Perturbative Unitarity of Strongly Interacting Massive Particle Models Ayuki Kamadaab Shin Kobayashic and Takumi Kuwaharad

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Perturbative Unitarity of Strongly Interacting Massive Particle

Models

Ayuki Kamadaa,b, Shin Kobayashic, and Takumi Kuwaharad

aInstitute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul.

Pasteura 5, PL-02-093 Warsaw, Poland

bKavli Institute for the Physics and Mathematics of the Universe (WPI), The

University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa

277-8583, Japan

cICRR, University of Tokyo, Kashiwa, Chiba 277-8582, Japan

dCenter for High Energy Physics, Peking University, Beijing 100871, China

Abstract

Dark pion is a promising candidate for the strongly interacting massive particle

dark matter. A large pion self-coupling mπ/fπtends to be required for correct relic

abundance, and hence the partial-wave amplitudes can violate the perturbative

unitarity even for the coupling within na¨ıve perturbative regime. We improve the

partial-wave amplitudes in order to satisfy the optical theorem. We demonstrate

that the improvement is relevant only for semi-relativistic pions, and thus this does

not aﬀect the self-scattering cross section at the cosmic structures. We also discuss

the impact of the improvement of the πππ →ππ scattering process, and we ﬁnd

that there is an upper bound on mπat which the correct relic abundance is never

achieved even at large mπ/fπdue to the optical theorem.

arXiv:2210.01393v2 [hep-ph] 3 Mar 2023

1 Introduction

Little is known about particle nature of dark matter (DM) even though the existence of

DM has been ﬁrmly conﬁrmed by the astrophysical observations. Strongly interacting

massive particles (SIMPs) [1] are an interesting framework for the thermal relic of sub-

GeV DM: the thermal relic DM is determined by the freeze-out of 3 →2 processes.

Self-interactions of DM are generically sizable to get the correct relic abundance in this

framework, and it leads to a large 2 →2 self-scattering. The small-scale structure of the

Universe may indicate the sizable self-scattering of DM [2] (see Ref. [3] a review).

Dark sector is a hypothetical sector where DM resides, and has its own gauge dynamics.

As a consequence of dark strong dynamics, the dark sector would consist of composite

particles (dark hadrons) at the low-energy scale as with the SM hadrons [4–51] (see

Ref. [52] for a review). Ref. [53] proposed a model of the SIMP framework where the dark

pion is identiﬁed as DM. The dark pions arise as the pseudo-Nambu-Goldstone boson

(pNGB) from the strong dynamics in the dark sector. The dark pions have the 2 →2

self-interaction and the 3 →2 number-changing process induced by the Wess-Zumino-

Witten (WZW) term [54, 55].

The chiral perturbation theory (χPT) describes the interactions among dark pions.

The pion self-coupling, which is deﬁned by the ratio of the pion mass and decay constant

mπ/fπ, determines the size of the pionic scattering processes. The 2 →2 self-scattering

interaction arises at the leading order terms of O(m2

π/f2

π), while the 3 →2 number-

changing interaction via the WZW term appears as an O(m5

π/f5

π) term of χPT. The pion

self-coupling is found to be larger than unity in order to explain the relic abundance

and to evade constraints on the self-scattering cross section [53]. Meanwhile, we cannot

validate the perturbative expansion of χPT unless mπ/fπ.4π. We would not be able to

ignore contributions from resonances and the higher-order terms of the chiral Lagrangian

for the pion self-coupling near the na¨ıve perturbative bound. For the consistent treatment

of the chiral expansion, Ref. [56] has discussed the impact of the higher order of the chiral

Lagrangian on SIMP scenarios.

We may encounter the other bound on the self-scattering cross section of dark pions

in the SIMP models even when the perturbative expansion of χPT is valid. The unitarity

of the S-matrix imposes constraints on the partial-wave amplitude. The partial-wave

amplitude for the 2 →2 self-scattering is T'm2

π/(32πf2

π) at the tree-level, and the

perturbative unitarity places an upper bound 1/2βon ReTwith βcorresponding to

velocity of pions. The perturbative unitarity bound mπ/fπ.4√π/√βgets weaker at the

cosmic structures due to the DM velocity at maximum of ∼10−2, while this bound can be

important for the annihilation mechanism determining the current relic abundance since

dark pions are semi-relativistic. In other words, there are two bounds on the pion self-

couplings: one originates from the limitation of the perturbative expansion and another

is the perturbative unitarity of the scattering processes. In the chiral limit (mπ→0),

the perturbative unitarity violation will be cured by resumming multiple rescattering

processes, which is known as the “self-healing” mechanism [57]. In this paper, we will

propose the improvement of 2 →2 and 3 →2 partial-wave amplitudes in a similar way

to the “self-healing” mechanism in the non-chiral limit since we focus on the dark pion

DM.

This paper is organized as follows. We will show the chiral Lagrangian for the SIMP

models in Section 2. We discuss the perturbative unitarity of ππ →ππ scattering cross

section and thermally-averaged πππ →ππ cross section in Section 3, and we will see these

cross sections would violate the perturbative unitarity at large mπ/fπ. In Section 4, we

propose the improved amplitude, which satisﬁes the optical theorem automatically, and

then we will apply the procedure to the SIMP models. Section 5 is devoted to conclusions

of our work.

2 Chiral Lagrangian for SIMP

We discuss the SIMP model that is realized by the conﬁning gauge dynamics of a gauge

group Glocal. We consider Nf-ﬂavor quarks with the mass below the dynamical scale of

Glocal in the ultraviolet description of this model. This model possesses an approximate

global symmetry Gamong quarks that is broken by the mass terms. It is believed that

this model leads to the chiral symmetry breaking, and that chiral condensation breaks

the global symmetry into the subgroup H. In the following, we consider three classes of

the models: (i)Glocal =SU(Nc), G=SU(Nf)×SU(Nf), and H=SU(Nf); (ii)Glocal =

SO(Nc), G=SU(Nf), and H=SO(Nf); and (iii)Glocal =USp(Nc), G=SU(Nf), and

H=USp(Nf) (with even integers Ncand Nf[58–61]).

The dark pions, which are the pNGBs of the chiral symmetry breaking, are the fun-

damental degrees of freedom in the low-energy eﬀective theory of this model. We are

interested in the dark pions for realizing SIMP framework, and hence we focus on the

chiral expansion with typical momentum p2' O(m2

π) in the non-relativistic limit. The

coset space G/H is parameterized by NπpNGB ﬁelds πa[62,63], which corresponds to the

broken generators Tawith a= 1,··· , Nπ. The parameterization and the normalization

of the dark pion ﬁelds are the following.

Σ = exp 2iΠ

fπ,Π≡πaTa,Tr(TaTb) = 1

2δab ,(1)

and only for the case (iii), the non-linear sigma model ﬁeld Σ is

Σ = exp 2iΠ

fπJ , (2)

where Jis the symplectic metric that satisﬁes JT=−J , J2=−1. Here, fπdenotes the

pion decay constant. Under the residual symmetry H, the dark pion ﬁelds transform as

(i) adjoint representation, (ii) rank-2 symmetric tensor representation, and (iii) rank-2

antisymmetric tensor representation for each diﬀerent symmetry. The relevant Lagrangian

of the dark pions is given by

L=f2

4Tr (DµΣ)†DµΣ+Bf2

4Tr M†Σ + MΣ†+LWZW .(3)

Here, the ﬁrst-two terms are the leading order (LO) terms in the χPT and the soft

chiral symmetry breaking term, and the third term is the Wess-Zumino-Witten (WZW)

term [54,55]. We take the quark mass matrix Mto be invariant under H, and hence the

pion mass is universal m2

π=Bmqwith mqbeing the quark mass.

The relevant Lagrangian contains the four-point interactions among the dark pions,

which induce the self scattering of the dark pions. We obtain the four-point interaction

terms from the kinetic and mass terms by expanding Σ.

L ⊃ rabcd

4f2

πaπb∂µπc∂µπd+1

cabcdπaπbπcπd,(4)

where rabcd and cabcd are the coeﬃcients deﬁned by group-theoretical constants, which is

discussed in Appendix C. We will discuss the self-scattering cross section in detail in the

next section. Meanwhile, the ﬁve-point interaction among the dark pions arises from the

WZW term. The πππ →ππ scattering process arising from the WZW term determines

the relic abundance of the dark pions. The WZW term is given by1

LWZW =2k

15π2f5

µνρσTr [Π∂µΠ∂νΠ∂ρΠ∂σΠ]

⊃2k

15π2f5

T[abcde]µνρσπa∂µπb∂νπc∂ρπd∂σπe.

(5)

Here, k=NCfor SU (NC) and 2k=NCfor SO(NC) and USp(NC) [65]. T[abcde]denotes

the anti-symmetrization of ﬁve broken generators:

T[abcde]=1

5! XTr(T[aTbTcTdTe]).(6)

The four-point interactions arise at the order of m2

π/f2

π, while the ﬁve-point interaction is

at the order of m5

π/f5

π. In accordance with the standard order-counting of the χPT using

p2expansion in the chiral limit, the former is the LO contribution and the latter is the

next-to-leading order (NLO) contribution in the non-chiral limit.

A large coupling tends to be required for correct relic abundance in the SIMP sce-

narios. In the dark-pion realization, the number-changing process arises from the WZW

term, which is the higher order of the chiral expansion, with the velocity suppression.

Meanwhile, the χPT breaks down mπ'ΛχSB, and the cutoﬀ scale is expected to be

ΛχSB 'min 4πfπ

√Nc

,4πfπ

pNf!.(7)

This is known as the scale estimated by na¨ıve dimensional analysis (NDA) [66, 67] with

taking into account the large-Ncscaling and the large-Nfscaling. The large-Ncscaling of

fπand ΛχSB is known to be f2

π' O(Nc) and ΛχSB ' O(1) [68–73]. It is also known that

some of the partial-wave amplitudes for the scattering process of ππ →ππ is proportional

to Nfin the large Nflimit [74, 75]. 2Since the pion self-coupling is required to be close

1Our conventions are diﬀerent from some literature. The normalizations of both generators and fπ

diﬀer factor two from Refs [53, 55], and the resultant coeﬃcients are same as ours. Meanwhile, the

normalization of generators diﬀers, but the same convention is used for fπas ours in [64].

2This discussion is based on the chiral symmetry breaking of SU(Nf)×SU(Nf)→SU(Nf) in the

original literature. We ﬁnd the similar Nfdependence of the amplitude with the isospin singlet even for

SU(Nf)/SO(Nf) and SU(Nf)/USp(Nf), and hence we expect that we get the same cutoﬀ scale up to

constant of order unity.

to its na¨ıve perturbative bound mπ/fπ.ΛχSB/fπin the SIMP scenario, the higher-

order contributions of the χPT may aﬀect the predictions of the relic abundance and

the self-scattering cross section. Ref. [56] has discussed the impacts of the higher-order

contributions of the χPT in the context of the SIMP: in particular, the NLO and the

next-to-next-to-leading order (NNLO) contributions.

3 Perturbative Unitarity

We discuss the perturbative unitarity of ππ →ππ and πππ →ππ scattering processes.

The partial-wave decomposition of the invariant amplitude for elastic scattering process

is given by3

Mab;cd

2→2= 32πX

(2`+ 1)P`(cos θ)TR

`(s)Pab;cd

R.(8)

Here, P`is the Legendre polynomial with P`(1) = 1, sdenotes the collision energy,

and θis the scattering angle of the ﬁnal state pions with respect to the collision axis.

Pab;cd

Rdenotes the projection operators: the product of two pions is projected into the

irreducible representation Rof residual global symmetry H.TR

`(s) is the partial-wave

elastic amplitude for the channel of a representation R. The projection operators satisfy

c0,d0

Pab;c0d0

RPc0d0;cd

R0=δRR0Pab;cd

R,(Pab;cd

R)∗=Pcd;ab

R,X

a,b

Pab;ab

R=dR.(9)

Here, dRdenotes the dimension of the representation R. With this decomposition, the

total cross section for πaπb→πcπdtakes the form

σab;cd

2→2=32π

(2`+ 1)|TR

`(s)|2Pab;cd

R.(10)

The unitarity of the S-matrix imposes the optical theorem that relates the imaginary

part of the invariant amplitude M2→2for the forward scattering to the total cross section

σtotal. Below the four-pion threshold, the optical theorem is given by

ImM2→2= 2√sp(σ2→2+σ2→3).(11)

Here, p=√sβ/2 is the momentum of incoming particles with β= (1−4m2

π/s)1/2denoting

the velocity of the particles in the center of mass frame. We use the relativistic formula for

the two-body system (pand β), while taking the non-relativistic limit of the three-body

system. This is because the two-body system is semi-relativistic for the 2 →3 process.

We include the inelastic cross section σ2→3in the right-hand side since it is also sizable

in the SIMP models, and it vanishes below the inelastic threshold.

We deﬁne the partial-wave amplitude for the ππ →πππ process in the non-relativistic

limit of the three-body ﬁnal state. It is challenging to give the explicit form of the

decomposition of the three-body ﬁnal state into its irreducible representations, in other

3We use 32πas a normalization instead of 16πdue to the identical particles.

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PerturbativeUnitarityofStronglyInteractingMassiveParticleModelsAyukiKamadaa;b,ShinKobayashic,andTakumiKuwaharadaInstituteofTheoreticalPhysics,FacultyofPhysics,UniversityofWarsaw,ul.Pasteura5,PL-02-093Warsaw,PolandbKavliInstituteforthePhysicsandMathematicsoftheUniverse(WPI),TheUniversityofTokyoInstit...

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Perturbative Unitarity of Strongly Interacting Massive Particle Models Ayuki Kamadaab Shin Kobayashic and Takumi Kuwaharad

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