Higher-Dimensional QCD without the Strong CP Problem

QCD in a five-dimensional sliced bulk with chiral extra-quarks on the boundaries is generically free from the strong CP problem. Accidental axial symmetry is naturally present except for suppressed breaking interactions, which plays a role of the Peccei-Quinn symmetry to make the strong CP phase sufficiently small.


Introduction
The standard model of elementary particles has two apparent fine-tuning problems which are hard to be undertaken directly by additional (gauge) symmetries: the cosmological constant and the strong CP [1] problems. The presence of extra dimensions might serve as an alternative to symmetry in obtaining such fine-tuned parameters in a natural way.
In this paper, as a concrete example, we consider QCD in a five-dimensional sliced bulk with chiral extra-quarks on the boundaries and point out that it is generically free from the strong CP problem.
For definiteness, let us suppose that there is a pair of extra-quarks in addition to the standard-model quarks: a left-handed colored fermion ψ L and a right-handed one ψ R . We assume an extra-dimensional space which separates them from each other along the extra dimension. If the distance between them is sufficiently large, the theory possesses an axial approximately, whose breaking is suppressed at a fundamental scale. This accidental global symmetry, which is actually broken by a QCD anomaly, plays a role of the Peccei-Quinn symmetry [2], making the effective strong CP phase to be sufficiently small.
The point is that the presence of such an approximate symmetry is not an artificial requirement, but a natural result stemming from the higher-dimensional space, which might well be stable even against possible quantum gravitational corrections. Note that axion models in the four-dimensional spacetime suffer from a fine-tuning problem: operators that break the axial symmetry should be highly suppressed without any dynamical reasons. The present model provides a solution to this problem.

Bulk color gauge theory
Let us consider a four-dimensional Minkowski spacetime M 4 along with one-dimensional extra-space S 1 , whose coordinate y extends form −l to l (that is, two points at y = l and y = −l are identified). The SU(3) C gauge field is assumed to propagate on the whole spacetime M 4 × S 1 . The action of the five-dimensional gauge field is given by where Here, M * is supposed to be a cutoff scale in the higher-dimensional gauge theory and g(y) and h(y) are gauge and Chern-Simons coupling functions, respectively.
Kaluza-Klein reduction to the four-dimensional spacetime, however, yields a massless color-octet scalar which is undesirable in the low-energy spectrum. Hence we consider an where µ = 0, · · · , 3. Then, we have only a vector field at low energies without the scalar field after the Kaluza-Klein reduction. In order to define the theory on the orbifold consistently, the action eq.(2) on the S 1 should be invariant under the Z 2 transformation.
This invariance is achieved as long as g(y) = g(−y) and h(y) = −h(−y). We take the g(y) to be y-independent and the h(y) as where c is a constant to be determined in the next section. The gauge symmetries are left unbroken with the infinitesimal SU(3) C gauge transformation parameter restricted to satisfy ε(x, y) = ε(x, −y).

Boundary extra-quarks
There are two fixed points in the S 1 /Z 2 orbifold: y = 0 and y = l. Let us put chiral extra-quarks 1 on the fixed-point boundaries: 2 a left-handed extra-quark ψ L at y = 0 and a right-handed extra-quark ψ R at y = l. The action of the extra-quarks contains Under an infinitesimal gauge transformation, this fermionic sector provides a gauge anomaly due to its chirality, though the fermion content is vector-like from a four-dimensional perspective.
On the other hand, the bulk action yields under the gauge transformation. Gauge anomaly cancellation with the fermionic sector implies c = i 48π 2 . (8)

Anomalous quasi-symmetry
The gauge-invariant theory is given by the total action S = S A + S ψ . Then, there is an approximate axial U(1) symmetry given by eq.(1). Indeed, interactions between the chiral extra-quarks (in the four-dimensional effective theory) such as The extra-quarks should be decoupled from the low-energy spectrum to escape from detection. Thus, we introduce hypercolor gauge interactions in the bulk to confine the extra-quarks at high energies. Such new gauge interactions at the same time induce a chiral condensate ψ L ψ † R , break down the axial U(1) symmetry [1] and provide a corresponding Nambu-Goldstone (NG) boson called an axion [3]. Non-vanishing anomaly U(1)[SU(3) C ] 2 induces a potential of the axion field.
Let us adopt SU (3) Two color-singlets would remain massless if it were not for anomalies. In this case, one of them does play a role of the axion that makes the effective strong CP phase to be sufficiently small.
The accidental chiral symmetry discussed above is broken by effective operators of the chiral fermions. The operators involving both 'ψ L or χ L ' and 'ψ † R or χ † R ', such as eq.(9), are highly suppressed. 4 However, the operators involving either 'ψ L and χ L ' or 'ψ † R and χ † R ' are expected to be suppressed only by powers of 1/M * . Thus, such operators induce an additional potential of the axion, but this correction is not too large to render the Peccei-Quinn mechanism ineffective: Axial-symmetry breaking on each boundary may have coupling coefficients of order one. Such operators of the lowest mass dimension 5 are given by Integration of heavy particles of masses of order F a induces an additional potential of the axion field a as where f (a/F a ) is a function of a/F a , whose minimum is generically different from that of the potential induced exclusively by the QCD effects. The resulting shift in the θ eff parameter of the QCD is expected to be of order The present experimental bound θ eff < 10 −9 is satisfied, for instance, with M * ≃ 10 18 GeV and F a ≤ 10 12 GeV, which is phenomenologically viable.
We note that the axial-symmetry breaking operators on each boundary can be made to have higher mass dimensions if we adopt a larger hypercolor gauge group instead of the SU(3) H . Then, the θ eff will be more suppressed. acknowledgment T.W. would like to thank the Japan Society for the Promotion of Science for financial support. This work was partially supported by "Priority Area: Supersymmetry and Unified Theory of Elementary Particles (# 707)" (T.Y.).