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In asymptotically free theories with collinear divergences it is sometimes claimed that these divergences are cancelled if one sums over initial and final state degenerate cross-sections and uses an off-shell renormalisation scheme. We show for scalar

Infrared (IR) divergences plague the extraction of physical cross-sections in gauge theories. In practice most calculations are restricted to infrared safe quantities. However, according to the Lee-Nauenberg (LN) theorem [

The indistinguishable processes which are summed over depend upon experimental resolutions. In this paper we will consider two such thresholds. There is an energy resolution,

We should also emphasise that, following on the work of Yennie and collaborators [

It is sometimes argued that soft divergences may be removed solely by summing over the soft emissions (BN) while the collinear divergences are separately cancelled by summing over emission and absorbtion processes. If one includes semihard absorbtion, but not soft absorbtion, one is left with

Below we will investigate massless scalar

We will study the process of two particle elastic scattering in

At one loop diagrams such as those in Figure

In the context of this model, the Bloch-Nordsieck trick, summing over final states, does not lead to a collinear finite cross-section and thus one is naturally led to follow Lee and Nauenberg and sum over initial and final states. This introduces diagrams like in Figure

A widespread response [

Of course there are still collinear logarithms in the S-matrix. They now arise through the LSZ correction factor, which we denote by

In our process this introduces four factors, one per external leg, which generate the following collinear logarithmic contribution to the cross-section:

In the spirit of the LN theorem, we now add the contribution of collinear emission from the two outgoing lines and collinear absorbtion on all incoming lines. These contributions are usually taken to be identical. (It is not immediately obvious that this should be the case and we feel that a satisfactory solution to the IR problem would not require such a strong requirement on the initial state. However, for the moment we will also make this assumption.) The four contributions, one per leg, of these real, collinear processes yield

Recall that the LN theorem says that we should sum over

However, the situation is different if a particle is emitted from an incoming line and is parallel to that same line or if an incoming particle parallel to an outgoing line is absorbed by that outgoing line (Figure

This is because the intermediate propagator can now generate a collinear divergence. Such processes might be thought to be distinguishable, and indeed they are for semihard emission/absorbtion, but if such a particle is soft, that is, has an energy below the experimental energy resolution

The result of calculating these four processes (soft collinear emission from incoming lines and absorbtion on outgoing lines) is another collinear divergence in the experimentally indistinguishable cross-section

The above soft-collinear processes include either a (soft) final state particle emerging collinear with an incoming particle or a (soft) particle being absorbed collinear with an outgoing particle. Following the initial paper of Lee and Nauenberg [

It is known since the Lee-Nauenberg paper that such disconnected lines in S-matrix elements can produce a connected interference term in the cross-section (see Appendix D of LN's paper and the discussion in [

The result of calculating the connected interference of emission and absorbtion on each of the four lines with a disconnected line is an additional collinear logarithm

It should be noted that these divergences only arise in diagrams where the emission and absorbtion take place on the same line. Those diagrams with emission on one line and absorbtion on another such as Figure

The next set of diagrams we might consider are in Figure

The above issues are not exclusive to this simple model and neither are they restricted to off-shell schemes. In this section we will consider Coulomb scattering in on-shell massless QED (though retaining a small fermion mass as a collinear regulator) and investigate infrared divergent structures of the form considered above. One-loop virtual diagrams yield soft divergences (apparent as

The BN trick says one should, at the level of probabilities, add the effects of emitting soft photons from the incoming and outgoing fermion lines. This cancels almost all the divergences leaving the tree-level cross-section multiplied by a subleading collinear logarithm

Massless electrons are argued to be indistinguishable from such fermions accompanied by semihard photons inside a cone with angular resolution

As we have repeatedly stressed, the LN argument says that one should add all possible degenerate processes. In particular, LN included the absorbtion of semihard photons on the incoming fermion line but not the absorbtion of soft photons on these lines. However, including only semihard absorbtion means imposing a nonzero lower limit on the emitted energy. Thus this reintroduces the singularities proportional to

Following instead the fully inclusive LN approach for all IR divergences, one should include all possible indistinguishable processes: soft and semihard absorbtion and emission including interference effects from disconnected diagrams. It has been previously shown [

In this paper we have considered an asymptotically free theory,

The origin of these divergent structures is that, for example, emission of collinear particles from an outgoing line should include both soft and semihard particles, that is, one integrates over the energy resolution,

We have pointed out that there are still more processes including disconnected lines which generate connected contributions at the level of the cross-section. However, due again to the masslessness of the fields, there are infinitely many diagrams at fixed order in perturbation theory and the series does not converge.

Such non-convergence has also been seen in QED and we have pointed out in this paper that soft emission/absorbtion also introduce collinear divergences multiplied by energy resolution in gauge theories.

This means that there are different sums of collinear divergences: those with and without a dependence on the energy resolution,

If one recalls that the origin of infrared divergences is that the asymptotic particles are not free [

T. Steele is grateful for the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and thanks the Plymouth high-energy physics group for their generous hospitality.