Chiral SUSY Theories with a Suppressed SUSY Charge

The well-known Chiral and Gauge SUSY Actions realize the SUSY charge in terms of transformations among the Fields. These transformations are included in the Master Equation by coupling them to Sources. Here we show that there are new local SUSY Actions where the Chiral SUSY transformations are realized in terms of transformations among both Fields and Sources. These Actions can be easily obtained from the Chiral case by a very simple and local `Exchange Transformation', which carries along all the interactions without difficulty. For these new SUSY Actions, the SUSY charge does not exist in the relevant sector, because Sources do not satisfy Equations of Motion. Nevertheless, the `Exchange Transformation' ensures that the new Master Equation is true for the new Action. As a consequence, the Master Equation also is true for the new 1PI Generating Functional. This implies that a `Suppressed SUSY Charge' version of SUSY is still present. SUSY certainly becomes more obscure and less constrained in this case. But it is still very restrictive. The new theories can be obtained from the old theories by using a special technique, but it is not true that they are a sort of `broken version of supersymmetry'. They are simply a new type of theory that is governed by Supersymmetry, but without the use of Supercharges (except perhaps in some sectors). In particular the number of physical Bosonic and Fermionic degrees of Freedom are not equal for these new (sub)-Actions, although there is still Boson/Fermion mass degeneracy in a (sub)-Action, so long as there is still a Boson present. Notably, there is even a SUSY (sub)-Action where the physical Scalars are not present, so that the (sub)-Action contains physical Fermions only. In this theory the degeneracy of Bosonic and Fermionic masses is obviously not present.

The new theories can be obtained from the old theories by using a special technique, but it is not true that they are a sort of 'broken version of supersymmetry'. They are simply a new type of theory that is governed by Supersymmetry, but without the use of Supercharges (except perhaps in some sectors).
In particular the number of physical Bosonic and Fermionic degrees of Freedom are not equal for these new (sub)-Actions, although there is still Boson/Fermion mass degeneracy in a (sub)-Action, so long as there is still a Boson present. Notably, there is even a SUSY (sub)-Action where the physical Scalars are not present, so that the (sub)-Action contains physical Fermions only. In this theory the degeneracy of Bosonic and Fermionic masses is obviously not present, and this happens without the spontaneous or explicit breaking of SUSY. No such 'Exchange Transformation' has been found for SUSY gauge Actions.

Introduction:
In spite of a large amount of work, there is still much that is unknown about SUSY [1,2,3,4,5,6]. Even its representation theory is filled with unanswered questions [7]. Much recent work has concerned itself with geometric and duality issues in supersymmetric theories. Some of this is associated with Branes, M theory and AdS CFT [8,9,10,11]. A large body of SUSY work has, understandably, focussed on phenomenology and experimental signals, while leaving the problems of the origin of the spectrum of mass splitting for future work [12,13,14]. The progress reported in this paper is the direct result of a study of the algebraic problem of the local BRST cohomology of the Chiral Multiplet [15,16,17], which examined the solution to the tachyon problem. Previous work, troubled by the tachyon problem, was in [18,19,20,21,22,23] 3. This paper is organized as follows: We introduced some names for the new theories in Section 2. Next, in Section 4, we will pose and answer some questions that are bound to arise in the mind of a reader who is familiar with SUSY. Then the way that these new Multiplets arose from the study of the BRST cohomology is described in Section 5. The Chiral Multiplet and the integration of its auxiliary field is in Sections 6 to 8. The Majorana version of the Half-Chiral Multiplet is explained starting with Section 9 to 14. Section 15 dicusses how and why these Exchange Transformations are special. Then in Section 16, we show how to add a mass term to the Half-Chiral Multiplet. One starts by adding a mass term to the Chiral Multiplet, which was already done in Section 6 . Then the Exchange Transformation is used to convert that Action to the Half-Chiral Multiplet Action. The result is interesting, because it is easy to see how the SUSY Charge is modified, as is explained in Section 18. Interactions in the Half-Chiral Multiplet are discussed in Section 19. Then we introduce the Un-Chiral Multiplet in Section 20 . Again we introduce a mass term by starting with the massive Chiral Multiplet, and ending with the massive Un-Chiral Multiplet in Section 21. This case is simpler than the Half-Chiral Multiplet case, because there are only fermions left in the Un-Chiral Multiplet. Next the well known derivation of the Master Equation for the Chiral Multiplet is recalled in Section 22. This is followed by the parallel derivation for the Un-Chiral Multiplet in Section 23. This latter derivation depends crucially on the invariance of the Master Equation under the Exchange Transformation, as shown in Section 24. Some remarks about the Hamiltonian and the Cohomology are made in Section 25. The Conclusion summarizes the results in section 26. This is followed by a summary of the Dirac version of the Half-Chiral Multiplet in the Appendix, which occupies Sections 27 to 37.

Surely the Chiral Multiplet is irreducible, so how can it be reduced?
And surely the SUSY algebra implies Degenerate Super-Multiplets [36], so how can there be field theories which do not have them unless SUSY itself is spontaneously or explicitly broken? And if the SUSY charge is not conserved, how can there be any SUSY left at all? Answer: • To see in detail how the theory gets around these very reasonable objections, the reader (if he or she is impatient) can look at Section 21 for the Un-Chiral Multiplet case, which is rather simple. The Half-Chiral Multiplet case, which is a little more complicated, is explained in Sections 16, 17 and 18.
• The result of the Exchange Transformation is not a new representation of SUSY. It is a representation of the SUSY algebra in terms of Fields and Zinn Sources, which is physically not the same at all as a representation in terms of Fields alone. Fields are physically realizable, and Zinn Sources are not. When the SUSY algebra spreads to the Zinn Sources, it is no longer realized in the same way on the physical states.
• • The relation of this 'Suppressed SUSY Charge' mechanism to the BRST cohomology is rather obscure. This is true even though the examination of BRST cohomology is how it was found, as will be explained below in Section 5. One could say that the SUSY chiral algebra is still present, although the conserved SUSY charge is not. This puzzling topic, and the question of the Hamiltonian, are discussed a little more in Section 25.

BRST Cohomology of the Chiral Multiplet:
The BRST cohomology of the Chiral Multiplet is immense, but much of it has unsaturated Lorentz Spinor indices [18]. By and large, this has been taken to mean that 'this cohomology cannot appear in a Lorentz invariant Action and so it is of no interest'. But this pessimistic view is not valid. The cohomology is very important, because it can only be relevant if one couples the unsaturated indices to something new, so that the cohomology can appear in an Action.
The point is that this opens up a new view on SUSY. The simplest such new object is evidently a Chiral Dotted Spinor Superfield [15]. But then the question is whether such an object makes sense by itself, and the answer has been discouraging for a long time. The most obvious Action for the Chiral Dotted Spinor Superfield has higher derivatives and also tachyons. But recently some progress was made on this problem [15,16,17]. A tachyon free Action for the Irreducible Chiral Dotted Spinor Superfield was found and used in a free theory 5 . These Exchange Transformations were found while trying to make that theory an interacting one. In fact they go beyond the results from the cohomology. We will return to the Majorana version of the Irreducible Chiral Dotted Spinor Superfield in Section 9. The Dirac version of the Irreducible Chiral Dotted Spinor Superfield is in the Appendix starting with Section 27.
6. The Chiral Scalar Superfield: We start with the well-known Chiral Multiplet theory. It has the total Action: where the free massless kinetic Action is: and mass and interaction terms look like 6

A Chiral Mass and Interaction
and the Zinn Action is: where Here A is a complex Scalar Field, ψ α is a two-component Weyl Spinor Field, and F is a complex auxiliary Scalar Field. The Zinn Action has the form d 4 x {ΓδA + Y δψ + ΛδF + * }, which is a sum of Zinn Sources coupled to the SUSY variations of the Fields.
The SUSY invariance can be summarized by the fact that the above Action A Total yields zero for the Master Equation: 5 That progress was not very exciting however, because it dealt only with free theories. Moreover, efforts to make those theories interacting have been fraught with obstructions of the kind mentioned in [16]. However this changes if one integrates the auxiliary W , as will be seen below. 6 Here + * means 'add the Complex Conjugate of the previous terms'.
We define The following terms 7 must always be added to every Master Equation of SUSY theories: These terms take account of the fact that the SUSY algebra closes onto a translation. The BRST ghost of that translation for rigid SUSY is the constant anticommuting vector ξ αα . Equation (9) summarizes the fact that two SUSY transformations with the SUSY parameter C α act like a spacetime translation.
7. The Chiral Scalar Theory after Integration of the Auxiliary F After integrating the F auxiliary Field in Section 6, and dropping the Source Λ for its variation, one gets 8 .
which yields zero for the smaller Master Equation: This is the Master Equation for a Chiral Multiplet where the auxiliary F has been integrated out. We will see this form again below. The Zinn Sources Y appear quadratically, and they keep the invariance intact. Now we shall put this derivation in a theorem that we will often need: 8. Theorem about Auxiliary Fields and Master Equations: The technique in the above example is frequently used in this paper, and it is worth making it into a theorem. We will use this theorem repeatedly in this paper and the paper [34].
Theorem 1: Given an Action A that satisfies a given Master Equation P, then 1. Suppose that there is a Field F in that Action 9 which has an algebraically invertible quadratic term, and a linear term 10 in F , so that the total Action has the form: and that the Master Equation has the form 7 One cannot implement the BRST method unless one closes the algebra. But there is a genuine problem here. Although there are identities which arise from this Master Equation that are true in perturbation theory for this rigid SUSY theory, those that involve ξ are not true, because ξ is not integrated in the Feynman path integral, although it is more like a Field than a Zinn Source. This problem can be resolved by embedding the rigid theory in Supergravity. 8 Insert the Action into a Feynman path integral with Sources for the Fields, as in Section 22 below, and derive the Master Equation in the usual way. Then complete the quadratic in F and F and perform the same exercise, after dropping the Source Λ. Then shift the F and integrate it, which just leaves a number. This leaves the terms and Zinn Action shown. 9 Here we assume that F is real. In Section 7, F was complex. 10 Auxiliary Fields generally satisfy this condition.
The proof is simple. Write the relevant terms in the Action (15) in the form  [15,16,17]. We will use the notation of [15]. The Action is quite simple: where In the above, G is a real Scalar Field, φα is a two component complex Weyl Spinor, W µ is a real vector Field 12 , and it turns out to be auxiliary (no kinetic term), η is a real Grassmann odd Antighost Scalar Field and C α is again the Grassmann even space-time constant Weyl Spinor ghost Field. The Zinn Action that we need is: 11 This could be derived in exactly the same way as in [15,16,17], using BRST recycling, starting with the U(1) Gauge theory in this Majorana case. We start with this simplest case but the other case is also needed and we return to it below in Section 27 12  10. Integrate the auxiliary out of the Irreducible Chiral Dotted Spinor Supermultiplet: After [15,16,17] were written, the main problem for the new version of the Irreducible Chiral Dotted Spinor Superfield was whether it could be put into an interacting Action. That seemed very difficult at first. However, it turns out that interactions can be generated easily if we first integrate the auxiliary vector Field W αβ out of the Action for the Majorana Irreducible Chiral Dotted Spinor Superfield in Section 9. We will give the result of that a new name: the Half-Chiral Multiplet Action. We use that name for the Half-Chiral Multiplet because it has half the Scalar degrees of freedom that a Chiral Multiplet has.
Once the integration of W is done, it is fairly easy to recognize that the resulting Half-Chiral Multiplet Action is really the result of an Exchange Transformation acting on a Chiral Multiplet that has had its auxiliary Field F integrated. This is the Exchange Transformation that we will be using. Because it will turn out that this theory can be obtained from an Exchange Transformation acting on a Chiral Multiplet, we can couple the Chiral Multiplet using known methods, and then using the inverse Exchange Transformation, we can deduce the interactions of the Half-Chiral Multiplet. (20) in Section 9, and integrate W out of the Action 13 . This yields a closely related Action, which we will dignify by a new name, the Majorana Half-Chiral Multiplet:

The Majorana Half-Chiral Multiplet: Drop the Σ terms in the action in Equation
which we can rearrange to 13 Insert the Action into a Feynman path integral with Sources for the Fields, and derive the Master Equation in the usual way. Then complete the quadratic in W and perform the same exercize, while leaving its Source out, and shifting the W to integrate it. This leaves the terms and Zinn Action shown. This is an application of the theorem in Section 8. Now this yields zero for: 12. Remarkable Symmetries of the Action A MHC : The above Action A MHC in Equation (25) has a remarkable symmetry which was not obvious before we integrated the auxiliary W . The Field η and the Source Υ appear in similar ways. The term η 1 φδCδ looks like a Zinn Source coupled to a variation, except that η is a Field. The Field G and the Source J also appear in similar ways. The term Our new Action will have 'a new complex Field' (S, S) and 'a new complex Zinn Source' (Γ, Γ). These will replace the 'old real Fields' (η, G) and the 'old real Zinn Sources' (J, Υ). We will choose a Generating Functional of the new Zinn Sources (Γ, Γ) and the old Field G and the old Zinn Source J.
and the Exchange Transformations are: These have the following inverses: 14. New Action after Exchange Transformation: It looks like the Chiral Action The new Action expressed in terms of the new variables is: The new Master Equation is: The invariance of the new Action A CM is expressed by: Note that the expressions in this Section are identical to the results for the Chiral Multiplet in Section 7 above, if one changes the names of the Fields and Zinn Sources, and sets m 1 = g 1 = 0 in Section 7. Here is the mapping from Section 7 to this Section 14.
and their Complex Conjugates.

Poisson Brackets, Canonical Transformations, Exchange Transformations, and the Master Equation.
The Master Equation [26,27,28,29,30,31] has the same form as a Poisson Bracket 14 in classical mechanics [37,38]. There is no essential distinction between coordinates and momentum for classical mechanics, but there is one for the Master Equation.
The reason is that the Fields are quantized and the Zinn Sources are not. Nevertheless, Canonical Transformations play a role for both kinds of Poisson Brackets, because they leave the Poisson Bracket invariant [37,38]. Multiplet with a mass term and an interaction term has the following Action, once the auxiliary has been integrated: We are using the notation in Section 14, rather than the notation in Sections 6 and 7. This is done to agree with the notation in [34].

The Half-Chiral Multiplet Action with Mass and Interaction
It is elementary to use the Exchange Transformation from Section 13 on the expression in Section 16. The result is the Action for the massive interacting Majorana Half-Chiral Multiplet: The Master Equation is a Poisson Bracket, except that it uses Grassmann anticommuting quantities (these actually simplify things somewhat).

−(
18. Details for the Majorana Half-Chiral Multiplet with mass, and the Loss of the SUSY Charge Let us set g 1 → 0 in Section 17. Then the expression in Section 17 is the Half-Chiral Multiplet with just a Majorana mass term. The only difference from the massless case is: We see that indeed there is a mass term here for the Scalar, namely But note that there is also a 'mass' term for the Zinn Source J in Equation (40), and then there are extra terms that are all in the Zinn Action So here is what we have discovered: The Half-Chiral Multiplet Action has one Scalar G and a Majorana Spinor φ and also the Source J. When we generate the Half-Chiral Multiplet mass term from the massive Chiral Multiplet plus the Exchange Transformation, we get a massive Spinor and a massive Scalar G. We also get an object that looks like a mass term for J, but J is a Zinn Source.
Because this is all a result of the Exchange Transformation, we are guaranteed that it will satisfy the Half-Chiral Master Equation in Equation (26).
But just looking at it we can see that it describes a Multiplet of SUSY that has one Scalar and a Spinor-but this is clearly not a proper mass multiplet that forms a representation of the SUSY algebra-that needs two Scalars and there is only one here. And yet there is some mass degeneracy here, as though the SUSY algebra is 'half-present'.
And that is the essential point of this entire paper! The Exchange Transformation has enabled us to build a SUSY theory that does not have a nice conserved Noether Charge-the physical theory here is not the proper one we would expect from a theory with a conserved Noether current-it has this Source J where the Scalar should be. And because of the Exchange Transformation, it has the right set of Zinn Source terms to satisfy the Half-Chiral Master Equation.
19. Details for the Half-Chiral Multiplet with Mass and Interactions Now consider the case where g 1 = 0 in Section 17. The Action there has both mass and interactions. Note the complicated way that the Zinn Source is intertwined with the Scalar Field. We could do the same with a Chiral Action that also includes any other kind of interactions, with Gauge, other Chiral Multiplets, even Supergravity. We would end up with a lot of Zinn Source terms and an Action that only goes half-way towards a representation of the SUSY algebra.
The Action in Section 17 is probably the simplest possible Half-Chiral massive interacting theory, and it would be worth while to examine its nilpotent BRST operator δ BRST (this is the 'square root' of the Master Equation), and its one loop diagrams to get a feel for how this Half-Chiral Multiplet works at one loop.
These Half-Chiral Multiplets will be used in [34] for the Higgs Multiplets. We will use a Dirac Half-Chiral Multiplet and a Majorana Half-Chiral Multiplet there.

Un-Chiral Multiplets
If we take the Exchange Transformation that goes all the way, to generate an Un-Chiral Multiplet, we get a theory with no Scalar and some interesting Zinn terms, and a mass just for the Spinor. That case is actually simpler than the above, and we write it down in Section 21. In that case the interaction term would just add to the Zinn Source sector, and the theory would still be a free massive theory. To get interactions there requires Gauge theory.
The Exchange Transformation can be applied twice, so that both Scalars are removed from the Lagrangian. The new Lagrangian gains two terms with Antighosts while the Fermions remain as quantized Fields. Start with the Chiral Multiplet with the Action in Equation (31). Now consider using an Exchange Transformation generated as follows. Instead of the Exchange Transformation in Section 13 we now note that the old 'Fields' (S, S) are conjugate to the old 'Zinn Sources' (Γ, Γ), and we want an Exchange Transformation that takes us to the new 'Fields' (η, η) which are conjugate to the new 'Zinn Sources' (J, J). We choose a generating functional of the new Zinn Sources (J, J) and the old 'Zinn Sources' (Γ, Γ). This is and we have and We get the following transformed Action: This looks very similar to (31), but the theory is not at all the same. The J are not quantized and the η are quantized. So the complex quantized Scalar Field (S, S) is gone from the theory along with its Zinn Source (Γ, Γ), while the quantized Fermion (φ, φ) remains, and the new Zinn Sources (J, J) and quantized Antighosts (η, η) appear. We are assured that the new Action satisfies the new Master Equation, because the old Action satisfied the old Master Equation.
This procedure does not correspond to any known starting Action like the Half-Chiral Multiplet discussed above in section 11. Equation (46) is a new Action. We started with the Half-Chiral Multiplets found by BRST recycling, and then found the Exchange Transformations that took those theories to Chiral Multiplets. Now by generalizing those Exchange Transformations we have discovered new theories that have no physical Scalars at all, just physical Fermions.

Mass Term for the Un-Chiral Multiplet
In Section 18 above, we discussed the mass term for the Half-Chiral Multiplet. It has a mass term that only goes half-way towards that of a Chiral Multiplet. Now let us start again with the Chiral Multiplet with the Action in Equation (31), with a mass term, so that we get Equation (35), and then use the Exchange Transformation in Section 20 on Equation (35). This yields the following Action (we are setting g 1 → 0): The Scalar Mass terms S ( − m 1 m 1 ) S in Equation (35) have become the Zinn Source term J ( − m 1 m 1 ) J in Equation (48). So Equation (48) is an Action which has a Fermionic Mass term and no Scalars. Note that it is quite a lot simpler than the Half-Chiral Multiplet. We are guaranteed that it will satisfy the appropriate Master Equation for the Un-Chiral Multiplet, and so SUSY is still preserved. But the SUSY Charge has been suppressed in this Un-Chiral Multiplet Quantum Field theory.
It is clear that this is not a representation of the SUSY algebra on the physical states.
This is the kind of SUSY multiplet that we will use for the Quarks and Leptons in [34], except that we need to have the Dirac version for that. A similar exercise to the above would show that there is only a massive Fermion left in the Dirac Unchiral Multiplet, if one adds a Dirac type mass term to Section 36. where the variables are Gauge and Ghost Fields, as defined in [15]. Then one shifts all the fields as follows (λ is an anticommuting parameter so that the Grassmann nature of the fields is not violated.)

A Derivation of the
We follow the derivation of Zinn-Justin [28] by defining the Generator of Connected Diagrams: and the Generator of One-Particle-Irreducible Diagrams: where the Legendre transform is defined by: and the Zinn Sources bring in variations which summarizes all the BRST identities here.

A Derivation of the Master Equation for the Interacting Un-Chiral Multiplet that arises from the Above Chiral Multiplet:
Here we derive the identity for the Un-Chiral Multiplet. Exactly the same kind of reasoning applies to Half-Chiral Multiplets, but the notation is more complicated for that case. This is similar to the derivation in Section 22, but it has one step that is not immediately obvious. The Un-Chiral Multiplet is defined by simply changing variables in the Chiral Multiplet in Section 22: Now we want to derive a new Master Equation using this Action. We have a different set of variables in the Feynman path integral over all the fields, because S is gone and η has appeared: G UCM [Z, Z, j η , j η , J, J, j φ , j φ ; Gauged] = δφδφδηδη; δ (Gauge) 15 There is a problem with the ξ terms here, that can be solved by coupling to Supergravity as explained in footnote 7 on page 5. e i{AUCM+ d 4 x(jη η+j φ φ+ * +Gauge Terms)} (63) So now the integration variables are the Scalar Antighost Fields η and Fermion Fields φ, and also Gauge field integration variables. Now we do the usual shift of fields plus Complex Conjugates with similar terms for the Gauge Fields. Now for the moment for the Un-Chiral Multiplet, we assume the following: This is the crucial point, and we will return to this below in Section 24. Now we simply shift all the fields in the integrand, with the result and defining as usual and and their Complex Conjugates, plus Gauge Terms, we get 16 the Master Equation just as we did above in Section 22: This shows that this Action also generates a supersymmetric theory, although there are plenty of questions that we need to understand about the nature of that theory.

The Invariance Equation δ
Fields A UCM = 0 : The only step that was tricky in the above was Equation But the latter can be written in the form: The point is that one can view the equation (75) either as a variation of the fields or the Zinn Sources or any combination, but there is no need to include both. So the fact that the original Action in Equation (49) is invariant under transformations of the fields alone, means that the derived Action in Equation (62) is invariant under transformations of the fields alone, even though they are different fields in the two cases.
Exactly the same kind of reasoning applies to Half-Chiral Multiplets.
25. Remarks about the Hamiltonian and the Cohomology: In [39], for example, it was shown that the Hamiltonian is the sum of the SUSY charges H = Q 2 i and that the conservation of the SUSY charge implies thatQ i = [H, Q i ] = 0. These imply that there is a set of states with the same mass and different spins. So how can these 'Suppressed SUSY Charge' theories exist without this structure?
The present mechanism gets around this reasoning because it is simply impossible to construct a SUSY charge Q i in these new theories. Nevertheless, a large amount of the power of SUSY is maintained by the fact that the new Master Equation is true for the new theories. Furthermore, the algebra of the Chiral SUSY transformations (acting on the Zinn Sources as well as the Fields, in the new theories) is still used to satisfy the new Master Equation at the Lagrangian level.
It appears to be, more or less, a coincidence that the Chiral Action and the Half-Chiral Action are related by cohomology considerations 17 . These cohomology considerations come from the nilpotent BRST cohomology operator δ BRST which is the 'square root' of the Master Equation.
There might be a deeper connection and a clearer explanation, but, if those explanations exist, they are presently quite obscure to the author.
It is much easier to simply use the mechanism of 'Suppressed SUSY Charge', by noting that the SUSY charge is not present in some parts of the new actions, and that the new actions are constrained by the new Master Equation, than it is to explain 'why' this mechanism exists. It just does exist, and it is simple to use, and its use solves many outstanding problems of the SSM, as will be seen in [34]. It is in the Chiral form that it is easy to write down the couplings. Then one simply implements the appropriate set of Exchange Transformations, which result in a new Action and a new Master Equation.
The new Action yields zero for the new Master Equation. This Exchange Transformation takes all or part of the Scalar Fields S and replaces them with Zinn Sources J, and also takes the related Zinn Sources Γ and replaces them with Antighost Fields η. These Exchange Transformations are closely related to Canonical Transformations, as is explained in Section 15.
These new Actions were discovered by generating the Irreducible Chiral Dotted Spinor Superfield in [15], in the hope of coupling the BRST cohomology of the Chiral Multiplet to it. Once the auxiliary W was integrated in that theory, it was noticed that the resulting Half-Chiral Action could be generated by an Exchange Transformation from a Chiral Action that has its auxiliary F integrated. So, in a sense, that coupling of the cohomology has now been done, and the result is that we have discovered new ways to realize SUSY in local Actions, and those new Half-Chiral Actions can be coupled to Gauge theory and each other (and even Supergravity).
But the result here goes farther, because we also have discovered Un-Chiral Multiplets that do not arise by way of the Irreducible Chiral Dotted Spinor Superfield of [15]. The Un-Chiral Multiplet Action arises simply by taking the full version of the Exchange Transformation that was suggested by the existence of the Half-Chiral Multiplet.
Supercharges are not constructible in the new theories (except in some sectors), because when the Zinn Sources become involved, no divergenceless SUSY current exists. This happens because the Zinn sources are not quantized, and they do not satisfy Equations of Motion.
The new theory has new Fields and Sources, and so it has a different generating functional for the Feynman path integral, as shown in Sections 22 and 23. But because the Actions and Master Equations are related through an Exchange Transformation, it was easy to prove, in Sections 23 and 24 that the new Action satisfies the new Master Equation for its 1PI Generating Functionals G 1PI .
In Sections 16, 17 and 18 we set out the details for the massive Half-Chiral Multiplet case, which shows explicitly how the effect of a SUSY Charge acts like it is 'half-present'. In Section 19 we observed the interactions of the Half-Chiral Multiplet.
Section 21 discusses mass for the Un-Chiral Multiplet case and shows that the SUSY Charge is completely gone there. In that case we need to couple the theory to SUSY Gauge Theory to get an interaction.
The result is that that these new theories, when calculated in the renormalized Feynman expansion, iteratively, loop by loop, should be as valid as is the original purely Chiral theory [40]. But the theories are very different, of course, and the Half-Chiral and Un-Chiral theories are not subject to the SUSY algebra, as described in [36], because they do not have conserved SUSY Charges. They are lacking in Scalar Fields compared to Chiral Multiplets. We shall use some of these new Half-Chiral Multiplets and Un-Chiral Multiplets, coupled to SUSY Gauge theory and to each other, to write down a new kind of SSM, in [34].
In the Appendix at Section 27, we review the Dirac case. This is necessary to describe Leptons and Quarks, since they have conserved Baryon and Lepton numbers. This is not very different from the Majorana case which is discussed above.
thank Xerxes Tata who took the trouble to insist on the importance of the tachyon problem for the Chiral Dotted Spinor Superfield, and also on the questions surrounding the non-conservation of the Supercharge. These were the origin of the Half-Chiral Multiplet, which gave rise in turn to the Exchange Transformation, which is the foundation of this paper and its companion on the SSM. Some of this work was done at the Mathematical Institute at Oxford University, and I thank Philip Candelas and the Mathematical Institute for hospitality.  [15,16,17]. The progress reported there was to find and use the following free massless kinetic Action for a generic Dirac (ie Complex) Irreducible Chiral Dotted Spinor Superfield 19 : 28 The Dirac Irreducible Chiral Dotted Spinor Superfield has the following kinetic Action. We will use the notation of [15]. Here we also will add a group representation 20 index i:

29.
Here is the Zinn Action that goes with this Action, also taken from [15], with indices added.

30.
Here is the Master Equation that goes with this Action, also taken from [15], with indices added.  31. Integrate the Auxiliary W to form the Half-Chiral Multiplet Action: There is little difference between this Dirac version and the Majorana version analyzed above, except that the Dirac version can admit a conserved quantum number like Lepton number. After integrating the auxiliary out, the resulting Action can be written in the form:

Master Equation for
Half-Chiral Multiplet: We termporarily dropped the indices i in the above and now we will restore them to their proper places. The Action above in Section 31 yields zero 21 for the following Master Equation 22 : 33. Remarkable Symmetries of the Action A DHC : As was remarked for the Majorana case in Section 31 above, the above Dirac type Action A DHC has a remarkable symmetry which was not obvious before we integrated the auxiliary W . The Field η and the Source Υ appear in similar ways to each other. The Field H and the Source J also appear in similar ways to each other. The term 1 2 ∂ αα J∂ αα J looks like a kinetic term for J, except that J is a Source. The term η 1 √ 2 φ Lδ C δ + 1 √ 2 φ Rδ Cδ looks like a Zinn Source coupled to a variation, except that η is a Field.
34. This symmetry can be exploited with the following Generator for an Exchange Transformation of the Action which will leave the Master Equation invariant, but of a new form. Our new Action will have 'new Fields' (H L , H R ) and 'new Sources' (Γ L , Γ R ). These will replace the 'old Fields' (H, η) and the 'old Zinn Sources' (J, Υ). We choose a generating function G DHC of the new Zinn Sources (Γ L , Γ R ) and the old Field H and the old Zinn Source J: 35. Equations from the Generator and their inverses Now using the Generator defined in Section 34 above, we get 21 We have dropped the Source Σ αα for the auxiliary Field here of course. This is an application of the theorem in Section 8. 22 We have simply removed the terms δA δΣ αα i δA δW i αα from 80 in accord with the theorem in Section 8.
36. Chiral Action for Dirac Multiplet Using the above definitions, the Action can be written in terms of the new variables as follows: plus a similar term with L → R: We must add +A SUSY as usual to the above. This is a pair of Chiral Multiplets that we can put together to make a Dirac Chiral Multiplet.

New Form of Master Equation
We must add + ∂A ∂hαα ∂A ∂ξ αα as usual to the above. These add to make a Dirac Master Equation for the Chiral Multiplet pair.