††institutetext: Enrico Fermi Institute, University of Chicago, Chicago, IL
60637, U.S.A.
Oehme Postdoctoral Fellow<EMAIL_ADDRESS>they/him

# A Cosmological Lithium Solution from Discrete Gauged Baryon Minus Lepton
Number

Seth Koren

###### Abstract

The cosmological lithium problem—that theory predicts a primordial abundance
far higher than the observed value—has resisted decades of attempts by
cosmologists, nuclear physicists, and astronomers alike to root out
systematics. We reconsider this problem in the setting of the Standard Model
extended by gauged baryon minus lepton number, which we spontaneously break by
a scalar with charge $2N_{c}N_{g}$. Cosmic strings from this breaking can
support interactions converting three protons into three positrons, and we
argue that an ‘electric’-‘magnetic’ interplay can give this process an
amplified, strong-scale cross-section in an analogue of the Callan-Rubakov
effect. We suggest such cosmic strings have disintegrated $\mathcal{O}(1)$ of
the primordial lithium nuclei, and lay out what is necessary for this scheme
to succeed. To our knowledge this is the first new physics mechanism with
microphysical justification for the abundance of lithium uniquely to be
modified after Big Bang Nucleosynthesis.

Figure 0: Photo of a commissioned abstract artist’s impression in acrylic on
canvas by Bailey Lingen, a Chicago-based artist and trans rights activist who
produces music under the name LiTHiUM THiEF.

###### Contents

  1. 1 The Lithium Problem
  2. 2 Schema
  3. 3 Conclusion
  4. 4 Dedication and Acknowledgements
  5. A Theory Motivation
  6. B Discrete Gauge Symmetries
    1. B.1 Gauging SM Accidental Symmetries
    2. B.2 $N$-Charged Abelian Higgs
    3. B.3 Abelian Higgs Cosmic Strings
    4. B.4 Discrete Aharonov-Bohm Effect
    5. B.5 $\mathbb{Z}_{N}$ Gauge Theory
  7. C Cosmic String Dynamics
    1. C.1 Cosmic String Evolution
    2. C.2 Intercommutation
    3. C.3 Diminution
    4. C.4 Scaling from Simulations
  8. D Topological Defect Catalysis
    1. D.1 Instantons and Fermi Zero-Modes
      1. D.1.1 Infrared Inelastic Interactions
    2. D.2 Monopoles and Callan-Rubakov
      1. D.2.1 Infrared Inelastic Interactions
      2. D.2.2 Into the Defect
    3. D.3 Cosmic Strings
      1. D.3.1 Into the Defect
      2. D.3.2 Infrared Inelastic Interactions?

##  1 The Lithium Problem

| $\left({}^{7}\text{Li}/\text{H}\right)$
---|---
Observation | $\left(1.6\pm 0.3\right)\times 10^{-10}$
Theory | $\left(4.7\pm 0.7\right)\times 10^{-10}$
Table 1: For reference, the status of the number density of lithium-7 relative
to that of hydrogen, taken from the recent review and status update
Fields:2019pfx .

Big Bang Nucleosynthesis (BBN) is the theory of the formation of light nuclei
a few minutes after the hot big bang, and is the earliest epoch of the
universe for which we have direct evidence. In the framework of the Standard
Models (SM) of particle physics and cosmology, BBN is essentially a parameter-
free cross-check on our understanding of the universe. And a powerful one at
that: BBN is sensitive to all four known forces, as it pits the strong force’s
ability to bind nucleons against the gravitational expansion of the universe,
the weak decays of the neutron, and the electromagnetic repulsion of protons.
The status is given by the comparison of theoretical predictions and empirical
observations of just three quantities Fields:2019pfx ; Iocco:2009a ;
Coc:2017pxv ; Mathews:2017xht : The primordial abundances of deuterium with
one proton, of helium-4 with two protons, and of lithium-7 with three protons.
It’s these three values that characterize our only semi-direct understanding
of the universe far before recombination.

While theory and observation agree well for hydrogen and helium, there has
been a consistent mystery regarding the abundance of lithium Fields:2011 ;
Iocco:2012a  beginning with observations in 1982 by Spite and Spite
Spite:1982a ; Spite:1982b , and the current best values are shown in Table 1.
The concern, of course, is what sort of systematic effects might be
responsible. Early on there was a possibility of unseen effects in the nuclear
data Cyburt:2009cf , but the relevant transition rates are thought to have now
all been measured, with by 2015 a status update stating “a ‘nuclear option’ to
the lithium problem is essentially excluded” Cyburt:2008kw ; Cyburt:2015mya ;
Iliadis:2020a . Continued measurement and modelling has agreed Pitrou:2018cgg
; Rijal:2019a ; nTOF:2018vnr ; Bystritsky:2017a  and ever-more-inventive
channels are being studied by nuclear physicists and discarded as
inconsequential Voronchev:2009zz ; Kirsebom:2011vp ; Li:2018a ; Hartos:2018tpy
; Damone:2018a .

The remaining worry arises since the empirical data on ‘primordial’ lithium
comes primarily from measurements of its abundance in the atmospheres of
metal-poor, population II stars in the Milky Way. While we must necessarily
use nearby sources due to the scarcity of lithium, the argument that we can
access the primordial abundance follows from elementary stellar dynamics
considerations and was given by Spite & Spite Spite:1982a ; Spite:1982b . In
relatively cool stars, the full volume of the star undergoes convection, such
that lithium from the atmosphere may be convected to the core and destroyed.
However, hot-enough stars develop an inner radiative zone which is
convectively stable, such that convection is confined to an outer layer and
the lithium abundance in the atmosphere does not get reprocessed. Then if one
measured the lithium abundance in atmospheres of stars whose lithium is
primordial, as a function of temperature—with all of the proper astronomical
care taken to look at the right sorts of stars—for cooler stars there should
be less and less of it, but on the other side at some temperature the plot
should level off to a constant value. Indeed, this is the famous ‘Spite
plateau’ which was observed first by Spite and Spite in 1982 Spite:1982a ;
Spite:1982b , suggesting that this plateau abundance indeed grants us an
observational window to the primordial abundance. The next few decades of
observational progress (e.g. Rebolo:1988a ; Balachandran:1990a ; Ryan:1996a ;
Ryan:1998a ; Ryan:1999a ; Boesgaard:2005a ; Charbonnel:2005a ;
Bonifacio:2006au ; Asplund:2006a ; Aoki:2009a ; Hosford:2009a ; Sbordone:2010a
; Schaeuble:2012a ) in measuring stellar lithium are reviewed in Spite:2012 ,
overall showing further support for a Spite plateau discrepant with the BBN
predictions. This has now included observations of the Spite plateau in
environments as diverse as stellar streams Aguado:2021a , multiple globular
clusters Bonifacio:2002a ; Mucciarelli:2011a ; Schiappacasse-Ulloa:2022 , the
Sculptor dwarf spheroidal galaxy Hill:2019a , the Gaia-Enceladus galaxy
Molaro:2020a ; Simpson:2021a , and Messier 54 Mucciarelli:2014a . A recent
study of galactic chemical evolution models for these data concluded that the
observed abundances were well-understood only if the primordial lithium
abundance was set to the Spite plateau value Matteucci:2021 .

In recent years some ever-larger datasets have showed slightly higher
variability Bonifacio:2006au ; Bonifacio:2007a ; Gonzalez:2009a ; Monaco:2012a
; Crandall:2014cca ; Melendez:2010a ; Howk:2012a  in the pleateau star
abundances and even some anomalously low abundances in extremely metal-poor
stars Sbordone:2010a ; Bonifacio:2015top , which has raised doubts in some
quarters (see e.g. Deal:2021a ), but the implications of these new data are
unclear Aguado:2019bac ; Matteucci:2021  and the situation is still under
active investigation by many scientists in a wide variety of fields across
physics and astronomy. On the theoretical side, there is no well-accepted new
stellar dynamics mechanism that could be responsible, though there have been
proposals Richard:2005a ; Piau:2006a ; Korn:2006a ; Prantzos:2012a ;
Ouyed:2013a ; Marrder:2020a ; Tognelli:2020a ; Grisoni:2019a . On the
observational side, astronomers continue to further probe the Spite plateau in
a variety of innovative directions Prodanovic:2004a ; Ruchti:2011a ;
Caffau:2011a ; Mucciarelli:2012 ; Hansen:2014a ; Spite:2015 ; Fu:2018a ;
Bensby:2018a ; Lambert:2004a ; Pinto:2021a .***It is also important to note
when reading earlier literature that claims about positive detections of
lithium-6 have now been called into question Fields:2022mpw .

Of course, any time there is an astronomical or cosmological anomaly our
Bayesian priors should be peaked toward learning something new about stars.
But after so long with the prospects for an astrophysical resolution still
unclear, some broader exploration is surely well worth doing. And indeed,
there have been some new physics mechanisms explored—mainly fundamental
constants varying to modify early-universe nuclear rates Dent:2007a ;
Dmitriev:2004a ; Flambaum:2008a ; Berengut:2010a ; Cheoun:2011yn ;
Landau:2006a ; Coc:2007a ; Mori:2019a ; Franchino-Vinas:2021nsf ;
Clara:2020efx ; Fung:2021wbz  or decaying dark matter disrupting reactions
Pospelov:2006sc ; Jedamzik:2007cp ; Ellis:2005ii ; Olive:2012hig ; Coc:2014gia
; Yamazaki:2017uvc ; Cyburt:2006a ; Kaplinghat:2006a ; Bird:2008a ;
Kohri:2007a ; Kusakabe:2010a ; Kusakabe:2011a ; Anchordoqui:2020djl ;
Alcaniz:2019kah ; Sato:2016len ; Cyburt:2013fda ; Goudelis:2015wpa ;
Pospelov:2010cw ; Bailly:2008yy ; Cumberbatch:2007me ; Jedamzik:2005dh ;
Cyburt:2012kp ; Jittoh:2011ni ; Kawasaki:2010yh ; Kusakabe:2017brd ;
Kusakabe:2014ola . See also Mathews:2019hbi ; Luo:2019a ; Makki:2019a ;
Malaney:1993a ; Mathews:2005a ; Hou:2017a ; Nakamura:2017a ; Lu:2022a ;
Flambaum:2018ohm ; Pospelov:2012pa ; Bailly:2010hh ; Cyburt:2010vz ;
Olive:2012xf ; Coc:2013a ; Pospelov:2010hj  for further ideas. In our
diagnosis, these approaches have been stymied by not uniquely picking out
lithium to be affected, and they generally encounter severe difficulties with
putting other observables into tension.

We are motivated by the surprisingly long lifetime of the proton and its exact
stability in the SM to take advantage of the exact SM selection rule

$\displaystyle\hskip 28.45274pt\Delta(\\#\text{ of baryons})$
$\displaystyle=\Delta(\\#\text{ of leptons})$ $\displaystyle=0\ (\text{mod
number of generations}).$

Since the SM has $N_{g}=3$ generations of particles and the BBN predictions
first fail for lithium with three protons, this symmetry could explain why
elements with one or two protons are unaffected. This allows us to
construct—to our knowledge—the first new physics mechanism for the lithium
abundance to be discrepant which does not require tuning to avoid modifying
the abundances of other elements.

We must go only slightly beyond the SM to find our effect of interest. In
brief, we will break gauged baryon minus lepton number to a discrete subgroup
which will produce a large abundance of cosmic string loops. These topological
defects amplify the $B+N_{c}L$ violation, and we introduce a model in which
leptoquarks communicate this violation in a way which picks out solely
processes in which three protons are turned into three positrons. We propose
that such cosmic strings have disintegrated lithium nuclei in the early
universe, reducing the primordial abundance to the level inferred from
astronomical observations. In this Letter we will put together the various
pieces of field theory and cosmology behind this mechanism, and estimate the
rate to demonstrate its plausibility. Precise predictions will require much
further work.

##  2 Schema

| $\quad Q\quad$ | $\quad\bar{u}\quad$ | $\quad\bar{d}\quad$ | $\quad L\quad$ | $\quad\bar{e}\quad$ | $\quad\bar{\nu}\quad$
---|---|---|---|---|---|---
$SU(3)_{C}$ | $3$ | $\bar{3}$ | $\bar{3}$ | – | – | –
$SU(2)_{L}$ | $2$ | – | – | $2$ | – | –
$U(1)_{Y}$ | $+1$ | $-4$ | $+2$ | $-3$ | $+6$ | –
$U(1)_{B}$ | $+1$ | $-1$ | $-1$ | – | – | –
$U(1)_{L}$ | – | – | – | $+1$ | $-1$ | $-1$
Table 2: Representations of the left-handed SM Weyl fermions (and the sterile
neutrino) under the classical symmetries of the SM.

##### New physics setting

We extend the Standard Model by gauging $U(1)_{B-N_{c}L}$, which is the
continuous, generation-independent, Adler-Bell-Jackiw (ABJ) Adler:1969gk ;
Bell:1969ts  anomaly-free ‘accidental’ global symmetry of the SM. †††We
normalize the charges such that the least charged particles in the unbroken
phase of the SM (the quarks) have unit charge, which alleviates confusion when
properly counting integer invariants. After QCD confines, it is
$(B-N_{c}L)/N_{c}$ that counts baryons and leptons. The fermion charges are
given in Table 2. We address the ’t Hooft anomaly tHooft:1979rat  and neutrino
oscillations by adding right-handed neutrinos, but this choice does not impact
our mechanism. Though note that $U(1)_{B-N_{c}L}$ does forbid Majorana masses
for neutrinos. To Higgs this gauge symmetry we introduce a scalar $\Phi$ with

$\left[\Phi\right]_{B-N_{c}L}=2N_{c}{N_{g}},\qquad\left[\Phi\right]_{B+N_{c}L}=0,\qquad\langle\Phi\rangle=v,$
(1)

which condenses in the early universe at a temperature $T_{c}\sim v$. This
breaks the baryon minus lepton number symmetry without breaking the additional
discrete anomaly-free global $\mathbb{Z}^{L}_{{N_{g}}}$ subgroup of lepton
number enjoyed by the SM Koren:2022bam . The global symmetry-breaking pattern
providing selection rules is

$U(1)_{B-N_{c}L}\times\mathbb{Z}^{L}_{{N_{g}}}\rightarrow\mathbb{Z}_{2N_{c}{N_{g}}}^{B-N_{c}L}\times\mathbb{Z}^{L}_{{N_{g}}}\supset\mathbb{Z}_{2N_{c}{N_{g}}}^{B+N_{c}L},$
(2)

where the anomaly-free subgroup of baryon plus lepton number is generated by
$(1,1)\in\mathbb{Z}_{2N_{c}{N_{g}}}^{B-N_{c}L}\times\mathbb{Z}^{L}_{{N_{g}}}$.
We emphasize that only the former factor is gauged in our model, though it
would be interesting to consider a model in which the discrete lepton number
is gauged as well. While this Higgsing is ‘non-minimal’ in the sense that
$\left[\Phi\right]>1$, this choice of $\left[\Phi\right]$ preserves the SM
selection rule which stabilizes the proton Tong:2017oea ; Koren:2022bam ;
Anber:2021upc ; Davighi:2022qgb ; Wang:2022eag . That is, the SM constraint of
$\Delta B/N_{c}=\Delta L=N_{g}$, already familiar from electroweak sphaleron
processes, will remain exact in our theory.

##### Discrete gauge theory

The nontrivial Higgs charge $\left[\Phi\right]_{B-N_{c}L}$ means its
condensation preserves an unbroken discrete
$\mathbb{Z}_{2N_{c}{N_{g}}}^{B-N_{c}L}$ gauge symmetry Krauss:1988zc ;
Alford:1989ch ; Preskill:1990bm ; Coleman:1991ku ; Banks:1989ag  and stable
cosmic string solutions exist with tension $\mu\sim v^{2}$. The transverse
slices of a static, $z$-independent string are Abrikosov-Nielsen-Olesen
vortices Abrikosov:1956sx ; Nielsen:1973cs  and have asymptotics

$\Phi(\rho,\theta)\rightarrow
ve^{ik\theta},\qquad\vec{A}(\rho,\theta)\rightarrow\frac{k}{g}\frac{1}{2{N_{g}}}\frac{\hat{\theta}}{\rho},$
(3)

with $\vec{A}$ the $B-N_{c}L$ gauge boson, $g$ the $U(1)_{B-N_{c}L}$
‘electric’ charge, and $k\in\mathbb{Z}$ the winding number of $\Phi$. With
nonzero winding, continuity in the interior demands a zero in the Higgs
field,‡‡‡By applying Brouwer’s fixed point theorem to $\Phi(x)$ on each
transverse disk. and indeed the $B-N_{c}L$ symmetry is unbroken in a string
‘core’ of radius $R\sim v^{-1}$. These solutions have discrete
$U(1)_{B-N_{c}L}$ ‘magnetic’ fluxes

$\Phi_{B}=\frac{2\pi}{g}\frac{k}{2N_{c}{N_{g}}}$ (4)

confined to their cores. This magnetic flux is in ‘fractional’ units of
$1/(2N_{c}{N_{g}})$ that of the Dirac magnetic flux quantum $\Phi_{0}=2\pi/g$
Dirac:1931kp , so the string is stable and interacts via the Aharonov-Bohm
(AB) effect Aharonov:1959fk ; Aharonov:1984zza  for the unbroken discrete
symmetry. The cosmic strings undergo elastic AB scattering with all the SM
fermions of differential cross-section per unit length (for a plane wave of
$B-N_{c}L$ charge $q$ and momentum $p$ incident on a string) Alford:1988sj

$\frac{1}{\ell}\frac{d\sigma}{d\theta}=\frac{\sin^{2}(\pi
q\Phi_{B}/\Phi_{0})}{2\pi p\sin^{2}(\theta/2)},$ (5)

which, importantly for the phenomenology, is independent of any high scales or
small couplings. §§§This is nicely understood by dualizing the infrared
physics to the topological BF theory Horowitz:1989ng ; Horowitz:1989km , which
is a generalized, covariant theory of this discrete AB effect. In this dual
description, these are cosmic BF strings which are charged under an emergent
two-form $\mathbb{Z}_{2N_{c}{N_{g}}}$ gauge symmetry Teitelboim:1985ya ;
Banks:2010zn . The interaction between point charges and strings then must
depend solely on the product of electric and magnetic charges of the
$\mathbb{Z}_{2N_{c}{N_{g}}}$ BF theory, $kq\text{ (mod }2N_{c}{N_{g}})$
Teitelboim:1985yc .

##### Strings in cosmology

During an early universe phase transition, topological defects are produced
proportional to the correlation length $\xi$ of the order parameter
fluctuations. Causality ensures this is no larger than the Hubble distance
$\xi\lesssim d_{h}$, so that at a critical temperature $T_{c}\sim v$,
$\mathcal{O}(1)$ horizon-crossing strings of length $d_{h}=2H^{-1}(T_{c})\sim
M_{\text{pl}}/v^{2}=(G\mu)^{-1}\ell_{pl}$ per Hubble volume are formed
Kibble:1976sj ; Kibble:1980mv , $d_{h}$ being the Hubble size, $H$ the Hubble
parameter, and $G\mu\sim v^{2}/M_{\text{pl}}^{2}$ a dimensionless measure of
the gravitational effects of the strings. At formation, these long strings
contain energy density

$\rho_{string}\sim\frac{\mu d_{h}}{d_{h}^{3}}\sim(G\mu)^{+1}\rho_{tot},$ (6)

a small fraction of the total. The elastic AB scattering with the SM plasma
provides a total transverse cross-section per unit length of
$\sigma_{\perp}/\ell\sim 1/T$, which leads to a long epoch of friction-
dominated evolution until the temperature drops below $T_{f}\lesssim
v^{2}/M_{\text{pl}}$ Vilenkin:1991zk .

When cosmic strings encounter each other, they ‘intercommute’ with probability
approaching unity Manton:1981mp ; Shellard:1987bv ; Matzner:1988qqj ;
Ruback:1988ba ; Shellard:1988ki ; Atiyah:1985dv ; Gibbons:1986df ;
Rosenzweig:1990ea ; Samols:1991ne ; Abdelwahid:1993fg ; MacKenzie:1995np ;
Gauntlett:2000ks ; Hanany:2005bc ; Hashimoto:2005hi . This results in the
production of small-scale structure on the strings Blanco-Pillado:2015ana ;
Kibble:1990ym ; Copeland:2009dk ; Kibble:1990ym ; Siemens:2001dx ;
Siemens:2002dj  and many closed loops from strings crossing themselves. String
loops oscillate and shrink via gravitational radiation Vilenkin:1981kz ;
Vilenkin:1981bx ; Vilenkin:2000jqa , which dissipates energy and prevents the
strings from dominating the density budget Turok:1984db .

After decades, we now understand the behavior of the cosmic string network at
late times and on large scales: There is an attractor ‘scaling’ solution where
the length spectrum of loops depends only on the ratio with the horizon
distance $\ell/d_{h}$ Albrecht:1984xv ; Bennett:1987vf ; Shellard:1987bv ;
Shellard:1988zx ; Matzner:1988qqj ; Moriarty:1988em ; Myers:1991yh . The small
scale behavior of this ‘scaling’ population is assumed to continue down to the
‘gravitational cutoff’ scale $\ell\sim(G\mu)d_{h}$, at which length a loop
evaporates within a Hubble time, which leads to a large number of ‘small’ such
loops, $n\sim(G\mu)^{-0.7}$ Blanco-Pillado:2013qja .

However, the regime relevant for their effects on lithium requires
understanding cosmic strings far before the infrared fixed point is reached.
In addition to being very difficult to study, this is sensitive to details
such as the strength of the phase transition, since the correlation length of
Higgs fluctuations may in fact be quite small compared to the horizon length
$\xi\ll d_{h}$, resulting in a far greater initial number density of defects
Guth:1982pn ; Zurek:1985qw ; Murayama:2009nj . The small-scale behavior is
also sensitive to the details of intercommutation dynamics, and furthermore
the presence of currents on the strings may stabilize them against decay and
result in a large portion of energy density in cosmic string ‘vortons’
Davis:1988ij ; Brandenberger:1996zp ; Davis:1996zg ; Carter:1999an ;
Peter:2013jj ; Auclair:2020wse ; Battye:2021kbd . For these reasons, rather
than attempting to model the evolution of the phase space of cosmic strings,
below we will simply parametrize their effects on lithium disintegration in
terms of the fraction of the energy density which is in cosmic strings.

| $\quad\Phi\quad$ | $\quad\chi\quad$ | $\quad\omega_{\ell}\quad$ | $\quad\omega_{q}\quad$ | $\quad\omega_{s}\quad$
---|---|---|---|---|---
$SU(3)_{C}$ | – | – | $3$ | $3$ | –
$SU(2)_{L}$ | – | – | – | – | –
$U(1)_{Y}$ | – | – | $-8$ | $-8$ | –
$U(1)_{B}$ | $+9$ | $+9$ | $+1$ | $-2$ | $+3$
$U(1)_{L}$ | $-3$ | $+3$ | $+1$ | $0$ | $+1$
Table 3: Representations of the scalars added in our benchmark model. Cosmic
strings appear from the winding of $\Phi$ in the vacuum, the condensation of
$\chi$ on the string allows it to freely exchange $B+N_{c}L$, and the scalars
$\omega$ communicate $B+N_{c}L$-breaking by $\chi$ to the right-handed SM
fields.

##### Topological defect catalysis

We wish to consider interactions which destroy lithium nuclei by turning three
protons into three positrons, thereby violating $B+N_{c}L$ by $2N_{c}{N_{g}}$
units. We utilize a stringy analogue of the Callan-Rubakov effect where grand
unified monopoles ‘catalyze’ proton decay at strong scale rates
$\sigma(p^{+}+\text{ monopole}\rightarrow e^{+}+\text{
monopole})\sim\Lambda_{\text{QCD}}^{-2}$. For monopoles, the inevitability of
such inelastic interactions can be seen purely from the infrared Dirac:1931kp
; Dirac:1948um ; Goldhaber:1965cxe ; Weinberg:1965rz ; Zwanziger:1968ams ;
Peres:1968umt ; Lipkin:1969ck ; Wu:1976qk ; Schwinger:1966nj ;
Zwanziger:1968rs ; Zwanziger:1970hk ; Zwanziger:1972sx ; Kazama:1976fm ;
Callan:1983ed ; Csaki:2020inw ; Csaki:2020yei . An ultraviolet description for
the monopole tHooft:1974kcl ; Polyakov:1974ek ; Arafune:1974uy ; Prasad:1975kr
; Julia:1975ff ; Hasenfratz:1976vb ; Tomboulis:1975qt ; Callan:1975yy ;
Boulware:1976tv  allows one to resolve the microphysical interactions and see
$p^{+}\rightarrow e^{+}$ arising from interactions with leptoquarks in the
core and the anomalous violation of $B+N_{c}L$ Jackiw:1975fn ;
Hasenfratz:1976gr ; Goldhaber:1976dp ; Goldhaber:1977xw ; Callan:1982ac ;
Rubakov:1981rg ; Rubakov:1982fp ; Callan:1982ac ; Callan:1982au ;
Callan:1982ah ; Callan:1983tm ; Dawson:1983cm ; Sen:1984qe ; Polchinski:1984uw
; Isler:1987xn ; Rubakov:1988aq ; Affleck:1993np ; Brennan:2021ewu ;
Brennan:2021ucy ; Csaki:2021ozp .

The cosmic string case has seen far less study, and no infrared argument for
this catalysis effect is yet known. To provide an explicit benchmark—and to
give a microphysical model targeting this mechanism toward the cosmological
lithium problem—we introduce ultraviolet physics which lets us map our case on
to a model studied by Alford, March-Russell, & Wilczek in 1989 Alford:1989ie
in which a leptoquark condenses inside cosmic strings of some broken gauged
$\widetilde{U(1)}\rightarrow\mathbb{Z}_{N}$. In that model the leptoquark
allows $p^{+}\rightarrow e^{+}$ violation of $B+N_{c}L$, but here we respect
the exact SM symmetry $\mathbb{Z}_{N_{g}}^{L}$, so rather than the
condensation of a leptoquark we instead introduce a gauge-singlet scalar
$\chi$ with

$\left[\chi\right]_{B-N_{c}L}=0,\qquad\left[\chi\right]_{B+N_{c}L}=2N_{c}{N_{g}},\qquad\langle\chi\rangle=0,$
(7)

but which condenses on the string core so long as

$\left.\partial_{\chi}^{2}V(\Phi,\chi)\right|_{\Phi=0}<0\rightarrow\langle\chi\rangle_{\Phi=0}=v_{\chi}.$
(8)

This breaks solely the global approximate $U(1)_{B+N_{c}L}$ on the string core
down to the subgroup preserved by the SM, and we take the symmetry-breaking
scales equal for simplicity, $\langle\Phi\rangle=v$ around the vacuum and
$\langle\chi\rangle_{\Phi=0}=v$ in the string. This condensate realizes a
global baryon plus lepton number analogue of Witten’s bosonic
superconductivity Witten:1984eb , which allows the string core to exchange
$2N_{c}{N_{g}}$ units of $B+N_{c}L$ charge with the outside world
Witten:1984eb .

Figure 1: A Feynman diagram portraying our benchmark for how the
$B+N_{c}L$-breaking effects of the $\chi$ vev get communicated to the SM
fermions at low energies.

The infrared symmetries forbid interactions of $\chi$ with fewer than twelve
SM fermions, but we can explicitly generate one such interaction by
introducing the scalar leptoquark $\omega_{\ell}$ with hypercharge $8$ and its
analogue diquark $\omega_{q}$, as well as the gauge singlet scalar
$\omega_{s}$ with $B=N_{c}L=3$. This choice of scalars (see Table 3) allows
the nontrivial interactions

$\mathcal{L}\supset\lambda_{\ell}\omega_{\ell}\bar{d}\bar{e}+\lambda_{q}\omega_{q}^{\dagger}\bar{u}\bar{u}+\kappa\omega_{\ell}^{\dagger}\omega_{q}\omega_{s}+\lambda_{\chi}\omega_{s}^{3}\chi^{\dagger}.$
(9)

Around the vacuum the Feynman diagram portrayed in Figure 1 generates the
operator

$\mathcal{L}\sim\frac{1}{v^{15}}\mathcal{\chi}(\bar{u}\bar{u}\bar{d}\bar{e})^{3},$
(10)

and $B+N_{c}L$-violation on the string core is communicated only to electrons
and protons in our model. This appears naturally due to our choice of
leptoquark representations, but it would be interesting to consider relaxing
this and examining the possibility of interactions with helium as well. We
further take the mediators $\omega$ to be light in the string core, so this
suppression by $v$ disappears when the SM fermions are interacting with the
$\chi$ vev in the core, ¶¶¶Microphysically, this requires quartic interactions
with $\chi$, $V\propto|\chi|^{2}|\omega_{i}|^{2}$ to be much smaller than
those with $\Phi$, $V\propto|\Phi|^{2}|\omega_{i}|^{2}$. Our intention here is
just to evince some model in which the effect of interest occurs, but one
interesting model-building direction is to sharpen this microphysical
model—perhaps with unbroken supersymmetry in the string core which can leave
these scalars light. and we’ve set all couplings to unity and scales equal for
simplicity.

In the model of Alford, March-Russell, & Wilczek Alford:1989ie , the Yukawa
interaction allows an incoming quark to inelastically scatter into a lepton by
exciting the leptoquark condensate, and the $\mathbb{Z}_{N}$ discrete gauge
field background enhances the cross-section. When the discrete gauge charge of
the incoming quark is maximal, the cross-section is unsuppressed by
ultraviolet scales, as in the Callan-Rubakov effect. In our case, we instead
have an interaction of many SM partons with the string condensate.

However, in the cosmological case of interest, a ${}^{7}\text{Li}$ nucleus
delivers a state of three protons $\psi_{3p}\sim(\bar{u}\bar{u}\bar{d})^{3}$
incident upon the string. To map our model to Alford:1989ie , we rewrite
Equation 10 as a Yukawa interaction between $\psi_{3p}$, the nine quarks in
three protons, and $\psi_{3e}$, the three positrons. Adapting their results
while ignoring many subtleties about this matching, we infer that an incoming
plane wave of $B-N_{c}L$-charge $q$ and momentum $p$ undergoes enhanced
inelastic scattering with a long cosmic string as

$\frac{\sigma}{\ell}\sim\frac{1}{p}\sin^{2}\left(\pi\alpha\right)\left(\frac{p}{v}\right)^{4\left|\alpha-\frac{1}{2}\right|},\qquad\alpha\equiv\frac{kq}{2N_{c}{N_{g}}},$
(11)

where we are generalizing Eqns 18-21 of Alford:1989ie . The inelastic
scattering is unsuppressed by ultraviolet scales for an incoming state with
$B-N_{c}L$ charge $q=N_{c}{N_{g}}$ in the case of a single-winding string. We
note also that topological defect interactions can violate crossing symmetry,
as may be understood simply for monopoles Csaki:2020inw , so for example
crossing one proton to an outgoing antiproton results in a drastically
suppressed cross-section because the incoming state no longer has maximal
$\mathbb{Z}_{2N_{g}N_{c}}$ charge nor half-integer spin, which is also crucial
for catalysis Alford:1989ie . This suppression is useful for ensuring
consistency with observations of primordial helium, and it would be
interesting to understand precisely the subleading rate at which helium might
be affected.

##### Parameter space

In the early universe after BBN, just such a $q=N_{c}{N_{g}}$ state of three
protons may be delivered to the string within the nuclear potential well of
${}^{7}\text{Li}$.∥∥∥In fact BBN initially produces ${}^{7}\text{Be}$ which
then captures an electron to produce ${}^{7}\text{Li}$ Khatri:2010ed . While
in much of the parameter space it is ${}^{7}\text{Be}$ which is being
destroyed, in any case these are the only elements with more than 2 protons,
so our selection rule still serves to distinguish them. We assume string loops
have a cross-section per unit length

$\frac{\sigma}{\ell}\left(3p^{+}+\text{ string}\rightarrow 3e^{+}+\text{
string}\right)\sim\Lambda_{\text{QCD}}^{-1},$ (12)

but note again that there are necessarily large theoretical uncertainties
here.

Figure 2: A contour plot of the string tension scale required for
$\mathcal{O}(1)$ lithium to be disintegrated as a function of the temperature
and the fraction of energy density in cosmic strings from Equation 14. The
hatched region is constrained in simple models of gauged $B-N_{c}L$ with a
massive $Z^{\prime}$, taking $\sqrt{\mu}\sim v\sim m_{Z^{\prime}}/g>100\text{
GeV}$ as a benchmark Heeck:2014zfa . The dashed line is an approximate upper
bound on the energy density in strings near the Hubble length from CMB and
gravitational wave constraints Sanidas:2012ee ; Rybak:2021scp .

However a first question is just whether such interactions can plausibly
destroy an order-one fraction of lithium nuclei after BBN. Due to the
complications discussed above, we will not attempt to model the evolution of
the phase space of string loops, but instead content ourselves to parametrize
their effects as a function of the fraction of energy density in cosmic
strings. We simply ask that the rate for a lithium nucleus to encounter a
cosmic string be greater than Hubble,

$\Gamma\simeq\int
d\ell\frac{dn}{d\ell}\sigma\simeq\Lambda_{\text{QCD}}^{-1}\int
d\ell\frac{dn}{d\ell}\ell=\frac{\rho_{\text{string}}}{\Lambda_{\text{QCD}}\mu}\gtrsim
H,$ (13)

where we have begun with a general differential number density of cosmic
strings $\frac{dn}{d\ell}$, and assumed the interaction is characterized as
$\sigma/\ell=\Lambda_{\text{QCD}}^{-1}$ which makes the rate proportional to
the energy density. Then introducing $f\equiv\rho_{\rm string}/\rho_{\rm tot}$
we require

$f\gtrsim\frac{\Lambda_{\rm QCD}}{M_{\rm pl}}\frac{\mu}{T_{\star}^{2}},$ (14)

where $\Lambda_{\rm QCD}/M_{\rm pl}\sim 10^{-20}$, and $T_{\star}$ is the
temperature at which disintegration occurs. We note that this is a greater
fraction than that eventually expected in long strings $f\sim G\mu$,
underscoring the necessity of understanding the nonequilibrium behavior of
strings after the phase transition. Along similar lines, we emphasize also
that the precise requirement to destroy the correct fraction of lithium will
depend on the duration over which these interactions are efficient. We plot
this relation in Figure 2, where we assume that $\mu\gtrsim T_{\rm BBN}^{2}$
such that the $B-N_{c}L$ phase transition takes place before BBN, and $T_{\rm
CMB}\lesssim T_{\star}\lesssim T_{\rm BBN}$ to avoid severe constraints on
exotic energy injection which are sensitive even down to the level of
$\mathcal{O}(10^{-10})$ at the time of recombination Poulin:2016nat ;
Acharya:2019uba ; Dienes:2018yoq ; Slatyer:2016qyl ; Acharya:2019owx . If the
$B-N_{c}L$-breaking scale is low then one requires relatively small gauge
coupling Heeck:2014zfa ; Ilten:2018crw ; Bauer:2018onh , but the
‘electric’-‘magnetic’ interplay of the Callan-Rubakov effect means that this
does not suppress the cross-section, as above. Nevertheless if the scale is so
low that the string width is larger than nuclear scales, the benchmark cross-
section, which does not account for the string’s internal structure, seems
less applicable. In Figure 2 we also show a benchmark constraint on gauged
$B-N_{c}L$ with massive $Z^{\prime}$ from Heeck:2014zfa , assuming the tension
goes as $\sqrt{\mu}\sim m_{Z}^{\prime}/g$, though we note that this constraint
does assume a particular right-handed neutrino sector and it is possible these
constraints are modified either with non-standard neutrino interactions (see
e.g. Miranda:2015dra ) or exotic neutrino sectors (see e.g. Wang:2020mra ;
Wang:2020xyo ; Wang:2020gqr ).

##### Complementary signatures

Cosmic strings may generally be searched for through their gravitational
effects either in affecting the CMB anisotropies or in contributing to the
stochastic gravitational wave background. However, the details of the spectra
and the ensuing constraints depend strongly on the size distribution of
strings. To guide the eye, we offer in Figure 2 the rough constraint
applicable to the scaling population of strings. For discussions on how the
constraints depend on some characteristics of the spectrum we refer to
Caldwell:1991jj ; Sanidas:2012ee ; Blanco-Pillado:2013qja ; Blasi:2020mfx ;
Chang:2021afa ; Blanco-Pillado:2021ygr  and Contaldi:1998mx ; Lazanu:2014eya ;
Lazanu:2014xxa ; Lizarraga:2016onn ; Charnock:2016nzm ; Rybak:2021scp .

The non-trivial dynamics of cosmic strings means there may be a non-negligible
number density at any time, so one may hope to constrain their existence with
direct detection experiments. General $B-N_{c}L$ cosmic strings may be probed
through their unitarity-limited elastic Aharonov-Bohm scattering, but this
requires a detailed understanding of the kinematics of such interactions in
direct detection experiments. The inelastic effect is more striking, but one
must also understand the cross-section for general nuclei, such as the many
oxygen in Super-Kamiokande. Experimentally, dedicated analysis or new searches
may be required for the ‘hard’ signature of three outgoing positrons, their
ensuing showers, and possible delayed decays of the leftover nucleus.

While such effects are too rare with solely the scaling population of large
cosmic strings, in scenarios with increased number density such as with vorton
end-states this would be an interesting signature to understand. Alternatively
it would be interesting to integrate over cosmic volumes and look for indirect
effects of $B-N_{c}L$ strings in astrophysical settings. Regardless of their
role in producing the lithium discrepancy, such strings have enormous elastic
scattering cross-sections with SM matter from the AB effect.

##  3 Conclusion

In this work we have stirred together a mélange of ingredients from particle
theory and cosmology to write down a mechanism through which the lithium
problem is an entrée to fundamental physics beyond the SM. This scenario
motivates various important directions of inquiry, which are interesting more
generally than in their application here:

  * •

It is clearly important to better conceptually understand the cosmic string
analogue of the Callan-Rubakov effect, which likely requires understanding BF
theory with light fermions.

  * •

Any precise rate predictions will require understanding the evolution and
interactions of small cosmic string loops, which is a difficult problem.

  * •

It would be useful to think about other interesting applications for
unitarity-limited interactions of fermions with cosmic strings—perhaps in
effecting baryogenesis or dark matter production.

  * •

For any such applications it will be necessary to understand better how to
translate from many-parton interactions with cosmic strings to nuclear
effects.

Theoretical physicists have been captivated by the interplay of electric and
magnetic effects for centuries now, from Maxwell Maxwell:1865zz  up to its
grandest form in Montonen-Olive duality Montonen:1977sn ; Goddard:1976qe ;
Osborn:1979tq  with maximal supersymmetry. The case of less symmetry, as with
our discrete $\mathbb{Z}_{N}$, has received far less attention. In addition to
their fascinating topological character, these interactions may have
interesting, heretofore-underappreciated phenomenological applications in the
early universe.

##  4 Dedication and Acknowledgements

I am grateful to many colleagues for relevant conversations, including Wen Han
Chiu, Clay Cordova, Brian Fields, Christina Gao, Andrew Long, Ryan Plestid,
Carlos Wagner, Liantao Wang, and Yiming Zhong, and I thank especially T.
Daniel Brennan and Sungwoo Hong for discussion and many useful comments. I
further thank Samuel Alipour-fard and Sungwoo Hong for helpful comments on
earlier drafts of this manuscript, and an anonymous referee for pointing out
an inconsistency among assumptions in a prior draft. I would like to also
thank Bailey Lingen for their artistic creativity and enthusiasm.

This work was supported by an Oehme Postdoctoral Fellowship from the Enrico
Fermi Institute at the University of Chicago, and I am sincerely grateful for
the freedom this has granted me to think in offbeat directions. I also thank
organizers at the Galileo Galilei Institute, Stony Brook University, the
University of Chicago, and the Aspen Center for Physics who allowed me to
present this work in progress, and the participants therein for interesting
questions.

This paper is dedicated to my mother, whose death during its writing was
premature and preventable. At cause is my parents’ mistaken understanding of
the nature of reality, as encapsulated by the fact that they have dedicated
their lives to promoting anti-vaccine misinformation. After I understood how
incorrect their worldview was, in protest explicitly thereof I cut off all
contact with them back in 2015.

Nonetheless, it is also the case that my mother meant well, and even my
parents’ considerable medical abuse was doled out with loving intentions.
Whatever substantial damage this did me, it is responsible also for my need to
understand how the universe works and what is possible within it. And if I
have here newly envisaged some interesting connection which has been lurking
amongst thousands of papers since before my birth, it is surely because of my
experiences having had neuroendocrine tumors rather than in spite of that
brain injury and the many related struggles.

I write in the hopes that my parents’ sincere motivations, transmuted as they
have been toward my contributions to our understanding of reality, may morally
counterbalance their misguided actions.

##  Appendix A Theory Motivation

Following the theoretical success of the SM half a century ago, brilliant and
inventive physicists turned their attention and curiosity to higher energies.
They discovered remarkably interesting field theories and constructed
beautiful UV pictures suggesting how the universe should work. Maybe an
orbifold compactification provides super-gauge-Higgs-unification, and
everything we know is brought together at high energies. Maybe supergravity
mediates SUSY breaking from a dark sector. Maybe we get right-handed neutrinos
to fill out our spinor $SO(10)$ representation, maybe dark matter is the
lightest superpartner, maybe leptons are the fourth color of quarks. And
flavor this, and CP that, and left-right so-and-so. Everything fit together as
so many puzzle pieces.

So where is all of the new physics that should be there? Where are the $Z$
primes, the flavons, the leptoquarks? The KK partners, the superpartners, the
mirror fermions? Where is the electron EDM, flavor-changing neutral currents,
neutrinoless double beta decay, dark matter annihilation or custodial symmetry
violation? And why haven’t we observed proton decay?

With each additional, amazingly precise, null finding from our experimental
colleagues, it becomes more clear that something is not quite right with our
grand picture. The question then becomes which cherished principles to hold
fast, and which to give up. Perhaps weak scale SUSY giving $SU(5)$ gauge
coupling unification is merely accidental. Maybe understanding the
dimensionful parameters in the SM requires ‘UV/IR mixing’ effects past the
paradigm of local effective field theory (e.g. Koren:2020biu ; Craig:2018yvw ;
Craig:2019fdy ; Craig:2019zbn ). Perhaps tiny Dirac neutrino masses shouldn’t
concern us.

Or maybe there’s some hint we’ve overlooked, or should give greater weight.
Perhaps it’s of fundamental importance that we live near
$\Omega_{m}=\Omega_{\Lambda}$, or that $\Lambda\sim m_{\nu}^{4}$ allows the
existence of an $\text{AdS}_{3}\times S_{1}$ minimum in the SM Arkani-
Hamed:2007ryu , or that supersymmetry in three dimensions does not require
Bose-Fermi degeneracy Witten:1995rz , or that we have ${N_{g}}=3$ generations
of fermions. We do not yet know whence a new understanding may arise. The
silver lining of this nebulous outlook is the opportunity it provides to
explore novel scenarios and apply fun field theory ideas.

In this work we explore an interesting consequence of second-guessing an
implicit assumption of most new physics models which we have recently
emphasized in Koren:2022bam . Most of these grand pictures explicitly break an
underappreciated $\mathbb{Z}_{2N_{c}{N_{g}}}$ subgroup of baryon plus lepton
number which is part of the exact global symmetry group of the SM. In
contrast, we shall make especial efforts to ensure it remains a good symmetry.

##  Appendix B Discrete Gauge Symmetries

In this section we recall the anomaly-free, generation-independent global
symmetries of the SM, before considering upgrading a subset to discrete gauge
symmetries in the IR. We then discuss finding such a discrete gauge theory
from the Higgsing of a continuous gauge symmetry. As we have emphasized in a
companion note Koren:2022bam , the SM anomaly-free global symmetries include
not only the familiar continuous baryon minus lepton number $U(1)_{B-N_{c}L}$
but also a discrete $\mathbb{Z}_{2N_{c}{N_{g}}}$ subgroup of baryon plus
lepton number. This exact symmetry is responsible for ensuring that the proton
cannot decay in the SM Byakti:2017igi ; Tong:2017oea .

If we go beyond the marginal operators composed of SM fields, we can look for
higher-dimensional operators in the SM EFT which do not break the SM
symmetries. Just imposing the continuous exact symmetry $U(1)_{B-N_{c}L}$
still allows $B+N_{c}L$-violating proton decay at dimension six, as commonly
seen in grand unified theories. However, if one imposes all the global
symmetries of the quantum SM, the lowest-dimension operators which break
$B+N_{c}L$ are instead

$\mathcal{L}\sim\frac{1}{\Lambda^{14}}(\bar{u}\bar{u}\bar{d}\bar{e})^{3}$ (15)

at dimension eighteen, and similar operators involving neutrons and neutrinos,
as well as other generations. The point being that the SM structure in itself
already privileges processes in which three baryons turn into three leptons,
but any such operator effect in the SM is ridiculously suppressed and
certainly negligible Helset:2021plg . We must go beyond the SM to find our
effect of interest, but in fact not too far.

###  B.1 Gauging SM Accidental Symmetries

We now wish to consider the possibility that some of these SM global
symmetries are in fact gauge symmetries at low energies. In Koren:2022bam  we
have discussed that the above symmetries have no ‘ABJ-type’ Adler:1969gk ;
Bell:1969ts  global-gauge-gauge anomalies, meaning that the SM gauge fields do
not violate the global $U(1)_{B-N_{c}L}\times\mathbb{Z}_{N_{g}}^{L}$. However,
to consistently gauge these we must also check for the existence of a ’t Hooft
anomaly tHooft:1979rat , namely the cubic anomaly $U(1)^{3}_{B-N_{c}L}$.

If we hope to integrate this into a theory of quantum gravity, we should
consider also the mixed anomaly with gravity, $\text{grav}^{2}\times
U(1)_{B-N_{c}L}$, which just sums over all $B-N_{c}L$-charged degrees of
freedom since gravity couples universally. In the SM both of these are nonzero

$\displaystyle\sum B-N_{c}L$ $\displaystyle=-N_{g}N_{c},$
$\displaystyle\sum(B-N_{c}L)^{3}$ $\displaystyle=-N_{g}N_{c}^{3},$ (16)

which is often interpreted as due to the lack of sterile neutrinos, which
indeed would cancel both identically.

So in the SM by itself, we may not gauge $U(1)_{B-N_{c}L}$—restricting to the
SM means only the subgroup
$\mathbb{Z}_{N_{c}N_{g}}^{B-N_{c}L}\times\mathbb{Z}^{L}_{N_{g}}$ may be
consistently gauged. We are within our rights as effective field theorists to
simply extend the SM by the discrete gauge theory of a gauge boson $A$ which
satisfies $\exp(i\oint
A)\in\mathbb{Z}_{N_{c}N_{g}}^{B-N_{c}L}\times\mathbb{Z}^{L}_{N_{g}}$
Tachikawa:2017gyf . This theory SM+ has the same local physics as the SM, but
disallows even otherwise-conjecturally-inevitable quantum gravitational decays
of the proton, since it imposes $\Delta B=N_{c}\Delta L=N_{c}N_{g}$
nonperturbatively. So there is some sense in which better lower bounds on the
lifetime of the proton provide strictly positive Bayesian evidence toward
preferring SM+ over SM at low energies. But I digress.

Here we wish to begin with the full continuous $B-N_{c}L$ symmetry gauged in
the ultraviolet, so we have two options for how to deal with the nonzero ’t
Hooft anomalies of Equation 16. It is possible that there are light right-
handed neutrinos, meaning that our counting above is wrong as they must be
included in our infrared SM. Then there is no barrier to gauging the full
$U(1)$, and indeed we take this as the benchmark to keep in the back of our
minds.

Alternatively, the degrees of freedom which cancelled that anomaly in the UV
theory may have gained a large mass during the $B-N_{c}L$ symmetry breaking
$U(1)\rightarrow\mathbb{Z}_{2N_{c}N_{g}}$. Once this is upgraded to gauge
theory, due to the low-energy anomaly the heavy $B-N_{c}L$ degrees of freedom
do not fully decouple. Instead, they have effects in the infrared which are
universally captured by the construction of Wess & Zumino Wess:1971yu  and
Witten Witten:1983tw . While this would generally seem to be less-UV-
sensitive, it poses a challenge to standard neutrino mass mechanisms.******We
note here some recent work on exotic ways of addressing this anomaly
Wang:2020mra ; Wang:2020xyo ; Wang:2020gqr ; Wang:2021ayd . In any case the
details will not be relevant for us here, so we postpone further exploration
to future model-building work.

###  B.2 $N$-Charged Abelian Higgs

We wish to study the Higgsing of $U(1)_{B-N_{c}L}$ down to
$\mathbb{Z}_{2N_{c}N_{g}}^{B-N_{c}L}$, and have the $\mathbb{Z}_{N_{g}}^{L}$
factor as an unbroken global symmetry. For aesthetic reasons we discuss here a
general $U(1)$ Abelian Higgs and a scalar $\Phi$ with some non-minimal charge
$N>1$. The Lagrangian is

$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\left|(\partial_{\mu}-ieNA_{\mu})\Phi\right|^{2}-V(\Phi),$
(17)

where the potential is minimized for some nontrivial vev
$\langle\left|\Phi\right|\rangle=v$. At low energies, we expand the Higgs
field around this vev in terms of a radial mode $\rho$ and a Goldstone mode
$\phi$ as

$\Phi(x)=\frac{1}{\sqrt{2}}(v+\rho(x))e^{i\phi(x)}.$ (18)

In terms of these fields, a gauge transformation $A_{\mu}\rightarrow
A_{\mu}+\partial_{\mu}\alpha(x)$ then effects a nonlinear change in the
Goldstone mode $\phi\rightarrow\phi+N\alpha(x)$.

As a result of the factor $N$, a subgroup of gauge transformations satisfying
$e^{iN\alpha(x)}=1$ leaves the Higgs field unchanged. Since $\Phi$ is not
charged under these symmetries, they remain unbroken when $\Phi$ condenses.
While this locally requires trivially constant gauge transformations
$\alpha(x)\equiv 2\pi k/N$ for $k=0,\dots,N-1$, we shall nonetheless find
nontrivial effects from the unbroken $\mathbb{Z}_{N}$ factor.

We go to low energies by imagining $v\rightarrow\infty$, in which case we
should integrate out the radial mode $\rho$ of the Higgs field around the
broken vacuum, and we see that the relevant coupling of the gauge and
Goldstone fields dominates

$\mathcal{L}=-v^{2}\left(\partial_{\mu}\phi-eNA_{\mu}\right)^{2},$ (19)

where corrections are of order $\mathcal{O}(v^{0})$ and it is clear that in
the infrared the equations of motion simply set $A=\partial\phi/N$. The gauge
boson has no local degrees of freedom at low energies and is pure gauge, as is
to be expected for a Higgsed vector field.

The ground state of the theory has $\langle\Phi\rangle=v$ everywhere, but our
interest is in a slightly broader class of solutions.

###  B.3 Abelian Higgs Cosmic Strings

We search for time-independent, $z$-independent solutions to the Higgs-Yang-
Mills equations of motion with finite energy per unit length, which simplifies
things to a $2$d problem. Of course the ground state with uniform vev
satisfies this, but it is not the only one. In general the cross-sections look
like Abrikosov-Nielsen-Olesen vortices Abrikosov:1956sx ; Nielsen:1973cs .

Figure 3: Cartoon of a cosmic string solution with the winding of the Higgs
phase plotted on a transverse circle.

With the scalar potential $V(\Phi)$ minimized for some $|\Phi|=v\neq 0$,
finite energy requires toward infinity in the transverse directions

$\Phi(r,\theta)\rightarrow ve^{i\phi(\theta)}$ (20)

where we may view $e^{i\phi(\theta)}$ as a map from the $S^{1}$ at transverse
infinity to the $S^{1}$ of the gauge group. Such maps can be topologically
classified by their winding number $k\in\mathbb{Z}$, the canonical example of
which is $e^{ik\theta}$. Since $\Phi$ has charge $N$, the transformation
$\phi\mapsto\phi+N\theta$ is a gauge symmetry and winding is only physical mod
$N$.

The equations of motion for the gauge field then dictate setting at long
distances

$\vec{A}(r,\theta)\rightarrow\frac{\phi^{\prime}(\theta)}{eNr}\hat{\theta}.$
(21)

With this we may directly relate the magnetic flux in the string to the
winding number of the scalar using Stokes’ theorem. Starting with the
definition of the winding number, we have

$\displaystyle k=\frac{eN}{2\pi}\oint_{\gamma}\vec{A}\cdot d\vec{x}$
$\displaystyle=\frac{eN}{2\pi}\int\int_{S}\vec{\nabla}\times\vec{A}\cdot
d\vec{S}=\frac{eN}{2\pi}\Phi_{B}$ (22)
$\displaystyle\Rightarrow\Phi_{B}=\frac{2\pi}{eN}k,$ (23)

where we have simply used Stokes’ theorem to relate the winding number to the
magnetic flux $\Phi_{B}$ through the string. We see the flux carried by the
string is fractionally quantized in $1/N$ of the Dirac flux quantum,
$g_{D}=2\pi/e$ Dirac:1931kp ; Dirac:1948um . The strings then contain only a
fraction of the flux around a Gaussian sphere of a fundamental magnetic
monopole.

With nonzero winding number of the scalar field, these are then cosmic string
solutions. The continuity of the field demands that somewhere in the interior
$\Phi=0$ and the symmetry is unbroken. While the ultraviolet form of the
solutions may be found in a given theory without much difficulty, we will here
only need what could be guessed by dimensional analysis: there is a core of
size $\sim v^{-1}$ inside which the Higgs is pushed up to the origin and the
$U(1)$ is unbroken, and the strings have tension $E/L\equiv\mu\sim v^{2}$.

###  B.4 Discrete Aharonov-Bohm Effect

The cosmic string solutions we’ve described are cylindrical configurations
with magnetic flux confined within a radius $R\sim v^{-1}$. This is a nearly-
idealized solenoid, and indeed our long-distance solution for the vector
potential in Eqn 21 is precisely that of the elementary Aharonov-Bohm setup
Aharonov:1959fk . As they did, we consider shooting a charged fermion from
point $p$ with initial wavefunction $\psi_{p}$ past a solenoid to an
interferometer at point $r$, and we can write the resulting wavefunction as an
integral over paths $\gamma$ from $p$ to $r$

$\psi_{r}=\int_{p}^{r}\mathcal{D}\gamma e^{iS(\gamma)}\psi_{p}$ (24)

where $S(\gamma)$ is the action evaluated for the path $\gamma$. We diagram
this in Figure 4. The charged fermion has an electric current which in the
nonrelativistic limit leads to an action contribution $\int
d^{4}xA_{\mu}J^{\mu}\rightarrow q\int_{\gamma}A$. We assume no other terms are
relevant. If the path integral has two saddle points $\gamma_{1},\gamma_{2}$,
for example due to a screen with two slits between $p$ and $r$, then at lowest
order $\psi_{r}$ is the superposition of these two paths

$\psi_{r}\propto\left(e^{iq\int_{\gamma_{1}}A}+e^{iq\int_{\gamma_{2}}A}\right)\psi_{p},$
(25)

and the probability density is then proportional to

$\displaystyle\left|\psi_{r}\right|^{2}$
$\displaystyle\propto\frac{1}{2}\left(2+e^{iq\int_{\gamma_{1}}A-iq\int_{\gamma_{2}}A}+e^{iq\int_{\gamma_{2}}A-iq\int_{\gamma_{1}}A}\right)\left|\psi_{p}\right|^{2}$
(26) $\displaystyle\propto\left(1+\text{Re}\ e^{iq\oint_{\gamma}A}\right),$
(27)

where $\gamma$ is the loop consisting of the path $\gamma_{1}$ composed with
the path $\gamma_{2}^{-1}$. We see that the two paths interfere to the extent
that $\exp iq\oint_{\gamma}A$ differs from one—that is, due to the presence of
magnetic flux in the region encircled by $\gamma$.

Figure 4: Cartoon of Aharonov-Bohm effect setup with two slits on either side
of a solenoid providing inequivalent paths for electrons toward an
interferometer.

In other language, the existence of the magnetic flux has induced a holonomy
in the gauge field $A$, $\text{Hol}_{\gamma}(A)\equiv\exp i\oint_{\gamma}A\neq
0$. This is precisely the observable of the Aharonov-Bohm effect. ††††††I
refrain from fiber bundle language in this manuscript, but I cannot resist
mentioning the explanations by Socolovsky and collaborators of the Aharonov-
Bohm effect in full geometric power Socolovsky:2002a ; Socolovsky:2007a . A
particle of electric charge $e$ transported around the solenoid then picks up
a phase $\exp i\left(e\Phi_{B}\right)$. In the case of electromagnetism this
is not quantized, since the magnetic flux through an experimentalist’s
solenoid is ultimately proportional to the velocity of the electrons in the
wires, which is a continuous parameter. But when electromagnetism is broken,
as in a superconductor, or as above from an elementary Higgs field, magnetic
flux is quantized into vortices. The AB effect for the Abelian Higgs with
unbroken $\mathbb{Z}_{N}$ symmetry results in a phase from Eqn 23 of

$\exp{ie\oint_{\gamma}A}=\exp{ie\frac{2\pi}{Ne}}=\exp\frac{2\pi ik}{N},$ (28)

and we see that for a fundamental $N=1$ Higgs the Aharonov-Bohm effect
vanishes in the broken phase. The $U(1)$ charge is completely screened and the
vortices have magnetic flux $\Phi_{0}\equiv 2\pi/e$, which we recognize also
as the flux through a sphere around a monopole with the Dirac charge quantum.

However, when the $U(1)$ symmetry is broken by a Higgs of charge $N>1$—such
that the discrete $\mathbb{Z}_{N}$ gauge symmetry remains unbroken—the
magnetic flux tubes are fractionally quantized with fundamental unit
$\Phi_{B}=\Phi_{0}/N$. Then the Higgs condensate can only screen electric
field lines in $N$-fold units, such that the charge $(\text{mod }N)$ is
unscreened and a long-range $\mathbb{Z}_{N}$ gauge field remains Krauss:1988zc
; Alford:1989ch ; Preskill:1990bm ; Coleman:1991ku ; Banks:2010zn . A
discretized version of the Aharonov-Bohm effect must therefore persist in the
infrared. And indeed, we have seen that the winding of the Higgs field
provides for dynamical solitons with discrete magnetic flux—precisely a
fundamental particle version of the idealized solenoid.

Importantly this remaining discrete gauge symmetry implies their infrared
stability even in the presence of magnetic monopoles. If we consider chopping
off the top half of our solenoid to have a semi-infinite solenoid running off
along the negative $z$ axis, at the origin we have a source of $\Phi_{B}$
magnetic flux. If $\Phi_{B}=\Phi_{0}$, then in the limit when the radius
vanishes $R\rightarrow 0$, the solenoid becomes the unphysical Dirac string of
Dirac’s $U(1)$ monopole. Then along a cosmic string one may imagine nucleating
a monopole-anti-monopole pair which are pulled apart by the cosmic string
tension and eat it up, as in Figure 5. But these monopoles simply cannot
remove any fractional flux $\Phi_{B}/\Phi_{0}=k/N\ (\text{mod }1)$.

Figure 5: The so-called ‘pac-man’ instability of the minimal Abelian Higgs
cosmic string loop to the nucleation of a monopole-antimonopole pair. With
inspiration from R. Lichtenstein.

While the above experiment with solely two paths for the charged particle is a
conceptually simple demonstration of the Aharonov-Bohm effect, the
cosmologically relevant problem is simply of an incident charged particle on
the solenoid. Building off earlier studies of AB scattering Aharonov:1984zza ;
Rohm:1985aa ; deSousaGerbert:1988kpn ; deSousaGerbert:1988qzd ; Alford:1988sj
; Wilczek:1981du , including the original result of Aharonov and Bohm for a
general solenoid Aharonov:1959fk , Alford & Wilczek looked at an example of
cosmic strings from the breaking $SO(10)\rightarrow SU(5)\times\mathbb{Z}_{2}$
Alford:1988sj . The generalization of their result is a differential cross-
section per unit length of

$\frac{d\sigma}{d\theta}=\frac{\sin^{2}(\pi q\Phi_{B}/\Phi_{0})}{2\pi
p\sin^{2}(\theta/2)},$ (29)

with $p$ the incoming particle momentum, $q$ its charge, and
$\Phi_{B}/\Phi_{0}$ the magnetic flux in fundamental units. There is then a
large cross-section for elastic scattering due to the nontrivial gauge
potential. The fundamental cosmic string has $\Phi_{B}/\Phi_{0}=1/N$, so that
the Aharonov-Bohm scattering affects any particle with $q\neq 0\ (\text{mod
}N)$.

The maximum is reached for $q=N/2$, the largest possible charge in
$\mathbb{Z}_{N}$, but in our model with $N=2N_{c}{N_{g}}$ the suppression of
elastic scattering from the $\sin^{2}$ is minor. For quarks with $q=1$, we
have $\sin^{2}(\pi/18)\sim 3\times 10^{-2}$, and proton or lepton scattering
is only suppressed by $\sin^{2}(\pi/6)=1/4$. Indeed, the cosmic strings have a
large elastic scattering cross-section with all of the SM fermions. Such
scattering affects neither the identity or energy of the scattered particles,
but will lead to a drag force on cosmic strings moving in a dense medium.

###  B.5 $\mathbb{Z}_{N}$ Gauge Theory

In the far infrared of the $N$-Abelian Higgs when only the discrete
$\mathbb{Z}_{N}$ gauge theory remains, the physics can be made more
transparent by dualizing $\phi$ to a two-form gauge field $B_{\mu\nu}$, as
first done in Banks:1989ag ; Coleman:1991ku  and later emphasized in
Banks:2010zn . Beginning with the infrared Lagrangian of Equation 19, we first
rewrite it as

$\mathcal{L}=-iJ^{\mu}(\partial_{\mu}\phi-
eNA_{\mu})-\frac{1}{2v^{2}}J^{\mu}J_{\mu},$ (30)

which can be seen to produce the same Lagrangian upon setting $J^{\mu}$ to its
equations of motion $J_{\mu}=-iv^{2}(\partial_{\mu}\phi-eNA_{\mu})$. If we
instead integrate the first term by parts, then we can interpret $\phi$ as
merely a Lagrange multiplier enforcing the constraint
$\partial_{\mu}J^{\mu}=0$.

We can trivialize this constraint on the dual current by changing variables as

$J_{\mu}=-\frac{i}{2\pi}\epsilon_{\mu\nu\rho\sigma}\partial^{\nu}B^{\rho\sigma}$
(31)

where $\epsilon_{\mu\nu\rho\sigma}$ is the Levi-Civita symbol and $B$ is an
antisymmetric two-tensor field. The two-tensor $B_{\mu\nu}$ is invariant under
a gauge transformation by an exact two-form $B_{\mu\nu}\rightarrow
B_{\mu\nu}+\partial_{\left[\mu\right.}\lambda^{(1)}_{\left.\nu\right]}$
parametrized by a one-form $\lambda^{(1)}$ in a higher-dimensional analogue of
the one-form gauge symmetry of electromagnetism $A_{\mu}\rightarrow
A_{\mu}+\partial_{\mu}\lambda^{(0)}$.

When we then rewrite the Lagrangian in terms of this two-form and look at low
energies we find

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{eN}{2\pi}A^{\mu}\partial^{\nu}B^{\rho\sigma}\epsilon_{\mu\nu\rho\sigma}$
(32) $\displaystyle=-\frac{N}{2\pi}B\wedge F,$

the so-called ‘BF theory’ first introduced by Horowitz Horowitz:1989ng ;
Blau:1989bq ; Blau:1989dh ; Wu:1990ci , where in the second line we’ve cleaned
up the indices by instead writing a wedge product of two-forms (and also put
the coupling back into $A$). In the IR the kinetic terms for the $A,B$ fields
disappear, meaning that there are no local degrees of freedom, as expected.

Nonetheless, the theory is not entirely trivial: The BF theory is a
topological theory of an Abelian one-form gauge field $A$ and an Abelian two-
form gauge field $B$. With just our two gauge fields, we have as observables
the Wilson lines $\exp in_{A}\int_{\Sigma_{1}}A$ and Wilson surfaces $\exp
in_{B}\int_{\Sigma_{2}}B$, where $\Sigma_{d}$ is a $d$-dimensional surface and
$n_{A},n_{B}$ are integers. These capture the effects of the gauge field on
the phase of the wavefunction for a probe particle which has no other
dynamics. As effective field theorists, we may think of these as the infrared
remnants of massive charge-$n_{A}$ particles with worldline $\Sigma_{1}$ or
massive strings of winding $n_{B}$ with worldsheet $\Sigma_{2}$. Indeed, the
one-dimensional objects charged under the two-form gauge symmetry are none
other than the cosmic strings we studied above. Our absorption of the charge
$e$ into the gauge field $A$ puts the Wilson operators on equal footing in
terms of solely the integers $n_{A},n_{B}$ appearing. Alternatively we could
have rescaled $B$ to make explicit its coupling, which is $v^{-1}$.

Figure 6: Example configuration for a two-observable correlator in the BF
theory with $\Sigma_{1}$ spacelike and $\Sigma_{2}$ at $x=y=0$.

Furthermore, the topological interaction gives precisely the appropriate
Aharonov-Bohm phases as these wind around each other, and ensures that only
$n_{A}n_{B}\ (\text{mod }N)$ is observable. This is the signature of the
$\mathbb{Z}_{N}$ gauge symmetry at low energies. One can calculate the
expectation value of a Wilson loop and a Wilson surface to find

$\left\langle e^{in_{A}\int_{\Sigma_{1}}A}\
e^{in_{B}\int_{\Sigma_{2}}B}\right\rangle=\exp{\left[2\pi
i\frac{n_{A}n_{B}}{N}\text{link}(\Sigma_{1},\Sigma_{2})\right]},$ (33)

where $\text{link}(\Sigma_{1},\Sigma_{2})$ is the linking number of these two
manifolds and in some sense this can be used to define generalized linking
numbers quantum field theoretically Oda:1989tq ; Horowitz:1989km . This can be
understood by recognizing that BF theories are naturally theories of de Rham
cohomology groups of fiber bundles, but we can convince ourselves of this in a
more elementary fashion with inspiration from our above discussion of the AB
effect.

Our strategy is to push the Wilson surface operator into the action, and
reinterpret this as the expectation value of the Wilson loop in a background
of a probe string. We let $X^{a}$ be coordinates on $\Sigma_{2}$, and
$x^{\mu}=\xi^{\mu}(X^{a})$ the position of the surface in spacetime, and set
$n_{A}=n_{B}=1$ for cleanliness. We have

$\left\langle
e^{i\int_{\Sigma_{1}}A}e^{i\int_{\Sigma_{2}}B}\right\rangle=\int\mathcal{D}A\mathcal{D}Be^{i\int_{\Sigma_{1}}A}e^{i\left(S_{BF}+\int_{\Sigma_{2}}B\right)},$
(34)

where the action has been modified to

$S_{BF+B}=\int d^{d}x\
\left(-\epsilon_{\mu\nu\rho\sigma}\frac{N}{2\pi}\partial^{\mu}A^{\nu}B^{\rho\sigma}+B_{\mu\nu}\delta^{(2)}\left(x-\xi(X)\right)\frac{\partial\xi^{\mu}}{\partial
X^{a}}\frac{\partial\xi^{\nu}}{\partial X^{b}}\epsilon^{ab},\right)$ (35)

and we can reinterpret this as adding to our Lagrangian an operator to create
an extended object, which the delta function localizes to a submanifold
Brennan:2021a . We can make the simple gauge choice
$\xi^{\mu}(X)=\delta^{\mu}_{a}X^{a}$ to make the Jacobian trivial. Now by
examining the equation of motion from varying $B$, we see that this term
induces a fractional flux for the gauge field $A$,

$\frac{\delta\mathcal{L}}{\delta
B^{\sigma\rho}}=-\epsilon_{\mu\nu\rho\sigma}\frac{N}{2\pi}\partial^{\mu}A^{\nu}+\delta^{(2)}\left(x-X\right)\delta_{\rho}^{a}\delta_{\sigma}^{b}\epsilon_{ab}=0.$
(36)

To make the physical picture clearer, let’s specialize to the case of
$\Sigma_{2}$ the worldsheet of a static cosmic string at $X^{a}=(t,z)$, and
$\Sigma_{1}$ a spatial loop which encircles the $z$-axis, as in the Aharonov-
Bohm setup. Then the equation of motion for $\rho=t,\sigma=z$ gives us

$\vec{\nabla}\times\vec{A}=\frac{2\pi}{N}\delta\left(x\right)\delta\left(y\right)\hat{z},$
(37)

and this is nothing other than an induced fractional holonomy for the gauge
field around the string. If we integrate this over a spatial surface $D$ with
$\partial D=\Sigma_{1}$, then it is clear that if $\Sigma_{1}$ loops around
the spatial $S_{1}$ of $\Sigma_{2}$ we have

$\oint_{\Sigma_{1}}A=\int_{D}\nabla\times A=\frac{2\pi}{N}.$ (38)

Then we can use this modified equation of motion for $A$ in evaluating the
expectation value and find

$\left\langle e^{i\int_{\Sigma_{1}}A}\right\rangle_{BF+B}=e^{2\pi i/N},$ (39)

precisely the holonomy we saw in Appendix B.4 appearing for a unit-charged
particle transported around a fundamental cosmic string in the non-minimal
Abelian Higgs. The general result of Equation 33 should then be thought of as
the relativistic generalization of the discrete Aharonov-Bohm effect to
arbitrary line and surface operators.

Indeed, the cosmic strings of the Abelian Higgs theory source the two-form
gauge field $B_{\mu\nu}$, and the Wilson surfaces are their remnant in the IR
$\mathbb{Z}_{N}$ gauge theory as non-dynamical sources. Just as in the Abelian
Higgs description, the strings induce a holonomy $e^{2\pi
i\Phi_{B}/\Phi_{0}}\in\mathbb{Z}_{N}$ for the gauge field $A$. We will return
to the BF theory in Appendix D, but its full application to our scenario will
be further studied elsewhere. In the meantime, reviews of discrete gauge
theories include deWildPropitius:1995hk ; Alford:1991vr ; Birmingham:1991ty .

##  Appendix C Cosmic String Dynamics

In this section we review some qualitative features of the formation and
evolution of cosmic strings. Ultimately our interest will be in an early-time
and small-scale regime where the details and not yet known, so we will resort
simply to parametrizing the string number density, as above. But understanding
the features below is still a useful guide to their behavior.

###  C.1 Cosmic String Evolution

In the cosmological case of a dynamical phase transition in the early
universe, the production of cosmic strings is guaranteed by the finite size of
the cosmic horizon. The field $\Phi$ condenses with independent phases in
different Hubble patches, so line defects necessarily form. The string
formation may be thought of as a random walk, which in three spatial
dimensions need not return to its starting point. So an order one number of
super-Hubble length cosmic strings per Hubble volume is expected to form by
simple dimensional analysis. This was first studied by Kibble Kibble:1976sj ;
Kibble:1980mv  and the formation of such topological defect solutions is
sometimes known as the ‘Kibble mechanism’. Numerical simulations thereafter
Vachaspati:1984dz  confirmed that an order-one fraction of the energy density
in strings during formation is in infinite, horizon-crossing strings.

Resultingly, the exotic solutions to the Abelian Higgs equations of motion
that we constructed in Sec B.3 are not merely of formal interest, but are
necessarily realized in the early universe. In the following section we will
discuss the evolution of these objects after the phase transition in which
they are formed. The strings are not charged under any unbroken continuous
gauge symmetries, so there are no massless gauge bosons mediating long-range
forces. As a result, the long-range dynamics of an isolated string are purely
gravitational and controlled by the classical Nambu-Goto action of a
gravitating string Nambu:1970a ; Goto:1971ce ; Forster:1974ga .

We note that there is lots of physics in the details of the Kibble mechanism
and the spectrum of defects formed, and this connects closely to condensed
matter realizations thereof. In particular, the correlation length of Higgs
fluctuations may in fact be quite small compared to the horizon length $\xi\ll
d_{h}$, resulting in a far greater initial number density of defects
Guth:1982pn ; Zurek:1985qw ; Murayama:2009nj . Our scenario will need to rely
on this early-time overabundance of strings, but this is not yet well-
understood.

The late-time behavior is still non-trivial and quite interesting. There is an
infamous result that cosmic strings reach an attractor solution where the
string network has a scaling symmetry—the distribution of string loop lengths
depends solely on the length as a fraction of the Hubble radius. This was
first argued for on energetic grounds by Turok and collaborators
Albrecht:1984xv ; Turok:1984db , with the only other possibility being
overclosing the universe Kibble:1984hp . Simulations by Bennett & Bouchet
provided numerical evidence for the scaling regime near the horizon scale
shortly thereafter Bennett:1987vf ; Bennett:1989ak ; Bennett:1989yp ;
Sakellariadou:1990nd ; Allen:1990tv . Analytic arguments for the scaling
solution were later given by Kibble & Copeland Martins:1996jp ; Kibble:1990ym
; Copeland:1991kz ; Austin:1993rg , but confusion reigned about the situation
on small scales, with some simulations finding copious production of small
loops Vincent:1996rb  and uncertainty about the role of ‘kinks’ and other
structure on the strings Kibble:1990ym ; Siemens:2001dx ; Siemens:2002dj .
Increasingly powerful simulations have further confirmed the scaling regime
Moore:2001px ; Ringeval:2005kr ; Martins:2005es ; Vanchurin:2005pa ;
Olum:2006ix ; Blanco-Pillado:2011egf ; Blanco-Pillado:2013qja ;
Hindmarsh:2008dw ; Auclair:2019zoz . A microphysical explanation in terms of
the classical string theory was finally provided by Polchinski & Rocha
Polchinski:2006ee ; Polchinski:2007rg ; Rocha:2007ni ; Dubath:2007mf ;
Lorenz:2010sm , with Polchinski:2007qc  giving a nice overview of the
difficulties of this problem.

What this means is that the late-time behavior is controlled by strings
chopping each other up into loops and ensmallening by emitting gravitational
radiation. We will discuss these effects in the next two sections.

However, before the late-time attractor dynamics can take over, there is a
period when friction from elastic Aharonov-Bohm interactions with the SM
plasma dominates. That cosmic strings dominantly interact with matter through
the Aharonov-Bohm effect was first appreciated only in 1989 by Alford &
Wilczek Alford:1988sj . As we saw in Sec B.4, the string elastically scatters
charges at an enormous rate $\sigma/\ell\sim 1/p$, with $p$ the momentum. This
scattering from the gauge potential is far more important than the friction
due to gravitational effects which had previously been considered
Everett:1981nj ; Vilenkin:1984ib ; Silva:2021www .

Vilenkin then showed Vilenkin:1991zk  that in the rest frame of a straight
section of string moving at $\vec{v}$ with respect to the background gas of
particles with charge $q$ at temperature $T$, the frictional force is

$\vec{F}\simeq-T^{3}\frac{\vec{v}}{\sqrt{1-v^{2}}}\sin^{2}(\pi\frac{qk}{N}).$
(40)

where I’ve gotten rid of some numerical factors. The effect on the string is
to damp out motion on a timescale $\propto\mu/T^{3}$. This frictional force
turns out to severely damp the dynamics until $T\lesssim(G\mu)M_{\text{pl}}$,
when the strings can begin free-streaming. See also Garriga:1993gj ;
Martins:1995tg .

While a single, long, straight string is easy enough, an understanding of the
dynamics when they free-stream quickly becomes very complicated. There are an
enormous range of scales involved in the problem—from the Hubble scale, to the
interstring distance, to the curvature length along the string, down to the
fundamental transverse size of the string. For fundamental cosmic strings, the
interactions are controlled by a string coupling which may be small. However,
our strings are semi-classical and have no free limit. As a result, the
strings strongly self-interact. The Polyakov dynamics Polyakov:1981rd  break
down entirely when strings intersect each other, or themselves, and the
results of these interactions are crucial for understanding the network
dynamics.

For completeness, I briefly note here the interesting program of understanding
the cosmic string behavior using the worldsheet theory derived from the
Abelian Higgs theory (e.g. Gregory:1988qv ; Akhmedov:1995mw ; Gregory:1990pm ;
Silveira:1993am ; Orland:1994qt ; Arodz:1995dg ; Baker:1999xn ; Baker:2000ci ;
Aharony:2013ipa ; Szabo:1998ej ; Bergeron:1994ym ; Zee:1994qw ; Sato:1994vz ).

###  C.2 Intercommutation

Figure 7: Cartoon of intercommutation dynamics whereby colliding string
exchange partners,and self-intersecting strings chop off loops.

There is a counter-intuitive effect in string scattering which turns out to be
crucial to our phenomenology: colliding Abelian Higgs cosmic strings always
‘intercommute’ (or ‘reconnect’ or ‘trade ends’ or ‘exchange partners’), as
diagrammed in Figure 7. This effect has been seen numerically to apply quite
broadly, but as far as I am aware there is analytic understanding only in the
case of supersymmetric defects, so we discuss that case to gain some
qualitative conceptual insight.

That understanding begins with Ruback Ruback:1988ba , who found in $2+1$d that
identical vortices undergo right-angle scattering by following the moduli
space approximation introduced by Manton Manton:1981mp  for monopoles and
eventually rigorously justified by Stuart Stuart:1994tc ; Stuart:1994yc , see
also Atiyah:1985dv ; Gibbons:1986df ; Rosenzweig:1990ea ; Samols:1991ne ;
Abdelwahid:1993fg ; MacKenzie:1995np ; Gauntlett:2000ks . Topological defects
at low energy follow geodesics on the moduli space, with a metric derived from
the kinetic terms in the Lagrangian.

In the case of two identical vortices colliding at low energies, the only
relevant coordinate on the two-vortex moduli space is the relative position
$\vec{r}$ pointing from the zero in the Higgs field of one to the other. So
the moduli space in the winding number $k=2$ sector looks locally like the
plane $\mathbb{R}^{2}$. But despite being semi-classical objects, two vortices
may be identical and so indistinguishable, so $-\vec{r}$ describes the same
configuration. Then the moduli space has the geometry of a cone with deficit
angle $\pi$—at least at large separations. But it’s more difficult to reason
about the behavior near the tip, since at small distances $|r|\lesssim v^{-1}$
the vortices lose their separate identities and our above argument fails.

Figure 8: Cartoon of motion in the two-vortex moduli space. Since the vortices
are indistinguishable, the space is a cone, though the tip is smoothed out.
The radial geodesic then clearly passes straight over the cone, which in
physical space corresponds to right-angle scattering.

In fact the moduli space is known to be smooth Taubes:1979tm  so that the cone
must be rounded off, and indeed in solvable UV-completions one can explicitly
see the tip of the cone smoothed out Hanany:2005bc ; Hashimoto:2005hi . Then
the intuitive picture is clearer, and we’ve diagrammed the situation in Figure
8. An ingoing radial geodesic goes straight over the cone, and this
corresponds in physical space to a scattering angle of $\pi/2$, whereas
forward scattering would correspond to a geodesic ‘bouncing back’ at the same
angle after hitting $\rho=0$. We try to diagram this logic in Figure 8.

The $2+1$d right-angle scattering can be uplifted to cosmic string
intercommutation by projecting the motion of the $3+1$d strings down to the
motion of the cores in various planes Shellard:1988ki ; Hashimoto:2005hi ;
Tong:2005un , as in Figure 9. The fluxes pass through some such planes in the
same direction, in which case the motion on the plane is of vortices
undergoing right-angle scattering, but the fluxes pass oppositely through
other planes on which the interaction is of vortex-antivortex annihilations.
Intercommutation necessarily follows from the consistency of these lower-
dimensional interactions.

Figure 9: Cartoon of an interaction between cosmic strings, zoomed in near the
collision point. Orange arrows give the string velocities, which are initially
into/out of the page. The configuration can be projected onto the red plane
through which the string fluxes pass in different directions, such that the
motion of the zeros of the Higgs field on this plane looks like vortex-
antivortex annihilation. On the other hand, projecting onto the green plane
the motion is of right-angle vortex-vortex scattering. Consistency with these
2+1d dynamics implies that after the collision the strings have exchanged
ends—they must intercommute. Heavily inspired by Figure 1 of Hashimoto:2005hi
.

Numerical simulations of the underlying field theory dynamics have also been
used to investigate the string dynamics during collisions Shellard:1987bv ;
Shellard:1988zx ; Matzner:1988qqj ; Moriarty:1988em ; Myers:1991yh , and
indeed are in agreement with the intercommutation probability being near
unity. The string network simulations we will match to use the Nambu-Goto
action to evolve their numerical strings and then merely implement
intercommutation by hand whenever a collision is detected. Given the enormous
range of scales in the problem it seems untenable to resolve the microphysical
interaction, though we note recent work on axion strings using adaptive mesh
refinement methods which may help also in this case Buschmann:2021sdq .

This intercommutation is crucial to prevent the string network from dominating
the energy budget of the universe Albrecht:1984xv ; Turok:1984db . Such
interactions chop strings into smaller pieces, and lead to a distribution of
lengths of string loops being created at any time, as we will see in Sec C.4.
The many loops chopped off in such processes allow the network to dissipate
energy by emitting gravitational radiation as they oscillate.

We note also that each intercommutation leads to the formation of
‘kinks’—small-scale structure originating at the interaction point and then
travelling in left- and right-moving modes (see e.g. Blanco-Pillado:2015ana ;
Kibble:1990ym ; Copeland:2009dk ; Kibble:1990ym ; Siemens:2001dx ;
Siemens:2002dj ; Rosenzweig:1990ea ). We defer further understanding of the
effects of this small-scale structure to future work—if anything, it seems
sensible to expect this to result in a larger number of small loops produced
by the network (e.g. Dubath:2007mf ), so that ignoring this is in some sense
conservative.

###  C.3 Diminution

Energy loss by string loop radiation is then an important effect for
understanding the network dynamics Vilenkin:1981bx ; Vilenkin:1981kz . In the
string rest frame the power emitted into gravitational waves from a string
loop of length $\ell$ oscillating at a frequency $\omega$ is given by the
‘quadrupole formula’ and estimated as Vilenkin:2000jqa

$\frac{dM}{dt}\sim G\left(\frac{d^{3}Q}{dt^{3}}\right)^{2}\sim
GM^{2}\ell^{4}\omega^{6}\sim G\mu^{2}$ (41)

where $Q\sim M\ell^{2}$ is the approximate quadrupole moment of the string
loop and $\omega\sim\ell^{-1}$ is the characteristic frequency. This results
in an energy loss which is independent of the overall size of the string, and
simulations have found $\dot{M}=\Gamma G\mu^{2}$ with $\Gamma$ a numerical
coefficient of order $\Gamma\sim 50$. We note also that strings may well
radiate more than just gravitons Vilenkin:1986zz ; Cui:2008bd ; Long:2014lxa ;
Long:2014mxa ; Hindmarsh:2017qff ; Long:2019lwl ; Auclair:2019jip ;
Vachaspati:2009jx ; Hyde:2013fia ; Lunardini:2012ct ; Vachaspati:2009kq ;
Damour:2001bk , and this fact may be used for interesting phenomenological
ends (see e.g. Bhattacharjee:1982zj ; Brandenberger:1991dr ;
Brandenberger:1992ys ; Davis:1992fm ; Ma:1992ky ; Lew:1993bb ;
Brandenberger:1994mq ; Mohazzab:1994xj ; Sato:1995ea ; Davis:1996zg ;
Sahu:2004ir ; Sahu:2005vu ).

On large distances the cosmic strings are expected to behave as Nambu-Goto
strings, and to have universal gravitational effects which offer the best
observational prospects for general cosmic strings. The gravitational effects
are dominated by the long strings, which carry an order one fraction of the
energy density, so our understanding of this limit has seen intense study. But
our interest is in the total number density, which will be dominated by small
loops. And so the behavior which has yet been understood precisely does not
suffice for our interests, malheureusement. As to smaller scales, Eqn 41
motivates the definition of a dimensionless ‘gravitational cutoff’ or
‘gravitational backreaction scale’ $\xi\sim G\mu$, such that at time $t$, a
loop of length $\ell\sim G\mu t$ would disappear entirely in a Hubble time.
The behavior of such small loops remains the subject or debate.

The evolution at the smallest scales further depends on the stability
properties of the strings. While in the low-energy theory the cosmic strings
are topologically stable as a result of the remnant $\mathbb{Z}_{N}$ symmetry
discussed above, this stability is challenged when the radial mode can jump
over the barrier at the origin of field space or tunnel through it. Thus once
a string loop has shrunk in size to near the symmetry-breaking scale $\ell\sim
v^{-1}$, the remnant discrete gauge symmetry does not prevent it from further
shrinking to nonexistence.

However, these loops may be stabilized by the presence of superconducting
currents Witten:1984eb ; Davis:1988ip ; Davis:1988jq  if there are appropriate
degrees of freedom living on their cores. These ‘vortons’ are then stabilized
against further decay after reaching a small size where the repulsive effect
of angular momentum prevents further dissipation Carter:1990sm .
Interestingly, the stability properties may be affected by degrees of freedom
which are heavy in vacuum, but have massless excitations on the string
worldsheet. Indeed, depending on the UV field content some relic vortons are
inevitable Auclair:2020wse , which places an upper limit on the string tension
$G\mu$ so the vortons do not overclose the universe Peter:2013jj . In the case
$G\mu\lesssim 10^{-15}$, stable vortons may comprise some part of the dark
matter density.

Our benchmark scenario is of strings which are ultimately unstable, but below
we will also calculate the number density for the case in which strings are
stable. Of course a realistic model may lie somewhere between these, with a
fraction of strings ending up as vortons and others decaying.

###  C.4 Scaling from Simulations

As we discussed in Appendix C.1, we know now with confidence that the cosmic
string network has a scaling solution at distances near the horizon length.
The interplay of long strings entering the horizon, intersections of strings
chopping off loops, and gravitational radiation emission leads to an attractor
solution with self-similar dynamics. As this is the only regime in which the
evolution is under control, we will make the simplifying assumption that we
enter it instantaneously at some time $t_{\star}$ when the scale factor is
$a_{\star}$.

But it is important to note that many uncertainties remain. We will assume the
scaling regime extends down to the gravitational backreaction scale, which is
further than numerical simulations have been able to test. We note also that
some simulations have found a lengthy ‘transient’ regime before scaling is
reached, so it is possible that $t_{\star}$ may be long after the phase
transition. During this approach to scaling, it has been found that small
loops are overproduced (e.g. Olum:2006ix ), but as far as I am aware this
transient regime has not been well-characterized.

We will follow the formalism of Blanco-Pillado:2013qja  and describe the
distribution of strings using the scaling length variable $\alpha\equiv
m/(d_{h}\mu)$ with $m$ the mass of the string and $\mu$ the string tension.
The scaling regime implies that the loop production as a function of cosmic
time and loop mass depends only on the ratio $\alpha\propto m/t$. In the
following section we will sum up the total number density of loops in the
scaling regime.

The behavior can be qualitatively understood quite simply: In scaling, the
network produces a constant number of loops per Hubble spacetime volume. In a
cosmology with equation of state $w>-1/3$ (e.g. radiation or matter
domination), the Hubble size grows more quickly than does the physical volume.
Then the same loop production is spread over a larger physical volume at later
times, so it is the early production of loops that matters most.

We use the formalism and numerical results of Blanco-Pillado:2013qja  as a
benchmark. We define a momentum-integrated form on the string loop phase space
as $f(t,m)=\int f(t,m,p)\text{d}p$, and then $f(t,m)\Delta t\Delta m$ gives a
comoving number density of loops produced in time $t\dots t+\Delta t$ with
mass between $m\dots m+\Delta m$. We wish to sum up the production of loops
over time using the production function found numerically.

The results of simulations in the scaling regime are characterized in terms of
a scaling production function $f(\alpha)\Delta\alpha$. At a fixed time $t$,
this may be interpreted as the number of loops produced in a Hubble volume
$d^{3}_{h}$ over a Hubble time $d_{h}$ with mass $m=d_{h}\mu\alpha\dots
d_{h}\mu(\alpha+\Delta\alpha)$. We can then relate the two functions as

$\frac{f(\alpha)\Delta\alpha}{d^{4}_{h}}=\frac{f(t,m=d_{h}\mu\alpha)\Delta
m}{a^{3}}$ (42)

and in the limit $\Delta m/\Delta\alpha\rightarrow\left.\frac{\partial
m}{\partial\alpha}\right|_{t}=d_{h}\mu$ we have

$\left.f(\alpha)\right|_{t}=\frac{\mu d^{5}_{h}}{a^{3}}f(t,m=d_{h}\mu\alpha).$
(43)

Simulations have found power-law scaling behavior of the sort

$f(\alpha)\text{d}\alpha\sim\frac{\text{d}\alpha}{\alpha^{p}}$ (44)

with $p>1$, so the production is peaked toward small loops. Though simulations
are limited by their resolution to explore only a few decades below the Hubble
scale, we will extrapolate this production down to the gravitational
backreaction scale $\xi\sim G\mu$, taking $m_{\text{min}}(t,t)=\xi\mu d_{h}$.

The coefficient is by definition independent of $\alpha$, which is the only
relevant parameter by the scaling symmetry, so a coefficient of order unity is
indeed borne out by simulations. As for the exponent $p$, we will use $p=1.7$
as a benchmark, which is the value determined during the matter-dominated era
by Blanco-Pillado:2013qja . As our purpose is just to evince the parametrics
and show the number density of string loops may be large enough, we will not
delve into the full body of work determining the precise form of scaling, work
on which may be found in e.g. Allen:1990tv ; Bennett:1985qt ; Bennett:1986zn ;
Bennett:1987vf ; Bennett:1989ak ; Bennett:1989yp ; Albrecht:1989mk ;
Hindmarsh:2008dw ; Polchinski:2006ee ; Polchinski:2007rg ; Lorenz:2010sm ;
Auclair:2019zoz ; Vanchurin:2005pa ; Ringeval:2005kr ; Martins:2005es ;
Blanco-Pillado:2011egf ; Blanco-Pillado:2013qja ; Martins:1996jp ;
Martins:1995tg ; Vincent:1996rb ; Copeland:1991kz ; Blanco-Pillado:2019tbi ;
Rocha:2007ni ; Almeida:2021ihc .

##### Small Loop Enhancement

If we have a loop production function in terms of the comoving number density,
then we can find the total number of loops simply by adding up the loops
produced at each time. We must translate the constant production per Hubble
spacetime volume to the comoving volume.

$\displaystyle n(t)$
$\displaystyle=\int^{t}_{t_{\star}}dt^{\prime}\int_{m_{\text{min}}(t^{\prime},t)}dm^{\prime}f(t^{\prime},m^{\prime})$
(45)
$\displaystyle=\int^{t}_{t_{\star}}dt^{\prime}\int_{\alpha_{\text{min}}(t^{\prime},t)}\frac{a^{\prime
3}}{d^{\prime 5}_{h}\mu}f(\alpha^{\prime})d\alpha^{\prime}d^{\prime}_{h}\mu$
(46)
$\displaystyle=\int^{t}_{t_{\star}}\frac{dt^{\prime}}{d^{\prime}_{h}}\frac{a^{\prime
3}}{d^{\prime
3}_{h}}\int_{\alpha_{\text{min}}(t^{\prime},t)}f(\alpha^{\prime})d\alpha^{\prime}$
(47)

If loops are stable, we can simply sum over all loops produced at each time,
taking $\alpha_{\text{min}}(t^{\prime},t)\equiv\xi$, the gravitational cutoff
scale. But if loops are unstable, we must only sum over those which will not
have disappeared by $t$. Then we should take

$\displaystyle m_{\text{min}}(t^{\prime},t)$
$\displaystyle=\xi\mu\left(d_{h}+t-t^{\prime}\right)$ (48)
$\displaystyle\alpha_{\text{min}}(t^{\prime},t)$
$\displaystyle=m_{\text{min}}(t^{\prime},t)/(\mu d^{\prime}_{h})$ (49)

for which a loop produced with scaling mass $\alpha_{\text{min}}$ at time
$t^{\prime}$ has shrunk to the cutoff scale $\xi$ at time $t$. Then with a
power-law loop production function we have

$n(t)=\int^{t}_{t_{\star}}\frac{dt^{\prime}}{d^{\prime}_{h}}\frac{a^{\prime
3}}{d^{\prime 3}_{h}}\xi^{1-p}$ (50)

in the stable case, and

$n(t)=\int^{t}_{t_{\star}}\frac{dt^{\prime}}{d^{\prime}_{h}}\frac{a^{\prime
3}}{d^{\prime
3}_{h}}\xi^{1-p}\left[\frac{\left(d_{h}+t-t^{\prime}\right)}{d^{\prime}_{h}}\right]^{1-p}$
(51)

when loops are ultimately unstable. When solely one sort of background
evolution is relevant, we have $d^{\prime}_{h}=t^{\prime}/(1-\nu)$ and
$a^{\prime}=a_{\star}(t/t_{\star})^{\nu}$. During radiation domination,
$\nu=1/2$, and during matter domination, $\nu=2/3$.

In the stable case the integral may be done immediately, to find

$\displaystyle n(t)\simeq\xi^{1-p}\frac{a_{\star}^{3}}{t_{\star}^{3}}$ (52)

where we have left off some order-one factors involving $\nu$ for readability,
and ignored a term which vanishes as $t_{\star}/t\rightarrow 0$. We see that
indeed the early loop production matters most so that the physical loop number
density just dilutes as $1/t^{3\nu}$, as for a gas of particles.

Since early loop production is most important, the number density in the
unstable case may be computed easily noting that at early times
$\frac{\left(d_{h}+t-t^{\prime}\right)}{d^{\prime}_{h}}\rightarrow\frac{2-\nu}{1-\nu}\frac{t}{t^{\prime}}$.
One finds

$n(t)\simeq\xi^{1-p}\frac{a_{\star}^{3}}{t_{\star}^{3}}\left(\frac{t_{\star}}{t}\right)^{p-1}$
(53)

and we see that the suppression is by a factor
$\left(\frac{t_{\star}}{t}\right)^{p-1}$ with respect to the stable case. Now
the physical number density dilutes as $1/t^{3\nu+p-1}$.

As is clear from Equations 52 & 53, our estimate for the total string loop
number density depends upon our assumptions about the behavior of the cosmic
string network at early times $t_{\star}$ and at small scales $\xi$. But the
conclusion we take from this to use in our benchmark is that there may be a
quite large abundance of small cosmic string loops—for a symmetry-breaking
scale of order a TeV, the enhancement in the number density of small cosmic
strings from $\xi^{1-p}$ may be upwards of $10^{20}$.

##  Appendix D Topological Defect Catalysis

We will discuss the specific example of cosmic string catalysis borne out by
our model in Sec D.3. However, the conceptual physical understanding of the
catalysis effect is rather opaque in that case as yet. This is such an exotic
effect—in the sense of being far from free field theory—that to build
intuition and aid further understanding we begin with exploring similar, more-
familiar effects for topological defects of lower dimensionality.

Topological defects are a rich topic in field theory going back all the way to
Dirac, who first puzzled over the interactions of charged fermions with
magnetic monopoles.‡‡‡‡‡‡Of course even Dirac stood on the shoulders of giants
and I refer the reader to KT McDonald’s manuscript mcDonald:2015a  for an
interesting review of the prehistory. It is from this setting that we adopt
the term ‘catalysis’, which comes from the Callan-Rubakov effect of GUT
monopoles catalyzing proton decay. In this example the need for inelastic
interactions can be seen elementarily from the far infrared. There has been
much progress in the last century of understanding both the infrared and
ultraviolet pictures of the electric and magnetic charges.

While it is not usually referred to with this language, the nontrivial
intertwining of charged fermions with instantons has also been appreciated
since ’t Hooft. The physics of these zero-dimensional topological defects is
naturally understood in fully Euclidean space, and this makes it easy to see
the role and peculiarity of fermion zero modes and the violation of anomalous
symmetries. The drawback and disambiguation with the other cases is that
instantons do not have a relic abundance and do not truly undergo scattering,
so an assertion of ‘unitarity-limited’ interactions is not immediately
sensible. But we nonetheless distinguish between some large action cost for
producing the instanton, and the fermion interactions which come along with
unit probability.

Taking these together, we find it natural to conjecture that all of the
phenomena discussed admit some sort of unified description—perhaps in the
language of generalized global symmetries—but the analogy has thus far not
been made precise. We will below try to call attention to the similarities,
but a full exploration is postponed to future work.

At the very least, we can speak about the defects in the general case in
topological language. In this section our scenario of interest is the
classical, static Yang-Mills(-Higgs) equations of motion solved on a sphere at
the compactification of asymptotic infinity of some number of spatial
dimensions $S^{k}$. The configuration space in this problem for a gauge theory
of the group $G$ (possibly spontaneously broken to the subgroup $H$) consists
of the gauge boson $A$ (and scalar $\Phi$) fields, which are locally defined
maps from this sphere at infinity into the gauge group $S^{k}\rightarrow G$
(or $S^{k}\rightarrow G/H$). Such maps are classified by the homotopy group
$\pi_{k}(G)$ (or $\pi_{k}(G/H)\simeq\pi_{k-1}(H)$) which counts the
topologically distinct sectors—a generalized notion of ‘winding numbers’.
These classify the topological defects in a given theory.

Figure 10: Cartoon of catalysis processes for topological defects of different
dimensions.

###  D.1 Instantons and Fermi Zero-Modes

Instantons in Yang-Mills theories are dimension-zero topological defects. Such
solutions exist when there are nontrivial maps from the boundary of
Euclideanized spacetime to the gauge group. In the unbroken electroweak case,
maps $W:S^{3}\rightarrow SU(2)\cong S^{3}$ are classified by a winding
$\mathbb{Z}$. Instanton configurations are those with nonzero topological
charge $\int W\tilde{W}/(16\pi^{2})\neq 0$, and the field non-trivially winds
around spacetime.

####  D.1.1 Infrared Inelastic Interactions

The inclusion of fermion fields in the path integral can qualitatively affect
the instanton solutions through the presence of fermionic zero-modes. ‘Zero-
modes’ refers to zero-energy eigenmodes of the Hamiltonian, and in the bosonic
case these are the ‘collective coordinates’ of the object—be they orientation
in the physical space or in an internal space as a result of Higgsing.
However, Pauli-Fermi-Dirac exclusion leads fermionic zero-modes to behave
quite differently. With a given background of the (Euclidean) metric and gauge
fields, we follow Fujikawa Fujikawa:1979ay  and diagonalize the fermions’
quadratic action using the Dirac eigenspinors $\varphi_{n}(x)$

$\gamma^{\mu}D_{\mu}\varphi_{n}=\lambda_{n}\varphi_{n}$ (54)

where $\gamma^{\mu}D_{\mu}\equiv\not{D}$ is the Hermitian Dirac operator for
that background and $\lambda_{n}$ are the real eigenvalues. We can write
Grassmann-valued Dirac fields $\psi,\bar{\psi}$ in terms of Grassmann
coefficients $a_{n},\bar{b}_{m}$ of these eigenspinors

$\psi(x)=\sum_{n}a_{n}\varphi_{n}(x),\qquad\bar{\psi}(x)=\sum_{m}\varphi_{m}^{\dagger}(x)\bar{b}_{m},$
(55)

where the conjugate spinors have the decomposition into
$\varphi_{m}^{\dagger}(x)\overleftarrow{(\not{D})}=\lambda_{m}\varphi_{m}^{\dagger}(x)$.
We take the eigenspinors to be orthnormalized as
$\int_{M}\varphi_{m}^{\dagger}\varphi_{n}=\delta_{mn}$. Recall the chirality
operator $\gamma_{5}\equiv
i\epsilon^{\mu\nu\rho\sigma}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}/4!$,
and note that since $\\{\gamma_{5},\gamma_{\mu}\\}=0$,
$(\gamma_{5}\varphi_{n})$ is also an eigenspinor with eigenvalue
$-\lambda_{n}$. These are then linearly independent, as can be seen by taking
$\int_{M}\varphi^{\dagger}_{n}\not{D}(\gamma_{5}\varphi_{n})$ and acting with
the derivative to either side.

But in the case of a zero-mode this no longer holds as the eigenvalues are the
same, so $\gamma_{5}\varphi_{0}\propto\varphi_{0}$. Since
$\gamma_{5}^{2}=\mathds{1}$, we conclude
$\gamma_{5}\varphi_{0}=\pm\varphi_{0}$ and the zero-mode is chiral. If the
zero-mode satisfies $\gamma_{5}\varphi_{0}=+\varphi_{0}$ it is called right-
handed and one with $\gamma_{5}\varphi_{0}=-\varphi_{0}$ is called left-
handed. On the other hand for an eigenvalue $\lambda\neq 0$ there are
independent left-handed and right-handed solutions we can construct by
projecting onto chiral subspaces $\varphi_{\pm n}\equiv
P_{\pm}\varphi_{n}=\frac{1}{2}(1\pm\gamma_{5})\varphi_{n}$.

The Dirac-Yang-Mills action is then

$\displaystyle-S=\int_{M}\bar{\psi}i\not{D}\psi$
$\displaystyle=\sum_{ij}\int_{M}(\varphi_{m}^{\dagger}\bar{b}_{m})i\not{D}(a_{n}\varphi_{n})$
(56) $\displaystyle=\sum\limits_{\begin{subarray}{c}n\text{ with}\\\
\lambda_{n}\neq 0\end{subarray}}i\lambda_{n}\bar{b}_{n}a_{n}$ (57)

as a sum over Grassmann coefficients of the Dirac eigenspinors. The zero modes
do not appear—there is no action cost to exciting them. The path integral over
the fermion fields can be re-expressed in terms of integration over the
Grassmann coefficients as

$\int\mathcal{D}\bar{\psi}\mathcal{D}\psi
e^{\int_{M}\bar{\psi}i\not{D}\psi}=\int\left(\prod\limits_{\begin{subarray}{c}\text{
nonzero}\\\ \text{modes }n\end{subarray}}d\bar{b}_{n}da_{n}\right)\int da_{0}\
e^{\sum\limits_{n}i\lambda_{n}\bar{b}_{n}a_{n}}.$ (58)

where there is only one Grassmann integral for the zero-mode since it is
chiral and has half the number of independent components. Now we recall that
for a Grassmann variable $\xi$, integration behaves like differentiation

$\int 1\ d\xi=0,\qquad\int\xi\ d\xi=1.$ (59)

From which can evaluate the nonzero mode integrals directly

$\int d\bar{b}_{n}da_{n}e^{i\lambda_{n}\bar{b}_{n}a_{n}}=\int
d\bar{b}_{n}da_{n}(1+i\lambda_{n}\bar{b}_{n}a_{n})=i\lambda_{n}$ (60)

or just recognize that such a functional integral of fermionic fields leads
quite generally to $\det(i\not{D})=\prod i\lambda_{n}$. But if the Dirac
operator has a zero-mode, since its Grassmann coefficient does not appear in
the action, the path integral above vanishes manifestly.

In other words, the integration over the Grassmann coefficient of the zero-
mode causes any correlation function to vanish unless enough fermion fields
are inserted to saturate all the zero-modes. So for a given background
$\mathcal{I}$ of instantons in which the fermion species $\\{\psi_{i}\\}_{i}$
have zero modes, the only nonzero correlation functions have an insertion of
each of these fermion fields,

$\left\langle\mathcal{O}\prod\limits_{i}\psi_{i}\right\rangle_{\mathcal{I}}=\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{O}\left(\prod\limits_{i}\psi_{i}\right)e^{-S_{\mathcal{I}}},$
(61)

where $\mathcal{O}$ are whatever other operators and $S_{\mathcal{I}}$ is the
action in the instanton background. So the functional integral over the
fluctuations around an instanton solution must be accompanied by a fermion
field for each zero-mode.

Now as for why there should be such zero-modes, there is a celebrated and
beautiful result linking anomalous global symmetries to fermi zero-modes of
instantons known as the Atiyah-Singer index theorem atiyah1963index . This
relates the ‘index’ of the theory’s Dirac operator—just meaning the number of
zero eigenvalues, which is invariant under small deformations of the
background—to the topological density of the gauge field. Defining
$J^{\mu}_{X}$ as a global $U(1)_{X}$ current in a theory with an ABJ anomaly
with the gauge group G with field strength $F$.

$\text{index}(\not{D})=\int\frac{F\tilde{F}}{16\pi^{2}}=\partial_{\mu}J^{\mu}_{X}.$
(62)

The number of zero modes of the fermions in an instanton background in this
theory are controlled by the mixed anomaly of $U(1)_{X}$ and $F$, which
results in violation of the otherwise-conserved current.

In the infrared, ’t Hooft showed that we can interpret the necessity of
fermion zero-modes in instanton background as inducing interactions among the
fermions that are controlled by the structure of anomalies tHooft:1976rip ;
tHooft:1976snw  also Kiskis:1977vh ; Brown:1977bj ; Bitar:1977wy ;
Christ:1979zm . In the case of the SM electroweak sector, the Higgsing of the
symmetry gives the instantons a characteristic size $\rho\sim v_{EW}^{-1}$,
and at low energies these may be accounted for by matching onto so-called ’t
Hooft vertices. We refer to Morrissey:2005uza  for a lucid discussion.

Such vertices are heavily suppressed in the infrared just because the
instantons are rare in the infrared. This differs from the cases of monopoles
and cosmic strings, which are not localized in time. So while the occurrence
of an instanton is exponentially suppressed, we want to separate this
conceptually from the relationship between the defect and charge violation. We
suggest that the fact that each instanton must come along with a violation of
the charge $\Delta Q_{X}$ has some interpretation as a ‘unitarity-limited’
effect of the fermi zero-modes.

###  D.2 Monopoles and Callan-Rubakov

Monopoles are dimension-1 topological defects in a spontaneously broken gauge
theory $G\rightarrow H$ where the quotient can non-trivially wrap the spatial
sphere $\pi_{2}(G/H)\simeq\pi_{1}(H)\neq 0$. In particular, the SM case is of
electromagnetic monopoles of $U(1)$ QED, which has $\pi_{1}(U(1))=\mathbb{Z}$.
Many sorts of complementary approaches have been taken toward studying the
interactions between electrically charged fermions and magnetic monopoles over
the decades. The literature on the topic of monopoles is enormous, and
introductions can be found in the classic reviews Preskill:1984gd ;
Preskill:1986kp ; Rubakov:1988aq  and recent notes Tong:2018aa . I will for
now resist the temptation to outline and synthesize all of this literature,
and content myself to present some basic conceptual pictures.

####  D.2.1 Infrared Inelastic Interactions

Starting from the bottom up, an important strain of analysis has been to show
directly in the infrared, $U(1)$ gauge theory description that inelastic,
helicity-violating interactions must occur. Indeed, the recognition that QED
with both electric and magnetic charges leads to novel effects goes back to
Dirac Dirac:1931kp ; Dirac:1948um , who first described the singular magnetic
monopole solution which exists in QED (also Tamm Tamm:1931dda ). Novel
electric-magnetic interplay was found in many guises amenable to elementary
arguments Goldhaber:1965cxe ; Zwanziger:1968ams ; Peres:1968umt ;
Lipkin:1969ck ; Wu:1976qk  even before attempting a quantum field theoretic
treatment Schwinger:1966nj ; Zwanziger:1968rs .

Historically, it was this endeavor which sparked high energy theorists’
enduring love affair with topology Wu:1975es ; Wu:1976ge ; Greub:1975mi .
Fibre bundles appear as the natural structure through which to define maps
that look locally like functions but have nontrivial global structure, and
indeed the fermion wavefunction may be thought of as a ‘wave-section’ of a
bundle. But we will here stick to familiar particle theory terminology and
defer further formal mathematical description to future work.

It all begins with the nonvanishing electromagnetic angular momentum in the
presence of an electric charge $e$ and a magnetic charge $g$—even if they are
stationary. It is not hard to derive this from the Lagrangian, but we take a
shortcut to see this via Schwinger’s elementary argument for charge
quantization Schwinger:1969ib  using the Lorentz force.***The idea for this
elementary argument was invented many times independently, with Saha
mentioning it in 1936 in the Indian Journal of Physics Saha:1936mn  and Fierz
in 1944 giving it in German in the Swiss journal Helvetica Physica Acta
Fierz:1944zt . Americans caught on late, with the Physical Review publishing
it in a short letter by H. Wilson in 1949 Wilson:1949zz ; Saha:1949np . We
imagine a nonrelativistic test particle of mass $m$ and electric charge $e$
moving with velocity $\vec{v}$. In the field of a monopole with magnetic
charge $g$, $\vec{B}=g\vec{r}/r^{3}$, the Lorentz force law is

$m\frac{d\vec{v}}{dt}=e\left(\vec{E}+\vec{v}\times\vec{B}\right)=eg\vec{v}\times\frac{\vec{r}}{r^{3}}.$
(63)

Considering the moment of this equation by taking the cross product with the
radial vector $\vec{r}$, and with some linear algebra providing
$d\hat{r}/dt=\vec{r}\times(\vec{v}\times\vec{r})/r^{3}$ for the unit vector
$\hat{r}\equiv\vec{r}/r$, one finds a modified conserved angular momentum
vector

$0=\frac{d\vec{J}}{dt}=\frac{d}{dt}\left[\vec{r}\times
m\vec{v}-eg\hat{r}\right],$ (64)

with an extra radial component proportional to both charges! The orbital
component of the angular momentum is not quantized, but in light of quantum
mechanics we may sensibly postulate the quantization of the perpendicular,
radial component of the angular momentum Mcintosh:1970gg . If this is forced
to take integer or half-integer values, we get Dirac’s quantization condition

$eg\in\mathbb{Z}/2.$ (65)

The minimal monopole with $g=1/2e$ produces a magnetic flux through a Gaussian
surface of $\Phi_{0}=4\pi g=2\pi/e$, matching the flux quantization condition
of the minimal Abelian Higgs as we saw in Appendix B.3 above.

In any case, the result that the electromagnetic field configuration stores
$\vec{L}=-eg\hat{r}$ persists in the relativistic theory, and immediately
poses a puzzle. If we consider shooting a charged fermion at a monopole,
$\hat{r}$ pointing from the electric to the magnetic charge would switch sign
discontinuously as one moves past the other. This necessary change in the sign
of the electromagnetic angular momentum as the fermion passes through the
monopole means that the scattering boundary conditions must pair incoming
fermions of one helicity with outgoing fermions of the other, such that the
spin flip makes up for the electromagnetic contribution Kazama:1976fm ;
Callan:1983ed . There must be an inelastic interaction, and the purely IR
description means that the effect does not vanish as the symmetry-breaking
scale is taken to infinity.

Recent work taking advantage of on-shell techniques has explicitly constructed
the $S$-matrix elements describing the scattering of electric and magnetic
charges from the infrared Csaki:2020inw ; Csaki:2020yei . This beautifully
overcomes the impossibility of a local, Lorentz-invariant Langrangian
description as shown by Dirac and Zwanzinger Dirac:1948um ; Zwanziger:1970hk ;
Zwanziger:1972sx ; Weinberg:1965rz . Here the catalysis effect may simply be
stated as the absence of forward scattering between these states, leading to
inelastic interactions at the unitarity limit. With no forward scattering
present, note that we can speak only of the $S$-matrix element and not of an
amplitude, whose definition assumes a decomposition heuristically as
$S_{ab}=\delta_{ab}+\mathcal{M}_{ab}$. This unusual structure is also closely
related to the violation of crossing symmetry of electric-magnetic scattering.
Physically this results from the fact that a configuration with an incoming
electric and magnetic charge has an angular momentum contribution from the
electromagnetic field that is entirely absent if one of the initial state
particles is crossed to an outgoing antiparticle.

####  D.2.2 Into the Defect

While the effect may be seen purely in the infrared theory, further insight
may be gained from how this appears out of the microphysics. Once the UV
completion of Dirac’s monopole had been found from Higgsed non-Abelian gauge
theory by ’t Hooft and Polyakov tHooft:1974kcl ; Polyakov:1974ek (see also
Arafune:1974uy ; Prasad:1975kr  and dyons shortly thereafter Julia:1975ff ;
Hasenfratz:1976vb ; Tomboulis:1975qt ; Callan:1975yy ; Boulware:1976tv ), one
could delve into the monopole to attempt to understand the microphysics of
scattering. With the full solution in hand, one can observe that the monopole,
as a map $\varphi^{a}(x)$ from physical space to internal space, breaks the
separate rotational and gauge orientational symmetries. In the monopole
background, the correct conserved angular momentum-like quantum number is the
unbroken sum, which sums the contributions

$\vec{J}=\vec{L}+\vec{T},$ (66)

where the normal angular momentum is combined with the generator $\vec{T}$ of
a broken $SU(2)$ subgroup of the gauge group, leading to a sort of mixing of
internal and external ‘spin’ factors Jackiw:1975fn ; Hasenfratz:1976gr ;
Goldhaber:1976dp ; Goldhaber:1977xw . This can easily be seen to imply that
the chirality-flipping boundary conditions should also flip the $\vec{T}$
quantum numbers Sen:1984qe .

From the ultraviolet we can also see that the monopole has a dyonic (or
‘phase’ or ‘rotor’ or ‘rotator’) mode—an electrically-charged degree of
freedom living on the monopole core—resulting from the orientation of the
Higgs in the internal space. It arises physically because the “little group”
of the Higgs is parametrized by the coset of the $U(1)$ within the broken
gauge group. The importance of this mode in scattering off monopoles was
powerfully emphasized by Polchinski Polchinski:1984uw  who demonstrated that
the essential physics can be seen just in the quantum mechanics of $2d$ Weyl
fermions interacting with this mode. Recently, Brennan has thoroughly explored
the Callan-Rubakov physics from this perspective through explicit calculation
Brennan:2021ewu ; Brennan:2021ucy , following earlier work including
Isler:1987xn ; Affleck:1993np . The dyonic degree of freedom living in the
monopole core is identified as arising from the longitudinal mode of the
broken gauge bosons.

The physics in the spherically-symmetric sector simplifies to two dimensions,
and an incoming fermion can be seen to undergo inelastic scattering by
interacting with this mode. If we integrate it out, we can derive the boundary
conditions for fermions in the $1+1$-dim theory of $s$-wave scattering. And
indeed, Brennan shows explicitly that the boundary conditions violate those
global symmetries that have an ABJ anomaly with the broken gauge group.
Heuristically, the intuition is that exciting the dyonic mode turns on an
electric field which activates the ABJ anomaly with, in the Callan-Rubakov
case, $B+N_{c}L$ as

$\partial_{\mu}J^{\mu}_{B+N_{c}L}\propto\int
F\tilde{F}\propto\int\vec{E}\cdot\vec{B},$ (67)

where these are the familiar electric and magnetic fields of electromagnetism,
and this leads to violation of $B+N_{c}L$. The Callan-Rubakov effect is then
that a monopole passing through a proton, by means of interacting with the
dyonic mode, realizes this violation by turning the proton into a positron
with unitarity-limited probability Rubakov:1982fp .

To gain further conceptual understanding it is useful to bosonize the physics
of 2d fermions in the $(r,t)$ half-plane Callan:1982ac ; Callan:1982au ;
Callan:1982ah ; Callan:1983tm ; Dawson:1983cm  in the spirit of the original
duality between the scalar ‘sine-Gordon’ theory and the ‘Thirring model’ of
interacting fermions Coleman:1974bu ; Mandelstam:1975hb . See also
Rubakov:1988aq  for an early review. Heuristically, one realizes each fermion
flavor bilinear as a compact bosonic degree of freedom
$\bar{\psi}_{i}\psi_{i}\sim e^{i\phi_{i}}$. Now this is still perturbatively a
theory of $2d$ scattering, but we can actually solve the bosonic theory to
find the ground states. The bosonized fields have the interpretation that
$\phi_{i}(r=0)$ is the $i$th particle number on the monopole core. One has a
Lagrangian of the form

$\mathcal{L}_{\text{int}}\sim\sum\limits_{\text{species}\
i}\mu_{i}^{2}\cos(2\pi\phi_{i})+\sum\limits_{\begin{subarray}{c}\text{unbroken}\\\
U(1)_{k}\end{subarray}}\frac{g_{k}^{2}}{r^{2}}\left(\sum\limits_{i}q_{i}\phi_{i}\right).$
(68)

In the first term $\mu_{i}$ is related to the fermion mass, and the periodic
potential imposes that the theory has vacua with different numbers of fermions
lodged on the core. The second ‘Coulomb’ term, which includes sums over $U(1)$
subgroups of unbroken non-Abelian symmetries, enforces these symmetries such
that only neutral combinations of fermions appear together on the core.

In the SU(5) grand unified case, one finds that the monopole core contains an
integer multiple of the combination $uude$, which is gauge invariant and
furthermore has $B-N_{c}L=0$, but $B+N_{c}L=-2N_{c}$. Since transitions
between the vacua have finite action, the ground state of the monopole is a
superposition $\sum\left|n\right\rangle$ with indefinite baryon plus lepton
number. So when the monopole interacts with the external world, it plainly
violates $B+N_{c}L$ Callan:1982ac ; Csaki:2021ozp . We expect that transitions
between degenerate vacua with different numbers of fermions in this reduced
model correspond to fermi zero modes of the monopole in the full 4d theory, as
a sort of parallel to the instanton case.

Finally, we note there are a variety of exotic effects arising from the
inclusion of fermions—the basic fact being that fermion number may fail to be
a good symmetry in the vicinity of a monopole. Jackiw & Rebbi Jackiw:1975fn
suggest that a monopole with a fermi zero-mode has fractional fermion number
$\frac{1}{2}$, and Goldstone & Wilczek Goldstone:1981kk  made this a bit more
intuitive by studying kinks in a $1+1$d model of polyacetylene and showing the
same effect. Goldhaber Goldhaber:1976dp  resolved its seeming violation of
spin-statistics Hasenfratz:1976gr ; Hasenfratz:1976vb  by careful analysis
(see also Jackiw:1975fn ; Rossi:1977im ; Callias:1977kg ; Callias:1977cc ;
Blaer:1981ui ; Wilczek:1981dr ; Yamagishi:1982wp ; Matveev:1988pj ).

###  D.3 Cosmic Strings

The strings we studied above are an example of dimension-2 topological defects
in a spontaneously broken gauge theory $G\rightarrow H$ where the quotient can
non-trivially wrap the transverse circle $\pi_{1}(G/H)\simeq\pi_{0}(H)\neq 0$
Schwarz:1982ec . $\pi_{0}(H)$ is simply the number of connected components of
the infrared symmetry group. For reviews see e.g. Hindmarsh:1994re ;
Vilenkin:2000jqa ; Copeland:2009ga .

Now in contrast to the prior cases, the scattering of fermions off cosmic
strings has seen much sparser study. Though aspects of the interactions of
fermions with $2+1$d Abelian Higgs Nielsen-Olsen Nielsen:1973cs ;
deVega:1976xbp  vortices have been studied beginning with Nohl:1975jg ;
deVega:1976rt ; Jackiw:1981ee ; Weinberg:1981eu . For cosmic strings, the
literature begins with Alford:1989ie ; Alford:1988sj  which we will use as our
model of the scattering in which charged fermions interact with a scalar
condensate in the string. Other related work on fermions and cosmic strings
includes Brandenberger:1991dr ; Hill:1987qk ; Ganoulis:1989hz ;
deSousaGerbert:1988kpn ; Semenoff:1987ki ; Perkins:1990uv ; Widrow:1988rc ;
Hill:1986ts .

The string case has only been studied from a UV perspective for specific
models of the string core, so in this section we will begin there. We will
write down our own microphysical model which maps to the previously-studied
case of Alford:1989ie  such that we may appropriate their results. We note,
however, that our understanding of the ultraviolet picture here is far less
well-developed than in the monopole case. Afterwards, we’ll offer some
comments about the infrared picture, but a full analysis is postponed to
future work.

####  D.3.1 Into the Defect

An explicit calculation of catalyzed baryon number violation by a cosmic
string was performed by Alford, March-Russell, and Wilczek Alford:1989ie .
They have cosmic strings of some gauged
$\widetilde{U(1)}\rightarrow\mathbb{Z}_{N}$, and they introduce a scalar
$\phi$ which condenses on the cosmic string and couples to a lepton and a
quark as

$\mathcal{L}\supset\lambda\left(\phi\bar{\psi}_{q}\psi_{\ell}+\phi^{\star}\bar{\psi}_{\ell}\psi_{q}\right).$
(69)

So $\phi$ is a genuine leptoquark carrying color and electromagnetic charge
and $B+N_{c}L$, and its condensation breaks all these symmetries. It should
also provide bosonic zero modes which can carry off the imbalance of charges
when the string catalyzes $p^{+}\rightarrow e^{+}$, as with the monopole core.
They compute that an incoming quark wavefunction is amplified towards the core
of the string and undergoes an inelastic interaction at the unitarity limit

$\frac{d\sigma}{d\theta}\sim\frac{1}{p}\sin^{2}(\pi\alpha)\left(\frac{p}{v}\right)^{4\left|\alpha-\frac{1}{2}\right|},$
(70)

where I am generalizing Eqns 18-21 of Alford:1989ie  and have $\alpha=kq/N$,
with $k$ the winding number of the string, $q$ the $\widetilde{U(1)}$ charge
of the incoming quark, and $N$ the order of the infrared discrete symmetry. We
recognize this form from the elastic Aharonov-Bohm scattering, but now with
additional suppression by ultraviolet scales except for the case $\alpha=1/2$.
Then it is only configurations with maximal BF charge, which for $k=1$ is
$q=N/2$, for which the scattering is entirely unsuppressed by ultraviolet
scales as in the Callan-Rubakov effect. We note that while their string
superconducts in the electromagnetic sense, the SM gauge fields are explicitly
ignored and manifestly not required for the catalysis.

Now our purpose here is slightly different—in particular we wish to preserve
the SM selection rule which stabilizes the proton, so we instead add a gauge-
singlet scalar $\chi$ with solely $B+N_{c}L$ charge equal to $2N_{c}{N_{g}}$.
This condenses on the core of the string and breaks the classical global
$U(1)_{B+N_{c}L}\rightarrow\mathbb{Z}^{B+N_{c}L}_{2N_{c}{N_{g}}}$, down to the
exact global symmetry in the SM. We must include a broader scalar sector to
communicate the $U(1)_{B+N_{c}L}$-breaking to the SM fermions while respecting
the unbroken symmetries. As our benchmark example, we take the scalars listed
in Table 3 and the mediation is ultimately through the leptoquark and diquark
$\omega_{\ell},\omega_{q}$ both with hypercharge $+8$ and $B-N_{c}L$ charge
$-2$ but different global charges $B+N_{c}L=+4,-2$, see Figure 1. We imagine
the mediators have $v$-scale masses from $\left\langle\Phi\right\rangle$, so
integrating out these mediators around the vacuum generates an operator

$\mathcal{L}\sim\frac{1}{v^{15}}\chi\left(\bar{u}\bar{u}\bar{d}\bar{e}\right)^{3}+\text{h.c.}$
(71)

The situation of cosmological interest is of three protons incident on a
cosmic string, so we interpret this by shuffling the quarks and leptons into
spinors of three protons and three electrons,

$\mathcal{L}\sim\lambda\left(\chi\bar{\psi}_{3p}\psi_{3e}+\chi^{\star}\bar{\psi}_{3e}\psi_{3p}\right),$
(72)

where we are ignoring many subtleties. This equation is really just a
heuristic for us, the importance of which is to evince the boundary condition
at the string core for an incoming charged fermion. We aren’t actually
interested in the operator of Equation 71 near the vacuum; the important
interaction with $\chi$ will take place in the core of the string, where we
assume the mediating fields $\omega_{i}$ are light and so do not heavily
suppress the interaction.

In the analogue of this setup in Alford:1989ie , Alford, March-Russell, and
Wilczek computed the two-dimensional quantum mechanics problem of an incoming
$\psi_{3p}$ particle incident on the cosmic string and scattering
inelastically into an outgoing $\psi_{3e}$. Now clearly the three-proton state
$\psi_{3p}={}^{3}\text{Li}$ is not a stable bound state, but in the
cosmological case of interest lumps of three protons are bound in the
${}^{7}\text{Li}$ nuclear wells. It is these lithium nuclei which are incident
on the string, but without further apology we assume that we can just use the
calculation for an incident $\psi_{3p}$ and ignore the neutron background they
live in.

They set out to solve the Aharonov-Bohm scattering problem with an incident
plane wave and an outgoing scattering wave

$\lim\limits_{\rho\rightarrow\infty}\begin{pmatrix}\psi_{3p}\\\
\psi_{3e}\end{pmatrix}=\begin{pmatrix}\psi^{i}_{3p}\\\
0\end{pmatrix}+\begin{pmatrix}\psi^{s}_{3p}\\\ \psi^{s}_{3e}\end{pmatrix}$
(73)

where the fermions obey equations of motion in the string background which in
our case have approximately the form

$\begin{pmatrix}\not{\partial}-iqg\not{A}&\lambda\langle\chi\rangle^{\star}\\\
\lambda\langle\chi\rangle&\not{\partial}-iqg\not{A}\end{pmatrix}\begin{pmatrix}\psi_{3p}\\\
\psi_{3e}\end{pmatrix}=0,$ (74)

where $A$ here is the $U(1)_{B-N_{c}L}$ gauge boson, g the gauge coupling, and
$q=N_{c}{N_{g}}$ the $B-N_{c}L$ charge of the incoming three proton state.
$\not{A}$ is the discrete Aharonov-Bohm potential at long distances as in
Equations 21, 3, and $\langle\chi\rangle$ vanishes except in the string core.
The wavefunctions in and outside the core must be carefully matched, and the
cross-section is proportional to the magnitude of the outgoing three-positron
current.

Then by mapping our case to that studied in Alford:1989ie , we may appropriate
the cross section they computed, Equation 70, with $N=2N_{c}{N_{g}}$ the
infrared discrete subgroup of $B-N_{c}L$ under which three protons have charge
$q=N_{c}{N_{g}}$. By the fact that we’ve chosen $\chi$ to preserve the SM
proton stability selection rule, we have unsuppressed inelastic scattering in
our case of $\psi_{3p}\rightarrow\psi_{3e}$. That is, our choice of
leptoquarks and Higgs charges has picked out exactly the process of interest

$\frac{\sigma}{\ell}\left(3p^{+}+\text{string}\rightarrow
3e^{+}+\text{string}\right)\sim\Lambda_{QCD}^{-1}.$ (75)

where $\frac{\sigma}{\ell}$ is the cross-section per unit length, as computed
for a long string. We note that as we saw above with monopoles, crossing
symmetry may be violated for topological defects, and in particular for us
Alford:1989ie  show the cross-section behaves differently for bosons, as if we
crossed one of the initial protons to a final antiproton.

Let me make a couple further comments on the properties of this model.
Firstly, the condensation of $\chi$ on the string core allows it to
superconduct bosonic global $B+N_{c}L$ currents, as first pointed out by
Witten Witten:1984eb . Such modes presumably carry off the $U(1)_{B+N_{c}L}$
charge which appears from the IR to be violated by $2N_{c}{N_{g}}=0\text{ (mod
}2N_{c}{N_{g}})$ units, but this has not been studied in detail. This behavior
would parallel the Callan-Rubakov effect, where $2N_{c}$ units of $B+N_{c}L$
charge are deposited onto the monopole core.

Secondly, the $B-N_{c}L$ Higgs $\Phi$ also admits low-energy many-fermion
interactions with the SM fermions, for example

$\mathcal{L}\supset\frac{1}{v^{15}}\Phi(QL\bar{u}^{\dagger}\bar{d}^{\dagger})^{3},$
(76)

which could be implemented with additional leptoquarks. With some similar
arrangement of these products of fundamental fermions into Dirac fermions,
this could imply that the string has fermionic zero-modes of the sort that
were explicitly constructed by Callan & Harvey Callan:1984sa . But here the
fact that such collections of fermions are not good bound states would seem to
pose a larger problem, and it is at least nonobvious whether there could
really be electromagnetically-charged multi-fermion zero modes on the string.
It is not clear what role there could be for these interactions, if any.

####  D.3.2 Infrared Inelastic Interactions?

Our explicit example above is an ultraviolet description, and despite it being
speculated long ago that this feature should be quite general, it has not had
the sort of intense study that has informed our understanding of the Callan-
Rubakov effect. Agreeing with the speculation that this is a feature of
infrared $\mathbb{Z}_{N}$ gauge theory, it should have a natural description
in the dual BF theory.

When we studied the $\mathbb{Z}_{N}$ BF theory above we studied operators
which create probe particles or strings, and we found that they have a long-
range topological interaction. The $\mathbb{Z}_{N}$ symmetry is there
reflected in the periodicity of the Aharonov-Bohm phases from the long-range
interaction—we can only detect charges $(\text{mod }N)$. But then we should
also be able to create charges in $N$-fold batches, since these are
collectively uncharged. However, the BF theory does not include any such
short-range interactions.

Such an interaction of line or surface operators should be described by
operators which need not be integrated over a closed loop, but rather have a
boundary where the operators meet to interact. Then this must be a line
operator which is gauge-invariant, and we came across just such an object
above—the combination $(\partial_{\mu}\phi-NA_{\mu})$—which when exponentiated
indeed provides the further line operator

$e^{i\int_{\gamma_{p}}(\partial_{\mu}\phi-NA_{\mu})dx^{\mu}}\rightarrow
e^{i\left[\phi(p)-N\int_{\gamma_{p}}A_{\mu}dx^{\mu}\right]},$ (77)

where $\gamma_{p}$ is a line from infinity to the point $p$. This is a sort of
’t Hooft operator for the two-form $B$, but comes along with $N$ Wilson lines
as a result of the BF interaction. In general these lines need not be
coincident, and they can describe e.g. a process where $N/2$ charge-1 fermions
have a point interaction with the Goldstone and scatter inelastically into
another $N/2$ fermions. And of course there is a similar operator describing
$N$ strings meeting along a line.

However, in the pure BF theory these ’t Hooft operators are trivial in that
they vanish on the support of the equations of motion. It is clear that one
missing ingredient is light charged fermions, but their addition is
nontrivial. We plan to pursue this further in future work.

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