Phenomenology of Strongly Coupled Chiral Gauge Theories

A sector with QCD-like strong dynamics is common in models of non-standard physics. Such a model could be accessible in LHC searches if both confinement and big-quarks charged under the confining group are at the TeV scale. Big-quark masses at this scale can be explained if the new fermions are chiral under a new $U(1)^\prime$ gauge symmetry such that their bare masses are related to the $U(1)^\prime$-breaking and new confinement scales. Here we present a study of a minimal GUT-motivated and gauge anomaly-free model with implications for the LHC Run 2 searches. We find that the first signatures of such models could appear as two gauge boson resonances. The chiral nature of the model could be confirmed by observation of a $Z^\prime \gamma$ resonance, where the $Z^\prime$ naturally has a large leptonic branching ratio because of its kinetic mixing with the hypercharge gauge boson.

with Φ 24 ∼ O(M GUT ). In order for the resulting fermion masses to be close to but below the confinement scale, these two contributions must either have very small coefficients, or cancel out very finely to arrange a TeV-scale mass. While this is true in GUT models, it can also be the case for example in models of right-handed neutrinos. Technical naturalness protects the fermion masses from large loop corrections, but they can still receive large contributions to their masses when coupled to any singlet which acquires a large vacuum expectation value (VEV).
The SM QCD sector evades this puzzle and indeed has non-decoupled pions close to the scale of strong dynamics. The puzzle is resolved through a combination of factors: small Yukawa couplings, but also a chiral gauge symmetry structure. It is therefore plausible that the new strong dynamics sector operates in a similar way. Since the fermions cannot be chiral under the GUT gauge group and be consistent with constraints on a chiral fourth generation [2][3][4][5][6], they can only be chiral under a new gauge group. For simplicity, we study the case where this new gauge group is a U (1) ′ Abelian group.
If the fermions are charged under this additional gauge group, then the terms of Eq. (1) are forbidden without additional insertions of a U (1) ′ -breaking VEV. Given a U (1) ′ -breaking sector near the scale Λ, the lightest composite particles will also be near Λ and will have large couplings to the SM. While there is still a coincidence of scales, it is far less acute. The fermion masses will no longer be near the GUT scale, but rather the scale at which the hierarchy problem is ultimately resolved.
In this paper, we consider a model with an SU (N b ) (big-color) confining gauge group and a U (1) ′ Abelian group. We also take the fermions in the big-color sector (big-quarks) to transform as fundamentals of an SU (5) GUT symmetry. 1 Their charge under the Abelian group is chiral, such that the big-quark masses will be forbidden without additional structure. To give the big-quarks mass, we introduce a scalar ϕ charged under U (1) ′ that couples to the big-quarks. It develops a VEV through its coupling to the confining sector, lifting the big-pion masses to be close to, but below, the mass of ϕ. In this way, we construct a chiral model that not only has an experimentally-accessible spectrum, but also that alleviates the tension caused by the coincidence of scales.
The addition of the Abelian gauge group with chiral structure has important and interesting phenomenological consequences. In particular, there is an additional massive Z ′ gauge boson coupling to the big-quarks and mixing with the SM γ and Z. This opens new decay modes for the big-pions, as well as the possibility of direct production of the Z ′ . The chiral structure also leads to additional accidentally approximate discrete and continuous symmetries that lead to three-body decays and potentially long-lived particles.
The remainder of the paper is structured as follows. In Section 2, we present the detailed structure of the chiral composite model. Within that section, we study the chiral symmetry breaking pattern in subsection 2.1, the big-pion spectrum in subsection 2.2, the properties of the Z ′ boson in subsection 2.3, and the properties of the big-pions in subsection 2.4. We conclude in Section 3. In Appendix A, we discuss the perturbativity of models with the elementary real scalar field as a light resonance. We study U (1) ′ gauge coupling running in Appendix B and the Yukawa coupling running in Appendix C.
Kinetic mixing and decay properties of the Z ′ are given in Appendix D.

The Chiral Composite Pseudoscalar Model
As described above, we study a model with a confining gauge group SU (N b ) with a confinement scale Λ b at O(TeV). We introduce big-quarks charged under both the SM gauge group and SU (N b ). Due to the electroweak constraints mentioned above, the big-quarks cannot be chiral under the SM gauge group, so we introduce an Abelian U (1) ′ gauge symmetry with chiral charges. In order for the U (1) ′ group to be gauge anomaly-free, we must have two sets of big-quarks. The chiral structure of the charges prevents large quark mass contributions from UV scales such as the GUT scale. As in the SM, we introduce a scalar field ϕ charged under U (1) ′ that can develop a VEV to spontaneously break U (1) ′ and give a mass to the corresponding gauge boson Z ′ . We show the content of our model as well as the gauge symmetries in Table 1. The new fermions, ψ 1,2 , transform as fundamentals or anti-fundamentals Table 1: Field content of a chiral model with a confining QCD-like gauge group SU (N b ) and U (1) ′ .
Here, q 1 = q 2 . The SM model fermions are neutral under U (1) ′ and are not listed here. The U (1) ′ charge assignment shown here to achieve anomaly cancellation is not unique; one may also assign charges of q 2 and q 1 for ψ 2,L and ψ 2,R , respectively.
under the SU (5) GUT gauge group. In terms of the SM gauge interactions, [SU (3) c , SU (2) W ] U (1) Y , we separate them into the QCD color-triplet ψ T 1 = (3, 1) −1/3 and weak-doublet ψ D 1 = (1, 2) 1/2 , and similarly for ψ 2 . To have both the SU (N b ) and SU (3) c gauge couplings asymptotically free in the UV, we require 2 ≤ N b ≤ 5. The case with N b = 2 differs from the other allowed values because it has enhanced global symmetry due to the fact that the fundamental representation of SU (2) is pseudo-real. 2 It also requires the choice of q 2 (q 1 ) for ψ 2,L (ψ 2,R ) to forbid bare mass terms of the form ψ T 1,L C ψ 2,L with C as the charge-conjugation operator. The remainder of this paper will focus on the cases with 3 ≤ N b ≤ 5. 3 The U (1) ′ charge for the scalar field ϕ is chosen such that it can have renormalizable interactions with the big-quarks in this model. Because of chirality under the additional U (1) ′ gauge symmetry, there are no bare big-quark masses. On the other hand, the complex scalar field ϕ can have Yukawa 2 In the case where N b = 2, the global symmetry breaking pattern would be SU (20) × U (1)ϕ → Sp(20) and we would expect (399 -210) + 1 = 190 PNGB's. The decomposition is 190 = 4 × 24 + 3 × (10 + 10) + 15 + 15 + 1 + 1 + 1 + 1. Additionally, the N b = 2 case has a perturbative infrared fixed point and is likely to have an approximate conformal symmetry in the IR [7]. 3 There are some debates about whether N b = 3 with N f = 10 is inside the conformal window or not [8][9][10][11][12]. In the later part of our paper, we assume confinement for N b = 3 and N f = 10.
couplings to some of new fermions. The allowed renormalizable Yukawa interactions are For simplicity, we choose identical, real Yukawa couplings such that y T 1 = y T 2 = y T and y D 1 = y D 2 = y D . The most general renormalizable potential for ϕ is with H the Higgs doublet in the SM. The Yukawa coupling of λ ϕh can potentially modify the SM Higgs boson properties by introducing additional decay channels. In light of the good agreement of the Higgs boson properties with the SM, there could be a stringent constraint on λ ϕh . Absorbing the electroweak VEV correction, we define a new mass for the ϕ field as m 2 If m 2 ϕ < 0, similar to the Higgs field in the SM, the scalar field can develop a non-zero VEV independent of the strong dynamics sector. For λ ϕh = 0, the VEV is ϕ = (−m 2 ϕ /λ ϕ ) 1/2 / √ 2, which could be far above the TeV scale. For this case, we need to have small Yukawa couplings, y T,D , to have small big-pion masses. The situation is very similar to the light flavors in the SM QCD sector, except that for the new Z ′ gauge boson to be within the low energy spectrum below around 1 TeV its gauge coupling should be small. However, if m 2 ϕ > 0 the situation is even more interesting. Because the big-quark condensate can generate a tadpole term for ϕ through the Yukawa interactions, ϕ can still develop a VEV, which is triggered by the SU (N b ) confinement scale Λ b . As a result, the Z ′ gauge boson mass should be related to and likely below Λ b . Some big-pions could decay into this Z ′ , which could be a smoking gun to test our model. In our paper, we will mainly concentrate on the case with m 2 ϕ > 0.

Symmetry Breaking and Counting PNGB's
At the confinement scale of Λ b = O(TeV), the gauge coupling of SU (N b ) becomes large such that the bi-fermion operator develops a nonzero VEV. For a weak U (1) ′ gauge interaction 4 , following the vacuum alignment argument in Ref. [13], we anticipate the fermion condensate to spontaneously break the U (1) ′ gauge symmetry but preserve the SU (5) GUT symmetry. Defining Q L = (ψ 1,L , ψ 2,L ) and similarly for the right-handed fermions, one has which spontaneously breaks the U (1) ′ gauge symmetry. After Q L Q R develops a VEV, the existence of a tadpole potential term for ϕ also generates a nonzero VEV defined as ϕ ≡ v ϕ / √ 2, which also breaks U (1) ′ . Specifically, we can write the bare fermion mass matrix Q L M Q (ϕ)Q R as Using a non-linear parametrization for the big-pions, we have with the big-pion generators normalized such that Tr[T A T B ] = 1 2 δ AB . Thus, the full potential for the scalar field ϕ and the big-pions is In the limit of f Π ≪ m ϕ , the VEV for ϕ induced by the fermion condensation is After fields develop their VEV's, the spontaneous global symmetry breaking pattern is where we have ignored the U (1) A symmetry that is broken by the SU (N b ) instanton effects. Altogether we anticipate a total of 100 pseudo Nambu-Goldstone bosons (PNGBs), the big-pions.
In addition to these continuous global symmetries, our model contains two approximate discrete symmetries. Before turning on SM and U (1) ′ gauge interactions, one can identify the following two transformations: Here, C denotes charge conjugation and the SM gauge fields are denoted by A A µ . All SM fermions are invariant under both discrete transformations. The first symmetry, P m , is just a simple matter parity and is a good symmetry when y T,D 1 = y T,D 2 . Under it, the SM gauge fields transform as by charge conjugation. The SM fermion electroweak gauge interactions explicitly break P m . The second discrete symmetry, G d , is a new G-parity for the new strong dynamics sector [14][15][16][17]. Because ψ T,D 1,L(R) has a different absolute U (1) ′ charge from ψ T,D 2,R(L) , the U (1) ′ gauge interaction explicitly breaks this discrete symmetry.
The big-pions charged under the SM and U (1) ′ gauge groups become massive after gauge quantum corrections. We present a calculation of their masses later. Of the remaining four gauge singlets, (1, 1) 0,0 , three of them are odd under both P m and G d , while the remaining one is even under both parities. We label the parity-even singlet as This parity-even singlet will be the lightest state in the spectrum that couples through triangle anomalies to the SM gauge bosons. The other three parity-odd states are , One linear combination of Π 1 C and φ I will be eaten by the Z ′ and become its longitudinal component.
For the remaining three gauge singlet PNGB's, the Yukawa couplings in Eq. (2) explicitly break global U (1)'s associated with the PNGB's, making all gauge singlet PNGB's massive.

The Big-pion Spectrum
We first calculate the gauge singlet PNGB spectrum. We expand Eq. (7) and find that the (1, 1) ++ 0,0 singlet Π 1 A has no mass mixing with other states; its mass is given by with R y ≡ y D /y T . For the three (1, 1) −− 0,0 singlets, we note that the linear combination of Π 1 C and φ I eaten by Z ′ is given by its mixing term with Z ′ via The Z ′ gauge boson mass is related to the combined VEV from fermion condensation and the VEV of ϕ, and is Defining the orthogonal combination to the one eaten by the Z ′ as φ ′ I = cos θ Z ′ φ I − sin θ Z ′ Π 1 C , we find the square of the mass mixing matrix in the basis of ( The rotation matrix from the flavor basis to the mass-eigenstate basis, defined as Π 1 β andφ ′ I , is calculated to be The corresponding state masses are For the gauge charged big-pions, we can use the electromagnetic correction to π ± in the SM to estimate the radiative corrections from gauge interactions to the big-pion masses. Specifically, we with ∆m 2 π = m 2 π ± − m 2 π 0 and f π = 93 MeV. This formula may not work for non-QCD-like strong dynamics. Additional uncertainties on our spectrum calculation would apply in this case. Here, C 2 (r i ) is the quadratic Casimir of the representation r i under the SM i'th and Z ′ gauge groups. To calculate the bare big-quark mass contribution to big-pion masses, we use the Dashen formula Big-pions  The results of various gauge charged big-pion masses are shown in Table 2.
Searches for diphoton resonances put the strongest constraints on the model parameters [18,19].
To discuss the features of this spectrum, we choose a benchmark mass of m Π 1A = 1.5 TeV for Π 1 A .
The ATLAS and CMS collaborations place constraints on the diphoton production cross section for a 1.5 TeV scalar at ∼ 0.2 fb, so we target a production cross section of σ = 0.1 fb, which fixes the value of the big-pion decay constant at f Π = (1040, 1380, 1730) GeV for N b = (3,4,5). Then there is only one additional independent combination of Yukawa couplings which we take to be R y = y D /y T . As a further benchmark choice, we fix the values of R y for different N b by choosing an identical boundary condition with y D = y T at the GUT scale. As shown in Appendix C, the ratio R y is insensitive to the actual boundary values and ranges from 0.51 for N b = 3 to 0.36 for N b = 5. Fixing the preferred R y values and adding bare big-quark mass and gauge loop contributions to the big-pion masses together, we show the benchmark mass spectra in Fig. 1 for N

Interactions and Properties of Z ′
As shown in Eq. (17), the Z ′ mass is related to the chiral symmetry breaking scale of the new strong dynamics sector. For a weak gauge coupling of g ′ on the order of electromagnetic interaction strength, the Z ′ mass is anticipated to be ∼ 500 GeV for f Π fitting our benchmark parameters. There are no tree-level interactions of the Z ′ with SM particles. At loop-level, we have found that the kinetic mixing of the Z ′ with SM hypercharge gauge boson determines its interactions with the SM and decay properties. The relevant kinetic mixing term is defined as in the flavor basis. The mixing parameter sin χ is scale-dependent. Above the GUT scale, y D = y T and Tr(T Y T Z ′ ) = 0, so this coefficient is zero. Below the GUT scale, due to different coupling running of y D and y T (see Appendix C), a non-zero value of sin χ is generated from a vacuum polarization "bubble digram" withB andẐ ′ as external fields. At one-loop, the resulting mixing angle is [20] sin In Appendix D, we diagonalize both the kinetic and mass mixings of the Z ′ , γ and Z boson system. gauge boson. Unless its mass is close to the SM Z boson value, the leptonic branching ratio of this Z ′ is a factor of 4 − 5 larger than the SM Z boson. For the benchmark point of m Z ′ = 620 GeV, the leptonic branching ratio of the Z ′ is 25%.
In the right panel of Fig. 2, we show the bounds on the mass and kinetic mixing of a Z ′ boson from precision measurements of SM observables [21]. Low mass Z ′ bosons are most strongly constrained by the muon anomalous magnetic moment [22] and BaBar searches for dark gauge bosons in four-lepton final states [23], whereas high mass Z ′ bosons are constrained by e + e − collider measurements [24]. Not shown are additional narrow-Z ′ enhanced constraints from various e + e − experiments where m Z ′ ≃ √ s that do not affect our conclusion. We conclude that our model parameter region is unconstrained by low-energy experiments.

Interactions and Properties of Big-Pions
Starting with Π 1 A , we list the most relevant interactions for all big-pions in our model spectrum. Since the triangle anomaly mediated interactions become the leading interactions for the big-pions with real representations under the SM and U (1) ′ gauge groups, we first introduce a general formula for these types of interactions. Additional higher-dimensional operators are required to induce decays of the QCD triplet and sextet, as well as U (1) Y or U (1) ′ charged big-pions.
For big-pions charged under real representations of SM gauge group, the general form for the triangle anomaly mediated interaction is where the group structure constant is d ABC = 2 Tr T C {T A , T B } . Here, the normalization for the big-pion generators is canonical with Tr[T C T C ′ ] = 1 2 δ CC ′ . The SM gauge group generators are reducible and have 10 × 10 representations as T A = diag(t A , −t A * ), with Tr(T A T B ) = δ AB , in the space of the big-pions.

Interactions of Parity-even Π 1 A
Using the general triangle anomaly interaction formula Eq. (25), we have interactions for Π 1 A , which is even under both the P m and G discrete symmetries, shown in Table 3. Based on the interactions of Here, t W ≡ tan θ W and θ W is the Weinberg angle.
Π 1 A with gauge bosons, we calculate its various partial widths. For instance, one has There also exist additional interactions of Π 1 A with other big-pions. For instance, the following charge radius transition operator can mediate a sub-dominant decay of Π 1 A into (off-shell) Π 1 β and Z ′ .
Numerically, we show the various branching ratios and the total width of Π 1 A in Table 4 for a fixed gauge coupling of g ′ = 0.2 with a corresponding Z ′ mass of 620 GeV. We can see that the branching ratio into Z ′ γ is comparable to the diphoton one. Depending on the subsequent decays of the Z ′ , one could search for Z ′ γ resonances to confirm this model.
Experimental signatures of this model may first appear in the diphoton channel. Using its couplings to two gluons and two photons, we compute the production cross section for gg → Π 1 A → γγ, which in the narrow width approximation is given by  Here,ŝ means the center-of-mass energy of partons. Integrating this cross section with the MSTW2008 NNLO central parton distribution function set [25] and an NNLO K-factor of 2.5 [26] for a 1500 GeV big-pion at the 13 TeV LHC, we have the required f Π as  In our chiral composite model, Π 1 A decays to Z ′ γ with a similar branching fraction to γγ. For our benchmark point with g ′ = 0.2 and m Z ′ = 620 GeV, the production cross section times branching ratio of the final state of ℓ + ℓ − γ is at the 13 TeV LHC. While this channel is unlikely to provide the first hints of new physics from this model, observation of such a decay mode serves as a key indicator of new chiral dynamics in the big-color sector.

Interactions of Parity-odd Π 1 β
A single parity-odd big-pion, Π 1 β , cannot couple to two gauge bosons because of the discrete symmetries. From a box diagram at one loop, we expect it to couple to three gauge bosons. For instance, one can have the following dimension-9 operator and similar interactions with the W i and B gauge bosons. So, the leading decay channel of Π 1 β is Π 1 β → ggZ ′ . The dimension-7 operator like Π 1 β G a µρ G a ρ σ Z ′ µσ can be shown to be zero. There also exist operators containing two Π 1 β 's, for instance with c GG β coming from the strong dynamics and of order unity. This operator provides the dominant interaction for producing Π 1 β at the LHC. For c GG β = 1, f Π = 1040 GeV, and m Π 1 β = 1400 GeV, the production cross section of pp → Π 1 β Π 1 β is 0.005 fb at the 13 TeV LHC, which is unlikely to be observed at the LHC Run 2. After both Π 1 β 's decay, we have a very interesting signature with four jets plus one or two leptonic Z ′ .

Interactions of Parity-even Color-octet Π 8
The parity-even color-octet can also couple to two SM gauge bosons through triangle anomalies.
Because of the QCD gauge invariance, it couples to two gluons or one gluon plus one hypercharge boson. After electroweak symmetry breaking, the relevant interactions are After summing color factors, the partial widths of Π 8 are given by Numerically, we show the various decay branching ratios and the total width in Table 5.  At the LHC, the color-octet big-pion can be singly produced from two gluons. The parton-level production cross section is It is also interesting to compare the above formula to the color-singlet production in Eq. (28). The ratio of the two cross sections is which is independent of f Π and N d .
Integrating this parton-level cross section with the MSTW2008 NNLO central parton distribution function set and an NNLO K-factor of 3.0 [27] for the benchmark mass 3580 GeV (f Π ∼ 1380 GeV and N b = 4) color-octet at the 13 TeV LHC, we find σ(gg → Π 8 → gZ ′ ) ∼ 0.05 fb for g ′ = 0.2. For different masses, we show the color-octet single production cross section times branching ratio in Fig. 4 for gγ (left panel) and gZ ′ (right panel). As already studied in Ref. [28], the dijet and jet plus photon resonance searches have already imposed stringent constraints on the color-octet production cross sections. However, color-octet pions from the benchmark points considered here are unconstrained by such searches. Different from the pure vector-like models in Ref. [28], one can also search for the color-octet big-pion as a three-body resonance of j ℓ + ℓ − with a mass between 3 TeV to 4.3 TeV. The signal cross section times branching ratio is 0.013 fb for a 3580 GeV color-octet pion which, while unlikely to be observed in the near future, provides a relatively clean signal for high luminosity LHC. The color-octet big-pions can also be pair produced via their QCD interactions (see Refs. [29][30][31] for similar phenomenology studies). The leading discovery channel is a pair of dijet resonances, but the semi-weak decays of one octet big-pion into gγ and gZ ′ should be visible as well.

Interactions of Parity-odd Color-octet Π odd 8
The discrete-symmetry-odd color-octet big-pion Π odd 8 can decay into three gauge bosons via the following dimension-7 operator Furthermore, it can also decay into the lighter discrete-symmetry-odd singlet Π 1 β via the dimension-6 operators which should provide the leading decaying channels Π odd 8 → Π 1 β + 2g/gγ/gZ. Similarly, Π odd 8 can also which provides the subdominant decay channel Π odd At the 13 TeV LHC, Π odd 8 could be pair-produced via its QCD interactions. The tree-level production cross section is around 2.3 × 10 −4 (7.8 × 10 −6 ) fb for a 3.0(3.6) TeV Π odd 8 , which is too small to be observed at the LHC.

Interactions of Weak Triplets Π 3 and Π odd 3
The discrete-symmetry-even weak-triplet Π 3 contains both electric charged big-pions Π ± 3 and a neutral big-pion Π 0 3 . Through the triangle anomaly, they can couple to two gauge bosons via We show numerical values for the various branching ratios in Table 6 for the benchmark model point with a mass of 970 GeV. At the LHC, the weak-triplet big-pions can be singly produced from vectorboson fusion with two forward jets and small cross sections. They can also be produced in pairs from their weak interactions. While the weak triplets are unlikely to be observed at the LHC, a future 100 TeV collider could be capable of probing these states.  Table 6: The decay branching ratios and total width for Π 0 3 (left panel) and Π ± 3 (right panel) with a mass of 1960 GeV. For decays involving the massive Z ′ gauge boson, g ′ = 0.2 was used.
For the discrete-symmetry-odd weak-triplet Π odd 3 and similar to the color-octet case, the leading decaying operators are So, the main decay channels for the charged states are Π odd ± 3 → Π 1 β W ± Z/γ and Π 1 A W ± Z ′ .

Additional UV Interactions of Complex Big-Pions
Additional interactions in the UV physics are required to make the big-pions with complex representations under SM gauge groups decay [32]. In general, there are two classes of operators, depending on whether additional ϕ insertions are needed or not. If we write down such operators in a GUTpreserving form, then the gauge structure of the operator is fixed. The big-pions with complex SM representations can come from either the decomposition of 5 × 5 (and 5 × 5 which we neglect as it can be treated analogously by conjugation) or 5 × 5. To allow all complex big-pions to decay, we need to introduce operators in which the SM fields transform as 10, 15 and 24. The 10 can come from 5 × 5 or 5 × 10. Both cases lead to one big-pion that decays as a leptoquark and one big-pion that decays as a diquark. The 15 must come from a product of 5 × 5 SM fermions, such that the (6, 1) −2/3,q 1 +q 2 decays as down-type diquarks, while the (3, 2) 1/6,q 1 +q 2 behaves as a leptoquark. The QCD-neutral complex big-pions decay to leptons. The operator with SM fields in a 24 can come from a 5 × 5 or a 10 × 10. In either case, it decays as a diquark. For complex big-pions that also have U (1) ′ charge, additional insertions of the ϕ field are required. If we choose q 1 = 1 and q 2 = 0, then only one insertion is required and the UV operators inducing the decays are dimension-7. In order to have (3, 2) −5/6,0 decay, one also needs to add a dimension-6 operator without ϕ insertion. For example, one could introduce decays for all complex big-pions with the following operators, where i, j are indices of SU (5) and we suppress Lorentz and flavor indices. The new particles required to UV-complete the above two operators may change gauge coupling running if they are not SU (5) singlets.
These operators break the discrete symmetries of the theory and so induce decays for both even and odd big-pions. To have these big-pions decay before Big Bang Nucleosynthesis (∼ 1 s), we estimate that the cutoff scales should be less than O(10 7 GeV) and O(10 10 GeV), for Λ 1 and Λ 2 respectively. If the abundance of the complex big-pions is small, their lifetime could be longer and leads to weaker constraints on cutoffs [33]. Searches for stopped long-lived particles at the 8 TeV LHC place a bound on the color triplet mass of to be above around 470 GeV, for a wide range of decay times, 10 −6 s τ 10 4 s [34]. Similarly for the color sextet, the bound is that the mass should be above around 690 GeV. Furthermore, for τ above O(10 ns), the searches for long-lived charged particles have imposed a more stringent bound, which requires the color triplet complex scalar mass above around 900 GeV [35]. This constraint can be easily satisfied in our model, as can be seen in Fig. 1. Since the colored big-pions in our model are much heavier, we do not anticipate any constraints coming from these bounds. We show the summary table of all big-pion decays in Table 7.
Big-Pion Decay Modes Big-Pion Decay Modes

Discussion and Conclusions
In The model also predicts a rich spectrum of pseudo-scalars accompanying the potential digamma resonance. Among the unique features of this model, it predicts two very interesting decay channels involving the Z ′ gauge boson. One is the decay of Π 1 A to Z ′ γ and the other is the decay of the color-octet big-pion to gZ ′ . For our benchmark model point with g ′ = 0.2 and m Z ′ = 620 GeV, the Z ′ has roughly a 25% branching fraction to a di-lepton final state, giving signatures of ℓ + ℓ − γ or ℓ + ℓ − j. If a scalar resonance is confirmed in the future, these signals could be considered smoking gun signatures for chiral composite models.

Acknowledgments
We would like to thank Vernon Barger for discussion. This work is supported by the U. S. Department of Energy under the contract DE-FG-02-95ER40896.

A Light Elementary Scalar Field
The model presented in the main paper includes a chiral symmetry-breaking scalar whose mass is expected to be out of reach of near-future experiments. Here we present a brief discussion of models where the real component of the scalar ϕ field is light. The strong SU (N b ) is demoted to a global flavor symmetry, and m 2 ϕ < 0 such that the scalar field develops a non-zero VEV via the Higgs mechanism. For λ ϕh = 0, we have Expanding about the minimum, ϕ = (v ϕ + φ R + iφ I )/ √ 2, the imaginary component φ I is eaten by the Z ′ gauge boson and the remaining scalar degree of freedom has mass We set the real scalar φ R to be at the benchmark mass of 1.5 TeV, and require a cross section σ(pp → φ R → γγ) ≃ 0.1 fb. The gluon fusion production cross section is given bŷ and the branching fraction to two photons is approximately Br(φ R → γγ) ≈ 2α 2 /3α 2 s = 4 × 10 −3 . Using the MSTW2008 NLO PDFs [25] and including a K-factor of 2.5, we find the diphoton production cross section to be Requiring a phenomenologically-motivated cross section of 0.1 fb, we anticipate v ϕ ∼ 2 − 3 TeV for The bare fermion masses are given by m ψ 1,2 = y 1,2 v ϕ / √ 2. The strongest bounds come from the QCD triplet masses, which are constrained to be m ψ T 1 TeV [36] for  Furthermore, the tree level coupling φ R Z ′ Z ′ will dominate the decay processes unless it is kinematically forbidden, which provides the additional constraint m Z ′ > m φ R /2. The Z ′ mass is given by For N b = 4, this suggests values of the U (1) ′ gauge coupling of g ′ |q 1 − q 2 | 0.28 at m φ R .

B U (1) ′ Gauge Coupling Running
The one-loop beta function for the U (1) ′ gauge coupling running is for complex scalars s and Weyl fermions f . For the matter content given in Table 1, we have Solving this equation, we find To have the Landau pole occur at or below M GUT ≈ 3 × 10 16 GeV, we need to have (for q 1 = 1 and which is

C Doublet and Triplet Yukawa Coupling Running
The beta functions for the doublet and triplet Yukawa couplings are where N f = 2N b ; C F 3 = 4/3 is the quadratic Casimir of SU (3) c ; C F 2 = 3/2 is the quadratic Casimir of SU (2) W ; the sub-leading gauge interactions are neglected. We impose the GUT scale boundary condition that y D (Λ GUT ) = y T (Λ GUT ) = y 0 . However, the running of the ratio R y ≡ y D /y T is nearly independent of the value of y 0 . To see this, we re-write the differential equations for y T and y D in terms of R y and y T as follows where the new GUT scale boundary conditions are R y (Λ GUT ) = y D (Λ GUT )/y T (Λ GUT ) = 1 and y T (Λ GUT ) = y 0 . We see that the initial running for R y does not depend on the value of y 0 ; it is determined only by the GUT scale gauge couplings. Numerical integration of these equations is shown in Fig. 6. The numerical values for y D /y T at µ = 1.5 TeV are shown in Table 8. Additionally, to avoid a Landau pole at or before µ = Λ GUT ∼ 3 × 10 16 GeV, we require y T (µ = 1.5 TeV) < (0.54, 0.50, 0.49) (N b = 3, 4, 5) .  Table 8: Numerical values for the ratios of the doublet and triplet Yukawa couplings at µ = 1.5 TeV generated from a GUT scale boundary condition of y D (Λ GUT )/y T (Λ GUT ) = 1.

D.2 Decay of the Z ′ Gauge Boson to Standard Model Fermions
After diagonalization of the gauge boson kinetic and mass terms, we find that the Z ′ gauge boson has a milli-charge coupling to the SM neutral current, given as with Θ AZ ′ = −χ c W cos ξ , Θ ZZ ′ = χ s W cos ξ − sin ξ .
Inserting the Standard Model fermion current, the part of the Lagrangian relevant for Z ′ decay can be cast in a familiar form with the vector and axial milli-couplings defined as From this, we find that the Z ′ decay width to Standard Model fermions is D.3 Decay of the Z ′ Gauge Boson to Zh and W + W − If the Z ′ gauge boson is heavy enough, there are also decay channels for Z ′ → Z h and Z ′ → W + W − .
The decay to Zh comes from the hZZ vertex in the Standard Model, which has a coupling of g hZZ = m 2 Z /v EW , where v EW = 246 GeV is the electroweak VEV. After diagonalization of the gauge boson kinetic and mass terms, we have the following term in the Lagrangian at O(χ), which allows the decay Z ′ → Z h. The decay width is [38] Γ(Z ′ → Z h) = g 2 hZZ ′ m Z ′ 192π λ(1, r 2 Z , r 2 h ) λ(1, r 2 Z , r 2 h ) + 12 r 2 Z , where r Z = m Z /m Z ′ , r h = m h /m Z ′ , and λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2ac − 2bc. The Z ′ → W + W − decay is mediated by the W W A and W W Z vertices in the Standard Model, which have couplings of g W W A = e and g W W Z = e cot θ W . The Z ′ boson will therefore couple to W + W − with strength g W W Z ′ = e [cot θ W (s W χ cos ξ − sin ξ) − c W χ cos ξ] = −e cot θ W sin ξ , which leads to a decay width of [39] Γ where r W = m W /m Z ′ .