arXiv:1407.2392v2 [hep-ph] 16 Mar 2016 inmaueet ae lc ttelvlof level the preci- at with the place predictions of takes sector measurements theoretical EW sion of the agreement in and the draw particular, flavor SM to In order (i.e. in conclusions. sectors) sectors preci- final boson SM of gauge level other (EW) the in electroweak to achieved sec- increased tests SM noticeably sion Higgs be sym- the should fundamental in description new tor precision effective a experimental an of The fact consequence metry. in a or as SM-like appearing actually whether is Namely, SM. it the electroweak in of (EWSB) mechanism breaking Higgs symmetry the initiates what Na- is in that ture symmetries questions of understanding immediate current the our challenge of yet the One at mean sym- level. understood electroweak fundamental completely not and is boson does preci- (EWSB) Higgs breaking high consistency the metry of rather observed nature a the the that to So field energies effective low the sion. an matches at emerge. and as reality results to perceived physical accurate began widely quite providing consistency has being theory of is (SM) picture SM Model rough The Standard interactions a the its [3], with of matter studies known precision with follow-up and LHC h M[–] aey tiiitsteES dynam- (or EWSB techniquark the strongly-interacting initiates of in it means EWSB Namely, by for responsible ically [5–8]. be SM to the considered often is scale sector LHC. 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T-baryon scalar been have indirect measurements and direct for prospects osvate-ee ig oo xhnei the spin-independ in dominating exchange The boson Higgs tree-level threshold. via the goes to close small bet tte ee,wieoelo ag oo eitdc mediated boson gauge one-loop while level, tree at absents itn ihnieepcain rmtelgttcnpo c technipion light the recent from the expectations from naive extracted with been sistent has coupling T-baryon–Higgs 1 opst clrDr atrfo vector-like from Matter Dark scalar Composite eateto srnm n hoeia hsc,Ln Univ Lund Physics, Theoretical and Astronomy of Department o-oe with toy-model A 2 .INTRODUCTION I. eerhIsiueo hsc,Suhr eea Universit Federal Southern Physics, of Institute Research 3 nttt o ula eerho usa cdm fScienc of Academy Russian of Research Nuclear for Institute oa Pasechnik, Roman SU (2) 1 TC iayBeylin, Vitaly yaiscnnda ihsae Λ scales high at confined dynamics ∼ 10 − 2 3 ldmrKuksa, Vladimir in t B canl h orsodn on nteeffective the on bound corresponding The -channel. 0 = opst tts ietcnpos(rTposbelow), T-pions (or technipions techni- like pseudo-Goldstone light states, relatively composite new of latter com- plenty the (UV) a state, Higgs-like predict ultraviolet composite its the Besides and [13], pletion. theory struc- fermions underlined group-theoretical SM of the of ture on partners depend and properties 12] whose a [11, Higgs- predicts composite particles unavoidably importantly, like most dynamics states, Higgs con- new new of T-quark effective a plenty non-diagonal Such the a that by densate. difference initiated is only sponta- mechanism with the chi- QCD is non-perturbative in ral effect breaking this symmetry chiral to neous analogy scales straightforward energy A low at condensation T-quark) oeooia motneo h etrlk confinement phe- vector-like The the of scenario. completion importance (VLC) UV nomenological confinement (Dirac) vector-like a vector-like the with is – dy- constraints models of current of scenarios with class appealing consistent has observations. most EWSB however, see particle the namical variety, Higgs-like among review, big SM present, tests recent a At precision a and (EW) Such (for 15] electroweak severe [14, far 10]). by reduced [9, so got Refs. literature been the e.g. have “Tech- as in scenarios, dubbed “compositeness” proposed commonly or scale, (TC) TeV nicolor” a at dynamics for Run topics LHC priority the the at of searches II. one a physics is invariant strongly-coupled for GeV with new search 200 states The below (pseudo)scalar light masses particles. new SM of to family backgrounds couplings SM weak large by and limited strongly is regions mass UD ag ubro aiu elstoso uhanew a such of realisations various of number large A nti ok eivsiaeaptnilof potential a investigate we work, this In . ;Dr atrdrc eeto;low-energy detection; direct Matter Dark n; oupled ihmass with ase itn etrlk aueo hi weak their of nature vector-like a dicting mto eoeadatrteelectroweak the after and before rmation η t ic t etrculn to coupling vector its since nts techni- , si srpyisa ela ncollider in as well as astrophysics in es nrbto ssont evanishingly be to shown is ontribution n S)Tbro–ulo scattering T-baryon–nucleon (SI) ent ,349 otvo-o,Russia Rostov-on-Don, 344090 y, te D)cniae eso that show We candidate. (DM) atter m U aaadtrsott econ- be to out turns and data LUX 2 smercD,rsetvl.Such respectively. DM, asymmetric TC π ˜ SU n rgr Vereshkov Grigory and riy E236 ud Sweden Lund, 62 SE-223 ersity, ≪ ≫ s 132Mso,Russia Moscow, 117312 es, (2) m Λ K 0 e nbe oconstruct to enables GeV 100 B TC SU TC t,woesac nlwinvariant low in search whose etc, , & h atrpoie the provides latter The . yaiss a.Future far. so dynamics (2) e rdce ythe by predicted TeV 1 confinement ,3 2, Z boson UT 14-21 LU-TP ∼ 0 GeV. 100 2 has been broadly discussed in e.g. Ref. [16] without re- However, confined even SU(2n)TC, n = 1, 2,... sym- ferring to its implication to the dynamical EWSB. The metries giving rise to scalar T-baryon B = QQ (diquark- simplest realisation of the VLC scenario of the EWSB like) states instead are void of this problem. Indeed, the with two vector-like or Dirac techniflavors and a SM-like elastic scattering of scalar T-baryons off nucleons occurs Higgs boson has been studied in Refs. [17–19] and very mainly via the Higgs boson exchange at tree level and recently has emerged in composite Higgs scenarios with is strongly suppressed compared to stable Dirac com- confined SU(2)TC [20, 21]. In this paper, we discuss im- posites. As was advocated recently in Refs. [31, 32] the plications of the vector-like confinement to Dark Matter light scalar T-baryons (or T-diquarks) can play a role of (DM) astrophysics. pseudo-Goldstone bosons under global SU(4) symmetry Besides the dynamical nature of EWSB in the SM and such that the lightest neutral UD T-baryon state could a possible compositeness of the Higgs boson, another very become a new appealing composite asymmetric or mixed important prediction of QCD-like TC scenarios based DM candidate. In this Letter, we are focused primarily on a simpler global chiral SU(2) SU(2) symmetry upon SU(NTC)TC confined symmetry is the existence L ⊗ R of heavy composite baryon-like states possessing an ad- acting on complex vector-like (Dirac) UV completion in local gauge U(1) SU(2) SU(2) symmetry which ditional conserved quantum number. The lightest neu- Y ⊗ W ⊗ TC tral technibaryon (or T-baryon) state thus appears to be possesses an additional conserved T-baryon charge [17– stable and weakly interacting with ordinary matter. If 19]. Continuing earlier line of studies, now we discuss a new strong dynamics exists in Nature just above the important phenomenological implications of heavy scalar EW scale MEW 200 GeV and if there is a mecha- T-baryons mB & 1 TeV for direct DM searches in as- nism for T-baryon≃ asymmetry generation analogical to trophysics and collider measurements. Specifically, we that of baryon asymmetry, such new particles could be demonstrate that the heavy scalar T-baryon is a good abundantly produced in early Universe and survived un- candidate for self-interacting symmetric DM which is til today in the form of DM [23]. These ideas is widely within a projected few-year reach at direct detection ex- discussed in the literature during past two decades. periments. So far, a number of different models of composite DM candidates and hypotheses about their origin and inter- II. SCALAR T-BARYON INTERACTIONS actions has been proposed. Generic DM signatures from TC-based models with stable T-baryons were discussed e.g. in Refs. [24–28] (for a review see also Ref. [29] and A. Vector-like confinement and Dirac T-quarks references therein). In particular, well-known minimal dynamical EWSB mechanisms predict relatively light In the considering version of the model, the large T-baryon states as pseudo Nambu-Goldstone bosons of scalar T-baryon mass terms explicitly break global chi- the underlying gauge theory [30, 31, 33]. The latter ral SU(4), so in this case it suffices to work within the can naturally provide partially-asymmetric or asymmet- global chiral SU(2)R SU(2)L symmetry which classifies ⊗ ric DM (ADM) candidates if one assumes the existence the lightest TC states only, similarly to that in hadron of a T-baryon asymmetry in Nature similarly to ordinary physics. This in variance with models of Refs. [31, 32] baryon asymmetry [31, 33] (for a review on ADM models, where the pseudo-Goldstone T-baryon states can be ar- see e.g. Ref. [34] and references therein). Having similar bitrarily light and are typically considered to be around mechanisms for ordinary matter and DM formation in the electroweak scale. early Universe one would expect the DM density to be Consider the simplest vector-like TC model with single of the same order of magnitude as that of baryons. De- SU(2)W doublet of Dirac T-quarks confined under a new pending on a particular realization of dynamical EWSB strongly-coupled gauge symmetry SU(NTC)TC at the T- mechanism such composite DM candidates may be self- confinement scale ΛTC & 1 TeV interacting which helps in avoiding problematic cusp-like DM halo profiles [35]. The ongoing search for the DM U 0, if NTC =2 , Q˜ = , Y = (2.1) in both direct and indirect measurements can thus pro- Q˜ D (1/3, if NTC =3 . vide further tight constraints on possible TC scenarios additional to those coming from the LHC. where the T-quark doublet hypercharges are chosen to To this end, in Ref. [18] it has been demonstrated provide integer-valued electric charges of corresponding explicitly that the TC scenarios with an odd confined bounds states. The case of NTC = 3 has been stud- SU(2n + 1)TC, n = 1, 2,... symmetry are most likely ied in Refs. [17, 18], and here we are focused primarily ruled out by recent constraints on the spin-independent on NTC = 2 theory where phenomenologically consistent DM-nucleon scattering cross section [36, 37]. In particu- vector-like weak interactions of an underlined UV com- lar, stable Dirac T-neutron DM predicted by the confined pletion (i.e. Dirac T-quarks) can be naturally obtained QCD-like SU(3)TC symmetry is excluded due to its large from a conventional chiral one (for NTC = 3 this is not tree-level vector gauge coupling to the Z boson unless it the case). is not directly coupled to weak isospin SU(2)W sector, In order to demonstrate this fact explicitly, let us start only via a small mixing. with two generations (A =1, 2) of left-handed T-quarks 3
Qaα transformed under gauge SU(2) SU(2) as (LσM) L(A) W ⊗ TC ′ i 1 µ 1 µ ¯ ˜aα ˜aα ab ˜bα LσM = ∂µS ∂ S + DµPa D Pa + iQ˜DˆQ˜ QL(A) = QL(A) + gW θkτk QL(A) 2 (2.2) L 2 2 i αβ aβ ˜¯ ˜ ˜¯ ˜ + gT C ϕ τ Q˜ , gTCQ(S + iγ5τaPa)Q gTC S QQ k k L(A) − − h i 2 1 λ 4 λ (S2 + P 2)2 + λ 2(S2 + P 2) where a =1, 2 is the index of fundamental representation − HH − 4 TC H of weak isospin SU(2)W group, α = 1, 2 in the index of 1 + µ2 (S2 + P 2)+ µ2 2 , (2.7) fundamental representation of T-strong SU(2)TC, and 2 S HH YQ˜ = 0. Now, let us keep the first generation of T- quarks unchanged and apply the charge conjugation to with a particular choice of the “source” term linear in ¯ the second generation such that T-sigma where Q˜Q˜ < 0 is the diagonal T-quark con- 2 h † i 2 0 0 + − densate, = , P PaPa =π ˜ π˜ +2˜π π˜ , and ˆ aα Caα H HH ≡ CQL(2) = QL(2) , the EW-covariant derivatives are ′ Caα Caα i ab ∗ Cbα Q = Q gW θ (τ ) Q iY ˜ L(2) L(2) k k L(2) (2.3) ˆ ˜ µ Q ′ i a ˜ − 2 DQ = γ ∂µ g Bµ gWµ τa Q, i αβ ∗ Caβ − 2 − 2 gT C ϕk(τ ) Q . −2 k L(2) b DµPa = ∂µPa + gǫabcWµPc . (2.8) The charge conjugation of a chiral fermion changes its The Higgs boson doublet in Eq. (2.7) acquires an in- chirality. This fact enables us to define the corresponding H right-handed field as terpretation as a composite bound state of vector-like T- quarks e.g. in the model extended by an extra SU(2)W- 0 1 ˜ ˜ ˜¯ Qaα εabεαβQCbβ , εab = εαβ = . (2.4) singlet Dirac S T-quark such that = QS (for other R(2) ≡ L(2) 1 0 possibilities, see also Refs. [20, 21]).H Note, however that − at the moment the question about a particular UV con- Starting from the gauge group transformation property tent of the Higgs boson doublet is not of primary impor- (2.3) and applying SU(2) defining relations like δab = ac bc tance for the effective low-energy description of scalar ε ε and T-meson interactions described by the phenomenologi- εab(τ bc)∗εcf = τ af , εαβ(τ βγ )∗εγµ = τ αµ , cal LσM Lagrangian (2.7) and thus will not be further k k k k discussed here. it is rather straightforward to show that Note, the chiral symmetry implies the equality of con- stituent Dirac masses MU = MD MQ˜ at tree level. In aα′ aα i ab bα ≡ Q = Q + gW θ τ Q the limit of small current T-quark masses m ˜ compared R(2) R(2) 2 k k R(2) Q (2.5) to the constituent ones M ˜ , i.e. m ˜ M ˜ ΛTC, i αβ aβ Q Q ≪ Q ∼ + gT C ϕkτ QR . in analogy to ordinary QCD the conformal symmetry is 2 k (2) approximate such that the µ-terms can be suppressed By a comparison of Eq. (2.5) with Eq. (2.2) one notices µS,H mπ˜ , which will be employed below throughout that the transformation properties of the right-handed T- this work.≪ Then the spontaneous EW and chiral sym- quark field obtained by charge conjugation and transpo- metry breakings are initiated dynamically by the Higgs sition of the left-handed field of the second generation v 246GeV and T-sigma u vevs coincide with the transformation properties of the left- ≃ 1 √2iφ− handed field of the first generation. Therefore, starting = , H v , S u & v , initially with two chiral (left-handed) T-quark genera- H √2 H + iφ0 h i≡ h i≡ tions we arrive at one vector-like generation of (Dirac) H = v + hcθ σs˜ θ , S = u + hsθ +˜σcθ , (2.9) T-quarks, namely − respectively, by means of T-quark condensation, namely, aα aα aα aα ab αβ Cbβ Q = QL(1) + QR(2) = QL(1) + ε ε QL(2) . (2.6) g λ 1/3 u = TC H Q˜¯Q˜ 1/3 , As was argued for the first time in Ref. [17], practically δ |h i| any simple Dirac UV completion with chirally-symmetric 1/2 1/3 λ gTCλH ¯ 1/3 weak interactions easily evade the most stringent elec- v = | | Q˜Q˜ , (2.10) troweak constraints which is the basic motivation for the λH δ |h i| VLC scenario. and the T-pions acquire a mass The phenomenological interactions of the constituent Dirac T-quarks and the lightest T-hadrons, namely, the g Q˜¯Q˜ scalar SM-singlet T-sigma S field, and the SU(2) - m2 = TCh i . W π˜ − u adjoint triplet of T-pion fields Pa, a = 1, 2, 3, are de- scribed by the (global) chiral SU(2)R SU(2)L invari- In the above expressions, sθ sin θ, cθ cos θ, δ = ant low-energy effective Lagrangian in the⊗ linear σ-model λ λ λ2, g > 0 and λ ≡> 0. The minimal≡ choice H TC − TC H 4
1 ~ hσ mixing v/u ratio 1 | θ 0.1 v/u
|sin 0.1
m~π = 80 GeV m~π = 80 GeV m~π = 150 GeV m~π = 150 GeV m~π = 300 GeV m~π = 300 GeV
0.01 0.01 -40 -20 0 20 40 -40 -20 0 20 40 ∆mσ~ (GeV) ∆mσ~ (GeV)
FIG. 1: The absolute value of sine of the hσ˜-mixing angle sin θ (left) and the ratio of the chiral and EW breaking scales u/v | | (right) as functions of ∆mσ˜ mσ˜ √3mπ˜ for three different values of the T-pion mass mπ˜ = 80, 150 and 300 GeV. Here and below, the nearly-conformal≡ limit has− been imposed. of the “source” term in the LσM Lagrangian (2.7) is natu- Dirac T-quark condensation due to a presence of λ 2S2 ral since it simultaneously (i) sets up a pseudo-Goldstone term in the potential (2.7). While the Peskin-TacheuchiH mass scale for T-pions, (ii) allows to link all the incident S and U parameters [14, 15] are strongly suppressed vevs u and v, and hence the constituent Dirac T-quark for all relevant model parameters S, U . 0.01 0.001, − mass scale MQ˜ = gTCu, to the T-confinement scale, the T-sigma–Higgs mixing angle θ is bounded by the T - and (iii) describes the Yukawa interactions of the T- parameter and the SM Higgs decay constraints provided sigma with the diagonal T-quark condensate Q˜¯Q˜ which that sθ . 0.2 [17]. In general, such a phenomenologically h i consistent small hσ-mixing limit s 0 corresponds to is the only dimensionfull nonperturbative parameter in θ → the model at low energy scales. The Nambu-Goldstone a decoupling of the TC dynamics from the SM up to ± 0 higher energy 1 TeV scales, hence, to a suppressed d.o.f.’s φ , φ originating from the Higgs doublet do not ∼ † 2 ratio v/u 1 as well as to relatively weak TC cou- appear in the potential since = H /2, where ≪ and H are defined in Eq. (2.9).HH They, thus, can not mixH plings gTC, λ, λTC . 1 compared to analogical couplings with pseudoscalar T-pions and get absorbed by the gauge in QCD. The latter property will be further employed in bosons in normal way giving rise to longitudinal polarisa- the T-baryon sector. tions of W ±,Z bosons, respectively. So, T-pions do not In Fig. 1 the dependencies of sine of the hσ˜-mixing angle s (left panel) and the Higgs and T-sigma vevs have tree-level couplings to SM fermions, and can only | θ| be produced in vector-boson fusion channels [17, 19]. v/u (right panel) on ∆mσ˜ mσ˜ √3mπ˜ are shown for three different T-pion mass≡ values− m = 80, 150 and In the VLC approach, one could distinguish two natu- π˜ 300 GeV. Consequently, in the case of small s 0 and ral physical scales associated with two chiral u and EW θ v/u 1, or equivalently, ∆m 0, the deviations→ of the v symmetry breaking scales, which can, in principle, be σ˜ Higgs≪ properties from those in the→ SM are small while the very different from each other although are related to the dynamical nature of the Higgs mechanism as a theoreti- single T-confinement scale, or the T-quark condensate cally favorable possibility is preserved. The physical La- Q˜¯Q˜ . In a phenomenologically natural hierarchy u v, h i ≫ grangian of the VLC model can be found in Refs. [17, 19] the upper scale would then roughly be associated with and we do not repeat it here. the mass scale of constituent T-quarks and T-baryons, In the currently favorable phenomenological situation whereas the lower one – with the mass scale of lightest with the SM-like Higgs boson, what would be the basic pseudo-Goldstone states, e.g. T-pions. By a convention, phenomenological signature for dynamical EW symme- in effective field theory approach one would then consider try breaking? Besides the light Higgs boson, in the VLC only the lightest states (˜π,σ ˜) which propagate at short model described above the T-pions are among the lightest distances and contribute to vacuum polarisations of the physical T-hadron states which should be searched for in SM gauge bosons, while a net effect of all the heavy states vector boson V V and photon γγ fusion channels, prefer- (e.g. T-baryons, T-rho etc) can be effectively accounted ably, in the low invariant mass region mπ˜ 80 200 GeV. for by constituent T-quark loops. This is a natural con- The T-sigma state σ is expected to be somewhat∼ − heavier sequence of imposing an upper cut-off in loop momentum since the small Higgs–T-sigma mixing limit sθ 1 cor- scale µ ΛTC M ˜ in the low-energy effective descrip- ≪ ∼ ∼ Q responds to mσ √3mπ˜ (Fig. 1). Besides, the T-sigma tion suitable for e.g. oblique corrections calculation [17]. interactions with∼ gauge bosons are strongly suppressed. In the framework of VLC model, the SM-like Higgs So, one of the most straightforward ways to search for mechanism has an effective nature and is initiated by the the new strongly-coupled dynamics and dynamical EW 5 symmetry breaking in collider measurements is to look For simplicity, at the first step in this work we assume for T-pion signatures in γγ-fusion channel [19]. that energetically favored states for such composites are Here, we discuss another source of constraints on such the ones with J = 0, although the lightest state with a new strong dynamics possibly coming from astrophysics J = 1 is not completely excluded and will be studied else- measurements at direct DM detection experiments. For where. The low-energy effective Lagrangian of T-baryon this purpose, let us consider the T-baryon spectrum of interactions with SM gauge bosons, T-sigma S and T- a SU(2)TC two-flavor theory. pion P , a = 1, 2, 3 invariant under local SM symmetry group additional to the LσM Lagrangian (2.7) reads 1 1 B. Scalar T-baryon Lagrangian ∆ = D B¯ DµB + µ2 BB¯ + g (BB¯ )2 LB µ a a 2 B 2 BB 2 2 † + gBS(S + P )(BB¯ )+ gBH( )(BB¯ ) Extra bound states of SU(2)TC theory possessing an ¯ H H additional conserved (T-baryon) number analogous to + gBP(BP )(BP ) . (2.15) the usual baryon number are given by scalar (anti)T- ¯ ¯ After the EW and CS breaking, the first four terms in baryon multiplets QiQj and QiQj. As was previously the scalar T-baryon potential give rise to the T-baryon studied in Refs. [31, 32], these states can play a role of mass term m BB¯ , where pseudo-Goldstone bosons originating from global SU(4) B 2 2 2 2 multiplets. In the phenomenologically relevant TC de- m = 2µ 2gBSu gBHv , (2.16) B − B − − coupling limit u v the T-baryons can have a large mass 2 ≪ where, in general, µB < 0 and gBS,gBH couplings can m & M ˜ m ,m significantly exceeding the EW B Q ≫ π˜ H be positive or negative. In the nearly-conformal limit breaking scale. In this case, just above the EW scale one µ u, v, one notices that the T-baryon mass should employ the global chiral SU(2) SU(2) sym- | B| ≪ L ⊗ R metry in the T-meson sector whose effective Lagrangian mB & MQ˜ u , (2.17) is supplemented by an additional phenomenological SM- ∼ is naturally large in the TC decoupling limit u v and group invariant Lagrangian of heavy T-baryons. This g < 0. In practice, one arrives at the model with≫ five case has not been discussed in the literature and deserves BS physical parameters controlling the properties of scalar a special attention. And this is in variance with other T-baryons: the T-baryon mass scale m & Λ 1 somewhat similar UV completions not participating in B TC TeV and four T-baryon–scalar dimensionless couplings∼ the dynamical EW symmetry breaking such as those in g g , g , g , g . Remind, the masses ofπ ˜, Ref. [22]. B,i BB BH BS BP σ˜ and≡h { bosons remain light} and are placed at the EW In order to describe consistently the EW and effective scale in the heavy T-baryon limit under discussion [17], interactions of scalar T-baryons, let us start with two real thus, validating the suggested scenario. adjoint (composite) representations of the SM SU(2) W The meson-meson and baryon-meson interactions gauge group G and F , a = 1, 2, 3, i.e. transforming in a a in ordinary QCD are usually very strong and non- the weak basis as perturbative i.e. the corresponding couplings are typi- ′ b ′ b cally much larger than unity. Naively, this happens since Ga = Ga gǫabcθ Gc , Fa = Fa gǫabcθ Fc . (2.11) − − the confinement scale is turned out to be comparable to Introducing a complex adjoint representation B with the the mass scale of pseudo-Goldstone modes in QCD. One unit T-baryon charge and YB = 0 such as sometimes refers to a very dense pion cloud surrounding nucleons as to a cause for such large nonperturbative cou- 1 ∗ ¯ Ba = (Ga + iFa) , B Ba = Ba , (2.12) plings. In the case of relatively light (possibly, composite) √2 a ≡ 6 Higgs boson and T-pions mπ˜ 100 150 GeV compared ′ b ∼ − Ba = Ba gǫabcθ Bc , & − to a large T-confinement scale ΛTC 1 TeV, it may be more natural to expect a very different situation in T- one ends up with the gauge interactions of the scalar T- hadron interactions – the T-pion cloud surrounding T- baryons B set by the corresponding covariant derivative a baryons is likely to be rather loose or sparse for v/u 1 b ≪ DµBa = ∂µBa + gǫabcWµBc . (2.13) and sθ 1. Thus, the T-hadron interactions at low ener- gies may≪ not be as intense as in QCD and the correspond- In the charge basis, the physical T-baryon states are ing couplings should be smaller in this case although a more dedicated (e.g. lattice) analysis, of course, is nec- B1 iB2 B¯1 iB¯2 B± = ∓ , B¯± = ± , B0 = B .(2.14) essary. The latter argument is in full agreement with the √ √ 3 2 2 consistent small hσ˜-mixing limit mentioned above where Desirably, these states now have a definite partonic repre- the scalar self-couplings turn out to be small or even van- sentation as diquark–like bound states of U,D T-quarks ishing, and thus one expects the T-baryon–T-meson cou- with conserved T-baryonic number TB, namely, plings to be small as well, i.e. gB,i . 1. The low-energy effective Lagrangian (2.15) appears to B+ = UU, B− = DD, B0 = UD,T = +1 , B have an extra exact global U(1)TB symmetry correspond- + − 0 B¯ = U¯U,¯ B¯ = D¯D,¯ B¯ = U¯D,T¯ ¯ = 1 . ing to the T-baryon number conservation in analogy to B − 6
0 Z γ W +
B+ B+ B+ B+ B+ B+ ∆MB = + + + B+ B+ B0
0 Z γ W ±
+ + + + B+ B+ B+ B+ B B −
W W ∓ ± B0 B0 0 0 B± − B B
FIG. 2: Diagrams contributing to the EW mass splitting ∆mB in the scalar T-baryon spectrum. ordinary baryon symmetry in the SM. Note, this symme- level with an intermediate charged T-baryon, and thus try has not been imposed forcefully. Instead, it emerged are strongly suppressed. The scalar T-baryon DM in automatically due to the structure of T-baryon multi- this sense has a certain similarity to the electroweak- plet in the complex SU(2)-adjoint representation im- interacting DM scenario discussed e.g. in Ref. [38]. posed by the chosen Dirac UV completion. In analogy The scalar T-baryon annihilation processes in early to the usual baryon abundance, however, the present DM Universe are largely dominated by effective T-baryon in- abundance may originate via generation of a sufficient T- teractions with light composites (˜σ andπ ˜) and the Higgs baryon asymmetry e.g. by means of the well-known EW boson given by sphaleron mechanism at high energies in early Universe, in accordance to the popular ADM scenario [31, 33, 34]. 1 = BB¯ g S2 +2uS + P 2 + g S2 +2vS LBS BS 1 1 2 BH 2 2 0 0 0 0 + + − − + − The physical Lagrangian derived from Eq. (2.15) de- + gBPB¯ B π˜ π˜ + (B¯ B + B¯ B )˜π π˜ scribes the T-baryon interactions with gauge W ±,Z,γ + (B¯−π˜− + B¯+π˜+)B0π˜0 + (B+π˜− + B−π˜+)B¯0π˜0 bosons, scalar Higgs h and T-sigmaσ ˜ and T-pionsπ ˜0, π˜±, − + − − + − + + as well as their self-interactions. The full expression is + B¯ B π˜ π˜ + B¯ B π˜ π˜ , (2.20) rather lengthy, and here we show the terms most rele- where S = hs +σc ˜ and S = hc σs˜ . The latter vant for T-baryon phenomenology only. In particular, 1 θ θ 2 θ − θ the parameter-free gauge interactions critically impor- Lagrangian is thus critical for the Higgs-induced tree- tant for possible T-baryon production mechanisms at the level elastic scattering of scalar T-baryons off nucleons LHC and T-baryon scattering off nucleons are given by in direct DM detection measurements as well as for the formation of the relic T-baryon abundance. − 0 − 0 − 0 + 0 + ¯ = ig W (B B¯ B B¯ + B¯ B B¯ B ) LVBB µ ,µ − ,µ ,µ − ,µ h + ¯+ ¯− − + (sW Aµ + cW Zµ)(B,µB + B,µB ) + c.c. , (2.18) III. SCALAR T-BARYON DARK MATTER
2 0 0 + + + µi− ¯ = g (B¯ B + B¯ B ) W W LVVBB µ A. T-baryon mass splitting ¯+ + h 2 ¯+ − + µ+ + B B (sW Aµ + cW Zµ) B B Wµ W (2.19) − As was mentioned above, the T-baryon states in the ¯+ 0 ¯0 − µ+ (B B + B B )(sW Aµ + cW Zµ)W + c.c. , strongly coupled SU(2) theory are scalar composite − TC i di-T-quark states and can be rather heavy in the TC with the SM parameters g and θ fixed at the T-baryon decoupling limit, u v. W ≫ mass scale µ = mB as a low-energy limit of the T- At temperatures above the EW phase transition scale baryonic gauge form factors. Note, vector B¯0B0Z-type but below the chiral symmetry breaking scale, u T & couplings which determine the SI scattering of DM par- v, the UD state remains degenerate with other compo-≫ ticles off nucleons at the Born level do not exist in this nents of the SU(2) triplet Ba. This is valid in the frame- scenario. The scalar T-baryon–quark scattering, in fact, work of considering low-energy effective theory where enters via an exchange by W -boson pair at one-loop one-loop self-energies with virtual T-baryons, T-pions, level while the T-baryon–gluon scattering – at two-loop Higgs boson and T-sigma are the same for B0 and B± 7
B0
B¯0
0 0 a b FIG. 3: Typical topologies for scalar T-baryon annihilation in the high-symmetry (HS) phase, B B¯ W W , , SS, PaPb, 0 0 + − →±,0 HH a,b = 1, 2, 3 (upper row), and low-symmetry (LS) phase, B B¯ W W , S1S2, where S1,2 =π ˜ , σ,˜ h (both upper and lower rows). →
1000 - - + - B0B0 -> W+W- annihilation B0B0 annihilation W W Z0Z0 LS phase, ∆m = 10 GeV - 1000 LS phase B tt 100
100 10 (pb) (pb) ann ann v) v)
σ σ 1 ( (
10 mB = 0.2 TeV mB = 1 TeV mB = 3 TeV 0.1
1 0.01 10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 2 2.5 3 ∆ mB (GeV) mB (TeV)
FIG. 4: The T-baryon LS annihilation cross section in the B0B¯0 W +W − channel as a function of the T-baryon mass → 0 0 + − 0 0 splitting ∆mB (left) and the largest annihilation channels into SM final states B B¯ W W , Z Z ,tt¯ as functions of the → T-baryon mass scale mB (right). In the left panel, the results are shown for three different mass scales mB = 0.2, 1, 3 TeV, in + − the right panel the W W contribution is shown for ∆mB = 10 GeV. Here, mπ˜ = 150 GeV and ∆mσ˜ = 20 GeV are fixed. and thus do not contribute to the mass splitting between ting between B0 and B± at low temperatures get well- neutral and charged T-baryons. defined contribution from the EW corrections depicted At lower temperatures, T < v, corresponding to the in Fig. 2. The latter are given by the one- and two-point the EW-broken phase the di-T-quark state with zeroth self-energies with gauge bosons in the loop which are not electric charge UD is expected to be energetically favor- cancelled in the mass difference according to Eqs. (2.18) able. Together with exact global T-baryon symmetry, and (2.19). The result for EW-induced mass splitting the latter suggests that the neutral di-T-quark B0 UD reads state is also the lightest and thus stable. The mass≡ split-
2 2 2 EW g2mW mZ 2 2 ∆mB = 2 ln 2 β (µZ ) ln(µZ )+ β (µW ) ln(µW ) 16π mB mW − 3 n 4β (µ ) √µZ 2 µ Z arctan + arctan − Z − √µZ 2β(µZ ) 2√µZ β(µZ ) 3 h i 4β (µ ) √µW 2 µ + W arctan + arctan − W , (3.1) √µW 2β(µW ) 2√µW β(µW ) h io where µ = m2 /m2 and β(x) = 1 x2/4. In the realistic limits, ∆m m and m m , the Z/W Z/W B − B ≪ B Z/W ≪ B p 8
G ± EW mass splitting may be estimated as B , GeV
2 EW g2 0.01 ∆mB mW (1 cW ) 0.17 GeV . (3.2) ≃ 8π − ≈ 0.001
Note, at the perturbative level the T-strong-induced -4 mass splitting vanishes in both EW-unbroken and EW- 10 - broken phases in the considering model. However, in 10 5 analogy to the di-quark spectrum QCD, one may discuss - potentially large non-perturbative T-strong effects in the 10 6 mass splitting in the chiral/EW symmetry broken phase. - 10 7 The situation is close to what we have in ordinary QCD when ud di-quark is split down in the di-quark mass spec- DmB, GeV trum by as much as 70 MeV due to pion exchanges. 20 40 60 80 100 ∼ Such non-perturbative effects, should be a proper sub- ± FIG. 5: Total decay width of charge T-baryons B B0 + ject for lattice studies and are not discussed here. In any ± → W ( lν¯l, qiq¯j ) as a function of the mass splitting ∆mB case, the EW-induced splitting above (3.2) can be safely → for different T-baryon mass scales mB = 100 GeV (dashed), treated as a conservative lower bound. So for the sake mB = 500 GeV (dotted) and mB = 10 TeV (solid). of generality in our numerical analysis we consider the T-baryon DM implications in the EW-broken phase over the following wide range of allowed mass splittings: participate in annihilation processes. For example, the total leptonic B± 3-body decay width reads EW ∆mB < ∆mB
- - B0B0 annihilation B0B0 annihilation
LS phase LS phase, mB=1 TeV 100 100
(pb) (pb) 10 ann ann v) v) σ σ ( 10 ( ~π0~π0 ~π0~π0 ~π+~π- ~π+~π- 1 hh hh σ~ σ~ h h σ~σ~ σ~σ~ 1 0.1 0.5 1 1.5 2 2.5 3 -40 -20 0 20 40
mB (TeV) ∆mσ~ (GeV)
FIG. 6: The T-baryon LS annihilation cross sections in the (pseudo)scalar B0B¯0 π˜0π˜0, π˜−π˜−, hh, hσ,˜ σ˜σ˜ channels as functions → of the T-baryon mass splitting mB for fixed ∆mσ˜ = 20 GeV (left) and ∆mσ˜ mσ˜ √3mπ˜ for fixed mB = 1 TeV (right). ≡ − Here, mπ˜ = 150 GeV is fixed.
In the case of large mass splitting ∆mB ml,q, the scalar self-couplings are equal to gTC, i.e. the partial B± decay widths into light fermions≫ are not M ˜ sensitive to the fermion masses ml,q, and can be ac- Q mB g = g = ,M ˜ , (3.6) counted for by a multiplicative factor in the total decay B,i TC u Q ≃ 2 width Γ ± . The latter is shown in Fig. 5 as a func- B which contains basics features of the general case. In this tion of ∆mB = [10 ... 100] GeV for three distinct T- simplified scenario, mB, mπ˜ , and ∆mσ˜ mσ˜ √3mπ˜ baryon mass scales mB =0.2, 1, 10 TeV by dashed, dash- ≡ − dotted and solid lines, respectively. There is a rather are the only variable independent parameters. strong dependence of the decay rate on the T-baryon Dominating topologies for EW and T-strong anni- mass splitting, whereas almost no sensitivity is found hilation channels are shown in Fig. 3. In the high- symmetry phase (or HS phase) valid for heavy T-baryons w.r.t. the mass scale mB variations. For TeV-mass T- mB & 2 TeV, the weak-induced T-baryon annihilation baryons, it is straightforward to estimate the realistic 0 0 ± goes through the process B B¯ W1W1 + W2W2 (with mean lifetime of charge B in the cosmological plasma a → −1 −23 −22 massless W ) whose cross section reads τ = Γ ± 10 10 s for a realistic splitting B B ∼ − ∆mB 100 GeV which is about ten orders of magni- ∼ 2g4 4m2 4m2 tude smaller than the Hubble time in the beginning of σW,HS = 2 5+ B 1 B (3.7) −1 −12 ann 2 DM annihilation epoch H (Ti) 10 s. This means 4mB s r − s ∼ πs 1 s h that at temperatures below Ti the charge T-baryons do − q 2 not present in the plasma and, thus, no co-annihilation 4mB 6m2 64m4 1+ 1 reactions B0B¯± X, B∓B¯± X etc contribute to the + 1 B + B ln − s . 2 2 → → 0 0 − s s q 4m formation of DM relic density. So, only B B¯ X pro- 1 1 B → − − s i cess should be considered which simplifies the subsequent q calculations. In the non-relativistic limit v 1, however, the corre- sponding kinetic cross section ≪ In analogy to ordinary baryons, the existence of a suf- 4 ficient initial T-baryon asymmetry can be critical for a W,HS 13g2 (σv)ann 2 v (3.8) non-negligible amount of the scalar T-baryon DM in the ≃ 2πmB case of large T-baryon–scalar couplings. So the T-baryon asymmetry appears to be an important parameter which vanishes. So, in this case the total annihilation cross th completely determines the amount of such DM in the section (σv)ann which enter Eq. (3.3) will be essentially determined by the TC-induced B0B¯0 , SS, P P present Universe in the case of too fast annihilation rates. → HH a a Various types of mixed (partially ADM) scenarios refer channels (with massless final states), i.e. to an intermediate case of noticeable amounts of both 2 particles and antiparticles although in not exactly equal th TC,HS gHS (σv)ann (σv)ann 2 , (3.9) amounts. A large existing freedom in the choice of T- ≃ ≃ 16πmB baryon–scalar couplings gB,i makes it possible to con- sider both ADM and mixed DM scenarios. Below, for where simplicity in our numerical analysis we consider a naive 1 g2 = g2 +3g2 + (g + g )2 . VLC scenario (in the near-conformal regime) where all HS 4 BH BS BS BP 10
Here, the diagrams with a single 4-particle vertex dom- GeV for three different mass scale mB = 0.2, 1, 3 TeV. inate, others are strongly suppressed by extra powers of Indeed, in the small ∆m 0 limit one observes gross B → mB in propagators. Note, an inclusion of extra possi- cancellations between the contact and t,u-channel con- ble composites to the theory, e.g. in extended chiral tributions (s-channel terms are always suppressed) while symmetries and composite Higgs scenarios, as well as at larger ∆mB the cancellations are less precise leading (pseudo)vector T-mesons may only increase the annihi- to such a characteristic shape of the cross section. We lation cross section. can immediately conclude from this result that the case It is worth to mention that the inequality (3.3), ΩTB . of symmetric T-baryon DM which undergoes its annihi- ΩCDM, applied for the T-baryon annihilation in the HS lation mainly in the LS phase of the cosmological plasma phase (3.9) imposes a constraint on the chiral symmetry can not be realized for large T-baryon mass splittings breaking scale ∆mB & 10 GeV and relatively light T-baryons mB < 2 TeV. This is thus the particular version of ADM whose u & 4.3 TeV v, m 2ug , (3.10) ≫ B ≃ TC relic abundance is fully characterized by the initial T- baryon asymmetry providing a stable relic remnant of which is valid under the naive scaling of the scalar self- particles with the same T-baryon number and no strict couplings with the T-baryon mass (3.6). This constraint constraints on T-baryon interaction rates analogical to is consistent with the TC decoupling limit, and hence that in the HS phase (3.13) can be drawn in this case. with the T-parameter constraint and the SM-like Higgs In Fig. 4(right) the largest T-baryon annihilation cross boson as discussed above in Sect. IIA. Together with sections into SM particles WW,ZZ,tt¯are shown as func- Eq. (3.10), the requirement for DM annihilation in the tions of the T-baryon mass scale mB = [0.2 ... 3] TeV. HS phase pushes up the T-baryon mass scale mB Here, only W W channel is sensitive to ∆mB which in the m latter figure has been fixed to 10 GeV, for comparison. T B & T , m & 2 TeV , (3.11) f ≃ 20 EW B The TC-induced annihilation channels into h, σ,˜ π˜ final states approximately add up to leading to a simple lower bound on scalar self-interaction rates g2 (σv)TC,LS LS , (3.14) 1 GeV ann ≃ 16πm2 g = g & gmin , gmin . 0.23 . (3.12) B B,i TC TC ≃ u TC altogether where in the case of large coupling gBS 1 Then, for the saturated inequality (3.3) when all the DM (or equivalently m u) we have ≫ B ≫ is made of the scalar T-baryons only, ΩTB ΩCDM, one ≃ 2 2 obtains 2 4mh 1 2 2 gLS = 1 2 gBHcθ + gBSsθ − mB 2 gB,i = gTC & 0.23 . (3.13) 2 4mπ˜ 2 2 + 1 2 2gBS + (gBS + gBP) This bound is consistent with the initial hypothesis about − mB the weakly-interacting heavy T-baryons in the TC de- 1 ¯ 2 2 2 2 2 2 coupling limit Λ & 1 TeV M formulated in + λ(mh,mσ˜; mB)(2gBS gBH) sθcθ TC ≫ EW 2 − Sect. IIB. So, the heavy symmetric scalar T-baryon DM 2 2 4mσ˜ 2 1 2 scenario with the relic abundance formation before the + 1 2 gBScθ + gBHsθ , − mB 2 EW phase transition appears to be a feasible and ap- pealing option. Of course, for a more precise quantitative corresponding to the contact diagrams in Fig. 3 only. The analysis one would need to know the exact dependence exact result applicable also for the case of small effective of non-perturbative T-baryon couplings on T-sigma vev couplings, e.g. gBS . 1, and hence for relatively light u (or, equivalently, on Q˜¯Q˜ ) going beyond the naive as- h i T-baryons mB . u, is more complicated since it includes sumption (3.6) which is an important further subject for additional s,t,u-channel diagrams; it can be found in a lattice analysis (for a recent lattice study of effective Appendix A. Note, in the small hσ˜-mixing limit sθ 1, Higgs–T-baryon interactions in SU(4) gauge theory, see ≪ or u v, and heavy T-baryon limit mB u, or gTC Ref. [32]) 1, ≫ ≫ ≫ The T-baryon annihilation in the low-symmetry phase (or LS phase) B0B¯0 W W which should be relevant g g g . → LS ≃ HS ≡ eff for not very heavy particles mB . 1 TeV, the weak- induced annihilation rate qualitatively change its energy as expected. dependence such that it does not disappear close to the The B0B¯0 annihilation cross sections in the TC- threshold any longer. In opposite, it becomes enhanced induced (pseudo)scalarπ ˜0π˜0, π˜−π˜−,hh,hσ,˜ σ˜σ˜ channels in the heavy T-baryon limit mB mW , especially in the are presented in Fig. 6 as functions of the T-baryon mass 0 ± ≫ case of a large B B mass splitting ∆mB & mW . This scale mB (left panel) and ∆mσ˜ mσ˜ √3mπ˜ (right − 0 0 ≡ − effect can be seen in Fig. 4(left), where the B B¯ W W panel). The mB dependence flattens out at large mB ex- cross section is shown as a function of ∆m = [5 →... 100] cept for the hσ˜ channel since g = g m (c.f. B B,i TC ∼ B 11
- - B0B0 annihilation B0B0 annihilation
LS phase, mB=250 GeV LS phase, mB=1 TeV 100 100 (pb) (pb) ann ann
v) v) ~0~0 σ σ π π ( 10 ( 10 ~π+~π- ~0~0 hh π π ~ ~π+~π- hσ σ~σ~ hh ~ hσ σ~σ~ 1 1 80 100 120 140 160 180 200 220 240 260 100 150 200 250 300 m~π (GeV) m~π (GeV)
FIG. 7: The T-baryon LS annihilation cross sections in the (pseudo)scalar B0B¯0 π˜0π˜0, π˜−π˜−, hh, hσ,˜ σ˜σ˜ channels as functions → of the T-pion mass mπ˜ for light mB = 250 GeV (left) and heavy mB = 1 TeV (right) T-baryons. Here, ∆mσ˜ = 20 GeV is fixed.
B0 B0 h, σ˜ h, σ˜ 0 B0 B
FIG. 8: Diagrams contributing to the elastic scattering of non-relativistic scalar T-baryons.
Eq. 3.6). In the TC decoupling limit u v, corre- ≫ 0 0 sponding to sθ 1 and hence to ∆mσ˜ 0, all the B B elastic scattering ≪ → 100 (pseudo)scalar annihilation channels vanish which is an mB=1 TeV important peculiarity of the considering scenario since g = g 1/u 0 while there is no exact symmetry B,i TC ∼ → of the cross sections w.r.t. ∆mσ˜ ∆mσ˜. In Fig. 7 10 the same cross sections are plotted↔ as − functions of the (pb) el
T-pion mass mπ˜ for two different T-baryon mass values σ mB = 250 GeV (left panel) and 1 TeV (right panel). 1 Consequently, one finds a disappearance of the total TC- m~π = 80 GeV induced cross section in the heavy T-pion limit m & m m~π = 150 GeV π˜ B m~π = 300 GeV which is otherwise peaked in intermediate regions. It is worth to summarize that the total T-baryon anni- 0.1 hilation rate is generally lower in the HS phase than that -40 -20 0 20 40 ∆m~ (GeV) in the LS one not only by means of a higher T-baryon σ mass scale mB, but also due to vanishing weak-induced contribution (3.8). In opposite, the weak-induced chan- FIG. 9: The T-baryon elastic scattering cross section σel as a nels, especially into W W , appear to dominate the T- function of ∆mσ˜ for three different values of the T-pion mass baryon annihilation cross section in the LS phase for a mπ˜ = 80, 150 and 300 GeV. Here, mB = 1 TeV is fixed. not too small T-baryon mass splitting. The bottom line of this study is that it is impossible to accommodate the symmetric T-baryon DM in the LS annihilation scenario C. Elastic T-baryon scattering unless the TC decoupling limit ∆mσ˜ 0 and, simul- taneously, the heavy T-baryon and low→ mass splitting limits are realized effectively approaching the HS anni- One of the important inputs for the cosmogonic DM hilation scenario developed in Eqs. (3.7)–(3.13). There- evolution and Structure Formation in late Universe is fore, we encounter two rather different HS/LS annihila- the elastic WIMP-WIMP scattering cross section. In tion scenarios which thus lead to symmetric/asymmetric the case of non-relativistic T-baryon DM, the elastic T-baryon DM, respectively. The DM of an intermediate B0B0 B0B0 scattering is described by five diagrams or mixed type can also be accommodated in the HS phase – one contact→ diagram, two t-channel and corresponding depending on the amount of initial T-baryon asymmetry. cross (u-channel) diagrams via h andσ ˜ exchanges shown 12 in Fig. 8. The non-relativistic T-baryon elastic scattering %) branching ratios in the considering 2-flavor SU(2)TC cross section is given by VLC scenario are
2 2 2 G c s 0,± 0,∓ 0 ± ± σ B , G g +8u2g2 θ + θ (3.15). σ˜ π˜ π˜ , π˜ Zγγ , π˜ W γγ , el 2 B BB BS 2 2 → → → ≃ 32πmB ≡ mσ˜ mh respectively. We conclude that a key distinct indirect sig- As a reference estimate, for G 1 10 and 1 TeV B nature of the symmetric scalar T-baryon DM at a TeV T-baryon, one obtains σ 15 ∼(1 −100) pb, respec- mass scale will be its annihilation into predominantly tively. In Fig. 9 the elastic∼ cross× section− is illustrated light T-pions decaying into a vector boson and into two as a function of ∆m under the setting (3.6). So, the σ˜ energetic photons which is a straightforward subject for elastic T-baryon scattering appears to be weaker than a promising multi-GeV γ-lines search at FERMI [42]. usual elastic nucleon scattering but significantly stronger Finally, in the very strong TC decoupling limit when than DM particles’ scattering in ordinary WIMP-based ∆m 5 GeV, the only B0B¯0 W ±W ∓ annihila- scenarios. Therefore, in fact we deal with a particular σ˜ tion channel≪ survives while the ZZ→and ff¯ ones vanish case of self-interacting DM which may be useful for DM since the T-baryon–Higgs boson coupling is expected to astrophysics (see e.g. Ref. [41]). decrease with the TC scale, e.g. gBH 1/u under the setting (3.6). The similar indirect observational∼ conse- IV. DARK MATTER DETECTION PROSPECTS quences can be drawn for the T-baryon DM of a mixed type, with a lower density of particles and antiparticles capable of mutual annihilation, which may diminish cor- A. Indirect detection responding detection rates. This situation means that even in the highly theoretically constrained VLC sce- At later stages of the Universe evolution, namely, af- nario, there is phenomenologically enough freedom to ter termination of T-baryon annihilation epoch, the T- accommodate the heavy scalar symmetric T-baryon DM baryon interactions are described by the effective low- scenario predicting rather interesting and potentially de- energy Lagrangians (2.7) and (2.15) where the chiral and tectable hard multi-γ signatures. EW symmetries breaking should be performed according As was advocated in the previous section, if the T- to Eqs. (2.9) and (2.10). In the case of symmetric T- baryon mass splitting is large ∆mB > 10 GeV and/or baryon DM mB & 2 TeV, its late time LS annihilation is the mass scale is lower mB < 2 TeV, one deals with described by the cross sections shown in Figs. 4, 6 and the purely asymmetric T-baryon DM (other possibilities 7. Here, one could distinguish two distinct cases: small would be highly fine-tuned and are thus less likely). This ∆mB < 10 GeV and large ∆mB > 10 GeV T-baryon is a pessimistic scenario for indirect DM detection mea- mass splitting. surements looking for T-baryon annihilation products – in the absence of a significant amount of anti-T-baryons B0 B0 and exact T-baryon conservation in present Universe, one does not expect to see any DM annihilation signatures. Only, direct ADM detection is relevant, so does in the h, σ q symmetric DM case, considered above.
N N B. Direct detection
FIG. 10: The elastic scattering of neutral scalar T-baryon B0 off a nucleon target N at tree level via h and σ exchanges Consider now the scalar T-baryon implications for the in the t-channel relevant for direct DM detection close to the direct DM detection experiments looking for nuclear re- threshold √sth mB + mN . coils due to elastic non-relativistic WIMP-nucleon scat- ≡ tering. At tree level, the elastic scalar T-baryon–nucleon scattering is mediated by the scalar Higgs boson and T- In the first case, the B0B¯0 W W annihilation chan- sigma exchanges in the t-channel only as shown in Fig. 10. nel is suppressed, and WW,→ ZZ and tt¯ channels alto- gether contribute at most 10 pb or less to the total cross section, which is therefore∼ dominated by T-strong Consider the consistent TC decoupling limit, sθ 1. In this case, the elastic scattering of non-relativistic≪ T- channels for not too small ∆mσ˜ & 5 GeV, specifically, by baryons off nucleons in an underground DM detector is B0B¯0 π˜0π˜0 , π˜−π˜− , σ˜σ˜ , dominated essentially by the Higgs boson exchange with → a very small q2 1KeV2. For such small q2 values ∼ whose total contribution exceeds e.g. 100 pb for mB =2 (slightly above the threshold √sth mB +mN ) one deals ≡ TeV and ∆mσ˜ = 20 GeV. Then, the dominant decay the effective Higgs-nucleon form factor previously dis- modes of T-sigma and T-pions with rather large (over 90 cussed in Ref. [43]. Then, the effective spin-independent 13
0 B0 B 0 0 B0 B0 B B B∓
B± W W W ± W ∓ W ± W ∓ ± ∓
qj q qj j q q qi qi qi qi i i
0 FIG. 11: The elastic scattering of neutral scalar T-baryon B off a light quark target qi at one loop level. Only the dominant topologies are shown.
T-baryon–nucleon elastic cross section reads 10-10 qB0 -> qB0 elastic cross section ∆ κ2 m4 MB = 1 TeV, mB = 10 GeV nucl N 2 10-12 σSI 2 4 FN m~π = 150 GeV, ∆mσ~ = 20 GeV ≃ π mBmh 1 GeV 2 -14 5.45 κ2 10−38 cm2 , (4.1) 10
≃ mB · (pb) SI
q σ 10-16 where mN is the nucleon mass, and FN parameterizes up, tree the effective tree-level matrix element for the Higgs in- down, tree teraction with the nucleon target incorporating the gluon 10-18 up, 1-loop down, 1-loop anomaly [43, 44] 10-20 1 mN 0 0.02 0.04 0.06 0.08 0.1 N m qq¯ N F NN . (4.2) ∆ v h | q | i≡ N v h i B (GeV) q X In Eq. (4.1), we used the SM lattice result F 0.375 FIG. 12: The elastic T-baryon–quark scattering cross section N 0 following the corresponding analysis of Ref. [45]≃ where of neutral scalar T-baryon B off a light (u, d) quark target the uncertainty is dominated by the strange quark con- at one loop level as a function of ∆B . The corresponding tribution, and κ is the effective T-baryon–Higgs coupling tree-level cross section is shown for comparison. which in the generic VLC scenario reads 0 0 2m2 to the elastic B q B q scattering cross section has = κvBBh,¯ κ g + h g . been found as a function→ of LBBh ≃ BH m2 m2 BS σ˜ − h 2mN lab ∆B √s √sth = EB , (4.3) At one-loop level, one encounters an appearance of ≡ − mB gauge boson mediated contributions to the elastic T- lab 2 baryon scattering off a quark in the nucleon target. where EB = mBv /2 (v 0.001) is the T-baryon ki- Thus, it is meaningful to consider first the elementary netic energy in the laboratory∼ frame. The result shown in B0q B0q subprocess for such a short-range loop- Fig. 12 clearly demonstrates a strong suppression of the induced→ reaction. one-loop correction compared to tree-level B0q B0q We have performed a dedicated analysis of the full scattering cross section in the non-relativistic scalar→ T- one-loop order correction to the B0q B0q (q = u, d) baryon limit, very close to the threshold. This is the elastic cross section close to the corresponding→ thresh- case of thermal relic T-baryons in a Sun neighborhood. old √sth. The dominant diagrams are shown in Fig. 11, Namely, individual u and d quark scatterings at one loop while other possible topologies are found to be strongly level are found to be over three orders of magnitude suppressed. For this purpose, the effective Lagrangian smaller then the corresponding tree-level results. There- of T-baryon interactions (2.15) together with the VLC fore, the cross section given by t-channel Higgs boson model (2.7) has been implemented into the FeynRules exchange (4.1) is considered to be accurate enough and framework [46] whose output was then used by FeynArts sufficient for a comparison to the DM direct experimen- [47] to calculate the respective one-loop amplitudes. The tal constraints. Note, for energetic T-baryons away from latter were processed by the FormCalc package [48] into the threshold the one-loop induced cross section becomes a Fortran code, together with necessary VLC parameter large and dominate the elastic T-baryon scattering off a relations and mass formulae. The one-loop master inte- quark. grals were evaluated by the LoopTools package [48], and In Ref. [32], the effective scalar T-baryon–Higgs cou- the final cross section has been evaluated for relevant sets pling has been constrained by lattice simulations in the of physical parameters. Namely, the one-loop correction strongly-coupled SU(4)TC model. While a similar analy- 14 sis in the VLC model with confined SU(2)TC symmetry EW one (in a few TeV region) then the EW precision con- yet has to be performed, in this first study we treat κ as straints. The bound (4.5) should be tested against lattice a free parameter which can be constrained by the direct simulations in the considered strongly-coupled SU(2)TC DM detection data, together with scalar T-baryon mass theory. scale mB. At the moment, the LUX experiment [37] pro- nucl vides the most stringent limit on σSI (per nucleon in the case of a xenon target) which is roughly C. Collider signatures
nucl σSI The composite DM studies, and the new SU(2) log 44.2 43.7 , mB 0.5 2 TeV(4.4), TC − 10 cm2 ≃ − ≃ − strong dynamics searches, in general, should be accompa- nied by respective scalar T-baryon searches at the LHC. providing the following bound on effective coupling In order to digest this possibility, let us briefly discuss the corresponding signatures of scalar T-baryon production κ . 0.17 1.2 , (4.5) − in pp collisions at 14 TeV. respectively, and somewhat weaker constraints for larger 0 10 0 - + mB. This is in agreement with the lower bound on T-baryon production in Drell-Yan process pp -> B Β + X T-strong couplings from the heavy symmetric T-baryon LHC, Ec.m.s. = 14 TeV constraint (3.13) and with our expectations for relatively 10-2 weak T-baryon–T-meson interactions in the small hσ- mixing limit noticed above. Indeed, in Fig. 13 the recent 10-4 LUX bound has been plotted vs T-baryon mass scale ∆m = 1 GeV
(pb) B ∆m = 100 GeV mB together with the effective scalar T-baryon–nucleon DY B σ -6 SI cross section given by Eq. (4.1). The latter have been 10 obtained for three distinct ∆mσ˜ values by using the naive scaling condition for scalar TC couplings (3.6). The cross 10-8 section is less sensitive to the T-pion mass so it has been fixed to mπ˜ = 150 GeV. 10-10 0.5 1 1.5 2 2.5 3
mB (GeV) 10-6 SI cross section LUX data vs T-baryon-nucleon scattering FIG. 14: The T-baryon B0B¯+ pair production cross section -7 10 via Drell-Yan channel in pp collisions at the LHC, Ec.m.s. = 14 TeV, for two distinct T-baryon mass splitting values, ∆mB = 1 and 100 GeV.
(pb) 10-8 SI N σ One of the characteristic signals can be expected from the Drell-Yan charged-current reaction 10-9 LUX constraint ∆m~ = 0.1 GeV ± ∗ 0 ± ± 0 ± σ qiq¯j (W ) B B¯ , B¯ B¯ (W fif¯j ) , ∆mσ~ = 1 GeV → → → → ∆mσ~ = 10 GeV 10-10 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 with deeply virtual W boson in the s channel. Here, 0 ¯0 mDM (TeV) heavy B and B leave the detector unnoticed giving rise to a very large (a few TeV) missing mass in the miss missing ET spectrum, while the tagging on charged FIG. 13: The spin-independent T-baryon–nucleon cross sec- debris from the final W decay may be very problematic tion vs LUX constraint [37]. for a relatively small mass splitting ∆mB . 50 70 GeV. Also, this is due to a very dramatic dependence− on the ± We note that the direct DM detection constraint from B width on ∆mB shown in Fig. 5, such that at small ± LUX in practice excludes large ∆mσ˜ & 10 GeV. It leaves ∆mB the charged B “lives” longer before it decays into the space only for a narrow a few GeV region in the vicin- a W and invisible B0 (a possibility for a displaced W dec ity of the TC decoupling limit ∆mσ˜ 0. Therefore, emission accompanied by heavy invisibles). Note, the the direct DM constraint is complimentary≡ to the corre- corresponding B0B¯± production cross section shown in sponding constraints on hσ˜ mixing angle from the Higgs Fig. 14 depends strongly on mass scale mB only and decays and from the T-parameter. It is worth to stress does not depend on any other TC parameters and cou- that the DM direct bound in Fig. 13 has turned out to plings, and only weakly depends on the T-baryon mass be the most stringent constraint on new SU(2) confined splitting ∆mB. Further more detailed studies of the T- dynamics among other ones. Indeed, it pushes the ac- baryon detection capabilities including realistic detector ceptable SU(2)TC scale ΛTC even further away from the constraints should be performed elsewhere. 15
V. SUMMARY AND CONCLUSIONS Higgs boson is not critical for cosmological evolution of the T-baryon DM analysis in the TC decoupling limit, Λ 100 GeV, when the light technipion limit m In this work, we have performed a detailed theoreti- TC ≫ π˜ ≪ cal and phenomenological study of scalar T-baryon sec- ΛTC is concerned as long as scalar T-baryon–Higgs bo- tor, in particular, with respect to its possible important son coupling is within the perturbative limit. Indeed, role for DM astrophysics. The corresponding scenario of as we have advocated above in detail the couplings of new gauge dynamics is based upon the consistent vector- heavy T-baryons with light composites (e.g. with pseudo- Goldstone states) are suppressed in the decoupling limit. like TC model with SU(2)TC group which allows to ob- tain Dirac T-quarks in confinement from original chiral As we have demonstrated above, the existing direct DM fermion multiplets. Starting from the phenomenological detection bounds set a limit on the T-baryon-Higgs bo- arguments provided by (i) EW precision tests and (ii) son coupling which should be understood as an impor- SM Higgs-like couplings, the T-baryon states turn out to tant constraint on the existing variety of composite Higgs be split and pushed up towards higher scales, away from models. An account for the Higgs compositeness will the dynamical EW symmetry breaking scale (the TC de- not change this situation and, specifically, does not af- fect our basic conclusions neither for asymmetric, nor for coupling limit, ΛTC MEW 100 GeV). In this regime, the scalar neutral T-baryon≫ (T-diquark)∼ B0 = UD state symmetric DM scenarios. This is a motivation for the (the lightest among other T-diquarks) possessing a con- study of the cosmological T-baryon evolution indepen- served quantum number can serve as an appealing DM dently on aspects of possible Higgs compositeness which will be thoroughly discussed in our forthcoming works. It candidate at a TeV mass scale, mB & 0.5 1 TeV. In the consistent TC decoupling limit, the scalar− self-couplings is worth to stress here that the LUX experiment bound of heavy T-baryons with light T-pions/T-sigma states are [37] in the TC decoupling limit is consistent with cosmo- expected to decrease. logical evolution and, in particular, with the freeze out of heavy T-baryons in the high-symmetry phase of the For mB > 2 TeV one expects that the relic T-baryon abundance has been formed mainly before the EW phase cosmological plasma setting an important stage for fur- transition. We have shown that in this case the total ther development of composite DM scenarios in the TC T-baryon annihilation cross section is rather weak and decoupling limit. enables symmetric DM formation, whereas lighter T- An incorporation of the Sommerfeld enhancement ef- baryons annihilating mostly in the low-symmetry phase fect into the cosmological evolution of scalar T-baryons of the cosmological plasma could only remain if there was could be a nice development of this work. However, in a significant T-baryon asymmetry generation. A specific this first study we prefer not to incorporate this effect astrophysical signature of symmetric T-baryon DM is its and hence to introduce an extra freedom into our anal- annihilation into hard multi-γ final states via interme- ysis naively assuming a sharp cutoff in self-annihilation diate T-pion/T-sigma states, relevant for indirect DM rates after the T-baryon freeze-out temperature. In fact, detection measurements. it is believed that the freeze out happens at relative ve- locities of about v 0.1 0.3 where the Sommerfeld en- We have shown that the elastic T-baryon–quark scat- ∼ − tering is induced at tree level by the Higgs/T-sigma hancement factor is of the order of unity for rather weak bosons exchanges in the t-channel only, since the vector T-baryon-Higgs couplings suggested by both the TC de- tree-level Z-boson is absent. We have calculated com- coupling limit and direct DM detection constraints. A plete one-loop correction to the elastic scalar T-baryon proper analysis of the Sommerfeld enhancement effect scattering and found that it is strongly suppressed com- could be a good point for further studies. pared to the tree-level result. Most importantly, the di- rect DM detection constraints, e.g. those from LUX, Acknowledgments on spin-independent elastic T-baryon–nucleon scattering Stimulating discussions and helpful correspondence cross section impose further ever stringent constraint on with Johan Bijnens, Johan Rathsman and Torbj¨orn the T-baryon–Higgs coupling and hence on the chiral Sj¨ostrand are acknowledged. V. B. and V. K. were par- symmetry breaking scale. Namely, it imposes a much tially supported by Southern Federal University grant stronger suppression to the hσ˜ mixing angle (and hence No. 213.01-.2014/-013VG. R. P. is grateful to the “What the stronger hierarchy between the chiral and EW break- is Dark Matter?” Program at Nordita (Stockholm) for ing scales) than the corresponding EW precision and SM support and hospitality during completion of this work. Higgs bounds. R. P. was supported in part by the Swedish Research Finally, the search for heavy scalar T-baryons in the Council, contract number 621-2013-428. Drell-Yan production process with a trigger on a few TeV missing mass and, possibly, on accompanying W emis- sion from a displaced vertex, is advised. A further more Appendix A: T-baryon annihilation into T-mesons dedicated analysis of T-baryon implications at the LHC, in particular, in vector-boson fusion channels would be In Eq. (3.14) only contact diagrams dominating in the desirable. limit mB u have been included. Here, we list the com- Note, the question about compositeness of the light plete results≫ for partial annihilation cross sections includ- 16 ing all diagrams in Fig. 3 and hence applicable for any where s 4m2 (1 + v2/4) close to the threshold v 1. ≃ B ≪ hierarchy between mB and u, mh, mσ˜ . These read
2 XY,LS gXY ¯ 2 2 2 (σv)ann 2 λ(mX ,mY ,mB) , (A1) ≃ 16πmB where λ¯(a,b,c) is defined in Ref. (3.5), and effective cou- plings are 1 g = g c2 + g s2 hh 2 BH θ BS θ 2 2 + 2ug s + vg c s 2m2 BS θ BH θ − h gσhh˜ 2ug c vg s − s m2 BS θ − BH θ − σ˜ ghhh 2ug s + vg c , − s m2 BS θ BH θ − h 1 g = g s2 + g c2 σ˜σ˜ 2 BH θ BS θ 2 2 + 2ug c vg s s 2m2 BS θ − BH θ − σ˜ ghσ˜σ˜ 2ug s + vg c − s m2 BS θ BH θ − h gσ˜σ˜σ˜ 2ug c vg s , − s m2 BS θ − BH θ − σ˜ √2 g = (2g g )s c σh˜ BS − BH θ θ 4(s m2 m2 ) + − h − σ˜ 2ug s + vg c (s 2m2 )(s 2m2 ) BS θ BH θ − h − σ˜ 2ug c vg s × BS θ − BH θ ghσ˜σ˜ 2ug c vg s − s m2 BS θ − BH θ − σ˜ gσhh˜ 2ug s + vg c − s m2 BS θ BH θ − h gσ˜π˜0π˜0 g 0 0 = g + g 2ug c vg s π˜ π˜ BS BP − s m2 BS θ − BH θ − σ˜ g 0 0 hπ˜ π˜ 2ug s + vg c , − s m2 BS θ BH θ − h gσ˜π˜+π˜− √2 g + − = 2g 2ug c vg s π˜ π˜ BS − s m2 BS θ − BH θ − σ˜ g + − hπ˜ π˜ 2ug s + vg c , − s m2 BS θ BH θ − h
[1] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716, [7] S. Dimopoulos and L. Susskind, Nucl. Phys. B155, 237 1 (2012) [arXiv:1207.7214 [hep-ex]]. (1979). [2] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B [8] E. Eichten and K. D. Lane, Phys. Lett. B90, 125 (1980). 716, 30 (2012) [arXiv:1207.7235 [hep-ex]]. [9] C. T. Hill and E. H. Simmons, Phys. Rept. 381, 235 [3] ATLAS Collaboration, ATLAS-CONF-2014-009 and (2003) [Erratum-ibid. 390, 553 (2004)] [hep-ph/0203079]. ATLAS-CONF-2014-010, CERN Geneva, March 2014 ; [10] F. Sannino, Acta Phys. Polon. B 40, 3533 (2009). S. Chatrchyan et al. [CMS Collaboration], [11] L. Vecchi, arXiv:1304.4579 [hep-ph]. arXiv:1401.6527 [hep-ex]. [12] D. Barducci, A. Belyaev, M. S. Brown, S. De Curtis, [4] S. Chatrchyan et al. [CMS Collaboration], JHEP 1306, S. Moretti and G. M. Pruna, JHEP 1309, 047 (2013) 081 (2013) [arXiv:1303.4571 [hep-ex]]. [arXiv:1302.2371 [hep-ph]]. [5] S. Weinberg, Phys. Rev. D13, 974 (1976). [13] A. De Simone, O. Matsedonskyi, R. Rattazzi and [6] L. Susskind, Phys. Rev. D20, 2619 (1979). A. Wulzer, JHEP 1304, 004 (2013) [arXiv:1211.5663 17
[hep-ph]]. [32] T. Appelquist, E. Berkowitz, R. C. Brower, M. I. Buchoff, [14] M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 G. T. Fleming, J. Kiskis, G. D. Kribs and M. Lin et al., (1990). arXiv:1402.6656 [hep-lat]. [15] M. E. Peskin, T. Takeuchi, Phys. Rev. D 46, 381 (1992). [33] A. Belyaev, M. T. Frandsen, S. Sarkar and F. Sannino, [16] C. Kilic, T. Okui and R. Sundrum, JHEP 1002, 018 Phys. Rev. D 83, 015007 (2011) [arXiv:1007.4839 [hep- (2010) [arXiv:0906.0577 [hep-ph]]. ph]]. [17] R. Pasechnik, V. Beylin, V. Kuksa and G. Vereshkov, [34] K. Petraki and R. R. Volkas, arXiv:1305.4939 [hep-ph]. Phys. Rev. D 88, 075009 (2013) [arXiv:1304.2081 [hep- [35] C. Kouvaris, Phys. Rev. D 88, no. 1, 015001 (2013) ph]]. [arXiv:1304.7476 [hep-ph]]. [18] R. Pasechnik, V. Beylin, V. Kuksa and G. Vereshkov, [36] E. Aprile et al. [XENON100 Collaboration], Phys. Eur. Phys. J. C 74, 2728 (2014) [arXiv:1308.6625 [hep- Rev. Lett. 109, 181301 (2012) [arXiv:1207.5988 [astro- ph]]. ph.CO]]. [19] P. Lebiedowicz, R. Pasechnik and A. Szczurek, Nucl. [37] D. S. Akerib et al. [LUX Collaboration], Phys. Rev. Lett. Phys. B 881, 288 (2014) [arXiv:1309.7300 [hep-ph]]. 112, 091303 (2014) [arXiv:1310.8214 [astro-ph.CO]]. [20] A. Hietanen, R. Lewis, C. Pica and F. Sannino, [38] J. Hisano, K. Ishiwata, N. Nagata and T. Takesako, arXiv:1404.2794 [hep-lat]. JHEP 1107, 005 (2011) [arXiv:1104.0228 [hep-ph]]. [21] G. Cacciapaglia and F. Sannino, arXiv:1402.0233 [hep- [39] G. Steigman, B. Dasgupta and J. F. Beacom, Phys. Rev. ph]. D 86, 023506 (2012) [arXiv:1204.3622 [hep-ph]]. [22] M. R. Buckley and E. T. Neil, Phys. Rev. D 87, no. 4, [40] G. Hinshaw et al. [WMAP Collaboration], Astrophys. J. 043510 (2013) [arXiv:1209.6054 [hep-ph]]. Suppl. 208, 19 (2013) [arXiv:1212.5226 [astro-ph.CO]]. [23] S. Nussinov, Phys. Lett. B 165, 55 (1985); [41] D. N. Spergel and P. J. Steinhardt, Phys. Rev. Lett. 84, S. M. Barr, R. S. Chivukula, and E. Farhi, Phys. Lett. B 3760 (2000) [astro-ph/9909386]. 241, 387 (1990). [42] M. Ackermann et al. [Fermi-LAT Collaboration], Phys. [24] S. B. Gudnason, C. Kouvaris and F. Sannino, Phys. Rev. Rev. D 88, 082002 (2013) [arXiv:1305.5597 [astro- D 74, 095008 (2006) [hep-ph/0608055]. ph.HE]]. [25] S. B. Gudnason, C. Kouvaris and F. Sannino, Phys. Rev. [43] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, D 73, 115003 (2006) [hep-ph/0603014]. Phys. Lett. B 78, 443 (1978). [26] M. Y. Khlopov and C. Kouvaris, Phys. Rev. D 78, 065040 [44] J. Giedt, A. W. Thomas and R. D. Young, Phys. Rev. (2008) [arXiv:0806.1191 [astro-ph]]. Lett. 103, 201802 (2009) [arXiv:0907.4177 [hep-ph]]. [27] M. Y. Khlopov, A. G. Mayorov and E. Y. Soldatov, Int. [45] R. J. Hill and M. P. Solon, Phys. Lett. B 707, 539 (2012) J. Mod. Phys. D 19, 1385 (2010) [arXiv:1003.1144 [astro- [arXiv:1111.0016 [hep-ph]]. ph.CO]]. [46] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr [28] E. Del Nobile and F. Sannino, Int. J. Mod. Phys. A 27, and B. Fuks, Comput. Phys. Commun. 185, 2250 (2014) 1250065 (2012) [arXiv:1102.3116 [hep-ph]]. [arXiv:1310.1921 [hep-ph]]. [29] F. Sannino, Acta Phys. Polon. B 40, 3533 (2009) [47] T. Hahn, Comput. Phys. Commun. 140, 418 (2001) [arXiv:0911.0931 [hep-ph]]. [hep-ph/0012260]. [30] T. A. Ryttov and F. Sannino, Phys. Rev. D 78, 115010 [48] T. Hahn and M. Perez-Victoria, Comput. Phys. Com- (2008) [arXiv:0809.0713 [hep-ph]]. mun. 118, 153 (1999) [hep-ph/9807565]. [31] R. Lewis, C. Pica and F. Sannino, Phys. Rev. D 85, 014504 (2012) [arXiv:1109.3513 [hep-ph]].