Diphotons, New Vacuum Angles, and Strong CP

The Standard Model contains a well-understood, natural, spin-0 diphoton resonance: the $\pi^0$. Numerous studies have pointed out that the hint of a new diphoton resonance at 750 GeV could be a pion analog, identified with the pseudo-Nambu-Goldstone boson of a chiral symmetry spontaneously broken by new strong dynamics at the TeV scale. These"hypercolor"models are generically expected to violate parity through a topological angle $\tilde\theta$. We discuss the physics of $\tilde\theta$ and its impact on the phenomenology of the new sector. We also describe some of the theoretical implications of a nonzero $\tilde\theta$. In particular, $\tilde\theta$ can generate an ${\cal O}(1)$ threshold correction to the QCD vacuum angle $\theta$ near the TeV scale, sharply constraining ultraviolet solutions to the strong CP problem. Alternatively, finding that $\tilde\theta$ is small may be interpreted as evidence in favor of UV solutions to strong CP, particularly those based on spontaneously broken P or CP symmetries.


Introduction
The ATLAS and CMS collaborations have both reported modest excesses in diphoton resonance searches near m γγ 750 GeV [1,2]. The appearance of the bump in both experiments in a regime where the background is expected to be featureless is certainly one of the most exciting hints of physics beyond the Standard Model (SM) to date.
An attractive candidate for the diphoton excess at 750 GeV is a neutral pion-like state of a new strongly coupled gauge theory, termed "hypercolor" in earlier work on vectorlike confinement (VC) [3,4] (see also related studies [5][6][7][8]). The neutral hyperpionπ 0 couples to the QED and QCD topological charge densities through a chiral anomaly, allowing resonant production and decay at the LHC via gg →π 0 → γγ . (1.1) Like the ordinary π 0 of QCD, due to its composite nature, no scalar mass parameters have to be fine-tuned in order for the hyperpion to remain light. A number of groups have studied VC models, new pion-like states, and other pseudo-Nambu-Goldstone boson (PNGB) interpretations of the diphoton excess [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Most studies invoking a new hypercolor sector have been performed in the simplifying limit that the model preserves parity, and in this case the candidate 750 GeV resonance is a pseudoscalar meson.
However, a priori, we expect that the new strong dynamics should violate parity through an O(1) hypercolor vacuum angle,θ. Here we will study theθ-dependence of physics in the hypercolor sector. (Insofar as models of new strong dynamics are interesting for LHC phenomenology apart from the diphoton excess, this question is also of independent interest, even if the excess is not confirmed by future data.) In addition to its implications for hypercolor phenomenology,θ has interesting consequences for the strong CP problem. The same flavor anomalies with QCD that give rise to the production channel (1.1) imply that the phases that generateθ feed directly into the QCD vacuum angle θ. Since there is no a priori reason forθ in particular to be small, the contribution to θ is generically ten orders of magnitude larger than the bound from electric dipole moment measurements [24].
These new contributions to θ indicate that either there is a new "hyper-CP problem," or that the strong CP problem must be solved by new physics further in the infrared, such as via the Peccei-Quinn mechanism and its associated axion [25][26][27][28]. Thus, θ is a discriminator between solutions to strong CP: the observation of a largeθ would disfavor ultraviolet solutions, while boundingθ to be small would lend support to models where both θ andθ are suppressed by the same UV mechanism.
In the caseθ ∼ 1, there is a direct analogy with the electroweak hierarchy problem. The knowledge of the existence of high energy scales like M p , the scale of neutrino masses, and others, through their quantum corrections to the electroweak scale, tells us that the hierarchy problem is real and must be solved by dynamics around or below those scales. Likewise, a detection of a large CP-violating phase in a new sector at the LHC may indicate that strong CP is not solved through dynamics at still higher scales, but instead takes place in the infrared.
This paper is organized as follows. In Sec. 2 we study the chiral Lagrangian and hyperpion phenomenology of a benchmark model in the presence ofθ, including the vacuum structure and existence of Dashen phases [29], the spectrum, and the couplings relevant for collider physics. While parity-preserving couplings are the dominant source of the diphoton signal, parity-violating couplings can lead to large decay rates of heavier hyperpions into pairs of lighter hyperpions in some regions of parameter space, providing an interesting observable signature of nonzeroθ. In Sec. 3 we discuss the impact ofθ on θ in ordinary QCD and the manner in whichθ can be viewed as a discriminator between UV and IR solutions to the strong CP problem. In Sec. 4 we summarize and conclude.
We note that Ref. [23], which appeared as this paper was being finished, has some overlap with our study.

Generalities
The ingredients of VC models [3,4] are similar to those of QCD: a new asymptotically free gauge group, "hypercolor," which we take here to be SU (Nc), and new vectorlike fermions carrying charges under both hypercolor and the other SM gauge groups. The masses of some of the new fermions are assumed to be less than the strong scale of the SU (Nc), triggering chiral symmetry breaking and the appearance of light pseudo-Nambu-Goldstone "hyperpions," among other resonances. Because the SM gauge groups are (gauged) subgroups of the approximate flavor symmetries of the hypercolor sector, typically some hyperpions are charged under SM gauge groups, while others are neutral. Some of the neutral states may decay through the anomaly to pairs of SM gauge bosons, analogously to the decay π 0 → γγ in ordinary QCD. A neutral state near the bottom of the spectrum with anomaly-induced couplings to QCD (allowing production through gluon fusion) and QED (allowing decay to diphotons) can provide a candidate for the putative resonance at 750 GeV.
In general the hypercolor sector may possess an arbitrary vacuum angleθ analogous to the QCD vacuum angle θ. In the presence of vectorlike fermions, the microscopic Lagrangian contains the terms where G and H are the color and hypercolor field strengths. 1 Here q and ψ (q andψ) are left-handed Weyl fermions taken to be in the fundamental (anti-fundamental) representations of color and hypercolor, respectively. In terms of these parameters, the invariant CP-violating parameters are θ andθ are defined as These expressions are easily generalized in models where some of the vectorlike fermions carry both color and hypercolor, as we will use in this paper. θ explicitly breaks parity in the hypercolor sector and has several interesting consequences for the phenomenology of the light hyperpions, including modifying the spectrum and generating parity-violating triple-hyperpion couplings. Also, whereas in the parity-conserving limit, anomaly diagrams lead to pseudo-Goldstone couplings to SM gauge bosons in the form Tr(πGG), in the presence ofθ, there are additional couplings of the form Tr(πGG).
In the next subsection we discuss these features concretely in a specific VC benchmark model.

Benchmark Model
Perhaps the simplest model that can accommodate the diphoton excess was studied in [14]. The model contains two vectorlike fermions, one hypercolor-fundamental QCD-singlet, and the other bifundamental under QCD and hypercolor. Both carry ordinary hypercharge, and neither are charged under SU (2) L . The angleθ was set to zero in [14]. Let us briefly recall the sense in which this model is "simplest." The easiest way to resonantly produce a neutral pion-like field is through gluon fusion, and an anomaly-driven coupling to GG can be generated if some of the new fermions are colored. Likewise the decay to diphotons may proceed through an anomaly with QED. A model with just one color-triplet fermion produces a color octet hyperpion and a heavy hyper-η (η ), but no neutral hyperpion. (Theη is not a good candidate for the diphoton excess because in that caseΛ is also of order 750 GeV, and the color octets are too light.) We could add a second color-triplet fermion, in which case the lightest degree of freedom is a neutral hyperpion. However, in this case the QCD anomaly is carried entirely by theη , and the light state has to mix with it in order to be resonantly produced through gluon fusion. This mixing is possible, but incalculable in chiral perturbation theory for small Nc, and furthermore pushes some of the quark masses to rather large values, since the mixing is suppressed in the chiral expansion [22]. In the model of [14], with one singlet and one triplet vectorlike fermion, there is a light neutral hyperpioñ π 0 and it naturally possesses an unsuppressed anomaly coupling to GG.
This simple model would be sufficient to exhibit the physics ofθ we wish to discuss, including aθ-dependent mass for the light state, parity-violating couplings and decays, and an O(1) contribution to θ. However, the type of parity-violating decays we will consider proceeds in this model throughη →π 0π0 , so the relevant coupling is incalculable in chiral perturbation theory. Phenomenologically this is not a problem, but for analytical purposes it is more convenient to discuss a benchmark model with one additional hypercolor-fundamental QCDsinglet flavor. This model contains another light neutral hyperpion,η, which has calculable parity-violating couplings that permit decays toπ 0π0 .
The elementary fields of our benchmark model and their charges are summarized in Table 1. The approximate flavor group of the model is SU (5) V × SU (5) A . The axial symmetries are spontaneously broken by chiral condensates ψ ψ ∼ 4πf 3 π , and we parametrize the resulting hyperpion Goldstone fieldsΠ as where the T generate SU (5) A .
Ordinary color corresponds to gauging the SU (5) generators under which the 24 hyperpion fields decompose into one color octet, living in block-diagonal elements ofΠ; two complex color triplets, living in off-diagonal components ofΠ; and four color singlets, two living in diagonal and two living in off-diagonal elements ofΠ. 2 For our purposes, we can restrict our attention entirely to the two diagonal singlet hyperpions. These we refer to asΠ A andΠ B , corresponding to the axial SU (5) generators (2.5) The axial transformations generated by T B are anomalous under QCD. Also, the axial transformations generated by are anomalous with both QCD and hypercolor. The leadingθ-dependent terms in the chiral lagrangian are generated by the hyperpion mass terms, µ is a scale parameter expected to be of orderΛ ∼ 4πfπ/ √ Nc. Without using the axial transformations anomalous under color, the mass matrix in our benchmark model may be 2 Gauging subgroups of the vector flavor symmetry has two other important effects. First, the gauging explicitly breaks some of the spontaneously broken axial symmetries, leading to 1-loop masses for the charged hyperpions. Colored hyperpions thus obtain masses that are typically an order-1 factor below the cutoff. Second, the gauging breaks most of the ungauged elements of the vector flavor group, since a general element mixes gauged with ungauged generators. However, some generators may accidentally commute with the gauged elements. In the benchmark model, the two ungauged Cartan elements of SU (5)V commute with the SU (3)c generators, yielding an accidental U (1) 2 "species symmetry" that is preserved at the renormalizable level [3,4]. The off-diagonal hyperpions transform under this species symmetry, and the lightest in each species is stable unless higher dimension operators are added that explicitly break the symmetries (alternatively, if the lightest state is neutral, it may provide a DM candidate [22]). The hypercharges in the benchmark model here are chosen to allow the triplets to decay through dimension-6 operators of the sort discussed in [14]. For further discussion of the complications and phenomenology associated with species symmetry, see [3,4,14]. Since thẽ θ-dependent physics we will study can be illustrated with neutral diagonal hyperpions, we will not need to consider species symmetry, its breaking, or the charged hyperpion states further.

brought into the form
where M 1,2,3 are real. Each sub-block corresponds to fields that form a representation under a gauged subgroup of the diagonal flavor symmetry. Because the axial transformation generated by T B − 2/3Tη is anomalous with hypercolor, but not color, we may assume that the hypercolor vacuum angle has already been moved to reside entirely in arg det M 0 without shifting the QCD θ term. Thusθ Subsequently, using T B and T A transformations, we may take the mass matrix into the form which we will use to obtain the hyperpion Lagrangian terms in Eq. (2.7). Since the T B transformation that brings the matrix (2.8) into the form (2.10) is anomalous with QCD, it shifts θ by an amount Another way to say it is we have a new contribution to arg det M q , where M q is the colored fermion mass matrix, coming from ψ 3 . Eq. (2.11) is one of two contributions to ∆θ from the hypercolor sector. We discuss the second and their implications further in Sec. 3.

Neutral Sector Phenomenology for
As mentioned above, we can use the two neutral diagonal hyperpionsΠ A andΠ B to illustrate various effects ofθ. In this section we will study the physics of the neutral hyperpions in simplifying limits amenable to analytic treatment, in particular the "QCD-like" limit In the next section we perform precise numerical analysis on a broader range of parameter space, but the analysis here in the limit (2.12) will help us understand qualitative features. The potential forΠ A andΠ B arising from (2.7) with mass matrix (2.10) is given by Forθ = 0, this potential is minimized for nonzeroΠ A ,Π B . To analyze the vacuum structure, we may eliminateΠ A with its equation of motion, after which the equation forΠ B reduces to with the upper sign (-) corresponding to the solutions with the lowest energy. The vacuum structure reflected by Eq. (2.15) is a highly nontrivial function of the input parameters. In certain regimes of the hyperquark masses, (2.15) has multiple solutions, analogous to the Dashen phenomenon of QCD at θ = π [29] and Witten's generalization to other values of θ [30]. In QCD, the global minimum of V (π 0 , η) is a non-analytic function of θ when multiple vacua exist, with the energies of different vacua crossing at θ = π [30]. Structure similar to this "multi-branched" behavior may also arise in our hypercolor theory, and is easiest to exhibit atθ = π. Here Eq. (2.15) always has at least one solution, Factoring out this root, we may rearrange theΠ B equation to read where we have defined α ≡ 2Π B / √ 15fπ. In the simplifying regime (2.12), in order for Eq. (2.17) to exhibit new solutions, we evidently require that the splitting between M 1 and M 2 is small compared to M 1,2 , in which case new solutions may appear at small α. In the regime a new pair ofΠ B /fπ vacua exists when Condition (2.19) is completely analogous to the Dashen phase condition m u m d > m s |m u −m d | in QCD at θ = π, and when the nonzero solutions exist, they are (degenerate) global minima in which parity is spontaneously broken. For other values of θ andθ, the levels are split and there is one global minimum. The global and local minima exchange roles asθ crosses π. This behavior is illustrated in Fig. 1, where we plot the energy of the global minimum as a function ofθ for values of the M i both satisfying and violating Eq. (2.19). In the former case, the energy is cuspy atθ = π + 2πk, corresponding to the crossing of branches; in the latter case, the energy is smooth. For generalθ = π, the phenomenology in which we are interested is qualitatively the same regardless of whether Eq. (2.15) has exactly one or multiple solutions. Therefore, for simplicity we will now focus on the limit (2.12) without strong degeneracy between M 1 and M 2 . In this regime there is only one solution, and it is convenient to reanalyze the potential, starting by integrating out theΠ B . To first order in M 1,2 , the solution forΠ B is reflecting the fact that mixing between theΠ B andΠ A states is controlled by in the limit (2.12). Eq. (2.20) generates an effective potential for the light field, Plugging back into Eq. (2.20), the corresponding vev for theΠ B field is . (2.23) In general, the statesΠ A andΠ B undergo mass mixing. We will refer to the lighter mass eigenstate asπ 0 and the heavier asη. Unlike QCD, the states may be heavily mixed. However, in the limit analyzed in this section, the spectrum is insensitive to mixing at first order. The masses are given by While forθ = 0 theπ 0 mass grows with M 1 and M 2 , forθ of order π, the mass is controlled by the difference |M 1 − M 2 |.
In the above approximations, the diagonal octet mass where a 0.3 parametrizes the effects of loop corrections from QCD. m 2 8 is sensitive toθ only at subleading order in M 1 /M 3 , and then only through the expectation value forΠ B .
A particularly interesting feature of nonzeroθ is the appearance of a large number of parity-violating cubic couplings in the hyperpion potential. Again is it sufficient study the potential (2.13) in the limit M 1,2 M 3 . To zeroth order, the cubic couplings are: V cubic allows the parity-violating decayη →π 0π0 when kinematically allowed. At the LHC, the most important couplings for the neutral hyperpions are to the QCD and QED anomalies, which allow production through gluon fusion and decay to diphotons even in the absence of parity violation. Before mixing, only theΠ B field couples to the QCD GG and QED FF , 3 These couplings leading to the resonant process pp →η → γγ at the LHC, offering a discovery mode forη when the diphoton branching ratio is unsuppressed. Mixing induced by (2.20) also generatesπ 0 GG andπ 0 FF couplings whenη is integrated out, There are also similar couplings to ZZ and Zγ.
These couplings can be large if either M 1 or M 2 is not substantially smaller than M 3 , and lead to the resonant process (1.1).
For nonzeroθ, we expect the effective theory should also contain other parity violating couplings allowed by the symmetries. For example, for smallθ, we expect a coupling of the form where G is the QCD field strength and M is a characteristic hyperquark mass. Unlike the anomaly-generated couplingπ 0 Tr(GG), we cannot compute c in Eq. (2.29). Relative to the anomaly coupling it is also chirally suppressed, so (2.28) still plays the dominant role inπ 0 resonant production.

Numerical Analysis and Diphoton Rates
In the previous section we discussed the impact ofθ and the quark mass parameters on the properties of the benchmark model in a special limit amenable to analytic treatment. Here we illustrate some of these features quantitatively and extend the analysis numerically to more general parameter regimes. (However, we postpone one phenomenological question -that of the parity-violating decaysη →π 0π0 -for Sec. 2.5.) Fixing µ =Λ (which can be taken as a definition of the overall scale of the hyperfermion masses) and choosingÑ = 3, there are five free parameters in the benchmark model, given by fπ, the three M i , andθ. We fix fπ by requiring that the lightest neutral state has mass 750 GeV for each value of the hyperfermion masses andθ, and we analyze the potential (2.13) in four different parameter scenarios.
1. Scenario 1: M 1 = 100 GeV, M 2 = 300 GeV. As M 3 becomes larger than M 1,2 , we approach the regime analyzed in the previous section, with a "mostly-Π A "π 0 state and a "mostly-Π B "η state. Much of the phenomenology in each of the four scenarios is governed by the cutoffΛ = 4πfπ/ Ñ , which in turn is fixed by the requirement mπ0 = 750 GeV. In Fig. 2 we plotΛ. In most cases the cutoff increases asθ approaches π, reflecting the fact that terms in mπ0 begin to cancel against each other when cos(θ) < 0. This behavior is evident in Eq. (2.24) in the  regime M 1 ∼ M 2 M 3 . In Scenario 2, the cancellation is particularly efficient for a range of masses around M 3 ∼ 1 TeV. In Fig. 3 we plot the mass of the heavier neutral Goldstone, which largely tracks the features ofΛ. Octet masses (not shown) exhibit similar behavior and are of the order 1-2 TeV in all scenarios. Fig. 4 shows the pp →π 0 → γγ cross section at 13 TeV. We compute the cross section at leading order and apply a K-factor of 1.6 [31] in each of the four scenarios. In most cases the rate decreases asθ → π due to the increase inΛ, which controls the dimension-5 coupling to GG. In Scenario 2 the rate becomes small and insensitive toθ at low M 3 , where M 2 is approaching M 1 and the mixing angle is suppressed. For fixedθ, the rate in Scenario 1 decreases with increasing M 3 again because of mixing angle suppression. In the other scenarios, the rate mostly increases with M 3 ; since two or three masses are becoming large together, mπ0 increases unless the cutoff is lowered, leading to larger anomaly-type couplings. We see that on each slice of parameter space, there are sizable regions consistent with the observed diphoton excess of order 1-10 fb, and in particular in each case there are viable regions for all values ofθ.

Parity violating Hyper-meson Decays
Parity-violating triple-meson couplings can give rise to new decay channels that are absent whenθ = 0. In the simplest model of one color triplet and one color singlet hyperfermion studied in [14], a natural candidate isη →π 0π0 (this channel was also noted recently in [23]). This mode is also present in our five-flavor benchmark model whenθ = 0. However, in both cases the coupling can only be studied in chiral perturbation theory at large Nc, and moreover theη may be too heavy to produce at the LHC ifΛ is large. Alternative channels in our benchmark model areη →π 0π0 , allowed if mη > 2mπ0, and η →π 0π0 * →π 0 gg, relevant when mη < 2mπ0. 4 Due to the large tree-level cubic coupling V cubic ⊃ Aηπ 0π0 (where A is a dimension-1 coefficient, given in Eq. (2.26) in the limit M 1,2 M 3 ), and the fact that the next 2-bodyη decay mode is the loop-suppressed decay into gluons through (2.27), theη →π 0π0 channel is expected to be dominant when it is kinematically accessible. The rate for this decay is The subsequent decays of theπ 0 lead to the final states (gg)(gg), (gg)(γγ), and (γγ)(γγ), where the parentheses indicate that the dijets or diphotons reconstruct theπ 0 mass of 750 GeV. The invariant mass of the two pairs peaks at theη mass. Compared to the paired dijets, the (gg)(γγ) final state avoids combinatoric backgrounds and offers increased resolution on theπ 0 mass using the diphotons, but has branching fraction suppressed by the electromagnetic coupling. For a detailed discussion of these issues in the context of Higgs boson decays, see [32].
If mη < 2mπ0, the parity-violating decay is 3-body and is heavily suppressed by the off-shellπ 0 . The rate is To get a sense of the magnitude of this suppression, note that as the mass splitting ∆m ≡ mη − mπ0 decreases, the 3-body rate falls off rapidly, where we have ignored terms of order Γ 2 π 0 /m 2 π 0 . Because the 3-body decay is generally negligible, it is most interesting to focus on cases where the 2-body decays to on-shellπ 0 's can proceed in regions of parameter space overlapping with aπ 0 → γγ cross section compatible with the observed excess. Comparison of Figs. 3 and 4 indicates this overlap is most likely to occur in scenarios 2 and 3, where the relationship M 2 M 3 leads to large mixing between the Π states.
It is worth a note of explanation why we have not taken the strict "simplifying" limit M 2 = M 3 in scenarios 2 and 3. In the exact M 2 = M 3 limit, for M 1 < M 3 , theη carries charge -1 under a discrete symmetry which is a hypercolor analog of G-parity in QCD (for further discussion and application of such symmetries, see [7,19]). Thus this "isospin-like" limit forbids couplings in the hyperpion potential with an odd number ofη particles, including the cubic couplingηπ 0π0 we wish to study. Therefore, we keep a modest M 3 − M 2 splitting in the benchmarks. In Fig. 5 we plot the 13 TeV cross sections for pp →η → γγgg in these scenarios, restricting the plots to points where σ(pp →π 0 → γγ) is in the range 1-10 fb. We find that in the parameter space consistent with the diphoton excess, there are sizable regions in which pp →η → γγgg may be observable at the LHC with O(100) fb −1 of integrated luminosity.

Otherθ-dependent Phenomenology
We conclude our discussion of the benchmark model with some brief comments on other probes ofθ.
Apart from parity-violating hyperpion decays, another test of nonzeroθ at colliders arises in the angular distribution of leptons in the decay modeπ 0 → ZZ when both Z's decay leptonically, as has been discussed in the case of the Higgs (see, e.g., [33]). In principle, such a measurement could determine the parity-violating coupling of theπ 0 to a pair of Z's. In the benchmark model we have described, this coupling comes from the termθπ 0 B µν B µν where B µν is the U (1) Y field strength, in analogy with Eq. (2.29). As mentioned above, this parity-violatingπ 0 coupling to the gauge boson kinetic term is chirally suppressed relative to the parity-conserving coupling to the topological charge density (corresponding to the fact thatθ is unphysical in the limit of a vanishing hyperquark mass). 5 Therefore, sensitivity must be high to disentangle the subleading contribution. Furthermore, this measurement is challenging because of the small branching ratio for leptonic Z decays.
Parity violation also permits the hyperpion states to mix with the Higgs boson, which in principle might be observable at colliders through newπ 0 decay channels. This mixing can be generated in the UV by dimension-5ψψ|H| 2 operators, or in the IR through gauge boson loops sensitive toθ. The former contributions are model-dependent and may be negligible if the scale suppressing the higher-dimension operators is large, while the latter appear at 3-loop order in our benchmark model and are likewise negligible. Therefore, at least in models of the type studied here, Higgs mixing is expected to be unobservable in practice.
A potential low-energy probe of parity violation comes from searches for the neutron electric dipole moment (EDM). As pointed out in the appendix of [23], the leading contribution comes from the generation of the three-gluon Weinberg operator [34] and could plausibly be tested by next generation searches. Direct contributions to quark (chromo)EDMs from diagrams involvingπ 0 exchange occur at two loops, but are effectively four-loop in magnitude since the couplings to gluons and photons are generated at one loop, and are thus well below current or near future experimental sensitivity.

Strong CP
In VC models, the hypercolor sector typically gives new O(1) contributions to θ. In this section we illustrate the shift in θ in the benchmark model and discuss the implications of this effect for solutions to the strong CP problem.

3.1θ and θ
We have already seen an obvious contribution to θ, Eq. (2.11), from the new set of quarks ψ 3 . There is also a contribution from theΠ B vev. From its coupling to the QCD topological charge density, Eq. (2.27), we see thatΠ B gives a threshold correction to θ, Together, the two contributions give a total shift in θ of in the limit studied in Sec. 2.3 and using Eq. (2.23). More generally, we expect ∆θ to receive the shift from φ 3 , as well as a dynamical shift of orderθ that reduces to the second term in Eq. (3.2) in the appropriate limit.
In the absence of other sources of chirality violation, the phase φ 3 can always be moved completely into the QCD θ angle. However even if φ 3 = 0, there is still an independent contribution to θ fromθ, and as we have discussed,θ has in principle a number of other observable effects. Theθ contribution is generic, although it arises in different ways in different models. For example, in the minimal model of [14], there is noη state, but theπ 0 couples directly to the QCD anomaly and carries a vev in the presence ofθ. In both cases, the effect is unsuppressed by couplings or loop factors.
Crudely speaking, a shift in θ near the TeV scale tells us that the strong CP problem "has yet to be solved" by dynamics at lower energies. We will make this assertion more precise below.

Solutions to Strong CP: UV vs. IR
Proposed solutions to the strong CP problem fall broadly into two categories. The first type of solution deals with infrared physics, and leaves infrared signatures of its presence. Two examples are the Peccei-Quinn (PQ) solution and its signature axion, and the m u = 0 solution, disfavored by lattice data. The second type of solution uses ultraviolet physics (compared to QCD), and from the infrared point of view, largely appears to be a miracle. Examples include the Nelson-Barr (NB) models of spontaneous CP violation [35][36][37][38], leftright models with spontaneous P violation [39][40][41][42][43][44][45][46][47][48], and models with a new massless colored fermion confined by a new gauge group [49].
Although axions are very weakly coupled, there exist a variety of experimental probes. Comparatively, UV solutions to strong CP are usually difficult to test. In general the relevant scales can be vastly higher than the TeV scale, leaving little trace at low energies, other than the value of θ. In specific cases, some new states might be accessible at colliders [43,49]. Alternatively, the study of phases in other new TeV-scale dynamics, such asθ in VC models, might be used to discriminate whether strong CP is solved by ultraviolet or infrared physics.
Solving strong CP in the UV is a delicate matter: it relies on the curious fact that the renormalization of θ within the SM alone is tiny. If microscopic physics with scale Λ U V can explain why θ = 0 is the right UV boundary condition for the EFT below Λ U V , then as long as the EFT is not too different from the SM, θ ≈ 0 will be preserved. On the other hand, if there is still substantial BSM physics below Λ U V , it can easily spoil the solution to strong CP through radiative contributions to θ. 6 The detection of new pion-like states coupling to gluons, and a large new vacuum angleθ, is a clear example: threshold corrections like Eq. (3.2) generically provide a large shift in θ. The threshold correction is innocuous ifθ itself is tiny, but then we must solve a second strong CP problem. From a model-building perspective, this is most natural if θ andθ are suppressed in the UV by the same mechanism.
Below we briefly review specific UV and IR solutions to the strong CP problem and how they are affected by the addition of a VC sector.

Nelson-Barr
In NB models [35][36][37][38], CP is taken to be a good underlying symmetry, so in the ultraviolet θ andθ both vanish by assumption. Since CP must be broken at low energies, a sector is added to spontaneously break it at some intermediate scale M CP through a complex vev for a field σ (in general, a set of fields). The particle content, interactions, and symmetries are arranged so that the σ vev is communicated to the CKM phase in an unsuppressed way, while θ is not generated, at least at tree level.
Without specifying the full structure of the NB sector, let us add a VC sector near the TeV scale and take M CP > TeV. In the absence of additional symmetries on the VC sector, θ is generated when CP is spontaneously broken, for example, by renormalizable couplings of the form In this case,θ feeds in to θ in an O(1) way near the TeV scale, reintroducing the strong CP problem.
On the other hand, it is not difficult to forbid couplings like (3.3) with discrete symmetries, for example a Z 2 under which σ, ψ, andψ i are all odd. Indeed, such symmetries are a necessary ingredient of NB models, even without VC sectors, in order to forbid other problematic renormalizable couplings involving σ. The symmetries may be extended to the hyperfermion couplings, and it is conceivable thatθ is sufficiently small at M CP to preserve the NB solution.
We will not attempt to build a complete model exhibiting both NB and VC sectors here, but simply note that the presence of the VC sector in NB models requiresθ to be as wellprotected as θ. A signature of this case is thatθ will not be observable. If, on the other hand, θ is observed, we may conclude that strong CP must solved in another, more infrared way.

Parity Models
A similar but distinct class of UV solutions to strong CP, based on parity, was first studied in [39][40][41][42][43]. In these models, a (generalized) parity symmetry is enforced in the ultraviolet theory. The simplest implementation expands the SM gauge group to SU (3) × SU (2) L × SU (2) R ×U (1) Y , and parity exchanges SU (2) L ↔ SU (2) R [43] (see also the recent study [52]). To symmetrize the fermion content, mirror fermions are added. For example, the ordinary left-handed electroweak doublet Q, transforming as (3, 2, 1, 1/6), is matched with a mirror lefthanded fieldQ transforming as (3, 1, 2, −1/6). Parity then exchanges Q ↔Q * , and requires θ = 0 in the UV. In the VC extensions of the SM like the benchmark model studied here, parity can act on the new vectorlike hyperfermions as ψ i ↔ψ * i . With this transformation, parity also requiresθ = 0 in the UV.
Like CP in NB models, parity must be spontaneously broken at low scales. Again there are typically couplings that reintroduceθ at tree level. For example, if parity is broken by a vev for a pseudoscalar a, then is parity-invariant for hermitian y and contributes toθ when parity is broken. We draw the same conclusion as in the case of NB: the presence of the VC sector in left-right models requiresθ to be as well-protected as θ, which is plausible, at least at treelevel, with the addition of symmetries to forbid couplings between the hyperfermions and the parity-breaking sector.θ will not be observable if such models are realized in nature. 7 If, on the other hand,θ is observed, we conclude as before that strong CP is not solved by a P symmetry of the UV theory.

The QCD Axion
We conclude this section by commenting briefly on the most plausible IR solution to strong CP, the Peccei-Quinn mechanism [25][26][27][28], and its interplay with a new hypercolor sector.
As we have emphasized, unlike the UV solutions to strong CP, the cancellation of θ by a vev for an axion coupling to GG is unspoiled by any threshold corrections to θ down to very low scales. The vacua of the QCD-induced axion potential are simply shifted to relax whatever value θ takes in the IR. This is not to say, however, that the Peccei-Quinn solution is automatic in any theory with an axion, a hypercolor sector, and a generic value ofθ. If PQ symmetry is anomalous under hypercolor, there is a new contribution to the axion potential that swamps the QCD contribution, resulting in the relaxation ofθ, but not θ. In the case of field theory axions, avoiding this contribution amounts to constraints on the field content such that the PQ anomaly with hypercolor vanishes.

New Massless Quarks
There is another category of solutions to the strong CP problem that bears mention and does not fall neatly into the UV/IR classification discussed above: the possibility that the sector responsible for the diphoton resonance is itself complicit in the solution to strong CP. In the case of a new strongly coupled hypercolor sector, an example of such a solution can arise when there is a new massless colored and hypercolored quark. A model of this type was studied prior to the diphoton excess in Ref. [49], which proposed that strong CP might be solved if there is a newÑ = 3 hypercolor sector, a color-and hypercolor-fundamental fermion with vanishing mass, and a Z 2 mirror symmetry that fixes θ =θ to high precision. Then, the same anomalous chiral rotation may be used to simultaneously eliminateθ and θ from the theory. The Z 2 is spontaneously broken at very high scales by a very large vev for the mirror Higgs field, so that the mirror partners of the SM fermions are all very heavy and the hypercolor group runs strong before QCD. From a top-down perspective such models appear to face fine-tuning challenges [51], but from a bottom-up point of view it is interesting to study their compatibility with the diphoton excess and their further predictions.
At low energies, the field content of our benchmark model and the parameter limits θ → θ, M 3 → 0 are almost sufficient to realize the structure required of this type of solution. In addition, we have to add the mirror partners of the color singlet hyperfermions, which become ordinary hypercolor-singlet vectorlike quarks in the fundamental of QCD, with masses set by M 1 and M 2 . These degrees are freedom are likely to be long-lived, since higher-dimension operators must be added to permit their decay.
In Fig. 6 we plot the diphoton cross section in the M 3 → 0 limit. We see that compatibility with the observed rate indicates an O(100) GeV mass for one of the two singlet hyperfermions. Consequently, the model predicts a new light colored vectorlike fermion in addition to the hyperpion sector. The phenomenology of this state is model-dependent due to the freedom in the extra structure that must be added to allow it to decay, but it is likely to be severely constrained.

Conclusions
New QCD-like sectors provide attractive and natural candidates for the diphoton excess observed by ATLAS and CMS. In the presence of light fermions charged under the new strong gauge group, a neutral composite pseudo-Goldstone stateπ 0 , analogous to the π 0 , may couple to QCD and QED through chiral anomalies. However, unlike ordinary QCD, the new sector may exhibit strong parity violation through a large vacuum angleθ. We have studied the impact ofθ on the physics of the new pseudo-Goldstone sector and the importance ofθ as a probe of the strong CP problem.
Varyingθ reveals a rich vacuum structure and has substantial impact on the pseudo-Goldstone spectrum. Furthermore, in all models of this type,θ controls parity violating decays of the formη →π 0π0 . However, because of the axial anomaly in the new sector, these decays are typically not calculable in chiral perturbation theory. We have instead considered a larger benchmark model with an additional pseudo-Goldstone stateη, analogous to the η of QCD, and studied the processη →π 0π0 in ChPT. We find that in the benchmark model, the ggγγ final state for this process can be probed at the LHC in sizable regions of parameter space consistent with the diphoton excess.
There are other potential experimental probes ofθ deserving of dedicated analysis, in particular whether angular distributions inπ 0 → ZZ → 4 offer sufficient sensitivity to disentangle the CP-conserving from the CP-violating contributions. We reserve this question for future work.
Chiral anomalies with QCD allow resonant production of the new pseudo-Goldstones at the LHC through gluon fusion, pp →π 0 ,η. The same couplings generate threshold corrections to θ of orderθ near the TeV scale. Thus,θ is an efficient discriminator of whether the strong CP problem is solved by ultraviolet or infrared physics. Ifθ is small, the most plausible explanation is that a UV symmetry like P or CP protects both θ andθ. If, on the other hand, θ is large, the threshold correction implies that θ must be eliminated by an IR mechanism like the axion.