {Physical Review D},

number = 7,

volume = 99,

place = {United States},

year = {2019},

month = {4}

}

https://doi.org/10.1103/PhysRevD.99.074515

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