From Faddeev-Kulish to LSZ. Towards a non-perturbative description of colliding electrons

In a low energy approximation of the massless Yukawa theory (Nelson model) we derive a Faddeev-Kulish type formula for the scattering matrix of $N$ electrons and reformulate it in LSZ terms. To this end, we perform a decomposition of the infrared finite Dollard modifier into clouds of real and virtual photons, whose infrared divergencies mutually cancel. We point out that in the original work of Faddeev and Kulish the clouds of real photons are omitted, and consequently their scattering matrix is ill-defined on the Fock space of free electrons. To support our observations, we compare our final LSZ expression for $N=1$ with a rigorous non-perturbative construction due to Pizzo. While our discussion contains some heuristic steps, they can be formulated as clear-cut mathematical conjectures.


Introduction
Infrared problems enjoyed recently a revival, triggered by works of Strominger et al. on relations between soft photon theorems, asymptotic symmetries and memory effects (see [St17] for a review). One line of developments consisted in reformulating this 'infrared triangle' in terms of modified asymptotic dynamics in the sense of Faddeev and Kulish [GS16,GP16,GS17,Pa17,MP16,KPRS17]. Given the ambitions of these recent advances, reaching quantum gravity and black-hole physics, we have to point out that the mathematical and conceptual basis of the Faddeev-Kulish approach is not very solid, not even in its original context. First of all, both in the original work [FK70] and in the recent references, the Faddeev-Kulish approach is justified at best by working out some test cases in perturbation theory. The question if the infrared finite S -matrix has any nonperturbative meaning is left completely open. Secondly, the relation between the Faddeev-Kulish approach to the more standard LSZ scattering theory has never been clarified. While a naive application of the LSZ ideas clearly fails in the presence of infrared problems, a careful LSZ description of a bare electron accompanied by real and virtual photons is in fact possible [Fr73,Pi05,CFP07]. In the present work we outline a bridge from the Faddeev-Kulish formalism to this LSZ description in the massless Nelson model.
The Nelson model has been used for many decades for non-perturbative discussions of infrared problems (see e.g. [Fr73,Fr74,Pi05,AH12,DP13.1]). Its Hamiltonian, stated in Section 2 below, can be obtained as a low energy approximation of the massless Yukawa theory with the interaction Lagrangian L I = λψφψ. Here ψ is the massive Dirac field, whose excitations will be called electrons/positrons, and φ is the massless scalar field whose excitations will be called photons (although they are spinless). We fix an ultraviolet cut-off κ and approximate the dispersion relation of the massive particles by the non-relativistic formula p → p 2 /(2m), where m = 1 for simplicity. As the creation and annihilation processes of electron-positron pairs can be neglected in the low-energy regime, we can restrict attention to the zero-positron sector and include only the electron-photon interactions in the Hamiltonian H of the Nelson model. This Hamiltonian commutes with the total number of electrons and we denote by H (N) the N-electron Hamiltonians. Furthermore, by the translation invariance of the model, H (N) commutes with the respective total momentum operator P (N) and thus this family of operators can be diagonalized simultaneously. For N = 1 the lower boundary of their joint spectrum is the physical (renormalized) energy-momentum relation of the electron which we denote p → E p (see Figure 1). This dispersion relation has been a subject of study for many decades and it is relatively well understood [AH12,Fr74,Pi03,DP13.2]. Two comments about its properties are in order, since they anticipate our discussion in the later part of this paper: (a) In the presence of interaction the physical dispersion relation p → E p differs from the bare one p → p 2 2 appearing in the free Hamiltonian (2.2). This is caused by certain photon degrees of freedom 'sitting' on the bare electron, which are responsible, in particular, for radiative corrections to its mass. We will refer to these photons as 'clouds of virtual photons', to distinguish them from 'clouds of real photons' described in (b) below. In the following discussion these virtual photons will appear in the step from the bare creation operator b * (p) to the renormalized creation operatorb * σ (p) of the electron (cf. formula (6.6) below).
(b) It is also well known that there are no normalizable states in the Hilbert space of the model, that would 'live' exactly at the lower boundary of the spectrum from Figure 1. In other words, it is not possible to find normalizable states describing just the physical electron (including its cloud of virtual photons) and no other particles. Hence, the electron is always accompanied by some 'cloud of real photons', moving to lightlike infinity. This cloud, denoted W p,σ (t), will also appear naturally in our discussion below, see (5.3).
An early discussion of the Faddeev-Kulish formalism in the Nelson model is due to Fröhlich [Fr73, Chapter 5], who was quite pessimistic about its rigorous mathematical justification. Our work still contains some heuristic steps, but they have a form of plausible, clear-cut conjectures (see Sections 5 and 6). As one can expect, we start in Section 3 below from the concept of the Dollard modifier U D p (t), which comes from quantum mechanical long-range scattering. It does not suffer from any infrared divergencies and thus does not require infrared regularization. Such divergencies appear only in Section 4 when we start rewriting the Faddeev-Kulish scattering states in LSZ terms. This is completed in Section 5, where we express the quantity U D p (t) as a product of infrared divergent objects of two types: the clouds of real photons W p,σ (t) and the renormalized creation operatorsb * σ (p), both of which are well-defined only in the presence of an infrared cut-off σ > 0. From this perspective it is completely clear, that the two types of infrared divergencies, discussed in (a) and (b) above, must mutually cancel as σ → 0. In Section 6 we indicate that the resulting LSZ formula in the case N = 1 reproduces, up to minor technical differences, a rigorous formula for oneelectron scattering states in the Nelson model due to Pizzo [Pi05]. We conclude our discussion with several clear-cut mathematical conjectures concerning the convergence of N-electron scattering state approximants in the Nelson model.
Strangely, the original work of Faddeev and Kulish misses the central point above, namely the cancellation of infrared divergences coming from the clouds of real and virtual photons. In fact, the omission of the lower boundary of integration in formula (9) of [FK70] (which corresponds to dropping term (4.2) below) ensures commutation of the S -matrix with the total momentum of charged particles. Consequently, there is no room for clouds of real photons and the S -matrix is ill-defined on the Fock space of free electron states. Faddeev and Kulish try to cure this problem by a contrived construction of the asymptotic Hilbert space, based on singular coherent states. While this strategy may work in some test-cases in perturbation theory, to our knowledge it has never matured into a non-perturbative argument.
Some aspects of this problem have recently been noticed in [GP16], but the modification of the Faddeev-Kulish ansatz in this reference is somewhat ad hoc. Our solution is very natural: we apply the Dollard formalism according to the rules of the art [DG], without tampering with the lower boundary of integration. The resulting S -matrix may not commute with the total momentum of the electrons, but it acts on the usual Fock space. As mentioned above, the resulting scattering state can be given a solid LSZ interpretation in terms of electrons dressed with clouds of virtual photons and accompanied by clouds of real photons. It should be pointed out, that a similar picture of the electron is behind the well-tested Yennie-Frautschi-Suura algorithm for inclusive cross-sections [YFS61].

The model
The Hilbert space of the Nelson model is given by H = F e ⊗F ph , where F e , F ph are the Fock spaces of the electrons and photons with creation and annihilation operators denoted b ( * ) , a ( * ) , respectively. The Hamiltonian of this model is given by (2.1) where H 0 involves the free evolution of the electrons and photons, V is the interaction, κ is a fixed ultraviolet cut-off and χ [0,κ] (|k|) = 1 for 0 ≤ |k| ≤ κ and χ [0,κ] (|k|) = 0 otherwise. As the Fermi statistics and the spin degrees of freedom of the electron will not play any role in the following discussion, we suppress the latter in the notation.
Since this Hamiltonian commutes with the total number N of electrons, we can consider the Hamiltonians H (N) on the N-electron subspace H (N) := F (N) e ⊗ F ph , given by where x ℓ is the position operator of the ℓ-th electron and F (N) e is the N-particle subspace of F e . This quantum-mechanical representation will facilitate the application of the Dollard prescription in Section 3.

The Dollard formalism
As we are primarily interested in electron collisions, we treat all photons in the model as 'soft' and do not introduce any division of the range of photon energies [0, κ] into a soft and hard part. Our starting point is the interaction V, which is given on (3.1) According to the Dollard prescription, we construct the asymptotic interaction as follows: We substitute x ℓ → ∇E p ℓ t, where ∇E p ℓ is the velocity of the ℓ-th electron moving with momentum p ℓ along the ballistic trajectory, as expected for asymptotic times. Thus we have where p := (p 1 , . . . , p N ) are momenta of the electrons. As the physical dispersion relation of the electron is not p → p 2 /2 appearing in H 0 but rather the lower boundary p → E p of the energymomentum spectrum, we define the renormalized free Hamiltonian: Here Ω p (k) := |k| − k · ∇E p and the choice of the normalization constant C p will be justified a posteriori in Section 5. (The need to renormalize the free Hamiltonian was noted already in [Fr73]). Thus the asymptotic interaction in the interaction picture is Now we define the Dollard modifier where the second step above is standard [FK70]. For any family of functions h ℓ ∈ C ∞ 0 (R 3 ), ℓ = 1, . . . , N, of the electron momenta we define the corresponding scattering state approximant as follows: where in the second step we introduced some obvious short-hand notation. We note that all quantities above are well defined without infrared regularization. But a need for infrared regularization will arise in the next subsection, where we start reformulating states (3.6) in terms of the LSZ asymptotic creation operators of photons and electrons, whose approximating sequences are given schematically by (3.7) As we will see in (6.6)-(6.7) below, b * (p) will actually require renormalisation.
To conclude this section, we define the wave-operators Ω in/out : F e → H for the electron scattering as follows so that the corresponding scattering matrix S := (Ω out ) * Ω in is an operator on F e . The existence of the limit in (3.8) is not settled, but seems to be a feasible functional-analytic problem, as we discuss in Section 6.

Infrared regularization
Let us consider the exponential in the Dollard modifier (3.5) and perform the time integral Since the l.h.s. of (4.1) is manifestly infrared finite, the same is true for the r.h.s. of this expression. However, terms (4.2) and (4.3) considered separately, coming from the lower and upper boundary of the τ-integration, are infrared singular. Indeed, they involve a ( * ) (k) integrated with functions which have a non-square-integrable singularity at zero momentum. This division of a regular expression into two singular parts, which will be needed to express the approximating vector (3.6) in the LSZ fashion, is the source of infrared divergencies, which must mutually cancel. As we pointed out above, in the work of Faddeev and Kulish [FK70] the counterpart of (4.2) is omitted.

Clouds of real and virtual photons, phases
We now rewrite formula (4.11) in the LSZ fashion to facilitate its interpretation in terms of real and virtual photon clouds. By shifting the term e −iH ren 0;σ t to the right and noting the cancellation of the constants C p ℓ ,σ (cf. (4.6)) we get where h t (p) := N ℓ=1 e −iE p ℓ ,σ t h ℓ (p ℓ ) is the (renormalized) free evolution of h. In the bracket in (5.1) we recognize the LSZ approximants of the clouds of real photons. For future reference we set It is more difficult to recast the expression in (5.2) as LSZ approximants pertaining to the electrons. For this purpose we reverse the Dollard prescription in the expression e −ik·∇E p,σ t in (5.2) that is we make a substitution e −ik·∇E p ℓ ′ ,σ t → e −ik·x ℓ . This leads us to the new family of approximating vectors Although we do not have a rigorous proof that lim t→∞ Ψ σ h,t −Ψ σ h,t = 0, it is intuitively clear, that the position x of the freely evolving electron behaves asymptotically as ∇E p,σ t. To simplify (5.4), we define the following (tentative) renormalized creation operator of the electroñ Thus, intuitively,b * σ (p) creates from the vacuum the electron with its cloud of virtual photons. Consequently, we can rewrite (5.4) in the LSZ form: The real-valued functions γ p,σ and θ p,σ , appearing above, have the following explicit form γ p,σ (t) := γ 1;p,σ (t) + γ 2;p,σ (t), , (5.9) Recalling that Ω p,σ (k) = |k| − ∇E p,σ · k and therefore Ω p ℓ ,σ (k) − Ω p ℓ ′ ,σ (k) = (∇E p ℓ ′ ,σ − ∇E p ℓ ,σ ) · k we expect that the above contributions facilitate the asymptotic decoupling between the following particles: • (5.9): the ℓ-th electron and a photon from the ℓ-th cloud.
Expression (5.11) corresponds to the Coulomb phase and it is easy to show that it behaves as log t for large t and σ = 0. The remaining terms do not have counterparts in many-body quantum mechanical scattering.

Comparison with a rigorous LSZ approach
For N = 1 formula (5.8) is very similar to the single-electron state approximants obtained by Pizzo in [Pi05]. To obtain these latter states from (5.8) one has to make the following modifications: 1. Cell partition: The region of p-integration in (5.8) has to be divided into time-dependent cubes. Suppose, for convenience, that this region is a cube of volume equal to one, centered at zero. At time 1 ≤ |t| the linear dimension of each cell is 1/2 n , where n ∈ N is s.t.
2. Photon clouds: The photon cloud W p,σ (t) from (5.8) should be replaced with the cloud W σ (v j , t), defined in (6.3) below, associated with the cube Γ (t) j containing p and depending on the velocity v j := ∇E p j ,σ in the center of the cube Γ (t) j . Thus one makes the following substitution where v j := ∇E p j ,σ is the velocity in the center of the cube Γ (t) j andk := k/|k|. Clearly, the difference |∇E p,σ − v j | tends to zero as t → ∞ and the size of each cube Γ (t) shrinks to zero, so it should not be difficult to justify this substitution.
3. Phases: The phase γ p,σ (t) from (5.8) should be replaced with the phase defined in (6.5) below. Thus in view of (5.9) and the definition above Ω p,σ (k) := |k| − k · ∇E p,σ , we make the substitution where dω(k) := sin θˆkdθˆkdφˆk is the measure on the unit sphere, and τ → σ S τ = κτ −α , 1/2 < α < 1, is the slow infrared cut-off. (As stated in 5. below, the cut-off σ will tend to zero with t much faster). Since the region of momenta |k| ≥ σ S τ affected by the above change is well separated from the infrared singularity, it is easy to justify the above step using stationary phase arguments.

Renormalized creation operators:
The tentative renormalized creation operator of the electron (5.5)-(5.6) should be replaced with the actual renormalized creation operator, given by (6.7) below. That is, we make the following replacement: where the functionsf m p,σ are given by (5.6) and f m p,σ are wave-functions of the normalized ground states ψ p,σ of the fiber Hamiltonians H p,σ . These latter Hamiltonians are defined via the direct integral decomposition  (k 1 , . . . , k m ) + · · · , (6.10) where the omitted terms are either of order λ or more regular near zero thanf m p,σ , at least in some variables k i . Thus in the weak coupling regimef m p,σ captures the leading part of the infrared singularity of f m p,σ . Further analysis in this direction is needed to justify the substitution (6.6) → (6.7), which takes correlations between the virtual photons dressing the electron into account.
After the above changes, we obtain from (5.8) the following approximating sequencê It was rigorously proven by Pizzo in [Pi05] that the outgoing and incoming single-electron stateŝ Ψ in/out h := lim t→−/+∞Ψh,t exist and are non-zero. Given the above considerations, there is hope for proving convergence of the Faddeev-Kulish type approximating sequence (3.6) in the single-electron case by estimating the norm distance to the Pizzo state (6.12). The most difficult parts will be the partial reversal of the Dollard prescription (5.2) → (5.4) and the step from the tentative to the actual renormalized creation operator of the electron (6.6) → (6.7). A more ambitious strategy consists in proving the existence of the limit of (3.5) directly, e.g. via an application of the Cook's method. Also here it seems necessary to make contact with the renormalized creation operatorb * (p), in order to exploit the key property (6.9). We hope to come back to these problems in future publications.
So far there is no counterpart of the result of Pizzo for two or more electrons. Actually, it is not even clear how the approximating sequence (6.12) should look like in this case. As scattering of two electrons in the Nelson model is currently under investigation [DP13.1, DP13.2, DP17], it is worth pointing out that the Faddeev-Kulish type analysis from previous sections gives a reasonable candidate. In fact, let us simply apply the modifications 1.-5. listed above to the approximating vector (5.8) in the case N = 2. We obtain h,t := e iHt j 1 , j 2 ∈Γ (t) j 1 ×Γ (t) j 2 d 3 p 1 d 3 p 2 e iγ 2;p,σ t (t) e −θ p,σ t (t) × (6.13) × e −iE p 1 ,σ t t e iγ σ t (v j 1 ,t)(p 1 ) h 1 (p 1 )b * σ t (p 1 ) e −iE p 2 ,σ t t e iγ σ t (v j 2 ,t)(p 2 ) h 2 (p 2 )b * σ t (p 2 ) |0 , (6.14) where γ 2;p,σ , θ p,σ are given by (5.10)-(5.12) and may require some small modifications, akin to (6.4)→(6.5). We are confident that the above observations will facilitate mathematically rigorous research on scattering of two electrons in the Nelson model.

Conclusion
In this paper we revisited the Faddeev-Kulish approach to electron scattering in the context of the massless Nelson model. In contrast to the original paper of Faddeev and Kulish, we applied the Dollard formalism according to the rules of the art, without dropping the lower boundary of integration. This led us to a scattering matrix which is meaningful on the usual Fock space of free electrons, but does not commute with the total electron momentum. This latter point was clarified in the later part of our analysis, where we reformulated this scattering matrix in LSZ terms: The lower boundary of integration gives rise to clouds of real photons which always carry some momentum. Furthermore, we checked that the resulting LSZ formula at the one-electron level reproduces single-electron states constructed rigorously by Pizzo, up to minor technical differences. Our observations provide clear-cut mathematical conjectures, which will facilitate rigorous research of N-electron scattering in the massless Nelson model. Our findings may also provide a more solid basis for heuristic discussions of scattering theory in QED, which is a popular topic in current physics literature.