A circuit consists of two loops connected between two nodes, \(a\) and \(b\). The left branch contains an ideal battery of EMF \(\mathcal{E}_1\) in series with a resistor of resistance \(R\), with its positive terminal directed toward node \(a\). The central branch contains a single resistor of resistance \(R\). The right branch contains an ideal battery of EMF \(\mathcal{E}_2\) in series with a resistor of resistance \(2R\), with its positive terminal also directed toward node \(a\). In terms of \(\mathcal{E}_1\) and \(\mathcal{E}_2\), what is the electric potential difference \(V_a - V_b\) across the central branch?