Gaugino mass in heavy sfermion scenario
The heavy sfermion scenario is naturally realized when supersymmetry breaking fields are charged under some symmetry or are composite fields. There, scalar partners of standard model fermions and the gravitino are as heavy as TeV while gauginos are as heavy as TeV. The scenario is not only consistent with the observed higgs mass, but also is free from cosmological problems such as the Polonyi problem and the gravitino problem. In the scenario, gauginos are primary targets of experimental searches. In this thesis, we discuss gaugino masses in the heavy sfermion scenario. First, we derive the so-called anomaly mediated gaugino mass in the superspace formalism of supergravity with a Wilsonian effective action. Then we calculate gaugino masses generated through other possible one-loop corrections by extra light matter fields and the QCD axion. Finally, we consider the case where some gauginos are degenerated in their masses with each other, because the thermal relic abundance of the lightest supersymmetric particle as well as the the strategy to search gauginos drastically change in this case. After calculating the thermal relic abundance of the lightest supersymmetric particle for the degenerated case, we discuss the phenomenology of gauginos at the Large Hadron Collider and cosmic ray experiments.
- I Introduction
II Heavy sfermion scenario
- II.1 Higgs mass and scalar mass
- II.2 SUSY breaking without singlet
- II.3 Compatibility with cosmology
III Anomaly mediated gaugino mass
- III.1 Approximate super-Weyl symmetry in classical action
III.2 Anomaly of the super-Weyl symmetry and Gaugino Mass
- III.2.1 Wilsonian effective action
- III.2.2 Super-diffeomorphism invariance
- III.2.3 Anomaly of the super-Weyl symmetry
- III.2.4 Gaugino mass in the Wilsonian effective action
- III.2.5 Radiative corrections from path-integration
- III.2.6 Decoupling effects of massive matter
- III.2.7 Anomaly mediation a la Pauli-Villars regulalization
- III.3 Fictitious super-Weyl gauge symmetric formulation
- III.4 Non-abelian gauge theory
- IV Gaugino mass by additional matter fields
- V Gaugino coannihilation
- VI Summary and discussion
- A MSSM Higgs
B Review on supergravity
- B.1 Gravity theory from local Lorentz symmetry
- B.2 Global supersymmetry
- B.3.1 super-diffeomorphism and super local Lorentz symmetry
- B.3.2 Vielbein, connection and torsion constraint
- B.3.3 Supergauge transformation
- B.3.4 Gauge fixing and supergravity transformation
- B.3.5 Chiral and real multiplet
- B.3.6 Super-diffeomorphism and super local Lorentz invariant action
- B.3.7 Chiral representation
- C Super-Weyl transformation
- D Super-diffeomorphism invariant measure
- E Anomaly mediated gaugino mass in SUSY breaking with singlet
- F Annihilation of gluino
- G Wino annihilation cross sections
Supersymmetry (SUSY) has been intensively studied as a fundamental law of nature for following reasons: SUSY reduces degrees of divergences in quantum field theories Grisaru et al. (1979), and hence stabilizes the vast separation between the electroweak scale and the Planck scale or the grand unification scale Maiani (1979); Veltman (1981); Witten (1981); Kaul (1982). If SUSY is broken by some strong gauge dynamics, the smallness of the electroweak scale is explained by dimensional transmutation. In the minimal SUSY extension of the standard model (MSSM), three gauge coupling constants unify, which is consistent with the prediction of the simplest grand unified theory (GUT), namely the GUT Georgi and Glashow (1974). With an parity preserved, the lightest SUSY particle (LSP) is a good candidate of dark matter Goldberg (1983); Ellis et al. (1984).
The discovery of the standard-model-like higgs as heavy as GeV Aad et al. (2012); Chatrchyan et al. (2012) strongly constrains the mass spectrum of MSSM particles. In the MSSM, quartic couplings of higgs fields are given by term potentials of standard model gauge interactions in the SUSY limit. Thus the mass of the standard-model-like higgs is lighter than the mass of the boson at the tree level.111We do not consider the case where the discovered higgs is not the lightest higgs. The observed large higgs mass indicates that quantum corrections from SUSY breaking effects to the higgs mass are significant Okada et al. (1991a, b); Ellis et al. (1991); Haber and Hempfling (1991). To explain the observed higgs mass, soft scalar masses of scalar top quarks are required to be larger than about TeV Hahn et al. (2014).
We note that gravity mediated soft scalar masses larger than TeV are compatible with physics in the early universe. Planck scale-suppressed mediation, where the gravitino mass is as large as soft scalar masses, with the gravitino mass larger than TeV, is free of the constraint from the Big-Bang-Nucleosynthesis even for a high reheating temperature Weinberg (1982); Pagels and Primack (1982); Khlopov and Linde (1984); Kawasaki and Moroi (1995); Kawasaki et al. (2005); Jedamzik (2006); Kawasaki et al. (2008a).
With scalar masses and the gravitino mass larger than TeV, one may naively expect that the LSP mass is also of the same order, which leads to the overproduction of the LSP from thermal bath in the early universe.222If the reheating temperature of the universe is smaller than the mass of the LSP, a correct abundance of the LSP can be obtained even if the LSP mass is TeV Chung et al. (1999); Giudice et al. (2001); Allahverdi and Drees (2002a, b); Harigaya and Mukaida (2014); Harigaya et al. (2014a, b). However, if SUSY breaking fields are charged or composite, gaugino masses are far smaller than soft scalar masses due to an approximate classical super-Weyl symmetry. Actually, this is the case with many dynamical SUSY breaking models (see e.g. Refs. Affleck et al. (1984a, b); Affleck et al. (1985); Luty and Terning (1998)).333For a dynamical SUSY breaking model with a singlet SUSY breaking field, see Ref. Harigaya et al. (2013a) for example. It should be noted that SUSY breaking models with charged or composite SUSY breaking fields are free from the Polonyi problem Coughlan et al. (1983); Ibe et al. (2006).
In this thesis, we consider the SUSY breaking scenario where the gravitino mass is larger than TeV and SUSY breaking fields are charged or composite, which we refer to as the “heavy sfermion scenario”. The PeV SUSY Wells (2003, 2005), the pure gravity mediation model Ibe et al. (2007); Ibe and Yanagida (2012); Ibe et al. (2012), the minimal split SUSY model Arkani-Hamed et al. (2012) and the spread SUSY model Hall and Nomura (2012) belong to this scenario. In Sec. II, we review the theoretical framework of the heavy sfermion scenario, including its cosmology.
Further, we assume that the Dirac mass term of the higgsino, so-called the term, is also as large as the gravitino mass. Indeed, the term as large as the gravitino mass is naturally explained if any charges of the up and the down type higgs chiral multiplets add up to zero Inoue et al. (1992); Casas and Munoz (1993) (see also Ref. Giudice and Masiero (1988)). The heavy sfermion scenario with this origin of the term is called the pure gravity mediation model Ibe et al. (2007); Ibe and Yanagida (2012); Ibe et al. (2012).
Assuming that the higgsino is heavy, we pay attention to gaugino masses in the heavy sfermion scenario. In the heavy sfermion scenario, gaugino masses are at least given by the anomaly mediation Randall and Sundrum (1999); Giudice et al. (1998). The anomaly mediation yields gaugino masses proportional to the gravitino mass and the beta functions of the corresponding gauge coupling constants. The LSP is the neutral wino with a mass of TeV for the gravitino mass of TeV.444If the higgsino threshold correction is large, the bino can also be the LSP. Due to its large (co)annihilation cross-section, the thermal abundance of the neutral wino is smaller than the observed dark matter abundance as long as the wino mass is smaller than TeV Hisano et al. (2007).
Sec. III is devoted to the derivation of the anomaly mediated gaugino mass in the superspace formulation of supergravity with a Wilsonian effective action Harigaya and Ibe (2014). In the superspace formulation of supergravity Wess and Bagger (1992), the anomaly mediated gaugino mass in the Wilsonian effective action invariant under the super-diffeomorphism was not known de Alwis (2008); D’Eramo et al. (2013). This is because the graivitino mass, which is the origin of the anomaly mediated gaugino mass, is the vacuum expectation value (VEV) of the scalar component of the supergravity multiplet. Couplings of the supergraivity multiplet to the gauge multiplet are strongly constrained by the super-diffeomorphism invariance. Thus, it is difficult (actually impossible) to write down the anomaly mediated gaugino mass in the Wilsonian effective action in a manifestly super-diffeomorphism invariant way. We give a super-diffeomorphism invariant expression of the anomaly mediated gaugino mass by taking the path-integral measure into account.
In Sec. IV, we discuss the deviation of gaugino masses from the prediction of the anomaly mediation in the MSSM. In the presence of a flat direction coupling to gauge charged matters, gaugino masses receive corrections as large as the anomaly mediation Pomarol and Rattazzi (1999). Actually, this is the case with a KSVZ-type QCD axion Kim (1979); Shifman et al. (1980). The KSVZ-type QCD axion couples to vector-like matter fields charged under the standard model gauge interaction. Thus, gaugino masses in general deviate from the prediction of the anomaly mediation in the MSSM if the KSVZ-type QCD axion exists Nakayama and Yanagida (2013).
Gaugino masses also receive corrections as large as the anomaly mediation if there are vector-like matter fields whose masses are smaller than the gravitino mass Nelson and Weiner (2002); Hsieh and Luty (2007); Gupta et al. (2013). Actually, vector-like matter fields as heavy as the gravitino are predicted in models with an anomaly free discrete symmetry Kurosawa et al. (2001); Harigaya et al. (2013b).
With the above two corrections, the gaugino mass spectrum drastically changes. For example, the gluino mass can be lighter than the prediction of the anomaly mediation in the MSSM, which enhances the detectability of the gluino at the Large Hadron Collider (LHC). We pay attention to the case where some gauginos are degenerated in their masses with each other, because the thermal relic abundance of the LSP as well as the the strategy to search gauginos drastically change in this case. We refer to this region of gaugino masses as the “gaugino coannihilation region”. In Sec. V, we calculate the thermal relic abundance of the LSP in the gaugino coannihilation region, and discuss the phenomenology of gaugino searches at the Large Hadron Collider (LHC) and cosmic ray experiments.
Ii Heavy sfermion scenario
In this section, we review the heavy sfermion scenario. We first review the relation between sfermion masses and the observed higgs mass, and show that heavy sfermion masses are suggested. Then we discuss the mass spectrum of SUSY particles in the heavy sfermion scenario, where SUSY breaking fields are charged or composite fields. As we will see, gaugino masses are one-loop suppressed in comparison with soft scalar masses. Finally, we investigate the compatibility of the heavy sfermion scenario with cosmology. We show the upper bound on the wino mass from the thermal relic abundance of the wino. We also show that the gravitino problem and the saxion/axino problem are is considerably relaxed in the heavy sfermion scenario.
ii.1 Higgs mass and scalar mass
ii.1.1 Tree level higgs mass
We first calculate the tree level higgs mass. At the tree level in the MSSM, the potential of the up-type higgs and the down-type higgs is given by
where , , , , and are the supersymmetric higgsino mass, the soft scalar squared masses of the up-type and the down-type higgs, the holomorphic quadratic soft mass, the and gauge coupling constants, respectively.
The minimum of the potential in Eq. (1) is calculated in Appendix A. In this section we assume the decoupling limit, where higgs bosons except for the standard-model like one are far heavier that the boson. Then the standard-model like higgs is given by
where . The potential of is given by
where GeV is the VEV of the higgs determined by quadratic mass terms. Remembering that the mass of the boson is given by , the mass of , , is given by
At the tree level, the mass of the standard-model like higgs is lighter than the mass of the boson, GeV.
ii.1.2 Quantum correction by SUSY breaking
The tree level mass of the higgs is bounded from above because the quartic coupling of higgs is given by the term potential, whose value is restricted by SUSY. SUSY breaking effects can raise the quartic coupling through quantum corrections. At the one-loop level, corrections are dominated by the following two effects.
where , , and are the top yukawa, the soft trilinear coupling of higgses and stops, the soft squared masses of the left-hand and right-hand stops, respectively. Another is the running of the quartic coupling. For simplicity, we assume that SUSY particles have the same mass and hence the running is determined by the standard model interaction from the soft mass scale down to the electroweak scale,
where is the renormalization scale. The correction in Eq. (6) should be added at the soft mass scale and the renormalization equation (8) should be solved with the corrected boundary condition at the soft mass scale.
In Fig. 2, we show the higgs mass obtained from Eqs. (6) and (8) with and . Here, we have also included the threshold correction at the electroweak scale summarized in Ref. Degrassi et al. (2012). The blue band shows the observed higgs mass, GeV Aad et al. (2014). It can be seen that GeV is obtained for sufficiently large . For example, for , the observed higgs mass requires TeV. For the recent accurate calculation, see Ref. Hahn et al. (2014).
ii.2 SUSY breaking without singlet
In most of dynamical SUSY breaking models (see e.g. Refs. Affleck et al. (1984a, b); Affleck et al. (1985); Luty and Terning (1998)), SUSY breaking fields are charged under some symmetry or are composite fields. We refer to this type of SUSY breaking as “SUSY breaking without singlet”. In this subsection, we discuss the mass spectrum of SUSY particles expected in SUSY breaking without singlet.
We denote the SUSY breaking field as and assume the following simplest effective superpotential,
where is the SUSY breaking scale and is a constant term.555When the SUSY breaking field have a charge under some symmetry, has a charge so that the superpotential is invariant under the symmetry. That is, the SUSY breaking is associated with the breaking of the symmetry. The gravitino mass is given by , where is the reduced Planck mass. Vanishing of the cosmological constant requires that , where we have normalized so that it is canonical.
ii.2.1 Soft squared scalar mass
Let us first discuss soft squared scalar mass terms of chiral multiplets in the MSSM. The SUSY breaking field in general couples to through the Kahler potential,
We assume that the SUSY breaking sector couples to the standard model sector only through Planck scale suppressed interactions and hence is at largest . The soft squared mass term of is given by
Thus, in general, the soft squared scalar mass term is as large as the gravitino mass. Among MSSM higgses, only the standard model-like higgs is light while other higgses have the same mass as large as the gravitino mass.
We note that generic without any structures induce flavor changing neutral currents, CP violations and lepton flavor violations. For discussions on these issues in the heavy sfermion scenario, we refer to Refs. Sato et al. (2012); Moroi and Nagai (2013); Moroi et al. (2014); Sato et al. (2013); Altmannshofer et al. (2013); Fuyuto et al. (2013); Tanimoto and Yamamoto (2014).
ii.2.2 term and term
In order for the electroweak symmetry to be broken by VEVs of MSSM higgses, the term must be as large as or smaller than soft scalar mass terms of higgses (see Appendix A). The origin of the correct magnitude of the term is one of the key issues in SUSY model-building, which is dubbed as the “ problem”.
A trivial way to obtain the term is to assume that the combination has an charge of 2 while is neutral under other symmetries.666We use a normalization of the charge where the superpotential has an charge of 2. Then the term of any magnitude is allowed. The natural value would be as large as the fundamental scale such as the Plack scale and the GUT scale. Indeed, this is the case with the minimal GUT model Witten (1981); Dimopoulos et al. (1981); Dimopoulos and Georgi (1981); Sakai (1981). In the landscape point of view Bousso and Polchinski (2000); Kachru et al. (2003); Susskind (2003); Denef and Douglas (2004), the small term may be selected from the landscape by the anthropic principle Weinberg (1987). is obtained by the supergravity effect.
Here, instead of readily adopting the anthropic principle, we show two ways to naturally obtain a small term. In both cases, the term is forbidden by a symmetry and is given by the breaking of the symmetry.777 A solution to the problem in this way requires the GUT group to be a product one Witten (2001); Harigaya et al. (2015a). We do not discuss the origin of the smallness of the breaking scale in detail here. The smallness is naturally explained, for example, if the breaking scale is generated by dimensional transmutation.
The first model assume the Peccei-Quinn (PQ) symmetry Peccei and Quinn (1977a, b). The PQ charge of MSSM fields is given in Tab. 1. With this charge assignment, yukawa couplings are allowed while the term is forbidden. The term is provided by the following coupling to a PQ breaking field with a PQ charge Kim and Nilles (1984),
For , the term is of TeV for the PQ breaking scale of GeV. This range is consistent with the lower bound from the burst duration of SN1987A (Ref. Raffelt (2008) and references therein) and the “upper bound” from the cosmic abundance of the axion by an initial misalignment angle Preskill et al. (1983); Abbott and Sikivie (1983); Dine et al. (1981).
The term is given by the VEV of the scalar component of the supergravity multiplet and the term of as large as (see Sec. IV.2). Both contributions result in the term of . If , is smaller than the soft squared masses of sfermions, and hence is large unless cancellation occurs (see Eq. (136)). Then the gravitino mass must be less than TeV to explain the observed higgs mass. In this case, however, gaugino masses discussed later are too small that it is inconsistent with constraints from the LHC. Thus, the term must be as large as the gravitino mass.
The second model assumes the symmetry. We assume that the combination has vanishing charges under any symmetries.888If is a fundamental field with an charge of and without any other charges, an order one yukawa coupling between and in the superpotential is possible. Then higgs fields obtain large vacuum expectation values. We assume that is a composite field, has an charge other than , or charged under non- symmetries. Then following terms are allowed in the Kahler potential,
Then the and terms are given by Ibe et al. (2007)
The term of can be understood by observing that the chiral higgsino pair has an charge of while the gravitino mass has an charge of . Since is as large as the soft squared mass term, the natural value of is (see Eq. (136)). Then the observed higgs mass requires TeV. The term can be far smaller than the gravitino mass by tuning , but we assume a natural value, .
ii.2.3 Gaugino mass
If is charged under some symmetry, the coupling between and a gauge multiplet through the gauge kinetic function is forbidden because the gauge kinetic function is neutral under any symmetries. If is a singlet composite field in a dynamical SUSY breaking model, the following coupling may be possible,
where is the dynamical scale of the SUSY breaking model. Here, we assume that corresponds to a composite field composed of chiral multiplets. In both cases, the tree-level gaugino mass is far smaller than the gravitino mass.
where and are the gauge coupling constant and the beta function of , respectively. The anomaly mediated gaugino mass is derived in Sec. III.
At the one-loop level in the MSSM, bino, wino, and gluino masses (, , and ) are given by
where , and are the gauge coupling constants of , and , respectively. Here, we use the GUT normalization for the gauge coupling constant, .
In addition to the anomaly mediation, electroweak gauginos (bino and wino) receive threshold corrections from the higgsino via the diagram in Fig. 3. The corrections are evaluated as Giudice et al. (1998) (the calculation is the essentially same as the one in Sec. IV.1),
where is the mass of heavy higgses. The contributions are comparable to those of anomaly mediated contributions when and .
In the MSSM, physical masses of the gauginos are obtained by adding the contributions in Eqs. (17) and (18), and also considering the effect of renormalization group running of the masses down to those scales from ,
In Fig. 4, the gaugino masses are shown as a function of assuming the phase of the higgsino threshold corrections to be zero (), and TeV. Unless the higgsino threshold correction is large, the wino is the LSP.
The wino LSP is constrained by the disappearing track search at the LHC as GeV Aad et al. (2013). Also, the search for jets with missing energy at the LHC put the constraint on the gluino mass, TeV Chatrchyan et al. (2014) unless the gluino is degenerated with the LSP. Thus, the gravitino mass larger than TeV is required, which is consistent with the observed higgs mass for .
Let us briefly mention the term. As is the case with gaugino masses, term is suppressed in SUSY breaking without singlet. However, the term is generated through the conformal anomaly. For a superpotential , the corresponding term is given by the wave function normalization as Randall and Sundrum (1999); Giudice et al. (1998)
which is one-loop suppressed in comparison with the gravitino mass.
ii.3 Compatibility with cosmology
We have shown that the gravitino mass of TeV is consistent with the observed higgs mass and the constraint from the LHC in the heavy sfermion scenario. Here, we investigate the compatibility of the large gravitino mass with cosmology. We discuss the thermal relic abundance of the LSP, the gravitino problem, and the saxion/axino problem. For the compatibility of the large gravitino mass with inflation models, see Refs. Buchmuller et al. (2012a, b); Nakayama and Takahashi (2012); Nakayama et al. (2012); Harigaya et al. (2013c); Takahashi (2013); Harigaya et al. (2014c, d); Nakayama et al. (2014); Harigaya and Yanagida (2014).
ii.3.1 Thermal abundance of the wino LSP
Unless the higgsino threshold correction is large, the wino is the LSP. The neutral and the charged wino masses degenerate with each other at the tree-level. Through quantum corrections by electroweak interactions, the neutral wino becomes lighter than the charged wino by MeV Cheng et al. (1999); Feng et al. (1999); Gherghetta et al. (1999); Yamada (2010); Ibe et al. (2013).
The thermal abundance of the neutral wino LSP has been calculated in Ref. Hisano et al. (2007), including the coannihilation between the charged and the neutral wino as well as the Sommerfeld effect (see Sec. V). In Fig. 5, we show the thermal abundance of the wino for a given wino mass. The blue band shows the observed dark matter abundance by the Planck experiment Ade et al. (2014). For TeV, the thermal abundance of the wino is consistent with the observed dark matter abundance.
In the present universe, the neutral wino annihilates into a pair of bosons at the tree level and a pair of photons at the one-loop level. The former mode yields gamma-rays, positrons and light elements with spread spectra while the latter yields gamma-rays with a line spectrum. For the wino search with cosmic-rays, see Refs. Moroi and Randall (2000); Ibe et al. (2012); Bhattacherjee et al. (2014).
ii.3.2 Gravitino problem
Let us consider the production of the gravitino from thermal bath in the early universe. Since the gravitino interacts with MSSM particles through higher dimensional interactions, the production of the gravitino from thermal bath is more efficient for higher temperature. Therefore, the number density of the gravitino is determined by the reheating temperature as Kawasaki and Moroi (1995); Kawasaki et al. (2008a),
where it the entropy density and we have used the approximation that gaugino masses are far smaller than the gravitino mass.
The produced gravitino eventually decays into the LSP. The abundance of the LSP produced in this way is given by
It is known that the thermal leptogenesis Fukugita and Yanagida (1986) requires the reheating temperature larger than about GeV Giudice et al. (2004); Buchmuller et al. (2005). Then, the thermal leptogenesis puts an upper bound on the mass of the LSP, TeV.
Next, let us discuss the constraint from the Big Bang Nucleosynthesis (BBN). The gravitino decays with a rate , where we have assumed that sfermions and the higgsino are heavier than the gravitino.999Inclusion of the decay into sfermions and higgsino does not change the following discussion. The lifetime of the gravitino is given by
Note that the gravitino dominantly decays into the gluino and the gluon. In this case, if the lifetime of the gravitino is longer than about sec, decay products of the gravitino hadronically interact with light elements and hence spoil the success of the BBN Kawasaki et al. (2005). For TeV, the constraint from the BBN is absent.
ii.3.3 Saxion/axino problem
In the heavy sfermion scenario, the saxion/axino problem is relaxed. In general PQ mechanisms, there is a pseudo-Nambu-Goldstone boson associated with the spontaneous breaking of the PQ symmetry, which is called the axion. The axion couples to the Pontryagin density of the QCD through the anomaly of the PQ symmetry. The strong dynamics of the QCD generate the potential of the axion. At the minimum of the potential, the angle of the QCD vanishes, which solves the strong CP problem Peccei and Quinn (1977a, b); Weinberg (1978); Wilczek (1978).
In SUSY models, the axion is associated with a light scalar called the saxion and a light fermion called the axino. , and form a chiral multiplet, which we refer to as the axion multiplet . The saxion and the axino are as light as the gravitino. Then their dynamics in the early universe cause various problems. Here, we discuss two problems which are important for TeV. For a rigorous discussion on generic gravitino masses, see Ref. Kawasaki et al. (2008b) and references therein.
Dark radiation from saxion
The saxion produced in the early universe in general decays into the axion at the tree level through the Kahler potential,
where is the decay constant of the axion. Here, we identify the imaginary part of the lowest component of as the axion. The axion produced in this way has a large momentum and hence works as radiation. The extra radiation component, which is called the dark radiation, changes the expansion history of the universe and hence is constrained e.g. by the BBN, the cosmic microwave background and the structure formation.
There are two sources of the saxion production in the early universe. One is the coherent oscillation of the saxion. The mass of the saxion is given by SUSY breaking, which is dominated by an inflaton sector and the SUSY breaking sector during and after inflation, respectively. Thus, the minima of the saxion potential during and after inflation are in general different and typically separated by the decay constant of the axion. The saxion starts its oscillation with an initial amplitude as the Hubble scale of the universe becomes comparable to the mass of the saxion . Assuming that the inflaton decays after the saxion starts its oscillation, the energy density of the saxion before it decays is
The other source is the thermal scattering. The saxion interacts with thermal bath with a rate
where and are the thermal number density of the gluon and the quark. The saxion is in thermal bath if the interaction rate is sufficiently larger than the Hubble rate, that is,
The energy density of the saxion before it decays is
The constraint on the dark radiation is conventionally expressed by the effective neutrino number defined by
where , , and are the energy density of the total radiation, the photon, the temperature of the neutrino and the photon, respectively. The standard cosmology predicts and the deviation is constrained.
As the saxion dominantly decays into the axion, is given by
where is the effective degree of freedom of the entropy density and is the temperature at which the saxion decays,
We adopt the limit by the Planck experiment Ade et al. (2014).
LSP overproduction from axino
The axino is produced from thermal bath and eventually decays into the LSP. If the temperature at which the axino decay ,
is smaller than the freeze-out temperature of the LSP, , the LSP produced in this way contributes to the dark matter density.
Here, we assume that the axino is not in thermal bath (). Otherwise, the LSP produced from the axino over-closes the universe. From Eq. (33), the abundance of the LSP is given by
In Fig. 6, we show the upper bound on the reheating temperature for a given gravitino mass and the axion decay constant. Here, we assume that and TeV. In the blue-shaded region, the saxion produces too much dark radiation. In the red-shaded region, the axino over-produces the LSP. It can be seen that the constraint is weaker for larger gravitino masses. This is because the saxion and the axino decay faster for larger masses. For example, for GeV and TeV, the saxion/axino problem is absent.
Iii Anomaly mediated gaugino mass
Anomaly mediated gaugino mass Randall and Sundrum (1999); Giudice et al. (1998) is the essential ingredient of the heavy sfermion scenario. In this section, we derive the anomaly mediated gaugino mass in the superspace formalism of supergravity with a Wilsonian effective action. For instructive discussions on the anomaly mediation, we refer to Refs. Bagger et al. (2000); Boyda et al. (2002); Dine and Seiberg (2007); Jung and Lee (2009); Sanford and Shirman (2011); D’Eramo et al. (2012, 2013); Dine and Draper (2014); Di Pietro et al. (2014).
iii.1 Approximate super-Weyl symmetry in classical action
Before discussing the anomaly mediated gaugino mass, let us first clarify the gaugino mass expected in the local supergravity action at the classical level.101010Here, we assume that the classical action consists of local interactions. If the classical action is allowed to be non-local, an arbitrary gaugino mass of can be introduced by using the non-local term in Eq. (77) without conflicting with the super-diffeomorphism invariance. In our discussion, we concentrate ourselves on a situation where SUSY is dominantly broken by some charged fields under some symmetries or by some composite fields. Otherwise direct interactions between the SUSY breaking fields and gauge multiplets lead to the “tree-level” gaugino mass of the order of the gravitino mass, . Under this assumption, direct interactions between the SUSY breaking fields and the gauge supermultiplets are suppressed at least by a second power of the Planck scale, and hence resultant gaugino masses from those interactions are negligible. By the same reason, we also assume that no SUSY breaking field obtains a VEV of the order of the Planck scale.111111These assumptions also eliminate contributions to gaugino masses from the Kahler and sigma-model anomalies (see Refs. Bagger et al. (2000); D’Eramo et al. (2012) and Appendix E).
Once we assume that gaugino masses from couplings to the SUSY breaking sector are highly suppressed, remaining sources of gaugino masses are couplings to the supergravity multiplets. As is well known, however, gaugino masses from tree level interactions to the supergravity multiplets are also suppressed in spite of the apparent -term VEVs of in the supergravity multiplets. As we shortly discuss, the absence of gaugino masses from the supergravity multiplets is due to an approximate super-Weyl symmetry, which is the key to understand the origin of the anomaly mediated gaugino mass in the next subsection. For the time being, we restrict ourselves to the gaugino mass generation in a gauge theory with a pair of vector-like matter fields.
iii.1.1 Classical supergravity action
In this thesis, we follow the notation and the formulation in Ref. Wess and Bagger (1992) (see Appendix B), except for the notation of complex conjugate (we use ) and for the normalization of gauge supermultiplets that we adopt in Ref. Arkani-Hamed and Murayama (2000). For a simple model with charged chiral multiplets and , and a gauge multiplet , the classical supergravity action is given by,
where , , , , , and are the chiral fermionic coordinate, the chiral density, the superspace covariant derivative, the superspace curvature, the Kahler potential, and the gauge coupling constant, respectively. Here, we have assumed that the chiral multiplets and are massless. By expanding with respect to the chiral multiplets, we can extract relevant interactions,
from which we can extract gauge interactions and kinetic terms. Other interactions are suppressed by the Planck scale.
Now, let us expand , , and in terms of component fields;
Here, , , and are the gaugino, the determinant of the vielbein, and the auxiliary scalar component of the supergravity multiplet, respectively. Ellipses denote terms which are irrelevant for our discussion on the gaugino mass. The auxiliary field is fixed by the equation of motion as
where we have omitted contributions from terms of SUSY breaking fields which are negligible under the assumption we have made at the beginning of this section.
Since the chiral density has a non-vanishing term, it might look non-trivial why a gaugino mass of does not appear from the interaction in Eq. (37).121212Technically speaking, the contribution from the term cancels with the contribution from in ; is given by covariant derivatives of and hence the supergravity multiplet is included in . In the rest of this section, we show that the absence of the gaugino mass in the classical action is understood by an approximate super-Weyl symmetry.
iii.1.2 Approximate super-Weyl symmetry
Let us consider the super-Weyl transformation parameterized by a chiral scalar (see Ref. Wess and Bagger (1992) and Appendix C),131313In this thesis we define an infinitesimal transformation of a superfield by .
where ellipses denote terms which are irrelevant for our discussion. denotes the lowest component of a superfield .
A parameter is the Weyl weight of .141414If is not a chiral scalar but a chiral density with a density weight , the super-Weyl transformation is given by, (41) From Eqs. (III.1.1) and (III.1.2), the transformation laws of , and are given by
From the transformation laws of the component fields in Eq. (III.1.2), it is clear that the possible origin of a gaugino mass of ,
is not invariant under the super-Weyl transformation. This shows that the gaugino mass is generated only through terms which break the super-Weyl symmetry.
As we immediately see, the kinetic term of the gauge multiplet in Eq. (37) is invariant under the super-Weyl transformation, and hence, does not contribute to the gaugino mass. Higher dimensional terms omitted in Eq. (III.1.1) are, on the other hand, not invariant under the super-Weyl transformation. Contributions from such terms to the gaugino mass are, however, at the largest of , and hence are negligible. Altogether, we find that there is no gaugino mass of originated from couplings to the supergravity multiplets due to the approximate super-Weyl symmetry.151515The term in Eq. (43) is invariant under the -symmetry and the dilatational symmetry parts of the super-Weyl symmetry, which are parameterized by the lowest component of . Thus, the gaugino mass from the couplings to the supergravity multiplets cannot be forbidden by the -symmetry nor the dilataional symmetry.
For later convenience, let us also note that the terms of massless matter fields in Eq. (36) are also invariant under the super-Weyl symmetry. That is, for , it can be shown that
Finally, let us stress that interaction terms of the gauge supermultiplets which are unsuppressed by the Planck scale is uniquely determined to the form of Eqs. (36) and (37) by the super-diffeomorphism invariance and by the gauge invariance. Thus, one may regard the approximate super-Weyl symmetry as an accidental one. Due to this accidental symmetry, the gaugino mass of is suppressed at the classical level.
iii.2 Anomaly of the super-Weyl symmetry and Gaugino Mass
In the last subsection, we have shown that no gaugino mass of is generated through couplings to the supergravity multiplets even though the chiral density has a non-zero term, due to the approximate super-Weyl symmetry. However, the approximate super-Weyl symmetry is in general broken by quantum effects. In this subsection, we investigate effects of quantum violation of the approximate super-Weyl symmetry by Fujikawa’s method Fujikawa (1979) in a Wilsonian effective action.
iii.2.1 Wilsonian effective action
To discuss quantum effects on the super-Weyl symmetry, we take the local classical action in the previous section (Eq. (III.1.1)) as a Wilsonian effective action with a cutoff at the Planck scale. Here, let us remind ourselves that effective quantum field theories suffer from ultraviolet divergences, and hence, they are well-defined only after the divergences are properly regularized. In our arguments, we presume an ultraviolet regularization such that the “tree-level” action at the cutoff scale is manifestly invariant under the super-diffeomorphism and the gauge transformations. We refer to this super-diffeomorphism invariant tree-level action at the cutoff scale as the Wilsonian effective action.161616Although we fix the cutoff scale to the Planck scale for a while, the following discussion is essentially unchanged as long as the cutoff scale is far larger than the gravitino mass. We also discuss effects of the change of the cutoff scale later.
The Wilsonian effective action in general includes higher dimensional interactions than those in Eq. (III.1.1) suppressed by the cut off scale. As we have discussed, however, contributions from those terms to the gaugino mass are highly suppressed by the cutoff scale and hence negligible. One concern is whether non-local interaction terms appear in the Wilsonian effective action at the cutoff scale, which could lead to the gaugino mass of . In our argument, we presume that such non-local interactions do not show up in the Wilsonian effective action, which is reasonable because we are dealing with effective field theories after integrating out only ultraviolet modes.
iii.2.2 Super-diffeomorphism invariance
In the above definition of the super-diffeomorphism invariant theory, there is a missing ingredient, the measure of the path-integral. As elucidated in Ref. Fujikawa (1979), the path-integral measure plays a crucial role in treating quantum violations of symmetries. Moreover, the definition of the “tree-level” interactions in the Wilsonian effective action depends on the choice of the path-integral measure, which we will encounter shortly. To clarify these issues, let us first discuss which path-integral measure we should use in conjunction with the “tree-level” Wilsonian action.
Under the infinitesimal (chiral) super-diffeomorphism transformation,171717 In the formulation given in Ref. Wess and Bagger (1992), gauge symmetries are fixed except for the diffeomorphism invariance, the local Lorentz symmetry and the supergravity symmetry. Thus, is restricted so that it parameterizes only the diffeomorphism and the supergravity transformation. We retain the word “super-diffeomorphism” to simplify our expression. The word “super-diffeomorphism” used in Appendix B, on the other hand, refers to the full super-diffeomorphism. and transform as
where denotes the indices of the chiral super coordinate , parameterizes the super-diffeomorphism, and for . As is shown in Appendix D, naive path-integral measures of chiral fields are not invariant under the super-diffeomorphism due to the anomaly of the gauge interactions, i.e.
Instead, anomaly free measures are given by
For a later purpose, we define weighted chiral fields () which are no more chiral scalar fields but chiral density fields with density weights .
In our discussion, we take the super-diffeomorphism invariant Wilsonian effective action. Therefore, in order to obtain a super-diffeomporphism invariant quantum theory, we inevitably use the super-diffeomorphism invariant path-integral measure in Eq. (47). If we use different measures instead, we need to add appropriate super-diffeomorphism variant counter terms to the tree-level Wilsonian action so that the super-diffeomorphism is restored in the quantum theory.
iii.2.3 Anomaly of the super-Weyl symmetry
Once we choose appropriate path-integral measures for the charged fields, we can now discuss quantum violation of the super-Weyl symmetry. Here, since we are interested in the gaugino mass, we only look at the breaking of the super-Weyl symmetry by the anomaly of the corresponding gauge interaction.
Before proceeding further, let us comment on a technical point. As in Eq. (III.1.2), the super-Weyl transformation is accompanied by a super-diffeomorphism parameterised by , so that the super-Weyl transformation is expressed in terms of the component fields defined in the chiral superspace spanned by . The accompanied super-diffeomorphism, however, makes it complicated to discuss the quantum violation of the super-Weyl symmetry. To avoid such a complication, we only consider a subset of the super-Weyl transformation where has only an -term, i.e.
Here, is an arbitrary function of the space-time. Under this restricted super-Weyl transformation, we find , and hence, no super-diffeomorphism is accompanied. We refer this type of the super-Weyl transformation as an “-type” super-Weyl transformation. It should be noted that the -type super-Weyl transformation is sufficient to forbid the gaugino mass in the discussion of Sec. III.1. In the followings, we concentrate on the anomalous breaking of the -type super-Weyl symmetry.
Now let us examine the invariance of the path-integral measures in Eq. (47) under the -type super-Weyl transformation. Under the transformation, and are not invariant but transform by
Here, we have used the fact that the super-Weyl weight of the massless chiral fields are so that the kinetic term of the chiral fields in Eq. (36) is invariant under the super-Weyl symmetry. Thus, due to the Konishi-Shizuya anomaly Konishi and Shizuya (1985), we find that the super-diffeomorphism invariant measure is not invariant under the -type super-Weyl transformation. Instead, the -type super-Weyl invariant measures are given by
where and are invariant under the the -type super-Weyl transformation. Here, the weighted chiral superfields and have density weights .
It should be commented that the component fields of () defined by
have canonical kinetic terms, in a sense that kinetic terms does not contain the supergravity multiplets in the flat limit: For a generic chiral scalar superfield, , the chiral projection of its complex conjugate is given by
where the ellipses denote higher dimensional terms. Then, by remembering that the component fields of are related to those of via
we find that the kinetic terms of the component fields of are canonical and decouple from .18