Symmetries and Feynman Rules for Ramond Sector in Heterotic String Field Theory

Examining the symmetries of the pseudo-action, we propose a prescription for the new Feynman rules for the Ramond sector of the WZW-like heterotic string field theory. The new rules are an analog of that recently proposed for the open superstring field theory and respect all the gauge symmetries including those provided we impose the constraint after transformation. It is shown that the new rules reproduce the well-known on-shell tree-level amplitudes for four and five external strings including fermions.


§1. Introduction
In the previous paper, 1) we examined the gauge symmetries of the pseudo-action, the action supplemented by the constraint, in Wess-Zumino-Witten (WZW)-like open superstring field theory. 2), 3) It was found that the pseudo-action has a new kind of symmetry provided we impose the constraint after the transformation. We proposed a prescription for the new Feynman rules for the Ramond (R) sector so as to respect all these symmetries. It was shown that the new rules reproduce the well-known on-shell tree-level amplitudes in the case of four and five external states, including those that cannot be reproduced by the self-dual Feynman rules which had already been proposed. 4), 5), 6) The aim of this paper is to extend these arguments to the heterotic string field theory and to propose a similar prescription providing the new Feynman rules.
Similar to the open superstring field theory, the heterotic string field theory can also be constructed utilizing the large Hilbert space, 7), 8) which is WZW-like in the sense that the Neveu-Schwarz (NS) action is constructed as a WZW-type action. 8) In spite of this success in the NS sector, it is difficult to construct a covariant action including the R sector, which is a disadvantage of the formulation. Without introducing any extra degrees of freedom, only the equations of motion have been constructed in a covariant manner. 9), 10) Alternatively, however, we can define the pseudo-action by introducing an auxiliary R string field. The pseudo-action of the heterotic string field theory is non-polynomial in both the NS and R string fields, which is required so as to reproduce the correct amplitudes, 11), 12), 13), 14) and was constructed at some lower order in the fermion expansion, the expansion with respect to the number of the R string fields. 9) The self-dual Feynman rules were also proposed in a parallel way to the open superstring case and shown to reproduce the on-shell four-point amplitudes. 9) It was pointed out, however, that these rules contain some ambiguity, which appears when we calculate the amplitudes with five or more external states including the fermions.
We will examine, in this paper, the gauge symmetries of the pseudo-action in detail. It will be found, at some lower order in Ψ , that the missing gauge symmetries, which have been considered the symmetries of only the equations of motion, are realized as a new kind of symmetry under which the pseudo-action is transformed into the form proportional to the constraint. We will then improve the self-dual Feynman rules to those which respect all these gauge symmetries and have no ambiguity. We will show that the new Feynman rules reproduce the correct on-shell amplitudes at the tree level, at least for the case of four and five external states including fermions.
This paper is organized as follows. In §2, we will first summarize the known basic properties of the WZW-like heterotic string field theory. After fixing the linearized gauge symmetries, we will introduce the self-dual Feynman rules proposed previously. Then the symmetries of the pseudo-action will be studied at lower-order levels in the fermion expansion. It will be found that the pseudo-action is invariant under the missing gauge symmetries if we suppose it to be subject to the constraint after the transformation. The new Feynman rules will be proposed without ambiguity so as to respect all the gauge symmetries. The on-shell tree-level amplitudes for the case of the four and the five external states including fermions will be calculated in §3 and shown to agree with those obtained in the first quantized formulation. The final section §4 is devoted to the conclusion and discussion. Some lengthy results of the missing gauge symmetries at a higher order will be given in Appendix. The higher-order corrections to the constraint, which do not exist in the case of the open superstring, first become important at this order. §2. WZW-like heterotic string field theory and the self-dual Feynman rules In this section, after introducing the WZW-like heterotic string field theory including the R sector, we will recall the self-dual Feynman rules. Examining the gauge symmetries of the pseudo-action, we will propose a prescription for the new Feynman rules, which respects all the gauge symmetries.

WZW-like heterotic string field theory
We denote the Neveu-Schwarz (NS) string as V , which is Grassmann odd and has the ghost and picture numbers (G, P ) = (1, 0). The action for the NS sector of the heterotic string field theory is given by a WZW-type action, where the pure-gauge string field G(tV ) is defined as by integrating the gauge transformation of the bosonic closed string field theory. 8) The BRST charge Q and the string products satisfy the algebraic relation 14) 0 = Q[B 1 , B 2 , · · · , B n ] + n i=1 (−1) B 1 +···+B i−1 [B 1 , · · · , QB i , · · · , B n ] + {i l ,j k } l+k=n σ(i l , j k )[B i 1 , · · · , B i l , [B j 1 , · · · , B j k ]], (2 . 3) where σ(i l , j k ) is a sign factor defined to be the sign picked up when one rearranges the sequence {Q, B 1 , · · · , B n } into the order {B i 1 , · · · , B i l , Q, B j 1 , · · · , B j k }. The arbitrary variation of the integrand of the action becomes the total derivative, and is integrated as where B δ (V ) is a function of V and δV defined by a solution of some specific ordinary differential equation, 3) whose first few terms are given by * ) (2 . 5) The pseudo-action for the R sector is constructed by introducing two R strings, Ψ and Ξ, which are both Grassmann odd and have the ghost and picture numbers (G, P ) = (1, 1/2) and (1, −1/2), respectively. The fermion bilinear term of the pseudo-action is then given by a straightforward extension of that of the open superstring field theory as where the shifted BRST charge Q G is defined by the operator acting on a general string field B as (2 . 7) From simple consideration, however, one can easily see that the pseudo-action has to be non-polynomial not only in the NS string field but also in the R string fields to reproduce the on-shell fermion amplitudes. 9) The explicit form of such a pseudo-action can in principle be obtained order by order in the fermions, the number of the R string fields, starting from (2 . 6), where each S R[2n] contains n Ψ and n Ξ. In particular, the next-leading (four-fermion) action, which is necessary for calculating the four-and five-point amplitudes in the next section, is given by (2 . 9) * ) This relation is invertible and solved by δV as Here the shifted string product [·] G is defined by (2 . 10) for general n string fields {B 1 , · · · , B n }. The equations of motion derived from the variation of S = S N S + S R agree with those obtained without introducing the auxiliary field, 9), 10) if we impose the constraint In this sense, the pseudo-action (2 . 8) describes the R sector of the heterotic string field theory.

Gauge fixing and the self-dual Feynman rules
Let us next explain how tree-level amplitudes are calculated in this formulation. For the NS sector, the Feynman rules can be derived from the action (2 . 1) in a conventional way.
Expanding the action in the power of the coupling constant κ, the kinetic term of the NS string is given by (2 . 14) Since this is invariant under the gauge transformations we have to fix these symmetries to obtain the propagator. If we impose the simplest gauge conditions, the NS propagator is given by This Ω is denoted as B −1/2 in Ref. 10), which can be determined order by order in Ψ .
The three and four NS string vertices, which are necessary for the calculation in the next section, are given by Note that the first term in the four-point vertices (2 . 19) contains the integration over two parameters (moduli) realized by the restricted tetrahedron, 11), 12) and corresponding antighost insertion. 13), 14) The second term in (2 . 19), on the other hand, is integrated over one parameter, the twist angle of the collapsed propagator. For the R sector, however, the Feynman rules cannot be uniquely derived from the pseudoaction (2 . 8) since it is not the true action. We can only propose some plausible Feynman rules and confirm whether they reproduce the correct physical on-shell amplitudes. In the previous paper, we proposed the Feynman rules, which we refer to as the self-dual Feynman rules and confirmed that they actually reproduce the well-known four-point amplitudes with external fermions. 9) We first summarize the self-dual Feynman rules. Similar to the NS case, we can expand the pseudo-action in the power of the coupling constant κ as (2 . 20) The kinetic term of the R string, the propagator of the R sector in this gauge is given by For the R sector, in addition, the constraint (2 . 11) has to be taken into account. For the on-shell external states, this is naturally implemented by simply restricting them to those satisfying the linearized constraint, QΞ = ηΨ . In contrast, however, the prescription for the off-shell (propagating) states is not unique. The self-dual Feynman rules are defined by adopting a prescription in which only the self-dual part ω = (QΞ + ηΨ )/2 of the R strings propagates through the effective propagator Although the fermion interaction vertices can be obtained by replacing the R string fields with their self-dual part, we need some preparation since, unlike the case of the open superstring field theory, the R string fields do not appear only in the form of QΞ or ηΨ . For example, the terms with three, four and five string fields needed in the next section are given as does not affect the on-shell physical amplitudes, as with the point transformation in the conventional quantum field theory, we can rewrite (2 . 26) so that theΞ always appears in the form of QΞ thanks to the relation (2 . 28) Then we can replace QΞ with ω in the alternative expression. Contrary to this, the prescription for Ψ is not unique but depends on the gauge condition in general. In the simplest gauge (2 . 23), we can replace Ψ with ξ 0 ω since Ψ = {η, ξ 0 }Ψ = ξ 0 ηΨ . However, we have two choices in replacing ηΨ ; either we simply replace it with ω, or η(ξ 0 ω) in accordance with the above prescription for Ψ . Since ω = ηξ 0 ω for the off-shell states, this is an ambiguity in the self-dual Feynman rules, which does not appear in the four-point amplitudes. If we take the former choice, the interaction vertices for the self-dual rules becomẽ after the replacements. It was shown that these self-dual Feynman rules reproduce the well-known on-shell tree-level amplitudes for the case of four external states including the fermions. 9)

Gauge symmetries and the new Feynman rules
In order to revise the Feynman rules, let us examine the gauge symmetries in detail. As was pointed out in Ref. 9), the total action, S = S N S + S R , is invariant under the gauge transformations We can define the next-order transformation, so that the total action is invariant up to the higher-order corrections: At the next-next-order, however, we cannot keep the action invariant. Instead, we can find the transformations, The right-hand side vanishes, up to the higher-order corrections, if we impose the constraint (2 . 11). We can also construct the nonlinear transformation generated by Λ 1/2 . The leadingorder transformation, This can be extended to the next order as which transforms the pseudo-action in the form proportional to the constraint as δ [4] Λ 1/2 S N S +δ The remaining two gauge symmetries in (2 . 22) generated by Λ 3/2 andΛ 1/2 can similarly be found order by order in Ψ . The transformation This is also a new kind of symmetry, which is shown in Appendix by constructing the nextorder correction. The last gauge transformation, defined at the linearized level by We can find the next-order transformation, From these considerations, it is natural to expect that these new types of gauge symmetries can be constructed order by order in Ψ , although we cannot yet prove it. We give the next-order results as a further evidence in Appendix. They are also nontrivial in the sense that the higher-order correction of the constraint is included.
Since all these gauge symmetries, including those provided by imposing the constraint, must be important to reproduce the unitary amplitudes, we assume that they have to be respected by the new Feynman rules and propose the following alternative prescription: • Use the off-diagonal propagator (2 . 24) for the R string.
• Use the vertices (2 . 26) as they are without any restriction.
• Add two possibilities, Ξ and Ψ , for each external fermion, and impose the linearized constraint, QΞ = ηΨ , on the on-shell external states.
Our claim is that this prescription respecting all the gauge symmetries is more suitable for the Feynman rules suggested by the pseudo-action (2 . 8). This is supported by the fact that there is no ambiguity, associated with the self-dual−anti-self-dual decomposition already mentioned, in the new Feynman rules. The new prescription, in addition, has an advantage that it does not require any special preparation like the field redefinition (2 . 27). §3. Amplitudes with external fermions Using the new Feynman rules, we will explicitly calculate in this section the on-shell fourand five-point amplitudes with external fermions. It will be shown that the results agree with the well-known amplitudes obtained in the first quantized formulation.

Four-point amplitudes
The on-shell four-point amplitudes with external fermions were already calculated using the self-dual Feynman rules and shown to agree with the well-known amplitudes obtained in the first quantized formulation. 9) We first have to confirm that the new Feynman rules also reproduce the same results.
Let us start from the calculation of the four-fermion amplitude A F 4 . The contributions come from the s-, t-, and u-channel diagrams constructed using two three-string vertices, and also a contact-type diagram containing a four-string vertex. * ) In this paper, we denote for example the s-channel diagram, schematically depicted by Fig. 1

(a), as (AB|CD), where
A, B , C, and D are labels which distinguish external strings. Since the order of strings A and B, or C and D, has no meaning in the heterotic (closed) string theory, this has as much information as this type of Feynman diagram. The t-and u-channel diagrams are denoted by (AC|BD) and (AD|BC) in this notation, respectively. Using the new Feynman rules, (3) the s-channel contribution is written as where the correlation is evaluated as the conformal field theory on the corresponding string diagram. The insertions ξ c , b − c , and b + c are the corresponding fields integrated along the contour winding around the propagator. The numbers in the parentheses are the labels which distinguish each leg of the diagram, but they are redundant if we always arrange the * ) The corresponding string diagrams are depicted in Ref. 9). external states in order of the numbers from the left as in (3 . 1). We omit them hereafter by taking this convention. * ) The t-and u-channel contributions can similarly be written as where we used the shorthand notation Unlike the open superstring case, a contact-type diagram also gives the contribution integrated over a region of the moduli space not covered by those from these three diagrams.
It was shown that such a contribution can be realized using the four-string interaction represented by the restricted tetrahedron, 11) or n-faced polyhedra for general n-string contact interactions, 12) parametrized by θ I (I = 1, · · · , 2(n − 3)) in the notation in 13). Then the contribution from the contact-type diagram (ABCD) is given by Here the definition of the parameters θ 1 and θ 2 , their integration region and the corresponding contours C 1 and C 2 , along which the anti-ghost insertions are integrated, are given in Ref. 13); their explicit forms are not necessary here. Adding all these contributions and imposing the linearized constraint QΞ = ηΨ on each external state, the on-shell four-fermion amplitude eventually becomes where · · · W represents the correlation in the small Hilbert space: where O 1 , · · · , O n are the operators in the small Hilbert space. The ξ on the right-hand side can either be local or integrated. The correlation is independent of its position or contour since only the zero mode gives the non-vanishing contribution. Although we can, in principle, map this expression (3 . 6) to the well-known form in the first quantized formulation evaluated on the complex plane, 15), 16), 17) it is not necessary if we notice that each term has the same form as that in the bosonic closed string field theory with the identification of ηΨ and the bosonic string fields, both of which have the same ghost number, G = 2. Using the fact that the bosonic closed string field theory reproduces the correct perturbative amplitudes, we can conclude that the amplitude (3 . 6) agrees with that obtained in the first quantized formulation. We can similarly calculate the two-boson-two-fermion amplitude. After a little manipulation, the contributions from the s-, t-and u-channel diagrams become respectively. The contribution from the contact-type diagram consists of two parts coming from the two vertices in (2 . 26b): is the integration over the twist angle of the collapsed propagator, and b − θ is the corresponding anti-ghost insertion. Although these four contributions other than the second term of (3 . 11) cover the whole moduli space, they are not smoothly connected at each boundary since the external states in each contribution appear in different forms (pictures). This gap is canceled by the remaining contribution, the second term in (3 . 11). * ) We can show this by aligning the * ) These discrepancies can be interpreted as coming from the difference of the positions of the picturechanging operators. 18) The second term in (3 . 11) corresponds to the contribution from the vertical integration introduced in Ref. 19). external bosons in the four contributions to the same form, say (QV C , ηV D ). This is possible by integrating by parts with respect to η and Q, but the latter produces extra boundary contributions appearing through the relation and the similar relation for the anti-ghost insertions in the tetrahedron vertex, which can be read from the algebraic relation (2 . 3) satisfied by the corresponding string products. After such an alignment, each contribution becomes We can easily see that the boundary contributions are completely canceled, and the total amplitude becomes and can be rewritten as after imposing the constraint. Similarly to the case of the four-fermion amplitude, this final expression agrees with that in the bosonic closed string field under the identification of the external bosonic strings and the external strings in (3 . 19), that is, ηΨ , QV and ηV . * ) Thus, we can again conclude that the well-known amplitude in the first quantized formulation is correctly reproduced.

Five-point amplitudes
Let us next calculate the on-shell five-point amplitudes with external fermions. We follow the convention in the previous subsection; we label the five external strings by A, B, C, D, and E arranged in order of the number assigned to the legs as depicted in Figs. 2 and 3. There are three types of diagrams contributing to the five-point amplitudes, which we refer to as the two-propagator (2P), one-propagator (1P), and no-propagator (NP) diagrams corresponding to the number of propagators to be included. The 2P diagrams contain three three-string vertices and two propagators as depicted in Fig. 2, which we simply denote as (BC|A|DE). The 1P diagram contains one three-string vertex, one four-string vertex, and (3) one propagator as depicted in Fig. 3. We denote this diagram as (AB|CDE).
There are two types of five-point amplitudes including external fermions: the fourfermion-one-boson (F 4 B) and two-fermion-three-boson (F 2 B 3 ) amplitudes. Let us first calculate the former, F 4 B, amplitude. Suppose that the strings A, B, C, and D are fermions and the string E is a boson. We begin with the calculation of the contributions from the fifteen, ( 5 C 1 × 4 C 2 )/2, 2P diagrams. For example, the contribution of the diagram (BC|A|DE) is calculated as where the inserted operators, ξ c i or b ± c i , are integrated along the contour winding around the i-th propagator. We moved, by integrating by parts without exchanging the order of Q and ξ, the operators Q and η in a way that acts on the external states. This produces the boundary contributions, in which one of the two propagators collapsed. Eleven of the remaining fourteen diagrams are obtained by simply relabeling the external fermions: The last three contributions, coming from the diagrams including the boson in the center, are obtained by calculating one of them, for example, and relabeling its external fermions as Note that the external boson appears in the same form QV E in all the dominant contributions integrated over (a part of) the full moduli space.
There are ten, 5 C 2 , 1P diagrams classified two categories by whether the external boson is attached to the three-string vertex or the four-string vertex. It is enough to calculate only one of the contributions in each category, and the others can be obtained by relabeling the external fermions. The amplitudes in the first category are given by and those in the second category are The contributions from the last (NP) diagram can also be divided into two parts; the dominant part integrated over the whole moduli space and the boundary part coming from the first and the second four-string vertices in (2 . 26e), respectively: The total amplitude is obtained by summing up all these contributions. Almost all the boundary contributions are canceled, except for a small portion given by which vanishes if we impose the constraint QΞ = ηΨ . In consequence, the total amplitude can be written as the sum of the dominant contribution of each diagram, which can be evaluated as the correlations in the small Hilbert space as after imposing the constraint. The first, second and third lines come from the 2P, 1P, and NP diagrams, respectively. Each of these contributions has the same form as that in the bosonic closed string field theory if we identify the bosonic string fields with ηΨ or QV . Hence the four-fermion-one-boson amplitude calculated by the new Feynman rules agrees with the well-known amplitude in the first quantized formulation.
We can similarly calculate the two-fermion-three-boson, F 2 B 3 , amplitude. The 2P diagram (BC|A|DE) is, for example, given by using the new Feynman rules. We can move Q, by integrating by parts, so as to act on Ξ, and align the external bosons as (QV C , QV D , ηV E ), which are uniquely realized by requiring not to exchange the order of Q and ξ: According to this recipe, the contributions from the other fourteen diagrams are similarly calculated as The contributions from the 1P diagrams are also calculated in the same manner, for example: The external bosons in the dominant contribution, the first term, are again aligned as (QV C , QV D , ηV E ). The contributions from the other nine 1P diagrams are also calculated as The last contribution from the NP diagram can be divided into three parts: those integrated by four, three, and two moduli parameters, respectively. After a little calculation to align the bosons in the first part, the dominant contribution, we obtain: The total amplitude is given by summing all these contributions. One can show that the boundary contributions integrated over less (two or three) moduli parameters are canceled, and consequently the total amplitude becomes the sum of the dominant contribution of each diagram: Each contribution again has the same form as that in the bosonic closed string field theory after imposing the constraint if we identify the external bosonic strings and ηΨ , QV or ηV .
Hence the two-fermion-three-boson amplitude is also reproduced by the new Feynman rules. §4. Conclusion and discussion In this paper we have reconsidered the symmetries of the pseudo-action of the heterotic string field theory. It has been found, at some lower order in the fermion expansion, that the missing gauge symmetries, which were considered to be present only in the equations of motion, are realized as the symmetries provided we impose the constraint after the transformation. Respecting also this type of gauge symmetry, we have proposed a prescription for the new Feynman rules and shown that they actually reproduce the correct tree-level amplitudes in the case of the four-and five-external strings including fermions.
An important remaining task is to prove that the new Feynman rules actually reproduce an arbitrary on-shell amplitude at the tree level. For this purpose, it is necessary to complete the pseudo-action, which has only been obtained at some lower order in the number of fermions or string products.

Acknowledgments
This work was initiated at the workshop on "String Field Theory and Related Aspects VI" held, from 28 July to 1 August 2014, at SISSA in Trieste, Italy . The author would like to thank the organizers, particularly Loriano Bonora, for their hospitality and providing a stimulating atmosphere.
The gauge symmetries provided by the constraint given in § §2.3 have only been shown to exist at some lower order in the fermion expansion. Up to the order discussed in the text, however, the transformation of the pseudo-action is proportional to the constraint in the lowest order of the fermion expansion: Q G Ξ = ηΨ . It is therefore worthwhile to show that the transformation including the next-order corrections properly transforms the pseudo-action to the form proportional to the constraint correctly including the next-order corrections. Including the next-order pseudo-action, 9) we can find that the next-order Λ 1 -transformation has to be δ [4] Then the transformation of the pseudo-action at this order is given by where the first four terms give the O(Ψ 3 ) corrections to the constraint in the previous order result (2 . 43).
Last of all, theΛ 1/2 -transformation can be found as: The first two terms give the correction to the constraint in (2 . 62).