CALT-TH/2024-007

Generalized eikonal identities for charged currents

Ryan Plestid

Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA, 91125 USA

We discuss QED radiative corrections to contact operators coupling two heavy
fields and one light field. New eikonal identities are derived in the static
limit that demonstrate the equivalence of a class of ladder graphs to an
equivalent theory with a single heavy-light vertex and a background Coulomb
field which communicates exclusively with the light field. We apply these new
identities to nuclear beta decays and demonstrates that the “independent
particle model” used by Jaus, Rasche, Sirlin & Zucchini is closely related,
though not identical, to a model independent EFT calculation.

###### Contents

  1. I Introduction
  2. II Eikonal identities for charged currents
  3. III Effective theory of beta decay in the point-like limit
  4. IV Comparison with Jaus, Rasche, Sirlin & Zucchini
  5. V Conclusions
  6. A Gauge invariant sub classes for beta decay
  7. B Coulomb gauge & the static limit

##  I Introduction

The heavy particle limit of gauge theories is dramatically simplified by
eikonal identities Korchemsky and Radyushkin (1992); Isgur and Wise (1989,
1990); Georgi (1990); Falk et al. (1990); Bauer et al. (2002); Becher and
Neubert (2009); Collins et al. (1989); Grozin (2022). Perhaps the most famous
example is the result of Yennie Frautschi & Suura (YFS) Yennie et al. (1961)
regarding the factorization of soft-radiation in QED. Splitting functions in
QCD and QED make heavy use of eikonal algebra Gribov and Lipatov (1972);
Altarelli and Parisi (1977); Dokshitzer (1977). The same eikonal properties
underlie the simplifications inherent to Wilson lines Korchemsky and
Radyushkin (1992); Grozin (2022). A related identity allows one to demonstrate
the emergence of classical background, e.g. Coulomb, fields sourced by heavy
particles Brodsky (1971); Neghabian and Gloeckle (1983); Weinberg (2005).
Suffice to say, eikonal algebra is a key tool in the study of soft limits for
gauge theories.

Surprisingly little is known about the eikonal structure of gauge theories in
the presence of external charged currents. Perhaps the most relevant example
is nuclear beta decay Hardy and Towner (2020) which involves a heavy-heavy-
light vertex formed by a nucleus of charge $Z$, a nucleus of charge $Z\pm 1$,
and a single electron/positron. The same vertex appears in charged-current
neutrino nucleus scattering. Precision theory for both processes are important
for modern experimental programs in fundamental physics Branca et al. (2021);
Tomalak et al. (2022); Seng et al. (2018); Czarnecki et al. (2019); Hardy and
Towner (2020). More complicated scenarios are furnished in e.g. double-beta
decay Dolinski et al. (2019) where the vertex would be heavy-heavy-light-light
and the charge exchange from the Wilson line would be $\pm 2$.

In this work we derive new eikonal identities for charged current processes.
We apply these identities to nuclear beta decay and find that their
application results in substantial simplifications in the analysis of QED
corrections at high loop order. We identify new gauge invariant sub-classes of
diagrams that emerge in the heavy-particle limit. We use these gauge invariant
sub-classes to show that the “independent particle model” (IPM) introduced by
Jaus and Rasche Jaus and Rasche (1970); Jaus (1972) and used by Sirlin and
Zuchini Sirlin and Zucchini (1986); Sirlin (1987) for outer radiative
corrections in beta decay, is nearly (but not exactly) equivalent to a model
independent effective field theory (EFT). We conclude with a discussion of
potential future applications.

##  II Eikonal identities for charged currents

Consider an external current $\mathcal{J}$ which induces a charge-changing
reaction between two heavy particles $A\rightarrow B$. We work in a heavy
particle EFT with eikonal propagators, $1/(v\cdot p+{\rm i}0)$ minimally
coupled to the photon field, $\mathcal{L}\supset h_{v}^{\dagger}v\cdot Dh_{v}$
with $D_{\mu}=\partial_{\mu}-{\rm i}QeA_{\mu}$ where $Q$ is the particle’s
charge; we work in the static limit where $v^{\prime}=v$. We will write
$Z_{A}=Z+z$ and set $Z_{B}=Z$, such that $z=Z_{A}-Z_{B}$. We use
$G_{\mu_{1},...,\mu_{n}}^{(\rm el)}(q_{1},...,q_{n})$ to denote the sum over
all $n$-photon dressings of the bare matrix element
$\mathcal{J}_{AB}=\matrixelement{B}{\mathcal{J}}{A}$. We may simplify our
analysis considerably by using the standard soft-photon identity Yennie et al.
(1961); Weinberg (1965)

$\sum_{\rm perms}\frac{1}{v\cdot q_{1}+{\rm
i}0}\frac{1}{v\cdot(q_{1}+q_{2})+{\rm
i}0}\times\ldots\times\frac{1}{v\cdot(\sum_{i=1}^{n}q_{i})+{\rm
i}0}=\prod_{i=1}^{n}\frac{1}{v\cdot q_{i}+{\rm i}0}~{}.$ (1)

We may partition the set of crossed ladders into those with no photons to the
right $\mathcal{J}_{AB}$, one photon to the right of $\mathcal{J}_{AB}$ etc.
Let us introduce the set $\mathcal{N}=\\{1,2,...,n\\}$ and the set
$\mathcal{N}_{i,j,k}=\mathcal{N}/\\{i,j,k\\}$ (i.e. the set “not $i$, $j$, or
$k$”). For a set of integers $S$ let us denote the symmetrized product of soft
photon emissions from the initial state by $L(S)$ and by the final state by
$R(S)$, i.e.

$\displaystyle L(S)$ $\displaystyle=\prod_{i\in S}^{n}\frac{1}{v\cdot
q_{i}+{\rm i}0}~{},$ (2) $\displaystyle R(S)$ $\displaystyle=\prod_{i\in
S}^{n}\frac{1}{-v\cdot q_{i}+{\rm i}0}~{}.$ (3)

We may then write the result for general $z$ as (with $i,j\in\mathcal{N}$)

$\begin{split}G^{(\rm
el)}_{\mu_{1},...,\mu_{n};\nu}(q_{1},...,q_{n})=\quantity[\prod_{i}v_{\mu_{i}}]\times\bigg{[}&(Z+z)^{n}R(\mathcal{N})\\\
&+Z(Z+z)^{n-1}\sum_{i}R(\mathcal{N}_{i})L(\\{i\\})\\\
&+Z^{2}(Z+z)^{n-2}\sum_{i>j}R(\mathcal{N}_{i,j})L(\\{i,j\\})\\\ &+\ldots\\\
&+Z^{n-1}(Z+z)\sum_{i}R(\\{i\\})L(\mathcal{N}_{i})\\\
&+Z^{n}L(\mathcal{N})\bigg{]}~{}.\end{split}$ (4)

This can be written diagrammatically using a square for the external current,
drawing the parent, $A$, with a double line, and drawing the daughter state,
$B$ with a dashed double line. The resulting diagrams at two-loop order are
given by,

Setting $z=0$ (i.e. $Z_{A}=Z_{B}$) we obtain the standard result in the static
limit Brodsky (1971); Dittrich (1970); Neghabian and Gloeckle (1983); Weinberg
(2005),

$\begin{split}\bigg{[}R(\mathcal{N})+\sum_{i\in\mathcal{N}}R(\mathcal{N}_{i})&L(\\{i\\})+\sum_{i\in\mathcal{N}}\sum_{j\in\mathcal{N}_{i}}R(\mathcal{N}_{i,j})L(\\{i,j\\})\\\
&+\ldots+\sum_{a\in\mathcal{N}}R(\\{a\\})L(\mathcal{N}_{a})+Z^{n}L(\mathcal{N})\bigg{]}=\prod_{i\in\mathcal{N}}(2\pi{\rm
i})\delta(v\cdot q_{i})~{}.\end{split}$ (5)

The general result can be organized in a series in $Z^{n-m}z^{m}$ for $0\leq
m\leq n$. The $Z^{n}$ contributions match the equal charge limit, and are
given by the expressions presented above. Let us consider the contributions
proportional to $Z^{n-1}z$. We pick up a binomial coefficient from the
expansion of $(Z+z)^{n}$ which we will write explicitly as ${n\choose 1}$. We
then have

$\begin{split}Z^{n-1}z\bigg{[}{n\choose 1}R(\mathcal{N})+{n-1\choose
1}\sum_{i}R(\mathcal{N}_{i})&L(\\{i\\})+{n-2\choose
1}\sum_{i>j}R(\mathcal{N}_{i,j})L(\\{i,j\\})\\\ &\hskip
43.36464pt+\ldots+{1\choose
1}\sum_{i}R(\\{i\\})L(\mathcal{N}_{i})\bigg{]}~{}.\end{split}$ (6)

The binomial factors can all be reproduced by introducing an additional “dummy
sum” to each term, i.e. ${n\choose
1}R(\mathcal{N})=\sum_{a\in\mathcal{N}}R(\mathcal{N})$ and ${n-1\choose
1}\sum_{i\in\mathcal{N}}R(\mathcal{N}_{i})L(\\{i\\})=\sum_{i}\sum_{a\in\mathcal{N}_{i}}R(\mathcal{N}_{i})L(\\{i\\})$.
The indices of the original sum in each term of Eq. 6 are always excluded from
the second sum. This gives us

$\begin{split}&Z^{n-1}z\bigg{[}\sum_{a}R(\mathcal{N})+\sum_{i}\sum_{a\neq
i}R(\mathcal{N}_{i})L(\\{i\\})+\sum_{i>j}\sum_{a\neq
i,j}R(\mathcal{N}_{i,j})L(\\{i,j\\})+\ldots\bigg{]}~{}.\end{split}$ (7)

Using the property that for $a\in S$ we have $R(S)=\frac{1}{v\cdot q_{a}+{\rm
i}0}R(S/\\{a\\})$ we obtain

$\begin{split}&Z^{n-1}z\bigg{[}\sum_{a}\frac{1}{v\cdot q_{a}+{\rm
i}0}R(\mathcal{N}_{a})+\sum_{i}\sum_{a\neq i}\frac{1}{v\cdot q_{a}+{\rm
i}0}R(\mathcal{N}_{a,i})L(\\{i\\})\\\ &\hskip
173.44534pt+\sum_{i>j}\sum_{a\neq i,j}\frac{1}{v\cdot q_{a}+{\rm
i}0}R(\mathcal{N}_{a,i,j})L(\\{i,j\\})+\ldots\bigg{]}~{},\\\
&=Z^{n-1}z\sum_{a}\frac{1}{v\cdot q_{a}+{\rm
i}0}\bigg{[}R(\mathcal{N}_{a})+\sum_{i}R(\mathcal{N}_{a,i})L(\\{i\\})\\\
&\hskip
173.44534pt+\sum_{i>j}R(\mathcal{N}_{a,i,j})L(\\{i,j\\})+\ldots\bigg{]}~{},\\\
&=Z^{n-1}z\sum_{a\in\mathcal{N}}\frac{1}{v\cdot q_{a}+{\rm
i}0}\prod_{i\in\mathcal{N}_{a}}(2\pi{\rm i})\delta(v\cdot
q_{a})~{},\end{split}$ (8)

where in the second line the sums over $i$ and $j$ are for
$i,j\in\mathcal{N}_{a}$. This may be recognized as the Feynman rules for
$n-1$-photons coupling to a Coulomb field, and one photon coupling to a heavy-
particle of charge $z$.

The analysis presented above (for $z^{1}Z^{n-1}$) generalizes readily to
$z^{m}Z^{n-m}$ with ${n\choose 1}$ replaced by ${n\choose m}$. The dummy sum
that must be introduced is appropriately modified; for example ${n\choose
2}=\sum_{i>j}1$ and ${n\choose 3}=\sum_{i>j>k}1$. The rest of the analysis
proceeds identically. The final result is that

$\begin{split}G^{(\rm
el)}_{\mu_{1},...,\mu_{n};\nu}(q_{1},...,q_{n})&=\quantity[\prod_{i=1}^{n}v_{\mu_{i}}]\bigg{[}Z^{n}\prod_{i=1}^{n}(2\pi{\rm
i})\delta(v\cdot q_{i})\\\ &\hskip 65.04034pt+zZ^{n-1}\sum_{a}\frac{1}{v\cdot
q_{a}+{\rm i}0}\prod_{i\neq a}(2\pi{\rm i})\delta(v\cdot q_{i})\\\ &\hskip
65.04034pt+z^{2}Z^{n-2}\sum_{a<b}\frac{1}{v\cdot q_{a}+{\rm
i}0}\frac{1}{v\cdot q_{b}+{\rm i}0}\\\ &\hskip 182.1196pt\times\prod_{i\neq
a,b}(2\pi{\rm i})\delta(v\cdot q_{i})\\\ &\hskip 65.04034pt+\hskip
43.36464pt\ldots\hskip 151.76964pt\bigg{]}\end{split}$ (9)

This is equivalent to the coherent sum of amplitudes from a heavy particle of
charge $z$ transitioning to a heavy particle with vanishing charge and a
static background Coulomb field $V(\mathbf{r})=Z\alpha/|\mathbf{r}|$. The
static Coulomb field couples to all charged particles in the diagram except
not to the heavy particle of charge $z$. The sum over $a$ and $b$ accounts for
crossed diagrams between Coulomb modes and the soft photons emitted by the
initial state heavy particle.

This result may have been expected Szafron (2023) on the basis on the Abelian
exponentiation theorem Yennie et al. (1961); Grozin (2022). Since the webs for
a particle of charge $Z_{A}$ and $Z_{B}$ will be linear in the charge such
that the product of the Wilson line soft-functions would be given be
proportional to $Z_{A}-Z_{B}$. This neglects the subtlety that accounts for
the Coulomb field in the $Z_{A}=Z_{B}$ limit, and the analysis above is a
direct demonstration via combinatorics that the intuition from the
exponentiation theorem is indeed correct.

##  III Effective theory of beta decay in the point-like limit

Let us now consider beta decay in an EFT where nuclei appear as point-like
heavy particles. In the EFT we have two heavy particle fields, $h_{A}$ and
$h_{B}$, and two relativistic fermions, $e$ and $\nu$. The relevant Lagrangian
is given by

$\begin{split}\mathcal{L}_{\rm
HPET}&=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+h_{B}^{\dagger}\quantity[{\rm
i}v_{\mu}(\partial^{\mu}+{\rm
i}Z_{B}A^{\mu})]h_{B}+h_{A}^{\dagger}\quantity[{\rm
i}v_{\mu}(\partial^{\mu}+{\rm i}Z_{A}A^{\mu})]h_{A}\\\ &\hskip
32.52017pt+\bar{\nu}{\rm i}\gamma_{\mu}\partial^{\mu}\nu+\bar{e}\quantity[{\rm
i}\gamma_{\mu}(\partial^{\mu}-{\rm
i}eA^{\mu})-m]e+\quantity[-G_{F}\bar{e}\Gamma_{\ell}\nu
h_{B}^{\dagger}\Gamma_{h}h_{A}+{\rm c.c.}]~{},\end{split}$ (10)

where $\Gamma_{\ell}$ and $\Gamma_{h}$ are the relevant spin structures for
the weak charged current.

A decoupling transformation can be performed on Eq. 10 by introducing a field
redefinition in terms of Wilson lines. The key is to shift $h_{A}$ and $h_{B}$
to

$\displaystyle h_{A}\rightarrow\mathfrak{h}_{A}(x)$
$\displaystyle=S(x)h_{A}(x)~{},$ (11) $\displaystyle
h_{B}\rightarrow\mathfrak{h}_{B}(x)$ $\displaystyle=\bar{S}(x)h_{B}(x)~{}.$
(12)

where $S(x)$ and $\bar{S}(x)$ are Wilson lines appropriate for particles in
the initial and final state with charge $Ze$,

$\displaystyle S(x)$ $\displaystyle=\exp\quantity[{\rm
i}Ze\int_{-\infty}^{0}\differential s~{}v\cdot A(x+sv)]~{}.$ (13)
$\displaystyle\bar{S}(x)$ $\displaystyle=\exp\quantity[-{\rm
i}Ze\int_{0}^{\infty}\differential s~{}v\cdot A(x+sv)],$ (14)

In terms of the new fields the Lagrangian assumes the form,

$\begin{split}\mathcal{L}_{{\rm
HPET}^{\prime}}&=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\mathfrak{h}_{B}^{\dagger}\quantity[{\rm
i}v_{\mu}(\partial^{\mu})]\mathfrak{h}_{B}+\mathfrak{h}_{A}^{\dagger}\quantity[{\rm
i}v_{\mu}(\partial^{\mu}+{\rm i}zA^{\mu})]\mathfrak{h}_{A}\\\ &\hskip
32.52017pt+\bar{\nu}{\rm i}\gamma_{\mu}\partial^{\mu}\nu+\bar{e}\quantity[{\rm
i}\gamma_{\mu}(\partial^{\mu}-{\rm
i}eA^{\mu})-m]e+\quantity[-G_{F}\bar{e}\Gamma_{\ell}\nu\mathfrak{h}_{B}^{\dagger}\Gamma_{h}\mathfrak{h}_{A}S\bar{S}^{\dagger}+{\rm
c.c.}]~{},\end{split}$ (15)

Now background Coulomb diagrams arise from the matrix element of the Wilson
lines $\langle S\bar{S}^{\dagger}\rangle$. The differing boundary conditions
in position space for the Wilson lines $S$ and $\bar{S}$ reproduce the
differing causal regulators in momentum space. This can be seen explicitly as

$\bar{S}^{\dagger}(x)S(x)=\exp\quantity[{\rm
i}Ze\int_{-\infty}^{\infty}\differential sv\cdot A(x+sv)]=\exp[2\pi{\rm
i}\delta(v\cdot\partial)v\cdot A]~{}.$ (16)

In the decoupled theory there is a single heavy particle with residual charge
$z$, $\mathfrak{h}_{A}$ and a background field with charge $Z$.

Diagrams involving Coulomb exchanges with the heavy field do not contribute to
amplitudes. This can be seen in two ways: First, consider working diagram by
diagram, one can note that these diagrams can always be canceled by a mass
counter term which enforces a vanishing residual mass for the heavy
particle111The diagrams vanish in dimensional regularization unless a photon
mass is included as an IR regulator. In this case the counter term enforces
zero residual mass order-by-order in $Z\alpha$. Borah et al. (2024). Second,
one can show that these diagrams belong to a gauge invariant sub-class (see
Appendix A), and that this subclass vanishes.

A direct evaluation of diagrams using Eq. 10 is conceptually straightforward,
but tedious and eventually unwieldy at high orders in perturbation theory.
Current extractions of $|V_{ud}|$ from superallowed beta decay require
$O(Z^{2}\alpha^{3})$ input Sirlin and Zucchini (1986); Hardy and Towner
(2020). At this order, without considering counter terms, there are 120
diagrams which contribute to the amputated vertex function without vacuum
polarization, and 144 diagrams after including vacuum polarization. The
eikonal identities presented above drastically reduces the number of diagrams
which must be computed. For example at three loops, including vacuum
polarization, one only needs to compute ten graphs for the amputated amplitude
using the background-field Feynman rules derived above Borah et al. (2024).

##  IV Comparison with Jaus, Rasche, Sirlin & Zucchini

Let us now consider outer radiative corrections in beta decay. Calculations
are have historically been performed in the IPM Jaus and Rasche (1970); Jaus
(1972); Sirlin and Zucchini (1986); Jaus and Rasche (1987); Sirlin (1987).
This model corresponds to that defined by $\mathcal{L}_{\rm bkg}$ in Eq. 18
except that in place of a heavy particle, the authors use a soft-photon or YFS
approximation, making the replacement

$\frac{1}{v\cdot q+{\rm i}0}\rightarrow\frac{2m_{p}}{2p\cdot q+q^{2}+{\rm
i}0}~{},$ (17)

in their diagrams. This renders the diagrams UV convergent but introduces a
dependence on the “proton mass”. In the calculations of Sirlin and Zucchini
Sirlin and Zucchini (1986); Sirlin (1987) vacuum polarization diagrams are
added to Coulomb modes in agreement with the argument sketched above.

One may treat $m_{p}$ as a new hard scale in the problem and separate scales
using the method of regions Beneke and Smirnov (1998); Jantzen (2011). The
hard region supplies a contribution to the Wilson coefficients that depends on
$m_{p}$. This dependence is unphysical, since the propagating degrees of
freedom at low momenta in are the atomic nuclei of charge $Z$ and $Z\pm 1$. In
other work by Sirlin Sirlin (1987), a charge form factor is included for the
Coulomb field, which can in certain cases eliminate sensitivity to $m_{p}$,
replacing it by the physical scale of nuclear structure. The soft region of
the IPM reproduces amplitudes computed with $\mathcal{L}_{{\rm
HPET}^{\prime}}$. We therefore conclude that, upon separating scales in the
IPM, one will obtain amplitudes which have the correct long-distance but may
contain spurious short-range contributions.

##  V Conclusions

We have derived new eikonal identities that are relevant for problems with
heavy particles whose charge is modified by an external charged current e.g.,
for semi-leptonic weak interactions. The identities which give rise to Coulomb
fields in the static limit are substantially modified. We have obtained a
simple expression involving uncorrelated photon exchange between either a
background Coulomb field, or a charge heavy particle of charge $Z_{A}-Z_{B}$.

We have applied these identities to the EFT relevant for nuclear beta decay
and identified new gauge invariant sub-classes of diagrams. The results
presented above can be used to simplify calculations of the anomalous
dimension and matrix elements of operators that mediate beta decays. Detailed
calculations are presented elsewhere Hill and Plestid (2023); Borah et al.
(2024).

## Acknowledgments

This work benefited greatly from collaboration with Richard Hill on related
projects, and I am specifically grateful for suggestions related to the
decoupling transformation used in Eq. 15. I thank Robert Szafron, Michele
Papucci for useful discussions. I thank Richard Hill, Andreas Helset and Julia
Para-Martinez for providing feedback on early versions of this manuscript.

This work is supported by the Neutrino Theory Network under Award Number
DEAC02-07CHI11359, the U.S. Department of Energy, Office of Science, Office of
High Energy Physics, under Award Number DE-SC0011632, and by the Walter Burke
Institute for Theoretical Physics.

##  Appendix A Gauge invariant sub classes for beta decay

In this Appendix we use of the eikonal identities derived in Section II, an
auxiliary background field Lagrangian, and certain useful properties of
Coulomb gauge to identify gauge invariant sub-classes of diagrams. This
simplifies the analysis of beta decay amplitudes at high perturbative order,
since some of these sub-classes vanish. This section is complimentary to the
discussion of the decoupling tranform in Eq. 15.

Equation 9 implies that the Feynman rules generated by Eq. 10 for ladder
graphs in which all photon attachments to the heavy composite lead to a
leptonic line, can be reproduced order-by-order in perturbation theory by
using the Lagrangian,

$\begin{split}\mathcal{L}_{\rm
bkg}&=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+h_{B}^{\dagger}\quantity[{\rm
i}v_{\mu}\partial^{\mu}]h_{B}+h_{A}^{\dagger}\quantity[{\rm
i}v_{\mu}(\partial^{\mu}+{\rm i}zA^{\mu})]h_{A}+\bar{\nu}{\rm
i}\gamma_{\mu}\partial^{\mu}\nu\\\ &\hskip 32.52017pt+\bar{e}\quantity[{\rm
i}\gamma_{\mu}(\partial^{\mu}-{\rm i}eA^{\mu}-{\rm
i}Z\alpha\mathscr{A}^{\mu})-m]e+\quantity[-\sqrt{2}G_{F}\bar{e}\Gamma_{\ell}\nu
h_{B}^{\dagger}\Gamma_{h}h_{A}+{\rm c.c.}]~{}.\end{split}$ (18)

We will refer to these graphs as “dynamically dressed” in that they have a
ladder skeleton but may be dressed by dynamical photons e.g. vertex
corrections. The theory has a fixed classical background field
$\mathscr{A}^{\mu}(x)=\phi(x)v^{\mu}$ with $\phi(x)=1/|\mathbf{r}|$ where
$\mathbf{r}^{\mu}=x^{\mu}-v^{\mu}(v\cdot x)$.

Amplitudes computed using Eq. 18 can be written in the form

${\rm i}\mathcal{M}_{\rm
bkg}=\sqrt{\mathcal{Z}_{A}(z)}\prod_{i=1}^{n}\sqrt{\mathcal{Z}_{i}}\times\quantity({\rm
connected~{}diagrams})_{\rm bkg}$ (19)

where $\mathcal{Z}_{A}$ is the wavefunction renormalization of the heavy field
of charge $z$. This amplitude is invariant under two separate gauge groups
$U_{\rm QED}(1)\otimes U_{\rm bkg}(1)$, since in the background model the
background field and gauge field are independent of one another.

Figure 1: Examples of graphs in the dynamic subset (left) and the static
subset (right). Heavy fields are drawn with double lines, while light fields
(i.e. an electron) are drawn as single lines. The weak-current vertex is shown
as a grey square (the neutrino is not drawn as it is electrically neutral).
The photon drawn as a dashed line on the right causes the graph to vanish in
Coulomb gauge.

Let us consider an amplitude computed using the Feynman rules of Eq. 10. We
will group dynamically dressed graphs together and then add and subtract the
wavefunction renormalization for the charge $z$ heavy particle order-by-order
in perturbation theory222One could equivalently appeal to the Abelian
exponentiation theorem, since the product of Wilson lines in the initial and
final state will multiply such that the overall effect is only proportional to
$Z_{A}-Z_{B}$; we find the above argument cleaner.. An example of a graph from
this subset is shown on the left of Fig. 1. This subset of diagrams, which we
will refer to as the “dynamic subset” is precisely equivalent to
$\mathcal{M}_{\rm bkg}$ order-by-order in perturbation theory and therefore
gauge invariant. The remaining diagrams, with the contribution from
wavefunction renormalization subtracted, then form a separate gauge invariant
class. These diagrams involve dressings of the heavy particles by photons
(e.g. vertex corrections, self energies etc.). Since we work in the static
limit, we refer to this set of diagrams as being the “static subset”. An
example of a graph in the static subset is shown on the right of Fig. 1. Since
the sum of both subsets is gauge invariant, and the dynamic subset is gauge
invariant, it follows that the static subset is separately gauge invariant.

We now evaluate the static subset in Coulomb gauge (see Appendix B for a
discussion). All diagrams contain either heavy-particle wavefunction
renormalization, or heavy-heavy vertex corrections and these vanish diagram-
by-diagram in Coulomb gauge. Since the static subset is gauge invariant, this
statement is true for the sum of all diagrams for arbitrary gauge. Therefore,
an amplitude computed with Eq. 10 agrees order by order in perturbation theory
with an amplitude computed using Eq. 18.

##  Appendix B Coulomb gauge & the static limit

Let us work in the static limit $v=v^{\prime}$. We may then work in Coulomb
gauge choosing $v_{\mu}=(1,0,0,0)$ such that transverse photons explicitly
decouple from the heavy particles and Coulomb propagators, $D_{00}={\rm
i}/\mathbf{q}^{2}$, contain no energetic poles. For a heavy particle in the
initial and final state any sub-graph involving a photon which connects two
heavy particle lines will result in an integrand with all of its poles on one
side of the complex plane. The contour can then be closed in the opposite
direction, and the integral will vanish. For example, let us take the
following loop graph

$\vspace{3pt}\raisebox{-25.0pt}{ \fmffile{new-name} \fmfgraph*(75,45)
\fmfbottom{i1,o1} \fmftop{t1} \fmf{double}{i1,vl,t1}
\fmf{dbl_dashes}{t1,vr,o1} \fmf{photon,tension=0}{vl,vr}
\fmfv{d.sh=square,d.si=3mm}{t1}
}\quad=\quad\int\frac{\differential^{d}L}{(2\pi)^{d}}\quantity(\frac{1}{L_{0}+{\rm
i}0})^{2}\frac{{\rm i}}{\mathbf{L}^{2}}\quad=\quad 0~{}.\vspace{3pt}$ (20)

A corollary of Eq. 20, and the fact that the heavy particle self-energies also
vanish in Coulomb gauge, is the well known fact that the cusp anomalous
dimension vanishes at zero recoil Grozin (2022).

This result also holds in the presence of dynamical fermions. Fermion loops
can be performed first and do not mix longitudinal and transverse modes. As a
result any graph containing a photon connecting two heavy lines will vanish.
The only remaining graphs which are non-vanishing are ladder graphs, and
ladder graphs dressed by sub-graphs on the electron lines. As discussed above
these can be combined into the dynamic subset, and after adding wavefunction
renormalization for heavy particle with charge $z$, form a gauge invariant
subset. They may then be evaluated separately in whatever gauge is most
convenient.

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# Constructing abelian varieties from rank 3 Galois representations with real
trace field

Raju Krishnamoorthy<EMAIL_ADDRESS>and  Yeuk Hay Joshua Lam
<EMAIL_ADDRESS>Humboldt Universität Berlin, Institut für Mathematik-
Alg.Geo., Rudower Chaussee 25 Berlin, Germany

###### Abstract.

Let $U/K$ be a smooth affine curve over a number field and let $L$ be an
irreducible rank 3 $\overline{\mathbb{Q}}_{\ell}$-local system on $U$ with
trivial determinant and infinite geometric monodromy around a cusp. Suppose
further that $L$ extends to an integral model such that the Frobenius traces
are contained in a fixed totally real number field. Then, after potentially
shrinking $U$, there exists an abelian scheme $f\colon B_{U}\rightarrow U$
such that $L$ is a summand of $R^{2}f_{*}\overline{\mathbb{Q}}_{\ell}(1)$.

The key ingredients are: (1) the totally real assumption implies $L$ admits a
square root $M$; (2) the trace field of $M$ is sufficiently bounded, allowing
us to use [KYZ24] to construct an abelian scheme over $U_{\bar{K}}$
geometrically realizing $L$; and (3) Deligne’s weight-monodromy theorem and
the Rapoport-Zink spectral sequence, which allow us to pin down the
arithmetizations using the total degeneration.

######  Setup 1.

Let $X/K$ be a smooth projective geometrically connected curve over a number
field, let $D\subset X$ be a reduced divisor, and set $U:=X\setminus D$. Fix a
prime $\ell$ and an algebraic closure $\overline{\mathbb{Q}}_{\ell}$ of
$\mathbb{Q}_{\ell}$. Let $L$ be a rank $r\geq 1$ irreducible
$\overline{\mathbb{Q}}_{\ell}$-local system on $U$ with trivial determinant.

It is well known that there exists an integral model
$(\mathfrak{X},\mathfrak{D})$, defined over $\mathcal{O}:=\mathcal{O}_{K,S}$
for $S$ a finite set of primes of $\mathcal{O}_{K}$, of $(X,D)$, such that $L$
naturally extends to a $\overline{\mathbb{Q}}_{\ell}$-local system
$\mathscr{L}$ on $\mathfrak{U}:=\mathfrak{X}\setminus\mathfrak{D}$, still with
trivial determinant (see e.g. [Pet23, Proposition 6.1]).

The relative Fontaine–Mazur conjecture, as reformulated by Petrov, predicts
that local systems in Setup 1 occur in the cohomology of a smooth projective
family after possible restricting to an open $V\subset U$ and twisting by a
character of $G_{K}$. In this article, we make some progress towards this
conjecture when the rank is 3.

######  Theorem A.

Suppose $r=3$ and $L$ has infinite geometric monodromy around at least one
point of $D$. Suppose further there exists a totally real number subfield
$E\subset\overline{\mathbb{Q}}_{\ell}$ such that the field generated by
Frobenius traces of $\mathscr{L}$ is contained in $E$. Then after potentially
increasing $D$, there exists an abelian scheme $f\colon B_{U}\rightarrow U$
such that $L$ is a summand of $R^{2}f_{*}\overline{\mathbb{Q}}_{\ell}(1)$.

######  Remark 2.

Note that if we do not make any assumption on the trace field being totally
real, then the conclusion of Theorem A is false. Indeed, there exist rank 3
hypergeometric local systems (with monodromy Zariski dense in
$\text{SL}(3,\overline{\mathbb{Q}}_{\ell})$) with bounded Frobenius trace
field that do not come from abelian varieties.

An auxiliary result, which may be of indepedent interest, is the following.

######  Lemma 3.

Let $U/K$ be a smooth curve over a number field with compactification $X$, let
$f\colon Y\rightarrow U$ be a smooth projective morphism, and let $Q$ be a
rank $r$ summand of $R^{r-1}f_{*}\overline{\mathbb{Q}}_{\ell}$. Further
suppose that $Q$ is totally degenerating at some $\infty\in X\setminus U$,
i.e., the local inertia around $\infty$ acts with a maximal Jordan block. Then
$Q$ has determinant isomorphic to $\overline{\mathbb{Q}}_{\ell}(-r(r-1)/2)$ up
to a finite order character.

######  Remark 4.

Our assumptions in 3 include that of $Q$ being of rank $r$ and appearing in
weight $r-1$. Note that there must be some relation between these quantities
for the conclusion of the lemma, namely that the determinant is a power of
cyclotomic (up to a finite order character), to hold. For example, for any $Q$
as in 3, and $\chi$ a CM character appearing in $H^{1}$ of a CM abelian
variety $A_{0}/K$, $Q\otimes\chi$ now has rank $r$ and weight $r$, and no
power of the determinant is a power of cyclotomic.

###### Proof of A.

We break the proof up into steps.

  1. (1)

First of all, we claim that it is sufficient to prove the theorem after
replacing $K$ by a finite extension and $U_{K}$ with a (geometrically
connected) finite étale cover. Indeed, suppose we have a finite étale map
$V\rightarrow U$ and an abelian scheme $g\colon B_{V}\rightarrow V$ such that
$L(-1)|_{V}$ is a summand of $R^{2}g_{*}\overline{\mathbb{Q}}_{\ell}$. Then we
claim that the Weil restriction $f\colon
A_{U}:=\mathfrak{Res}^{V}_{U}B_{V}\rightarrow U$, an abelian scheme on $U$,
will satisfy the conclusion of the theorem. The two relevant facts we need:
$R^{2}g_{*}\overline{\mathbb{Q}}_{\ell}$ is semi-simple by [Fal+92, Ch. 6,
Sec. 3, Theorem I on p. 211], and if $H\subset G$ is a finite index subgroup
and $V$ a representation of $H$, then
$\text{Ind}^{G}_{H}(\bigwedge^{2}V)\hookrightarrow\bigwedge^{2}\text{Ind}^{G}_{H}(V)$.

  2. (2)

Fix $\mathfrak{p}\in\text{Spec}(\mathcal{O})$ with residue field
$\mathbb{F}_{q}$ and set
$\mathscr{L}_{\mathfrak{p}}:=\mathscr{L}|_{\mathfrak{U}_{\mathfrak{p}}}$.
First of all, we claim that the monodromy of $\mathscr{L}_{\mathfrak{p}}$ is
contained in ${\rm SO}(3,\overline{\mathbb{Q}}_{\ell})$. To see this, fix an
isomorphism $\iota\colon\overline{\mathbb{Q}}_{\ell}\rightarrow\mathbb{C}$ and
set $\sigma\in\text{Aut}(\overline{\mathbb{Q}}_{\ell})$ to be the transport of
structure of complex conjugation.

[Laf02, Théorème VII.6] implies that every Frobenius eigenvalue of
$\mathscr{L}_{\mathfrak{p}}$ is a $q$-Weil number of weight 0 and is contained
in a CM number field. Therefore the Frobenius trace field (see [KL23,
Definition 1.1.4] for the definition) of $\mathscr{L}_{\mathfrak{p}}$ is
contained in a CM number field. Hence $\sigma$ acts by complex conjugation on
the traces of Frobenius conjugacy classes. Let
${}^{\sigma}(\mathscr{L}_{\mathfrak{p}})$ denote the $\sigma$-companion of
$\mathscr{L}_{\mathfrak{p}}$, which exists again by [Laf02, Théorème VII.6].
Then ${}^{\sigma}(\mathscr{L}_{\mathfrak{p}})\cong\mathscr{L}_{\mathfrak{p}}$
by the totally real assumption. On the other hand, as
$\mathscr{L}_{\mathfrak{p}}$ is pure of weight 0, it follows that
${}^{\sigma}(\mathscr{L}_{\mathfrak{p}})\cong\mathscr{L}^{\vee}_{\mathfrak{p}}$.
Therefore there is an equivariant pairing
$\mathscr{L}_{\mathfrak{p}}\otimes\mathscr{L}_{\mathfrak{p}}\rightarrow\overline{\mathbb{Q}}_{\ell}$,
which is moreover non-degenerate as $\mathscr{L}_{\mathfrak{p}}$ is
irreducible. As $\mathscr{L}_{\mathfrak{p}}$ has odd rank, this must be a
symmetric pairing, i.e. the monodromy lies in ${\rm
SO}(3,\overline{\mathbb{Q}}_{\ell})$.

  3. (3)

There is an isogeny
$\text{SL}(2,\overline{\mathbb{Q}}_{\ell})\rightarrow\text{SO}(3,\overline{\mathbb{Q}}_{\ell})$,
given by the second symmetric square of the defining representation.
Therefore, after possibly extending $K$ and replacing $U$ by a finite étale
cover, we may assume $L\cong\text{Sym}^{2}M$, with $M$ an irreducible rank 2
$\overline{\mathbb{Q}}_{\ell}$-local system on $U$, with trivial determinant.
Note that $M$ still has infinite monodromy around a point of $D$. Moreover, by
[Pet23, Proposition 6.1], after potentially enlarging $S$, $M$ canonically
extends to a $\overline{\mathbb{Q}}_{\ell}$ local system $\mathscr{M}$ on
$\mathfrak{U}$ with trivial determinant.

  4. (4)

Enlarge $S$ to contain 2. We claim that there exists a finite étale map
$\mathcal{O}\rightarrow\mathcal{O}^{\prime}$, a curve
$\mathfrak{V}/\mathcal{O}^{\prime}$, and a finite étale morphism
$\mathfrak{V}\rightarrow\mathfrak{U}$ such that the stable trace field
(defined in [KL23, Definition 1.1.4]) of $\mathscr{M}|_{\mathfrak{V}}$ is
contained in $E$. Indeed, let $V\rightarrow U$ be a cover that trivializes the
2-torsion of the Jacobian. Let us prove that $\mathscr{M}|_{\mathfrak{V}}$ has
stable trace field contained in $E$.

Fix a prime $\mathfrak{p}^{\prime}$ of $\mathcal{O}^{\prime}$ lying over a
prime $\mathfrak{p}$ of $\mathcal{O}$, let the trace field of
$\mathscr{M}_{\mathfrak{p}}$ be $F\supset E$, and consider the embeddings
$\mathcal{S}:=\\{\sigma\colon F\hookrightarrow\overline{\mathbb{Q}}_{\ell}\\}$
that fix $E$ elementwise. Then, for each $\sigma\in\mathcal{S}$, the companion
${}^{\sigma}(\mathscr{M}_{\mathfrak{p}})$ has second symmetric square
isomorphic to $L$, as the operation of companions commutes with tensorial
constructions [KP22, Proof of Lemma 2.5]. Hence for each
$\sigma\in\mathcal{S}$, there exists a 2-torsion rank 1 local system $R$ on
$\mathfrak{U}_{\mathfrak{p}}$ such that
${}^{\sigma}(\mathscr{M}_{\mathfrak{p}})\cong\mathscr{M}_{\mathfrak{p}}\otimes
R$. Therefore we have
${}^{\sigma}(\mathscr{M}_{\mathfrak{p}})|_{\mathfrak{V}_{\mathfrak{p}^{\prime}}}\cong\mathscr{M}_{\mathfrak{p}}|_{\mathfrak{V}_{\mathfrak{p}^{\prime}}}\otimes\psi$,
with $\psi$ being a 2-torsion rank 1 local system of the base ${\rm
Spec}(\mathcal{O}^{\prime}/\mathfrak{p}^{\prime})$.

Therefore, all of the local systems
${}^{\sigma}(\mathscr{M}_{\mathfrak{p}})|_{\mathfrak{V}_{\mathfrak{p}^{\prime}}}$
become isomorphic after a quadratic extension of the base
$\mathcal{O}^{\prime}/\mathfrak{p}^{\prime}$, which implies that the stable
trace field of $\mathscr{M}|_{\mathfrak{V}}$ is contained in $E$. The upshot
is that, replacing $U$ by $V$, we may assume that the stable trace field of
$M$ is bounded.

  5. (5)

The boundedness of the stable trace field of $M$ implies, using the argument
of [KYZ24] (using the moduli space $\mathcal{H}$, not the more refined moduli
spaces $\mathcal{H}_{i}$ or $\mathcal{H}_{\infty}$ of _loc. cit._) that there
exists a principally polarized abelian scheme $\bar{f}\colon
A_{U_{\bar{K}}}\rightarrow U_{\bar{K}}$ such that
$R^{1}\bar{f}_{*}\overline{\mathbb{Q}}_{\ell}$ has $M|_{U_{\bar{K}}}$ as a
summand.111This argument is also indicated in Remark 1.9 of _loc. cit._ Here
is a sketch. After inverting finitely many primes, $\mathcal{H}/\mathcal{O}$
of Definition 3.1 of _loc. cit._ is finite flat by Section 4. Moreover,
$\mathcal{H}$ and has mod $p$ points for infinitely many $p$ by Section 2;
hence has a characteristic 0 point. Then the prime-to-$p$ specialization
isomorphism of $\pi_{1}$ implies the desired result. See also Footnote 5 of
_loc. cit._ As the abelian scheme $A_{U_{\bar{K}}}\rightarrow U_{\bar{K}}$ is
rigid (see Section 4 of _loc. cit._), it follows from [Fal83, Theorem 2] that
the summand $M|_{U_{\bar{K}}}$ is cut out by an element of
$End(A_{U_{\bar{K}}})\otimes\overline{\mathbb{Q}}_{\ell}$. Therefore, after
replacing $K$ by a finite extension (using that the endomorphism ring of an
abelian scheme is a finitely free $\mathbb{Z}$-module), it follows that there
exists an abelian scheme $f:A\rightarrow U$ such that the cohomology has a
rank 2 factor $M^{\prime}$ with the following property: $M_{U_{\bar{K}}}\cong
M^{\prime}_{U_{\bar{K}}}$. It now suffices to show that there exists a finite
étale cover $V\rightarrow U$ such that $M|_{V}\simeq M^{\prime}|_{V}(1/2)$ for
some choice of $\overline{\mathbb{Q}}_{\ell}(1/2)$: indeed, this implies that
$L|_{V}\simeq({\rm Sym}^{2}M)|_{V}\simeq{\rm Sym}^{2}(M^{\prime}|_{V})(1)$,
and by construction the latter appears in
$R^{2}f_{*}\overline{\mathbb{Q}}_{\ell}(1)$.

  6. (6)

By 3, $M^{\prime}$ has determinant equal to $\overline{\mathbb{Q}}_{\ell}(-1)$
up to a finite order character, and we conclude that

$M\simeq M^{\prime}(1/2)\otimes\chi$

for any choice of $\overline{\mathbb{Q}}_{\ell}(1/2)$, and $\chi$ some finite
order rank one local system on $U$. Passing to the finite cover $V\rightarrow
U$ which trivializes $\chi$ gives $M|_{V}\simeq M^{\prime}|_{V}$ as desired.

∎

###### Proof of 3.

Since the conclusion of our lemma is insensitive to passing to finite covers
of $U$, we may assume that $Q$ has maximal unipotent degeneration around some
cusp and moreover the map $f\colon Y\rightarrow U$ has strict semistable
reduction.

By Cebotarev density, it suffices to prove the claim after reduction mod
$\mathfrak{p}$ for almost all primes 222using also the fact that, for an
$\ell$-adic local field $F$ and a number field $K$, an almost everywhere
unramified character $\lambda:G_{K}\rightarrow\mathcal{O}_{F}^{\times}$ is of
finite order iff the Frobenius elements at almost all primes have finite
(though a priori unbounded) orders. Let $\infty$ be a cusp of
$U_{\mathfrak{p}}$, around which $Q$ is totally degenerating, with
uniformizing parameter $z_{\infty}$. It suffices to address the formal local
picture around $k((z_{\infty}))$, where $k$ is the field of definition of
$\infty$. More precisely, if $V$ is the $G_{k((z_{\infty}))}$-representation
corresponding to $Q|_{{\rm Spec}(k((z_{\infty})))}$, we will show that ${\rm
det}V$ is isomorphic to $\overline{\mathbb{Q}}_{\ell}(-r(r-1)/2)$, up to a
finite order character.

The maximal unipotent degeneration assumption implies the monodromy filtration
${\rm Fil}^{N}_{i}$ (see e.g. [Sch12, Definition/Proposition 9.2] for a
definition) has associated graded satisfying

$\dim_{\overline{\mathbb{Q}}_{\ell}}{\rm
gr}^{N}_{-(r-1)+2k}=1,\dim_{\overline{\mathbb{Q}}_{\ell}}{\rm
gr}^{N}_{-(r-1)+2k+1}=0$

for $k=0,\cdots,r-1$; furthermore, the monodromy operator

(1) $N:{\rm gr}^{N}_{-(r-1)+2k+2}\rightarrow{\rm gr}^{N}_{-(r-1)+2k}$

is a $G_{k}$-equivariant isomorphism.

Moreover $Q^{I}={\rm Fil}^{N}_{-(r-1)}$ is one dimensional, where $I\subset
G_{k((z_{\infty}))}$ denotes the inertia subgroup. Let $\Phi\in G_{k}$ denote
a geometric Frobenius element. The $\ell$-adic Steenbrink spectral sequence of
Rapoport-Zink ([RZ82, Satz 2.10]) implies that $\Phi$ acts on $Q^{I}$ via an
algebraic integer; on the other hand, the weight-monodromy conjecture, in this
setting a theorem of Deligne [Del80, Corollaire 1.8.5], implies that it is
pure of weight 0, and we deduce that $\Phi$ acts on $Q^{I}$ by a root of unity
$\zeta$. By Equation (1) and the commutation relation $N\Phi=q\Phi N$, the
$\Phi$-action on ${\rm gr}^{N}_{-(r-1)+2k}$ is given by $q^{k}\zeta$, and
hence the $\Phi$-action on ${\rm det}V$ is given by
$\zeta^{r}q^{1+\cdots+(r-1)}=\zeta^{r}q^{\frac{(r-1)r}{2}}$, as desired. ∎

######  Remark 5.

The hypothesis of Theorem A may be weakend to the following: the Frobenius
trace field of $L$ is bounded and there exists a prime $\mathfrak{p}$ of
$\mathcal{O}$ such that the Frobenius trace field of
$\mathscr{L}_{\mathfrak{p}}$ is totally real. Indeed, the hypothesis will
imply that the geometric monodromy of $L$ lands in
$\text{SO}(3,\overline{\mathbb{Q}}_{\ell})\subset\text{SL}(3,\overline{\mathbb{Q}}_{\ell})$.
Then we claim that for any other prime $\mathfrak{p}^{\prime}$, the arithmetic
monodromy of $\mathscr{L}_{\mathfrak{p}^{\prime}}$ must also be orthogonal;
indeed, this follows from the fact that the geometric monodromy group is a
normal subgroup of the arithmetic monodromy group and that the group
$\text{SL}(3,\overline{\mathbb{Q}}_{\ell})$ is almost simple.

######  Remark 6.

Step (5) of the proof of A, and in particular 3 obviates the use of the more
delicate moduli spaces $\mathcal{H}_{k}$ in [KYZ24, Proof of Theorem 1.4],
c.f. Definition 3.5 and Lemma 3.6 of _loc. cit._

## Acknowledgments

During the course of this work, Lam was supported by a Dirichlet Fellowship
and Krishnamoorthy was supported by the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation program, grant
agreement no. 101020009, project TameHodge.

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