aa r X i v : . [ phy s i c s . g e n - ph ] J u l Invisible decays of neutral hadrons
Wanpeng Tan ∗ Department of Physics, Institute for Structure and Nuclear Astrophysics (ISNAP),and Joint Institute for Nuclear Astrophysics -Center for the Evolution of Elements (JINA-CEE),University of Notre Dame, Notre Dame, Indiana 46556, USA (Dated: September 5, 2020)
Abstract
Invisible decays of neutral hadrons are evaluated as ordinary-mirror particle oscillations usingthe newly developed mirror matter model. Assuming equivalence of the CP violation and mirrorsymmetry breaking scales for neutral kaon oscillations, rather precise values of the mirror mattermodel parameters are predicted for such ordinary-mirror particle oscillations. Not only do these pa-rameter values satisfy the cosmological constraints, but they can also be used to precisely determinethe oscillation or invisible decay rates of neutral hadrons. In particular, invisible decay branchingfractions for relatively long-lived hadrons such as K L , K S , Λ , and Ξ due to such oscillations arecalculated to be . × − , . × − , . × − , and . × − , respectively. These significantinvisible decays are readily detectable at existing accelerator facilities. ∗ [email protected] . INTRODUCTION Searching for new physics beyond the Standard Model (BSM), such as the puzzle of darkmatter, has stimulated studies on invisible decays of heavy hadrons like D [1], B [2], andHiggs particles [3, 4]. But the attention to possible invisible decays in hadrons made oflight quarks [5] has been much more scarce than one would think. Early measurementsof invisible decays on short-lived light mesons of π [6] and η ( η ′ ) [7] were attempted.Search for invisible decays of ω and φ were also conducted by the BESIII collaboration[5]. However, no experiment has been conducted for invisible decays of relatively long-livedneutral hadrons like K [8], which present the most significant CP -violation effect to ourknowledge. Insufficient experimental work in light hadrons may miss the discovery of BSMphysics that could very well be revealed in their invisible decays.Invisible decays via neutrino emission ¯ νν were calculated for heavy mesons of B and D with negligible branching fractions (on the order of − or lower) [9] while the four-neutrino( ¯ νν ¯ νν ) decay channel can be much more enhanced due to lack of helicity suppression witha still low branching fraction (on the order of − for B ) [10]. Calculations for lightmeson decays via ¯ νν and ¯ νν ¯ νν have also shown very low branching ratios [11]. Therefore,these invisible neutrino decay channels will not hide new physics if a much larger branchingfraction (e.g., ≫ − ) due to BSM physics is predicted. Indeed, unexpectedly large rates ofordinary-mirror particle oscillations of long-lived neutral hadrons predicted under the newlydeveloped mirror-matter model [12–15] can manifest as significant invisible decays of theseparticles.In this work, we will present a rather exact model of neutral hadron-mirror hadron os-cillations inferred from the similarity between CP violation and mirror symmetry breaking.Assuming equivalence of the CP -violation and mirror symmetry breaking scales, one can pindown the model parameters very precisely leading to precise predictions of large invisibledecay branching fractions for long-lived neutral hadrons that can be detected at existingand planned facilities. 2 I. MIRROR MATTER MODEL
The mirror matter idea was originated from the discovery of parity violation in the weakinteraction by Lee and Yang [16]. It has subsequently been developed into an intriguingmirror-matter theory by various efforts [17–24]. The general picture of the theory is that aparallel sector of mirror particles (denoted by the prime symbol below) exists as an almostexact mirrored copy of the known ordinary particles and the two worlds can only interactwith each other gravitationally. Nevertheless, many of previous models [21–24] attemptedto add some explicit feeble interaction between the two sectors. On the contrary, in thenewly proposed mirror-matter model [12–15], no explicit cross-sector interaction is intro-duced, namely, the two parallel sectors share nothing but the same gravity before the mirrorsymmetry is spontaneously broken.The new mirror-matter model has been applied to consistently and quantitatively solve awide variety of puzzles in physics and cosmology: dark matter and neutron lifetime anomaly[12], evolution of stars [25], matter-antimatter imbalance [13], ultrahigh energy cosmic rays[26], unitarity of CKM [14], neutrinos and dark energy [15]. The model has also been ex-tended into a set of supersymmetric mirror models under dimensional evolution of spacetimeto explain the arrow of time and big bang dynamics [27, 28] and to understand the natureof black holes [29]. Last but not least, various feasible laboratory experiments are proposedto test the new theory [14].An immediate result of this model is the probability of ordinary-mirror neutral particleoscillations in vacuum [12] assuming no decay or a much shorter time scale than its decaylifetime, P ( t ) = sin (2 θ ) sin ( 12 ∆ t ) (1)where θ is the mixing angle, sin (2 θ ) denotes the mixing strength (on the order of − – − for the most significant n − n ′ and K − K ′ oscillations [12]), t is the proper propagationtime, and ∆ is the small mass difference of the two mass eigenstates (on the order of − eV for both K − K ′ and n − n ′ [12]). Here natural units are used for simplicity.Under the new model, the spontaneous mirror symmetry breaking results in one extended3 ABLE I. Adopted or predicted element values of the extended unitary mixing matrix in Eq. (2)under the new mirror-matter model are listed. In particular, V ud is determined from the “beam”lifetime approach [32, 33] while V us is taken from the semileptonic K l decay measurements [31]and V ub is from Ref. [30]. The cross-sector mirror mixing elements are predicted using the unitarityand the n − n ′ and K − K ′ mixing strengths as discussed in the text. | V ud | | V us | | V ub | | V uu ′ | | V dd ′ | | V ss ′ | | V cc ′ | | V bb ′ | | V tt ′ | quark mixing matrix as follows [14, 15], V qmix = V CKM ... V uu ′ ... V cc ′ ... V tt ′ · · · · · · · · · · · · · · · · · · · · · V dd ′ ... V ′ CKM V ss ′ ... V bb ′ ... (2)where the ordinary × CKM matrix V CKM and its mirror counterpart V ′ CKM are no longerunitary. Using the results from neutron lifetime measurements via the “beam” approachor superallowed nuclear + → + decays for the matrix element V ud , the deviation fromunitarity of the ordinary CKM matrix was demonstrated at a significance level of > σ [14].The missing strength from unitarity is supplied by quark-mirror quark mixing elements V qq ′ .The unitarity condition for the first row of the extended matrix can then be written as, | V ud | + | V us | + | V ub | + | V uu ′ | = 1 . (3)The matrix element | V ub | = 0 . [30] contributes little to the unitarity. The bestdirect constraint on matrix element V us can be set from measurements of meson decays as | V us | = 0 . under the new scenario [14, 31]. Using the neutron β -decay lifetimeof τ n = 888 . ± . s from the averaged “beam” values [32, 33], we can obtain the matrixelemment | V ud | = 0 . [14]. As shown in Table I, the unitarity requirement of Eq. (3)then results in | V uu ′ | ≃ . with the known values of the above matrix elements.4o estimate the strengths of the other quark-mirror quark mixing elements, we can applythe following relationship between the mixing strength of a neutral hadron and the mirror-mixing matrix elements for its corresponding constituent quarks, sin (2 θ ) ≃ Y i | V q i q ′ i | (4)where the mixing angle θ is assumed to be small. Using Eq. (4), more mirror-mixing elementscan be estimated from the known mixing strengths of ordinary-mirror hadron oscillations.For example, the n − n ′ mixing strength sin (2 θ nn ′ ) = | V uu ′ | | V dd ′ | ∼ × − leads to | V dd ′ | ∼ . [14]. The study of K − K ′ oscillations in the early universe for the originof baryon asymmetry [13] supports the mixing strength sin (2 θ KK ′ ) = | V dd ′ | | V ss ′ | ∼ − resulting in | V ss ′ | ∼ . [14]. Unfortunately, these mixing strengths without betterexperimental constraints still carry large uncertainties and we shall present a better way topin down these parameters precisely. III. CP VIOLATION AND MIRROR SYMMETRY BREAKING
By analyzing the published neutron lifetime measurements via the “bottle” approach,a probable range of × − – × − for the n − n ′ mixing strength is obtained [12].According to a recent simulation result for the UCN τ setup [34], the mean free flight timefor neutrons in the trap is estimated to be . ± . s [35]. This gives an estimate of sin (2 θ ) = 8 . ± . × − for n − n ′ oscillations, which is effectively a lower limit on themixing strength as higher energy neutrons tend to disappear more easily in the trap via n − n ′ oscillations and the effect was not taken into account in the simulation. On the otherhand, an anomaly of ± s in neutron lifetime measured in a He-filled magnetic trapat NIST [36] in combination with a new simulation study [37] presents another estimate of sin (2 θ ) ≃ × − . This in fact provides an upper limit of the mixing strength since therecould exist other unaccounted processes (e.g., from He contamination) that may contributeto the apparently short lifetime value. For such a mixing strength range of . – × − ,one can obtain a range of | V dd ′ | = 0 . – . using Eq. (4) as discussed above. Withoutfurther magnetic trap measurements, however, could we constrain V dd ′ better? One way isto infer it from the mass splitting parameter as presented below.For ordinary CP -violating neutral kaon oscillations, its mixing and mass splitting param-5ters have been well measured. It is natural to postulate that the CP violation scale is thesame as the mirror oscillation scale, in particular, for the case of neutral kaon oscillations[15]. This may very well be the case if we consider CP violation as a direct result of sponta-neous mirror symmetry breaking at staged quark condensation [15]. Therefore, we assumethat the mixing angle and mass splitting scale are the same for both symmetry breakings asfollows, ∆ K K ′ = ∆ K L K S = 3 . × − eV sin θ K K ′ = sin θ K L K S = 2 . × − (5)where the well-measured CP violation parameter values are taken from the PDG compilation[30].As a particle’s mass originates from the same Higgs mechanism, we assume that the masssplitting parameter scales as the particle’s mass, i.e., the relative mass splitting scale of ∆ /m ≃ × − derived from Eq. (5) is constant under the mechanism of spontaneousmirror symmetry breaking [15]. Therefore, we can obtain a rather precise value of ∆ nn ′ =6 . × − eV for n − n ′ oscillations from the well known kaon data, which is similar tothe value estimated under the cosmological consideration in the new mirror matter model[12, 13].For consistent origin of dark (mirror) matter and baryon asymmetry of the universe, wecan obtain an approximate relationship of the mixing strength and mass splitting parametersusing the dark-to-baryon density ratio of 5.4 [13], sin (2 θ nn ′ ) = (cid:18) . × − eV ∆ nn ′ (cid:19) . (6)which gives sin (2 θ nn ′ ) = 1 . × − for ∆ nn ′ = 6 . × − eV assuming equivalence of the CP violation and mirror symmetry breaking scales. Amazingly, this mixing strength valuefalls right within the range of . – × − constrained by the above-discussed neutron life-time measurements. Using Eq. (4), the n − n ′ mixing strength sin (2 θ nn ′ ) = | V uu ′ | | V dd ′ | ≃ . × − then leads to | V dd ′ | ≃ . as shown in Table I.Under the equivalence of the CP and mirror symmetry breaking scales in Eq. (5), wecan also find the following relationship between the mixing strength and mass splittingparameters for K − K ′ oscillations, sin (2 θ K K ′ )∆ K K ′ = 2 . × − eV (7)6hich is remarkably close to what is required to account for the baryon asymmetry of theuniverse using K − K ′ oscillations [13]. From its mixing strength of sin (2 θ K K ′ ) ≃ × − and Eq.(4), one can derive another mirror mixing matrix element of | V ss ′ | ≃ . as shownin Table I.There is no adequate data to directly determine the mirror mixing matrix elements forheavier quarks. For simplicity, we can assume that these matrix elements follow a geometricsequence with common ratio of r = | V ss ′ /V dd ′ | ≃ . corresponding to the quark hierarchy,e.g., | V cc ′ | = r | V ss ′ | . Then we can easily calculate these matrix elements for heavier c-,b-, and t-quarks as shown in Table I. This will provide a convenient way to estimate theinvisible decay branching ratios of heavier neutral hadrons as discussed below. Obviously,the uncertainty goes much larger when heavier quarks are involved. IV. INVISIBLE DECAYS OF NEUTRAL HADRONS
For a neutral hadron h during the process of free decay, we can calculate its h − h ′ oscillation rate using Eq. (1) as follows, λ hh ′ = 1 τ Z ∞ P ( t ) exp( − t/τ ) dt = 12 τ sin (2 θ ) (∆ τ ) τ ) (8)where τ is the hadron’s mean lifetime and ∆ is the mass splitting parameter of h − h ′ oscillations. The corresponding invisible decay branching ratio due to h − h ′ oscillations canthen be written as, B inv = 12 sin (2 θ ) (∆ τ ) τ ) . (9)It is easy to see in Eq. (9) that a neutral hadron has to be relatively long-lived, i.e.,with lifetime of τ & / ∆ , for its invisible decay branching ratio to be significant. Withthe universal relative mass splitting scale of ∆ /m ≃ × − as discussed above, onecan calculate mass splitting parameters of long-lived hadrons as listed in Table II. Thelongest-lived neutral hadrons are neutrons and we can obtain B inv ( n ) ≃ . × − usingthe above-discussed parameters. Such an effect is greatly amplified by wall scattering andreadily observed for neutrons confined in a trap [12, 14].Using the rather precise values of the mirror mixing matrix elements listed in Table Iunder the assumption of equivalence of the CP / mirror breaking scales, we can calculate7 ABLE II. Branching fractions of invisible decays via ordinary-mirror hadron oscillations and cor-responding mirror symmetry breaking parameters for long-lived neutral hadrons are listed. Lifetimevalues are taken from the PDG compilation [30].Hadrons K L K S Λ Ξ D B ∆ hh ′ [eV] . × − . × − . × − . × − . × − . × − sin (2 θ hh ′ ) 2 × − × − . × − . × − . × − . × − lifetime [s] . × − . × − . × − . × − . × − . × − B inv . × − . × − . × − . × − . × − . × − the mirror mixing strengths and the corresponding invisible decay branching fractions ofneutral hadrons with significant ordinary-mirror particle oscillations. As shown in Table II,the results for relatively long-lived neutral mesons and baryons are B inv ( K L ) ≃ . × − , B inv ( K S ) ≃ . × − , B inv (Λ ) ≃ . × − , B inv (Ξ ) ≃ . × − , B inv ( D ) ≃ . × − ,and B inv ( B ) ≃ . × − . In particular, the large branching fractions for K L , K S , Λ , and Ξ should be detectable at current and planned facilities. Note that the predicted invisibledecay branching fractions for K L and K S are consistent with the indirect experimental upperlimits of < . × − and < . × − , respectively, set from existing visible decay data [8].For comparison, very short-lived neutral hadrons have negligible branching ratios due tothe invisible oscillations. For example, π with a lifetime of . × − s, though its mixingstrength of . × − is quite large, has a very small B inv ≃ × − . The Higgs particleas a top-quark condensate [15] can be estimated to have an even smaller B inv ∼ − .Based on the known observational and experimental evidence, these large yet realisticestimates of the oscillation effects under the new mirror matter model may help motivatemore experimental efforts in search of invisible decays of long-lived light neutral hadrons.New physics could be revealed in such experiments and it is also one of the promisinglaboratory approaches to test the unique predictions of the new mirror matter model [14]. V. FURTHER DISCUSSIONS
Recent observation of anisotropy of the universe [38–40] could be well understood underthe new framework of mirror matter theory, in particular, with the new supersymmetric mir-8or models [15, 27]. Using the new result of a large dipole component of cosmic accelerationin a reanalysis of type Ia supernova data by Colin et al. [38], i.e., the monopole q m = − . and the dipole q d = − . of the cosmic deceleration parameter, we can further discuss itseffects on the parameters of the new mirror matter model.In this scenario, the monopole component q m dominates at large cosmic scales that shouldbe defined by the universe contents of ordinary matter, dark (mirror) matter, and dark(vacuum) energy. Under the standard cosmic model Λ CDM assuming a flat universe, wecan estimate that the content of dark energy would be Ω Λ = (1 − q m ) / ≃ %. Assuming5% for the content of ordinary matter, dark (mirror) matter takes up about 51%. Therefore,taking into account the possible cosmic anisotropy of Colin et al. [38], the dark-to-baryondensity ratio is about 10:1, which seems to agree better with observation of galactic darkmatter. This almost doubled ratio will modify the relationship between the n − n ′ massdifference and its mixing strength slightly from Eq. (6) as follows, sin (2 θ nn ′ ) = (cid:18) . × − eV ∆ nn ′ (cid:19) . (10)which leads to sin (2 θ nn ′ ) = 1 . × − for ∆ nn ′ = 6 . × − eV. Intriguingly, this n − n ′ mixing strength value is still within the experimental limits of . – × − . It modifies n − n ′ oscillations slightly but has no effect on the invisible decay branching fractions of K L , K S , and Λ . The predictions on the other neutral hadrons are changed slightly to B inv (Ξ ) ≃ . × − , B inv ( D ) ≃ . × − , and B inv ( B ) ≃ . × − .One caveat in the assumption of a universal relative mass splitting scale for mirror sym-metry breaking could be that the role of the strong interaction may complicate the massgeneration mechanism in hadrons. There may be some quark-dependent effects making therelative mass splitting scale not exactly constant. In the case of D , its mass splitting scaleis . +3 . − . × − eV from a global fit of all experimental data by the HFLAV collaborationassuming CP violation is negligible [41], which deviates just by a little over one sigma fromthe scale inferred from kaon oscillations as shown in Table II. This indicates that a constantrelative mass splitting scale is at least a good approximation. For the B system, on theother hand, the deviation is about one order of magnitude. But it could be a result ofcontamination from B s . 9 CKNOWLEDGMENTS
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