Note on the strong CP problem from a 5-dimensional perspective
aa r X i v : . [ h e p - ph ] A p r IFT-07-02UCRHEP-T434
Note on the strong CP problem from a 5-dimensional perspective – the gauge-axion unification –
Bohdan GRZADKOWSKI ∗ Institute of Theoretical Physics, University of Warsaw, Ho˙za 69, PL-00-681 Warsaw, Poland
Jos´e WUDKA † Department of Physics, University of California, Riverside CA 92521-0413, USA
We consider 5 dimensional gauge theories where the 5th direction is compactified on an interval.The Chern-Simons (CS) terms (favored by the naive dimensional analysis) are discussed. A simplescenario with an extra U (1) X gauge field that couples to SU (3) color through a CS term in the bulk isconstructed. The extra component of the Abelian gauge field plays a role of the axion (gauge-axionunification), which in the standard manner solves the strong CP problem easily avoiding most ofexperimental constraints. Possibility of discovering the gauge-unification at the LCH is discussed. PACS numbers: 11.10.Kk, 11.15.-q, 12.10.-g, 14.80.Mz
I. INTRODUCTION
In the Standard Model (SM), the Higgs mechanism is responsible for generating fermion and vector-boson masses.Although the model is renormalizable and unitary, it has severe naturality problems associated with the so-called“hierarchy problem”. At loop-level this problem reduces to the fact that the quadratic corrections tend to increasethe Higgs boson mass up to the UV cutoff of the theory. Extra dimensional extensions of the SM offer a novel approachto gauge symmetry breaking in which the hierarchy problem could be either solved or at least reformulated in termsof the geometry of the higher-dimensional space.Other inherent problems of the SM could also be addressed in extra-dimensional scenarios. For instance, withinthe SM the amount of CP violation is not sufficient to explain the observed baryon asymmetry [1], the gauge-Higgsunification scenario offers a possible solution since in such models the geometry can be a new source of explicit andspontaneous CP violation [2]. In this note we shall prove that the strong CP problem could be solved introducingan appropriate Chern-Simons (CS) terms in 5D . The scenario leads to an attractive possibility of gauge-axionunification. II. HIERARCHY OF EFFECTIVE OPERATORS
We will first consider models in D = 5 dimensions with fermions, gauge bosons and scalars propagating throughoutthe D -dimensional bulk, and some unspecified matter localized on lower dimensional manifolds (branes). Thoughthese models are non-renormalizable it is possible to define a hierarchy of possible terms in the Lagrangian thatallows for a proper perturbative expansion; the procedure is a simple application of the arguments used in the naivedimensional analysis (NDA) [4], see the Appendix. This hierarchy is specified by assigning to each gauge invariantoperator an index s = d c + b ′ + (3 f / −
4, ( d c is the number of covariant derivatives, f and b ′ the number of fermionand scalar fields). As it is shown in the Appendix the least suppressed operators are those that have the index s = 0: F ; ¯ ψDψ ; | Dφ | ; ¯ ψφψ ; φ , (1)where F denotes the generic gauge tensor, φ a generic scalar, and ψ generic fermions.The s = 1 operators not containing scalar fields are ( A denotes a generic gauge field) AF ; ¯ ψF ψ , (2) ∗ Electronic address: [email protected] † Electronic address: [email protected] For other attempts to solve the strong CP problem by 5 dimensions see [3]. whose coefficients are naturally suppressed by 1 / (24 π ), together with all brane terms, presumably including the SMLagrangian multiplied by l − δ ( y − y o ). The first operator in (2) corresponds to the 5-dimensional Chern-Simons (CS)term, while the second includes all magnetic-type couplings. Operators of index s = 1 containing φ are of the form D φ , D φ , or D ¯ ψψφ .The NDA argument favors the presence of a CS term (if only 5D vector bosons are present the CS term is the onlybulk operator with index s = 1), with a coefficient as large as 1 / (24 π ). Of course, it is still possible that there existadditional symmetries that forbid this term, however if present, the CS term can generate interesting effects.Hereafter we shall consider a 5D model containing U (1) X and SU (3) color bulk gauge fields, denoted by X and G respectively. Application of the NDA for this case (where there are no bulk fermions) yields the following action upto index s = 1 S = Z X d x (cid:26) − X MN X MN −
12 Tr (cid:2) G MN G MN (cid:3) + − π ǫ LMNP Q h c g ′ g X L Tr ( G MN G P Q ) + c g ′ X L X MN X P Q ++ c g Tr (cid:18) G L G MN G P Q + i G L G M G N G P Q − G L G M G N G P G Q (cid:19)(cid:21)(cid:27) + 116 π S brane (3)where X MN and G MN are, respectively, the field strength tensors for the Abelian and non-Abelian groups withthe 5D gauge couplings respectively denoted by g ′ and g ; c , , are undetermined numerical constants, presumablyof O (1). In our specific applications we will consider models constructed on the space-time X = M × [0 , R ], andwe will concentrate on the “mixed” Chern-Simons term proportional to g ′ g . We will assume that all SM fields areneutral under U (1) X . Hereafter, whenever possible, in order to make the analysis as model independent as possible,we will avoid referring to any details of the embedding of the SM into 5D. The only assumption we make is that theSM is localized on one or perhaps both ends of the interval [0 , R ]. III. SOLVING THE STRONG CP PROBLEM FROM A 5D PERSPECTIVE
As shown above, the NDA favors the CS term as an operator of index s = 1. We will argue that the presence ofthis term allows for a simple solution to the strong CP problem.As it is well known, in a basis where the Yukawa matrices are diagonal, the phases of the Kobayashi-Maskawa matrixare responsible for all electroweak CP violation effects. There is, however, an additional (“strong”) CP-violating termallowed by the symmetries of the 4D SM Lagrangian: L QCD CP = θ α s π Tr (cid:16) G µν ˜ G µν (cid:17) , (4)where G µν is the QCD field strength tensor, ˜ G µν = ǫ µναβ G αβ /
2, and α s ≡ g / (4 π ) for g the SM 4D QCD gaugecoupling constant. In the process of diagonalizing the Yukawa matrices, quark fields undergo a chiral rotation, whichgenerates the same structure as in (4) (within the path-integral formulation this results from a non-trivial Jacobianfor the fermionic measure [5]); therefore the total effect of the strong CP violation is parameterized by the effectivecoefficient θ eff ≡ θ + θ weak . The experimental data (EDM of the neutron) indicates that | θ eff | ∼ < − [6]; this is referredto as the strong-CP “problem” since none of the symmetries of the SM requires such a strong suppression.Models in extra dimensions offer new possibilities to solve this problem due to a possibility of constructing theChern-Simons terms. Specifically, we will assume that the color gauge fields G aN propagate in the bulk, but that therest of the SM fields are confined to one or two branes located at y = 0 and y = R . In addition we assume thepresence of an Abelian gauge field X N also propagating in the bulk. For the 5D models being considered here, theQCD strong-CP term (4) can be written as follows: S brane = α s π Z d x [ θ L δ ( y ) + θ R δ ( y − R )] Tr (cid:16) G µν ˜ G µν (cid:17) , (5)where θ R,L are constant parameters. The convention for the antisymmetric tensors which we follow is such that ǫ = ǫ = 1 for the metric tensor η MN =diag(1 , − , − , − , −
1) and η µν = diag(1 , − , − , − T a are Hermitian andnormalized according to Tr T a T b = 2 − δ ab . Among the various terms in (3) we will concentrate on the effects of the mixed CS term: S CS = − g ′ g c π Z X d x dy ǫ LMNP Q X L Tr ( G MN G P Q ) . (6)The action (6) is not automatically gauge invariant under the U (1) X . However, using the Bianchi identity ǫ NMQP R D Q G P R = 0, one can show that under the Abelian transformation X L → X ′ L = X L + ∂ L λ X (7)the change in S CS is localized on the boundary of the space . δS CS = g ′ g c π Z M d x λ X ǫ µναβ Tr ( G µν G αβ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = Ry =0 (8)There are various ways of insuring that this vanishes. One can, for example, add an appropriate set of chiral fermionson the two branes; in this case the anomaly generated by these fermions can be adjusted so that it cancels (8), see e.g.[7]. Brane scalars can be also arranged to have the same effect [3], [7] provided they couple to ǫ µναβ Tr( G µν G αβ ). Asimpler alternative, which we will adopt here, is to impose appropriate boundary conditions such as λ X Tr( G ) | y =0 = λ X Tr( G ) | y = L .Variation of the total action (3) with c = c = 0 and c = 1 leads to the following equations of motion for thegauge fields: D B G BA = J A + brane terms and ∂ B X BA = j A + brane terms , (9)with the following Chern-Simons currents J A = g ′ g π ǫ ABCDE X BC G DE ; j A = g ′ g π ǫ ABCDE
Tr ( G BC G DE ) (10)The brane terms in (9) originate from possible couplings of the bulk gauge fields to the fields localized on the branes.For the extremum of the action the following boundary conditions (BC) must be fulfilled:tr " G µ − g ′ g π X ν ˜ G µν ! δG µ y = Ry =0 = 0 and X µ δX µ (cid:12)(cid:12) y = Ry =0 = 0 (11)Here we will restrict ourselves to theories containing massless zero-modes (gluons) of the non-Abelian gauge field.This implies a unique choice of BC for SU (3) color : ∂ y G aµ | y =0 ,R = 0 , G a | y =0 ,R = 0 . ; (12)these conditions imply G a µ | y =0 ,R = 0. For the Abelian field we require X µ | y =0 ,R = 0 , ∂ y X | y =0 ,R = 0 , (13)so that X µν | y =0 ,R = 0. It follows that the BC (11) are satisfied .The resulting Kaluza-Klein (KK) expansions read G aµ ( x, y ) = R − / P n =0 d n G a ( n ) µ ( x ) cos m n y G a ( x, y ) = R − / √ P n =1 G a ( n )4 ( x ) sin m n yX µ ( x, y ) = R − / √ P n =1 X ( n ) µ ( x ) sin m n y X ( x, y ) = R − / P n =0 d n X ( n )4 ( x ) cos m n y (14)where m n = πn/R and d n = 2 (1 − δ n, ) / . The zero-mode G a (0) µ ( x ) is the standard 4D gluon; it is also clear that themodel also contains a massless 4D scalar X (0)4 ( x ). This assumes that λ X is not a constant. We thank Kin-ya Oda for a discussion at this point.
Let’s focus now on the Abelian gauge transformations. In order to preserve the BC, the gauge function λ X ( x, y )must satisfy the following constraints: ∂ µ λ X | y =0 ,R = 0 , ∂ y λ X | y =0 ,R = 0 (15)That implies a corresponding KK expansion for the Abelian gauge function λ X ( x, y ) = X n =1 λ ( n ) X ( x ) sin m n y + βy (16)where β is a constant. The 4D vector and scalar fields transform as X ( n ) µ → X ( n ) µ + 1 √ ∂ µ λ ( n ) X X ( n )4 → ( X (0)4 + β for n = 0 X ( n )4 + m n √ λ ( n ) X for n > . (17)In the following we will take β = 0, which is the simplest condition ensuring the gauge symmetry of the CS action .In order to discuss phenomenological predictions of the model let us expand the CS action into KK modes: S CS = R π g ′ R / g R c Z d x " X (0)4 Tr G (0) µν ˜ G µν (0) + 2 ∂ µ X (0)4 ∞ X n =1 Tr G ( n ) ν D ρ G ( n ) σ ǫ µνρσ − G µν (0) ∞ X n =1 Θ ( n ) µν + · · · (18)where D µ ≡ ∂ µ + ig h G (0) µ , · i G (0) µν ≡ ∂ µ G (0) ν − ∂ ν G (0) µ + ig h G (0) µ , G (0) ν i (19)for g = g / √ R andΘ ( n ) µν ≡ "(cid:16) ∂ µ X ( n )4 G ( n ) ν − ∂ ν X ( n )4 G ( n ) µ (cid:17) − (cid:16) ∂ µ X ( n ) ν G ( n )4 − ∂ ν X ( n ) µ G ( n )4 (cid:17) − m n (cid:16) X ( n ) µ G ( n ) ν − X ( n ) ν G ( n ) µ (cid:17) (20)Ellipsis in (18) stands for terms (irrelevant for any practical applications) that involve four non-zero KK modes.Expanding the kinetic terms of (3), one can verify that indeed G (0) µν corresponds to the SM QCD gluon (which ispresent due to our having adopted (12)), while X (0)4 ( x ) = a ( x ) can play the role of the axion. The lowest-order termsconform the usual QCD action, the axion kinetic term and the axion-gluon interactions : S (0)low = Z M (cid:26) −
12 Tr ( G µν G µν ) + 12 ∂ µ a∂ µ a + α s π (cid:18) af a + θ eff (cid:19) Tr (cid:16) G µν ˜ G µν (cid:17)(cid:27) , (21)where θ eff ≡ θ L + θ R and we dropped the (0) superscript in G . Adopting the NDA estimation of the CS coefficientone obtains for the axion decay constant f − a = 16 g ′ π R (22)where g ′ is the 4D Abelian gauge coupling, g ′ = g ′ / √ R , and α s = g / (4 π ). Note that for this mechanism of axiongeneration to work, the extra Abelian gauge symmetry must be broken by the boundary conditions (Scherk–Schwarzbreaking) so that no additional massless vector boson associated with X µ is present. The only low-energy remnantof X M is the axion a ( x ). The crucial advantage of the model presented here is the unification of the axion and the U (1) 5D gauge field. There are serious attempts to construct in 5D a realistic gauge-unification theory [8]. Thosemodels combined with the scenario discussed here could provide an interesting alternative for a theory of electroweakinteractions that offers the scalar sector of 4D theory fully unified with a gauge fields (solving the hierarchy problem [8] This is also a natural choice for S /Z orbifold models since it insures that X µ ( x, − y ) = − X µ ( x, y ), X ( x, − y ) = X ( x, y ) and X N ( x, y + 2 R ) = X N ( x, y ) are preserved under gauge transformations. It turns out that each term in the KK expansion of (5) is a total derivatives (as they emerge form the full derivative Tr[ G µν ˜ G µν ]).Only the zero-mode contribution will be relevant as it contributes to the effective non-perturbative axion potential, other terms couldbe dropped. and the strong CP problem at the same time). As it will be discussed below the gauge-axion unification is consistentwith the existing experimental constraints and there is a chance to test the scenario at the LHC.As in the standard Peccei-Quinn scenario the effective axion coupling ( a/f a + θ eff ) relaxes to zero through instantoneffects, solving the strong CP problem dynamically. The axion mass is generated in a standard manner [9] m a = f π m π f a √ m u m d m u + m d = 0 . GeV f a , (23)and no strictly massless scalars remain in the spectrum.Let us discuss consequences of the remaining interactions in the 5D CS term (6) that consists of terms quadraticand quartic in the non-zero KK modes. We will focus (for obvious phenomenological reasons) on the quadratic termsshown explicitly in (18). Of course, there are other terms involving the heavy fields generated by the kinetic part ofthe action (3), those have have been considered previously in the literature, see e.g. [10] .Because of its relatively large coupling, the very last term ( ∝ m n ) in (18), will produce the most noticeable effectsat the LHC. Therefore let us consider the production of heavy gluons G ( n ) µ and vector bosons X ( n ) µ (with n ≥
1) at theLHC. At the partonic level the leading contributions are the following: GG → G ⋆ → G ( n ) X ( n ) and GG → G ( n ) X ( n ) .Since the SM fields do not carry U (1) X quantum numbers, the X ( n ) µ bosons are stable at the tree level; on the otherhand, heavy gluons G ( n ) µ couple to SM quarks located on a brane. Therefore the experimental signature for the abovereactions would be missing energy and momentum (carried away by the stable X ( n ) µ ) and two jets from the G ( n ) µ decays. Let us compare the amplitude strength for this process with the standard QCD two jet production amplitude.Adopting the estimate of the CS coupling from the NDA in (18) we find that the ratio of the X ( n ) µ G ( n ) ν G α couplingto the SM triple gluon vertex is of the order of g ′ g α s π n ∼ g ′ g − n (24)Since n ∼ G ( n ) X ( n ) over the two-jet QCD background. Nevertheless it should be noticed, that the huge amount ( ∼ TeV) of missing energy(carried away by the stable and heavy X ( n ) µ ) may enhance the signal relative to the QCD background very efficiently,and that the large gluon luminosity of the LHC could be sufficient to provide enough events to test the scenario.Though these expectations are supported by the results for similar processes at the Tevatron [11], a dedicated MonteCarlo study would be needed to resolve this issue definitively; this, however, lies beyond the scope of this note.Other possible signature of the axion being the 4th component of 5D gauge field could be the heavy gluon productionprocess through a virtual axion exchange: GG → a ⋆ → G ( n ) G ( n ) for n ≥
1. The amplitude for this process is generatedby the first two terms in (18). It is straightforward to find that the order of magnitude for the amplitude normalizedto two gluon ( GG ) production is the following: α ′ π α s n ∼ − α ′ n , (25)where α ′ ≡ g ′ / (4 π ). If α ′ ∼ α s then for small n the amplitude is suppressed by the factor 10 − . Since both G ( n ) G ( n ) and GG states decay roughly the same way (the signature is n ≥ .Let us assume that the axion mass m a (or equivalently the decay constant f a ) is known. Then the definite test ofthe model discussed here would be a verification of the gauge-axion unification that is caused by the fact that the axionis a component of the 5D gauge field X M . The important consequence of the unification is that the total cross sectionfor G ( n ) X ( n ) production is predicted including the normalization. Therefore the measurement of σ tot ( G ( n ) X ( n ) ) shallprovide the definite experimental test of the model.Concluding the review of various possible experimental tests of gauge-axion unification discussed here, one can saythat, because of a hudge missing energy ( ∼ TeV), the process GG → G ( n ) X ( n ) provides the cleanest signature, thatmakes the observation of the signal plausible.For the model being considered here the axion decay constant f a is determined by the geometrical scale R − (ifthe NDA arguments are applied), therefore experimental limits on f a constrain the size of the compact dimensiondimension. However, it should be emphasized that most of these constraints rely on effects produced by the coupling Note also that the amplitude receives contributions from the other terms in the action. of the axion to two photons, and this coupling is absent in our model (to leading order). (For a review of experimentalconstraints see [6].) Nevertheless there exists a bound that should be obeyed also by our photofobic axion; this is theso called “misalignment” lower axion mass limit that originates from the requirement that the contribution to thecosmic critical density from the relaxation of the axion field ( θ eff →
0) does not overclose the universe. The resultingconstraint [6], m a > − eV, leads to R − ∼ < GeV, having used (22-23) and taken g ′ = O (1). Note that the NDAestimate of the CS coupling was crucial to derive the limit on R . IV. CONCLUSIONS
We shown that an extension of naive dimensional analysis to 5D gauge theories naturally allows relatively largecoefficients in front of Chern-Simons (CS) terms. The strong CP problem was discussed within a simple scenariocontaining a new U (1) X gauge field and the SU (3) color gauge fields propagating in the bulk, and interacting througha a mixed CS term. Adopting appropriate boundary conditions, the CS term was shown to be gauge invariant (withoutany need for brane matter). The zero mode of the extra component of the new Abelian gauge field was seen to playa role of the axion (gauge-axion unification), which in the standard manner receives the instanton-induced potential,so that the strong CP problem (localized on the branes) disappears while the axion receives a mass. In the effectivelow-energy regime, the axion couples only to gluons, therefore most of the limits on the axion decay constant do notapply in the context of this model. It was shown that the most promising test of the gauge-axion unification is theprocess of G ( n ) X ( n ) production: GG → G ( n ) X ( n ) . The hudge missing energy ( ∼ TeV) carried away by the stable andheavy X ( n ) µ is believed to provide a sufficiently clean signature of the final state. APPENDIX
In this appendix we provide, for completeness, a summary of the application of Naive Dimensional Analysis (NDA)to higher-dimensional models. The NDA allows to determine the scale Λ at which the theory becomes stronglyinteracting. For that purpose let us compare two graphs with the same number of external legs, one of which has anadditional gauge-boson propagator. This second graph will be suppressed with respect to the first by the factorΛ δ g l − δ ; l D = (4 π ) D/ Γ( D/ , (26)where g denotes the gauge coupling constant, and l D is the geometric loop factor obtained form integrating overmomentum directions (note that in D = 4 + δ dimensions g has a mass dimension of − δ/ ∼ (cid:0) l δ g − (cid:1) /δ . (27)The same NDA requirement allows an estimate of the coefficients in front of effective operators. For this we considera generic vertex of the form V = λ Λ D (2 π ) D δ D ( X p i ) g ψ Λ / ψ ! f (cid:16) p Λ (cid:17) d (cid:18) g A M Λ (cid:19) b (cid:18) g φ Λ φ (cid:19) b ′ ; (28)where scale appropriate for the vector fields and derivatives (they enter together through the covariant derivative)was chosen to be Λ, while the coefficient λ , the fermionic scale (Λ ψ ) and the scalar scale (Λ φ ) are to be determined.The requirement to reproduce the starting operator by radiative corrections determines the maximal value of λ andminimal scales Λ ψ , Λ φ that are allowed by perturbativity λ = l − δ and Λ ψ = Λ φ = Λ . (29)Let us now restrict ourselves to 5d theories, δ = 1, and define the “index” of a vertex by s = d c + b ′ + 32 f − d c = d + b . (30)where d c is the number of covariant derivatives present in the vertex V . If an L -loop graph contains V n vertices withindices s n , then the vertex corresponding to this graph has an index s = L + X n V n s n . (31)In terms of s the coefficient of a given operator is (see also [12]) (cid:18) π (cid:19) s × (the powers of g needed to get a dimension 5 object) ; (32)and Λ = 24 π /g .If the indices of all vertices are non-negative, then it follows from (31) that s ≥ s n for all n . This implies that if V has index s , then only operators with indices ≤ s can renormalize the coefficient of V and we can then define ahierarchy according to the value of s in the sense that we can consistently assume that operators with higher indicesare generated only by higher orders in the loop expansion . This would be spoiled if the theory has vertices withnegative indices, (as an addition of an internal line attached by vertices with s n < s , so an extra loopleads to less suppressed operator) which corresponds to the case d c = f = 0 , b ′ = 3, according to the definition (30).In order to define a hierarchy one should accordingly require that all terms cubic in the scalar fields be absent dueto an additional symmetry such as a discrete Z under which the φ are odd, by gauge invariance, (as in the SM) orjust by an absence of scalar fields (as in this note where we are considering only vector bosons in 5D therefore thecubic scalar interactions cannot be constructed and the hierarchy of operators is given just by (32) without any otherconstraints). Fermion fields are assumed to transform appropriately under this symmetry, so as to allow all desirablescalar-fermion couplings.In order to include consistently possible brane terms in the hierarchy we note that this type of interactions arenaturally generated by the bulk terms in a compactified space at the one loop level [13]. It is then natural add 1 to s whenever a localizing factor of the form δ ( y − y o ) is present. In addition the geometric suppression factor for theseterms equals l = 16 π that replaces l = 24 π present in (32); see also [14]. Acknowledgments
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