Light Cold Dark Matter in a Two-Singlet Model

Understanding the nature of dark matter (DM) as well as the origin of baryon asymmetry are two of the most important questions in both Cosmology and Particle Physics. The failure of the Standard Model (SM) of the electroweak and strong interactions in accommodating such questions motivates the search for an explanation in the realm of new physics beyond it. The most attractive strategy so far has been to look into extensions of the SM that incorporate electrically neutral and colorless weakly interacting massive particles (WIMPs), with masses from one to a few hundred GeV, coupling constants in the milli-weak scale and lifetimes longer than the age of the Universe.


Introduction
Understanding the nature of dark matter (DM) as well as the origin of baryon asymmetry are two of the most important questions in both Cosmology and Particle Physics.The failure of the Standard Model (SM) of the electroweak and strong interactions in accommodating such questions motivates the search for an explanation in the realm of new physics beyond it.The most attractive strategy so far has been to look into extensions of the SM that incorporate electrically neutral and colorless weakly interacting massive particles (WIMPs), with masses from one to a few hundred GeV, coupling constants in the milli-weak scale and lifetimes longer than the age of the Universe.
The most popular extension of the SM is the minimal supersymmetric standard model (MSSM) in which the neutral lightest supersymmetric particle (LSP) is seen as a candidate for dark matter.Indeed, neutralinos are odd under R -parity and are only produced or destroyed in pairs, thus making the LSP stable [1].They can annihilate through a t-channel sfermion exchange into Standard Model fermions, or via a t-channel chargino-mediated process into W + W − , or through an s-channel pseudoscalar Higgs exchange into fermion pairs.They can also undergo elastic scattering with nuclei through mainly a scalar Higgs exchange [2].
However, most particularly in light of the recent signal reported by CoGeNT [3], which favors a light dark matter (LDM) with a mass in the range 7 − 9GeV and nucleon scattering cross section σ det ∼ 10 −4 pb, having a neutralino as a LDM candidate can be challenging.Indeed, systematic studies show that an LSP with a mass around 10GeV and an elastic scattering cross-section off a nucleus larger than ∼ 10 −5 pb requires a very large tan β and a relatively light CP-odd Higgs [4].This choice of parameters leads to a sizable contribution to the branching ratios of some rare decays, which then disfavors the scenario of light neutralinos [5].Also, in the next-to-minimal supersymmetric standard model (NMSSM) with 12 input parameters [6], realizing a LDM with an elastic scattering cross section capable of generating the CoGeNT signal is possible only in a finely-tuned region of the parameters where the neutralino is mostly singlino and the light CP-even Higgs is singlet-like with a mass below a few GeV.In such a situation, it is very difficult to detect such a light Higgs at the collider.It is clear then that other alternative scenarios for LDM are needed [7].
The simplest of scenarios is then to extend the Standard Model by a real Z 2 symmetric scalar field, the dark matter, which has to be a SM gauge singlet that interacts with visible particles via the Higgs field only.Such an extension was first proposed in [8] and further studied in [9] where the Z 2 symmetry is extended to a global U(1) symmetry, more extensively in [10].Specific implications on Higgs detection and LHC physics were discussed in [11] and one-loop vacuum stability and perturbativity bounds discussed in [12].However, the work of [13] uses constraints from the experiments XENON10 [14] and CDMSII [15] to exclude DM masses smaller than 50, 70 and 75GeV for Higgs masses equal to 120, 200 and 350 GeV respectively.Also, the Fermi-LAT data on the isotropic diffuse gamma-ray emission can potentially exclude this one-singlet dark-matter model for masses as low as 6GeV, assuming a NFW profile for the dark-matter distribution [16].
Therefore, a two-singlet extension of the Standard Model was proposed in [17] as a simple model for light cold dark matter.Both scalar fields are Z 2 symmetric with one undergoing spontaneous symmetry breaking.The behavior of the model under the DM relic-density constraint and the restrictions from experimental direct detection was studied.It was concluded that the model was capable of bearing a light dark-matter WIMP.
The present chapter describes how we can further the study of the two-singlet scalar model (2SSM) by discussing some of its phenomenological implications.We limit ourselves to small DM masses, from 0.1GeV to 10GeV.We discuss the implications of the model on the meson factories and the Higgs search at the LHC.In fact, it is pertinent at this stage to mention that there are more than one motivation for scalar-singlet extensions of the SM.Indeed, besides providing a possible account for the dark matter in the Universe consistent with the CoGeNT signal, they also provide a solution to the mu problem in the supersymmetric standard model.They can explain the matter-anti matter asymmetry in the Universe [18], and account for the possible occurrence of a light Higgs with a mass less or equal to 100GeV while still in agreement with the electroweak precision tests [19] and potential signatures at B-factories [20].This chapter is organized as follows.The next section introduces briefly the model and summarizes the results of [17] regarding relic-density constraint and direct detection.The two following sections investigate the rare decays of Υ (section three) and B (section four) mesons, most particularly their invisible channels.Section five looks into the decay channels of the Higgs particle.In each of these situations, we try, when possible, to deduce preferred regions of the parameter space and excluded ones.The last section is devoted to concluding remarks.Results presented here are found in [20].

The 2SSM Model
The Standard Model is extended with two real, spinless and Z 2 -symmetric fields.One is the dark-matter field S 0 with unbroken symmetry to ensure the stability of the dark matter, the other is an auxiliary field χ 1 for which the symmetry is spontaneously broken.Both fields are Standard-Model gauge singlets and can interact with SM particles only via the Higgs doublet H.This latter is taken in the unitary gauge such that , where h ′ is a real scalar.We assume all processes calculable in perturbation theory.The potential function that incorporates S 0 , h ′ and χ 1 is: The mass parameters squared m2 0 , µ 2 and µ 2 1 and all the coupling constants are real positive numbers.Electroweak spontaneous symmetry breaking occurs for the Higgs field with the vacuum expectation value v = 246GeV.The field χ 1 will oscillate around the vacuum expectation value v 1 which we take at the electroweak scale 100GeV.Writing h ′ = v + h and χ 1 = v 1 + S1 , the potential function becomes, up to an irrelevant zero-field energy: where the quadratic terms are given by: with the mass-squared coefficients related to the original parameters of the theory by the following relations: Clearly, we need to diagonalize the mass-squared matrix.
Denoting the physical mass-squared field eigenmodes by h and S 1 , we rewrite: Written now directly in terms of the physical fields, the cubic interaction terms are expressed as follows: where the cubic physical coupling constants are related to the original parameters via the following relations: In the same way, in terms of the physical fields too, the quartic interactions are given by: with the physical quartic coupling constants written as: Finally, after spontaneous breaking of the electroweak and Z 2 symmetries, the part of the Standard Model Lagrangian that is relevant to dark matter annihilation writes, in terms of the physical fields h and S 1 , as follows: The quantities m f , m w and m z are the masses of the fermion f , the W and the Z gauge bosons respectively, and the above coupling constants are given by the following relations:

Dark Matter relic density constraint
The field S 0 is odd under the unbroken Z 2 symmetry, and so is a stable relic and can therefore constitute the dark matter of the Universe.Its relic density can be obtained using the standard approximate solutions to the Boltzmann equations [21]: where h is the normalized Hubble constant,M Pl = 1.22 × 10 19 GeV is the Planck mass, g * the number of relativistic degrees of freedom at the freeze-out temperature T f , and x f = m 0 /T f which, for m 0 in the range 1 − 20GeV, is in the range 18.2 − 19.4.The quantity υ 12 σ ann is the thermally averaged annihilation cross section of S 0 into fermion pairs f f for m f < m 0 /2, and into S 1 S 1 for m 1 < m 0 /2.The annihilation cross-section into fermions proceeds via s-channel exchange of h and S 1 and is given by: and the annihilation into S 1 pairs given by: 01 η (3) 01 λ (3) 01 η (3) Solving (14) with the current value for the dark matter relic density Ω D h2 = 0.1123 ± 0.0035 [22] translates into a relation between the parameters of a given theory entering the calculated expression of v 12 σ ann , hence imposing a constraint on these parameters which will limit the intervals of possible dark matter masses.This constraint is exploited to examine aspects of the theory like perturbativity, while at the same time reducing the number of parameters by one.
Indeed, the model starts with eight parameters.The spontaneous breaking of the electroweak and Z 2 symmetries introduces the two vacuum expectation values v and v 1 respectively, which means we are left with six.Four of the parameters are the three physical masses m 0 (dark-matter singlet S 0 ), m 1 (the second singlet S 1 ) and m h (Higgs h), plus the mixing angle θ between h and S 1 .We will fix the Higgs mass to m h = 125GeV [23, 24], except in the section discussing Higgs decays where we let m h vary in the interval 100GeV − 200GeV.We will take both m 0 and m 1 in the interval 0.1GeV − 10GeV.For the purpose of our discussions, it is sufficient to let θ vary in the interval 1 o − 40 o .The last parameters are the two physical mutual coupling constants λ (4) 0 (dark matter -Higgs) and η (4) 01 (dark matter -S 1 particle).But η (4) 01 is not free as it is the smallest real and positive solution to the dark-matter relic density constraint (14), which will be implemented systematically throughout [17].Thus we are left with four parameters, namely, m 0 , m 1 , θ and λ (4) 0 .To ensure applicability of perturbation theory, the requirement η (4) 01 < 1 will also be imposed throughout, as well as a choice of rather small values for λ (4) 0 .
Studying the effects of the relic-density constraint for large ranges of the parameters through the behavior of the physical mutual coupling constant η (4) 01 between S 0 and S 1 as a function of the DM mass m 0 shows that, apart from forbidden regions and others where perturbativity is lost, viable solutions in the small-moderate mass ranges of the DM sector exist for most values of the parameters [17].Forbidden regions are found for most of the ranges of the parameters whereas perturbativity is lost mainly for larger values of m 1 .

Direct detection
On the other hand, experiments like CDMS II [15], XENON 10/100 [14,25], DAMA/LIBRA [26] and CoGeNT [3] search directly for a dark matter signal, which would typically come from the elastic scattering of a dark matter WIMP off a non-relativistic nucleon target.However, throughout the years, such experiments have not yet detected an unambiguous signal, but rather yielded increasingly stringent exclusion bounds on the dark matternucleon elastic-scattering total cross-section σ det in terms of the DM mass m 0 .Any viable theoretical dark-matter model has to satisfy these bounds.In the 2SSM, σ det is found to be given by the relation [17]: in which m N is the nucleon mass and m B the baryon mass in the chiral limit [13,27,28].This relation was compared against the experimental bounds from CDMSII and XENON100.We found that strong constraints were imposed on m 0 in the range between 10 to 20GeV.We found also that for small values of m 1 , very light dark matter is viable, with m 0 as small as 1GeV.

Upsilon decays
We now further the analysis of the two-singlet model and start by looking at the constraints on the parameter space of the model coming from the decay of the meson Υ in the state nS (n = 1, 3) into one photon γ and one particle S 1 .For m 1 8GeV, the branching ratio for this process is given by the relation: In this expression, x n ≡ 1 − m 2 1 /m 2 Υ ns with m Υ 1(3)S = 9.46(10.355)GeVthe mass of Υ 1(3)S , the branching ratio Br (µ) ≡ Br Υ 1(3)S → µ + µ − = 2.48(2.18)× 10 −2 [29], α is the QED coupling constant, α s = 0.184 the QCD coupling constant at the scale m Υ nS , the quantity G F is the Fermi coupling constant and m b the b quark mass [22].The function f (x) incorporates the effect of QCD radiative corrections given in [30].
But the above expression is not sufficient because a rough estimate of the lifetime of S 1 indicates that this latter is likely to decay inside a typical particle detector, which means we ought to take into account its most dominant decay products.We first have a process by which S 1 decays into a pair of pions, with a decay rate given by: In the above decay rate, m π(K) is the pion (kaon) mass.Also, chiral perturbation theory is used below the kaon pair production threshold [31,32], and the spectator quark model above up to roughly 3GeV, with the dressed u and d quark masses M u = M d ≃ 0.05GeV.Note that this rate includes all pions, charged and neutral.Above the 2m K threshold, both pairs of kaons and η particles are produced.The decay rate for K production is: In the above rate, M s ≃ 0.45GeV is the s quark-mass in the spectator quark model [33,34].
For η production, replace m K by m η and 9  13 by 4  13 .The particle S 1 can also decay into c and b quarks (mainly c).Including the radiative QCD corrections, the corresponding decay rates are given by: The dressed quark mass mq ≡ m q (m 1 ) and the running strong coupling constant ᾱs ≡ α s (m 1 ) are defined at the energy scale m 1 [35].Gluons can also be produced, with a corresponding decay rate given by the relation: Open Questions in Cosmology Here, α ′ s = 0.47 is the QCD coupling constant at the spectator-quark-model scale, between roughly 1GeV and 3GeV.
We then have the decay of S 1 into leptons, the corresponding rate given by: where m ℓ is the lepton mass.Finally, S 1 can decay into a pair of dark matter particles, with a decay rate: The branching ratio for Υ nS decaying via S 1 into a photon plus X, where X represents any kinematically allowed final state, will be: In particular, X ≡ S 0 S 0 corresponds to a decay into invisible particles.The best available experimental upper bounds on 1S-state branching ratios are: (i) Br (Υ 1S → γ + ττ) < 5 × 10 −5 for 3.5GeV < m 1 < 9.2GeV [36]; (ii) Br Υ 1S → γ + π + π − < 6.3 × 10 −5 for 1GeV < m 1 [37]; (iii) Br Υ 1S → γ + K + K − < 1.14 × 10 −5 for 2GeV < m 1 < 3GeV [38].Figure 1 displays the corresponding branching ratios of Υ 1S decays via S 1 as functions of m 1 , together with these upper bounds.Also, the best available experimental upper bounds on Υ 3S branching ratios are: (i) Br (Υ 3S → γ + µµ) < 3 × 10 −6 for 1GeV < m 1 < 10GeV; (ii) Br (Υ 3S → γ + Invisible) < 3 × 10 −6 for 1GeV < m 1 < 7.8GeV [39].Typical corresponding branching ratios are shown in figure 2. When scanning the parameter space, we see that the Higgs-dark-matter coupling constant λ (4) 0 and the dark-matter mass m 0 have little effect on the shapes of the branching ratios, apart from excluding, via the relic density and perturbativity constraints, regions of applicability of the model.This shows in figures 1 and 2 where the region m 1 1.9GeV is excluded.Also, the onset of the S 0 S 0 channel for m 1 ≥ 2m 0 abates sharply the other channels as this one becomes dominant by far.The effect of the mixing angle θ is to enhance all branching ratios as it increases, due to the factor sin 2 θ.Furthermore, we notice that the dark matter decay channel reaches the invisible upper bound already for θ ≃ 15 o , for fairly small m 0 , say 0.5GeV, whereas the other channels find it hard to get to their respective experimental upper bounds, even for large values of θ.

B meson decays
Next we look at the flavor changing process in which the meson B + decays into a K + plus invisible.The corresponding Standard-Model mode is a decay into K + and a pair of neutrinos, with a branching ratio Br SM B + → K + + ν ν ≃ 4.7 × 10 −6 [40].The experimental upper bound is Br Exp B + → K + + Inv ≃ 14 × 10 −6 [41].Here too, the most prominent B invisible decay is into S 0 S 0 via S 1 .The process B + → K + + S 1 has the following branching ratio: In the above relation, m ± = m B ± m K where m B is the B + mass, τ B its lifetime, andV tb and V ts are flavor changing CKM coefficients.The function f 0 (s) is given by the relation: The different S 1 decay modes are given in ( 19) -( 24) above.The branching ratio of B + decaying into K + + S 0 S 0 via the production and propagation of an intermediary S 1 will be: Figure 3 displays a typical behavior of Br (S 1 ) B + → K + + S 0 S 0 as a function of m 1 .As we see, the branching ratio is well above the experimental upper bound, and a mixing angle θ as small as 1 o will not help with this, no matter what the values for λ (4) 0 and m 0 are.So, we conclude that for m 1 4.8GeV, this process excludes the two-singlet model for m 0 < m 1 /2.For m 1 4.8GeV or m 0 ≥ m 1 /2, the decay does not occur, so no constraints on the model from this process.
Another process involving B mesons is the decay of B s into predominately a pair of muons.The Standard Model branching ratio for this process is Br SM B s → µ + µ − = (3.2± 0.2) × 10 −9 [42], and the experimental upper bound is Br Exp B s → µ + µ − < 1.08 × 10 −8 [43].In the present model, two additional decay diagrams occur, both via intermediary S 1 , yielding together the branching ratio: In this relation, τ B s is the B s life-time, m B s = 5.37GeV its mass, and f B s a form factor that we take equal to 0.21GeV.The quantity Γ 1 is the total width of the particle S 1 [17].A typical behavior of Br (S 1 ) (B s → µ + µ − ) as a function of m 1 is displayed in figure 4. The peak is at m B s .All three parameters λ (4) 0 , m 0 and θ combine in the relic density constraint to exclude few regions of applicability of the model.For example, for the values of figure 4, the region m 1 < 2.25GeV is excluded.However, a systematic scan of the parameter space shows that outside the relic density constraint, λ (4) 0 has no significant direct effect on the shape of Br (S 1 ) (B s → µ + µ − ).As m 0 increases, it sharpens the peak of the curve while pushing it up.
This works until about 2.7GeV, beyond which m 0 ceases to have any significant direct effect.Increasing θ enhances the values of the branching ratio without affecting the width.Also, for all the range of m 1 , all of Br (S 1 ) + Br SM stays below Br Exp as long as θ < 10 o .As θ increases beyond this value, the peak region pushes up increasingly above Br Exp , like in figure 4, and hence gets excluded, but all the rest is allowed.

Higgs decays
We finally examine the implications of the model on the Higgs different decay modes.In this section, we allow the Higgs mass m h to vary in the interval 100GeV − 200GeV.First, h can decay into a pair of leptons ℓ, predominantly τ's.The corresponding decay rate Γ h → ℓ + ℓ − is given by the relation (23) where we replace m 1 by m h and sin θ by cos θ.It can also decay into a pair of quarks q, mainly into b's and, to a lesser degree, into c's.Here too the decay rate Γ(h → q q) is given in (21) with similar replacements.Then the Higgs can decay into a pair of gluons.The corresponding decay rate that includes the next-to-next-to-leading QCD radiative corrections is given by: where the sum is over all quark flavors q.A systematic study of the double integral above shows that, with m h in the range 100GeV -200GeV, the t quark dominates in the sum over q, but with non-negligible contributions from the c and b quarks.
For m h smaller than the W or Z pair-production thresholds, the Higgs can decay into a pair of one real and one virtual gauge bosons, with rates given by: In this expression, m V is the mass of the gauge boson V, the factor A V = 1 for W and For a heavier Higgs particle, the decay rates into a V pair is given by: with B V = 1 for W and 1 2 for Z.While all these decay modes are already present within the Standard Model, the two-singlet extension introduces two additional (invisible) modes, namely a decay into a pair of S 0 's and a pair of S 1 's.The corresponding decay rates are: where λ i = λ (3) 0(2) for S 0(1) given in (9).The total decay rate Γ (h) of the Higgs particle is the sum of these partial rates.The branching ratio corresponding to a particular decay will be Br (h Typical behaviors of the most prominent branching ratios are displayed in figure 5.A systematic study shows that for all ranges of the parameters, the Higgs decays dominantly into invisible.The production of fermions and gluons is comparatively marginal, whereas that of W and Z pairs takes relative importance towards and above the corresponding thresholds, and more significantly at larger values of the mixing angle θ. However, the decay distribution between S 0 and S 1 is not even.The most dramatic effect comes from the coupling constant λ (4) 0 .When it is very small, the dominant production is that of a pair of S 1 .This is exhibited in figure 5 for which λ (4) 0 = 0.01.As it increases, there is a gradual shift towards a more dominating dark-matter pair production, a shift competed against by the increase in θ. Figure 6 displays the branching ratios for λ (4) 0 = 0.1 and figure 7 for the larger value λ (4) 0 = 0.7.In general, increasing θ smoothens the crossings of the WW and ZZ thresholds, and lowers the production of everything except that of a pair of S 1 , which is instead increased.Like in the Standard Model, the production of a pair of b quarks dominates over the production of the other fermions, and all fermions are not favored by increasing λ (4) 0 .Changes in m 0 and m 1 have very little direct effects on all the branching ratios except that of S 0 S 0 production where, at small θ, increasing m 1 (m 0 ) increases (decreases) the branching ratio, with reversed effects at larger θ.Note though that these masses have indirect impact through the relic density constraint by excluding certain regions [17].

Concluding remarks
Understanding light dark matter is one of the challenges facing popular extensions of the Standard Model.In this chapter, we have furthered the study of a two-singlet extension of the SM we proposed as a model for light dark matter by exploring some of its phenomenological aspects.We have looked into the rare decays of Υ and B mesons and studied the implications of the model on the decay channels of the Higgs particle.In brief, for both Υ and B decays, the Higgs-DM coupling constant λ (4) 0 and the DM mass m 0 have little effect on the shapes of the branching ratios, apart from combining with the other two parameters in the relic-density and perturbativity constraints to exclude regions of applicability of the model.Also, the effect of increasing the h − S 1 mixing angle θ is to enhance all branching ratios.For Υ decays, the DM channel dominates over the other decay modes in regions where kinematically allowed.It reaches the experimental invisible upper bound for already fairly small values of θ and m 0 .From B + decays, we learn that our model is excluded for m 1 < 4.8GeV (= m B − m K ) and m 0 < m 1 /2.From B s decay into muons, we learn that for the model to contribute a distinct signal to this process, it is best to restrict 4GeV m 1 6.5GeV with no additional constraint on m 0 [20].Also, in general, keeping λ (4) 0 0.1 to avoid systematic exclusion from direct detection for all these processes is safe.
Before closing the chapter, it is useful to comment briefly on how light dark matter in our model affects Higgs searches.Since m h ≫ 2m 0 , the process h → S 0 S 0 is kinematically allowed and, for a large range of the parameter space, the ratio R (b) can be larger than one for m h < 120GeV as can be seen in figure 6.In this situation, the LEP bound on the Higgs mass can be weaker.Also, in our model, the Higgs production at LEP via Higgstrahlung can be smaller than the one in the Standard Model, and so the Higgs can be as light as 100GeV.Such a light Higgs would be in good agreement with the electroweak precision tests.As to the Higgs searches at the LHC, the ATLAS and CMS collaborations have reported the exclusion of a Higgs mass in the interval 145 -460 GeV [23, 24], which seems to suggest that we should have limited our analysis of the Higgs branching ratios to m h < 145GeV.However, it is important to note that these experimental constraints apply to the SM Higgs and can not therefore be used as such if the Higgs interactions are modified.In our model, the mixing of h with S 1 will result in a reduction of the statistical significance of the Higgs discovery at the LHC.Indeed, the relevant quantity that allows one to use the experimental limits on Higgs searches to derive constraints on the parameters of the model is the ratio: In this expression, X SM corresponds to all the Standard Model particles, X inv = S 0 S 0 and S 1 S 1 , σ is a cross-section, Br SM (h → X) the branching fraction of the SM Higgs decaying into any kinematically allowed mode X, and Γ SM h the total Higgs decay rate in the Standard Model.To open up the region m h > 140GeV requires the ratio R X SM to be smaller than 0.25 [23,24], a constraint easily fulfilled in our model.By comparison, the minimal extensions of the Standard Model with just one singlet scalar or a Majorana fermion, even under a Z 2 symmetry, are highly constrained in this regard [44].Finally, if the recent data from ATLAS and CMS turn out to be a signal for a SM-like Higgs with mass about 125GeV, then this will put a very strong constraint on the mixing angle θ.Indeed, only for θ 0.5 0 will the ratio Γ(h → b b)/Γ (h → inv) 1.For larger values, the model is ruled out, independently of the dark matter mass, but as long as m 1 50GeV, which we are assuming in this work.Finally, it is important to find the bounds on the mass SM Higgs that can satisfy the triviality and perturbativity constraints on the coupling constants in the scalar sector of this model up to a scale higher then 1TeV.This requires studying the renormalization group equations of these coupling constants [45].

Figure 1 .
Figure 1.Typical branching ratios of Υ 1S decaying into τ's, charged pions and charged kaons as functions of m 1 .The corresponding experimental upper bounds are shown.

Figure 2 .
Figure 2. Typical branching ratios of Υ 3S decaying into muons and dark matter as functions of m 1 .The corresponding experimental upper bounds are shown.

Figure 3 .
Figure 3.Typical branching ratio of B + decaying into dark matter via S 1 as a function of m 1 .The SM and experimental bounds are shown.

Figure 4 .
Figure 4. Typical behavior of Br (S 1 ) (B s → µ + µ − ) as a function of m 1 , together with the SM and experimental bounds.

7 12 − 10 9 sin 2 θ w + 40 9 sin 4 θ
w for Z with θ w the Weinberg angle, and we have the definition: