One Loop Tadpole in Heterotic String Field Theory We compute the off-shell 1-loop tadpole amplitude in heterotic string field theory. With a special choice of cubic vertex, we show that this amplitude can be computed exactly. We obtain explicit and elementary expressions for the Feynman graph decomposition of the moduli space, the local coordinate map at the puncture as a function of the modulus, and the b-ghost insertions needed for the integration measure. Recently developed homotopy algebra methods provide a consistent configuration of picture changing operators. We discuss the consequences of spurious poles for the choice of picture changing operators.
1 Introduction 1 2 Quantum Closed String Field Theories 2 2.1 Closed Bosonic String Field Theory. 2 2.2 Heterotic String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 One Loop Tadpole in Closed Bosonic String Field Theory 10 3.1 Elementary Cubic Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Propagator Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Elementary One Loop Tadpole Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 One Loop Tadpole in Heterotic String Field Theory 22 4.1 General Construction of Cubic and Tadpole Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 Case I: Local PCO insertions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Spurious Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.4 Case II: PCO contours around punctures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5 Concluding Remarks 33 1 Introduction In this paper we compute the off-shell, one-loop tadpole amplitude in heterotic string field theory. The purpose is twofold: (1) First, we would like to show that the amplitude can be computed exactly. Our success in this regard is largely due to a nonstandard choice of cubic vertex defined by SL(2, C) local coordinate maps.4 We will take special care to provide explicit results concerning the Feynman-graph decomposition of the moduli space, the local coordinate map as a function of the modulus, and the b-ghost insertions needed for the integration measure. Actually, these data are primarily associated with closed bosonic string field theory, but they also represent the most significant obstacle to explicit results for the heterotic string. Importantly, our string vertices differ from the canonical ones defined by minimal area metrics [1]. The minimal area vertices are cumbersome for elementary calculations, though some analytic results are available at tree level up to 4 points [2] and numerical calculations have been performed up to 5 points [3, 4].5 (2) Second, we would like to see how recent homotopy-algebraic constructions of classical superstring field theories [6, 7, 8, 9, 10, 11, 12] may be extended to the quantum level. A significant issue is the appearance of spurious poles in βγ correlation functions at higher genus [13]. We find that the general methodology behind tree level constructions still functions in loops. Spurious poles are mainly important for the choice of picture changing operators (PCOs), which must ensure that loop vertices are finite and unambiguously defined. We consider two approaches to inserting PCOs. In the first approach, PCOs appear at specific points in the surfaces defining the vertices, in a manner similar to [14]. In the second approach, PCOs appear as contour integrals around the punctures, paralleling the construction of classical superstring field theories. The second approach has a somewhat different nature in loops, however, due to the necessity of specifying the PCO contours precisely in relation spurious poles. A naive treatment can lead to inconsistencies. In this paper we are not concerned with computing the 1-loop tadpole in a specific background or for any particular on- or off-shell states. So when we claim to “compute” this amplitude, really what we mean is that we specify all background and state-independent string field theory data that goes into the definition of this amplitude. It will remain to choose a vertex operator of interest, compute the correlation functions, and integrate over the moduli space to obtain a final expression. Ultimately, however, one would like to use string field theory to compute physical 4The possibility of using SL(2, C) maps in defining vertices was pointed out to us by A. Sen. We thank him for this suggestion. 5A new approach to the computation of off-shell amplitudes has recently been proposed based on hyperbolic geometry [5]. It would be interesting to approach the computation of the heterotic tadpole from this perspective. 1 amplitudes in situations where the conventional formulation of superstring perturbation theory breaks down. The computations of this paper can be regarded as a modest step in this direction. The paper is organized as follows. In section 2 we briefly review the definition of bosonic and heterotic closed string field theories. To obtain the data associated to integration over the bosonic moduli space, in section 3 we compute the 1-loop tadpole in closed bosonic string field theory. This requires, in particular, choosing a suitable cubic vertex, computing the contribution to the tadpole from gluing two legs of the cubic vertex with a propagator, and defining an appropriate fundamental tadpole vertex to fill in the remaining region of the moduli space. In section 4 we compute the 1-loop tadpole in heterotic string field theory. This requires dressing the amplitude of the bosonic string with a configuration of PCOs. Homotopy algebraic methods constrain the choice of PCOs to be consistent with quantum gauge invariance, but some freedom remains. We discuss two approaches: one which inserts PCOs at specific points in the surfaces defining the vertices, and another which inserts PCOs in the form of contour integrals around the punctures. We discuss the consequences of spurious poles for both approaches. We also present the amplitudes in a form which is manifestly well-defined in the small Hilbert space. We end with concluding remarks. Note Added: While this work was in preparation, we were informed of the review article [15] which contains a schematic discussion of spurious poles in the 1-loop tadpole amplitude in the limit of large stub parameter. Our work contains a more complete calculation of the tadpole amplitude, but considering this issue our conclusions are in agreement.
