Monopoles and Fermions in the Standard Model

Since their theoretical inception Dirac:1931kp , magnetic monopoles have fascinated particle physics community thanks to their unique reach and links with distinct theoretical physics areas and applications. These include an explanation for quantisation of the electric charge, applications to cosmology, baryon and lepton number violation, potential applications in cosmic ray physics, also strong dynamics, confinement and duality.

What is most relevant for this paper is the subtle role monopoles play in scattering processes with electrically charged particles. In particular, if a proton finds a monopole to scatter on in a head-on collision with kinetic energy of the order of few GeV, the proton would decay. The characteristic scale for this reaction is set by strong interactions, the proton decay is rapid. At the parton level there are Callan–Rubakov scattering processes of the type Rubakov:1981rg ; Callan:1982ah ,

$$u^{1}\,+\,u^{2}\,+\,M\,\to\,\bar{d}^{3}\,+\,\bar{e}\,+\,M$$

(1)

which result in the monopole catalysis of proton decay. Since the process is insensitive to the monopole mass, it is unsuppressed Rubakov:1981rg ; Rubakov:1982fp ; Callan:1982ah ; Callan:1982au by large scales of the microscopic UV theory.

If we consider a single fermion in the initial state scattering on a monopole in the lowest partial wave, the outcome is even more surprising and it appears to necessarily involve final states with fractional fermion numbers Callan:1982au ; Callan:1983tm ,

$$e\,+\,M\,\to\,\frac{1}{2}\left(e\,\bar{u}_{1}\,\bar{u}_{2}\,\bar{d}_{3}\right)\,+\,M$$

(2)

The final state consists of four semitons, each with a fermion number 1/2. These states cannot be described by the usual asymptotic Fock states in perturbation theory, they exist only in the massless limit and disintegrate at distances further away from the monopole where mass effects become important.

In a recent paper the authors of vanBeest:2023dbu provided an explanation of what the final states Fock space is and why it is different from the standard one. The Fock space of (massless) final states lives in a twisted sector related to the perturbative one by an action of a discrete symmetry. According to vanBeest:2023dbu , the outgoing radiation carries integer fermion numbers, while the fractional charges become the property of the vacuum state.

Our main motivation is to apply and investigate this construction to the Standard Model. It was pointed out in Ref. vanBeest:2023mbs that the chiral nature of fermion representations creates an obstacle for applying the twisted sector formalism vanBeest:2023dbu for all but the simplest monopoles in the Standard Model. We find that these restrictions are circumvented when we require that the monopoles respect the electroweak symmetry breaking (EWSB). One could of course argue that at very short distances, near the GUT-scale monopole core, the electroweak scale masses are negligible and one can ignore the effects of EWSB. But at such short scales the concept of asymptotic states is suspect and without good asymptotic states we cannot talk about scattering. Our conclusion is that for all of the SM monopoles in the theory after EWSB, the application of the twisted sector method works out well and there are no apparent obstacles, except for the usual interpretational issues of precisely how the twisted sector vacuum or the semitons disintegrate when fermion masses become important.

The second aim of the paper is to link these formal developments with analytic expressions for scattering amplitudes and to provide estimates for the cross-sections that can be used in phenomenological studies.

The paper is organised as follows. In section 2 we consider fermions scattering on a monopole in pure QED and summarise key facts about partial-waves and the Kazama, Yang and Goldhaber (KYG) result Kazama:1976fm for the cross-section.

In section 3 we classify monopoles in the Standard Model before and after EWSB. The monopoles are fully characterised by the magnetic fluxes of the SM gauge groups which do not depend on the specific UV completion of the SM or details of the SM embedding. Tables 1 and 2 provide this classification of SM monopoles and also map it to the monopoles in the minimal GUT theory.

In section 4 we discuss the selection rules for the $j_{0}$-wave scattering where helicity-flip and $(B+L)$-violating processes occur. Section 5 contains a detailed discussion of scattering on the minimal GUT monopole $M_{1}$. In section 6 we turn what we have leaned so far into phenomenological cross-sections and provide estimates for different types of processes. This section contains some of our main results, and these will be re-used also in the following sections considering the rest of the SM monopoles.

Section 7 considers $M_{3}$ and $M_{3}^{\prime}$ monopoles which carry no chromomagnetic fluxes. We find in sections 7-8 that for all SM monopoles, $M_{1},M_{2}^{\,\prime},\ldots\,M_{6}^{\,\prime}$, that respect EWSB there are no obstacles in describing fermion scattering on any of the SM monopoles after EWSB which is one of our main observations. Section 9 presents a summary and conclusions.

A note on conventions

In our notation the anti-particle of a left-handed fermion $f_{L\,\alpha}$ is described by the fermion field of the right-handed chirality, $(\overline{f_{L}})_{R}^{\,\,\dot{\alpha}}$ and vice versa, for example,

$\displaystyle(u_{L}^{\,i})^{\dagger}\,=\,(\overline{u_{L}})_{R\,i}\qquad{\rm and}\qquad(e_{R})^{\dagger}\,=\,(\overline{e_{R}})_{L}$

(3)

The raised index $i$ indicates that the field is in the fundamental representation (in this case of $SU(3)_{c}$); after hermitian conjugation symmetry group representations change to the conjugate representations which is represented by the lowered index.

In this section we summarise some of the well-known facts related to magnetic monopoles and their interactions with electrically charged fermions in QED. This is a warm-up to the scattering of fermions on monopoles in the Standard Model (SM).

Magnetic monopoles in QED are described by the Dirac gauge field configuration Dirac:1931kp ,

$${\cal A}_{\mu}^{\,\,(q_{m}){\rm Dirac}}\,=\,\frac{q_{m}}{2}\,(1-\cos\theta)\,\partial_{\mu}\phi\,,$$

(4)

where $q_{m}$ is the monopole’s $U(1)$ magnetic charge and we have used conventional spherical polar angles $\theta$ and $\phi$ for the monopole placed at the origin $r=0$. To facilitate subsequent applications to the SM, the gauge fields are rescaled by the corresponding coupling constant(s), ${\cal A}_{\mu}\,=\,gA_{\mu}$, in our case $g=e$. Given a matter field $\psi_{q_{e}}$ with an electric charge $q_{e}$, the Dirac quantisation condition requires that the quantity,

$$q_{J}\,:=\,\frac{q_{e}q_{m}}{2}\,\in\,\frac{1}{2}{\mathds{Z}}\,,$$

(5)

is quantised in half-integer units, and each pair of electric and magnetic states in QED with charges $(q_{e},q_{m})$ is characterised by a half-integer parameter $q_{J}$.

Scattering amplitudes of electrically charged particles in QED on magnetic monopoles, are conventionally described the partial-wave decomposition approach pioneered of Kazama:1976fm . In this approach the wave-functions of charged states are expanded in the basis of generalised spherical harmonics in the monopole background Wu:1976ge . These spherical harmonics are the eigenfunctions of ${\mathbf{J}}^{2}$ and $J_{z}$, where ${\mathbf{J}}$ is the conserved total angular momentum operator,

$${\mathbf{J}}\,=\,{\bf r}\times({\bf p}-q_{e}{\mathbf{\cal A}})\,+\,{\mathbf{S}}\,-\,q_{J}\,\hat{\bf r}\,.$$

(6)

The first term on the right hand side is the orbital momentum momentum in the background ${\cal A}_{\mu}$ of the Dirac monopole (4), the second term is the spin of the electrically charged field, for Weyl fermions it is given by Pauli matrices, ${\mathbf{S}}={\mathbf{\sigma}}/2$. Notably, the final term in (6) shifts the overall expression by the unit radius-vector times $q_{J}$. This means that even at infinite separations between the electric charges and the monopole, there is a non-vanishing angular momentum contribution$~{}\propto q_{J}$, and these states are pairwise entangled Zwanziger:1972sx . The half-integer-valued parameter $q_{J}$ that characterises this entanglement is also the effective charge quantifying the strength of the electric-magnetic interaction in the quantum-mechanical scattering amplitudes and cross-sections that was constructed in Kazama:1976fm and is shown below in Eq. (12) .

Wave-functions of fermion states in QED, that are being scattered on a monopole, are expanded in the basis of the generalised spherical functions, $\psi(x)=\sum_{j,s}\psi_{j,s}(t,r)\,f_{j,s}(\Omega)$, which are simultaneous eigenfunctions of ${\mathbf{J}}^{2}$ and $J_{z}$ operators,

$${\mathbf{J}}^{2}\,f_{j,s}(\Omega)\,=\,j(j+1)\,f_{j,s}(\Omega)\,\,,\quad J_{z}\,f_{j,s}(\Omega)\,=\,s\,f_{j,s}(\Omega)\,,$$

(7)

and, as such, are labelled by two indices, $j=j_{0},j_{0}+1,\ldots,\infty$ and $s=-j,-j+1,\ldots j-1,j$, and so are the the wave-functions $\psi_{j,s}(t,r).$ For a given $j$, a 1-particle electric state $\psi_{j={\rm fixed},s}(t,r)$ forms a $(2j+1)$-dimensional representation of the $SU(2)_{J}$ rotation group of the total angular momentum (6). The minimal value of $j$ (i.e. the lowest partial-wave) is $j_{0}$, which for a fermion of electric charge $q_{e}$ scattered on a monopole of magnetic charge $q_{m}$ is,

$$j_{0}=|q_{J}|-1/2\,.$$

(8)

Kazama, Yang and Goldhaber (KYG) Kazama:1976fm computed scattering amplitudes of fermions on a static (i.e. infinitely heavy) Dirac monopole in non-relativistic quantum mechanics (QM). In the lowest partial wave, $j_{0}$, they found that the only processes allowed involved a $L\leftrightarrow R$ helicity flip of the scattered fermion, specifically:

$\displaystyle j=j_{0}:\quad\left\{\begin{aligned} \,\,q_{J}>0:\qquad&\psi_{L\,j_{0},s}\,+\,M\,\to\,\psi_{R\,j_{0},s}\,+\,M\\ \,\,q_{J}<0:\qquad&\psi_{R\,j_{0},s}\,+\,M\,\to\,\psi_{L\,j_{0},s}\,+\,M\,,\end{aligned}\right.$

(9)

while for all higher partial-waves the fermion helicity does not change from the incoming to the outgoing fermion,

$$j\geq j_{0}+1:\quad\psi_{L\,j,s}\,+\,M\,\to\,\psi_{L\,j,s}\,+\,M\ \quad{\rm and}\quad\psi_{R\,j,s}\,+\,M\,\to\,\psi_{R\,j,s}\,+\,M.$$

(10)

A simple way to understand the selection rule (9) is to assemble a pair of Weyl fermions of the same QED electric charge into a single Dirac fermion and notice that $j_{0}$-wave solution of the Dirac equation in the monopole background has the form,

$$\psi_{D}\,=\,\begin{pmatrix}\psi_{L}\\ \psi_{R}\end{pmatrix}\,=\,\frac{1}{r}\begin{pmatrix}\psi_{s}(t+r)\\ \tilde{\psi}_{s}(t-r)\end{pmatrix}f_{j_{0},s}(\Omega)\,,\qquad{\rm for}\,q_{J}>0\,.$$

(11)

This means that the left-handed Weyl fermion with positive $q_{J}$ in the lowest $j_{0}$-wave can be only incoming and that the corresponding right-handed fermion has only the outgoing component when scattered on a monopole.¹¹1If $q_{J}<0$, the upper and lower entries on the right-hand side of (11) interchange, and the right-handed fermions become incoming and vice versa. Hence the helicities of the incoming and outgoing fermions have to flip in accordance with the KYG selection rule (9).

The $j_{0}$ partial-wave differential cross-section for both processes in (9) is Kazama:1976fm ,²²2Note that we assume that incoming fermions have fixed polarisation. The result quoted in Kazama:1976fm has an extra factor of $1/2$ because they take their cross-section to be unpolarised. Our expressions in (12) and (13) are for polarised incoming fermions.

$$\frac{d\sigma_{j_{0}}}{d\Omega}\,\,=\,\frac{q_{J}^{2}}{p^{2}}\,\left(\sin\theta/2\right)^{4|q_{J}|-2}$$

(12)

where $p$ is the 3-momentum and $\theta$ is the polar angle of the scattered fermion in the laboratory frame where the monopole is at rest.

Cross-sections for all higher-partial-wave processes (9) have also been computed, summed over $j$ and tabulated in Kazama:1976fm . In distinction with (9), the higher-$j$ cross-sections have a Rutherford-type behaviour $\sim 1/\left(\sin\theta/2\right)^{4}$ at small angles,

$$\frac{d\sigma_{j>j_{0}}}{d\Omega}\,\,\sim\,\,\frac{q_{J}^{2}}{2p^{2}}\frac{1}{\left(\sin\theta/2\right)^{4}}$$

(13)

see Ref. Kazama:1976fm for details of the angular dependence for general values of $\theta$. We also note that the actual (unpolarised) Rutherford cross-section for all electrically charged particles, e.g. electrons scattered on muons,

$$\frac{d\sigma_{R}}{d\Omega}\,=\,\,\frac{\alpha^{2}}{4p^{2}\beta^{2}}\,\frac{1}{\left(\sin\theta/2\right)^{4}}$$

(14)

contains an additional $1/\beta^{2}$ enhancement factor relative to (13). The reason it is absent on the right hand side of (13) is that the Lorentz force between the electron and the monopole is also proportional to $\beta=v/c$ and the resulting factors of $\beta^{2}$ cancel in the numerator and the denominator.

The coupling constant factor comparison between the electric-electric Rutherford scattering (14) and the electric-magnetic KYG formulae (13) relates the fine structure constant squared $\alpha^{2}$ to the pairwise helicity factor $q_{J}^{2}$, which is what it should be since,

$$q_{J}^{2}\,=\,\alpha\,\alpha_{M},$$

(15)

where the magnetic coupling constant is

$$\alpha_{M}\,=\,\frac{g_{M}^{2}}{4\pi}\,,\quad{\rm and}\quad g_{M}\,:=\,q_{J}\,\frac{4\pi}{e}$$

(16)

Note that the KYG result (12) must also be correct in a fully relativistic theory for elastic $2\to 2$ processes in the limit where the monopole mass $M_{M}\gg m_{\psi}$. This is indeed the case, as was demonstrated in Csaki:2020inw , using the pairwise helicity formalism for electric-magnetic scattering amplitudes developed by these authors, and the requirement that the helicity-flip amplitudes for (9) should saturate unitarity of the $S$-matrix in the $2\to 2$ sector.

There are two points that we would like to note:

1. All the monopoles that may or may not be present in the Standard Model must descend from some underlying theory in the UV. Since only finite-energy non-singular field configurations are admissible as monopole avatars, their existence necessarily requires a UV extension of the SM.

2. Fortunately, neither the knowledge of the specific UV theory selection nor the technical details of the SM embedding are needed in order to classify all possible magnetic monopole states in the Standard Model. The monopoles are fully characterised by the quantum numbers of magnetic fluxes of the SM gauge groups, their topological magnetic charge, and the requirement that these configurations are stable within the Standard Model. This information is contained in the Standard Model itself.

The Dirac monopole configuration in QED (4) is singular, its energy density diverges at $r\to 0$ and, if taken literally, would result in an infinite-mass state. It should be seen instead as a low-energy effective theory representation of a finite-mass monopole solution of the underlying UV theory. For an appropriate UV completion of QED that would support such solutions, the monopole would have a core of radius $R_{M}\sim 1/{\cal M}$ given by the inverse mass scale of the UV theory with the energy density convergent inside the core. Principal examples of finite-energy monopoles are the ’t Hooft-Polyakov monopole solutions tHooft:1974kcl ; Polyakov:1974ek in a non-Abelian gauge theory with a Higgs field in the adjoint representation, and the monopoles Kim:1999bd of the Dirac-Born-Infeld (DBI) extensions of QED and/or other sectors of the SM. ’t Hooft-Polyakov monopoles exist in all UV extensions of the SM where the $U(1)_{Y}$ gauge factor is compact, i.e. originates from a non-Abelian gauge factor. This occurs in all Grand Unified Theories (GUT) settings. The ’t Hooft-Polyakov monopole mass, $M_{M}\sim{\cal M}/\alpha$, and the non-Abelian core, $R_{M}\sim 1/{\cal M}$, are set by the masses ${\cal M}$ of the heavy GUT vector bosons. Other than GUT non-Abelian embeddings of $U(1)_{Y}$, e.g. into a separate $SU(2)_{R}$ group, also give rise to ’t Hooft-Polyakov monopoles. The second category where finite-mass, regular-core monopole solutions are present Kim:1999bd is when the $U(1)_{Y}$ theory is embedded into a Dirac-Born-Infeld (DBI) formulation. In this case genuine regular monopoles also known to exist and their core is set by the DBI mass scale. In all these scenarios, at distances greater than the monopole core $r\gg R_{M}$ any spherically symmetric monopole configuration is gauge-equivalent to a Dirac monopole times a constant matrix from the Lie algebra of the SM gauge group Coleman:1982cx ; Preskill:1984gd ,³³3In our notation the gauge field is Lie algebra-valued, ${\cal A}_{\mu}=g_{s}A_{\mu}^{a}T^{a}_{c}+g_{w}A_{\mu}^{b}T^{b}_{L}+g^{\prime}B_{\mu}$, where $g_{s}$, $g_{w}$ and $g^{\prime}$ are the SM gauge coupling constants, and $T^{a}_{c}$ and $T^{b}_{L}$ are the generators of $SU(3)_{c}$ and $SU(2)_{L}$.

$${\cal A}_{\mu}\,=\,\,Q_{\rm mg}\,{\cal A}_{\mu}^{\,{\rm Dirac}}\,=\,\,Q_{\rm mg}\,\,\frac{1}{2}\,(1-\cos\theta)\,\partial_{\mu}\phi\,,$$

(17)

where ${\cal A}_{\mu}^{\,{\rm Dirac}}$ is the Dirac monopole and $Q_{\rm mg}$ is a constant diagonal matrix from the Cartan subalgebra of the gauge group that is constrained by the Dirac quantisation condition.

It is well-known that monopoles are sensitive to the global structure of the SM gauge group Tong:2017oea .⁴⁴4For a recent discussion of possible applications of the $G_{SM}$ global structure to SM phenomenology see e.g. Alonso:2024pmq ; Li:2024nuo . In what follows, we will concentrate on the SM group with the largest allowed center (${\mathds{Z}}_{p}={\mathds{Z}}_{6}$) that is consistent with the SM filed content and which allows for the largest selection of monopole states Tong:2017oea ,

$$G_{SM}\,=\,\frac{SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}}{{\mathds{Z}}_{6}}\,.$$

(18)

It is worthwhile to point out that the general monopole classification described above is actually equivalent to what can be obtained from the minimal embedding of the SM into the $SU(5)_{GUT}$ theory. It does not matter whether the minimal $SU(5)_{GUT}$ was the physically realised UV theory of the SM at any stage in the early universe. In so far as our goal is to identify all possible UV monopoles that are stable against radiative decay into massless SM degrees of freedom, and which are characterised by their magnetic fluxes of the low-energy theory (18), we can equally well use $SU(5)_{GUT}$ as formal book-keeping device for SM gauge and matter fields. To see this we recall that for all spherically-symmetric $SU(5)_{GUT}$ monopoles their magnetic are of the general form Gardner:1983uu ,

$$Q_{{\rm mg}}\,=\,n_{8}T_{c}^{8}\,+\,n_{3}T_{L}^{3}\,+\,pY\,,$$

(22)

where we introduce the conventional colour-, weak- and $Y$-hypercharge generators of the $SU(5)_{GUT}$,

	$\displaystyle T_{c}^{8}$	$\displaystyle=$	$\displaystyle{\rm diag}(1/3,1/3,-2/3,0,0)\,,$
	$\displaystyle T_{L}^{3}$	$\displaystyle=$	$\displaystyle{\rm diag}(0,0,0,1/2,-1/2)\,,$		(23)
	$\displaystyle Y$	$\displaystyle=$	$\displaystyle{\rm diag}(-1/3,-1/3,-1/3,1/2,1/2)\,,$

and in this notation the electric charge generator $Q_{\rm el}$ satisfies the usual Gell-Mann–Nishijima formula,

$$Q_{\rm el}\,=\,T_{L}^{3}\,+\,Y\,=\,{\rm diag}(-1/3,-1/3,-1/3,1,0)\,.$$

(24)

There is a on-to-one correspondence between (22) and the magnetic fluxes in the second column in the Table 1 with $n_{3}=m$, $p=p$ and $n$ being equal to the coefficient of the $SU(3)_{c}$ matrix $Q_{{\rm mg}\,c}^{(1,1)}$. We conclude that the monopoles resulting from (22) are identical to those specified by (19)-(20), as can be seen by comparing the second and the third column in Table 1.

Note that there is a caveat to the general model-independent classification of SM monopoles presented above. Apart from the SM quantum numbers/fluxes, one can also ask more detailed questions about the monopole mass spectrum and whether all of the $M_{p}$ monopoles are stable against separation into pairs of more elementary monopoles with lower values of $p$. These are dynamical questions that do depend on the specifics of the UV-theory, especially when it contains a hierarchy of different UV-scales and Higgs fields contributing to the symmetry breaking patterns and monopole production. For example, it was demonstrated in Gardner:1983uu that in the minimal $SU(5)_{GUT}$ model all the monopoles in Table 1, except $M_{5}$, are stable against decay into pairs of more elementary monopoles, while $M_{5}$ was unstable against a decay into a double and a triple monopole. These consideration a based on a computation of the monopole binding energy and are sensitive to the Higgs potential as well as the gauge degrees of freedom. The conclusions depend on a specific model. In the minimal GUT model the lightest monopole is the monopole with the minimal (single unit) of $U(1)_{Y}$ magnetic charge $M_{1}$, but as explained in section 5 of Ref. Preskill:1984gd , in $SO(10)_{\rm GUT}$ the doubly charged monopole $M_{2}$ can be much lighter than $M_{1}$. In general, a UV theory with a complicated symmetry-breaking hierarchy would contain stable monopoles with widely different masses with the minimal $M_{1}$ monopole often being the heaviest. Keeping these points in mind, it is still the case that any magnetic monopole that is stable and has retained at least some of the SM magnetic fluxes (and thus can interact directly with SM particles) must given by one of the configurations in Table 1.⁵⁵5This statement also assumes that the monopole in the UV theory was spherically symmetric up to gauge transformations. This is required to justify (17).

We now have to account for the electroweak symmetry breaking (EWSB) of $G_{SM}$ to

$$G_{SM}^{\,\prime}\,=\,\frac{SU(3)_{c}\times U(1)_{QED}}{{\mathds{Z}}_{3}}\,,$$

(25)

induced by the SM Higgs vev. This is not merely an academic point as the EWSB modifies the long-range fields of SM monopoles by screening their non-Abelian magnetic fluxes associated with the now massive $Z^{0}$ bosons (i.e. the screening of magnetic neutral currents). At distances $r>1/M_{Z}\sim 1/(100~{}{\rm GeV})$ monopole gauge field configurations cannot contain a non-vanishing component along the $Z^{0}$ vector boson direction, i.e. along the $T^{3}_{L}-Y$ generator.

The requirement that the monopole gauge field has a vanishing $Z_{\mu}^{0}$ implies that $m$ must be set equal to $p$ in (19),

$$G_{SM}^{\,\prime}:\quad Q_{\rm mg}\,=\,\begin{pmatrix}n_{1}-\frac{n_{1}+n_{2}}{3}&&\\ &n_{2}-\frac{n_{1}+n_{2}}{3}&\\ &&-\frac{n_{1}+n_{2}}{3}\end{pmatrix}\oplus\begin{pmatrix}\frac{p}{2}&\\ &-\frac{p}{2}\end{pmatrix}\oplus\,p\,,$$

(26)

This of course is not a matter of choice, but an automatic consequence of satisfying classical equations for the monopole in the presence of the SM Higgs field with the vev$\neq 0$.

The monopole stability conditions (19) are modified accordingly. The non-Abelian $SU(2)_{L}$ bosons are no longer massless and the monopole with $m=p$ is now automatically stable in the $SU(2)_{L}$ sector. At the same time, the decay mode into massless gluons are continued to be avoided by restricting $(n_{1},n_{2})$ in the same way as was done in (19). In summary, after EWSB the monopoles surviving in the Standard Model are the modified monopoles $M_{p}^{\,\prime}$ which we detail in Table 2. Note that only the minimal monopole configuration remains the same, $M_{1}^{\,\prime}=M_{1}$, with all other monopoles having to adjust their $T^{3}_{L}$ flux to change the Abelian flux $pY$ into $p\,Q_{\rm el}$ in accordance with (26), see the last column in Table 2. The third column of the Table proves the equivalence of the full set of general stable $M_{p}^{\,\prime}$ monopoles with the monopoles in the minimal $SU(5)_{\rm GUT}$ realisation of monopoles after EWSB Dokos:1979vu .

In the subsequent sections we will study interactions of the SM monopoles $M_{p}$ and $M_{p}^{\,\,\prime}$ with SM fermions before and after the electroweak symmetry breaking. Most interesting are the scattering amplitudes in the lowest partial wave $j_{0}$ which can generate baryon and lepton number $(B+L)$-violation and lead to unsuppressed proton decay and proton generation processes catalysed by magnetic monopoles Rubakov:1981rg ; Rubakov:1982fp ; Callan:1982ah ; Callan:1982au . As in the case of QED, the long-range interactions of SM monopole with charged matter fields are characterised by the total angular momentum operator ${\mathbf{J}}$ given in (6). The pairwise helicity variable $q_{J}$ can be determined for each (monopole, fermion)-pair by generalising the QED expression (5) to the full non-Abelian SM case by summing over the strong, weak and $Y$-hypercharge pairs of the corresponding electric and magnetic charges. Specifically, for a SM fermion $\psi_{(\boldsymbol{3},\boldsymbol{2})_{Y}}$ (such as the quark doublet $q_{L}$ in Table 3) transforming in the fundamental representations of $SU(3)_{c}$ and $SU(3)_{L}$ with the hypercharge $Y$, and the monopole with the magnetic flux matrix in (19) that we abbreviate as,

$$Q_{\rm mg}\,=\,Q_{{\rm mg}\,c}\oplus Q_{{\rm mg}\,L}\oplus p\,,$$

(27)

the pairwise helicity is given by

$$q_{J}\,=\,\frac{1}{2}\left(q_{{\rm mg}\,c}+q_{{\rm mg}\,L}\,+Yp\right)\,,$$

(28)

where the magnetic charges $q_{{\rm mg}\,c}$ and $q_{{\rm mg}\,L}$ are the eigenvalues of the non-Abelian magnetic matrices $Q_{{\rm mg}\,c}$ and $Q_{{\rm mg}\,L}$ and we have used the fact that non-Abelian electric charges are all =1 for the field in the fundamental representation, while the $U(1)_{Y}$ electric charge of $\psi_{(\boldsymbol{3},\boldsymbol{2})_{Y}}$ is $Y$.

For a SM fermion that is a singlet under $SU(3)_{c}$ or $SU(3)_{L}$ or both, the corresponding non-Abelian magnetic charge $q_{{\rm mg}}$ on the right hand side of (28) would be zero.

Since the magnetic flux matrices $Q_{\rm mg}$ and the electric hypercharges of the SM matter fields are consistent with Dirac quantisation conditions, the values of $q_{J}$ on the right hand side of (28), evaluated using Table 3, will be automatically half-integer-valued. This can be checked directly for each combination of fermions and SM monopoles – and will also follow from our results below.

The sign of $q_{J}$ for each (monopole, fermion)-pair determines whether a given Weyl fermion is an incoming or an outgoing state in its $j_{0}$-wave scattering, and the absolute value of $q_{J}$ tells us how the fermion transforms under the action of the $SU(2)_{J}$ rotation of the pair. The dimension of the $SU(2)_{J}$ representation is $2j_{0}+1\,=\,2|q_{J}|$ according to (8). If $q_{J}$ vanishes for a particular (fermion, monopole) pair, the fermion does not scatter on the monopole (the corresponding $T$-matrix is trivial) and the fermion is removed from the scattering states analysis.

In summary, we proceed as follows: first select a monopole $M_{p}$ or $M_{p}^{\,\prime}$. Next compute pairwise helicities $q_{J}$ for all SM fermions and project on the subspace of fermions with non-vanishing $q_{J}$. If we can assemble all Weyl fermions with the same value of $q_{j}$ and in the same flavour representation into Dirac doublets – as we have done in QED in (11) – we can assign the incoming $\psi$ and the outgoing $\tilde{\psi}$ states accordingly, and find a simple $2\to 2$ scattering helicity-flip process in the $j_{0}$-wave,

$$\psi^{\,I}+M\,\to\,\tilde{\psi}^{\,I}+M\,,$$

(29)

where the index $I$ is the flavour index. This is analogous to the process (9) inn QED. This scenario corresponds to a vector-like theory after the non-vanishing $q_{J}$-projection, and the corresponding selection rules are summarised in Table 4.

Such simple flavour-symmetry-preserving fermion pairing $\psi^{\,I}\leftrightarrow\tilde{\psi}^{\,I}$ indicated by the process (29), however, is not guaranteed and is often impossible for SM monopoles $M_{p}$ as was emphasised in vanBeest:2023mbs . This is because in a general SM monopole background the resulting fermion representations are not vector-like. We now want to check how this works for $M_{p}$ or $M_{p}^{\,\prime}$ monopoles in the SM before/after EWSB in a case by case basis.

The SM monopole with the minimal value of the topological charge $p$ (a.k.a. the 1-form charge) is the $p=1$ monopole $M_{1}$, also known as the minimal GUT monopole. As already noted, this monopole, shown in the top row in Tables 1 and 2, has no magnetic flux along the $Z_{\mu}^{0}\sim T_{L}^{3}-Y$ direction. This particular monopole is the same before and after EWSB.

To establish how $M_{1}$ interacts with the SM fermions we compute its pairwise helicity $q_{J}$ with each of the SM quarks and leptons using the formula (28) along with the magnetic fluxes in Table 1 and the charges of fermions listed in Table 3. We find that the subset of SM fermions with a non-vanishing $q_{J}$ is given for a single generation by $e_{L/R}$, $u_{L/R}^{1,2}$ and $d_{L/R}^{3}$ (where superscripts denote the $SU(3)_{c}$ colour indices of SM quarks surviving the $q_{J}\neq 0$ projection) and list their charges in Table 5 below. The incoming/outgoing status of the fermions is determined by the sign of $q_{J}$ following the general rule of Table 4, and the dimension of the rotation group $SU(2)_{J}$ representation is $2|q_{J}|$. Since all SM fermions interacting the the $M_{1}$ monopole turn out to be singlets of $SU(2)_{J}$, this rotation symmetry will play no further role in the $M_{1}$ case.

We now need to understand what the formal expressions for $j_{0}$-wave scattering on the monopole, tell us about scattering of physical SM states and derive cross-sections that are measurable quantities.

We have learned that there are three stages of intermediate would-be-asymptotic final states. Gradually increasing the distance $r$ from the monopole, we first have semitons or the twisted-sector states, then SM partons, and finally colour-neutral states - hadrons and leptons.