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Candidates for HyperCharge Axion in Extensions of the Standard Model

Many theoretically well-motivated extensions of the Standard Model contain heavy pseudoscalars that couple to hypercharge topological density. The cosmological dynamics of such hypercharge axions could, under certain conditions, lead to generation of a net baryon number in a sufficient amount to explain the observed baryon asymmetry in the universe. We examine the Minimal Supersymmetric Standard Model and string/M-theory models and determine specific conditions which heavy axion-like pseudoscalars must satisfy to successfully drive baryogenesis. We find that all candidates in the Minimal Supersymmetric Standard Model fail to obey some of the constraints, and that only in special string/M-theory models some axions may be adequate.


I. INTRODUCTION AND SUMMARY
Topologically non-trivial configurations of hypercharge gauge fields can play a relevant role in the electroweak (EW) scenario for baryogenesis [1][2][3][4][5]. A hypothetical heavy pseudoscalar field that couples to hypercharge topological number density, the hypercharge axion (HCA), can exponentially amplify primordial hypermagnetic fields in the unbroken phase of the EW plasma, while coherently rolling or oscillating. The coherent motion provides the three Sakharov's conditions [6] and can lead, under certain conditions, to generation of a net hypercharge topological number that can survive until the phase transition and then be converted into a net baryon number in a sufficient amount to explain the origin of the baryon asymmetry in the universe [7,8].
In [7,8] we have focused on a simple model with an extra singlet HCA, a, whose only coupling to Standard Model (SM) fields is through the following effective operator where Y µν is the U(1) Y hypercharge field strength and Y µν is its dual. The constant M Y has units of mass. The HCA is massless at very high energies and gets a mass m ∼ Λ 2 /M A from a generic potential generated at an energy scale Λ. The potential V is a bounded periodic function and M A is a normalization mass scale.
Unless some fine-tuning mechanism is effective when the axion potential is generated, the pseudoscalar is trapped far from the minimum of its potential and then starts to coherently roll or oscillate around this minimum until the condensate finally decays. Typically, the cosmological misalignment of a, a c , is of the same order of magnitude as M A . The axion rolls if the Hubble time at the scale of potential generation t H ∼ M p /Λ 2 (M p being the Planck mass) is shorter or comparable to the characteristic time t s ∼ m −1 for a coherent oscillation. Otherwise, the axion will oscillate a few times before the topological condensate decays. While the axion rolls it generates a net topological number which is stored in very long wavelength modes, k ≪ Λ, that remains frozen in the plasma until the EW phase transition. Oscillations should happen just before or during the EW phase transition so that the generated topological number, stored in shorter wavelength modes k ∼ Λ, does not diffuse in the highly conducting plasma once the oscillations have stopped [7,8].
A crucial element in the scenario of HCA driven baryogenesis is the requirement that the misalignment a c ∼ M A is larger than the decay constant of the HCA into two hypercharge photons, M Y [7,8]. If the axion oscillates, it is enough that < a > c is somewhat larger than but if it rolls, it is necessary that The heavy HCA differs significantly from the original axion proposed by Peccei and Quinn (PQ) as an elegant solution to the strong CP problem [9][10][11]. The PQ axion couples to the nonabelian SU(3) topological density which generates a potential at the QCD scale Λ g ∼ 200 MeV through instanton effects. The QCD axion should get its mass mainly from potential (6) in order to fix θ QCD = 0. In this case, the normalization scale for the axion potential M A , is determined by the axion coupling constant 1/M g such that M A ∼ M g . This is not the case for the HCA. The axion coupling (1) to the abelian hypercharge topological density does not generate any potential for the HCA, which gets its mass from some additional sector. In general the scales M Y and M A in the hypercharge sector (1) and in the mass generation sector (2) are not related.
In this paper we try to identify HCA candidates in theoretically well motivated extensions of the SM. Heavy pseudoscalars that couple to topological gauge densities appear in lowenergy supersymmetric models with an extended higgs sector. We analyze here the Minimal SM group, and with coupling 1/M H to the topological density of a "hidden" gauge group which interacts only through gravitational strength interactions with the observable sector: Axions usually get their potentials mainly from instanton effects in the nonabelian hidden sector whose field strength is G, while the abelian hypercharge group U(1) Y is contained in the observable sector whose field strength is F . In this case, the scale M A is given by the inverse axion coupling to the hidden gauge topological density, M H , (see eq. (5) and (6)), while M Y is given by M O . Conditions (3) and (4), that axions couple to the observable sector more strongly than they do to the hidden sector.
In 4D models the ratio M H /M O is determined by the compactification scheme. As we will show, standard compactifications restrict parameter space of string candidates for HCA to regions where it is difficult to satisfy in a natural way all the necessary conditions that allow them to successfully drive baryogenesis.

II. HCA IN THE MINIMAL SUPERSYMMETRIC STANDARD MODEL.
The higgs sector of the MSSM contains two complex SU(2) L doublets that give masses to the up and down components of the three families of SM fermions. The coupling structure of the MSSM of relevance to us here appears in each one of the two higgs Yukawa sectors, and for each one of the fermion families separately. Therefore, instead of discussing the full and complicated MSSM we first discuss, for the sake of simplicity, a linear σ-model with a U(1) Y abelian gauge group, one fermion flavour ψ, and a singlet complex higgs field φ, which will allow us to analyze relevant issues. We then apply the results to the full MSSM.
The σ-model (8) is invariant, up to anomalies, under a global PQ chiral rotation FIG. 1. HCA decay into two hypercharge photons through a fermion loop.
which is spontaneously broken by the vacuum expectation value of the complex scalar field We are interested in the pseudoscalar component a of the complex higgs that couples axially to the fermion through the Yukawa sector, Here m = λf P Q is the chiral mass of the fermion, and m a is the pseudoscalar mass. In the linear σ-model (8) the chiral symmetry (9) protects the pseudoscalar from getting a perturbative mass, but in the MSSM it gets a mass in the TeV range from soft SUSYbreaking terms or supersymmetric µ-term, and therefore we have introduced it in this simple model as an extra free parameter.
The anomalous coupling of the massive axion to the hypercharge topological density (1) can be obtained directly in a standard way from the invariant amplitude for the decay process of a into two hypercharge photons a → 2γ Y shown in Fig. 1, and therefore where k 1 ,k 2 are the 4-momenta of the two outgoing hypercharge photons in the center of mass system, and ǫ 1 ,ǫ 2 are their polarizations vectors. The scalar function I is given by The integral in eq. (13) is finite and can be computed using standard techniques: If 2 < ξ = ma m < ∞, where τ = (1 − 4 ξ 2 ) 1/2 . The function M Y (ξ) reaches its global minimum at ξ = 2, when the fermion circulating in the loop is on-shell, The function decreases slowly from its value at ξ = 0 (massless axion) until it reaches its minimum and then grows fast towards infinity when ξ ≫ 2.
The smaller M Y is, the stronger the axion coupling (1) to hypercharge topological density.
If we assume the axion mass m a is in the TeV range, we can conclude that fermions with masses around a few hundreds GeV or larger contribute significantly to the anomalous coupling, but much lighter fermions do not.
At this point we would like to make two comments: 1) The anomalous axion coupling to gauge topological densities is closely associated with the spontaneous or explicit breaking of chiral symmetry in the fermionic mass sector. In the case of a massless fermion the trace of γ matrices in (11) vanishes, and therefore the coupling M vanishes.
2) The scale M Y is fixed by the scale f P Q at which chiral symmetry is broken.
The effects of high temperature and chiral symmetry restoration on the anomalous vertex have been studied in the simplest case of a perturbatively massless axion, ξ → 0 [12][13][14][15].
We report the result obtained there for the invariant amplitude: where m(T ) is the chiral fermion mass that now depends on the temperature T . At high temperature the scalar function I T is proportional to 1 x(T )T , where x(T ) is the infrared cut-off. Far below the phase transition, the infrared cut-off is provided by the mass of the fermion  (16), The factor 1 3 takes into account three different colors for the quark. The heavy pseudoscalar higgs has additional axial coupling to charginos and neutralinos,  (19), if charginos or neutralinos masses m f are not much heavier than the top mass.
The picture described above changes drastically at temperatures above the EW phase transition. Both pseudoscalar mass eigenstates are massive but, as we discussed above eq. (18), the contribution from SM fermions to their anomalous coupling to topological densities vanishes because the fermion chiral masses vanish.
Charginos and neutralinos, on the other hand, remain massive in the symmetric phase and their contributions could be nonvanishing. We will describe here the chargino sector [16], but the situation is very similar for neutralinos. There are two charginos which contain four Weyl spinors, λ + , λ − (W-inos), H − 1 , H + 2 (charged higgsinos). The mass term in this sector in the symmetric phase of the MSSM is: where we have defined mass eigenstates Dirac spinors, χ 1 = can guess that the anomalous coupling will vanish because of chiral symmetry restoration.
In fact, this is what happens: if we write the pseudoscalar-higgsino-Wino axial vertex, where a 1 ,a 2 are the two massive pseudoscalars, in terms of chargino chiral mass eigenstates in the symmetric phase of the plasma χ 1 ,χ 2 we obtain only non-diagonal χ 1 − χ 2 vertices, while the hyperphoton-chargino-chargino only vertex, Y µχ1 γ µ χ 1 , is diagonal. The loop shown in Fig. 1 cannot be closed. Therefore, although charginos and neutralinos are massive in the symmetric phase of the plasma, their contributions to the anomalous 1-loop coupling of pseudoscalars to hypercharge topological density also vanish because of chiral symmetry restoration.
The mass matrix (20) gets thermal contributions that we have not written explicitly. It is not clear if non-diagonal thermal terms appear in the mass matrix but, in any case, they do not break the restored chiral symmetry and our conclusion is not expected to be modified: the MSSM extended higgs sector does not contain an HCA capable to drive baryogenesis while coherently rolling or oscillating.
The mechanism described in [7,8] for HCA-driven baryogenesis through coherent amplification of hypercharge fields in the symmetric phase of the EW plasma could possibly serve to amplify primordial ordinary electromagnetic fields in the broken phase of the plasma [17].
From our previous analysis, it may seem that the heavy pseudoscalar of the MSSM could have a relevant role: if, as it is generally assumed, the SUSY breaking terms that give the pseudoscalar its mass are generated at an intermediate scale ∼ 10 8 GeV , the coherent motion of the pseudoscalar when its potential is generated could have a typical misalignment a c , which is several orders of magnitude larger than the inverse coupling M Y ∼ 50 TeV (see (19)), as required by conditions (3) and (4) for coherent amplification of the magnetic modes.
But, if the potential is generated at the intermediate SUSY breaking scale and gives the pseudoscalar a mass in the TeV range, the typical time for a coherent oscillation is much shorter than the Hubble time at the epoch of potential generation. In [7,8] we have considered a singlet HCA that couples only to hypercharge photons through the operator (1).
The heavy pseudoscalar of the MSSM, on the contrary, couples to many other fields so that the topological condensate decays incoherently after a few oscillations at temperatures much above the electroweak phase transition and does not survive until the EW phase transition when the coupling (1) is generated. So our conclusion is that the pseudoscalar higgs of the MSSM cannot amplify ordinary electromagnetic fields.
The MSSM contains additional pseudoscalar neutral fields, i.e. pseudoscalar components of sneutrinos, which could, in principle, play a similar role to HCA in amplification of hypercharge topological number. But for sneutrinos, a dimension five operator similar to (1) is forbidden by R-parity symmetry. Extensions of the MSSM without R-parity in which sneutrinos couple to topological gauge densities have been discussed in [18] but, as in the case of the pseudoscalar higgs, the coupling to hypercharge topological number is generated through chiral symmetry breaking and therefore it also vanishes in the unbroken phase of the plasma.

III. HCA IN STRING/M-THEORY MODELS
In this section we consider possible candidates for HCA in low energy 4D effective field theory of weakly coupled heterotic E 8 × E 8 string theory (HE) and Horava-Witten (HW) theory [19,20]. Axions in such theories were studied in detail in [21][22][23][24] and we recall here the necessary results to be able to discuss their relevance and application to the subject at hand: HCA's. The model-independent axion a M I coupling to topological gauge density in HE theory can be obtained directly from the Bianchi identity of the 3-index antisymmetric tensor, dH = We will focus on the linear combination of model-dependent axions, the overall model- The normalization mass scale M 2 is determined by the detailed properties of the compactification scheme. In HW compactifications the coupling of a M D can be obtained by treating HW theory as a strongly coupled HE theory, and varying the coupling continuously from weak to strong coupling. The argument is that the axion coupling is determined by some topological invariants and does not change in the process. Typically, M 2 < ∼ M 1 . In HE models we are considering l i = +1, and therefore l = +1, is required by the fact that the three-index antisymmetric tensor H is globally defined, dH = 0. In HW models l = +1 is also required for similar reasons [24], but we keep l here as a free parameter for the sake of generality.
The mass generation scale Λ in string/M-theory models depends on hidden sector matter content and interactions in addition to its dependence on compactification details. It is therefore not so well determined, and can be anywhere below M p . But in many models, transition: Λ ∼ M EW . Additionally, the HCA mass m a ∼ Λ 2 M H , has to be about the EW scale, m a ∼ M EW . It follows that the HCA coupling to the hidden sector has to satisfy M H ∼ M EW , which can be fulfilled only if at least one of the mass scales M 1 , M 2 is about the EW scale. This is not expected in typical string/M-theory models, but may perhaps be viable in models with a very low string scale [25], or in models in which HCA is protected by some symmetry. In this range of parameters, M Y < ∼ M EW and the HCA would be directly detectable in future colliders through its decay into 2 photons [26].

IV. CONCLUSIONS.
We have studied the MSSM and 4D string/M-theory low energy effective field theories, looking for suitable candidates for HCA.
The natural candidate in the MSSM is the heavy higgs pseudoscalar that couples in the broken phase of EW theory to topological gauge densities through 1-loop triangle diagrams.
But, as we have shown in section II, this coupling vanishes in the symmetric phase of the EW theory due to chiral symmetry restoration in the fermionic mass sector of the theory.
The coupling of other possible candidates, i.e. pseudoscalar components of sneutrinos, to topological gauge densities is forbidden by R-parity symmetry. We conclude that the MSSM does not contain an HCA that can successfully drive baryogenesis. In some models with broken R-parity, sneutrinos do couple to topological gauge densities through triangle diagrams but also fail to serve as HCA's due to chiral symmetry restoration in the unbroken phase of the EW theory.
Stringy axions couple directly to hypercharge topological density at the compactification scale, which is typically much higher than the EW scale. But, we have concluded in section III that in generic compactifications the specific conditions for successful baryogenesis are violated in one way or another. We have outlined requirements for more elaborate models which may lead to HCA's which obey all the conditions. Such models may be realized in some special compactification schemes, or in scenarios of very low string scale.