Cosmological implications of the KOTO excess
Wolfgang Altmannshofer, Benjamin V. Lehmann, Stefano Profumo
CCosmological implications of the KOTO excess
Wolfgang Altmannshofer, ∗ Benjamin V. Lehmann, † and Stefano Profumo ‡ Department of Physics, University of California Santa Cruz,1156 High St., Santa Cruz, CA 95064, USA andSanta Cruz Institute for Particle Physics,1156 High St., Santa Cruz, CA 95064, USA
The KOTO experiment has reported an excess of K L → π ν ¯ ν events above thestandard model prediction, in tension with the Grossman–Nir bound. The GN boundheavily constrains new physics interpretations of an excess in this channel, but an-other possibility is that the observed events originate from a different process entirely:a decay of the form K L → π X , where X denotes one or more new invisible species.We introduce a class of models to study this scenario with two light scalars play-ing the role of X , and we examine the possibility that the lighter of the two newstates may also account for cosmological dark matter. We show that this species canbe produced thermally in the presence of additional interactions apart from thoseneeded to account for the KOTO excess. Conversely, in the minimal version of themodel, dark matter must be produced non-thermally. In this case, avoiding over-production imposes constraints on the structure of the low-energy theory. Moreover,this requirement carries significant implications for the scale of reheating in the earlyuniverse, generically preferring a low but observationally-permitted reheating tem-perature of O (10 MeV). We discuss astrophysical and terrestrial signatures that willallow further tests of this paradigm in the coming years. PACS numbers: 14.40.Df, 95.35.+d, 98.80.-kKeywords: rare decays; particle dark matter; cosmology ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] J un CONTENTS
I. Introduction 3II. Model 5A. Effective interactions of the scalars and meson decay rates 6B. Events at the KOTO experiment 9C. Simplified UV models 141. Vector-like quark model 142. Inert Higgs doublet model 15III. Astrophysical and terrestrial constraints 17A. Supernova constraints 17B. Beam dump constraints 18C. Direct dark matter detection 20IV. Cosmological production 21A. Computing the yield 24B. Determining the reheating temperature 27V. Discussion 31VI. Conclusions 37Acknowledgments 38A. KOTO simulation 38References 39
I. INTRODUCTION
The rare kaon decays K + → π + ν ¯ ν and K L → π ν ¯ ν are widely recognized as very sensitiveprobes of new physics (NP). In the Standard Model (SM), the branching ratios of thesedecays are strongly suppressed, and can be precisely predicted [12, 14] to beBR( K + → π + ν ¯ ν ) SM = (8 . ± . × − , (1)BR( K L → π ν ¯ ν ) SM = (3 . ± . × − . (2)On the experimental side, several K + → π + ν ¯ ν candidate events have been observed by theE787/E949 experiment [2, 5, 6] and the NA62 experiment [18], but a discovery of K + → π + ν ¯ ν has still to be established. The current best limit on the branching ratio is from apreliminary analysis of NA62 data and reads [47]BR( K + → π + ν ¯ ν ) exp < . × − (95% C.L.) , (3)not far above the SM prediction. The NA62 experiment aims to measure the SM branchingratio with O (10%) uncertainty. In the case of K L → π ν ¯ ν , the current most stringentbound on the branching ratio comes from the KOTO experiment [3], and is still two ordersof magnitude above the SM prediction:BR( K L → π ν ¯ ν ) exp < . × − (90% C.L.) . (4)Interestingly, in the latest status update by KOTO [49], 4 events are seen in the signal box,with an expected number of 0 . ± .
01 SM K L → π ν ¯ ν events and 0 . ± .
02 backgroundevents. One of the events has been identified as likely background. If the remaining eventsare interpreted as signal, one finds a branching ratio of BR( K L → π ν ¯ ν ) ∼ × − [40].A branching ratio of this size would be a spectacular discovery. Not only does it imply NP,it also violates the Grossman-Nir (GN) bound [31], BR( K L → π ν ¯ ν ) (cid:46) . × BR( K + → π + ν ¯ ν ) (cid:46) − , when combined with the NA62 constraint in eq. (3). The GN bound is veryrobust in models where the K → πν ¯ ν decays are modified by heavy new physics well abovethe kaon mass. However, in the presence of light new physics, the GN bound can be violatedand the observed events at KOTO may find an explanation [19, 21, 23, 25, 30, 36–38, 40–42, 52].Here we focus on a new physics scenario first discussed in [45]. Two new light scalars S and P , neutral under the SM gauge interactions, are introduced such that K L can decay SK L P Pπ FIG. 1. Decay chain accounting for the KOTO signal in our scenario. into a pair of the new particles, K L → SP . If the decay S → π P is allowed and P is stableon the relevant experimental scales, then the decay chain K L → SP → π P P can mimicthe K L → π ν ¯ ν signature (see fig. 1). The corresponding chain of two-body decays doesnot exist for the charged kaon. A possible decay K + → π + SP is suppressed by three-bodyphase space or may be forbidden entirely by kinematics.If P is absolutely stable, it is also a candidate for cosmological dark matter. In theminimal setup that can provide a NP explanation of the KOTO events, P couples to theSM very weakly, implying that annihilation cross sections into SM states are too small forproduction by freeze-out. We therefore investigate alternative scenarios for cosmologicalproduction, and interpret overproduction of P as a cosmological constraint on the structureof the low-energy theory. We show that P is readily produced non-thermally if the scale ofreheating is low, close to but safely above the current observational bound. We also showthat this class of models can account for the KOTO excess without requiring a low reheatingtemperature, but only in the presence of additional interactions. We investigate prospectsfor testing this model with future experiments and with additional data from KOTO, andshow that much of the parameter space will be probed in the near future.This paper is organized as follows: in section II, we present the model and discuss howit can explain the KOTO events. In section III, we evaluate astrophysical and terrestrialconstraints on the parameter space of our model. In section IV, we consider cosmologicalproduction of P , and relate the production of P to the scale of reheating. We discuss theimplications of our results in section V and conclude in section VI. II. MODEL
We start with very simple kinematical considerations concerning the masses of the twoscalars S and P . Figure 2 shows the plane of the two scalar masses m S and m P . Asdescribed in the introduction, we are interested in regions of parameter space where thedecay K L → π P P , which mimics K L → π ν ¯ ν , can be realized as a sequence of the two-body decay K L → SP followed by S → π P . For m S too large, the decay K L → SP iskinematically forbidden, while for m S too small, the S → π P decay is not open, excludingthe dark gray regions in the plot. In the light gray region one faces potential constraints fromthe charged kaon decay K + → π + SP that is generically expected in the models discussedbelow. In the white region, however, this decay is kinematically forbidden, while K L → π ν ¯ ν remains open.The plot also indicates two other interesting kinematical boundaries. If m P < m π /
2, theexotic pion decay π → P P is possible which, as we will discuss in section IV, can impactcosmological production considerably. If m S > m P , the decay S → P can be allowed,thus modifying the lifetime of S , which is a crucial parameter for beam dump constraints.Note that low P masses may be subject to constraints from supernova cooling, which wewill discuss further in section III A. A weaker lower bound on the P mass also follows fromassuming a particular thermal history, a point to which we shall return in section V.In the following sections, we will discuss four benchmark parameter points covering themost interesting regimes:BM1: m S = 400 MeV , m P = 10 MeV , BM2: m S = 350 MeV , m P = 100 MeV , BM3: m S = 300 MeV , m P = 125 MeV , BM4: m S = 200 MeV , m P = 10 MeV . (5)Next we discuss in detail the interactions of S and P with SM quarks. We first focus onnon-renormalizable effective couplings and identify viable regions of parameter space. Thenwe comment on simplified UV models that map onto the effective couplings. m P ( MeV ) m S ( M e V ) m P = m π / m S = m P BM1 BM2BM4 BM3
FIG. 2. The plane of the scalar masses m S vs. m P . In the dark gray region the K L → π P P decay cannot be realized as a sequence of 2-body decays. In the light gray region the K + → π + SP decay is open. The black dots indicate four benchmark scenarios that we consider later (eq. (5)). A. Effective interactions of the scalars and meson decay rates
We assume that the scalars S and P interact with SM particles via the effective couplings L int ⊃ iSP (cid:18) g SPdd Λ NP ( ¯ dd ) + ˜ g SPdd Λ NP ( ¯ diγ d ) + g SPss Λ NP (¯ ss ) + ˜ g SPss Λ NP (¯ siγ s ) (cid:19) + iSP (cid:18) g SPsd Λ NP (¯ sd ) + ˜ g SPsd Λ NP (¯ siγ d ) + h . c . (cid:19) . (6)The factors of i in the above Lagrangian are reminiscent of considering S to be a CP-evenscalar and P to be a CP-odd pseudoscalar, a notational pattern that we will retain whenmatching onto low-energy QCD later on. The coefficients g SPdd , g SPss , ˜ g SPdd , and ˜ g SPss are purelyimaginary (by hermiticity of the Lagrangian) while the g SPsd and ˜ g SPsd coefficients can have anarbitrary complex phase. There could also be interactions involving b quarks, but as longas they are not considerably larger than the interactions with the light quarks, their impacton phenomenology will be negligible.In the following, we will also entertain the possibility of additional interactions involving P and S , of the form L int ⊃ P (cid:32) g P dd Λ NP ( ¯ dd ) + ˜ g P dd Λ NP ( ¯ diγ d ) + g P ss Λ NP (¯ ss ) + ˜ g P ss Λ NP (¯ siγ s ) (cid:33) + P (cid:32) g P sd Λ NP (¯ sd ) + ˜ g P sd Λ NP (¯ siγ d ) + h.c. (cid:33) . (7)While the interactions in eq. (7) are not directly relevant for the KOTO signal, they dohave important implications for other meson decays and in particular for the dark matterphenomenology as we will discuss in section IV below.The decays relevant for an enhanced KOTO signal, K L → SP and S → π P are inducedby the couplings Re(˜ g SPsd ) and Im(˜ g SPdd ), respectively. For the corresponding decay rates wefind Γ( K L → SP ) = 18 π f K m K L m s (cid:18) Re(˜ g SPsd )Λ NP (cid:19) η QCD (cid:113) λ (cid:0) , m S /m K L , m P /m K L (cid:1) , (8)Γ( S → π P ) = 1128 π f π m π m S m d (cid:18) Im(˜ g SPdd )Λ NP (cid:19) η QCD (cid:113) λ (cid:0) , m π /m S , m P /m S (cid:1) , (9)with the phase space function λ ( a, b, c ) = a + b + c − ab + ac + bc ). The down andstrange quark masses in the above expressions should be interpreted as the MS masses ata renormalization scale of µ = 2 GeV. Leading-log QCD corrections are then taken intoaccount through the factor η QCD η QCD = (cid:18) α s ( m t ) α s ( M ) (cid:19) / (cid:18) α s ( m b ) α s ( m t ) (cid:19) / (cid:18) α s (2 GeV) α s ( m b ) (cid:19) / , (10)where M is the scale of new physics that is responsible for the effective interactions of S and P with the SM quarks. Because of SU (2) L invariance we expect M ∼ √ Λ NP v , where v = 246 GeV is the vacuum expectation value of the SM Higgs. Note that including the η QCD factor is equivalent to evaluating the down and strange masses in eqs. (8) and (9) atthe scale M .The coupling | g SPsd | can lead to the decay K + → π + SP , if kinematically allowed. Thedifferential 3-body decay rate of K + → π + SP is given by d Γ( K + → π + SP ) dq = 1256 π m K + m s (cid:18) | g SPsd | Λ NP (cid:19) η QCD (cid:18) − m π + m K + (cid:19) × (cid:113) λ (1 , m S /q , m P /q ) (cid:113) λ (cid:0) , m π + /m K + , q /m K + (cid:1) , (11)where we estimated the relevant scalar form factor as (cid:104) π + | ¯ sd | K + (cid:105) (cid:39) ( m K + − m π + ) /m s and q is the invariant mass of the SP system, with ( m P + m S ) < q < ( m K + − m π + ) .Similarly to the K L → SP decay, the interactions in eq. (6) also lead to the exotic etadecay η → SP , which has been identified as a possible source of the scalar S at beam dumpexperiments [37]. Neglecting η – η (cid:48) mixing, we findΓ( η → SP ) = 3512 π f η m η m s (cid:18) g SPss ) − Im(˜ g SPdd )Λ NP (cid:19) η QCD (cid:113) λ (cid:0) , m S /m η , m P /m η (cid:1) . (12)For completeness, we also provide the expression for the decay K S → SP :Γ( K S → SP ) = 132 π f K m K S m s (cid:18) Im(˜ g SPsd )Λ NP (cid:19) η QCD (cid:113) λ (cid:0) , m S /m K S , m P /m K S (cid:1) . (13)In the presence of the P interactions in eq. (7), there are additional exotic meson decays, π → P P , η → P P , K L/S → P P , and K + → π + P P , with the following decay rates:Γ( π → P P ) = 164 π f π m π m d (cid:32) Re(˜ g P dd )Λ NP (cid:33) η QCD (cid:115) − m P m π , (14)Γ( η → P P ) = 3256 π f η m η m s (cid:32) g P ss ) − Re(˜ g P dd )Λ NP (cid:33) η QCD (cid:115) − m P m η , (15)Γ( K L → P P ) = 14 π f K m K L m s (cid:32) Im(˜ g P sd )Λ NP (cid:33) η QCD (cid:115) − m P m K L , (16)Γ( K S → P P ) = 14 π f K m K S m s (cid:32) Re(˜ g P sd )Λ NP (cid:33) η QCD (cid:115) − m P m K S , (17)dΓ( K + → π + P P )d q = 1128 π m K + m s (cid:32) | g P sd | Λ NP (cid:33) η QCD (cid:18) − m π + m K + (cid:19) × (cid:115) − m P q (cid:113) λ (cid:0) , m π + /m K + , q /m K + (cid:1) , (18)In the K + → π + P P decay width, q denotes the P P invariant mass, which lies in the range4 m P < q < ( m K + − m π + ) .The interactions of S and P with quarks that we have introduced preserve a Z symmetryunder which S and P are odd, while all SM particles are even. We assume that the Z symmetry is also respected by the scalar potential, such that P is an absolutely stable darkmatter candidate. Among the allowed Z symmetric terms in the scalar potential, the SP interaction L int ⊃ λ SP SP , (19)will turn out to be relevant. When kinematically allowed, this interaction leads to the decay S → P with rate Γ( S → P ) = 3256 π λ SP m S f ( m P /m S ) , (20)where f is the three-body phase space integral, f ( y ) = 2 (cid:90) (1 − y ) y d x (cid:112) λ (1 , x, y ) λ (1 , y /x, y /x ) , (21)which is normalized to 1 in the limit y →
0. The S → P rate will modify the lifetimeof S and can therefore have a crucial impact on possible constraints from beam dumpexperiments. B. Events at the KOTO experiment
The model introduced in the previous section will lead to K L → π P P events at theKOTO experiment. We now identify the regions of parameter space in which this decay canmimic the KOTO signal.The number of events that can be expected to be detected at KOTO can be written as N = BR( K L → SP ) × BR( S → π P )BR( K L → π ν ¯ ν ) SM × R × N SM , (22)where BR( K L → π ν ¯ ν ) SM = (3 . ± . × − is the SM prediction for the K L → π ν ¯ ν branching ratio [12, 14], N SM = 0 . ± .
01 is the expected number of SM signal events atKOTO [49], and R = A ( K L → SP → π P P ) A ( K L → π ν ¯ ν ) (23)is the ratio of acceptances of the considered model signal and the SM signal at the KOTOdetector. As has been pointed out before [30, 37, 40], an exotic contribution to the KOTOsignal (in our case K L → SP → π P P ) can have a considerably different acceptance.We determine the acceptance ratio R using a Monte Carlo simulation. Details are pro-vided in appendix A. The result is given in fig. 3, which shows R as a function of the S lifetime for our four benchmark points (eq. (5)). For prompt decays, τ S →
0, we find { R BM1 , R
BM2 , R
BM3 , R
BM4 } (cid:39) { , , , } . Once the lifetime of S becomes com-parable to the size of the KOTO detector, τ S ∼ R starts to decrease as more and more S leave the detector before decaying.0 τ S ( cm ) R ( % ) BM2 BM4BM1BM3
FIG. 3. The acceptance ratio R of the K L → SP → π P P signal over the SM K L → π ν ¯ ν signalat KOTO as a function of the S lifetime τ S for the four benchmark scenarios. In our setup, the lifetime of S is determined by the S → π P and S → P decays. Inthe four benchmark cases for the scalar masses defined above we find (cid:110) Γ( S → π P ) BM1 , Γ( S → π P ) BM2 , Γ( S → π P ) BM3 , Γ( S → π P ) BM4 (cid:111) (cid:39) (cid:26) . , . , . , . (cid:27) × (cid:18) GeVΛ dd (cid:19) (cid:18) α s (10 GeV) α s ( M ) (cid:19) / , (24) (cid:110) Γ( S → P ) BM1 , Γ( S → P ) BM2 , Γ( S → P ) BM4 (cid:111) (cid:39) (cid:26) . ,
149 cm , . (cid:27) × (cid:18) λ SP − (cid:19) , (25)where in the S → π P decay width we have defined Λ dd = Λ NP / Im(˜ g SPdd ). Note that S → P is not kinematically allowed in benchmark BM3. The S → π P branching ratio is given byBR( S → π P ) = Γ( S → π P ) / [Γ( S → π P ) + Γ( S → P )].Finally, we find the following K L → SP branching ratios (cid:110) BR( K L → SP ) BM1 , BR( K L → SP ) BM2 , BR( K L → SP ) BM3 , BR( K L → SP ) BM4 (cid:111) (cid:39) (cid:110) . , . , . , . (cid:111) × − × (cid:18) GeVΛ sd (cid:19) (cid:18) α s (10 GeV) α s ( M ) (cid:19) / , (26)1where we have defined Λ sd = Λ NP / Re(˜ g SPsd ).Figures 4 and 5 show the number of expected events in the Λ sd –Λ dd plane for our bench-mark cases in the absence of the S → P decay (fig. 4) and in the presence of the S → P decay induced by a coupling λ SP = 10 − (fig. 5). Along the solid green lines one expects3 events, in the dark green regions one expects 2–4 events, and in the light green regionsone expects 1–5 events. In the gray regions labeled “ K L → π inv.”, the number of pre-dicted events exceeds the limit from KOTO (see eq. (4)). The right vertical axis showsthe lifetime of S corresponding to Λ dd . In fig. 5, the lifetime is approximately constant forΛ dd > GeV, as in this region of parameter space, the lifetime is set by the S → P decaywidth.For S lifetimes of τ S (cid:38) S lifetimes as low as τ S (cid:38) λ SP =10 − , the S lifetimes are short enough throughout the parameter space that existing beamdump constraints are avoided.In fig. 5 we also show additional constraints from other meson decays. The known K L branching fractions add up to a value compatible with 1 with very high precision. Anyadditional K L branching ratio, in particular K L → SP , is thus bounded above as BR( K L → SP ) < . × − [28]. In fig. 5 the gray regions left of the dashed vertical lines denoted“ K L → inv.” are excluded by this constraint. Note that this gives an absolute lower boundΛ sd (cid:38) few × GeV.The other meson decay constraints shown in fig. 5 are less robust as they depend oncouplings that are in principle unrelated. If we assume that the coupling g P sd (correspondingto (¯ sd ) P ) is of the same order as the coupling ˜ g SPsd (corresponding to (¯ siγ d ) SP ), we findrelevant constraints from the searches for K + → π + ν ¯ ν . To evaluate the constraints wecompare the predicted K + → π + P P branching ratio with the bound from NA62 givenin eq. (3). We correct for the different signal acceptances of K + → π + P P comparedto K + → π + ν ¯ ν that arise due to kinematical cuts on the missing mass and the chargedpion momentum. For the three P masses relevant to our benchmarks, we find the boundsBR( K + → π + P P ) < . × − for m P = 10 MeV, BR( K + → π + P P ) < . × − for m P = 100 MeV, and BR( K + → π + P P ) < . × − for m P = 125 MeV. SettingΛ NP / | g P sd | = Λ NP / Re(˜ g SPsd ) = Λ sd , we find that in fig. 5, the regions left of the dotted2 × × × × Λ sd ( GeV ) Λ dd ( G e V ) m S =
400 MeV, m P =
10 MeV, λ SP = τ S ( c m ) K O T O p r e f e rr e d K L →π inv. SeaQuestbeam dumpconstraints × × × × Λ sd ( GeV ) Λ dd ( G e V ) m S =
350 MeV, m P =
100 MeV, λ SP = τ S ( c m ) K O T O p r e f e rr e d K L →π inv. SeaQuestbeam dumpconstraints × × × × Λ sd ( GeV ) Λ dd ( G e V ) m S =
300 MeV, m P =
125 MeV, λ SP = τ S ( c m ) K O T O p r e f e rr e d K L →π inv. SeaQuestbeam dumpconstraints × × × × Λ sd ( GeV ) Λ dd ( G e V ) m S =
200 MeV, m P =
10 MeV, λ SP = τ S ( c m ) K O T O p r e f e rr e d K L →π inv. SeaQuestbeam dumpconstraints FIG. 4. Number of expected K L → SP → πP P events at KOTO in the Λ sd –Λ dd plane for fourbenchmark points of the S and P masses. The SP coupling is set to zero. The right vertical axisindicates the S lifetime. One expects 3 events along the solid dark green line, 2–4 events in the darkgreen region, and 1–5 events in the light green region. In the gray regions labeled “ K L → π inv.”,the number of predicted events exceeds the limit from KOTO. The dashed lines show constraintsfrom existing beam dump experiments and the potential reach of the SeaQuest upgrade. vertical lines are excluded.If we assume that the coupling ˜ g P dd (corresponding to ( ¯ diγ d ) P ) is of the same orderas the coupling ˜ g SPdd (corresponding to ( ¯ diγ d ) SP ), we find relevant constraints from theinvisible branching fraction of the neutral pion, BR( π → inv.) < . × − [47]. Setting3 Λ sd ( GeV ) Λ dd ( G e V ) m S =
400 MeV, m P =
10 MeV, λ SP = - τ S ( c m ) K O T O p r e f e rr e d K L → π i n v . K + → π + i n v . π → inv. K L → i n v . Λ sd ( GeV ) Λ dd ( G e V ) m S =
350 MeV, m P =
100 MeV, λ SP = - τ S ( c m ) K O T O p r e f e rr e d K L → π i n v . K + → π + i n v . K L → i n v . Λ sd ( GeV ) Λ dd ( G e V ) m S =
200 MeV, m P =
10 MeV, λ SP = - τ S ( c m ) K O T O p r e f e rr e d K L → π i n v . K + → π + i n v . π → inv. K L → i n v . FIG. 5. Number of expected K L → SP → πP P events at KOTO in the Λ sd –Λ dd plane for threebenchmark points of the S and P masses. The SP coupling is set to λ SP = 10 − . The rightvertical axis indicates the S lifetime, which is approximately constant for Λ dd > GeV. Oneexpects 3 events along the solid dark green line, 2–4 events in the dark green region, and 1–5 eventsin the light green region. The gray regions are excluded by the KOTO limit on K L → π inv. orthe bound on the invisible K L branching ratio. The dotted lines show the generic location of otherconstraints that depend on additional model parameters. Benchmark BM3 is not shown, as the S → P decay is kinematically forbidden. NP / Re(˜ g P dd ) = Λ NP / Im(˜ g SPdd ) = Λ dd in the benchmarks BM1 and BM4, the regions belowthe dotted horizontal lines are excluded. For benchmarks BM2 and BM3, the P mass is toolarge for the π → P P decay, so the couplings are therefore completely unconstrained byBR( π → inv.). C. Simplified UV models
The higher dimensional interactions in eq. (6) that lead to the exotic meson decays canbe UV completed by simplified models in various ways. In this section, we discuss brieflytwo possibilities: (1) vector-like quarks and (2) an inert Higgs doublet.
1. Vector-like quark model
We introduce two sets of heavy vector-like quarks D and Q which have quantum numbersof the right-handed down quark singlets, D = ( , ) − , and of the left-handed quark doublets Q = ( , ) , respectively. These quantum number assignments admit the following terms inthe Lagrangian: L ⊃ m Q ¯ Q L Q R + m D ¯ D L D R + Y QD ( ¯ Q L D R ) h + Y DQ ( ¯ D L Q R ) h c + h.c.+ X Dd ( ¯ D L d R ) S + X Ds ( ¯ D L s R ) S + Z Qd ( ¯ Q R d L ) iP + Z Ds ( ¯ Q R s L ) iP + h.c. . (27)The first line contains the masses m Q and m D for the vector-like quarks, as well as inter-actions with the SM Higgs doublet h . The masses m Q , m D and the couplings Y QD , Y DQ are in general complex parameters. However, not all of their phases are observable. Usingthe freedom to re-phase the vector-like quark fields, we will choose real m Q , m D and Y QD without loss of generality. The second line in eq. (27) contains couplings of the SM downand strange quarks with S and the vector-like quark D as well as with P and the vectorlikequark Q . The couplings X Dd , X Ds , Z Qd , and Z Qs contain physical phases.Note that the above Lagrangian is invariant under a Z symmetry under which all SMparticles are even, while the vector-like quarks as well as S and P are odd. Thus P remainsan absolutely stable dark matter candidate. In addition to the couplings shown, the modelcould also contain Z invariant couplings involving S and Q or P and D . However, such5 DQs L , d L s R , d R PS (cid:104) h (cid:105) Hs L , d L s R , d R PS (cid:104) h (cid:105) FIG. 6. Feynman diagrams that show the matching of the vector-like quark model (left) and theinert Higgs model (right) onto the effective
SP qq (cid:48) interactions in eq. (6). couplings are not required to generate the desired low energy interactions and we will neglectthem in the following.Integrating out the vector-like quarks at tree level (see fig. 6, left diagram), and matchingonto the effective Lagrangian of eq. (6), we find g SPdd Λ NP = − iY QD v √ m Q m D Im( X Dd Z ∗ Qd ) , ˜ g SPdd Λ NP = iY QD v √ m Q m D Re( X Dd Z ∗ Qd ) , (28) g SPss Λ NP = − iY QD v √ m Q m D Im( X Ds Z ∗ Qs ) , ˜ g SPss Λ NP = iY QD v √ m Q m D Re( X Ds Z ∗ Qs ) ,g SPsd Λ NP = Y QD v √ m Q m D
12 ( Z Qs X ∗ Dd − X Ds Z ∗ Qd ) , ˜ g SPsd Λ NP = Y QD v √ m Q m D i Z Qs X ∗ Dd + X Ds Z ∗ Qd ) . As required by SU (2) L invariance, the effective interactions g SPij / Λ NP and ˜ g SPij / Λ NP areproportional to the SM Higgs vev v (cid:39)
246 GeV. If all couplings X ij , Y ij , Z ij are of O (1), wecan expect vector-like quark masses m Q,D ∼ √ Λ NP v ∼ GeV. The couplings above arenot all independent but obey the relation | ˜ g SPsd | − | g SPsd | + 2 i Re( g SPsd ˜ g SP ∗ sd ) = ˜ g SPdd ˜ g SP ∗ ss − g SPdd g SP ∗ ss + i (˜ g SPdd g SP ∗ ss + ˜ g SP ∗ ss g SPdd ) . (29)One therefore expects that the flavor changing couplings are of the order of the geometricmean of the flavor conserving couplings.The vector-like quarks also give 1-loop contributions to kaon mixing. We checked explic-itly that those contributions scale as v / ( m Q m D ) and are completely negligible.
2. Inert Higgs doublet model
In a second scenario, we introduce an inert Higgs doublet H with mass m H , which couplesto down and strange quarks, the SM Higgs, and the scalars S and P through the following6interactions: L ⊃ m H H † H + λ SP ( H † h + h † H ) SP + Y dd ( ¯ d L d R ) H + Y ds ( ¯ d L s R ) H + Y sd (¯ s L d R ) H + h.c. . (30)As in the vector-like quark scenario, this inert Higgs Lagrangian is invariant under a Z symmetry: S and P are odd, while all other particles are even. Additional Z symmetricquartic couplings of the inert Higgs involving e.g. S or P are also possible but are notrequired to generate the low energy interactions in eq. (6), and we neglect them in thefollowing.Integrating out the inert Higgs at tree level (see fig. 6, right diagram), and matching ontothe effective Lagrangian of eq. (6), we find g SPdd Λ NP = iλ SP v √ m H Re( Y dd ) , ˜ g SPdd Λ NP = iλ SP v √ m H Im( Y dd ) , (31) g SPds Λ NP = λ SP v √ m H i Y ds + Y ∗ sd ) , ˜ g SPds Λ NP = λ SP v √ m H
12 ( Y ds − Y ∗ sd ) . (32)In addition, integrating out the inert Higgs gives 4-fermion contact interactions of the type( ¯ d L s R )( ¯ d R s L ) that modify kaon oscillations. We find the following contributions to the kaonmixing matrix element: M = m K f K m s m H η QCD B Y sd Y ∗ ds , (33)where B (cid:39) .
78 [15] (see also [17, 26]) and η QCD is the QCD correction factor given ineq. (10), with M = m H . Modifications to the mixing matrix alter the neutral kaon oscillationfrequency ∆ M K and the observable (cid:15) K that measures CP violation in kaon mixing. Theabove contribution to M modifies these two quantities as∆ M K = ∆ M SM K + 2 Re( M ) , (cid:15) K = (cid:15) SM K + Im( M ) √ M K . (34)Taking into account the SM predictions ∆ M SM K and (cid:15) SM K from [11, 13], and the correspondingexperimental values from [50], we find the boundsRe( Y sd Y ∗ ds ) < . × − × (cid:16) m H (cid:17) (cid:18) α s ( m H ) α s (1 TeV) (cid:19) / , (35)Im( Y sd Y ∗ ds ) < . × − × (cid:16) m H (cid:17) (cid:18) α s ( m H ) α s (1 TeV) (cid:19) / . (36)Assuming | Y ds | (cid:39) | Y sd | and O (1) CP violating phases, the kaon mixing bounds are compatiblewith Λ sd (cid:38) × GeV. Also, note that the bounds are entirely avoided if either of Y sd or Y ds is set to zero.7 III. ASTROPHYSICAL AND TERRESTRIAL CONSTRAINTS
We now consider extant astrophysical and terrestrial constraints that may apply to ourmodel.First, anticipating our treatment of P as a dark matter candidate, we note that directdetection, indirect detection, and self-interaction constraints are not relevant for our modelin its minimal configuration (see eq. (6)). If our P is the cosmological dark matter, but theSM is only coupled to the current SP , then direct detection is only sensitive to the inelasticscattering process P + SM → S + SM, which is kinematically forbidden unless the darkmatter is boosted. Similarly, indirect detection and self-interaction processes require twovertices, and thus the cross sections are suppressed by Λ .Extensions of our minimal model containing couplings to P (see eq. (7)) may be subjectto these constraints due to the presence of additional interactions. However, we first treatconstraints from supernova cooling and beam dump experiments, which apply directly tothe minimal model. A. Supernova constraints
Supernova cooling provides powerful constraints on new weakly-coupled light particles.Evaluating these bounds properly requires a detailed analysis that lies beyond the scope ofthis work, but we can perform an order-of-magnitude estimate to determine the regions ofour parameter space that are likely to be subject to such constraints.In the case of axions, the cross section for axion production
N N → N N a is constrainedby SN1987A to lie in the range [46]3 × − (cid:46) σ GeV − (cid:46) − . (37)Below the lower limit, axions are not produced in sufficient numbers to affect the coolingprocess. Above the upper limit, the produced axions are trapped within the supernovaenvironment, and are unable to cool the system more effectively than neutrinos. Manydetails of the calculation for axions should be modified in our case, but we will make a crudeestimate of the constraints by requiring our production cross section to lie in the same range.Since P is stabilized by a Z symmetry, it can only be produced in pairs, or in associationwith S . The process N N → N N P P is mediated at the loop level in the minimal model,8involving two insertions of the effective interaction vertex. Since T SN (cid:39)
30 MeV [46], weestimate the cross section as σ NN → NNP P ∼ π T Λ dd (cid:39) × − GeV − (cid:18) T SN
30 MeV (cid:19) (cid:18) GeVΛ dd (cid:19) , (38)lying below the constrained range of cross sections, even neglecting exponential suppressionwhen m P (cid:38) T SN . In the case of SP production, N N → N N SP , since m S (cid:29) T SN , weestimate the cross section as σ NN → NNSP ∼ π Λ dd e − ( m S + m P ) /T SN (cid:39) × − GeV − exp (cid:20) (cid:18) − m S + m P
350 MeV 30 MeV T SN (cid:19)(cid:21) (cid:18) GeVΛ dd (cid:19) . (39)While parts of our parameter space are thus expected to be unconstrained by supernovalimits, it is important to note that if m P is small, or if Λ dd (cid:46) GeV, the estimatedproduction cross section enters the prohibited range. In particular, if Λ dd = 10 GeV, thenavoiding the bound requires m S + m P (cid:38)
450 MeV, favoring the larger P masses in fig. 2.However, in this naive projection of supernova constraints, our model remains viable in awide region of the parameter space. B. Beam dump constraints
In minimal form, our model of the KOTO excess is potentially subject to constraintsfrom long-lived particle searches: the partial decay width of S → π P is bounded frombelow by the observed KOTO event rate, so in the absence of additional interactions, the S lifetime can be O (m) or larger. Such lifetimes are probed very effectively by beam-dumpexperiments with O (100 m) baseline lengths. In such an experiment, a proton beam isdirected at a target, potentially producing a large number of S particles. The S particlestravel unimpeded through shielding and earth over a distance L B , reaching an instrumenteddecay volume with length L D . The S → π P events within the decay volume can betypically detected with an O (1) efficiency E . Thus, the strength of the constraints is mainlydetermined by two factors: (1) how many S particles are produced, and (2) what fractionof these undergo S → π P within the decay volume.First we estimate the number of S particles produced. There are at least two channelsto consider: direct production from nucleon-nucleon scattering, and secondary production9from kaon and other meson decays. Observe, however, that the fraction of proton-protoncollisions that produce an SP pair is of order ( s/ Λ NP ) /α S , which is much smaller than thebranching ratios BR( K L → SP ) and BR( K S → SP ) implied by our interpretation of theKOTO excess. We also checked that the number of S from eta decays η → SP is smallcompared to those coming from the kaon decays in our scenarios.Given N p protons on target, we expect that of order N K ∼ − N p kaons are produced[30], and this is sufficient for kaon decays to dominate production. However, of these kaons,most will be stopped or scattered away from the axis of the beam before they decay. Thedynamics of kaon energy loss and deflection in materials are complicated, but the nuclearinteraction length for relativistic kaons in most materials is L nuc ∼ O (10 cm) [50], so wewill assume that any kaons traveling this far before decaying are sufficiently slowed downor deflected such that only a negligible fraction of the S particles are directed towards thedetector. Thus, the number of S particles produced and directed towards the detector is oforder N S ∼ (cid:88) X = L,S − N p Γ( K X → SP )Γ K X (cid:20) − exp (cid:18) − Γ K X L nuc γ K X (cid:19)(cid:21) , (40)where γ is the boost factor. Now, accounting for the fraction of S particles which decay inthe decay volume, and accounting for the efficiency of the detector, the number of events isgiven by N E ∼ (cid:88) X = L,S − N p BR( K X → SP ) BR( S → π P ) E× (cid:20) − exp (cid:18) − Γ K X L nuc γ K X (cid:19)(cid:21)(cid:124) (cid:123)(cid:122) (cid:125) avoid kaon deflection exp (cid:18) − Γ S L B γ S (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) reach decay volume (cid:20) − exp (cid:18) − Γ S L D γ S (cid:19)(cid:21)(cid:124) (cid:123)(cid:122) (cid:125) decay in decay volume . (41)In the minimal scenario, with no additional interactions, BR( S → π P ) = 1.We now estimate the event counts in the CHARM [7] and NuCal [10] experiments.CHARM conducted a search for decays of axion-like particles with 2 . × protons in-cident on a copper target at 400 GeV, a baseline length of 480 m, and a 35 m-long decayvolume. The detector efficiency is approximately 0 .
5. No candidate events were observed.NuCal conducted a similar search, with 1 . × protons incident on an iron target at70 GeV, a baseline length of 64 m, and a 23 m-long decay volume. One candidate event wasobserved with an expected standard model background of 0.3. To estimate the event counts0that would be produced by our model, we set γ K X = γ S = 10 for CHARM and reduce theseproportionally for NuCal’s lower beam energy.Assuming BR( S → π P ) = 1, the resulting event count is shown as a function of the S lifetime in fig. 7. The minimum expected number of events at long S lifetime is large unless τ S (cid:38) m, and lifetimes as large as 10 m may be excluded. This potentially rules out asignificant portion of our parameter space, as indicated in fig. 4. On the other hand, theevent rate cuts off sharply for τ S (cid:46) τ S ∼ SP interactionin our model is non-zero, which can shorten the S lifetime significantly if m P is small (seefig. 5). The presence of this additional interaction greatly extends the parameter spaceconsistent with the null results at CHARM and NuCal.Looking towards future prospects, most proposed beam-dump experiments are competi-tive in the same regime of S lifetimes. However, it has been suggested [8] that the SeaQuestexperiment [4] may be modified to serve as a short-baseline beam dump experiment, withthe instrumented area starting only ∼ SP couplingis unconstrained, the S lifetime can be shortened by many orders of magnitude, potentiallyevading even these experiments. C. Direct dark matter detection
Direct detection of P can occur in the extended model via the interactions in eq. (7).While the interaction terms containing (¯ qiγ q ) P give rise to suppressed velocity-dependentcross sections off of nucleons, the operators (¯ qq ) P with q = d, s produce potentially de-tectable scattering off of nucleons. We define the integrated nucleon form factors B Nq ≡ (cid:104) N | ¯ qq | N (cid:105) = m N m q f Nq , (42)where f Nq are the form factors for nucleon N of quark q [39]. The direct detection crosssection can be cast as σ = (cid:88) q = d,s (cid:32) m N m P + m N g P qq Λ NP B Nq (cid:33) ≈ (cid:104) ( B Nd ) ( g P dd ) + ( B Ns ) ( g P ss ) (cid:105) . (43)1 − − τ S [m] − − − − E v e n t c o un t NuCalCHARMSeaQuest
FIG. 7. Estimated event counts at CHARM and NuCal and prospective event counts at SeaQuestas a function of the S lifetime. The top curve fixes Γ( K L → SP ) to saturate the experimentalbound on the invisible K L width. The bottom curve fixes Γ( K L → SP ) such that BR( K L → SP ) isequal to the ratio inferred from the KOTO excess, i.e., Γ( K L → SP ) is the smallest width for whichthis model can account for the excess. Both curves assume that Γ( K L → SP ) = Γ( K S → SP ) andthat BR( S → π P ) = 1. Under these conditions, 1 m (cid:46) τ S (cid:46) m is ruled out. SeaQuest mayeventually probe lifetimes as short as τ S ∼ Using the central values B pd ≈ .
77 and B ps ≈ .
50, it is clear that the dominant effect isscattering off of d quarks if g P ss (cid:39) g P dd . The scattering cross section off of protons is then σ p ≈ × − cm ( g P dd ) (cid:18) GeVΛ NP (cid:19) , (44)i.e., close to 0 . IV. COSMOLOGICAL PRODUCTION
We now turn to the question of cosmological production of the dark matter candidate P :which scenarios allow P to be produced with the observed dark matter density?The standard thermal freeze-out paradigm is not viable in our minimal scenario. Esti-2mating the freeze-out temperature by n P σ ( P P → SM) ∼ H ( T ), we have T FO ∼ π Λ m Pl ∼ (cid:18) Λ NP MeV (cid:19) MeV , (45)where Λ NP is the scale of new physics in question—for practical purposes, the lesser of Λ sd and Λ dd . For typical values of Λ NP consistent with the KOTO excess, T FO (cid:29) m P , so P freezes out as a hot relic, with relic abundanceΩ P h ∼ m P keV (cid:18) g (cid:63) | T FO (cid:19) ∼ . (cid:16) m P
80 eV (cid:17) . (46)Thus, for the masses and couplings considered in this work, P is generically overproducedin the freeze-out scenario. If the P mass were small enough to be produced with the rightrelic abundance, then P would be ruled out as a dark matter candidate because of structureformation constraints on relativistic relics.Departing from the minimal scenario outlined above opens up the possibility that an additional effective interaction with SM species keeps P in thermal equilibrium, and thatthe P relic abundance is set by thermal decoupling (freeze-out). Since generally thermaldecoupling happens at temperatures T ∼ m P /
25, in order to avoid possible constraints fromBBN, one can assume that the effective interaction only involves SM neutrinos:
L ⊃ νν ¯ ννP P. (47)For the effective dimension five operator in the equation above, we find that the zero-velocitythermally averaged product of the pair-annihilation cross section and relative velocity islim v → (cid:104) σv (cid:105) = 14 π νν . (48)A standard treatment of the relic abundance for the pair-annihilation cross section aboveindicates that P would be produced in the right amount if Λ νν (cid:39) P were in equilibrium at high temperatures, an effectiveinteraction with SM neutrinos—which, incidentally, can be quite naturally embedded in theUV completions described above—could suppress the P abundance to an acceptable relicdensity in agreement with observations.In the absence of the additional neutrino portal described in the paragraph above, theonly alternative is production via freeze-in [32]. Here the dark species is produced out of3equilibrium by some standard model species, and the abundance increases until cosmologicalexpansion halts production. It is thus possible to avoid overproduction of dark matter withextremely small couplings. Note that while other mechanisms might allow for additionalproduction of P , the freeze-in contribution is unavoidable in the range of temperatureswhere our effective theory is valid.Typically, freeze-in is applied to a UV-complete theory, where the dark matter productionrate can be computed starting at very high temperatures. In the context of a renormalizablemodel, it can be shown that dark matter is produced primarily at lower temperatures, sothe details of the UV physics are unimportant. Thus, freeze-in can be used to consistentlycalculate the non-thermal relic abundance, even though a formal dependence on initial con-ditions remains. Note that this is in contrast to the freeze-out paradigm, where equilibriumwith the standard model bath erases any non-trivial initial conditions in the dark sector.However, in our scenario, the dark matter is produced through non-renormalizable in-teractions, and the standard freeze-in mechanism cannot be directly applied: our effectivetheory cannot be applied at scales above some O (Λ NP ) cutoff. At first, this does not seemto be a problem: in standard freeze-in, production is IR-dominated, and we can apply oureffective theory in this regime. But for higher-dimension operators, production is no longerIR-dominated, and it is no longer possible to self-consistently estimate the relic abundanceunless an initial condition is fixed at a temperature where the effective theory is valid.Naively, one can place a lower bound on the relic abundance by fixing the dark matterabundance to zero at T ∼ Λ NP and computing the amount of dark matter produced at lowertemperatures, where the effective theory is valid. However, as we shall see in the followingsection, this still leads to overproduction of P . Thus, in our model, it would seem thatdark matter is overproduced in the freeze-in scenario, even with the most favorable initialconditions.There is, however, a significant loophole in this argument: setting the dark matter abun-dance to zero at T ∼ Λ NP is in fact not the most favorable initial condition. If reheatingoccurs at a temperature T rh (cid:28) Λ NP , then the dark matter abundance should be set to zeroat this lower temperature, allowing for a much lower relic abundance. There is nothingparticularly unnatural about this scenario: in general, freeze-in production of dark matterdepends on the reheating temperature. This dependence is weak if the reheating scale hap-pens to be much higher than any scale in the theory, but the convenience of this arrangement4does not constitute evidence for it. Moreover, if T rh (cid:28) Λ NP , then our effective theory canbe used to self-consistently compute the dark matter relic abundance independently of anyUV completion. This paradigm is known as UV freeze-in [24]. A. Computing the yield
First, we briefly review the computation of the dark matter relic abundance in the stan-dard freeze-in paradigm. The basic technology of UV freeze-in is identical to that of standardfreeze-in, but the initial condition is fixed at the reheating temperature T rh , which becomesan important free parameter of the theory. In certain scenarios, the dark matter yield isquite sensitive to temperatures near T rh , and decreasing T rh can significantly reduce the relicabundance.The starting point is the Boltzmann equation,˙ n χ + 3 Hn χ = (cid:88) I,F [ N χ ( F ) − N χ ( I )] (cid:90) d n I Π I d n F Π F (2 π ) δ ( p I − p F ) |M I → F | (cid:89) i ∈ I f i . (49)Here n χ denotes the number density of a dark species χ , I and F index initial and final states, N χ ( S ) denotes the number of χ particles in the state S , dΠ i = g i d p i / (2 π ) E i , |M I → F | isthe spin-averaged squared matrix element, and f k is the phase space density of the species k .We assume Maxwell-Boltzmann statistics, and by conservation of comoving entropy density,we rewrite the left-hand side of eq. (49) as ˙ n χ + 3 Hn χ = S ˙ Y χ , where S = (2 π / g (cid:63)S T isthe entropy density and Y χ ≡ n χ /S . In turn, since ˙ T ≈ − HT , we have S ˙ Y χ ≈ xHSY (cid:48) χ ( x ),where x = µ/T for any fixed mass µ .In freeze-in, one assumes that the phase space density of the dark species is always small,so that any initial state with N χ ( I ) > f i can be replacedwith equilibrium distributions e − E i /T . Now eq. (49) reads Y (cid:48) χ ( x ) = 1 xHS (cid:88) I (cid:54)(cid:51) χ,F N χ ( F ) (cid:90) d n I Π I d n F Π F (2 π ) δ ( p I − p F ) |M I → F | exp ( − xE I /µ ) . (50)We will be interested in two types of processes: 1 → → → i → χf , we set µ = m i , i.e., x = m i /T . We recognize thedecay width Γ i → χf in eq. (50), which becomes Y (cid:48) χ ( x ) = 12 π g i m i x HS N χ ( F )Γ i → χf K ( x ) , (51)5where K is a modified Bessel function of the second kind, and now N χ ( F ) is either 1 or 2,depending on whether f = χ . Substituting H = 1 . g / (cid:63) x − m i m − , the total yield can nowbe computed by performing a 1-dimensional integration of eq. (51), as Y χ ( ∞ ) = 45 N χ ( F ) g i m Pl Γ i → χf . × π m i (cid:90) ∞ x min d x x K ( x ) g / (cid:63) g (cid:63)S . (52)In particular, suppose that f = χ , m χ (cid:28) m i , and |M i → χχ | = λ . If production mainlytakes place during an epoch when g (cid:63) and g (cid:63)S are not changing rapidly, then we can estimatethe yield as Y χ ( ∞ ) (cid:39) N χ ( F ) g i m Pl λ . × π ) g / (cid:63) g (cid:63)S m i x min (cid:28) (cid:113) π x / exp ( − x min ) x min (cid:29) . (53)Similarly, if the abundance of χ is set by 2 → ij → χf , then theintegrals over the final-state phase space produce the cross section σ ij → χf , and eq. (50)becomes Y (cid:48) χ ( x ) = N χ ( F ) g i g j xHS (cid:90) d p i (2 π ) d p j (2 π ) σv exp ( − xE i /µ ) exp ( − xE j /µ ) . (54)This remaining integrals can be reduced to a single 1d integral, following e.g. [29]. Integrat-ing in x , the yield is then Y χ ( ∞ ) = µN χ ( F ) g i g j π ) (cid:90) ∞ x min d xx HS (cid:90) ∞ s min d s σ ( s ) r − r + ×× (cid:110) m + m − s (cid:16) µx + √ s (cid:17) exp (cid:0) − x √ s/µ (cid:1) + r − r + √ s K (cid:0) x √ s/µ (cid:1)(cid:111) , (55)where m ± = | m i ± m j | , r ± = (cid:0) s − m ± (cid:1) / , and s min = min( m i + m j , m χ + m f ) . As in the1 → ii → χχ when m i (cid:28) m χ and the evolution of g (cid:63) and g (cid:63)S is negligible. If |M ii → χχ | = λ , then the result is Y χ ( ∞ ) (cid:39) N χ ( F ) g i m Pl λ . × π g / (cid:63) g (cid:63)S m i (3 π/ m i /m χ x min (cid:28) x min exp ( − x min m χ /m i ) x min (cid:29) , (56)where x min = m i /T max . The analogous expression for m χ (cid:28) m i is obtained by interchanging m i and m χ and taking µ = m χ (i.e. x min = m χ /T max ). However, in our model, 2 → NP , so we should6instead set |M ii → χχ | = s/ Λ . In this case, the result is Y χ ( ∞ ) (cid:39) N χ ( F ) g i m Pl m χ . × π g / (cid:63) g (cid:63)S m i Λ π ( m χ /m i ) − x − x min (cid:28) x min exp ( − x min m χ /m i ) x min (cid:29) . (57)This demonstrates a key difference between standard freeze-in and UV freeze-in: a naiveextrapolation of the production rate to arbitrarily high temperatures (small x min ) diverges.Of course, one should not expect to accurately compute the production rate in the effectivetheory at T (cid:29) Λ NP . But even so, if Λ NP (cid:29) T max (cid:29) max { m χ , m i } , then production candominated by 2 → → m χ and m i are MeV-scale, while Λ NP (cid:38) GeV. Thus, production by2 → Y χ ( ∞ ) ≈ × − ( m χ / MeV), we can estimate the ranges of parameterswhich account for all of dark matter—or, at least, those which do not overclose the universe.If dark matter in our model is produced dominantly by quark annihilation via an interactionof the form Λ − dd d ( iγ ) ¯ dSP , then the only important parameters are Λ dd and x min . Note thatif this is the only interaction at work, there is no contribution from decays.First, suppose that x min (cid:28)
1. Then the scale Λ dd must satisfyΛ dd (cid:38) (cid:18) g (cid:63) | T rh (cid:19) − / (cid:18) T rh GeV (cid:19) / × GeV . (58)Per the analysis in section II B, this is too large to account for the KOTO excess—and thisestimate accounts for only one production channel! In particular, if T rh > Λ dd , dark matteris dramatically overproduced. At the very least, one requires T rh (cid:46)
100 MeV, where theapproximations made for this estimate are no longer trustworthy. However, suppose insteadthat reheating indeed takes place near the MeV scale, so that x min (cid:29)
1. Then the situationis quite different: neglecting the difference between m S and m P , we haveΛ dd (cid:38) (cid:18) g (cid:63) | T rh (cid:19) − / (cid:18) T rh
10 MeV (cid:19) exp (cid:20) − (cid:18) m S T rh − (cid:19)(cid:21)
300 GeV . (59)This bound poses no obstacle to accounting for the KOTO excess. When combined, thesetwo estimates naively suggest that our model can account for all of dark matter if reheatingtakes place between 100 MeV and 10 MeV. While the scale of reheating is often assumed tobe much higher, the strongest observational lower bound on the reheating temperature is in7fact only T rh (cid:38) → T rh <
100 MeV, thenquarks are confined into hadrons during the entire production period. One must then mod-ify the effective couplings to account for hadronic scattering, and since the initial and finalstates are all (pseudo)scalars, the matrix elements no longer carry any s -dependence. Addi-tionally, since single hadrons can now decay to S and P , hadronic decays can dominate therelic abundance, and must be included in the calculation of the yield.In the following section, we treat these issues in detail and calculate the relic densitynumerically. B. Determining the reheating temperature
Our estimates in the previous section suggest that P can be produced non-thermally,and can account for all of dark matter, if the initial temperature of the SM bath is between100 MeV and 10 MeV. We now refine our estimate of the yield to account for confinementand hadronic decays, and then numerically compute the yield to establish the requiredreheating temperature in our model.At T (cid:46)
200 MeV, quarks are confined into hadrons, and the effective interactions ofthe hadrons with S and P are well described by chiral perturbation theory (chiPT). Theeffective couplings of hadrons to S and P are built from a combination of the new physicsscales and QCD parameters. Since the couplings in the quark-level effective Lagrangianare proportional to Λ − , and the hadron-level 1 → / Λ NP , where Λ chiPT is some scale associated with low-energy QCD. Similarly, in the 2 → (cid:48) chiPT / Λ NP . As we will see momentarily, Λ ( (cid:48) )chiPT is a combination of two constants, f π ≈
92 MeV and B ≈ s and p , respectively a scalar and pseudoscalar.These take the form L QCD [ s, p ] = − ¯ q ( s ( x ) − iγ p ( x )) q . (60)Interactions of hadrons with these currents enter the chiPT Lagrangian via the current χ = 2 B ( s + ip ). At lowest order, we have L ⊃ f π (cid:0) χU † + U χ † (cid:1) , U = exp (cid:32) i √ f π Φ (cid:33) , (61)where Φ is the PNGB matrix [see e.g. 44]. Now consider a quark-level interaction of theform L ⊃
12 ¯ q i (cid:0) g O S ij − i ˜ g O S ij γ (cid:1) q j O S + i q i (cid:0) g O P ij − i ˜ g O P ij γ (cid:1) q j O P + h . c . (62)where O S is a scalar (CP-even) and O P is a pseudoscalar (CP-odd). We can then identify s ij = − (cid:0) g O S ij + g O S ∗ ji (cid:1) O S − (cid:0) g O P ij − g O P ∗ ji (cid:1) O P , (63) p ij = − i (cid:0) ˜ g O S ij − ˜ g O S ∗ ji (cid:1) O S − i (cid:0) ˜ g O P ij + ˜ g O P ∗ ji (cid:1) O P . (64)Substituting these expressions into eq. (61) with O S = S , P , and O P = SP gives theinteractions of S and P with the PNGBs. For instance, the interactions of S and P with π are specified by L ⊃ B f π π (cid:16) SP Im g SPdd − S Im ˜ g S dd − P Im ˜ g P dd (cid:17) − B ( π ) (cid:16) SP Re ˜ g SPdd − S Re ˜ g S dd − P Re ˜ g P dd (cid:17) + · · · , (65)where the ellipsis denotes a series of higher-dimensional operators. We include all termsup to second order in the PNGB fields in our analysis, and the form of the hadron-levelLagrangian is as expected from dimensional analysis. Note that it is essential to considercomplex-valued g ij and ˜ g ij , without which some interactions will vanish.We can now determine the reheating temperature required to produce the observed darkmatter density as a function of our model parameters. First, using the normalization factorsas they appear in eq. (65), we can now estimate the relative significance of decays andscattering, starting with eqs. (54) and (56). Assuming that all dimensionless couplings are O (1), we set the coupling λ for 3-point vertices equal to B f π / Λ NP , and we set the coupling9for 4-point vertices to B / Λ NP . In this regime, we typically have m i (cid:29) max { m P , T rh } , andin this limit, Y → P ( ∞ ) Y → P ( ∞ ) (cid:39) (cid:18) f π m i (cid:19) m P (cid:28) T rh (cid:28) m i π ( T rh /m i ) exp (2 m i /T rh ) T rh (cid:28) m P (cid:28) m i . (66)Our parameter space includes 1 MeV (cid:46) m P (cid:46)
200 MeV, so the ratio above can be largeor O (1) depending on the choice of the P mass, but it is never small. Note, however,that increasing m P can also close certain decay channels. In particular, if there exist in-teractions allowing the decay π → P P , this channel naively dominates production at lowtemperatures, but is closed for 2 m P > m π .Since decays dominate in most of the parameter space, we can make a first estimate ofthe yield by considering only production via K L → SP , the same decay process which isnecessary to account for the KOTO excess. Neglecting the distinction between m S and m P ,the yield is Y K L → SPP ( ∞ ) (cid:39) . × π ) / g / (cid:63) g (cid:63)S (cid:18) Bf π m K Λ sd (cid:19) m Pl T rh exp ( − m S /T rh ) , (67)and the resulting upper bound on Λ sd isΛ sd (cid:38) (cid:18) g (cid:63) | T rh (cid:19) − / (cid:18) T rh
15 MeV (cid:19) / exp (cid:20) − (cid:18) m S T rh − (cid:19)(cid:21) × GeV . (68)For the typical parameter values selected above, this upper bound is towards the lower edgeof our parameter space of interest for the KOTO excess. Thus, although hadronic decayssignificantly enhance production relative to the prediction of eq. (59), this channel on itsown does not pose an obstacle to accounting for the KOTO excess.However, in general, it is necessary to numerically evaluate the yield to determine the ex-tent of the viable parameter space—and, in particular, to identify the reheating temperaturethat produces the observed relic density at each parameter point. The resulting reheatingtemperatures are shown in fig. 8, and are of order 10 MeV throughout the parameter spaceof interest. The required reheating temperature is mainly controlled by the smaller of Λ sd and Λ dd , with a slight bias towards Λ sd , since production by η decays is suppressed comparedto production by K decays due to their relative masses. Note that all couplings except for g SPsd and g SPdd are neglected in fig. 8, so, in particular, π → P P does not contribute to the0relic density even when 2 m P < m π . If we suppose that all of the couplings in the effectivetheory are of similar order, the viable parameter space can change significantly.We can estimate this effect by taking g S q q = g SPq q = g P q q and setting g O sd = (cid:0) g O ss g O dd (cid:1) / tofix g O ss . The resulting reheating temperatures are shown in fig. 9. With these choices for thecouplings, our two benchmark points with m P = 10 MeV are incompatible with freeze-in asa production mechanism, since the required reheating temperature is below observationalbounds throughout the relevant parameter space. This is due to the open π → P P decay,which is kinematically closed for the other two benchmark points with m P = 100 MeV and m P = 125 MeV. For these points, the required reheating temperature is again of order10 MeV throughout the relevant parameter space. At the top-left of the correspondingpanel of fig. 9, the required reheating temperature decreases with increasing Λ dd . This isjust because of our assumption that g O sd is the geometric mean of g O dd and g O ss : increasing Λ dd corresponds to decreasing g O dd , so if g O sd is held fixed, then g O ss must increase to compensate.This increases the relic density, forcing a lower reheating temperature.Finally, we note that the reheating temperatures shown in figs. 8 and 9 are potentiallyimprecise, and should be viewed as lower bounds. Our calculation of the yield assumes thatall of the initial-state species are thermalized, but the mesons freeze out at temperatures ofthe same order considered here. In particular, π , K , and η freeze out at 3 MeV, 10 . . T rh (cid:38)
15 MeV, in which case all of the relevant mesonsare thermalized, so this is an upper bound on T rh . Likewise, we can see that DM wouldbe underproduced for T rh below a particular value even if the mesons have their equilib-rium number densities. This lower threshold is O (7 MeV) if π → P P is forbidden, and Since production is dominated by decays, the dark matter relic abundance is mainly determined only Λ sd [GeV] Λ dd [ G e V ] m S = 400 MeV , m P = 10 MeV . . . . . . . . Λ sd [GeV] Λ dd [ G e V ] m S = 350 MeV , m P = 100 MeV . . . . . . . . Λ sd [GeV] Λ dd [ G e V ] m S = 300 MeV , m P = 125 MeV . . . . . . . . FIG. 8. Reheating temperature in MeV to produce the observed DM relic density, including allproduction channels with no DM in the initial state. The couplings g SPsd and g SPdd are taken to bepurely imaginary, while all other couplings are set to zero, corresponding to the minimal scenarioto account for the KOTO excess. In the leftmost panel (BM1), all decay channels are open. In themiddle panel (BM2), S → P is kinematically closed, so there are no number-changing interactionsin the dark sector: S decays via S → π P . In the rightmost panel (BM3), S → P and π → P P are both closed, so there is no contribution to the relic density from π decays. O (2 MeV) if it is not. The only qualitative importance of out-of-equilibrium effects is thatit may be possible to construct a cosmologically-viable model in which dark matter is notoverproduced even if π → P P is open. However, such a model would depend on the detailsof reheating, and this analysis lies beyond the scope of this work.
V. DISCUSSION
In the foregoing sections, we have introduced a model to account for the KOTO excessand explored the cosmological effects. We now discuss the implications of our results andfuture experimental prospects.If the KOTO excess is interpreted at face value, this suggests apparent violation of theGN bound. As has been discussed by several authors [19, 21, 23, 25, 30, 36–38, 40–42, 52],such a signal at KOTO can be mimicked by a decay of the form K L → π X , where X denotes one or more invisible species. In contrast to most studies, we focus on a new physics by the number density of the parent mesons, and is fairly insensitive to other details of the phase spacedistribution. Λ sd [GeV] Λ dd [ G e V ] m S = 400 MeV , m P = 10 MeV . . . Λ sd [GeV] Λ dd [ G e V ] m S = 350 MeV , m P = 100 MeV . . . . . . . . . . Λ sd [GeV] Λ dd [ G e V ] m S = 300 MeV , m P = 125 MeV . . . . . . . . . . FIG. 9. Reheating temperature (in MeV) to produce the observed DM relic density, including allproduction channels with no DM in the initial state, as in fig. 8. Here it is assumed that S , P ,and SP couple equally to light quark bilinears, and that g O sd is the geometric mean of g O ss and g O dd .The real and imaginary parts of all couplings are taken to be equal. In the first panel (BM1), alldecay channels are open, and production is dominated by π decays. In the middle panel (BM2), π → P P is closed, but S → P is still open. In the rightmost panel (BM3), both π → P P and S → P are closed, so S decays only via S → π P . In the leftmost panel, since productionis dominated by π → P P , the relic abundance is controlled exclusively by Λ dd . In this case, therequired reheating temperatures are observationally inviable throughout the parameter space. Inthe other two panels, production is dominated by K and η decays, and their relative importancedepends on Λ sd and Λ dd . scenario where the decay K L → π inv. is realized through a sequence of two-body decays K L → SP → π P P , where S and P are light neutral scalar particles. Similar scenarios werealso studied in [37] where the light particles interact with the SM through a vector or scalarportal. Here we instead analyze a setup where S and P are coupled to the SM througheffective operators at a characteristic new physics scale of Λ NP ∼ –10 GeV. We havestabilized P with a Z symmetry under which SM species are even and our new species areodd, and we have entertained the possibility of other interactions consistent with such Z invariance, including an SP term that could mediate the decay of S → P . Our effectivetheory is readily UV-completed by e.g. very heavy vector-like quarks or a TeV-scale inertHiggs doublet. Such UV completions can realize a minimal case in which only interactionsbetween SM quarks and SP are present at low energies, as well as more generic cases that3include interactions with S and P .If the KOTO excess persists, the GN bound heavily constrains new physics interpreta-tions. A model of the type we consider, with new light scalars, is one of the simplest andmost elegant solutions. Since the scale Λ NP ∼ –10 GeV indicated by the KOTO excessis so large, most other experiments are not substantially constraining (with the notable ex-ception of beam dump experiments, to which we will return shortly). In particular, in ourscenario, there is a large region of parameter space which can account for the KOTO excesswhile still unconstrained by other rare meson decays. However, it is important to considerastrophysical constraints. Supernova cooling limits can potentially rule out lower P masses:as discussed in section III A, supernova temperatures are high enough, at tens of MeV, toprobe the lightest S and P masses that we consider in fig. 2. These constraints are mostsignificant for Λ dd (cid:46) GeV, and it is important to note that establishing firm constraintsfrom supernova cooling requires a much more detailed analysis beyond the scope of thiswork. However, the simplistic expectation is that P masses of O (10 MeV) and below aredisfavored, making our scenario easier to test.Since the KOTO excess motivates the introduction of new feebly-coupled particles, it isnatural to speculate that these new species might contribute to cosmological dark matter—and indeed, we have shown that S and P can constitute all of DM even in the most minimalscenarios needed to explain the KOTO signal. Nevertheless, this comes at a cost: in theabsence of additional interactions, there is no mechanism to reduce the DM abundance,and cosmological reheating must take place at very late times, at a temperature of order10 MeV. This requirement should be interpreted as a cosmological constraint on our modeland similar models accounting for the KOTO excess. The scale of the preferred reheatingtemperature originates mainly from the masses of the new scalars: since the DM abundanceis exponentially suppressed in m DM /T rh , the required reheating temperature depends onlylogarithmically on the couplings and other scales of new physics.Such a thermal history is necessary because the effective coupling lies in an intermediateregime: it is too small for freeze-out to deplete the DM abundance, but large enough thatUV freeze-in generically overproduces DM. Thus, an additional feature is needed to preventoverproduction. The simplest mechanism to accomplish this, without any modificationto the model, is to make a judicious choice of the reheating temperature. Since we areworking with an effective theory, the DM relic density is inherently sensitive to the reheating4temperature—indeed, if T rh (cid:38) Λ NP , we cannot consistently calculate the relic density, butonly bound it below. Thus, since T rh is necessarily a parameter of our model, T rh ∼
10 MeV isas natural as any other choice. As we have discussed, observational constraints are ineffectiveat temperatures above ∼ T rh (cid:46) T rh (cid:38)
10 MeV. However, it is important to point out that the reheating scenario mightinclude features that could manifest themselves when more stringent probes of CMB becomeavailable in the future [1]. For instance, the field driving reheating might actually be an ensemble of fields, with different masses; the S and P particles might be directly producedin the decay of the field(s) driving reheating, changing the predictions for T rh made above;or new physics in the neutrino sector could make reheating temperatures in the 10s of MeVvisible once constraints on N eff significantly improve.There are other mechanisms which prevent the overproduction of dark matter withoutrequiring a particular temperature for reheating. One possibility is to add an interaction withthe SM to restore freeze-out as a viable thermal history, as we discussed briefly in the contextof a neutrino portal. This would be a heartening scenario: reheating can still take place at avery high temperature, and the coupling to leptons might allow for additional experimentalprobes. However, there are several other possibilities. In particular, it is possible thatthe DM abundance is depleted by additional interactions within the dark sector. This isnot possible in our effective theory, but one can consider extensions which keep the DM in5thermal equilibrium long after decoupling from the SM bath, or which allow other number-changing processes at a sufficient rate to allow for freeze-out at high temperatures. Weemphasize again that our results imply cosmological constraints on models of the KOTOexcess: cosmology requires either a restricted range of reheating temperatures or additionalfeatures of the low-energy theory, regardless of what fraction of cosmological DM is composedof P .Of course, one can also consider constraints which only apply if P makes up a significantfraction of DM. The simplest of these is the Lyman- α constraint on warm DM [51], whichrequires the P population to be non-relativistic at temperatures of O (keV). If P is producednon-thermally via decays at 10 MeV, typical energies will be of order the masses of theparent states, i.e., O (100 MeV). Thus, in order for P to be non-relativistic when T γ ∼ keV,we require that m P (cid:38)
10 keV. This is a somewhat weaker bound than one expects fromsupernovae, but it is not subject to the complicated physics involved in such constraints.The annihilation cross section into visible states is much too small ( ∼ − cm ) forindirect detection to be viable, nor is there any significant self-interaction in the dark sector.However, the scattering cross section with nuclei could be as large as ∼ . SP , and not P P . Since any DM accounted for byour model is composed entirely of P , this means that any diagrams contributing to indirectdetection must be suppressed by Λ − . Moreover, at lowest order, direct detection is onlysensitive to the inelastic scattering process N P → N S , which is kinematically prohibitedfor non-relativistic DM. It is thus challenging to conclusively establish that P makes upcosmological DM through direct observational means.However, it is potentially much easier to determine whether a model like ours accountsfor the KOTO excess. If the excess persists at its present size, then as KOTO reaches itsdesign sensitivity, hundreds of events will be observed. With a sample of this size, it ispossible to distinguish our model from SM three-body decays kinematically in much of ourparameter space, simply by measuring the pion’s transverse momentum. In fig. 10, we show6 K L →π νν K L → SP →π PPprompt S K L → SP →π PPdisplaced S ( τ S = )
50 100 150 200 250 300 350 400 p T ( MeV ) m S =
400 MeV, m P =
10 MeV K L →π νν K L → SP →π PPprompt S K L → SP →π PPdisplaced S ( τ S = )
50 100 150 200 250 300 350 400 p T ( MeV ) m S =
350 MeV, m P =
100 MeV K L →π νν K L → SP →π PPprompt S K L → SP →π PPdisplaced S ( τ S = )
50 100 150 200 250 300 350 400 p T ( MeV ) m S =
300 MeV, m P =
125 MeV K L →π νν K L → SP →π PPprompt S K L → SP →π PPdisplaced S ( τ S = )
50 100 150 200 250 300 350 400 p T ( MeV ) m S =
200 MeV, m P =
10 MeV
FIG. 10. Pion p T distributions for the K L → π ν ¯ ν decay and the K L → SP → π P P decay inour benchmark points. The distributions are shown both for prompt S decays and S decays witha lifetime of 30 cm. the transverse momentum distributions expected at KOTO in the SM and in our model. Bysampling from these distributions and applying the Kolmogorov–Smirnov test, we find thatthe p T distribution in our model can be distinguished from the SM three-body decay at 5 σ with O (100) events in much of our parameter space. Sensitivity is lost when m P is smalland m S ∼ m K L , and the distributions may also be too close to distinguish at smaller m S ifthe S lifetime is shorter than O (10 cm). Still, there are good prospects for making such adetermination within the next several years, as KOTO continues to collect data.There are also discovery prospects for S particles with meter- and centimeter-scale life-times at future beam-dump experiments. In particular, as discussed in section III B, theSeaQuest experiment can probe much shorter lifetimes than those to which CHARM andNuCal are sensitive. Backgrounds are relatively easy to control for experiments of this type,7and they remain sensitive even in our minimal scenario. The figure of merit is the S lifetime,which is at least O (cm) in our minimal scenario. This can be reduced by enhancing the SP interaction in our effective theory, but nonetheless, searches for long-lived particles promiseto be a powerful probe of our scenario in the coming decade. VI. CONCLUSIONS
Taken together, the anomalous KOTO events and the Grossman–Nir bound provide astrong hint for light new physics. In this work, we have introduced an effective theory thataccounts for the excess in the K L → π inv. channel with a metastable scalar S ; a lighter,stable pseudoscalar P ; and effective dimension-5 operators that mediate interactions between S , P and the d and s quarks. We provided two UV-complete models that would producean effective theory consistent with our assumptions. We then investigated the implicationsof our effective theory for cosmology and vice versa. In particular, we have shown thatcosmological overproduction of P places important constraints on the structure of the low-energy theory.At face value, in our minimal scenario, P cannot account for either dark matter or theKOTO excess unless the reheating temperature is close to 10 MeV. While it is possible toescape this conclusion by augmenting the model, e.g. with couplings of P to neutrinos, a lowreheating temperature is unavoidable in the model’s simplest incarnation. However, unless P is very light, the required reheating temperature is compatible with current constraintsfrom BBN and CMB, possibly even offering an observational handle on the model once CMBStage IV experiments further probe the effective number of relativistic species.Finally, we have discussed three experimental tests of our scenario. First, we have shownthat portions of our parameter space are within reach of future dark matter direct detectionexperiments. Second, our metastable S may be discovered by upcoming long-lived particlesearches, particularly the planned SeaQuest upgrade. Finally, if P is in our favored massrange, future KOTO data alone can discriminate between our decay chain and the SM three-body decay on the basis of the neutral pion p T distribution. There are thus strong discoveryprospects for P dark matter within the next decade.8 ACKNOWLEDGMENTS
The research of WA is supported by the National Science Foundation under Grant No.NSF 1912719. BVL and SP are partly supported by the U.S. Department of Energy grantnumber de-sc0010107. We thank Maxim Pospelov for introducing us to this class of modelsfor the KOTO excess, and for subsequent discussions. We are grateful to James Unwin forvaluable conversations regarding the UV freeze-in paradigm. We thank Stefania Gori forpointing us to relevant beam dump constraints, and we thank Natalie Telis for valuableconversations concerning statistical methodology.
Appendix A: KOTO simulation
In this appendix, we provide details of our calculation of the quantity R introduced ineq. (23). R is the acceptance of the K L → SP → π P P signal relative to the SM K L → π ν ¯ ν acceptance at KOTO. Our calculation is based on a Monte Carlo simulation following stepssimilar to the ones described in [37, 40].The layout of the KOTO beamline and the KOTO detector is described e.g. in [43]. Westart by generating K L momenta, p K L , and K L decay vertex locations, z K L , based on thedistribution f ( p K L , z K L ) ∝ g ( p K L ) × exp (cid:18) − ( z K L − z exit ) m K L τ K L p K L (cid:19) , (A1)where z exit = 20 m is the distance of the beam exit from the target and g ( p K L ) is themeasured K L momentum distribution at the beam exit from [43]. We include a smalltransverse component of the K L momentum such that the beam profile at the beam exit isconstant within an 8 . × . K → π form factorfrom [16]. In the case of the K L → SP → π P P decay, we first generate momenta for S ,based on the fixed energy of S in the K L rest frame, E S = ( m K L + m S − m P ) / (2 m K L ).We then decay S with a decay length distribution that is determined by the S → π P and S → P partial widths. The pion momentum is generated based on the known pion energyin the S rest frame, E π = ( m S + m π − m P ) / (2 m S ).Both in the SM case and the NP case, we let the pion decay promptly into two photons,each with energy E γ = m π / . .
507 m after the beamexit), as they would be rejected by photon veto collar counters. All other photons arepropagated to the calorimeter located 6 .
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