Thermal right-handed neutrino production rate in the relativistic regime
aa r X i v : . [ h e p - ph ] S e p Thermal right-handed neutrino production ratein the relativistic regime
M. Laine
Institute for Theoretical Physics, Albert Einstein Center, University of Bern,Sidlerstrasse 5, CH-3012 Bern, Switzerland
Abstract
The production rate of right-handed neutrinos from a Standard Model plasma at a tem-perature above a hundred GeV is evaluated up to NLO in Standard Model couplings. Theresults apply in the so-called relativistic regime, referring parametrically to a mass M ∼ πT ,generalizing thereby previous NLO results which only apply in the non-relativistic regime M ≫ πT . The non-relativistic expansion is observed to converge for M > ∼ T , but thesmallness of any loop corrections allows it to be used in practice already for M > ∼ T . In thelatter regime any non-covariant dependence of the differential rate on the spatial momen-tum is shown to be mild. The loop expansion breaks down in the ultrarelativistic regime M ≪ πT , but after a simple mass resummation it nevertheless extrapolates reasonably welltowards a result obtained previously through complete LPM resummation, apparently con-firming a strong enhancement of the rate at high temperatures (which facilitates chemicalequilibration). When combined with other ingredients the results may help to improve uponthe accuracy of leptogenesis computations operating above the electroweak scale.August 2013 . Introduction The neutrino sector is arguably the least precisely charted among the different parts ofexperimentally accessible particle physics. Whereas it is well established that at least twodominantly left-handed neutrinos have masses, and that the mass eigenstates are misalignedwith the weak interaction eigenstates, already the absolute value of the mass scale remainspoorly constrained. There is also feeble empirical handle on the dynamics of neutrino massgeneration, although a see-saw mechanism involving Majorana masses of gauge-singlet right-handed neutrinos is a natural candidate. Of course right-handed neutrinos can be introducedin any case, but then a large parameter space of Yukawa couplings and Majorana masses, thelatter unbounded from above, remains available to phenomenological consideration.The big volume of the parameter space suggests seeking for cosmological constraints onneutrino properties. Apart from the well-studied significance of left-handed neutrinos to theoverall expansion rate of the Universe through the pressure and energy density that theyexert, it is also possible that right-handed neutrinos have cosmological significance. Forinstance, they could play a role in explaining two outstanding cosmological mysteries, theexistence of a matter-antimatter asymmetry [1] (for reviews, see e.g. refs. [2, 3]) as well asthe existence of particle dark matter [4] (for a review see e.g. ref. [5]).The present paper is related to developing theoretical tools for studying right-handed neu-trinos within a cosmological environment. It has been understood recently [6, 7, 8] that inthe so-called ultrarelativistic regime πT ≫ M , where T denotes the temperature and M aright-handed neutrino Majorana mass parameter, the thermal loop expansion breaks downand needs to be resummed with techniques analogous to those that were previously developedin the context of photon production from a hot QCD plasma [9, 10, 11]. In contrast, in theso-called non-relativistic regime πT ≪ M , next-to-leading order (NLO) corrections can becomputed and are in general small [12, 13], in accordance with expectations based on theOperator Product Expansion (OPE) [14]. This leaves open the question of how these twoqualitatively very different regimes interpolate to each other.The purpose of the present paper is to present a theoretically consistent computation ofthe right-handed neutrino production rate in the so-called relativistic regime, πT ∼ M .The principal tools needed for this have been developed in ref. [15], and here we assemblethe full results. We also inspect under which conditions the results go over to those of thelimiting ultrarelativistic and non-relativistic cases. In addition the structure of the differentialproduction rate is analyzed with the goal of suggesting a numerically affordable and yetrelatively accurate approximation scheme that may be used in practical applications (howeverthe study of practical applications is postponed to future work). Another project with partlysimilar goals has recently been outlined in ref. [16].After summarizing the setup in sec. 2, the basic theoretical results, together with compar-1sons with the non-relativistic and ultrarelativistic regimes, are presented in sec. 3. Examplesof numerical results for differential production spectra are shown in sec. 4, whereas in sec. 5the total production rate is considered. Some conclusions and an outlook are offered in sec. 6.
2. Setup
Solving the Liouville - von Neumann equation for the time evolution of the density matrixof a coupled system of Standard Model particles and right-handed neutrinos to leading non-trivial order in neutrino Yukawa couplings (but to all orders in Standard Model couplings),for times sufficiently small that the right-handed neutrinos do not chemically equilibrate [17],it is found that their “differential production rate” can be written asd N + ( K )d X d k ≡ d N ( K )d X d k (cid:12)(cid:12)(cid:12)(cid:12) d N ( K )d3 x d3 k ≈ = 2 n F ( k )(2 π ) Γ( K ) . (2.1)Here a “width” has been defined asΓ( K ) ≡ k Im Π R ( K ) = 1 k Im (cid:8) Π E ( K ) (cid:9) k n →− i [ k + i + ] , (2.2)where Π E is a gauge-invariant and Lorentz-singlet 2-point correlation function of the “cur-rents” that right-handed neutrinos couple to,Π E ( K ) ≡ | h ν B | Tr n i /K hZ /T d τ Z x e iK · X D ( ˜ φ † a L ℓ )( X ) (¯ ℓa R ˜ φ )(0) E T io , (2.3)and Π R is the corresponding retarded real-time correlator (its imaginary part equals the spec-tral function). Moreover, h ν B denotes a bare neutrino Yukawa coupling (or, more generally,elements of a Yukawa matrix); Euclidean variables are denoted by X ≡ ( τ, x ), K ≡ ( k n , k );the corresponding Minkowskian ones by X ≡ ( t, x ), K ≡ ( k , k ); the metric conventions are K = k n + k , K = k − k , with k ≡ | k | ; k n stands for a fermionic Matsubara frequency,reflecting the fact that spin- fields are antiperiodic across the Euclidean time direction ofextent 1 /T . (Definitions of other variables appearing in eq. (2.3) can be found in ref. [13].)A corresponding integrated “total production rate” is γ + ( T ) ≡ d N + d X = Z d k (2 π ) n F ( k ) Γ( K ) . (2.4)We note in passing that an opposite case of a “differential decay rate”, defined by taking thethermal average of a spin-summed rate for the disappearance of a right-handed neutrino ofmomentum k , can be expressed in terms of the same function Γ( K ) that appears in eq. (2.2):d N − ( K )d X d k = − − n F ( k )](2 π ) Γ( K ) . (2.5)2he observables above are particularly simple because they involve a sum over the spinstates of the right-handed neutrinos. A more general problem concerns the determinationof the self-energy matrix of the right-handed neutrinos, given to leading order in neutrinoYukawa couplings by eq. (2.3) without a Dirac contraction with i /K . An NLO discussion ofthis observable in the non-relativistic regime can be found in ref. [18].Returning to eq. (2.3), one of the strengths of the imaginary-time formulation of thermalfield theory is that the expression obtained for Π E can be significantly simplified throughsubstitutions of loop momenta before taking the cut leading to Γ. In fact, as shown inref. [13], Π E can be represented in terms of a small number of “master” sum-integrals. Forthe specific case of naive dimensional regularization of the γ -matrix, the expression readsΠ E | h ν B | = 2 (cid:16) e J a − J a − J b (cid:17) + 12 λ B (cid:16) −I b + I c + I d (cid:17) + 2 h t B N c h (cid:16)e I b − e I c − e I d (cid:17) + e I e − e I f + e I h i + g B + 3 g B h −I b + 2 (cid:16)e I e − I e + I g + I j (cid:17) − (cid:16) I h + b I h (cid:17) + ( D − (cid:16) I c + I d (cid:17) + ( D − (cid:16) I c − I d − e I b − b I c + b I d + b I h’ (cid:17)i , (2.6)where λ B , h t B , g B , g B denote the bare Higgs, top Yukawa, U(1) gauge, and SU(2) gaugecouplings, respectively; N c = 3 is the number of colours; and D ≡ − ǫ is the space-timedimensionality. The definitions of the independent master sum-integrals J a , ... are listed inappendix A. Renormalization of this bare expression is achieved through | h ν B | = | h ν (¯ µ ) | µ ǫ Z ν , Z ν ≡ π ) ǫ h h t N c −
34 ( g + 3 g ) i , (2.7)where µ is a scale parameter related to dimensional regularization (in the following inconse-quential factors µ ± ǫ are omitted); the MS scale is defined as ¯ µ ≡ πµ e − γ E ; and h t , g , g denote the renormalized top Yukawa, U(1) gauge, and SU(2) gauge couplings, respectively.Taking a cut like in eq. (2.2) leads to what we term master spectral functions: ρ I x ≡ Im[ I x ] k n →− i [ k + i + ] . (2.8)Numerical results for all of these are listed in appendix B, apart from ρ I j ; the case ρ I j ,together with the general methodology used, were discussed in ref. [15]. In the next sectionwe collect the results obtained after inserting the master spectral functions into the imaginarypart of eq. (2.6). 3 . Main results Each of the master spectral functions can be split into two parts: ρ I x = ρ vac I x + ρ T I x . (3.1)The first term must include all divergences, and may be chosen to include finite parts as well.We note that although denoted by ρ vac I x , this structure does have an overall temperature de-pendence, of the same functional form as the leading-order (LO) result which it renormalizes.The purely thermal part ρ T I x is, in contrast, finite and of a more complicated functional form.The divergences of the vacuum parts cancel against those in Z ν , eq. (2.7). Subsequently,with the choices of ρ vac I x explained in appendix B, we obtain a finite renormalized expressionfor the imaginary part of the retarded correlator:Im Π R | h ν (¯ µ ) | = M T πk ln (cid:20) sinh( k + /T )sinh( k − /T ) (cid:21) + 12 λ n − ρ T I b + ρ T I d o + 2 h t N c (cid:26) h ρ T e I b − ρ T e I d i − ρ T e I f + ρ T e I h − πM (4 π ) k Z k + k − d p n F ( k − p ) n B ( p ) n F ( k ) (cid:20) ln ( k + − p )( p − k − )¯ µ k M + 112 (cid:21)(cid:27) + g + 3 g (cid:26) − ρ T I b + 2 h − ρ T e I b + ρ T b I d − ρ T I d i + 3 ρ T I d + 2 h ρ T I g + ρ T b I h’ + ρ T I j i − h ρ T I h + ρ T b I h i + 3 πM (4 π ) k Z k + k − d p n F ( k − p ) n B ( p ) n F ( k ) (cid:20) ln ( k + − p )( p − k − )¯ µ k M + 416 (cid:21)(cid:27) . (3.2)Here we have defined k ± ≡ k ± k , M ≡ K = 4 k + k − > , (3.3)and n B , n F are the Bose and Fermi distributions.A numerical evaluation of this expression is shown in fig. 1 (parameters and the renor-malization scale are chosen as explained in appendix C). Results are displayed at severalmomenta on both sides of k = 3 T . It is clear that the naive loop expansion breaks down for M ∼ T . Based on this plot one might conclude that loop corrections decrease the productionrate but, as will become apparent in sec. 3.3, such a conclusion is premature.4 -1 M / T -3 -2 -1 I m Π R / | h ν | T k = 0.3Tk = 1.5Tk = 3Tk = 6Tk = 9T LO NLO
Figure 1:
The expression from eq. (3.2), in units of T , for k ≡ k + M . The couplings and therenormalization scale are fixed as specified in appendix C. The loop expansion breaks down at M ∼ T . In the non-relativistic limit M ≫ ( πT ) eq. (3.2) can be represented in more explicit form:up to and including O ( T /M ), this “OPE” expression reads [13]Im Π R | h ν (¯ µ ) | = M π (cid:26) − λM Z p n B p (3.4) − h t N c (cid:20) π ) (cid:18) ln ¯ µ M + 72 (cid:19) + k + k / M Z p p n F (cid:21) + ( g + 3 g ) (cid:20) π ) (cid:18) ln ¯ µ M + 296 (cid:19) + k + k / M Z p p (17 n F − n B )3 (cid:21)(cid:27) , where the integrals over the phase space distributions have elementary forms: Z p n B p = T , Z p n F p = T , Z p p n B = π T , Z p p n F = 7 π T . (3.5)An interesting question is how low the temperature should be in order for eq. (3.5) to yieldan accurate representation of the full result. It has been pointed out in ref. [15] that for aparticular 2-loop master spectral function, the non-relativistic approximation is only accuratefor M > ∼ T . However, in eq. (3.2) the 2-loop contributions are suppressed by couplings andloop factors, whereas for the 1-loop term the thermal corrections are exponentially small for k + , k − ≫ πT . Therefore, a somewhat better convergence may be expected.5 -1 M / T -3 -2 -1 I m Π R / | h ν | T NLOOPE ( M ) OPE ( M +T ) OPE ( M +T +T / M ) k = 3T -1 M / T -10-505 [NLO - OPE( M )] / OPE( T ) [NLO - OPE( M +T )] / OPE( T / M ) k = 3T Figure 2:
Left: The “OPE” expression from eq. (3.4), up to three consecutive orders as indicatedin the parentheses, compared with the “NLO” result from eq. (3.2), for k ≡ k + M . Right: Therelative difference between eqs. (3.2), (3.4). The couplings and the renormalization scale are fixed asspecified in appendix C. On the resolution of the logarithmic plot it seems that the OPE expressionis accurate for all M > ∼ T , however as shown by the right panel discrepancies with respect to theunexpanded expression are correctly represented only for M > ∼ T . The full and non-relativistic results are compared in fig. 2 for k = 3 T . We observe thatthe OPE-asymptotics appears to join the full expression at M > ∼ T . This is somewhat of anoptical illusion, however; as shown by the right panel, even the sign of the thermal correctionis correctly reproduced only for M > ∼ T , and the relative error decreases only for M > ∼ T .In any case the convergence is better in the full result than in the particular individual 2-loopmaster spectral function studied in ref. [15]. Let us return to the breakdown of the loop expansion, as illustrated in fig. 1. What happensis that for k ∼ πT but M ≪ πT , i.e. 2 k − = √ k + M − k ≈ M / (2 k ) ≪ πT , the LOterm is of magnitude Im Π LOR / | h ν | ∼ M ln( T /M ) whereas the NLO term is of magnitudeIm Π
NLOR / | h ν | ∼ g M T /k − ∼ g T /M , where g denotes a generic coupling. The relativemagnitude of the correction is ∼ g T /M , and consequently the loop expansion requiresresummation for M < ∼ g T . The most divergent NLO correction comes with a negative sign;it is related to Higgs mass thermal resummation, as will be discussed presently.For an even smaller M < ∼ gT , further resummations are needed. A Hard Thermal Loop(HTL) resummation was presented in ref. [19], however HTL resummation alone does not6ead to a consistent weak-coupling expansion for the present observable in the ultrarelativisticregime [6]. Indeed a systematic computation requires a Landau-Pomeranchuk-Migdal (LPM)resummation [7, 8]. This amounts to a solution of a Schr¨odinger-type equation with a light-cone potential, which implements a resummation of ladder diagrams, representing multiplesoft scatterings taking place within the average “formation time” of the ultrarelativistic right-handed neutrino being produced.In the present study, we will not implement LPM resummation (comments on this arehowever made in sec. 6). Rather, we follow the convention of capturing a sub-series of higherorder corrections through the assignment of thermal masses to otherwise massless particles.The concept of a thermal mass is ambiguous for particles of non-zero spin, depending e.g. onwhether soft ( k ≪ πT ) or hard ( k > ∼ πT ) excitations are considered. For the present problemthe latter kinematics is the relevant one, and then the thermal masses are those sometimescalled the “asymptotic” ones (for a concise summary see ref. [20]). It turns out that the Higgsmass thermal resummation (where no ambiguities appear) indeed consistently removes thedominant divergences from Im Π R in the regime M ∼ g T , as we now show.Let us start by simply inserting the masses m φ , m ℓ for the Higgs and for the left-handedleptons, respectively, and subsequently define a Euclidean correlator throughΠ tree E ( K ) ≡ | h ν | PZ P K · ( P − K )[ P + m φ ][( P − K ) + m ℓ ] . (3.6)This 1-loop sum-integral is labelled a “tree-level” contribution because of frequent conventionsin literature: its cut corresponds to 1 ↔ M > m φ + m ℓ , m φ > M + m ℓ , and m ℓ > M + m φ ,respectively. For the actual values relevant for the Standard Model, m φ = T (cid:16) g + 3 g + 43 h t N c + 8 λ (cid:17) , (3.7) m ℓ = T (cid:0) g + 3 g (cid:1) , (3.8)only the first two channels can get realized. In each channel, the angular integral between thedirections of p and k can be carried out by taking E pk ≡ q ( p − k ) + m ℓ as an integrationvariable, and subsequently the integral over the radial direction can also be performed, bytaking E p ≡ q p + m φ as a variable. For k = √ k + M the result reads Im Π tree R | h ν | = ( M − m φ + m ℓ ) T πk ln sinh (cid:16) E max T (cid:17) cosh (cid:16) k − E min T (cid:17) sinh (cid:16) E min T (cid:17) cosh (cid:16) k − E max T (cid:17) × h θ (cid:0) M − m φ − m ℓ (cid:1) − θ (cid:0) m φ − m ℓ − M (cid:1) − θ (cid:0) m ℓ − m φ − M (cid:1)i , (3.9) Equivalent expressions can be found in literature. E max(min) ≡ k ( M + m φ − m ℓ ) ± k ∆( M, m φ , m ℓ )2 M , (3.10)∆( M, m φ , m ℓ ) ≡ q M − M ( m φ + m ℓ ) + ( m φ − m ℓ ) . (3.11)Suppose now that we are in the regime m φ , m ℓ ≪ M , and expand to first order in thesmall masses. Then E min ≈ k − + m φ k − − m ℓ k + , E max ≈ k + + m φ k + − m ℓ k − , (3.12)and eq. (3.9) becomesIm Π treeR | h ν | ≈ ( M − m φ + m ℓ ) T πk ln (cid:26) sinh( k + /T )sinh( k − /T ) (cid:27) + M πk (cid:26) m φ (cid:20) n B ( k + ) − n F ( k − )4 k + − n B ( k − ) − n F ( k + )4 k − (cid:21) + m ℓ (cid:20) n B ( k − ) − n F ( k + )4 k + − n B ( k + ) − n F ( k − )4 k − (cid:21)(cid:27) + O ( m φ,ℓ ) . (3.13)Inserting the expressions from eqs. (3.7), (3.8) this is seen to agree exactly with the sum of all ρ I b ’s and ρ I d ’s in eq. (3.2), cf. eqs. (B.15), (B.22). These master spectral functions includethe only quadratically divergent structures in the limit k − ≪ πT as can be deduced from theright panels of figs. 6–11. The most divergent terms are ∼ M n B ( k − ) /k − and are related tothe Higgs mass resummation as is clearly visible from the second row of eq. (3.13).We now define a “resummed” result by accounting for all ρ I b ’s and ρ I d ’s of eq. (3.2) throughthe thermal masses:Im Π resum R | h ν (¯ µ ) | ≡ Im Π tree R | h ν (¯ µ ) | + 2 h t N c (cid:26) − ρ T e I f + ρ T e I h − πM (4 π ) k Z k + k − d p n F ( k − p ) n B ( p ) n F ( k ) (cid:20) ln ( k + − p )( p − k − )¯ µ k M + 112 (cid:21)(cid:27) + g + 3 g (cid:26) h ρ T I g + ρ T b I h’ + ρ T I j i − h ρ T I h + ρ T b I h i + 3 πM (4 π ) k Z k + k − d p n F ( k − p ) n B ( p ) n F ( k ) (cid:20) ln ( k + − p )( p − k − )¯ µ k M + 416 (cid:21)(cid:27) , (3.14)where Im Π tree R is the tree-level result from eq. (3.9). Note that the remaining master spectralfunctions continue to be evaluated without masses (in these spectral functions masses amountto higher-order corrections). 8 -1 M / T -3 -2 -1 I m Π R / | h ν | T k = 0.3Tk = 1.5Tk = 3Tk = 6Tk = 9T resum tree Figure 3:
The mass-resummed expression from eq. (3.14) (thick lines), in units of T , for k ≡ k + M ,versus the tree-level result from eq. (3.9) (thin lines). The couplings and the renormalization scale arefixed as specified in appendix C. The cusp is expected to be removed through higher-order corrections,but the overall magnitude of the mass-resummed result is already in qualitative agreement with theultrarelativistic results of refs. [7, 8] (cf. fig. 5). The tree and mass-resummed spectral functions are shown in fig. 3. It is apparent thatthe downwards divergence seen in fig. 1 is a reflection of thermal mass generation; after thiseffect has been taken into account, the other NLO terms show an enhancement. However themass resummation implemented does not capture all the terms that need to be resummed for
M < ∼ gT ; indeed, as has been demonstrated with the case of hot QCD [21] and more recentlywith the problem at hand [7, 8], cusps such as those seen in fig. 3 are also removed througha systematic resummation of all corrections pertinent to the ultrarelativistic regime. It isnevertheless interesting that the order of magnitude of the mass-resummed result is not unlikethat found in refs. [7, 8] (cf. sec. 5).
4. Spectra and spectral functions
We have already observed that, for a fixed K = M , the non-covariant dependence of Im Π R on the spatial momentum k is small, cf. fig. 3. This is illustrated again in fig. 4(left), for a The mass-resummed result of the current study still diverges logarithmically for
M/T →
0; this originatesfrom the master spectral function ρ T b I h’ , cf. fig. 11(right). The divergence is also removed by resummations. -1 k / T -3 -2 -1 I m Π R / | h ν | T tree-levelmass-resummed M = 10TM = 5TM = 2TM = 1TM = 0.5T -1 k / T -10 -9 -8 -7 -6 -5 -4 -3 -2 d k γ + / | h ν | T M = 10TM = 5TM = 2TM = 1TM = 0.5T
Figure 4:
Left: The k -dependence of eq. (3.14) (thick lines), for selected M and k ≡ k + M . Thinlines indicate the “tree-level” result from eq. (3.9). In the regime of validity of the computation, i.e. M > ∼ πT , k -dependence is quite modest. Right: The corresponding production spectra, ∂ k γ + fromeq. (4.1). The couplings and the renormalization scale are fixed as specified in appendix C. number of different M/T . Subsequently we plot the whole spectra according to eq. (2.4), i.e. ∂ k γ + ≡ k n F ( √ k + M ) π √ k + M Im Π R , (4.1)for a few selected M/T , in fig. 4(right). Obviously the latter results display a much stronger k -dependence than Im Π R , however this emerges through the trivial “kinematic” structuresshown in eq. (4.1), rather than complicated plasma physics determining Im Π R .
5. Total production rate
According to eq. (2.4), the total right-handed neutrino production rate reads γ + = Z ∞ d k ∂ k γ + , (5.1)where ∂ k γ + is the differential production rate from eq. (4.1). Given the small k -dependenceas seen in fig. 4(left), a good approximation for the total production rate can be obtained byevaluating the “expensive” Im Π R only at some typical momentum, for instance k ∼ T . Theaccuracy of this approximation is illustrated in fig. 5, and found to be in general excellent.It should be noted that in the non-relativistic regime the actual average momentum is k ∼ -1 M / T -6 -5 -4 -3 -2 γ + / | h ν | T tree-levelmass-resummed with k = 3Tmass-resummed with full k-dep.LPM-resummed Figure 5:
The total production rate based on evaluating Im Π R at k = 3 T and taking it as a constantotherwise (red dashed line), compared with results obtained with the full k -dependence included (blackcircles). The couplings and the renormalization scale are fixed as specified in appendix C. For M < ∼ T we also compare with the complete LPM-resummed result from ref. [8] (dash-dotted line). √ M T rather than k ∼ T , but the approximation does not lose its accuracy, because the k -dependence of Im Π R becomes even less significant for M ≫ πT (cf. fig. 4(left)).In fig. 5 the total production rate is also compared with the LPM-resummed result fromref. [8]. Although our expression is not reliable for M ≪ πT and the result of ref. [8] is notreliable for M > ∼ πT , it is remarkable how well the two appear to extrapolate towards eachother. (Our results are closer to the systematic analysis of ref. [8] than the phenomenologicalapproach of ref. [22].) In principle it should also be possible to combine the two results intoan expression applicable for a general M/πT (some comments are made in sec. 6), howeverimplementing this in practice necessitates a dedicated separate study.
6. Conclusions and outlook
The purpose of this paper has been to extend previous NLO results for the right-handedneutrino production rate up to higher temperatures, into the so-called relativistic regime inwhich the temperature is of a similar magnitude as the mass of the right-handed neutrinos. Inthe so-called strong washout scenario of leptogenesis, the right-handed neutrinos equilibrateinitially, whereby no lepton asymmetry exists in a certain temperature range; it is generated atlow temperatures when the right-handed neutrinos chemically decouple and can subsequently11ecay. It is conceivable that most of the decays take place in a non-relativistic regime, howeverit is not clear a priori how accurate computations based on the non-relativistic approximationare, because thermal effects are only power-suppressed [14]. The results obtained here suggestthat corrections are substantial for
T > ∼ M/
4. If a significant contribution arises from thisrange, then the results of the current study may be used for a more precise analysis.One finding of the current investigation is that in the relativistic and non-relativisticregimes, the non-covariant dependence of the retarded correlator denoted by Im Π R on thespatial momentum with respect to the heat bath is quite modest (cf. fig. 4(left)). Thereforethe “expensive” part of the computation needs to be carried out only for a specific chosen k , for instance k = 3 T , in order to determine the overall magnitude of Im Π R . When in-serted into the proper overall relations, this information is sufficient for determining the totalproduction rate with good accuracy (cf. fig. 5).Apart from the strong washout scenario of leptogenesis, another possibility is that theright-handed neutrinos never equilibrate chemically; one then speaks of a weak washoutscenario (cf. e.g. ref. [23]). In this case even ultrarelativistic temperatures play a role, andresummations are needed for consistent results [7, 8]. Although the results of the currentpaper lose their validity when approaching the ultrarelativistic regime, they do allow usto anticipate some features of the corresponding expressions, such as that there is no gapbetween the two possible 1 ↔ Nevertheless theproblem may be worth giving a go; for practical applications a result valid for all temperatureswould clearly be quite convenient. (Ultimately NLO corrections should also be worked outfor the LPM regime; they are likely to be suppressed only by p g there [25].) Schematically, omitting Lorentz-violating structures, the NLO computation we have carried out is of theform Im Π
R,UV ∼ M [ φ ( T /M ) + g φ ( T /M ) + ... ] whereas any resummed result contains all orders in g :Im Π R,IR ∼ M [ χ ( T /M , g )+ ... ]. For combining the two, the resummed result has to be re-expanded in g ,in order to cancel the terms from Im Π R,UV that it resums: Im Π
R,full ∼ M { χ ( T /M , g ) + φ ( T /M ) − χ ( T /M ,
0) + g [ φ ( T /M ) − ∂ g χ ( T /M , ... } . The determination of the latter subtraction termis not quite trivial, because χ ( T /M , g ) contains parts only available through a numerical solution of aninhomogeneous Schr¨odinger-type equation and is otherwise complicated as well. Acknowledgements
I am grateful to Dietrich B¨odeker for helpful discussions. This work was partly supported bythe Swiss National Science Foundation (SNF) under grant 200021-140234.
Appendix A. Definitions of master sum-integrals
Denoting by Σ R P and Σ R { P } sum-integrals over bosonic and fermionic Matsubara four-momenta,the master sum-integrals entering the computation are defined as follows [13]: J a ≡ PZ P P , (A.1) e J a ≡ PZ { P } P , (A.2) J b ≡ PZ P K P ( P − K ) , (A.3) I b ≡ PZ P Q Q P ( P − K ) , (A.4) e I b ≡ PZ P { Q } Q P ( P − K ) , (A.5) I c ≡ PZ P Q Q P , (A.6) e I c ≡ PZ P { Q } Q P , (A.7) b I c ≡ PZ { P } Q Q P , (A.8) I c ≡ PZ { P Q } Q P , (A.9) I d ≡ PZ P Q K Q P ( P − K ) , (A.10)13 I d ≡ PZ P { Q } K Q P ( P − K ) , (A.11) b I d ≡ PZ { P } Q K Q P ( P − K ) , (A.12) I d ≡ PZ { P Q } K Q P ( P − K ) , (A.13) I e ≡ PZ P Q Q P ( P − Q ) , (A.14) e I e ≡ PZ P { Q } Q P ( P − Q ) , (A.15) I f ≡ lim λ → PZ P Q Q [( Q − P ) + λ ]( P − K ) , (A.16) e I f ≡ lim λ → PZ P { Q } Q [( Q − P ) + λ ]( P − K ) , (A.17) I g ≡ PZ P Q K P ( P − K ) Q ( Q − K ) , (A.18) I h ≡ lim λ → PZ P Q K Q P [( Q − P ) + λ ]( P − K ) , (A.19) e I h ≡ lim λ → PZ P { Q } K Q P [( Q − P ) + λ ]( P − K ) , (A.20) b I h ≡ lim λ → PZ { P } Q K Q P [( Q − P ) + λ ]( P − K ) , (A.21) b I h’ ≡ lim λ → PZ { P } Q K · QQ P [( Q − P ) + λ ]( P − K ) , (A.22) I j ≡ lim λ → PZ P Q K Q P [( Q − P ) + λ ]( P − K ) ( Q − K ) . (A.23)In order to handle the different statistics simultaneously, we introduce a generic labelling oflines (with individual propagators omitted or doubled in some cases): σ ,K σ ,Kσ ,Qσ , Q − Pσ ,P − Kσ ,P σ ,Q − K . (A.24)14he labels σ , ..., σ equal +1 for bosons and − σ σ σ ) has been chosenfor this task; subsequently σ = σ σ , σ = σ σ , σ = σ σ . (A.25)The spectral functions are obtained from eq. (2.8) which is commensurate with the spectralrepresentation Z ∞−∞ d k π ρ I x k − ik n = I x . (A.26)In practice, the spectral function can be read from an imaginary-time correlator by partialfractioning its dependence on k n and by then replacing1 − ik n + C → πδ ( − k + C ) . (A.27)Note that structures containing no or polynomial k n -dependence yield a vanishing spectralfunction according to eq. (2.8); in these cases eq. (A.26) needs to be modified by “contactterms” (a discussion can be found e.g. in ref. [14]). Appendix B. Results for master spectral functions in time-like domain
In the following spectral functions are listed for the range k > k >
0. Results for k → − k follow from antisymmetry, whereas the space-like domain 0 < k < k has not been workedout here, although thermal spectral functions can be non-zero there as well. B.1. ρ J a Since J a and e J a are independent of the external momentum, there is no cut: ρ J a = ρ e J a = 0 . (B.1) B.2. ρ J b For the case in eq. (A.3) we get, after carrying out the Matsubara sum, J b = Z p K ǫ p ǫ pk (cid:26)(cid:20) ik n + ǫ p + ǫ pk + 1 − ik n + ǫ p + ǫ pk (cid:21)h n σ ( ǫ p ) + n σ ( ǫ pk ) i + (cid:20) ik n − ǫ p + ǫ pk + 1 − ik n − ǫ p + ǫ pk (cid:21)h n σ ( ǫ p ) − n σ ( ǫ pk ) i(cid:27) , (B.2)where we used the labelling of eq. (A.24), denoted ǫ p ≡ p ≡ | p | , ǫ pk ≡ | p − k | , (B.3)15nd defined n σ ( ǫ ) ≡ σe ǫ/T − σ , n − σ ( ǫ ) = σe ǫ/T − , Z ǫ d ǫ ′ n σ ( ǫ ′ ) = T ln (cid:0) − σe − ǫ/T (cid:1) , (B.4)which satisfies n + = n B and n − = − n F .Taking the cut, only one of the four channels contributes for k > k , and the spectralfunction reads ρ J b = − Z p π K ǫ p ǫ pk δ ( k − ǫ p − ǫ pk ) h n σ ( ǫ p ) + n σ ( ǫ pk ) i . (B.5)It is often convenient to employ the alternative representation1 + n σ ( ǫ p ) + n σ ( k − ǫ p ) = n − σ ( k ) n σ ( k − p ) n σ ( p ) , (B.6)where we made use of σ σ = σ .Because the leading-order contribution gets multiplied by a counterterm, we need to de-termine it up to O ( ǫ ). The integration measure reads, in d = 3 − ǫ spatial dimensions, Z p = (4 π ) ǫ π Γ(1 − ǫ ) Z ∞ d p p − ǫ Z +1 − d z (1 − z ) − ǫ , (B.7)where z ≡ p · k /pk . The integral over z can be converted into one over ǫ pk throughd z = − ǫ pk d ǫ pk pk , (B.8)and the Dirac- δ gets realized for k − < p < k + , with k ± defined according to eq. (3.3).Recalling the constraint δ ( k − p − ǫ pk ), the function appearing in the angular integration isconveniently expressed as 1 − z = K ( k + − p )( p − k − ) k p . (B.9)Introducing the MS scheme scale parameter, ¯ µ , by inserting1 = µ − ǫ ¯ µ ǫ e ǫγ E (4 π ) ǫ , (B.10)and suppressing the inconsequential µ − ǫ , we thereby obtain ρ J b = − π K (4 π ) k e ǫγ E Γ(1 − ǫ ) Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) (cid:20) ¯ µ k K ( k + − p )( p − k − ) (cid:21) ǫ = − π K (4 π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) × (cid:20) ǫ ln ¯ µ K + ǫ ln k ( k + − p )( p − k − ) + O ( ǫ ) (cid:21) . (B.11)16he remaining integral is easily carried out in the term of O ( ǫ ): ρ J b = − π K (4 π ) k (cid:26) T ln (cid:18) e k + /T + σ e − k + /T − σ − σ e k − /T + σ e − k − /T − σ − σ (cid:19)(cid:20) ǫ (cid:18) ln ¯ µ K + 2 (cid:19)(cid:21) + ǫ Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) (cid:20) ln k ( k + − p )( p − k − ) − (cid:21)(cid:27) + O ( ǫ ) . (B.12)For k + , k − ≫ πT , the second row vanishes up to exponentially small corrections so that, inaccordance with ref. [13], ρ J b k + ,k − ≫ πT ≈ − K π (cid:20) ǫ (cid:18) ln ¯ µ K + 2 (cid:19) + O (cid:0) ǫ (cid:1)(cid:21) . (B.13)The specific statistics relevant for the current paper are J b ⇔ ( σ σ σ | σ ) = (+ − + |− ) . (B.14)Here and in the following, the values of selected non-independent indices as obtained fromeq. (A.25) have also been indicated to the right of the vertical line. B.3. ρ I b The spectral function corresponding to eq. (A.4) can be written generically as ρ I b = − ρ J b K Z q n σ ( q ) q = T πk ln (cid:18) e k + /T + σ e − k + /T − σ − σ e k − /T + σ e − k − /T − σ − σ (cid:19) Z q n σ ( q ) q . (B.15)For k + , k − ≫ πT this goes over into [13] ρ I b k + ,k − ≫ πT ≈ π Z q n σ ( q ) q , (B.16)up to exponentially small corrections. The value of the remaining integral is given in eq. (3.5).The specific statistics for the current problem are I b ⇔ ( σ σ σ | σ σ ) = (+ − + | + − ) , (B.17) e I b ⇔ ( σ σ σ | σ σ ) = (+ − −| − − ) . (B.18)Defining a typical thermal momentum through k ( M ) ≡ R ∞ d k k exp( − √ k + M T ) R ∞ d k k exp( − √ k + M T ) = 3 M T K ( MT ) K ( MT ) , (B.19)numerical results for ρ I b are shown in fig. 6. 17 -1 k / T ρ I b / T T k / T = 0.0k / T > 0.0 (+++)0.1 0.3 1 3 864 -1 M / T -0.005 ρ I b / T T (+ -- )(+ - +) exactOPE ( T + T /M ) k = k av (M); k = √ (k + M ) Figure 6:
Left: The spectral function ρ T I b ≡ ρ I b with the purely bosonic statistics ( σ σ σ ) = (+++),for k ≥ k + 0 . T , compared with the zero-momentum limit determined in ref. [24]. Right: Thespectral function ρ I b with the momentum of eq. (B.19) and statistics of eqs. (B.17), (B.18) as afunction of M/T , compared with the OPE-asymptotics from eq. (B.16).
B.4. ρ I c Since all versions of I c are independent of the external momentum, there is no cut: ρ I c = ρ e I c = ρ b I c = ρ I c = 0 . (B.20) B.5. ρ I d The derivation of the spectral function corresponding to eq. (A.10) follows from that for ρ J d in sec. B.2; we simply give the line with momentum P a mass, λ , and take a derivative withrespect to the mass. If we change variables from p to E p ≡ p p + λ , (B.21)then λ only appears in the boundaries of the E p -integration. No terms of O ( ǫ ) are needed,so the general result can be expressed as ρ I d = π K (4 π ) k Z q n σ ( q ) q dd λ (cid:26)Z k + + λ k + k − + λ k − d E p h n σ ( E p ) + n σ ( k − E p ) i(cid:27) λ =0 = π K n − σ ( k )(4 π ) k (cid:20) n σ ( k + ) n σ ( k − )4 k + − n σ ( k − ) n σ ( k + )4 k − (cid:21) Z q n σ ( q ) q . (B.22)18 -1 k / T -0.10-0.08-0.06-0.04-0.020.00 ρ I d / T T k / T = 0.0k / T > 0.0 (+++)0.1 0.3 1 3 864 -1 M / T -0.005 ρ I d / T T (+ -- )(+ - +)( - ++)( - + - ) exactOPE ( T + T /M ) k = k av (M); k = √ (k + M ) Figure 7:
Left: The spectral function ρ I d ≡ ρ T I d with the purely bosonic statistics ( σ σ σ ) = (+++),for k ≥ k + 0 . T , compared with the zero-momentum limit determined in ref. [24]. Right: Thespectral function ρ I d with the momentum of eq. (B.19) and statistics of eqs. (B.24)–(B.27) as afunction of M/T , compared with the OPE-asymptotics from eq. (B.23).
The value of the remaining integral is given in eq. (3.5). For k + , k − ≫ πT the asymptoticsreads [13] ρ I d k + ,k − ≫ πT ≈ − π Z q n σ ( q ) q , (B.23)with exponentially small corrections. The specific statistics for the current problem are I d ⇔ ( σ σ σ | σ σ ) = (+ − + | + − ) , (B.24) e I d ⇔ ( σ σ σ | σ σ ) = (+ − −| − − ) , (B.25) b I d ⇔ ( σ σ σ | σ σ ) = ( − + −| + − ) , (B.26) I d ⇔ ( σ σ σ | σ σ ) = ( − + + | − − ) . (B.27)A numerical evaluation in shown in fig. 7. B.6. ρ I e Since both versions of I e are independent of the external momentum, there is no cut: ρ I e = ρ e I e = 0 . (B.28)19 .7. ρ I f After carrying out the Matsubara sums, the expression for I f reads I f = lim λ → Z p , q ǫ q ǫ pk E qp (cid:26) − ik n + ǫ pk + ǫ q + E qp (cid:16)(cid:2) n σ ( ǫ pk ) + n σ ( ǫ q ) (cid:3)(cid:2) n σ ( E qp ) (cid:3) + n σ ( ǫ pk ) n σ ( ǫ q ) (cid:17) + 1 − ik n − ǫ pk + ǫ q + E qp (cid:16) n σ ( ǫ pk ) (cid:2) n σ ( ǫ q ) + n σ ( E qp ) (cid:3) − n σ ( ǫ q ) n σ ( E qp ) (cid:17) + 1 − ik n + ǫ pk − ǫ q + E qp (cid:16) n σ ( ǫ q ) (cid:2) n σ ( ǫ pk ) + n σ ( E qp ) (cid:3) − n σ ( ǫ pk ) n σ ( E qp ) (cid:17) + 1 − ik n + ǫ pk + ǫ q − E qp (cid:16) n σ ( E qp ) (cid:2) n σ ( ǫ pk ) + n σ ( ǫ q ) (cid:3) − n σ ( ǫ pk ) n σ ( ǫ q ) (cid:17) (cid:27) + ( ik n → − ik n ) . (B.29)The corresponding spectral function is obtained from eq. (A.27). In addition to the energiesof eq. (B.3), a further variable appears here which contains the infrared regulator λ : E qp ≡ p ( q − p ) + λ . (B.30)The four channels of eq. (B.29) represent real corrections and were labelled (r1)–(r4) inref. [15]; the 2 ↔ ↔ σ , σ shrunk to points.In the notation of ref. [15], eq. (4.28) now reads D Φ r1 ( k − p | q | p − q |· ) E = − D Φ r2 ( p − k | q | p − q |· ) E = − D Φ r3 ( k − p | − q | p − q |· ) E = − D Φ r4 ( k − p | q | q − p |· ) E = n σ ( k − p ) n σ ( q ) n σ ( p − q )2 n σ ( k ) . (B.31)The integration over p , with ranges as specified in ref. [15], is trivial. Factoring out π n σ ( k − p ) n σ ( q ) n σ ( p − q )(4 π ) k n σ ( k ) , (B.32)20he λ → p − q ) , (B.33)(b) : 4( k + − q ) , (B.34)(b) : 4( p − k − ) , (B.35)(c) = − (h) = − (h) = − (j) : 4( k + − k − ) , (B.36)(˜c) : 4 k − , (B.37)(d) : 4( k + + q − p ) , (B.38)(e) = (f) : 4( k + + k − − p ) , (B.39)(e) = (f) : 4 q , (B.40)(g) = (g) : 4( k − + q − p ) , (B.41)(i) = (k) : 4( k − − q ) , (B.42)(i) = (k) : 4( p − k + ) . (B.43)For k + , k − ≫ πT , the ultraviolet asymptotics of the spectral function reads [13] ρ I f k + ,k − ≫ πT ≈ π K (4 π ) ×
12 + Z p (cid:26) n σ + n σ + n σ πp (cid:27) . (B.44)The specific statistics for the current problem are I f ⇔ ( σ σ σ | σ σ ) = (+ − + | + − ) , (B.45) e I f ⇔ ( σ σ σ | σ σ ) = (+ − −| − − ) . (B.46)For numerical evaluation, we have reflected the final integral to the domain defined in fig. 6of ref. [15]. The corresponding integrand is not shown explicitly, since no substantial cancel-lations take place in the reflections. In addition, we always separate a “vacuum-like” part asin eq. (3.1), with a coefficient given by the leading term in eq. (B.44): ρ vac I f ≡ π K T (4 π ) k ln (cid:18) e k + /T + σ e − k + /T − σ − σ e k − /T + σ e − k − /T − σ − σ (cid:19) × . (B.47)It is the thermal part which is plotted numerically in fig. 8, and compared with the OPEasymptotics from eq. (B.44) as well as with the k → -1 k / T ρ I f / T T k / T = 0.0k / T > 0.0 (+++)0.1 0.3 1 3 864 -1 M / T -0.005 ρ I f / T T (+ -- )(+ - +) exactOPE ( T + T /M ) k = k av (M); k = √ (k + M ) Figure 8:
Left: The thermal part of ρ I f with the purely bosonic statistics ( σ σ σ ) = (+ + +), for k ≥ k + 0 . T , compared with the zero-momentum limit determined in ref. [24]. Right: The thermalpart of ρ I f with the momentum of eq. (B.19) and statistics of eqs. (B.45), (B.46) as a function of M/T ,compared with the OPE-asymptotics from eq. (B.44).
B.8. ρ I g For I g of eq. (A.18), Matsubara sums lead to I g = Z p K ǫ p ǫ pk (cid:26)(cid:20) ik n + ǫ p + ǫ pk + 1 − ik n + ǫ p + ǫ pk (cid:21)h n σ ( ǫ p ) + n σ ( ǫ pk ) i + (cid:20) ik n − ǫ p + ǫ pk + 1 − ik n − ǫ p + ǫ pk (cid:21)h n σ ( ǫ p ) − n σ ( ǫ pk ) i(cid:27) × Z q ǫ q ǫ qk (cid:26)(cid:20) ik n + ǫ q + ǫ qk + 1 − ik n + ǫ q + ǫ qk (cid:21)h n σ ( ǫ q ) + n σ ( ǫ qk ) i + (cid:20) ik n − ǫ q + ǫ qk + 1 − ik n − ǫ q + ǫ qk (cid:21)h n σ ( ǫ q ) − n σ ( ǫ qk ) i(cid:27) . (B.48)Taking the cut like in eq. (A.27), the result factorizes into a product of a structure like ineq. (B.5), and a principal value integral over the other part; the latter corresponds to a virtualloop correction, of the type illustrated in fig. 3 of ref. [15] but with the line carrying the index σ shrunk to a point. It has a divergent vacuum contribution, Z q ǫ q ǫ qk P (cid:20) k + ǫ q + ǫ qk + 1 − k + ǫ q + ǫ qk (cid:21) = 1(4 π ) (cid:18) ǫ + ln ¯ µ K + 2 (cid:19) , (B.49)22 -1 k / T ρ I g / T T k / T = 0.0k / T > 0.0 (+++)0.1 0.3 1 3 864 -1 M / T -0.005 ρ I g / T T (+ - +) exactOPE ( T ) OPE ( T + T /M ) k = k av (M); k = √ (k + M ) Figure 9:
Left: The thermal part of ρ I g (eq. (B.51)) with the purely bosonic statistics ( σ σ σ ) =(+ + +), for k ≥ k + 0 . T , compared with the zero-momentum limit determined in ref. [24]. Right:The thermal part of ρ I g with the momentum of eq. (B.19) and statistics of eq. (B.53) as a functionof M/T , compared with the OPE-asymptotics from eq. (B.52). as well as a finite thermal part. In the latter the angular integral is doable if we substi-tute integration variables so as to always have ǫ p or ǫ q as the argument of the phase spacedistribution. Recalling the O ( ǫ )-part from eq. (B.12), we get ρ vac I g ≡ − π K T (4 π ) k ln (cid:18) e k + /T + σ e − k + /T − σ − σ e k − /T + σ e − k − /T − σ − σ (cid:19)(cid:18) ǫ + 2 ln ¯ µ K + 4 (cid:19) + ( σ ↔ σ , σ ↔ σ ) , (B.50) ρ T I g = π K (4 π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) (cid:26) ln ( k + − p )( p − k − ) k + 2+ Z ∞ d qk (cid:0) n σ + n σ (cid:1) ( q ) ln (cid:12)(cid:12)(cid:12)(cid:12) ( q − k + )( q + k − )( q + k + )( q − k − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) + ( σ ↔ σ , σ ↔ σ ) . (B.51)For k + , k − ≫ πT , the ultraviolet asymptotics reads [13] ρ I g k + ,k − ≫ πT ≈ − K π ) (cid:18) ǫ + 2 ln ¯ µ K + 4 (cid:19) + Z p X i =1 (cid:26) n σ i πp + p n σ i π k + k K (cid:27) . (B.52)The specific statistics for the current case are I g ⇔ ( σ σ σ | σ σ σ ) = (+ − + | + −− ) . (B.53)23he thermal part of ρ I g is plotted numerically in fig. 9, and compared with the OPE-asymptotics from eq. (B.52) as well as with the k → B.9. ρ I h The spectral functions corresponding to I h , e I h , and b I h can be handled simultaneously withthe labelling of eq. (A.24), where now the line with the index σ is absent. The expressionafter carrying out the Matsubara sums reads I h = lim λ → Z p , q K ǫ q ǫ pk E qp (cid:26) − ik n + ǫ pk + ǫ q + E qp (cid:2) n σ ( ǫ pk ) + n σ ( ǫ q ) (cid:3)(cid:2) n σ ( E qp ) (cid:3) + n σ ( ǫ pk ) n σ ( ǫ q ) ǫ p − ( ǫ q + E qp ) + 1 − ik n − ǫ pk + ǫ q + E qp n σ ( ǫ pk ) (cid:2) n σ ( ǫ q ) + n σ ( E qp ) (cid:3) − n σ ( ǫ q ) n σ ( E qp ) ǫ p − ( ǫ q + E qp ) + 1 − ik n + ǫ pk − ǫ q + E qp n σ ( ǫ q ) (cid:2) n σ ( ǫ pk ) + n σ ( E qp ) (cid:3) − n σ ( ǫ pk ) n σ ( E qp ) ǫ p − ( ǫ q − E qp ) + 1 − ik n + ǫ pk + ǫ q − E qp n σ ( E qp ) (cid:2) n σ ( ǫ q ) + n σ ( ǫ pk ) (cid:3) − n σ ( ǫ q ) n σ ( ǫ pk ) ǫ p − ( ǫ q − E qp ) (cid:27) + lim λ → Z p K ǫ p ǫ pk (cid:20) n σ ( ǫ p ) + n σ ( ǫ pk ) − ik n + ǫ p + ǫ pk + n σ ( ǫ p ) − n σ ( ǫ pk ) ik n − ǫ p + ǫ pk (cid:21) × Z q (cid:20) n σ ( ǫ q ) + n σ ( E qp )4 ǫ q E qp (cid:18) ǫ p + ǫ q + E qp − ǫ p − ǫ q − E qp (cid:19) + n σ ( ǫ q ) − n σ ( E qp )4 ǫ q E qp (cid:18) ǫ p − ǫ q + E qp − ǫ p + ǫ q − E qp (cid:19) (cid:21) + ( ik n → − ik n ) . (B.54)Taking the cut like in eq. (A.27), the first four structures here are real corrections, the lasttwo are virtual corrections. The corresponding scattering processes can be depicted like infig. 3 of ref. [15], with internal propagators carrying the index σ shrunk to points.For the real corrections, eq. (4.28) of ref. [15] now reads D Φ r1 ( k − p | q | p − q |· ) E = − D Φ r2 ( p − k | q | p − q |· ) E = − D Φ r3 ( k − p | − q | p − q |· ) E = − D Φ r4 ( k − p | q | q − p |· ) E = n σ ( k − p ) n σ ( q ) n σ ( p − q ) n σ ( k ) P (cid:26) K p − p ) (cid:27) . (B.55)24he integration over p , with ranges as specified in ref. [15], is readily carried out, leading tosimple logarithms. Renaming subsequently p → p , and factoring out π K n σ ( k − p ) n σ ( q ) n σ ( p − q )(4 π ) k n σ ( k ) , (B.56)the λ → p ln (cid:12)(cid:12)(cid:12)(cid:12) p ( p − q ) λ q (cid:12)(cid:12)(cid:12)(cid:12) , (B.57)(b) : 1 p ln (cid:12)(cid:12)(cid:12)(cid:12) k + ( p − q ) q ( p − k + ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.58)(b) : 1 p ln (cid:12)(cid:12)(cid:12)(cid:12) p ( p − q )( p − k − ) λ k − (cid:12)(cid:12)(cid:12)(cid:12) , (B.59)(c) = − (h) = − (h) = − (j) : 1 p ln (cid:12)(cid:12)(cid:12)(cid:12) k + ( p − k − ) k − ( p − k + ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.60)(˜c) : 1 p ln (cid:12)(cid:12)(cid:12)(cid:12) k − p ( p − q ) λ ( p − k − ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.61)(d) : 1 p ln (cid:12)(cid:12)(cid:12)(cid:12) qk + ( p − q )( p − k + ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.62)(e) = (f) : 1 p ln (cid:12)(cid:12)(cid:12)(cid:12) k + k − ( p − k + )( p − k − ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.63)(e) = (f) : 1 p ln (cid:12)(cid:12)(cid:12)(cid:12) pqλ (cid:12)(cid:12)(cid:12)(cid:12) , (B.64)(g) = (g) : 1 p ln (cid:12)(cid:12)(cid:12)(cid:12) qk − ( p − q )( p − k − ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.65)(i) = (k) : 1 p ln (cid:12)(cid:12)(cid:12)(cid:12) k − ( p − q ) q ( p − k − ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.66)(i) = (k) : 1 p ln (cid:12)(cid:12)(cid:12)(cid:12) p ( p − q )( p − k + ) λ k + (cid:12)(cid:12)(cid:12)(cid:12) . (B.67)As far as the virtual corrections go, they include a divergent part: ρ I h ∋ Z p π K δ (cid:0) k − ǫ p − ǫ pk (cid:1) n σ ( k − p ) n σ ( p )4 ǫ p ǫ pk n σ ( k ) Z q P (cid:26) ǫ q E qp ǫ q + E qp ǫ p − ( ǫ q + E qp ) (cid:27) = ρ J b × Re Z Q Q [( Q − P ) + λ ] (cid:12)(cid:12)(cid:12)(cid:12) p n = − iǫ p . (B.68)Here we made use of the fact that the vacuum integral is independent of p :Re Z Q Q [( Q − P ) + λ ] (cid:12)(cid:12)(cid:12)(cid:12) p n = − iǫ p = 1(4 π ) (cid:18) ǫ + ln ¯ µ λ + 1 (cid:19) . (B.69)25t is helpful, however, not to separate the vacuum integral from the outset, but rather to treatthe structures + n σ ( ǫ q ) and + n σ ( E qp ) identifiable on the last two rows of eq. (B.54) assingle entities for as long as possible. They can then be combined with the real corrections,cancelling all λ -dependence, which appears at moderate values of | p | , | q | < ∼ k . Only the large- q range requires a more careful treatment, and we return to this presently.In order to implement this strategy, we first carry out angular integrals and substitutevariables, obtaining [the virtual correction part is denoted by ρ (v) I h , and ≃ is a reminder of thedivergences appearing at large q and of the omission of terms of O ( ǫ )] ρ (v) I h ≃ − π K (4 π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) × (cid:26)Z ∞−∞ d qp h
12 + n σ ( q ) i ln (cid:12)(cid:12)(cid:12)(cid:12) λ + 4 pqλ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:20)Z p − λ −∞ + Z ∞ p + λ (cid:21) d qp (cid:12)(cid:12)(cid:12)
12 + n σ ( q − p ) (cid:12)(cid:12)(cid:12) ln (cid:12)(cid:12)(cid:12)(cid:12) ( p ( p − q ) − λ + p ) − q ( p ( p − q ) − λ − p ) − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) . (B.70)The first “weight function” + n σ ( q ) has a potential singularity at q = 0 (if σ = +1), thelatter at q = p (if σ = +1); however, noticing that the combination in eq. (B.56) can bere-expressed as n σ ( k − p ) n σ ( q ) n σ ( p − q ) = n σ ( k − p ) n σ ( p ) h
12 + n σ ( q ) − − n σ ( q − p ) i , (B.71)these terms cancel exactly against the corresponding real corrections within the domains (e)and (f) as well as (a) and (l), respectively, which are adjacent to the singular lines. Anapproximate form of the cancellation can be seen be rewriting eq. (B.70) in the limit λ → ρ (v) I h ≈ − π K (4 π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) × Z ∞−∞ d qp (cid:26)h
12 + n σ ( q ) i ln (cid:12)(cid:12)(cid:12)(cid:12) pqλ (cid:12)(cid:12)(cid:12)(cid:12) + h
12 + n σ ( q − p ) i ln (cid:12)(cid:12)(cid:12)(cid:12) λ q p ( p − q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) . (B.72)Summing this together with eqs. (B.57)–(B.67) all λ ’s cancel, and the result is integrable inthe small- q domain.It remains to deal with the ultraviolet divergence from the large- q domain. The idea is toinsert 0 = Z | q | > Λ q − Z | q | > Λ q (B.73)inside the integrand representing virtual corrections. The individual terms have the samedivergence as eq. (B.69): Z | q | > Λ q = 1(4 π ) (cid:18) ǫ + ln ¯ µ + 2 (cid:19) . (B.74)26eparating the 1 /ǫ -part hereof, together with finite terms chosen according to the vacuumresult (cf. eqs. (B.17-19) of ref. [13]), and recalling the contribution from the O ( ǫ )-term ineq. (B.11), we obtain ρ (v) I h = − π K (4 π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) (cid:18) ǫ + 2 ln ¯ µ K + 5 (cid:19) + π K (4 π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) × (cid:26) ln 4( k + − p )( p − k − )Λ k K + 3 + (cid:18)Z −∞− Λ + Z ∞ Λ (cid:19) d qq − (4 π ) Z q [ ... ] (cid:27) , (B.75)where [ ... ] refers to the original integrand from eq. (B.54). In addition it must be realizedthat going over to the shifted variables of eq. (B.70) has introduced an “error” (because wehave carelessly handled logarithmically divergent integrals) which must now be compensatedfor. Indeed, eq. (B.74) and an infrared (IR) part from | q | < Λ only add up to the correcteq. (B.69) if the IR part yields 1(4 π ) (cid:18) λ − (cid:19) . (B.76)Yet the corresponding contribution from the vacuum parts of eq. (B.72) reads1(4 π ) Z Λ − Λ d q p (cid:26) sign( q ) ln (cid:12)(cid:12)(cid:12)(cid:12) pqλ (cid:12)(cid:12)(cid:12)(cid:12) + sign( q − p ) ln (cid:12)(cid:12)(cid:12)(cid:12) λ q p ( p − q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) = 1(4 π ) (cid:18) λ + 1 (cid:19) . (B.77)The difference of eqs. (B.76) and (B.77) needs to be cancelled from the integrand of eq. (B.75)if we use the shifted variables. Thereby the final expression reads ρ (v) I h = ρ vac I h + π K (4 π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) × (cid:26) ln 4( k + − p )( p − k − )Λ k K + 5 + Z ∞−∞ d qp (cid:20) p θ ( | q | − Λ) | q | + (cid:16)
12 + n σ ( q ) (cid:17) ln (cid:12)(cid:12)(cid:12)(cid:12) λ pq (cid:12)(cid:12)(cid:12)(cid:12) − (cid:16)
12 + n σ ( q − p ) (cid:17) ln (cid:12)(cid:12)(cid:12)(cid:12) λ q p ( p − q ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21)(cid:27) , (B.78)where a vacuum part has been defined as ρ vac I h ≡ − π K T (4 π ) k ln (cid:18) e k + /T + σ e − k + /T − σ − σ e k − /T + σ e − k − /T − σ − σ (cid:19)(cid:18) ǫ + 2 ln ¯ µ K + 5 (cid:19) . (B.79)There is no dependence on Λ in eq. (B.78), and the 1 /q -tails at | q | > Λ cancel as well so that,when combined with the real corrections, the expression is integrable.For k + , k − ≫ πT , the ultraviolet asymptotics of ρ I h reads [13] ρ I h k + ,k − ≫ πT ≈ − K π ) (cid:18) ǫ + 2 ln ¯ µ K + 5 (cid:19) + Z p (cid:26) n σ − ( n σ + n σ )16 πp + p [3 n σ − ( n σ + n σ )]12 π k + k K (cid:27) . (B.80)27 -1 k / T -0.010-0.0050.000 ρ I h / T T k / T = 0.0k / T > 0.0 (+++)0.1 0.3 1 3 864 -1 M / T -0.015 -0.010-0.0050.0000.005 ρ I h / T T ( - + - )(+ - +)(+ -- ) exactOPE ( T ) OPE ( T + T /M ) k = k av (M); k = √ (k + M ) Figure 10:
Left: The thermal part of ρ I h with the purely bosonic statistics ( σ σ σ ) = (+ + +),for k ≥ k + 0 . T , compared with the zero-momentum limit determined in ref. [24]. Right: Thethermal part of ρ I h with the momentum of eq. (B.19) and statistics of eqs. (B.81)–(B.83) as a functionof M/T , compared with the OPE-asymptotics from eq. (B.80).
The specific statistics needed in the present case are I h ⇔ ( σ σ σ | σ σ ) = (+ − + | + − ) , (B.81) e I h ⇔ ( σ σ σ | σ σ ) = (+ − −| − − ) , (B.82) b I h ⇔ ( σ σ σ | σ σ ) = ( − + −| + − ) , (B.83)where only the first three indices are independent. For numerical evaluation, we have reflectedthe final integral to the domain defined in fig. 6 of ref. [15]. The corresponding integrand isnot shown explicitly, since no substantial cancellations take place in the reflection. Results ofnumerical evaluations (after subtracting the vacuum part, cf. eq. (B.79)) are shown in fig. 10,and are seen to agree with the OPE-asymptotics from eq. (B.80) as well as with the k → B.10. ρ I h’ After carrying out the Matsubara sums for I h’ , we get I h’ = lim λ → Z p , q ǫ q ǫ pk E qp (cid:26) − ik n ǫ q + k · q − ik n + ǫ pk + ǫ q + E qp (cid:2) n σ ( ǫ pk ) + n σ ( ǫ q ) (cid:3)(cid:2) n σ ( E qp ) (cid:3) + n σ ( ǫ pk ) n σ ( ǫ q ) ǫ p − ( ǫ q + E qp ) − ik n ǫ q + k · q − ik n − ǫ pk + ǫ q + E qp n σ ( ǫ pk ) (cid:2) n σ ( ǫ q ) + n σ ( E qp ) (cid:3) − n σ ( ǫ q ) n σ ( E qp ) ǫ p − ( ǫ q + E qp ) + + ik n ǫ q + k · q − ik n + ǫ pk − ǫ q + E qp n σ ( ǫ q ) (cid:2) n σ ( ǫ pk ) + n σ ( E qp ) (cid:3) − n σ ( ǫ pk ) n σ ( E qp ) ǫ p − ( ǫ q − E qp ) + − ik n ǫ q + k · q − ik n + ǫ pk + ǫ q − E qp n σ ( E qp ) (cid:2) n σ ( ǫ pk ) + n σ ( ǫ q ) (cid:3) − n σ ( ǫ pk ) n σ ( ǫ q ) ǫ p − ( ǫ q − E qp ) (cid:27) + lim λ → Z p ǫ p ǫ pk (cid:20) n σ ( ǫ p ) + n σ ( ǫ pk ) − ik n + ǫ p + ǫ pk + n σ ( ǫ p ) − n σ ( ǫ pk ) ik n − ǫ p + ǫ pk (cid:21) × Z q (cid:20) n σ ( ǫ q ) + n σ ( E qp )4 ǫ q E qp (cid:18) ik n ǫ q + k · q ǫ p + ǫ q + E qp − − ik n ǫ q + k · q ǫ p − ǫ q − E qp (cid:19) + n σ ( ǫ q ) − n σ ( E qp )4 ǫ q E qp (cid:18) − ik n ǫ q + k · q ǫ p − ǫ q + E qp − ik n ǫ q + k · q ǫ p + ǫ q − E qp (cid:19) (cid:21) + ( ik n → − ik n ) . (B.84)Taking the cut like in eq. (A.27), the first four structures are real corrections, the last twoare virtual corrections. The corresponding scattering processes can be depicted like in fig. 3of ref. [15], with internal propagators carrying the index σ shrunk to points.For the real corrections, eq. (4.28) of ref. [15] now reads D Φ r1 ( k − p | q | p − q |· ) E = − D Φ r2 ( p − k | q | p − q |· ) E = − D Φ r3 ( k − p | − q | p − q |· ) E = − D Φ r4 ( k − p | q | q − p |· ) E = n σ ( k − p ) n σ ( q ) n σ ( p − q ) n σ ( k ) × P (cid:28) k q − k · q p − p (cid:29) ( k − p | q | p − q |· ) , (B.85)where the arguments ( ... | ... | ... |· ) on the last line refer to ǫ pk , ǫ q , and E qp , respectively. Theazimuthal average, as defined in ref. [15], yields (cid:10) k · q (cid:11) ( ǫ pk | q | E qp ) = (cid:0) p + k − ǫ pk (cid:1)(cid:0) p + q + λ − E qp (cid:1) p . (B.86)The subsequent p -integral leads to logarithms and fractions. Renaming finally p → p andfactoring out π n σ ( k − p ) n σ ( q ) n σ ( p − q )(4 π ) k n σ ( k ) , (B.87)29he λ → p − q ) (cid:20) p − k − )( p − k + ) p (cid:21) + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) p ( p − q ) λ q (cid:12)(cid:12)(cid:12)(cid:12) , (B.88)(b) : 4 k − ( k + − q ) p + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) k + ( p − q ) q ( p − k + ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.89)(b) : 2( p − k − ) (cid:20) p − q )( p − k + ) p (cid:21) + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) p ( p − q )( p − k − ) λ k − (cid:12)(cid:12)(cid:12)(cid:12) , (B.90)(c) = − (h) = − (h) = − (j) : 4 q ( k + − k − ) p + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) k + ( p − k − ) k − ( p − k + ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.91)(˜c) : 2 k − (cid:20) − ( p − q )( p − k + ) p (cid:21) + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) k − p ( p − q ) λ ( p − k − ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.92)(d) : 4( p − k − )( k + − p + q ) p + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) qk + ( p − q )( p − k + ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.93)(e) = (f) : 4( p − q )( k + + k − − p ) p + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) k + k − ( p − k + )( p − k − ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.94)(e) = (f) : 2 q (cid:20) − ( p − k − )( p − k + ) p (cid:21) + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) pqλ (cid:12)(cid:12)(cid:12)(cid:12) , (B.95)(g) = (g) : 4( p − k + )( k − − p + q ) p + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) qk − ( p − q )( p − k − ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.96)(i) = (k) : 4 k + ( k − − q ) p + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) k − ( p − q ) q ( p − k − ) (cid:12)(cid:12)(cid:12)(cid:12) , (B.97)(i) = (k) : 2( p − k + ) (cid:20) p − q )( p − k − ) p (cid:21) + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) p ( p − q )( p − k + ) λ k + (cid:12)(cid:12)(cid:12)(cid:12) . (B.98)As far as the virtual corrections go, they include a divergent part: ρ I h’ ∋ Z p πδ (cid:0) k − ǫ p − ǫ pk (cid:1) n σ ( k − p ) n σ ( p )4 ǫ p ǫ pk n σ ( k ) × Z q P (cid:26) ǫ q E qp (cid:18) k ǫ q + k · q ǫ p + ǫ q + E qp + − k ǫ q + k · q − ǫ p + ǫ q + E qp (cid:19)(cid:27) = − ρ J b K × Re Z Q K · QQ [( Q − P ) + λ ] (cid:12)(cid:12)(cid:12)(cid:12) k n = − ik , p n = − iǫ p , ǫ pk = k − ǫ p . (B.99)Here we made use of the fact that the vacuum integral is independent of p :Re Z Q K · QQ [( Q − P ) + λ ] (cid:12)(cid:12)(cid:12)(cid:12) k n = − ik , p n = − iǫ p , ǫ pk = k − ǫ p = − K π ) (cid:18) ǫ + ln ¯ µ λ + 12 (cid:19) . (B.100)Like with ρ I h it is helpful, however, not to separate the vacuum integral from the outset, butrather to treat the structures + n σ ( ǫ q ) and + n σ ( E qp ) identifiable on the last two rowsof eq. (B.84) as single entities for as long as possible.30n order to implement this, we first carry out angular integrals and substitute integrationvariables, obtaining [the virtual correction part is denoted by ρ (v) I h’ , and ≃ is a reminder ofthe divergences appearing at large q and of the omission of terms of O ( ǫ )] ρ (v) I h’ ≃ π (4 π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) × (cid:26)Z ∞−∞ d q h
12 + n σ ( q ) i(cid:20) q ( K − pk ) p + 2 qp K + λ ( K − pk )2 p ln (cid:12)(cid:12)(cid:12)(cid:12) λ λ + 4 pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) − (cid:20)Z p − λ −∞ + Z ∞ p + λ (cid:21) d q (cid:12)(cid:12)(cid:12)
12 + n σ ( q − p ) (cid:12)(cid:12)(cid:12)(cid:20) K − pk ) p ( q − p ) − λ p + 2 qp K + λ ( K − pk )2 p ln (cid:12)(cid:12)(cid:12)(cid:12) ( p ( p − q ) − λ + p ) − q ( p ( p − q ) − λ − p ) − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) . (B.101)The first “weight function” + n σ ( q ) has a potential singularity at q = 0 (if σ = +1), thelatter at q = p (if σ = +1); however, making use of eq. (B.71), it can be seen that these termscancel (apart from a harmless ∼ p ( q − p ) − λ in the latter case) against the correspondingreal corrections within the domains (e) and (f) as well as (a) and (l), respectively, whichare adjacent to the singular lines. An approximate form of the cancellation can be seen berewriting eq. (B.101) in the limit λ → ρ (v) I h’ ≈ π (4 π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) × Z ∞−∞ d q (cid:26)h
12 + n σ ( q ) i(cid:20) q ( K − pk ) p + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) λ pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) − h
12 + n σ ( q − p ) i(cid:20) q − p )( K − pk ) p + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) λ q p ( p − q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) . (B.102)Summing together with eqs. (B.88)–(B.98), all λ ’s cancel, and the remainder is integrable inthe IR domain | p | , | q | < ∼ k .It remains to deal with the ultraviolet divergence. We add0 = − K Z | q | > Λ q + K Z | q | > Λ q (B.103)in the integrand of the virtual corrections. As seen from eq. (B.74) the individual termshave the same divergence as eq. (B.100). Separating the 1 /ǫ -part hereof, together with finiteterms chosen according to the vacuum result (cf. eq. (B.20) of ref. [13]), and recalling the31ontribution from the O ( ǫ )-term in eq. (B.11), we obtain ρ (v) I h’ = − π K π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) (cid:18) ǫ + 2 ln ¯ µ K + 92 (cid:19) + π K π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) × (cid:26) ln 4( k + − p )( p − k − )Λ k K + 52 + (cid:18)Z −∞− Λ + Z ∞ Λ (cid:19) d qq + 4(4 π ) K Z q [ ... ] (cid:27) , (B.104)where [ ... ] refers to original integrand in eq. (B.84). In addition it must be realized thatgoing over to the shifted variables of eq. (B.101) has introduced an “error” (because wehave carelessly handled logarithmically divergent integrals) which must now be compensatedfor. In order for the ultraviolet contribution − K R | q | > Λ 14 q (cf. eq. (B.74)) and the infraredcontribution from | q | < Λ to add up to the correct result in eq. (B.100), the vacuum termsof the latter should yield − K π ) (cid:18) λ − (cid:19) . (B.105)Explicit integration shows, however, that they yield12(4 π ) Z Λ − Λ d q (cid:26) sign( q )2 (cid:20) q ( K − pk ) p + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) λ pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) − sign( q − p )2 (cid:20) q − p )( K − pk ) p + q K p ln (cid:12)(cid:12)(cid:12)(cid:12) λ q p ( p − q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) = 1(4 π ) (cid:20) k p − K (cid:18) ln 4Λ λ + 12 (cid:19)(cid:21) . (B.106)The difference of eqs. (B.106) and (B.105) needs to be cancelled from the integrand ofeq. (B.104) if we employ the shifted variables. This finally yields ρ (v) I h’ = ρ vac I h’ + π K π ) k Z k + k − d p n σ ( k − p ) n σ ( p ) n σ ( k ) × (cid:26) ln 4( k + − p )( p − k − )Λ k K + 92 − k p K + Z ∞−∞ d q (cid:20) θ ( | q | − Λ) | q | + (cid:16)
12 + n σ ( q ) (cid:17)(cid:18) q ( K − pk ) p K + 2 qp ln (cid:12)(cid:12)(cid:12)(cid:12) λ pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) − (cid:16)
12 + n σ ( q − p ) (cid:17)(cid:18) q − p )( K − pk ) p K + 2 qp ln (cid:12)(cid:12)(cid:12)(cid:12) λ q p ( p − q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:21) (cid:27) , (B.107)where the vacuum part has been defined as ρ vac I h’ ≡ − π K T π ) k ln (cid:18) e k + /T + σ e − k + /T − σ − σ e k − /T + σ e − k − /T − σ − σ (cid:19)(cid:18) ǫ + 2 ln ¯ µ K + 92 (cid:19) . (B.108)32 -1 k / T -0.004-0.0020.0000.002 ρ I h ’ / T T k / T = 0.0k / T > 0.0 (+++)0.1 0.3 1 3 864 -1 M / T -0.005 ρ I h ’ / T T ( - + - ) exactOPE ( T ) OPE ( T + T /M ) k = k av (M); k = √ (k + M ) Figure 11:
Left: The thermal part of ρ I h’ with the purely bosonic statistics ( σ σ σ ) = (+ + +),for k ≥ k + 0 . T , compared with the zero-momentum limit determined in ref. [24]. Right: Thethermal part of ρ I h’ with the momentum of eq. (B.19) and statistics of eq. (B.110) as a function of M/T , compared with the OPE-asymptotics from eq. (B.109).
There is no dependence on Λ in eq. (B.107), and the 1 /q -tails at | q | > Λ cancel as well, so thatthe expression is integrable once combined with the real corrections. (In fact the integrandalso has a constant part at large | q | , but this cancels due to its antisymmetry in q → − q .)For k + , k − ≫ πT , the ultraviolet asymptotics of the spectral function reads [13] ρ I h’ k + ,k − ≫ πT ≈ − K π ) (cid:18) ǫ + 2 ln ¯ µ K + 92 (cid:19) + Z p (cid:26) n σ πp + p [3( n σ − n σ ) + n σ ]24 π k + k K (cid:27) . (B.109)The specific statistics needed in this paper are b I h’ ⇔ ( σ σ σ | σ σ ) = ( − + −| + − ) . (B.110)For numerical evaluation, we have reflected the final integral to the domain defined in fig. 6of ref. [15]. The corresponding integrand is not shown explicitly, since no substantial cancel-lations take place in the reflection. Results of numerical evaluations are shown in fig. 11, andit can be seen that the OPE-asymptotics from eq. (B.109) as well as the k → σ = σ we have also checked that the identity ρ I h’ = ( ρ I f + ρ I h ),obtained by substitutions of sum-integration variables, is satisfied.33 ppendix C. Choice of parameters For illustration we choose M = 10 GeV for the numerics like in refs. [7, 8]. The physicalHiggs mass is set to m H = 126 GeV. In order to convert pole masses and the muon decayconstant to MS scheme parameters at a scale ¯ µ = ¯ µ ≡ m Z we employ 1-loop relationsspecified in ref. [26]; subsequently, 1-loop renormalization group equations determine therunning of the couplings to a scale ¯ µ ref ≡ max( M, πT ) . (C.1)Within this approximation the U(1), SU(2) and SU(3) gauge couplings g , g , g have explicitsolutions (we have set N c = 3 and considered 3 families), g (¯ µ ) = 48 π
41 ln(Λ / ¯ µ ) , g (¯ µ ) = 48 π
19 ln(¯ µ/ Λ ) , g (¯ µ ) = 24 π
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