A Candidate for a Supergravity Anomaly
John A<EMAIL_ADDRESS><EMAIL_ADDRESS>
University of Calgary
Calgary, Alberta, Canada

Abstract

Using elementary BRS cohomology theory, this paper describes a supergravity
anomaly analogous to, but very different from, the well known gauge and
gravitational anomalies. The form of the anomaly is simply the ghost charge
one expression ${\cal A}^{1}_{\rm Supergravity}=\int
d^{4}x\;\varepsilon^{\lambda\mu\nu\sigma}\left\\{c_{1}C_{\alpha}\partial_{\lambda}\Psi_{\mu}^{\alpha}\partial_{\nu}V_{\sigma}+c_{2}\partial_{\lambda}\Psi_{\mu}^{\alpha}\partial_{\nu}\Psi_{\sigma\alpha}\;\omega\right\\}$.
Here $\Psi_{\mu\alpha}$ and $V_{\mu}$ are the gravitino and a gauge boson, and
$C_{\alpha}$ and $\omega$ are their Faddeev Popov ghost fields. There is one
boundary generator here, namely the ghost charge zero expression ${\cal
A}^{0}_{\rm Counterterm}=\int
d^{4}x\;\varepsilon^{\lambda\mu\nu\sigma}\left\\{\Psi_{\lambda\alpha}\partial_{\mu}\Psi_{\nu}^{\alpha}V_{\sigma}\right\\}$,
but there are two cycles in ${\cal A}^{1}_{\rm Supergravity}$, so the anomaly
is present if the coefficients $c_{1},c_{2}$ are not in the ratio that can
arise from the boundary generator. A model that is likely to generate this
supergravity anomaly is described. The coefficient of this anomaly, in
perturbation theory with unbroken supergravity, appears to be zero, because no
relevant diagrams are linearly divergent. However, when, and only when, there
is spontaneously broken supergravity, there are counterterms needed in the
action, and these counterterms are just right to contribute to linearly
divergent diagams that can contribute to the anomaly. We conjecture that the
coefficient of the anomaly, for a general gauge group, is of the form
$\kappa^{3}M_{G}\left<D^{a}\right>$, where $M_{G}$ is the mass of the
gravitino, $\kappa$ is the Planck length, and $\left<D^{a}\right>$ is the VEV
of the gauge auxiliary field. This paper also discusses gauge parameters for
the gravitino and the massive vector boson. It is possible that this anomaly
might provide significant constraints to characterize viable theories that are
based on supergravity or superstring theory.

1. Supergravity and Superstring Theory: These theories appeared likely to explain how we could generate a Grand Unified Theory, including supergravity, and then explain why the Standard Model looks so special and peculiar [1, 2, 3, 4, 5, 6]. But the initial promise of these theories has yielded disappointment, because there does not seem to be any way to understand why one set of particles and groups is better than another. At the same time supersymmetry [7, 8, 9, 10, 11] has also been a disappointment, because no plausible superpartners have been observed in the many experiments that have looked for them. Anomalies in higher dimensional gravity theories generated much of the interest in superstring theory [12, 13]. Gauge anomalies play a very important role in the Standard Model [14].

It could be argued that adding supergravity has not really helped in the
search for a Grand Unified Theory. Adding supergravity is very complicated.
Incorporating it might seem worthwhile, if only it explained something. But,
up to now, it has been generally believed that there are no further anomalies
that are special to supergravity. However, this paper claims that there are
such anomalies.

2. The main result of this paper is in paragraph *, and it is very simple to describe, as follows: There is at least one expression, with ghost charge one, in the BRS cohomology of supergravity, which is not simply the supersymmetric version of any known gauge or gravitational anomaly, and it is in equation (14) below. It is also true that the methods needed for the calculation of the coefficient of this anomaly are very different from those for any other known anomaly, as will be shown below, particularly in paragraphs * and *.

3. General Remarks about the BRS Cohomology of Supergravity: The questions regarding supersymmetry and BRS symmetry [15, 16, 17, 18, 19, 20, 21] are popular topics. The subject, in general, is immense and confusing. A particularly important issue is that there are various claims that the BRS cohomology of supergravity and related theories have been examined, and that no ghost charge one objects, special to supersymmetry, exist. It seems to have been generally accepted that supergravity does not have any anomaly that is not simply the supersymmetric version of a well known non-supersymmetric anomaly. This paper shows that this viewpoint is not correct. At the end of this paper, there are some tables which summarize some of the notation and results in this paper.

4. BRS and ZJ and the SU(3) and U(1) Anomalies: The power of the Becchi Rouet Stora (BRS) and Zinn-Justin (ZJ) Master Equation technique is that it summarizes all the symmetries of an action, and its transformations, in a compact and powerful way. Originally BRS used the mathematical fact that if one takes

$\delta=\omega^{a}T^{a}-\frac{1}{2}f^{abc}\omega^{b}\omega^{c}\frac{\partial}{\partial\omega^{a}}$
(1)

where $T^{a}$ are the antihermitian generators of the SU(3) group on some
field, satisfying

$\left[T^{a},T^{b}\right]=f^{abc}T^{c}$ (2)

then this generates cohomology because $\delta^{2}=0$. They noticed that the
anomaly of the SU(3) Yang Mills theory in four dimensions arises from the
totally antisymmetric tensor with five ghosts (here $d^{abc}$ is the symmetric
tensor with three indices in the real adjoint representation of SU(3)):

$d^{abc}f^{bde}f^{cpq}\omega^{a}\omega^{d}\omega^{e}\omega^{p}\omega^{q}$ (3)
$\rightarrow{\cal A}^{1}_{\rm SU(3)\;Anomaly}=\int
d^{4}xd^{abc}\left\\{\varepsilon^{\mu\nu\lambda\sigma}\partial_{\mu}V_{\nu}^{a}\partial_{\lambda}V_{\sigma}^{b}\omega^{c}+\cdots\right\\}$
(4)

The $+\cdots$ in the above are higher order terms in the coupling constant and
we can ignore them when computing the anomaly. The notation ${\cal A}^{1}_{\rm
SU(3)\;Anomaly}$ is to remind us that this expression looks like a piece of
the action, except that it has ghost charge one.

5. Similarly the usual U(1) anomaly arises from the expression:

${\cal A}^{1}_{\rm U(1)\;Anomaly}=\int
d^{4}x\left\\{\varepsilon^{\mu\nu\lambda\sigma}\partial_{\mu}V_{\nu}\partial_{\lambda}V_{\sigma}\omega\right\\}$
(5)

The Grassmann odd ghost fields $\omega^{a},\omega$ are the Faddeev Popov (FP)
Ghosts. Early on, ZJ introduced a way of incorporating the nilpotent $\delta$
of BRS into the path integral technique. Later on, Batalin and Vilkovisky (BV)
named these results the ‘Master Equation’ [22, 23].

6. Gauge Transformations: These anomalies arise from the following nilpotent gauge transformations:

$\delta V^{a}_{\mu}=\partial_{\mu}\omega^{a}+\cdots$ (6) $\delta
V_{\mu}=\partial_{\mu}\omega+\cdots$ (7)

These simple terms are nilpotent because they are Grassmann odd. When the
higher order terms are included, it gets more complicated.

7. Why is the expression an anomaly? Here is the reason. The expressions like (5) can be obtained, in perturbation theory, by taking the BRS variation of some Feynman diagram one-particle-irreducible vertex, as follows, for example (see (19) below for an example of the calculation):

${\cal A}^{1}_{\rm U(1)\;Anomaly}=\delta{{\cal G}}^{0}_{\rm Feynman}$ (8)

This vertex ${{\cal G}}^{0}_{\rm Feynman}$ is a complicated non-local
expression and the calculation is an interesting and subtle process, which
necessarily uses the linear divergence of the amplitude ${{\cal G}}^{0}_{\rm
Feynman}$. See [15] for a simple discussion of this process. Clearly one can
try to remove the above expression (5) as follows:

${\cal A}^{1}_{U(1)}=\delta{\cal A}^{0}_{\rm U(1)\;Counterterm}$ (9)

where one chooses a local counterterm in the action, of the form:

${\cal A}^{0}_{\rm U(1)\;Counterterm}=\frac{1}{2}\int
d^{4}x\left\\{\varepsilon^{\mu\nu\lambda\sigma}V_{\mu}V_{\nu}\partial_{\lambda}V_{\sigma}\right\\}$
(10)

The problem is that this particular expression ${\cal A}^{0}_{\rm
U(1)\;Counterterm}=0$. We are antisymmetrizing a symmetric expression
$V_{\mu}V_{\nu}$. In fact there is no appropriate local expression that can
satisfy equation (9). Exactly the same problem arises for expression (4). Our
example of a supergravity anomaly (sugranomaly) below in (14) has exactly the
same features, but it is trickier because it involves the gravitino.

8. BRS realized that anomalies can be identified as objects in the local BRS ghost charge one cohomology: The reason for this is simple. We observe that for example, it is true that for the relevant nilpotent BRS operators $\delta$, the objects in equations (5) and (4) satisfy the equations

$\delta{\cal A}^{1}_{\rm SU(3)\;Anomaly}=\delta{\cal A}^{1}_{\rm
U(1)\;Anomaly}=0$ (11)

but there do not exist any local polynomial objects such that

${\cal A}^{1}_{\rm SU(3)\;Anomaly}=\delta{\cal A}^{0}_{\rm
SU(3)\;Counterterm}$ (12) ${\cal A}^{1}_{\rm U(1)\;Anomaly}=\delta{\cal
A}^{0}_{\rm U(1)\;Counterterm}$ (13)

In practice, the expressions ${\cal A}^{1}_{\rm SU(3)\;Anomaly}$ do arise in
perturbation theory, by taking the variation of some Green’s function. If the
anomaly is in the local BRS cohomology space, it cannot be removed by any
terms added to the Lagrangian, since the latter is local and the 1PI vertices
are not local. This is true at the lowest level, and it continues to be true
at all levels in the relevant coupling constants.

9. Supergravity Anomaly: This paper will argue that supergravity has an anomaly of its own, of the form

${\cal A}^{1}_{\rm Supergravity}=\int
d^{4}x\;\varepsilon^{\lambda\mu\nu\sigma}\left\\{c_{1}C_{\alpha}\partial_{\lambda}\Psi_{\mu}^{\alpha}\partial_{\nu}V_{\sigma}+c_{2}\partial_{\lambda}\Psi_{\mu}^{\alpha}\partial_{\nu}\Psi_{\sigma\alpha}\;\omega+\cdots\right\\}$
(14)

where the numerical coefficients $c_{i},i=1,2$ are discussed below. In the
above formula, $\Psi_{\mu\alpha}$ is the Grassmann odd spin $\frac{3}{2}$
gravitino of supergravity (in the complex two-component formulation), and
$V_{\mu}$ is a (real) vector boson in a gauge multiplet coupled to chiral
multiplets and to supergravity. The field $C_{\alpha}$ is the Grassmann even
FP ghost for the supersymmetry and these two fields have the nilpotent
transformations:

$\delta\Psi_{\mu\alpha}=M_{P}\partial_{\mu}C_{\alpha}+\cdots,\;\delta
V_{\mu}=\partial_{\mu}\omega+\cdots$ (15)

where $M_{P}$ is the Planck mass, and $\omega$ is again the Grassmann odd
scalar FP ghost, as in the above examples. For the purposes of calculating the
coefficient of the anomaly the higher order terms $+\cdots$ in equation (14)
and equation (15) can be ignored, and that is very important for a complicated
calculation like this.

10. The counterterm situation for supergravity is more complicated than for the gauge theories: The relevant possible counterterm like (10) would be

${\cal A}^{0}_{\rm Supergravity\;Counterterm\;1}=\int
d^{4}x\left\\{\varepsilon^{\mu\nu\lambda\sigma}\Psi_{\mu}^{\alpha}\Psi_{\nu\alpha}\partial_{\lambda}V_{\sigma}\right\\}=0$
(16)

and again this is zero, which is just like the situation for (10), except that
here there is a triple antisymmetrization which makes this zero, whereas in
(10) it is a single antisymmetrization. However, there is another possible
counterterm in this supergravity case. This takes the form

${\cal A}^{0}_{\rm Supergravity\;Counterterm\;2}=\int
d^{4}x\left\\{\varepsilon^{\mu\nu\lambda\sigma}\partial_{\lambda}\Psi_{\mu}^{\alpha}\Psi_{\nu\alpha}V_{\sigma}\right\\}\neq
0$ (17)

An effort to show that this is zero results in a quadruple antisymmetrization
(including integration by parts), and so it does not succeed in showing this
is zero. This expression generates objects that look exactly like (14):

$\delta{\cal A}^{0}_{\rm Supergravity\;Counterterm\;2}=\int
d^{4}x\;\varepsilon^{\lambda\mu\nu\sigma}\left\\{c^{\prime}_{1}C_{\alpha}\partial_{\lambda}\Psi_{\mu}^{\alpha}\partial_{\nu}V_{\sigma}+c^{\prime}_{2}\partial_{\lambda}\Psi_{\mu}^{\alpha}\partial_{\nu}\Psi_{\sigma\alpha}\;\omega\right\\}$
(18)

where we have integrated by parts. The values are
$M_{P}c^{\prime}_{1}=-c^{\prime}_{2}=1$. Note that this combination of these
two factors occurs with a definite and unchangeable linear combination with
definite numbers $c^{\prime}_{1},c^{\prime}_{2}$ in the above expression (18).
An effort to show that the second term on the right hand side in (18), is
zero, results in a double antisymmetrization, and so the term is non-zero. The
term in (17) can generate one particular linear combination of these numbers,
but not two. Both of the expressions in ${\cal A}^{1}_{\rm Supergravity}$ in
(14) separately yield zero when acted upon by $\delta$. So there is no way, in
general, to obtain expression (14) from the BRS variation of a local object
with ghost charge zero in the theory. If it arises in perturbation theory with
coefficients that do not have the same ratio as in the above expression (18),
it will spoil the SUSY type gauge symmetry of supergravity in an incurable
way. In quantum field theory, we expect that this will happen in general–the
rule there is that what can happen, usually does happen, because the
perturbation expansion tends to explore all the corners.

11. Equations of motion of the gravitino and the vector boson: We will not worry here about the issues that relate to the higher levels in the coupling. However, it is crucial to observe that a genuine anomaly must not vanish using the equations of motion. Sometimes this is called invariance under deformations of the transformation rules [24]. This requirement is equivalent to ensuring that the anomaly is in the cohomology space (a cocyle, but not a coboundary) of the entire $\delta$ derived in paragraph * here. The equation of motion for the gravitino is discussed below in paragraph *. The expression (14) does not vanish using any of the forms of the equations of motion of either the gravitino or the vector boson.

12. The supergravity anomaly requires the spontaneous breaking of supersymmetry in order to appear in the theory: Later in this paper, starting with paragraph *, we will write down the detailed components of a theory which we believe will generate this anomaly. Based on this theory, we present a diagram for the anomaly in (19). The surprising thing here is that this anomaly of supergravity does not make its appearance in perturbation theory in the simple way familiar from the two anomalies of gauge theory above. In fact, a preliminary examination of the diagrams indicates that there are no linearly divegent diagrams for the relevant amplitude, which excludes any possibility of generating it with a non-zero coefficient. I have tried to find a linearly divergent diagram for the relevant amplitude in the theory here, before introducing the terms described in paragraph *, and it does not seem possible, but I have no proof that it does not happen.

13. Conjecture for the supergravity anomaly: My conjecture is, that for this anomaly to appear, SUSY and gauge symmetry must both be spontaneously broken. This means that the gravitino mass, and the vector boson mass, must both be present, and must arise from the spontaneous breaking of local supersymmetry and local gauge symmetry, using the well known Higgs mechanisms [25]. The reasoning behind this conjecture follows from the diagram (22) in paragraph * below. At the present time, this claim remains a conjecture.

14. A diagram for the sugranomaly: Here is an example of a diagram that appears likely to generate this anomaly in supergravity. It is a linearly divergent triangle diagram, which is a necessary feature for a diagram to generate an anomaly, by analogy with the known anomalies discussed above. However, there is a very serious problem in this diagram. It needs a propagator from ${\overline{\lambda}}^{\dot{\gamma}}$ to $\chi_{1}^{\delta_{1}}$ in the top left of the triangle (we are using a circle, actually), but no such propagator exists in the theory! This problem appears, at first sight, to be an insuperable difficulty for generating the anomaly. As we shall see below, this problem is actually a hint about how the anomaly can be generated.

(19)

The vertices for the above diagram can be found in the chiral action in
paragraph *, and in the gauge action in paragaph *. However the new mixed
kinetic term $c_{\rm
new}\lambda^{\alpha}\sigma^{\mu}_{\alpha\dot{\beta}}\partial^{\mu}{\overline{\chi}}_{1}^{\dot{\beta}}$
arises from the new counterterm (23) in the action, discussed in paragraph *.
The massive $VV$ propagator at the bottom comes from the terms in paragraph *.
Its mass could be dropped here, since only the linear divergence contributes
to the anomaly.

Note that there is no problem in getting the propagator $M_{\rm
1,mix}\overline{\lambda}^{\dot{\alpha}}{\overline{\chi}}_{1\dot{\alpha}}$
here, since it arises immediately from the term
$ig\overline{\lambda}^{\dot{\alpha}}{\overline{\chi}}_{1\dot{\alpha}}\left<A_{1}\right>$
which is in the last line (91) of the chiral action below. It brings in a
power of mass from the VEV. It is, of course, unusual to find a power of mass
in an anomaly, but this diagram clearly has a power of $gM_{O}v=M_{\rm 1,mix}$
from one propagator.

15. Calculation of the sugranomaly: We now recall that the sugranomaly is present in the theory if the following equation holds true in perturbation theory:

$\delta{\cal G}\left[\Psi,\Psi,V\right]=\int
d^{4}x\;\varepsilon^{\mu\nu\lambda\sigma}\left\\{c_{1}\partial_{\lambda}\Psi_{\mu}^{\alpha}C_{\alpha}\partial_{\nu}V_{\sigma}+c_{2}\partial_{\lambda}\Psi_{\mu}^{\alpha}\partial_{\sigma}\Psi_{\nu\alpha}\;\omega\right\\}$
(20)

where the vertex is the sum of all relevant diagrams, as usual:

${\cal G}\left[\Psi,\Psi,V\right]=\sum_{n}{\cal
G}_{n}\left[\Psi,\Psi,V\right]$ (21)

Here $c_{1}$ and $c_{2}$ are coefficients, calculated from perturbation
theory. In order for this to be an anomaly, it is necessary that these
coefficients not be in a ratio such that they can arise from equation (18). It
is quite remarkable that this requires us to have a propagator resulting from
a counterterm that arises from diagrams like (22) below. Note that by the
usual interpretation of quantum field theory, this term appears with a new
arbitrary coefficient $c_{\rm new}$, as further explained below in paragraph
*.

16. The Counterterms for the theory with spontaneously broken SUSY: As we mentioned above, the problem with the diagram in paragraph * is that there are no propagators that take ${\lambda}^{\gamma_{1}}$ to ${\overline{\chi}}_{1}^{\dot{\delta}_{1}}$ in the theory. But consider the following diagram:

(22)

The vertex at the left comes from the action ${\cal A}_{\rm Gauge}$ in
paragraph *. Similarly, the vertex at the right comes from the action ${\cal
A}_{A_{1}}$ in paragraph * below after the VEV $<A_{1}>=M_{O}v$ described in
paragraph * appears. There is a similar term and diagram from ${\cal
A}_{A_{2}}$. The propagator at the top comes from the action in paragraph *.
There is still some work needed to convert that to the massive case. The
propagator at the bottom comes from the terms in paragraph *. This diagram
clearly vanishes if the Gravitino mass $M_{G}$ vanishes.

17. The Diagram (22) can exist only if the gravitino is massive: Otherwise the propagator ${\overline{\Psi}}^{\nu_{1}\dot{\gamma}_{1}}{\overline{\Psi}}^{\nu_{2}\dot{\gamma}_{2}}$ does not exist. This diagram is linearly divergent, and we can expect that it will need a counterterm in our action. But this counterterm will be needed only if the SUSY symmetry is spontaneously broken. We expect that the calculations in our theory will satisfy the Master Equation (see below in equation (77)), which reflects the complicated symmetry of this action. What that means is that if there are counterterms needed in the theory, like the following

${\cal A}_{\rm A,1\;new\;counterterm\;needed}=c_{\rm new}\int
d^{4}x\;\left\\{\lambda^{\alpha}\sigma^{\mu}_{\alpha\dot{\beta}}\partial_{\mu}{\overline{\chi}}_{1}^{\dot{\beta}}\right\\}$
(23)

where $c_{\rm new}$ is the divergent coefficient calculated from diagrams like
(22), then there will also be other counterterms needed with the same
coefficient. These can all be generated together as a boundary, as will be
discussed below in paragraph *, after we have discussed more of the details of
the Master Equation and actions here.

Note that $c_{\rm new}$ can indeed be dimensionless, because the counterterm
with a divergence in the diagram above in (22) for ${\cal
G}_{1}\left[\lambda,\overline{\chi}\right]$ is proportional to $\int
d^{4}x\kappa^{2}M_{O}M_{G}\lambda^{\alpha}\sigma^{\mu}_{\alpha\dot{\beta}}\partial_{\mu}{\overline{\chi}}_{1}^{\dot{\beta}}$
times a dimensionless (but divergent) constant, so that the whole term is of
dimension zero. On the other hand, we also note that the counterterm with a
divergence in the diagram for ${\cal G}_{1}\left[\Psi,\Psi,V\right]$ has
dimension $\int d^{4}x\kappa^{2}M_{\rm 1,mix}\Psi\partial\Psi V$, which is
also of dimension zero. This means that we can expect to calculate the
sugranomaly coefficients in the form $c_{1}=\kappa M_{\rm 1,mix}{\rm
N}_{1},c_{2}=\kappa^{2}M_{\rm 1,mix}{\rm N}_{2}$, where ${\rm N}_{i}$ are
finite, but mass-dependent, numbers. See paragraph * for more comments on
these issues.

18. The counterterms like (23), calculated from diagrams like (22), which fall into groups as discussed in paragraph *, must be added to the action when the gravitino is massive: They then allow the diagrams like (19) to appear in the theory, and so they allow the sugranomaly to arise. That is why we said in paragaph * that the strange propagators needed for that diagram are a hint as to how it arises.

19. In a more detailed analysis of this situation, one would want to diagonalize the various terms, so that one is dealing with propagators that are diagonal in the masses and in the kinetic terms. Here there is all sorts of mixing going on, and it takes a lot of effort to sort it out. Our claim is that we can still find a lot of information from the action without doing all that work. But that work now needs to be done to determine how these anomalies restrict the set of viable supergravity theories, particularly in the context of unification [29, 30, 31].

20. Is this a real supergravity anomaly? The reader might wonder whether this is just an example of one of the usual anomalies, but made supersymmetric. It seems clear that this is actually quite different from the known anomalies, although it certainly has a family resemblance, as noted in this paper. We can expect that eliminating it from theories will give rise to interesting new restrictions on viable theories. However, getting its general form, performing the Feynman calculations accurately, and figuring out what those restrictions are, are complicated and demanding tasks, given the complicated nature of the action and transformations.

21. Conjecture for the Coefficient of the Anomaly in General:

For the general case where we have an arbitrary gauge group and arbitrary
representations for matter and Higgs fields coupled to that group and to
supergravity, here is a conjecture about the form and coefficient of this
anomaly.

${\cal A}^{1}_{\rm Supergravity}=\int
d^{4}x\;\varepsilon^{\lambda\mu\nu\sigma}\left\\{c_{1}C_{\alpha}\partial_{\lambda}\Psi_{\mu}^{\alpha}\partial_{\nu}V^{a}_{\sigma}+\kappa
c_{2}\partial_{\lambda}\Psi_{\mu}^{\alpha}\partial_{\nu}\Psi_{\sigma\alpha}\;\omega^{a}\right\\}g^{3}\kappa^{3}M_{G}\left<D^{a}\right>$
(24)

Here we have written the coefficients $c_{1},c_{2}$ so that they are
dimensionless. The reasoning behind this conjecture resides simply in the
above diagrams (19) and (22), as follows:

If we add group matrices to this paper rather than leave it in the Abelian
notation, then we get the following group matrix in (19):

$\kappa^{3}g^{3}M_{G}M^{2}v^{i}(T^{a}T^{b}T^{a})_{i}^{\;\;j}{\overline{v}}_{j}=\left(\frac{1}{2}C_{2}(G)+C_{1}(G)\right)\kappa^{3}g^{3}M_{G}M^{2}<D^{a}>$
(25)

where $T^{a}$ are the representation matrices for a Lie group, and

$<D^{b}>=M^{2}v^{i}T_{i}^{b\;\;j}{\overline{v}}_{j};\;\left<A^{i}\right>=Mv^{i},$
(26)

and we are taking into account what we get for the mixed propagator from (22).
We note that the vector propagator is diagonal in group space, if we drop the
mass terms, and we can do that because we are interested only in the linearly
divergent part of the diagram (19). But then we note that

$T^{a}T^{b}T^{a}=\left([T^{a},T^{b}]+T^{b}T^{a}\right)T^{a}=f^{abc}T^{c}T^{a}+T^{b}T^{a}T^{a}$
(27)
$=f^{abc}\frac{1}{2}f^{cad}T^{d}+T^{b}T^{a}T^{a}=\left(\frac{1}{2}C_{2}(G)+C_{1}(G)\right)T^{b}$
(28)

and so we get the form (24), where we absorb this constant
$\left(\frac{1}{2}C_{2}(G)+C_{1}(G)\right)$, made from Casimir coefficients of
the group, into the coefficients $c_{i}$. In [2, 8] there are discussions
about the question whether the auxiliary field $D^{a}$ can have a VEV, and
[32] is also useful.

22. Calculating the Diagrams: It would be nice to calculate the diagrams like (22) and (19). The diagram (22) needs to be non-zero in order for the diagram (19) to even exist. But that requires us to have a proper version for both massive propagators, with gauge parameters, in (22). That is why some effort has been made to start the discussion of the terms in (73) and (74) for the gravitino propagator in paragraph * below.

23. The Action for the model: We take supergravity coupled to a U(1) super gauge theory, coupled to two chiral multiplets ${\widehat{A}}_{i}=A_{i}+\theta^{\alpha}\chi_{i\alpha}+\frac{1}{2}(\theta^{2})F_{i},i=1,2$, with opposite charges, and two uncharged chiral multiplets ${\widehat{B}}_{i}=B_{i}+\theta^{\alpha}\phi_{i\alpha}+\frac{1}{2}(\theta^{2})G_{i},i=1,2$ . We choose a simple superpotential of the O’Raiffertaigh [1] form, so that the theory will have spontaneous breaking of supergravity. This part can be done as if we were using merely rigid SUSY, and we use $M_{O}$ for a mass parameter here:

${\cal A}_{\rm Scalars\;from;Chiral\;Superpotential}$ (29) $=\int
d^{4}xd^{2}\theta\left\\{g_{1}{\widehat{B}}_{1}{\widehat{A}}_{1}{\widehat{A}}_{2}+{\widehat{B}}_{2}\left(g_{2}{\widehat{A}}_{1}{\widehat{A}}_{2}-M_{O}^{2}\right)\right\\}+*$
(30) $=\int
d^{4}x\left\\{g_{1}G_{1}A_{1}A_{2}+g_{1}B_{1}A_{1}F_{2}+g_{1}B_{1}F_{1}A_{2}\right.$
(31)
$\left.+g_{1}B_{1}\chi_{1}^{\alpha}\chi_{2\alpha}+g_{1}\phi_{1}^{\alpha}\chi_{1\alpha}A_{2}+g_{1}\phi_{1}^{\alpha}A_{1}\chi_{2\alpha}\right.$
(32)
$\left.+g_{2}G_{2}A_{1}A_{2}+g_{2}B_{2}A_{1}F_{2}+g_{2}B_{2}F_{1}A_{2}\right.$
(33)
$\left.+g_{2}B_{2}\chi_{1}^{\alpha}\chi_{2\alpha}+g_{2}\phi_{2}^{\alpha}\chi_{1\alpha}A_{2}+g_{2}\phi_{2}^{\alpha}A_{1}\chi_{2\alpha}-G_{2}M_{O}^{2}\right\\}$
(34)

and we add to this the purely scalar terms relating to the auxiliary fields
and scalars in the other parts of the actions:

${\cal A}_{\rm Scalars\;from\;Action}=\int
d^{4}s\left\\{\sum_{n=1}^{2}\left\\{F_{n}{\overline{F}}_{n}+G_{n}{\overline{G}}_{n}\right\\}+D^{2}+DgA_{1}{\overline{A}}_{1}-DgA_{2}{\overline{A}}_{2}\right\\}$
(35)

We put these together into

${\cal A}_{\rm SP}={\cal A}_{\rm Scalars\;from\;Action}$ (36) $+{\cal A}_{\rm
Scalars\;from;Chiral\;Superpotential}+{\overline{\cal A}}_{\rm
Scalars\;from;Chiral\;Superpotential}$ (37)

24. The bosonic parts of the various auxiliary fields here are obtained from the bosonic parts of their equations of motion:

$\frac{\delta{\cal A}_{\rm SP}}{\delta
G_{1}}={\overline{G}}_{1}+g_{1}A_{1}A_{2}=0;\ \frac{\delta{\cal A}_{\rm
SP}}{\delta G_{2}}={\overline{G}}_{2}+g_{2}A_{1}A_{2}-M_{O}^{2}=0$ (38)
$\frac{\delta{\cal A}_{\rm SP}}{\delta
F_{1}}=\overline{F}_{1}+g_{1}B_{1}A_{2}+g_{2}B_{2}A_{2}=0;\;\frac{\delta{\cal
A}_{\rm SP}}{\delta F_{2}}=\overline{F}_{2}+g_{1}B_{1}A_{1}+g_{2}B_{2}A_{1}=0$
(39) $\frac{\delta{\cal A}_{\rm SP}}{\delta
D}=D+g\left(A_{1}{\overline{A}}_{1}-A_{2}{\overline{A}}_{2}\right)=0$ (40)

25. We integrate the auxiliary fields in the path integral, which replaces them with the negative of their values from their equations of motion, and this yields the following potential in the usual way:

$V=-{\cal L}_{\rm
Scalar\;Potential}=\sum_{n=1}^{2}\left\\{F_{n}{\overline{F}}_{n}+G_{n}{\overline{G}}_{n}\right\\}+DD$
(41)

which is, using the equations in paragraph *:

$V=\left|g_{1}A_{1}A_{2}\right|^{2}+\left|g_{2}A_{1}A_{2}-M_{O}^{2}\right|^{2}+\left|g_{1}B_{1}A_{2}+g_{2}B_{2}A_{2}\right|^{2}$
(42)
$+\left|g_{1}B_{1}A_{1}+g_{2}B_{2}A_{1}\right|^{2}+\left|g\left(A_{1}{\overline{A}}_{1}-{\overline{A}}_{2}A_{2}\right)\right|^{2}$
(43)

26. The minimum value of this expression occurs for Vacuum Expectation Values (VEVs) as follows:

$\left<B_{1}\right>=\left<B_{2}\right>=\left<{\overline{F}}_{1}\right>=\left<{\overline{F}}_{2}\right>=0$
(44)

These zero results all follow from the fact that setting them to zero just
contributes zero to the VEV of the expression (43). The VEVs of the auxiliary
fields $G_{1},G_{2},D$ may be non-zero, which means that the gravitino gains a
mass, and that the supersymmetry here is spontaneously broken.

27. Values of the VEVs: These VEVs need to satisfy the equations obtained by taking the derivatives of the expression (43) with respect to the four scalar fields $A_{1},A_{2},B_{1},B_{2}$ and setting the result to zero, so as to find an extremum. Then one chooses the minimum value for V among the extrema. We can choose real values for the VEVs here. The solution is unique for present purposes, and is

$\left<A_{1}\right>=\left<A_{2}\right>=M_{O}v=M_{O}\sqrt{\frac{g_{2}}{g_{1}^{2}+g_{2}^{2}}}$
(45)
$\Rightarrow\left<{\overline{G}}_{1}\right>=-M_{O}^{2}\frac{g_{1}g_{2}}{g_{1}^{2}+g_{2}^{2}};\;\left<{\overline{G}}_{2}\right>=+M_{O}^{2}\frac{g_{1}^{2}}{g_{1}^{2}+g_{2}^{2}};\;\left<D\right>=0$
(46)

and it yields a minimum value for V, with those VEVs, of:

$V=V\left[A_{1}\rightarrow\left<A_{1}\right>,A_{2}\rightarrow\left<A_{2}\right>,B_{1}\rightarrow
0,B_{2}\rightarrow
0\right]=M_{O}^{4}\frac{g_{1}^{2}}{\left(g_{1}^{2}+g_{2}^{2}\right)}$ (47)

28. We expect to need a non-zero cosmological constant counterterm in any gravitational theory: In supergravity theories this is of the anti-de-Sitter kind, negative in value, whereas the value (47) is positive. We can and do ‘fine tune’ the above value to be equal to the cosmological constant, and so the result has zero cosmological constant corresponding to asssymptotically flat spacetime. That is the origin of the mass of the gravitino.

29. This is a bit different from the well known way that the mass of the vector boson appears in such theories: In this case that mass occurs in the form

$-D_{\mu}A_{1}{\overline{D}}^{\mu}{\overline{A}}_{1}-D_{\mu}A_{2}{\overline{D}}^{\mu}{\overline{A}}_{2}\rightarrow$
(48)
$-igM_{O}v\partial_{\mu}V^{\mu}{\overline{A}}_{1}+igM_{O}v\partial_{\mu}V^{\mu}A_{1}$
(49)
$+igM_{O}v\partial_{\mu}V^{\mu}{\overline{A}}_{2}-igM_{O}v\partial_{\mu}V^{\mu}A_{2}$
(50) $-2g^{2}M_{O}^{2}v^{2}V_{\mu}V^{\mu}$ (51)

30. Cosmological term and gravitino mass: In this model we need to suppose we start with a cosmological term. To preserve the BRS invariance of supergravity we also need to add a gravitino mass term and a term that changes the gravitino transformation.

${\cal A}_{\rm Cosmological}=\int d^{4}x\left\\{eM_{\rm
cc}^{4}-e\left(M_{G}\Psi\Psi-\frac{1}{e}M_{\rm
New}^{2}{\tilde{\Psi}}^{\mu}\sigma_{\mu}{\overline{C}}+*\right)\right\\}$ (52)
$=\int
d^{4}x\left(1+\frac{\kappa}{6}h^{\sigma}_{\;\;\sigma}+\cdots\right)\left\\{M_{\rm
cc}^{4}-\left(M_{G}\Psi\Psi-\frac{1}{e}M_{\rm
New}^{2}{\tilde{\Psi}}^{\mu}\sigma_{\mu}{\overline{C}}+*\right)\right\\}$ (53)

and in order to have invariance we need

$M_{G}M_{\rm New}^{2}=\frac{\kappa}{6}M_{\rm cc}^{4}$ (54)

Now suppose that we have, using (47)

$M_{O}^{4}\frac{g_{1}^{2}}{\left(g_{1}^{2}+g_{2}^{2}\right)}=M_{\rm cc}^{4}$
(55)

so that the cosmological term $\int d^{4}x\;M_{\rm cc}^{4}e$ is cancelled by
the VEV of V. These terms have the right signs to accomplish this because the
(uncancelled) cosmological term in supergravity corresponds to an anti de
Sitter universe [2]. We still have the gravitino mass term and we still have
the new term in the transformation of the gravitino, but we no longer have the
cosmological term in terms of $h$. How does the supergravity symmetry survive?

Strictly speaking we really should integrate the auxiliary terms out at this
point. But for present purposes we can proceed as follows. We get several new
kinds of terms from the VEV of the auxiliary terms at the same time. We get
the mixing terms from the lines that correspond to (89) in ${\cal A}_{\rm
Chiral\;Kinetic\;B\;1}+{\cal A}_{\rm Chiral\;Kinetic\;B\;2}$. It is

${\cal A}_{\rm Goldstino\;mixing}=\int
d^{4}x\left\\{\overline{\phi}_{1}\overline{\sigma}^{\mu}\kappa<G_{1}>+\overline{\phi}_{2}\overline{\sigma}^{\mu}\kappa<G_{2}>\right\\}\Psi_{\mu}$
(56) $=\int d^{4}xM_{\rm New,3}\;\overline{\phi}_{\rm
Goldstino}\sigma^{\mu}\Psi_{\mu}$ (57)

Also we get the terms in the BRS transformations of the $\phi$ fields in
${\cal A}_{\rm ZJ\;Chiral;B\;1}+{\cal A}_{\rm ZJ\;Chiral;B\;2}$ from the lines
that correspond to (119). They are:

$\int
d^{4}x\left\\{{\widetilde{\phi}}_{1}^{\alpha}\kappa<G_{1}>C_{\alpha}+{\widetilde{\phi}}_{2}^{\alpha}\kappa<G_{2}>C_{\alpha}+*\right\\}$
(58) $=\int d^{4}x\;M_{\rm New,4}\left\\{{\widetilde{\phi}}_{\rm
Goldstino}^{\alpha}C_{\alpha}\right\\}+*$ (59)

So now we have no cosmological term but we have the following special terms in
the action

$\int d^{4}x\left\\{M_{G}\Psi\Psi+M_{\rm New,3}\;\overline{\phi}_{\rm
Goldstino}\overline{\sigma}^{\mu}\Psi_{\mu}\right\\}+*$ (60)

and special terms in the ZJ action too:

$\int d^{4}x\left\\{-M_{\rm
New}^{2}{\widetilde{\Psi}}^{\mu}\sigma_{\mu}{\overline{C}}+M_{\rm
New,4}{\widetilde{\phi}}_{\rm Goldstino}^{\alpha}C_{\alpha}\right.$ (61)
$\left.+{\widetilde{G}}_{1}{\overline{C}}_{\dot{\beta}}{\overline{\sigma}}^{\mu\alpha\dot{\beta}}\kappa<G_{1}>\Psi_{\mu\alpha}+{\widetilde{G}}_{2}{\overline{C}}_{\dot{\beta}}{\overline{\sigma}}^{\mu\alpha\dot{\beta}}\kappa<G_{2}>\Psi_{\mu\alpha}\right\\}+*$
(62)

There would also be a term
$-i\kappa{\overline{\lambda}}^{\dot{\beta}}{\overline{\sigma}}^{\mu}_{\dot{\beta}\alpha}\left<D\right>\Psi_{\mu\alpha}+*$
from line (95) in the gauge action ${\cal A}_{\rm Gauge}$ except that in this
model, from paragraph *, we have $\left<D\right>=0$ The Goldstino then is

${\phi}_{\rm
Goldstino}=\frac{\kappa}{2M_{G}}\left(\left<{\overline{G}}_{1}\right>\phi_{1}+\left<{\overline{G}}_{2}\right>\phi_{2}\right)=\frac{\kappa
M_{O}^{2}}{M_{G}}\frac{g_{2}}{g_{1}^{2}+g_{2}^{2}}\left(g_{2}\phi_{1}+g_{1}\phi_{2}\right)$
(63)

This needs more attention to see exactly how the invariance remains, but we
will not do that here for now.

31. We can eliminate the Goldstino mixing term ${\cal A}_{\rm Goldstino\;mixing}$ in (57) by choosing a special form of the Ghost and Gauge Fixing Action: This technique was invented by ’t Hooft for the case of the spontaneous breaking of the massive gauge vector boson, and later it was updated using the BRS transformations [15]. This also allows us to keep a free (and useful) gauge parameter. We choose the following Ghost and Gauge Fixing (GGF) action for the gravitino:

${\cal A}_{\rm Gravitino\;GGF}=\int
d^{4}x\delta\left\\{E^{\alpha}\left(\Lambda_{\alpha}+\left(\sqrt{m}\sigma^{\mu}_{\alpha\dot{\beta}}{\overline{\Psi}}_{\mu}^{\dot{\beta}}+\sqrt{\frac{M_{G}^{2}}{m}}\phi_{\alpha;\rm
Goldstino}\right)\right)+*\right\\}$ (64)

where $E^{\alpha}$ is a Grassmann even antighost for the $C_{\alpha}$ SUSY
ghost, and $\Lambda^{\alpha}$ is a Grassmann odd ‘ghost auxiliary field’, and
these are taken to have the BRS variations:

$\delta E^{\alpha}=\Lambda^{\alpha},\;\delta\Lambda^{\alpha}=0$ (65)

After the auxiliary $\Lambda_{\alpha}$ and its complex conjugate are
integrated out of the path integral this becomes:

${\cal A}_{\rm Gravitino\;GGF}\Rightarrow\int
d^{4}x\left\\{-\left(\sqrt{m}\sigma^{\mu}_{\alpha\dot{\beta}}{\overline{\Psi}}_{\mu}^{\dot{\beta}}+\sqrt{\frac{M_{G}^{2}}{m}}\chi_{\alpha;\rm
Goldstino}\right)\right.$ (66)
$\left.\varepsilon^{a\gamma}\left(\sqrt{m}\sigma^{\mu}_{\gamma\dot{\delta}}{\overline{\Psi}}_{\mu}^{\dot{\delta}}+\sqrt{\frac{M_{G}^{2}}{m}}\chi_{\gamma;\rm
Goldstino}\right)+E^{\alpha}\delta\left(\sqrt{m}\sigma^{\mu}_{\alpha\dot{\beta}}{\overline{\Psi}}_{\mu}^{\dot{\beta}}+\sqrt{\frac{M_{G}^{2}}{m}}\chi_{\alpha;\rm
Goldstino}\right)\right\\}+*$ (67)

The cross term in line (67) cancels the mixing term (57), yielding a gauge
parameter dependent mass term for the Goldstino, and a gauge fixing term for
the Gravitino, along with a ghost action that involves the BRS variations of
the Gravitino and the Goldstino. The gauge parameter is $m$ and the Gravitino
mass is $M_{\rm G}$. Both have the dimension of mass.

All of this is exactly like what happens for the massive vector boson and its
Goldstone boson and its gauge parameter, except that the gauge parameter of
the Gravitino has to have the dimension of mass. The old style of gauge fixing
the gravitino [34] did not accomodate any gauge parameter.

32. The Gravitino Propagator with a gauge parameter Calculating the form of this with a gauge parameter is not simple. There are ten ‘spinor-tensor-derivative’ operators that must be considered. Here is the kinetic action which results after the mixing term (57) has been removed using the methods in paragraph *:

${\cal A}^{0}_{\rm Kinetic\;Gravitino}=\int
d^{4}x\;\Psi_{\mu}\gamma^{0}K^{\mu\nu}\Psi_{\nu}$ (68) $=\int
d^{4}x\;\Psi_{\mu}\gamma^{0}\left(\varepsilon^{\mu\nu\rho\kappa}\gamma_{\rho}\gamma_{5}\partial_{\kappa}-M_{G}\eta^{\mu\nu}-m\gamma^{\mu}\gamma^{\nu}\right)\Psi_{\nu}$
(69)

Here we have switched to real gamma matrices as used in [33]. This term is a
Lorentz covariant tensor in the Lorentz indices $\mu\nu\cdots$, and a Lorentz
covariant matrix in the implicit spinor indices that go with
$\gamma_{\mu}\cdots$. and we want to find the propagator, which is the
inverse:

$P^{\mu\nu}K_{\nu\lambda}=\delta^{\mu}_{\;\;\lambda}$ (70)

To find this inverse we need to write down the general Lorentz covariant form
of this tensor-matrix. This turns out to be:

$P^{\mu\nu}=\left\\{\left(a_{1}\varepsilon^{\mu\nu}_{\;\;\;\alpha\beta}\gamma^{\alpha}\partial^{\beta}\gamma_{5}+a_{2}\partial^{\mu}\gamma^{\nu}+a_{3}/\partial\eta^{\mu\nu}+a_{4}\gamma^{\mu}\partial^{\nu}+\frac{c_{1}}{m^{2}}\partial^{\mu}/\partial\partial^{\nu}\right)\right.$
$\left.+\left(\frac{b_{1}}{m}\partial^{\mu}\partial^{\nu}+\frac{b_{2}}{m}\Box\eta^{\mu\nu}+\frac{b_{3}}{m}\Box\gamma^{\mu}\gamma^{\nu}+\frac{b_{4}}{m}\gamma^{\mu}/\partial\partial^{\nu}+\frac{b_{5}}{m}\partial^{\mu}/\partial\gamma^{\nu}\right)\right\\}$
(71)

In (71) we need to assume that the ten coefficients $a_{1},\cdots
c_{1},b_{1}\cdots$ are functions of the ratio

$R=\frac{\Box}{m^{2}}$ (72)

which has mass dimension zero.

So far, this work has only been done for the case where $M_{G}=0$. For that
case, the solution is:

$P^{\mu\nu}=\frac{1}{2\Box}\left\\{Q^{\mu\nu}+\partial^{\mu}\partial^{\nu}\frac{2}{m}\right\\}$
(73)

where

$Q^{\mu\nu}=\varepsilon^{\mu\nu}_{\;\;\;\alpha\beta}\gamma^{\alpha}\partial^{\beta}\gamma_{5}+\partial^{\mu}\gamma^{\nu}-/\partial\eta^{\mu\nu}+\gamma^{\mu}\partial^{\nu}-\partial^{\mu}\partial^{\nu}\frac{4/\partial}{\Box}$
(74)

Note that the propagator has a simple form:

  * •

When $m\rightarrow 0$ this becomes singular. There is no inverse when $m=0$.

  * •

There are no $\frac{1}{\Box+m^{2}}$ poles. The only place that $m$ appears in
the propagator is in the term
$\frac{1}{m}\frac{\partial^{\mu}\partial^{\nu}}{\Box}$ which looks and acts
like a gauge parameter term.

  * •

The part $Q^{\mu\nu}$ satisfies $\gamma_{\mu}Q^{\mu\nu}=0$

  * •

The part
$\varepsilon^{\mu\nu\rho\kappa}\gamma_{\rho}\gamma_{5}\partial_{\kappa}$
satisfies

$\varepsilon^{\mu\nu\rho\kappa}\gamma_{\rho}\gamma_{5}\partial_{\kappa}\partial_{\nu}=0$
(75)

For the case where $M_{G}\neq 0$ this still needs to be completed. Obviously,
this propagator needs to be used so that one can calculate diagrams like
(22)–the significant physics of that diagram needs to be gauge invariant under
two different gauge parameters. The physical consequences, including the
anomaly coefficient, of the diagram (19) also should be gauge invariant under
the gauge parameter for the massive vector boson.

33. The Action and the BRS transformations for the model used in this paper: At this point we need to write down the complicated and long action and transformations for the theory used in this paper. These are needed to understand the remarks in * to see what is going on. The invariance of the action and the nilpotence of the BRS transformations are all summarized in the Master Equation, found in the next section.

34. The Master Equation: This summarizes all the consequences of the BRS symmetry of the theory. For the total action ${\cal A}$, which is the sum of all the actions and all the ZJ actions, it takes the form:

${\cal M}_{\rm Total}={\cal M}_{\rm Sugra}+{\cal M}_{\rm Gauge}+{\cal M}_{\rm
A\;Chiral;\;1}$ (76) $+{\cal M}_{\rm A\;Chiral;\;2}+{\cal M}_{\rm
B\;Chiral;\;1}+{\cal M}_{\rm B\;Chiral;\;2}=0$ (77)

where

${\cal M}_{\rm Sugra}=\int d^{4}x\left\\{\frac{\delta{\cal A}}{\delta h^{\mu
a}}\frac{\delta{\cal A}}{\delta{\widetilde{h}}_{\mu a}}+\frac{\delta{\cal
A}}{\delta\Psi^{\mu\alpha}}\frac{\delta{\cal
A}}{\delta{\widetilde{\Psi}}_{\mu\alpha}}+\frac{\delta{\cal
A}}{\delta{{\overline{\Psi}}}^{\mu\dot{\alpha}}}\frac{\delta{\cal
A}}{\delta{\widetilde{\overline{\Psi}}}_{\mu\dot{\alpha}}}\right.$ (78)
$\left.+\frac{\delta{\cal A}}{\delta{w}_{\mu ab}}\frac{\delta{\cal
A}}{\delta{\widetilde{w}}_{\mu ab}}+\frac{\delta{\cal
A}}{\delta\rho^{ab}}\frac{\delta{\cal
A}}{\delta{\widetilde{\rho}}_{ab}}+\frac{\delta{\cal
A}}{\delta\xi^{\mu}}\frac{\delta{\cal
A}}{\delta{\widetilde{\xi}}_{\mu}}+\frac{\delta{\cal A}}{\delta
C^{\alpha}}\frac{\delta{\cal
A}}{\delta{\widetilde{C}}_{\alpha}}+\frac{\delta{\cal
A}}{\delta{\overline{C}}^{\dot{\alpha}}}\frac{\delta{\cal
A}}{\delta{\widetilde{\overline{C}}}_{\dot{\alpha}}}\right\\}$ (79)

${\cal M}_{\rm Gauge}=\int d^{4}x\left\\{\frac{\delta{\cal A}}{\delta
V^{\mu}}\frac{\delta{\cal A}}{\delta{\widetilde{V}}_{\mu}}+\frac{\delta{\cal
A}}{\delta\lambda^{\alpha}}\frac{\delta{\cal
A}}{\delta{\widetilde{\lambda}}_{\alpha}}+\frac{\delta{\cal
A}}{\delta\overline{\lambda}^{\dot{\alpha}}}\frac{\delta{\cal
A}}{\delta{\widetilde{\overline{\lambda}}}_{\dot{\alpha}}}+\frac{\delta{\cal
A}}{\delta D}\frac{\delta{\cal A}}{\delta{\widetilde{D}}}+\frac{\delta{\cal
A}}{\delta\omega}\frac{\delta{\cal A}}{\delta{\widetilde{\omega}}}\right\\}$
(80)

${\cal M}_{\rm A\;Chiral;\;n=1,2}=\int d^{4}x\left\\{\frac{\delta{\cal
A}}{\delta A_{n}}\frac{\delta{\cal
A}}{\delta{\widetilde{A}}_{n}}+\frac{\delta{\cal
A}}{\delta{\overline{A}}_{n}}\frac{\delta{\cal
A}}{\delta{\widetilde{{\overline{A}}}}_{n}}\right.$ (81)
$\left.+\frac{\delta{\cal A}}{\delta\chi_{n}^{\alpha}}\frac{\delta{\cal
A}}{\delta{\widetilde{\chi}}_{n\alpha}}+\frac{\delta{\cal
A}}{\delta{{\overline{\chi}}_{n}}^{\dot{\alpha}}}\frac{\delta{\cal
A}}{\delta{\widetilde{\overline{\chi}}}_{n\dot{\alpha}}}+\frac{\delta{\cal
A}}{\delta{F}_{n}}\frac{\delta{\cal
A}}{\delta{\widetilde{F}}_{n}}+\frac{\delta{\cal
A}}{\delta{\overline{F}}_{n}}\frac{\delta{\cal
A}}{\delta{\widetilde{\overline{F}}}_{n}}\right\\}$ (82)

${\cal M}_{\rm B\;Chiral;\;n=1,2}=\int d^{4}x\left\\{\frac{\delta{\cal
A}}{\delta B_{n}}\frac{\delta{\cal
A}}{\delta{\widetilde{B}}_{n}}+\frac{\delta{\cal
A}}{\delta{\overline{B}}_{n}}\frac{\delta{\cal
A}}{\delta{\widetilde{{\overline{B}}}}_{n}}\right.$ (83)
$\left.+\frac{\delta{\cal A}}{\delta\phi_{n}^{\alpha}}\frac{\delta{\cal
A}}{\delta{\widetilde{\phi}}_{n\alpha}}+\frac{\delta{\cal
A}}{\delta{{\overline{\phi}}_{n}}^{\dot{\alpha}}}\frac{\delta{\cal
A}}{\delta{\widetilde{\overline{\phi}}}_{n\dot{\alpha}}}+\frac{\delta{\cal
A}}{\delta{G}_{n}}\frac{\delta{\cal
A}}{\delta{\widetilde{G}}_{n}}+\frac{\delta{\cal
A}}{\delta{\overline{G}}_{n}}\frac{\delta{\cal
A}}{\delta{\widetilde{\overline{G}}}_{n}}\right\\}$ (84)

The Master Equation actually applies to the 1PI vertex functional ${\cal
G}={\cal A}+\hbar{\cal G}_{1}+O(\hbar^{2})$, and the action ${\cal A}$ is just
the local part of that. The next sections will contain the various actions
which sum to the total action ${\cal A}$ . There are also some simple
adjustments needed for complications that relate to the gauge fixing terms and
ghost actions. We will ignore those complications for now.

The BRS ZJ $\delta$ operator is the ‘square root’ of the above, by which we
mean

$\delta=\int d^{4}x\left\\{\frac{\delta{\cal A}}{\delta h^{\mu
a}}\frac{\delta}{\delta{\widetilde{h}}_{\mu a}}+\frac{\delta{\cal
A}}{\delta{\widetilde{h}}_{\mu a}}\frac{\delta}{\delta h^{\mu
a}}+\cdots\right\\}$ (85)

Note that the nilpotence of $\delta$ and the vanishing of ${\cal M}_{\rm
Total}$ imply each other:

${\cal M}_{\rm Total}=0\Leftrightarrow\delta^{2}=0$ (86)

It should also be remembered that ${\cal M}$ is rather like a Poisson Bracket,
and so it is invariant under transformations that are like canonical
transformations [26, 27]. That is what is happening with the generation of the
new counterterms in paragraph *, as was explained in [28].

35. The Action and the BRS transformations for the Four Chiral Kinetic Multiplets, Coupled to Gauge Theory, and to Supergravity: Here is the action for the theory used in this paper:

${\cal A}_{\rm Chiral\;Kinetic\;A\;1}\equiv{\cal A}_{A_{1}}=$ (87) $\int
d^{4}xe\left\\{\left(\partial_{\mu}A_{1}-igV_{\mu}A_{1}+\kappa\Psi_{\mu}\cdot\chi_{1}\right)\left(\partial^{\mu}{\overline{A}}_{1}+igV^{\mu}{\overline{A}}_{1}+\kappa{\overline{\Psi}}^{\mu}\cdot\overline{\chi}_{1}\right)\right.$
(88)
$\left.+\overline{\chi}_{1}\overline{\sigma}^{\mu}\left[\partial_{\mu}\chi_{1\alpha}-igV_{\mu}\chi_{1\alpha}+w_{\mu
ab}(\sigma^{ab})_{\alpha\beta}\chi_{1}^{\beta}+\kappa
F_{1}\Psi_{\mu\alpha}\right.\right.$ (89)
$\left.\left.+\kappa\sigma^{\tau}_{\alpha\dot{\beta}}\overline{\Psi}_{\mu}^{\dot{\beta}}\left(\partial_{\tau}A_{1}-igV_{\tau}A_{1}+\kappa\Psi_{\tau}\cdot\chi_{1}\right)\right]+*-F_{1}{\overline{F}}_{1}\right.$
(90)
$\left.+g\left(DA_{1}{\overline{A}}_{1}+\lambda^{\alpha}\chi_{1\alpha}{\overline{A}}_{1}+\overline{\lambda}^{\dot{\alpha}}{\overline{\chi}}_{1\dot{\alpha}}A_{1}\right)\right\\}$
(91)

The action for ${\cal A}_{A_{2}}$ is obtained, from 1 above, by simply
changing $1\rightarrow 2$ and also changing the sign of g throughout. The
actions for ${\cal A}_{B_{1}}$ and ${\cal A}_{B_{2}}$ are the same except for
the change of notation (the superfields are discussed above, and indicate the
notation) and setting $g\rightarrow 0$, since they are neutral under the gauge
group. In the above we are using ‘supercovariant derivatives’, as discussed in
[1]. We have written these out in full in the first line (88) as
${\widehat{D}}_{\mu}A_{1}{\widehat{D}}^{\mu}{\overline{A}}_{1}$, for example.

36. The Action for Supergravity: Here is the Action for supergravity as used here:

${{\cal L}}_{\rm
Supergravity}=M_{P}^{2}eR+\varepsilon_{\mu\nu\lambda\sigma}\Psi^{\mu\alpha}\sigma^{\nu}_{\alpha\dot{\beta}}\partial^{\lambda}{\overline{\Psi}}^{\sigma\dot{\beta}}+\varepsilon_{\mu\nu\lambda\sigma}\Psi^{\mu}\sigma^{\nu}w^{\lambda
cd}\sigma_{cd}{\overline{\Psi}}^{\sigma}$ (92)

We use the formulation by Zumino and Deser in [35]. There is an important
feature here that relates to the spectral sequence and to the quantization of
these theories which we discuss in paragraph *.

37. The Action for the U(1) Gauge Theory, Coupled to Supergravity:

${\cal A}_{\rm Gauge}=\int
d^{4}x\left\\{-\frac{1}{4}\left(F_{\mu\nu}+\kappa\Psi_{[\mu}^{\gamma}\sigma_{\nu]\gamma\dot{\delta}}\overline{\lambda}^{\dot{\delta}}+\kappa\overline{\Psi}_{[\mu}^{\dot{\gamma}}\overline{\sigma}_{\nu]\dot{\gamma}\delta}\lambda^{\delta}\right)\right.$
(93)
$\left.\left(F^{\mu\nu}+\kappa\Psi^{[\mu}_{\varepsilon}\sigma^{\nu]\varepsilon\dot{\zeta}}\overline{\lambda}_{\dot{\zeta}}+\kappa\overline{\Psi}^{[\mu}_{\dot{\epsilon}}\overline{\sigma}^{\nu]\dot{\epsilon}\zeta}\lambda_{\zeta}\right)\right\\}$
(94) $+\int
d^{4}x\left\\{{\overline{\lambda}}^{\dot{\beta}}{\overline{\sigma}}^{\mu}_{\dot{\beta}\alpha}\Bigl{[}\partial_{\mu}\lambda^{\alpha}+w_{\mu}^{ab}({\sigma}_{ab})^{\alpha\beta}\lambda_{\beta}+\kappa(\sigma_{\sigma\tau})^{\alpha\beta}F^{\sigma\tau}\Psi_{\mu\beta}-iD\kappa\Psi_{\mu\alpha}\Bigr{]}\right.$
(95) $\left.+({\rm*\;of\;previous\;term})-D^{2}\right\\}$ (96)

where we define

$F_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$ (97)

38. The ZJ part of the Supergravity Action: This is also needed for the theory. It describes the nilpotent BRS transformations that leave the action invariant.

${{\cal L}}_{\rm ZJ,Supergravity}={\widetilde{h}}^{\mu
a}\left\\{-M_{P}(\partial_{\mu}\xi_{a})+M_{P}(\rho_{\mu
a})+\rho_{a}^{\;\;b}h_{\mu b}-h_{\mu\sigma}\partial^{\sigma}\xi_{a}\right.$
(98) $\left.+\xi^{\sigma}\partial_{\sigma}h_{\mu
a}+{\overline{C}}\sigma_{a}\Psi_{\mu}+C\overline{\sigma}_{a}\overline{\Psi}_{\mu}+\kappa\left(\Psi_{\mu}\sigma^{\nu}{\overline{C}}+C\sigma^{\nu}\overline{\Psi}_{\mu}\right)h_{\nu
a}\right\\}+{\widetilde{w}}^{\mu ab}\left\\{\partial_{\mu}\rho_{ab}\right.$
(99) $\left.+[\rho,w_{\mu}]_{ab}+\xi^{\sigma}\partial_{\sigma}w_{\mu
ab}+\kappa B_{\mu ab}+\kappa{\overline{B}}_{\mu
ab}+\kappa^{2}\left(B^{c}_{\;\;\;ac}h_{\mu b}-B^{c}_{\;\;\;bc}h_{\mu
a}\right)\right.$ (100)
$\left.+\kappa^{2}\left({\overline{B}}^{c}_{\;\;\;ac}h_{\mu
b}-{\overline{B}}^{c}_{\;\;\;bc}h_{\mu
a}\right)-\kappa\left(\Psi_{\mu}\sigma^{\nu}{\overline{C}}+C\sigma^{\nu}\overline{\Psi}_{\mu}\right)w_{\nu
ab}\right\\}$ (101)
$+{\widetilde{\Psi}}^{\mu}\left\\{M_{P}(\partial_{\mu}C+w_{\mu
ab}\sigma^{ab}C)+\rho^{ab}\sigma_{ab}\Psi_{\mu}\right.$ (102)
$\left.+h_{\mu\nu}\partial^{\nu}C+\xi^{\sigma}\partial_{\sigma}\Psi_{\mu}-\kappa\left(\Psi_{\mu}\sigma^{\nu}{\overline{C}}+C\sigma^{\nu}\overline{\Psi}_{\mu}\right)\Psi_{\nu}\right\\}$
(103)
$+{\widetilde{\xi}}^{a}\left\\{-C^{\alpha}\sigma_{a\alpha\dot{\beta}}{\overline{C}}^{\dot{\beta}}+\kappa
C\sigma^{\mu}{\overline{C}}h_{\mu
a}+\xi^{\sigma}\partial_{\sigma}\xi_{a}\right\\}$ (104)
$+{\widetilde{\rho}}^{ab}\left\\{(\rho\rho)_{ab}+\xi^{\sigma}\partial_{\sigma}\rho_{ab}+\kappa
C\sigma^{\mu}{\overline{C}}w_{\mu ab}\right\\}$ (105)
$+{\widetilde{C}}_{\alpha}\left\\{\rho^{ab}(\sigma_{ab})^{\alpha\beta}C_{\beta}+\xi^{\sigma}\partial_{\sigma}C^{\alpha}+\kappa
C\sigma^{\mu}{\overline{C}}\Psi_{\mu}^{\;\;\alpha}\right\\}$ (106)
$+{\widetilde{\overline{C}}}_{\dot{\beta}}\left\\{\rho^{ab}\overline{\sigma}_{ab}{\overline{C}}+\xi^{\sigma}\partial_{\sigma}{\overline{C}}+\kappa
C\sigma^{\mu}{\overline{C}}\overline{\Psi}_{\mu}\right\\}^{\dot{\beta}}$ (107)

where following Zumino and Deser in [35] we put

$B_{a}^{\;\;\lambda\mu}=\varepsilon^{\lambda\mu\nu\rho}{\overline{C}}\sigma_{a}D_{\nu}\psi_{\rho}$
(108)

39. The ZJ part of the Gauge Action: This describes the nilpotent BRS transformations of the gauge ZJ sources, that leave the action invariant.

${\cal A}_{\rm ZJ,Gauge}=\int
d^{4}x\left\\{{\widetilde{V}}^{\mu}\left(\partial_{\mu}\omega+C^{\alpha}\sigma_{\mu\alpha\dot{\beta}}{\overline{\lambda}}^{\dot{\beta}}\right.\right.$
(109)
$\left.\left.+{\overline{C}}^{\dot{\alpha}}\overline{\sigma}_{\mu\dot{\alpha}\beta}{\lambda}^{\beta}+\xi^{\sigma}\partial_{\sigma}V_{\mu}\right)\right.$
(110)
$\left.+{\widetilde{\lambda}}^{\alpha}\left(\sigma_{\alpha\beta}^{\mu\nu}{\widehat{F}}_{\mu\nu}C^{\beta}+\rho_{ab}\sigma^{ab}_{\alpha\beta}\lambda^{\beta}+iDC_{\alpha}+\xi^{\mu}\partial_{\mu}\lambda^{A}_{\alpha}\right)\right.$
(111)
$\left.+{\widetilde{\overline{\lambda}}}^{\dot{\alpha}}\left({\overline{\sigma}}_{\dot{\alpha}\dot{\beta}}^{\mu\nu}{\widehat{F}}_{\mu\nu}{\overline{C}}^{\dot{\beta}}+\rho_{ab}\overline{\sigma}^{ab}_{\dot{\alpha}\dot{\beta}}\lambda^{\dot{\beta}}-iD{\overline{C}}_{\dot{\alpha}}+\xi^{\mu}\partial_{\mu}{\overline{\lambda}}_{\dot{\alpha}}\right)\right.$
(112)
$\left.+{\widetilde{D}}\left(i(C^{\alpha}\sigma_{\alpha\dot{\beta}}^{\mu}{\widehat{\overline{D}}}_{\mu}{\overline{\lambda}}^{\dot{\beta}}-{\overline{C}}^{\dot{\alpha}}\overline{\sigma}_{\dot{\alpha}\beta}^{\mu}{\widehat{D}}_{\mu}{\lambda}^{\beta})+\xi^{\mu}\partial_{\mu}D\right)\right.$
(113)
$\left.+{\widetilde{\omega}}\left(C^{\alpha}\sigma_{\alpha\dot{\beta}}^{\mu}{{\overline{C}}}^{\dot{\beta}}V_{\mu}+\xi^{\mu}\partial_{\mu}\omega\right)\right\\}$
(114)

In the above we are using ‘supercovariant derivatives’, indicated with the
notation ${\widehat{D}}_{\mu}{\lambda}^{\beta}$, for example, as discussed in
[1] and as used above in paragraph *.

40. The ZJ part of the Four Chiral Actions: This describes the nilpotent BRS transformations of the four kinds of chiral ZJ sources, that leave the action invariant.

${\cal L}_{\rm ZJ\;Chiral\;A\;1}=$ (115) ${\widetilde{A}}_{1}\left[-ig\omega
A_{1}+C\cdot\chi_{1}+\xi^{\mu}\partial_{\mu}A_{1}\right]$ (116)
$+{\widetilde{\overline{A}}}_{1}\left[+ig\omega
A_{1}+{\overline{C}}\cdot{\overline{\chi}}_{1}+\xi^{\mu}\partial_{\mu}{\overline{A}}_{1}\right]$
(117)
$+{\widetilde{\chi}}_{1}^{\alpha}\left[-ig\omega\chi_{1\alpha}+\rho^{ab}(\sigma_{ab})_{\alpha\beta}\chi_{1}^{\beta}+F_{1}C_{\alpha}+\xi^{\mu}\partial_{\mu}\chi_{1\alpha}\right.$
(118)
$\left.+\sigma^{\tau}_{\alpha\dot{\beta}}{\overline{C}}^{\dot{\beta}}\left(\partial_{\tau}A_{1}-igV_{\tau}A_{1}+\kappa\Psi_{\tau}^{\beta}\chi_{1\beta}\right)\right]+*$
(119) $+{\widetilde{F}}_{1}\left[-ig\omega
F_{1}+\xi^{\mu}\partial_{\mu}F_{1}\right]+*$ (120)
$+{\widetilde{F}}_{1}{\overline{C}}_{\dot{\beta}}{\overline{\sigma}}^{\mu\alpha\dot{\beta}}\left[\partial_{\mu}\chi_{1\alpha}-igV_{\mu}\chi_{1_{\alpha}}+\kappa
F_{1}\Psi_{\mu\alpha}\right]+*$ (121)
$+{\widetilde{F}}_{1}{\overline{C}}_{\dot{\beta}}{\sigma}^{\mu\alpha\dot{\beta}}\kappa\sigma^{\tau}_{\alpha\dot{\gamma}}\overline{\Psi}_{\mu}^{\dot{\gamma}}\left[\partial_{\tau}A_{1}-igV_{\tau}A_{1}+\kappa\Psi_{\tau}^{\beta}\chi_{1\beta}\right]+*$
(122)

The expressions for ${\cal L}_{\rm ZJ\;Chiral\;A\;2}$ and ${\cal L}_{\rm
ZJ\;Chiral\;B\;n},n=1,2$ are obtained from the above in the same way that was
described above for the actions.

41. The Diagram (22) can exist only if the gravitino is massive: All these counterterms will be expected to be obtained from an expression like this (the meaning of the notation is given below). Note that the ghost charge of this expression is $N_{\rm Ghost}=-1$: it is linear in ZJ sources:

${\cal A}^{-1}_{\rm Generator}=\int
d^{4}x\left\\{if_{1}{\widetilde{D}}\left(F_{1}-{\overline{F}}_{1}\right)+if_{2}{\widetilde{D}}\left(F_{2}-{\overline{F}}_{2}\right)\right.$
(123)
$\left.+f_{3}\left({\widetilde{\chi}}_{1}^{\alpha}\lambda_{\alpha}+{\widetilde{\overline{\chi}}}_{1}^{\dot{\alpha}}{\overline{\lambda}}_{\dot{\alpha}}\right)+f_{4}\left({\widetilde{\chi}}_{2}^{\alpha}\lambda_{\alpha}+{\widetilde{\overline{\chi}}}_{2}^{\dot{\alpha}}{\overline{\lambda}}_{\dot{\alpha}}\right)\right.$
(124)
$\left.+if_{5}\left({\widetilde{F}}_{1}-{\widetilde{\overline{F}}}_{1}\right)D+if_{6}\left({\widetilde{F}}_{2}-{\widetilde{\overline{F}}}_{2}\right)D+\right.$
(125)
$\left.+f_{7}\left({\widetilde{\lambda}}^{\alpha}\chi_{1\alpha}+{\widetilde{\overline{\lambda}}}^{\dot{\alpha}}{\overline{\chi}}_{1\dot{\alpha}}\right)+f_{8}\left({\widetilde{\lambda}}^{\alpha}\chi_{2\alpha}+{\widetilde{\overline{\lambda}}}^{\dot{\alpha}}{\overline{\chi}}_{2\dot{\alpha}}\right)+\cdots\right\\}$
(126)

To get the counterterms one takes

$\delta{\cal A}^{-1}_{\rm Generator}={\cal A}^{0}_{\rm Counterterms}$ (127)

where the operator $\delta$ comes from the ‘square root’ of the Master
equation, found in equation (77). It is easy to verify that there are plenty
of diagrams that appear when the gravitino is massive, and we expect them all
to be related in this way. It is interesting to note that these counterterms
change the scalar interaction in paragraph * in a simple way. That does not
appear to spoil the general argument or the VEVs here, however.

42. Errors: Anyone who works with SUSY makes plenty of these, and the author is no exception. There are so many minus signs to take care of that it is exhausting, and short cuts must be taken. If the reader thinks there is a missing minus sign or factor or term, he may well be right. Some features are crucial of course, but they are usually simple. The crucial and simple issues in this paper are in paragraphs * and *, and in the diagrams (19) and (22). The author apologizes for not yet having the energy to carefully and accurately check all the terms in the actions and transformations. I believe that these are pretty close to correct, but there are hundreds of terms to check, and the task is not easy. There are two stages in finding the action and transformations here. The first stage is to make sure that there are enough terms of the right kinds, with indices in the right places, to satisfy each identity. The second stage is to make sure that the signs and factors are correct. I believe that I have accomplished the first stage, but I have currently not yet had the energy to perform the second stage. In the case of $\delta w_{\mu ab}$, I am still working on the first stage. Much of the necessary material there is already in the paper [35]. However the current results reported here are, I think, sufficient to justify the conjecture that spontaneous breaking is needed to generate these sugranomalies. What difference does this make to this paper? I think that the fundamental result is not affected at all. It involves only the simple terms discussed above. The calculation of the anomaly coefficient and the question whether it requires spontaneous breaking of supersymmetry are crucially important, of course. To calculate the anomaly coefficient we need the exact action and BRS transformations. We need to be sure that the Master equation is correct and that $\delta^{2}=0$ is correct, and that no mistakes are made in the calculation. That is a large undertaking.

43. First Order and Second Order Formalism: Most work in supergravity has been done using the second order formalism, where the spin connection $w_{\mu ab}$ is replaced by its expression in terms of the vierbein $e_{\mu a}=\eta_{\mu a}+\kappa h_{\mu a}$. We do not do that here for two reasons. The first is that it seems easier to formulate the BRS formalism in terms of the first order formalism. The second is that it seems very strange to try to quantize the theory when $w_{\mu}^{ab}$ is no longer present. It is the gauge field of the ghost $\rho^{ab}$. This transformation is the usual sort of gauge transformation: $\delta w_{\mu}^{ab}=\partial_{\mu}\rho^{ab}+\cdots$. The absence of $w_{\mu}^{ab}$ means that in some sense, in the second order formalism, one needs to have two different antighosts for the gauge fixing term for the field $h_{\mu a}$, one for the general coordinate ghost $\xi_{\mu}$ and the other for the Lorentz ghost $\rho^{ab}$. From the spectral sequence point of view, this is particularly disturbing and it looks wrong. We also prefer the Weyl two component complex formalism because the Fierz transformations are so much simpler there. It seems clear that supersymmetry is more natural, and simpler, in that formalism. For example the anomaly (14) discussed here is very chiral–its complex conjugate

${\overline{\cal A}}^{1}_{\rm Supergravity}=\int
d^{4}x\;\varepsilon^{\lambda\mu\nu\sigma}\left\\{{\overline{c}}_{1}{\overline{C}}_{\dot{\alpha}}\partial_{\lambda}{\overline{\Psi}}_{\mu}^{\dot{\alpha}}\partial_{\nu}V_{\sigma}+{\overline{c}}_{2}\partial_{\lambda}{\overline{\Psi}}_{\mu}^{\dot{\alpha}}\partial_{\nu}{\overline{\Psi}}_{\sigma\dot{\alpha}}\;\omega+\cdots\right\\}$
(128)

is also an anomaly form.

44. Auxiliary Fields: This paper is composed using the first order formalism and in terms of Weyl two component complex spinors. The reader might wonder whether we need additional auxiliary fields to close the algebra of supergravity, for example in the expressions in paragraph *. Auxiliary fields are useful for generating a calculus where multiplets can be multiplied to make other multiplets and invariants. However they are not needed to create the nilpotent BRS transformations. The reason is that one can imagine having the auxiliary fields and integrating them out using the path integral [38]. That changes the terms in the Master Equation, but the derived $\delta$ is still nilpotent, because it must be nilpotent, given that it is the ‘square root’ of the Master Equation. As is well known, auxiliary fields are used to close transformations that do not close, except using the field equations. In the Master Equation approach, exactly the same function is served by the Zinn sources. It appears to me that there is no need in the first order formalism for any auxiliary fields in the present paper–they would show up as a need for terms with doubled Zinn sources. But it seems likely that they would start to appear at higher orders, in particular because of the terms in paragraph *.

45. Spectral Sequence: The results in this paper were obtained by the author using spectral sequences [36, 37, 39, 40, 41]. These methods provided the insight needed to deduce that there had to be an anomaly in supergravity. When a suitable spectral sequence can be used, the space $E_{\infty}$ represents all the possible invariants of the theory–but their explicit construction is still needed in that case. In the present situation, a spectral sequence indicated that there ought to be an anomaly, but its construction from that took a very long time. Unfortunately, those methods are quite difficult and confusing, though powerful. There are still lots of unsolved issues there when those methods are applied to supergravity, and indeed other theories too.

46. Future Steps: The action and the transformations, and their closure, need to be completed and rechecked. The work here on the massless gravitino propagator with a massive gauge parameter needs to be extended to the case where the gravitino is also massive. The supergravity anomalies need to be calculated, in many theories. The values of their coefficients should be independent of all gauge parameters. The significance of these supergravity anomalies needs to be understood. Is the conjecture about their coefficients in equation (24) correct? If it is correct, does that mean that the viable theories must be those which do have some $\left<F^{i}\right>\neq 0$, but which also have the sugranomaly coefficient, or perhaps $\left<D^{a}\right>$ itself, arranged to vanish, somehow? Is that easy? What kinds of theories are those? The spectral sequence work also needs to be advanced.

47. Supergravitational fields in this model:

Symbols |  | Ghost
---
Number
| Reality and
---
Grassman
| Dim $m^{d}$
---
$d=$
Name
$w_{\mu}^{\;\;ab}$ | 0 | Real, Even | 1 | Lorentz Gauge Field
$\rho^{ab}$ | 1 | Real, Odd | $0$ | Lorentz Tensor Ghost
$\zeta^{ab}$ | -1 | Real, Odd | $2$ | Lorentz Tensor AntiGhost
$R^{ab}$ | 0 | Real, Odd | $2$ | Lorentz Ghost Auxiliary
$\Psi_{\mu}^{\;\;\alpha},{\overline{\Psi}}_{\mu}^{\;\;\dot{\alpha}}$ | 0 | Complex, Odd | $\frac{3}{2}$ | Gravitino Gauge Field
$C^{\alpha},{\overline{C}}^{\dot{\alpha}}$ | 1 | Complex, Even | $-\frac{1}{2}$ | Gravitino Spinor Ghost
$E^{\alpha},{\overline{E}}^{\dot{\alpha}}$ | -1 | Complex, Even | $\frac{5}{2}$ | Gravitino Spinor AntiGhost
$\Theta^{\alpha},{\overline{\Theta}}^{\dot{\alpha}}$ | 0 | Complex, Even | $\frac{5}{2}$ | Gravitino Ghost Auxiliary
$e_{\mu a}\rightarrow\eta_{\mu a}+\kappa h_{\mu a}$ | 0 | Real, Even | $0$ | Vierbein
$h_{\mu a}$ | 0 | Real, Even | 1 | Gravitational Field
$\xi_{\mu}$ | 1 | Real, Odd | $-1$ | General Coordinate Ghost
$\zeta_{\mu}$ | -1 | Real, Odd | $3$ | General Coordinate AntiGhost
$H_{\mu}$ | 0 | Real, Odd | $3$ | General Coordinate Ghost Auxiliary

Note that we define:

$e_{\mu a}=\eta_{\mu a}+\kappa h_{\mu a};g_{\mu\nu}=e_{\mu a}\eta^{ab}e_{\nu
b}$ (129)

which means that the inverse of $e$ is an infinite series in $h$, and I am
ignoring that for now.

48. Gauge Particles for the Abelian Gauge Theory:

Symbols |  | Ghost
---
Number
| Reality and
---
Grassman
| Dim $m^{d}$
---
$d=$
Name
$V_{\mu}$ | 0 | Real, Even | 1 | Gauge Vector Boson
$\omega$ | 1 | Real, Odd | 0 | Gauge Ghost Field
$\eta$ | -1 | Real, Odd | 2 | Gauge AntiGhost Field
$H$ | 0 | Real, Odd | 2 | Gauge Ghost Auxiliary Field
$\lambda^{\alpha},\overline{\lambda}^{A\dot{\alpha}}$ | 0 | Complex, Odd | $\frac{3}{2}$ | Gauge Fermion Field
$D$ | 0 | Real, Even | 2 | Gauge Auxiliary Field

49. Higgs Representations in this Model

Symbols |  | Reality and
---
Grassman
| Dim $m^{d}$
---
$d=$
Name
$A_{n},{\overline{A}}_{n}$ | Complex, Even | 1 | Charged Chiral Scalar Pair: n=1,2
$\chi_{n}^{\alpha},\overline{\chi}_{n}^{\dot{\alpha}}$ | Complex, Odd | $\frac{3}{2}$ | Charged Chiral Spinor Pair :n=1,2
$F_{n},{\overline{F}}_{n}$ | Complex, Even | 2 | Charged Chiral Auxiliary Pair:n=1,2
$B_{n},{\overline{B}}_{n}$ | Complex, Even | 1 | Singlet Chiral Scalars: n=1,2
$\phi_{n}^{\alpha},\overline{\phi}_{n}^{\dot{\alpha}}$ | Complex, Odd | $\frac{3}{2}$ | Singlet Chiral Spinors :n=1,2
$G_{n},{\overline{G}}_{n}$ | Complex, Even | 2 | Singlet Auxiliary Scalars: n=1,2

50. Some Constants in this Model

Symbols |  | Paragraph
---
| Dim $m^{d}$
---
$d=$
Name
$\kappa=\frac{1}{M_{{}_{P}}}$ |  | $-1$ | Planck Length
$M_{{}_{P}}=\frac{1}{\kappa}$ |  | $+1$ | Planck Mass
$M_{O}$ | * | $+1$ | 0’Raiffeartaigh Mass
$M_{G}$ |  *,* | $+1$ | Gravitino Mass
$m$ | * | $+1$ | Gravitino Gauge Parameter
$M_{{}_{\rm CC}}^{4}$ | * | $+4$ | Cosmo Mass
$g_{1}$ | * | $0$ | Chiral Coupling
$g_{2}$ | * | $0$ | Chiral Coupling
$g$ | * | $0$ | Gauge Coupling
$v=\sqrt{\frac{g_{2}}{g_{1}^{2}+g_{2}^{2}}}$ | * | $0$ | VEV parameter
$M_{\rm 1,mix}=g<A_{1}>=gvM_{O}$ |  *,* | $1$ | Mass term from VEV
$\left<A_{1}\right>=M_{O}v$ | * | $1$ | VEV of $A_{1}$
$\left<A_{2}\right>=M_{O}v$ | * | $1$ | VEV of $A_{2}$
$\left<G_{1}\right>=-M_{O}^{2}\frac{g_{1}g_{2}}{g_{1}^{2}+g_{2}^{2}}$ | * | $2$ | VEV of $G_{1}$
$\left<G_{2}\right>=M_{O}^{2}\frac{g_{1}^{2}}{g_{1}^{2}+g_{2}^{2}};$ | * | $2$ | VEV of $G_{2}$

51. Some Important Concepts in the paper

Symbol |  | Paragraph
---
Name and Comment
${\cal M}$ | * | Master Equation
$\delta$ | * | BRS Nilpotent Transformation
$\widetilde{\rm F}$ | * | ZJ Source for the BRS Variation $\delta{\rm F}$ of the Field F:
${\cal A}^{1}_{\rm Supergravity}$ | * | Supergravity Anomaly
${\cal A}^{0}_{\rm Supergravity\;Counterterm\;2}$ | * | Generates boundary for Supergravity Anomaly
$V=-{\cal L}_{\rm Scalar\;Potential}$ | * | Scalar Potential
${\phi}_{\rm Goldstino}$ | * | Goldstino Fermion
${\cal A}^{-1}_{\rm Generator}$ | * | Generator for new terms in action
$P_{\mu\nu},Q_{\mu\nu}$ | * | (73) , (74) Gravitino Propagator

Acknowledgments

I thank Doug Baxter, Carlo Becchi, Philip Candelas, James Dodd, Mike Duff,
Richard Golding, Dylan Harries, Pierre Ramond, Graham Ross, Peter Scharbach,
Kelly Stelle, Xerxes Tata, J.C. Taylor, and Peter West for stimulating
correspondence and conversations.

## References

  * [1] P. C. West, “Introduction to supersymmetry and supergravity,” Singapore, Singapore: World Scientific (1990) 425 p
  * [2] D. Z. Freedman and A. Van Proeyen, “Supergravity”, Cambridge, UK (2012): ISBN: 9781139368063 (eBook), 9780521194013 (Print) (Cambridge University Press).
  * [3] Many of the original papers on SUSY and Supergravity are collected in “Supersymmetry”, Vols. 1 and 2, ed. Sergio Ferrara, North Holland, World Scientific, (1987).
  * [4] S. J. Gates, M. T. Grisaru, M. Rocek and W. Siegel, “Superspace”, Benjamin, 1983.
  * [5] J. Wess and J. Bagger, “Supersymmetry and Supergravity”, Second Edition, Princeton University Press (1992).
  * [6] I. L. Buchbinder and S. M. Kuzenko, “Ideas and methods of supersymmetry and supergravity: Or a walk through superspace,” Bristol, UK: IOP (1998) 656 p
  * [7] H. Baer and X. Tata, “Weak scale supersymmetry: From superfields to scattering events,” Cambridge, UK: Univ. Pr. (2006)
  * [8] Steven Weinberg: “The Quantum Theory of Fields” Volume 3, Cambridge University Press, ISBN 052155002.
  * [9] P.A. Zyla et al. [Particle Data Group], “Review of Particle Physics, PTEP 2020, no.8, 083C01 (2020) doi:10.1093/ptep/ptaa104; “Supersymmetry Theory”: Revised August 2019 by B.C. Allanach (DAMTP, Cambridge U.) and H.E. Haber (UC Santa Cruz).
  * [10] P.A. Zyla et al. [Particle Data Group], “Review of Particle Physics, PTEP 2020, no.8, 083C01 (2020) doi:10.1093/ptep/ptaa104; “Supersymmetry Part II: Experiment”: Revised September 2019 by O. Buchmuller (Imperial Coll. London) and P. de Jong (NIKHEF).
  * [11] S. Ferrara and A. Sagnotti, “Supergravity at 40: Reflections and Perspectives,” Riv. Nuovo Cim. 40, no. 6, 1 (2017) doi:10.1393/ncr/i2017-10136-6 [arXiv:1702.00743 [hep-th]].
  * [12] M. B. Green, J. H. Schwarz and E. Witten, “SUPERSTRING THEORY. VOL. 2: LOOP AMPLITUDES, ANOMALIES AND PHENOMENOLOGY”, Cambridge, Uk: Univ. Pr. ( 1987) 469 P. (Cambridge Monographs On Mathematical Physics)
  * [13] M. B. Green, J. H. Schwarz and E. Witten, “SUPERSTRING THEORY. VOL. 1: INTRODUCTION”, Cambridge, Uk: Univ. Pr. ( 1987) (Cambridge Monographs On Mathematical Physics)
  * [14] Any text on the Standard Model contains this. See for example [15].
  * [15] J. C. Taylor, “Gauge Theories of Weak Interactions,” Cambridge 1976, 167p
  * [16] The Master Equation goes back to [17] and Zinn-Justin’s early contribution is set out in a later textbook [18]. An early and pithy introduction was in [15]. A bit of history and Zinn Justin’s involvement is in [19]. A more recent treatment is in [23].
  * [17] C. Becchi, A. Rouet and R. Stora, “Renormalization of Gauge Theories,” Annals Phys. 98, 287 (1976).
  * [18] J. Zinn-Justin, “Quantum Field Theory and Critical Phenomena”, Oxford Science Publications, Reprinted 1990.
  * [19] A summary and some history can be found in J. Zinn-Justin, “From Slavnov-Taylor identities to the ZJ equation,” Proc. Steklov Inst. Math. 272, 288 (2011).
  * [20] C. Becchi, “Slavnov–Taylor and Ward identities in the electroweak theory,” Theor. Math. hys. 182, no. 1, 52 (2015) [Teor. Mat. Fiz. 182, no. 1, 65 (2014)] doi:10.1007/s11232-015-0244-8 [arXiv:1407.3960 [hep-th]].
  * [21] C. Becchi, “Becchi- Rouet- Stora- Tyutin symmetry”, http://www. scholarpedia .org/article/Becchi-Rouet-Stora-Tyutin symmetry
  * [22] I. A. Batalin and G.A. Vilkovisky, “Gauge Algebra and Quantization”, Physics Letters B 102.1 (1981): 27-31
  * [23] Steven Weinberg: “The Quantum Theory of Fields” Volume 2, Cambridge University Press, ISBN 052155002. This contains an introduction to the Master Equation and BRS.
  * [24] M. B. Fröb, C. Imbimbo and N. Risso, “Deformations of Supergravity and Supersymmetry Anomalies,” [arXiv:2107.03401 [hep-th]].
  * [25] For the Higgs mechanism, see for example [15].
  * [26] Herbert Goldstein,“Classical Mechanics” (Second Edition) Addison Wesley ISBN 0-201-02918-9
  * [27] L.D. Landau and E.M. Lifschitz, “Mechanics”, Pergamon Press (1960)
  * [28] J. A. Dixon, “Field Redefinition and Renormalization in Gauge Theories,” Nucl. Phys. B 99, 420 (1975).
  * [29] Pran Nath, “Supersymmetry, Supergravity, and Unification” (Cambridge Monographs on Mathematical Physics) ,Dec 15, 2016
  * [30] Ross, Graham G., “GRAND UNIFIED THEORIES”, Reading, USA: Benjamin/Cummings ( 1984) 497 P. ( Frontiers In Physics, 60), (1985)
  * [31] P.A. Zyla et al. [Particle Data Group], “Review of Particle Physics”, PTEP 2020, no.8, 083C01 (2020) doi:10.1093/ptep/ptaa104: Grand Unified Theories, Revised August 2019 by A. Hebecker (Heidelberg U.) and J. Hisano (KMI, Nagoya U.).
  * [32] K. Intriligator and N. Seiberg, “Lectures on supersymmetry breaking”, arXiv hep-ph/070269 Classical and Quantum Gravity 24 (2007)
  * [33] J. Wess and B. Zumino, “A Lagrangian model invariant under supergauge transformations” Phys. Lett. B 49: 52–54 (1974).
  * [34] R. E. Kallosh, “Modified Feynman Rules in Supergravity,” Nucl. Phys. B 141, 141 (1978). doi:10.1016/0550-3213(78)90340-1
  * [35] S. Deser and B. Zumino, “Consistent Supergravity”, Phys. Lett. B 62, 335 (1976)
  * [36] J.A. Dixon, “Supersymmetry is full of holes”, Class. Quant. Grav. 7, pp 1511–1521, (1990)
  * [37] Ibid, “Calculation of BRS cohomology with spectral sequences”, Commun. Math. Phys., 139 pp 495–526 (1991)
  * [38] Ibid, “Closure of the algebra and remarks on BRS cohomology in D = 10 superYang-Mills”, Unpublished but available at Inspire HEP, UTTG-14-91, (1991)
  * [39] Ibid, “BRS cohomology of the chiral superfield”, Commun. Math. Phys. 140”, pp 169–201 (1991)
  * [40] Dixon, John A. and Minasian, R., “BRS cohomology of the supertranslations in D = 4”, arXiv hep-th/9304035, Commun. Math. Phys. 172, pp 1–12 (1995). The appendix in this paper has some errors that crept in somehow.
  * [41] Dixon, John A. and Minasian, R. and Rahmfeld, J., “Higher spin BRS cohomology of supersymmetric chiral matter in D = 4”, arXiv hep-th/9308013, Commun. Math. Phys. 171 pp 459–474 (1995)

JB178 Candidate