The Bethe/Gauge Correspondence

54 Chapter 3. GaugeLooking at the terms in the Lagrangian (3.61) we see that the two terms on the right-hand sidecan be represented by the Feynman diagramsOur task is to evaluatewhere the ‘blobs’ may be filled with any diagram containing the component fields of the mattersuperfield Φ that we’re integrating out. Supersymmetry ensures that the vacuum energyvanishes, so we only have to consider connected diagrams.Due to the absence of superpotential terms the Lagrangian is quadratic in Φ, and we seefrom (3.61) that the only correction to r bare is given by a tadpole diagram in φ:The ‘interaction’ parameter of the vertex is the charge Q. As we are interested in the Coulombbranch, we may assume that σ varies slowly. Therefore, we get a mass term for φ in (3.61), withmass |σ|. In order to ensure that Φ is massive and can be integrated out we suppose that |σ| issufficiently large.To compute the tadpole diagram we have to evaluate〈 D〉δe 2 = Q∫R 1,1d 2 k(2π) 2∫ik 2 + |σ| 2 = QR 2d 2 k 1(2π) 2 k 2 + |σ| 2 ,where we have performed a Wick rotation in the second equality. The integral diverges logarithmically.We use Pauli-Villars regularization and introduce a reference mass scale µ:〈 D〉 ∫δe 2 = Qd 2 ()k 1(2π) 2 k 2 + |σ| 2 − 1k 2 + µ 2 .Performing the integral we find that the correction to the vev of D/e 2 is given by − Q 2πHence, the result for the effective FI-parameter islog|σ|µ .r eff (µ, σ) = r bare − Q 2πlog|σ|µ .This correction to r is compatible with the following correction to the effective twistedsuperpotential:δ ˜W eff = Q 2π σ (log σ µ − 1 )= Q 2π σ (log |σ|µ + i (arg σ + 2πm) − 1 ). (3.62)Since the argument σ/µ of the logarithm is complex, in the second equality we have chosen abranch cut. A calculation of the corrections for the theta angle precisely gives the part of thisexpression proportional to arg σ (see p. 384–5 of [13]). The branching of the complex logarithm,encoded by m ∈ Z, does not influence the twisted superpotential: since Q ∈ Z it can be absorbedin the shift (3.55), and the actual value of m is such that the energy density (3.54) is minimal.