blusrgb0.1,0.1,0.8 \definecolorGreenYellowcmyk0.15,0,0.69,0 \definecolorYellowcmyk0,0,1,0 \definecolorGoldenrodcmyk0,0.10,0.84,0 \definecolorDandelioncmyk0,0.29,0.84,0 \definecolorApricotcmyk0,0.32,0.52,0 \definecolorPeachcmyk0,0.50,0.70,0 \definecolorMeloncmyk0,0.46,0.50,0 \definecolorYellowOrangecmyk0,0.42,1,0 \definecolorOrangecmyk0,0.61,0.87,0 \definecolorBurntOrangecmyk0,0.51,1,0 \definecolorBittersweetcmyk0,0.75,1,0.24 \definecolorRedOrangecmyk0,0.77,0.87,0 \definecolorMahoganycmyk0,0.85,0.87,0.35 \definecolorMarooncmyk0,0.87,0.68,0.32 \definecolorBrickRedcmyk0,0.89,0.94,0.28 \definecolorRedcmyk0,1,1,0 \definecolorOrangeRedcmyk0,1,0.50,0 \definecolorRubineRedcmyk0,1,0.13,0 \definecolorWildStrawberrycmyk0,0.96,0.39,0 \definecolorSalmoncmyk0,0.53,0.38,0 \definecolorCarnationPinkcmyk0,0.63,0,0 \definecolorMagentacmyk0,1,0,0 \definecolorVioletRedcmyk0,0.81,0,0 \definecolorRhodaminecmyk0,0.82,0,0 \definecolorMulberrycmyk0.34,0.90,0,0.02 \definecolorRedVioletcmyk0.07,0.90,0,0.34 \definecolorFuchsiacmyk0.47,0.91,0,0.08 \definecolorLavendercmyk0,0.48,0,0 \definecolorThistlecmyk0.12,0.59,0,0 \definecolorOrchidcmyk0.32,0.64,0,0 \definecolorDarkOrchidcmyk0.40,0.80,0.20,0 \definecolorPurplecmyk0.45,0.86,0,0 \definecolorPlumcmyk0.50,1,0,0 \definecolorVioletcmyk0.79,0.88,0,0 \definecolorRoyalPurplecmyk0.75,0.90,0,0 \definecolorBlueVioletcmyk0.86,0.91,0,0.04 \definecolorPeriwinklecmyk0.57,0.55,0,0 \definecolorCadetBluecmyk0.62,0.57,0.23,0 \definecolorCornflowerBluecmyk0.65,0.13,0,0 \definecolorMidnightBluecmyk0.98,0.13,0,0.43 \definecolorNavyBluecmyk0.94,0.54,0,0 \definecolorRoyalBluecmyk1,0.50,0,0 \definecolorBluecmyk1,1,0,0 \definecolorCeruleancmyk0.94,0.11,0,0 \definecolorCyancmyk1,0,0,0 \definecolorProcessBluecmyk0.96,0,0,0 \definecolorSkyBluecmyk0.62,0,0.12,0 \definecolorTurquoisecmyk0.85,0,0.20,0 \definecolorTealBluecmyk0.86,0,0.34,0.02 \definecolorAquamarinecmyk0.82,0,0.30,0 \definecolorBlueGreencmyk0.85,0,0.33,0 \definecolorEmeraldcmyk1,0,0.50,0 \definecolorJungleGreencmyk0.99,0,0.52,0 \definecolorSeaGreencmyk0.69,0,0.50,0 \definecolorGreencmyk1,0,1,0 \definecolorForestGreencmyk0.91,0,0.88,0.12 \definecolorPineGreencmyk0.92,0,0.59,0.25 \definecolorLimeGreencmyk0.50,0,1,0 \definecolorYellowGreencmyk0.44,0,0.74,0 \definecolorSpringGreencmyk0.26,0,0.76,0 \definecolorOliveGreencmyk0.64,0,0.95,0.40 \definecolorRawSiennacmyk0,0.72,1,0.45 \definecolorSepiacmyk0,0.83,1,0.70 \definecolorBrowncmyk0,0.81,1,0.60 \definecolorTancmyk0.14,0.42,0.56,0 \definecolorGraycmyk0,0,0,0.50 \definecolorBlackcmyk0,0,0,1 \definecolorWhitecmyk0,0,0,0 \definecolormygrrgb0,0.6,0 \definecolormygreyrgb0,0.1,0.2 \definecolormybluergb0,0.5,0.9 \definecolormyblue2rgb0,0.5,0.5 \definecolormyorangergb1,0.5,0 \definecolormypurplergb0.6,0,1 \definecolormygoldenrgb1,0.8,0.2


July 15, 2019

On axionic dark matter in Type IIA string theory

Gabriele Honecker and Wieland Staessens

PRISMA Cluster of Excellence & Institut für Physik (WA THEP), Johannes-Gutenberg-Universität, D-55099 Mainz, Germany


[2ex] We investigate viable scenarios with various axions in the context of supersymmetric field theory and in globally consistent D-brane models. The Peccei-Quinn symmetry is associated with an anomalous symmetry, which acquires mass at the string scale but remains as a perturbative global symmetry at low energies. The origin of the scalar Higgs-axion potential from F-, D- and soft breaking terms is derived, and two Standard Model examples of global intersecting D6-brane models in Type II orientifolds are presented, which differ in the realisation of the Higgs sector and in the hidden sector, the latter of which is of particluar importance for the soft supersymmetry breaking terms.

1 Introduction

The nature of dark matter and dark energy remains one of the biggest puzzles of modern physics. Collider searches as well as direct and indirect detection experiments rule out more and more scenarios Beyond the Standard Model. Most of these searches rely on the requirement that new particles have sizable cross-sections involving Standard Model gauge interactions. String theory on the other hand contains a variety of particles without such gauge interactions, including in particular the neutral axionic pseudo-scalars in the closed string sector that complexify the string coupling and compactification moduli. On the other hand, axions carrying some charge under a global symmetry were proposed in 1977 to solve the strong CP-problem [1, 2, 3, 4], and experimental searches for axions have been intensified recently, see e.g. [5, 6, 7, 8, 9]. Over the past decades, a combination of astrophysical observations, laboratory experiments and cosmological considerations, shrunk the parameter space for the axion decay constant to the region GeV GeV, the so-called ‘axion window’. The lower bound of the axion window is set by the stellar evolution of red giants, white dwarfs and hot neutron stars. And the higher bound follows from cosmological considerations, when the axion is treated as a dark matter candidate. Current and future experiments are designed to probe (parts of) the axion window, see e.g. [10] for an up-to-date overview.

While global continuous symmetries have been argued to be inconsistent with gravity [11, 12, 13, 14, 15, 16], string theory generically contains Abelian gauge symmetries whose anomalies are cancelled by the generalised Green-Schwarz mechanism, which also produces string scale mass terms for the ’s. Within the context of Type II superstring theories, such massive gauge symmetries and associated charged states arise in the open string sector, see e.g. [17, 18] for reviews. remains as a perturbative global symmetry at low energies, which is broken by non-perturbative effects such as D-brane instantons [19]. The Green-Schwarz mechanism associates a closed string axion with the longitudinal mode of an open string vector. It is thus natural to combine axions from the closed and open string sector such that one axion solves the strong CP problem, while others account for the dark sector of the universe, see e.g. [20, 21]. In contrast to compactifications of the heterotic string, within Type II string theory different energy scales can easily be decoupled due to the affiliation of gravity to the closed string sector and gauge symmetries to the open string sector.

There exist various supersymmetric or string inspired versions of field theoretic axion models in the literature, see e.g. [22, 23, 24, 25, 26] and references therein, and axions have been discussed before in the context of the heterotic string, see e.g. [27]. Within the Type IIB string theory context, previous work in the LARGE volume scenario concentrated on closed string axions, see e.g. [28, 29, 30, 31, 32, 33, 34, 35], and the idea of open string axions in intersecting D-brane inspired scenarios has been put foward recently in [36], while to our knowledge little is known about the combined effect of closed and open string axions in the context of globally consistent D-brane configurations with the (supersymmetric) Standard Model or GUT spectrum.

This article aims at closing this apparent gap by first discussing the appearance of various types of axion within field theory and Type II string theory. We then proceed to investigate two globally consistent D6-brane models with Standard Model spectrum in detail in view of their Higgs-axion potentials and possibilities of supersymmetry breaking. The D-brane set-up and massless spectrum of the first model on has been constructed in [37, 38, 39] and contains a ‘hidden’ gauge factor, while the set-up and spectrum of the second model on with hidden gauge group have been presented in [40, 39] with a discussion on superpotential couplings in [41, 42]. Throughout the present work, particular attention will be paid to the occurrence of different energy scales such as the Peccei-Quinn, electroweak and supersymmetry breaking scales in dependence of their prevailing origin from the closed or open string sector.

This article is organised as follows: in section 2 we briefly review field theoretical axion models and provide a Type II string theory motivated extension to their supersymmetric version. The origin of couplings within global supersymmetry including soft supersymmetry breaking terms is discussed. Section 3 contains a discussion of the various types of closed and open string axions in Type IIA string theory and associated massive Abelian gauge symmetries. The concept is illustrated by two examples of Type IIA orientifold compactifications on and with supersymmetric Standard Model spectrum, and the possibility to break supersymmetry by a hidden sector gaugino condensate is addressed. Section 4 contains our conclusions. Technical details on the four-dimensional field theory are collected in appendices A and B, and appendix C contains the full light matter spectrum of the two examples.

2 Axions in Field Theory

In this section, we first briefly review the well-known DFSZ axion model and then proceed to discuss a supersymmetric D-brane inspired extension as well as the origin of soft supersymmetry breaking terms in the scalar Higgs-axion potential.

2.1 The DFSZ model

In 1977, Peccei and Quinn proposed the existence of a spontaneously broken global symmetry to solve the strong CP-problem [1, 2]. The associated pseudo Nambu-Goldstone boson in the original model consists of an axion that arises from two Higgs doublets [1, 2, 3, 4]. In this original PQWW axion model, the breaking scales of the electro-weak and global symmetry coincide, and consequently non-negligible contributions of axions to hadronic decay products involving heavy quarks in the initial state are expected. Due to non-observation of such effects, the most simple PQWW axion model is ruled out to date, see e.g. [43, 44, 45] and references therein.

The breaking scale of the global symmetry can be decoupled from the electroweak scale by introducing an additional scalar field, which is only charged under , but neutral under the Standard Model group . One distinctive model by Kim, Schifman, Vainshtein and Zakharov (KSVZ) uses a vector-like heavy quark pair to which this neutral scalar couples [46, 47]. In contrast to the original PQWW model, the KSVZ model contains only one Higgs doublet, and therefore we do not expect any immediately obvious generalisation to supersymmetric field theory as low-energy limit of some Type II string theory compactification.

Another prominent axion model by Dine, Fischler, Sredenicki and Zhitnitsky (DFSZ) contains two Higgs doublets as well as a Standard Model-singlet complex scalar field [48, 49]. Since this model can be generalised to supersymmetric field theory in a natural way, we give here a brief review of the distinctive features such as the scalar potential, axion decay constant and axion mass. The two Higgs doublets schematically couple to quarks and leptons via


and the coupling to the complex scalar singlet field is contained in the scalar Higgs-axion potential,

where the abbreviation has been used as well as the standard decomposition of the Higgs doublets into charged and uncharged components and vevs,


in equation (2.1) denote all possible four-point couplings. Due to the term , also the Higgs doublets have to be charged under the global symmetry if the scalar singlet transforms non-trivially,


From the Yukawa couplings (1), one can then also read off relations among the transformations of the quarks and leptons and Higgs bosons under ,


There are two inequivalent consistent choices of charge assignments with either left-handed quarks uncharged and right-handed quarks charged or vice versa. The discussion for leptons is completely analogous. In both cases, the (up to overall sign flip) unique choice is for the charge of the Higgses. Since in stringy D-brane models such as the examples below in section 3, an anomalous and massive gauge symmetry constitutes a natural candidate for , we summarise the associated second choice of charge assignments with neutral right-handed particles in table 1.

Table 1: Standard Model particles and axion field charge assignment for the choice of uncharged right-handed particles. refers to the axion in the original DFSZ model of equation (2.1), to the axion superfield in the supersymmetric D-brane inspired DFSZ model of section 2.2.

The coupling constants in equation (2.1) are constrained by measurements as follows:

  • The experimentally observed value of the parameter is close to one. In the Standard Model, this is true at tree-level, whereas in D-brane inspired models, at tree-level can deviate from one, see e.g. [50], in which case the scale at which new physics appears is firmly constrained, unless higher order or non-perturbative effects become important.

  • The axion remains invisible at low energies if is broken at a much higher scale than the electro-weak symmetry, implying a hierarchy of vevs,


The physical axion is generically a mixture of the argument of the complex scalar and the neutral CP-odd Higgs bosons. However, imposing the hierarchy (6) results in the physical axion stemming primarily from the singlet . The axion mass is determined using Bardeen-Tye methods [51],


with the number of quark doublets, the masses of up- and down-type quarks and pions and the pion decay constant. The coupling of the axion to matter is determined by the axion decay constant , which also sets the strength of the axion coupling to gluons,


With respect to the PQWW axion model (for which the axion decay constant is equal to ), the axion-gluon coupling as well as the couplings to ordinary matter are suppressed by a factor , implying that the production of axions is reduced by a factor . For this reason, the DFSZ axion has been dubbed an ‘invisible axion’.

2.2 Supersymmetrising the DFSZ model

Since the DFSZ axion model contains two Higgs doublets in conjugate representations, it can easily be promoted to supersymmetric field theory. The axion and Higgs scalar potential can then be decomposed into three different components,


One sublety arises concerning the axion-Higgs coupling in the second line of equation (2.1). On the one hand, any four-point coupling is non-renormalisable and thus suppressed by the cut-off scale, which might be in field theoretical models or in string models. Such suppression has been related to the smallness of the -term e.g. in [52]. On the other hand, a renormalisable three-point coupling can be engineered if the axion is replaced by a chiral superfield containing the axionic scalar with twice the charge,


such that . This choice is motivated by D-brane models as follows: since in Type II string theory all charged particles arise from open strings, their gauge representations are defined by their two end points to be of the type , plus in the presence of O-planes , or some hermitian conjugate thereof under . In particular, the Higgs fields have one endpoint on a stack of D-branes supporting either a or gauge group, and the other endpoint lies on a single D-brane with gauge group111 In many models, arises from a spontanous breaking of a right-symmetric group or ., which contributes to the hypercharge . Demanding that both the Higgs field and some singlet carry charge under an anomalous and massive gauge symmetry boils down to two options:

  1. The Peccei-Quinn symmetry is identified with the anomalous and massive symmetry, and the Standard Model singlet with charge arises as the antisymmetric representation of or its conjugate. This non-Abelian singlet obeys the charge assignment of in table 1 under the standard decomposition of non-Abelian/Abelian representations (see e.g. [18])


    This choice of thus leads to a three-point coupling among charged states.

  2. The symmetry involves some massive linear combination of Abelian gauge factors including . This case can only occur if cannot be continuously connected to any right-symmetric group or since such a connection enforces to be a massless and anomaly-free gauge symmetry by itself. There are again two different options:

    1. contains the combination , which is orthogonal to the massless hypercharge defined below in equation (40). In this case, the axion can be identified with the scalar superpartner of the neutrino, and it again obeys the charge assignment of in equation (10).

    2. If contains some combination of with an additional gauge factor of a D-brane, that is not required to obtain the Standard Model chiral spectrum, yet is a part of the definition for the hypercharge, the axion can arise from strings stretched between D-branes and . In this case, the axion carries charge in accordance with in table 1.

In section 3.2, two supersymmetric sample DFSZ models are presented, which display typical features in the context of intersecting D6-brane model building. In both cases, the models are intrinsically left-right symmetric with a spontaneously broken right-symmetric gauge group . Hence, in the remainder of the article we will focus on case 1 with and the rôle of the singlet fulfilled by a chiral supermultiplet in the antisymmetric representation of .

After choosing this configuration, the next step consists in mapping the various terms in the DFSZ potential from equation (2.1) to a specific origin in supersymmetric field theory based on the decomposition into F-terms, D-terms and soft supersymmetry breaking terms as in equation (9):

  1. F-term potential: The terms of the form , and all arise from a single superpotential term of the form:


    where the various coupling constants are now unified, i.e. , due to supersymmetry. Note that this cubic coupling already requires the Higgses to be charged under the Peccei-Quinn symmetry if the Standard Model singlet superfield with scalar component transforms non-trivially under .

  2. D-term potential: The quartic terms , and arise from the Kähler potential , and more explicitly from the gauge-invariant coupling of the chiral Higgs superfields to the vector superfields associated with the gauge symmetry, with takes the following, generic form in global supersymmetry:


    where represents a generic chiral superfield and a vector superfield. The Peccei-Quinn symmetry originates from a massive gauge boson in string theory, as explained below in detail in section 3.1.2. This implies that the Kähler potential will also contain gauge-invariant couplings between the vector multiplet associated to and the matter multiplets charged under the symmetry. In this way, also the quartic term will be generated in the D-term potential.

  3. Soft supersymmetry breaking terms: The remaining terms have (in field theory) to be added by hand as soft supersymmetry breaking terms that are invariant under the Standard Model gauge group and the Peccei-Quinn symmetry, such as the mass terms , and . The cubic coupling can be realised through trilinear -terms. The soft supersymmetry breaking terms can be written in a manifestly supersymmetric way through the introduction of a spurion superfield . The trilinear -term can then be captured by the superpotential:


    The soft supersymmetry breaking mass terms on the other hand can be generated through a Kähler potential of the following form:


    In section 2.3 we briefly discuss how these soft supersymmetry breaking terms can arise through gravity mediation of spontaneous supersymmetry breaking in a hidden sector.

2.3 Gravity mediation and gaugino condensation

For realistic globally supersymmetric models, soft supersymmetry breaking terms form a hands-on way to break supersymmetry without spoiling its abilities to solve the hierarchy problem and the related naturalness problem concerning the Higgs mass in the Standard Model. One way to generate these soft supersymmetry breaking terms is by coupling the supersymmetric field theory to gravity and allowing gravity to mediate spontaneous supersymmetry breaking in a hidden sector to the visible sector. For a Type II orientifold compactification, the low energy effective field theory of the massless string modes reduces to an supergravity theory, so that gravity mediated supersymmetry breaking appears as a natural way to generate the soft supersymmetry breaking terms for the supersymmetrized DFSZ model.

Coupling the supersymmetrised DFSZ model to gravity leads to an supergravity theory with chiral multiplets and vector multiplets, whose bosonic sector is characterised by the following generic action, cf. e.g. [17, 53]:


where denote the bosonic components of the chiral multiplets, the field strength associated to a gauge group and the holomorphic gauge kinetic function. The F-term scalar potential can be expressed in terms of the supergravity Kähler potential and its derivatives, the Kähler metrics and the superpotential :


while the D-term scalar potential is given by:


where represent the auxiliary fields in the vector multiplets. An explicit expression for is not given here, as they will play no rôle in the following.

From a low energy perspective, the massless chiral multiplets can be split up into a set of observable chiral matter () charged under the Standard Model group and a set of hidden matter (). This decomposition allows for an expansion of the Kähler potential and superpotential about the stabilised vacua for the hidden matter fields:


By inserting these expansions into the F-term scalar potential of equation (17) and replacing the hidden scalars by their vacuum expectation values, one obtains an effective field theory for the visible sector, which in the flat limit (i.e.  while keeping the gravitino mass fixed) reduces to a globally supersymmetric theory with soft supersymmetry-breaking terms [54, 55] of the following form for the scalar fields:


with soft supersymmetry-breaking parameters given by:


with the tree-level cosmological constant and the gravitino mass :


and the auxiliary field associated to the hidden field given by:


When one of the hidden chiral matter fields acquires a non-vanishing -term (), not only supersymmetry will be spontaneously broken, but also the soft supersymmetry breaking terms for the visible sector are expected to emerge by virtue of the gravitational coupling between the hidden and visible sector.

These considerations shift the spotlight to the mechanism by which the hidden chiral matter is stabilised and its auxiliary field simultaneously obtains a non-zero vev. If the hidden sector contains a strongly coupled gauge sector, condensing gaugini [56, 57] are expected to induce a non-perturbative superpotential for (some of) the hidden chiral matter, such that the latter are stabilised and no longer correspond to flat directions in the scalar potential. More explicity, assume that the model contains a hidden chiral superfield that couples to the supergauge-invariant field strength of the strongly coupled gauge symmetry through the superpotential term:


The auxiliary field (23) of should then be generalised to:


where represents the gauge kinetic function and the gaugino of the strongly coupled gauge theory. At a mass scale , the gauge symmetry becomes strongly coupled, and the gaugini condense with the characteristic scale . The auxiliary field then acquires a non-zero vev that sets the supersymmetry breaking scale . The gravitino will absorb the Weyl-fermion of the superfield through the super-Higgs effect and acquire a mass of the order .

In type II superstring compactifications, the rôle of the hidden matter is played both by truly hidden gauge sectors as well as by the complex structure and Kähler moduli of the internal Calabi-Yau orientifold. In type IIA compactifications gaugino condensation will generate a non-perturbative superpotential for the complex structure moduli, provided that the hidden sector comes with a strongly coupled gauge symmetry as in the examples below in section 3.2.6.

3 Axions in Type IIA String Theory

3.1 Massive gauge symmetries and open and closed string axions

In Type II superstring theory, axions and axion-like particles as well as Abelian gauge bosons can arise from various sectors. We first briefly discuss axions arising in the closed string sector in section 3.1.1 and then focus on axions from the open string sector and massive Abelian gauge symmetries in section 3.1.2.

3.1.1 Closed string axions

The most explored scenarios focus on closed string axions emerging from the Kaluza-Klein reduction of massless -forms appearing in the closed string spectrum such as the NS-NS 2-form and the RR-forms, see e.g. [58] for an overview. The shift symmetry of these CP-odd real scalars is a remnant of the gauge invariance of the -forms, while the axion decay constant is set by the non-canonical prefactor in the kinetic term of the axion appearing in the low energy effective action upon dimensional reduction. For D-brane models in Type II superstring theory, the linear coupling of the axion to the topological QCD charge density follows from the dimensional reduction of the D-brane Chern-Simons action, and more explicitly from the term (see e.g. [18, 17]):


with the -cycle wrapped by the D-brane along the internal directions and the -form from the RR-sector.

Let us briefly review some of these aspects for intersecting D6-brane scenarios in Type IIA string theory, where the closed string axions arise from the dimensional reduction of the RR 3-form .222The external and internal degrees of freedom of the RR 1-form are projected out on a Calabi-Yau orientifold compactification (see e.g. [59]), and expanded along orientifold invariant -forms provides four-dimensional vectors related to isometries of the compact space. The first step to obtain the low energy effective action for the closed string axions consists in expanding this 3-form with respect to a basis of real -even 3-forms ,


where denote the coordinates along and the coordinates on the internal Calabi-Yau orientifold. The dimensional reduction of the kinetic term for the four-form flux will then reduce to the kinetic terms for the closed string axions , while the coupling in the Chern-Simons action (26) for the D6-brane reduces to the instanton term:


In this expression, is the volume of the six-dimensional internal space and the volume of the compact three-cycle wrapped by the D6-brane. A simple rescaling of the closed string axion brings the low energy effective action to the conventional form of equation (8), from which we can read off the axion decay constant333Similar relations exist for other D-brane model building scenarios involving .:


One immediately notices that the axion decay constant for closed string axions is proportional to , which makes it challenging to identify the QCD axion as a closed string axion if the string scale is too high or too low. This consideration extends to other model building scenarios in string theory.

Another obstacle [60] for closed string axions to solve the strong CP-problem appears when involving moduli stabilisation of their CP-even scalar partners within multiplets, i.e. the dilaton and complex structure and Kähler moduli (the ‘saxions’). For a saxion stabilised supersymmetrically by non-perturbative corrections, its associated axion is also stabilised with the same mass. The no-go theorem in [60] further indicates that the presence of massless axions implies tachyonic directions in the scalar potential. However, if some of the saxions are stabilised by perturbative effects in or , the no-go theorem can be circumvented, as realised explicitly in the context of the LARGE Volume Scenario in [61]. Yet for unfixed closed string axions, their axion decay constant is still proportional to , such that their appropriateness to serve as the QCD axion is strongly correlated with an intermediate string scale ( GeV).

In order to disentangle the axion decay constant from , one has to turn to other axion sources than the closed string sector. In section 2.2 we argued that there exists a natural way to embed a supersymmetric DFSZ-type axion model into D-brane model building scenarios, in which case the axion then resides in a chiral matter multiplet from the open string sector. Moreover, as the open string saxion is assumed be stabilisable through standard field theory mechanisms (i.e. spontaneous symmetry breaking), open string axions represent an alternative loophole to the no-go theorem of [60].

3.1.2 The Green-Schwarz mechanism and open string axions

In D-brane model building scenarios, symmetries appear as the centers of unitary gauge groups supported on the respective D-brane worldvolumes, and gauge anomalies are canceled by the generalized Green-Schwarz mechanism, in which the local symmetry acquires a Stückelberg mass proportional to by eating a closed string axion. In that case, the survives perturbatively as a global anomalous symmetry that is only broken by non-perturbative effects [19] with at most a discrete Abelian symmetry as remnant [62, 63, 64, 65, 66]. Such perturbative global symmetries are very suitable to serve as Peccei-Quinn symmetries, as the discussion in the preceding section 2.2 and the two explicit examples in section 3.2 below clearly show.

In section 2.2, it was already pointed out that the open string axion resides in an =1 matter multiplet arising at some D-brane intersection involving the D-brane supporting the anomalous symmetry. More explicitly, an open string axion corresponds to the phase of a complex scalar field charged under the anomalous symmetry, similarly to the field theory set-up discussed in section 2. At the string scale, the bosonic part is described by the following Lagrangian:


where denotes the complex scalar field with charge under the anomalous symmetry with gauge potential , while represents the closed string axion eaten by in the Stückelberg mechanism. The open string axion arises as the phase of the complex field :


where describes the fluctuations of the open string saxion about its vacuum expectation value . After inserting the expression for back into the Lagrangian, one obtains the following action by keeping only track of the CP-odd scalars:


This action can be brought back to the standard form:


by an transformation on the two CP-odd scalars :


Hence, it is in fact the linear combination that turns into the longitudinal component of the gauge boson with mass


while the orthogonal linear combination remains as massless axion.

Analogously to field theory axions, the open string axion couplings to Higgses and matter can be chirally rotated away in favour of the linear coupling to , see appendix A for technical details. Thus, both open and closed string axions will provide for an axion coupling to the topological QCD charge density:


with the closed string axion decay constant and the open string axion decay constant .444The axion-gluon couplings are proportional to the anomaly coefficient due to the standard string theoretic anomaly cancellation via the generalised Green-Schwarz mechanism. Performing the rotation also in this part of the Lagrangian yields the axion-gluon couplings in the -basis:


for which the decay constants are now given by


For models where the Stückelberg mass is much heavier than the scale at which acquires a vev, i.e. , the axion eaten by the gauge boson consists primarily of the closed string axion . The orthogonal massless state on the other hand will then be mostly composed of the open string axion . Notice that this will also be reflected in the decay constants of the respective axions: and . Hence, in this configuration the string scale can be much higher than GeV, as the presence of an open string axion provides another candidate for the QCD axion.

Up to this point, both closed and open string axions have been considered, but only symmetries from the open string sector have been taken into account. Generically, orientifold compactifications of Type II string theory contain some number of closed string vectors, also called ‘RR photons’. In Type IIA string theory, these arise from the dimensional reduction with orientifold-even (1,1)-forms . However, as shown in [67], these closed string ’s are completely decoupled from the open string sector unless closed string fluxes significantly distort the Calabi-Yau geometry.

3.2 Standard Models on and orbifolds

Two supersymmetric intersecting D6-brane models, in which the distinguished features presented in the previous sections are explicitly realised, have been constructed on  [37] and  [40]. Both models are globally defined (i.e. they satisfy RR tadpole cancellation and K-theory constraints) and contain a spontaneously broken right-symmetric group,


with the following and hypercharge assignments in terms of original charges,


On the ‘hidden’ gauge group is , and it can be broken to its center in the same way as the right-symmetric group is broken by switching on a of the complex scalar encoding the displacement and Wilson line of the respective D6-brane. The hidden gauge group on on the other hand cannot be broken by such a vev since the corresponding stack of D6-branes is perpendicular to the O6-plane orbits while the other special D6-branes with gauge groups in both models are parallel to some O6-plane orbit.

The chiral spectra of both models are listed in tables 6 and 8 in appendix C, and the corresponding non-chiral spectra are given in tables 7 and 9 for and , respectively. In both cases, the three left-handed quarks stem from sectors, where denotes the orbifold image of , which in turn denotes the orientifold image of D6-brane . The right-handed down-type quarks are localised in the sectors, and the right-handed up-type quarks in the sectors. The charges under the anomalous and massive in the quark sector thus coincide with those given for the global in supersymmetric field theory in table 1. The sectors produce the right-handed electrons , while the right-handed neutrinos emerge from intersections in the sectors. In both models, four vector-like pairs of states in the antisymmetric representation of can be found in the sectors. These states provide exactly the degrees of freedom for the singlet superfield required to establish the supersymmetric version of the DFSZ axion-model as discussed in section 2.2. Moreover, due to the spontaneously broken underlying left-right symmetric gauge structure in both models, the symmetry provides the only viable realisation of a symmetry.

The first noteworthy difference between the massless spectra of the two models occurs in the left-handed leptonic sector: the model on has exactly three left-handed leptons in the sectors, while the chiral part of the spectrum on contains six left-handed leptons in the sectors and three anti-leptons in the sectors. A second difference concerns the Higgs-sector for the models: in the case, the Higgses arise as vector-like pairs with opposite charge, while on the Higgses have identical charge and belong to the chiral sector. As a consequence, the way in which the supersymmetric DFSZ model of section 2.2 is realised for both D-brane models will differ. Further details are given in sections 3.2.1 and 3.2.4 for the and model, respectively. Last but not least, the model contains the weakly coupled ‘hidden’ gauge group , whereas for the model on the hidden gauge group runs to strong coupling below the string scale, and therefore a hidden sector gaugino condensate can form and break supersymmetry spontaneously. As anticipated in section 2.3, this will by gravity mediation result in soft breaking terms in the Standard Model sector as further discussed in section 3.2.6.

3.2.1 A supersymmetric DFSZ model on

We briefly review here the Higgs sector of the global 5-stack D6-brane model on , that was first constructed in [37, 38] with the non-chiral spectrum, beta function coefficients and 1-loop gauge threshold corrections explicitly given in [39], before proceeding to the new discussion of the Higgs-axion potential. The complete massless chiral and vector-like matter states are for convenience listed in appendix C in tables 6 and 7 after spontaneous breaking of the right-symmetric group .

In the left-right symmetric phase, the Higgs-sector of this model consists of a non-chiral pair of bifundamental states located at the intersection points of the -sector. Under a continuous displacement of (or a Wilson line along) the -brane on the third two-torus without action, the gauge group is broken to an Abelian gauge group , and the non-chiral pair of splits up into a non-chiral pair of