BPHZ renormalization and its application to non-commutative field theory

Abstract

In a recent work a modified BPHZ scheme has been introduced and applied to one-loop Feynman graphs in non-commutative ϕ ⁴-theory. In the present paper, we first review the BPHZ method and then we apply the modified BPHZ scheme as well as Zimmermann’s forest formula to the sunrise graph, i.e. a typical higher-loop graph involving overlapping divergences. Furthermore, we show that the application of the modified BPHZ scheme to the IR-singularities appearing in non-planar graphs (UV/IR mixing problem) leads to the introduction of a 1/p ² term and thereby to a renormalizable model. Finally, we address the application of this approach to gauge field theories.

Appendix

The purpose of this appendix is to illustrate problems arising for the one-loop polarization (88) associated to the naïve non-commutative gauge field model (85) when applying the s-trick described in Sect. 2.10. Thus, we introduce a mass term involving an auxiliary mass M and an auxiliary variable s in all internal propagators,

$$ \frac{1}{k^2} \; \leadsto\; \frac{1}{k^2 + M_s^2}, \quad \mathrm{with}\ M_s \equiv(1-s) M, $$

(A.1)

and apply the Taylor expansion operator $t_{p,s}^{\delta(\varGamma)}$ around both p=0 and s=0, while setting s=1 at the end of the calculation so as to discard artificial IR-problems.

For the quadratically divergent integral (88) we have

$$\begin{aligned} &{\int d^4k \bigl(t_{p,s}^2 I_{\varGamma\mu\nu} \bigr) (p,\tilde {p},k,s)} \\ &{\quad{}=\int d^4k \biggl( I_{\varGamma\mu\nu} (0,\tilde{p},k,0) + \cdots} \\ &{\qquad{}+ \frac{p^{\rho} p^{\sigma}}{2} \frac{\partial^2 I_{\varGamma\mu\nu}}{\partial p^{\rho} \partial p^{\sigma}}(0,\tilde{p},k,0)} \\ &{\qquad{} + \frac{s^2}{2} \frac{\partial^2 I_{\varGamma\mu\nu } }{\partial s^2} (0,\tilde{p},k,0) \biggr)} \\ &{\quad{}={2g^2}\int\frac{d^4k}{(2\pi)^4} \frac{1-\cos(k\tilde {p})}{(k^2+M^2)^2} \biggl\{ 4k_\mu k_\nu-3p_\mu p_\nu} \\ &{\qquad{}+2\delta_{\mu\nu} \bigl(p^2-k^2 \bigr)+2 \frac{2{k_\mu k_\nu}- \delta_{\mu\nu} k^2 }{k^2+M^2}} \\ &{\qquad{}\times\biggl[ 4 \frac{ (pk)^2+3 s^2 M^4 }{k^2+M^2}} \\ &{\qquad{}+ 2 s(2-s) M^2 -p^2 \biggr] \biggr\},} \end{aligned}$$

(A.2)

where we took into account the fact that the integral over an odd function of k vanishes upon integration over symmetric intervals. Thus, the finite part of the vacuum polarization reads

$$\begin{aligned} \varPi_{\mu\nu}^{\mathrm{finite}}(p) &= \lim_{s\to1} \int d^4k \bigl(1-t_{p,s}^2 \bigr) I_{\varGamma\mu\nu} (p, \tilde{p},k,s) \\ &={2g^2} \lim_{s\to1} \int\frac{d^4k}{(2\pi)^4} {\bigl[ 1-\cos(k\tilde{p})\bigr]} \\ &\quad{}\times \biggl\{-2 \frac{2{k_\mu k_\nu}- \delta_{\mu\nu} k^2 }{(k^2+M^2)^3} \\ &\quad{}\times \biggl[ 4 \frac{ (pk)^2+3 s^2 M^4 }{k^2+M^2} + 2 s(2-s) M^2 -p^2 \biggr] \\ &\quad{}+ \bigl[ 4k_\mu k_\nu-3p_\mu p_\nu+2\delta_{\mu\nu}\bigl(p^2-k^2\bigr) \bigr] \\ &\quad{}\times \biggl[ \frac{1}{(k^2+M_s^2) ((k+p)^2+M_s^2 )} \\ &\quad{}-\frac{1}{ (k^2+M^2 )^2} \biggr] \biggr\}. \end{aligned}$$

(A.3)

Evaluation of the resulting integral using Schwinger’s parametrization leads to an expression whose planar part is UV-finite, and whose non-planar part involves modified Bessel functions of the second kind. Expansion of the latter around small values of $\tilde{p}^{2}$ enables us to perform the final parametric integral (i.e. the integral over the Schwinger parameter ξ already considered in (64)). After an expansion around small mass M we finally get

$$\begin{aligned} &{\varPi_{\mu\nu}^{\mathrm{finite}}(p)} \\ &{\quad{}= \frac{g^2 M^2}{96 \pi^2} \biggl\{ 2p^2 \tilde{p}_{\mu} \tilde{p}_{\nu} \biggl[ \ln \biggl( \frac{1}{4} M^2 \tilde{p}^2 \biggr)+2 \gamma_E -1 \biggr]} \\ &{\qquad{}+5 \tilde{p}^2 \bigl(p_{\mu} p_{\nu}-p^2\delta_{\mu\nu}\bigr) \biggl[ \ln \biggl( \frac{1}{4} M^2 \tilde{p}^2 \biggr)+2 \gamma_E -2 \biggr]} \\ &{\qquad{}+p^2\tilde{p}^2 \delta_{\mu \nu} \biggl[ 1-2 \gamma _E- \ln\biggl(\frac{1}{4} M^2 \tilde{p}^2 \biggr) \biggr] \biggr\}} \\ &{\qquad{}-\frac{g^2 p^2\tilde{p}^2}{4800\pi^2}\bigl(p_\mu p_\nu-p^2 \delta _{\mu\nu}\bigr) \biggl[ 45\ln \biggl(\frac{1}{4}p^2 \tilde{p}^2 \biggr)} \\ &{\qquad{}+90\gamma_E-163 \biggr]} \\ &{\qquad{}-\frac{g^2 (p^2 )^2}{7200 \pi^2} \tilde{p}_{\mu} \tilde{p}_{\nu} \biggl[ 15 \ln \biggl(\frac{1}{4} p^2 \tilde{p}^2 \biggr)+30 \gamma_E -46 \biggr]} \\ &{\qquad{}+\mathcal{O} { \bigl(\bigl(\tilde{p}^2 \bigr)^2,M^4 \bigr)}.} \end{aligned}$$

(A.4)

The mass dependent parts are not transversal, but the limit M→0 exists,

$$\begin{aligned} &{\lim_{M\to0} \varPi_{\mu\nu}^{\mathrm{finite}}(p)} \\ &{\quad{}=-\frac{g^2 p^2\tilde{p}^2}{4800\pi^2}\bigl(p_\mu p_\nu-p^2 \delta_{\mu \nu}\bigr) \biggl[ 45\ln \biggl(\frac{1}{4}p^2 \tilde{p}^2 \biggr)} \\ &{\qquad{}+90\gamma_E-163 \biggr]} \\ &{\qquad{}-\frac{g^2 (p^2 )^2}{7200 \pi^2} \tilde{p}_{\mu} \tilde{p}_{\nu} \biggl[ 15 \ln \biggl(\frac{1}{4} p^2 \tilde{p}^2 \biggr)+30 \gamma_E -46 \biggr]} \\ &{\qquad{}+\mathcal{O} { \bigl(\bigl(\tilde{p}^2 \bigr)^2 \bigr)},} \end{aligned}$$

(A.5)

and this expression is indeed transversal. These results show already at one-loop level that the introduction of a regulator mass explicitly breaks gauge invariance and violates Slavnov–Taylor identities such as the transversality of the vacuum polarization, see also [39] and references therein. In our specific example, the limit M→0 exists and restores gauge symmetry, but this need not be the case for other graphs. Fortunately, as discussed in Sect. 4, we do not have to consider an infrared regularization using an auxiliary mass for the non-commutative gauge field model of Ref. [54].