PACs

December 29, 2018

Infrared finiteness of theories with bino-like

[0.3cm] dark matter: II. Finite temperature

[0.5cm] Pritam Sen111pritamsen@imsc.res.in and D. Indumathi222indu@imsc.res.in

[0.2cm] The Institute of Mathematical Sciences, Chennai and Homi Bhabha National Institute, Mumbai

[0.5cm] Debajyoti Choudhury333debajyoti.choudhury@gmail.com

[0.2cm] Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India

[1cm]

Abstract

Models incorporating moderately heavy dark matter (DM) typically need charged (scalar) fields to establish admissible relic densities. Since the DM freezes out at an early epoch, thermal corrections to the cross sections can be important. In a companion paper [1] we established that, at zero temperature, the infrared (IR) divergences, accruing from both fermion-photon and scalar-photon interactions cancel to all orders in perturbation theory. Here, we study the IR behaviour at finite temperatures, which potentially contains both both linear and sub-leading logarithmic divergences. We prove that the theory is IR-finite to all orders with the divergences cancelling when both absorption and emission of photons from and into the heat bath are taken into account. While 4-point interaction terms are known to be IR finite, their inclusion leads to a neat exponentiation. The calculation follows closely the technique used for the zero temperature theory.

PACs

: 11.10.−z, 11.10.Wx, 11.15.−q, 11.30.Pb, 12.60.−i, 95.35.+d

1 Introduction

Astrophysical and cosmological observations spanning over a multitude of length scales, beginning with rotation curves, lensing, on to galactic and cluster collisions, and finally the origin of large scale structure in the Universe as also the power spectrum of the cosmic microwave background radiation, all point to the existence of a mysterious Dark Matter (DM) that overwhelms ordinary matter in the Universe. And while all the evidence so far has come from the study of gravitational effects, simple modifications in the theory of gravity have, so far, failed to account for all the observations. The conundrum, apparently, can be resolved only by postulating a particulate DM that, of necessity, must be immune to strong interactions and, preferably, neutral111Although milli-charged DM is still allowed, such models tend to be somewhat contrived. The inclusion of this possibility, though, would not change the central thesis of this paper, other than adding a layer of complication to the calculations..

If we posit that the DM particle has no interactions at all with the Standard Model (SM) particles, other than the gravitational, there would be virtually no hope of ever observing it directly in a controlled experiment. Consequently, it is assumed that the DM must have sufficient interactions with at least some of the SM particles, presumably with a strength comparable to weak interactions, or at worst, a couple of order of magnitudes weaker. Such an assumption has a further ramification. A standard assumption in explaining the evolution of the Universe is that all particles—whether those within the SM, or the DM—were created during the (post-inflation) reheating phase. The subsequent number densities are supposed to have been determined by the expansion of the Lemaitre-Friedmann-Robertson-Walker Universe, augmented by a set of coupled Boltzmann equations that are operative when the particles are in equilibrium. If the DM particle does have such interactions, then it can stay in equilibrium with the SM sector via interactions of the form,

(1)

where is a particle corresponding to some arbitrary SM field. This equilibrium phase would last until the interaction rate falls below the Hubble expansion rate. With the Universe cooling as it expands, the DM must fall out of equilibrium by the time its mass exceeds the temperature. With large scale structure formation liable to be destroyed in the presence of a dominantly hot DM (i.e., one where the DM decoupled well before the temperature fell down to ), the favourite scenario is that of a dominantly cold DM (i.e., one which had become nonrelativistic at the decoupling era). Interestingly, if the interactions governing Eq. 1 typically have a strength comparable to the weak interaction, then for a wide range of masses, GeV, the relic density is of the required order.

Given the high precision to which the DM contribution to the energy budget has been measured by the wmap [2] and, subsequently, the planck [3] collaborations, an order of magnitude estimation is no longer acceptable, and precise predictions need to be made. Indeed, the measurements have imposed rather severe constraints on several well-motivated models for the DM, to the extent of ostensibly even ruling out some of them. Such drastic conclusions, though, need to be treated with caution, for many of the theoretical estimates have resulted from lowest-order calculations alone. More importantly, the effect of non-zero temperatures are rarely considered. Together, these effects can alter the predictions to a significant degree.

Initial efforts to include thermal corrections to the relevant processes were made in Ref. [4], wherein it was shown, albeit to only the next-to-leading order (NLO), that infra-red divergences (both soft and collinear) cancel out in processes involving both charged scalars and charged fermions. However, no proof to all orders exists so far and we provide one in this paper. In an earlier paper [1], referred to henceforth as Paper I, we had discussed, in detail, the infra red (IR) behaviour, via factorisation and resummation, of theories associated with bino-like DM particles, inspired by supersymmetric (MSSM) models. In doing this, we proved the IR finiteness of such theories to all orders of perturbation theory, at zero temperature. In this second paper, we set out the analogous proof of the IR finiteness of this model, but now at non-zero temperatures. While IR finiteness is important to establish the consistency of the theory, the inclusion of such corrections is also crucial for an accurate calculation of the relic density of the DM.

Although it might seem that a bino-like DM candidate is a very specific choice, it actually captures the essence of a wide class of models. Whereas the MSSM spectrum would include, apart from the SM particles, the entire gamut of their supersymmetric partners, only a handful of them play a significant role in determining the relic density. Apart from the DM candidate itself, these are some charged scalars (typically, close to the DM in mass), and occasionally, depending on the details of the supersymmetry breaking scheme, others as well. And while the DM itself is a linear combination of the bino, the neutral wino and the two neutral higgsinos, for a very large class of supersymmetry breaking scenarios, the higgsino mass parameter is much larger than the soft terms () for the gauginos, thereby suppressing the higgsino component to negligible levels. And since wino-bino mixing is pivoted by , a large value for the latter also suppresses the wino-component in the DM. The assumption of a bino-like DM further simplifies the calculations as we may safely neglect additional diagrams, e.g., with -channel gauge bosons or Higgs222For pure binos, such couplings arise only at one-loop level, and are of little consequence.. It should also be appreciated that no new infrared divergence structures would appear even on the inclusion of such additional mediators333The only caveat to this is presented by the diagrams involving the , as photons could also radiate off the latter. The structure of the ensuing IR divergences, however, are quite analogous to those that we would encounter here, and can be analysed similarly.. In other words, restricting ourselves to the particular case of the bino does not represent the neglect of subtle issues while allowing for considerable simplifications, both algebraic and in bookkeeping.

The Lagrangian density relevant to this simplified scenario is given by an extension of the Standard Model containing left handed fermion doublets, , with an additional scalar doublet, , namely the supersymmetric partners of , along with the singlet Majorana fermion which is the dark matter candidate. We have,

(2)

We assume that the bino is a TeV scale DM particle so that freeze-out occurs after the electro-weak transition; hence, only electromagnetic interactions are relevant for the IR finiteness at these scales444Such an approximation is a very good one for . For , again, one could proceed in an entirely analogous fashion, replacing the photon by the entire set of four electroweak gauge bosons. For an intermediate mass bino, on the other hand, the analysis is rendered much more complicated and is beyond the scope of this paper.. In other words, the interacts only with fermions and sfermions, and , and not with the photon. Thus, only the charged (s)fermion interactions with are shown in Eq. 2 since it is the resummation of the radiative photon diagrams which are of interest here.

The simplest process for DM annihilation (or, equivalently, DM scattering off a SM particle), as driven by the Lagrangian of Eq. 2, is illustrated in Fig. 1. Higher order electromagnetic corrections to such diagrams involve, apart from real photon emissions from either or , virtual photon exchanges as well. The latter include

  1. the two vertices of the virtual photon being on different fermion lines,

  2. both vertices of the virtual photon on either of the fermion lines,

  3. both vertices on the intermediate scalar line, and

  4. one of the vertices on either of the fermion lines and the other on the scalar line.

Figure 1: A typical dark matter annihilation/scattering process.

Such contributions can be calculated in a real time formulation of the thermal field theory [5]. In Ref. [6], the eikonal approximation was used and the interaction of photons with a semi-classical current was analysed within the framework of thermal field theory. In Ref. [7], 1-loop corrections to thermal QED were computed for both fermions and scalars. It was found that the finite temperature mass shift for scalar QED when the temperature is less than the scalar mass () is identical to the corresponding fermion case as is the IR divergent piece of the wave function and vertex renormalisation constants. However, the contribution to the plasma screening mass was twice that for the fermion loop due to the difference in the form of the thermal distributions (boson versus fermion). In Ref. [8], dynamical renormalisation group resummation of finite temperature IR divergences to 1-loop was done to study relaxation as well as out-of-equilibrium damping in real time for a scalar thermal field theory. This was applied to study IR divergences in scalar QED (to the lowest order in the hard-thermal loop (HTL) resummation). It was found that the infrared divergences in this theory are similar to those found in QED and in lowest order in QCD. Results in thermal scalar QED have also been applied to study Schwinger pair-production [9].

Here we use a similar approach to address the issue of IR finiteness of the thermal field theory of dark matter, thereby combining and extending the results of the earlier work on thermal fermions [10] with the zero temperature results for charged scalars as discussed in Paper I [1].

In Section 2, we set up the real-time formulation of the thermal field theory corresponding to the Lagrangian of Eq. 2 and use the approach of Grammer and Yennie [11] (henceforth referred to as GY), by defining the so-called and photon insertions. In Section 3, we show that the photon insertions contain the IR divergence and that the photon insertions are IR finite; the latter is the most complex part of the calculation. In Refs. [6, 10, 12] it was shown that pure fermionic thermal QED has both a linear divergence and a logarithmic subdivergence in the infra red compared to the purely logarithmic divergence encountered in the zero temperature theory, owing to the nature of the thermal photon propagator. The same is true here as well. We factorise and exponentiate the divergences and show that they cancel order by order between virtual and real photon contributions (the latter include both emission and absorption terms) and hence prove the IR finiteness of thermal scalar QED to all orders. The IR finiteness of thermal fermionic QED has already been proved in Ref. [10]. We use these two results to finally prove the IR finiteness of the dark matter thermal field theory in Section 4. Section 5 contains the discussions and conclusions. The appendices are used to set up the Feynman rules (Appendix A) for thermal field theories and to list some useful identities (Appendix B) that are used to factorise the photon contributions. Appendix C shows the details of the factorisation of the photon insertions while Appendix D shows that the photon insertions are IR finite in a case by case manner.

2 Real-time formulation of the thermal field theory with dark matter

IR divergences in virtual corrections are known to cancel against those from additional real emissions from the lower order diagrams wherein the real photons are so soft or collinear that they cannot be distinguished as separate entities. We begin by briefly outlining the approach of GY to address the IR finiteness of zero temperature QED with fermions, and subsequently generalize it here.

Starting with an arbitrary -order graph, another photon, real or virtual, is added, considering all symmetric permutations, that is, all possible insertions. GY rewrote the photon field so that the photon propagator term can be re-written as a sum over and photon contributions:

(3)

Here (to be defined later below) depends on the momentum of the inserted photon as well as the momenta , where the final and initial vertices are inserted, and is designed so that the so-called -photon terms in the matrix element with photons are IR finite while the -photon terms contain all the IR divergent terms. As Paper I shows, such a separation, and the consequent isolation of singularities, could be performed here as well, despite a complication due to the presence of the 4-point scalar-photon vertex.

In the case of a thermal field theory, there is an additional complication which can be understood in a real-time formulation [5] where the integration in the complex time plane is over a contour that includes the temperature, chosen so that correct thermal averages of the -matrix elements [13] are obtained. The fields satisfy the periodic or anti-periodic boundary conditions,

(4)

where the sign corresponds to boson and fermion fields respectively and , where is the temperature of the heat bath. This results in the well-known field-doubling, so that fields are of type-1 (physical) or type-2 (ghosts), with propagators acquiring matrix forms. Only type-1 fields can occur on external legs while fields of both types can occur on internal legs, with the off-diagonal elements of the propagator allowing for conversion of one type into another.

In particular, the photon propagator corresponding to a momentum can be expressed (in the Feynman gauge) as,

(5)

where the information on the field type is contained in ; see Appendix A for its definition. Note that the factor occurs in all components of the thermal photon propagator, enabling a separation into and type photons, just as before, with a similar definition for , viz.555Slightly different from that used by GY, this definition is more suitable for thermal field theory [10].,

(6)

Note that is not a gauge transformation because of its dependence. In addition, the scalar and fermion field propagators also assume matrix forms (see Appendix A for details) with the (11) and (22) terms having both and finite temperature contributions. Finally, the vertices, both 3-point and 4-point ones, are modified in the thermal theory. Details are again in Appendix A; we only note here that all the fields at a given vertex must be of the same type. We can therefore apply the technique of GY to the case of thermal fields in equilibrium with a heat bath at temperature . There are two major differences in this case, firstly, that the relevant part of the thermal photon propagator is proportional to,

(7)

where the first term corresponds to the contribution and the second to the finite temperature part. In contrast to the fermionic number operator, viz.,

(8)

the bosonic number operator contributes an additional power of in the denominator in the soft limit, since

(9)

Hence, it can be seen that the leading IR divergence in the finite temperature part is linear rather than logarithmic as was the case at zero temperature. Consequently, there is a residual logarithmic subdivergence that must also be shown to cancel at finite temperatures, thus making the generalisation to the thermal case non-trivial.

Secondly, it turns out that the inclusion of thermal matter fields adds another layer of complexity to the analysis, since, not only is the propagator structure now different from the zero temperature case, but, in contrast to the case of fermions, the number operator corresponding to charged scalars is bosonic and hence can potentially give rise to divergences as well.

In summary, the major differences between this and the earlier works are as follows.

  1. The scalar theory has additional vertices, including the 4-point seagull vertices; see Fig. 7 in Appendix A. This contributes additional terms to both the and photon insertions compared to the thermal theory with fermions only.

  2. The thermal theory has additional field types; in particular, the thermal charged scalar legs add more complications compared to the results with thermal fermions.

The object of this paper is to obtain an analogous proof of IR finiteness for a thermal field theory of dark matter interacting with both charged scalars and charged fermions with these additional complications. The dark matter particle is charge-neutral, while the thermal field theory of charged fermions has already been shown to be IR finite. We therefore focus first on the charged scalar sector.

3 IR finiteness of the thermal theory of charged scalars

We briefly review the notation and list the contributing higher order corrections to the process , as discussed in Paper I. Consider the insertion of an additional photon into a graph containing (real or virtual) photon vertices on charged (external) lines at vertices and . The out-going and in-coming charged lines are separated by the (arbitrary) hard vertex, ; see Fig. 2. We begin with the insertion of virtual photons.

Figure 2: Schematic of an order graph of , with vertices on the leg and on the leg, . Here labels the special but arbitrary photon–scalar vertex.

3.1 Insertion of virtual photons

Consider the insertion of one of the virtual photon vertices, say , on an external line. As per the Feynman rules listed in Appendix A, there can be two types of vertices, with one or two photon lines at each vertex, corresponding to 3-point or 4-point vertices respectively. (In addition, these fields carry a thermal index, , depending on the field type at the vertex). Hence there are two types of photon insertions possible; one where the insertion is at a new vertex, forming a new 3-point vertex, or one where the photon is inserted on an already existing vertex, thus forming a 4-point vertex. The total set of all possible insertions of the photon on the line can be grouped into sets having the new vertex as a 3-point or 4-point vertex, just as in Paper I, and as shown in Figs. 3 and 4 respectively. In contrast, note that only the set of graphs shown in Fig. 3 contributes if the -leg is a fermion line.

Figure 3: Set of diagrams showing all possible trilinear insertions of a virtual photon at vertex on the leg of a scalar/fermion.
Figure 4: Set of diagrams showing all possible insertions of a virtual photon at vertex which is one of the already existing vertices on the leg of a scalar particle, thus giving rise to a 4-point vertex. Analogous diagrams for fermions do not exist.

As shown in Paper I, it is convenient to group 3- and 4-point vertices to obtain “circled vertices”: for instance, consider the insertion of the vertex to the right of a generic vertex or at the vertex . The corresponding two diagrams are shown in Fig. 5 and the contribution from the sum of these is shown in the figure as a circled vertex and denoted by .

Figure 5: Combining sets of two possible insertions of the virtual photon at vertex on the leg to give a single circled vertex, ; see text for details. The photon lines have been suppressed for clarity.

The contributions from the leg, to the two terms on the LHS of Fig. 5 were shown to simplify in Paper I due to the structure of the scalar propagators and were shown to be expressible as a difference of two terms. In the thermal case, the propagators contain more than just the part and appear more complex. However, they satisfy generalised identities, analogous to the zero temperature case, as shown in Appendix B, which can be used to simplify and factor these contributions to obtain a similar result. Retaining only the factor in the part of the photon propagator (the factor will be similarly included when the other vertex is inserted on the leg, and is an overall factor), and omitting other terms in the photon propagator for clarity, we have (denoting a scalar propagator from vertex of thermal type to vertex with thermal type as ),

(10)
(11)

Here denote the thermal indices, , with the thermal index of the inserted photon at the vertex . There should be no confusion between the Lorentz index and the thermal index. Notice that apart from an overall factor of as in the case, all the thermal powers of match and there is no sign dependence between the contributions of the two terms, which is independent of the thermal field type. Hence the two can be combined to give,

(12)

This is the thermal generalisation of the corresponding result obtained in Paper I at . This combination of differences of terms from photon insertion helps in pair-wise cancellation and hence simplification and factorisation of the IR divergent part even at finite temperature.

We now apply this simplification to all sets of diagrams. We have the following possibilities:

  1. The inserted photon vertices are on different external lines, in-coming and out-going.

  2. The two vertices are on the same lines.

We will address them one by one. All details are to be found in Appendix C.

3.1.1 The two insertions on different lines

The case where the vertices are on different lines is straightforward. Start with a lower order diagram that contains only 3-point vertices; we will relax this condition later. Terms cancel pair-wise as shown in detail in Appendix C, leaving behind just one term that is proportional to the order matrix element, as in the scalar/fermionic case at .

The calculation can be extended to the case when there are both 3- and 4-point vertices in the -photon graph. Graphs with the same number of photons rather than the same number of vertices are grouped together, so that the overall charge factors (powers of ) are the same for the entire set of diagrams. Hence the corresponding -photon graph has fewer than vertices, and in fact will have vertices if of the photons participate in a 4-point vertex.

For such diagrams there is an additional constraint since it is obvious that the additional photon cannot be added at an already existing 4-point vertex. It turns out that the factorisation outlined above and detailed in Appendix C goes through in an identical fashion for such graphs as well: the simplification and cancellation occurs between the same sets of graphs as before and the fragment(s) of matrix element arising from the 4-point vertices in the lower order graph are simply carried through and do not spoil the result.

A similar result is obtained when the vertex of the virtual photon is inserted on the (distinct) leg. Hence, the total contribution from the insertion in all possible ways of an -photon (contributing a factor ) into a set of graphs with photons containing an arbitrary number of 3- or 4-point vertices, is given by,

(13)

and hence is proportional to the lower order matrix element .

The major difference between the case and the thermal case is the presence of the thermal indices. Crucially, there are additional delta functions, and , arising from matching the field types at the special scalar-photon vertex . Since the hard photon is observable, so , as well; hence the photon thermal propagator is constrained to be of type alone. This is a crucial requirement for the cancellation to occur between real and virtual photon contributions to the lower order diagram.

3.1.2 Both insertions on the leg alone

Just as at zero temperature, the case where both vertices of the photon are inserted on the leg is more complex due to the presence of 4-point tadople vertices. Again, double-counting is avoided by insisting that the vertex is always to the right of the vertex. Various contributions can be combined in order to simplify the calculation so that the term-by-term cancellation is more easily seen; details can be found in Appendix C.

As before, the case with only 3-point vertices in the lower order graph is first considered; this condition is then relaxed to prove the general case. The diagrams obtained when the photon is inserted in all possible ways can be grouped into sets labelled I, II, III, and IV, as shown666Note that the same diagrams contribute as in the case; however, it is convenient here to group them differently since the factorisation is more complex now due to the presence of thermal factors. in Figs. 9 to 12. While Set I (Fig. 9) has circled vertices at both and insertions, Set II (Fig. 10) has circled vertices only at , with to the right of the special vertex. Set III (Fig. 11) has all 4-point vertex insertions at , with immediately adjacent to . Finally, Set IV (Fig. 12) is a set of circled vertices that includes all tadpole insertions, , as shown in Fig. 13.

Generalised identities are used as before to get term by term cancellations; as in the zero temperature case, there are many left-over terms due to the presence of the additional 4-point vertices, in contrast to the fermionic case. It is shown in Appendix C that left-over terms of Sets cancel against corresponding terms in Set IV, leaving behind a term proportional to the lower order matrix element , as required, and towers of terms linear in . Since the rest of the matrix element, including the volume element, the photon propagator , as well as , are symmetric under the exchange , these terms, being odd in , have only a vanishing contribution.

Note that there are no terms in the fermionic case; such terms are present in the two contributing graphs corresponding to each term in the typical circled vertices of Set IV and exactly cancel against one another. While the terms, and hence, the tadpole contributions are IR finite and do not pose any problems for the theory, it is not just simply a preference that these be included with the IR finite photon contributions; was designed to isolate the IR singular terms and resum them; hence the presence of such terms in addition to the term proportional to the lower order matrix element, precludes the factorisation and resummation of the photon contributions to all orders; hence it is a matter of satisfaction that such terms cancel exactly.

The sum of the contributions from all four sets of diagrams with all possible insertions of the photon, with both vertices on the leg, is a term that contains the IR divergence and is proportional to the lower order matrix element with no additional finite contributions, as is the case with the zero temperature theory and fermionic QED:

(14)

Since necessarily, it depends only on the photon propagator, as before.

3.1.3 Both insertions on the leg alone

A similar analysis can be done for the case when both the vertices of the inserted photon are on the leg. As discussed in GY, the outermost self energy insertion graph is neglected here to compensate for wave function renormalisation, due to which the sum of contributions for all possible insertions on the leg adds up to zero. Since this compensation could have been included in either of the legs, we symmetrise over the two possibilities, thus giving us the contributions:

(15)

The contribution is once more proportional to the lower order matrix element and depends on the part of the inserted photon propagator alone.

3.1.4 Inclusion of ‘disallowed diagrams’

Certain ‘disallowed diagrams’ may contribute at higher orders. For instance, the outermost self-energy insertion graph is removed at a certain order to account for wave function renormalisation. However, while making or photon insertions at the next higher order, these lower order diagrams must be included, as these can give rise to allowed graphs at the next order. As in the case of the zero temperature theories, these terms add to zero. There is an additional disallowed diagram in the thermal case that must be similarly included: these are lower order graphs with ‘outermost’ vertices next to the or external legs that are of thermal unphysical type with . A calculation shows that these diagrams also do not contribute at the next higher order.

The total contribution from the insertion of the virtual photon therefore is,

(16)

where

(17)

Hence the structure of the contribution from virtual photon insertion is the same as in the case; however, note that, due to the thermal contributions in the photon propagator, there are both linear and logarithmic divergences in these terms.

3.2 Insertion of virtual photons

In Paper I, it was shown that insertion of a virtual photon into the vertex graph with only 3-point vertices gives finite contributions. The key point was that the photon contribution at was proportional to

(18)

Since the leading divergence for the theory is a logarithmic one, terms proportional to powers of in the numerator are IR finite; hence the photon contribution was IR finite.

At finite temperatures, there are two major modifications: one due to the thermal part of the photon propagator and the other due to the thermal part of the scalar propagator. We start by considering the contribution due to the thermal part of the photon propagator. Although there are different types of thermal fields and hence four different photon propagators, , all of them have the same leading IR behaviour: the divergence is a linear one due to the presence of the term in the photon propagator that is proportional to

(19)

This is cancelled for the photons in exactly the same way as the case. However, there is also a logarithmic subdivergence arising from terms linear in in the numerator whereas these terms are IR finite in the case. Proving the IR finiteness of these contributions is the central result of this paper. A detailed case-by-case analysis can be found in Appendix D.

We start by ignoring the parts of the propagators and concentrate on the thermal parts alone. Since the thermal part of the photon propagator includes an overall , there are two simplifications that result. First, the coefficient factor simplifies to

(20)

In addition, we can ignore terms in the scalar propagators. The complete structure of this matrix element can be written as,

(21)

where the terms in the first two square brackets correspond to terms in the definition of the photon propagator, with the relative sign in the first being determined by the thermal field indices, . The last term represents the contribution from the and virtual photon insertions on the scalar legs, and , and are products of the vertex and propagator factors. Combining the second term in Eq. 21 with the vertex factors at only the and vertices (assuming them to be 3-point for now) in the third term, we get,

(22)

where we have used and . In the soft limit, replacing , , and substituting for from Eq. 20, we get the last line of Eq. 22. We see that the leading term vanishes (indeed, was chosen for this very reason) and the term in the square brackets is exact with no further corrections. The ellipses refer to the contribution from the remaining vertices and propagators, some of which (the set of vertices and propagators that lie between the and vertices) also depend on . Substituting this back in Eq. 21, we have,

where the slashes on and indicate that the contribution from these vertices have been removed and simplified as per Eq. 22 and,

(24)

where we have indicated the powers of in the numerator of the matrix element from the scalar contribution above.

We know that the part is logarithmically divergent while the leading thermal divergence is linear. The factor is so chosen so that the term, obtained by combining Eqs. LABEL:eq:MGstrsimp and 24, vanishes. Note that this term gives rise to the leading log divergence at (from the term in the photon propagator) as well as the leading linear divergence at (from the term). The remaining part is IR finite since any power of in the numerator renders the term finite.

At , in addition, the logarithmic subdivergence arising from the term from Eqs. LABEL:eq:MGstrsimp and 24, also vanishes since the coefficient of this term is zero. But there is a term arising from the factor in the thermal part, that appears to be a logarithmic subdivergence. We however observe that the terms in the scalar part are symmetric under the interchange ; since the term is linear in , the entire contribution is odd under this interchange, so that this potential subdivergent log contribution vanishes. Higher order terms arising from even powers of in the integrand are IR finite. Hence the photon insertions are IR finite.

We have implicitly assumed that there are no divergences associated with the photon momenta in the lower order graphs (that is, from ). This is not necessarily true; divergences can potentially arise from any of the soft photons in the graph. Here, the procedure, as shown by GY, is to separate out the photon momenta into groups that cause an IR divergence and those that do not. It is then possible to ignore the latter group and construct so-called “skeletal graphs” where the divergence arises only when each of the controlling momenta, , simultaneously vanish. It was shown in Refs. [11, 10] that photon insertions are finite with respect to all such controlling momenta for a theory of charged fermions at zero and finite temperature and this was extended to a zero temperature theory of charged scalars in Paper I. In Appendix D we show that this holds for scalars at finite temperature as well.

This result also holds when we extend the analysis to include the possibility that the and vertex insertions are of 4-point type, or even that some or all of the vertices in the lower order graph are of 4-point type as well; each of these cases is dealt with in detail in Appendix D. The final generalisation is when we include thermal effects in the scalar propagator as well (those in the vertices are quite trivial to deal with).

3.2.1 Effect of including thermal scalars

When the scalar field is also thermal, it is not sufficient to consider the part of the scalar propagator. There are factors of the scalar number operator, , that can cause a potential divergence since the scalar fields are bosons with,

(25)

in contrast to the fermionic case where the number operator is finite, , as ; so we need to check that this result holds when the scalars are thermal as well. We begin as usual by considering graphs with only 3-point vertices.

The numerator factors arising from the scalar-photon vertices acquire only irrelevant modifications when temperature effects are included; hence the structure of the vertices, that were crucial in obtaining the cancellation of the leading divergence of the photon contributions between the and terms in Eq. 22, still holds. We need to consider only the terms linear in that can give rise to subleading logarithmic divergences as discussed above.

We, therefore, examine the finite temperature dependence of the scalar propagators. In contrast to the case of thermal photons, the momentum (or ) flows through all the scalar lines and this controls the behaviour in the soft limit. The pure dependence at is replaced by a sum of and terms. Hence, none, some, or all the scalar propagators can have thermal contributions. The case where all scalar propagators correspond to is the case that we have studied so far.

While the two propagators have the same dimensional dependence on , contains a delta-function dependence which either makes the term finite or else leads to a constraint where is related to combinations of the remaining (controlling) momenta and hence there is no (logarithmic sub)divergence associated with this term. This holds even when more than one of the scalar propagators is a thermal type. The detailed analysis for adding a photon to a lower order graph with thermal electrons, and having one or more momenta in the controlling set, can be found in Ref. [10] and applies to the case of charged scalars as well. Hence the photon insertion is IR finite when we consider the entire thermal structure of the theory, both for charged scalars and photons, and even if the charged particles are fermions. More details are found in Appendix D. Finally, the cases when some of the vertices are photons or real photon vertices is also discussed in Appendix D.

As before, we have to verify that when we “flesh out” skeletal graphs and include self-energy or other terms, the graph remains IR finite; this is also shown in Appendix D. This concludes the proof that the entire virtual photon insertions of the full finite temperature theory (with both charged fermions and scalars) are in general IR finite.

3.3 The final matrix element for virtual photons

We have obtained the familiar result that the (IR divergent) contribution of the photon insertions is proportional to the lower order matrix element, while the photon insertions are finite. We proceed as in the case of scalar QED or thermal fermionic QED and consider the contribution of the order graph with virtual photons and virtual photons. Hence and there are at most vertices (since some can be seagulls or tadpoles). As a consequence of the Bose symmetry for the photons, each distinct graph can arise in ways, so that the total matrix element can be expressed as a sum of all possible individual contributions,

(26)

Summing over all orders, we get

(27)

and we use the result that the photon contribution is proportional to the lower order matrix element to obtain:

(28)

where as defined in Eq. 17 is the contribution from each -photon insertion and can be isolated and factored out, leaving only the IR finite -photon contribution, . Re-sorting and collecting terms, we obtain the requisite exponential IR divergent factor:

(29)

Again we highlight that this factorisation was made possible since the -photon insertions gave precisely one term and no additional pieces, IR-finite or otherwise; this occurred due to the presence of both 3-point and 4-point vertices in the theory. The resulting cross section including only the virtual photon contributions to all orders is,

(30)

where is the phase space factor corresponding to the final state scalar with momentum and a(n irrelevant) flux factor in the denominator has been suppressed. The IR-finite part is contained in the last term and the IR divergent part is contained in the exponent,

(31)

and will be shown to cancel against a corresponding contribution from real (soft) photon emission/absorption with respect to the heat bath, thus indicating that thermal scalar electrodynamics is also IR finite at all orders. This then sets the stage for the proof of the IR finiteness of the dark matter thermal field theory of interest.

3.4 Emission/absorption of real photons

There is a major difference in the thermal case: real photons can be emitted into or absorbed from the heat bath. Again, the real photon vertex can be either on the or leg, and the contributions of the two can be independently calculated. The insertion can be a 3-point vertex (photon inserted on the or leg at a new vertex ) or a 4-point vertex (photon inserted on an already existing vertex, giving seagull but not tadpole diagrams since a real photon is actually emitted/absorbed).

Unlike the virtual photon insertions, physical momentum is carried away or brought in by the real photon. Without loss of generality, this can be accounted for by retaining the momenta of the external scalar legs to be and and adjusting the momentum at the special vertex to maintain energy-momentum conservation. Hence the factors are somewhat different from the virtual photon case: when the photon is emitted from the leg, the momentum of the scalar/fermion to the right of the insertion is where is the vertex immediately to the left of ; here are the photon momenta emitted/absorbed at the vertex. Similarly, for an emission from the leg, the momentum of the scalar/fermion to the left of is , where is the vertex immediately to the right of . If the photon is absorbed rather than emitted, the sign of is reversed.

Since the real photon insertions contribute to , that is, to the cross section, we need to consider thermal modifications to the phase space. The thermal phase space element corresponding to the real photon with momentum is given by,

(32)

Here emission of photons corresponds to and absorption to , thus giving the correct statistical factors of for photon emission into, and for photon absorption from, the heat bath at temperature . Again, the presence of the thermal number operator worsens the divergence in the case of real photon emission/absorption as well, giving a leading IR dependence that is linear, since in the soft limit. Note that the presence of the same term acts as an UV cut off when .

We proceed as in Paper I, re-writing the polarisation sum in the cross section and separating it into a part that potentially contains the entire IR divergent part and an IR finite photon part:

(33)

with

(34)

where the tildes have been used to distinguish the real from the virtual photon contributions. Since for both real photon emission and absorption, we define,

(35)

where () corresponds to the initial (final) momentum of the hard scalar in () where the real photon of momentum is inserted.

3.4.1 Emission/absorption of real photons

The proof that the contribution from photons is IR divergent and can be factored is much simpler than the corresponding case of virtual photons. The key point to note is that real photons, whether emitted or absorbed, correspond to thermal type 1 photons, so that the inserted vertex (either or ) is of type 1 alone. This is critical in obtaining a cancellation against the virtual photon contribution and the significance of this virtual contribution being proportional to alone, as shown in Eqs. 13 and 15, is now clear.

The calculation for photon emission proceeds exactly as in the case of virtual photon vertex insertion on a or leg (see diagrams shown in Fig. 8). Again, there is a term-by-term cancellation, leading to a factor proportional to the matrix element of the photon diagram, . Similar insertions on the leg give a result proportional to ; the difference in sign with the case of insertion of virtual photons is due to the fact that the real photon momentum is always out-going for emitted photons; while the virtual momentum enters/leaves at the vertex. The overall sign is reversed in the case of photon absorption; however, this is irrelevant and unobserved in the cross section. Adding the two terms and squaring gives the contribution of the real photon insertion to be an overall factor multiplying the order cross section, proportional to,

(36)

The result holds even when some vertices of the lower order graph are 4-point ones, or correspond to virtual photon insertions as well; this follows from the arguments given for the virtual photon insertions in Appendix C.

Before discussing the cross section, we will first complete the discussion on insertions of real photons, which, as expected, will be IR finite.

3.4.2 Emission/absorption of real photons

The proof of IR finiteness of the real photon cross section follows from the same argument as for the virtual -photon insertion and is not repeated here in detail. Specifically, the case where the insertions are on different legs ( and ) is relevant for the real photon insertions. All the cases such as including both 3- and 4-point vertices, including thermal effects in both photon and scalar propagators, etc., hold here; there are no tadpole diagrams in this case and also no quadratic contribution that needs to be cancelled.

The key point to note here is the -dependence of the thermal part. The leading divergence (logarithmic in the zero temperature case and linear in the finite temperature case) cancels as before, between the and the parts of , owing to the definition of . We are thus concerned only with terms with powers of in the numerator which potentially give logarithmic subdivergences.

The main difference between virtual and real photon insertion is that the phase space factor is not symmetric under because of the presence of the theta function, as seen from Eq. 32. However, the finite temperature part of the phase space is symmetric under this exchange since it includes both photon emission and absorption. These are anyway the only contributions of interest since any powers of in the numerator are finite with respect to the part. This symmetry enables us to symmetrise the integrand with respect to and obtain the analogous result that real photon insertions are IR finite. Notice that application of the symmetry requires the presence of both soft photon emission and absorption terms.

Again, the result holds when one of the photons with momentum contributes through its part; in this case, its corresponding momentum cannot be flipped since its phase space is not symmetric under this exchange. We apply the same logic as with skeletal graphs in the virtual photon case: if this photon is not a part of the controlling set, there is no divergence associated when it vanishes and this gives us no trouble. If it is a part of the controlling set, then the sub-divergence occurs only when all (including this) momenta vanish simultaneously; however, any power of in the numerator renders the contribution finite since it contributes through its part and so again its contribution is finite. The analysis holds when arbitrary number of these photons contribute through their parts; also when some of these are virtual photons, since their contribution is always symmetric in the loop momentum.

Hence, photon emissions give a finite contribution to the cross section.

3.4.3 The total cross section from real photon emission/absorption

Consider an order graph with an arbitrary number of and photon insertions. Now and real photon emission/absorption can occur in ways; , and each real photon carries away/brings in a physical momentum from/to the process. Dividing by due to identical photons in the final state, to this order, we have,

(37)

where the factor corresponds to depending on whether the photon with momentum is emitted/absorbed. Here the phase space factor is given by Eq. 32 and the factor contains the IR divergent part. The dependence in the energy-momentum conserving delta function is removed by the usual trick of redefining the delta-function:

(38)

where the sign of in the last term depends on whether the real photon was emitted or absorbed; furthermore, we separate out the photon contribution in the last term:

(39)

The terms that depend on the photon momenta are then combined with the (common) factor for every insertion. Then the total contribution from each photon is:

(40)

The total contribution from real photons in Eq. 37 can now be factored as,

(41)

and hence can be exponentiated as . We will use this factor and compute the total cross section for the process to all orders.

3.5 The total cross section to all orders

The all-order corrections to the tree-level cross section for arising from both virtual and real (soft) photon insertions yields the total cross section for this process:

(42)

where contains the finite and p