Heavy octets and Tevatron signals with three or four b jets

Hypothetical color-octet particles of spin 0, pair-produced at hadron colliders through their QCD coupling, may lead to final states involving three or four b jets. We analyze kinematic distributions of the 3b final state that differentiate the scalar octets from supersymmetric Higgs bosons. Studying the scalar sector that breaks an SU(3) \times SU(3) gauge symmetry down to the QCD gauge group, we find that the scalar octet is resonantly produced in pairs via a spin-1 octet (coloron). A scalar octet of mass in the 140 - 150 GeV range can explain the nonstandard shape of the b-jet transverse energy distributions reported by the CDF Collaboration, especially when the coloron mass is slightly above twice the scalar mass. The dominant decay mode of the scalar octet is into a pair of gluons, so that the production of a pair of dijet resonances is large in this model, of about 40 pb at the Tevatron. Even when a W boson is radiated from the initial state, the inclusive cross section for producing a dijet resonance near the scalar octet mass remains sizable, around 0.15 pb.

3 Tevatron phenomenology of scalar octets Hadron colliders, such as the Tevatron and the LHC, allow the production at a high rate of new particles carrying QCD color. The subsequent decays of such particles often involve only QCD jets, which are hard to separate from the background. However, when the jets originate from b-quark decays, the backgrounds are substantially reduced. Among the hypothetical particles leading to signatures involving several b jets are color-octet bosons. The simplest of those is the weak-singlet particle of spin 0, generically referred to here as the G H scalar, which is present in various theories including Technicolor ("octet technipion" [1]), the 6D standard model (the Kaluza-Klein modes of the gluon polarized along extra dimensions [2]), vectorlike confinement [3], weakly-interacting metastable pion models [4], or certain supersymmetric models ("sgluons" [5]). The G H scalar can be produced in pairs through its QCD couplings to gluons, and may decay through higherdimensional operators into a pair of heavy quarks. The signature is four b jets, forming two bb resonances of same mass [6,7].
Here we point out that an alternative way of searching for scalar octets is to require only three jets to pass the basic cuts and to be b tagged. We show that in this case the shapes of certain kinematic distributions may allow the separation of the signal from the background, and the differentiation between various extensions of the standard model.
For example, compared to the Minimal Supersymmetric Standard Model (MSSM) at large tan β, where a Higgs boson is produced in association with a b quark and then decays to a bb pair, the octet pair production leads to a peak at a larger invariant mass of the leading and third jets.
Even more dramatic deformations of the distributions occur when the pair of G H scalars is produced through an s-channel resonance. We study these effects within a renormalizable theory that includes an extension of the QCD gauge group, SU(3) × SU(3), as proposed in [8,9]. The scalar sector responsible for spontaneously breaking this symmetry down to SU(3) c includes a G H scalar as well as two gauge singlet real scalar fields. We refer to this as the Renormalizable Coloron Model (ReCoM). The heavy gauge boson arising from the extended symmetry, labelled by G ′ µ and referred to as the coloron (or more generally gluon-prime), is a color-octet particle of spin 1.
We show that the coloron couples to a pair of G H scalars so that s-channel production of G ′ µ leads to a 4b signal. This is an example of nested resonances [10]: two pairs of b jets form each a resonance and the combination of the two pairs also forms a resonance.
These multiple resonant features allow an efficient rejection of the background. However, for that it is necessary to have four b-tagged jets, which requires large data sets. If only three b-tagged jets are required to pass the cuts, then the signal is larger and a detailed analysis could separate it from the background within smaller data sets.
The CDF Collaboration has searched for a resonance in the invariant mass distribution of the two leading b jets, produced in association with a third b jet. Preliminary results [11] suggest the existence of a resonance with mass of about 140 GeV, that could be attributed to a fluctuation of the standard model background at the 6% confidence level.
A more puzzling feature of these CDF results is that the measured transverse energy (E T ) distributions of the b jets have shapes that differ notably from the standard model predictions [12]. The shapes of the E T distributions of the two leading b jets may be due to a fluctuation of the standard model only at around 1% confidence level. We demonstrate that these shapes are nicely explained within the ReCoM when the G H mass is 140 GeV and the G ′ µ mass is slightly above the G H G H threshold. We describe the interactions and decays of the spin-0 octet in Section 2. Then we analyze the Tevatron phenomenology of QCD-produced G H pairs in Section 3. There we also discuss the boundstate effects due to gluon exchange between the two G H produced, and we compare the kinematic distributions of the multi-b-jet final state due to the scalar octet with those due to supersymmetric Higgs bosons. In Section 4 we derive the ReCoM predictions, compare them with the CDF data for several 3b kinematic distributions, and discuss implications for other final states, such as those arising from associated production of a coloron and a W boson. We summarize strategies to distinguish different models with multi b jets in Section 5.
2 Spin-0, weak-singlet, color-octet particle The theory considered in this section is the standard model plus only one particle, G H , which is a real field of spin 0, transforming as an octet under the QCD gauge group SU(3) c and as a singlet under the electroweak gauge group SU(2) W ×U(1) Y . These gauge charges imply that G H is electrically neutral, and does not have any renormalizable interactions with the quarks and leptons (note that G HbL b R is not invariant under SU(2) W ; the opposite is true in the case of weak-doublet scalar octets [13,14]). We refer to G H as a scalar octet, independently of whether it is a composite (as in technicolor) or elementary particle.

Renormalizable interactions
The renormalizable couplings of G H to gluons are fixed by SU(3) c gauge invariance: where f abc is the anti-symmetric SU(3) c tensor, g s is the QCD coupling, and G µ is the gluon field. The only other renormalizable couplings of G H are to the Higgs doublet (H) and to itself: where λ HG and λ G > 0 are dimensionless parameters, µ G is a parameter of mass dimension one, and d abc is the totally-symmetric SU(3) c tensor.
The first term in Eq. (2.2) contributes to the mass squared of G H after electroweak symmetry breaking. We take the sum (M 2 G H ) of this contribution and the mass squared from the Lagrangian to be positive. As a result, G H does not have a vacuum expectation value (VEV) provided there is an upper limit on the cubic coupling, We are primarily interested in the case where the physical mass of G H , M G H , is above around 100 GeV.
Curly lines represent gluons, while dashed lines represent scalar octets. Figure 2: Scalar octet decay to gluons, due to the trilinear G H interaction of Eq. (2.2). A diagram similar with the left one but with interchanged end points for the gluon lines is not shown.
The production of G H at hadron colliders occurs mainly in pairs, due to the couplings (2.1) to gluons, via the tree-level diagrams shown in Fig. 1. Single G H production is possible at one-loop through a cubic interaction [the last term in Eq. (2.2)], but it is suppressed enough to be neglected.

Decays of the scalar octet
The only renormalizable coupling of G H that violates the Z 2 invariance under the G H → −G H transformation is the cubic term in Eq. (2.2). Thus, the only decays of G H allowed by the renormalizable couplings shown above occur at one or more loops and involve at least one vertex proportional to µ G . At one-loop, divergent terms from the triangle and bubble diagrams shown in Fig. 2 cancel each other, and the finite result for the width of where α s is the QCD coupling evaluated at M G H . This width is accidentally suppressed by the small numerical coefficient given in the paranthesis (see [14] for a similar case).
Decays of G H into quarks induced by nonrenormalizable couplings may have large branching fractions because the above decay width into gluons is loop-suppressed. Dimension-5 operators of this type involve either a Higgs doublet, or a covariant derivative, There are also dimension-5 operators contributing to the G H → gg decay, and to the G H → gZ or gγ decays, Here, B µν is the field tensor of the U(1) Y gauge boson (− sin θ W Z µ + cos θ W A µ , where A µ is the photon field) and θ W is the weak mixing angle. These operators are generated by loops involving some new particle of mass m ψ whose coupling to G H determines the dimensionless coefficients ξ G , ξ ′ G , ξ B and ξ ′ B . The decay of a scalar octet into a gluon and a photon has been studied in Ref. [15].
Operators that allow G H to decay into a Higgs boson and gluons arise at dimension-7 or higher. For example, the operator (D µ G a H )G a µν H † D ν H induces the G H → g g h decay (the G H → g h process is forbidden by angular momentum conservation). The operators that couple G H to leptons also appear only at dimension-7 or higher, for example, Given that the operators (2.4) involve the Higgs doublet, their coefficients are typically proportional to the quark mass. It is natural to assume that the dominant decay GeV ∼ < M G H ∼ < 350 GeV. However, the relative normalization of the C d ij and C u ij coefficients depends on the underlying mechanism responsible for generating these operators.
To be concrete, we consider the following renormalizable model as an origin for the dimension-5 operators. A vectorlike quark, ψ, having the same gauge charges as b R may mix with the down-type quarks via the following terms in the Lagrangian: where y ij and λ i are Yukawa couplings, µ iψ is a mass mixing parameter, and m ψ is the ψ fermion mass in the small µ iψ limit. The scalar octet also has Yukawa couplings to ψ: The above interactions lead to effective couplings of G H to standard model quarks, as shown in the first diagram of Fig. 3. Upon integrating ψ out, the operators (2.4) are induced with coefficients where we assumed for simplicity that m ψ ≫ µ iψ . The coefficients C d ij are defined in the gauge eigenstate basis. Assuming that the two unitary matrices that diagonalize the down-type quark mass matrix are close to the identity matrix, the G H couplings to down-type quarks in the mass eigenstate basis are approximately equal to C d ij . For |λ 3 | ≫ |λ 1 |, |λ 2 | and |η 3 | ≫ |η 1 |, |η 2 | the width for G H → bb is much larger than for any other fermion final states, and is given by where v = 246 GeV is the electroweak scale.
The ψ fermion also induces some of the dimension-5 operators given in Eqs. (2.6) and (2.7), as shown in Fig. 3. The ensuing decay width into gluons, which adds to Γ 0 given where the function A arises from a momentum integral given in [16]: , the function has the value A ≈ 4/3, and the coefficients of the operators (2.6) are 14) The ratio of the decay widths into bb and gg, neglecting the interference of the two diagrams contributing to G H → gg in Figures 2 and 3, is given by (2.15) Note that m ψ may be significantly above the TeV scale, so that the vectorlike quark might not be produced even at the LHC. Thus, the branching fraction of G H → bb can be very small if µ G m ψ ≫ M G H . Alternatively, if ψ is not much heavier than G H , and η 0 = O(1), then the branching fraction of G H → bb can be large only for |λ 3 η 3 | ∼ > 10 −3 .
The scalar octet has a very narrow width (several orders of magnitude less than its mass), as its contributions arise from loops or higher dimensional operators. Its decays are usually prompt, unless µ G is very small and m ψ is very large. For example, µ G = 0, η 0 = 0 and m ψ /|λ 3 η 3 | ∼ > 2.5 × 10 5 TeV give a decay length cτ ∼ > 100 µm so that G H decays off the beam line. We will not investigate further these signatures involving dijet and bb resonances originating from displaced vertices.
The decays G H → gγ, gZ have small branching fractions due to coupling and color suppressions [the coefficients of the operators (2.7) are comparable with those in Eq. (2.14): Nevertheless, a photon-plus-jet resonance that arises together with a dijet resonance of equal mass, is an interesting signature that follows from the decays of a G H pair.

CP violation in B s mixing from scalar octet exchange
The operators (2.4) can induce tree-level flavor-changing neutral-current (FCNC) processes mediated by the scalar octet. In particular, integrating out the octet gives the following ∆B = 2 terms in the effective Lagrangian: where the quark fields shown here are mass eigenstates, and η ′ i and λ ′ i are the Yukawa couplings of Eqs. (2.8) and (2.9) transformed to the mass eigenstate basis.
Tree-level exchange of octet bosons has been proposed before [17] as a possible explanation for the anomalous dimuon charge asymmetry reported by the D0 Collaboration [18].
The novel feature in our case is that the scalar octet has dimension-5 couplings to the standard model quarks. Nevertheless, we now show that the effect of G H is large enough if the G H mass is below a few hundred GeV.
Using the relation T a ij T a kl = −δ j i δ l k /6 + δ l i δ k j /2, and substituting the matrix elements of the ensuing operators from Ref. [19], we find the matrix element of the Hamiltonian associated with operator (2.17): The decay constant, computed on the lattice with 2+1 flavors [20], is f Bs = 231±15 MeV.

Tevatron phenomenology of scalar octets
Scalar octets are produced in pairs at hadron colliders (see Fig. 1) and decay with a large branching fraction into bb. The partonic cross sections are given by [7,13].
and p T (G 2 ) are the transverse momenta of the two G H produced in each event. This cross section depends only on M G H , and is consistent with the result shown in Fig. 3 of Ref. [6].

3b versus 4b signals
The QCD background is dramatically reduced by selecting events containg four jets, all b tagged, and then imposing that two of them have an invariant mass close to that of the other two b jets [6]. We point out, however, that the sensitivity to the pair of octets could be improved if, instead of requiring four b jets, only the three jets of largest transverse momenta are b tagged and required to pass the basic cuts. In this case only two of the b jets will form a resonance, and therefore the background is substantially higher than in the case of a pair of resonances of same mass. However, the signal is also larger, such that the ratio S/ √ B where S and B are the number of signal and background events, respectively, may be increased.
To see this, first note that the b tagging efficiency is about 50%, leading to a factor of 2 suppression of the 4b signal compared to the 3b signal. The 4b signal is further suppressed because the 4th jet is rather soft, especially for low M G H , and does not always pass the basic cuts, such as p T > 20 GeV, imposed by the CDF [11] and D0 [27] searches.
To estimate the ensuing suppression of the 4b signal, we use MadGraph/MadEvents to generate parton-level events at tree level for pp → G H G H → bbbb at a center-of-mass energy of 1.96 TeV. Imposing the basic cuts used by CDF [11], namely p T > 20 GeV and |η| < 2 for each b jet, we obtain the result shown in columns 2 and 3 of Table 1.
Initial and final state radiation further soften the 4th jet. We estimate this effect by processing the parton-level events with Pythia [24] for showering and hadronization, and PGS [26] (with the default CDF detector card) for detector simulation. The result is shown in columns 4 and 5 of Table 1. The b-tagging efficiency also decreases for softer jets, so that the efficiency for four b jets to pass the basic cuts is significantly smaller than that for three b jets, especially for low M G H (see the last column of Table 1). In Fig. 4 we show the signal cross sections for 3 and 4 b-tagged jets after basic kinematic cuts.

Properties of the 3b signal
Given that the 3b signal is larger than the 4b signal by a factor ranging between 9 and 5 when M G H varies between 100 and 350 GeV (see last two columns of Table 1), it is useful to focus on the 3b final state and analyze its special features, which may be used for reducing the background. The most clear feature of this signal is that two of the three b jets form a narrow resonance of mass close to M G H . Let us label the three b jets by b i , i = 1, 2, 3. Within each event we take b 1 , b 2 , b 3 to be the jets of largest, 2nd-largest and 3rd-largest transverse momenta (p T ), respectively. In the limit where M G H is very large, close to √ s/2, the two octets are produced mostly at rest and the p T of a jet is fixed by the angle between the two b jets from the decay of an octet such that b 1 and b 2 form a mass peak.
For masses of interest at the Tevatron, between 100 and 400 GeV, the p T 's of the octets are comparable with the p T 's of the jets in the rest frame of the corresponding octet. As a result, the mass peak is formed in some events by b 1 and b 2 while in other events by b 1 and b 3 , or even by the b 2 and b 3 jets. We refer to these three types of events as To estimate the ratios between the numbers of events falling into these three categories, we impose an invariant mass cut    So far there have been no collider searches for the pair-produced octet. However, the CDF and D0 Collaborations have searched for a 3b signal predicted in the MSSM at large tan β: a b quark from the proton (in the 5-flavor PDF scheme) radiates off a heavy Higgs boson which then decays into a bb pair. This is the same final state as the 3b one due to octets discussed here, but we will show that the kinematic distributions are different.
The CDF search [11], with 2.2 fb −1 of data, shows an intriguing excess of events with m 12 in the 125-155 GeV range. The probablity for this excess to arise from a fluctuation of the standard model background is 0.9%, and increases to 5.7% when the whole range of invariant masses is taken into account.
The D0 search [27], with 5.2 fb −1 of data, rules out the presence of MSSM Higgs bosons with couplings large enough to account for the CDF excess. The tension between the CDF and D0 results may be due to a statistical fluctuation. An alternative explanation, however, is that the D0 search is less sensitive to the octet-induced signal; this is a consequence of the optimization of the signal within the D0 search for the MSSM Higgs bosons through the use of a likelihood discriminant. Some of the kinematic variables included in the likelihood discriminant (e.g., the angular separation ∆φ between the two b jets that are most likely to originate from the decay of the new particle) may discriminate against events due to G H G H production. which is narrower than the m 12 one. The m 13 peak is at a lower invariant mass than the m 12 peak, because radiation effects make the 3rd jet softer. The transverse energy distributions of the two leading jets (see the right panel of Fig. 6) also have clear peaks. However, those peaks appear to be too broad to provide a convincing explanation for the excess of the leading two jet transverse energy distributions at CDF [12]. We return to this issue in Section 4.2.
To study the octet discovery limit at the Tevatron, we impose high p T cuts on the three leading b jets, of 150, 120, and 100 GeV, respectively. For M G H = 300 GeV, the signal is simulated to have an acceptance around 3%. Further imposing the invariant mass cut |m(b 1 , b 2 ) − M G H | < 0.2M G H , there are approximate 6 signal events for 20 fb −1 with approximately zero background. Therefore, using only the three leading b jets, we estimate that the Tevatron can discover this octet particle up to a mass of ∼ 300 GeV.

Effects of boundstates
Since the octet is much heavier than the QCD confinement scale, the QCD interaction approximately generates an attractive Coulomb-like potential between two octets. We anticipate various boundstates named octonium from two octets. From group theory, there are only three attractive channels: 1, 8 A (antisymmetric) and 8 S (symmetric) from 8 ⊗ 8, because they have positive differences of the quadratic Casimir between the initial and the final states. Defining C f = [C 2 (8) + C 2 (8) − C 2 (f )]/2, we have C 1 = 3 and The potential from the one-gluon exchange is where α s is evaluated at the energy scale of the Bohr radius of the boundstate. In this paper, we will evaluate the QCD coupling running only at one loop. The gauge coupling in terms of the Bohr radius is given by .
We solve the Schrödinger equation numerically for the potential (3.3) using the above running gauge coupling. The Bohr radius is r 0 ≈ 0.07 GeV −1 , the binding energy is E b (8) ≈ 2.0 GeV and the wavefunction squared at the origin is |ψ 8 (0)| 2 ≈ (9.2 GeV) 3 .
To compare with the pure Coulomb potential, we fix the gauge coupling to be α s (1/r 0 ) and find 3 . So, the effects of these two potentials differ by about 10%. In following, we will use the more precise running potential. We also report the binding energy and wavefunction squared for the singlet channel as E b (1) ≈ 6.5 GeV and |ψ 1 (0)| 2 ≈ (16.9 GeV) 3 .
In our calculation of the boundstate wavefunction and the binding energy, we have neglected the width of the constituents. If the octet width is much larger than the binding energy [as is the case for m ψ /|λ 3 η 3 | ∼ 100 GeV in Eq. (2.11)], the boundstate's effects on the octet production can be neglected.
For a narrower octet, with Γ(G H ) ≪ E b , we neglect the width of the constituents. The S-wave boundstates of two scalar fields have the quantum numbers, J P C , of 0 ++ for the singlet, 0 +− for 8 A and 0 ++ for 8 S . Their couplings to the quarks should be suppressed by the quark mass because of the chirality, so the dominant production cross section should be from gluons at colliders. Since the production cross section is related to the decaying widths via the Breit-Wigner formula, we first calculate the decaying widths of those three boundstates to gg and 2b + 2b. The falling apart width is Γ 4b ≈ 2 Γ G . The decay widths of the boundstate B f to two gluons are calculated using the non-relativistic relation at the low-velocity limit, Γ(B f → X) = vσ(G H G H → X)|ψ(0)| 2 , as [28] Γ(1 → gg) = 9 π α 2 In the narrow width approximation, we have the following partonic production cross section of those boundstateŝ Using the gluon PDF, we obtain the production cross section at the Tevatron with √ s = For M G H = 150 GeV, using the Mathematica MSTW 2008 PDFs [29], the resonance production cross sections are σ(pp → 1) ≈ 190 fb and σ(pp → 8 S ) ≈ 57 fb, which is a few percent of the octet pair production cross section and can be neglected in the collider searches.

Supersymmetric Higgs bosons
Let us now compare the kinematic distributions of the multi-b-jet final states arising from scalar octet production with those due to the Higgs bosons of the MSSM. In Two-Higgsdoublet models, such as the Higgs sector of the MSSM, there are two color-singlet and electrically-neutral spin-0 particles (the heavy Higgs bosons H 0 and A 0 ) that may have large couplings to the b quark. In the 5-flavor PDF scheme, where the proton includes a b-quark, the emission of a heavy Higgs boson from a b quark line leads to a 3b final state (see diagram next to Table 2). Equivalently, in the 4-flavor PDF scheme, where there is no b inside the proton, production of a bb pair followed by emission of a heavy Higgs boson gives four b jets that may appear as a 3b final state when one of the jets does not  [30]. In order to study the multi-b Higgs signal, we impose the same cuts as in the case of scalar octets: p T > 20 GeV and |η| < 2 for each b jet. Compared to the octet case, fewer events pass these cuts because one of the jets does not arise from the decay of a heavy particle. For example, the fourth b jet has ∼3 times smaller chance to pass the basic cuts than in the octet case (see rows 2 and 3 of Table 2). The invariant mass of the leading and the third jets is also less likely to be within the mass window |m ij − M G H | < 0.2M G H than in the octet case (see last two rows of Table 2). Thus, the m 13 distribution is a good discriminant between the octet and the MSSM Higgs bosons, especially when the resonance is heavy. We show the invariant mass distributions for the b 1 , b 2 and b 1 , b 3 jet pair in Fig. 7 for heavy Higgs bosons of mass 140 GeV and a leading-order production cross section (summed over A 0 and H 0 ) corresponding to tan β = 40. Comparing Figs. 7 and 6, we can see that the MSSM Higgs bosons lead to an m 13 distribution with a peak that is both lower and located at a smaller value than the peak due to the scalar octet.
If a large excess of events over the standard model background will be established in the 3b final state, then an MSSM interpretation will typically require an excess of events in the bτ + τ − final state due to the decays of H 0 and A 0 into tau leptons. By contrast, G H can not decay into leptons-only final states because it is a color octet, so that no bτ + τ − excess is predicted in the octet models.  Figure 7: Same as Fig. 6, but for MSSM Higgs production at tan β = 40 (the leading-order production cross section is around 5.4 pb).

Renormalizable Coloron Model
Let us now analyze a minimal renormalizable model that includes a spin-1 color-octet particle [8,9], called "coloron". We will show that this model includes a scalar octet, identical with the G H studied in sections 2 and 3 except that here it can be resonantly

Interactions and masses
The most general renormalizable potential of Σ is where without loss of generality we take µ > 0. Note that the second term above is We assume m 2 Σ > 0 so that Σ acquires a VEV: where I 3 is the unit 3×3 matrix. The potential is bounded from below provided 3λ+κ > 0.
Expanding Σ around this vacuum, we find that its 18 degrees of freedom (Σ is a 3 × 3 complex matrix) are grouped into four real scalar fields: two octets (G a H and G a G , a = 1, ..., 8) and two singlets (φ R and φ I ) under SU(3) c , For µ → 0, there is a U(1) 1 × U(1) 2 global symmetry broken by Σ down to the diagonal U(1) subgroup, and the associated Nambu-Goldstone boson is φ I , which becomes massless. For any µ, the squared mass of φ I is given by For κ → 0 and µ → 0 the potential (4.1) has a global SO(18) symmetry, which is spontaneously broken by the Σ VEV down to SO (17), so that G H and φ I are massless Nambu-Goldstone bosons. Hence, the quartic term in V (Σ) proportional to λ does not contribute to the G H mass. For any κ and µ, the squared mass of G H is The mass of φ R also follows from Eq. (4.1): Eq. (4. 3) implies f Σ ≥ √ 6 µ/(κ + 3λ), so that M 2 φ R ≥ 0. The kinetic term of Σ, normalized such that all its component fields shown in Eq. (4.4) have canonical kinetic terms, is given by The kinetic term leads to a mass-square matrix for the two gauge fields, G µ a 1 and G µ a 2 , proportional to f 2 Σ . Upon diagonalization, one linear combination becomes the massless QCD gluon, while the orthogonal linear combination is a massive spin-1 octet, the coloron: The mixing angle depends only on the SU(3) 1 × SU(3) 2 gauge couplings, h 1 and h 2 : The QCD gauge coupling is given by g s = h 1 cos θ = h 2 sin θ, and the mass of the coloron is If all standard model quarks belong to the fundamental representation of SU(3) 1 , then G ′ couples to them as g s tan θ qγ µ T a G ′ a µ q . (4.17) The width of the coloron decay into quark pairs is where we have not summed over quark flavors. The kinematic suppression in the last The coloron effects on top-quark physics are rather sensitive to new quarks that mix with the top [32] and to the top charges under QCD corrections to the above decay widths may be large, perhaps of the order of 50%, but computing them is beyond the scope of this paper. Assuming that the QCD corrections to G ′ µ → G H G H and G ′ µ → qq decays are of the same size, the parameters given in Eq. (4.13) lead to branching fractions of 61% and 39% for G H G H and jj, respectively.

ReCoM and the CDF excess in the 3b final state
The total width of G ′ µ is rather small, typically less than one percent of its mass for tan θ ∼ < 0.3 and for a G ′ µ mass not more than a few percent above the G H G H and G H φ I thresholds. We can then use the narrow width approximation to estimate the cross section for producing a G ′ µ in the s-channel: Convoluting this partonic cross section with the MSTW [29] PDFs, and then multiplying by the branching fractions derived from Eqs. (4.16) and (4.18) we find the total cross sections shown in Fig. 8. The QCD corrections to these processes are also likely to be sizable, and need to be computed in the future.
The process pp → G ′ µ → jj is constrained by the CDF search for dijet resonances [33]: for M G ′ ≈ 290 GeV, the limit on the cross section times acceptance is about 100 pb.
Given that the G ′ µ production is proportional to tan 2 θ, Fig. 8 indicates that the CDF limit implies tan θ ∼ < 0.2 for a G H mass of 140 GeV.
The decays of G H into bb or gg proceed through higher-dimensional operators, similar to the case discussed in Section 2. For example, a vectorlike down-type quark ψ, which transforms as a triplet under SU(3) 2 , couples to Σ and a b quark: (4.20) Integrating out ψ we obtain at tree level the first operator of Eq. (2.5), which mediates the G H → bb decay, suppressed by two powers of the η 3 coupling. A 1-loop diagram as in Fig. 3 leads to G H → gg. This decay also occurs at one loop independently of ψ, as shown in Fig. 2, due to the det Σ term from Eq. (4.1).
Compared to the QCD pair production of two G H scalars, the kinematic distributions of b jets arising from G ′ → G H G H → 4b (see diagram in Fig. 9) are changed dramatically. Especially when the mass of G ′ is close to twice of the G H mass, those two G H 's are mainly produced at rest and the jet E T distributions are more peaked, as can be seen by comparing the right-hand panels of Figs. 6 and 10. As shown in Fig. 8, the resonant production of a G H pair is an order of magnitude larger than the QCD G H G H cross section (of about 4.6 pb for M G H = 140 GeV, as can be seen in Fig. 4). Hence, a smaller branching fraction (around 20%) of G H → bb gives a number of 4b events comparable to that due to the supersymmetric Higgs bosons for a mass of 140 GeV and tan β = 40.
The CDF 3b search [11] shows an excess in the m 12 distribution, as mentioned in Section 3.2, and in addition the shapes of the E T distributions of the b jets are shifted  values with similar splitting, such as 305 GeV and 150 GeV, work equally well). Fig. 8 shows that the production cross section is insensitive to tan θ for 0.1 < tan θ < 0.2; for illustration, we fix tan θ = 0.12. Taking into account the unknown overall normalization of the background, we fit the m 12 (first 10 bins) and E T (first 9 bins for each of the The fitted G H → bb branching fraction implies that the G H → gg branching fraction is large, around 78%. Thus, ReCoM predicts that the number of bbjj events with equal bb and jj invariant masses is about 7 times larger than the number of 4b events satisfying the similar equal-mass condition. Another prediction is that the invariant mass of the four jets has a peak close to the coloron mass of 290 GeV.

Dijet resonances plus a W boson and other signals
Another test of our ReCoM is associated production of G ′ µ with weak gauge bosons. Feynman diagrams for producing G ′ µ plus W + are shown in Fig. 12. We find that the partonic cross section for this process is where The leading-order cross section for the pp → G ′ µ W process at the Tevatron, computed using MadGraph, is shown in Fig. 13 for tan θ = 0.12, and is around 150 fb for M G ′ = 290 GeV.
Since the coloron can cascade decay into four jets via two G H scalars, or directly decay into two jets, it is useful to analyze the invariant mass distribution of the two leading jets in the final states that include a W decaying to ℓν, with ℓ = e, µ. Following the CDF search in this channel [34], we impose the p T (jets)> 15 GeV, p T (ℓ) > 20 GeV, / E T > 25 GeV cuts on the signal events, using the MadGraph/MadEvents to Pythia to PGS chain as in Section 3. For the signal events from G ′ µ → G H G H → 4j, the acceptance for passing those cuts is around 63% for inclusive 2 jets or 3 jets in the final state. For the signal events from G ′ µ → 2j, the acceptances are around 59% for inclusive 2-jet events and 28% for inclusive 3-jet events.
In Fig. 14, we show the invariant mass distributions of a pair of jets in the multijet-plus-ℓν events resulting from a simulation of the ReCoM signal. As expected, the Figure 12: Representative diagrams for W boson production in association with a coloron (G ′ µ ) decaying to a pair of scalar octets (G H ), each giving rise to a dijet resonance. invariant mass of the two leading jets (m 12 ) has two peaks corresponding roughly to the masses of G H and G ′ µ , chosen to be 140 GeV and 290 GeV, respectively. The height of the first peak is relatively insensitive to tan θ, is fairly constant for a range of tan θ values, which is similar to the case of single G ′ µ production shown by the solid (red) line in Fig. 8. The height of the second peak is strongly sensitive to tan θ; especially for small values of tan θ, the cross section times the B(G ′ µ → 2j) branching fraction is proportional to tan 4 θ. The invariant mass of the leading and third jets (m 13 ) has a single peak (see the dashed line in Fig. 14), near the G H mass, because the direct G ′ µ → 2j decay does not contribute at leading order to this distribution. The peak in m 13 is a distinctive feature of ReCoM, allowing to differentiate it from the low-scale technicolor model [35] that also predicts final states involving a dijet resonance and a W boson. Even more dramatic would be the observation of two different dijet resonances of equal mass in association with a W , as shown in Fig. 12, but in that case it is likely that a computation of next-to-leading order effects is necessary, in order to include the case where an extra jet is radiated with a p T larger than that of the third or fourth jet originating from G H decays.
G ′ µ production in association with a Z boson or photon may also be interesting. For example, Z-plus-jets events at the LHC, with the leading jets forming a resonance and the Z decaying into charged leptons, could allow a sufficient separation of the ReCoM  Figure 14: Invariant mass distributions for the two leading jets (m 12 , solid line) and for the first and third jet (m 13 , dashed line), in events arising from G ′ µ + W production at the Tevatron. All coloron decay channels and only the W → eν, µν decays are included. The first peak is mostly due to the G ′ µ → G H G H → 4j decay (its location is sensitive only to the G H mass), while the second peak in m 12 is due to the direct G ′ µ → 2j decay. The branching fractions for those two channels are 61% and 39%, respectively, for M G H = 140 GeV, M G ′ = 290 GeV and tan θ = 0.12. A slightly larger G ′ µ mass increases the first peak, and a larger tan θ increases the second peak.
signal from the large standard model background.
Throughout this section we have assumed for simplicity that φ I is too heavy to be produced in coloron decays. However, for a range of ReCoM parameters [see Eqs. (4.5) and (4.6)] φ I is lighter than G H so that G ′ µ → G H φ I is the dominant decay mode of the coloron. In that case, a larger coloron production cross section (i.e., a larger tan θ) is allowed by the dijet searches because the branching fraction for G ′ µ → qq is suppressed. If φ I decays predominantly into light jets, than the G ′ µ → G H φ I decay contributes to the 4j and W + 4j signals (increasing the height of the first peak in Fig. 14) without necessarily affecting the multi-b signal.

Conclusions
Four-jet final states at the Tevatron and the LHC are predicted in various theories for physics beyond the standard model. Even though the QCD background is very large, the presence of invariant mass peaks allows the separation of the signal from the background [7,6]. When the new particles decaying into jets have spin 0, it is natural to expect that a significant fraction of the jets comes from b quarks because the spin-0 couplings to standard model fermions are typically proportional to mass. Thus, b-tagging can further decrease the background.
In this paper we have investigated the properties of weak-singlet, color-octet scalars ("scalar octets"). Electroweak gauge invariance prevents these scalars from coupling at renormalizable level to standard model fermions, but dimension-5 couplings to quarks are allowed and lead to decays mostly into bb and gluon pairs. The exchange of a scalar octet of mass below a few hundred GeV could lead to large CP violation in B s mixing, as indicated by the D0 dimuon asymmetry (see section 2.3), if a vectorlike quark that induces the dimension-5 couplings has mass below the TeV scale.
The scalar octets are pair produced through their coupling to gluons, which is fixed by QCD gauge invariance, or via an s-channel resonance due to a spin-1 color-octet particle. We have shown that the simplest gauge invariant origin of such a resonance, namely the renormalizable version ("ReCoM") of the coloron model based on the SU(3) 1 × SU(3) 2 gauge extension of QCD [8,9], automatically includes a large coupling of a coloron to a pair of scalar octets. The parameters in this model that affect the 4b signal are the ratio tan θ of gauge couplings, which controls both the production and branching fractions of the coloron, the masses of the coloron and scalar octet, and the branching fraction of the scalar octet into bb. We have focused on the case where only three b jets pass some basic cuts, so that a large part of the signal is preserved, at the expense of not being able to reduce the background as efficiently as when both bb are present.
Both D0 and CDF have searched for 3b final states present in the MSSM at large tan β. The D0 search involves a likelihood discriminant which is optimized for the MSSM topology, and therefore cannot be applied to our 3b signal. The CDF preliminary result in the 3b final state, using the invariant mass distribution m 12 of the two leading jets, has some excess of events consistent with a new particle of mass in the 140 − 150 GeV range and decaying to bb. The background normalizations are taken as free parameters, and are fitted so that the deviation in m 12 is minimized. However, the shapes of transverse energy distributions of the leading and second jets do not fit well the standard model background [12]. We have shown here that the ReCoM changes the shapes of these distributions such that they agree well with the data (see Fig. 11). The best fit of the ReCoM to the CDF preliminary results indicates that the branching fraction of the scalar octet into two gluons is approximately 3 times larger than the branching ratio into bb.
Perhaps this good fit is only an accident, and the background modeling performed by CDF can be modified such that the standard model fit to the data improves. Nevertheless, the success of the ReCoM in describing the CDF data is intriguing enough to warrant its experimental study. Specifically, the CDF Collaboration could fit the background plus the ReCoM signal to the 3b data for the m 12 , E T 1 and E T 2 distributions. The D0 Collaboration could use the scalar octet kinematics to define the likelihood discriminant and to check whether the ReCoM is consistent with their observables. Both collaborations could look for a 4b signal exhibiting a pair of resonances of equal mass. They could also search for the bbjj signal from two resonances as a further test of the ReCoM.
The cross section for coloron production is large enough so that even the process where a W boson is radiated from the initial state (see Fig. 12), which has smaller backgrounds, leads to a sufficient number of events to be tested at the Tevatron. The signatures include a W decaying leptonically and four jets, with the invariant mass distribution for two jets peaking at the same location as for the other two jets, near the scalar octet mass of 140 − 150 GeV. When the two leading jets come from the same scalar octet, the signature is a W boson plus a dijet resonance, which may explain an excess in the CDF data from an inclusive search in this channel [34]. ReCoM predicts the presence of a second mass peak of the two leading jets, due to the direct decay of the coloron into quarks (see Fig.   14).
The ATLAS and CMS experiments can search for scalar octets with masses between a few hundred GeV and a couple of TeV, impresively extending the Tevatron reach. In the case of a scalar octet with mass of about 140 − 150 GeV, relevant for the CDF excess, it is hard to overcome the QCD background at the LHC, so that the Tevatron experiments should attempt to discover or rule out its existence.