IPMU 13-0152
ICRR 657-2013-6

The Peccei-Quinn Symmetry
from a Gauged Discrete R Symmetry
Keisuke Harigaya1 , Masahiro Ibe2,1 , Kai Schmitz1 and Tsutomu T. Yanagida1
1

Kavli IPMU (WPI), University of Tokyo, Kashiwa 277-8583, Japan
2

ICRR, University of Tokyo, Kashiwa 277-8582, Japan

Abstract
The axion solution to the strong CP problem calls for an explanation as to why the
Lagrangian should be invariant under the global Peccei-Quinn symmetry, U (1)PQ , to sucha high degree of accuracy. In this paper, we point out that the U (1)PQ can indeed survive
as an accidental symmetry in the low-energy effective theory, if the standard model gauge
R , forbidding all dangerous
group is supplemented by a gauged and discrete R symmetry, ZN

operators that explicitly break the Peccei-Quinn symmetry. In contrast to similar approaches,
R symmetry be anomaly-free forces us, in general, to extend thethe requirement that the ZN

supersymmetric standard model by new matter multiplets. Surprisingly, we find a large
landscape of viable scenarios that all individually fulfill the current experimental constraints
on the QCD vacuum angle as well as on the axion decay constant. In particular, choosing
R symmetry can takethe number of additional multiplets appropriately, the order N of the ZN

any integer value larger than 2. This has interesting consequences with respect to possible
solutions of the µ problem, collider searches for vector-like quarks and axion dark matter.

Contents
1 Introduction

32 Minimal extension of the MSSM with a PQ symmetry

5

2.1

Supersymmetric Standard Model Sector. . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Extra matter sector required by a non-anomalous ZNR symmetry . . . . . . . . . . .

6

2.2.1

Gauge anomalies of the ZNR symmetry . . . . . . . . . . . . . . . . . . . . . .

7

2.2.2

Constraints on the R charges of the MSSM fields . . . . . . . . . . . . . . .

8

2.2.3

Anomaly cancellation owing to new matter fields . . . . . . . . . . . . . . . 10

2.3

2.4

Extra singlet sector required to render the extra matter massive . . . . . . . . . . . 12
2.3.1

Coupling of the extra matter fields to a new singlet sector . . . . . . . . . . 12

2.3.2

Superpotential of the extra singlet sector . . . . . . . . . . . . . . . . . . . . 13

2.3.3

Identification of the PQ symmetry . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.4

Spontaneous breaking and colour anomaly of the PQ symmetry . . . . . . . 16

2.3.5

Mass scale of the extra matter sector . . . . . . . . . . . . . . . . . . . . . . 18

Generation of the MSSM µ term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1

µ term from spontaneous R symmetry breaking . . . . . . . . . . . . . . . . 20

2.4.2

Contributions to the µ term from spontaneous PQ breaking . . . . . . . . . 20

2.4.3

Singlet extension of the MSSM Higgs sector . . . . . . . . . . . . . . . . . . 21

2.4.4

Decay of the extra matter fields into MSSM particles . . . . . . . . . . . . . 24

3 Phenomenological constraints

28

3.1

Gauge coupling unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2

Shifts in the QCD vacuum angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1

PQ-breaking operators in the superpotential . . . . . . . . . . . . . . . . . . 31

3.2.2

PQ-breaking operators in the Kähler potential and the effective potential . . 34

3.2.3

Upper bounds on the axion decay constant . . . . . . . . . . . . . . . . . . . 35

3.2.4

Phenomenologically viable scenarios . . . . . . . . . . . . . . . . . . . . . . . 37

4 Conclusions and discussion

41

A Possible R charges of the MSSM fields

43

B Solution to the axion domain wall problem

47

2

1

Introduction

The Peccei-Quinn (PQ) symmetry, U (1)PQ , provides us with a very attractive mechanism to solve
the strong CP problem in quantum chromodynamics (QCD) [1, 2]. Up to now, a convincing
explanation for the origin of the PQ symmetry is, however, still pending, since it is a global
symmetry and any global symmetry is believed to be broken by quantum gravity effects [3, 4, 5].
In order for the PQ symmetry to accidentally survive in the low-energy effective theory, one thus
has to arrange for a sufficient suppression of all unwanted operators that explicitly break it. The
tight experimental upper bound on the QCD vacuum angle, θ̄ . 10−10 [6], necessitates in particularthat this suppression be extremely efficient. One natural way to protect the PQ symmetry is to
invoke some gauge symmetry that accidentally forbids all the operators that would break it too
severely. In this paper, we point out that, in the context of the supersymmetric standard model,
the role of this protective gauge symmetry could be played by a gauged discrete R symmetry, ZNR .
Given only the particle content of the minimal supersymmetric standard model (MSSM), any
ZNR symmetry, except for Z3R and Z6R , is anomalously broken by SU (3)C and SU (2)L instanton
effects [7, 8].1 On the supposition that a different ZNR symmetry, other than Z3R or Z6R , might
account for the protection of the U (1)PQ , we are hence naturally led to introduce an extra matter
sector canceling the MSSM contributions to the ZNR anomalies. For a particular value of N as
well as k additional pairs of vector-quark superfields charged under the MSSM gauge group, therequirement that the shift in the QCD vacuum angle induced by PQ-breaking operators be lessthan 10−10 then implies an upper bound on the axion decay constant fa . By identifying thoseextensions of the MSSM that yield an upper bound on fa above the astrophysical lower bound of
fa & 109 GeV [10], we are thus able to single out the values of N and k that are phenomenologically
viable. Surprisingly, for each integer value of N larger than 2, a variety of k values is admissible.
Here, k can in particular always be chosen such that the unification of the gauge coupling constants
still occurs at the perturbative level. Moreover, for k = 5, 6 and k ≥ 8, it is possible to protect
the PQ symmetry by means of a Z4R symmetry. As we will discuss, this is an especially interesting
case, since a Z4R may not only explain the origin of the PQ symmetry, but at the same time also
allow for a simple solution of the MSSM µ problem.
The very idea to protect the PQ symmetry against gravity effects by means of a gauge symmetryis, of course, not new. Many authors have, for instance, considered extensions of the standard
model gauge group GSM = SU (3)C × SU (2)L × U (1)Y by some continuous symmetry. Early
1R
We restrict ourselves to generation-independent ZN
symmetries, where N > 2, that commute with SU (5) anddo not consider anomaly cancellation via the Green-Schwarz mechanism [9] coming from string theory, cf. Sec. 2.2.

3

examples of such attempts include models based on the gauge group GSM × U (1)0 [4] or on thegroup E6 × U (1)0 [5]. Also extensions of the gauge group by a continuous and a discrete symmetry,such as GSM × SU (4) × ZN [11], SU (3)C × SU (3)L × U (1)Y × Z13 × Z2 [12] or SU (5) × SU (N ) ×
ZN [13], have been studied in the literature. Likewise, next to these field-theoretic models, string
constructions have been shown to give rise to accidental PQ symmetries. By compactifying theheterotic string on Calabi-Yau manifolds [14] or on Z6 -II orbifolds [15], it is, for example, feasible toretain accidental global symmetries in the low-energy effective theory as remnants of exact stringy
discrete symmetries. All of these approaches, however, rely on rather speculative assumptions
about the UV completion of the standard model (SM). In particular, they require in many cases
an ad hoc extension of the particle content of the standard model that is motivated by the intention
to eventually end up with a global PQ symmetry in the first place. In view of this situation, itis thus of great interest to assess what a minimal extension of the standard model or the MSSM
would look like that still accomplishes a successful protection of the PQ symmetry. The model
presented in Ref. [16] might, for instance, be considered a step into this direction. It forgoes any
additional continuous symmetry, but only extends GSM by a discrete Z13 × Z3 . Still, it comes witha multi-Higgs sector that, while being certainly interesting from a phenomenological point of view,lacks a decisive reason for its origin from a fundamental perspective.
Now, invoking nothing but a discrete ZNR symmetry in order to protect the PQ symmetry rests,
by contrast, on a very sound conceptional footing. A discrete R symmetry is an often importantand sometimes even imperative ingredient to model building and phenomenology in supersymmetry(SUSY). It allows for a solution to the µ problem [17, 18, 19], prevents too rapid proton decay [20,21] and forbids a constant term in the superpotential of order the Planck scale which, in scenariosof low-scale SUSY breaking, would otherwise result in a huge negative cosmological constant [22].
The existence of an R symmetry and its potential spontaneous or explicit breaking is furthermore
closely linked to the spontaneous breaking of SUSY, irrespectively of whether our present nonsupersymmetric vacuum corresponds to a true [23] or merely metastable ground state [24]. Finally,
it is interesting to observe that higher-dimensional supergravity theories such as superstring theory
always feature an R symmetry, which might be naturally broken down to its discrete subgroup
ZNR upon the compactification of the extra dimensions [25]. This last point may again be regarded
to be rather speculative, but it does not alter the fact that discrete R symmetries surely play an
preeminent role among all conceivable symmetries by which GSM could possibly be extended. In
this sense, the main result of this paper is that nothing but the arguably simplest and most natural
extra gauge symmetry, namely a gauged and discrete R symmetry ZNR , could be responsible forshielding the PQ symmetry from the dangerous effects of gravity.
4After having outlined why we are particularly interested in enlarging GSM by a gauged ZNRsymmetry, we shall present in the next section our minimal extension of the MSSM and explain(i) how the colour and weak anomalies of the discrete ZNR symmetry force us to introduce newmatter multiplets, (ii) how these new matter multiplets acquire masses as well as (iii) how a µterm of order of the soft masses can be generated dynamically. In Secs. 3, we will then study
the phenomenological constraints on our model and identify the viable combinations of N and k
along with upper and lower bounds on the axion decay constant fa . Finally, we conclude with asummary of our model and a short overview of its phenomenological implications. Two appendicesdeal with the R charges of the MSSM fields and a slight modification of our model that manages
to avoid the axion domain wall problem, respectively.

2

Minimal extension of the MSSM with a PQ symmetry

We shall now demonstrate how an anomaly-free discrete R symmetry ZNR in combination with an
extra matter sector automatically gives rise to a global PQ symmetry. As a preparation, let us
first summarize our conventions and assumptions regarding the MSSM sector.

2.1

Supersymmetric Standard Model Sector

We take the renormalizable MSSM superpotential to be of the following form,
1
WMSSM = huij 10i 10j Hu + hdij 5∗i 10j Hd + hνij 5∗i 1j Hu + Mi 1i 1i ,2

(1)where we have arranged the MSSM chiral quark and lepton superfields into SU (5) multiplets,
10 = (q, uc , ec ) and 5∗ = (dc , `). Throughout this paper, we shall assume that the tiny massesof the SM neutrinos are accounted for by the seesaw mechanism [26]. That is why we have also
introduced neutrino singlet fields, 1 = (nc ), in Eq. (1), next to the actual matter content of the
MSSM.2 Moreover, Hu and Hd is the usual pair of MSSM Higgs doublets, hu , hd and hν are Yukawa
matrices and M denotes the diagonalized Majorana mass matrix for the heavy neutrinos involvedin the seesaw mechanism. i and j finally label the three different generations of quarks and leptons,
i.e. i, j = 1, 2, 3.
We assume the MSSM quark and lepton fields to be unified in SU (5) representations in order
to allow for an embedding of the MSSM into a grand unified theory (GUT). Note, however, thattaking SU (5) alone to be the full GUT gauge group is problematic. The minimal supersymmetric2

As this sometimes falls victim to bad jargon, we emphasize that the fermions contained in nc are left-handed.
In fact, they are the hermitian conjugates of the right-handed neutrinos required for the seesaw mechanism.

5SU (5) GUT model [20, 27] namely fails to give GUT-scale masses to the coloured Higgs tripletsthat are expected to pair up with the MSSM Higgs doublets in complete SU (5) multiplets. This
results in too rapid proton decay and represents what is known as the infamous doublet-triplet
splitting problem [28]. In addition to that, the standard way to break SU (5) to GSM by meansof a 24-plet is not compatible with the assumption of an unbroken R symmetry below the GUT
scale.3 Because of that, we shall assume that SU (5) is merely a proper subgroup of the full GUTgroup, SU (5) ⊂ GGUT . An attractive possibility in this context is unification based on the productgroup SU (5) × U (3)H , which can be formulated in an R-invariant fashion [22, 30], while solving
the doublet-triplet splitting problem in a natural way [31].
Finally, we point out that we define the MSSM to conserve matter parity, PM , so as to forbid
all dangerous baryon and lepton number-violating operators in the renormalizable superpotential.
This renders the actual gauge group of the MSSM slightly larger than the one of the standard
model, GMSSM = GSM × PM . One possibility to account for the origin of matter parity is tointerpret it as the remnant discrete subgroup of a local U (1)B−L symmetry that is spontaneously
broken above the electroweak scale [32]. Here, B−L stands for the difference between baryon
number B and lepton L. Assuming the presence of an additional Abelian factor U (1)X in the
GUT gauge group orthogonal to SU (5), it can be expressed in terms of the Abelian GUT charge
X and the weak hypercharge Y through the relation X + 4Y = 5 (B−L), cf. also Appendix A.

2.2

R
Extra matter sector required by a non-anomalous ZN
symmetry

As outlined in the introduction, a discrete R symmetry ZNR represents a unique choice when
considering possible extensions of the MSSM gauge group. We now perform just such an extension,
such that the full gauge group G of our model also features a ZNR factor. A priori, we allow N , theorder of the ZNR symmetry, to take any integer value larger than 2. We disregard the case N = 2
since a Z2R symmetry, i.e. R parity, is not an R symmetry in the actual sense. By including a
Lorentz rotation, it can always be reformulated as an ordinary Z2 parity [19]. On top of that, given
only a Z2R symmetry, we would also be unable to forbid a constant term in the superpotential,
which would result in a cosmological constant of order the Planck scale. On the other hand, we
point out that, in the case of even N , the ZNR symmetry contains R parity as a subgroup, ZNR ⊃ Z2R
for N = 4, 6, 8, ... Depending on the details of the R charge assignments to the particles of ourmodel, this R parity coincides in some cases with the ordinary matter parity PM . In these cases,
we then do not need to additionally impose matter parity by hand, as it is already included in the3
If we managed to break SU (5) without breaking the R symmetry, we would be left with potentially interestingor dangerous GSM -charged exotics whose masses would only receive soft SUSY-breaking contributions [29].

6

ZNR factor of the gauge group. In all other cases, we rely on the assumption that a spontaneously
broken U (1)B−L gauge symmetry gives rise to matter parity at low energies. In summary, thegauge group of our model is, hence, given by(
ZNR × PMG = SU (3)C × SU (2)L × U (1)Y ×
ZNR
2.2.1

; ZNR 6⊃ PM.
; ZNR ⊃ PM

(2)

R
Gauge anomalies of the ZN
symmetry

We attribute the origin of the ZNR factor in the gauge group to the presence of a continuous gauged
R symmetry at high energies, after the breaking of which ZNR remains as a discrete subgroup. Thus
being part of the gauge group, it is crucial that the ZNR symmetry be anomaly-free. The relevant
anomaly cancellation conditions are those related to the colour as well as to the weak anomaly of the
ZNR , i.e. the ZNR [SU (3)C ]2 and the ZNR [SU (2)L ]2 anomaly, respectively. The anomaly coefficients
(C)

(L)

for these two anomalies, AR and AR , are given by [30, 33](C)

AR = 6 + Ng (3 r10 + r5∗ − 4) ,(L)

AR = 4 + Ng (3 r10 + r5∗ − 4) + (rHu + rHd − 2) .

(3)Here, r10 , r5∗ , r1 , rHu and rHd denote the R charges of the MSSM matter multiplets and Higgs
doublets and Ng = 3 is the number of fermion generations in the MSSM.4 Note that we haveassumed the R charges of the matter fields to be generation-independent. Otherwise, i.e. in thecase of generation-dependent R charges, the R symmetry would suppress some of the entries in
the Yukawa matrices hu and hd too heavily. We also remark that the R charges are normalized
such that the anti-commuting superspace coordinate θ carries R charge rθ = 1. By choosing a
different value for the R charge of θ, say, rθ0 6= 1, we always have the option to collectively rescale
all R charges by the common factor rθ0 /rθ .
Besides the colour and the weak anomaly, all further anomalies involving at least one ZNR factor
also have to vanish in order to render the ZNR symmetry fully anomaly-free. The anomalies non 3 2
linear in ZNR , such as ZNR or ZNR U (1)Y , are, however, sensitive to heavy, fractionally charged
states at high energies [34]. Similarly, the gravitational anomaly, ZNR [gravity]2 , also receives contributions from light sterile fermions as well as from hidden-sector fermions acquiring large massesof order the SUSY-breaking scale in the course of spontaneous SUSY breaking [7]. All of theseanomalies hence highly depend on the particle spectrum in the UV and, thus, do not allow us toderive further constraints on our model. In general, the ZNR [U (1)Y ]2 anomaly also does not yielda useful condition because the SM hypercharge is not quantized [34, 35]. Only if the GUT group
4

The authors of Ref. [8] have recently made the interesting observation that Ng ≥ 3 is a necessary condition forR
consistently extending the MSSM gauge group by an anomaly-free discrete R symmetry ZN
with N > 2.

7

is semi-simple, such that the normalization of the hypercharge is dictated by the gauge structure,
the ZNR [U (1)Y ]2 anomaly provides a meaningful constraint on the ZNR symmetry as well as on theset of particles charged under it.5
Obviously, we only included the contributions from the MSSM sector to the anomaly coefficients
(C)

(L)

(C)

(L)

in Eq. (3). AR and AR could, however, still receive corrections ∆AR and ∆AR due to newcoloured or weakly interacting fermions with masses at or above the electroweak scale. This
extra matter would need to be assembled in complete SU (5) multiplets in order not to spoil
the unification of the gauge coupling constants. Consequently, extra fermions ought to equally(C)

contribute to AR(C)∆AR = ∆AR

(L)

and AR , such that the corresponding corrections are equal to each other,
(L)

(C)

= ∆AR , and such that the difference between AR(L)

and AR ends up being

independent of the properties of the extra matter sector. A minimal necessary condition for
rendering the ZNR symmetry anomaly-free is hence that
(L)

(C)

(N )

AR − AR = rHu + rHd − 4 = 0 ,

(4)

(N )

where we have introduced the symbol = as a shorthand notation to denote equality modulo N ,
(N )

a = b

⇔

a mod N = b mod N⇔∃! ` ∈ Z : a = b + ` N .

(5)

(N )

The condition in Eq. (4) is equivalent to rHu + rHd = 4. We therefore see that an anomaly-free
ZNR symmetry automatically suppresses the µ term for the MSSM Higgs doublets.
2.2.2

Constraints on the R charges of the MSSM fields

Next to Eq. (4), the requirement that the first two terms in the superpotential WMSSM , cf. Eq. (1),be in accordance with the ZNR symmetry provides us with two further constraints on the R chargesof the MSSM fields,(N )2r10 + rHu = 2 ,(N )r5∗ + r10 + rHd = 2 .

(6)

(N )

The combination of all three conditions then implies 3r10 +r5∗ = 0, which automatically forbids thedangerous dimension-5 operator 10 10 10 5∗ in the superpotential, which would otherwise induce
too rapid proton decay [21]. Together with matter parity, the anomaly-free ZNR symmetry thusbans all baryon and lepton number-violating operators up to dimension 5 except for the operator
5∗ Hu 5∗ Hu , which we, of course, want to retain to be able to explain the small neutrino masses [36].
5

We mention in passing that neither of the previously discussed GUT gauge groups, i.e. neither SU (3) × U (3)Hnor SU (5) × U (1)X , is semi-simple. Assuming one of these two groups to correspond to the GUT gauge group, we2R
are hence not able to make use of the anomaly cancellation condition for the ZN[U (1)Y ] anomaly.

8

Finally, in the seesaw extension of the MSSM, we also have to ensure that the last two terms in
WMSSM respect the ZNR symmetry, which translates into(N )r5∗ + r1 + rHu = 2 ,(N )2r1 = 2 .

(7)Here, as for the second condition, we have assumed zero R charge for the Majorana neutrino mass
M , which is to say that we consider its origin to be independent of the mechanism responsible forthe spontaneous breaking of R symmetry.
In conclusion, we find that extending the particle content of the MSSM by three neutrino
singlets, the five R charges r10 , r5∗ , r1 , rHu and rHd are determined by the five conditions in
Eqs. (4), (6) and (7). However, due to the fact that all of these conditions only constrain the
MSSM R charges up to integer multiples of N , they do not suffice to fix the values of r10 , r5∗ , r1 ,rHu and rHd uniquely. Instead, for each value of N , there exist exactly ten different possibilities
to assign R charges to the MSSM fields. In Appendix A, we derive and discuss these solutions
in more detail. In particular, we show that, for any given value of N , the different R charge
assignments are related to each other by gauge transformations. First of all, in consequence of theSU (5) invariance of the MSSM Lagrangian, the ten solutions split into two equivalence classes ofrespectively five solutions. As shown in Appendix A, these two classes are generated by the actionof Z5 transformations on the following two R charge assignments,
(N )

r10 =

1
N
+` ,
5
2

3
N
(N )
r5∗ = − + ` ,
5
2

(N )

r1 = 1 + `

N
,
2

(N )

rHu = 2 −

2
,
5

(N )

rHd = 2 +

2
,
5

(8)

where ` = 0, 1 and where Z5 ⊂ SU (5) is the center of SU (5). Furthermore, if matter parity stems
from a U (1)X symmetry that is part of the gauge group at high energies, i.e. if PM ⊂ U (1)X ,these two solutions are in turn related to each other by a PM transformation, such that eventually
all ten R charge assignments end up being physically equivalent. On the other hand, if matterparity is a subgroup of the ZNR symmetry, i.e. if PM ⊂ ZNR , the two solutions in Eq. (8) cannot
be related to each other and we are left with two inequivalent classes of solutions. Last but notleast, we remark that, for all values of ` and N , all of the R charges in Eq. (8), expect for r1
in some cases, are fractional. In Appendix A, we however show that, for each N 6= 5, 10, 15, ..,there exists at least one R charge assignment that is equivalent to one of the two assignments in
Eq. (8) and which only involves integer-valued R charges. But not only that, we also demonstrate
that, in a U (1)X -invariant extension of our model, all R charges in Eq. (8) can always be rendered
integer-valued by means of a U (1)X transformation.

92.2.3Anomaly cancellation owing to new matter fields

Irrespectively of the concrete R charges in Eq. (8), the anomaly constraint in Eq. (4) in combination
with the two conditions in Eq. (6) immediately implies for the anomaly coefficients in Eq. (3)(C) (N )AR(L) (N )(N )= AR = 6 − 4Ng = −6 .

(9)As this result does not rely on any of the two conditions in Eq. (7), it is independent of the fact that
we extended the MSSM particle content by three right-handed neutrinos. It rather equally applies
in the MSSM as well as in its seesaw extension. But more importantly, it leads us to one of the key
observations of this paper: as long as the order N of the ZNR symmetry is different from N = 3 orN = 6, we are forced to introduce a new matter sector in order to cancel the MSSM contributions
to the colour and the weak ZNR anomaly. In this sense, the introduction of new coloured and weaklyinteracting states in our model is not an ad hoc measure, but rather a natural consequence of therequirement of an anomaly-free discrete R symmetry.6
The simplest way to cancel the MSSM anomalies in Eq. (9) without spoiling the unificationof the gauge coupling constants is to introduce k pairs of vector-quarks and anti-quarks, Qi andQ̄i , where i = 1, .., k, that respectively transform in the 5 and 5∗ of SU (5).7 As they transform incomplete SU (5) multiplets, the extra quarks and anti-quarks yield equal non-MSSM contributions
(C)

(L)

∆AR and ∆AR to the colour and weak anomaly coefficients of the ZNR asymmetry. According
to Eq. (9), we must require that
(C)

(L)

(N )

∆AR = ∆AR = k (rQ + rQ̄ − 2) = 6 ,(N )rQQ̄ = rQ + rQ̄ = 2 +1(6 + `Q N ) ,k`Q ∈ Z , (10)where rQ and rQ̄ denote the generation-independent R charges of the extra quarks and anti-quarks,
respectively, and rQQ̄ is the common R charge of the bilinear quark operators QQ̄ i = Qi Q̄i . Just
like all other R charges, the R charge rQQ̄ is only defined up to the addition of integer multiples
of N . Hence, all inequivalent solutions to the condition in Eq. (10) lie in the interval [0, N ). In
addition, we observe that, varying `Q in integer steps, the R charge rQQ̄ changes in steps of Nk .
Consequently, for each pair of values for N and k, there are k inequivalent choices for rQQ̄ ,(N )rQQ̄ = 2 +1(6 + `Q N ) ,k

6`Q = 0, .., k − 1 .

(11)Again, this statement can be defined down by allowing for anomaly cancellation via the Green-Schwarz mechanism, in the case of which not only Z3R and Z6R can be rendered anomaly-free solely within the MSSM, but alsoR
R
Z4R , Z8R , Z12
and Z24[7, 37]. Moreover, it is worth noting that, in the context of a two-singlet extension of theR
MSSM, the anomaly-free Z24
symmetry can be used to successfully protect the PQ symmetry [7].
7Transforming in the 5 and 5∗ of SU (5), the new multiplets Qi and Q̄i , of course, also contain lepton doublets.
From a phenomenological point of view and with regard to the PQ solution of the CP problem, these are howeverless interesting as compared to the corresponding quark triplets. Because of that, we will refer to Qi and Q̄i as the
new quark and anti-quark superfields in the following.

10

A crucial implication of this result is that, in most cases, the extra quarks and anti-quarks aremassless as long as the ZNR is unbroken. Only for rQQ̄ = 2, a supersymmetric and R-invariant
mass term is allowed for the extra quark fields in the superpotential. An R charge rQQ̄ of 2 can,however, only be obtained in the case of a Z3R or a Z6R symmetry,rQQ̄ = 2 :(N, `Q ) = (3, −2) ,(N, `Q ) = (6, −1) ,k = 1, 2, 3, .. ,(12)

which are just the two ZNR symmetries that do not require an extension of the MSSM particle
content in the first place. As soon as the introduction of a new matter sector is mandatory inorder to render the ZNR symmetry anomaly-free, the new quark fields are therefore guaranteed tobe massless. This observation also reflects the self-consistency of our result in Eq. (11). If we had
started with the requirement of non-zero contributions from the new quarks to the ZNR anomalies
and we had found that the new quarks could possibly be massive, our derivation of rQQ̄ would befaulty, since massive quarks would not contribute to the ZNR anomalies to begin with.
Of course, the requirement of massless quarks at high energies does not say anything about themasses of the new quarks at low energies, where the ZNR is spontaneously broken. The spontaneous
breaking of the ZNR symmetry might, in fact, even generate masses for the extra quarks of theorder of the gravitino mass [38], cf. also Sec. 2.4.1. A necessary condition for this to happen isthat rQQ̄ = 0, which can be fulfilled for each value of N as long as k and `Q are chosen such(N )that −2k = 6. This means in particular that, for k = 1 and N = 4, 8, the R charge rQQ̄ isalways zero. For N = 3, 4, 6, 8 and only one pair of extra quark fields, k = 1, the generation of a
sufficiently large mass for the new quark flavour is therefore not an issue. The new quark either
exhibits a supersymmetric mass from the outset or it acquires a mass in the course of R symmetry
breaking [38]. For completeness, we also mention that, provided rQQ̄ = 0, the new quarks could
equally acquire masses of the order of the gravitino mass in the course of spontaneous SUSY
breaking via the Giudice-Masiero mechanism [17]. A further necessary prerequisite in this casewould then be that there exists a coupling of the extra quark fields to the SUSY breaking sector
in the Kähler potential.
In our following analysis, we will disregard the two exceptional choices for N and `Q in Eq. (12)as well as all combinations of N , k and `Q yielding rQQ̄ = 0. Instead, we shall focus on R charges
rQQ̄ that imply vanishing masses for the new quarks and anti-quarks before and after R symmetry
breaking as long as no further fields are introduced. The absence of a supersymmetric mass termis then equivalent to the statement that the renormalizable superpotential of the extra quark
sector vanishes completely. This is because all SM singlets solely composed out of the fields Q
and Q̄ must be combinations of the operator products QQ̄, Q5 and Q̄5 , such that QQ̄ is the only
11

conceivable operator which could potentially show up in the renormalizable superpotential. At therenormalizable level, the global flavour symmetry of the extra matter sector by itself, i.e. neglecting
its interactions with the other sectors of our model for a moment, is therefore maximally large,
VA
U (k)Q × U (k)Q̄ ∼= SU (k)VQ × SU (k)A
Q × U (1)Q × U (1)Q ,(13)with U (k)Q and U (k)Q̄ accounting for the flavour rotations of the left-chiral superfields Qi and Q̄i ,respectively. As we will see in Secs. 2.3.3, the axial Abelian flavour symmetry U (1)A
Q will play an important role in the identification of the PQ symmetry. Finally, we remark that higher-dimensional
operators as well as couplings of the new quarks and anti-quarks to other fields explicitly break theflavour symmetry. In order not to spoil the PQ solution to the strong CP problem, these explicit
breaking effects must be sufficiently suppressed by means of a protective gauge symmetry. We will
return to this point in Sec. 3.

2.3

Extra singlet sector required to render the extra matter massive

In the previous section, we have seen how the requirement of an anomaly-free ZNR symmetry forces
us to extend the MSSM particle content by new quark fields, Qi and Q̄i . Except for some special
cases, these quark fields are, however, massless as long as the ZNR symmetry is unbroken. Extra
massless coloured and weakly interacting particles are, of course, in conflict with observations,which is why we have to extend our model once more, so as to provide masses to the new quarks
and anti-quarks.
2.3.1

Coupling of the extra matter fields to a new singlet sector
In order to generate sufficiently large mass terms for the quark pairs QQ̄ i in the superpotential,
we are in need of a SM singlet that acquires a vacuum expectation value (VEV) at least above
the electroweak scale. No such singlet exists in the MSSM or its seesaw extension, so that we are
required to introduce another new field. Let us refer to this field as P and demand that it couples
to the quark pairs QQ̄ i in the following way,8
WQ =

1

k
X

n−1
MPl
i=1

λi P n (QQ̄)i .

(14)Here, MPl = (8πG)−1/2 = 2.44 × 1018 GeV is the reduced Planck mass and the λi denote dimensionless coupling constants, which we assume to be of O(1). The power n can, a priori, be any
8Note that the field P might be part of the hidden sector responsible for the spontaneous breaking of SUSY [39].

12integer number, n = 1, 2, ... Moreover, the coupling in Eq. (14) fixes the R charge rP of the singlet
field P . In order to ensure that it is indeed allowed in the superpotential, we require that(N )n rP + rQQ̄ = 2 + `P N ,

rP =


1
2 − rQQ̄ + `P N ,n`P ∈ Z .

(15)

Making use of our result for the R charge rQQ̄ for the quark pairs, cf. Eq. (11), we then find(N )rP = −6N+ (k `P − `Q ) .
nk
nk

(16)

Similarly as in the case of the extra quark fields, the R charge rP is not uniquely determined.
For each combination of values for N , n and k, there are instead nk inequivalent solutions to thecondition in Eq. (15). These are all of the form given in Eq. (16), with (k `P − `Q ) = 0, 1, .., nk − 1.
2.3.2

Superpotential of the extra singlet sector

So far, the field P does not possess any interactions that would endow it with a non-vanishing
VEV. We thus introduce another singlet field X and couple it to the field P , in order to generatea non-trivial F -term potential for the scalar component of P , 2Λ
WP = κ X− f (P, ..) ,2(17)

where κ is a coupling constant, Λ denotes some mass scale and f stands for a function of P andprobably other fields. We assume the scale Λ to carry zero R charge, which directly entails thatthe singlet field X and the function f must have R charges 2 and 0, respectively. Besides that,
we also assume a value for rP such that none of the operators P , P 2 , P 3 , XP , XP 2 and X 2 P isallowed in the superpotential WP , i.e. we require rP to fulfill all of the following relations at once,
(N )

(N )

(N )

(N )

(N )

rP 6= 2 ,2rP 6= 2 ,3rP 6= 2 ,2rP 6= 0 ,rP 6= −2 .

(18)

As we will see shortly, these conditions ensure that WP ends up featuring a flat direction which
can be identified with the axion and its superpartners.
Now, if the function f were merely composed out of powers P m of the field P , where m = 3, 4, ..,the R charge of f would only vanish for particular values of rP and m in the case of particular
ZNR symmetries. We, however, wish to be able to give masses to the new quarks and anti-quarks,
irrespectively of the concrete value of N . For that reason, we have to introduce a singlet field P̄carrying the opposite R charge of the field P ,
(N )

(N )

rP̄ = −rP =6N− (k `P − `Q ) ,
nk
nk
13

(19)

such that we are able to render the function f an R singlet by taking it to be a function of thesinglet pair P P̄ . The superpotential in Eq. (17) can then be fixed to be of the following form,

 2

 2
Λ
Λ
− f (P P̄ ) = κ X− P P̄ + .. ,(20)
WP = κ X
2
2
with the dots after the plus sign indicating higher-dimensional non-renormalizable terms and where,
similarly as above, we have assumed that rP̄ = −rP is such that none of the operators P̄ , P̄ 2 , P̄ 3 ,X P̄ , X P̄ 2 and X 2 P̄ is allowed in the superpotential WP . In addition to the five conditions in
Eq. (18), we therefore also have to require that
(N )

(N )

2rP 6= −2 ,3rP 6= −2 .

(21)

In total, we hence impose seven conditions on the R charge rP , which, depending on N , allow
us to forbid as many as 14 different values for rP .9 We now also see that each of the combinationsof N , `Q and k that either result in rQQ̄ = 0 or rQQ̄ = 2 violates exactly one of these conditions. If(N )rQQ̄ = 0, we know that n rP = 2, such that either P or P 2 is allowed. Similarly, rQQ̄ = 2 implies(N )n rP = 0, such that XP and/or XP 2 is allowed. This means that, in those cases in which we
do not depend on an extra singlet sector to generate masses for the extra quarks, Qi and Q̄i , wewould not even succeed in doing so, if we attempted it nonetheless. Finally, we emphasize that,by construction, XΛ2 and XP P̄ end up being the only renormalizable operators in WP that arecompatible with the ZNR symmetry for any value of N . In the following, we shall now show thatthe new singlet sector consisting of the fields X, P and P̄ has the potential to accommodate theinvisible axion and its superpartners and hence provide a solution of the strong CP problem via
the PQ mechanism.
2.3.3

Identification of the PQ symmetry

Evidently, the superpotential in Eq. (20) exhibits a global U (1) symmetry, viz. it is invariant undera global phase rotation of the fields P and P̄ . Let us refer to this symmetry as U (1)P and stipulate(P )that the two singlets P and P̄ respectively carry charge qP(P )= 1 and qP̄= −1 under it. The

U (1)P symmetry is explicitly broken by the coupling of the singlet operator P n to the quark pairsQQ̄ i in the superpotential in Eq. (14). At the same time, this coupling also breaks the U (1)AQ
symmetry in the extra quark sector. Altogether, the coupling between the new quark sector andthe new singlet sector reduces the number of global Abelian symmetries from three to two,U (1)P × U (1)VQ × U (1)AQ
9→U (1)PQ × U (1)VQ .

(22)

To see this, note that all of our conditions can be written as rP 6= a/q + `/qN , where q ∈ {1, 2, 3}, a ∈ {−2, 0, 2}
and ` ∈ Z. The number of different rP values forbidden by some condition therefore corresponds to its value for q.

14
The operators QQ̄ i are invariant under U (1)VQ transformations, which is why the global vectorial
symmetry in the quark sector survives the introduction of the superpotential in Eq. (14). The otherglobal symmetry leaving the coupling in Eq. (14) invariant corresponds to some linear combinationof U (1)P , U (1)VQ and U (1)A
Q . It is this symmetry that we shall identify with the PQ symmetry.
In the remainder of this paper, we will now investigate under which circumstances it may besuccessfully protected against the effects of higher-dimensional operators.
Before continuing, let us, however, reiterate once more for clarity: U (1)P , U (1)VQ and U (1)AQ are
accidental global symmetries of the new singlet and quark sectors at the renormalizable level thatarise due to our particular choice of R charges. Neither of them manages to survive as an exact
symmetry in the full low-energy effective theory. To begin with, the coupling between the two newVsectors in Eq. (14) breaks U (1)P × U (1)VQ × U (1)AQ to its subgroup U (1)PQ × U (1)Q . This residual

symmetry is, in turn, explicitly broken by other higher-dimensional operators. The dimension-6
operators Q5 and Q̄5 , for instance, explicitly break the vectorial Abelian symmetry in the newquark sector. The crucial question which we will have to address in the following therefore is how
severe the explicit breaking of the PQ symmetry turns out to be and whether it remains sufficiently
small enough, so that our model can still explain a QCD vacuum angle θ̄ of less than O (10−10 ).
Up to now, we are unable to specify the PQ charges of the new quark and anti-quark fields
separately, as the superpotential in Eq. (14) only contains the quark product operators QQ̄ i .
Demanding that the PQ charges of the singlet fields P and P̄ coincide with their U (1)P charges,

all we can say is that the operators QQ̄ i must carry a total PQ charge of −n. For the time
being, we may thus work with the following PQ charges,
qP = 1 ,qP̄ = −1 ,qQ ∈ R ,qQ̄ = −n − qQ ,qQQ̄ = −n .

(23)

The PQ charges of the MSSM fields qi , where i now runs over i = q, uc , dc , `, ec , nc , Hu , Hd ,are subject to constraints deriving from the Yukawa couplings in the superpotential WMSSM in
Eq. (1). The first two terms in WMSSM yield the following three conditionsquc + qq + qHu = 0 ,qdc + qq + qHd = 0 ,qec + q` + qHd = 0 ,(24)

the first two of which combine to give quc + qdc + 2qq + qHu + qHd = 0. As we will see in Sec. 2.4,
the PQ charges of the two MSSM Higgs doublets must sum to zero, qHu + qHd = 0, implying thatquc + qdc + 2qq = 0 .

(25)

The total PQ charge of all MSSM quark fields hence vanishes, such that the colour anomaly of the
PQ symmetry ends up receiving contributions only from the extra matter sector and none from
the MSSM sector, cf. also Eq. (29) further below.
15

Having derived this important result, we would still like to know which values the MSSM
PQ charges can actually take. Forgetting for a moment about the neutrino singlets required forthe seesaw mechanism, the answer is clearly all values compatible with the three conditions in
Eq. (24). The PQ charges qi can then, for instance, be parametrized in terms of qq , q` , qHu ∈ R.
Moreover, we note that in the course of electroweak symmetry breaking the Yukawa couplings
in WMSSM turn into mass terms for the MSSM matter fields, breaking the PQ symmetry unless
qHu = −qHd = 0. In this particular case, the PQ symmetry can be identified as a linear combinationof U (1)B and U (1)L , the global Abelian symmetries associated with baryon number B and lepton
number L. This result is a useful crosscheck, since U (1)B and U (1)L are the unique accidental
global symmetries of the standard model. In the seesaw extension of the MSSM, the conditions in
Eq. (24) are supplemented by two further conditions deriving from the last two terms in WMSSM ,qnc + q` + qHu = 0 ,2qnc = 0 ,(26)eliminating the PQ charge q` as a free parameter. Upon extending the MSSM by three neutrino
singlet fields, the PQ charges qi can therefore be parametrized by only two charges, qq , qHu ∈ R.
Setting qHu to zero now renders the PQ symmetry proportional to U (1)B , which is, of course,
expected, since the U (1)L is explicitly broken by the Majorana mass term in WMSSM . The only
relation among the PQ charges qi relevant for our further analysis is Eq. (25). Without loss ofgenerality, we are thus free to take qq , q` and qHu to be zero, so that qi = 0 for all fields i. The
field content of our model as well as our assignment of the PQ charges are hence similar as in the
KSVZ axion model proposed by Kim [40] as well as by Shifman, Vainshtein and Zakharov [41].
2.3.4Spontaneous breaking and colour anomaly of the PQ symmetry

In the true vacuum of the scalar potential corresponding to the superpotential WP in Eq. (20), thesinglet field X vanishes and the PQ symmetry is spontaneously broken,10
 


Λ
A
Λ
A
1
hXi = 0 , hP i = √ exp, hP̄ i = √ exp −, φ ⊂ A , φ = √ (b + ia) , (27)
Λ
Λ
2
2
2
where the chiral superfield A represents the axion multiplet, which consists of the pseudo-scalar√axion a, the scalar saxino b and the fermionic axino ã. The various factors of 2 in Eq. (27) serve
two purposes. First, they render the kinetic term of the axion canonically normalized; second, they
ensure that the scalar mass eigenstate that actually breaks the PQ symmetry, p+ = √12 (p + p̄∗ ),where p and p̄ are the complex scalars contained in P and P̄ , acquires a VEV hp+ i = Λ.
10

Spontaneous R symmetry breaking results in a tadpole term for X in the scalar potential, V ⊃ −κ m3/2 Λ2 X.
Besides that, X also couples to other fields of our model, cf. Sec. 2.4.3, such that its VEV eventually turns out tobe of order the gravitino mass rather than zero, hXi ∼ m3/2 .

16

Before continuing, we remark that, in the special case of a Z4R symmetry, also a cubic term inthe singlet field X is allowed in the superpotential WP , 2
Λ− P P̄ − λX X 3 + .. ,N = 4 : W = κX
2

(28)

where λX is some dimensionless coupling constant of O(1). In this case, the field configurationin Eq. (27) no longer represents the unique vacuum of the scalar potential corresponding to WP .
p
At hP i = hP̄ i = 0 and hXi = κ/(6λX )Λ, the scalar potential exhibits another local minimum.
Because of the linear term in X in the scalar potential, V ⊃ −κ m3/2 Λ2 X, this vacuum thenhas a negative energy density, the absolute value of which is much larger than the energy densityof the PQ-breaking vacuum in Eq. (27). There exists, however, no flat direction connecting thealternative vacuum with our PQ-breaking vacuum, which is why we do not have to worry about
the stability of the latter one. We merely have to assume that, in the course of the cosmologicalevolution, our universe has settled in the vacuum in Eq. (27) rather than in the alternative vacuum.
In fact, this is a very plausible assumption, if we believe that the field X is stabilized at hXi = 0
during inflation due to a large Hubble-induced mass.
In order to solve the strong CP problem, it is necessary that the PQ symmetry has a colour
anomaly. Thanks to our derivation of the PQ charges of all coloured matter fields in the previousµ
subsection, we are now able to calculate the anomalous divergence of the axial PQ current JPQ

and show that it is non-zero,µ∂µ JPQ
= APQ

i
αs h
Tr Gµν G̃µν ,8π

APQ = k qQQ̄ + quc + qdc + 2qq = −nk ,(29)

where we have introduced APQ as the anomaly coefficient of the U (1)PQ [SU (3)C ]2 anomaly. This
colour anomaly of the PQ symmetry induces an extra term in the effective Lagrangian [1, 42],
√


i
2Λ
a αs hµνeff, fa =Tr Gµν G̃
,(30)
LQCD ⊃ θ̄ −fa 8π
|APQ |
with fa denoting the axion decay constant. In consequence of this coupling of the axion a to thegluon field strength Gµν , an effective non-perturbative potential for the axion is generated,



a
eff
4
Va = ΛQCD 1 − cos θ̄ −
,
fa

(31)

the minimum of which is located at hai = fa θ̄. Shifting a by its VEV hai then cancels the θ̄ termin Eq. (30), thereby rendering the QCD Lagrangian CP -invariant. Our singlet sector consisting of
the fields X, P and P̄ hence entails a manifestation of the PQ solution to the strong CP problem.
An important detail to note is that it is the scale fa , rather than Λ, which determines thestrength of all low-energy interactions of the axion [43]. This is also the reason why experimental
17

constraints on the axion coupling are always formulated as bounds on fa and not on Λ. Requiring,for instance, that astrophysical objects such as supernovae or white dwarfs do not lose energy too
fast due to axion emission allows one to put a lower bound of O (109 ) GeV [10] on fa . Meanwhile,cosmology restricts the possible range of fa values from above. In order to prevent cold axions
from overclosing the universe, fa must be at most of O (1012 ) GeV [44, 45], hence leaving open thefollowing phenomenologically viable window for the axion decay constant,
109 GeV . fa . 1012 GeV .

(32)

Furthermore, as evident from the effective axion potential in Eq. (31), the non-perturbative
QCD instanton effects break the PQ symmetry to a global and discrete ZNDW symmetry, where
NDW = |APQ | = nk, commonly referred to as the domain wall number, counts the number ofdegenerate axion vacua. If the breaking of the PQ symmetry occurs after inflation, this vacuum
structure of the axion potential implies the formation of axion domain walls during the QCD
phase transition, thereby leading to a cosmological disaster [45, 46]. One obvious solution to thisdomain wall problem is to impose that inflation takes place after the spontaneous breaking of the
PQ symmetry, such that the axion field is homogenized across the entire observable universe.11
Alternatively, one may attempt to construct an axion model with NDW = 1, in which case theaxion domain walls collapse under their boundary tension soon after their formation [48]. In
Appendix B, we present a slight modification of our model that just yields NDW = 1 and which
hence allows for a solution of the axion domain wall problem even if the spontaneous breaking of
the PQ symmetry takes place after inflation.
2.3.5

Mass scale of the extra matter sector

As anticipated, the spontaneous breaking of the PQ symmetry furnishes the extra quarks andanti-quarks with Dirac masses mQi , which can be read off from the superpotential WQ in Eq. (14)after expanding the singlet field P around its VEV,
2.0 × 1010 GeVn    n 
n 
6.6 × 102 GeV
λi
Λ
λi
k
fa
mQi = n−1 √
'
×

1
4
1010 GeV
MPl
3.6 × 10−5 GeV
2



···

; n=1
; n=2
.
; n=3
; n=4

(33)

11
In this case, perturbations in the axion field amplified during inflation may result in too large isocurvature
contributions to the temperature fluctuations seen in the cosmic microwave background. A variety of solutions tothis isocurvature perturbation problem have however been proposed in the literature, cf. for instance Ref. [47] andreferences therein, which is why we will not consider it any further.

18

n=1
n=2

Z3R

Z4R

2, 3, 6, 9
2, 6

2, 3

Z5R

Z6R

Z8R
2, 3

2

2, 3, 6, 9
2, 6

Z9R

R
Z10

R
Z12

R
Z18

3, 9
3

2

3

Table 1: Values of k leading to unwanted operators in WP , the superpotential of the new singlet
sector, cf. Eq. (20). This table does not indicate for which ZNR symmetries only one extra quark
pair is problematic. For n = 1, these are the symmetries with N = 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 20;
for n = 2, it is the symmetries with N = 5, 7, 10, 11, 14, 22. In addition, independently of n, wedisregard the possibility of only one extra quark pair for N = 3, 4, 6, 8 in any case, cf. Sec. 2.2.3.

For n ≥ 3, our model thus predicts k new quark multiplets with masses below the electroweak scale,
which is, of course, inconsistent with experiments. Hence, the only viable values for n are n = 1
and n = 2. From a phenomenological point of view, the n = 2 case is certainly more interesting as
it features new coloured states with masses possibly within the range of collider experiments. On
the other hand, if no heavy quarks should be found at or above the TeV scale, our model would
not automatically be ruled out. Falling back to the n = 1 case, the extra vector-quarks can always
be decoupled from the physics at the TeV scale, thereby leaving still some room for the realizationof our extension of the MSSM.
For both viable values of n, we can now ask how many new quark flavours we are allowed to
introduce, i.e. which values k can possibly take. Recall that in Sec. 2.3.2 we required the R chargerP to fulfill all of the seven conditions in Eqs. (18) and (21). Given the explicit expression for rP
in terms of n and k in Eq. (16), this requirement then directly translates into a set of k values
that, depending on the values of n and N , we are not allowed to employ, cf. Tab. 1. In Sec. 3.1,
we will derive further restrictions on the set of allowed k values based on the requirement that theunification of the SM gauge couplings ought to occur at the perturbative level.

2.4

Generation of the MSSM µ term

In absence of any new physics beyond the MSSM, one might expect the supersymmetric mass of
the MSSM Higgs doublets to be of order the Planck scale, µ ∼ MPl . Such a large µ value would
then require a miraculous cancellation between the supersymmetric and the soft SUSY-breaking
contributions to the MSSM Higgs scalar potential, given that one ought to end up with Higgs VEVs
hHu,d i = vu,d close the electroweak scale. This puzzle, i.e. the question why µ should be of the sameorder as the soft Higgs masses, represents the infamous µ problem. As we have seen in the previous
section, an anomaly-free discrete R symmetry ZNR forbids the µ term in the MSSM superpotential,

19thus solving the µ problem halfway through. What remains to be done is to demonstrate how theµ term emerges with the right order of magnitude once the ZNR has been spontaneously broken.
2.4.1µ term from spontaneous R symmetry breaking
(4)

(4)

In the special case of a Z4R symmetry, the R charges of Hu and Hd sum to zero, rHu + rHd = 4 = 0,
cf. Eq. (4), such that a µ term of the correct magnitude can be easily generated in the courseof spontaneous R symmetry breaking [38]. This mechanism is based on two ingredients: (i) the
observation that, for rHu + rHd = 0, the operator Hu Hd can be accommodated with some O(1)00coefficient gH
in the Kähler potential, K ⊃ gH
Hu Hd as well as (ii) the fact that, during spontaneous
2
R symmetry breaking, the superpotential acquires a non-zero VEV hW i = W0 ,12 where W0 /MPl2
can be identified with the gravitino mass, m3/2 = W0 /MPl
. At low energies, the Higgs operator in0
m3/2 Hu Hd , which is nothing
the Kähler potential then induces an effective superpotential Wµ = gH0
but the desired µ term with µ = gH
m3/2 . Besides that, an additional contribution to the µ term

may be generated in the course of spontaneous SUSY breaking, if the Kähler potential should
contain a coupling between the operator Hu Hd and the hidden SUSY breaking sector [17]. In theremainder of this section, we will now mostly focus on ZNR symmetries with N 6= 4.
2.4.2

Contributions to the µ term from spontaneous PQ breaking

Next, we note that sometimes already the spontaneous breaking of the PQ symmetry entails thegeneration of a supersymmetric mass term for the MSSM Higgs doublets Hu and Hd , which,however, turns out to be too small in all viable cases. The origin for this contribution to the
MSSM µ term are the following higher-dimensional operators in the tree-level superpotential,p̄p(p̄) P̄
(p) P
Wµ = CµHu Hd ,
(34)
p−1 + Cµ
p̄−1
MPl
MPl
(p)

(p̄)

with Cµ and Cµ denoting dimensionless coupling constants of O(1). Of course, these couplings
are only allowed if they are compatible with the ZNR symmetry, which is the case given that
(N )

(N )

p rP = −2 and/or p̄ rP̄ = −2 .

(35)

We shall now assume for a moment that at least one of these two conditions can be satisfied. Incase only the first or the second condition can be fulfilled, let q denote the corresponding value of
12

R
Since W carries R charge rW = 2, the VEV hW i breaks the ZN
completely; R parity, which potentially remainsR
as an unbroken subgroup of the ZN , is not an actual discrete R symmetry, cf. Sec. 2.2. A possible mechanism togenerate a constant term in the superpotential is the condensation of hidden gauginos, such that W0 = hWα W α i [49].
e suchAlternatively, the VEV of the superpotential might originate from the condensation of hidden-sector quarks Q,neethat W0 = h(QQ) i. In Appendix A of Ref. [50], we present an exemplary model illustrating how such a quark
condensate could potentially be generated by means of strong gauge dynamics in some hidden sector.

20

p or p̄. If both conditions can be satisfied simultaneously, q shall denote the smaller of the two
possible powers, q = min {p, p̄}. The spontaneous breaking of the PQ symmetry then induces asupersymmetric mass µ for Hu and Hd , which looks very similar to the Dirac masses mQi for the
extra quarks and anti-quarks in Eq. (33),
2.0 × 1010 GeV











(q)
(q)
q
q
q
6.6 × 102 GeV
Cµ
Cµ
Λ
k
fa
'
µ = q−1 √×3.6 × 10−5 GeV
1
41010 GeV
MPl
2



···

; q=1
; q=2
.
; q=3
; q=4

(36)

For q = 1, the generated µ term is, hence, dangerously large; for q = 2 it is of the desired order ofmagnitude; and for q ≥ 3 it is drastically too small. On the other hand, given our restrictions on
the R charge rP in Eqs. (18) and (21), we know that q has to be at least q = 4. This means that,for q = 1, 2, 3, all possible R charges rP fulfilling at least one of the two conditions in Eq. (35) leadto an unwanted operator in WP , the superpotential of the extra scalar sector, cf. Eq. (20). We
thus conclude that the spontaneous breaking of the PQ symmetry does not suffice to generate a µterm of the right order of magnitude. For the last time, we are therefore led to extend our model.
2.4.3

Singlet extension of the MSSM Higgs sector

Extensions of the MSSM aiming at generating the µ term dynamically usually couple the MSSM
Higgs doublets to another chiral singlet S, which acquires a VEV of order of the soft Higgs masses
in the course of electroweak symmetry breaking. We will now adopt this approach and introduce
a chiral singlet field S with R charge rS = −2, in order to allow for the operator S Hu Hd in the
superpotential. As we will see in the following, this operator usually indeed yields a µ term of theright order of magnitude.
Given the fact that the superpotential carries R charge rW = 2, the relation between the2gravitino mass and the VEV of the superpotential, m3/2 = W0 /MPl
, implies that m3/2 should

be regarded as a spurious field also carrying R charge r3/2 = 2. After spontaneous R symmetry
breaking, the superpotential of the field S hence contains the following terms,
N 6= 4 :WS ⊃ gH Hu Hd S + m23/2 S + mS S 2+λS S 3 ,(37)

where mS denotes a supersymmetric mass for the singlet field S and where the term in parenthesis
is only allowed in the case of a discrete Z8R symmetry. In this section, we explicitly exclude thepossibility of a Z4R symmetry, because in this case the µ term is already generated in the course of
R symmetry breaking, cf. Sec. 2.4.1. Besides that, for a Z4R symmetry, the R charge of the field S
would be equivalent to rS = 2, such that a tadpole term of order the Planck scale would be allowed21

in the superpotential. Such a large tadpole would then severely destabilize the electroweak scale.
By contrast, all other ZNR symmetries successfully prevent the appearance of a dangerously large
tadpole term. In fact, the only tadpole that we are able to generate for N 6= 4 arises from thespontaneous breaking of R symmetry and is of the size of the gravitino mass, cf. Eq. (37).
Assuming a discrete Z3R or Z6R symmetry, the R charge of S 2 is equivalent to rs = 2 and mS is
expected to be very large, mS = gS MPl , where gS is a dimensionless constant of O(1) in general.
For all other ZNR symmetries, the S mass term is only allowed if, similarly as for the gravitino
mass, mS is interpreted as a spurious field, now with R charge 6 instead of 2. On the supposition
that only a single dynamical process is responsible for the generation of m3/2 and mS , the S massthen turns out to be heavily suppressed,
N = 3, 6 :mS = gS MPl ,gS ∼ 1 ;N 6= 3, 6 :

mS ∼

m33/2
2
MPl

.

(38)

For N = 3, 6, the large supersymmetric mass mS hence leads to a very small VEV of the field S,
thereby causing our attempt to dynamically generate the MSSM µ term to fail. Only in case that,for one reason or another, the parameter gS is severely suppressed, gS  1, such that mS  MPl ,a Z3R or a Z6R symmetry may still be considered viable. Otherwise, ZNR symmetries with N = 3, 6
should be regarded disfavoured within the context of our model.13 By contrast, in the case of allother symmetries, i.e. ZNR symmetries with N = 5 or N ≥ 7, the mass of the field S is completely
negligible, which is why we will omit from now on. Thus, as far as the generation of the µ term in
our model is concerned, we will assume the following terms in the superpotential,
gH Hu Hd S + m23/2 S + mS S 2 ; N = 3, 6

g 0 m3/2 Hu Hd; N =4
H
WS ⊃
.
2
3

S
+
λ
S
;
N
=
8
g
H
H
S
+
m

S
H
u
d
3/2



g H H S + m2 S
; N 6= 3, 4, 6, 8
H u d
3/2

(39)

Together with the scalar masses and couplings in the soft SUSY-breaking Lagrangian, theinteractions in Eq. (39) result in a scalar potential that is minimized for hHu,d i = vu,d and hSi =µ/gH , whereby our solution to the µ problem is completed. The actual value of the µ parameter
depends in a complicated way on the couplings in the superpotential WS as well as on the soft
parameters for the fields Hu,d and S. For our purposes, it will however suffice to treat µ as aneffectively free parameter that is allowed to vary within some range.
13

R
Interestingly, these are just the two anomaly-free ZN
symmetries of the MSSM. Now we see that they are most
likely not compatible with the generation of the µ term by means of an additional singlet field S. This justifies
once more our approach to extend the particle content of the MSSM by a new quark sector in such a way that theRgauge anomalies of the ZN
symmetry are always canceled, independently of the value of N .

22

Some of the expressions for WS in Eq. (39) are reminiscent of the superpotential of otherextensions of the MSSM that successfully generate the µ term by means of a singlet field S. For
instance, assuming a discrete Z8R symmetry and neglecting the tadpole term, the superpotential in
Eq. (37) corresponds to the Higgs superpotential of the next-to-minimal supersymmetric standard
model (NMSSM).14 Conversely, assuming a ZNR symmetry with N 6= 3, 4, 6, 8 and taking thetadpole term into account, the superpotential WS coincides with the effective Higgs superpotentialof the new MSSM (nMSSM) [52] as well as with the effective Higgs superpotential of the PQinvariant extension of the NMSSM (PQ-NMSSM) [53]. While in the nMSSM, the shape of the
S superpotential is fixed by means of a discrete R symmetry, similarly as in our model, the PQNMSSM invokes a PQ symmetry by hand in order to ensure the absence of further couplings ofthe singlet field S. On the other hand, the PQ-NMSSM features a PQ singlet field, similar toour singlet fields P and P̄ , which couples to the field S. By contrast, such a PQ-breaking field
is absent in the nMSSM. But as the mixing between the singlet S and the PQ-breaking sector
is always suppressed by powers of the PQ scale Λ, this has basically no effect on the low-energy
phenomenology of the Higgs and neutralino sectors.
As for the expected low-energy signatures of these two sectors, our model thus makes thesame predictions as the PQ-NMSSM and the nMSSM. This means, in particular, that our model
predicts a fifth neutralino mostly consisting of the singlino, which only receives a small mass frommixing with the neutral Higgsinos. Among all superparticles that either directly belong to the
MSSM or that at least share some renormalizable interaction with it, the singlino-like neutralino
is hence expected to be the lightest. Furthermore, at small values of tan β, the decay of thestandard model-like Higgs boson into two singlino-like neutralinos might represent the dominant
Higgs decay mode. Such a scenario is already constrained by the search for invisible Higgs decays
by the ATLAS experiment at the LHC [54] and will be further tested as data taking at the LHC
is resumed. Another interesting feature of our model is that, independently of tan β, the Higgs
boson mass receives positive corrections of the order of a few GeV from singlino loops, provided
that the Higgsinos are lighter than all other superparticles of the MSSM. Finally, we mention that
our model features a series of interesting implications for cosmology [53, 55].
The operators on the right-hand side of Eq. (37) are the only terms in the superpotential of thesinglet field S playing a role in the generation of the µ term. Besides that, the field S participates,of course, also in a series of other interactions. As the field X carries the same R charge as thegravitino mass, rX = r3/2 = 2, the tadpole term in Eq. (37) has, in particular, to be supplemented
14

For reviews of the NMSSM, cf. for instance Ref. [51].

23

by the operators m3/2 XS and X 2 S. The full superpotential of the field S thus reads
N 6= 4 :WS = gH Hu Hd S + m23/2 S + gX m3/2 XS + gX 2 X 2 S+mS S 2+λS S 3+ .. . (40)Here, gX and gX 2 are again dimensionless coupling constants of O(1) and the dots after the plus
sign indicate higher-dimensional non-renormalizable terms. Given the fact that the field X does
not carry any PQ charge, the coupling between X and S immediately implies that the field S also
does not transform under PQ rotations, qS = 0. This proves in turn our statement in Sec. 2.3.3 that
the PQ charges of Hu and Hd must sum to zero, qHu + qHd = 0. Another important consequenceof the operators m3/2 XS and X 2 S in Eq. (40) is that, at the supersymmetric level, the scalar
field VEVs in Eq. (27) no longer represent the unique vacuum configuration. The PQ-breaking
vacuum, in which hP P̄ i = Λ2 /2, is now continuously connected to a family of degenerate vacua, allof which are characterized by the fact that they fulfill the condition hP P̄ i − gX /κ m3/2 hSi = Λ2 /2.
However, this vacuum degeneracy is fortunately lifted by the soft SUSY breaking masses for thescalar fields P , P̄ and S, such that, also in the presence of the operators m3/2 XS and X 2 S, thevacuum configuration of interest, i.e. hP P̄ i = Λ2 /2 together with hSi ∼ m3/2 and hXi ∼ m3/2 ,corresponds to a local minimum. Besides that, the new interactions between S and X also lead
1/3

to a second local minimum at hXi ∼ m3/2 Λ2/3 , hXSi ∼ Λ2 and hP P̄ i = 0. The energy of thisvacuum is, however, much higher than the one of the PQ-breaking vacuum and hence, we expect
the fields P , P̄ , S and X to settle in the PQ-breaking vacuum at low energies,Λ
hP i = √ eA/Λ ,
2
2.4.4

Λ
hP̄ i = √ e−A/Λ ,2

hSi ∼ m3/2 ,hXi ∼ m3/2 .

(41)

Decay of the extra matter fields into MSSM particles

The extension of the MSSM Higgs sector by the singlet field S completes the field content of ourmodel. We are therefore almost ready to turn to the phenomenological constraints on our modeland discuss which values of N , n and k allow for a sufficient protection of the PQ symmetry. But
before we are able to do so, we have to take care of one last detail: the new quarks and anti-quarks
are thermally produced in the early universe, which potentially results in serious cosmologicalproblems. If the extra quarks are stable, they might be produced so abundantly that they overclose
the universe. On the other hand, if they are unstable, their late-time decays might alter theprimordial abundances of the light elements produced during big bang nucleosynthesis (BBN), sothat these are no longer in accordance with the observational data. To avoid these problems, we
require a coupling between the extra quark sector and the MSSM fields, such that the extra quarks
quickly decay after their production. So far, we only had to fix the R charge rQQ̄ of the quark24

pair operator QQ̄, cf. Eq. (11). The R charges rQ and rQ̄ = rQQ̄ − rQ of the individual quarks andanti-quarks have by contrast remained unspecified up to now. By choosing a particular value for
the R charge rQ , we are therefore now able to pinpoint the operator by means of which the extra
quarks shall couple to the MSSM.
Under the SM gauge group, the anti-quark fields Q̄i transform in the same representation as the
MSSM 5∗i multiplets. An obvious possibility to couple the new quarks to the MSSM thus is to allow
for the operator Q̄i 10j Hd in the superpotential, in which case the anti-quarks ought to carry thesame R charge as the 5∗i multiplets, rQ̄ = r5∗ . The only way in which the extra anti-quark fields thendistinguish themselves from the MSSM 5∗i multiplets is their coupling to the extra quark fields Qi .

More precisely, starting out with a superpotential containing the operators P n Qi Q̄0i + 5∗0
and
i



0
∗0
Q̄0i + 5∗0→ Q̄i , 5∗i , such that,i 10j Hd , we can always perform a field transformation Q̄i , 5i
by definition, the MSSM 5∗i multiplets do not couple to the extra quark fields Qi and only theoperators P n Qi Q̄i and Q̄i + 5∗i 10j Hd remain in the superpotential. The operator Q̄i 10j Hd thenmixes the quarks and leptons respectively contained in the Q̄i and 5∗i multiplets, which potentiallygives rise to dangerous flavour-changing neutral-current (FCNC) interactions. In the case of very
heavy extra quarks, i.e. for n = 1, we however do not have to worry about FCNC processes asthese are always automatically suppressed by the large quark masses. Only for n = 2, we haveto pay attention that the mixing between the MSSM fermions and the new matter fields does not
become too large. For extra quarks with masses around 1 TeV, we have for instance to require that
the Yukawa coupling constants belonging to the operator Q̄i 10j Hd are at most of O (10−2 ) [56].
This is a rather mild constraint, which may be easily satisfied in a large class of flavour models.
Coupling the new quark sector to the MSSM via the operator Q̄i 10j Hd is therefore certainly aviable option. The R and PQ charges of the extra quarks and anti-quarks are then given by(N )rQ = rQQ̄ − rQ̄ ,(N )rQ̄ = r5∗ ;qQ = qQQ̄ − qQ̄ ,qQ̄ = q5∗ .

Making use of our results for r5∗ and rQQ̄ in Eqs. (8) and (11), we find for rQ and rQ̄ ,


6
`Q
`
3
N
(N ) 13
(N )
rQ =
+ +
−
N , rQ̄ = − + ` .
5
k
k
2
5
2

(42)

(43)

Likewise, employing our results for qQQ̄ in Eq. (23) and setting q5∗ to 0, we obtain for qQ and qQ̄ ,qQ = −n ,qQ̄ = 0 .

(44)

Finally, we also note that the operator Q̄i 10j Hd explicitly breaks the vectorial global symmetry in
the extra quark sector, such that the PQ symmetry remains as the only global Abelian symmetry,
U (1)PQ × U (1)VQ
25→

U (1)PQ .

(45)

Given our choice for the MSSM PQ charges in Sec. 2.3.3, we are now eventually able to determine
the relation between the generators of the three global Abelian symmetries U (1)P , U (1)VQ andU (1)AQ on the one hand and the PQ generator on the other hand. Denoting these generators by
P , V , A and P Q, respectively, we find
PQ = P −n(V + A) .
2(46)

In order to avoid the above constraint on the Yukawa couplings associated with Q̄i 10j Hd inthe case n = 2, one may alternatively consider couplings of the new quarks fields to the MSSM
via higher-dimensional operators. Naively, there are three different choices for such an operator,
namely S Q̄i 10j Hd , P Q̄i 10j Hd and P̄ Q̄i 10j Hd . Replacing the singlet fields S, P and P̄ in these
operators by their respective VEVs, all of them turn again into Q̄i 10j Hd , now, however, withcoupling constants that are naturally suppressed compared to unity. Allowing for any of these
operators rather than Q̄i 10j Hd , we therefore do not have to fear dangerous FCNC processes dueto the mixing between the Q̄i and 5∗i multiplets. Meanwhile, S Q̄i 10j Hd and P Q̄i 10j Hd do not
represent viable operators by means of which the new quarks could couple to the MSSM after all.
In the case of S Q̄i 10j Hd , the extra quarks do not decay sufficiently fast in the early universe. The
operator S Q̄i 10j Hd furnishes the new quarks with two-body and three-body decay channels, thepartial decay rates of which can roughly be estimated as(2
102 s−11µ/gHΓ Q̄i → qj Hd , ecj Hd ∼mQi ∼8π MPl10−5 s−1(3m1014 s−11Qi
Γ Q̄i → Sqj Hd , Secj Hd ∼∼2128π 3 MPl
10−7 s−1; mQi = 1010 GeV,; mQi = 1 TeV(47); mQi = 1010 GeV.
; mQi = 1 TeVHere, we have set the VEV of the scalar field S to hSi = µ/gH = 1 TeV. For n = 2, the extra quarks
thus decay only after BBN, which begins at a cosmic time of around 1 s and lasts for roughly 103 s.
Moreover, if we choose the R charge rQ , such that P Q̄i 10j Hd is contained in the superpotential,
also P n−1 Qi 5∗j is allowed. Unlike in our first case, in which we considered Q̄i 10j Hd , this operator
cannot be simply eliminated by a field re-definition. Together with P n QQ̄ i , it instead leads toan unacceptably strong mixing between the Q̄i and 5∗i multiplets.
The only remaining option therefore is to allow for P̄ Q̄i 10j Hd . In this case, the superpotential
also features P n+1 Qi 5∗j , which cannot be transformed away as well, but which fortunately results inthe mixing between the Q̄i and 5∗j multiplets being suppressed by a factor of O (Λ/MPl ). Furthermore, P̄ Q̄i 10j Hd gives rise to two-body decays of the extra quarks at a fast rate. After replacing26

SU (5) PM

ZNR

U (1)PQ(q, uc , ec )

10

−

1
+ ` N250(dc , `)

5∗−

− 35 + ` N2

0

(nc )

1

−

1 + ` N2

0

Hu

2L

+

2 − 25

0

Hd

2L

+

0

Q

5

−

−n − 1Q̄

5∗−

2 + 25hi(n+1)`Q(n+1)6
`P
`
13
+
−
+
−
N
5
nk
2
n
nk
h
i
`Q
6
− 35 − nk+ 2` + `nP − nk
N

P

1

+

6
N
− nk
+ (k `P − `Q ) nk

1

P̄

1

+

6
N
− (k `P − `Q ) nk
nk

−1

X

1

+

2

0

S

1

+

−2

0

1

Table 2: Summary of the possible charge assignments in our model assuming that the extra quarks
couple to the MSSM via the operator P̄ Q̄i 10j Hd . If the extra quarks should instead couple to
the MSSM via Q̄i 10j Hd , the values given in Eqs. (43) and (44) must be used for the R and PQ
charges of the fields Qi and Q̄i . The 2L in the column indicating the SU (5) representations denote
SU (2)L doublets. All R charges are only defined up to the addition of integer multiples of N . The
MSSM R charges can additionally be changed by acting on them with Z5 transformations. N ≥ 3;
n = 1, 2; k ≥ 1; `; `P and `Q are all integers.
√
the scalar field P̄ by Λ/ 2, we obtain
Γ Q̄i → qj Hd , ecj Hd



1
∼
16π



Λ
MPl

2

(
1016 s−1mQi ∼1010 s−1; mQi = 1010 GeV,; mQi = 1 TeV(48)

where we have chosen the PQ-breaking scale Λ such that it respectively results in mQi = 1010 GeV
or mQi = 1 TeV, if n is set to 1 or 2, cf. Eq. (33). Similarly to S Q̄i 10j Hd , the operator P̄ Q̄i 10j Hdalso entails three-body decays, which, however, always proceed at a slower rate than the corresponding two-body decays, cf. Eq. (47). A coupling of the extra quarks to the MSSM via P̄ Q̄i 10j Hd
is hence a viable alternative to the coupling via Q̄i 10j Hd . A particular advantage of this couplingis that we do not have to require suppressed Yukawa couplings, if n = 2. On the other hand, thecharges of the extra anti-quarks now do not coincide any more with the charges of the MSSM 5∗i27multiplets. The R and PQ charges of the new quark and anti-quark fields are instead given by(N )rQ = rQQ̄ − rQ̄ ,(N )rQ̄ = r5∗ − rP̄ ;qQ = qQQ̄ − qQ̄ ,qQ̄ = −qP̄ .

Our results for r5∗ , rQQ̄ and rP̄ in Eqs. (8), (11) and (19) therefore provide us with




3
(n + 1)6` `P(n + 1)`Q
6
` `P
`Q
(N )
(N ) 13
rQ =
+
−
+
−
N , rQ̄ = − −
+
+
−
N
5
nk
2
n
nk
5 nk
2
n
nk

(49)

(50)

Similarly, making use of the fact that qP̄ = −1 and qQQ̄ = −n, cf. Eq. (23), we find for qQ and qQ̄ ,qQ = −n − 1 ,qQ̄ = 1 .

(51)

Combining this result with our choice for the MSSM PQ charges in Sec. 2.3.3, the relation between
the four Abelian generators P , V , A and P Q now turns out to be
PQ = P − V −n(V + A) .
2(52)

These findings complete the construction of our model. To sum up, in this section, we haveintroduced (i) the field content of the MSSM along with three generations of right-handed neutrinos,(ii) k pairs of extra quarks and anti-quarks in order to render the discrete R symmetry anomalyfree, (iii) an additional singlet sector in order to provide masses to the new quarks and anti-quarksand (iv) a singlet field S in order to dynamically generate the MSSM µ term. The charges of allthese fields are summarized in Tab. 2.

3

Phenomenological constraints

The MSSM extension presented in the previous section is subject to a variety of phenomenologicalconstraints. As we have already seen in Sec. 2.3.5, the positive integer n can, for instance, only
be 1 or 2, since otherwise the extra quarks would always have masses below the electroweak scale.
Besides that, i.e. besides the lower bound on the masses of the new quarks, we also have to ensure(i) that, despite our extension of the MSSM particle content, the unification of the SM gauge
coupling constants still occurs at the perturbative level, (ii) that operators explicitly breaking thePQ symmetry do not induce shifts in the QCD vacuum angle larger than 10−10 as well as (iii) thatthe axion decay constant takes a value within the experimentally allowed window, cf. Eq. (32). In
the next two subsections, we will now discuss these constraints in turn and show how they allow us
to single out the phenomenologically viable combinations of N , n and k along with corresponding
upper and lower bounds on fa .
28

3.1

Gauge coupling unification

The new quark and anti-quark fields contribute to the beta functions of the SM gauge coupling
constants and thus cause a change in the value gGUT at which these coupling constants unify athigh energies. The more extra quark pairs we add to the MSSM particle content, the higher gGUT
turns out to be, which provides us with a means to constrain the allowed number of extra quark
pairs k from above. For given masses mQi of the new quarks, we define the maximal viable numberof extra quark pairs kmax such thatgGUT (mQi , k = kmax ) ≤√4π ,gGUT (mQi , k = kmax + 1) >

√

4π ,

kmax = kmax (mQi ) .

(53)

In order to determine kmax in dependence of the heavy quark mass spectrum, we make thesimplifying approximation that all new quark flavours have the same mass, MQ = mQi , where√n
MQ = Λ/ 2/MPl MPl , cf. Eq. (33). At the same time, we assume that all superparticles
share a common soft SUSY breaking mass MSUSY of 1 TeV. When solving the renormalizationgroup equations of the SM gauge couplings for energy scales µ ranging from the Z boson mass
MZ = 91.2 GeV to the GUT scale MGUT = 2 × 1016 GeV, we then have to distinguish between two
different scenarios:• If MQ > MSUSY , we use the SM one-loop beta functions for MZ ≤ µ < MSUSY , the MSSM
one-loop beta functions for MSUSY ≤ µ < MQ and the two-loop beta functions of the MSSM
plus the extra quark multiplets in the NSVZ scheme [57] for MQ ≤ µ ≤ MGUT .
• If MQ ≤ MSUSY , we use the SM one-loop beta functions for MZ ≤ µ < MQ , the one-loop
beta functions of the standard model plus the extra fermionic quarks for MQ ≤ µ < MSUSY
and the two-loop beta functions of the MSSM plus the extra quark multiplets, i.e. plus theextra fermionic and scalar quarks, in the NSVZ scheme for MSUSY ≤ µ ≤ MGUT .
Given the solutions of the renormalization group equations, we are able to determine kmax as afunction of MQ according to Eq. (53). The relation between the PQ scale Λ and the axion decay
constant fa in Eq. (30) then provides us with kmax as a function of fa . The result of our calculation
is presented in Fig. 1, which displays kmax as a function of fa for n = 1 and n = 2, respectively.
Moreover, we note that collider searches for heavy down-type quarks are capable of placing alower bound MQmin on the quark mass scale MQ . As MQ decreases with fa and k, cf. Eq. (33), thislower bound on MQ readily translates into a lower bound kmin on k,
MQ (fa , n, k = kmin ) ≥ MQmin ,MQ (fa , n, k = kmin − 1) < MQmin
29kmin = kmin (fa , n) ,(54)

Astrophysical lower
bound on fa

15

kmin Hn = 2L

kmax Hn = 1L

Cosmological upperbound on fa

Allowed number of extra quark pairs k

20

10

5

0

kmax Hn = 2L

kmin Hn = 1L

109

1010

1011

Axion decay constant fa @GeVD

1012

Figure 1: Constraints on the number of extra quark pairs k for n = 1 and n = 2, respectively.
The lower bounds are due to the experimental lower bound on the mass of new heavy down-type
quarks; the upper bounds derive from the requirement of perturbative gauge coupling unification.

Assuming that the new quarks primarily couple to the SM quarks of the third generation via theoperator Q̄i 10j Hd , such as in the model discussed in Ref. [58], the ATLAS experiment at the
LHC has recently reported a lower bound of 590 GeV on the heavy quark mass scale [59]. In thefollowing, we will adopt this value for MQmin , although we remark that smaller values of MQ might
still be viable, if the new quarks should predominantly couple to the first or second generation of
the SM quarks rather than to the third generation. Conversely, an even larger mass range couldin principle be excluded using the present data, if the new quarks should couple to the MSSM via
the operator P̄ Q̄i 10j Hd rather than via the operator Q̄i 10j Hd . In this case, the new quarks would
be long-lived, thereby leaving very distinct signatures in collider experiments. In this section, we,however, assume a coupling via the operator Q̄i 10j Hd and set MQmin to 590 GeV. Solving Eq. (54)for kmin , we then find kmin as a function of fa , cf. Fig. 1. For n = 1 and all values of fa of interest,
kmin is always 1. On the other hand, for n = 2 and fa . 3×1010 GeV, the minimal possible numberof quark pairs rapidly grows as we go to smaller and smaller values of fa .
In summary, we conclude that, for each value of fa , the requirements of perturbative gauge
coupling unification as well as the lower bound on the mass of heavy down-type quarks provide us
with a range of possible k values, cf. Eqs. (53) and (54),kmin (fa , n) ≤ k ≤ kmax (fa , n) .
30

(55)

Turning this statement around, we can say that, for given values of n and k, our two phenomenological constraints imply a lower bound on fa ,
fa ≥ max famin,p , famin,m ,(56)

where famin,p and famin,m are defined such
 √gGUT famin,p , n, k = 4π ,
MQ famin,m , n, k = MQmin ,famin,i = famin,i (k, n) ,i = p, m . (57)

In addition to that, we know from astrophysical and cosmological observations that the axion
decay constant must not be smaller than O (109 ) GeV and not be larger than O (1012 ) GeV, cf.
Eq. (32), so that we are eventually led to imposing the following lower and upper bounds on fa ,famin ≤ fa ≤ 1012 GeV ,3.2

famin = max 109 GeV, famin,p , famin,m .

(58)

Shifts in the QCD vacuum angle

Given the particle content and charge assignments of our model, it is easy to construct operatorsthat explicitly break the PQ symmetry. Instead of an exact symmetry, the PQ symmetry therefore
merely ends up being an approximate symmetry, which poses a threat to the PQ solution of thestrong CP problem. Most PQ-breaking operators induce a shift in the VEV of the axion field,
such that the θ̄ term in the QCD Lagrangian is no longer completely canceled. The magnitudesof these shifts in hai differ from operator to operator and depend in addition on the axion decay
constant fa , the gravitino mass m3/2 as well as on the scalar VEVs hSi and hXi in some cases. In
this section, we will now investigate for which ZNR symmetries, which choices of n and k as well as
which values of fa the total shift in the axion VEV remains small enough, such that the shifted θ̄angle does not exceed the upper experimental bound, θ̄ . 10−10 .
3.2.1

PQ-breaking operators in the superpotential

All PQ-breaking operators in the superpotential inducing a shift in hai are of the following form15
W ⊃C P p P̄ p̄sx(Hu Hd )h mm
3/2 S X ,cp! p̄! h! s! x! MPlc = p + p̄ + h + m + s + x − 3 ,p 6= p̄ ,(59)

where C is a O(1) constant and where the powers of the various fields have to be chosen such that,(N )rP (p − p̄) + 4h + 2m − 2s + 2x = 2 .

(60)

15
PQ-breaking operators that do not involve any power of P or P̄ (for instance, Q5 or Q̄5 ) do not induce a shift
in the axion VEV and are therefore irrelevant for our purposes.

31

Our intention behind explicitly dividing the operators in Eq. (59) by the factorials of the powers p,
p̄, h, s and x is to eventually obtain maximally conservative bounds on the axion decay constant.
Fortunately, we do not have to consider all possible combinations of p, p̄, h, m, s and x in thefollowing. For instance, if some operator involving powers (p, p̄) with max{p, p̄} > min{p, p̄} > 0
is allowed in the superpotential, the same operator with (p, p̄) being either replaced by (p − p̄, 0) or(0, p̄−p) is also allowed. The shift in hai induced by this second operator is then enhanced comparedq
to the shift induced by the original operator by a factor of O (MPl/Λq ), where q = 2 min{p, p̄}.

Consequently, we are allowed to solely focus on PQ-breaking operators in the following that either
involve some power of P or some power of P̄ . For a similar reason, we do not have to care about
operators involving some power of Hu Hd . Given an operator with powers h ≥ 1 and m ≥ 0, we canalways write down a similar operator in which (h, m) is replaced by (0, m + 2h). This is possible
because (Hu Hd )h and m2h
3/2 have the same R charge up to an integer multiple of N . Now assumingthat m23/2 is larger than hHu Hd i = vu vd , the operator with powers (0, m + 2h) always yields alarger shift in hai than the operator with powers (h, m). Furthermore, the same game as withthe fields P and P̄ can also be played with m3/2 and the fields S and X. Operators with powers(m, s, x) satisfying the relation s > m + x ≥ 0 can always be traded for operators with powers(0, s − m − x, 0). The shift in hai due to these alternative operators is then enhanced compared
2(m+x)
m+x
hXix . In theto the shift due to the original operators by a factor of O MPl/ mm
3/2 hSiend, we therefore only have to consider the following set of PQ-breaking operators,C Pp 1 s 1 m x
W ⊃S , m3/2 X| (P, p) ↔ (P̄ , p̄) .
c
p! MPls!
x!

(61)

Each of the operators in Eq. (61) results in PQ-breaking terms in the scalar potential. Amongthese PQ-breaking contributions to the scalar potential, one class of terms derives from the F -termsof the fields S and X,s p s−1s p̄ s−1
CP̄ S+ FS0 ,
FS = cP S ,MPl p! s!
p̄! s!
C
x p m x−1
x p̄ m x−1
FX = cP m3/2 X,P̄ m3/2 X+ FX0 ,
MPl p! x!
p̄! x!

(62)

where we have introduced FS0 and FX0 to denote the contributions to FX and FS deriving from
PQ-invariant operators in the superpotential. Given the superpotential in Eq. (40) and taking
into account the various supergravity effects induced by the constant term in the superpotential,
W0 = m3/2 MPl , we are able to estimate of what order of magnitude we expect FS0 and FX0 to be,FS0 = O m23/2 , vu vd , m3/2 hXi , X 2 , .. ,

FX0 = O m23/2 , m3/2 hSi , hXSi , .. ,
32

(63)

with the dots denoting further contributions to FS0 and FX0 that only arise in the case of certain
ZNR symmetries. The VEVs of the fields S and X are both of the order of the gravitino mass, suchthat the leading contributions to FS0 and FX0 can eventually be estimated asFS0 = O m23/2 ,
FX0 = O m23/2 .

(64)

The mixing between FS0 and FX0 and the PQ-breaking contributions to FS and FX in Eq. (63) thengives rise to the following PQ-breaking terms in the scalar potential,
C P p h s s−1 x m x−1 i+ h.c. | (P, p) ↔ (P̄ , p̄) .
S , m3/2 X
V ⊃ m23/2
cp! MPls!
x!

(65)

A second important class of PQ-breaking terms in the scalar potential are the A-terms which
derive from the mixing between the operators in Eq. (61) and the VEV of the superpotential W0 ,

W0 C P p (p + s − 3) s (p + x − 3) m x
V ⊃ 2
S ,
m3/2 X + h.c. | (P, p) ↔ (P̄ , p̄) .
(66)c
MPl p! MPls!
x!
For a given operator in the superpotential with powers (p, s) or (p̄, s), the largest PQ-breaking
term in the scalar potential hence corresponds toV ⊃ m3/2

C Pp
max s m3/2 , |p + s − 3| S S s−1 + h.c. | (P, p) ↔ (P̄ , p̄) .
c
p! s! MPl

(67)

Similarly, the largest term induced by an operator with powers (m, p, x) or (m, p̄, x) is given by16
V ⊃ m3/2


C Pp
x−1
max x m3/2 , |p + x − 3| X mm
+ h.c. | (P, p) ↔ (P̄ , p̄) .
3/2 Xc
p! x! MPl

Next, we replace all scalar fields in these two operators by their VEVs,




a
a
Λ
Λ
P → √ exp i √, P̄ → √ exp −i √, S → hSi ,
2
2Λ
2
2Λ

X → hXi .

(68)

(69)

This provides us with contributions to the axion potential all of which are of the following form,






1 4
pa
a
4
+ h.c. = M cos p √,(70)∆Va = M exp i √22Λ2Λ
where, for the terms in the scalar potential in Eqs. (67) and (68), the mass scale M is respectively
to be identified as17


2 C m3/2 Λ p√
P S : M →MS hSis−1 , MS = max s m3/2 , |p + s − 3| hSi ,(71)
c
p! s! MPl2

2 C m3/2 Λ p
p m
x
4
s−1
√
P m3/2 X : M →MX mm, MX = max x m3/2 , |p + x − 3| hXi ,3/2 hXicp! x! MPl
2
p

16
17

s

4

Note that, in Eqs. (67) and (68), we have implicitly absorbed the sign of (p + s − 3) and (p + x − 3) in C.
x
The expressions for M 4 corresponding to the operators P̄ p̄ S s and P̄ p̄ mm
3/2 X look exactly the same.

33

3.2.2

PQ-breaking operators in the Kähler potential and the effective potential

Next to the PQ-breaking operators in the superpotential, we also have to take into account the
PQ-breaking contributions to the Kähler potential K. It is, however, easy to show that the PQbreaking terms in the scalar potential induced by the Kähler potential can at most be as large
as the terms induced by the superpotential. Given some PQ-breaking term K ⊂ K, its largest
PQ
contribution to the scalar potential is given by
V ⊃

C0
0
2
|W0 |2 K, = C m3/2 K
PQ
PQ
2
MPl

C 0 ∼ O(1) .

(72)

The operator K is either holomorphic from the outset or it is accompanied by a holomorphic
PQ
0term in the Kähler potential K by performing the following replacements,
PQ that follows from K
PQP † → P̄ ,P̄ † → P ,S† → X ,X† → S ,(Hu Hd )† → S 2 .

(73)Furthermore, we know that, in order to be consistent with the ZNR symmetry, the R charge of K

PQ
(0)

must be zero. As the gravitino mass carries R charge 2, the holomorphicity of K in combination
PQ(0)

with its vanishing R charge thus directly implies that m3/2 K is one of the allowed operators in the
PQ(0)superpotential. The A-term deriving from m3/2 K is then exactly of the same order of magnitude
PQ
as the term in the scalar potential induced by K, cf. Eq. (66). We therefore do not have to take
PQ
care of the PQ-breaking terms in the Kähler potential explicitly. By studying the effects on theaxion VEV related to the PQ-breaking operators in the superpotential, we automatically cover all
relevant effects on the axion VEV related to the Kähler potential.
So far, we have only discussed PQ-breaking terms in the tree-level scalar potential. Below theheavy quark mass threshold, interactions at the loop level give rise to further PQ-breaking terms in
the effective scalar potential. These higher-dimensional terms are then no longer solely suppressedby the Planck scale, but partly also by the heavy quark mass scale MQ ,Veff ⊃

1
C Pp m s x
m S X ,cMPl
MQd p! s! x! 3/2

c + d = p + m + s + x − 4,

d>0

| (P, p) ↔ (P̄ , p̄) , (74)where the coupling constant C is in general now also field-dependent. We might therefore worry
that some of these effective operators could yield larger shifts in the axion VEV than the actual
tree-level operators that we have considered up to now. By imposing the requirement that theradiatively induced terms in the scalar potential must vanish in the limit MQ → 0,
MQ → 0 :Veff ⊃

1
C Pp m s x
m S X → 0,cMPl
MQd p! s! x! 3/2

34| (P, p) ↔ (P̄ , p̄) ,(75)

one can however show that the factor MQ−d in Eq. (74) is always canceled by factors contained in
the coupling constant C, such that all of the effective terms can eventually be rewritten as0

1 C0 P p0s 0 x0000
000Veff ⊃ c0 0 0 0 mm
3/2 S X , c = p + m + s + x − 4 , C ∈ R ,MPl p ! s ! x !

| (P, p0 ) ↔ (P̄ , p̄0 ) . (76)The radiative corrections in the effective potential are hence not enhanced with respect to theterms in the tree-level scalar potential. We conclude that, for our purposes, it will suffice to only
consider the PQ-breaking terms in the scalar potential induced by the superpotential. A separate
treatment of Kähler-induced effects or radiative corrections is not necessary.
3.2.3

Upper bounds on the axion decay constantThe ∆Va terms in the scalar potential, cf. Eq. (70), disturb the effective QCD instanton-induced
potential Vaeff , cf. Eq. (31), such that the axion potential is no longer minimized by fa θ̄,d (Va + ∆Va )
= 0,
da
a=hai


hai = fa θ̄ + ∆θ̄ .

(77)This shift in the axion VEV directly translates into a non-zero value ∆θ̄ of the QCD vacuum angle.
Making use of our results for Vaeff and ∆Va in Eqs. (31) and (70), we obtain for ∆θ̄∆θ̄ = ∆θ̄0 sin





2 
p
θ̄ + O ∆θ̄0,|APQ |∆θ̄0 =p
M4.
|APQ | Λ4QCD(78)

According to the experimental upper bound on the QCD vacuum angle, ∆θ̄0 must not be larger
than 10−10 ,∆θ̄0 ≤ ∆θ̄0max = 10−10 ,M4 ≤|APQ |∆θ̄0max Λ4QCD ,p(79)which results in an upper bound on the mass scale M . Combining this constraint with our expressions for M in Eq. (71), we are able to derive an upper bound on the axion decay constant fa foreach of the PQ-breaking operators in the superpotential,
"
p

s

P S :

famax,S =

p−1

2

c
∆θ̄0max |APQ |1−p Λ4QCD MPl
p! s!
C
p
MS hSis−1 m3/2

"

#1/p

c
Λ4QCD MPl∆θ̄0max |APQ |1−p
x
max,X
p−1
P p mm
X
:
f
=
2
p!
x!
3/2
a
C
p
MX hXix−1 mm+1
3/2

,

(80)

#1/p
,

x
with the bounds corresponding to P̄ p̄ S s and P̄ p̄ mm
3/2 X being of exactly the same form.

For given values of N , n, k and rP , a multitude of different PQ-breaking operators might beallowed in the superpotential, all of which imply an upper limit on fa . Let us denote the most
35

k

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

#

(0, 4)(0, 38)(16, 54)(4, 36)

49

54

36

62

101

36

120

110

56

132

160

Table 3: Numbers of viable scenarios for individual values of k, including as well all scenarios witha Z3R or a Z6R symmetry. For k ≤ 6, the respective numbers of solutions for the two cases n = 1
and n = 2 are indicated in the format (#|n=1 , #|n=2 ).

restrictive among these upper limits by famax,θ̄ ,famax,θ̄ = min all famax,S , all famax,X

.

(81)

Together with the constraints on fa in Eq. (58), we thus find the following total lower and upper
limits on the axion decay constant fa ,famin ≤ fa ≤ famax , famin = max 109 GeV, famin,p , famin,m , famax = min 1012 GeV, famax,θ̄ . (82)
By virtue of this result, we are now able to identify the phenomenologically viable combinationsof N , n, k and rP . The corresponding criterion is nothing but the requirement that there has tobe an allowed window of possible values for fa ,famin < famax⇒(N, n, k, rp ) viable .

(83)

To determine the allowed combinations of N , n, k and rP , we compute famin and famax for
N = 3, 4, .., 12 ;n = 1, 2 ;
k = 1, 2, .., kmax 1012 GeV, n ;rP = rP (N, n, k, `Q , `P ) ,(84)

where kmax (1012 GeV, 1) = 17 and kmax (1012 GeV, 2) = 6 and where rP as a function of N , n, k,`Q and `P is given in Eq. (16),18 and check whether or not the criterion in Eq. (83) is fulfilled. In
doing so, we set all dimensionless coupling constants to 1 and use a common value of 1 TeV forthe gravitino mass and the scalar VEVs,
m3/2 = 1 TeV ,hSi = 1 TeV ,hXi = 1 TeV .

(85)

Larger values of m3/2 , hSi and hXi would lead to more stringent bounds on fa , which means thatthe bounds that we obtain should be regarded as conservative. For the non-perturbative scale of
QCD, we employ the MS value above the bottom-quark mass threshold, ΛQCD ' 213 MeV [60].
18

In total, we thus scan 1950 different combinations of N , n, k and rP . Out of these combinations, 1530 belong
to the case n = 1, whereas 430 belong to the case n = 2.

36

N

3

4

5

6

7

8

9

10

#

(76, 8)(104, 16)

(91, 13)(76, 8)(103, 14)(104, 16)

(90, 6)(91, 13)

N

11

12

#

(109, 22)

(92, 16)

Table 4: Numbers of viable scenarios for individual values of N in the format (#|n=1 , #|n=2 ).

3.2.4

Phenomenologically viable scenarios

Restricting ourselves to the parameter values specified in Eqs. (84) and (85), we find in total 1068
viable combinations of N , n, k and rP , where 936 of these solutions belong to the case n = 1 and132 solutions to the case n = 2. In Tabs. 3 and 4, we indicate how many solutions we respectively
obtain for the individual values of k and N under study. In Tab. 5, we list all viable combinations of
N , n, k and rP for all k values up to k = 6. In summary, we conclude that our minimal extensionof the MSSM apparently gives rise to a large landscape of viable scenarios. It is in particularsurprising and intriguing that the order N of the ZNR symmetry can take any value, as long as thenumber of extra quark pairs k is chosen appropriately. A comprehensive phenomenological studyof this landscape of possible solutions is beyond the scope of this paper. In the following, we shallthus restrict ourselves to a few interesting observations, illustrating what kind of questions one
might be able to answer based on the full numerical data describing the landscape.
We observe for instance that, for all possible combinations of N , n and k, there exists either
no viable rP value at all or at least two different values. It is therefore interesting to ask which of
the various possible rP values for given N , n and k yields the least stringent upper bound on fa ,
n
o
max,0
max,θ̄
fa(N, n, k) = max fa(N, n, k, rP ) .
(86)
rP

This maximal upper bound can then be regarded as the most conservative constraint on fa for therespective combinations of N , n and k. The two panels of Fig. 1 present famax,0 as a function of N
and k for n = 1 and n = 2, respectively. Apart from four exceptions, famax,0 interestingly always
exceeds famin as long as it is larger than 109 GeV,
(N, n, k) 6= (4, 2, 4), (8, 2, 4), (9, 2, 5), (12, 1, 15) :famax,0 ≥ 109 GeV ⇒ famax,0 > famin .

(87)

Only for (N, n, k) = (4, 2, 4), (8, 2, 4), (9, 2, 5), (12, 1, 15), famax,0 is smaller than famin , which renders
these four cases phenomenologically unviable. This is indicated in Fig. 1 by the diagonal blacklines crossing out the respective squares.
37

(n, k)

Z3R

Z4R

Z5R

Z6R

(2, 4){3, 9, 15, 21}{2, 6, 10, 14, 18, 22, 26, 30}{9, 19, 29, 39}{6, 18, 30, 42}(2, 5){3, 9, 21, 27}{2, 6, 14, 18, 22, 26, 34, 38}{9, 19, 29, 39, 49}{6, 18, 42, 54}{2, 10, 14, 22, 26, 34, 38, 46}{19, 29, 49, 59}

Z8R

Z9R

(2, 6)(n, k)

Z7R

R
Z10(1, 5){1, 8, 22, 29}(1, 6){1, 29}(2, 3){1, 29}(2, 4){1, 15, 29, 43}{2, 10, 18, 26, 34, 42, 50, 58}{3, 21, 39, 57}{14, 34, 54, 74}(2, 5){1, 29, 43, 57}{2, 18, 26, 34, 42, 58, 66, 74}{3, 12, 21, 39, 48, 57, 66, 84}{14, 34, 54, 74, 94}(2, 6){1, 29, 43, 71}{2, 10, 26, 34, 50, 58, 74, 82}{3, 57}{14, 34, 74, 94}(n, k)

R
Z11

R
Z12

(1, 5){16, 27, 38, 49}{6, 18, 42, 54}(1, 6){5, 49}(2, 3){5, 49}(2, 4){5, 27, 38, 49, 71, 82}{6, 18, 30, 42, 54, 66, 78, 90}(2, 5){16, 27, 38, 49, 71, 82, 93, 104}

{6, 18, 42, 54, 66, 78, 102, 114}

(2, 6){5, 38, 49, 71, 82, 115}

{3, 12, 21, 39}

Table 5: Viable values of rP in units of 1/(nk) and in dependence of N , n and k for all k values up
to k = 6. We also include the rP values for (N, n, k) = (4, 2, 4), (8, 2, 4), (9, 2, 5), which are actually
phenomenologically unviable if we believe in the perturbative unification of the gauge coupling
constants. For these combinations of N , n and k, we namely find 109 GeV ≤ famax,0 ≤ famin , cf.
Eq. (87) and Fig. 1.

38

Number of extra quark pairs k

n = 1 : W É P IQ QMi

17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1

1016
1015
1014
1013
1012
1011
1010
109

<109
Z3R

Z4R

Z5R

Z6R

Z7R

Z8R

R
R
R
Z9R Z10Z11Z12n = 2 : W É P2 IQ QMi
6
Number of extra quark pairs k

fa @GeVDfa @GeVD
1016
1015

5

1014
4

1013
1012

3
1011
1010

2

109
1

<109
Z3R

Z4R

Z5R

Z6R

Z7R

Z8R

R
R
R
Z9R Z10
Z11
Z12

Figure 2: Upper bounds famax,0 on the axion decay constant fa according to the requirementthat the shift in the QCD vacuum angle θ̄ induced by PQ-breaking operators not be larger than
10−10 , cf. Eqs. (81) and (86). Both plots are based on m3/2 = 1 TeV, hSi = µ/gH = 1 TeV andhXi = 1 TeV. At the same time, all dimensionless coupling constants have been set to 1. The
black diagonal lines indicate that 109 GeV ≤ famax,0 ≤ famin , cf. Eq. (58).

39

(n, N, k)(1, 4, 12)(1, 5, 9)(1, 7, 7)(1, 8, 12)(1, 9, 7)1010 θ̄0

1 × 10−4

3 × 10−1

2 × 10−4

1 × 10−4

2 × 10−4(n, N, k)(1, 10, 9)(1, 11, 7)(1, 12, 7)(2, 4, 6)(2, 8, 6)1010 θ̄0

3 × 10−1

2 × 10−4

2 × 10−4

1 × 10−4

1 × 10−4

Table 6: All combinations of N , n and k which, in the case of axion dark matter, i.e. for fa =1012 GeV, result in a lower bound θ̄0 on the theta angle between 10−15 and 10−10 , cf. Eq. (89).

A further question that one might be interested in is which of the viable scenarios are compatible with the assumption of axion dark matter. In case inflation takes place after the spontaneous
breaking of the PQ symmetry, the only contribution to the relic axion density stems from thevacuum realignment of the zero-momentum mode of the axion field during the QCD phase transition [44]. The present value of the axion density parameter Ω0a h2 can then be estimated as [45]
Ω0a h2 ∼ 0.50θ̄i2
π 2 /3

fa1012 GeV

7/6
,

(88)

where θ̄i ∈ (−π, π] denotes the initial misalignment angle of the axion field before the onset of
the QCD phase transition, θ̄i = a (ti ) /fa . In the derivation of Eq. (88), it is assumed that θ̄i isconstant across the entire observable universe as well as that the axion relic density is not diluted
after its generation by some form of late-time entropy production. If all possible values of θ̄i areequally likely, we expect that θ̄i2 = π 2 /3. By comparing the expression for Ω0a h2 in Eq. (88)with the density parameter of cold dark matter (CDM), which has recently been determined very
precisely by the PLANCK satellite, Ω0CDM h2 ' 0.1199 [61], we see that, for an axion decay constant
fa of O(1012 ) GeV, cold axions may completely account for the relic density of dark matter. For
fa & 1012 GeV, the axion density exceeds the measured abundance of dark matter, which is nothing
but the cosmological upper bound on fa which we introduced in Sec. 2.3.3.
Setting fa to 1012 GeV, we can now ask how large a QCD vacuum angle θ̄ we expect to be
induced by the PQ-breaking operators in the respective viable scenarios. In the case of thosescenarios for which we found that famin < famax,θ̄ < 1012 GeV, the induced QCD vacuum angle, ofcourse, turns out to be larger than 10−10 , i.e. only scenarios in which famin < 1012 GeV < famax,θ̄
are compatible with the requirement of axion dark matter. In total, we find 861 of such scenarios.
Among these, 763 belong to the case n = 1 and 98 to the case n = 2. Analogously to the upperbounds on the axion decay constant, for which we introduced famax,0 , cf. Eq. (86), we would also

40

like to know which R charge rP for given N , n and k yields the smallest QCD vacuum angle,
fa = 1012 GeV :θ̄0 (N, n, k) = min θ̄(N, n, k, rP ) ,rPθ̄(N, n, k, rP ) = max{all ∆θ̄0 }(89)

where the shifts in the QCD vacuum angle ∆θ̄0 are to be calculated according to Eq. (78). The
angles θ̄0 then represent the most conservative lower bounds on θ̄ for the respective combinationsof N , n and k. For 96 combinations of N , n and k, splitting into 79 combinations corresponding to
n = 1 and 17 combinations corresponding to n = 2, the angle θ̄0 does not exceed 10−10 . But only
for a few of these solutions, θ̄0 falls into a range that might be experimentally accessible in the notso far future. For instance, only for 10 solutions we find values of θ̄0 between 10−15 and 10−10 , cf.
Tab. 6. Provided that dark matter is really composed out of axions, these 10 scenarios can thenbe tested in experiments aiming at measuring a non-zero value of the QCD vacuum angle.

4

Conclusions and discussion

The PQ solution of the strong CP problem requires an anomalous global Abelian symmetry,
U (1)PQ . On the other hand, any global symmetry is expected to be explicitly broken by quantum
gravity effects. In this paper, we have pointed out that imposing a gauged and discrete R symmetry,
ZNR , one is able to retain a PQ symmetry of high enough quality as an approximate and accidental
symmetry in the low-energy effective theory. The reasoning behind the construction of our model
was the following: In order to render the ZNR symmetry anomaly-free, it is, in general, necessary
to extend the particle content of the MSSM by new matter multiplets. Except for some special
cases, these new particles are a priori massless, which calls for a further extension of the spectrum
by an extra singlet sector that is capable of generating masses for the new particles. As we wereable to show, the new matter and singlet sectors then exhibit several global Abelian symmetries,
a linear combination of which can be identified as the PQ symmetry. In addition to that, for all
ZNR symmetries apart from Z4R , we supplemented the MSSM Higgs sector by an additional chiralsinglet S, so as to allow for a dynamical generation of the MSSM µ term.
The presence of the extra matter multiplets and the singlet S in our model entail a potentiallyrich phenomenology in collider experiments. Depending on the nature of the coupling between theextra matter and singlet sectors, the new particles might either have masses in the TeV or multiTeV range or they might be very heavy, with their masses being close to the scale of PQ symmetry
breaking. In the former case, our model is being directly probed by searches for heavy vector-likequarks at the LHC. At the same time, the phenomenology of the Higgs sector of our model issimilar to the one in the PQ-NMSSM or in the nMSSM. Next to the four ordinary neutralinos,
41

we expect a fifth, very light neutralino, the singlino S̃, which receives its mass only from mixing
with the neutral Higgsinos. The singlino may play an important role in the decay of the standard
model-like Higgs boson and contribute to the relic density of dark matter.
In order to single out the phenomenologically viable variants of our model, we imposed four
phenomenological constraints. We required (i) the masses of the new quarks to exceed the lower
experimental bound on the mass of heavy down-type quarks, MQmin = 590 GeV, (ii) the unification√of the standard model gauge couplings to still occur at the perturbative level, gGUT ≤ 4π, (iii) theshift in the QCD vacuum angle induced by higher-dimensional PQ-breaking operators to remain
below the upper experimental bound, θ̄ < 10−10 , as well as (iv) the axion decay constant to takea value within the experimentally allowed window, 109 GeV . fa . 1012 GeV. To our surprise,
we found a large landscape of possible scenarios, all compatible with these four constraints. In
particular, we showed that, for an appropriately chosen number of extra matter multiplets, theorder N of the ZNR symmetry can take any integer value larger than 2. Besides that, for each
viable scenario, we derived an upper bound on the axion decay constant based on the requirementthat QCD vacuum angle must not exceed 10−10 . In many cases, these upper bounds turned out
to be larger than 1012 GeV, thereby rendering the corresponding scenarios compatible with theassumption of axion dark matter. For these scenarios, we then estimated the expected value of the
QCD vacuum angle, in case dark matter should really be composed out of axions. A measurementof a non-zero theta angle in combination with a confirmation of axion dark matter would therefore
allow for a highly non-trivial experimental test of our model.
We also emphasized the virtues of the special case of a Z4R symmetry. In the case of a Z4R
symmetry, the MSSM µ term can be easily generated in the course of spontaneous R symmetry
breaking, such that there is no need to introduce an additional chiral singlet. As a consequence ofthat, the scalar potential does not exhibit a flat direction in the supersymmetric limit, so that we
do not have to rely on the soft SUSY breaking masses to stabilize the PQ-breaking vacuum, as is
the case for all other ZNR symmetries. Moreover, a Z4R is the only discrete R symmetry that allows
for MSSM R charges consistent with the assumption of SO(10) unification, cf. Appendix A.
Finally, we mention that our study needs be extended into several directions. First of all, it isnecessary to embed our extension of the MSSM into a grander model that explains the origin of
the ZNR symmetry and provides some guidance as to the number of extra matter multiplets andthe exact nature of their couplings. Likewise, it is important to further explore the cosmologicalimplications of our model. One open question, for instance, is the generation and compositionof dark matter in terms of axions, saxions, neutralinos and/or gravitinos in dependence of ourmodel parameters. Besides that, it would be interesting to make contact between our model
42

and R-invariant scenarios of inflation [62]. Altogether, our model promises to give rise to a richphenomenology that can be probed at colliders and in astrophysical and cosmological observations.
Future experiments will thus be able to test the intriguing possibility that the PQ symmetry,
required for the PQ solution of the strong CP problem, is indeed an accidental consequence of agauged and discrete R symmetry.

A

Possible R charges of the MSSM fields

In Sec. 2.2, we derive five constraints on r10 , r5∗ , r1 , rHu and rHd , the R charges of the MSSM
matter and Higgs multiplets, cf. Eqs. (4), (6) and (7). As these conditions only hold up to theaddition of integer multiples of N , they do not suffice to fix the values of the MSSM R charges
uniquely. In this appendix, we now show that, for each value of N , there exist exactly ten different
R charge assignments for the MSSM fields that comply with all constraints. Moreover, we also
discuss under which circumstances these solutions are equivalent to each other.

R charge assignments consistent with all constraints
To begin with, let us rewrite the five conditions in Eqs. (4), (6) and (7) as follows,
rHu + rHd = 4 + `1 N ,r5∗ + r1 + rHu = 2 + `4 N ,2r10 + rHu = 2 + `2 N ,r5∗ + r10 + rHd = 2 + `3 N ,(90)

2r1 = 2 + `5 N ,where we have made use of the relation in Eq. (5) and with `i ∈ Z for all i = 1, .., 5. Solving thissystem of linear equations for the R charges r = (r10 , r5∗ , r1 , rHu , rHd )T yields
 
 
 1 
  
`1
r10
0
0
0 0 0
1
5


 r5∗  −3 
−3 −1 1
1 0 0 `2   (N )  5    r1  =  1  + `˜ N  5  +  1 −2 −1 1 0 `3  N ,




 




10   rHu 
2 − 2  `4 
0
1
0
0
0
−2
5
`5
1 −1 0 0 0
rHd
2 + 252(91)

with `˜ = −2`1 + 4`2 + 2`3 − 2`4 + `5 ∈ Z. As indicated by the = symbol in Eq. (91), all R charges(N )are only defined modulo N . Thus, after picking explicit values for the `i , we always have to takeall R charges modulo N , such that 0 ≤ ri < N for all fields i. At the same time, the last summandon the right-hand side of Eq. (91) does nothing but shifting the charges r5∗ , r1 , rHu and rHd by
integer multiples of N . Its effect is hence always nullified by the modulo N operation, allowing us
to omit it in the following. Furthermore, we observe that the entries of the second column vector

43

on the right-hand side of Eq. (91) correspond to the X charges of the MSSM multiplets.19 We
shall therefore denote this vector by X, such that

T
1 3
2
2
N
(N )
˜
X , r0 =, − , 1, 2 − , 2 +,r = r0 + `
10
5 5
5
5

X = (1, −3, 5, −2, 2)T .

(92)

This result illustrates that, for any given N , there are indeed ten different possible R charge
assignments r. Independently of the concrete value of N , the assignment r 0 always represents asolution to the conditions in Eqs. (4), (6) and (7). All other solutions can be constructed from r 0
Nby adding multiples of 10
X to it. Here, the fact that the R charges ri are only defined modulo N
implies that all R charge assignments corresponding to values of `˜ that differ from each other byinteger multiples of 10 are equivalent to each other. The ten possible solutions for the MSSM R
charges then follow from Eq. (92) by setting `˜ to `˜ = 0, 1, 2, .., 9.
Among all viable R charge assignments that can be obtained from Eq. (92), there are several
which are particularly interesting. For instance, for N = 4, it is possible to assign R charges to the
MSSM fields in such a way that they are consistent with the assumption of SO(10) unification. In
this case, the GUT gauge group contains SO(10) as a subgroup, GGUT ⊃ SO(10) ⊃ SU (5), andthe MSSM matter and Higgs fields are unified in SO(10) multiplets, such that r10 = r5∗ = r1 and
rHu = rH . For N = 4 and `˜ = 2, 7, these two relations can indeed be realized,d

N =4 :`˜ = 2 : r = (1, 1, 1, 0, 0) ,`˜ = 7 : r = (3, 3, 3, 0, 0) .

(93)

In the case of N = 4, the R charge of the superpotential is equivalent to −2. Hence, given
any viable R charge assignment, reversing the signs of all R charges and applying the modulo N
operation, so that all R charges lie again in the interval [0, N ), provides one with another viable
R charge assignment. The two solutions for r in Eq. (93) are related to each other in just thisway, implying that they are in fact equivalent. In Refs. [7, 37], the discrete Z4R symmetry with
R charges r = (1, 1, 1, 0, 0) has been discussed in more detail. Allowing for anomaly cancellation
via the Green-Schwarz mechanism, this symmetry has in particular been identified as the unique
discrete R symmetry of the MSSM that may be rendered anomaly-free without introducing any
new particles and which, at the same time, commutes with SO(10) and forbids the µ term in the
superpotential. Finally, we point out that the two R charge assignments in Eq. (93) only feature
integer-valued R charges. We mention in passing that, in fact, for each value of N that is not an
integer multiple of 5 there is at least one viable R charge assignment that only involves integervalued R charges. This is a direct consequence of our result for r in Eq. (92) and the fact that all
R charges in r 0 are integer multiples of 51 .
19

X denotes the charge corresponding to the Abelian symmetry U (1)X , which is the subgroup of U (1)B−L ×U (1)Ythat commutes with SU (5). In terms of B−L and the weak hypercharge Y , it is given as X = 5(B−L) − 4Y .

44

Relationship between the different R charge assignments
The form of our result for r in Eq. (92) reflects the symmetries of the MSSM superpotential that
commute with SU (5). Among these symmetries, there is in particular a Z10 subgroup of U (1)X . To
see this, notice that the MSSM superpotential without the Majorana mass term for the neutrino
singlets 1i is invariant under U (1)X transformations. The Majorana mass term, however, carries
X charge 10 and thus breaks the U (1)X symmetry to its Z10 subgroup. Our solutions for the
MSSM R charges are therefore related to each other by Z10 transformations, which also explains
why we have found exactly ten different solutions for each value of N . This result is independentof the question of whether or not we assume the U (1)X symmetry to be part of the gauge group
above some high energy scale. We will address this question shortly, but before we do that, weremark that the Z10 subgroup of U (1)X is not the only symmetry of the MSSM superpotential that
commutes with SU (5). By definition, the center of SU (5), a discrete Z5 symmetry, also commuteswith all SU (5) elements. Under this Z5 symmetry, the MSSM multiplets 10i , 5∗i , Hu and Hd carrycharges 1, 2, 3 and 2, while all SM singlets have zero charge. At the same time, all SM singletsof our model transform trivially under the Z5 subgroup of the Z10 contained in U (1)X . The Z5
center of SU (5) is hence equivalent to this U (1)X subgroup,
SU (5) ⊃ Z5 ∼= Z5 ⊂ Z10 ⊂ U (1)X .

(94)

Therefore, independently of whether U (1)X is gauged or not, the Z5 subgroup of Z10 always has
to be treated as a gauge symmetry, as it is also contained in SU (5). Under the action of thisgauged Z5 symmetry, the R charge assignments in Eq. (92) split into two equivalence classes ofrespectively five solutions. The R charge assignments corresponding to `˜ = 2, 4, 6, 8 can all be
generated by acting with Z5 transformations on the R charge assignment corresponding to `˜ = 0.
Similarly, the R charge assignments corresponding to `˜ = 1, 3, 7, 9 can all be generated by actingwith Z5 transformations on the R charge assignment corresponding to `˜ = 5. All viable R charge
assignments are hence physically equivalent to one of the following two solutions, cf. Eq. (8),
(N )

r10 =

1
N
+` ,
5
2

3
N
(N )
r5∗ = − + ` ,
5
2

(N )

r1 = 1 + `

N
,
2

(N )

rHu = 2 −

2
,
5

(N )

rHd = 2 +

2
,
5

(95)

where ` = 0, 1. These two remaining R charge assignments are related to each other by transformations under the quotient group Z10 /Z5 , which is nothing but a simple Z2 parity.
Whether the two solutions in Eq. (95) are also equivalent to each other depends on the natureof this Z2 parity. If U (1)X is part of the gauge group at high energies, its Z10 subgroup is a gauge
symmetry at low energies. Dividing the center of SU (5) out of this Z10 , we are then left with agauged Z2 parity, which can be identified as matter parity, PM = Z10 /Z5 . The transformations
45

relating the two solutions in Eq. (95) to each other are then gauge transformations and both
solutions end up being equivalent. On the other hand, if U (1)X is not gauged and matter parity
is contained in the ZNR symmetry, PM ⊂ ZNR , the Z10 subgroup of U (1)X is also only a global
symmetry. The Z2 parity transformations relating the two solutions in Eq. (95) to each other arethen global transformations, rendering these two R charge assignments physically inequivalent. In
conclusion, we hence arrive at the following picture,
PM = Z10 /Z5 ⊂ U (1)X : all 10 solutions equivalent ,(96)

PM ⊂ ZNR : 2 equivalence classes containing respectively 5 solutions .

R charges in a U (1)X -invariant extension of the MSSM
If matter parity is a subgroup of the U (1)X , our model as presented in Sec. 2 is not yet complete,
as it still lacks an explanation for the spontaneous breaking of U (1)X at some high energy scale.
In the last subsection of this appendix, we shall thus illustrate by means of a minimal example
how our model could be embedded into a U (1)X -invariant extension of the MSSM.
The seesaw extension of the MSSM is not invariant under U (1)X transformations because of the
lepton number-violating Majorana mass term in the superpotential WMSSM , cf. Eq. (1). Imagine,
however, that this Majorana mass term derives from the Yukawa interaction of the neutrino singlets√
1i with some chiral singlet Φ carrying B−L charge −2 that acquires a non-vanishing VEV ΛB−L / 2
at the GUT scale,1
WMSSM ⊃ Mi 1i 1i
2→

1√ hni Φ1i 1i + λ T
2



Λ2B−L
− ΦΦ̄
2


.

(97)

Here, T and Φ̄ are two further SM singlets with B−L charges 0 and 2, respectively. The field T
carries R charge rT = 2, while Φ and Φ̄ have opposite R charges, rΦ = −rΦ̄ . The diagonal matrix
hn denotes a fourth Yukawa matrix and λ is a dimensionless coupling constant. This replacementof the Majorana mass term evidently renders the superpotential U (1)X -invariant, which allows us
to enlarge the gauge group of our model by a U (1)X factor. Above the GUT scale, the gauge grouphence contains the following subgroup,
GGUT ⊃ SU (5) × U (1)X × ZNR /Z5 ,(98)

where the Z5 symmetry dividing SU (5) × U (1)X corresponds to the center of SU (5) and, at thesame time, to a subgroup of U (1)X , cf. Eq. (94). To prevent it from appearing twice in the gauge
group, it has to be divided out once. At energies around ΛB−L , the Higgs fields Φ and Φ̄ acquire46

non-vanishing VEVs, whereby they spontaneously break U (1)X /Z5 to matter parity PM ,hΦi , hΦ̄i →

ΛB−L
,
2

U (1)X → Z10 ,

PM = Z10 /Z5 .

(99)

Having introduced the fields Φ, Φ̄ and T and modified the superpotential as in Eq. (97), we
have successfully embedded our model into a U (1)X -invariant extension of the MSSM. Let us
now discuss the set of possible R charge assignments in this extended model. Next to the five R
charges of the MSSM fields, rΦ , the R charge of the Higgs field Φ, now represents a further, sixth
independent R charge. All six R charges are again subject to five constraints, which are almost
identical to those in Eq. (90). The only difference now is that the condition deriving from theneutrino Majorana mass term has to modified, so as to account for the presence of the field Φ,
2r1 = 2 + `5 N→

2r1 + rΦ = 2 + `5 N .

(100)

This replacement entails a shift of all viable R charge assignments, cf. Eq. (92), proportional torΦ , which itself remains undetermined, in the direction of the vector X,

N
1 
(N )
(N )
r = r 0 + `˜ X → r = r 0 +−rΦ + `˜N X .
10
10

(101)

For rΦ = 0, we hence recover exactly the same solutions as in Eq. (92). On the other hand,for rΦ 6= 0, all solutions are shifted by − r10Φ X. The universal solution r 0 , which always satisfies
the conditions in Eq. (90), irrespectively of the value of N , turns in particular into r 0 − r10Φ X.
Given the fact that the field Φ carries X charge −10, these shifts are readily identified as U (1)Xgauge transformations acting on the MSSM R charges as well as on the R charge rΦ . The formof our result in Eq. (101) is hence a direct consequence of the U (1)X invariance of our extended
model. As expected, any R charge assignment is only uniquely defined up to arbitrary U (1)X gaugetransformations. Before closing this section, we remark that we are able to use this observation torender all R charges of the universal solution integer-valued. Performing a U (1)X transformationsuch that rΦ = −8, we obtain for the MSSM R chargesrΦ = −8 :

B

r0 −

rΦ
X = (1, −3, 5, 0, 4)T .
10

(102)

Solution to the axion domain wall problem

If the PQ-breaking sector only exhibits a single vacuum, i.e. if NDW = 1, the axion domain wall
problem [46] does not exist from the outset, whatever the thermal history of the universe is [48].
In this appendix, we now illustrate how our model may be easily modified in such a way that itends up having a unique PQ-breaking vacuum.
47

Q1

Q̄1

Qi (i > 1) Q̄i (i > 1)

ZNR

rQ1

6 + 2k − rQ1

rQi

U (1)PQ

qQ−1 − qQ

0

P

P̄

−rQi

−4 − 2k

4 + 2k0

1−1

Table 7: R and PQ charge assignments in a model with a single PQ-breaking vacuum. All R
charges are only defined up to the addition of integer multiples of N . As far as the PQ mechanism
is concerned, the R charges of the extra quark fields Qi can be chosen arbitrarily. They may,
however, be further constrained by requiring appropriate couplings between the new quarks andanti-quarks and the fields of the MSSM, cf. Sec. 2.4.4.

The simplest way to have a unique vacuum is to couple only one pair of additional quarks tothe singlet field P , as is done in the original KSVZ axion model. For n = 1, we may for instance
impose the following superpotential, cf. Eq. (14),
WQ = λ1 P QQ̄ 1 ,λ1 ∼ O(1) .

(103)

The other k − 1 quark pairs are then supposed to obtain masses in consequence of the spontaneous
breaking of R symmetry [38]. Given the coupling in Eq. (103) and requiring vanishing R charges
for all quark pairs that do not couple to the singlet field P , the R and PQ charges of P , P̄ as well
as of the new quarks and anti-quarks can be fixed as listed in Table 7.
Let us now discuss whether the PQ symmetry can be a good accidental symmetry. First ofall, in the continuous R symmetry limit, the global U (1)P , U (1)VQ and U (1)AQ symmetries arealmost exact accidental symmetries of the extra singlet and extra quark sectors, respectively. The
U (1)P symmetry is, however, always explicitly broken by the operator P mk+1
3/2 in the superpotential.
Likewise, given the charge assignments in Tab. 7, we see that the operator P̄ S k+1 is always allowedin the superpotential, even in the continuous R symmetry limit. Therefore, the number of extra
quark pairs k should be large enough in order to ensure that the PQ symmetry is not broken too
severely. After performing an analysis similar to the one Sec. 3, we find that




m3/2
Λ
hSi
+ 0.028 ln
k & 3.3 + 0.12 ln
1 TeV
1 TeV 1012 GeV(104)

is required in order to keep the QCD vacuum angle below 10−10 . Consequently, at least k = 4 extra
quark pairs are needed. With this setup, the PQ symmetry becomes a good accidental symmetry
for sufficiently large N , as is the case in the model discussed in the main body of this paper.

48

Acknowledgments
This work is supported by Grant-in-Aid for Scientific Research from the Ministry of Education,
Science, Sports, and Culture (MEXT), Japan, No. 22244021 (T.T.Y.), No. 24740151 (M.I.), andby the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
The work of K.H. is supported in part by a JSPS Research Fellowship for Young Scientists. K.S.
would like to thank Patrick Vaudrevange and Taizan Watari for helpful discussions pertaining
Appendix A.

References[1] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977);[2] R. D. Peccei and H. R. Quinn, Phys. Rev. D 16, 1791 (1977); S. Weinberg, Phys. Rev. Lett.
40, 223 (1978); F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
[3] M. Kamionkowski and J. March-Russell, Phys. Lett. B 282, 137 (1992) [hep-th/9202003];
T. Banks and N. Seiberg, Phys. Rev. D 83, 084019 (2011) [arXiv:1011.5120 [hep-th]].
[4] S. M. Barr and D. Seckel, Phys. Rev. D 46, 539 (1992).
[5] R. Holman, S. D. H. Hsu, T. W. Kephart, E. W. Kolb, R. Watkins and L. M. Widrow, Phys.
Lett. B 282, 132 (1992) [hep-ph/9203206].
[6] C. A. Baker, D. D. Doyle, P. Geltenbort, K. Green, M. G. D. van der Grinten, P. G. Harris,
P. Iaydjiev and S. N. Ivanov et al., Phys. Rev. Lett. 97, 131801 (2006) [hep-ex/0602020];
J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D 86, 010001 (2012).
[7] H. M. Lee, S. Raby, M. Ratz, G. G. Ross, R. Schieren, K. Schmidt-Hoberg and P. K. S. Vaudrevange, Nucl. Phys. B 850, 1 (2011) [arXiv:1102.3595 [hep-ph]].
[8] J. L. Evans, M. Ibe, J. Kehayias and T. T. Yanagida, Phys. Rev. Lett. 109, 181801 (2012)[arXiv:1111.2481 [hep-ph]].
[9] M. B. Green and J. H. Schwarz, Phys. Lett. B 149, 117 (1984).
[10] G. G. Raffelt, Lect. Notes Phys. 741, 51 (2008) [hep-ph/0611350].
[11] M. Dine, In Cincinnati 1992, Proceedings, Quantum gravity and beyond 157-169 and Calif.
Univ. Santa Cruz - SCIPP 92-027 (92/07,rec.Oct.) 14 p [hep-th/9207045].
49[12] A. G. Dias, V. Pleitez and M. D. Tonasse, Phys. Rev. D 67, 095008 (2003)[hep-ph/0211107].
[13] L. M. Carpenter, M. Dine and G. Festuccia, Phys. Rev. D 80, 125017 (2009)[arXiv:0906.1273 [hep-th]].
[14] G. Lazarides, C. Panagiotakopoulos and Q. Shafi, Phys. Rev. Lett. 56, 432 (1986).
[15] K. -S. Choi, H. P. Nilles, S. Ramos-Sanchez and P. K. S. Vaudrevange, Phys. Lett. B 675,
381 (2009) [arXiv:0902.3070 [hep-th]].
[16] A. G. Dias, V. Pleitez and M. D. Tonasse, Phys. Rev. D 69, 015007 (2004)[hep-ph/0210172].
[17] G. F. Giudice and A. Masiero, Phys. Lett. B 206, 480 (1988).
[18] T. Yanagida, Phys. Lett. B 400, 109 (1997) [hep-ph/9701394].
[19] M. Dine and J. Kehayias, Phys. Rev. D 82, 055014 (2010) [arXiv:0909.1615 [hep-ph]].
[20] S. Dimopoulos and H. Georgi, Nucl. Phys. B 193, 150 (1981).
[21] N. Sakai and T. Yanagida, Nucl. Phys. B 197, 533 (1982); S. Weinberg, Phys. Rev. D 26, 287(1982).
[22] K. I. Izawa and T. Yanagida, Prog. Theor. Phys. 97, 913 (1997) [hep-ph/9703350].
[23] A. E. Nelson and N. Seiberg, Nucl. Phys. B 416, 46 (1994) [hep-ph/9309299].
[24] K. A. Intriligator, N. Seiberg and D. Shih, JHEP 0604, 021 (2006) [hep-th/0602239].
[25] Y. Imamura, T. Watari and T. Yanagida, Phys. Rev. D 64, 065023 (2001)[hep-ph/0103251].
[26] P. Minkowski, Phys. Lett. B 67, 421 (1977); T. Yanagida, Conf. Proc. C 7902131, 95 (1979);
M. Gell-Mann, P. Ramond and R. Slansky, Conf. Proc. C 790927, 315 (1979); S. L. Glashow,
NATO Adv. Study Inst. Ser. B Phys. 59, 687 (1980); R. N. Mohapatra and G. Senjanovic,
Phys. Rev. Lett. 44, 912 (1980).
[27] N. Sakai, Z. Phys. C 11, 153 (1981).
[28] E. Witten, Nucl. Phys. B 188, 513 (1981).
50

[29] M. Fallbacher, M. Ratz and P. K. S. Vaudrevange, Phys. Lett. B 705, 503 (2011)[arXiv:1109.4797 [hep-ph]].
[30] K. Kurosawa, N. Maru and T. Yanagida, Phys. Lett. B 512, 203 (2001) [hep-ph/0105136].
[31] T. Yanagida, Phys. Lett. B 344, 211 (1995) [hep-ph/9409329]; T. Hotta, K. I. Izawa andT. Yanagida, Phys. Rev. D 53, 3913 (1996) [hep-ph/9509201]; J. Hisano and T. Yanagida,Mod. Phys. Lett. A 10, 3097 (1995) [hep-ph/9510277].
[32] S. P. Martin, Phys. Rev. D 46, 2769 (1992) [hep-ph/9207218]; S. P. Martin, Phys. Rev. D 54,
2340 (1996) [hep-ph/9602349].
[33] L. E. Ibanez, Nucl. Phys. B 398, 301 (1993) [hep-ph/9210211].
[34] T. Banks and M. Dine, Phys. Rev. D 45, 1424 (1992) [hep-th/9109045].
[35] L. E. Ibanez and G. G. Ross, Phys. Lett. B 260, 291 (1991).
[36] S. Weinberg, Phys. Rev. Lett. 43, 1566 (1979); T. Yanagida, Proceedings, Quark and Lepton
Physics, 233-237, Tokyo (1981).
[37] H. M. Lee, S. Raby, M. Ratz, G. G. Ross, R. Schieren, K. Schmidt-Hoberg and P. K. S. Vaudrevange, Phys. Lett. B 694, 491 (2011) [arXiv:1009.0905 [hep-ph]].
[38] K. Inoue, M. Kawasaki, M. Yamaguchi and T. Yanagida, Phys. Rev. D 45, 328 (1992);
J. A. Casas and C. Munoz, Phys. Lett. B 306, 288 (1993) [hep-ph/9302227].
[39] B. Feldstein and T. T. Yanagida, Phys. Lett. B 720, 166 (2013) [arXiv:1210.7578 [hep-ph]].
[40] J. E. Kim, Phys. Rev. Lett. 43, 103 (1979).
[41] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 166, 493 (1980).
[42] R. D. Peccei, Lect. Notes Phys. 741, 3 (2008) [hep-ph/0607268].
[43] H. Georgi, D. B. Kaplan and L. Randall, Phys. Lett. B 169, 73 (1986).
[44] J. Preskill, M. B. Wise and F. Wilczek, Phys. Lett. B 120, 127 (1983); L. F. Abbott and
P. Sikivie, Phys. Lett. B 120, 133 (1983); M. Dine and W. Fischler, Phys. Lett. B 120, 137(1983).
[45] P. Sikivie, Lect. Notes Phys. 741, 19 (2008) [astro-ph/0610440].
51

[46] P. Sikivie, Phys. Rev. Lett. 48, 1156 (1982).
[47] M. Kawasaki, T. T. Yanagida and K. Yoshino, arXiv:1305.5338 [hep-ph].
[48] A. Vilenkin and A. E. Everett, Phys. Rev. Lett. 48, 1867 (1982).
[49] G. Veneziano and S. Yankielowicz, Phys. Lett. B 113, 231 (1982); V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 229, 407 (1983); E. Cohen and
C. Gomez, Phys. Rev. Lett. 52, 237 (1984).
[50] K. Harigaya, M. Ibe, K. Schmitz and T. T. Yanagida, Phys. Lett. B 721, 86 (2013)[arXiv:1301.3685 [hep-ph]].
[51] M. Maniatis, Int. J. Mod. Phys. A 25, 3505 (2010) [arXiv:0906.0777 [hep-ph]]; U. Ellwanger,
C. Hugonie and A. M. Teixeira, Phys. Rept. 496, 1 (2010) [arXiv:0910.1785 [hep-ph]].
[52] C. Panagiotakopoulos and K. Tamvakis, Phys. Lett. B 469, 145 (1999) [hep-ph/9908351];
C. Panagiotakopoulos and A. Pilaftsis, Phys. Rev. D 63, 055003 (2001) [hep-ph/0008268].
[53] K. S. Jeong, Y. Shoji and M. Yamaguchi, JHEP 1204, 022 (2012) [arXiv:1112.1014 [hep-ph]].
[54] [ATLAS Collaboration], ATLAS-CONF-2013-011.
[55] A. Menon, D. E. Morrissey and C. E. M. Wagner, Phys. Rev. D 70, 035005 (2004)[hep-ph/0404184]; C. Balazs, M. S. Carena, A. Freitas and C. E. M. Wagner, JHEP 0706,
066 (2007) [arXiv:0705.0431 [hep-ph]]; J. Cao, H. E. Logan and J. M. Yang, Phys. Rev. D 79,
091701 (2009) [arXiv:0901.1437 [hep-ph]].
[56] G. Cacciapaglia, A. Deandrea, L. Panizzi, N. Gaur, D. Harada and Y. Okada, JHEP 1203,
070 (2012) [arXiv:1108.6329 [hep-ph]]; J. A. Aguilar-Saavedra, R. Benbrik, S. Heinemeyer and
M. Perez-Victoria, arXiv:1306.0572 [hep-ph].
[57] V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 229, 381(1983), Phys. Lett. B 166, 329 (1986) [Sov. J. Nucl. Phys. 43, 294 (1986)] [Yad. Fiz. 43,459 (1986)]; J. Hisano and M. A. Shifman, Phys. Rev. D 56, 5475 (1997) [hep-ph/9705417];
N. Arkani-Hamed and H. Murayama, JHEP 0006, 030 (2000) [hep-th/9707133].
[58] J. A. Aguilar-Saavedra, JHEP 0911, 030 (2009) [arXiv:0907.3155 [hep-ph]].
[59] [ATLAS Collaboration], ATLAS-CONF-2013-051.

52

[60] J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D 86, 010001 (2012).
[61] P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5076 [astro-ph.CO].
[62] K. Kumekawa, T. Moroi and T. Yanagida, Prog. Theor. Phys. 92, 437 (1994)[hep-ph/9405337]; K. -I. Izawa and T. Yanagida, Phys. Lett. B 393, 331 (1997)[hep-ph/9608359]; M. Ibe, K. -I. Izawa, Y. Shinbara and T. T. Yanagida, Phys. Lett. B 637,
21 (2006) [hep-ph/0602192]; M. Ibe, Y. Shinbara and T. T. Yanagida, Phys. Lett. B 642, 165(2006) [hep-ph/0608127].

53