Symmetries and Feynman Rules for Ramond Sector
in Open Superstring Field Theory
Abstract
We examine the symmetries of the action suppelemented by the constraint in the WZWtype open superstring field theory. It is found that this pseudoaction has additional symmetries provided we impose the constraint after the transformation. Respecting these additional symmetries, we propose a prescription for the new Feynman rules for the Ramond sector. It is shown that the new rules reproduce the wellknown onshell four and fivepoint amplitudes with external fermions.
[3cm]YITP14106
1 Introduction
One of the first criteria to determine whether a string field theory is acceptable or not is that its onshell physical amplitudes are equivalent to the wellknown results in the firstquantized formulation. This is well studied for bosonic string field theories. It has been proved that an arbitrary amplitude at any loop order is correctly reproduced in the cubic open string field theory[1, 2, 3] and the nonpolynomial closed string field theory.[4, 5, 6] In contrast, such considerations are still not sufficient for superstring field theories.
The most promising open superstring field theory at present is the WessZuminoWitten (WZW)type formulation proposed by Berkovits utilizing the large Hilbert space.[7, 8] Its NeveuSchwarz (NS) sector is compactly described by the WZWtype gaugeinvariant action with the help of the puregauge string field.[7] It does not require explicit insertions of the picturechanging operators and thus does not suffer from the divergence coming from their collisions. In return for this advantage, the action becomes nonpolynomial, which obscures whether it gives the correct amplitudes or not. It has been confirmed so far that only four[9] and fiveboson[10, 11] amplitudes at the tree level are correctly reproduced.
On the other hand, the Ramond (R) sector of the formulation is less well understood. While the equations of motion can be given,[8] it is difficult to construct the covariant action. Then, as an alternative to an action in the usual sense, Michishita constructed an action supplemented by an appropriate constraint by introducing an auxiliary field.[12] The variation of this pseudoaction leads to the equations of motion that reduce to the desired ones after eliminating the auxiliary field by imposing the constraint. Although this is not a usual action from which we can uniquely derive the Feynman rules, it could be used as a clue to propose the Feynman rules^{2}^{2}2 We call them the selfdual rules in this paper since they allow only the selfdual part (the part satisfying the linearized constraint) of the R string fields to propagate. that reproduce the correct onshell fourpoint amplitudes with external fermions.[12] After a while, however, it was found that these selfdual rules do not lead to the correct fivepoint amplitudes with two external fermions.[10, 11]
In order to determine the reason why the selfdual rules do not reproduce the correct amplitudes, we will examine the gauge symmetries of the theory. It has been known that the pseudoaction of the R sector has fewer gauge symmetries than those at the linearized level.[12] We will find that the missing symmetries exist provided we impose the constraint after transforming it. While these are not symmetries in the usual sense, we will assume that they have to be respected in the calculation and propose a prescription for new Feynman rules. Then, by using these new rules, we will calculate the four and fivepoint amplitudes with external fermions. It will be shown that they are in fact equivalent to the amplitudes in the first quantized formulation.
This paper is organized as follows. In § 2, we will first summarize the basic properties of the WZWtype open superstring field theory. The symmetries of the pseudoaction will then be studied. It will be found that it is invariant under the additional gauge symmetries if we suppose it to be subject to the constraint after the transformation. Respecting these symmetries, we will propose a prescription for the new Feynman rules, in which the R propagator has an offdiagonal form. By using these new rules, we will explicitly calculate the onshell four and fivepoint amplitudes with external fermions in §3. During the process to confirm that the fourpoint amplitudes are reproduced as in the selfdual rules, it will be clarified what kind of diagrams produce differences between two results by the two sets of the Feynman rules. For the fivepoint amplitudes, such diagrams certainly appear in the calculation of twofermionthreeboson amplitudes that cannot be correctly reproduced by the selfdual rules. We will show that the results are actually improved, and in consequence all the fivepoint amplitudes come to be equivalent to those in the first quantized formulation. The final section, §4, is devoted to the conclusion and the discussion.
2 New Feynman rules in WZWtype open superstring field theory
After summarizing the known basic properties of the WZWtype open superstring field theory,[7, 12] we will recall the selfdual Feynman rules proposed in Ref. \citenMichishita:2004by in this section. Then, examining the gauge symmetries of the pseudoaction, we will propose a prescription for the new Feynman rules, which are more natural in the viewpoint of the symmetry.
2.1 WZWtype open superstring field theory
The conventional (small) Hilbert space of the firstquantized RNS superstring is in general described by a tensor product of those of an superconformal matter with central charge , the reparametrization ghosts with , and the superconformal ghosts with . The superconformal ghost system is known to be represented also by a chiral boson and a pair of fermions through the bosonization formula,[13]
(1) 
The large Hilbert space can be introduced by replacing the Hilbert space of the superconformal ghosts in by that of the bosonized fields , which is twice as large as due to the zeromode . The correlation function in is normalized as
(2) 
and can be nonzero if and only if the ghost and picture numbers in total.
The NS (R) string field, (), is defined using . Their free equations of motion are given by^{3}^{3}3 In this paper the zeromode of is denoted as , for simplicity.
(3a)  
(3b) 
which are invariant under the gauge transformations
(4a)  
(4b) 
Here the is the firstquantized BRST operator. The string fields and are Grassmann even and have and , respectively. The gauge parameter string fields are Grassmann odd and have . The onshell states satisfying (3) up to the gauge transformation (4) are equivalent to the conventional physical spectrum defined by the BRST cohomology in .
For the NS sector, the free equation of motion (3a) was ingeniously extended by Berkovits[7] to the nonlinear equation of motion
(5) 
derived from the variation of the the WZWtype action,
(6) 
This action (6) is invariant under the nonlinear extension of (4a),
(7) 
where . The shifted BRST operator is defined as an operator that acts on a general string field as
(8) 
We can show that this is also nilpotent, , due to the identity,
(9) 
In contrast, only the equations of motion (3) can be extended to the nonlinear form,
(10a)  
(10b)  
for the full theory including the interaction with the R sector. 
The gauge transformations (4) can also be extended to the nonlinear form,
(11a)  
(11b) 
so as to keep the equations of motion (10) invariant. However, we cannot construct an action, or even its quadratic term, so as to have unless we introduce the (inverse) picturechanging operator or an additional string field.
Then, as an alternative formalism, an action for the R sector,
(12) 
was proposed[12] by introducing an auxiliary Grassmann even R string field with . The equations of motion derived from the variation of are
(13a)  
(13b)  
(13c) 
where , which reduce to (10) if we eliminate the by imposing the constraint,
(14) 
In this sense, the action (12) is not an action in the usual sense but a pseudoaction supplemented by the constraint (14).[14]
2.2 Gauge fixing and the selfdual Feynman rules
We next review how treelevel amplitudes are calculated in this formulation. Let us first derive the Feynman rules for the NS sector from the action (6). Since its quadratic part,
(15) 
is invariant under
(16) 
we have to fix these symmetries to obtain the propagator. We take here the simplest gauge conditions:
(17) 
The NS propagator in this gauge becomes
(18) 
The interaction vertices can be read by expanding (6) in the power of . The three and four string vertices are
(19a)  
(19b) 
respectively, which are necessary for the calculation in the next section. It was confirmed that these Feynman rules reproduce the same onshell physical amplitudes with four[9, 15] and five[10, 11] external bosons as those in the first quantized formulation.
For the R sector, however, the Feynman rules are not logically derived from the pseudoaction (12) since it is not an action in the usual sense. We summarize here the Feynman rules proposed in \citenMichishita:2004by. We first suppose that the and are independent string fields. Then the propagator can be easily read from the quadratic term,
(20) 
as in the case of the NS sector. Fixing the gauge symmetries
(21a)  
(21b) 
by the same conditions as (17),
(22) 
we can obtain the (offdiagonal) R propagator in this gauge as
(23) 
The auxiliary field is eliminated from the external onshell states by the linearized constraint .^{4}^{4}4 Note that the linearized constraint is sufficient to impose on the external (asymptotic) onshell states. The rule not uniquely determined is how we take into account the constraint at the offshell. A prescription for the selfdual Feynman rules is to replace and in the vertices with their selfdual part , by which the part that vanishes under the (linearized) constraint is decoupled. From the cubic, quartic and quintic terms of the action (12),
(24a)  
(24b)  
(24c) 
the three, four and five string vertices in this prescription, needed to calculate the fivepoint amplitudes later, are obtained as
(25a)  
(25b)  
(25c) 
respectively. In particular, the twofermiontwoboson vertex (or generally twofermionevenboson vertices) vanishes in this prescription.[12] The propagator has the form
(26) 
from (23). It was shown that these selfdual Feynman rules reproduce the wellknown fourpoint amplitudes,[12] but unfortunately do not do the fivepoint amplitudes with two external fermions.[10, 11] The extra contributions including no propagator are not completely cancelled by those from the five string interaction (25c), and remain nonzero.
2.3 Gauge symmetries and the new Feynman rules
In order to find out the reason why the selfdual Feynman rules do not work well, let us examine the gauge symmetries in detail. The total (pseudo) action, , is invariant under the gauge transformations,
(27a)  
(27b) 
Since these symmetries are compatible with the selfdual antiselfdual decomposition of the R strings,
(28) 
they are also the symmetries of the constraint (14), and so respected by the selfdual Feynman rules.^{5}^{5}5 It is not clear whether it is sufficient to take into account the linearized constraint to define the selfdual part or not. Nevertheless, these symmetries do not include all the symmetries of the linearized level, (21). The missing transformations extended to the nonlinear form
(29a)  
(29b) 
transform the action to the form proportional to the constraint:
(30) 
In other words, the action is invariant under (29) provided we impose the constraint after the transformation. Their consistent part with the constraint, obtained by putting , reduce to the symmetries (11) of the equations of motion (10) if we eliminate the by the constraint. These are not the symmetries in the usual sense, but have to be important properties to characterize the action. Therefore, it is natural to consider that a reason why the selfdual rules do not work is because the replacement to the selfdual part of and breaks these symmetries. This leads us to propose the following alternative prescription for the (treelevel) Feynman rules:

Use the offdiagonal propagator (23) for the R string.

Use the vertices (24) as they are without restricting both of and to their selfdual part.

Add two possibilities, and , of each external fermion and impose the linearized constraint on the onshell external states.
We claim this prescription respecting all the gauge symmetries, including those in the above sense, is more appropriate for the Feynman rules read from the pseudoaction (12).
3 Amplitudes with external fermions
In this section, we will explicitly calculate the onshell four and fivepoint amplitudes with external fermions using the new Feynman rules. It will be shown that the equivalent amplitudes to those in the first quantized formulation are correctly reproduced.
3.1 Fourpoint amplitudes
The onshell fourpoint amplitudes with external fermions were already calculated using the selfdual Feynman rules, and shown to be equivalent to those obtained in the first quantized formulation.[12] We first show that the new Feynman rules also reproduce the same results.
Let us start from the calculation of the fourfermion amplitude with fixed color ordering. Since there is no fourfermion vertex in (24b), the contributions only come from the  and channel diagrams in Fig. 1. We take a convention that the fermion legs and propagators in the Feynman diagrams are colored with gray.
If we label each four external states , , and , the channel contribution is calculated as
(31) 
where the correlation is evaluated as the conformal field theory on the Witten diagram given in Fig. 2.
The and denote the corresponding fields integrated along the path depicted on the diagram. Each leg is numbered from 1 to 4, but this is redundant if we always arrange the external states in order of the numbers from the left as in (31). Taking this convention, we omit hereafter to indicate them. Then the channel contribution can similarly be calculated as
(32) 
These two contributions are essentially the same as those obtained using the selfdual rules[12] and are combined into the conventional fourpoint amplitude as
(33) 
where, in the last equality, we eliminate the auxiliary field from the onshell external states by imposing the linearized constraint . Recalling that the BRSTinvariant fermion vertex operator in the picture is included in in the form , we can explicitly map the last expression in (33) to that evaluated on the upper halfplane:[16]
(34) 
Here is the appropriate Beltrami differential for an dependent parametrization of the modulus.[9]
The twofermiontwoboson amplitude with colorordering has three contributions from channel, channel and fourstring interaction diagrams in Fig. 3.
The channel contribution is evaluated as
(35) 
where, in the second equality, we moved the and on the external bosons so that each of them has single forms and . Consequently, the extra contribution, which does not include the propagator (proper time integration), is produced from the boundary at when we exchange the order of the and , due to the relation
(36) 
The channel contribution is similarly calculated as
(37) 
which was deformed again so that each external boson has the same forms, and , as in the channel contribution. Adding the contribution from the fourstring interaction, which we can read from (24b) as
(38) 
the total amplitude becomes
(39) 
We again eliminated the in the last expression by the linearized constraint. It should be noted here that the extra contributions with no propagator in (35) and (37) are cancelled by that from the fourstring interaction diagram (38) without imposing the constraint. Using the fact that the BRSTinvariant NS vertex operator in the picture is included in the in a form such as , and so , we can again map the result (39) to the form
(40) 
This is equivalent to that obtained in the first quantized formulation.^{6}^{6}6The overall minus sign can be absorbed into the phase convention for how the fermion vertex operator is embedded in . This has to be fixed by imposing the reality condition on .
The last fourpoint amplitude is that for twobosontwofermion scattering with ordering . It also has three contributions from the three diagrams in Fig. 4, which can be given by
(41a)  
(41b)  
(41c) 
by deforming so that the external boson states have the common forms and . The extra contributions in (41a) and (41b) are again cancelled by that from the fourstring interaction (41c) at this stage. In consequence the amplitude becomes the wellknown form:
(42) 
In this way the new Feynman rules also give the same onshell fourpoint amplitudes as those obtained by the selfdual rules.
In general, one can see that the two sets of rules give different results in the contribution from the diagram with either (i) at least two fermion propagators or (ii) twofermionevenboson interaction at least one of whose fermions is connected to the propagator. The difference in case (i) comes from the form of the fermion propagators. If the diagram has two fermion propagators, the selfdual rule using the propagator (26) gives a contribution of the form
(43) 
If we follow the new Feynman rules, on the other hand, the contribution of the same diagram becomes
(44) 
using the fermion interactions,
(45) 
and the (offdiagonal) R propagator (23).^{7}^{7}7 Similarly, it is easy to see that the two rules give the same contributions if the diagram has only one R propagator. In case (ii), the difference is due to the fact that (45) can be rewritten as
(46) 
Therefore, the twofermionevenboson vertices for the selfdual rules vanish, as previously mentioned. In the new Feynman rules, in contrast, the twofermionevenboson interactions can contribute if at least one of the two fermions is connected to the propagator. We will next show that these differences in fact improve the discrepancy in the fivepoint amplitudes.
3.2 Fivepoint amplitudes with external fermions
Then we calculate the onshell fivepoint amplitudes with external fermions. We follow the convention above; i.e., we label the five external strings by , and , and omit to explicitly indicate the numbers, depicted in Figs. 5(a), 7(a), and 10, by arranging the external states in order of these numbers from the left.
3.2.1 Fourfermiononeboson amplitude
Let us begin with the calculation of the fourfermiononeboson amplitude. The dominant contributions come from the diagrams containing three threestring vertices and two propagators, which we call in this paper the twopropagator (2P) diagrams. There are five different channels for colorordered amplitudes as in Fig. 5.
The contribution from the first diagram Fig. 5(a) is given by
(47) 
where the correlation is evaluated as the conformal field theory on the Witten diagram depicted in Fig. 6. and denote the corresponding fields integrated along the paths and , respectively.
Under the onshell conditions, all the and can be moved so as to act on the external states (without exchanging the order of and ):
(48a)  
where we used a shorthand notation  
The extra terms with less (one) propagator were produced from the boundary at or through the relation (36) by exchanging the order of the and . The contributions from the other four channels depicted in Figs. 5(b)(e) can similarly be evaluated as  
(48b)  
(48c)  