Anatomy of real intermediate state-subtraction scheme

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Anatomy of real intermediate state-subtraction scheme © 2024 the Authors Published version Ala-Mattinen, Kalle; Heikinheimo, Matti; Tuominen, Kimmo; Kainulainen, Kimmo Ala-Mattinen, K., Heikinheimo, M., Tuominen, K., & Kainulainen, K. (2023). Anatomy of real intermediate state-subtraction scheme. Physical Review D, 108(9), Article 096034. https://doi.org/10.1103/PhysRevD.108.096034 2023 Anatomy of real intermediate state-subtraction scheme Kalle Ala-Mattinen ,*Matti Heikinheimo ,†and Kimmo Tuominen ‡ Department of Physics, University of Helsinki, P.O. Box 64, FI-00014 University of Helsinki, Finland and Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland Kimmo Kainulainen § Department of Physics, University of Jyväskylä and Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland (Received 2 October 2023; accepted 31 October 2023; published 29 November 2023) We study the origin of the real intermediate state subtraction problem and compare its different solutions. We show that the ambiguity in subtraction schemes arises from the on-shell approximation for the 2-point functions that reduces the Schwinger-Dyson equations to the Boltzmann limit. We also suggest a new subtraction scheme which, unlike the earlier definitions, never leads to negative scattering rates. This scheme also quantifies the validity of the on-shell limit in terms of an effective one-particle weight function RðΔÞ, where Δmeasures the region around the resonance associated with the real state. DOI: 10.1103/PhysRevD.108.096034 I. INTRODUCTION There is a long-standing issue in setting up consistent kinetic equations for particle distribution functions when relevant scattering processes involve real particles that also appear as intermediate states. The problem is that the resonant on-shell scattering processes overlap with the decay contributions in the kinetic equations, which results in double counting unless some subtraction procedure is established for the scattering rates. Such a procedure, referred to as the real intermediate state (RIS) subtraction, was first set up in [1–3]. The RIS subtraction has since been discussed in various different forms [4–30], and an alternative method that does not rely on the RIS subtraction was recently suggested in [31]. Typically the RIS subtraction is performed at the level of Boltzmann equations, implementing some formal way to isolate and remove the on-shell part, DonðsÞ, from the full Breit-Wigner (BW) propagator DBWðsÞ, thus replacing it with the off-shell part, DoffðsÞ, that is used for scattering amplitudes. However, this procedure is inherently ambiguous due to the ambiguity in the definition of a “real”particle with unstable states. As a result, there is large freedom in the definition of the RIS subtraction, and different RIS-subtraction schemes are even known to lead to apparently unphysical negative scattering cross sections [24]. In this article we study and compare different solutions to the RIS problem. We also study the emergence of the problem in the context of Boltzmann equations (BE) and then from a more fundamental perspective of the Schwinger-Dyson (SD) equations for the 2-point functions. SD equations are a fully consistent setup for studying unstable particles with no notion of the real intermediate states. We will show how the problem arises when the SD equations are reduced to the spectral limit. This also makes the effect of RIS subtraction to different reaction channels evident. We then suggest a new definition for the RIS subtraction, which does not suffer from the negative cross sections. The new scheme quantifies the ambiguity in the RIS subtraction in terms of an effective one-particle weight function RðΔÞ<1for the approximate real states, where Δ measures the size of the kinematic region around the resonance that is counted to contribute to the real particle. For the real-particle picture to be a good approximation, one should be able to choose Δ, which is much smaller than the characteristic energy scale in the system, such that RðΔÞ≈1. Failing to satisfy these conditions signals the breakdown of the on-shell limit, and a BE network with the decay channel should not be used, or should be interpreted with due care. The article is structured as follows: We begin by revisiting the main RIS-subtraction schemes in Sec. II, where we lay out the RIS subtraction at the level of propagators via formal propagator modifications. We then *[email protected] †[email protected] ‡[email protected] §[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 108, 096034 (2023) 2470-0010=2023=108(9)=096034(10) 096034-1 Published by the American Physical Society apply this method to an explicit example in Sec. III and demonstrate how the standard RIS contribution emerges from the s-channel interaction. In Sec. IV we study the origin of the RIS problem in the Schwinger-Dyson formalism in the case of the Yukawa theory. We show how the RIS problem arises in the on-shell limit, when the resonance overlaps with the kinematic region of interest for the problem at hand. This then motivates us to suggest our new RIS-subtraction scheme in Sec. V. In Sec. VI we perform numerical comparisons between different subtraction methods including our new proposal and conclude in Sec. VII. II. REVIEW OF THE RIS-SUBTRACTION SCHEMES The goal of the RIS-subtraction procedure is to somehow eliminate the on-shell part from the finite-width BreitWigner propagator: DBWðsÞ¼ 1 s−m2þimΓ →DoffðsÞ;ð1Þ where sis the Mandelstam variable, mis the mass of the propagating intermediate particle, and Γis its total decay width. The on-shell part DonðsÞ¼DBWðsÞ−DoffðsÞis then reduced to a delta function in the limit Γ→0.Itis associated with the decay processes, while the remainder DoffðsÞis thought to describe the off-shell scattering processes. For this task, different technical schemes have been proposed. Let us begin with the standard RISsubtraction (SRS) scheme that is applied at the level of the propagator function squared. The standard subtraction scheme is based on the simple observation that the squared BW propagator itself has a formal delta-function limit: jDBWðsÞj2¼1 mΓ mΓ ðs−m2Þ2þðmΓÞ2≈π mΓδðs−m2Þ;ð2Þ where the last equality holds approximately for small Γ. Interpreting the right-hand side of (2) as the on-shell propagator for a finite Γcan then be used as the definition for the squared off-shell propagator: jDSRS off ðsÞj2¼jDBWðsÞj2−π mΓδðs−m2Þ:ð3Þ This procedure works for the resonances isolated in a single channel. However, we may have for example a process with sand t-channel contributions where the s-channel is resonant. Then the collision integral is proportional to jMsj2þjMtj2þ2Re½M sMt:ð4Þ The rule (3) is not sufficient to deal with the resonant propagator in the mixing term, as it only tells how to remove the on-shell part from the propagator squared contributions. To avoid this issue, a subtraction procedure at the level of the propagator is needed. This was discussed in [4] but, to our knowledge, properly proposed later in [5]. Here one starts by writing DBWðsÞ¼ s−m2 ðs−m2Þ2þðmΓÞ2−imΓ ðs−m2Þ2þðmΓÞ2 ¼Re½DBWðsÞþiIm½DBWðsÞ →Re½DBWðsÞ−iπδðs−m2Þ:ð5Þ In the last line the limit Γ→0was assumed (only) in the imaginary part of the propagator. One can use this result to complete the SRS scheme for the interference term in (4), setting DSRS off ðsÞ≡DBWðsÞþiπδðs−m2Þð6Þ at the propagator level, whenever a single resonant propagator (as opposed to a squared one) is encountered. In the principal value subtraction (PVS) scheme suggested in [5] one reverses the logic and assumes that the off-shell propagator is just the real part of the Breit-Wigner propagator: DPVS off ðsÞ≡Re½DBWðsÞ:ð7Þ Also in the PVS scheme different definitions for the propagator and for the squared propagator are needed, which require some care. One might naively think that the on-shell part of the squared propagator would be just jDonj2≡jDBWj2−jRe½DBWj2¼jIm½DBWj2, but this is not the correct subtraction, because the imaginary part squared produces only half of the on-shell contribution: jIm½DBWj2→ π 2mΓδðs−m2Þ:ð8Þ As pointed out in [12–14], this issue is particularly relevant in the resonant leptogenesis literature. The problem is that when we extract a distribution from a function the remainder is also a distribution, and one must be careful when taking the square. The explicit case at point here is that PV1 x2 ≠PV1 x2;ð9Þ where PV refers to the principal value part of the function. Indeed, while 1=x has the principal value sequence PVð1=xÞ¼Reð1=ðxþiεÞÞ¼x=ðx2þϵ2Þ, the corresponding sequence for 1=x2is KALLE ALA-MATTINEN et al. PHYS. REV. D 108, 096034 (2023) 096034-2 PV1 x2¼Re1 ðxiεÞ2¼x2−ε2 ðx2þε2Þ2:ð10Þ In these expressions, the limit ϵ→0is of course assumed. From this exercise one infers that the correct squared propagator in the PVS scheme for a finite Γis DPVS off ðsÞ¼ ðs−m2Þ2−ðmΓÞ2 ððs−m2Þ2þðmΓÞ2Þ2ð11Þ as was indeed suggested in [6]. Subtracting this off-shell part from the full BW propagator one finds that jDPVS on ðsÞj2¼jDBWðsÞj2−jDPVS off ðsÞj2 ¼2ðmΓÞ2 ððs−m2Þ2þðmΓÞ2Þ2 → π mΓδðs−m2Þ:ð12Þ That is, the narrow-width limit in the PVS scheme requires the use of the delta sequence πδðxÞ¼2ε3=½x2þε22. The formulas for the SRS scheme (3) and (6) for the PVS scheme (7) and (11) are equivalent up to order Γ2. The PVS scheme also has a well defined Γ→0limit for the off-shell propagators: DΓ¼0 off ðsÞ→PV1 s−m2; jDΓ¼0 off ðsÞj2→PV1 ðs−m2Þ2:ð13Þ The limiting case (13) can be understood as yet another subtraction scheme. We stress that the subtraction scheme dependence affects only the off-shell propagators. All schemes were designed to have the same on-shell limits: DonðsÞ¼−iπδðs−m2Þ; jDonðsÞj2¼π mΓδðs−m2Þ:ð14Þ There is no a priori reason to prefer one scheme over the other, although the limiting scheme (13) is perhaps conceptually most consistent, as we shall argue later. It seems that most confusion related to RIS subtraction in the literature has resulted from a failure to realize that each scheme requires separate formulas for the off-shell propagator and the squared off-shell propagator functions. Finally, we point out that the above discussion is not restricted to s-channel processes. Resonances can appear also in tand u-channels. When this happens, the subtraction should be performed as in the s-channel case. A fundamental problem in all above schemes is that they can occasionally lead to negative reaction rates. This can happen because the off-shell propagators are negative near on-shell; if the scattering process is enhanced there, the integrated rates may also become negative. This is an apparently unphysical result, but it does not necessarily lead to a failure of the kinetic equation network. We shall return to these issues in Sec. VI. III. A MINIMAL EXAMPLE OF DOUBLE COUNTING Here we show that the decay contribution to kinetic equations indeed corresponds to the contribution from the on-shell propagator in the scattering channel. To this end we consider a setting with unspecified particles a,b, and c, and for demonstrative purpose focus only on the processes shown in Fig. 1. We want to track the distribution function of species c, whose kinetic equation can be written as L½fcðp1Þ ¼ Cc aa↔ccðp1ÞþCc b↔ccðp1Þ;ð15Þ where the Liouville operator L½fðpÞ≡ð∂t−Hp∂pÞfðpÞ. To allow for analytic calculation, we work in the MaxwellBoltzmann (MB) approximation, 1fi≈1and feq i¼ e−βEifor i¼a; b; c, and assume that particles aand b are in equilibrium. We will prove that when the intermediate bparticle is treated at the idealized resonant limit, the equilibrium contribution of the scattering term reduces exactly to the equilibrium decay term. We start by evaluating the equilibrium scattering term in the MB limit: Ccðaeqaeq ↔ccÞ ¼1 2Ec p1Zp2;k1;k2ð2πÞ4δð4Þðk1þk2−p1−p2Þ ×jMaa↔ccj2feq aðk1Þfeq aðk2Þ−fcðp1Þfcðp2Þ;ð16Þ where Rp≡Rd3p=½2Epð2πÞ3. The integral over momenta k1and k2is easily performed when one demands that the detailed balance holds: feq aðk1Þfeq aðk2Þ¼feq cðp1Þfeq cðp2Þ. Furthermore, factoring the squared matrix element as jMaa↔ccj2¼jMaa↔bj2jDbj2jMb↔ccj2we end up with ð17Þ FIG. 1. Minimal viable set of interaction processes demonstrating a double counting of process (b) when the propagator in (a) becomes on-shell. ANATOMY OF REAL INTERMEDIATE STATE-SUBTRACTION …PHYS. REV. D 108, 096034 (2023) 096034-3 where λðx; y; zÞ¼ðx−y−zÞ2−4yz. To compare with the decay term, we need to evaluate Ccðbeq ↔ccÞ¼ 1 2Ec p1Zp2;qð2πÞ4δð4Þðq−p1−p2Þ ×jMb↔ccj2feq bðqÞ−fcðp1Þfcðp2Þ: ð18Þ The detailed balance condition again allows us to replace feq bðqÞby feq cðp1Þfeq cðp2Þ. Writing the three-dimensional integral over momentum qas a four-dimensional integral with the measure δðq2−m2 bÞð2πÞ−3d4qand using δð4Þðq−p1−p2Þto carry out integration over d4q, the decay term then becomes ð19Þ Showing the equality of Eqs. (17) and (19) amounts to establishing equality of the boxed quantities. To this end, we write the jMaa↔bj2matrix element in Eq. (17) in terms of the decay width: 1 8πjMaa↔bj2¼2m3 b λ1=2ðm2 b;m 2 a;m 2 aÞBRb→aaΓ;ð20Þ where Γ≡PiΓb→figis the full decay width of the b particle and BRb→fig¼Γb→fig=Γis the branching ratio to final state i. Furthermore, writing the propagator jDb BWj2¼jDoffj2þjDonj2, where the offand on-shell results are given in (3) and (14), the boxed part of the scattering term (17) becomes 1 8πsλ1=2ðs; m2 a;m 2 aÞjMaa↔bj2jDb BWj2 →2hðsÞBRb→aaπδðs−m2 bÞþmbΓjDb offj2;ð21Þ where we defined hðsÞ¼m2 b s λ1=2ðs; m2 a;m 2 aÞ λ1=2ðm2 b;m 2 a;m 2 aÞ:ð22Þ In the on-shell limit s→m2 bthe function hðsÞ→1proving that the on-shell part of the scattering term is indeed equal to the decay term contribution: Ccðaeqaeq ↔bon ↔ccÞ¼BRb→aaCcðbeq ↔ccÞ:ð23Þ This is the doubly counted state which the RIS subtraction is devised to remove via Ccðaa ↔ccÞþCcðb↔ccÞ →Ccðaa ↔boff ↔ccÞþCcðb↔ccÞ;ð24Þ i.e. by dropping the on-shell part of the scattering term. Finally all 2-2 process with different initial states, including elastic channels, should be included so that their corresponding branching ratios, as in Eq. (23), sum up PiBRi¼1. IV. THE ORIGIN OF THE RIS PROBLEM The previous example verified the double counting between the decay and scattering channels, and showed that the on-shell s-channel propagator defined in (14) indeed corresponds to the decay contribution. In this section we shall take a look at the problem from a more general point of view. A. Spectral functions A simple interacting system where the RIS problem may arise is the Yukawa theory with a scalar field φand a fermion field ψ.Ifφis too light to decay into a fermion pair, isolated poles exist for both φand ψ, and no RIS problem exists, however. In this case the vacuum scalar spectral functions Aφand Aψcan be written as follows: Aφ¼πϵðk0ÞðRφδðs−m2 φÞþρφðsÞÞ; Aψ¼πϵðk0Þð = kþmψÞRψδðs−m2 ψÞþρψðsÞ;ð25Þ where ρφ;ψðsÞare the continuum (off-shell) parts to the spectral functions and the one-particle weight functions Rφ;ψ are constrained to be less than unity by the spectral sum rules. Indeed, 1 πRdk0k0Aφ¼1and 1 πRdk0Aψ¼γ0imply that Rφ;ψ¼1−1 2πZdsρφ;ψðsÞ<1:ð26Þ The continuum parts ρφ;ψðsÞmay to a good accuracy be computed perturbatively using spectral free-theory propagators. In this case one can also derive a good BE approximation for kinetic equations reducing theSD equations to theon-shell limit using the spectral propagators: iΔ< φ¼2πϵðk0ÞRφfφðkÞδðs−m2 φÞ; iS< ψ¼2πϵðk0Þð = kþmψÞRψfψðkÞδðs−m2 φÞ;ð27Þ and moreover ΔH φ¼PVRφ s−m2 φ; iΔ11 φ¼iΔH φþ2πϵðk0ÞRφfφðkÞþ1 2δðs−m2 φÞ;ð28Þ KALLE ALA-MATTINEN et al. PHYS. REV. D 108, 096034 (2023) 096034-4 where Δ11 φis the Feynman propagator, whose Hermitian component is given by ΔH φ. Similar equations hold for the fermion ψ. One can usually set Rφ;ψ¼1to a good approximation for perturbative couplings. We kept them here to emphasize that in an interacting theory the one-particle states do not exhaust the entire state space of the system. However, in the current setup where isolated poles exist, the on-shell formulas (27) and (28) provide a good parametrization for the system. When mφ>2mψ,φis unstable and no longer has an isolated pole. In this case the spectral solutions do not strictly speaking exist, and it is not possible to derive the BE limit for the φfield without further approximations. For example, if the energy scales one is interested in do not overlap with the φresonance, one may be able to derive a good BE limit for the problem based on the spectral parametrization for the fermion and treating φonly as an intermediate resonance. The RIS problem becomes acute only when the relevant energy scales do overlap with the resonance and one implements an on-shell kinetic equation also for φ, as we shall see in the next section. The on-shell limit can often be a good approximation also for unstable particles. When this is so, it would seem natural to use the spectral limit for all propagator functions, including the Hermitian parts. This would give rise to the limiting subtraction prescription (13), which in this sense is the most natural scheme to use. But this choice is not unique, as we shall see below. B. Schwinger-Dyson equations We now discuss the RIS problem in the context of SD equations. We emphasize that the full SD equations, which span the entire phase space of the 2-point functions, do not need RIS subtraction. The issue emerges only when the SD equations for unstable particles are reduced to the on-shell limit. The SD equations for the Yukawa theory can be schematically written as Δφ¼Δφ;0þΔφ;0⊗Πφ⊗Δφ; S¼Sψ;0þSψ;0⊗Σψ⊗Sψ;ð29Þ where Δφ;0and Sψ;0are the free propagators and Δφand Sψ are the full propagators. In the direct space representation the convolutions are defined as ðA⊗BÞðx; yÞ≡ Rd4zAðx; zÞBðz; yÞ, where z0lives on the complex time contour. In real time the SD equations split into Kadanoff-Baym (KB) equations for the pole functions and for the Wightman functions. In the bosonic case we get ðΔ−1 φ;0−Πp φÞ⊗Δp φ¼1; ðΔ−1 φ;0−ΠH φÞ⊗Δs φ¼Πs φ⊗ΔH φþCs φ;ð30Þ where p¼r,arefer to the retarded and advanced pole functions and s¼h;ito the statistical Wightman functions, and we defined the collision terms C< φ¼1 2Π> φ⊗Δ< φ−Π< φ⊗Δ> φð31Þ and C> φ¼−C< φ. Similar decomposition can be derived for the fermionic correlation functions Sp;s ψ; see e.g. [32]. The KB equations usually cannot be solved without further approximations, such as the spectral ansatz (27). The reduction of the KB equations to the spectral limit is a delicate task [32–34]. However, to understand the RIS problem, it is sufficient to concentrate on the collision terms C< φ;ψ, which in the end emerge as the collision integrals in Boltzmann equations. In Fig. 2we show the diagrams contributing to Eq. (29) up to two loops in the two-particle irreducible (2PI) expansion. Dashed lines correspond to φ-propagators and continuous ones to ψ-propagators, and thin (thick) lines correspond to the free (full) propagators. We emphasize that the two-loop diagrams shown in Fig. 3are not included in full SD equations. They are already accounted for by the one-loop diagrams, because the full SD equations sum all perturbative corrections associated with the diagrams included in the 2PI expansion. However, when simplifying approximations are imposed, the consistency of the 2PI expansion breaks down and some non-2PI diagrams may need to be inserted by hand. The simplest example is the case where we treat the φ-field as a resonance and drop it from the SD equation network. In this case the scalar decay diagrams are not summed by the SD equations and the last diagram in Fig. 3must be included perturbatively into the SD equation for the ψ-field. The spectral limit is more delicate. Here the SD equation for the φ-field is kept, but all collision integrals are reduced to the on-shell limit, where the one-loop terms reduce to the decay terms in the BEs. While the BEs still sum these FIG. 2. The diagrams contributing to the SD equations in the Yukawa theory to the two-loop order in the 2PI expansion. Thin lines indicate the free and thick lines the full propagators. Blue dashed lines express cuts defined similarly to Fig. 3. ANATOMY OF REAL INTERMEDIATE STATE-SUBTRACTION …PHYS. REV. D 108, 096034 (2023) 096034-5 on-shell contributions to all orders, this summation completely misses all off-shell contributions. The 2PI expansion is again broken and the off-shell parts of the diagrams shown in Fig. 3must be included perturbatively. The RIS problem has thus entered. The cuts in two-loop diagrams indicated by blue dashed lines in Fig. 2give only the interference terms (e.g. between the sand t-channels) in the collision integrals, while the squared matrix elements in each channel emerge from the cuts in the forbidden diagrams of Fig. 3. The summation argument applies to all noncut internal propagators, and so the need for subtraction concerns all channels and all matrix elements including the interference terms in collision integrals. In the present example the on-shell condition is kinematically forbidden from appearing in fermion lines, but it affects the scalar propagators in the last diagrams in Figs. 2and 3. The standard on-shell reduction of the SD equations to BEs explicitly uses the spectral propagators of Eqs. (27) and (28). However, one can replace the Hermitian principal value propagators by other off-shell propagators with no change to the reduction procedure. Indeed, the different subtraction schemes discussed so far are but different definitions for the Hermitian propagator functions in the spectral ansatz. From this point of view all schemes are equally good. In the next section we will introduce another subtraction scheme that is free of the issue of negative rates and provides a quantitative estimate for the validity of the BE limit. V. CUT-SUBTRACTION SCHEME In the previous section we saw that both the need for and the ambiguity of the RIS subtraction arise when the full solutions to the SD equations are approximated by spectral solutions. The validity of the spectral limit clearly depends on the resonance width Γ. For a small width, excitations around the resonance may give nearly identical contributions to physical processes, allowing them to be treated as an effective single-particle state. If Γis very small, the integrated weight of such states may even saturate the spectral sum rule. This is the case when the usual spectral reduction of the SD equations to the BE limit with free theory normalization R¼1is valid. If the physical process changes significantly over the resonance region, however, the spectral approximation starts to break down. We quantify these simple observations by defining the following cut-subtraction scheme: DΔ offðaÞ¼1−Θða−m2;ΔÞDBWðaÞ;ð32Þ where a¼s,t,oru, depending on the channel one is looking at, and the cut function Θsingles out a region around the resonance. The simplest choice is the top-hat function: Θðx; ΔÞ¼θðΔ−xÞθðxþΔÞ:ð33Þ The squared off-shell propagator does not need a separate rule in this scheme. The propagator in Eq. (32) is again assumed to replace the Hermitian parts in the Feynman and anti-Feynman propagators in spectral decomposition, e.g. in Eq. (28). Combined with this prescription, the statistical propagators in Eq. (27) (as well as the spectral parts of the Feynman and anti-Feynman propagators), are rescaled by a weight function: RðΔÞ≡ 1 πZdsΘðs−m2;ΔÞmΓjDBWðsÞj2:ð34Þ It should be obvious that none of these changes interfere with the on-shell reduction of the SD equations. The precise form of the cut function in Eq. (33) is not relevant, and the characteristic width Δwill depend on the problem. The weight function will give a quantitative measure for the validity of the spectral limit. To see how this works, consider some physical quantity F, which is an integral over, say, the Mandelstam variable s: F¼ZdsFðsÞjDBWðsÞj2:ð35Þ For example the vacuum contribution to the ψcollision integral from process ψ¯ ψ→ψ¯ ψ, coming from the cut in the last diagram in Fig. 3would have this form. We can now divide Finto onand off-shell contributions F¼Fon þFoff, where Foff ¼ZdsFðsÞjDΔ offðsÞj2ð36Þ and FIG. 3. Additional perturbative diagrams needed in the on-shell limit. The numbers are the closed time path indices singling out the embedded self-energy diagram Σ>¼Σ21, and blue dashed lines indicate cuts corresponding to these indices. Red dotted lines in the internal boson propagators Δ11 and Δ22 indicate that these propagators are included without their on-shell parts. KALLE ALA-MATTINEN et al. PHYS. REV. D 108, 096034 (2023) 096034-6 Fon ¼1 mΓZdsFðsÞΘðs−m2;ΔÞmΓjDSðsÞj2 ≈RðΔÞπFðm2 SÞ mSΓS :ð37Þ In the second line we assumed that FðsÞremains essentially a constant in the cut region and then used Eq. (34).Thisclearly corresponds to replacing jDBWðsÞj2inside the on-shell region with a weighted delta function: πRðΔÞδðs−m2Þ=ðmΓÞ. If Findeed described the process ψ¯ ψ→ψ¯ ψ, then the on-shell part in Eq. (37) would be exactly canceled by the on-shell vacuum decay term φ→ψ¯ ψarising from the oneloop diagram in the fermion SD equation in Fig. 2. Indeed, because of our rule for weighting the statistical propagators by RðΔÞ, the latter is given by the standard vacuum decay term multiplied by RðΔÞ. The exact calculation equating the two is essentially the same we presented in Sec. III, with b¼φand aa; cc →ψ¯ ψ. Our scheme in corresponds to setting ΔH φ→DΔ off and Rφ→RðΔÞin Eqs. (27) and (28). All collision integrals can then be computed from these functions following standard finite temperature field theory methods. One then recovers the standard tree level rules for computing collision integrals, augmented with the scaling of the phase space factors fφ→RðΔÞfφand replacing DBW →DΔ off for all intermediate scalar propagators. Extensions to more complicated theories where also fermions can be unstable should be obvious. We should emphasize one important difference between our scheme and the other subtraction schemes. Consider the case where Fis a constant. In this case all standard schemes give Foff ¼0, whereas in the cut scheme Foff ¼Fð1−RðΔÞÞ. The latter one is the qualitatively correct result. The standard schemes thus overestimate the weight of the effective one-particle states, which forces the flat-weight integrals of the off-shell parts to vanish in compensation. The effect underestimates the off-shell contributions also for a nonconstant FðsÞand may even lead to a negative cross section if the corresponding FðsÞis enhanced near the resonance. The cut-subtraction scheme does not suffer from these problems by construction. The particle picture makes sense only if the approximation in Eq. (37) holds with a large enough Δ, such as RðΔÞ≃1. Indeed, if FðsÞchanged rapidly in the scale mΓ, we would be forced to use very small Δ≪mΓto extract FðsÞoutside the integral. This would then give RðΔÞ≪1, showing that the one-particle contribution to the process is very small. Of course, the particle picture would not make sense in this limit. Nevertheless, the cut-subtraction procedure allows for a continuous mapping between the limit where the particle picture is valid [RðΔÞ≃1] and the resonance limit where φno longer is part of the thermal bath [RðΔÞ≃0]. VI. NUMERICAL EXAMPLES In this section we compare quantitatively the predictions of the SRS and PVS schemes and study the cut scheme as a function of Δin a physical system where RIS subtraction is needed, at least in principle. To be specific, and to keep the discussion as simple as possible, we consider the dark matter freeze-out in the singlet extension of the standard model (e.g. [35–39]), where the DM-abundance calculations have been recently studied to high precision [30,31,40–42]. The model is described by the Lagrangian L¼1 2ð∂μSÞ2−1 2λhsh2S2þ;ð38Þ where his the Standard Model Higgs, Sis a singlet scalar, and dots refer to other terms whose precise form is not relevant here, including the SM Lagrangian. We focus on the hand Sparticle densities governed by the following set of coupled Boltzmann equations: L½fhðp1Þ ¼ Ch h↔SSðp1ÞþCh h↔ðSMÞðp1ÞþCh other; L½fSðk1Þ ¼ CS SS↔hðk1ÞþCS SS↔ðSMÞðk1ÞþCS other:ð39Þ The scattering process SS ↔ðSMÞdescribes the annihilation of the Sscalars to the standard model (SM) particles via s-channel Higgs boson resonance. Other channels that could affect these densities, not relevant for the present discussion, are contained in Ch;S other. The RIS subtraction is necessary in this model, because the decay and inverse decay terms h↔SS and h↔ðSMÞ already account for the on-shell part of the scattering term as discussed in previous sections. To perform a full comparison we should solve Eq. (39) using the different schemes for the off-shell propagator in the scattering integral. In the momentum dependent case the collision integrals should be arranged to contain an integral over s, as was done e.g. in [41], to facilitate the cut regularization. We will instead concentrate on the simpler momentum integrated version of Eq. (39), where the relevant dynamics are given by the thermal averaged cross section [43]1 1In thermal equilibrium this rate is exact when Sscalars are treated in spectral approximation and we assume the MB statistics. Indeed, in thermal equilibrium the full solution to the Higgs SD equation is iΔ< h¼2feqAh, where Ah¼ ϵðk0ÞmΓDBWðsÞ. With these assumptions the one-loop collision term in the SDE for Sis given by Eq. (19), with the delta function replaced by ðmΓ=πÞDBWðsÞ. Tracing the derivation in Sec. III backwards, one can establish the equality of that result and (17) which, when integrated over the initial momentum, gives (40) with the BW propagator. From this result one then must subtract the on-shell part, which eventually gives (40). ANATOMY OF REAL INTERMEDIATE STATE-SUBTRACTION …PHYS. REV. D 108, 096034 (2023) 096034-7 hvMϕlσXi¼ 1 8m4 STK2 2ðmS TÞZ∞ 4m2 S dss3=2v2 SðsÞK1ffiffiffi s p TσXðsÞ ≡Z∞ 4m2 S dsFðsÞjDX hðsÞj2:ð40Þ Here v2 S¼1–4m2 S=s and KiðxÞare the modified Bessel functions of the second kind. In the second line we extracted the subtracted propagator function, letting FðsÞ describe the rest of the integrand, and the index X refers to the subtraction scheme. The s-channel cross section, from which FðsÞcan be read off using Eq. (40), is given by [39] σXðsÞ¼y2ðλhsvÞ2 vSffiffiffi s pjDX SðsÞj2Γhðffiffiffi s pÞ;ð41Þ where v¼246 GeV is the SM-Higgs vacuum expectation value and Γh¼ΓSS þΓSM is the total width of a virtual Higgs particle with an effective mass ffiffiffi s p. The rate for h→SS is ΓSS ¼ðλhsvÞ2vS=ð32πffiffiffi s pÞ. For the exact definition of ΓSM see Ref. [39]. Figure 4shows the s-channel thermal average hvMϕlσXi in different subtraction schemes for a particular set of parameters given in the figure caption. The solid red line shows the PVS-subtracted propagator as a function of a varying effective width ϵ. The dash-dotted black line is the SRS-subtracted result corresponding to Eq. (3), while the blue dashed line shows the result for the PVS propagator (11) with ϵ¼ϵos ≡mhΓos, and the green dashed line shows the spectral PV propagator, Eq. (14). We take the area between the dashed lines corresponding to two constant width PVS schemes as indicative of the fundamental ambiguity in the calculation of the scattering rate using the SRSand PVS-subtraction schemes. In Fig. 5we show the thermal averaged results in the cut scheme, Eqs. (32) and (37), as a function of the width of the cut area Δ. The solid purple line presents the off-shell scattering contribution, and the solid yellow line shows the on-shell part. The solid black line shows the full thermal integral, which here agrees with the sum of the offand on-shell cut contributions. Note that a relatively large Δ is needed to make the off-shell cut result agree with the standard subtraction schemes, or equivalently to have RðΔÞ≈1. Indeed, to get RðΔÞ¼0.99, one needs Δ¼70ϵos. But even then one finds 1−rð70ϵosÞ≈ 4×10−4, where rðΔÞis the sum of the two cut contributions divided by the full result. This shows that the effective particle approximation works very well. This is as expected since the resonance is very narrow Γos=T ≈0.01. The difference between the standard subtraction schemes is also not visible in this scale. In Fig. 6we show results of a computation with a larger coupling λhs ¼0.01 and at a higher temperature T¼50 GeV. The variation between different PVS and FIG. 4. Comparison of the off-shell parts of the thermal averaged cross section in Eq. (40) for the standard subtraction schemes. We used mh¼125 GeV, mS¼50 GeV, λhs ¼0.001, and T¼5GeV. FIG. 5. The solid purple curve corresponds to the off-shell averaged cross section in the cut scheme, Eq. (36), and the solid yellow curve shows the on-shell result, Eq. (37). The black line is the full integral computed with the BW propagator. We used the same parameters as in Fig. 4. FIG. 6. Same as in Fig. 5, but for parameters λhs ¼0.01 and T¼50 GeV. Note that the standard subtraction scheme results are all negative in this case. KALLE ALA-MATTINEN et al. PHYS. REV. D 108, 096034 (2023) 096034-8