IFT-P.006/2000

The Tachyon Potential in Open Neveu-Schwarz String Field Theory

Nathan Berkovits^{†}^{†} e-mail:

Instituto de Física Teórica, Universidade Estadual Paulista

Rua Pamplona 145, 01405-900, São Paulo, SP, Brasil

A classical action for open superstring field theory has been proposed which does not suffer from contact term problems. After generalizing this action to include the non-GSO projected states of the Neveu-Schwarz string, the pure tachyon contribution to the tachyon potential is explicitly computed. The potential has a minimum of which is of the predicted exact minimum of from D-brane arguments.

January 2000

1. Introduction

The Neveu-Schwarz (NS) sector of the non-GSO projected superstring has recently been reconsidered as part of a sensible physical theory [1][2]. Although this sector contains a tachyon, there have been proposals for removing the undesired properties of the the tachyon by assuming a tachyon potential which is bounded from below.

The most efficient method for computing the tachyon potential uses open string field theory[3] [4] [5], however the cubic action of [6] for open superstring field theory contains contact term problems which spoil gauge invariance[7]. Recently, a new action for open superstring field theory has been constructed [8] which does not suffer from contact term problems. This action resembles a Wess-Zumino-Witten action and can be naturally obtained by embedding the N=1 description of the superstring into an N=2 string [9].

In this paper, the pure tachyon contribution to the tachyon potential will be explicitly computed using this new action. The pure tachyon contribution is

which has a minimum of when
This value of the minimum is of the predicted
exact minimum of using D-brane arguments^{†}^{†} In the original version of this paper, the mass of the brane-antibrane
was incorrectly stated to be . This value of the mass
is only correct if one doubles the number of states in the string field theory
action to allow for strings which end on the brane or antibrane [10] .
where the mass of the brane-antibrane is [4][10].
It would be interesting to check if the remaining comes from
including contributions to
the effective tachyon potential from
non-tachyon fields, as was found for the bosonic string
tachyon potential in [5].

2. Neveu-Schwarz String Field Theory Action

Using the superstring field theory action of [8], the GSO-projected NS contribution is given by

where is defined by fermionizing the super-reparameterization ghosts as[11] and ,

and signifies the two-dimensional correlation function in the “large” RNS Hilbert space [11] where . The normalization of (2.1) has been fixed by requiring that the quadratic Yang-Mills contribution to the action is , which is the correct sign for the metric that is being used. String fields are multiplied using the midpoint interaction of [12] and is related to the NS string field of [6] by or .

In the GSO-projected sector, the NS string field is bosonic. Since the unprojected NS states are fermionic with respect to the projected NS states, it will be convenient to define where described the projected states, describes the unprojected states, is the identity matrix, and are the Pauli matrices [4]. Furthermore, it will be convenient to define

which satisfy and .

The complete non-GSO projected NS string field theory action is defined by

where the trace is over the matrices as well as the Chan-Paton matrices.

One can check that (2.2) is invariant under the WZW-like gauge transformation

where and are string fields of the form with being fermionic and projected while is bosonic and unprojected. One subtle point in proving this gauge invariance is that since when and are unprojected states, where the minus sign is if they are bosons and the plus sign is if they are fermions. This reversal of the usual statistics comes from square-root factors produced by the -integer conformal weight of unprojected NS states. Note that a similar subtlety occurs with unprojected states using the action of [6].

3. Computation of Tachyon Potential

Expanding the action of (2.2) in powers of , one obtains

To compute the term with ’s, one uses the map

from the disc to a wedge of the complex plane. Rotating this map by a factor allows each successive string field to get mapped to a different wedge. The center of the disc gets mapped to the point and, to obtain an SL(2,R)-invariant expression, the string field gets multiplied by a factor is the conformal weight of the string field and is [13]. where

The tachyon field appears in the string field as

where the factor of is needed to get the right sign for the kinetic term. One can easily compute that at zero momentum,

Since , the only pure tachyon contribution to the action of (2.2) comes from the quadratic and quartic terms of (3.1). The quadratic contribution to the action (which is minus the tachyon potential) is given by

The quartic contribution to the action is given by

So when . which has a minimum of

Acknowledgements: I would like to thank Oren Bergman, Ashoke Sen, Ion Vancea and Barton Zwiebach for useful discussions, Caltech for their hospitality, and CNPq grant 300256/94-9 for partial financial support.

References

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