# Nonperturbative β function of twelve-flavor SU(3) gauge theory

## Abstract

We study the discrete β function of SU(3) gauge theory with N _{f} = 12 massless fermions in the fundamental representation. Using an nHYP-smeared staggered lattice action and an improved gradient flow running coupling ${\stackrel{~}{g}}_{c}^{2}\left(L\right)$ we determine the continuum-extrapolated discrete β function up to g _{c} ^{2} ≈ 8.2. We observe an IR fixed point at g _{*} ^{2} = 7.3( _{ -2} ^{+ 8} ) in the $c=\sqrt{8t}/L=0.25$ scheme, and g _{*} ^{2} = 7.3( _{-3} ^{+ 6} ) with c = 0.3, combining statistical and systematic uncertainties in quadrature. The systematic effects we investigate include the stability of the (a/L) → 0 extrapolations, the interpolation of ${\stackrel{~}{g}}_{c}^{2}\left(L\right)$ as a function of the bare coupling, the improvement of the gradient flow running coupling, and the discretization of the energy density. In an appendix we observe that the resulting systematic errors increase dramatically upon combining smaller c ≲ 0.2 with smaller L ≤ 12, leading to an IR fixed point at g _{*} ^{2} = 5.9(1.9) in the c = 0.2 scheme, which resolves to g _{*} ^{2} = 6.9( _{-1} ^{+ 6} ) upon considering only L ≥ 16. At the IR fixed point we measure the leading irrelevant critical exponent to be γ _{g} _{*} = 0.26(2), comparable to perturbative estimates.

- Authors:

- Univ. of Colorado, Boulder, CO (United States). Dept. of Physics
- Syracuse Univ., NY (United States). Dept. of Physics; Univ. of Bern (Switzerland). AEC Inst. for Theoretical Physics

- Publication Date:

- Research Org.:
- Syracuse Univ., NY (United States); Univ. of Colorado, Boulder, CO (United States)

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

- OSTI Identifier:
- 1505130

- Grant/Contract Number:
- SC0008669; SC0009998; SC0010005; CNS-0821794

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Volume: 2018; Journal Issue: 2; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; lattice quantum field theory; renormalization group; technicolor and composite models

### Citation Formats

```
Hasenfratz, Anna, and Schaich, David.
```*Nonperturbative β function of twelve-flavor SU(3) gauge theory*. United States: N. p., 2018.
Web. doi:10.1007/jhep02(2018)132.

```
Hasenfratz, Anna, & Schaich, David.
```*Nonperturbative β function of twelve-flavor SU(3) gauge theory*. United States. doi:10.1007/jhep02(2018)132.

```
Hasenfratz, Anna, and Schaich, David. Wed .
"Nonperturbative β function of twelve-flavor SU(3) gauge theory". United States. doi:10.1007/jhep02(2018)132. https://www.osti.gov/servlets/purl/1505130.
```

```
@article{osti_1505130,
```

title = {Nonperturbative β function of twelve-flavor SU(3) gauge theory},

author = {Hasenfratz, Anna and Schaich, David},

abstractNote = {We study the discrete β function of SU(3) gauge theory with N f = 12 massless fermions in the fundamental representation. Using an nHYP-smeared staggered lattice action and an improved gradient flow running coupling g ~ c 2 ( L ) we determine the continuum-extrapolated discrete β function up to g c 2 ≈ 8.2. We observe an IR fixed point at g * 2 = 7.3( -2 + 8 ) in the c = 8 t / L = 0.25 scheme, and g * 2 = 7.3( -3 + 6 ) with c = 0.3, combining statistical and systematic uncertainties in quadrature. The systematic effects we investigate include the stability of the (a/L) → 0 extrapolations, the interpolation of g ~ c 2 ( L ) as a function of the bare coupling, the improvement of the gradient flow running coupling, and the discretization of the energy density. In an appendix we observe that the resulting systematic errors increase dramatically upon combining smaller c ≲ 0.2 with smaller L ≤ 12, leading to an IR fixed point at g * 2 = 5.9(1.9) in the c = 0.2 scheme, which resolves to g * 2 = 6.9( -1 + 6 ) upon considering only L ≥ 16. At the IR fixed point we measure the leading irrelevant critical exponent to be γ g * = 0.26(2), comparable to perturbative estimates.},

doi = {10.1007/jhep02(2018)132},

journal = {Journal of High Energy Physics (Online)},

issn = {1029-8479},

number = 2,

volume = 2018,

place = {United States},

year = {2018},

month = {2}

}