We consider six-dimensional supergravity with Abelian bulk flux compactified on an orbifold. The effective low-energy action can be expressed in terms of chiral moduli superfields with a gauged shift symmetry. The -term potential contains two Fayet-Iliopoulos terms which are induced by the flux and by the Green-Schwarz term canceling the gauge anomalies, respectively. The Green-Schwarz term also leads to a correction of the gauge kinetic function which turns out to be crucial for the existence of Minkowski and de Sitter vacua. Moduli stabilization is achieved by the interplay of the -term and a nonperturbative superpotential. Varying the gauge coupling and the superpotential parameters, the scale of the extra dimensions can range from the GUT scale down to the TeV scale. Supersymmetry is broken by - and -terms, and the scale of gravitino, moduli, and modulini masses is determined by the size of the compact dimensions.
de Sitter vacua and supersymmetry breaking
[8pt] in six-dimensional flux compactifications
[0pt] Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany
The ultraviolet completion of the Standard Model remains a challenging question. There are strong theoretical arguments for supersymmetry at high scales and, in connection with gravity and string theory, also for compact extra dimensions. But in the absence of any hint for supersymmetry from the LHC the scale of supersymmetry breaking is completely unknown, except for a lower bound of TeV.
In this connection higher-dimensional theories are of interest which relate the scale of supersymmetry breaking to the size of compact dimensions via quantized magnetic flux . In the context of the heterotic string it has been argued that five or six dimensions are a plausible intermediate step on the way from 10d string theory to a 4d supersymmetric extension of the Standard Model [2, 3, 4, 5], and compactifications to six dimensions (6d) are also very interesting from the perspective of type IIB string theory and F-theory [6, 7, 8]. Furthermore, 6d theories are interesting from a phenomenological point of view. They can naturally explain the multiplicity of quark-lepton generations as a topological quantum number of vacua with magnetic flux  and, when compactified on orbifolds, they provide an appealing explanation of the doublet-triplet splitting in unified theories [10, 11, 12, 13]. Orbifold compactifications with flux combine both virtues , leading to 4d theories reminiscent of “split” [15, 16] or “spread”  supersymmetry.
Many aspects of 6d supergravity theories have already been studied in detail in the past. This includes the complete Lagrangian with matter and gauge fields [18, 19], compactification of gauged supergravity with a monopole background , localized Fayet-Iliopoulos terms generated by quantum corrections , singular gauge fluxes at the fixed points  and the cancellation of bulk and fixed point anomalies by the Green-Schwarz mechanism [23, 24, 25, 26]. In particular, it has been shown in  how magnetic flux together with a nonperturbative superpotential can stabilize both dilaton and volume modulus.
The present paper extends our previous work  where we showed that the Green-Schwarz mechanism also cancels the anomalies due to the chiral zero modes induced by the magnetic flux. We now demonstrate that the low-energy effective Lagrangian takes the form of an supergravity model for the moduli superfields with a gauged shift symmetry. The corresponding Killing vectors are induced by the magnetic flux and also by the Green-Schwarz term, respectively. Furthermore, supersymmetry implies an important modification of the gauge kinetic function. As discussed already in , this allows for a new class of metastable de Sitter solutions.
The paper is organized as follows. In Section 2 we derive the Kähler potential, the gauge kinetic function and the -term potential of the 4d theory with special emphasis on the effects of the Green-Schwarz term. Minkowski and de Sitter vacua are analyzed in Section 3 and it is shown how dilaton, volume and shape moduli can be stabilized by the flux together with a nonperturbative superpotential. The vector boson mass, the charged scalar mass, the moduli masses, and the axion masses are evaluated for two examples of Minkowski vacua with different size of the compact dimensions. An important aspect of the model is the realization of the super-Higgs mechanism with combined - and -term breaking of supersymmetry. This, together with the modulini masses, is discussed in Section 4. Details of the search for de Sitter vacua and the super-Higgs mechanism are given in Appendix A and Appendix B, respectively. Appendix C contains more details about the example models, including numerical values for their mass spectra.
2 Effective supergravity action
Let us first consider the bosonic part of the six-dimensional supergravity action with a gauge
involving the Ricci scalar , the dilaton , the gauge field and the antisymmetric tensor field . The corresponding fields strengths are given by
is the 6d Planck mass and denotes the 6d gauge coupling of mass dimension . The 3-form is the difference between the Chern-Simons forms and for the spin connection and the gauge field , respectively. In the following we ignore since we will not discuss gravitational anomalies, i.e.,
We choose as background geometry the product space with the metric
where correspond to the 4d Minkowski space and to the internal space. It is convenient to use dimensionless coordinates for the compact space, , where is a fixed physical length scale. The rescaling by the dimensionless radion field in (4) leads to standard kinetic terms for the moduli. The shape of the internal space is parametrized by the two real shape moduli in the two-dimensional metric ,
and the orbifold projection acts as . The physical volume of the internal space is , where is the vacuum expectation value of the radion field .
Neglecting the gravitational backreaction on the geometry of the internal space, a constant bulk flux is a solution of the equations of motion,
where . Furthermore, we add to the gauge-gravity sector a bulk hypermultiplet containing a 6d Weyl fermion with charge and two complex scalars. The hypermultiplet can be decomposed into two 4d chiral multiplets with charges and , respectively. The two complex scalars, and , have gauge interactions and a scalar potential which, in 4d language, corresponds to an -term and a -term potential of the two chiral multiplets ,
Due to charge quantization the value of the background flux can only take discrete values,
where , , and
where the correspond to the four fixed points on the orbifold. From the first term in Eq. (9) it is obvious that the background flux contributes to the chiral anomaly. As shown in , all these anomalies are canceled by the Green-Schwarz term
It is now straightforward to compute the 4d effective action by means of dimensional reduction. Matching the Ricci scalars and the gauge kinetic terms yields for the tree level 4d Planck mass and the 4d gauge coupling, respectively,
The gauge part of the 4d effective action has been worked out in . The field strength of the antisymmetric tensor can be written as
where , and denote 4d scalar, vector and tensor fields. Trading for the dual scalar by means of the Lagrange multiplier term
replacing radion and dilaton by the moduli fields and ,
rescaling the lowest state of the matter field, , and dropping the ‘hat’ for the 4d vector field, one obtains
For convenience, we have introduced the dimensionless parameter
and we have dropped
For a vacuum expectation value a constant Weyl rescaling has to be performed such that the rescaled metric describes physical 4d distances. The Ricci scalar of the rescaled metric is then multiplied by . corresponds to the physical Planck mass and is related to by the physical volume, . Analogously, the physical coupling is related to by . Furthermore, we now rescale moduli and matter fields, , . The resulting final 4d bosonic action is identical to Eq. (16) except for a change of parameters,
where and are physical Planck mass and gauge coupling, respectively. The parameter is replaced by
and the scalar mass is given by
By construction, the rescaled moduli fields satisfy . We conclude that given a vacuum field configuration one can always perform a rescaling such that the new metric describes physical 4d distances and the new moduli fields satisfy . The length scale corresponds to the physical size of the extra dimensions in terms of the physical Planck mass. In the following we shall therefore directly search for vacua with , and we set .
The flux compactification on the orbifold should lead to a 4d theory with spontaneously broken supersymmetry. Indeed, introducing the complex fields
and comparing expression (16) with the standard supergravity
Lagrangian , one immediately confirms that the
kinetic terms of the moduli and the matter field are reproduced by the
with the Killing vectors
where we have used the relation (8) from the flux quantization ; the gauge interaction of the matter field corresponds to the Killing vector
From the coefficient of one reads off the gauge kinetic function
whose real part we denote by . At first glance this appears to be at variance with the coefficient of suggesting . Note, however, that with the Green-Schwarz term we have included only part of the one-loop corrections to the effective action. A complete calculation should preserve supersymmetry. Hence, we expect that there are further contributions such that the correct gauge kinetic function is indeed given by Eq. (26), which remains to be confirmed by an explicit calculation. The one-loop corrected gauge kinetic function depends now on the moduli fields and . Such a -dependence has previously been found in heterotic string compactifications as an effect of quantum corrections [33, 34].
Knowing the Killing vectors and the gauge kinetic function, the -term potential is given by 
This is the standard -term potential for a charged complex scalar with a field-dependent Fayet-Iliopoulos (FI) term . The scalar potential in Eq. (19) contains the classical part of the -term potential which is given by . It contributes linearly to the tree-level mass of the charged scalar and quadratically to the energy density. Supersymmetry requires the quantum correction in addition, analogously to the gauge kinetic function. We therefore keep in the -term potential, which again should be verified by direct calculation.
Fayet-Iliopoulos terms which are generated at the orbifold fixed points by quantum corrections have been discussed in  in the case of zero bulk flux. These FI terms are , their sum and therefore the effective 4d FI term vanishes (), and they are locally canceled by a dynamically generated flux that modifies the zero-mode wave functions. As we have shown, there is a non-vanishing 4d FI term , which follows from the Green-Schwarz term. Following , it would be interesting to analyze systematically the interplay between the classical bulk flux and the local quantum flux at the fixed points, and to study their joint effect on the zero-mode wave functions.
3 Moduli stabilization and boson masses
The explicit form of the supersymmetric effective action enables us to determine the masses of the bosons in the theory. For the gauge boson and the charged scalar field the results were already given in . However, the non-trivial effective gauge coupling additionally includes higher order terms
The vector boson mass can be extracted form the bilinear term of the 4d gauge field in Eq. (16) and reads
The lowest charged scalar mass originates from the -term potential and includes corrections due to anomaly cancellation and non-trivial gauge kinetic function. This modifies the classical scalar mass term (18) to
Without flux the negative second term, accounting for the quantum corrections induced by anomaly cancellation, leads to a vacuum expectation value for such that the total -term vanishes. For non-vanishing flux, however, and the first term tends to stabilize the scalar field at zero. Consequently, the charged scalars are stabilized at the origin as long as the first, flux induced, term in Eq. (31) dominates, i.e. for
Moreover, the real part of the gauge kinetic function, determining the effective gauge coupling, has to be positive for the theory to be consistent. Because of the negative prefactor of this leads to a restriction of the physical moduli space parametrized by and (see Eq. (26)),
Hence, for the charged fields are always stabilized at zero
for and in the physical region of the moduli
A superpotential for the moduli can arise at the orbifold fixed points
in the six-dimensional supergravity theory. The superpotential in the
4d effective action is the sum of the fixed point contributions.
Consistency requires this superpotential to be gauge invariant
The superpotential is a holomorphic function of and the gauge invariant modulus . Inspired by typical superpotentials induced by nonperturbative effects, such as gaugino condensation or instanton corrections, we assume the following functional dependence
where, without loss of generality, we choose the parameters , , and to be real. For an exponential suppression of these nonperturbative effects we further demand that and are real and positive. The -term scalar potential reads
Note that the Kähler potential (23) is of the no-scale form, and the contribution is therefore canceled. The F-term potential (36) also contributes to the scalar mass term . However, this contribution is , which is much smaller than the leading contribution , and it can therefore be neglected.
With the charged fields stabilized at zero the -term
and the linearly independent combination
can be used to rewrite the scalar potential in the convenient form
where the parameters of the superpotential are encoded in
These are worked out in App. A. We use an inverted procedure to obtain the superpotential parameters by solving Eqs. (41) after fixing the vacuum expectation values of the moduli fields and the energy density. Consequently, for vanishing or small cosmological constant the derived superpotential is fine-tuned to compensate the large positive -term depending on and . To solve Eqs. (41) we further have to fix one of the parameters, which we choose to be . However, one of the above combinations, , can be uniquely determined in terms of the moduli values at the minimum. Up to the prefactor the form is identical to the two moduli case discussed in ,
where is the ratio . Therefore, the arguments for the existence of vacua given in  carry over to the three moduli case. Importantly, one necessary condition is that the prefactor of one of the moduli in the gauge kinetic function is negative. In the above case this constrains the allowed moduli region to a regime where the -terms are positive definite. The additional negative contributions from the -terms of the gauge invariant superpotential allow to find Minkowski or de Sitter vacua with all moduli stabilized. In this way we can construct models with Minkowski or de Sitter vacua with a size of the internal space that ranges between GUT and TeV scale.
Given a vacuum with and the cancellation between - and -term contributions to the potential, we can derive the -dependence of the various parameters and masses. For parametrized by solutions to Eqs. (41) can be found for certain values of the parameter . Keeping this parameter combination fixed we then obtain models with different size of the extra dimensions. Accordingly, the scaling of the effective gauge coupling and -term potential is
This directly implies a scaling of the superpotential parameters
and allows to deduce the behavior of the bosonic masses
Superpotential and boson masses: two examples
In order to illustrate our general results and to get some intuition for the parameters and mass scales involved, we work out explicitly two models exhibiting Minkowski vacua with different size of the internal dimensions. As explained above we start with the choice of the gauge coupling , the size of the compact dimensions, and in the superpotential. The parameters of the two models read
The number of flux quanta in both vacua is set to , which ensures the stabilization of the charged scalar fields, see Eq. (32), and already hints at a multiplicity which can be used in grand unified model building . The complex shape modulus is stabilized at , which corresponds to the square torus assumed in .
To achieve in the vacuum we parametrize by as above. Therefore, the mass scale of the internal dimensions (in 4d Planck units) is given by
which corresponds to the GUT scale and an intermediate mass scale, respectively. The moduli are stabilized at , . As discussed above, the combination remains constant in both models. The minima in the two different models are plotted in the - and the - plane in Figures 1 and 2.
One immediate consistency check for the solutions is the value of the effective gauge coupling which has to be positive and small enough to allow a perturbative treatment
Again, the scaling with of Eq. (45) is realized. For the masses of the moduli fields and the axions we need to evaluate the superpotential parameters. From Eq. (44) we would expect a factor of and indeed the respective orders of magnitude are
The numerical values are given in App. C. The nonperturbative exponent in both vacua is . Knowing the superpotential, and after canonical normalization, the eigenvalues of the moduli masses matrix can be calculated,
The eigenvalues are all of the order of the vector boson mass and slightly larger than the gravitino mass. The scaling between the two models matches the one predicted in Eq. (45). The same is true for the two non-vanishing eigenvalues of the axion mass matrix
One massless axion gives mass to the vector boson via the Stückelberg mechanism. Their numerical values in both vacua are discussed in App. C, where the scaling is explicitly demonstrated.
It is instructive to compare the mass spectra of the two models. From Eq. (31) one reads off that in both cases . Hence, even the lightest charged scalar does not belong to the low-energy effective Lagrangian for fields with masses . Nevertheless, we included this scalar in the above discussion to check the stability of the vacuum, that is . The vector boson, moduli and axion masses scale as and become parametrically smaller than the size of the extra dimension for large .
4 Super-Higgs mechanism and fermion masses
The first step in calculating the fermion masses is to disentangle their mixing with the gravitino, i.e. to identify the Goldstino. Since supersymmetry is broken by - and -terms, the Goldstino is a mixture of the gaugino and modulini. Charged fermions do not contribute as long as their scalar partners have vanishing vacuum expectation values. We can therefore restrict our discussion to the gaugino and modulini, and we also assume a ground state with vanishing cosmological constant.
The bilinear fermionic Lagrangian involves kinetic, mass, and mixing terms for the gravitino, gaugino and modulini fields. All these terms are determined by the Kähler potential, the superpotential, the Killing vectors and the gauge kinetic function of the model. The terms involving the gravitino are 
where is a linear combination of the fermions in the theory, and the gravitino mass is given by
In the model under consideration this yields
The -term affects the value of the gravitino mass via the expectation values of the moduli fields determined in a vacuum with vanishing cosmological constant.
The fermion in Eq. (54) is the Goldstino. It is given by a linear combination of the gaugino and modulini , determined by - and -terms,
With this shift, a straightforward calculation yields for the Lagrangian, (54)
Note that the kinetic term of has the sign of a ghost. The kinetic term and the mass term of lead to modifications of the kinetic terms and the mass matrix of gaugino and modulini whereas the mass term for the gravitino remains unchanged.
Eq. (4) represents the gravitino Lagrangian in unitary gauge. Hence, the dependence on the Goldstino should completely disappear in the full Lagrangian, which we explicitly demonstrate in the following. The kinetic terms of gaugino and modulini read
Since the Kähler metric is hermitian, it can be diagonalized by a unitary transformation. Moreover, the real eigenvalues are all positive for the kinetic terms to be well-defined. Hence, one can rescale the eigenvalues to one by conjugation with a real diagonal matrix. The combined transformation corresponds to a vielbein for the Kähler metric,
It is convenient to redefine gaugino and modulini,
such that their kinetic terms are canonical,
the new index labels gaugino () and modulini (). One can now perform a unitary transformation which rotates the canonically normalized fermions into and three orthogonal Weyl fermions
One easily verifies the unitarity condition for the -component,
where we used . Clearly, in Minkowski space one has . Since all fermions are canonically normalized the kinetic terms for the orthogonal Weyl fermions and are given by
Combined with Eq. (4) the kinetic terms for cancel, as expected in the unitary gauge.
Furthermore, we expect the mass eigenvalue of the Goldstino and its mixing terms with the other fermions to vanish. In order to show this, we express the fermion bilinears in terms of the canonically normalized gaugino and modulini,
which indeed vanishes in Minkowski space for a gauge invariant superpotential. Analogously, one finds
which also vanishes in Minkowski space for an extremum of the scalar potential. Hence, the Goldstino indeed decouples from the mass matrix.
is the mass matrix of the fermions orthogonal to the Goldstino. Again, the cancellation of - and -terms for small cosmological constant allows to derive a general scaling behavior for the fermion masses. With Eqs. (43) and (44) we obtain
Gravitino and fermion masses: two examples
The modulini masses are evaluated numerically from Eq. (87). As expected, one finds one eigenvector with zero mass, the Goldstino. The three remaining fermion fields are massive with mass eigenvalues of order the gravitino mass
Again, the explicit numerical values and scaling behaviors are summarized in App. C. An overview over the mass spectrum in both vacua is provided in Fig. 3. Interestingly, in both of the vacua one modulini field remains lighter than the gravitino whereas the other two are slightly heavier.
5 Summary and Outlook
We have analyzed a 6d supergravity model compactified to 4d on an orbifold. Using bulk flux, a nonperturbative superpotential, and the Green-Schwarz term for anomaly cancellation we obtained 4d de Sitter or Minkowski vacua where all moduli are stabilized. This allows for an explicit computation of the masses of all particles in the effective low-energy theory.
In the model under discussion, supersymmetry is broken by both - and -terms. By analyzing the bosonic 6d effective action, we extracted the -term potential resulting from the FI parameter of the anomalous , which receives contributions from the Green-Schwarz term and from the bulk flux. From the Green-Schwarz term we also obtained an important correction to the gauge kinetic function. The -term potential results from our choice of the superpotential which is of the KKLT-type. Knowing the complete scalar potential we then calculated the boson masses which depend on the superpotential parameters and the flux.
For the discussion of the fermion masses we have studied the super-Higgs mechanism in the presence of - and -term breaking. Via a rotation in field space we extracted the Goldstino which is eaten by the gravitino in unitary gauge. The Goldstino indeed completely drops out of the Lagrangian, as we explicitly verified using the extremum conditions of the scalar potential for Minkowski space and the gauge invariance of the superpotential.
In order to find vacua of our effective theory that are de Sitter or Minkowski and are within a reasonable parameter range for the moduli, we inverted the problem: Choosing a gauge coupling and the size of the extra dimensions, and starting from a point in moduli space we derived equations for the superpotential parameters for which the scalar potential is minimized. Having obtained the parameters of our effective theory that way, we inserted the parameters back into the scalar potential which we then minimized. As a cross-check, we found the minimum exactly at the point in moduli space which we used to obtain the parameters in the inverted problem.
Finally, we discussed two example models with different parameters and evaluated numerically the masses of all particles in the model. In the first example, the extra dimensions are of order the GUT scale, , and moduli, axion and gauge boson masses are also , slightly larger than gravitino and modulini masses. In the second example, the size of the extra dimensions corresponds to an intermediate scale, and all masses scale as . The only exception is the charged scalar mass which is of the order of the compactification scale. The size of the extra dimensions is controlled by the gauge coupling, with . This dependence on the gauge coupling can be used to construct models whose size of the extra dimensions interpolates between the GUT scale and the TeV scale.
The constructed family of de Sitter vacua can easily be combined with higher-dimensional GUT models, and they also offer an interesting playground to study the interplay of moduli stabilization and inflation. Since the considered 6d flux compactifications contain all ingredients familiar from string models, i.e. compact dimensions with flux, the Green-Schwarz mechanism and a nonperturbative superpotential, it will be very interesting to see whether they can in fact be realized within a string theory construction.
We thank Emilian Dudas, Zygmunt Lalak, Jan Louis, Hans-Peter Nilles, and Alexander Westphal for valuable discussions. This work was supported by the German Science Foundation (DFG) within the Collaborative Research Center (SFB) 676 “Particles, Strings and the Early Universe”. M.D. also acknowledges support from the Studienstiftung des deutschen Volkes.
Appendix A Parameters for de Sitter and Minkowski vacua
For the evaluation of the superpotential parameters it is convenient to define new linear combinations of the derivative operators
In terms of this derivative operators the constraints (41) can be rewritten as (note that )
Using the specific form of the nonperturbative superpotential (35) the parameters can be identified as: