Unitarity of Superstring Field Theory

We complete the proof of unitarity of (compactified) heterotic and type II string field theories by showing that in the cut diagrams only physical states appear in the sum over intermediate states. This analysis takes into account the effect of mass and wave-function renormalization, and the possibility that the true vacuum may be related to the perturbative vacuum by small shifts in the string fields.


Introduction
In a previous paper [1] we derived the Cutkosky rules for superstring field theory under the assumption that the action for string field theory is real. This assumption was proved later [2].
Cutkosky rules derived in [1] establish that the T-matrix -related to the S-matrix via the relation S = 1 − iT -satisfies the relation i(T − T † ) = T † |n n|T , where the sum over |n runs over all states in the Siegel gauge. This would establish unitarity of the theory if all states in the Siegel gauge were physical states, However since string field theory is a gauge theory, Cutkosky rules do not automatically prove unitarity. A cut propagator representing |n n|, besides propagating physical on-shell intermediate states, also has unphysical and pure gauge states. Therefore in order to prove unitarity we need to prove that the sum over intermediate states in a cut diagram receives contribution from only the physical on-shell states, and the contribution from all other states cancel. This is what we shall show in this paper.
Earlier attempts [3,4] to prove unitarity of superstring theory in the covariant formulation relied on proving equivalence to light-cone string field theory [5,6]. However since light-cone superstring field theory encounters contact term divergences [7][8][9][10], it is not clear if this can be lifted to a valid proof after taking into account the various subtleties of the covariant formulation described in [11]. Some recent attempts to circumvent this difficulty can be found in [12]. Another systematic procedure for computing the imaginary part of the string theory amplitude is the iǫ prescription of [13,14], but it is not clear at this stage how this can be used to prove unitarity of the amplitude. The rest of the paper is organized as follows. In §2 we derive some useful properties of the quantum corrected propagator of string field theory and residues at its poles. In §3 we use the Ward identities of string field theory to show that only physical states contribute in the sum over intermediate states in a cut diagram. In §4 we discuss some open problems.

The propagator
The main tool in our analysis will be the full propagator computed from the one particle irreducible (1PI) effective action of superstring field theory. We shall follow the conventions of [15], and begin by collecting some basic results in 1PI effective string field theory as reviewed in [15]. This will be followed by a review of some basic results in Cutkosky rules derived in [1]..
Finally we shall combine these results to derive the general form of the contribution from a cut propagator as given in (2.30).
We shall work with the heterotic string theory for simplicity and will describe the generalization to type II string theories at the end of the section. We define H T to be the subspace of GSO even states in the matter ghost conformal field theory satisfying Here b n ,b n , c n ,c n are the modes of the usual b,b, c,c ghost fields, L n ,L n are the total Virasoro generators, and H n will denote the subspace of states in H T carrying picture number n. Even though we follow closely the formalism described in [15], there are two ways in which the action that we consider differs from the one analyzed in [15]: 1. We shall implicitly assume that the 1PI effective action we use comes from the sum of 1PI diagrams of the action described in [16] so that the Cutkosky rules hold [1]. This means that the class of actions we shall consider will be more restrictive than the ones used in [15]. But this does not prevent us from using the results derived in [15] since the latter describes a more general class of theories.
2. In the analysis of [15] we had restricted the interacting string field to carry ghost number 2 and picture numbers −1 or −1/2. 1 Here we shall keep the form of the action unchanged but allow the string field to carry all possible ghost numbers since we want to include in our analysis not only the matter fields but also the ghost fields as external states. 2 It is easy to verify that the results of [15] that we shall be using, namely eq.(2.8) below for the propagator and eq.(3.1) below for the truncated Green's function, are valid for these more general string fields.
The kinetic operator Q B of 1PI effective string field theory around the quantum corrected vacuum, and a related operator Q B introduced in [15] will play special roles in our analysis.
Q B and Q B are operators of ghost number 1, acting respectively on the states in producing states in H T and H T respectively. Q B and Q B have the form Here Q B is the nilpotent BRST operator. K is some operator that acts on states in H T and produces states in H T . It is related to the 1PI two point function and can be computed using perturbation theory. G is the identity operator in the Neveu-Schwarz (NS) sector and the zero mode of the picture changing operator (PCO) in the Ramond (R) sector, and satisfies and where γ A and γ B are grassmannalities of A and B, and Q B A| and Q B B| are respectively the BPZ conjugates of Q B |A and Q B |B . It follows from (2.4) that While the interacting string field takes value in H T , Siegel gauge condition further restricts the string field to be annihilated by b + 0 . The full Siegel gauge propagator ∆ for the interacting string field of the 1PI effective action was constructed in [15]. It acts on states in c − 0 H T and produces states in H T , and has the form Here β is a constant that depends on the normalization of the action and includes a factor of i for Lorentzian signature space-time background. ∆ satisfies Construction of ∆ requires inverting the operator L + 0 + b + 0 KG which is an infinite dimensional matrix. At generic momentum we can evaluate ∆ using perturbation theory in K leading to a sum over Feynman diagrams contributing to the off-shell two point function. However when the momentum is close to a value where L + 0 vanishes for some states, perturbation theory breaks down since diagrams of higher order will have poles of higher order. In this case a systematic procedure for computing ∆ was described in [18]. The strategy is to work at some fixed mass 2 level L (defined by the momentum independent contribution to L + 0 ) and 'integrate out' the contribution to ∆ from fields in the other mass levels. The latter operation can be carried out perturbatively. This generates a finite dimensional matrix in the space of states at the mass 2 level L which can then be inverted explicitly. More explicitly, this corresponds to using the identity where A and D are square matrices and B and C are rectangular matrices. In our case A denotes the kinetic operator at mass 2 level L and is a finite dimensional matrix, D is the kinetic operator for all mass 2 levels other than L and B and C are the mixing matrices between mass 2 level L and mass 2 level other than L. D −1 can be computed perturbatively to any given order, and we compute (A − BD −1 C) −1 by exact matrix inversion. This matrix then can be used e.g. to find the poles of ∆ in the k 2 plane for −k 2 near mass 2 level L.
We now briefly review the results of [1]. The analysis of [1] tells us that the string field theory amplitudes obey the Cutkosky cutting rules. These rules may be summarized as follows. 3 Here m 0 is the tree level mass, k is the momentum flowing through the cut propagator from the left to the right and θ denotes step function. Furthermore, the amplitude to the right of the cut is hermitian conjugated. However, naive application of this result will give divergent result from cut diagrams of the form shown in Fig. 1. The cut passing through the propagator P 2 forces the momentum passing through this to be on-shell, but this also forces the momentum passing through the uncut propagators P 1 and P 3 to be on-shell, making them diverge. The remedy suggested in [19,20] is to sum over all cuts of a propagator to express the result as the hermitian part of the full propagator. It is simplest to illustrate this through a scalar field propagator. Let us suppose that the full quantum corrected propagator has the form where m 0 is the tree level mass, k is the momentum flowing from the left to the right, and −iΓ(k) represents the contribution from the 1PI two point function. Then the sum over all cuts of the propagator can be expressed as There are three cases to be considered. If for the value of k of interest Γ(k) has an imaginary Full Full 1PI Full +2π Z δ(k 2 + m 2 )θ(k 0 ) = Figure 2: Diagrammatic representation of (2.13)-(2.15). The left hand side represents (2.13), the first term on the right hand side represents (2.14) and the second term on the right hand side represents (2.15).
part, then we can ignore the iǫ term and express (2.13) as (2.14) If Γ(k) is real and k 2 + m 2 0 + Γ(k) is away from 0, then (2.13) vanishes. Finally if Γ(k) is real and the full propagator has a pole on the real k 2 axis at k 2 + m 2 = 0 with residue −i Z, then near k 2 + m 2 = 0. Therefore (2.13) may be expressed as the sum of (2.14) and (2.15). This can be represented diagrammatically as in Fig. 2.
For the full superstring field theory this has the following consequence. Let us suppose that the full propagator ∆ has a pole at k 2 + m 2 = 0. Then near k 2 = −m 2 we have The iǫ determines the side of the integration contour on which the pole lies [1]. Even though ∆ is an infinite dimensional matrix, ∆ 0 is a matrix of finite rank since for given momentum we expect only a finite number of states for which the propagator develops a pole at k 2 = −m 2 . The rules for computing the contribution from a cut propagator can be summarized as follows: 1. If m 2 is real then the corresponding cut propagator should be replaced by where k is the momentum carried by the cut propagator from the left side of the cut to the right side. Furthermore we do not include any virtual self-energy corrections on either side of a cut propagator, e.g. the diagrams of the type shown in Fig. 3(a) are not (a) (b) Figure 3: Fig. (a) shows the example of a disallowed cut diagram and Fig. (b) shows the example of an allowed cut diagram.
In both examples the thick vertical line denotes the cut.
allowed since their contribution has already been included in the ∆ 0 factor in (2.17) and in the use of the renormalized mass m. However a cut can pass through a self energy diagram, e.g. a diagram of the form shown in Fig. 3(b) is allowed. These diagrams capture the imaginary part of the propagator other than the pole contribution (2.17), as represented by (2.14).
2. If m 2 has an imaginary part, i.e. if the pole corresponds to an unstable particle, then we do not have the contribution (2.17). This is consistent with the fact that unstable particles are not genuine asymptotic states and should not appear in the sum over intermediate states in a unitarity relation. However cuts passing through the self energy diagrams, like the ones shown in Fig. 3(b) are still allowed.
3. If the momentum k carried by a virtual uncut propagator is near a pole of the propagator, we must use the resummed propagator that includes repeated insertion of one particle irreducible (1PI) self-energy diagrams on the propagator. For our analysis this means that near k 2 + m 2 = 0 we should use the propagator ∆ constructed using (2.8), (2.11) which already has resummation built into it. 4 4 This rule is particularly important for unstable particles as can be illustrated using the example of Fig. 3(b). If the intermediate single particle state represented by the horizontal line corresponds to an unstable particle, it is kinematically possible for momentum flowing through the horizontal line to be near its classical on-shell value, and repeated insertion of self-energy diagrams on this will generate divergences of arbitrarily high order. Therefore we must use the resummed propagator for which we only have first order pole and the pole is shifted away from the real axis. In the limit of zero string coupling, the combined contribution from the resummed propagators on two sides of the cut and the contribution from the cut 1PI two point function approaches the delta function contribution given in (2.17).
Therefore for analyzing cut diagrams we need to focus on the properties of ∆ 0 associated with the poles that occur at real momenta. Multiplying both sides of (2.9), (2.10) by k 2 + m 2 and taking the limit k 2 → −m 2 , we get Let us now use a general ansatz where R is the rank of ∆ 0 and {|Φ m } and { Ψ m |} are a set of linearly independent states. Since ∆ 0 acts on states in c − 0 H T to produce states in H T , and BPZ inner product pairs states in H T with states in c − 0 H T , we have where in the last step we have used (2.7) and the fact that b − 0 Q B |Ψ m = 0. (2.23) is just a reflection of the fact that the poles of ∆ are associated with zero eigenvalues of the kinetic operator in the Siegel gauge.
We now classify the candidates for |Φ m satisfying these conditions near a particular pole.
1. Unphysical states: These are linearly independent states |U r satisfying for any choice of {a r } other than a r = 0 for every r. 3. Pure gauge states: These are states of the form Q B |U r . These are automatically annihilated by Q B due to (2.6) and by b + 0 due to (2.24).
Note that for any given momentum if there is an unphysical state |U r , there is also a pure gauge state Q B |U r . Generically we expect no other degeneracy but we shall proceed without making this assumption. A similar classification can be done for the candidates for Ψ m |.
Let us now suppose that at some given momentum at which ∆ has pole, there are a certain number of linearly independent physical states {|P a }, unphysical states {|U r } and pure gauge states { Q B |U r }. The normalization of these states is chosen arbitrarily. Then the general form of ∆ 0 is given by for some states |B a , |C r , |D r ∈ H T . Eq.(2.23) now gives Using this, and the fact that {|P a }, {|U r } and { Q B |U r } are linearly independent, we get Since according to (2.8) ∆ carries total ghost number −2 and since the BPZ inner product pairs states carrying total ghost number 6, we have n Pa + n Ba = 4, n Ur + n Dr = 3 , (2.31) Figure 4: A cut diagram in superstring field theory.
where for any state |A , n A denotes its ghost number.
The generalization of this analysis to type II string theories is straightforward [15]. The string field will now have four sectors satisfying NSNS, NSR, RNS and RR boundary conditions. G will be given by the identity operator in the NSNS sector, zero mode of the right-handed PCO in the NSR sector, zero mode of the left-handed PCO in the RNS sector and the product of the zero modes of the left-handed and right-handed PCO's in the RR sector. H T and H T will be defined as where H m,n denotes the subspace of H T carrying left-handed picture number m and righthanded picture number n. The rest of the analysis remains unchanged.

Unitarity
In this section we shall prove unitarity of the amplitudes of superstring field theory. This analysis will be valid for both heterotic and type II string theories. Let Γ (N ) (|A 1 , · · · |A N ) denote the truncated Green's function in which the external leg propagators are removed. Γ (N ) satisfies the following Ward identity [15]: where γ i is the grassmannality of A i . Now in a cut diagram like the one shown in Fig. 4, each cut propagator is replaced by the right hand side of (2.30) together with a 2 π δ(k 2 + m 2 ) θ(k 0 ) factor. Let us suppose that we have a cut diagram with N cut propagators. Using the superscript (i) to label the states associated with the i-th cut propagator and the operators acting on these states, we have a net factor of associated with all the cut propagators. 5 In this the ket states are inserted into the amplitude Γ (N 1 ) on the left side of the cut and the bra states are inserted into the amplitude Γ (N 2 ) on the right side of the cut. Besides these Γ (N 1 ) and Γ (N 2 ) have insertions of external incoming and outgoing states respectively, which are all annihilated by Q B .
We now expand (3.2) as a sum of 3 N terms. There is one term given by

Discussions
In this paper we have completed the proof of unitarity of covariant superstring field theory. Therefore this theory represents a manifestly Lorentz invariant, ultraviolet finite and unitary theory. Furthermore infrared divergences associated with tadpoles and mass renormalization can be dealt with using standard quantum field theory techniques.
We must note however that when the number of non-compact space-time dimensions D is 4 or less, the S-matrix suffers from the usual infrared divergences and we have to carry out the usual procedure of summing over final states and averaging over initial states to get a finite result for physical cross section [21][22][23][24]. This has not yet been worked out in superstring field theory. We hope to return to this problem in the future.