Super- and CP-symmetric QCD in Higher Dimensions

An extremely precise global symmetry is necessary in the Peccei--Quinn solution to the strong CP problem. Such symmetry arises when colored chiral fermions are localized in an internal space. We present a supersymmetric model that incorporates the above mechanism. Extra colored chiral multiplets around the supersymmetry-breaking scale are a generic prediction of the supersymetric model.


Introduction
The Peccei-Quinn mechanism [1] is a promising solution to the strong CP problem, yet it requires an extremely precise global symmetry. The explicit breaking of the symmetry gives rise to an extra scalar potential of the axion. The energy scale relevant to the mechanism has to be 10 10 -10 12 GeV, while the extra potential has to be 10 −10 times smaller than the potential generated by the QCD dynamics. What is the origin of such a precise global symmetry?
Suppose that the SU(3) C gauge field propagates in extra space dimensions, and that some of SU(3) C -charged chiral fermions are localized in the internal space. Then, an approximate chiral symmetry naturally arises when the chiral fermions are separated sufficiently in the internal space. Such a symmetry might well be stable even against possible quantum gravitational corrections. Our previous papers [2] identified it with the Peccei-Quinn symmetry.
Supersymmetry is a leading candidate to be discovered through experiments in no distant future. As such, it is tempting to extend the higher-dimensional QCD, which naturally realizes the strong CP invariance in the nonsupersymmetric case mentioned above, to a supersymmetric setup. In this article, we present an explicit model, and show that colored chiral multiplets are around the supersymmetry-breaking scale in addition to the particles of the minimal supersymmetric standard model (MSSM).

Accidental Axial Symmetry
In this section, we provide a basic supersymmetric setup for the accidental axial (Peccei-Quinn) symmetry.

Bulk color gauge theory
Let us consider four-dimensional Minkowski spacetime M 4 along with one-dimensional extra space S 1 , whose coordinate y extends from −l to l (that is, two points at y = l and y = −l are identified). The SU(3) C vector field is assumed to propagate on the whole spacetime M 4 ×S 1 . Thus, the SU(3) C vector field belongs to a vector multiplet Ψ of the minimal supersymmetry (SUSY) in five dimensions. One vector multiplet Ψ consists of one vector multiplet V and one chiral multiplet Φ of N = 1 SUSY in four-dimensional spacetime.
The minimal SUSY in five-dimensional spacetime corresponds to N = 2 SUSY in four dimensions. Thus, description of gauge theories based on four-dimensional N = 2 SUSY is useful in considering SUSY theories in five dimensions. The action of a U(1) vector multiplet Ψ = (V, Φ) of N = 2 in four dimensions is given in terms of a holomorphic function F (Ψ): where K and W denote the Kähler potential and superpotential, respectively. In five-dimensional spacetime, the imaginary part of the scalar component in Φ is the 4-th polarization of the five-dimensional vector field. The superfields (V (x), Φ(x)) are promoted to fields in five dimensions (V (x, y), Φ(x, y)). The Kähler potential and superpotential are now integrated over d 4 xdyd 2 θd 2θ and d 4 xdyd 2 θ, respectively. The coordinate y is integrated along S 1 . Let us consider a prepotential that is an at-most-cubic polynomial The quadratic term provides the gauge kinetic term, and the cubic term contains the Chern- where A 4 denotes the 4-th polarization of the gauge field and F µν the field strength in four dimensions. The Lorentz symmetry and gauge covariance can be restored by modifying the action [3,4]. The precise form of the Chern-Simons term in five dimensions in the case of nonabelian gauge group is found in Ref. [5] (with supergravity). We now turn to the SU(3) C gauge group. Kaluza-Klein reduction to the four-dimensional spacetime, however, yields an unwanted N =1 chiral multiplet in the SU(3) C -adj. representation in the low energy spectrum, and the N = 2 SUSY is left unbroken. Hence, we consider an S 1 /Z 2 orbifold instead of S 1 . The multiplets V (x, y) and Φ(x, y) of N = 1 SUSY are now under a constraint Then, we have only an SU(3) C vector multiplet of the N = 1 SUSY without the chiral multiplet Φ after the Kaluza-Klein reduction. In order to define the theory consistently, the action in M 4 ×S 1 should be invariant under the Z 2 transformation. The invariance is achieved as long as i) the coefficient of the quadratic term 1/g 2 is even under the Z 2 transformation, and ii) that of the cubic term h is odd. We take 1/g 2 to be y-independent and h(y) as where c is a constant that plays the same role as the corresponding one in the non-SUSY models [2](see below). 1

Boundary extra quarks
There are two fixed points in the S 1 /Z 2 orbifold: y = 0 and y = l. Let us put SU(3) Ccharged chiral multiplets as extra quarks on the fixed-point boundaries: chiral multiplets Q in the SU(3) C -3 representation at y = 0 and the same number of chiral multipletsQ in the SU(3) C -3 * representation at y = l. The action of the extra-quark multiplets contains These chiral multiplets by themselves give rise to a triangle anomaly of SU(3) C at each fixed point. However, the anomaly at y = 0 is the same as that at y = l with the opposite sign. Thus, the anomaly can be canceled through its flow between y = 0 and y = l implemented by the Chern-Simons interaction (3), provided the constant c is adequately chosen. 2 When the extra dimension is sufficiently large, interactions involving both Q andQ are highly suppressed. Let the cutoff scale (such as the grand unification or Planck scale) of the model be given by M. Then the effects of particles with masses of order M may generically induce such terms as e −M l MQQ (7) in the effective superpotential. We assume that Ml > ∼ 10 2 to suppress the effects of such terms. Then there is an accidental axial symmetry which is to be identified with the Peccei-Quinn symmetry. Although interactions comprised of only Q's orQ's are not expected to be suppressed by e −M l , they are higher-dimensional operators, and are irrelevant to the axion potential, as we explicitly see in the next section. The internal dimension is moderately large: we take Ml ≃ 10 2 -10 4 . It follows that M ≃ (10 −1 -10 −2 )M pl , and l −1 ≃ (10 −3 -10 −6 )M pl , where M pl ≃ 2.4 × 10 18 denotes the Planck scale. The setup described in this section reduces to a four-dimensional N = 1 SU(3) C gauge theory with an accidental axial symmetry below the Kaluza-Klein scale.

The Model
The superpotential virtually contains no terms involving both Q andQ, and hence the extra quarks are not forced to develop a chiral condensation. Thus, another vector multiplet is introduced in the five-dimensional bulk, so that the chiral condensation is formed dynamically, and that a composite axion is obtained through the spontaneous chiral symmetry breaking.
For definiteness, let us introduce an SU (5)  are at y = l, 5 where i = 1, 2, 3 and α = 1, · · · , 5. The SU (5) H gauge theory has four flavors of extra quarks. Thus, an effective superpotential is generated due to the SU(5) H interaction [6]: where Λ H denotes the dynamical scale of the SU(5) H gauge theory. The run-away potential from Eq. (9) is stabilized by supersymmetry-breaking effects such as V = m 2 |Q| 2 in the scalar potential. We assume gravity-mediated supersymmetry breaking in this article for simplicity. The Peccei-Quinn scale F P Q is of the order of .
The spectrum below the Peccei-Quinn scale F P Q consists of chiral multiplets in the SU(3) C -(adj.+3+3 * ) representations and two singlets, in addition to the particle contents of the MSSM. All the fermions and real scalars in the extra multiplets acquire masses of the order of the supersymmetry-breaking scale m. Pseudo-scalars in the SU(3) C -charged chiral multiplets receive radiative corrections at the one-loop level, and have masses at least of the order of √ α QCD m. They are pseudo-Nambu-Goldstone bosons and cannot remain exactly massless, because the SU(3) C interaction explicitly breaks the corresponding chiral global symmetry. One of the singlet pseudo-scalars also has a mass of the order of m due to the mixed anomaly with SU(5) H (and an explicit breaking of an R symmetry). Only one pseudo-scalar field remains massless below the supersymmetry-breaking scale, and that field plays the role of the QCD axion.
Although there exists an accidental Peccei-Quinn symmetry as advocated above, higherdimensional operators such as break it explicitly, where z 0 and z l are dimensionless coupling constants of order one, D denotes the covariant superderivative, and the implicit gauge indices are contracted so that the terms are gauge invariant. 6 Let us make a conservative estimate of the QCD axion effective potential [7] induced by the explicit breaking operators localized at the fixed points. On dimensional grounds, the dominant contributions to the axion potential turn out to be of order where the spurion charges of Λ 11 H for the selection rules are apparent from Eq.(9) and the supersymmetry-breaking scale factor m 3 originates from the superderivatives and superspace integrals. These corrections should not be too large to make the Peccei-Quinn mechanism ineffective: where m u and m d denote the masses of the up and down quarks, and Λ χSB is the energy scale of the QCD chiral symmetry breaking.

Cosmological Issue
Owing to the dynamical condensation QQ ∼ F 2 P Q , there exist colored particles in the pseudo-Nambu-Goldstone multiplets in the low-energy spectrum, as mentioned above. These particles, if they were to live too long, would constitute dark matter and lead to a cosmological difficulty [8].
We allow 7 such gauge-invariant terms as where z is a dimensionless coupling constant of order one,d denotes the down-quark chiral superfield at y = 0 (see footnote 4), and the implicit gauge indices are contracted. In the presence of this operator, all the colored particles have sufficiently short lifetime. We note that the presence of the above terms does not alter the conclusion in the previous section. In fact, the operator (14) does not contribute to the extra potential of the axion, as seen in the same analysis as that at the end of section 3 .

Flow of Gauge Coupling Constants
The colored extra multiplets in the low-energy spectrum has impacts not only in cosmology but also in the renormalization-group flow of gauge coupling constants. In particular, the SU(3) C coupling becomes asymptotically non-free, and the model described in section 3 no longer serves as a good description at very short distance scale, since the QCD interaction becomes non-perturbative. A typical renormalization-group flow of the gauge-coupling constants is shown in Fig. 1. 8 It shows that the Kaluza-Klein scale can be as high as 10 15 GeV, and the cutoff scale as high as 10 16 GeV, for m ≃ 10 3.5 GeV. Thus, some of the extra QCDcharged particles can be within the reach of LHC, while sufficient suppression F P Q /M < ∼ 10 −5 is obtained in Eq.(13), even if the terms Eq.(11) are not forbidden by the U(1) Y charge.

A Kink Configuration and Supersymmetry
It is shown in this appendix that the kink configuration of the coefficient of the Chern-Simons interaction in Eqs. (3) and (5) can be understood as a VEV of a background field. The origin of the kink solution is also explained. In six-dimensional spacetime, the interaction is consistent with the minimal SUSY, where G (1) is 1-form field strength of a scalar field C (0) in a linear hypermultiplet, and I (5) a 5-form that satisfies F is the field strength of a Yang-Mills field. This interaction is known in the context of string theories as a part of the Wess-Zumino interaction on Dp-brane world volumes, where C (p+1) is the Ramond-Ramond (p + 1)-form potential, G a collection of Ramond-Ramond field strengths, and I (0) a differential form that is related with the anomaly polynomial I through I = dI (0) [9]. When there is a magnetic source of the field C (0) , its Bianchi identity is given by where δ (2) denotes a 2-form supported only on a point y i (of a magnetic source), and N i the magnetic charge located there. The interaction (15) implements the inflow of anomaly: Let us introduce a 4-form I (4) which satisfy δ ǫ I (5) = dI (4) , where δ ǫ denotes the gauge variation. Then the variation of the action (15) is given by and hence the triangle anomaly flows into a singularity by the amount proportional to the charge localized there. Let us assume that the internal space of the two extra dimensions is T 2 /Z 2 . One can consider a torus which is long in one direction, and short in the other. Then, one has an effective description in five dimensions, by performing Kaluza-Klein reduction in the short direction. This five-dimensional description is what we need in the main text.
In the limit where the second extra dimension is small,G 4 is constant along y ∈ (−l, 0) as well as along y ∈ (0, l). Thus, the kink configuration h(y) is given byG 4 .