nLab differential renormalization

Idea

In perturbative quantum field theory, differential renormalization (Freedman-Johnson-Latorre 92) is a method of ("re"-)normalization which makes choices for defining time-ordered products/Feynman amplitudes at coincident interaction vertices.

Differential normalization follows directly (Prange 97) from Epstein-Glaser renormalization in the guise of extensions of distributions of time-ordered products/Feynman amplitude to coinciding interaction points (this prop.):

In the proof of that construction (this prop) over Minkowski spacetime $\mathbb{R}^{p,1}$, a choice of ("re"-)normalization is embodied at each order $k \in \mathbb{N}$ (number of vertices in Feynman diagrams) by a choice of projection operator

$$p_{\rho_k} \;\colon\; \mathcal{D}(\mathbb{R}^{(p+1)(k-1)}) \longrightarrow \mathcal{D}_{\rho_k}(\mathbb{R}^{(p+1)(k-1)})$$

(this equation) from all test functions to those that vanish to order $\rho_k$ at the origin. (This projction is often denoted “$W$”, see this remark).

Restricted to the image of this projector the relevant time-ordered product $T_k$ has a unique extension of distributions to the origin, and this defines the ("re"-)normalization.

This defines a new distribution $T_{k,R}$ by

$$\left\langle T_{k,R}, b \right\rangle \;\coloneqq\; \left\langle T_k, p_{\rho_k}(b)\right\rangle \,.$$

Differential renormalization focuses on manipulating theses expressions $T_{k,R}$ (Prange 97, section 1.1).

References

The concept is due to

• D. Z. Freedman, K. Johnson, J. I. Latorre, Differential regularization and renormalization: a new method of calculation in quantum field theory, Nucl. Phys. B 371 (1992) 353-414

• J. I. Latorre, X. Vilasis-Cardona, Systematic Differential Renormalization to All Orders, Ann. Phys. (N.Y.) 231 (1994) 149

Discussion in causal perturbation theory/relation to Epstein-Glaser renormalization is due to

• Dirk Prange, Epstein-Glaser renormalization and differential renormalization, J. Phys. A 32, 2225 (1999) (arXiv:hep-th/9710225)

• Jose Gracia-Bondia, Differential renormalization and Epstein-Glaser renormalization, Mod. Phys. Lett. A, 16, 281 (2001). (spire)

• Michael Dütsch, Klaus Fredenhagen, Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity, Reviews in Mathematical Physics, Volume 16, Issue 10, November 2004 (arXiv:hep-th/0403213)

• Michael Dütsch, Klaus Fredenhagen, Kai Johannes Keller, Katarzyna Rejzner, Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization, J. Math. Phy. 55(12), 122303 (2014) (arXiv:1311.5424)

• José Gracia-Bondía, Heidy Gutiérrez, Joseph C. Várilly, Improved Epstein-Glaser renormalization in $x$-space versus differential renormalization, Nuclear Physics B 886 (2014), 824-869 (arXiv:1403.1785)

Discussion of application to anomalous magnetic moment and supergravity:

• F. del Aguila, A. Culatti, R. Munoz-Tapia, M. Perez-Victoria, Supergravity corrections to $(g-2)_l$ in differential renormalization, Nuclear Physics B 504 (1997) 532-550 (arXiv:hep-ph/9702342)