# Operator mixing in the $\epsilon $ -expansion: Scheme and evanescent-operator independence

## Abstract

We consider theories with fermionic degrees of freedom that have a fixed point of Wilson–Fisher type in noninteger dimension $$d=4{-}2{\epsilon}$$. Due to the presence of evanescent operators, i.e., operators that vanish in integer dimensions, these theories contain families of infinitely many operators that can mix with each other under renormalization. We clarify the dependence of the corresponding anomalous-dimension matrix on the choice of renormalization scheme beyond leading order in $${\epsilon}$$-expansion. In standard choices of scheme, we find that eigenvalues at the fixed point cannot be extracted from a finite-dimensional block. We illustrate in examples a truncation approach to compute the eigenvalues. These are observable scaling dimensions, and, indeed, we find that the dependence on the choice of scheme cancels. As an application, we obtain the IR scaling dimension of four-fermion operators in QED in $$d=4{-}2{\epsilon}$$ at order $$\mathcal{O}({{\epsilon}}^{2})$$.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of Chicago, IL (United States)

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

- OSTI Identifier:
- 1425520

- Alternate Identifier(s):
- OSTI ID: 1498881

- Grant/Contract Number:
- SC0009924

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 97 Journal Issue: 6; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; conformal field theory; critical exponents; lower-dimensional field theories; quantum electrodynamics; quantum spin liquid; renormalization group; strongly correlated systems; Feynman diagrams

### Citation Formats

```
Di Pietro, Lorenzo, and Stamou, Emmanuel. Operator mixing in the ε -expansion: Scheme and evanescent-operator independence. United States: N. p., 2018.
Web. doi:10.1103/PhysRevD.97.065007.
```

```
Di Pietro, Lorenzo, & Stamou, Emmanuel. Operator mixing in the ε -expansion: Scheme and evanescent-operator independence. United States. doi:10.1103/PhysRevD.97.065007.
```

```
Di Pietro, Lorenzo, and Stamou, Emmanuel. Mon .
"Operator mixing in the ε -expansion: Scheme and evanescent-operator independence". United States. doi:10.1103/PhysRevD.97.065007.
```

```
@article{osti_1425520,
```

title = {Operator mixing in the ε -expansion: Scheme and evanescent-operator independence},

author = {Di Pietro, Lorenzo and Stamou, Emmanuel},

abstractNote = {We consider theories with fermionic degrees of freedom that have a fixed point of Wilson–Fisher type in noninteger dimension $d=4{-}2{\epsilon}$. Due to the presence of evanescent operators, i.e., operators that vanish in integer dimensions, these theories contain families of infinitely many operators that can mix with each other under renormalization. We clarify the dependence of the corresponding anomalous-dimension matrix on the choice of renormalization scheme beyond leading order in ${\epsilon}$-expansion. In standard choices of scheme, we find that eigenvalues at the fixed point cannot be extracted from a finite-dimensional block. We illustrate in examples a truncation approach to compute the eigenvalues. These are observable scaling dimensions, and, indeed, we find that the dependence on the choice of scheme cancels. As an application, we obtain the IR scaling dimension of four-fermion operators in QED in $d=4{-}2{\epsilon}$ at order $\mathcal{O}({{\epsilon}}^{2})$.},

doi = {10.1103/PhysRevD.97.065007},

journal = {Physical Review D},

number = 6,

volume = 97,

place = {United States},

year = {2018},

month = {3}

}

DOI: 10.1103/PhysRevD.97.065007

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