Similar analyses can be performed on systems with particles of different
helicities. Here, simple cases of mixed scalar, fermion and vector amplitudes
will be discussed in order to derive new constraints.

To begin with, the system will be restricted to consisting of a real scalar, a
photon and a chiral fermion. The full table has dimensions $25\times 25$ when
external helicity eigenstates are chosen as a basis, but most of the entries
vanish either altogether by conservation of angular momentum or, specifically
at dimension $8$, by incompatibility of Lorentz invariance and dimensional
analysis (see discussion below in Section 6.2). For this reason, only the
(few) important entries containing new information will be quoted here. The
full sum rule in $SO(2)$ form is given in the Appendix.

Restricting entirely to the bosons to begin with, the entries relevant for
photons are given in Table 2 in the section above, while the others of
relevance are:

$\displaystyle M^{\phi\phi\phi\phi}$
$\displaystyle=2|\boldsymbol{m}^{\phi\phi}|^{2}$ (130) $\displaystyle
M^{\gamma^{+}\phi\gamma^{+}\phi}$
$\displaystyle=2|\boldsymbol{m}^{\gamma^{+}\phi}|^{2}$ (131) $\displaystyle
M^{\gamma^{-}\phi\gamma^{-}\phi}$
$\displaystyle=2|\boldsymbol{m}^{\gamma^{-}\phi}|^{2}$ (132) $\displaystyle
M^{\phi\phi\gamma^{+}\gamma^{+}}$
$\displaystyle=\boldsymbol{m}^{\gamma^{+}\gamma^{+}}\cdot\boldsymbol{m}^{\phi\phi}+\boldsymbol{m}^{\gamma^{+}\phi}\cdot\boldsymbol{m}^{\phi\gamma^{-}}$
(133) $\displaystyle M^{\phi\phi\gamma^{-}\gamma^{-}}$
$\displaystyle=\boldsymbol{m}^{\gamma^{-}\gamma^{-}}\cdot\boldsymbol{m}^{\phi\phi}+\boldsymbol{m}^{\phi\gamma^{-}}\cdot\boldsymbol{m}^{\gamma^{+}\phi}$
(134)

(and others related by crossing). Simplification with $CPT$ and $Y$ (which
also equate
$|\boldsymbol{m}^{\gamma^{+}\phi}|=|\boldsymbol{m}^{\phi\gamma^{-}}|$ has been
invoked. The bounds are identified as:

$\displaystyle|M^{\phi\phi\gamma^{+}\gamma^{+}}|,|M^{\phi\phi\gamma^{-}\gamma^{-}}|\leq\frac{1}{2}M^{\phi\gamma^{+}\phi\gamma^{+}}+\sqrt{\frac{1}{2}M^{\phi\phi\phi\phi}M^{\gamma^{+}\gamma^{-}\gamma^{+}\gamma^{-}}}.$
(135)

Unlike the helicity-violating four-vector operators, these bounds are not
accessible by considering elastic forward scattering of a scalar with a
linearly polarised vector. Superpositions of vectors and scalars are instead
necessary. The bounds are also stronger by various factors of $2$ compared to
what would be anticipated from direct application of (11). This is because the
identity of the scalars is crossing symmetric, which simplifies the sum rule.
The analogous bounds with complex scalars, used below in Section 6.3, are
weaker.

The other constraint arises for a mixed spin amplitude. The relevant entries
are

$\displaystyle M^{\phi\psi^{+}\phi\psi^{+}}$
$\displaystyle=|\boldsymbol{m}^{\phi\psi^{+}}|^{2}+|\boldsymbol{m}^{\phi\psi^{-}}|^{2}$
(136) $\displaystyle M^{\gamma^{+}\psi^{+}\gamma^{+}\psi^{+}}$
$\displaystyle=|\boldsymbol{m}^{\gamma^{+}\psi^{+}}|^{2}+|\boldsymbol{m}^{\gamma^{-}\psi^{-}}|^{2}$
(137) $\displaystyle M^{\phi\psi^{-}\gamma^{+}\psi^{+}}$
$\displaystyle=2\boldsymbol{m}^{\gamma^{+}\psi^{+}}\cdot\boldsymbol{m}^{\phi\psi^{-}}$
(138) $\displaystyle M^{\phi\psi^{+}\gamma^{-}\psi^{-}}$
$\displaystyle=2\boldsymbol{m}^{\gamma^{-}\psi^{-}}\cdot\boldsymbol{m}^{\phi\psi^{+}},$
(139)

where $|\boldsymbol{m}^{\phi\psi^{+}}|=|\boldsymbol{m}^{\phi\psi^{-}}|$. The
resulting inelastic constraint is

$\displaystyle|M^{\phi\psi^{-}\gamma^{+}\psi^{+}}|,|M^{\phi\psi^{+}\gamma^{-}\psi^{-}}|\leq\sqrt{2M^{\phi\psi^{+}\phi\psi^{+}}M^{\psi^{+}\gamma^{+}\psi^{+}\gamma^{+}}}.$
(140)

Along with the postivity constraints given in Bellazzini:2016xrt , this
completes the causality bounds for the simple minimal, toy theories of
spinning particles. Similar arguments can be used to adapt these to more
complicated theories with more states. This will be partly done in the
supersymetric case below.

##  6 Supersymmetry

Having addressed the management of spin in the sum rule, it is natural to now
extend this to supermultiplets. Supersymmetry unifies states of different spin
and likewise their interactions. Supersymmetry is a consistent extension of
the spacetime symmetry algebra, so should not affect conclusions drawn from
the (rigid) causal structure of background flat Minkowski space. It is
therefore expected that causality constraints on scattering of a particular
set of component states should be shared by the other interactions related by
supersymmetry. Precisely these connections will be explored in this section.
For simplicity, attention will be restricted to EFTs with minimal particle
content.

###  6.1 Superamplitudes

Superspaces at the level of the effective action are generally arduous and
cumbersome to work with. As is very well appreciated, on-shell scattering
amplitudes cut-through the off-shell baggage of the effective action, not only
making computations substantially easier, but also clarifying the presence and
action of symmetries that are either not manifest or are convoluted in the
Lagrangian field theory. The “on-shell superspace”, to be employed here, makes
the super-Ward identities (SWIs) manifest as relations between scattering
amplitudes - see e.g. Elvang:2013cua  for review. Amplitudes between
individual states in a multiplet are unified into superamplitudes. This makes
transparent the relation between the unified effective interactions and their
component operators without recourse to an off-shell superspace. The
especially simple case of $2\rightarrow 2$ scattering amplitudes, under
discussion here, are highly constrained by fundamental principles. See
Elvang:2010jv  for numerous examples in supergravity.

The chiral superspace of Lal:2009gn , later used by Elvang:2011fx , will be
employed here, where the highest helicity state is selected as the Clifford
vacuum for the representation. This determines the little group representation
of the entire “superfield”. The massless multiplets for $\mathcal{N}=1$
theories are

$\displaystyle\begin{split}\Phi^{+}&=\psi^{+}+\eta\overline{\phi}\\\
V^{+}&=v^{+}+\eta\lambda^{+}\end{split}\begin{split}\Phi^{-}&=\phi+\eta\psi^{-}\\\
V^{-}&=\lambda^{-}+\eta v^{-}.\end{split}$

For $\mathcal{N}=2$, they are

$\displaystyle\begin{split}K&=\chi^{+}+\eta^{A}\phi_{A}-\frac{1}{2}\epsilon_{AB}\eta^{A}\eta^{B}\chi^{-}\\\
V^{+}&=v^{+}+\eta^{A}\lambda^{+}_{A}-\frac{1}{2}\epsilon_{AB}\eta^{A}\eta^{B}\overline{\phi}\end{split}\begin{split}\overline{K}&=\overline{\chi}^{+}+\eta^{A}\overline{\phi}_{A}-\frac{1}{2}\epsilon_{AB}\eta^{A}\eta^{B}\overline{\chi}^{-}\\\
V^{-}&=\phi+\eta^{A}\lambda^{-}_{A}-\frac{1}{2}\epsilon_{AB}\eta^{A}\eta^{B}v^{-}.\end{split}$

See Elvang:2011fx  for general explanation of notation. The multiplet $K$ is a
half-hypermultiplet and is usually paired with a conjugate multiplet of
antiparticles, $\overline{K}$. However, as both multiplets have identical
helicity structure, it will not be important here to continue to distinguish
between the two. The $\mathcal{N}=4$ vector is defined in Elvang:2013cua  and
will not be reproduced here.

All external states defining the superamplitudes will be taken to be outgoing.
This is the convention adopted in Elvang:2013cua . However, it will be
necessary to cross two of the states to be incoming. Crossing has been
discussed in the present context in Bellazzini:2016xrt  and will be performed
here on the component amplitudes. As mentioned in at the end of Section 2.1,
the ordering of the superfields in the correlator from which the amplitude is
derived is $\langle 0|4312|0\rangle$. This gives the order of the states and
the Grassmann variables in the superamplitude, which determines the order in
which the Grassmann derivatives should be applied to extract the components.
Note that the Feynman rules for external antifermion legs include a factor of
$-1$ that is frequently dropped Srednicki:2007qs , but is required here. This
implies that a single fermion leg must be accompanied by a factor of $-1$ when
crossed, in addition to the usual rules of reversing the momentum and
replacing the external polarisation.

###  6.2 Effective operators

It is of general interest to classify the effective contact interactions
combined together under various degrees of supersymmetry. Here these will be
systematically classified from dimension $5$ to dimension $8$ for operators
inducing contact interactions between three or four particles. Again, the
discussion will be restricted to helicities $h\leq 1$ (so no (super)gravity).
Different species of multiplets with the same superspin will not be
distinguished in order to emphasise the purely kinematical structure of the
allowed interactions, but no assumptions will be made about permutation
symmetries and internal quantum numbers (unrelated to the supersymmetry
algebra).

Most of the interactions considered here will be four-particle contact
interactions. These are severely constrained by consistency with dimensional
analysis, little group representation, Lorentz invariance, locality and
supersymmetry. The last condition is the requirement that the superamplitude
depend on the Grassmann variables exactly through $\delta^{(2)}(Q^{\dagger})$,
while the former conditions demand that the amplitudes be polynomials in
spinor bilinears with the required mass dimension and total helicity charge
for each leg.

While supersymmetry unifies interactions, it can also prohibit them. A common
reason for this is that the spectrum of effective interactions for higher spin
particles is sparser than for lower spin particles, so not all lower spin
interactions can be uniquely paired with a higher spin interaction.

####  6.2.1 Dimension $5$

An anomalous magnetic dipole moment (MDM)-like operator for matter fermions is
prohibited by supersymmetry (this is already prohibited by exchange
antisymmetry if the fermions are identical). However, the axion/dilaton
coupling is promoted to

$\displaystyle\mathcal{A}(\Phi^{+},V^{+},V^{+})\propto\frac{1}{\Lambda}\tilde{\delta}^{(1)}(Q)\left[23\bf\right],$
(141)

which also contains a mixed MDM-like interaction between the matter fermion
and the gaugino. This interaction can be further promoted to $\mathcal{N}=2$
in the superamplitude

$\displaystyle\mathcal{A}(V^{+},V^{+},V^{+})\propto\frac{1}{\Lambda}\tilde{\delta}^{(2)}(Q),$
(142)

which contains no further interactions. There are likewise conjugate
superamplitudes between the corresponding anti-multiplets. Notably, the
axion/dilaton cannot belong to a hypermultiplet.

The Weinberg operator (uniquely) supersymmetrises into itself:

$\displaystyle\mathcal{A}(\Phi^{+},\Phi^{+},\Phi^{+},\Phi^{+})\propto\frac{1}{\Lambda}\delta^{(2)}(Q^{\dagger})\frac{\left[12\bf\right]}{\left\langle
34\right\rangle}.$ (143)

The conjugate amplitude $\mathcal{A}(\Phi^{-}\Phi^{-}\Phi^{-}\Phi^{-})$ is
similar. Despite appearances, all spinor prefactors are equivalent. The
operator is altogether incompatible with $\mathcal{N}\geq 2$.

####  6.2.2 Dimension $6$

Cubic vector interactions are altogether prohibited by supersymmetry, so there
are no $3$-particle operators to consider.

Permissible $\mathcal{N}=1$ superamplitudes are

$\displaystyle\mathcal{A}(\Phi^{+},\Phi^{+},\Phi^{-},\Phi^{-})$
$\displaystyle\propto\frac{1}{\Lambda^{2}}\delta^{(2)}(Q^{\dagger})\left[12\bf\right]$
(144) $\displaystyle\mathcal{A}(V^{+},V^{+},\Phi^{+},\Phi^{+})$
$\displaystyle\propto\frac{1}{\Lambda^{2}}\delta^{(2)}(Q^{\dagger})\frac{\left[12\bf\right]^{2}}{\left\langle
34\right\rangle}$ (145)

and analogous conjugates. The first superamplitude combines dimension-$6$ four
fermion operators with scalar and mixed fermion-scalar operators. The
superamplitude must be helicity conserving, so helicity-violating matter
interactions are forbidden (such as those induced by the operators of the form
$\psi\psi\partial^{2}\psi\psi$). For the scalar interactions, “helicity
preserving” becomes charge preserving (each $\phi$ must be paired with a
$\phi^{\dagger}$ in the operator). The second superamplitude does allow for
helicity violation, provided that it involves a gaugino and a matter fermion.
It relates this to the bosonic operators of the form $F^{2}\phi^{2}$, where
the scalar must be charge-violating, as well as the MDM-like operator in
(141), dressed with an additional scalar.

Each of these superamplitudes may be respectively further enhanced to
$\mathcal{N}=2$,

$\displaystyle\mathcal{A}(K,K,K,K)\propto\frac{1}{\Lambda^{2}}\delta^{(4)}(Q^{\dagger})\frac{\left[12\bf\right]}{\left\langle
34\right\rangle}$ (146)
$\displaystyle\mathcal{A}(V^{+},V^{+},V^{+},V^{+})\propto\frac{1}{\Lambda^{2}}\delta^{(4)}(Q^{\dagger})\frac{\left[12\bf\right]^{2}}{\left\langle
34\right\rangle^{2}}$ (147)

(and conjugates), neither of which contains any new types of component
interactions. The chiral multiplets of (145) must descend from the vector
multiplets in (147), while the $\mathcal{N}=1$ matter interactions can only be
promoted to $\mathcal{N}=2$ matter interactions. Interestingly, the
$\mathcal{N}=2$ vectors still cannot couple to the hypermultiplets at this
order.

####  6.2.3 Dimension $7$

The $\mathcal{N}=1$ possibilities are

$\displaystyle\mathcal{A}(\Phi^{+},\Phi^{+},\Phi^{+},\Phi^{+})$
$\displaystyle\propto\frac{1}{\Lambda^{3}}\delta^{(2)}(Q^{\dagger})\\{\left[12\bf\right]\left[34\bf\right],\left[13\bf\right]\left[24\bf\right]\\}$
(148) $\displaystyle\mathcal{A}(V^{+},V^{+},\Phi^{-},\Phi^{-})$
$\displaystyle\propto\frac{1}{\Lambda^{3}}\delta^{(2)}(Q^{\dagger})\left[12\bf\right]^{2}$
(149)

The first superamplitude is the first example that admits multiple possible
independent terms, a particular basis for which is given inside the brackets.
These correspond to helicity-violating operators of the schematic form
$\psi\psi\partial^{2}\phi\phi$, where each term corresponds to a particular
distribution of the derivatives. Like the Weinberg operator, these
supersymmetrise into themselves. The second superamplitude describes operators
of the form $F^{2}\psi^{2}$ and its superpartners: the gaugino-scalar
coupling, similar to that in (148), but restricted to the term proportional to
the Mandelstam invariant of both scalars’ momenta, and a coupling of the
schematic form $F\lambda\psi^{\dagger}\cancel{\partial}\phi^{\dagger}$.

The only $\mathcal{N}=2$ possibility is

$\displaystyle\mathcal{A}(V^{+},V^{+},K,K)$
$\displaystyle\propto\frac{1}{\Lambda^{3}}\delta^{(4)}(Q^{\dagger})\frac{\left[12\bf\right]^{2}}{\left\langle
34\right\rangle}.$ (150)

This superamplitude unifies both of the $\mathcal{N}=1$ superamplitudes listed
above (although selecting-out only one of the terms (148) determined by which
chiral multiplets are embedded in the $\mathcal{N}=2$ vectors).

####  6.2.4 Dimension $8$

$\mathcal{N}=4$ compatible interactions become admissible at dimension $8$.
The only possible superamplitude consistent with dimensional analysis is

$\displaystyle\mathcal{A}(V,V,V,V)\propto\frac{1}{\Lambda^{4}}\delta^{(8)}(Q^{\dagger})\frac{\left[12\bf\right]^{2}}{\left\langle
34\right\rangle^{2}}.$ (151)

This is the supersymmetrisation of the helicity-preserving $F^{4}$ operator
and is (kinematically) unique.

For $\mathcal{N}<4$, more possibilities arise than for lower dimension, as
dimensional analysis permits more derivatives and therefore more ways that
they can be distributed, as well as new Lorentz-invariant combinations of
fermion chirality. However, supersymmetry still places stringent constraints
on the possible component interactions.

For $\mathcal{N}=1$, the possible superamplitudes are

$\displaystyle\mathcal{A}(\Phi^{+},\Phi^{+},\Phi^{-},\Phi^{-})$
$\displaystyle=\frac{1}{\Lambda^{4}}\delta^{(2)}(Q^{\dagger})\left[12\bf\right]\\{c_{\Phi^{4}s}s,c_{\Phi^{4}t}t\\}$
(152) $\displaystyle\mathcal{A}(V^{+},V^{-},\Phi^{+},\Phi^{-})$
$\displaystyle=\frac{c_{V^{2}\Phi^{2}}}{\Lambda^{4}}\delta^{(2)}(Q^{\dagger})\left[13\bf\right]\left[14\bf\right]\left\langle
24\right\rangle$ (153)
$\displaystyle\mathcal{A}(V^{+},V^{+},\Phi^{+},\Phi^{+})$
$\displaystyle=\frac{1}{\Lambda^{4}}\delta^{(2)}(Q^{\dagger})\left[12\bf\right]\\{d_{V^{2}\Phi^{2}s}\left[12\bf\right]\left[34\bf\right],d_{V^{2}\Phi^{2}t}\left[31\bf\right]\left[24\bf\right]\\}$
(154) $\displaystyle\mathcal{A}(V^{+},V^{+},V^{-},V^{-})$
$\displaystyle=\frac{c_{V^{4}}}{\Lambda^{4}}\delta^{(2)}(Q^{\dagger})\left[12\bf\right]^{2}\left\langle
34\right\rangle.$ (155)

The numerical Wilson coefficients are retained in these expressions to match
them with the operators to be given below.

Extending to $\mathcal{N}=2$, the permitted superamplitudes are

$\displaystyle\mathcal{A}(K,K,K,K)$
$\displaystyle\propto\frac{1}{\Lambda^{4}}\delta^{(4)}(Q^{\dagger})\\{\left[12\bf\right]\left[34\bf\right],\left[13\bf\right]\left[24\bf\right]\\}$
(156) $\displaystyle\mathcal{A}(V^{+},V^{-},K,K,)$
$\displaystyle\propto\frac{1}{\Lambda^{4}}\delta^{(4)}(Q^{\dagger})\left[13\bf\right]\left[14\bf\right]$
(157) $\displaystyle\mathcal{A}(V^{+},V^{+},V^{+},V^{+})$
$\displaystyle\propto\frac{1}{\Lambda^{4}}\delta^{(4)}(Q^{\dagger})\frac{\left[12\bf\right]}{\left\langle
34\right\rangle}\\{\left[12\bf\right]\left[34\bf\right],\left[13\bf\right]\left[24\bf\right]\\}$
(158) $\displaystyle\mathcal{A}(V^{+},V^{+},V^{-},V^{-})$
$\displaystyle\propto\frac{1}{\Lambda^{4}}\delta^{(4)}(Q^{\dagger})\left[12\bf\right]^{2}.$
(159)

These mostly just describe promotions of the respective $\mathcal{N}=1$
superamplitudes into $\mathcal{N}=2$. The superamplitudes (157) and (159) also
decompose into $\mathcal{N}=1$ components that include the $c_{\Phi^{4}s}$
term in (152).

Note that the $\mathcal{N}=1,2$ helicity-violating interactions (which have
been singled-out in (154) by having coupling labelled as $d$ rather than $c$),
which are the only examples of inelastic superamplitudes listed above, are
also the only type that do not appear when that $\mathcal{N}=4$ superamplitude
is decomposed into lower $\mathcal{N}$ components and are therefore not
$\mathcal{N}=4$ compatible.

For minimal field theories, it is interesting to consider the terms in an
effective action that would generate the superamplitudes listed above and
identify the Wilson coefficients united by supersymmetry. For a
$\mathcal{N}=1$ chiral multiplet, the dimension $8$ operators are

$\displaystyle\mathcal{L}_{EFT_{8}}$
$\displaystyle\propto\frac{c_{\Phi^{4}_{s}}}{\Lambda^{4}}\left(\frac{1}{4}\phi\phi(\partial^{2})^{2}(\phi^{*}\phi^{*})+\frac{1}{2}\psi\psi\partial^{2}(\psi^{\dagger}\psi^{\dagger})+2i\partial_{\mu}\phi^{\dagger}\partial^{\nu}\phi\partial_{\nu}\psi\sigma^{\mu}\psi^{\dagger}\right)$
(160)

(identical particle exchange symmetry rules-out the other possible operator
displayed above in (152)). For the vector multiplet,

$\displaystyle\mathcal{L}_{EFT_{8}}\propto\frac{c_{V^{4}}}{\Lambda^{4}}\left(\frac{1}{16}\left((F^{2})^{2}+(F\tilde{F})^{2}\right)+\frac{1}{2}\lambda\lambda\partial^{2}(\lambda^{\dagger}\lambda^{\dagger})+2i\lambda
F_{L}\sigma^{\mu}F_{R}\partial_{\mu}\lambda^{\dagger}\right).$ (161)

Here, $F_{L}=F_{\mu\nu}S_{L}^{\mu\nu}$ and $F_{R}=F_{\mu\nu}S_{R}^{\mu\nu}$.
The mixed interactions are:

$\displaystyle\mathcal{L}_{EFT_{8}}$
$\displaystyle\propto\frac{c_{V^{2}\Phi^{2}}}{\Lambda^{4}}\Big{(}2\text{tr}\left(F_{L}\sigma^{\mu}F_{R}\overline{\sigma}^{\nu}\right)\partial_{\mu}\phi^{\dagger}\partial_{\nu}\phi-\partial_{\mu}\psi\lambda\psi^{\dagger}\partial^{\mu}\lambda^{\dagger}$
$\displaystyle\qquad\qquad\qquad-2i\lambda^{\dagger}\overline{\sigma}^{\mu}\partial_{\nu}\lambda\partial_{\mu}\phi^{\dagger}\partial^{\nu}\phi+2i\psi
F_{L}\sigma^{\mu}F_{R}\partial_{\mu}\psi^{\dagger}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\sqrt{2}i\partial_{\mu}\psi\lambda
F^{\mu\nu}\partial_{\nu}\phi^{\dagger}+conj.\Big{)}$ (162)
$\displaystyle+\frac{d_{V^{2}\Phi^{2}s}}{\Lambda^{4}}\left(\frac{1}{8}\left(F^{2}+iF\tilde{F}\right)\partial^{2}\phi^{2}+\frac{1}{4}\psi\psi\partial^{2}\left(\lambda\lambda\right)+2\sqrt{2}i\partial_{\mu}\phi\lambda
F_{L}\partial^{\mu}\psi\right)+\text{conj}.$ (163)

The terms in the effective action above, or equivalently, the amplitudes that
they correspond to, can be compared to those listed in Section 5.4 for the two
fermion system. The notable differences are that supersymmetry forbids the
helicity-violating interactions involving only a single species $\psi$ or
$\lambda$ (i.e. $\psi^{2}\partial^{2}\psi^{2}$,
$\lambda^{2}\partial^{2}\lambda^{2}$ and their conjugates). For interactions
between two different species of fermion, helicity-violating interactions are
permitted. However, only the two interactions listed in (162) and (163) are
allowed. The others listed in (129) are forbidden by supersymmetry. Note that
$\partial\psi\lambda\cdot\partial\psi^{\dagger}\lambda^{\dagger}+\frac{1}{2}\psi\lambda\partial^{2}\left(\psi^{\dagger}\lambda^{\dagger}\right)=-\partial_{\mu}\psi\lambda\psi^{\dagger}\partial^{\mu}\lambda^{\dagger}$,
so supersymmetry only permits this single combination of mixed helicity
conserving interaction.

Before advancing on to the superymmetrised postivity constraints, I first
digress to make a comment on the application to supersymmetry breaking given
in Dine:2009sw . The general low-energy EFT of a goldstino and $R$-axion was
constructed in Dine:2009sw  to describe the breaking of $\mathcal{N}=1$
supersymmetry and its $R$-symmetry. This action included an interaction of the
form given by the mixed interaction in (160) (but with a real scalar), as well
as dimension $8$ helicity preserving and violating pure goldstino operators.
By demanding positivity of the mixed interaction, an upper bound on the vev of
the superpotential was derived from the product of the $R$-axion and goldstino
decay constants (all parameters determining the low energy constants in the
effective action). This same bound could be equivalently obtained by instead
applying the conclusions of Section 5.2 directly to the goldstino
interactions.

###  6.3 Unity of the positivity theorems and new bounds

To begin with, the positivity constraints on the scalar Adams:2006sv  and the
fermion Bellazzini:2016xrt  in (160) are unified when the fields are combined
in a supermultiplet. Note that these interactions are consistent with both a
Goldstone shift-symmetry for the scalar and a goldstino non-linear
supersymmetry for the fermion. These interactions are expected for a Goldstone
multiplet in a theory with extended SUSY spontaneously broken to
$\mathcal{N}=1$. Likewise, the interactions in (161) must each have positive
coefficients Bellazzini:2016xrt , which is consistent with their unification
under supersymmetry.

The mixed interactions are similar to those discussed in the previous Section
5.5. The first four terms of (162) induce elastic scattering of different
species off each other, so the positivity of the coefficient
$c_{V^{2}\Phi^{2}}>0$ is again expected. Interestingly however, the inelastic
partner operators of the form $\partial_{\mu}\psi\lambda
F^{\mu\nu}\partial_{\nu}\phi^{\dagger}$ seem to inherit this condition. It is
unclear how these operators would be constrained in the absence of
supersymmetry. Their on-shell contact amplitudes that vanish in the forward
limit in all channels, so a departure away from the forward scattering would
seem necessary to access them. This operator will remain a puzzle here.

Promoting to $\mathcal{N}=2$, the $V^{4}$ and $V^{2}\Phi^{2}$
(super)-operators unify further and the positivity of their Wilson
coefficients is combined into that of the ($\mathcal{N}=2$) $V^{4}$ operator.
Similarly, positivity of the $\mathcal{N}=1$ chiral multiplet interactions
(160) becomes positivity of the analogous $\mathcal{N}=2$ hypermultiplet
interactions. Promoting further to $\mathcal{N}=4$, all of these are unified
into the single positivity constraint on the dim-$8$ vector multiplet
interaction.

Finally, the new inelastic constraints derived in the previous section also
unify. With supersymmetry, the fermion constraints from Section 5.4 simplify,
as the amplitudes

$\displaystyle
A\left(\psi^{\pm}\psi^{\pm}\rightarrow\psi^{\mp}\psi^{\mp}\right)=A\left(\lambda^{\pm}\lambda^{\pm}\rightarrow\lambda^{\mp}\lambda^{\mp}\right)=A\left(\lambda^{\pm}\lambda^{\mp}\rightarrow\psi^{\pm}\psi^{\mp}\right)=A\left(\lambda^{\pm}\lambda^{\pm}\rightarrow\psi^{\pm}\psi^{\pm}\right)=0,$
(164)

while specifically for identical particles,
$M\left(\psi^{\pm}\lambda^{\pm}\rightarrow\psi^{\mp}\lambda^{\mp}\right)=0$.
This leaves only one type of inelastic amplitude and its parity conjugate and
the constraints can be stated as

$\displaystyle\frac{1}{2}|M^{\lambda^{-}\lambda^{-}\psi^{+}\psi^{+}}\pm
M^{\lambda^{+}\lambda^{+}\psi^{-}\psi^{-}}|<M^{\psi^{+}\lambda^{-}\psi^{+}\lambda^{-}}+\sqrt{M^{\psi^{+}\psi^{-}\psi^{+}\psi^{-}}M^{\lambda^{+}\lambda^{-}\lambda^{+}\lambda^{-}}}.$
(165)

The combinations appearing on the LHS correspond to the $P$ conserving and
violating interactions in the inelastic transitions (corresponding to, at
tree-level, the real and imaginary parts of the coupling $d_{V^{2}\Phi^{2}s}$
above). Notably, the amplitudes forbidden by the SWIs would all appear as
additional contributions to the left hand side of (165), strengthening the
lower bound on the elastic amplitudes.

The analysis of Section 5.5 can be easily extended to a complex scalar and two
fermion species. With the scalar complex, the bound (135) generalises to

$\displaystyle\frac{1}{2}|M^{\phi\phi\gamma^{+}\gamma^{+}}\pm
M^{\overline{\phi}\overline{\phi}\gamma^{-}\gamma^{-}}|<M^{\phi\gamma^{+}\phi\gamma^{+}}+\sqrt{M^{\phi\phi\phi\phi}M^{\gamma^{+}\gamma^{-}\gamma^{+}\gamma^{-}}}$
(166)

(after simplifying with $CPT$ and $Y$), which has the expected form resembling
(165). The last partner relation, involving the mixed fermion-boson
amplitudes, can also be found to be

$\displaystyle\frac{1}{2}|M^{\phi\psi^{-}\gamma^{+}\lambda^{+}}\pm
M^{\overline{\phi}\psi^{+}\gamma^{-}\lambda^{-}}|<\sqrt{M^{\phi\lambda^{+}\phi\lambda^{+}}M^{\psi^{+}\gamma^{-}\psi^{+}\gamma^{-}}}+\sqrt{M^{\phi\psi^{+}\phi\psi^{+}}M^{\lambda^{+}\gamma^{-}\lambda^{+}\gamma^{-}}}.$
(167)

SWIs imply that
$M^{\phi\lambda^{+}\phi\lambda^{+}}=M^{\psi^{+}\gamma^{-}\psi^{+}\gamma^{-}}$,
so this bound has identical structure to the previous two, completing the full
super-positivity constraint. Again, the inelastic components of (162) do not
appear in any of these bounds and seem only to be dragged into participation
by supersymmetry. For Wilson coefficients at tree-level, these bounds are
encapsulated by

$\displaystyle|\Re(d_{V^{2}\Phi^{2}s})|,|\Im(d_{V^{2}\Phi^{2}s})|<c_{V^{2}\Phi^{2}}+\sqrt{c_{\Phi^{4}s}c_{V^{4}}}.$
(168)

In contrast to the case from Section 5.2, the space of consistent
$P$-violating inelastic couplings is a square rather than a disc.

As mentioned above, the inelastic amplitudes are also the only type not
consistent with $\mathcal{N}=4$ supersymmetry. This indicates that the lower
bounds in these inequalities are minimised by requiring increasingly more
supersymmetry (where simple $\mathcal{N}=1$ is sufficient to rule-out many
possible inelastic amplitudes that may potentially appear, such as the other
fermionic helicity configurations in Section 5.4). This is the (expected)
consequence of symmetries imposing selection rules that prohibit inelastic
processes, but also illustrates how extreme symmetry breaking can be
prohibited by the positivity bounds.

##  7 Conclusion

The general implication of unitarity for the causality sum rule (8) is to
bound the size of inelastic scattering amplitudes by the size of the elastic
ones (11). This places fundamental limits on the extent to which hypothetical
symmetries, which manifest themselves in the $S$-matrix as selection rules,
can be broken by effective interactions. These constraints appear naively
invisible in the construction of a general effective action of local contact
interactions.

Employing the convex cone picture of Zhang:2020jyn , the general set of
positivity bounds for fundamental $SU(2)$ and $SU(3)$ fermions were derived,
including some that cannot be obtained from considering scattering of
factorised states. Separately, general constraints on flavour violation were
also derived for the cases in which the fermions are only non-singlets under
one symmetry group. Simple inelastic bounds for spinning particles were also
derived, in particular for two scalars and two vectors and the mixed case of
two fermions, a vector and a scalar, where each particle has the same-sign
helicity in the all outgoing convention. It was then shown that all of the
standard bounds for particles of different spin unify under supersymmetry.

In the examples discussed in 4.2, all states under consideration are related
by symmetries. When the particles transform under multiple symmetry groups,
the convex cone picture is needed for a complete characterisation of the
information in the sum rule. However, if transitions between multiple distinct
states are permitted by the symmetries, then the cone is non-polyhedral and
the standard results for polyhedral cones cannot be so simply applied. This
was analysed in Yamashita:2020gtt  for parity-symmetric weak boson operators,
where a set of necessary constraints were derived using analytic and numerical
methods. It is easy to apply (11) directly to obtain necessary conditions on
the couplings. However, finding the complete set of sufficient bounds remains
the most immediate open problem in applying the constraints from the sum rule
to EFTs of multiple species, in particular the SMEFT. Subsequent to the
release of this work, Li:2021cjv  were able to reformulate the the sum rule
(8) as a positive semi-definite statement in a dual space to the space of
external scattering states. Once appropriately crossing-symmetrised, a tensor
in external particle labels may be contracted with (8) to obtain an expression
of positive-definiteness. This dual space of positive-definite matrices was
identified as a “spectrahedron”, the geometry of which has been studied in
Ramana:1995xxx , and, as a space of positive-definite matrices, enabled
methods from semi-definite programming to be applied (see Simmons-
Duffin:2016gjk  for introduction of recent applications of these ideas to
solving the CFT bootstrap equations). See also Hebbar:2020ukp  for recent
progress in applying the $S$-matrix bootstrap to constraining Wilson
coefficients in EFTs directly from the full crossing constraints. This
leverages positive semi-definiteness of the $S$-matrix to utilise semi-
definite programming.

Generalisation to higher mass dimension, departure from the forward limit and
massive particles are obvious future directions deRham:2017zjm ,
Bellazzini:2020cot , Tolley:2020gtv , Caron-Huot:2020cmc . See e.g.
Green:2019tpt , Huang:2020nqy  for recent discussions in string theory.

The entire discussion of this work has been concentrated at the level of
dimension-$8$ order effective interactions. These have the minimum energy
scaling to ensure that the integral over the contour deformed to infinity
converges to zero and does not introduce an additional unknown UV ingredient
into the sum rule with an unknown impact on the structure. The results
described here at dimension $8$ readily extends to higher dimension $4n$ for
$n>2$. However, no statement about lower dimensional amplitudes has been made,
so it would seem that the remarks about symmetry violation would not extend to
them. However, lower dimensional interactions will typically contribute to
dimension-$8$ level interactions through multiple insertions and still appear
in the sum rule. The impact of loops in higher order constraints was recently
discussed in Bellazzini:2020cot . In agreement with the observations here,
these strengthen the positivity bounds that would naively apply at tree-level.
It would be of interest to further investigate the implications of these
constraints for RG flow, both in the context for the SMEFT and more generally.
As mentioned in Bellazzini:2020cot , this would require a treatment of IR
divergences and inclusive observables. Furthermore, as recent works have
shown, even operators appearing at higher dimension $4n+2$, for which the
standard sum rule does not imply positivity, are highly constrained away from
the forward limit. It would be interesting to establish precise points of
distinction between the rigid set of constraints applicable at dimension $8$
and above and freedom for lower dimensional (including renormalisable)
operators. Examination of UV completions may provide insight into this. Some
discussion of dimension $6$ operators has recently been given in
Remmen:2020uze , Gu:2020thj , Bellazzini:2014waa , Falkowski:2012vh , although
conclusions drawn for the IR interactions have been predicated on assumptions
about the amplitudes having extra-friendly high-energy scaling.

While most of the applications presented here have had an eye toward the
SMEFT, no analysis of the impacts of these constraints on tests of the SM has
been attempted here. See Bellazzini:2018paj , Gu:2020ldn , Bi:2019phv ,
Zhang:2018shp , Fuks:2020ujk  for some recent discussion of this. The bounds
discussed here activate at dimension $8$ order, which is expected to be
typically sub-leading to the (many) dimension $6$ interactions that pervade
the SMEFT. However, it may still be possible to access the affected dimension
$8$ interactions through non-interference effects and angular distributions
Alioli:2020kez , Azatov:2016sqh . While amplitudes vanishing in the forward
limit (possibly because of angular momentum conservation) appear naively
unaffected, this is not necessarily true in a crossed, inelastic channel
described by the same amplitude, and as result, such a process would not
escape constraint.

This entire work has relied upon the $S$-matrix formulation of causality in
order to derive constraints on effective interactions. However, it would also
be interesting to construct background solutions (such as was done in
Adams:2006sv ) in order to see precisely how such a breakdown arises if the
constraints are violated, especially for the fermionic and mixed spin
interactions. The validity and scope of the $S$-matrix formulation is also
dependent upon the analytic structure being established and the standard
derivation of the dispersion relations in Section 2.1 relied upon this. These
have been established for Wightman theories (at least for the analogous
correlation functions) and theories loosely satisfying the requirements of LSZ
reduction. However, this does not include applicability to perturbative gauge
theories. Clarification over this issue would be informative. Some possibly
related foundational questions that have practical implications are the
interpretation and use of the sum rule in the presence of IR divergences, as
well as possible extension away from the forward limit where some
understanding of the singularity structure may become necessary. The
assumption that all particles can be given a small mass and that the results
will apply to the exactly massless theory also needs to be validated (in
particular, the assumption that the forward and massless limits commute),
although no effort has been made in this direction here. This is least clear
for possible applications to gravitational systems, where other foundational
assumptions about locality in the UV completion are also uncertain.

It would also be interesting to find applications of these bounds to model
building. As mentioned above, the fact that dimension $8$ operators usually
contribute subleading effects naively poses an obstruction to the constraints
having widespread, leading-order consequences. Continuing the analysis of
Bellazzini:2018paj  for constraining massive higher spin particles and their
hypothetical coupling to the SM is another possible application with direct
consequence for the constraining the space of possible particle models of dark
matter or other hypothetically fields associated with other cosmological
mysteries. See e.g. Falkowski:2020mjq  for discussion of coupling of massive
gravitons to (regular) matter, or deRham:2018qqo  for various other theories
related to modified gravity. Alternatively, it would be of interest if these
results can be used to make general statements about the nature and scope of
symmetry breaking (such as of time-reversal) that is possible at low energies.

###### Acknowledgements.

Thank you to Grant Remmen for discussion, Isabel Garcia Garcia for comments on
a draft, Nathaniel Craig for assistance with funding and Cen Zhang for some
questions and comments. This work is supported by the US Department of Energy
under the grant DE-SC0011702.

##  Appendix A Sum Rules for Spinning Particles with $SO(2)$

###  A.1 Angular momentum projectors in $SO(2)$ form

The Clebsch-Gordan coefficients for general spinning particles were given in
(96). If one of the helicities is zero, then the Clebsch-Gordan coefficients
are

$\displaystyle C^{i0}_{h}=C^{0i}_{h}=\frac{1}{\sqrt{2}}\begin{cases}1,\qquad
i=1\\\ -i,\qquad i=2\end{cases}.$ (169)

The $0$ superscript denotes the scalar state. The parity-conjugate
coefficients are $C^{i0}_{-h}=C^{0i}_{-h}=(C^{i0}_{h})^{*}$.

The next step is to find the projectors. There are several special cases. For
the case in which one of the incoming and outgoing particles are scalars, the
projectors are simply

$\displaystyle P_{Ph}^{i0k0}$ $\displaystyle=\delta^{ik}\qquad
P_{Ph}^{0j0l}=\delta^{jl}\qquad P_{Ph}^{i00l}=\delta^{il}\qquad
P_{Ph}^{0jk0}=\delta^{jk}$ (170) $\displaystyle P_{\cancel{P}h}^{i0k0}$
$\displaystyle=i\epsilon^{ik}\qquad
P_{\cancel{P}h}^{0j0l}=i\epsilon^{jl}\qquad
P_{\cancel{P}h}^{i00l}=i\epsilon^{il}\qquad
P_{\cancel{P}h}^{0jk0}=i\epsilon^{jk}.$ (171)

More generally, if none of the particles are scalars and $h_{1}\neq h_{2}$,
$h_{3}\neq h_{4}$ (so that singlets do not appear in the product irreps), then
the possible projectors are

$\displaystyle P_{\cancel{P}\,h_{1}+h_{2}}^{ijkl}$
$\displaystyle=\frac{i}{2}\left(\delta^{ik}\epsilon^{jl}+\delta^{jl}\epsilon^{ik}\right)$
(172) $\displaystyle P_{P\,h_{1}+h_{2}}^{ijkl}$
$\displaystyle=\frac{1}{2}\left(\delta^{ik}\delta^{jl}+\delta^{il}\delta^{jk}-\delta^{ij}\delta^{kl}\right),\qquad$
$\displaystyle\text{if}\,h_{1}+h_{2}=h_{3}+h_{4},$ (173) $\displaystyle
P_{\cancel{P}\,|h_{1}-h_{2}|}^{ijkl}$
$\displaystyle=\frac{i}{2}\left(\epsilon^{ij}\delta^{kl}-\delta^{ij}\epsilon^{kl}\right)$
(174) $\displaystyle P_{P\,|h_{1}-h_{2}|}^{ijkl}$
$\displaystyle=\frac{1}{2}\left(\delta^{ij}\delta^{kl}+\delta^{ik}\delta^{jl}-\delta^{il}\delta^{jk}\right),\qquad$
$\displaystyle\text{if}\,\left(h_{1}-h_{2}\right)=\left(h_{3}-h_{4}\right),$
(175) $\displaystyle P_{\cancel{P}\,|h_{1}-h_{2}|}^{ijkl}$
$\displaystyle=\frac{i}{2}\left(\delta^{ij}\epsilon^{kl}+\epsilon^{ij}\delta^{kl}\right)$
(176) $\displaystyle P_{P\,|h_{1}-h_{2}|}^{ijkl}$
$\displaystyle=\frac{1}{2}\left(\delta^{ij}\delta^{kl}-\delta^{ik}\delta^{jl}+\delta^{il}\delta^{jk}\right),\qquad$
$\displaystyle\text{if}\,\left(h_{1}-h_{2}\right)=-\left(h_{3}-h_{4}\right),$
(177) $\displaystyle P_{\cancel{P}\,h_{1}+h_{2}}^{ijkl}$
$\displaystyle=\frac{-i}{2}\left(S^{ij}\delta^{kl}\pm
P^{ij}\epsilon^{kl}\right)$ (178) $\displaystyle P_{P\,h_{1}+h_{2}}^{ijkl}$
$\displaystyle=\frac{1}{2}\left(P^{ij}\delta^{kl}\mp
S^{ij}\epsilon^{kl}\right),\qquad$
$\displaystyle\text{if}\,h_{1}+h_{2}=\pm\left(h_{3}-h_{4}\right),$ (179)
$\displaystyle P_{\cancel{P}\,h_{3}+h_{4}}^{ijkl}$
$\displaystyle=\frac{i}{2}\left(\delta^{ij}S^{kl}\pm\epsilon^{ij}P^{kl}\right)$
(180) $\displaystyle P_{P\,h_{3}+h_{4}}^{ijkl}$
$\displaystyle=\frac{1}{2}\left(\delta^{ij}P^{kl}\pm\epsilon^{ij}S^{kl}\right),\qquad$
$\displaystyle\text{if}\,\pm\left(h_{1}-h_{2}\right)=h_{3}+h_{4}.$ (181)

The $u$-channel projectors all agree with the required $s$-channel projectors
that they should be related to under crossing. For example, if
$h_{1}+h_{2}=h_{3}+h_{4}$, it is easily verified that
$P_{{h_{1}-h_{4}}}^{ilkj}=P_{{h_{1}+h_{2}}}^{ijkl}$ (in either parity-
symmetric or violating cases). The identity

$\displaystyle\delta^{ij}\epsilon^{kl}+\delta^{kl}\epsilon^{ij}=\delta^{il}\epsilon^{kj}+\delta^{kl}\epsilon^{il}$
(182)

is useful in handling the parity-violating projectors.

For the special case in which exactly one of the particles is a scalar, the
projectors are

$\displaystyle P_{\cancel{P}\,h_{3}+h_{4}}^{0jkl}$
$\displaystyle=\frac{i}{\sqrt{2}}\left(\delta^{j1}S^{kl}-\delta^{j2}P^{kl}\right)$
(183) $\displaystyle P_{P\,h_{3}+h_{4}}^{0jkl}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\delta^{j1}P^{kl}+\delta^{j2}S^{kl}\right)$
(184) $\displaystyle P_{\cancel{P}\,h_{3}-h_{4}}^{0jkl}$
$\displaystyle=\frac{-i}{\sqrt{2}}\left(\delta^{j1}\epsilon^{kl}+\delta^{j2}\delta^{kl}\right)$
(185) $\displaystyle P_{P\,h_{3}-h_{4}}^{0jkl}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\delta^{j1}\delta^{kl}-\delta^{j2}\epsilon^{kl}\right)$
(186) $\displaystyle P_{\cancel{P}\,-h_{3}+h_{4}}^{0jkl}$
$\displaystyle=\frac{i}{\sqrt{2}}\left(\delta^{j1}\epsilon^{kl}-\delta^{j2}\delta^{kl}\right)$
(187) $\displaystyle P_{P\,-h_{3}+h_{4}}^{0jkl}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\delta^{j1}\delta^{kl}+\delta^{j2}\epsilon^{kl}\right).$
(188)

Then $P^{i0kl}=P^{0ikl}$ and $P^{ij0l}=P^{ijl0}=(P^{l0ij})^{*}$ for each
helicity configuration. These expressions encapsulate each of the possible
relations between the helicities of the scattered particles. Again, it can be
easily checked that these are consistent with crossing e.g. for the case
$h_{3}=0$ and $h_{1}+h_{2}=h_{4}$,
$P_{P\,h_{4}-h_{1}}^{il0j}=P_{P\,h_{1}+h_{2}}^{ij0l}$, while the parity-
violating projectors pick-up a negative sign
$P_{\cancel{P}\,h_{4}-h_{1}}^{il0j}=-P_{\cancel{P}\,h_{1}+h_{2}}^{ij0l}$.

For the special case in which the outgoing states are both scalars, the
projectors are simply the Clebsch-Gordan coefficients (or if the scalars are
incoming, their conjugates)

$\displaystyle P^{ij00}_{A}=\frac{1}{\sqrt{2}}\delta^{ij}\qquad
P^{00kl}_{A}=\frac{1}{\sqrt{2}}\delta^{kl}\qquad
P^{ij00}_{B}=\frac{-i}{\sqrt{2}}\epsilon^{ij}\qquad
P^{00kl}_{B}=\frac{i}{\sqrt{2}}\epsilon^{kl}.$ (189)

These have crossing relations

$\displaystyle P^{0lk0}_{Ph}=\sqrt{2}P_{A}^{00kl}\qquad\qquad
P^{0lk0}_{\cancel{P}h}=-\sqrt{2}P_{B}^{00kl}.$ (190)

When the particles are not scalars, then if $h_{1}=h_{4}$ and $h_{2}=h_{3}$,
the $u$-channel projectors decompose as

$\displaystyle P_{\cancel{P}\,h_{1}+h_{2}}^{ilkj}$
$\displaystyle=-P_{AB}^{ijkl}-P_{BA}^{ijkl}$ (191) $\displaystyle
P_{P\,h_{1}+h_{2}}^{ilkj}$ $\displaystyle=P_{AA}^{ijkl}+P_{BB}^{ijkl}$ (192)
$\displaystyle P_{\cancel{P}\,h_{1}-h_{2}}^{ilkj}$
$\displaystyle=P_{AB}^{ijkl}-P_{BA}^{ijkl}$ (193) $\displaystyle
P_{P\,h_{1}-h_{2}}^{ilkj}$ $\displaystyle=P_{AA}^{ijkl}-P_{BB}^{ijkl},$ (194)

while in the opposite channel,

$\displaystyle P_{AA}^{ilkj}$
$\displaystyle=\frac{1}{2}\left(P_{P\,h_{1}+h_{2}}^{ijkl}+P_{P\,h_{1}-h_{2}}^{ijkl}\right)$
(195) $\displaystyle P_{BB}^{ilkj}$
$\displaystyle=\frac{1}{2}\left(P_{P\,h_{1}+h_{2}}^{ijkl}-P_{P\,h_{1}-h_{2}}^{ijkl}\right)$
(196) $\displaystyle P_{AB}^{ilkj}$
$\displaystyle=\frac{1}{2}\left(P_{\cancel{P}\,h_{1}-h_{2}}^{ijkl}-P_{\cancel{P}\,h_{1}+h_{2}}^{ijkl}\right)$
(197) $\displaystyle P_{BA}^{ilkj}$
$\displaystyle=-\frac{1}{2}\left(P_{\cancel{P}\,h_{1}-h_{2}}^{ijkl}+P_{\cancel{P}\,h_{1}+h_{2}}^{ijkl}\right).$
(198)

The results given in Section 5.1 also apply to the case $h_{1}=h_{2}=h\neq
h_{3}=h_{4}$, except that only the transitions between singlet irreps are
possible.

###  A.2 Sum rules for two fermion and multispin theories

The sum rule entries for scattering in the two fermion theory of Section 5.4
are, in $SO(2)$ form:

$\displaystyle M\left(\lambda,\lambda\rightarrow\lambda,\lambda\right)$
$\displaystyle=\left(|\boldsymbol{m}_{1}^{\lambda\lambda}|^{2}+\frac{1}{2}\left(|\boldsymbol{m}_{A}^{\lambda\lambda}|^{2}+|\boldsymbol{m}_{B}^{\lambda\lambda}|^{2}\right)\right)P_{P1}^{ijkl}+\frac{1}{2}\left(2|\boldsymbol{m}_{1}^{\lambda\lambda}|^{2}+3|\boldsymbol{m}_{A}^{\lambda\lambda}|^{2}-|\boldsymbol{m}_{B}^{\lambda\lambda}|^{2}\right)P_{AA}^{ijkl}$
$\displaystyle\quad+\frac{1}{2}\left(2|\boldsymbol{m}_{1}^{\lambda\lambda}|^{2}-|\boldsymbol{m}_{A}^{\lambda\lambda}|^{2}+3|\boldsymbol{m}_{B}^{\lambda\lambda}|^{2}\right)P_{BB}^{ijkl}+2\boldsymbol{m}_{A}^{\lambda\lambda}\cdot\boldsymbol{m}_{B}^{\lambda\lambda}\left(P_{AB}^{ijkl}-P_{BA}^{ijkl}\right)$
(199) $\displaystyle M\left(\psi,\psi\rightarrow\psi,\psi\right)$
$\displaystyle=\left(|\boldsymbol{m}_{1}^{\psi\psi}|^{2}+\frac{1}{2}\left(|\boldsymbol{m}_{A}^{\psi\psi}|^{2}+|\boldsymbol{m}_{B}^{\psi\psi}|^{2}\right)\right)P_{P1}^{ijkl}+\frac{1}{2}\left(2|\boldsymbol{m}_{1}^{\psi\psi}|^{2}+3|\boldsymbol{m}_{A}^{\psi\psi}|^{2}-|\boldsymbol{m}_{B}^{\psi\psi}|^{2}\right)P_{AA}^{ijkl}$
$\displaystyle\quad+\frac{1}{2}\left(2|\boldsymbol{m}_{1}^{\psi\psi}|^{2}-|\boldsymbol{m}_{A}^{\psi\psi}|^{2}+3|\boldsymbol{m}_{B}^{\psi\psi}|^{2}\right)P_{BB}^{ijkl}+2\boldsymbol{m}_{A}^{\psi\psi}\cdot\boldsymbol{m}_{B}^{\psi\psi}\left(P_{AB}^{ijkl}-P_{BA}^{ijkl}\right)$
(200) $\displaystyle M\left(\psi,\lambda\rightarrow\psi,\lambda\right)$
$\displaystyle=\left(|\boldsymbol{m}_{1}^{\psi\lambda}|^{2}+\frac{1}{2}\left(|\boldsymbol{m}_{A}^{\psi\lambda}|^{2}+|\boldsymbol{m}_{B}^{\psi\lambda}|^{2}\right)\right)P_{P1}^{ijkl}+\frac{1}{2}\left(2|\boldsymbol{m}_{1}^{\psi\lambda}|^{2}+3|\boldsymbol{m}_{B}^{\psi\lambda}|^{2}-|\boldsymbol{m}_{A}^{\psi\lambda}|^{2}\right)P_{AA}^{ijkl}$
$\displaystyle\quad+\frac{1}{2}\left(2|\boldsymbol{m}_{1}^{\psi\lambda}|^{2}-|\boldsymbol{m}_{B}^{\psi\lambda}|^{2}+3|\boldsymbol{\lambda}|^{2}\right)P_{BB}^{ijkl}+2\boldsymbol{m}_{A}^{\psi\lambda}\cdot\boldsymbol{m}_{B}^{\psi\lambda}\left(P_{AB}^{ijkl}-P_{BA}^{ijkl}\right)$
(201) $\displaystyle M\left(\lambda,\psi\rightarrow\lambda,\psi\right)$
$\displaystyle=\left(|\boldsymbol{m}_{1}^{\lambda\psi}|^{2}+\frac{1}{2}\left(|\boldsymbol{m}_{A}^{\lambda\psi}|^{2}+|\boldsymbol{m}_{B}^{\lambda\psi}|^{2}\right)\right)P_{P1}^{ijkl}+\frac{1}{2}\left(2|\boldsymbol{m}_{1}^{\lambda\psi}|^{2}+3|\boldsymbol{m}_{B}^{\lambda\psi}|^{2}-|\boldsymbol{m}_{A}^{\lambda\psi}|^{2}\right)P_{AA}^{ijkl}$
$\displaystyle\quad+\frac{1}{2}\left(2|\boldsymbol{m}_{1}^{\lambda\psi}|^{2}-|\boldsymbol{m}_{B}^{\lambda\psi}|^{2}+3|\boldsymbol{m}_{A}^{\lambda\psi}|^{2}\right)P_{BB}^{ijkl}+2\boldsymbol{m}_{A}^{\lambda\psi}\cdot\boldsymbol{m}_{B}^{\lambda\psi}\left(P_{AB}^{ijkl}-P_{BA}^{ijkl}\right)$
(202) $\displaystyle M\left(\lambda,\lambda\rightarrow\psi,\psi\right)$
$\displaystyle=\frac{1}{2}\left(\boldsymbol{m}_{1}^{\psi\psi}\cdot\boldsymbol{m}_{1}^{\lambda\lambda}+\boldsymbol{m}_{-1}^{\psi\psi}\cdot\boldsymbol{m}_{-1}^{\lambda\lambda}+\boldsymbol{m}_{B}^{\psi\lambda}\cdot\boldsymbol{m}_{B}^{\lambda\psi}+\boldsymbol{m}_{A}^{\psi\lambda}\cdot\boldsymbol{m}_{A}^{\lambda\psi}\right)P_{P1}^{ijkl}$
$\displaystyle\quad+\frac{1}{2}\left(2\boldsymbol{m}_{A}^{\psi\psi}\cdot\boldsymbol{m}_{A}^{\lambda\lambda}+\boldsymbol{m}_{1}^{\psi\lambda}\cdot\boldsymbol{m}_{1}^{\lambda\psi}+\boldsymbol{m}_{-1}^{\psi\lambda}\cdot\boldsymbol{m}_{-1}^{\lambda\psi}+\boldsymbol{m}_{B}^{\psi\lambda}\cdot\boldsymbol{m}_{B}^{\lambda\psi}-\boldsymbol{m}_{A}^{\psi\lambda}\cdot\boldsymbol{m}_{A}^{\lambda\psi}\right)P_{AA}^{ijkl}$
$\displaystyle\quad+\frac{1}{2}\left(2\boldsymbol{m}_{B}^{\psi\psi}\cdot\boldsymbol{m}_{B}^{\lambda\lambda}+\boldsymbol{m}_{1}^{\psi\lambda}\cdot\boldsymbol{m}_{1}^{\lambda\psi}+\boldsymbol{m}_{-1}^{\psi\lambda}\cdot\boldsymbol{m}_{-1}^{\lambda\psi}-\boldsymbol{m}_{B}^{\psi\lambda}\cdot\boldsymbol{m}_{B}^{\lambda\psi}+\boldsymbol{m}_{A}^{\psi\lambda}\cdot\boldsymbol{m}_{A}^{\lambda\psi}\right)P_{BB}^{ijkl}$
$\displaystyle\quad+\frac{1}{2}\left(2\boldsymbol{m}_{A}^{\psi\psi}\cdot\boldsymbol{m}_{B}^{\lambda\lambda}-\boldsymbol{m}_{1}^{\psi\lambda}\cdot\boldsymbol{m}_{1}^{\lambda\psi}+\boldsymbol{m}_{-1}^{\psi\lambda}\cdot\boldsymbol{m}_{-1}^{\lambda\psi}+\boldsymbol{m}_{A}^{\psi\lambda}\cdot\boldsymbol{m}_{B}^{\lambda\psi}-\boldsymbol{m}_{B}^{\psi\lambda}\cdot\boldsymbol{m}_{A}^{\lambda\psi}\right)P_{AB}^{ijkl}$
$\displaystyle\quad+\frac{1}{2}\left(2\boldsymbol{m}_{B}^{\psi\psi}\cdot\boldsymbol{m}_{A}^{\lambda\lambda}-\boldsymbol{m}_{1}^{\psi\lambda}\cdot\boldsymbol{m}_{1}^{\lambda\psi}+\boldsymbol{m}_{-1}^{\psi\lambda}\cdot\boldsymbol{m}_{-1}^{\lambda\psi}-\boldsymbol{m}_{A}^{\psi\lambda}\cdot\boldsymbol{m}_{B}^{\lambda\psi}+\boldsymbol{m}_{B}^{\psi\lambda}\cdot\boldsymbol{m}_{A}^{\lambda\psi}\right)P_{BA}^{ijkl}.$
(203)

Here $\boldsymbol{m}_{-1}^{ff^{\prime}}$ and
$\boldsymbol{m}_{1}^{ff^{\prime}}$ for example represent the UV couplings of
the $f^{+}f^{\prime-}$ and $f^{-}f^{\prime+}$ helicity configurations
respectively for any fermions $f$ and $f^{\prime}$. For brevity, the amplitude
$M\left(\lambda,\psi\rightarrow\psi,\lambda\right)$ has not been stated as it
is entirely determined from crossing
$M\left(\lambda,\lambda\rightarrow\psi,\psi\right)$. The $Y$-symmetry implies
that
$|\boldsymbol{m}_{-1}^{\lambda\lambda}|=|\boldsymbol{m}_{1}^{\lambda\lambda}|$,
$|\boldsymbol{m}_{-1}^{\psi\psi}|=|\boldsymbol{m}_{1}^{\psi\psi}|$,
$|\boldsymbol{m}_{B}^{\lambda\psi}|=|\boldsymbol{m}_{B}^{\psi\lambda}|$,
$|\boldsymbol{m}_{A}^{\lambda\psi}|=|\boldsymbol{m}_{A}^{\psi\lambda}|$,
$|\boldsymbol{m}_{1}^{\psi\lambda}|=|\boldsymbol{m}_{-1}^{\psi\lambda}|=|\boldsymbol{m}_{1}^{\lambda\psi}|=|\boldsymbol{m}_{-1}^{\lambda\psi}|$,
which has been used to (partially) simplify the elastic amplitudes. As for the
example of Section 5.2, fundamental principles rule-out the existence of the
parity-violating, spinning component amplitudes in all configurations above
(although it is present in the omitted
$M\left(\lambda,\psi\rightarrow\psi,\lambda\right)$).

The sum rule entries for the simple multispin theory of Section 5.5 in $SO(2)$
form are given next. The terms can be classified by spin projection
$m_{z}=0,\frac{1}{2},1,\frac{3}{2},2$. They are given in Table 4.

$m_{z}=0$ |  $\phi\phi$ |  $\psi\psi$ |  $\gamma\gamma$
---|---|---|---
$\phi\phi$ |  $2|\boldsymbol{m}^{\phi\phi}|^{2}$ |  0 |  $\left(\boldsymbol{m}_{A}^{\gamma\gamma}\cdot\boldsymbol{m}^{\phi\phi}+\sqrt{2}\boldsymbol{m}_{1}^{\phi\gamma}\cdot\boldsymbol{m}_{1}^{\gamma\phi}+\sqrt{2}\boldsymbol{m}_{1}^{\gamma\phi}\cdot\boldsymbol{m}_{1}^{\phi\gamma}\right)P_{A}+\left(\boldsymbol{m}_{B}^{\gamma\gamma}\cdot\boldsymbol{m}^{\phi\phi}-\sqrt{2}\boldsymbol{m}_{1}^{\phi\gamma}\cdot\boldsymbol{m}_{1}^{\gamma\phi}+\sqrt{2}\boldsymbol{m}_{1}^{\gamma\phi}\cdot\boldsymbol{m}_{1}^{\phi\gamma}\right)P_{B}$
$\psi\psi$ |  . |  $\frac{1}{2}\left(2|\boldsymbol{m}_{1}^{\psi\psi}|^{2}+3|\boldsymbol{m}_{A}^{\psi\psi}|^{2}-|\boldsymbol{m}_{B}^{\psi\psi}|^{2}\right)P_{AA}+\frac{1}{2}\left(2|\boldsymbol{m}_{1}^{\psi\psi}|^{2}-|\boldsymbol{m}_{A}^{\psi\psi}|^{2}+3|\boldsymbol{m}_{B}^{\psi\psi}|^{2}\right)P_{BB}+2\boldsymbol{m}_{A}^{\psi\psi}\cdot\boldsymbol{m}_{B}^{\psi\psi}\left(P_{AB}-P_{BA}\right)$ |  0
$\gamma\gamma$ |  . |  . |  $\frac{1}{2}\left(2|\boldsymbol{m}_{2}^{\gamma\gamma}|^{2}+3|\boldsymbol{m}_{A}^{\gamma\gamma}|^{2}-|\boldsymbol{m}_{B}^{\gamma\gamma}|^{2}\right)P_{AA}+\frac{1}{2}\left(2|\boldsymbol{m}_{2}^{\gamma\gamma}|^{2}-|\boldsymbol{m}_{A}^{\gamma\gamma}|^{2}+3|\boldsymbol{m}_{B}^{\gamma\gamma}|^{2}\right)P_{BB}+2\boldsymbol{m}_{A}^{\gamma\gamma}\cdot\boldsymbol{m}_{B}^{\gamma\gamma}\left(P_{AB}-P_{BA}\right)$

$m_{z}=\frac{1}{2}$ |  $\phi\psi$ |  $\psi\phi$ |  $\psi\gamma$ |  $\gamma\psi$
---|---|---|---|---
$\phi\psi$ |  $|\boldsymbol{m}_{\frac{1}{2}}^{\phi\psi}|^{2}P_{P\frac{1}{2}}$ |  0 |  0 |  $\left(\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}\cdot\boldsymbol{m}_{\frac{1}{2}}^{\phi\psi}+\boldsymbol{m}_{\frac{1}{2}}^{\phi\psi}\cdot\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}\right)P_{P\frac{1}{2}}+\left(\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}\cdot\boldsymbol{m}_{\frac{1}{2}}^{\phi\psi}-\boldsymbol{m}_{\frac{1}{2}}^{\phi\psi}\cdot\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}\right)P_{\cancel{P}\frac{1}{2}}$
$\psi\phi$ |  . |  $|\boldsymbol{m}_{\frac{1}{2}}^{\phi\psi}|^{2}P_{P\frac{1}{2}}$ |  $\left(\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}\cdot\boldsymbol{m}_{\frac{1}{2}}^{\phi\psi}+\boldsymbol{m}_{\frac{1}{2}}^{\phi\psi}\cdot\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}\right)P_{P\frac{1}{2}}-\left(\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}\cdot\boldsymbol{m}_{\frac{1}{2}}^{\phi\psi}-\boldsymbol{m}_{\frac{1}{2}}^{\phi\psi}\cdot\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}\right)P_{\cancel{P}\frac{1}{2}}$ |  0
$\psi\gamma$ |  . |  . |  $\left(|\boldsymbol{m}_{\frac{3}{2}}^{\psi\gamma}|^{2}+|\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}|^{2}\right)P_{P\frac{1}{2}}$ |  0
$\gamma\psi$ |  . |  . |  . |  $\left(|\boldsymbol{m}_{\frac{3}{2}}^{\psi\gamma}|^{2}+|\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}|^{2}\right)P_{P\frac{1}{2}}$

$m_{z}=1$ |  $\psi\psi$ |  $\phi\gamma$ |  $\gamma\phi$
---|---|---|---
$\psi\psi$ |  $\left(|\boldsymbol{m}_{1}^{\psi\psi}|^{2}+\frac{1}{2}\left(|\boldsymbol{m}_{A}^{\psi\psi}|^{2}+|\boldsymbol{m}_{B}^{\psi\psi}|^{2}\right)\right)P_{P1}$ |  0 |  0
$\phi\gamma$ |  . |  $|\boldsymbol{m}_{1}^{\phi\gamma}|^{2}P_{P1}$ |  $\left(\boldsymbol{m}_{1}^{\gamma\phi}\cdot\boldsymbol{m}_{1}^{\phi\gamma}+\boldsymbol{m}_{1}^{\phi\gamma}\cdot\boldsymbol{m}_{1}^{\gamma\phi}+\frac{1}{\sqrt{2}}\boldsymbol{m}_{A}^{\gamma\gamma}\cdot\boldsymbol{m}^{\phi\phi}\right)P_{P1}+\left(\boldsymbol{m}_{1}^{\gamma\phi}\cdot\boldsymbol{m}_{1}^{\phi\gamma}-\boldsymbol{m}_{1}^{\phi\gamma}\cdot\boldsymbol{m}_{1}^{\gamma\phi}-\frac{1}{\sqrt{2}}\boldsymbol{m}_{B}^{\gamma\gamma}\cdot\boldsymbol{m}^{\phi\phi}\right)P_{\cancel{P}1}$
$\gamma\phi$ |  . |  . |  $|\boldsymbol{m}_{1}^{\gamma\phi}|^{2}P_{P1}$

$m_{z}=\frac{3}{2}$ |  $\psi\gamma$ |  $\gamma\psi$
---|---|---
$\psi\gamma$ |  $\left(|\boldsymbol{m}_{\frac{3}{2}}^{\psi\gamma}|^{2}+|\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}|^{2}\right)P_{P\frac{3}{2}}$ |  0
$\gamma\psi$ |  . |  $\left(|\boldsymbol{m}_{\frac{3}{2}}^{\psi\gamma}|^{2}+|\boldsymbol{m}_{\frac{1}{2}}^{\psi\gamma}|^{2}\right)P_{P\frac{3}{2}}$

$m_{z}=2$ |  $\gamma\gamma$
---|---
$\gamma\gamma$ |  $\left(|\boldsymbol{m}_{2}^{\gamma\gamma}|^{2}+\frac{1}{2}\left(|\boldsymbol{m}_{A}^{\gamma\gamma}|^{2}+|\boldsymbol{m}_{B}^{\gamma\gamma}|^{2}\right)\right)P_{P2}$

Table 4:  Sum rule for theory of spinning particles.

The vanishing entries correspond to amplitudes that do not have dimension $8$
order contributions. Amplitudes with the required mass dimension cannot be
constructed consistently respecting Lorentz invariance and possessing the
required little group scaling. Many of the processes, most notably the elastic
ones, are also accidentally parity symmetric as a consequence of $CPT$ (and
the fact that the particles are assumed to be self-conjugate in this theory).
Both $Y$ and $CPT$ can be invoked to simplify the entries. The $Y$ symmetry
further relates
$|\boldsymbol{m}_{1}^{\phi\gamma}|=|\boldsymbol{m}_{1}^{\gamma\phi}|$. The
couplings $\boldsymbol{m}_{2}^{\gamma\gamma}$, $\boldsymbol{m}_{1}^{\psi\psi}$
and $\boldsymbol{m}_{\frac{3}{2}}^{\psi\gamma}$ are redundant.

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