Scattering amplitudes of fermions on monopoles

We consider scattering processes involving massless fermions and ’t Hooft-Polyakov magnetic monopoles in a minimal SU(2) model and in the Grand Unified SU(5) theory. We construct expressions for on-shell amplitudes for these processes in the J = 0 partial wave using the spinor helicity basis consisting of single-particle and pairwise helicities. These processes are unsuppressed and are relevant for the monopole catalysis of proton decay. The amplitudes for the minimal processes involving a single fermion scattering on a monopole in the initial state and half-fermion solitons in the final state are presented for the first time and are used to obtain the amplitudes for processes involving more fermions in the initial state and integer fermion numbers in the final state. A number of such anomalous and non-anomalous processes, along with their amplitude expressions, are written down for the SU(5) GUT model.


Introduction
Studies of scattering processes of light electrically charged fermions on magnetic monopoles, pioneered in the early eighties by Rubakov and Callan [1][2][3], provide some remarkable insights into fundamental properties of quantum field theory.
It is well-known that no Lagrangian formulation exists which is both Lorentz-invariant and local, for a QFT with electric and magnetic degrees of freedom.Zwanziger's formulation [4] has the advantage of featuring a local Lagrangian which depends on two gauge potentials and on an external vector n µ associated with the Dirac string.The monopoles here are of Dirac type [5] given by external point-like U(1) magnetic sources with no nonsingular core unlike the finite-energy 't Hooft-Polyakov monopole solutions [6,7] in a non-Abelian theory.For certain applications Zwanziger's theory can be used successfully as a low-energy EFT description of the underlying fundamental non-Abelian theory, but it fails to describe scattering processes of light fermions on 't Hooft-Polyakov monopoles even at low energies.
When light elementary fermions scatter on a 't Hooft-Polyakov monopole, the fermions in the J = 0 partial wave can penetrate the monopole core even at asymptotically low energies and probe the short-distance non-Abelian dynamics of the underlying microscopic theory. 1 These J = 0 amplitudes saturate unitarity and the cross-sections for the corresponding s-channel processes behave as σ s ∝ 1/p 2 c for p c ≳ Λ QCD , where p c is the fermion 1 In general, for a fermion scattering on a monopole, the lowest partial wave in the partial wave expansion of the amplitude [8,9], is characterised by J = j0 where j0 = |q| − 1/2 and q is the product of the fermion's electric and the monopole's magnetic charge over 2π.For the SU (2) and GUT models considered in this paper we have j0 = 0, cf.Eq. (2.3).For brevity we in this paper will refer to them as J = 0 amplitudes.
3-momentum in the CoM frame [1][2][3]9].These results are to be contrasted with what one would have naively expected to hold in a perturbative QFT settings where an s-channel process would have a 1/s pole with the corresponding cross-section for a head on collision being suppressed by 1/M 2 X ∼ R 2 M at low momenta, where R M is the monopole core size and M X is the mass of vector bosons in the non-Abelian theory.It turns out instead that the fermion-monopole scattering in the spherical wave is in fact unsuppressed by the fundamental mass scale of the microscopic theory.
In particular, for monopoles in a Grand Unified Theory (GUT) there are unsuppressed anomalous scattering processes of the type [1,2], which lead to the monopole catalysis of proton decay.This is the seminal Rubakov-Callan effect.For GUT monopoles the natural scale of the J = 0 scattering process is the scale of strong interactions, some 15 orders of magnitude below the relevant GUT scale that gave rise to 't Hooft-Polyakov monopoles in the first place.
One can also consider a single fermion, for example a massless positron, scattering on a GUT monopole [3], Here the Callan bosonization formalism for J = 0 scattering implies that particles in the final state carry half-integer fermion numbers, which in the first instance appears to be highly counter-intuitive as these cannot be described by the usual asymptotic non-interacting Fock states in perturbation theory.We will study such processes in detail.
The fact that there exists no acceptable Lagrangian formulation describing QFT interactions of electric particles and magnetic monopoles should by itself not be an obstacle for studying their scattering amplitudes.In fact, the lack of the underlying Lagrangian along with the apparent strong-coupling regime for the magnetic monopole coupling implied by the Dirac-Schwinger-Zwanziger (DSZ) quantization condition [5,10,11], makes an excellent case for developing an on-shell S matrix formalism with no possible input from Feynman rules in a theory with electrically and magnetically charge particles.This programme was carried out in a series of very interesting recent papers by Csáki and collaborators [12][13][14].
At the foundation of this on-shell S matrix formalism is the observation by Zwanziger that asymptotic in-and out-states containing electrically charged particles and magnetic monopoles are not the tensor products of single particle states.Even at infinite separations between the electric charges and the monopole, there exists a non-vanishing angular momentum of the electromagnetic field [15] and the electrically and magnetically charged states are pairwise entangled [16].The formalism developed in [12,13] provides a compelling practical realisation of this electric-magnetic entanglement by identifying the orbital momentum carried by each pair of electrically and magnetically charged particles with a novel pairwise helicity variable associated with the pair.
Our goal in the this paper is to re-visit scattering processes of massless fermions on monopoles and to compute their amplitudes in the regime where they are unsuppressed i.e. in the J = 0 partial wave.For the first time we will construct the amplitudes for the elementary processes of the type (1.2) that involve a single fermion in the initial state and fractionally charged fermions in the final state.We will then combine such processes to re-derive scattering amplitudes for the Rubakov-Callan reactions with two fermions in the initial state (1.1) thus validating the process (1.2).Our results for the amplitudes of two fermions scattered on a monopole agree with the recent work [17] but we disagree with their results for the processes where a single initial fermion is scattered on a monopole.
There is a number of intriguing and, at first sight, unexpected in QFT features that emerge from studying fermion-monopole scattering processes, which explains the ongoing interest in these investigations: 1.There is no crossing symmetry.One can neither apply crossing to individual particles in the Rubakov-Callan processes (1.1)-(1.2),nor would it be allowed by the multiparticle electric-magnetic entanglement; 2. Forward scattering amplitudes trivially vanish for many such processes, and 3.The optical theorem does not apply; For example the complex conjugate amplitude for (1.1) is the amplitude for which involves anti-monopoles rather than monopoles while the fermion states are the same.
4. As noted already, there is no decoupling of heavy mass-scales from the low-energy physics in fermion-monopole scattering.Low-energy fermions in the lowest partial wave penetrate freely non-Abelian monopole cores and result in unsuppressed scattering rates that cannot be obtained from low-energy U(1) EFT formulations.
5. There are fermion number violating anomalous, as well as fermion number preserving non-anomalous processes on monopoles that are both unsuppressed.
6. Production of fractional fermion numbers is possible for massless fermions scattered on monopoles thus restructuring the perturbative Fock space.
The paper is organised as follows.In section 2 we start with a minimal SU(2) theory that supports 't Hooft-Polyakov monopoles and provides simple settings and convenient general notation for studying all relevant to us aspects of fermion-monopole scattering.Section 2.1 presents a brief overview of fermion-monopole interactions and the special role played in the scattering by the J = 0 partial wave.The simplest model with N f = 2 massless fermion flavours and the scattering processes therein are discussed in section 2.2.A more non-trivial case with N f = 4 flavours, which is also more relevant phenomenologically, is detailed in section 2.3.In section 2.4 we present our main results for the corresponding fermion-monopole scattering amplitudes using pairwise and single-particle helicity spinors.Section 3 summarises applications of our results to an SU(5) GUT theory with one generation of massless fermions and lists a number of scattering processes and their amplitudes.Section 4 outlines our conclusions.

Fermion-monopole scattering in the SU(2) Model
In this section we consider a basic model supporting 't Hooft-Polyakov monopoles: an SU(2) gauge theory with the Higgs field in the adjoint representation.We add N f flavours of Left-handed Weyl fermion doublets, which we take to be massless, Being elementary fields, the fermions are much lighter than the monopole, m ferm ≪ M X /α, where M X is the vector-boson mass2 and α = g 2 /4π in the SU(2) theory.Hence we can set m ferm = 0 is a good approximation.We follow Rubakov's notation [18] where Left-handed fermions a + and b − have respective U(1) electric charges e = ±1/2 in the units of the SU(2) gauge coupling g.Their corresponding anti-particles, (ψ i L ) = ψ i R , transform as Right-handed spinors and are represented by the SU(2) doublets, The Witten's SU(2) anomaly argument requires that N f must be even and we will consider in turn the N f = 2 and 4 cases, with the latter being particularly relevant to applications to Grand Unification in section 3. The 't Hooft-Polyakov monopole [6,7] is a topological soliton and has an integervalued topological charge g M ∈ Z.At distances much greater that the monopole core r ≫ R M ∼ 1/M X , the 't Hooft-Polyakov solution in the unitary gauge coincides with the Dirac monopole configuration with g M being twice the magnetic charge of the Dirac monopole in units of 4π/g.Throughout the paper we consider the minimal 't Hooft-Polyakov monopole with the positive minimal value of the magnetic charge, g M = +1, of the unbroken U(1). 3t is easy to see that our monopole magnetic charge, g M = 1, times the electric charge of the a + or b − fermions satisfies, in agreement with the standard Dirac quantization [5] condition, q el • q mg = 2πn, for the minimal value of the integer, n = 1, after one recovers the units of gauge coupling g and 4π/g for the electric and magnetic charges.

Unsuppressed s-wave interactions inside the monopole core
When massless fermions scatter on a static 't Hooft-Polyakov monopole, the fermions in the J = 0 partial wave can penetrate the monopole core, while the higher J > 0 harmonics experience a centrifugal barrier in their interaction potential with the monopole and scatter without being able to reach inside the monopole core.This implies that the fermionmonopole scattering in the spherical J = 0 wave saturates unitarity and is unsuppressed by the monopole core mass scale.This observation is the first of the two key features of fermion-monopole scattering that are at heart of the Rubakov-Callan discovery of monopole catalysis of proton decay [1][2][3].
The second key feature of the Rubakov-Callan approach is the observation that for Lefthanded Weyl fermions in the J = 0 wave only their a + components exist as incoming waves while their b − components give the outgoing states in the fermion-monopole scattering.For the Right-handed spinors, ā− are incoming and b+ are outgoing.These facts follows from truncating the theory to J = 0 waves for each fermion, and analysing solutions of the Dirac equation for massless Weyl fermions in the 't Hooft-Polyakov monopole background.The truncation to J = 0 partial waves is justified since the higher partial waves cannot penetrate the monopole core and result in subdominant contributions to the scattering suppressed by powers of M X (see e.g.comprehensive reviews [18][19][20] for details of the derivation).
For future convenience, in Table 1 we summarise the charges (2.3) and properties of massless Weyl fermions (2.1)-(2.2) scattered on a 't Hooft-Polyakov monopole in J = 0 partial wave.

Scattering in the SU (2) theory with N f = 2 flavours
In a scattering process in a gauge theory, electric charge must be conserved.For low-energy scattering processes we are interested in, that is when the energy carried by the initial state fermions is (much) lower than the monopole-dyon mass splitting, ∼ M X ∼ 1/R M , the electric charge cannot be deposited on the monopole core by turning it to a dyon.
Starting with a single fermion a 1 +L in the initial state, the J = 0 state amplitude on a monopole that satisfies the in/out selection rules of Table 1 and the electric charge conservation, allows for the process, in the N + f = 2 model.This is an anomalous process as the chirality is not conserved.In fact, by applying large gauge transformations to a static monopole configuration, one can construct a semiclassical monopole-state 'theta-vacuum' as a linear combination of monopole states with different Chern-Simons numbers n.Instanton tunnelling effects that change n → n + 1 also change the number of R-handed minus L-handed fermions by one unit for each fermion flavour.One can use a simple selection rule [18] which states, where ∆R i and ∆L i are the change in the Right-and Left-handed fermions between the final and the initial states for each flavour i, and n a fixed constant for all N f flavours.The process (2.4) gives ∆R 1 − ∆L 1 = 1 = ∆R 2 − ∆L 2 and is allowed by the triangle anomaly.
On the other hand, an alternative proposal for the scattering, a 1 +L + M → b 1 +R + M , is not allowed by the anomaly selection rule (2.5) since in this case n is not universal for different flavours: ∆R 1 − ∆L 1 = 2 is not equal to ∆R 2 − ∆L 2 = 0.This process can also be ruled out by the flavour symmetry of the massless 2-flavour model at hand.The process (2.4) can also be used as an elementary building block for constructing multi-fermion-monopole scattering reactions in the N f = 2 model.The process with two fermions in the initial state incident on the monopole, can be obtained from the process (2.4) combined with (i.e.followed by) the same process with flavours 1 and 2 interchanged, Similarly to (2.4), (2.6), there are also anomalous processes with n = −1 and n = −2, which are allowed by all selection rules and the eelectrc charge conservation in this model, They are the counterparts of the corresponding processes (2.4), (2.6) with positive n.One can also construct non-anomalous processes with n = 0 by combining for example (2.7) with (2.8), Thus we see that fermion-monopole scattering can be anomalous or non-anomalous and it can change the ∆R − ∆L fermion number (or (B + L) in Grand Unification) by positive, negative or zero amount.
The overall story, however, becomes more complicated and interesting for the model with N f = 4 flavours which is also relevant to the SU(5) GUT theory with a single family.

Scattering in the N f = 4 model
Let us first consider the process with two initial state fermions scattering on the monopole.Selecting the same initial state as in (2.6) now gives, The final state is unambiguously fixed by our selection rules and electric charge conservation.The process (2.11) is consistent with the in/out selection rules of Table 1, the anomaly selection rule (2.5) with n = 1 and the electric charge conservation.The expression above insures that all available fermion flavours participate in the process, as required by the anomaly (or equivalently by flavour symmetry).
In the context of the SU(5) GUT theory (2.11) is a key process for the monopole catalysis of the proton decay [1,2], as will be reviewed in section 3.But what about the more elementary constituent process with a single fermion in the initial state?
Callan [3] was the first to study such processes in the four-flavour model and found that the final state must include half-solitons of the bosonized truncated J = 0 theory in the monopole background.Such half-solitons (aka semitons) correspond to particles with fractional (in present context half-integer) fermion numbers.The scattering process consistent with all the selection rules in Table 1, Eq. (2.5) and charge conservation is [3] The question arises of how to interpret the half-fermion particles in the final state.Since such states cannot arise in perturbation theory, Callan proposed that their half-integer fermion numbers should be interpreted statistically with a 50% probability that the fermion number for a given flavour is zero ore one.The authors of [17] argued against this and rejected the scattering process (2.12) altogether based on the argument that if such massless half-fermion states existed, they would have to be true asymptotic states far from the monopole perturbation theory can be reliable applied.They have proposed instead the process [17] with a different final state that now includes three fermions and avoids half-integer fermion numbers.The final state fermion ā 4 −R in this process cannot be in the J = 0 state, since the selection rules in Table 1 would require it to be an incoming rather than the outgoing wave, hence the final state of the scattering in (2.13) cannot be made out of three individual J = 0 fermions.The authors of [17] have addressed this issue by arguing that there is a cross-entanglement between one of the final state fermions with the field angular momentum arising from one of the other fermions in the final state, so that the complete final state in (2.13) is a J = 0 state.To support this statement they presented an expression for the scattering amplitude for (2.13) in the J = 0 partial wave using pairwise helicities.
We disagree with the assertion of [17] that the Callan process (2.12) is invalid and that the 2D truncation in this case fails to capture the 4D physics of unitarity saturating fermion-monopole amplitudes.Our first objection is that even though the final 3-fermion state in (2.13) may be in the J = 0 wave, the individual fermions are not.As such it is hard to understand how the outgoing ā 4 −R fermion could be produced inside the monopole core, since it is not in a J = 0 single particle state and would experience a very strong Coulomb repulsion from the core.We posit that the scattering process (2.12), if it exists, is suppressed by powers of E/M X ≪ 1 where E is the energy carried by the incoming fermion. 4The point we are making does not invalidate the helicity-basis expression for the amplitude of (2.13) constructed in [17], instead it invalidates an assumption that it is unsuppressed by the monopole scale.
Our second point against rejecting the process (2.12) is that it can be iterated to derive the non-controversial scattering process (2.11) with two fermions in the initial state, in analogy to our earlier discussion (2.4)⊕(2.7)⇒(2.6) in the N f = 2 model.Indeed, combining two single-fermion scattering processes (2.12), we obtain, which reproduces correctly the process (2.11).
On the other hand, if we attempt to combine two of the alternative processes of the type (2.13), we would find that the desired final state on the right hand side of (2.11) is now contaminated by the ā 3 −R and ā 4 −R fermions and is not of the form (2.11) which allows only b fermions and bb pairs.Once again this points to the fact that the corresponding combined alternative process with two fermions in the initial state contains interactions outside of the monopole core as is suppressed by the monopole scale.
We would also like to comment on the apparent problem raised in [17] that final states with half-integer fermion numbers, and hence the entire process (2.12), should not be allowed as such asymptotic states cannot exist in perturbation theory.But we already know from the pairwise entanglement argument that any electrically charged states (with integer or fractional charges) cannot be considered decoupled from the monopole and cannot be described by tensor products of standard perturbative asymptotic states.In addition, it was also shown in [21] that the outgoing massless fermion excitations are attached to a topological surface that carries no charge or energy and which ends on the monopole.The presence of this topological surface means that the outgoing radiation does not have to have the same quantum numbers as states in the ordinary Fock space.Similar ideas that the 4 Another way to visualise this point is to consider the inverse process where the three fermions b 2 +R + b 3 +R + ā 4 −R form an initial state scattering on an anti-monopole M .Since the incoming fermion ā 4 −R in the M cannot be in the J = 0 single particle state, it must bounce from the monopole core due to the centrifugal barrier.
fermions from a 'perturbatively missing' final state live in a different Fock space have also been discussed in [22,23].
In a more realistic theory, the fermions are massive.But at the distances near and inside the monopole core masses of light fermions, m ferm ≪ M X , are irrelevant and can be safely neglected, since the Coulomb terms dominate over the mass terms at small r.Only at large distances outside the monopole core the effects of light masses5 become relevant and the outgoing states with fractional fermion number become unstable and are expected to decay to states with integer fermion number [3,[18][19][20]24].
Heavy fermion flavours with m ferm ≳ M X do not enter the monopole core and are decoupled in the bosonized formalism of Callan.

Scattering amplitudes with pairwise helicities
To provide some concrete evidence in favour of the scattering processes of the type (2.12) involving fractional fermions here we will construct their on-shell amplitude expressions using a combination of single particle and pairwise helicities.
When both electric e i and magnetic g M i charges are present the asymptotic state of the S matrix are not given by tensor products of single-particle states.More than 50 years ago Zwanziger [16] pointed out that each pair of particles (i, j) with non-vanishing q ij = e i g M j −e j g M i ̸ = 0 constitutes an entangled pairwise state that carries a non-vanishing angular momentum of the electromagnetic field for the pair (i, j).The DSZ quantization rule [5,10,11], which follows from the quantization of this angular momentum, implies that q ij is quantized in half integer units.In particular, for a fermion-monopole pair we have q f M = ±1/2 in agreement with Eq. (2.3).
The S matrix formalism for electric-magnetic scattering pioneered by Csáki etal in [12] identified q ij as the pairwise helicity -a half-integer variable that characterises each entangled (i, j) pair, and introduced the corresponding pairwise helicity spinors |p ♭± ij ⟩ and |p ♭± ij ].It then follows that that under Lorentz transformations asymptotic multi-particle states pick up an extra little group phase factor e iq ij ϕ ij for each electric-magnetic pair.For example, a two-particle out-state with with an electrically charged particle i and a monopole M with momenta p i , p M and spins s i , s M transforms as [12], where U (Λ) is the unitary representation of the Lorentz transformation Λ, the phase factor e iq iM ϕ iM is the action of the little group associated with the momentum pair (p i , p M ), and D s ′ i s i , D s ′ M s M are the individual single particle little group factors.For an in-state the pairwise little group phases are opposite, so that the amplitude satisfies [12], (2.18) The amplitude above is written for n particles scattering into m particles on a single monopole and we have also absorbed the single-particle little group D-factors into the expression Ã on the left hand side.To construct the amplitude Ã which transforms according to (2.18), we make use of the pairwise helicity spinors constructed in [12] (which for completeness we briefly review in the Appendix) and their Lorentz transformation properties, where Λ and Λ on the left hand sides represent Lorentz transformations in spinor-bases, and on the right -in momentum-basis.
We now proceed to construct an expression for the amplitude Ã for the minimal process (2.14) where a single fermion is scattering on a scalar monopole in the J = 0 partial wave.Using the standard all-outgoing conventions for amplitude momenta, the contribution to the amplitude Ã from the incoming state a 1 +L + M is given by, where the single-particle helicity spinor |a 1 +L ] is the standard wave-function representing an incoming a 1 +L fermion in the all-outgoing momentum convention. 6The second factor, |p ♭− a 1 M ], is the pairwise helicity spinor associated with the (a 1 , M ) pair.The expression on the right hand side of (2.21) is uniquely determined by the requirements that: its helicity spinors can involve only the initial states; all (Lorentz) spinor indices must be contracted as this is a J = 0 state; Lorentz transformations of the pairwise helicity spinor |p ♭− a 1 M ] should give the phase factor e iq a 1 M ϕ , to be consistent with (2.18), which is indeed the case thanks to (2.19) and the fact that q a 1 M = 1/2 according to Table 1.The expression (2.21) for the in-state contribution of a single-fermion-single-monopole pair is in agreement with the result in [17].
We next should determine the contribution to of the outgoing state to the amplitude Ã.For each half-fermion in the monopole background we take, (2.23) 6 An outgoing particle with momentum p µ i and helicity h contributes to an amplitude a factor of |i⟩ −2h or equivalently |i] 2h , see e.g.[25,26].For example, a negative-helicity h = −1 outgoing gluon contributes to the amplitude an overall factor of |i⟩ 2 and a positive-helicity gluon -a factor of |i] 2 , in agreement with the well-known Parke-Taylor MHV amplitudes.An outgoing left-handed fermion has h = −1/2 and gives |i⟩ 1 , while an incoming left-handed fermion in the all-outgoing convention counts for h = +1/2 and contributes 1/|i⟩ ∼ |i].
It is worthwhile to point out that when we combine two square roots, for example one factor of ([ b 3 +R | p ♭− b3 M ]) 1/2 from the amplitude (2.24) and another factor of ([ b 3 +R | p ♭− b3 M ]) 1/2 from the second amplitude (2.25), they each correspond to the same semiton but with potentially different on-shell momenta, p 1 and p 2 , as they occur in two independent processes.But since we are working with the J = 0 spherical waves, the four-momenta of the two semitons are actually proportional to each other.This implies that the spinor helicities We also note that processes with no bb pairs in the out state are given by omitting the second line in (2.26), thus we have (2.28) Finally, following analogous steps to the approach described above, we can construct amplitudes for the processes with negative or vanishing (i.e.non-anomalous) n, such as: that we will need in the following section.
Analytic expressions we have derived for the amplitudes above are presented with a proportionality signs.The so-far missing kinematic factors on the right hand side these amplitudes can nevertheless be fixed on dimensional grounds and from the requirement that the lowest partial wave anomalous amplitudes for scattering processes saturate unitarity bounds.

Scattering of fermions with SU(5) GUT monopoles
In the minimal GUT theory the 't Hooft-Polyakov monopole lives in the SU (2) M subgroup of the SU (5) GUT .We consider a single generation of massless fermions in this model.Left-handed Weyl fermions transform in the 5 and 10 representations of SU (5) GUT are represented by N f = 4 of SU (2) M doublets [18],

([ b 3 +R (p 1 )
| and ([ b 3 +R (p 2 )| are also the same up to a proportionality coefficient.It then follows that for an appropriately normalised amplitude we can combine the two square roots into a single expression [ b 3 +R | p ♭− b3 M ], which is what we have done in deriving the result for the amplitude (2.25).