The role of the Seiberg–Witten field redefinition in renormalization of noncommutative chiral electrodynamics

Abstract

It has been conjectured in the literature that renormalizability of the θ-expanded noncommutative gauge theories improves when one takes into account full nonuniqueness of the Seiberg–Witten expansion which relates noncommutative (‘high-energy’) with commutative (‘low-energy’) fields. In order to check this conjecture we analyze renormalizability of the θ-expanded noncommutative chiral electrodynamics by quantizing the action which contains all terms implied by this nonuniqueness. After renormalization we arrive at a different theory, characterized by different relations between the coupling constants: this means that the θ-expanded noncommutative chiral electrodynamics is not renormalizable.

Appendix: One-loop divergences

All calculations mentioned in the text are done actually using the Majorana spinors and the algebra of γ-matrices. The direct outcome of calculation for the Lagrangian containing only one spinor field is

$$\begin{aligned} {\varGamma}^{(1)}_{\rm div} =& \frac{1}{(4\pi)^2 \varepsilon} q^2 \int{\rm d}^4 x \biggl( {\rm i}\bar{\psi}\gamma^\mu(\partial_\mu- {\rm i} q \gamma_5 A_\mu) \psi \\ &{}-\frac{1}{3}F_{\mu\nu}F^{\mu\nu} + \theta^{\mu\nu} \biggl( \frac{\rm i}{12} {\varepsilon _{\mu\nu\rho\sigma}} \bar{\psi}\gamma^\sigma D^\rho D^\tau D_\tau\psi \\ & {}+q \biggl( \frac{2}{3} F_{\mu\rho}F_{\nu \sigma} F^{\rho\sigma} - \frac{1}{6} F_{\mu\nu}F_{\rho\sigma} F^{\rho\sigma} \\ &{}-\frac{5\rm i}{6} F_{\mu\rho} \bar{\psi}\gamma^\rho D_\nu \psi+ \frac{\rm i}{6} F_{\mu\rho} \bar{\psi}\gamma_\nu D^\rho\psi \\ & {}+ \frac{2\rm i}{3} F_{\mu\nu} \bar{\psi}\gamma^\rho D_\rho\psi+\frac{1}{6} {\varepsilon_{\mu\rho\sigma\tau}} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\nu\psi \\ &{}-\frac{7}{8} {\varepsilon_{\mu\nu \rho\sigma}} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\tau\psi \biggr) \\ &{}+\kappa_2 \biggl( \frac{\rm i}{12} { \varepsilon_{\mu\nu\rho \sigma}} \bar{\psi}\gamma^\sigma D^\rho D^\tau D_\tau\psi \\ & {}+q \biggl( \frac{2}{3} F_{\mu\rho}F_{\nu\sigma} F^{\rho \sigma} - \frac{1}{6} F_{\mu\nu}F_{\rho\sigma} F^{\rho\sigma} \\ &{}+\frac{\rm i}{2} F_{\mu\rho} \bar{\psi}\gamma^\rho D_\nu\psi - \frac{3\rm i}{2} F_{\mu\rho} \bar{\psi}\gamma_\nu D^\rho\psi \\ &{} +\frac{5}{36} { \varepsilon_{\mu\rho\sigma\tau}} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\nu\psi \\ & {} - \frac{1}{8} {\varepsilon_{\mu\nu\rho\sigma}} F^{\rho\sigma} \bar{\psi}\gamma _5 \gamma^\tau D_\tau\psi \biggr) \biggr) \\ & {}+\kappa_3 \biggl( -\frac{4 \rm i}{3} { \varepsilon_{\mu\nu \rho\sigma}} \bar{\psi}\gamma^\sigma D^\rho D^\tau D_\tau\psi \\ &{}+q \biggl(- \frac{16}{3} F_{\mu\rho}F_{\nu\sigma } F^{\rho\sigma} + \frac{4}{3} F_{\mu\nu}F_{\rho\sigma} F^{\rho\sigma} \\ & {}-\frac{10\rm i}{3} F_{\mu\rho} \bar{\psi}\gamma^\rho D_\nu \psi - \frac{34\rm i}{3} F_{\mu\rho} \bar{\psi}\gamma_\nu D^\rho \psi \\ &{}+ \frac{20\rm i}{3} F_{\mu\nu} \bar{\psi}\gamma^\rho D_\rho\psi-\frac{11}{3} {\varepsilon_{\mu\rho\sigma\tau}} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\nu\psi \\ & {}+2 {\varepsilon_{\mu\nu\rho\sigma }} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\tau\psi \biggr) \biggr) \\ & {}+\kappa_4 \biggl( -\frac{7 \rm i}{6} { \varepsilon_{\mu\nu \rho\sigma}} \bar{\psi}\gamma^\sigma D^\rho D^\tau D_\tau\psi \\ &{}+q \biggl(- \frac{8}{3} F_{\mu\rho}F_{\nu\sigma} F^{\rho\sigma} +\frac{2}{3} F_{\mu\nu}F_{\rho\sigma} F^{\rho\sigma} \\ & {}-\frac{10\rm i}{3} F_{\mu\rho} \bar{\psi}\gamma^\rho D_\nu \psi - \frac{13\rm i}{3} F_{\mu\rho} \bar{\psi}\gamma_\nu D^\rho \psi \\ &{}+ \frac{19\rm i}{6} F_{\mu\nu} \bar{\psi}\gamma^\rho D_\rho\psi \\ & {} +\frac{2}{3} {\varepsilon_{\mu\rho \sigma\tau}} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\nu\psi \\ &{} - \frac{1}{4} {\varepsilon_{\mu\nu\rho\sigma}} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\tau\psi \biggr) \biggr) \\ & {}+\kappa_5 \biggl( \frac{\rm i}{3} { \varepsilon_{\mu\nu\rho \sigma}} \bar{\psi}\gamma^\sigma D^\rho D^\tau D_\tau\psi \\ &{} +q \biggl(-\frac{4\rm i}{3} F_{\mu\rho} \bar{\psi}\gamma^\rho D_\nu\psi + \frac{2\rm i}{3} F_{\mu\rho} \bar{\psi}\gamma_\nu D^\rho \psi \\ & {}+ \frac{5\rm i}{3} F_{\mu\nu} \bar{\psi}\gamma^\rho D_\rho\psi+\frac{2}{3} { \varepsilon_{\mu\rho\sigma \tau}} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\nu\psi \\ &{} - \frac{1}{2} { \varepsilon_{\mu\nu\rho\sigma}} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\tau\psi \biggr) \biggr) \\ & {}+\kappa_6 \bigl(2 q {\varepsilon_{\mu\rho\sigma\tau}} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\nu\psi \bigr) \\ & {}+\kappa_7 \biggl( -\frac{4\rm i}{3} { \varepsilon_{\mu\nu\rho \sigma}} \bar{\psi}\gamma^\sigma D^\rho D^\tau D_\tau\psi \\ &{} +q \biggl(\frac{4\rm i}{3} F_{\mu\rho} \bar{\psi}\gamma^\rho D_\nu \psi+ \frac{4\rm i}{3} F_{\mu\rho} \bar{\psi}\gamma_\nu D^\rho \psi \\ & {} - \frac{8\rm i}{3} F_{\mu\nu} \bar{\psi}\gamma^\rho D_\rho\psi -\frac{2}{3} { \varepsilon_{\mu \rho\sigma\tau}} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\nu\psi \\ &{} + 2 {\varepsilon_{\mu\nu\rho\sigma}} F^{\rho\sigma} \bar{\psi}\gamma_5 \gamma^\tau D_\tau\psi \biggr) \biggr) \biggr) \biggr) . \end{aligned}$$

(A.1)

This result reduces to the result found before in [21] for κ _i =0. One immediately notices that, apart from the usual commutative divergences and the 3-photon term , all other terms are electron-photon interactions and they should be converted to the ‘canonical’ . Partial integrations and other transformations based on the spinor identities, succeeded by the change from the Majorana to the chiral form, give final expression (40).

In order to analyze the case with several matter fields let us first shortly discuss the corresponding noncommutative classical action. We assume that we have a set of fermions $\hat{\varphi}_{i}$, i=1,…,N with electric charges q _i . It is known [28] that in the θ-expanded theories each noncommutative field comes with its own noncommutative potential $\hat{A}_{i}^{\mu}$; all of them, however, for θ=0 reduce to the same A ^μ. The $\hat{A}_{i}^{\mu}$ are different because their corresponding SW maps differ: they depend on charges q _i . Therefore in order to obtain the action with the correct limit, in the sum

(A.2)

we have to rescale the gauge fields and the charges, $A^{\mu}_{i} \rightarrow\sqrt{c_{i}} A_{i}^{\mu}= \sqrt{c_{i}} A^{\mu}$, $\ q_{i} \rightarrow\frac{1}{\sqrt{c_{i}}}q_{i}$. We get

(A.3)

In order to associate (A.3) with the Lagrangian of the usual commutative theory we fix the sum of weights c _i to 1, $\ \sum_{i=1}^{N} c_{i}=1$. In the noncommutative U(1) case we are free to choose c _i =1/N; for other symmetries the analogous relations are more complicated, [17]. The same rescaling applied to the noncommutative part of the Lagrangian gives, for the boson vertex,

(A.4)

and clearly in an anomaly-safe model in which ∑ _i q _i =0 we find that this term vanishes, . Fermion terms are on the other hand unchanged as they are mutually independent for each field:

We can extract the value of the one-loop divergences from our previous result either using the same rescaling of charges q _i by c _i or straightforwardly, by repeating the calculation. We obtain for the renormalized Lagrangian:

with

$$\begin{aligned} \alpha_{i,2} =&\frac{1}{3}(-5+3\kappa_{i,2} - 20 \kappa_{i,3} - 20 \kappa_{i,4} - 8\kappa_{i,5} + 8 \kappa_{i,7}), \\ \alpha_{i,3} =&\frac{1}{12}(-1 -\kappa_{i,2} + 16 \kappa_{i,3} +14 \kappa_{i,4} -4\kappa_{i,5} +16 \kappa_{i,7}), \\ \alpha_{i,4} =&\frac{1}{6}(-1 + 9\kappa_{i,2}+ 68 \kappa_{i,3} + 26 \kappa_{i,4} - 4\kappa_{i,5} - 8 \kappa_{i,7}), \\ \alpha_{i,5} =&\frac{1}{4}(-1 - \kappa_{i,2} - 20 \kappa_{i,3} -6\kappa _{i,4} - 4 \kappa_{i,5} + 8 \kappa_{i,7}), \\ \alpha_{i,6} =&\frac{1}{36}(-6 -5 \kappa_{i,2}+ 132 \kappa_{i,3} -24 \kappa_{i,4}- 24 \kappa_{i,5}\\ &{} - 72 \kappa_{i,6} + 24 \kappa _{i,7}), \\ \alpha_{i,7} =&\frac{1}{8}(7 +\kappa_{i,2} -16 \kappa_{i,3} + 2\kappa _{i,4} + 4\kappa_{i,5} - 16 \kappa_{i,7}). \end{aligned}$$

While renormalization of fields and charges is standard,

$$\begin{aligned} &\varphi_{i,0}=\sqrt{Z_{i,2}}\varphi_i=\sqrt {1-2 \frac {q_i^2}{(4\pi)^2 \varepsilon}} \varphi_i, \\ &A^\mu_0=\sqrt{Z_3} A^\mu=\sqrt {1-\frac{4}{3} \frac{\sum_j q_j^2}{(4\pi)^2 \varepsilon}} A^\mu, \\ &q_{i,0}=\mu^{\frac{\varepsilon}{2}} Z_3^{-1/2}Z_{i,2}^{-1} \biggl(1-2 \frac{q_i^2}{(4\pi)^2 \varepsilon} \biggr) q_i\\ & \hphantom{q_{i,0}}= \mu^{\frac{\varepsilon}{2}} \biggl(1+\frac{2}{3} \frac{\sum_j q_j^2}{(4\pi)^2 \varepsilon} \biggr) q_i , \end{aligned}$$

the noncommutativity θ ^μν can renormalize arbitrarily. In order to try to use this fact we assume that it is of the form

$$\theta^{\mu\nu}_0= \biggl(1+\alpha \frac{\sum_j q_j^2}{(4\pi)^2 \varepsilon} \biggr) \theta^{\mu\nu}, $$

with an arbitrary coefficient α which is to be determined from some renormalizability constraint. Renormalization of the κ _i follows:

$$\begin{aligned} (\kappa_{i,2})_0 =& \kappa_{i,2} - \frac{\alpha}{(4\pi)^2 \varepsilon} \biggl( (1+\kappa_{i,2}) \sum _j q_j^2+\frac{1}{3} (1 + 9\kappa_{i,2} \\ & {} - 20\kappa_{i,3} -20 \kappa_{i,4} -8 \kappa_{i,5} +8 \kappa_{i,7})q^2_i \biggr), \\ (\kappa_{i,3})_0 =& \kappa_{i,3} - \frac{\alpha}{(4\pi)^2 \varepsilon} \biggl( \kappa_{i,3} \sum _j q_j^2+ \frac{1}{12}(-1- \kappa_{i,2} \\ &{} +40 \kappa_{i,3} +14 \kappa_{i,4} - 4\kappa_{i,5} +16 \kappa_{i,7})q^2_i \biggr), \\ (\kappa_{i,4})_0 =& \kappa_{i,4} - \frac{\alpha}{(4\pi)^2 \varepsilon} \biggl(\kappa_{i,4} \sum _j q_j^2+ \frac{1}{6} (-1+9 \kappa_{i,2} \\ & {} +68 \kappa_{i,3} +38 \kappa_{i,4} - 4\kappa_{i,5} -8 \kappa_{i,7})q^2_i \biggr), \\ (\kappa_{i,5})_0 =& \kappa_{i,5} - \frac{\alpha}{(4\pi)^2 \varepsilon} \biggl( \kappa_{i,5} \sum _j q_j^2+ \frac{1}{4}(-1- \kappa_{i,2} \\ &{} -20 \kappa_{i,3} -6 \kappa_{i,4}+ 4 \kappa _{i,5} +8 \kappa_{i,7})q^2_i \biggr), \\ (\kappa_{i,6})_0 = &\kappa_{i,6} - \frac{\alpha}{(4\pi)^2 \varepsilon} \biggl( \kappa_{i,6} \sum _j q_j^2+\frac{1}{36}(-6-5 \kappa_{i,2} \\ & {} +132 \kappa_{i,3} -24\kappa _{i,4}- 24 \kappa_{i,5} +24 \kappa_{i,7})q^2_i \biggr), \\ (\kappa_{i,7})_0 =& \kappa_{i,7} - \frac{\alpha}{(4\pi)^2 \varepsilon} \biggl( \kappa_{i,7} \sum _j q_j^2 + \frac{1}{8}(7+ \kappa_{i,2} -16 \kappa_{i,3}\\ &{} +2\kappa_{i,4} + 4 \kappa _{i,5} ) q^2_i \biggr). \end{aligned}$$

However, we easily observe that, in whatever way we fix α, the expressions

$$\begin{aligned} & (\kappa_{i,2})_0 -4 (\kappa_{i,3})_0 -2(\kappa_{i,4})_0 \\ &\quad = \biggl( 1- \frac{\alpha}{(4\pi)^2 \varepsilon} \sum _j q_j^2 \biggr) ( \kappa_{i,2}-4 \kappa_{i,3} -2\kappa_{i,4}) \\ &\qquad{} - \frac{\alpha}{(4\pi)^2 \varepsilon} \sum_j q_j^2 + \frac {1}{(4\pi)^2 \varepsilon}\frac{q_i}{3}( 3+19 \kappa_{i,2} \\ &\qquad {} -128 \kappa _{i,3}-72\kappa_{i,4}), \end{aligned}$$

cannot be zero.