On the geometry of operator mixing in massless QCD-like theories
Abstract We revisit the operator mixing in massless QCD-like theories. In particular, we address the problem of determining under which conditions a renormalization scheme exists where the renormalized mixing matrix in the coordinate representation, $$Z(x, \mu )$$ Z ( x , μ ) , is diagonalizable to...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-08-01
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| Series: | European Physical Journal C: Particles and Fields |
| Online Access: | https://doi.org/10.1140/epjc/s10052-021-09543-5 |
| _version_ | 1818435246002012160 |
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| author | Marco Bochicchio |
| author_facet | Marco Bochicchio |
| author_sort | Marco Bochicchio |
| collection | DOAJ |
| description | Abstract We revisit the operator mixing in massless QCD-like theories. In particular, we address the problem of determining under which conditions a renormalization scheme exists where the renormalized mixing matrix in the coordinate representation, $$Z(x, \mu )$$ Z ( x , μ ) , is diagonalizable to all perturbative orders. As a key step, we provide a differential-geometric interpretation of renormalization that allows us to apply the Poincaré-Dulac theorem to the problem above: We interpret a change of renormalization scheme as a (formal) holomorphic gauge transformation, $$-\frac{\gamma (g)}{\beta (g)}$$ - γ ( g ) β ( g ) as a (formal) meromorphic connection with a Fuchsian singularity at $$g=0$$ g = 0 , and $$Z(x,\mu )$$ Z ( x , μ ) as a Wilson line, with $$\gamma (g)=\gamma _0 g^2 + \cdots $$ γ ( g ) = γ 0 g 2 + ⋯ the matrix of the anomalous dimensions and $$\beta (g)=-\beta _0 g^3 +\cdots $$ β ( g ) = - β 0 g 3 + ⋯ the beta function. As a consequence of the Poincaré-Dulac theorem, if the eigenvalues $$\lambda _1, \lambda _2, \ldots $$ λ 1 , λ 2 , … of the matrix $$\frac{\gamma _0}{\beta _0}$$ γ 0 β 0 , in nonincreasing order $$\lambda _1 \ge \lambda _2 \ge \cdots $$ λ 1 ≥ λ 2 ≥ ⋯ , satisfy the nonresonant condition $$\lambda _i -\lambda _j -2k \ne 0$$ λ i - λ j - 2 k ≠ 0 for $$i\le j$$ i ≤ j and k a positive integer, then a renormalization scheme exists where $$-\frac{\gamma (g)}{\beta (g)} = \frac{\gamma _0}{\beta _0} \frac{1}{g}$$ - γ ( g ) β ( g ) = γ 0 β 0 1 g is one-loop exact to all perturbative orders. If in addition $$\frac{\gamma _0}{\beta _0}$$ γ 0 β 0 is diagonalizable, $$Z(x, \mu )$$ Z ( x , μ ) is diagonalizable as well, and the mixing reduces essentially to the multiplicatively renormalizable case. We also classify the remaining cases of operator mixing by the Poincaré–Dulac theorem. |
| first_indexed | 2024-12-14T16:49:50Z |
| format | Article |
| id | doaj.art-16064adf0a414764ac72ef586c24033d |
| institution | Directory Open Access Journal |
| issn | 1434-6044 1434-6052 |
| language | English |
| last_indexed | 2024-12-14T16:49:50Z |
| publishDate | 2021-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | European Physical Journal C: Particles and Fields |
| spelling | doaj.art-16064adf0a414764ac72ef586c24033d2022-12-21T22:54:04ZengSpringerOpenEuropean Physical Journal C: Particles and Fields1434-60441434-60522021-08-018181510.1140/epjc/s10052-021-09543-5On the geometry of operator mixing in massless QCD-like theoriesMarco Bochicchio0Physics Department, INFN Roma1Abstract We revisit the operator mixing in massless QCD-like theories. In particular, we address the problem of determining under which conditions a renormalization scheme exists where the renormalized mixing matrix in the coordinate representation, $$Z(x, \mu )$$ Z ( x , μ ) , is diagonalizable to all perturbative orders. As a key step, we provide a differential-geometric interpretation of renormalization that allows us to apply the Poincaré-Dulac theorem to the problem above: We interpret a change of renormalization scheme as a (formal) holomorphic gauge transformation, $$-\frac{\gamma (g)}{\beta (g)}$$ - γ ( g ) β ( g ) as a (formal) meromorphic connection with a Fuchsian singularity at $$g=0$$ g = 0 , and $$Z(x,\mu )$$ Z ( x , μ ) as a Wilson line, with $$\gamma (g)=\gamma _0 g^2 + \cdots $$ γ ( g ) = γ 0 g 2 + ⋯ the matrix of the anomalous dimensions and $$\beta (g)=-\beta _0 g^3 +\cdots $$ β ( g ) = - β 0 g 3 + ⋯ the beta function. As a consequence of the Poincaré-Dulac theorem, if the eigenvalues $$\lambda _1, \lambda _2, \ldots $$ λ 1 , λ 2 , … of the matrix $$\frac{\gamma _0}{\beta _0}$$ γ 0 β 0 , in nonincreasing order $$\lambda _1 \ge \lambda _2 \ge \cdots $$ λ 1 ≥ λ 2 ≥ ⋯ , satisfy the nonresonant condition $$\lambda _i -\lambda _j -2k \ne 0$$ λ i - λ j - 2 k ≠ 0 for $$i\le j$$ i ≤ j and k a positive integer, then a renormalization scheme exists where $$-\frac{\gamma (g)}{\beta (g)} = \frac{\gamma _0}{\beta _0} \frac{1}{g}$$ - γ ( g ) β ( g ) = γ 0 β 0 1 g is one-loop exact to all perturbative orders. If in addition $$\frac{\gamma _0}{\beta _0}$$ γ 0 β 0 is diagonalizable, $$Z(x, \mu )$$ Z ( x , μ ) is diagonalizable as well, and the mixing reduces essentially to the multiplicatively renormalizable case. We also classify the remaining cases of operator mixing by the Poincaré–Dulac theorem.https://doi.org/10.1140/epjc/s10052-021-09543-5 |
| spellingShingle | Marco Bochicchio On the geometry of operator mixing in massless QCD-like theories European Physical Journal C: Particles and Fields |
| title | On the geometry of operator mixing in massless QCD-like theories |
| title_full | On the geometry of operator mixing in massless QCD-like theories |
| title_fullStr | On the geometry of operator mixing in massless QCD-like theories |
| title_full_unstemmed | On the geometry of operator mixing in massless QCD-like theories |
| title_short | On the geometry of operator mixing in massless QCD-like theories |
| title_sort | on the geometry of operator mixing in massless qcd like theories |
| url | https://doi.org/10.1140/epjc/s10052-021-09543-5 |
| work_keys_str_mv | AT marcobochicchio onthegeometryofoperatormixinginmasslessqcdliketheories |