# An -Matrix for Massless Particles

###### Abstract

The traditional -matrix does not exist for theories with massless particles, such as quantum electrodynamics. The difficulty in isolating asymptotic states manifests itself as infrared divergences at each order in perturbation theory. Building on insights from the literature on coherent states and factorization, we construct an -matrix that is free of singularities order-by-order in perturbation theory. Factorization guarantees that the asymptotic evolution in gauge theories is universal, i.e. independent of the hard process. Although the hard -matrix element is computed between well-defined few particle Fock states, dressed/coherent states can be seen to form as intermediate states in the calculation of hard -matrix elements. We present a framework for the perturbative calculation of hard -matrix elements combining Lorentz-covariant Feynman rules for the dressed-state scattering with time-ordered perturbation theory for the asymptotic evolution. With hard cutoffs on the asymptotic Hamiltonian, the cancellation of divergences can be seen explicitly. In dimensional regularization, where the hard cutoffs are replaced by a renormalization scale, the contribution from the asymptotic evolution produces scaleless integrals that vanish. A number of illustrative examples are given in QED, QCD, and super-Yang-Mills theory.

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###### Contents

## 1 Introduction

The scattering matrix, or -matrix, is a fundamental object in physics. Intuitively, the -matrix is meant to transform an “in” state at into an “out” state at . Unfortunately, constructing an operator in quantum field theory which achieves this projection is far from trivial. To begin, one might imagine that . However, this operator does not exist, even in a free theory. For example, acting on states with energies , matrix elements of this operator would be infinitely-oscillating phases. The proper resolution in quantum mechanics was first understood by Wheeler [Wheeler], who defined the -matrix to project from a basis of metastable asymptotic states (a nucleus) to other states (other nuclei) . This idea was expanded for use in quantum field theory by Heisenberg, Feynman, and Dyson [Heisenberg, Feynman, Dyson1] for calculations in quantum electrodynamics (QED). In modern language, one must factor out the evolution due to the free Hamiltonian to make well-defined.

In the Wheeler-Heisenberg-Feynman-Dyson (henceforth “traditional”) approach, one assumes that in the far past, the “in” state is well approximated with a freely evolving state, i.e. a state that evolves with the free Hamiltonian : as . The interaction is assumed to occur during some finite time interval so that in the far future, the time evolution is again nearly free: as . The state is then related to the in and out states by Møller operators

(1) |

as and so where the traditional -matrix is defined as

(2) |

Unfortunately, this textbook approach has problems too: bare -matrix elements computed this way are both ultraviolet (UV) and infrared (IR) divergent.^{1}^{1}1In this paper, we use “IR divergences" to refer to any divergence that is not of short-distance origin. So IR divergences come from both soft and collinear regions.
Ultraviolet divergences are by now completely understood: they are an artifact of computing -matrix elements using unphysical fields in terms of unphysical (bare) parameters. When -matrix elements are computed with physical, renormalized, fields in terms of physical observable parameters, the UV divergences disappear. IR divergences, however, are not as well understood and remain an active area of research. In theories with massless charged particles, such as QCD, -matrix elements have IR divergences of both soft and collinear origin. Historically, three approaches have been explored to ameliorate the problem: the “cross section method”, the “dressed-state method” and the “modification-of- method”.

The first way of dealing with IR divergences, referred to as the cross section method (following [nelson1981origin, Contopanagos:1991yb]) is the most common. It argues that -matrix elements themselves are not physical; only cross sections, determined by the squares of -matrix elements integrated over sufficiently inclusive phase space regions, correspond to observables. Importantly, in this method, IR divergences cancel between virtual contributions and real emission contributions to different final states. The cancellation in QED was demonstrated definitively by Bloch and Nordsieck [Bloch:1937pw] in 1937. They showed that cross sections in QED (with massive fermions) are IR finite order-by-order in perturbation theory when processes with all possible numbers of final state photons with energies less than some cutoff are summed over. The proof of Bloch-Nordsieck cancellation [Yennie:1961ad, Weinberg:1965nx, Grammer:1973db] crucially relies on Abelian exponentiation [Yennie:1961ad]: the soft singularities at any given order in in QED are given by the exponential of the 1-loop soft-singularities. For theories with massless charged particles, such as QCD, Bloch-Nordsieck fails [Doria:1980ak].

In non-Abelian gauge theories, the theorem of Kinoshita, Lee and Nauenberg (KLN) [Kinoshita:1962ur, Lee:1964is] is often invoked to establish IR finiteness. The KLN theorem states that for any given process a finite cross section can be obtained by summing over all possible initial and final states for processes whose energy lies within some compact energy window around a reference energy , i.e. for a given . In fact, the KLN theorem is weaker and its proof more complicated than required. First of all, energy is conserved, so the cancellation must occur without the energy window. Second of all, one does not need to sum over initial and final states; the sum over only final states for a fixed initial state will do, as will the sum over initial states for a fixed final state. This stronger version of the KLN theorem was proven recently by Frye et al. [Frye:2018xjj]. The proof is one line: for a given initial state, the probability of it becoming anything is 1, which is finite to all orders in perturbation theory. Importantly, both the KLN theorem and its stronger version by Frye et al. generically require the sum of diagrams to include the forward scattering contribution, which is usually excluded from a cross section definition. Unless they happen to be IR finite on their own, the forward scattering diagrams are crucial to achieve IR finiteness. Multiple illustrative examples can be found in [Frye:2018xjj]. If one wants the cross section to be finite when summing over only a restricted set of final states, insights beyond Block-Nordsieck, KLN, and Frye et al. are required, such as those coming from factorization (e.g. [Collins:1988ig, Collins:1989gx, CATANI1989323, Bauer:2000ew, Bauer:2000yr, Beneke:2002ph, Beneke:2002ni, Feige:2013zla, Feige:2014wja]).

In the second approach to remedy IR divergences, the dressed-state method, the -matrix is defined in the traditional way, but it is evaluated between states that are not the usual few-particle Fock states . One of the first proposals in this direction was by Chung [Chung:1965zza], who argued that in QED one should replace single-particle electron states with dressed states of the form with defined as

(3) |

where is a photon polarization vector and is its corresponding creation operator. The idea behind this dressing is that the eikonal factors

While the coherent state approach is in some ways appealing, it has drawbacks. The main problem is that the IR divergences are just moved from the amplitudes to the states. That is, the coherent states themselves are IR divergent and therefore not normalizable elements of a Fock space (although they may be understood as living in a non-separable von Neumann space, as explained in a series of papers by Kibble [Kibble:1968sfb, Kibble:1969ip, Kibble:1969ep, Kibble:1969kd]). The IR divergence problem is therefore still present in this construction; it has merely been moved from the -matrix elements to the states of the theory. Additionally, generalizing beyond massive QED to theories like QCD with collinear divergences and color factors has remained elusive [Catani:1985xt, Gonzo:2019fai]. In particular, no prescription is given for how to go beyond the singular points (zero energy or exactly collinear). For example, the coherent states are sums over particles with different momenta, so they do not have well defined momenta themselves. Is momentum then conserved by the -matrix in the coherent-state basis? How does one integrate over coherent states to produce an observable cross section? These problems are not commonly discussed in the literature. As far as we know, no one has explicitly computed an -matrix element between coherent states. This defect gives the coherent-state literature a rather formal aspect.

The third approach to removing the IR divergences in scattering theory is to redefine the -matrix rather than the states. That the traditional -matrix inaccurately captures the asymptotic dynamics arises already in non-relativistic scattering of a charged particle off a Coulomb potential in non-relativistic quantum mechanics. The standard assumption that particles move freely at asymptotic times is not justified for non-square-integrable potentials, like the Coulomb potential, and leads to ill-defined -matrix elements. In modern language, the -matrix element for non-relativistic Coulomb scattering has the form

(4) |

We see that the leading term of order , corresponding to the first Born approximation, is not problematic: except in the exactly forward limit, there are no divergences in the tree-level scattering process. The logarithmic IR divergence (showing up as a pole in dimensions) first appears in the second Born approximation, where it is seen to be purely imaginary. Moreover, the IR divergent part exponentiates (as do all IR divergences in QED), into the Coulomb phase. Thus, in non-relativistic quantum mechanics, one can apply the cross section ideology even without the inclusive phase space integrals: the cross section for the scattering of a single electron off a Coulomb potential is well defined. However, the -matrix is not.

One of the first attempts to define an -matrix for potentials that are not square-integrable was made by Dollard [dollard1971quantum] in 1971. He noted that when incoming momentum eigenstates are evolved to late times with the Coulomb interaction , there is a residual logarithmic time dependence for large :

(5) |

The intuition for this form is that at large , the particle moves approximately on a classical trajectory with , which gives the logarithmic dependence on when integrated up to infinity. While the is removed by Wheeler’s factor, the other term is not and persists to generates the divergences in the -matrix. Dollard then proposed to replace to the factor with a factor, with defined with exactly the logarithmic time dependence needed to cancel the time dependence in Eq. (5). He then showed the a modified -matrix, defined with his asymptotic Hamiltonian replacing , exists for Coulomb scattering.

When the electron is relativistic, the IR divergence in the second Born approximation has real part that does not cancel at the cross section level. So first-quantized quantum mechanics is insufficient to produce an IR-finite cross section: QED is needed. Faddeev and Kulish [Kulish:1970ut] combined the aforementioned work of Chung in QED and Dollard’s in non-relativistic quantum mechanics. They observed that in QED, infrared divergences have both a real part (as Chung observed) and an imaginary part (the relativistic generalization of the Coulomb phase). These can be combined into a modified -matrix of the form

(6) |

where

(7) |

corresponds to the Coulomb phase (compare to the dependence in Dollard’s form, Eq. (5)). The factor is similar to Chung’s in Eq. (3) but with a power-expanded phase, and annihilation operators included as well:

(8) |

where

(9) |

is the electron-number operator. Acting on states, it pulls out the direction of each fermion and multiplies the contribution by 1 for electrons or -1 for positrons: . Faddeev and Kulish proceed to argue that has finite matrix elements between coherent states in QED. They argued that one should include the phase factors in a redefinition of the -matrix while including the factors in dressing the states. Although there are some suspicious orders-of-limit and signs in Faddeev and Kulish’s paper (see [Contopanagos:1991yb]), we believe their construction is essentially valid. Indeed, one goal of our paper is to translate this classic work in QED to modern language. As we will show in Section 2.3, both the real and imaginary parts in the factor are reproduced by the action of a single Wilson line.

In the 50 odd years since Faddeev and Kulish’s work, there has been intermittent progress on generalizing the coherent state construction from QED to non-Abelian theories. Early work [Catani:1985xt, Butler:1978rd, Nelson:1980qs] focused on trying to use coherent states to salvage the Bloch-Nordsieck theorem, following the QCD counterexamples given by Doria et al. [Doria:1980ak, Andrasi:1980qw]. Although soft divergences in QCD do not exponentiate into a compact form as they do in QED [Gatheral:1983cz, Frenkel:1984pz], they still have a universal form and factorize off of the hard scattering [Collins:1988ig, Feige:2014wja]. Using this observation, it has been argued using a frequency-ordered formalism that dressed states can be constructed between with -matrix elements in QCD are soft-finite [Catani:1985xt, Giavarini:1987ts]. Collinear divergences and the soft-collinear overlap in gauge theories were explored in [curci1978mass, Havemann:1985ra, DelDuca:1989jt, Contopanagos:1991yb]. An explicit check of the dressed formalism was performed by Forde and Signer [Forde:2003jt] who used explicit cutoffs to separate the regions and showed that the cross section for jets can be reproduced at leading power at order through finite -matrix elements. Ref. [DelDuca:1989jt] argued that if soft-collinear factorization holds in QCD, then the dressed state formalism should allow one to construct a finite -matrix in QCD to all orders. Collinear factorization was proven diagrammatically at large a decade later [Kosower:1999xi] and a full proof of collinear factorization and soft/collinear factorization for QCD to all orders in perturbation theory was given in [Feige:2013zla, Feige:2014wja], inspired by [Collins:1988ig, Collins:1989gx, CATANI1989323, Bauer:2000yr, Bauer:2002nz, Beneke:2002ph]. One goal of the current paper is to combine these various insights to provide, for the first time, an explicit construction of an IR-finite -matrix for QCD.

In all of this literature, there are a number of unresolved issues. First, there are essentially no results about the finite parts of a finite -matrix. Showing the cancellation of the IR singularities is one thing, but to evaluate one needs to deal with complications of momentum conservation, cutoffs, UV divergences, and to actually be able to compute the resulting integrals. A prescription to determine the finite parts of the modified -matrix is required if we are explore the -matrix’s properties. While some authors have suggested criteria such as that the dressed states should be gauge [Bagan:1999jf] or BRST invariant [catani1987gauge], or have asymptotic charges [Kapec:2017tkm, Strominger:2017zoo], or be compatable with decoherence [Carney:2017jut, Carney:2018ygh], the necessity of these choices is unclear. Certainly nothing goes wrong at the level of cross sections if we proceed using the cross section method. After the finite part is fixed, one must further explain how to relate modified -matrix elements to observables: what is the measure for integration over momenta in the von Neumann space of dressed states (if one goes that route)? To agree with data, the predictions had better reduce to what one calculates using the IR-divergent , but how that will happen in any of the approaches to dressed states is rarely discussed. In this paper, we attempt to raise the bar for constructing a finite -matrix by providing a motivated, calculable scheme, and give explicit expression for -matrix elements and observables in a number of cases in QED, QCD, and super Yang-Mills theory.

The organization of this paper is as follows. We start by motivating and defining a “hard" -matrix in Section 2. We show how to get finite answers, and connect to the previous work on QED using dressed states in Section 2.1. In Section 2.2, we discuss how to compute observables and show that the same predictions for infrared-safe differential cross sections results from as from the traditional . In Section 2.3 we connect our construction to the expressions of Faddeev and Kulish in QED. We then proceed to explicit calculations, working out the Feynman rules and some toy examples in Section 3. In Section 4 we demonstrate IR finiteness in the process in QED using cutoffs, and illustrate the relative simplicity when pure dimensional regularization is invoked. In Section 5 we discuss including the connection to the Coulomb phase and the Glauber operator as well as an explicit calculation of the thrust distribution, both exactly at NLO and to the leading logarithmic level using the asymptotic interactions. Section 5.2 makes explicit some of the general observations about exclusive measurements from Section 2.2. Section 6 gives some examples in super Yang-Mills theory, connecting to observations about remainder functions, renormalization and subtractions schemes. Concluding remarks and a summary of our main results are given in Section 7.

## 2 The hard -matrix

The intuition behind scattering is that one starts with some initial state, usually well approximated as a superposition of momentum eigenstates, which then evolves with time into a region of spacetime where it interacts, and then a new state emerges. The -matrix is meant to be a projection of this emergent final state on to a basis of momentum eigenstates. For scattering off a local (square-integrable) potential, this picture works fine. The -matrix is then defined as as in Eq. (2) with the Møller operators defined in Eq. (1). However, when the interactions cannot be confined to a finite-volume interaction region, as in Coulomb scattering or in a quantum field theory with massless particles, this picture breaks down: the states at early and late times continue to interact, so the momentum-eigenstate approximation is no longer valid.

As mentioned in the introduction, the simplest example with the traditional definition of breaks down is for non-relativistic scattering off a Coulomb potential. In this case, the Møller operators acting on momentum eigenstates generate an infrared divergent “Coulomb” phase. While the infrared divergence is a problem for a formal definition of the -matrix, it is not a problem for cross section calculations that depend only on squares of -matrix elements. In relativistic Coulomb scattering, or in QED, has both an infrared divergent Coulomb phase and an infrared divergent real part. A convenient feature (Abelian exponentiation [Yennie:1961ad]) of QED is that a closed form expression is known for the IR-divergent contribution to all orders in perturbation theory for any process. Indeed, the 1-loop divergences are given by where the cusp-anomalous dimension is (see [Chien:2011wz])

(10) |

with the cusp angle defined by and and are the 4-velocities of the incoming and outgoing electrons. To all orders, the IR divergences exponentiate as [Korchemsky:1987wg]. Thus, it is possible to factor out IR-divergent parts from the -matrix and redefine a new -matrix that is IR-finite order-by-order. This was done by Chung and Faddeev and Kulish, as discussed in the introduction. Note that the non-relativistic limit corresponds to in which case becomes the purely imaginary Coulomb phase.

When the charged particles are also massless, as in QED with , new IR divergences appear associated with collinear divergences. Soft-collinear divergences appear as double IR-poles. Indeed, in the limit, becomes lightlike, so . At large in the cusp angle diverges linearly with , so the -matrix now has double, poles. In QCD, or other non-Abelian theories, the cusp angle gets corrections beyond one loop and the IR divergences do not exponentiate into a closed form expression [Gatheral:1983cz, Frenkel:1984pz, Gardi:2013ita]. These complications have made it difficult to come up with a complete formulation of an IR-finite -matrix in general quantum field theories [DelDuca:1989jt, Contopanagos:1991yb, Forde:2003jt].

The approach we take in this paper is to construct an -matrix that is IR finite by replacing the free Hamiltonian in the definition of the traditional -matrix with an appropriate asymptotic Hamiltonian . That is, we can define new hard Møller operators

(11) |

and a hard -matrix as

(12) |

Ideally, we would want to choose so that the hard Møller operators exist, as unitary operators on the Hilbert space. Proving their existence is challenging, as even in a mass-gapped theory, where we can take , they do not exist by Haag’s theorem [Haag:1955ev]. From a practical point of view, we can be less ambitious and aim to choose so that the hard -matrix is free of IR divergences at each order in perturbation theory. If this was our only criteria, we could choose , so that .

A better criteria for defining is that, in addition to capturing long-distance interactions, the asymptotic Hamiltonian should be defined so that the asymptotic evolution of the states is independent of how they scatter. It is possible to define this way due to universality of infrared divergences in gauge theories. Using factorization [Collins:1988ig, Collins:1989gx, CATANI1989323, Bauer:2000ew, Bauer:2000yr, Beneke:2002ph, Beneke:2002ni, Feige:2013zla, Feige:2014wja], the soft and collinear interactions can be separated from the hard scattering process: Any -matrix element in gauge theories can be reproduced by the product of a hard factor, collinear factors for each relevant direction, and a single soft factor. See [Feige:2014wja] for a concise statement of factorization at the amplitude level.

In order to exploit factorization, we employ methods developed in Soft-Collinear Effective Theory (SCET). The theory provides a systematic power expansion of the QED or QCD Lagrangian, and reproduces all infrared effects. The leading power Lagrangian in SCET is [stewart2013lectures, Becher:2014oda]

(13) |

where and are soft and collinear labels respectively and the collinear covariant derivative is

(14) |

The last term describes Coulomb or Glauber gluon interactions [Rothstein:2016bsq] (see also [Schwartz:2017nmr]). Pedagogical introductions to SCET can be found in [Becher:2014oda, stewart2013lectures, Schwartz:2013pla].

We define the asymptotic Hamiltonian to be the SCET Hamiltonian appended with free Hamiltonians for massive particles. The hard -matrix is then defined in terms of using Eqs. (11) and (12).

Although the SCET Lagrangian looks complicated and non-local, much of the complication comes from being careful to include only leading-power interactions. In principle, for a theory to be valid at leading power, one could include any subleading power interactions one wants. Exploiting this flexibility, the collinear interactions in can be replaced simply with the full interactions of QCD: . The soft interactions, from the

In practice, when computing elements we will not use the explicit and cumbersome interactions in . Instead, we will take the method-of-regions approach [Beneke:1997zp, Becher:2014oda]. We start with a particular Feynman diagram and then expand to leading power based on the collinear or soft scaling associated with particles involved. In a sense, this is the most straightforward and foolproof way to compute amplitudes. Numerous examples are given in subsequent sections.

We also, in accord with the general principles of the method of regions, do not impose any hard cutoffs on the momenta of the soft and collinear particles that interact through . Imposing cutoffs is helpful for demonstrating explicit IR-divergence cancellation, and some examples are provided in Section 4.1. However, cutoffs generally lead to very difficult integrals, and moreover they break symmetries like gauge-invariance that we would like to respect. More precisely, it is only the finite, cutoff-dependent remainder terms that may depend on gauge – the IR divergence cancellation mechanism is gauge-independent. Since the cutoff-dependent finite parts are unphysical anyway, it is not a problem that they are also gauge-dependent. In general, however, the whole framework with cutoffs is rather unwieldy.

When using pure dimensional regularization, the diagrams involving interactions will lead to scaleless integrals. These integrals are both UV and IR divergent. The IR divergences cancel in other contributions to (as we will provide ample demonstration), but the UV divergences must be removed through renormalization. As a consequence, in pure dimensional regularization, -matrix elements are not guaranteed to be independent of renormalization scheme. Indeed, they are generally complex and will depend on the scale at which renormalization is performed. The -matrix is not scale independent: , in contrast to which does satisfy the Callan-Symanzik equation . This is unsatisfying, but not unsettling, as elements are not themselves observable. (To be fair, if -matrix elements are IR divergent, it is not clear what it means to say they are scale-independent). In any case, one should think of like one thinks about the strong coupling constant in . While is not observable, it is still an extraordinarily useful concept. The running coupling indeed encodes qualitatively and quantitatively a lot of important physics, such as unification and confinement. As with , when is used to compute an observable, the scale dependence will cancel. We demonstrate that in general in Section 2.2, and provide an explicit example in Section 5.

### 2.1 and dressed states

The usual way of calculating -matrix elements in perturbation theory is to work in the interaction picture, where one expands the interactions in terms of freely evolving fields. The propagators for free fields have a relatively simple form, and -matrix elements then become integrals over these propagators. One might try to work out Feynman rules for analogously, in an asymptotic interaction picture. Then propagators would correspond to non-perturbative Green’s functions for the soft and collinear fields in , including all of their interactions. Unfortunately, finding a closed-form expression for these propagators is not possible. In any case, it is not necessary, since if we want to work perturbatively in the coupling constants, we must do so consistently in both and .

To proceed, we note that the hard S-matrix can be written suggestively as

(15) |

where

(16) |

are asymptotic Møller operators and are the usual Møller operators. Inserting complete sets of states lets us write hard -matrix elements between a Heisenberg picture out-state and a Heisenberg picture in-state as

(17) |

Here the integral is over complete sets of Fock-space states and . The hard scattering matrix elements are written as a product of three terms. The middle term is the traditional -matrix and the outer terms correspond to evolution with the asymptotic Møller operators. The Feynman rules for these contributions closely resemble those of time-ordered perturbation theory and are derived in Section 3.1 below.

Another interpretation of the hard matrix elements can be obtained by defining dressed states as

(18) | ||||

Then,

(19) |

i.e. the matrix elements of the hard -matrix are equivalent to matrix elements of the traditional -matrix between dressed states. This connection was made in the context of QED in [Contopanagos:1991yb]. The role of the asymptotic evolution can then be viewed as transforming the in-state defined at into a dressed state at that scatters in the traditional way (with ). The role of dressed states is illustrated in Figure 1.

The dressed states and are not normalizable elements of the Fock space that and live in. Indeed, if we expand them perturbatively their coefficients in the Fock space basis contain infrared divergent integrals. For example, starting with an state

(20) |

in QED, the asymptotic Møller operator can add or remove soft photons with each factor of the coupling . Up to order the dressed state will be a superposition of the leading order state, states and Fock states. Explicitly,

(21) | ||||

Let us make a few observations about these dressed states. First, note that the Fock states being added have different 3-momenta. When has exactly zero momentum (the case almost exclusively considered in the literature), momentum is conserved. But if one really wants to take these dressed states seriously, must be allowed to have finite energy too, and then is not a momentum eigenstate.

Second, the coefficient at order is a UV and IR divergent integral. The IR divergence is expected; it is exactly the IR divergence that cancels the IR divergence in elements of to make elements of IR finite. Nevertheless, it makes hard to deal with as a state. The divergence requires an excursion from the traditional Fock space to a von Neumann space [Kibble:1968sfb, Kibble:1969ip, Kibble:1969ep, Kibble:1969kd]. The UV divergence is due to the fact a soft momentum is not sensitive to any hard scale in the problem, so there is no natural cutoff on the integrals. One could, of course, put in explicit hard cutoffs on the soft momenta, however, it is easier to simply renormalize the UV divergence by rescaling .

Third, it is not each separate electron that is being dressed. Rather it is the combination. Indeed, the IR divergence in the example above comes from loops connecting the two electrons. These loops are critical to cancelling the IR divergences in . In Chung’s original formulation (cf. Eq. (3)), a picture can be sketched for a coherent state as an electron moving with a cloud of photons around it. But this picture is too naive: the cloud depends on all the charged particles. This is even clearer in QCD, where the soft factors come with non-Abelian color matrices so one cannot rely on the crutch of Abelian exponentiation to move the dressing factors from state to state at will. A discussion of additional complications in QCD and the failure of Bloch-Nordsieck mechanism, can be found in [Catani:1985xt].

In conclusion, although the dressed state picture fits in naturally with the construction of we have presented, we doubt that thinking of the dressed states as physical states will ultimately be profitable.

We emphasize that for the purpose of having finite matrix elements, neither the in- and out-states and , nor the dressed states and , need to be eigenstates of the asymptotic Hamiltonian. In the examples to follow we will take and to be eigenstates of the free momentum operator with a finite number of particles, but in principle they can be taken to be any sensible linear combination of states in the relevant Hilbert space, i.e. with finite coefficients, in contrast to the usual coherent states which are an infinite linear superposition of Fock state elements. The -matrix elements between any such states are always finite.