Bulk dominated fermion emission on a Schwarzschild background

Bulk dominated fermion emission on a Schwarzschild background

H. T. Cho Department of Physics, Tamkang University, Tamsui, Taipei, Taiwan, Republic of China    A. S. Cornell Université de Lyon, Villeurbanne, F-69622, France; Université Lyon 1, Institut de Physique Nucléaire de Lyon    Jason Doukas School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia.    Wade Naylor Department of Physics, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
August 23, 2019
Abstract

Using the WKBJ approximation, and the Unruh method, we obtain semi-analytic expressions for the absorption probability (in all energy regimes) for Dirac fermions on a higher dimensional Schwarzschild background. We present an analytic expression relating the absorption probability to the absorption cross-section, and then use these results to plot the emission rates to third order in the WKBJ approximation. The set-up we use is sufficiently general such that it could also easily be applied to any spherically symmetric background in -dimensions. Our results lead to the interesting conclusion that for bulk fermion emission dominates brane localised emission. This is an example contrary to the conjecture that black holes radiate mainly on the brane.

pacs:
02.30Gp, 03.65Ge, 0470.Dy, 11.10.Kk
preprint: LYCEN 2007-13preprint: RITS-PP-013

I Introduction

In the last decade a great deal of attention has been focused on large extra-dimensional scenarios ADD , where the hierarchy problem can be shifted into a problem of the scale of the extra-dimensions. These scenarios have also led to the somewhat striking prediction that black holes (BHs) may be observed at particle accelerators such as the LHC BHacc . However, one poignant problem is that in order to suppress a rapid proton decay quarks and leptons need to be physically separated in the higher dimension(s). Such models are generically called split fermion models, see reference Split , for example. Note that in supersymmetric versions of this idea the localizing scalars and bulk gauge fields will also have fermionic bulk superpartners. In this respect it is important to consider the properties of bulk fermions, something which until now has been largely overlooked. For a treatment of brane localized Hawking emission in the case of static BHs see reference HiHawk ; Kanti , for rotating BHs, see reference Rotate as well as references BraneFerm .

In a previous work we applied conformal methods, which allowed us to separate the Dirac equation on a higher dimensional spherically symmetric background, to discuss the quasinormal modes (QNMs) for Schwarzschild BHsCCDN ; where such a method was previously used in reference Das to calculate low energy -wave absorption cross-sections. In this work we shall use this same method to calculate the greybody factors Das and emission rates for Dirac perturbations on a -dimensional Schwartzschild background. Our results may also be of some interest to the tense-less limit of the tense branes in six dimensions discussed in reference Dai .

Here we shall write the -dimensional Schwarzschild background as:

 ds2=−f(r)dt2+f−1(r)dr2+r2dΩ2d−2, (1)

with

 f(r)=1−(rHr)d−3, (2)

where the horizon is at and:

 rd−3H=8πMΓ((d−1)/2)π(d−1)/2(d−2). (3)

In reference CCDN we used a conformal transformation of the metric to separate the Dirac equation into a time-radial part and a -sphere. Moreover, the radial part was reduced to a Schrödinger-like equation in the tortoise coordinate :

 (−d2dr2∗+V1)G=E2G, (4)

where , and the potential is given by:

 V1(r) = κ2fr2+κfddr[√fr], (5)

where

 κ=ℓ+d−22=d2−1,d2,d2+1,… (6)

It may be worth mentioning that the above potential reduces to the brane-localized results found in reference JP when we set and therefore provides an alternate derivation of the brane localized potential.

In the next section, we shall compare the absorption probability for this potential as calculated under various approximation techniques, namely, the low energy WKBJ, the first to third order intermediate WKBJ and the low energy Unruh method. Then we shall relate these quantities to the absorption cross-section and calculate the Hawking emission rate. After this we will compare bulk emission with brane emission (to third order in WKBJ) for various dimensions, before presenting our concluding remarks in section IV.

Ii Absorption probabilities via the WKBJ approximation

In a recent work by two of the authors Cornell we applied the intermediate WKBJ approximation (up to first order) to evaluate the absorption probability of a graviton to a static BH. The WKBJ approximation can, however, be applied at all energies (including low energy) as has been discussed in reference ChoWKB . In Appendix A we calculate a different low energy approximation based on the method developed by Unruh Unruh . This ‘Unruh method’ leads to useful analytic results, but does not work well for high energy regimes.

Low Energy WKBJ

In terms of the WKBJ approximation, in general, it will be convenient to make a change of variables to Cornell . This leads to the following form for the potential:

 Q(x∗)=1−κ2fx2−κfddx[√fx],f(x)=1−(εx)d−3, (7)

where

 E2Q(x∗)=E2−V1, (8)

and .111Note that in reference Cornell we defined the horizon, , in terms of , however, by using this different parameterisation here, we may compare this result with the low energy Unruh approach. As such, the Schrödinger equation, equation (4), takes the form:

 (d2dx2∗+Q)G=0. (9)

The low energy absorption probability corresponds to the probability for a particle to tunnel through the potential barrier (the barrier penetration probability) and can be found in many standard quantum mechanics text books (a derivation can also be found in reference ChoWKB ). The result to first order in the low energy WKBJ approximation is given by:

 |Aκ(E)|2=exp[−2∫x2x1dx′f(x′)√−Q(x′)], (10)

where and are the turning points, or , for a given energy with potential . This approximation is valid for and as long as we can solve for the turning points in . Note that we can numerically integrate equation (10) for each energy to obtain the absorption probability as a function of .

Intermediate Energy: 3rd Order WKBJ

As discussed in reference IW , an adapted form of the WKBJ method can be employed to find the QNMs, or the absorption probability (which we are primarily interested in here), when the scattering takes place near the top of the potential barrier. In the following we shall use the same notation as reference WG , where we have confirmed their results to fourth order. However, for the purposes of this paper, we shall consider only up to and including third order, in which case we express the absorption probability as:

 |Aκ(E)|2=11+e2S(E), (11)

where

 S(E) = πk1/2[12z20+(1564b23−316b4)z40+(11551024b43−459128b4b23+9532b5b3+6764b24−2516b6)z60] (12) +O(k−3/2z00,k−1/2z40,k1/2z80).

Note that the first order result comprises of just the first term in this expression, while the second order result consists of the second term on the first line and the first term on the second line.

We would like to draw the readers attention to the fact that as we go to higher order the approximation becomes valid for lower energies. However, as can be seen from figure 1, even orders in the intermediate WKBJ method drop back down to zero for large energy. For this reason we shall work to third and not fourth order in our calculations, as odd orders have the nice property that for large energy, making numerical work with these orders easier. Also, note that the first order WKBJ under-predicts the absorption probability for as compared to the second order WKBJ result. We have compared these results up to fourth order in the WKBJ approximation WG , and the essential features of the energy absorption profile (such as dipping back down to zero for even orders) do not change when going from second to fourth order; nor do features of the first order change when going to third order, see figure 1.

In this method in equation (7) is expanded near the peak (), in terms of a new variable :

 Q(z)=k(z2−z20+∑n=3bnzn), (13)

where

 z20≡−2Q0Q′′0,k≡12Q′′0,bn≡2n!Q′′0dnQdrn∣∣∣0. (14)

Note that the subscript represents the maximum of , and that for our purposes it is more convenient to work in terms of , rather than . However, given that the WKBJ form of the potential is in terms of the parameter , including its derivatives, we can convert derivatives in terms of into via the equation . See reference CCDN for a discussion of how to locate the maximum.

High Energy

For high energies the absorption probability tends to unity, and the cross-section reduces to that of the classical cross-section, see reference Cornell . However, as discussed in reference ChoWKB , there will always be small corrections to the large energy limit. A high energy WKBJ approach can be applied in this limit, but for the purposes of this current study it will be sufficient to use (given that it reproduces the high energy geometric optics limit, see references Kanti ; Cornell ).

Absorption Probability Results

The results of these analyses have been shown in figure 1. In this figure we presented plots for the first, second and third order WKBJ Iyer and Will method IW , which is an intermediate WKBJ approximation, as well as the low energy WKBJ result (to first order) and the analytical method of Unruh Unruh . The WKBJ approximation, in general, is accurate for larger angular momentum channels, whereas the Unruh approach is valid for only the lowest angular momentum channels and (namely small BHs). However, it is interesting to note that although the low energy WKBJ result does not agree exactly with the Unruh result, they both tend to zero for . On the other hand, unlike the Unruh result, the low energy WKBJ is valid for energies up to , where it matches onto the intermediate WKBJ.222In order for the first order low energy WBKJ to match onto the Iyer and Will second/third order result we are really required to evaluate it to second/third order as well.

Iii Emission Rates

The emission rate for a massless fermion from a BH is related to the cross-section by a dimensional momentum integral times a fermionic thermal temperature distribution:

 dEdt=∑λ,Eσλ,EEeETH+1dd−1k(2π)d−1, (15)

where is the Hawking temperature, are the greybody factors and the sum is a generic sum over all angular momentum and momentum variables. In Appendix B using the method of Cardoso et al. Cardoso we relate the greybody factor to the absorbtion probability:

 σλ,E=12Ωd−2(2πE)d−2∑κDκ|Aκ(E)|2. (16)

Given that angular integration over the momentum for a massless field () leads to the Jacobian , the fermion emission rate can be expressed solely in terms of the absorption probability:

 dEdt=∑κ∫dE2πEeETH+1Dκ|Aκ(E)|2. (17)

In the above we have used as the degeneracy (as defined in equation (64)) and the sum over for , , , . However, since the integrand depends only on the absolute value of we only sum for and multiply by a factor of two. We then recover a result identical to that for a scalar field Kanti , except for a difference in sign due to fermion statistics.

After changing variables to , and using the fact that the Hawking temperature is , we obtain:

 d2EdEdt=1πrH∑κ>0εe4πεd−1+1Dκ|Aκ(ε)|2. (18)

As such, the evaluation of the emission rate is now a simple task, as can be obtained either numerically or via the WKBJ method in each appropriate energy regime. Note that we have presented an example of these emission rates for various values of in figure 2 (up to third order WKBJ), these results shall be discussed in the conclusion. Importantly, when we come to consider brane-localized emissions we simply set and , on assuming . These are also plotted in figure 2 for various dimensions .

Third Order WKBJ Emission Spectrum

Since the intermediate WKBJ is accurate for we plot the emission rates in this approximation, as shown in figure 2.

We have also calculated the total power by integrating over , see equation (18). The results are shown in Table 1. From these results we find that for the emission is predominantly into the bulk.

Note that in order to obtain convergence in equation (18) we must choose some value of and to ensure this we have taken .

From our results we should note that of interest is the region where in the bulk emission plots, where we see that the lines are crossing over. To be sure that this was not due to the semi-analytic approximation breaking down we computed the emission rates in the low energy regime using the low energy WKBJ and Unruh methods as shown in figure 3. From these plots we clearly see an opposite ordering of the lines to those from the high energy region in figure 2, verifying that a crossing does indeed take place.

Iv Concluding remarks

In this paper we have presented new results for the emission rate of a massless Dirac field on a bulk -dimensional Schwarzschild background, where the method we used in reference CCDN is sufficiently general and could be applied to other spherically symmetric backgrounds (and also for massive bulk Dirac fields). The main result is that fermions are mainly emitted into the bulk for , as we have shown, see Table 1, which is in contrast to the scalar field case Kanti and for bulk to brane photons Jung . This is an example contrary to the conjecture that BHs radiate mainly on the brane EHM . Furthermore, bulk dominated fermion emission is also consistent with the original motivation for split-fermions, namely that of a suppression of a rapid proton decay Split .

We also highlighted how semi-analytic results can be obtained by considering different versions of the WKBJ approximation, where we also compared this to the low energy analytic results derived from the method first developed in reference Unruh , see figure 1. We also used the low energy WKBJ approximation, where it should be stressed that this approximation extends further than the range of the Unruh result (right up to intermediate energies) in the plots of , again see figure 1. However, in terms of the emission rates for low energy, see figure 3, the range of validity of extends up to , due to this energy range, for the emissions, being dominated by the two lowest angular momentum channels and , essentially an -wave scattering.

In figure 2 we have plotted the third order WKBJ approximation of the emission rates for various . As can be seen from this figure, in the bulk emission case, there is an interesting behaviour around where the emission rates in different dimensions cross. That is, at a certain intermediate energy there is a region where the emission rates become approximately independent of the dimension, . We initially discovered this feature when plotting the first order WKBJ emission rates and found that this effect persisted to third order in the WKBJ approximation. The cross-over effect is confirmed by plotting the emission in the low energy region using the Unruh method, see figure 3, and plotting the geometric optics limit at high energy and observing an opposite ordering of lines. We are unaware of this curious feature being reported anywhere else in the literature.

The results, when compared with the brane-localized fermion results of references BraneFerm ; Kanti reveal that bulk fermion modes result in much larger emission rates (though in both cases larger results in greater emission rates). These results also agree qualitatively with our work for the QNMs on such a background CCDN , where the BH damping rate was found to increase with dimension. Interestingly, the cross-over feature is absent from the emission rates of the brane-localized fermions, as can be seen in the right panel of figure 2, also see figure 4 of reference Kanti .333Figure 4 of reference Kanti is a Log plot not a Linear plot, however, there is no observable cross-over behaviour at intermediate energies. Thus, this feature appears to be specific to bulk species as we have also verified the same behaviour for bulk, but not brane-localized, scalar fields (this can also be observed in plots made in reference Kanti ).

The next step would be to investigate rotating solutions for bulk/split fermions (which have yet to be found for even a single rotation parameter)444See reference Kanti2 for scalar bulk and brane results for low energy and rotation parameter.. However, in order to obtain these emissitivities the absorption cross-section would also need to be related to the absorption probability, but the method of Cardoso et al. Cardoso , see Appendix B, does not work. That said, the energy flux could be derived by using the energy-momentum tensor for a spin-half field and comparing the flux at spatial infinity with that of the horizon (for a nice discussion of massless scalar fields see reference LWY and the references therein). We are also currently applying our methods to investigate the effect of brane tension Dai on our results. Finally, in a forthcoming work we intend to present results for up to 6th order in the intermediate WKBJ approximation, which follows along the lines of the work by Konoplya for QNMs Konoplya6th , where we will compare it to exact numeric results.

Acknowledgments

HTC was supported in part by the National Science Council of the Republic of China under the Grant NSC 96-2112-M-032-006-MY3. JD wishes to thank Dr. G. C. Joshi for his advice and supervision during the production of this work. The authors would also like to thank Prof. M. Sasaki for his interesting and useful discussions on this topic.

Appendix A Low Energy and Momentum

The case for the lowest angular momentum channel, , has been considered for bulk fermions in reference Das . However, for completeness, we shall briefly discuss the low energy low angular momentum limit for general , which is valid for small BHs. Here we shall follow the method of Unruh Unruh , but as the generalization to bulk fermions follows almost identically to the brane-localized case discussed in reference JP we shall only briefly highlight the steps (referring the reader to reference JP for a fuller discussion). Finally, note that in the following we shall work in terms of the potential , not , see reference CCDN .

In terms of the component in equation (21) of reference CCDN , the near horizon limit (which corresponds to and ) becomes:

 d2GNHdr2∗+E2GNH=0, (19)

where some simple deliberation leads to the following outgoing wave solution:

 GNH=AIe−iEr∗≈AIe−iεrHd−3lnf. (20)

Next, in the intermediate region, we assume that is much smaller than the other terms and equation (21) of reference CCDN becomes (using the potential ):

 d2GIMdr2∗−(κ2fr2−κfddr[√fr])GIM=0. (21)

Following the approach of Unruh Unruh and defining:

 HIM≡dGIMdr∗+κ√frGIM, (22)

we can transform equation (21) into a first-order differential equation of the form:

 dHIMdr−κ√frHIM=0. (23)

The solution of equation (23) is then:

 HIM=BII(1−√f1+√f)−κd−3. (24)

Inserting the above result back into equation (23) leads to Unruh ; JP :

 GIM=AII(1−√f1+√f)−κd−3+BIIGIM, (25)

where is a particular solution of:

 dGIMdr+κ√frGIM=1f(1−√f1+√f)κd−3. (26)

In general we have different solutions for positive or negative , and in what follows we shall just discuss the solutions. As such, for the solution to is:

 GIM=rH(1−√f1+√f)−κd−3∫√f12(d−3)ρ(1−ρ)2|κ|−(d−2)d−3(1+ρ)2|κ|+(d−2)d−3dρ, (27)

with a slightly different solution for , see reference JP .

To determine the flux at infinity we also require the solution in the far field region, and given that in this region , or (see equation (21) of reference CCDN ), leads to the simplification:

 d2GFFdr2+[E2−κ(κ+1)r2]GFF=0. (28)

This solution can be expressed in terms of Bessel functions:

 GFF(r)=AIII√rrHJ|κ+12|(Er)+BIII√rrHY|κ+12|(Er). (29)

By the large argument expansion of Bessel functions we can formally express the absorption probability as:

 |Aκ(E)|2=2i(BIIIAIII−B∗IIIA∗III)∣∣1+iBIIIAIII∣∣2. (30)

Finally, all that is needed is to find the ratio of , which is done by matching the far-field, intermediate and near horizon regions. This leads to (for details see references Unruh ; JP ):

 BIIIAIII=−iπ2−4κd−3Γ2(12−κ)(ErH2)−2κ ,BIIIAIII=π2−4κd−3iΓ2(κ+12)(ErH2)2κ, (31)

for and respectively Unruh ; JP . It turns out, therefore, that the absorption probability is independent of the sign of :

 |Aκ(E)|2=⎛⎜ ⎜⎝4π2−4|κ|d−3(ErH2)2|κ|Γ2(|κ|+12)⎞⎟ ⎟⎠⎡⎢ ⎢ ⎢⎣1+π2−4|κ|d−3(ErH2)2|κ|Γ2(|κ|+12)⎤⎥ ⎥ ⎥⎦2≈4π2−4|κ|d−3(ErH2)2|κ|Γ2(|κ|+12). (32)

When these results are substituted into the relation for the cross-section this leads to the bulk generalization for the ratio between the bulk fermion and scalar cross-sections for general JP .555Given that this approximation is only valid for it can be expanded in a power series to give the approximate result on the right.

Appendix B Absorption Cross-section

In this appendix we shall discuss the relationship between and the total absorption cross-section . In what follows we shall follow the approach of Cardoso et al. Cardoso by integrating the total absorption cross-section over the direction of the incident plane wave and then dividing the result by the total solid angle (recall that the spacetime is spherically symmetric). Note that with this method one need not know the exact expansion of the incident plane wave in terms of the spherical ones.

As is done in reference Cardoso , we concentrate on the case at spatial infinity, that is, in a -dimensional Euclidean space, using a plane spinor wave:

 ψ(E,^k,s)plane(→x)=e−iEteiE^k⋅^xψ(^k,s)0. (33)

Here is the energy, the direction, and is the helicity of the wave. Note that as is a constant spinor it satisfies the equation:

 γμ∂μψ(E,^k,s)plane=0⇒γ0ψ(^k,s)0−^kiγiψ(^k,s)0=0. (34)

Note that the flux of the spinor wave is given by , where the minus sign is due to . For the plane spinor wave:

 ^kiJi=−^ki(ψ(^k,s)0)†eiEte−iE^k⋅→xγ0γie−iEteiE^k⋅→xψ(^k,s)0 = 1, (35)

where we have used equation (34) and have assumed that is normalized. This shows that the plane wave has unit flux along the direction.

Next, consider a spherical ingoing spinor wave:

 ψ(E,λ)spherical=ir(d−2)/2e−iEte−iEr(1/√2−1/√2)⊗χλ, (36)

where is an orthonormal spinor on the unit -sphere, that is:

 ∫dΩd−2(χλ)†χλ′=δλλ′. (37)

The radial flux of this ingoing spherical wave is given by:

 Jr = −(ψ(E,λ)spherical)†γ0γrψ(E,λ)spherical (38) = −1rd−2((1/√2−1/√2)(−iσ3)(σ2)(1/√2−1/√2))(χλ)†χλ = −1rd−2(χλ)†χλ.

In which case:

 N=−rd−2∫dΩd−2Jr=1. (39)

Note that this spherical wave represents one ingoing particle per unit time. Following Cardoso et al., we define such that:

 ψ(E,^k,s)plane=∑λα(E,^k,s,λ)ψ(E,λ)spherical+outgoing part. (40)

The total absorption cross-section for this particular incident plane wave is given by:

 σ(E,^k,s)=∑λ|α(E,^k,s,λ)|2|As=1/2(E,λ)|2, (41)

where is the transmission probability.

Since the BH is spherically symmetric is independent of and . Therefore, one can sum over and integrate over , and then divide the result by 2 times the total solid angle, to obtain the same absorption cross-section. That is:

 σ = 12Ωd−2∑s=±∫dΩd−2σ(E,^k,s) (42) = ∑λ(12Ωd−2∑s=±∫dΩd−2|α(E,^k,s,λ)|2)|As=1/2(E,λ)|2.

Next, we evaluate the quantity in the bracket. To do so we first consider the projection of the plane wave onto the spherical ones with different energies. That is, we integrate over the Euclidean -dimensional space:

 ∫dd−1x (ψ(E,λ)spherical(x))†ψ(E′,^k,s)plane(x) = ∫dd−1x∑λ′(ψ(E,λ)spherical(x))†α(E′,^k,s,λ′)ψ(E′,λ′)spherical(x) (43) = 2πδ(E−E′)α(E′,^k,s,λ).

Similarly:

 ∫dd−1x′ (ψ(E′,^k,s)plane(x′))†ψ(E′′,λ)spherical(x′) = ∫dd−1x′∑λ′(ψ(E′,λ′)spherical(x′))†α∗(E′,^k,s,λ′)ψ(E′′,λ)spherical(x′) (44) = 2πδ(E′−E′′)α∗(E′,^k,s,λ).

Multiplying these two expressions by , summing over , and integrating over , we have:

 ∑s=±∫dE′E′d−2dΩ(^k)d−2(2π)2δ(E−E′)δ(E′−E′′)|α(E′,^k,s,λ)|2 =(2π)2δ(E−E′′)Ed−2∑s=±∫dΩ(^k)d−2|α(E,^k,s,λ)|2.

We can also evaluate this expression another way:

 ∑s=±∫ dE′ E′d−2∫dΩ(^k)d−2∫dd−1x∫dd−1x′(ψ(E,λ)spherical(x))†ψ(E′,^k,s)plane(x)(ψ(E′,^k,s)plane(x′))†ψ(E′′,λ)spherical(x′) = ∫dd−1x∫dd−1x′(ψ(E,λ)spherical(x))†(∑s=±∫dE′E′d−2∫dΩ(^k)d−2ψ(E′,^k,s)plane(x)(ψ(E′,^k,s)plane(x′))†)ψ(E′′,λ)spherical(x′).

For the expression in the bracket:

 ∑s=±∫dE′E′d−2∫dΩ(^k)d−2ψ(E′,^k,s)plane(x)(ψ(E′,^k,s)plane(x′))†=∫dd−1kei→k⋅(→x−→x′)∑s=±ψ(^k,s)0(ψ^k,s)0)†, (47)

and the sum over (of the constant spinors) is actually a projection operator. From equation (34) we have:

 12(1+^kiγ0γi)ψ(^k,s)0=0 ⇒ 12(1+^kiγ0γi)∑s=±ψ(^k,s)0(ψ^k,s)0)†=0, (48)

hence:

 ∑±ψ(^k,s)0(ψ^k,s)0)†=12(1−^kiγ0γi)≡Λ+, (49)

where we have defined the projection operators for positive and negative energy solutions as:

 Λ±≡12(1∓^kiγ0γi), (50)

with , , , and . With this result we can write equation (47) as:

 ∫dd−1k 12(1−^kiγ0γi)ei→k⋅(→x−→x′) = 12⎛⎜ ⎜⎝1+i√−→∂2γ0γi∂i⎞⎟ ⎟⎠∫dd−1k ei→k⋅(→x−→x′) (51) = 12⎛⎜ ⎜⎝1+i√−→∂2γ0γi∂i⎞⎟ ⎟⎠(2π)d−1δ(→x−→x′).

Finally, returning to equation (LABEL:projection):

 ∫dd−1x∫dd−1x′(ψ(E,λ)spherical(x))†12⎛⎜ ⎜⎝1+i√−→∂2γ0γi∂i⎞⎟ ⎟⎠(2π)d−1δ(→x−→x′)ψ(E′′,λ)spherical(x′) =(2π)d−1∫dd−1x(ψ(E,λ)spherical(x))†12⎛⎜ ⎜⎝1+i√−→∂2γ0γi∂i⎞⎟ ⎟⎠ψ(E′′,λ)spherical(x). (52)

Since the spherical spinor waves also have positive energies, they satisfy:

 γμ∂μψ(E′′,λ)spherical=0 ⇒ γi∂iψ(E′′,λ)spherical=iE′′γ0ψ(E′′,λ)spherical (53) ⇒ −→∂2ψ(E′′,λ)spherical=E′′2ψ(E′′,λ)spherical.

In which case equation (52) becomes:

 (2π)d−1∫dd−1x(ψ(E,λ)spherical(x))†ψ(E′′,λ)spherical(x)=(2π)dδ(E−E′′). (54)

Equating this result to equation (LABEL:final) we have:

 ∑s=±∫dΩ(^k)d−2|α(E,^k,s,λ)|2=(2πE)d−2, (55)

with the result that the cross-section in equation (42) can be expressed as:

 σ=12Ωd−2(2πE)d−2∑λ|As=1/2(E,λ)|2. (56)

To express the cross-section more explicitly, we first note that the total volume of a unit -sphere is:

 Ωd−2=2π(d−1)/2Γ((d−1)/2), (57)

and that the Dirac operator eigenvalues of the eigenspinors on a -sphere are Camporesi :

 λl=±i(ℓ+d−22), (58)

where . Note that the degeneracies of the eigenvalues, which are equal to the dimension of the spinor representations Camporesi , are:

 Ds=1/2ℓ(d−2)=2(d−2)/2Γ(ℓ+d−2)Γ(ℓ+1)Γ(d−2), (59)

for even spheres, and

 Ds=1/2ℓ(d−2)=2(d−3)/2Γ(ℓ+d−2)Γ(ℓ+1)Γ(d−2), (60)

for odd spheres. One could write for even and odd spheres that:

 Ds=1/2ℓ(d−2)=2[(d−2)/2]Γ(ℓ+d−2)Γ(ℓ+1)Γ(d−2), (61)

where is the integral part of .

Instead of we have previously used another parameter, , where:

 κ=±(ℓ+d−22)=±(d2−1),±d2,±(d2+1),… (62)

or

 ℓ=|κ|−d2+1. (63)

Therefore:

 Ds=1/2κ(d−2)=2[(d−2)/2]Γ(|κ|+d2−1)Γ(