CERN-TH-2022-105

Spinning Sum Rules for the
Dimension-Six SMEFT

Grant N. Remmena,b and Nicholas L. Roddc

aKavli Institute for Theoretical Physics,
University of California, Santa Barbara, CA 93106, USA
bDepartment of Physics,
University of California, Santa Barbara, CA 93106, USA
cTheoretical Physics Department,
CERN, 1 Esplanade des Particules, CH-1211 Geneva 23, Switzerland ††e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> We construct new dispersive sum rules for the effective field theory of the
> standard model at mass dimension six. These spinning sum rules encode
> information about the spin of UV states: the sign of the IR Wilson
> coefficients carries a memory of the dominant spin in the UV completion. The
> sum rules are constructed for operators containing scalars and fermions,
> although we consider the dimension-six SMEFT exhaustively, outlining why
> equivalent relations do not hold for the remaining operators. As with any
> dimension-six dispersive argument, our conclusions are contingent on the
> absence of potential poles at infinity—so-called boundary terms—and we
> discuss in detail where these are expected to appear. There are a number of
> phenomenological applications of spinning sum rules, and as an example we
> explore the connection to the Peskin-Takeuchi parameters and, more
> generally, the set of oblique parameters in universal theories.

###### Contents

  1. 1 Introduction
  2. 2 Dispersion Relations at Dimension Six
  3. 3 Basis of SMEFT Operators
  4. 4 SMEFT Spinning Sum Rules
    1. 4.1 $h_{1}=h_{2}=0$
    2. 4.2 $h_{1}=\pm 1/2$, $h_{2}=0$
    3. 4.3 $|h_{1}|=|h_{2}|=1/2$
    4. 4.4 $h_{1}=\pm 1$, $h_{2}=0$
    5. 4.5 $|h_{1}|=|h_{2}|=1$
  5. 5 Boundary Terms and UV Completions
    1. 5.1 UV completions of $H^{4}$ operators
    2. 5.2 UV completions of $\psi^{4}$ operators
  6. 6 Oblique Parameters and Universal Theories
    1. 6.1 $W,Y,Z$
    2. 6.2 $\hat{T}$
    3. 6.3 $\hat{S}$
  7. 7 Conclusions

##  1 Introduction

The Standard Model Effective Field Theory (SMEFT) has emerged as the central
organizing framework in the search for new physics at the energy and precision
frontiers. Assuming only that the scale where new states appear is beyond
experimental energies, Wilsonian effective field theory (EFT) codifies the
possible infrared (IR) relics of the ultraviolet (UV). But the SMEFT parameter
space is vast. At mass dimension six, where the first deviations from the
standard model (SM) are expected to appear, the SMEFT basis requires 84
operators, even when considering just a single generation of fermions [1], and
3,045 operators for three generations.111Here we follow the convention in Ref.
[2], where operators and their hermitian conjugates are counted separately.
Within the context of specific completions, the full space of EFT operators
collapses to reveal a set of predictive correlations between the Wilson
coefficients and observables. In the fully UV-agnostic Wilsonian picture,
however, the parameter space remains unstructured.

Yet a middle ground exists between these two pictures: general assumptions
about the UV impose a detailed topography on the EFT landscape. The
restrictions arise as the higher-dimension operators in an EFT introduce
interactions that can mediate scatterings that manifestly violate unitarity,
locality, or causality for generic choices of the operator coefficients, which
can be revealed through dispersive arguments. These constraints were first
discovered in the context of chiral-perturbation theory [3, 4, 5] and then
outlined more generally in Ref. [6]. Given the primacy of EFT to the language
of modern physics, this “EFT consistency” program has been pursued in many
directions, a non-exhaustive list of which includes the chiral Lagrangian [7,
8, 9, 10, 11], electroweak effective theory [12, 13, 14, 15], various scalar
theories [8, 16, 17, 18, 19, 20, 21, 22]—including recent work on the
EFThedron [23] and other powerful new techniques [24, 25, 26, 27, 28]—along
with the $a$-theorem [29, 30], fermionic theories [31, 32, 33, 34, 35],
massive higher-spin particles and massive gravity [36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49], nonlinear electromagnetism [8, 6, 14, 50, 51],
cosmological settings [16, 52, 53, 54, 55, 56, 57], quantum corrections to the
Einstein equations [58, 59, 60, 61, 62, 63, 64], and Einstein-Maxwell theory
and its connections to the Weak Gravity Conjecture [65, 66, 67, 68, 69, 70,
71, 72, 73]. A focus of recent investigations has been on the application of
the technology of analyticity and unitarity to systematically constrain the
Wilson coefficients of the full EFT of the SM itself [74, 75, 76, 77, 14, 34,
12, 13, 15, 35, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89]. This is the
direction we pursue in this work.

The sharpest realization of these ideas arises at dimension eight. Consider,
for instance, the SMEFT operator ${\cal
O}_{e}^{(8)}=-c^{(8)}_{mnpq}\partial_{\mu}(\bar{e}_{m}\gamma_{\nu}e_{n})\partial^{\mu}(\bar{e}_{p}\gamma^{\nu}e_{q})$,
with $e=e_{R}$ the right-handed electron field in the unbroken SMEFT—a singlet
under color and weak isospin—and $m,n,p,q$ flavor indices. The traditional
Wilsonian picture is that any value of the coefficients is allowed, so long as
$|c_{mnpq}|\lesssim 1$, to ensure the EFT power counting is not violated.
Nevertheless, if the UV theory that generates ${\cal O}_{e}^{(8)}$ is unitary,
local, and causal, then the Wilson coefficients can be shown to satisfy a
dispersion relation that considerably limits their allowed structure.
Specifically, as shown in Ref. [34], consistency of two-to-two scattering of
pure electron-type states requires $c_{1111}>0$, whereas elastic scattering of
two arbitrary flavor superpositions, $\alpha$ and $\beta$, reveals that

$c^{(8)}_{mnpq}\alpha_{m}\beta_{n}\bar{\beta}_{p}\bar{\alpha}_{q}>0\hskip
8.5359pt\forall\;\alpha;\beta.$ (1)

The boundary defined by this relation contains considerable structure. The
inequality also imports physical consequences. At the level of this operator
both flavor and CP violation are strictly bounded by the size of the
equivalent interactions that conserve the effects; correlations between
operators and observables have emerged from bedrock UV tenets. However, this
is just a single operator, and considerable work has gone into providing a
more detailed map of the consistent boundary of the dimension-eight SMEFT [14,
34, 12, 13, 15, 79, 80, 81, 82, 83, 84, 85, 86, 87] (the full basis of which
has now been enumerated [90, 91]).

Progress at dimension six has been less rapid and is the focus of the present
work. As at higher dimensions, if the EFT emerged from a unitary, local, and
causal completion, then the Wilson coefficients can be shown to satisfy a
dispersion relation. The fundamental challenge is that as the forward
amplitude now scales linearly with Mandelstam $s$, as opposed to quadratically
as at dimension eight, the interpretation of the resulting dispersive
arguments is less straightforward. We will review the obstructions later in
this work. For the moment we simply note that while at dimension eight the
forward dispersion relation involves a sum of integrated cross sections, and
is therefore strictly positive, at dimension six it involves differences of
integrated cross sections and also a sign-indefinite contribution from a pole
at infinity. Conventionally, the dispersive relations at dimension six are
referred to as sum rules.

While there are challenges at dimension six, the phenomenological importance
of the SMEFT at this order mandates the determination of the structure imposed
by a consistent UV. This question was considered in the seminal work of Low,
Rattazzi, and Vichi [74], in the context of operators that modify observable
Higgs properties at the LHC, and has been taken up in a number of additional
works (see, for example, Refs. [75, 76, 77, 89]), including a recent
determination in Ref. [78] of the full set of sum rules that can be
established by studying the forward amplitude (obtained by taking Mandelstam
$t$ to zero). Sum rules derived from the forward amplitude only constrain the
coefficient of $s$. However, two-to-two scattering amplitudes also depend on
$t$, and constraints on these beyond-forward contributions can also be
established, as shown in Refs. [16, 92]. For example, Ref. [85] recently
demonstrated that the set of possible bounds can be considerably extended with
beyond-forward dispersion relations.

Combining forward and non-forward dispersion relations, Ref. [35] demonstrated
that a novel class of sum rules can be derived if one assumes that the UV
theory sufficiently improves the scaling of the forward amplitude at large $s$
in order to avoid poles at infinity. While intuitively one may expect UV
completions to reduce the momentum scaling of an amplitude—in many instances
they do—we emphasize that the assumption is far less fundamental than
causality, unitarity, and locality; indeed, tree-level UV completions that
involve massive states exchanged in the $t$ channel can violate the conditions
for poles at infinity to vanish, and we will discuss when we expect
obstructions in detail. Regardless, if the UV completion sufficiently improves
the asymptotic scaling of the amplitude, then the result in Eq. (1) can
effectively be replaced by a novel sum rule that implies that the Wilson
coefficients carry a memory of the UV state from which they were dominantly
generated, rather than direct positivity constraints. For instance, the lower-
dimension analogue of ${\cal O}_{e}^{(8)}$ is ${\cal
O}_{e}^{(6)}=-c^{(6)}_{mnpq}(\bar{e}_{m}\gamma_{\mu}e_{n})(\bar{e}_{p}\gamma^{\mu}e_{q})$,
and as shown in Ref. [35], it satisfies the following
inequality:222Importantly, note that as in Ref. [35], we use the mostly-plus
metric convention. As ${\cal O}_{e}^{(6)}$ contains the Lorentz contraction
$\sim g_{\mu\nu}\gamma^{\mu}\gamma^{\nu}$, changing signature would invert the
sign of the result in Eq. (2). The same is not true of Eq. (1): as ${\cal
O}_{e}^{(8)}$ contains two Lorentz contractions, the sign of the bound is
independent of signature. We emphasize more generally that the signature must
be considered carefully.

$c^{(6)}_{mnpq}\alpha_{m}\beta_{n}\bar{\beta}_{q}\bar{\alpha}_{q}\;\begin{array}[]{l}>0~{}\text{scalars~{}dominate~{}the~{}UV},\\\
<0~{}\text{vectors~{}dominate~{}the~{}UV},\end{array}$ (2)

so that the sign of the Wilson coefficient encodes the spin of the UV
completion, giving rise to spinning sum rules.333A similar phenomenon appears
in the “low-spin dominance” bounds on gravitational EFTs in Ref. [93].
Further, as long as either sign dominates, one still expects that CP and
flavor violation are bounded from above by the corresponding conservation
processes. Beyond the additional assumption invoked, this result relied on the
fermion spin in that it made use of the Jacob-Wick expansion [94], where
amplitudes are expanded in the total angular momentum (cf. the Legendre
partial-wave expansion, which is performed in orbital angular momentum).

Nevertheless, Ref. [35] only applied this new approach to fermions. In the
present work, we derive spinning sum rules for the full dimension-six SMEFT,
with a view to adding as much character to the space as possible. Again, the
spirit of our work is not to simply derive clear-cut positivity bounds, but
rather to consider any restrictions on combinations of the SMEFT Wilson
coefficients dictated by our assumptions. We will show that the use of the
Jacob-Wick expansion is not critical, and therefore spinning sum rules can be
established for the purely scalar operators involving the Higgs. In this way,
we will be able to explain several features of the results in Ref. [74], in
particular, why specific UV completions generated specific signs. More
generally, we find bounds on the four-Higgs and four-fermion operators with
sign dependent on whether the UV is dominated by states of low or high spin.
For the remaining operators, while spinning sum rules cannot be established,
we show that the dispersion relations are still informative of the UV; in
particular, a connection emerges between the form of the UV amplitude and the
presence of the poles at infinity, also referred to as boundary terms. For
Higgs-fermion scattering, if the boundary terms vanish, then the UV must be
dominated by a fermionic mode, whereas for Higgs-gauge boson or quartic gauge
boson operators a stronger statement is possible: if the boundary terms
vanish, then the UV amplitudes mediating interactions between these states
must be identically zero. While the main application of our results is to the
dimension-six SMEFT, we also demonstrate their utility by establishing a
connection to the electroweak oblique parameters ($S$, $T$, $W$, $Y$, $Z$)
describing universal theories. As these parameters can be connected to SMEFT
operators, we show how we can establish effective spinning sum rules for a
number of the observables in universal theories.

We will organize our analysis and discussion as follows. In Sec. 2 we
introduce and develop our formalism for deriving our spinning sum rules. After
summarizing the relevant basis of SMEFT operators in Sec. 3, we systematically
apply these bounds to various sectors of SM scattering processes in Sec. 4,
obtaining our main results. In Sec. 5, we perform an extended investigation of
where poles at infinity are expected to appear by considering a wide range of
UV completions and demonstrate that, at tree level, boundary terms are
associated with UV states exchanged in the $t$ channel. Finally, we apply our
results to the electroweak oblique parameters in Sec. 6 and conclude in Sec.
7.

##  2 Dispersion Relations at Dimension Six

In this section, we construct the dispersion relations that we will use to
derive spinning sum rules for dimension-six operators in the SMEFT. These sum
rules will allow us to write the Wilson coefficients of these operators as
sums over spins of states in the UV completion.444Wilson coefficients depend
on the scale at which they are evaluated. Throughout, we will always assume
the coefficients are evaluated at the matching scale, where the new physics is
integrated out, or at a lower scale only if the theory is sufficiently weakly
coupled that the running can be neglected. For a discussion of how the running
can impact the types of relations considered in this work, see, for example,
Refs. [10, 83, 24, 23, 95, 96]. In Sec. 4, we will be able to use these
relationships to associate the signs of certain Wilson coefficients with the
spin of the UV state. Our methods generalize those developed in Ref. [35]
(which considered exclusively four-fermion operators) to states of arbitrary
spin.

As already discussed, the standard methods of establishing positivity bounds
on EFT coefficients of quartic operators (see, e.g., Ref. [6]) do not
immediately extend to terms of mass dimension six. This can be seen by
straightforward power counting: two-to-two EFT amplitudes—functions of $s$ and
$t$, denoted ${\cal A}(s,t)$—mediated by dimension-six operators will go like
$p^{2}$, where $p$ is some generic external momentum. This scaling has the
result that applying a dispersion relation to the forward, $t\to 0$, amplitude
to extract the $\propto s$ part results in a mixed-sign relation:555The
integral over $s$ in the second line is associated with the discontinuity (due
to poles and branch cuts) of the forward amplitude in the complex $s$ plane.
If the theory allows for loops of massless states to be exchanged in the
scattering, then the discontinuity will run to $s=0$ on the real axis, as we
show on the second line. Formally, this would imply that the contour $\gamma$
cannot be defined, but this obstruction can always be removed by introducing
an IR regulator for the massless states, as discussed in Ref. [6].
Accordingly, technically the integral over $s$ here and in later similar
expressions should be viewed as starting at the scale associated with the IR
regulator.

$\displaystyle\lim_{s\rightarrow 0}\partial_{s}{\cal A}(s,0)$
$\displaystyle=\frac{1}{2\pi i}\oint_{\gamma}\frac{{\rm d}s}{s^{2}}{\cal
A}(s,0)$ (3)
$\displaystyle=-C_{\infty}^{(s)}+\frac{1}{\pi}\int_{0}^{\infty}\frac{{\rm
d}s}{s}[\sigma(s)-\overline{\sigma}(s)].$

Here, $\gamma$ is a small contour around the origin evaluated at sufficiently
small $|s|$ that the amplitude can be computed in the EFT, and therefore
related to the coefficient of $s$ in the EFT amplitude ${\cal A}(s,0)$, as we
have on the first line. In the second line, we expand the contour to large
$|s|$, so that we can rewrite this result in terms of UV quantities, and
exploit the optical theorem: $\sigma(s)=s\,{\rm Im}\,{\cal A}(s,0)$ is the
cross section associated with the scattering process on the left-hand side,
and $\overline{\sigma}(s)=s\,{\rm Im}\,\bar{\cal A}(s,0)$ is the cross section
for the crossed amplitude $\bar{\cal A}$, where particles 1 and 4 have been
exchanged under crossing symmetry, e.g., the fully inclusive cross section for
$e^{+}e^{-}$ versus $e^{-}e^{-}$ (where we have conjugated the charge, but in
general any additional internal quantum numbers will be conjugated
also).666While at general kinematics, states with spin transform nontrivially
under crossing, in the forward limit helicities behave just like an internal
quantum number and are simply conjugated under crossing [32, 35]. The residue
at infinity is determined as

$C_{\infty}^{(s)}={\rm Res}\left[\frac{{\cal
A}(s,0)}{s^{2}},s=\infty\right]=-\lim_{|s|\rightarrow\infty}\frac{{\cal
A}(s,0)}{s}.$ (4)

Heuristically, $C_{\infty}^{(s)}$ is the $\propto s$ coefficient in the
expansion of ${\cal A}(s,0)$ at large $s$ and as already noted is commonly
referred to as a boundary term. Observe that in order to go from the first to
the second line of Eq. (3), we needed to assume that the UV was causal, which
implies that the UV amplitude is an analytic function of $s$ except for branch
cuts and poles along the real $s$ axis. Using the sum rule in Eq. (3), one can
place bounds on Wilson coefficients of dimension-six operators under varying
assumptions about the UV, e.g., whether $\sigma(s)$ or $\overline{\sigma}(s)$
dominates; this was investigated systematically in Ref. [78], see also Refs.
[31, 74, 75, 76].

From Eq. (4), if the amplitude satisfies $\lim_{|s|\to\infty}|{\cal
A}_{\alpha\beta}(s,0)|<O(s)$ (i.e., it is super-Froissart in the language of
Ref. [85]), then $C_{\infty}^{(s)}=0$. As discussed in the introduction, while
one may expect the UV to soften the energy scaling of the amplitude, it is not
required. The fact that UV states exchanged in the $t$ channel generically
introduce a residue at infinity can be readily seen. Schematically, for an
amplitude that generates a dimension-six operator in the IR from UV
$t$-channel exchange, we will have ${\cal A}(s,t)\propto
p^{2}/(-t+m^{2}-i\epsilon)$, where $m$ is the mass of the UV state and
$p^{2}\sim s,t,u$. As the cross sections in Eq. (3) are determined by the
imaginary part of the amplitudes, for such a completion the integrand of the
dispersive integral contains ${\rm Im}\,{\cal A}(s,0)\propto
p^{2}\delta(m^{2})$, which vanishes since $m>0$. Consequently, the EFT results
can only be reproduced by $C_{\infty}^{(s)}$, which is nonzero if the $p^{2}$
in the amplitude’s numerator goes like $s$ in the forward limit.777If the
amplitude scales as $t/(-t+m^{2}-i\epsilon)$, as can arise from the
$t$-channel exchange of a derivatively coupled scalar, then as the amplitude
is constant in $s$ it will not generate $C_{\infty}^{(s)}$. Nevertheless, such
an amplitude can generate a boundary term in the beyond-forward dispersion
relation we consider in Eq. (5). Explicit UV completions demonstrating these
points are given in Sec. 5. We will discuss this point, and the contributions
from boundary terms more generally, in detail in Sec. 5.

Applying the optical theorem in the forward limit does not leverage all of the
information we can access about the structure of the amplitude. In particular,
in addition to extracting the $\propto s$ part of ${\cal A}(s,0)$ through Eq.
(3), one can also write a beyond-forward dispersion relation for the $\propto
t$ scaling of ${\cal A}(0,t)$. Setting $s=0$ with $t$ nonzero effectively
interchanges an in- and out-state, for instance a swapping of states 1 and
4.888To be explicit, if we have an amplitude where the in-states (1 and 2) and
out-states (3 and 4) have incoming and outgoing momenta, respectively, then we
can achieve $s\to 0$ by taking $p_{1}\to-p_{2}$ and $p_{4}\to-p_{3}$. The
minus signs then exchange the roles of the states so that state 4 is incoming
and 1 is outgoing. While such an exchange is trivial for a bosonic amplitude,
it induces an extra minus sign if we are exchanging fermions [32, 35].
Requiring $t\rightarrow 0$ to correspond to the forward limit of elastic
scattering, if the two initial particles have helicities $h_{1}$ and $h_{2}$,
then working in a convention with all particles incoming (as we do so
throughout), we must take $h_{4}=-h_{1}$ and $h_{3}=-h_{2}$. Running through a
generalization of the contour integral considered in Ref. [35], we arrive at
the following dispersion relation:

$\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{h_{1};h_{2}}(0,t)=-C_{\infty}^{(t)}+\frac{(-1)^{2h_{1}}}{\pi}\int_{0}^{\infty}{\rm
d}s\lim_{t\rightarrow 0}\partial_{t}\left[\frac{{\rm Im}\,{\cal
A}_{h_{1};h_{2}}(s,t)}{s}+\frac{{\rm Im}\,\bar{\cal
A}_{-h_{1};h_{2}}(s,t)}{s+t}\right].$ (5)

This relation introduces a distinct residue at infinity (i.e., a new boundary
term),

$C_{\infty}^{(t)}=(-1)^{2h_{1}}\,{\rm Res}\left[\frac{\lim_{t\rightarrow
0}\partial_{t}{\cal A}_{h_{1};h_{2}}(s,t)}{s},s=\infty\right]\\!.$ (6)

Before continuing, we note that our choice of achieving the $s\to 0$ limit by
exchanging the in and out nature of particles 1 and 4 was arbitrary, and we
could equally well have exchanged the properties of particles 2 and 3. The
sign of the right-hand sides in Eqs. (5) and (6) would then be determined by
$h_{2}$ rather than $h_{1}$. For purely fermionic amplitudes this exchange is
irrelevant, but when scattering a fermion and boson, consistency would require
$\lim_{t\rightarrow 0}\partial_{t}{\cal A}_{h_{1};h_{2}}(0,t)=0$, and indeed
this is the case. Taking the fermion to correspond to particles 1 and 4, the
kinematic dependence of the amplitude can only be supported by expressions
such as $\langle 12\rangle[24]$, $[12]\langle 24\rangle$, $\langle
13\rangle[34]$, or $[13]\langle 34\rangle$, all of which vanish as $s\to 0$.

Moving forward, our strategy is to expand the amplitudes that appear in the
integrands of Eqs. (3) and (5) in partial waves of total angular momentum. We
then apply unitarity at the level of each partial amplitude $a^{(j)}(s)$ of
fixed total angular momentum $j$, requiring that ${\rm Im}\,a^{(j)}(s)>0$.
Doing so will allow us to derive sum rules that access finer details of the UV
than Eq. (3). For initial particles with nonzero helicity, recasting the
amplitude in partial waves of the total angular momentum requires the Jacob-
Wick expansion [94] (see also Refs. [97, 98, 99]), in which the Legendre
polynomials are replaced by the Wigner small $d$-matrices. We have999The sum
runs over $j\in\mathbb{N}_{0}+\lambda$, so that if, e.g., $\lambda=3/2$,
$j\in\\{3/2,5/2,\ldots\\}$.

${\cal
A}_{\alpha,h_{1};\beta,h_{2}}(s,t)=16\pi\sum_{j=\lambda}^{\infty}(2j+1)a_{\alpha,h_{1};\beta,h_{2}}^{(j)}(s)\,d_{\lambda,\lambda}^{(j)}(z),$
(7)

where we have introduced $\lambda=|h_{1}-h_{2}|$, $z=\cos\theta=1+2t/s$ with
$\theta$ the scattering angle, and $\alpha$ and $\beta$ as labels denoting the
internal quantum numbers (for example, flavor) of the two scattered states. We
can similarly expand the crossed amplitude $\bar{\cal A}$, which in the
forward limit becomes ${\cal A}_{\bar{\alpha},-h_{1};\beta,h_{2}}$. When
implementing this expansion, it is convenient to express the Wigner small
$d$-matrices in terms of the Jacobi polynomials,

$d_{\lambda,\lambda}^{(j)}(z)=\left(\frac{1+z}{2}\right)^{\lambda}P_{j-\lambda}^{(0,2\lambda)}(z).$
(8)

In particular, in the forward limit, where $z=1$, we have
$P_{n}^{(0,\beta)}(1)=1$, and further to evaluate the derivative in Eq. (5),
we use $\lim_{z\rightarrow 1}{\rm d}P_{n}^{(0,\beta)}(z)/{\rm
d}z=n(\beta+n+1)/2$, so that

$\lim_{z\rightarrow 1}\frac{{\rm d}}{{\rm
d}z}d_{\lambda,\lambda}^{(j)}(z)=\frac{1}{2}\left[\lambda+(j-\lambda)(j+\lambda+1)\right]\\!.$
(9)

Combining these results, we can apply the Jacob-Wick expansion to the
$s$-channel dispersion relation in Eq. (3). Doing so yields

$\displaystyle\lim_{s\rightarrow 0}\partial_{s}{\cal
A}_{\alpha,h_{1};\beta,h_{2}}(s,0)+C_{\infty}^{(s)}$ (10) $\displaystyle=$
$\displaystyle 16\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\Biggr{\\{}\sum_{j=\lambda}^{\infty}(2j+1){\rm
Im}\,a_{\alpha,h_{1};\beta,h_{2}}^{(j)}(s)-\sum_{j=\bar{\lambda}}^{\infty}(2j+1){\rm
Im}\,a_{\bar{\alpha},-h_{1};\beta,h_{2}}^{(j)}(s)\Biggr{\\}},$

where we have defined $\bar{\lambda}=|h_{1}+h_{2}|$. Continuing, the partial-
wave expansion of Eq. (5) yields

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha,h_{1};\beta,h_{2}}(0,t)+C_{\infty}^{(t)}$ (11) $\displaystyle=$
$\displaystyle 16(-1)^{2h_{1}}\\!\\!\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\Biggl{\\{}\sum_{j=\lambda}^{\infty}(2j+1)\left[\lambda+\left(j-\lambda\right)\left(j+\lambda+1\right)\right]{\rm
Im}\,a_{\alpha,h_{1};\beta,h_{2}}^{(j)}(s)$ $\displaystyle\hskip
82.51282pt+\sum_{j=\bar{\lambda}}^{\infty}(2j+1)\left[\bar{\lambda}+\left(j-\bar{\lambda}\right)\left(j+\bar{\lambda}+1\right)-1\right]{\rm
Im}\,a_{\bar{\alpha},-h_{1};\beta,h_{2}}^{(j)}(s)\Biggr{\\}}.$

Finally, combining the two, we have an additional sum rule with different
weighting of the partial waves,

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha,h_{1};\beta,h_{2}}(0,t)+\lim_{s\rightarrow 0}\partial_{s}{\cal
A}_{\alpha,h_{1};\beta,h_{2}}(s,0)+C_{\infty}^{(s)}+C_{\infty}^{(t)}$ (12)
$\displaystyle=$ $\displaystyle
16(-1)^{2h_{1}}\\!\\!\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\Biggl{\\{}\sum_{j=\lambda}^{\infty}(2j+1)\left[\lambda+\left(j-\lambda\right)\left(j+\lambda+1\right)\\!+\\!(-1)^{2h_{1}}\right]{\rm
Im}\,a_{\alpha,h_{1};\beta,h_{2}}^{(j)}(s)$ $\displaystyle\hskip
82.51282pt+\sum_{j=\bar{\lambda}}^{\infty}(2j+1)\left[\bar{\lambda}+\left(j-\bar{\lambda}\right)\left(j+\bar{\lambda}+1\right)\\!-\\!1\\!-\\!(-1)^{2h_{1}}\right]{\rm
Im}\,a_{\bar{\alpha},-h_{1};\beta,h_{2}}^{(j)}(s)\\!\Biggr{\\}}.$

Depending on the helicities of the incoming states, either Eq. (11) or Eq.
(12) will be most useful in formulating informative sum rules. For example, if
$\bar{\lambda}\geq 1$, then the right-hand side of Eq. (11) is either
positive- or negative-definite, with sign $(-1)^{2h_{1}}$, while if
$(-1)^{2h_{1}}=-1$ and $\lambda\geq 1$, the right-hand side of Eq. (12) is
negative-definite. In Ref. [35], this latter observation was exploited by
considering the scattering of opposite-helicity fermions, so that $\lambda=1$
and $\bar{\lambda}=0$.

In Sec. 4, we will use the dispersion relations in Eqs. (11) and (12) to
derive the full set of spinning sum rules. As already mentioned, these
relations will connect the sign of the Wilson coefficients in the IR to the
physical characteristics of the UV in terms of which angular momentum
contributions dominate the sums over $j$ above. The spinning sum rules can be
contrasted with the result in Eq. (10), which is the expansion of the
conventional forward dispersion relation. The sign difference that enters in
that relation between the sum over $\lambda$ and that over $\bar{\lambda}$
obstructs similar arguments regarding which angular momentum states dominate
in the UV, before even considering boundary terms. Before deriving the main
results of this work, however, let us first review the basis of the SMEFT and
establish our conventions for it.

##  3 Basis of SMEFT Operators

$F^{3}$ operators |  $H^{4}$ operators
---|---
|  . |  |
${\cal O}_{G}$ |  $f^{abc}G^{a\;\nu}_{\mu}G^{b\;\rho}_{\nu}G^{c\;\mu}_{\rho}$ |  ${\cal O}_{H1}$ |  $(H^{\dagger}H)\Box(H^{\dagger}H)$
$\widetilde{\cal O}_{G}$ |  $f^{abc}G^{a\;\nu}_{\mu}G^{b\;\rho}_{\nu}\widetilde{G}^{c\;\mu}_{\rho}$ |  ${\cal O}_{H2}$ |  $(H^{\dagger}D_{\mu}H)^{*}(H^{\dagger}D^{\mu}H)$
${\cal O}_{W}$ |  $\varepsilon^{IJK}W^{I\;\nu}_{\mu}W^{J\;\rho}_{\nu}W^{K\;\mu}_{\rho}$ |  |
$\widetilde{\cal O}_{W}$ |  $\varepsilon^{IJK}W^{I\;\nu}_{\mu}W^{J\;\rho}_{\nu}\widetilde{W}^{K\;\mu}_{\rho}$ |  |
|  . |  |
|  . |  $\psi^{4}$ operators
$H^{2}F^{2}$ operators |  |
|  . |  ${\cal O}_{e}$ |  $-(\bar{e}_{m}\gamma_{\mu}e_{n})(\bar{e}_{p}\gamma^{\mu}e_{q})$
${\cal O}_{HB}$ |  $(H^{\dagger}H)B_{\mu\nu}B^{\mu\nu}$ |  ${\cal O}_{u}$ |  $-(\bar{u}_{m}\gamma_{\mu}u_{n})(\bar{u}_{p}\gamma^{\mu}u_{q})$
$\widetilde{\cal O}_{HB}$ |  $(H^{\dagger}H)B_{\mu\nu}\widetilde{B}^{\mu\nu}$ |  ${\cal O}_{d}$ |  $-(\bar{d}_{m}\gamma_{\mu}d_{n})(\bar{d}_{p}\gamma^{\mu}d_{q})$
${\cal O}_{HW}$ |  $(H^{\dagger}H)W^{I}_{\mu\nu}W^{I\mu\nu}$ |  ${\cal O}_{L}$ |  $-(\bar{L}_{m}\gamma_{\mu}L_{n})(\bar{L}_{p}\gamma^{\mu}L_{q})$
$\widetilde{\cal O}_{HW}$ |  $(H^{\dagger}H)W^{I}_{\mu\nu}\widetilde{W}^{I\mu\nu}$ |  ${\cal O}_{Q1}$ |  $-(\bar{Q}_{m}\gamma_{\mu}Q_{n})(\bar{Q}_{p}\gamma^{\mu}Q_{q})$
${\cal O}_{HG}$ |  $(H^{\dagger}H)G^{a}_{\mu\nu}G^{a\mu\nu}$ |  ${\cal O}_{Q2}$ |  $-(\bar{Q}_{m}\tau^{I}\gamma_{\mu}Q_{n})(\bar{Q}_{p}\tau^{I}\gamma^{\mu}Q_{q})$
$\widetilde{\cal O}_{HG}$ |  $(H^{\dagger}H)G^{a}_{\mu\nu}\widetilde{G}^{a\mu\nu}$ |  ${\cal O}_{eu}$ |  $-(\bar{e}_{m}\gamma_{\mu}e_{n})(\bar{u}_{p}\gamma^{\mu}u_{q})$
|  . |  ${\cal O}_{ed}$ |  $-(\bar{e}_{m}\gamma_{\mu}e_{n})(\bar{d}_{p}\gamma^{\mu}d_{q})$
|  . |  ${\cal O}_{ud1}$ |  $-(\bar{u}_{m}\gamma_{\mu}u_{n})(\bar{d}_{p}\gamma^{\mu}d_{q})$
$H^{2}\psi^{2}$ operators |  ${\cal O}_{ud2}$ |  $-(\bar{u}_{m}T^{a}\gamma_{\mu}u_{n})(\bar{d}_{p}T^{a}\gamma^{\mu}d_{q})$
|  . |  ${\cal O}_{LQ1}$ |  $-(\bar{L}_{m}\gamma_{\mu}L_{n})(\bar{Q}_{p}\gamma^{\mu}Q_{q})$
${\cal O}_{He}$ |  $(H^{\dagger}i\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}_{\mu}H)(\bar{e}_{m}\gamma^{\mu}e_{n})$ |  ${\cal O}_{LQ2}$ |  $-(\bar{L}_{m}\tau^{I}\gamma_{\mu}L_{n})(\bar{Q}_{p}\tau^{I}\gamma^{\mu}Q_{q})$
${\cal O}_{Hu}$ |  $(H^{\dagger}i\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}_{\mu}H)(\bar{u}_{m}\gamma^{\mu}u_{n})$ |  ${\cal O}_{Le}$ |  $(\bar{L}_{m}\gamma_{\mu}L_{n})(\bar{e}_{p}\gamma^{\mu}e_{q})$
${\cal O}_{Hd}$ |  $(H^{\dagger}i\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}_{\mu}H)(\bar{d}_{m}\gamma^{\mu}d_{n})$ |  ${\cal O}_{Lu}$ |  $(\bar{L}_{m}\gamma_{\mu}L_{n})(\bar{u}_{p}\gamma^{\mu}u_{q})$
${\cal O}_{HL1}$ |  $(H^{\dagger}i\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}_{\mu}H)(\bar{L}_{m}\gamma^{\mu}L_{n})$ |  ${\cal O}_{Ld}$ |  $(\bar{L}_{m}\gamma_{\mu}L_{n})(\bar{d}_{p}\gamma^{\mu}d_{q})$
${\cal O}_{HL2}$ |  $(H^{\dagger}i\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}\vphantom{D}^{I}_{\mu}H)(\bar{L}_{m}\tau^{I}\gamma^{\mu}L_{n})$ |  ${\cal O}_{Qe}$ |  $(\bar{Q}_{m}\gamma_{\mu}Q_{n})(\bar{e}_{p}\gamma^{\mu}e_{q})$
${\cal O}_{HQ1}$ |  $(H^{\dagger}i\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}_{\mu}H)(\bar{Q}_{m}\gamma^{\mu}Q_{n})$ |  ${\cal O}_{Qu1}$ |  $(\bar{Q}_{m}\gamma_{\mu}Q_{n})(\bar{u}_{p}\gamma^{\mu}u_{q})$
${\cal O}_{HQ2}$ |  $(H^{\dagger}i\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}\vphantom{D}^{I}_{\mu}H)(\bar{Q}_{m}\tau^{I}\gamma^{\mu}Q_{n})$ |  ${\cal O}_{Qu2}$ |  $(\bar{Q}_{m}T^{a}\gamma_{\mu}Q_{n})(\bar{u}_{p}T^{a}\gamma^{\mu}u_{q})$
|  . |  ${\cal O}_{Qd1}$ |  $(\bar{Q}_{m}\gamma_{\mu}Q_{n})(\bar{d}_{p}\gamma^{\mu}d_{q})$
|  . |  ${\cal O}_{Qd2}$ |  $(\bar{Q}_{m}T^{a}\gamma_{\mu}Q_{n})(\bar{d}_{p}T^{a}\gamma^{\mu}d_{q})$
Table 1: The subset of operators in the dimension-six SMEFT [1] considered in
this work: those containing quartic interactions allowing for two-to-two
elastic scattering of eigenstates of spin and the SM gauge group.

We wish to apply the sum rules in Eqs. (11) and (12) to the dimension-six
operators in the SMEFT. Let us now enumerate the operators in the basis of
Ref. [1] that we will consider. We write the SM field content as follows. We
have left-handed fermions ($L$, $Q$) and right-handed fermions ($e$, $d$, $u$)
comprising the quarks ($Q$, $u$, $d$) and leptons ($L$, $e$). The multiplet
$L$ is a fundamental of ${\rm SU}(2)_{L}$, $u$ and $d$ are fundamentals of
${\rm SU}(3)_{C}$, while $Q$ is a fundamental and $e$ a singlet of both. We
will define the hypercharges carried by the fermions under ${\rm U}(1)_{Y}$ as
$(Y_{Q},Y_{L},Y_{u},Y_{d},Y_{e})=(+1/6,-1/2,+2/3,-1/3,-1)$. We write the
generators of ${\rm SU}(2)_{L}$ and ${\rm SU}(3)_{C}$ as
$\tau^{I}=\sigma^{I}/2$ and $T^{a}=\lambda^{a}/2$, where $\sigma^{I}$ are the
Pauli and $\lambda^{a}$ the Gell-Mann matrices. We associate each of the
fermionic fields with a generation index $m,n,p,q$, running from $1$ to
$N_{f}$ ($=3$ in the SM), parameterizing flavor. We will write the SM bosons
as $B_{\mu}$, $W_{\mu}^{I}$, and $G_{\mu}^{a}$ for ${\rm U}(1)_{Y}$, ${\rm
SU}(2)_{L}$, and ${\rm SU}(3)_{C}$, with couplings $g^{\prime}$, $g$, and
$g_{s}$ (and Weinberg angle $\tan\theta_{\scriptscriptstyle W}=g^{\prime}/g$)
and defining the respective field strengths as $B_{\mu\nu}$, $W_{\mu\nu}^{I}$,
and $G_{\mu\nu}^{a}$. Finally, we write the full, complex Higgs doublet—a
fundamental of ${\rm SU}(2)_{L}$ carrying half a unit of hypercharge—as

$H_{i}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\phi_{1}+i\phi_{2}\\\
\phi_{3}+i\phi_{4}\end{array}\right)\\!,$ (13)

with $\phi_{1,2,3,4}\in\mathbb{R}$, where $i$ indexes the fundamental SU(2)L
representation, which we will contract using $\epsilon_{ij}$.

To apply our sum rules, we need to consider forward, two-to-two elastic
scattering amplitudes for states of definite helicity. For simplicity, we will
also restrict to the scattering of definite eigenstates of the SM gauge group
(i.e., we will not consider processes with superpositions of $e$ and $L$, $L$
and $Q$, $H$ and $u$, $G_{\mu}^{a}$ and $H$, etc.). Thus, we need the quartic
operators in the dimension-six SMEFT with two copies of one field and two
copies of another. A full, minimal basis at dimension-six was constructed in
Ref. [1]. The subset that we will need is given in Table 1, where we make use
of the following notation:

$\displaystyle(H^{\dagger}\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}_{\mu}H)$
$\displaystyle=H^{\dagger}(D_{\mu}H)-(D_{\mu}H)^{\dagger}H$ (14)
$\displaystyle(H^{\dagger}\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}\vphantom{D}^{I}_{\mu}H)$
$\displaystyle=H^{\dagger}\tau^{I}(D_{\mu}H)-(D_{\mu}H)^{\dagger}\tau^{I}H.$

All of the operators in Table 1, except for the four $F^{3}$ operators, have
tree-level UV completions, as shown in Ref. [100]. The full class of one-loop
completions of ${\cal O}_{G}$ and ${\cal O}_{W}$ are also known, and were
enumerated in Ref. [101].

##  4 SMEFT Spinning Sum Rules

We now apply the sum rules in Eqs. (11) and (12) directly to the SMEFT
operators of Sec. 3, considering each possible combination of helicities from
Table 1 in turn. Spinning sum rules will be established for purely scalar and
purely fermionic amplitudes. The obstruction in the remaining cases will be
that the EFT amplitudes we wish to constrain exactly vanish. Nevertheless,
even in these cases we will show that the dispersion relations carry
information about the structure of the UV.

At the outset, let us emphasize two points. Firstly, the sum rules we consider
fundamentally exploit the beyond-forward dispersion relation in Eq. (5), which
will be how we can obtain information beyond the widely-explored forward
relation of Eq. (3). Secondly, to establish a direct connection between the UV
spin and the sign of Wilson coefficients, we will require the absence of the
boundary terms $C_{\infty}^{(s)}$ and $C_{\infty}^{(t)}$: our sharpest
relations hold only for the subset of UV completions that do not generate such
poles, and we explore this point in detail in Sec. 5. We will, however, keep
track of the boundary contributions throughout our discussion in this section,
explicitly stating wherever we require the boundary terms to vanish and
otherwise including them in our discussion of what properties of the UV we can
infer.

###  4.1 $h_{1}=h_{2}=0$

We consider first the only case where the external particles carry no spin,
the scattering of Higgs bosons through the two $H^{4}$ operators in Table 1.
Using these operators, we wish to compute the amplitude and sum rule
associated with scattering an arbitrary superposition of the real $\phi_{i}$
fields, with the state of the particles given by

$\left|1\right\rangle=\alpha_{i}\left|\phi_{i}\right\rangle,\hskip
14.22636pt\left|2\right\rangle=\beta_{i}\left|\phi_{i}\right\rangle,\hskip
14.22636pt\left|3\right\rangle=\gamma_{i}\left|\phi_{i}\right\rangle,\hskip
14.22636pt\left|4\right\rangle=\delta_{i}\left|\phi_{i}\right\rangle.$ (15)

For elastic scattering, we require $\delta_{i}=\bar{\alpha}_{i}$ and
$\gamma_{i}=\bar{\beta}_{i}$.101010Throughout, we work in a convention where
the momenta of states involved in two-to-two scattering are all taken as
incoming and labeled cyclicly. Cyclic labeling implies that, with our metric
convention, we have Mandelstam invariants
$s=-(p_{1}+p_{2})^{2}=-(p_{3}+p_{4})^{2}$,
$t=-(p_{1}+p_{4})^{2}=-(p_{2}+p_{3})^{2}$, and
$u=-(p_{1}+p_{3})^{2}=-(p_{2}+p_{4})^{2}$. The amplitude is111111Canonical
normalization of the external state requires $|\alpha|=|\beta|=1$. However, we
choose to keep such terms unsimplified in order to leave expressions
homogeneous in both $\alpha$ and $\beta$.

$\displaystyle{\cal A}_{\alpha;\beta}=$ $\displaystyle
2c^{H1}\left(s|\alpha\cdot\beta|^{2}+t|\alpha|^{2}|\beta|^{2}+u|\alpha\cdot\bar{\beta}|^{2}\right)$
(16)
$\displaystyle+c^{H2}s\left[(\alpha\gamma^{1}\gamma^{3}\bar{\alpha})(\beta\gamma^{1}\gamma^{3}\bar{\beta})-\frac{1}{2}(\alpha\gamma^{1}\bar{\alpha})(\beta\gamma^{1}\bar{\beta})-\frac{1}{2}(\alpha\gamma^{3}\bar{\alpha})(\beta\gamma^{3}\bar{\beta})\right]$
$\displaystyle+\frac{1}{2}c^{H2}t\left(\varepsilon^{ijkl}\alpha_{i}\beta_{j}\bar{\beta}_{k}\bar{\alpha}_{l}-|\alpha|^{2}|\beta|^{2}+|\alpha\cdot\bar{\beta}|^{2}+3|\alpha\gamma^{1}\gamma^{3}\beta|^{2}\right)\\!,$

where $\epsilon^{ijkl}$ is the four-dimensional Levi-Civita tensor, and we
write $\gamma^{\mu}$ for the gamma matrices in the Dirac representation. (Note
the scattering amplitude is purely bosonic, the $\gamma$ matrices are simply
being used to capture the matrix structures that appear in the amplitude in a
more compact form.)

The amplitude simplifies considerably if we choose a less general
$\left|2\right\rangle$ state, taking $\beta_{i}=(1,0,0,0)$, i.e., we analyze
the scattering of an arbitrary superposition of the $\phi_{i}$ off of
$\phi_{1}$. Doing so, the amplitude in Eq. (16) reduces to

${\cal
A}_{\alpha}=t\left[2c^{H1}\left(|\alpha_{2}|^{2}+|\alpha_{3}|^{2}+|\alpha_{4}|^{2}\right)+\frac{1}{2}c^{H2}\left(2|\alpha_{2}|^{2}-|\alpha_{3}|^{2}-|\alpha_{4}|^{2}\right)\right]\\!.$
(17)

This amplitude is fully invariant under crossing the 1 and 4 states: it
depends only on $t$ and is invariant under
$\alpha_{i}\leftrightarrow\bar{\alpha}_{i}$. If this symmetry persists in the
UV, then we would have ${\rm Im}\,a_{\alpha}^{(j)}(s)={\rm
Im}\,a_{\bar{\alpha}}^{(j)}(s)$, and as $\partial_{s}{\cal
A}_{\alpha}(s,0)=0$, Eq. (10) implies that $C_{\infty}^{(s)}=0$.
Alternatively, so long as the UV does not generate a contribution to
$C_{\infty}^{(s)}$, then Eq. (10) would demand $\sum_{j}(2j+1){\rm
Im}\,a_{\alpha}^{(j)}(s)=\sum_{j}(2j+1){\rm Im}\,a_{\bar{\alpha}}^{(j)}(s)$.
In any case, remaining agnostic about the status of this symmetry in the UV,
we have the spinning sum rule from Eq. (11):

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha;\beta}(0,t)+C_{\infty}^{(t)}$ (18)
$\displaystyle=16\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{j=0}^{\infty}(2j+1)\left\\{\left[j\left(j+1\right)\right]{\rm
Im}\,a_{\alpha;\beta}^{(j)}(s)+\left[j\left(j+1\right){-}1\right]{\rm
Im}\,a_{\bar{\alpha};\beta}^{(j)}(s)\right\\}.$

From the sum over partial waves, we see that if the UV completion is dominated
by scalars or states with spin, then the above linear combinations are either
negative or positive, respectively. This statement is, of course, modulo the
boundary term $C^{(t)}_{\infty}$, and in Sec. 5 we will expand on when we
expect such contributions. We emphasize that the connection between sign and
spin is reversed as compared with the result we stated for fermion scattering
in Eq. (2), and we will see in Sec. 4.3 that the fermionic spinning sum rule
is positive for scalars and negative for vectors. For specific scattering
states, the same conclusion holds for the following combinations of the
operators:

$\displaystyle 2c^{H1}+c^{H2}$
$\displaystyle=\\!16\\!\int_{0}^{\infty}\\!\frac{{\rm
d}s}{s^{2}}\sum_{j=0}^{\infty}(2j{+}1)\left\\{\left[j\left(j{+}1\right)\right]{\rm
Im}\,a_{2}^{(j)}(s)+\left[j\left(j{+}1\right){-}1\right]{\rm
Im}\,\bar{a}_{2}^{(j)}(s)\right\\}-C^{(t)}_{\infty}$ (19) $\displaystyle
2c^{H1}-\frac{1}{2}c^{H2}$
$\displaystyle=\\!16\\!\int_{0}^{\infty}\\!\frac{{\rm
d}s}{s^{2}}\sum_{j=0}^{\infty}(2j{+}1)\left\\{\left[j\left(j{+}1\right)\right]{\rm
Im}\,a_{3}^{(j)}(s)+\left[j\left(j{+}1\right){-}1\right]{\rm
Im}\,\bar{a}_{3}^{(j)}(s)\right\\}-C^{(t)}_{\infty}$
$\displaystyle=\\!16\\!\int_{0}^{\infty}\\!\frac{{\rm
d}s}{s^{2}}\sum_{j=0}^{\infty}(2j{+}1)\left\\{\left[j\left(j{+}1\right)\right]{\rm
Im}\,a_{4}^{(j)}(s)+\left[j\left(j{+}1\right){-}1\right]{\rm
Im}\,\bar{a}_{4}^{(j)}(s)\right\\}-C^{(t)}_{\infty},$

where $a^{(j)}_{2,3,4}(s)$ are the UV partial-wave amplitudes for
$\phi_{1}+\phi_{2,3,4}\rightarrow X$, respectively.

These results are not exhaustive, however. Indeed, as ${\cal O}_{H2}$ violates
custodial symmetry—the ${\rm O}(4)$ symmetry associated with rotating the
$\phi_{i}$ among each other—we should not expect scattering off a single state
to fully determine the physics, that is to say, Eq. (17) does not contain the
full information of Eq. (16). As an explicit example, if we take
$\alpha=(1,-i,0,0)/\sqrt{2}$ and $\beta=(0,0,i,1)/\sqrt{2}$,121212Note that if
the full setup of the scattering experiment is crossing invariant, then
$\sigma=\bar{\sigma}$, the IR amplitude is proportional to $t$, and the sum
rule in Eq. (3) vanishes. Thus, in the $\phi_{i}$ basis, the only way to
obtain an amplitude that is not identical after crossing—i.e., to obtain a
term $\propto s$ in the IR—is to include complex values in the superposition.
As discussed in Ref. [78], a relation such as $\sigma=\bar{\sigma}$ can result
in the UV if custodial symmetry is preserved. the resulting amplitude is
${\cal A}(s,t)=2c^{H1}t+c^{H2}u$, which allows us to construct a spinning sum
rule for the combination $2c^{H1}-c^{H2}$. More generally, by allowing
arbitrary complex $\alpha_{i}$ and $\beta_{i}$, we can construct a spinning
sum rule for any choice of coefficients within the region shown in Fig. 1. In
particular, note that a spinning sum rule can be constructed for $c^{H1}$
alone, but not $c^{H2}$. We emphasize that the sign of the combination of
Wilson coefficients in this region is not positive, but instead dictated by
the dominant spin in the UV (unless the UV generates a pole at infinity). We
could, of course, obtain additional sum rules by subtracting various spinning
sum rules from each other, but doing so would not in general be compatible
with this connection between signs and UV spin.

Figure 1: The cone within which the coefficients of the two Higgs operators in
Table 1—${\cal O}_{H1}$ and ${\cal O}_{H2}$—can be chosen such that we can
construct a spinning sum rule: a relationship between the sign of the IR
coefficients and the dominant spin in the UV completion. The edges of the cone
extend from the origin out through $2c^{H1}\pm c^{H2}$, which are both
constructed explicitly in the text. The hexagons show the specific sum rules
considered in Eq. (19), and the figure highlights that these are just two
examples of a broad class of sum rules.

We can compare the spinning sum rules we have obtained to previous sum rules
used to study dimension-six Higgs operators. These were the operators
considered in the early work of Ref. [74]. In particular, accounting for sign
conventions, Ref. [74] finds via dispersion relations for quartic Higgs
scattering in the charged-state basis that a negative coefficient of ${\cal
O}_{H1}$ can only be generated at tree level via scalar exchange, which agrees
with what we find in the real Higgs basis in Eq. (18) above. A further notable
example those authors construct is the coupling of heavy vectors to the Higgs
to generate a positive $c^{H1}$, which is again consistent with Eq. (18).

More recently, Ref. [78] considered what sum rules can be determined on the
same Higgs operators using only the forward dispersion relation in Eq. (3).
The authors of that work adopt a different basis from what we outlined in
Table 1. Specifically, they consider ${\cal
O}_{H}=-(\partial_{\mu}|H|^{2})^{2}/2$ and ${\cal
O}_{T}=-(H^{\dagger}\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}_{\mu}H)^{2}/2$,
where we have introduced an overall sign to account for the opposite signature
convention employed in that work. It is straightforward to map between these
bases, and doing so reveals that the two sum rules they establish on
$c_{H}+3c_{T}$ and $-2c_{T}$ correspond to results for $2c^{H1}+c^{H2}$ and
$-c^{H2}$, respectively. The first of these lies on the boundary of the region
shown in Fig. 1, while the second lies outside it. There is no contradiction,
of course, as this is simply a result of the difference between traditional
sum rules as established in, e.g., Ref. [78] and the spinning sum rules we
consider here. For the combination $2c^{H1}+c^{H2}$, it so happens that both
types of sum rules can be established.

###  4.2 $h_{1}=\pm 1/2$, $h_{2}=0$

Next, we consider the case of fermion-scalar scattering (e.g., elastic
scattering of a lepton or quark with a Higgs), with $h_{1}=\pm 1/2$ and
$h_{2}=0$. Accordingly, we have $\lambda=\bar{\lambda}=1/2$, and then further
introducing a new summation index $n=j-1/2$, we can write Eq. (11)
as131313Where our results are independent of the helicity of the external
states, as is the case here, we will not explicitly flag the helicity in the
amplitude as a subscript. When it is, as is the case for $|h_{1}|=|h_{2}|=1/2$
or $1$, we will.

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha;\beta}(0,t)+C_{\infty}^{(t)}$ (20)
$\displaystyle=-32\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}(n+1)\Biggl{\\{}\left[n(n+2)+\frac{1}{2}\right]{\rm
Im}\,a_{\alpha;\beta}^{(n)}(s)+\left[n(n+2)-\frac{1}{2}\right]{\rm
Im}\,a_{\bar{\alpha};\beta}^{(n)}(s)\Biggr{\\}},$

where $a_{\bar{\alpha};\beta}$ denotes the partial waves for the helicity-
flipped process with $h_{1}=\mp 1/2$ and with any internal quantum numbers of
the fermion complex-conjugated.

As discussed in Sec. 2, when scattering a boson and fermion we expect
$\lim_{t\rightarrow 0}\partial_{t}{\cal A}_{\alpha;\beta}(0,t)=0$, and indeed
we will confirm this in explicit examples below. However, before any analysis
this observation implies that a spinning sum rule, which makes central use of
this EFT amplitude, cannot be constructed for fermion-scalar scattering.
Nevertheless, we show that even in this case the dispersion relations still
carry considerable information.

Let us now apply Eq. (20) to the $H^{2}\psi^{2}$ operators in Table 1. To
begin with, note that the Wilson coefficient tensors $c_{mn}$ for each of
these operators must be hermitian, $c_{mn}=\bar{c}_{nm}$, which follows from
the hermiticity of the operators themselves. Moreover, the operators conserve
CP if and only if all $c_{mn}$ are real. Expanding the Higgs in real scalars,
as in Eq. (13), we have

$H^{\dagger}i\overset{\text{\tiny$\bm{\leftrightarrow}$}}{\partial}_{\mu}H=\phi_{2}\partial_{\mu}\phi_{1}-\phi_{1}\partial_{\mu}\phi_{2}+\phi_{4}\partial_{\mu}\phi_{3}-\phi_{3}\partial_{\mu}\phi_{4}.$
(21)

Computing $\bar{e}^{-}H\rightarrow\bar{e}^{-}H$ scattering (where the sign
labels the helicity, not electric charge), with $\alpha_{m}$ and $\delta_{m}$
denoting the flavor of the $e_{m}$ and $\beta_{i}$ and $\gamma_{i}$ denoting
the superposition of $\phi_{i}$, we have

${\cal
A}=-i\alpha_{m}\delta_{n}c_{mn}^{He}(\beta_{1}\gamma_{2}-\beta_{2}\gamma_{1}+\beta_{3}\gamma_{4}-\beta_{4}\gamma_{3})\left([12]\langle
24\rangle-[13]\langle 34\rangle\right)\\!.$ (22)

For the case of elastic scattering, where we take $\alpha=\bar{\delta}$ and
$\beta=\bar{\gamma}$, we have

${\cal A}_{\alpha;\beta}=2\alpha_{m}\bar{\alpha}_{n}c_{mn}^{He}\left[{\rm
Im}(\beta_{1}\bar{\beta}_{2})+{\rm
Im}(\beta_{3}\bar{\beta}_{4})\right]\left([12]\langle 24\rangle-[13]\langle
34\rangle\right)\\!.$ (23)

At special kinematics,

$\displaystyle{\cal A}_{\alpha;\beta}(s,0)$
$\displaystyle=-4\alpha_{m}\bar{\alpha}_{n}c_{mn}^{He}\left[{\rm
Im}(\beta_{1}\bar{\beta}_{2})+{\rm Im}(\beta_{3}\bar{\beta}_{4})\right]s$ (24)
$\displaystyle{\cal A}_{\alpha;\beta}(0,t)$ $\displaystyle=0.$

Similar expressions hold for all of the other $H^{2}\psi^{2}$ operators. For
example, for the scattering of $L^{-}H$, we must specify both a flavor
superposition and an ${\rm SU}(2)_{L}$ charge vector for the lepton. Using the
completeness relation for the Pauli matrices, we find:

${\cal O}_{HL2,mn}=\frac{i}{2}(H^{\dagger
j}\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}_{\mu}H_{i})(\bar{L}_{m}^{i}\gamma^{\mu}L_{nj})-\frac{i}{4}(H^{\dagger
j}\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}_{\mu}H_{j})(\bar{L}_{m}^{i}\gamma^{\mu}L_{ni}).$
(25)

Without loss of generality, we can use the fact that the overall operator
${\cal O}_{HL2}$ is ${\rm SU}(2)$-neutral to rotate the $L$ into “up” isospin.
Doing so, we obtain:

${\cal
O}_{HL2,mn}\supset\frac{1}{4}(\phi_{2}\partial_{\mu}\phi_{1}-\phi_{1}\partial_{\mu}\phi_{2}-\phi_{4}\partial_{\mu}\phi_{3}+\phi_{3}\partial_{\mu}\phi_{4})(\bar{L}_{m}^{1}\gamma^{\mu}L_{n1}).$
(26)

The amplitude for $HL$ scattering is thus given by

$\displaystyle{\cal A}_{\alpha;\beta}=2\alpha_{m}\bar{\alpha}_{n}$
$\displaystyle\left[\left(c_{mn}^{HL1}+\frac{1}{4}c_{mn}^{HL2}\right){\rm
Im}(\beta_{1}\bar{\beta}_{2})+\left(c_{mn}^{HL1}-\frac{1}{4}c_{mn}^{HL2}\right){\rm
Im}(\beta_{3}\bar{\beta}_{4})\right]$ (27) $\displaystyle\times$
$\displaystyle\left(\langle 12\rangle[24]-\langle 13\rangle[34]\right)\\!.$

Again this amplitude vanishes for $s\to 0$.

The vanishing of the amplitudes as $s\to 0$ implies that according to Eq. (11)
the UV partial waves for $H\psi\rightarrow X$, where $\psi$ corresponds to any
(superposition of) SM fermion(s), satisfy a special sum rule:

$C_{\infty}^{(t)}=32\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}(n+1)\left\\{\left[n(n+2)+\frac{1}{2}\right]{\rm
Im}\,a_{\psi,H}^{(n)}(s)+\left[n(n+2)-\frac{1}{2}\right]{\rm
Im}\,a_{\bar{\psi},H}^{(n)}(s)\right\\}\\!.$ (28)

Since each of the partial waves has positive imaginary part, Eq. (28) implies
that either the boundary term is nonzero or that the scattering of
$H\psi\rightarrow X$ must be dominated by the $n=0$ mode (i.e., $X$ of spin
one-half), since this is the only contribution within the sum that can be
negative. If there is a symmetry in the UV such that
$a_{\alpha;\beta}=a_{\bar{\alpha};\beta}$ (for examples, see Refs. [102, 78]),
then Eq. (28) would imply that, in theories where $C_{\infty}^{(t)}$ vanishes,
only the $n=0$ term in the sum could be nonzero: the UV must be strictly
composed of only the lowest partial wave, a fermionic intermediate state.

Regardless, given that ${\cal A}_{\alpha;\beta}(0,t)=0$, the remaining
information will be conveyed by the expanded $s$-dispersion relation,

$\lim_{s\rightarrow 0}\partial_{s}{\cal
A}_{\alpha;\beta}(s,0)=32\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}(n+1)\left[{\rm
Im}\,a_{\alpha;\beta}^{(n)}(s)-{\rm
Im}\,a_{\bar{\alpha};\beta}^{(n)}(s)\right]-C_{\infty}^{(s)},$ (29)

which is a simple rewriting of Eq. (10), and therefore not a spinning sum
rule, as discussed in Sec. 2. If $a_{\alpha;\beta}=a_{\bar{\alpha};\beta}$,
then we should find that the EFT amplitude satisfies $\lim_{s\rightarrow
0}\partial_{s}{\cal A}_{\alpha;\beta}(s,0)=-C_{\infty}^{(s)}$ and therefore
can only be nontrivial if there is a boundary term. As an explicit example, we
can take the $\bar{e}^{-}H$ scattering considered above and imagine that the
SMEFT continues to describe physics in the UV, such that Eq. (23) describes
the scatterings at high energies. We can achieve
$a_{\alpha;\beta}=a_{\bar{\alpha};\beta}$ by taking CP-conserving coefficients
(i.e., $c_{mn}^{He}=c_{nm}^{He}$), and then from Eqs. (4) and (24), we have
$C_{\infty}^{(s)}=4\alpha_{m}\bar{\alpha}_{n}c_{mn}^{He}\left[{\rm
Im}(\beta_{1}\bar{\beta}_{2})+{\rm Im}(\beta_{3}\bar{\beta}_{4})\right]$, as
required to reproduce the IR result.

Explicitly, we can write the Wilson coefficients of the SMEFT operators in
terms of the sum rule in Eq. (29). For the $He$, $Hu$, and $Hd$ operators,

$\displaystyle\alpha_{m}\bar{\alpha}_{n}c_{mn}^{He}\left[{\rm
Im}(\beta_{1}\bar{\beta}_{2})+{\rm Im}(\beta_{3}\bar{\beta}_{4})\right]$
$\displaystyle=8\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}(n+1)\left[{\rm Im}\,a_{\bar{e},H}^{(n)}(s)-{\rm
Im}\,a_{e,H}^{(n)}(s)\right]-\frac{C_{\infty}^{(s)}}{4}$ (30)
$\displaystyle\alpha_{m}\bar{\alpha}_{n}c_{mn}^{Hu}\left[{\rm
Im}(\beta_{1}\bar{\beta}_{2})+{\rm Im}(\beta_{3}\bar{\beta}_{4})\right]$
$\displaystyle=8\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}(n+1)\left[{\rm Im}\,a_{\bar{u},H}^{(n)}(s)-{\rm
Im}\,a_{u,H}^{(n)}(s)\right]-\frac{C_{\infty}^{(s)}}{4}$
$\displaystyle\alpha_{m}\bar{\alpha}_{n}c_{mn}^{Hd}\left[{\rm
Im}(\beta_{1}\bar{\beta}_{2})+{\rm Im}(\beta_{3}\bar{\beta}_{4})\right]$
$\displaystyle=8\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}(n+1)\left[{\rm Im}\,a_{\bar{d},H}^{(n)}(s)-{\rm
Im}\,a_{d,H}^{(n)}(s)\right]-\frac{C_{\infty}^{(s)}}{4},$

while for $HL$ and $HQ$,

$\displaystyle\alpha_{m}\bar{\alpha}_{n}\left[\left(c_{mn}^{HL1}+\frac{1}{4}c_{mn}^{HL2}\right){\rm
Im}(\beta_{1}\bar{\beta}_{2})+\left(c_{mn}^{HL1}-\frac{1}{4}c_{mn}^{HL2}\right){\rm
Im}(\beta_{3}\bar{\beta}_{4})\right]$ (31)
$\displaystyle\qquad\qquad\qquad=8\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}(n+1)\left[{\rm Im}\,a_{\bar{L},H}^{(n)}(s)-{\rm
Im}\,a_{L,H}^{(n)}(s)\right]-\frac{C_{\infty}^{(s)}}{4}$
$\displaystyle\alpha_{m}\bar{\alpha}_{n}\left[\left(c_{mn}^{HQ1}+\frac{1}{4}c_{mn}^{HQ2}\right){\rm
Im}(\beta_{1}\bar{\beta}_{2})+\left(c_{mn}^{HQ1}-\frac{1}{4}c_{mn}^{HQ2}\right){\rm
Im}(\beta_{3}\bar{\beta}_{4})\right]$
$\displaystyle\qquad\qquad\qquad=8\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}(n+1)\left[{\rm Im}\,a_{\bar{Q},H}^{(n)}(s)-{\rm
Im}\,a_{Q,H}^{(n)}(s)\right]-\frac{C_{\infty}^{(s)}}{4}.$

As the above results are based on the conventional $s$-channel dispersion
relation, there should be a clear connection between our results and those of
Ref. [78]. After converting between bases, the authors of that work
established $s$-channel sum rules proportional to the following combinations
of Wilson coefficients (where we suppress flavor indices as that work
considered only a single generation): $c^{He}$, $c^{Hu}$, $c^{Hd}$,
$c^{HL1}\pm c^{HL2}/4$, and $c^{HQ1}\pm c^{HQ2}/4$. Each of these combinations
can be isolated from Eqs. (30) and (31) by appropriate choices of $\beta_{i}$.

###  4.3 $|h_{1}|=|h_{2}|=1/2$

For the next set of external helicities, we turn to four-fermion scattering.
This case was considered in Ref. [35], and as there we consider scattering
fermions of opposite helicity, which fixes $\lambda=1$ and $\bar{\lambda}=0$,
so that Eqs. (11) and (12) become

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha-;\beta+}(0,t)+C_{\infty}^{(t)}$ (32) $\displaystyle=$
$\displaystyle 16\int_{0}^{\infty}\frac{{\rm d}s}{s^{2}}\left\\{{\rm
Im}\,a_{\bar{\alpha}+;\beta+}^{(0)}(s)-\sum_{j=1}^{\infty}(2j+1)\left[j(j+1)-1\right]\left[{\rm
Im}\,a_{\alpha-;\beta+}^{(j)}(s)+{\rm
Im}\,a_{\bar{\alpha}+;\beta+}^{(j)}(s)\right]\right\\}$

and

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha-;\beta+}(0,t)+\lim_{s\rightarrow 0}\partial_{s}{\cal
A}_{\alpha-;\beta+}(s,0)+C_{\infty}^{(s)}+C_{\infty}^{(t)}$ (33)
$\displaystyle=$ $\displaystyle-16\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\left\\{\sum_{j=1}^{\infty}(2j\\!+\\!1)(j\\!-\\!1)(j\\!+\\!2){\rm
Im}\,a_{\alpha-;\beta+}^{(j)}(s)+\sum_{j=0}^{\infty}(2j\\!+\\!1)j(j\\!+\\!1){\rm
Im}\,a_{\bar{\alpha}+;\beta+}^{(j)}(s)\right\\}\\!.$

The right-hand side of the first relation is strictly negative except for the
contribution from ${\rm Im}\,a_{\bar{\alpha}+;\beta+}^{(0)}(s)$, and so this
relation allows for the construction of a spinning sum rule.141414For
identical-helicity scattering, so that $\lambda=0$ and $\bar{\lambda}=1$, the
sum rules we obtain are analogous to those above, but with the left-hand sides
in Eqs. (32) and (33) exchanged and with $a_{\alpha;\beta}$ and
$a_{\bar{\alpha};\beta}$ (and the appropriate helicities) exchanged on the
right-hand side. Thus, for identical-helicity scattering, we instead arrive at
a sum rule in which the only positive contribution arises from the lowest
partial wave of the uncrossed amplitude. By contrast, the right-hand side of
the second relation in Eq. (33) is of fixed sign. The definite sign suggests
the possibility of a positivity bound at a dimension-six (modulo boundary
terms), however, we will see next why this is not possible.

Let us consider a minimal example from Table 1, ${\cal O}_{e}$, and use this
operator to mediate elastic scattering between a positron and electron. We
take the positron and electron to be flavor superpositions denoted by
$\alpha_{m}$ and $\beta_{m}$ respectively. Under these conditions, the
amplitude at special kinematics takes the form (see Ref. [35] for details):

$\displaystyle{\cal A}_{\alpha-;\beta+}(s,0)$
$\displaystyle=-8\alpha_{m}\beta_{n}\bar{\beta}_{p}\bar{\alpha}_{q}c_{mnpq}^{e}s$
(34) $\displaystyle{\cal A}_{\alpha-;\beta+}(0,t)$
$\displaystyle=+8\alpha_{m}\beta_{n}\bar{\beta}_{p}\bar{\alpha}_{q}c_{mnpq}^{e}t.$

Accordingly, in this case our relation (33) becomes

$\displaystyle C_{\infty}^{(s)}+C_{\infty}^{(t)}$ (35)
$\displaystyle=-16\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\left\\{\sum_{j=1}^{\infty}(2j\\!+\\!1)(j\\!-\\!1)(j\\!+\\!2){\rm
Im}\,a_{\alpha-;\beta+}^{(j)}(s)+\sum_{j=0}^{\infty}(2j\\!+\\!1)j(j\\!+\\!1){\rm
Im}\,a_{\bar{\alpha}+;\beta+}^{(j)}(s)\right\\}\\!.$

The Wilson coefficients have vanished, removing the possibility of a
positivity bound. Indeed, as noted in Ref. [35], this is a property of the
kinematic structure of any four-fermion scattering amplitude at dimension six
and not specific to ${\cal O}_{e}$. Instead, the dispersion relation
constrains the poles at infinity in terms of the partial waves of the massive
states in the UV completion. If the UV completion is such that the boundary
contributions vanish, then as noted in Ref. [35], positivity of the partial
waves appearing in Eq. (35) implies that only the $j=1$ partial wave can
contribute to $a_{\alpha-;\beta+}^{(j)}$ and only the $j=0$ counterpart can
contribute to $a_{\bar{\alpha}+;\beta+}^{(j)}$; that is, if the boundary terms
vanish, the completion must be at tree level (at leading order in a weakly
coupled theory) and contain only massive scalars and vectors. In that case,
Eq. (32) becomes

$\alpha_{m}\beta_{n}\bar{\beta}_{p}\bar{\alpha}_{q}c_{mnpq}^{e}=2\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\left[{\rm Im}\,a_{\bar{\alpha}+;\beta+}^{(0)}(s)-3\,{\rm
Im}\,a_{\alpha-;\beta+}^{(1)}(s)\right]\\!,$ (36)

the sign of which is determined by whether scalars or vectors dominate the UV.
This result is the detailed version of the qualitative relation we gave in Eq.
(2), and again we emphasize that the sign is reversed as compared to the
scalar spinning sum rules of Sec. 4.1. Similar spinning sum rules and
positivity bounds apply to the other four-fermion operators listed in Table 1.
See Ref. [35] for details, as well as a discussion of the importance of these
sum rules for flavor and CP violation.

We can again derive a sum rule using the forward dispersion relation in Eq.
(10). This is exactly the case considered in Ref. [78], in which a sum rule is
established on $2c^{e}$, where again flavor indices are suppressed as it was
not considered in that work. Our approach for Eq. (10) would establish
identical relations, but with
$c^{e}\to\alpha_{m}\beta_{n}\bar{\beta}_{p}\bar{\alpha}_{q}c_{mnpq}^{e}$, so
that relations between different flavor structures can be derived.

###  4.4 $h_{1}=\pm 1$, $h_{2}=0$

Turning to the scattering of a gauge boson in a definite-helicity state off of
a Higgs, we can mechanically derive the appropriate relations from Eqs. (11)
and (12). Doing so results in a positivity bound and a mixed-sign sum rule (up
to boundary terms), namely,

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha;\beta}(0,t)+C_{\infty}^{(t)}$ (37)
$\displaystyle=16\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{j=1}^{\infty}(2j+1)\left\\{[j(j+1)-1]{\rm
Im}a_{\alpha;\beta}^{(j)}(s)+[j(j+1)-2]\,{\rm
Im}a_{\bar{\alpha};\beta}^{(j)}(s)\right\\}$ $\displaystyle>0$

and

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha;\beta}(0,t)+\lim_{s\rightarrow 0}\partial_{s}{\cal
A}_{\alpha;\beta}(s,0)+C_{\infty}^{(s)}+C_{\infty}^{(t)}$ (38)
$\displaystyle=16\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{j=1}^{\infty}(2j+1)\left\\{j(j+1)\,{\rm
Im}a_{\alpha\beta}^{(j)}(s)+[j(j+1)-3]{\rm
Im}a_{\bar{\alpha}\beta}^{(j)}(s)\right\\}\\!.$

In order to make use of these relations, however, we need to compute the
relevant SMEFT amplitudes. By direct calculation, the amplitude for the
hypercharge $B$ gauge field and a general superposition of the Higgs
$\phi_{i}$ via the operators in Table 1 is given by

${\cal A}=2\beta_{i}\gamma_{i}\left\\{c^{HB}\left[(e_{1}\cdot
e_{4})t+2(e_{1}\cdot p_{4})(e_{4}\cdot
p_{1})\right]+2\widetilde{c}^{HB}e_{1}^{\mu}e_{4}^{\nu}p_{1}^{\rho}p_{4}^{\sigma}\epsilon_{\mu\nu\rho\sigma}\right\\}\\!,$
(39)

where the gauge boson corresponds to states 1 and 4, $e_{i}$ and $p_{i}$ are
the polarization and momentum four-vectors for particle $i$, and $\beta_{i}$
and $\gamma_{i}$ dictate the superpositions of the initial and final Higgs.
Choosing definite-helicity states for the gauge field to correspond to elastic
scattering, so that $h_{1}=-h_{4}=\pm 1$, little-group scaling arguments imply
that the contact amplitude for dimension-six scalar-vector scattering must
vanish [78]. It is therefore not possible to constrain the $H^{2}B^{2}$ (and
similarly the $H^{2}W^{2}$ and $H^{2}G^{2}$) dimension-six Wilson coefficients
with either conventional or spinning sum rules.

As the elastic, fixed-helicity dimension-six amplitude vanishes, Eq. (37)
implies that

$\displaystyle C_{\infty}^{(t)}=\,16\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{j=1}^{\infty}(2j+1)\left\\{[j(j+1)-1]{\rm
Im}a_{\alpha;\beta}^{(j)}(s)+[j(j+1)-2]\,{\rm
Im}a_{\bar{\alpha};\beta}^{(j)}(s)\right\\}>\,0.$ (40)

If the pole at infinity is present, it must therefore be positive. Instead, if
it vanishes, then all partial cross sections of the amplitude must vanish
except ${\rm Im}\,a_{\bar{\alpha};\beta}^{(1)}$. If we further assume
$C_{\infty}^{(s)}=0$, so that the UV generates neither boundary term, then
Eqs. (37) and (38) imply that all imaginary parts of partial waves must
vanish. As an explicit example, a simple UV extension of ${\cal O}_{HB}$ is
given by $y_{1}\varphi H^{\dagger}H+y_{2}\varphi B_{\mu\nu}B^{\mu\nu}$ for a
real massive scalar $\varphi$. Integrating out $\varphi$ from the action, one
generates $(H^{\dagger}H)B_{\mu\nu}B^{\mu\nu}$ with Wilson coefficient
$c^{HB}=y_{1}y_{2}/m_{\varphi}^{2}$. The indefinite sign of the coefficient is
representative of the fact that there is no spinning sum rule for these
operators.

###  4.5 $|h_{1}|=|h_{2}|=1$

Finally, we consider scattering of gauge bosons via the four $F^{3}$ operators
in Table 1. For same-helicity scattering, $h_{1}=h_{2}=\pm 1$, we have the
spinning sum rules151515Here we use $\pm$ subscripts to indicate the
helicities of the two initial bosons.

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha,\pm;\beta,\pm}(0,t)+C_{\infty}^{(t)}$
$\displaystyle=16\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\Biggl{\\{}\sum_{j=0}^{\infty}(2j+1)j(j+1){\rm
Im}\,a_{\alpha,\pm;\beta,\pm}^{FF(j)}(s)$ (41) $\displaystyle\hskip
62.59596pt+\sum_{j=2}^{\infty}(2j+1)\left[j(j+1)-5\right]{\rm
Im}\,a_{\bar{\alpha},\mp;\beta,\pm}^{FF(j)}(s)\Biggr{\\}}$ $\displaystyle\,>0$

and

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha,\pm;\beta,\pm}(0,t)+\lim_{s\rightarrow 0}\partial_{s}{\cal
A}_{\alpha,\pm;\beta,\pm}(s,0)+C_{\infty}^{(s)}+C_{\infty}^{(t)}$ (42)
$\displaystyle=$ $\displaystyle 16\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\Biggl{\\{}\sum_{j=0}^{\infty}(2j+1)\left[j(j+1)+1\right]{\rm
Im}\,a_{\alpha,\pm;\beta,\pm}^{FF(j)}(s)$ $\displaystyle\hskip
46.37813pt+\sum_{j=2}^{\infty}(2j+1)\left[j(j+1)-6\right]{\rm
Im}\,a_{\bar{\alpha},\mp;\beta,\pm}^{FF(j)}(s)\Biggr{\\}}$
$\displaystyle\,>0.$

For opposite-helicity scattering, $h_{1}=-h_{2}=\pm 1$, we find the sum rules

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha,\pm;\beta,\mp}(0,t)+C_{\infty}^{(t)}$
$\displaystyle\,=16\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\Biggl{\\{}\sum_{j=2}^{\infty}(2j+1)\left[j(j+1)-4\right]{\rm
Im}\,a_{\alpha,\pm;\beta,\mp}^{FF(j)}(s)$ (43) $\displaystyle\hskip
64.30307pt+\sum_{j=0}^{\infty}(2j+1)\left[j(j+1)-1\right]{\rm
Im}\,a_{\bar{\alpha},\mp;\beta,\mp}^{FF(j)}(s)\Biggr{\\}}$

and

$\displaystyle\lim_{t\rightarrow 0}\partial_{t}{\cal
A}_{\alpha,\pm;\beta,\mp}(0,t)+\lim_{s\rightarrow 0}\partial_{s}{\cal
A}_{\alpha,\pm;\beta,\mp}(s,0)+C_{\infty}^{(s)}+C_{\infty}^{(t)}$ (44)
$\displaystyle=$ $\displaystyle 16\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\Biggl{\\{}\sum_{j=2}^{\infty}(2j+1)\left[j(j+1)-3\right]{\rm
Im}\,a_{\alpha,\pm;\beta,\mp}^{FF(j)}(s)$ $\displaystyle\hskip
46.37813pt+\sum_{j=0}^{\infty}(2j+1)\left[j(j+1)-2\right]{\rm
Im}\,a_{\bar{\alpha},\mp;\beta,\mp}^{FF(j)}(s)\Biggr{\\}}.$

We note that while Eqs. (41) and (42) establish relations of definite sign and
suggest a possible positivity relation (up to boundary terms), Eqs. (43) and
(44) are of indefinite sign.

The $F^{3}$ operators in the SMEFT generate both cubic and quartic vertices,
with the cubic vertices contributing to four-point scattering via tree
diagrams involving single-gluon exchange. Analogous diagrams exist in the pure
Yang-Mills sectors of the SM itself. Let us choose the color vectors
$t_{1,2,3,4}$ to be in an elastic configuration, $t_{1}^{a}=\bar{t}_{4}^{a}$
and $t_{2}^{a}=\bar{t}_{3}^{a}$. We avoid kinematic singularities from the
cubic exchange diagrams when the $s\rightarrow 0$ or $t\rightarrow 0$ limits
are taken by choosing colors that commute, i.e.,
$f^{abc}t_{1}^{a}t_{2}^{b}=0$, in which case both the SM amplitude and
corrected SMEFT amplitudes vanish identically (cf. Ref. [14]). As the EFT
amplitude vanishes, the dispersion relation then implies that either the poles
at infinity must be non-vanishing or else by Eqs. (41) and (42) all imaginary
parts of partial waves—i.e., the contributions to the amplitude from the UV
states—are zero. This is in keeping with the observation—made in Ref. [78] via
little-group scaling arguments and the on-shell formalism—that elastic vector-
vector scattering at dimension six must vanish.

##  5 Boundary Terms and UV Completions

Boundary terms are a fundamental aspect of dispersion relations at dimension
six and a fundamental obstruction to the clean UV-IR connection spinning sum
rules would otherwise provide. The complete classification of the conditions
required for a UV theory to generate a pole at infinity is not unknown. We
will not attempt an exhaustive analysis in this work. Instead, in this section
we will study various weakly coupled UV completions of our four-scalar and
four-fermion spinning sum rules, focusing exclusively on tree-level
completions; loop-level completions are left as a particularly interesting
open direction. Our goal is to build intuition for the role the boundary terms
that appear in Eqs. (11) and (12) play in the dispersion relations and the
properties of theories in which they can be neglected.

There have been a number of discussions of the boundary term associated with
the forward dispersion relation, $C_{\infty}^{(s)}$. For instance, Refs. [75,
76] have argued that the generic expectation is that such a pole does not
arise from strongly coupled UV theories, whereas for weakly coupled theories,
massive vectors exchanged in the $t$ channel have been identified as a
canonical source of a pole at infinity (see, e.g., Refs. [76, 78]). We have
already discussed the connection between $t$-channel exchanges and boundary
contributions in Sec. 2: if such terms contribute to the EFT, then as they
cannot be reproduced from the dispersive integrals, they must appear as poles
at infinity. This connection will extend to the beyond-forward boundary term
$C_{\infty}^{(t)}$. Indeed, we will see that a scalar exchanged in the $t$
channel can generate $C_{\infty}^{(t)}$, while leaving $C_{\infty}^{(s)}=0$,
whereas only a massive vector generates a non-vanishing $C_{\infty}^{(s)}$.

###  5.1 UV completions of $H^{4}$ operators

We begin by considering obstructions to the spinning sum rules introduced for
${\cal O}_{H1}$ and ${\cal O}_{H2}$ in Sec. 4.1. We do so by considering two
tree-level completions of these operators, drawing from the exhaustive list
enumerated in Ref. [100]. In particular, we will study a real singlet scalar
$S$ and a complex vector $V$ that is a singlet under ${\rm SU}(3)_{C}$ and
${\rm SU}(2)_{L}$, but carries unit hypercharge.161616In the notation of Ref.
[100], $S$ and $V$ correspond to ${\cal S}$ and ${\cal B}_{1}$, respectively.
The first of these will violate our spinning sum rules, whereas the latter
satisfies them, and we will isolate the violation as explicitly associated
with a $t$-channel exchange and boundary terms. More generally, we will use
these completions as an opportunity to demonstrate in detail how the
dispersion relations are satisfied and where the boundary terms play a key
role.

####  5.1.1 Scalar Completion

The singlet scalar can couple to the Higgs at the renormalizable and
dimension-five level as171717The theory in Eq. (45) is manifestly non-
renormalizable and therefore should be considered as a UV extension—a theory
that improves the momentum scaling of amplitudes in the UV and therefore
raises the cutoff—rather than a UV completion. Yet the theory is causal and
unitary, and therefore must be—and in fact is—consistent with our dispersion
relations.

${\cal L}\supset-\mu\,SH^{\dagger}H+y\,SD_{\mu}H^{\dagger}D^{\mu}H.$ (45)

Observe that $\mu$ and $y$ have unit and inverse mass dimensions,
respectively, and further that hermiticity requires that they both be real.
Other couplings, even at the renormalizable level, are allowed, but play no
role in the generation of the dimension-six Higgs operators we are
considering, so they are irrelevant to our discussion. Integrating out $S$ at
the level of the action generates ${\cal O}_{H1}$ and ${\cal O}_{H2}$ with the
Wilson coefficients

$c^{H1}=\frac{\mu^{2}}{2m_{\scriptscriptstyle S}^{4}}-\frac{\mu
y}{2m_{\scriptscriptstyle S}^{2}}\qquad{\rm and}\qquad c^{H2}=0.$ (46)

As $\mu$ and $y$ are of indefinite sign, so too is $c^{H1}$, and therefore
this completion cannot satisfy the expectations of the spinning sum rule,
which from Fig. 1 would require $c^{H1}<0$ for a scalar completion.

To study this case further, we require the UV amplitude for four-Higgs
scattering. The amplitude receives contributions from $s$-, $t$-, and
$u$-channel exchanges, where the vertices can be associated with either the
$\mu$ or $y$ couplings. Explicitly, choosing the scattering states in Eq.
(15), we obtain the following elastic amplitude:

${\cal A}(s,t)=\frac{(s\,\mu
y-\mu^{2})|\alpha\cdot\beta|^{2}}{s-m_{\scriptscriptstyle
S}^{2}+i\epsilon}+\frac{(t\,\mu
y-\mu^{2})|\alpha|^{2}|\beta|^{2}}{t-m_{\scriptscriptstyle
S}^{2}+i\epsilon}+\frac{(u\,\mu
y-\mu^{2})|\alpha\cdot\bar{\beta}|}{u-m_{\scriptscriptstyle
S}^{2}+i\epsilon}.$ (47)

Expanding this result for $m_{\scriptscriptstyle S}^{2}\gg p^{2}$ and
comparing with the EFT amplitude in Eq. (16) readily confirms the matching in
Eq. (46).

As an aside, note that in Eq. (47) we have not included terms at ${\cal
O}(y^{2})$. These contributions are present, and they can be trivially
included by modifying the $s$-channel numerator with $(s\,\mu
y-\mu^{2})\to-(s\,y/2-\mu)^{2}$ and similarly for other channels. We have
neglected the correction, however, as it only generates Higgs EFT operators at
dimension eight. Nevertheless, it is only after this contribution is added
that one would obtain a positive ${\rm Im}\,{\cal A}(s,0)$.

From Eq. (47), we can study the IR and UV sides of the dispersion relations.
First, consider the forward relation in Eq. (3). The IR contribution is given
by

$\lim_{s\to 0}\partial_{s}{\cal
A}(s,0)=\left(\frac{\mu^{2}}{m_{\scriptscriptstyle S}^{4}}-\frac{\mu
y}{m_{\scriptscriptstyle
S}^{2}}\right)(|\alpha\cdot\beta|^{2}-|\alpha\cdot\bar{\beta}|^{2}).$ (48)

Comparing this with the UV amplitude, it is clear that the $t$-channel
contribution has no low-energy remnant. As such, we would expect
$C_{\infty}^{(s)}=0$, which is confirmed by direct calculation from Eq. (4).
The UV then only needs to account for the $s$ and $u$ channels, which it does
so as

$\displaystyle{\rm Im}\,{\cal A}(s,0)$
$\displaystyle=\pi(\mu^{2}-m_{\scriptscriptstyle S}^{2}\mu
y)|\alpha\cdot\beta|^{2}\delta(s-m_{\scriptscriptstyle S}^{2})+\cdots$ (49)
$\displaystyle{\rm Im}\,\bar{\cal A}(s,0)$
$\displaystyle=\pi(\mu^{2}-m_{\scriptscriptstyle S}^{2}\mu
y)|\alpha\cdot\bar{\beta}|^{2}\delta(s-m_{\scriptscriptstyle S}^{2})+\cdots,$

where the ellipses contain only terms that do not have support in the
dispersive integrals over the positive $s$-axis.

We next consider the beyond-forward dispersion relation of Eq. (5), on which
the spinning sum rules were based. The EFT now records an explicit
contribution from the $t$ channel $\propto|\alpha|^{2}|\beta|^{2}$,

$\lim_{t\to 0}\partial_{t}{\cal
A}(0,t)=\left(\frac{\mu^{2}}{m_{\scriptscriptstyle S}^{4}}-\frac{\mu
y}{m_{\scriptscriptstyle
S}^{2}}\right)(|\alpha|^{2}|\beta|^{2}-|\alpha\cdot\bar{\beta}|^{2}).$ (50)

This contribution is—and can only be—reconstructed by a boundary term, which
from Eq. (6) is given by

$C_{\infty}^{(t)}=-\left(\frac{\mu^{2}}{m_{\scriptscriptstyle
S}^{4}}-\frac{\mu y}{m_{\scriptscriptstyle
S}^{2}}\right)|\alpha|^{2}|\beta|^{2}.$ (51)

To confirm the full dispersion relation, note that the integral can be
rewritten in terms of three terms,

$\displaystyle\frac{1}{\pi}\int_{0}^{\infty}{\rm d}s\lim_{t\rightarrow
0}\partial_{t}\left[\frac{{\rm Im}\,{\cal A}(s,t)}{s}+\frac{{\rm
Im}\,\bar{\cal A}(s,t)}{s+t}\right]$ (52) $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{0}{\rm d}s\left[\frac{\lim_{t\to
0}\partial_{t}{\rm Im}\,{\cal A}(s,t)}{s}+\frac{\lim_{t\to 0}\partial_{t}{\rm
Im}\,\bar{\cal A}(s,t)}{s}-\frac{{\rm Im}\,\bar{\cal
A}(s,0)}{s^{2}}\right]\\!.$

Given the range of integration, only terms proportional to
$\delta(s-m_{\scriptscriptstyle S}^{2})$ will contribute, but as in the
amplitude such terms are independent of $t$, the $t$ derivatives will cause
them to vanish. Accordingly, only the final term contributes, and using Eq.
(49) we see that this reproduces the $u$-channel contribution, so Eq. (5) is
indeed satisfied.

In summary, a scalar exchanged in the $t$ channel can generate the beyond-
forward boundary term, even when $C_{\infty}^{(s)}=0$, as claimed. The
presence of this boundary term, which is of indefinite sign, violates the UV
scaling assumptions needed for spinning sum rules, explaining why this
completion can generate a positive $c^{H1}$, which would otherwise be
suggestive of a vector in the UV.

####  5.1.2 Vector Completion

We next study a UV completion that satisfies the spinning sum rules: a vector
$V$ with unit hypercharge. The relevant interactions take the form

${\cal L}\supset\lambda\,V^{\mu\dagger}(i\partial_{\mu}H^{T})i\sigma_{2}H+{\rm
h.c.},$ (53)

such that when the vector is integrated out, we obtain

$c^{H1}=\frac{|\lambda|^{2}}{2m_{\scriptscriptstyle V}^{2}}\qquad{\rm
and}\qquad c^{H2}=-\frac{|\lambda|^{2}}{m_{\scriptscriptstyle V}^{2}}.$ (54)

These coefficients lead to a positive value for all the spinning sum rules
considered in Sec. 4.1, which is consistent with the prediction for the signal
a UV vector would leave.

Nonetheless, it is still illuminating to study how the dispersion relations
operate in detail, and we will see that this completion generates both
boundary terms and yet can still satisfy the conditions for the spinning sum
rules. Our starting point is the four-Higgs UV amplitude, which is now given
by

$\displaystyle{\cal A}(s,t)=\frac{|\lambda|^{2}}{4}\left[\frac{(u-t){\cal
C}_{s}^{\alpha\beta}}{s-m_{\scriptscriptstyle
V}^{2}+i\epsilon}+\frac{(u-s){\cal
C}_{t}^{\alpha\beta}}{t-m_{\scriptscriptstyle
V}^{2}+i\epsilon}+\frac{(s-t){\cal
C}_{u}^{\alpha\beta}}{t-m_{\scriptscriptstyle V}^{2}+i\epsilon}\right]\\!.$
(55)

We have introduced a shorthand notation for the Higgs flavor structure,
obtained by first defining

$\displaystyle{\cal C}(\alpha,\beta,\gamma,\delta)=$
$\displaystyle\,\Big{\\{}[(\alpha_{3}-i\alpha_{4})(\beta_{1}-i\beta_{2})-(\alpha_{1}-i\alpha_{2})(\beta_{3}-i\beta_{4})]$
(56)
$\displaystyle\;\;\;\;\;\;\times[(\delta_{3}+i\delta_{4})(\gamma_{1}+i\gamma_{2})-(\delta_{1}+i\delta_{2})(\gamma_{3}+i\gamma_{4})]\Big{\\}}$
$\displaystyle+\Big{\\{}(\alpha,\beta)\leftrightarrow(\delta,\gamma)\Big{\\}},$

in terms of which we can write

$\displaystyle{\cal C}_{s}^{\alpha\beta}={\cal
C}(\alpha,\beta,\bar{\beta},\bar{\alpha}),\hskip 14.22636pt{\cal
C}_{t}^{\alpha\beta}={\cal C}(\alpha,\bar{\alpha},\bar{\beta},\beta),\hskip
14.22636pt{\cal C}_{u}^{\alpha\beta}={\cal
C}(\alpha,\bar{\beta},\beta,\bar{\alpha}).$ (57)

These flavor structures satisfy a number of relations. First, all three
combinations are real. Under crossing, they transform as ${\cal
C}_{s}^{\bar{\alpha}\beta}={\cal C}_{u}^{\alpha\beta}$ and ${\cal
C}_{t}^{\bar{\alpha}\beta}=-{\cal C}_{t}^{\alpha\beta}$, so that the full
amplitude is crossing symmetric. Additionally, while ${\cal
C}_{t}^{\alpha\beta}$ is of indefinite sign, both ${\cal C}_{s}^{\alpha\beta}$
and ${\cal C}_{u}^{\alpha\beta}$ are positive for all $\alpha$ and $\beta$,
which ensures that Eq. (55) satisfies ${\rm Im}\,{\cal A}(s,0)>0$. Finally,
one can confirm that taking $m_{\scriptscriptstyle V}^{2}\gg p^{2}$ in Eq.
(55) and matching onto Eq. (16), the flavors are such that the Wilson
coefficients in Eq. (54) are reproduced.

Turning to the dispersion relations, for the forward expression, in the IR we
have

$\lim_{s\to 0}\partial_{s}{\cal
A}(s,0)=\frac{|\lambda|^{2}}{4m_{\scriptscriptstyle V}^{2}}\left({\cal
C}_{s}^{\alpha\beta}+2{\cal C}_{t}^{\alpha\beta}-{\cal
C}_{u}^{\alpha\beta}\right)\\!,$ (58)

which receives contributions from all three channels. In the UV, the $s$ and
$u$ channels are recreated from the dispersive integral using

$\displaystyle{\rm Im}\,{\cal A}(s,0)$
$\displaystyle=\frac{\pi|\lambda|^{2}m_{\scriptscriptstyle V}^{2}}{4}{\cal
C}_{s}^{\alpha\beta}\delta(s-m_{\scriptscriptstyle V}^{2})+\cdots$ (59)
$\displaystyle{\rm Im}\,\bar{\cal A}(s,0)$
$\displaystyle=\frac{\pi|\lambda|^{2}m_{\scriptscriptstyle V}^{2}}{4}{\cal
C}_{u}^{\alpha\beta}\delta(s-m_{\scriptscriptstyle V}^{2})+\cdots,$

whereas the $t$ channel is reconstructed from the boundary term,

$C_{\infty}^{(s)}=-\frac{|\lambda|^{2}}{2m_{\scriptscriptstyle V}^{2}}{\cal
C}_{t}^{\alpha\beta}.$ (60)

Consistency of the beyond-forward calculation requires a contribution from all
parts of the dispersion relation. Again the EFT result receives contributions
from all channels,

$\lim_{t\to 0}\partial_{t}{\cal
A}(0,t)=\frac{|\lambda|^{2}}{4m_{\scriptscriptstyle V}^{2}}\left(2\,{\cal
C}_{s}^{\alpha\beta}+{\cal C}_{t}^{\alpha\beta}+{\cal
C}_{u}^{\alpha\beta}\right)\\!.$ (61)

The integral recovers the non-$t$-channel contributions,

$\displaystyle\frac{1}{\pi}\int^{\infty}_{0}{\rm d}s\left[\frac{\lim_{t\to
0}\partial_{t}{\rm Im}\,{\cal A}(s,t)}{s}+\frac{\lim_{t\to 0}\partial_{t}{\rm
Im}\,\bar{\cal A}(s,t)}{s}\right]$
$\displaystyle=\frac{|\lambda|^{2}}{2m_{\scriptscriptstyle V}^{2}}\left({\cal
C}_{s}^{\alpha\beta}+{\cal C}_{u}^{\alpha\beta}\right)$ (62)
$\displaystyle-\frac{1}{\pi}\int^{\infty}_{0}\frac{{\rm d}s}{s^{2}}{\rm
Im}\,\bar{\cal A}(s,0)$
$\displaystyle=-\frac{|\lambda|^{2}}{4m_{\scriptscriptstyle V}^{2}}{\cal
C}_{u}^{\alpha\beta},\hskip 113.81102pt$

leaving consistency hinging on the pole at infinity, which is correctly
reproduced by

$\displaystyle C_{\infty}^{(t)}=-\frac{|\lambda|^{2}}{4m_{\scriptscriptstyle
V}^{2}}{\cal C}_{t}^{\alpha\beta}.$ (63)

Thus, the vector completion satisfies both dispersion relations and the
spinning sum rules, and yet does so while generating both boundary terms. This
naively seems inconsistent with the obstruction a pole at infinity provides to
the arguments in Sec. 4.1. The resolution is that the calculation above was
performed for general $\alpha$ and $\beta$, whereas the explicit spinning sum
rules derived always took specific values of the flavor coefficients. Indeed,
the calculation leading to Eq. (19) took $\beta=(1,0,0,0)$, which gives ${\cal
C}_{t}^{\alpha\beta}=0$ and therefore vanishing boundary terms. The choice
$\alpha=(1,-i,0,0)/\sqrt{2}$ and $\beta=(0,0,i,1)/\sqrt{2}$, which was used to
generate a spinning sum rule for $2c^{H1}-c^{H2}$, also leads to ${\cal
C}_{t}^{\alpha\beta}=0$. More generally, it is possible to choose flavors such
that the boundary terms do not vanish. For such cases, there are no guarantees
that $\lim_{t\to 0}\partial_{t}{\cal A}(0,t)$ will be positive. Nevertheless,
it turns out that $2\,{\cal C}_{s}^{\alpha\beta}+{\cal
C}_{t}^{\alpha\beta}+{\cal C}_{u}^{\alpha\beta}>0$ for all possible flavors,
so it is simply the case that for this UV completion, while the boundary terms
could impose an obstruction, in practice they do not.

###  5.2 UV completions of $\psi^{4}$ operators

We now turn to UV completions of dimension-six four-fermion operators, thereby
exploring the applicability of the spinning sum rules derived in Sec. 4.3. In
Ref. [35], where these fermionic spinning sum rules were first introduced, a
number of explicit UV completions were considered, and the expected result was
demonstrated that a violation of the UV-IR connection arises for $t$-channel
UV completions. Rather than consider explicit UV completions, in this section
we will be more general. In particular, we will consider how our formalism
applies to tree-level completions in the $s$, $t$, and $u$ channels, without
specifying the particular UV Lagrangian generating such completions. As we
restrict our attention to opposite-helicity scattering, there is an explicit
connection between the channel and the spin of the exchanged state; the $s$
and $t$ channels are associated with vectors, and the $u$ channel scalars.
Nevertheless, beyond what can be inferred from conservation of angular
momentum, we remain UV-agnostic. Doing so will allow us to disentangle the UV
behavior and boundary terms as general features of particular classes of
completions, independent of their details, and will also allow the discussion
to sidestep a number of sign-related subtleties associated with fermionic
scattering. Again, the purpose of the exercise is not only to demonstrate
where boundary terms can arise at tree level, but also to furnish several
explicit examples of a complete implementation of the sum rules, where both
sides of the expressions can be manifestly evaluated.

####  5.2.1 $s$-channel completion

Our focus in this section is tree-level completions of dimension-six four-
fermion operators. As appropriate for the sum rules considered in Sec. 4.3, we
focus on opposite-helicity scattering. To be explicit, let us consider the
scattering of a right-handed chiral fermion $\psi$ (an example in the SM would
be the right-handed electron). An amplitude to consider that satisfies all our
requirements is ${\cal
A}(\bar{\psi}^{-}_{1}\psi^{+}_{2}\bar{\psi}^{-}_{3}\psi^{+}_{4})$.181818Again,
we are working in the all-incoming convention. The helicity structure of the
external fermions ensures that any amplitude for this process will be
proportional to $[13]\langle 24\rangle$. Accordingly, we can write a generic
tree-level completion in the $s$ channel as

${\cal A}(s,t)=\frac{c_{s}\,[13]\langle 24\rangle}{s-m^{2}+i\epsilon}.$ (64)

Here, $c_{s}$ is an arbitrary constant that could depend on the additional
quantum numbers carried by the fermions (an explicit example can be found in
Ref. [35]). Nevertheless, we can fix the sign of $c_{s}$ as positive from the
optical theorem. Taking the forward limit $t\to 0$ ($p_{4,3}\to-p_{1,2}$), we
have

$\lim_{t\to 0}{\cal A}(s,t)=\frac{-c_{s}s}{s-m^{2}+i\epsilon},$ (65)

so that the imaginary part of the forward amplitude is $\pi
c_{s}s\,\delta(s-m^{2})$, and we require $c_{s}>0$, as claimed.

Let us now show explicitly the consistency of the dispersion relations used to
derive our sum rules for this completion, as we did for the Higgs completions
above. First, our forward dispersion relation in Eq. (3) gives

$\lim_{s\rightarrow 0}\partial_{s}{\cal
A}(s,0)=\frac{1}{\pi}\int_{s_{0}}^{\infty}\frac{{\rm d}s}{s^{2}}\left[{\rm
Im}\,{\cal A}(s,0)-{\rm Im}\,\bar{\cal A}(s,0)\right]-C_{\infty}^{(s)},$ (66)

where again $C_{\infty}^{(s)}=-\lim_{|s|\rightarrow\infty}{\cal A}(s,0)/s$.
The left-hand side of this expression is understood to be computed at low
energies, where $p^{2}\ll m^{2}$, so that the EFT description is appropriate.
In this simple example, our EFT amplitude is given by ${\cal
A}(s,t)=-(c_{s}/m^{2})[13]\langle 24\rangle$, so that ${\cal
A}(s,0)=(c_{s}/m^{2})s$, and the low-energy contribution to the dispersion
relation is

$\lim_{s\rightarrow 0}\partial_{s}{\cal A}(s,0)=\frac{c_{s}}{m^{2}}.$ (67)

Turning to the contributions that should be evaluated in the UV completion, we
see immediately from Eq. (65) that $C_{\infty}^{(s)}=0$. In order to evaluate
the integral, we require the amplitude with particles 2 and 3 crossed, given
by

$\bar{\cal A}(s,t)=\frac{c_{s}\,[12]\langle 34\rangle}{u-m^{2}+i\epsilon}.$
(68)

Note that we have ${\rm Im}\,\bar{\cal A}(s,0)=-\pi
c_{s}s\,\delta(s+m^{2})>0$, as required. We can further use this result to
evaluate the dispersive integral,

$\frac{1}{\pi}\int_{s_{0}}^{\infty}\frac{{\rm d}s}{s^{2}}\left[{\rm Im}\,{\cal
A}(s,0)-{\rm Im}\,\bar{\cal
A}(s,0)\right]=c_{s}\int_{s_{0}}^{\infty}\frac{{\rm
d}s}{s}\left[\delta(s-m^{2})+\delta(s+m^{2})\right]=\frac{c_{s}}{m^{2}},$ (69)

in agreement with Eq. (67).

We now move on to confirm the beyond-forward dispersion relation we exploited,
Eq. (5), which we restate for convenience, specializing to the fermion
scattering we are considering,

$\lim_{t\rightarrow 0}\partial_{t}{\cal
A}(0,t)=-C_{\infty}^{(t)}-\frac{1}{\pi}\int_{0}^{\infty}{\rm
d}s\lim_{t\rightarrow 0}\partial_{t}\left[\frac{{\rm Im}\,{\cal
A}(s,t)}{s}+\frac{{\rm Im}\,\bar{\cal A}(s,t)}{s+t}\right]\\!,$ (70)

where the boundary contribution is now given by $C_{\infty}^{(t)}=-{\rm
Res}\left[\lim_{t\rightarrow 0}\partial_{t}{\cal A}(s,t)/s,\,s=\infty\right]$.
From Eq. (64), and taking $m^{2}\gg p^{2}$, we can evaluate the EFT
contribution immediately as $\lim_{t\rightarrow 0}\partial_{t}{\cal
A}(0,t)=-c_{s}/m^{2}$. For the expressions evaluated in the UV completion, we
need an explicit form for the spinor-helicity expressions in terms of
Mandelstam invariants. These are provided by $[13]\langle 24\rangle=u\Phi_{u}$
and $[12]\langle 34\rangle=s\Phi_{s}$. We caution that the first of these
results cannot be evaluated for $s\to 0$ at finite $t$, as this corresponds to
a crossing that the scalar expression $u$ cannot account for. The little-group
phases of the fermions are encoded in
$\Phi_{u}=e^{i(\varphi_{1}-\varphi_{2}+\varphi_{3}-\varphi_{4})}$ and
$\Phi_{s}=e^{i(\varphi_{1}+\varphi_{2}-\varphi_{3}-\varphi_{4})}$, where
$\varphi_{i}$ is the little-group phase associated with particle $i$. In the
elastic forward limit, these reduce to $\Phi_{u}=\Phi_{s}=1$. Accordingly, we
have

$\displaystyle-\frac{1}{\pi}\int_{0}^{\infty}{\rm d}s\lim_{t\rightarrow
0}\partial_{t}\left[\frac{{\rm Im}\,{\cal A}(s,t)}{s}+\frac{{\rm
Im}\,\bar{\cal A}(s,t)}{s+t}\right]$ (71)
$\displaystyle=-c_{s}\int_{0}^{\infty}{\rm d}s\lim_{t\rightarrow
0}\partial_{t}\left[\frac{(s+t)\delta(s-m^{2})\Phi_{u}}{s}-\frac{s\delta(s+t+m^{2})\Phi_{s}}{s+t}\right]$
$\displaystyle=-\frac{c_{s}}{m^{2}}.$

Consistency then requires that the boundary condition cannot contribute, and
indeed for the UV amplitude $C_{\infty}^{(t)}=0$. Finally, as the boundary
terms both vanish, the negativity of $\lim_{t\rightarrow 0}\partial_{t}{\cal
A}(0,t)$ is indicative of the UV theory being dominated by vector exchange.
This is consistent with the fact that for opposite-helicity scattering, the
state exchanged in the $s$ channel must have unit angular momentum.

####  5.2.2 $u$-channel completion

The analogous case for a $u$-channel UV completion is almost identical. The
relevant amplitudes are now

$\displaystyle{\cal A}(s,t)$ $\displaystyle=\frac{-c_{u}\,[13]\langle
24\rangle}{u-m^{2}+i\epsilon}$ (72) $\displaystyle\bar{\cal A}(s,t)$
$\displaystyle=\frac{-c_{u}\,[12]\langle 34\rangle}{s-m^{2}+i\epsilon},$

where the sign of the constant $c_{u}>0$ can again be fixed by the optical
theorem. Once more, both boundary terms vanish, and an identical computation
reveals that the dispersion relations are satisfied, although this time the
dispersive integral receives a contribution from $\bar{\cal A}(s,t)$, rather
than ${\cal A}(s,t)$. In the EFT, we can compute $\lim_{t\rightarrow
0}\partial_{t}{\cal A}(0,t)=c_{u}/m^{2}>0$, consistent with a scalar exchange,
as expected for this channel.

####  5.2.3 $t$-channel completion

Finally, let us consider a $t$-channel completion, which will furnish an
explicit example that generates boundary contributions for a fermionic UV
completion. The amplitude is now given by

${\cal A}(s,t)=\frac{-c_{t}\,[13]\langle 24\rangle}{t-m^{2}+i\epsilon}.$ (73)

The imaginary part of the associated forward amplitude vanishes, as ${\rm
Im}\,{\cal A}(s,0)=-\pi c_{t}s\,\delta(m^{2})=0$ (since $m^{2}\neq 0$).
Consequently, the optical theorem has no bearing on $c_{t}$, indicating that
the constant is of indefinite sign. The effective theory contributions to the
dispersion relation are given by

$\lim_{s\rightarrow 0}\partial_{s}{\cal A}(s,0)=-\lim_{t\rightarrow
0}\partial_{t}{\cal A}(0,t)=-\frac{c_{t}}{m^{2}}.$ (74)

That these results are of indefinite sign already suggests the need for
boundary terms. This is confirmed by the fact that the integrals in both Eqs.
(66) and (70) vanish: ${\rm Im}\,{\cal A}(s,t)$ and ${\rm Im}\,\bar{\cal
A}(s,t)$ are proportional to $\delta(t-m^{2})$, which is only non-vanishing
for $t=m^{2}>0$, while for real momenta, $t<0$. The two relations then reduce
to

$\displaystyle\lim_{s\rightarrow 0}\partial_{s}{\cal A}(s,0)$
$\displaystyle=-C_{\infty}^{(s)}$ (75) $\displaystyle\lim_{t\rightarrow
0}\partial_{t}{\cal A}(0,t)$ $\displaystyle=-C_{\infty}^{(t)},$

and explicit computation shows
$C_{\infty}^{(s)}=-C_{\infty}^{(t)}=c_{t}/m^{2}$, consistent with Eq. (74).

The above calculation demonstrates that any $t$-channel tree-level completion
will induce boundary terms. The presence of such terms renders the arguments
leading to Eq. (36) inapplicable and would seemingly imply that such
completions could give rise to CP- or flavor-violating contributions that are
parametrically larger than their symmetry-conserving analogues. Nevertheless,
in practice engineering a violation of these constraints is less
straightforward. For instance, we could imagine a UV completion of ${\cal
O}_{e}$ that couples the UV state to a flavor-conserving electron vertex
$c_{11}$ and a flavor-violating electron-muon vertex $c_{12}$. An amplitude
involving both new couplings could mediate flavor-violating decays such as
$\mu\to 3e$, with a scaling in the UV couplings of $c_{11}c_{12}$. However,
the theory also allows flavor-conserving interactions proportional to
$c_{11}^{2}$ and $c_{12}^{2}$, which are of the same parametric order of the
flavor-violating amplitude. Achieving a separation, where flavor violation
dominates, requires a more complex completion containing both scalars and
vectors with masses and couplings tuned to suppress the conserving
interactions (exploiting the different signs they contribute to Eq. (36)). We
leave a detailed exploration of this point to future work; however, we note
that this argument suggests that the observation of flavor violation in a low-
energy experiment without concomitant flavor-conserving operators appearing in
the near-term at the LHC would signify the presence of both new scalars and
vectors.

##  6 Oblique Parameters and Universal Theories

An alternative to characterizing deviations from the SM with Wilson
coefficients is to instead consider parameters that are more directly
connected to precision measurements. In particular, the electroweak oblique
parameters $S$, $T$, $U$ of Peskin and Takeuchi [103] (see also Refs. [104,
105]), which parameterize deviation of vacuum polarization functions of the SM
bosons, are a traditional benchmark for comparing signatures of new physics.
The anomalous measurement of the $W$-boson mass reported by CDF in Ref. [106]
provides a recent reminder of their preeminence for electroweak precision
measurements, as the result can be interpreted as deviations in the $S$ and
$T$ parameters away from their SM values of zero. For our purposes, however,
the important detail is that both $S$ and $T$ correspond to deviations induced
by dimension-six operators in the SMEFT, so it is interesting to consider the
consequences of our dispersion relations for the oblique parameters.

In general, as discussed in Ref. [107], applying the oblique parameters is
appropriate only for particular beyond-SM scenarios called “universal
theories,” which are the subset of all theories for which the higher-dimension
operators in the SMEFT can, via field redefinitions, be rewritten entirely in
terms of bosonic operators. As described in Ref. [107], the set of universal
theories at dimension six is completely characterized by a subset of sixteen
Wilson coefficients: five oblique parameters, four anomalous triple gauge
couplings (TGC), three parameters describing the rescaling of the SM $h^{3}$
coupling, three non-SM Higgs-vector-vector $hVV$ couplings, and one additional
four-fermion coupling. Universal theories manifestly do not capture the most
general UV physics that could be seen at dimension six; rather, that is what
the full set of Wilson coefficients in the SMEFT describe.

To define the oblique parameters on which we will focus, we review the
essential notation. For a vector boson $V$, we can extract the transverse part
of the self-energy $\Pi_{VV}(p^{2})$ from

$\Pi^{\mu\nu}_{VV}=-\left(g^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}}\right)\Pi_{VV}(p^{2})+\frac{p^{\mu}p^{\nu}}{p^{2}}(\cdots),$
(76)

where we expand the self-energy function in a series in $p^{2}$, implicitly
defining $\Pi_{VV}(0)$, $\Pi^{\prime}_{VV}(0)$, and
$\Pi^{\prime\prime}_{VV}(0)$:

$\Pi_{VV}(p^{2})=\Pi_{VV}(0)-\Pi^{\prime}_{VV}(0)p^{2}+\frac{1}{2}\Pi_{VV}^{\prime\prime}(0)(p^{2})^{2}+\cdots,$
(77)

with primes denoting derivatives with respect to $p^{2}$.191919Relative to
Ref. [107], we have introduced minus signs to account for the difference in
metric convention (as we use mostly-plus signature), so that our
$\Pi_{VV}(0)$, $\Pi^{\prime}_{VV}(0)$, and $\Pi^{\prime\prime}_{VV}(0)$
numerically agree with theirs. That is, the notation $\Pi^{\prime}_{VV}(0)$ is
here defined to be $-\lim_{p^{2}\rightarrow 0}{\rm d}\Pi(p^{2})/{\rm d}p^{2}$.
In terms of these expanded self-energy functions, we can define the oblique
parameters as

$\displaystyle\hat{S}$
$\displaystyle=\frac{g^{2}}{16\pi}S=-\bar{\Pi}^{\prime}(0)_{W^{3}B}\cot\theta_{\scriptscriptstyle
W}$ (78) $\displaystyle\hat{T}$
$\displaystyle=\frac{g^{2}}{4\pi}T\sin^{2}\theta_{\scriptscriptstyle
W}=\frac{1}{m_{\scriptscriptstyle
W}^{2}}\left[\bar{\Pi}_{W^{+}W^{-}}(0)-\bar{\Pi}_{W^{3}W^{3}}(0)\right]$
$\displaystyle W$ $\displaystyle=-\frac{m_{\scriptscriptstyle
W}^{2}}{2}\bar{\Pi}_{W^{3}W^{3}}^{\prime\prime}(0)$ $\displaystyle Y$
$\displaystyle=-\frac{m_{\scriptscriptstyle
W}^{2}}{2}\bar{\Pi}_{BB}^{\prime\prime}(0)$ $\displaystyle Z$
$\displaystyle=-\frac{m_{\scriptscriptstyle
W}^{2}}{2}\bar{\Pi}_{GG}^{\prime\prime}(0).$

The bars indicate that we have chosen a definition of the SM fields and
parameters such that, as discussed in Ref. [107], only bosonic fields are
present, $B$ and $W^{\pm}$ have canonical kinetic terms, and
$\Pi_{W^{+}W^{-}}(0)$ vanishes, thus pinning $g$, $g^{\prime}$, and the Higgs
vev to fixed values. For a particular, concrete beyond-SM theory (i.e., a UV
completion of SMEFT operators by some massive degrees of freedom), the oblique
parameters are well-defined observables and thus invariant under whichever
SMEFT basis we choose; indeed, Ref. [107] demonstrates translations between
the Warsaw basis [1] and various other bases. For the present analysis, we
will use this freedom to work in whichever basis is most convenient.

In what follows, we will exploit the relation between the parameters in Eq.
(78) and the dimension-six SMEFT to draw connections to our spinning sum
rules. For all parameters except $\hat{S}$, a relation can be established with
no assumptions beyond those with which the spinning sum rules were derived.
For $\hat{S}$, however, a clean connection does not exist, and only after
invoking several additional assumptions about the UV will we be able to draw
any relation between the UV and IR. Given the widespread use of the oblique
parameters, we find it interesting to consider what it would take to draw such
a connection, however, particularly in the case of $\hat{S}$ a detailed
exploration of whether the relation can even be brought to bear on a realistic
UV completion is an interesting question that we leave to future work.

###  6.1 $W,Y,Z$

We first consider the second-derivative oblique parameters $W$, $Y$, and $Z$.
We will see how the sign of $Y$ can be explicitly derived, depending on the
spins of states in the UV, from the sum rule for $e^{+}e^{-}$ scattering and
unitarity of the partial-wave expansion.202020In Ref. [108], the Källén-
Lehmann representation of the two-point function was used to advocate for
positivity of $\Pi^{\prime\prime}(0)$, fixing the sign of $Y$ and $W$,
assuming that there are no extra gauge fields in the UV completion (see also
Ref. [109]). Note that Ref. [108] uses the opposite sign convention to that of
Eq. (78) and Ref. [107] in defining the oblique parameters relative to
$\Pi(0)$, $\Pi^{\prime}(0)$, and $\Pi^{\prime\prime}(0)$, but also that Refs.
[107] and [108] (which both use mostly-minus metric signature) differ from
each other in the definition of the overall sign of the self-energy
$\Pi(p^{2})$. For $W$ and $Z$, we use other sum rules to illustrate how the
magnitudes of these oblique parameters may be upper bounded in terms of $|Y|$.

In a universal theory in the Warsaw basis, four-fermion self-quartics (i.e.,
$e^{4}$, $L^{4}$, etc.) appear only through the combinations in the current-
squared operators, given in our mostly-plus metric signature by

$\displaystyle{\cal O}_{2JB}$ $\displaystyle=-J_{B\mu}J^{B\mu}$ (79)
$\displaystyle{\cal O}_{2JW}$ $\displaystyle=-J^{I}_{W\mu}J_{W}^{I\mu}$
$\displaystyle{\cal O}_{2JG}$ $\displaystyle=-J^{a}_{G\mu}J_{G}^{a\mu},$

where $J_{B}^{\mu}$, $J_{W}^{I\mu}$ and $J_{G}^{a\mu}$ are the currents
coupling to the respective SM gauge bosons, for example,
$J_{G}^{a\mu}=g_{s}\sum_{\psi}\bar{\psi}\gamma^{\mu}T^{a}\psi$, where $\psi$
runs over $u$, $d$, and $Q$. (The operator formed from the square of scalar
Yukawa currents contributes only to fermion cross-quartics, not self-quartics
[107]. The Wilson coefficient of this operator is one of those characterizing
a universal theory, although we will not need to consider it here.) We write
the Wilson coefficients of the operators in Eq. (79) as $c_{2JB}$, $c_{2JW}$,
and $c_{2JG}$. Then as shown in Ref. [107], the oblique parameters are

$\displaystyle W$ $\displaystyle=-\frac{1}{2}g^{2}c_{2JW}$ (80) $\displaystyle
Y$ $\displaystyle=-\frac{1}{2}g^{2}c_{2JB}$ $\displaystyle Z$
$\displaystyle=-\frac{1}{2}g^{2}c_{2JG}.$

To emphasize, in a universal theory there is a single Wilson coefficient for
each of the current-squared operators in Eq. (79). Accordingly, we can
determine the constraints on these coefficients by considering different
fermion scattering experiments, knowing that the coefficients of each are
related. To begin with, the $e^{4}$ four-fermi operator appears only in ${\cal
O}_{2JB}\supset-g^{\prime
2}Y_{e}^{2}(\bar{e}_{m}\gamma_{\mu}e_{m})(\bar{e}_{n}\gamma^{\mu}e_{n})$. As
in Sec. 4.3 and Ref. [35], we will consider opposite incoming helicities in
deriving sum rules for four-fermion scattering. Taking a superposition of
flavors, the amplitude for $\bar{e}^{-}e^{+}$ scattering via ${\cal O}_{2JB}$
is ${\cal A}=8\alpha_{m}\beta_{m}\gamma_{n}\delta_{n}g^{\prime
2}Y_{e}^{2}c_{2JB}[13]\langle 24\rangle$. Taking the flavor superposition
coefficients to satisfy $\alpha_{m}=\bar{\delta}_{m}$ and
$\beta_{m}=\bar{\gamma}_{m}$, we can apply our sum rules. We find that, if the
boundary terms vanish (which is true if we consider tree-level completions
without a vector exchanged in the $t$ channel, as we showed in Sec. 5), the
oblique parameter $Y$ has sign dictated by the spin of the UV states as in Eq.
(36):

$Y=\frac{\cot^{2}\theta_{\scriptscriptstyle
W}}{|\alpha\cdot\beta|^{2}Y_{e}^{2}}\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\left[3\,{\rm Im}\,a_{\alpha_{-};\beta_{+}}^{ee(1)}(s)-{\rm
Im}\,a_{\bar{\alpha}_{+};\beta_{+}}^{ee(0)}(s)\right]\\!.$ (81)

Looking to the universality of the Wilson coefficients, we can similarly
consider $d^{4}$ scattering, which contributes to the following operators:

$\displaystyle{\cal O}_{2JB}$ $\displaystyle\supset-g^{\prime
2}Y_{d}^{2}(\bar{d}_{m}\gamma_{\mu}d_{m})(\bar{d}_{n}\gamma^{\mu}d_{n})$ (82)
$\displaystyle{\cal O}_{2JG}$ $\displaystyle\supset-
g_{s}^{2}\left[-\frac{1}{6}(\bar{d}_{m}\gamma_{\mu}d_{m})(\bar{d}_{n}\gamma^{\mu}d_{n})+\frac{1}{2}(\bar{d}_{m}\gamma_{\mu}d_{n})(\bar{d}_{n}\gamma^{\mu}d_{m})\right]\\!.$

Writing the Wilson coefficient of
$(\bar{d}_{m}\gamma_{\mu}d_{n})(\bar{d}_{p}\gamma^{\mu}d_{q})$ as
$-c_{mnpq}^{d}$, we have

$c_{mnpq}^{d}=\left(g^{\prime
2}Y_{d}^{2}c_{2JB}-\frac{1}{6}g_{s}^{2}c_{2JG}\right)\delta_{mn}\delta_{pq}+\frac{1}{2}g_{s}^{2}c_{2JG}\delta_{mq}\delta_{np}.$
(83)

Computing the $d^{4}$ amplitude and applying the analogous sum rule from Eq.
(36), derived in Ref. [35], we have the bounds

$\displaystyle
c_{mnpq}^{d}(\alpha_{m}\beta_{n}\bar{\beta}_{p}\bar{\alpha}_{q}+\alpha_{m}\bar{\alpha}_{n}\bar{\beta}_{p}\beta_{q})$
$\displaystyle=4\int_{0}^{\infty}\frac{{\rm d}s}{s^{2}}\left[{\rm
Im}\,a_{\bar{\alpha}_{+};\beta_{+}}^{dd(0),\parallel}(s)-3\,{\rm
Im}\,a_{\alpha_{-};\beta_{+}}^{dd(1),\parallel}(s)\right]$ (84) $\displaystyle
c_{mnpq}^{d}\alpha_{m}\bar{\alpha}_{n}\bar{\beta}_{p}\beta_{q}$
$\displaystyle=4\int_{0}^{\infty}\frac{{\rm d}s}{s^{2}}\left[{\rm
Im}\,a_{\bar{\alpha}_{+};\beta_{+}}^{dd(0),\perp}(s)-3\,{\rm
Im}\,a_{\alpha_{-};\beta_{+}}^{dd(1),\perp}(s)\right]\\!,$

where the superscripts in parentheses indicate spin of the intermediate state
as always, and we have further denoted whether we are scattering incoming
states of the same ($\parallel$) or orthogonal ($\perp$) ${\rm SU}(3)_{C}$
charges. That is,

$\displaystyle\left(g^{\prime
2}Y_{d}^{2}Y+\frac{1}{3}g_{s}^{2}Z\right)\left(|\alpha|^{2}|\beta|^{2}+|\alpha\cdot\beta|^{2}\right)$
(85) $\displaystyle=2g^{2}\int_{0}^{\infty}\frac{{\rm d}s}{s^{2}}\left[3\,{\rm
Im}\,a_{\alpha_{-};\beta_{+}}^{dd(1),\parallel}(s)-{\rm
Im}\,a_{\bar{\alpha}_{+};\beta_{+}}^{dd(0),\parallel}(s)\right]$

and

$\displaystyle\left(g^{\prime
2}Y_{d}^{2}Y-\frac{1}{6}g_{s}^{2}Z\right)|\alpha|^{2}|\beta|^{2}+\frac{1}{2}g_{s}^{2}Z|\alpha\cdot\beta|^{2}$
(86) $\displaystyle=2g^{2}\int_{0}^{\infty}\frac{{\rm d}s}{s^{2}}\left[3\,{\rm
Im}\,a_{\alpha_{-};\beta_{+}}^{dd(1),\perp}(s)-{\rm
Im}\,a_{\bar{\alpha}_{+};\beta_{+}}^{dd(0),\perp}(s)\right]\\!,$

so in particular, considering $\alpha\parallel\beta$, we find that

$g^{\prime 2}Y_{d}^{2}Y+\frac{1}{3}g_{s}^{2}Z\gtrless 0,$ (87)

so that the combination is $<0$ ($>0$) for a completion dominated by scalars
(vectors). Similarly, considering $\alpha\perp\beta$, we find that

$g^{\prime 2}Y_{d}^{2}Y-\frac{1}{6}g_{s}^{2}Z\gtrless 0.$ (88)

Thus, if the completion is globally dominated exclusively by scalars or by
vectors regardless of flavor, then if the UV amplitudes do not generate a pole
at infinity, one finds that $|Z|$ must be bounded from above by ${\cal
O}(1)\times(g^{\prime 2}Y_{d}^{2}/g_{s}^{2})\times|Y|$ as a consequence of
unitarity. Moreover, while requiring vanishing boundary integrals, it is not
possible in a universal theory to realize a scenario where the amplitude for
$e^{+}e^{-}$ scattering is dominated by scalar exchange (which implies $Y<0$)
while requiring the amplitudes for $d^{+}d^{-}$ scattering to be dominated by
vector exchange, since then Eqs. (87) and (88), combined with Eq. (81), would
lead to contradictory requirements on the sign of $Z$. The same statement also
holds with signs and scalars/vectors swapped.

Analogously, $L^{4}$ scattering appears in a universal theory through the
operators

$\displaystyle{\cal O}_{2JW}$
$\displaystyle\supset-g^{2}\left[-\frac{1}{4}(\bar{L}_{m}\gamma_{\mu}L_{m})(\bar{L}_{n}\gamma^{\mu}L_{n})+\frac{1}{2}(\bar{L}_{m}\gamma_{\mu}L_{n})(\bar{L}_{n}\gamma^{\mu}L_{m})\right]$
(89) $\displaystyle{\cal O}_{2JB}$ $\displaystyle\supset-g^{\prime
2}Y_{L}^{2}(\bar{L}_{m}\gamma_{\mu}L_{m})(\bar{L}_{n}\gamma^{\mu}L_{n}).$

Our effective Wilson coefficient tensor is $c_{mnpq}^{L}=\left(g^{\prime
2}Y_{L}^{2}c_{2JB}-\frac{1}{4}g^{2}c_{2JW}\right)\delta_{mn}\delta_{pq}+\frac{1}{2}g^{2}c_{2JW}\delta_{mq}\delta_{np}$.
Computing the $L^{4}$ amplitude and applying the sum rule, we find

$\displaystyle
c_{mnpq}^{L}(\alpha_{m}\beta_{n}\bar{\beta}_{p}\bar{\alpha}_{q}+\alpha_{m}\bar{\alpha}_{n}\bar{\beta}_{p}\beta_{q})$
$\displaystyle=4\int_{0}^{\infty}\frac{{\rm d}s}{s^{2}}\left[{\rm
Im}\,a_{\bar{\alpha}_{+};\beta_{+}}^{LL(0),\parallel}(s)-3\,{\rm
Im}\,a_{\alpha_{-};\beta_{+}}^{LL(1),\parallel}(s)\right]$ (90) $\displaystyle
c_{mnpq}^{L}\alpha_{m}\bar{\alpha}_{n}\bar{\beta}_{p}\beta_{q}$
$\displaystyle=4\int_{0}^{\infty}\frac{{\rm d}s}{s^{2}}\left[{\rm
Im}\,a_{\bar{\alpha}_{+};\beta_{+}}^{LL(0),\perp}(s)-3\,{\rm
Im}\,a_{\alpha_{-};\beta_{+}}^{LL(1),\perp}(s)\right]\\!,$

that is,

$\displaystyle\left(2Y_{L}^{2}Y\tan^{2}\theta_{\scriptscriptstyle
W}+\frac{1}{2}W\right)\left(|\alpha|^{2}|\beta|^{2}+|\alpha\cdot\beta|^{2}\right)$
(91) $\displaystyle=4\int_{0}^{\infty}\frac{{\rm d}s}{s^{2}}\left[3\,{\rm
Im}\,a_{\alpha_{-};\beta_{+}}^{LL(1),\parallel}(s)-{\rm
Im}\,a_{\bar{\alpha}_{+};\beta_{+}}^{LL(0),\parallel}(s)\right]$

and

$\displaystyle\left(2Y_{L}^{2}Y\tan^{2}\theta_{\scriptscriptstyle
W}-\frac{1}{2}W\right)|\alpha|^{2}|\beta|^{2}+W|\alpha\cdot\beta|^{2}$ (92)
$\displaystyle=4\int_{0}^{\infty}\frac{{\rm d}s}{s^{2}}\left[3\,{\rm
Im}\,a_{\alpha_{-};\beta_{+}}^{LL(1),\perp}(s)-{\rm
Im}\,a_{\bar{\alpha}_{+};\beta_{+}}^{LL(0),\perp}(s)\right]\\!.$

In particular, for $\alpha$ and $\beta$ parallel, we find

$2Y_{L}^{2}Y\tan^{2}\theta_{\scriptscriptstyle W}+\frac{1}{2}W\gtrless 0,$
(93)

with $<$ ($>$) for a scalar- (vector-)dominated completion, and similarly, for
$\alpha$ and $\beta$ perpendicular,

$2Y_{L}^{2}Y\tan^{2}\theta_{\scriptscriptstyle W}-\frac{1}{2}W\gtrless 0.$
(94)

Hence, for UV completions that are globally scalar- or vector-dominated, we
have an upper bound on the size of the oblique parameter $W$,
$|W|<4Y_{L}^{2}|Y|\tan^{2}\theta_{\scriptscriptstyle W}$. Similarly, the
spinning sum rules imply that it is not possible, in a universal theory with
vanishing boundary integrals, for $e^{+}e^{-}$ scattering to be dominated by
scalar (vector) exchange while $L^{+}L^{-}$ scattering is dominated by vector
(respectively, scalar) exchange, since Eqs. (93) and (94), combined with Eq.
(81), would lead to contradictory predictions for the sign of $W$.

###  6.2 $\hat{T}$

Next, let us consider the $\hat{T}$ parameter. In terms of the basis used by
Elias-Miró, Grojean, Gupta, and Marzocca (EGGM) in Ref. [110], we define the
operators, in our mostly-plus metric, as

$\displaystyle{\cal O}_{T}$
$\displaystyle=-\frac{1}{2}(H^{\dagger}\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}_{\mu}H)^{2}$
(95) $\displaystyle{\cal O}_{H}$
$\displaystyle=-\frac{1}{2}(\partial_{\mu}|H|^{2})^{2},$

and write $E_{T,H}$ for the Wilson coefficients of ${\cal O}_{T,H}$,
respectively.212121Eq. (95) contains the same operators used by Ref. [78] and
discussed in Sec. 4.1. Nevertheless, we adopt a different symbol for the
Wilson coefficients here for consistency with the notation of Ref. [107].  No
other Higgs quartics appear in the EGGM basis. Relating these operators to the
oblique parameters, one has $\hat{T}=E_{T}$. Let us further define the
rescaling in the TGC for the $hWW$ vertex (in the field basis where the
vectors have been rescaled to have canonical kinetic terms) as
$1+\Delta\bar{\kappa}_{V}$. Then in the EGGM basis, we have
$\Delta\bar{\kappa}_{V}=-E_{H}/2$.

Expanding the operators in Eq. (95) in the basis of real scalars via Eq. (13),
the amplitudes are the same as those computed in Eq. (16), with the
identifications $E_{T}=c^{H2}/2$ and $E_{H}=(4c^{H1}-c^{H2})/2$. In
particular, from Eq. (17), we have the amplitudes for
$\phi_{1}+\phi_{2,3,4}\rightarrow\phi_{1}+\phi_{2,3,4}$:

$\displaystyle{\cal A}_{1,2}$ $\displaystyle=(3E_{T}+E_{H})t$ (96)
$\displaystyle{\cal A}_{1,3}={\cal A}_{1,4}$ $\displaystyle=E_{H}t.$

Applying the sum rules from Sec. 4.1, we thus have:

$\displaystyle 3\hat{T}-2\Delta\bar{\kappa}_{V}$
$\displaystyle=\\!16\\!\int_{0}^{\infty}\\!\frac{{\rm
d}s}{s^{2}}\sum_{j=0}^{\infty}(2j{+}1)\left\\{\left[j\left(j{+}1\right)\right]{\rm
Im}\,a_{2}^{(j)}(s)+\left[j\left(j{+}1\right){-}1\right]{\rm
Im}\,\bar{a}_{2}^{(j)}(s)\right\\}-C^{(t)}_{\infty}$ (97)
$\displaystyle\Delta\bar{\kappa}_{V}$
$\displaystyle=\\!-8\\!\int_{0}^{\infty}\\!\frac{{\rm
d}s}{s^{2}}\sum_{j=0}^{\infty}(2j{+}1)\left\\{\left[j\left(j{+}1\right)\right]{\rm
Im}\,a_{3}^{(j)}(s)+\left[j\left(j{+}1\right){-}1\right]{\rm
Im}\,\bar{a}_{3}^{(j)}(s)\right\\}+C^{(t)}_{\infty}$
$\displaystyle=\\!-8\\!\int_{0}^{\infty}\\!\frac{{\rm
d}s}{s^{2}}\sum_{j=0}^{\infty}(2j{+}1)\left\\{\left[j\left(j{+}1\right)\right]{\rm
Im}\,a_{4}^{(j)}(s)+\left[j\left(j{+}1\right){-}1\right]{\rm
Im}\,\bar{a}_{4}^{(j)}(s)\right\\}+C^{(t)}_{\infty}.$

As a result, for completions where $C^{(t)}_{\infty}=0$, if the UV is
dominated by scalar exchange, our dispersion relations predict
$\Delta\bar{\kappa}_{V}>0$ and $\hat{T}<2\Delta\bar{\kappa}_{V}/3$. On the
other hand, if the UV completion is dominated by higher-spin currents, then
$\Delta\bar{\kappa}_{V}<0$ and $\hat{T}>2\Delta\bar{\kappa}_{V}/3$.

###  6.3 $\hat{S}$

Finally, we consider what is required in order to draw a connection between
spinning sum rules and the first oblique parameter $\hat{S}$.222222We note
that a dispersive representation of $S$ was in fact constructed by Peskin and
Takeuchi in Ref. [103], where a sum rule for $S$ was established in terms of
the vector and axial isospin analogues of the $R$-ratio, in the spirit of the
Weinberg sum rules [111]. Their sum rule connected $S$ to the imaginary part
of self-energy functions, whereas our focus is a connection to two-to-two
scatterings that we can relate to dimension-six SMEFT coefficients. At the
outset, we note that we will not be able to establish a clean connection.
Additional assumptions will be required, and the study below should be
considered as in the spirit of determining what is needed in order to connect
$\hat{S}$ to the UV.

The first step is to relate $\hat{S}$ to Wilson coefficients in the SMEFT. To
do so, we again follow Ref. [107], but with an accounting for our differing
conventions, and define the following operators:

$\displaystyle{\cal O}_{HJB}$
$\displaystyle=\frac{ig^{\prime}}{2}(H^{\dagger}\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}_{\mu}H)J_{B}^{\mu}$
(98) $\displaystyle=\frac{1}{2}g^{\prime 2}\delta_{mn}(Y_{Q}{\cal
O}_{HQ1}+Y_{L}{\cal O}_{HL1}+Y_{u}{\cal O}_{Hu}+Y_{d}{\cal O}_{Hd}+Y_{e}{\cal
O}_{He})$ $\displaystyle{\cal O}_{HJW}$
$\displaystyle=ig(H^{\dagger}\overset{\text{\tiny$\bm{\leftrightarrow}$}}{D}\vphantom{D}^{I}_{\mu}H)J_{W}^{I\mu}$
$\displaystyle=g^{2}\delta_{mn}({\cal O}_{HL2}+{\cal O}_{HQ2})$
$\displaystyle{\cal O}_{HWB}$
$\displaystyle=2(H^{\dagger}\tau^{I}H)W^{I}_{\mu\nu}B^{\mu\nu},$

where, when possible, we have related the operators to those we listed in
Table 1. We can relate the Wilson coefficients of these operators, together
with those in Eq. (79), to specify

$\hat{S}=g^{2}\left(\frac{1}{gg^{\prime}}c_{HWB}+\frac{1}{4}c_{HJB}+\frac{1}{4}c_{HJW}-\frac{1}{2}c_{2JB}-\frac{1}{2}c_{2JW}\right)\\!.$
(99)

This relation immediately presents a challenge. As we saw in Sec. 4.4, our
dispersive arguments cannot constrain the Wilson coefficients of $H^{2}B^{2}$
and $H^{2}W^{2}$, and the same conclusion will hold for ${\cal O}_{HWB}$ via
the analogous sum rule one obtains from scattering a Higgs and a superposition
of $B$ and $W$ gauge bosonn: the obstruction is that the elastic, fixed-
helicity EFT amplitude vanishes by little-group scaling [78]. Accordingly, we
should expect that $c_{HWB}$ can take on any sign independent of any
properties of the UV, exactly as we saw for the simple scalar completion
considered in Sec. 4.4. To make progress, we will simply consider the case in
which the UV does not generate $c_{HWB}$.

From here, studying the form within Eq. (98) in which the fermion and Higgs
cross-quartics can appear in universal theories, we can then write $\hat{S}$
in terms of the lepton and Higgs Wilson coefficients, along with $W$ and $Y$
from Eq. (80):

$\hat{S}-Y-W=-\frac{\cot^{2}\theta_{\scriptscriptstyle
W}}{2Y_{e}}c^{He}_{mm}-\frac{1}{4}c^{HL2}_{mm}.$ (100)

Once more we have exploited the universality of the coefficients to replace
$c_{HJB}$ and $c_{HJW}$ solely with the coefficients $c^{He}_{mm}$ and
$c^{HL2}_{mm}$. We can now use our dispersion relations to study these SMEFT
coefficients. For the Higgs modes we are scattering parameterized by the
$\beta_{i}$, suppose we specifically choose a superposition such that
$H=H^{\dagger}\in\mathbb{R}$, with negative isospin, e.g., $\beta_{1}=1/2$,
$\beta_{2}=i/2$, $\beta_{3}=1/2$, $\beta_{4}=-i/2$; call this choice
$\boldsymbol{\beta}_{\downarrow}$. Then by Eq. (31) we have the sum rule

$c_{mm}^{HL2}=-64\sum_{\alpha}\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}(n+1)\left[{\rm
Im}\,a_{\alpha;\boldsymbol{\beta}_{\downarrow}}^{HL(n)}(s)-{\rm
Im}\,a_{\bar{\alpha};\boldsymbol{\beta}_{\downarrow}}^{HL(n)}(s)\right]\\!,$
(101)

where we have assumed $C_{\infty}^{(s)}=0$ and the sum on $\alpha$ is over an
orthonormal basis of the generation index vector for $L$. Similarly, we can
define another choice of Higgs superposition $\boldsymbol{\beta}_{\uparrow}$
as $\beta_{1}=1/2$, $\beta_{2}=i/2$, $\beta_{3}=1/2$, $\beta_{4}=i/2$, in
which $H=H^{\dagger}\in\mathbb{R}$, but with positive isospin. We then find by
Eq. (30) that when the pole at infinity can be neglected,

$c_{mm}^{He}=-16\sum_{\alpha}\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}\frac{{\rm d}s}{s^{2}}(n+1)\left[{\rm
Im}\,a_{\alpha;\boldsymbol{\beta}_{\uparrow}}^{He(n)}(s)-{\rm
Im}\,a_{\bar{\alpha};\boldsymbol{\beta}_{\uparrow}}^{He(n)}(s)\right]\\!,$
(102)

so that we have the sum rule

$\displaystyle\hat{S}-Y-W=$
$\displaystyle\;\frac{8\cot^{2}\theta_{\scriptscriptstyle
W}}{Y_{e}}\sum_{\alpha}\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}(n+1)\left[{\rm
Im}\,a_{\alpha;\boldsymbol{\beta}_{\uparrow}}^{HL(n)}(s)-{\rm
Im}\,a_{\bar{\alpha};\boldsymbol{\beta}_{\uparrow}}^{HL(n)}(s)\right]$ (103)
$\displaystyle\;+16\sum_{\alpha}\int_{0}^{\infty}\frac{{\rm
d}s}{s^{2}}\sum_{n=0}^{\infty}(n+1)\left[{\rm
Im}\,a_{\alpha;\boldsymbol{\beta}_{\downarrow}}^{HL(n)}(s)-{\rm
Im}\,a_{\bar{\alpha};\boldsymbol{\beta}_{\downarrow}}^{HL(n)}(s)\right]\\!.$

As discussed in Sec. 4.2, the forward dispersion relations we have used to
obtain the results for $c_{mm}^{HL2}$ and $c_{mm}^{He}$ do not provide clear-
cut sign constraints on the coefficients. Given this fact, and following the
discussion in that section, we invoke our last assumption: suppose there is a
symmetry in the UV such that in each case
$a_{\alpha;\beta}=a_{\bar{\alpha};\beta}$. Should this be true, then
$\hat{S}=Y+W$. Moreover, given the SM values of the Weinberg angle and
hypercharges, Eqs. (93), and (94) would then imply that for a theory whose
fermion scattering in the UV is globally dominated by scalars or vectors,
$|W|\lesssim 0.3\,|Y|$, so $Y+W$ has the same sign as $Y$, which by Eq. (81)
is negative (positive) in the scalar (respectively, vector) case. In such a
theory, we would then have a definite sign for $\hat{S}$:

$\hat{S}\gtrless 0,$ (104)

again with $<$ ($>$) obtaining for a scalar (vector) completion. Nevertheless,
we emphasize once more that such a result only holds for UV completions that
do not generate ${\cal O}_{HWB}$, that have vanishing boundary terms, and for
which the amplitude of $He$ and $HL$ scattering is crossing invariant.

##  7 Conclusions

The three-thousand-dimensional space describing the dimension-six SMEFT is
going to be probed extensively in the coming decades by an array of
experiments operating at the energy, intensity, and precision frontiers.
Deviations and discoveries could well be on our horizon. The focus of this
work has been to further chart where in this enormous space the UV may have
left distinctive relics.

The key findings of our exploration are the spinning sum rules: a connection
between the sign of the IR Wilson coefficients and the dominant UV spin. That
these relations reveal information beyond what has been considered so far at
dimension six can be traced to the use of the beyond-forward dispersion
relation we outlined in Sec. 2, which isolated the $t$ scaling of the EFT
amplitude, as opposed to the traditional focus on the $s$ coefficient. As
demonstrated in Sec. 4, spinning sum rules can be established for purely
bosonic and fermionic scattering. The obstruction to establishing such
relations more generally is that the required EFT amplitudes vanish, either
entirely or in the $s\to 0$ limit. Nevertheless, as shown in Sec. 6, the
relations we obtained are sufficient to establish connections to the oblique
parameters that define universal theories. In particular, while we were able
to draw a connection to the well-known Peskin-Takeuchi $S$ and $T$ parameters,
our statements for $S$ hinged on a number of strong assumptions about the UV
beyond what we required to establish the spinning sum rules themselves.

As we have emphasized throughout this work, the fundamental obstruction to a
sharper UV-IR connection at dimension six is the presence of poles at
infinity, which enter with arbitrary sign. The challenges for spinning sum
rules are only enhanced over previous studies, given the use of two dispersion
relations with distinct boundary terms. Section 5 focused on furthering our
understanding of the boundary contributions (in addition to providing explicit
demonstrations of how the dispersion relations are satisfied). As we showed,
at the level of tree-level completions, the boundary terms are associated with
states exchanged in the $t$ channel. The connection is straightforward: the
imaginary part of $t$-channel propagators vanishes in the $t\to 0$ limit, so
that these results cannot contribute to the dispersive integral and will only
be reproduced by the pole at infinity. The vanishing of the imaginary part of
the forward amplitude for $t$-channel propagators also mandates that the sign
of their coefficient is unconstrained by the optical theorem, and accordingly
we should expect boundary terms to take on arbitrary sign, exactly as they do
in explicit completions. Ultimately, while the spinning sum rules all carry
the boundary term caveat, the results remain one of the cleanest connections
between mass dimension six and the UV.

This paper opens many promising directions for future study. Our results
extend what is known at dimension six, but will not be the final word.
Additional operators could be constrained by scattering states in
superpositions of SM representations, and further by applying the generalized
optical theorem as considered in Ref. [79] to move beyond elastic scattering
altogether. Boundary terms are a topic that require further study. The
examples we considered were purely tree-level completions, a subset of
possible weakly coupled UV theories. Extending our analysis to consider
completions where loops play an important role and establishing the expected
behavior of the boundary terms in these scenarios would further help elucidate
the class of theories for which the spinning sum rules are expected to
hold.232323We note that loop completions for conventional sum rules have
already been considered in Ref. [78]. Finally, the ultimate goal of these
relations is to open a direct connection to experiment. Determining exactly
where spinning sum rules will be most impactful—particularly in the near-
term—remains a critical next step. In our current era of particle physics,
where the first hints of the laws of nature beyond the SM may come from
detection of operators in the SMEFT, pursuing a full understanding of how to
leverage the axioms of quantum field theory to turn observables like Wilson
coefficients into concrete statements about properties of the UV is a crucial
enterprise.

Acknowledgments

G.N.R. is supported at the Kavli Institute for Theoretical Physics by the
Simons Foundation (Grant No. 216179) and the National Science Foundation
(Grant No. NSF PHY-1748958) and at the University of California, Santa Barbara
by the Fundamental Physics Fellowship.

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