A Novel String Derived Z' With Stable Proton, Light-Neutrinos and
  R-parity violation
arXiv:0704.1256v2  [hep-ph]  10 Oct 2007
LTH–742
arXiv:????.????
A Novel String Derived Z ′ With Stable Proton,
Light–Neutrinos and R–parity violation
Claudio Corianò†1, Alon E. Faraggi♦2 and Marco Guzzi†3
†Dipartimento di Fisica, Universita’ di Lecce,
I.N.F.N. Sezione di Lecce Via Arnesano, 73100 Lecce, Italy
♦Department of Mathematical Sciences
University of Liverpool, Liverpool, L69 7ZL, United Kingdom
Abstract
The Standard Model indicates the realization of grand unified structures in
nature, and can only be viewed as an effective theory below a higher energy
cutoff. While the renormalizable Standard Model forbids proton decay medi-
ating operators due to accidental global symmetries, many extensions of the
Standard Model introduce such dimension four, five and six operators. Fur-
thermore, quantum gravity effects are expected to induce proton instability,
indicating that the higher energy cutoff scale must be above 1016 GeV. Quasi–
realistic heterotic string models provide the arena to explore how perturbative
quantum gravity affects the particle physics phenomenology. An appealing ex-
planation for the proton longevity is provided by the existence of an Abelian
gauge symmetry that suppresses the proton decay mediating operators. Addi-
tionally, such a low–scale U(1) symmetry should: allow the suppression of the
left–handed neutrino masses by a seesaw mechanism; allow fermion Yukawa
couplings to the electroweak Higgs doublets; be anomaly free; be family uni-
versal. These requirements render the existence of such U(1) symmetries in
quasi–realistic heterotic string models highly non–trivial. We demonstrate the
existence of a U(1) symmetry that satisfies all of the above requirements in a
class of left–right symmetric heterotic string models in the free fermionic for-
mulation. The existence of the extra Z ′ in the energy range accessible to future
experiments is motivated by the requirement of adequate suppression of proton
decay mediation. We further show that while the extra U(1) forbids dimension
four baryon number violating operators it allows dimension four lepton number
violating operators and R–parity violation.
1E-mail address: Claudio.Coriano@le.infn.it
2E-mail address: faraggi@amtp.liv.ac.uk
3E-mail address: Marco.Guzzi@le.infn.it
http://arxiv.org/abs/0704.1256v2
1 Introduction
The Standard Model of particle physics successfully accounts for all observations
in the energy range accessible to contemporary experiments. Despite this enormous
success the Standard Model can only be viewed as an effective low energy field theory
below a higher energy cutoff. In the least, the existence of a Landau pole in the
hypercharge sector, albeit at an enormously high scale, unequivocally demonstrates
the formal inconsistency of the Standard Model. In this regard, the renormalizability
of the Standard Model is an approximate feature and effects of nonrenormalizable
operators, suppressed by powers of the high scale cutoff, must be considered.
The high precision analysis of the Standard Model parameters, achieved at LEP
and other particle physics experiments, indicates that the Standard Model remains
an approximate renormalizable quantum field theory up to a very large energy scale.
Possibly the grand unification scale, or the Planck scale. The logarithmic evolution
of the Standard Model parameters is in agreement with the available data, and is
compatible with the notion of unification at a high energy scale in the gauge and heavy
matter sectors of the Standard Model. Preservation of the logarithmic evolution in
the scalar sector necessitates the introduction of a new symmetry between bosons
and fermions, dubbed supersymmetry.
Perhaps the most important observation indicative that the Standard Model cutoff
scale is a very high scale is the longevity of the proton. Renormalizability insures
that baryon and lepton violating operators are absent in the perturbative Standard
Model. Hence, in the renormalizable Standard Model baryon and lepton numbers are
accidental global symmetries. However, at the cutoff scale dimension six operators
are induced and the proton is in general expected to decay. The observed proton
lifetime implies that the cutoff scale is of order 1016GeV. The problem is exacerbated
in supersymmetric extensions of the Standard Model that allow dimension four and
five baryon and lepton violating operators [1]. Indeed, one would expect proton
decay mediating operators to arise in most extensions of the Standard Model. In the
Minimal Supersymmetric Standard Model one imposes a global symmetry, R–parity,
which forbids the dimension four baryon and lepton number violating operators.
The difficulty with dimension five operators can only be circumvented if one further
assumes that the relevant Yukawa couplings are suppressed. However, as global
symmetries are not expected to survive quantum gravity effects [2], the proton lifetime
problem becomes especially acute in the context of theories that unify the Standard
Model with gravity. This question has been examined extensively in the context of
quasi–realistic heterotic string models. In this context, the most appealing suggestion
is that the suppression of the proton decay mediating operators is a result of a gauged
U(1) symmetry, under which the undesired nonrenormalizable dimension four and
five operators are not invariant. If the U(1) symmetry remains unbroken down to
sufficiently low scales the problematic operators will be suppressed by at least the
VEV that breaks the additional U(1) symmetry over the cutoff scale.
The free fermionic heterotic string models are among the most realistic string
models constructed to date [3, 4, 5, 6, 7, 8]. The issue of proton stability was sporad-
ically explored in these models [9, 10, 11, 12, 13], as well as explorations of possible
U(1) symmetries that can ensure proton longevity [9, 11, 12, 13]. However, non of
the current proposals is satisfactory. The U(1) symmetry of ref. [9] is the U(1)
combination of B − L and T3R which is embedded in SO(10) and is orthogonal to
the electroweak hypercharge. However, this U(1) symmetry in general needs to be
broken to allow for the suppression of the left–handed neutrino masses by a seesaw
mechanism. Similarly, the U(1) symmetries studied in ref. [11, 12, 13], that arise
in the string models from combinations of the U(1) symmetries that are external
to SO(10) are flavour dependent U(1) symmetries that in general must be broken
near the string scale to allow for generation of fermion masses. In ref. [12] it was
concluded that non of the symmetries suggested in ref. [11] can remain unbroken
down to low energies and provide for the suppression of the proton decay mediat-
ing operators. Furthermore, a family non–universal U(1) symmetry is restricted by
constraints on flavour changing neutral currents, and cannot exist in energy range
accessible to forthcoming experiments.
The proton longevity, together with the Standard Model multiplet structure,
therefore provide the most important clues for the origin of the Standard Model
particle spectrum. These favour the embedding of Standard Model in a Grand Uni-
fied Theory, possibly broken to the Standard Model at the string level. The GUT
embedding of the Standard Model, and its supersymmetric extension, leads to pro-
ton decay mediating operators. The most robust and economical way to suppress
the dangerous operators is by the existence of an additional Abelian gauge symmetry
which is broken above the electroweak scale and does not interfere with the other
phenomenological constraints. Such a U(1) symmetry should fulfill the following
requirements:
• Forbid dimension four, five and six proton decay mediating operators.
• Allow suppression of left–handed neutrino masses by a seesaw mechanism.
• Allow the fermion Yukawa couplings to electroweak Higgs doublets.
• Be family universal.
• Be anomaly free.
This list of requirements render the existence of such a U(1) symmetry in string
models highly nontrivial. For example, in models with an underlying SO(10) GUT
embedding the U(1)B−L symmetry is gauged. It forbids the dimension four baryon
and lepton number violating operators, but not the dimension five operator. Further-
more, suppression of left–handed neutrino masses by a seesaw mechanism in general
necessitates that the symmetry is broken near the GUT scale. Hence, it cannot
remain unbroken down to low energies, and in general fast proton decay from dimen-
sion four operators is expected to ensue. Similarly, the U(1)A symmetry external
to SO(10) in E6 → SO(10) × U(1)A is anomalous in many of the quasi–realistic
string models constructed to date [14] and is broken by a generalised Green–Schwarz
mechanism. The additional U(1)s investigated in refs. [11, 12, 13] are either flavour
non–universal or constrain the fermion Yukawa mass terms and must therefore be
similarly broken near the Planck scale. Thus, of all the extra U(1)’s investigated to
date non seems to remain viable down to low energies, and to provide the coveted
proton protection symmetry.
In this paper we therefore explore further the possibility that quasi–realistic string
models give rise to Abelian gauged symmetries that can play the role of the proton
lifetime guard. We demonstrate the existence of a U(1) symmetry satisfying all of
the above requirements in the class of left–right symmetric string–derived models of
ref. [7]. The key to obtaining the U(1) symmetry satisfying the above requirements is
the SO(10) symmetry breaking pattern particular to the left–right symmetric models
[7]. The key distinction is that in these models the U(1)A, which is external to the
unbroken SO(10) subgroup, is anomaly free, and may remain unbroken down to low
energies. It is does not restrict the charged fermion mass terms, and it allows for the
suppression of the left–handed neutrino masses by a seesaw mechanism. Its existence
at low energies is motivated by the longevity of the proton lifetime. Furthermore, as
we discuss below, while it forbids the supersymmetric dimension four and five baryon
number violating operators, it allows the dimension four lepton number violating
operator. Hence, while proton decay from dimension four operators does not ensue,
lepton number and R–parity violation do arise. This observation has far reaching
implications in terms of the phenomenology and collider signatures of the models.
2 The structure of the free fermionic models
In this section we describe the structure of the quasi–realistic free fermionic models
and the properties of the proton protecting U(1) symmetry. The free fermionic for-
mulation the 4-dimensional heterotic string, in the light-cone gauge, is described by
20 left–moving and 44 right–moving two dimensional real fermions [16]. The models
are constructed by specifying the phases picked up by the world–sheet fermions when
transported around the torus non-contractible loops. Each model corresponds to a
particular choice of fermion phases consistent with modular invariance that can be
generated by a set of basis vectors vi, i = 1, . . . , n, vi = {αi(f1), αi(f2), αi(f3)) . . .} .
The basis vectors span a space Ξ which consists of 2N sectors that give rise to the
string spectrum. The spectrum is truncated by a Generalised GSO (GGSO) projec-
tions [16].
The U(1) charges, Q(f), with respect to the unbroken Cartan generators of the
four dimensional gauge group, which are in one to one correspondence with the U(1)
currents f ∗f for each complex fermion f , are given by:
Q(f) =
α(f) + F (f), (1)
where α(f) is the boundary condition of the world–sheet fermion f in the sector
α. F (f) is the fermion number operator counting each mode of f once (and if f is
complex, f ∗ minus once). For periodic fermions, α(f) = 1, the vacuum is a spinor
in order to represent the Clifford algebra of the corresponding zero modes. For each
periodic complex fermion f there are two degenerate vacua |+〉, |−〉 , annihilated by
the zero modes f0 and f0
∗ and with fermion numbers F (f) = 0,−1, respectively.
The two dimensional world–sheet fermions are divided in the following way: the
eight left–moving real fermions ψ1,2 and χ1,···,6 correspond to the eight Ramond–
Neveu–Schwarz fermions of the ten dimensional heterotic string in the light–cone
gauge; the twenty–four real–fermions {yi, ωi|ȳi, ω̄i}, i = 1, . . . , 6 correspond to the
fermionized internal coordinates of a compactified manifold in a bosonic formulation;
the complex right–moving fermions φ̄1,···,8 generate the rank eight hidden gauge group;
ψ̄1,···,5 generate the SO(10) gauge group; η̄1,2,3 generate the three remaining U(1)
generators in the Cartan sub-algebra of the observable rank eight gauge group. A
combination of these U(1) currents will play the role of the proton lifetime guard.
The free fermionic models are defined in terms of the basis vectors and one–
loop GGSO projection coefficients. The quasi–realistic free fermionic heterotic–string
model are typically constructed in two stages. The first stage consists of the NAHE–
set, {1, S, b1, b2, b3} [17, 18]. The gauge group at this stage is SO(10)×SO(6)
3×E8,
and the vacuum contains forty–eight multiplets in the 16 chiral representation of
SO(10). The second stage consists of adding three or four basis vectors to the NAHE–
set, typically denoted by {α, β, γ}. The additional basis vectors reduce the number
of generations to three, with one arising from each of the basis vectors b1, b2 and
b3. Additional non–chiral generations may arise from the basis vectors that extend
the NAHE–set. This distribution of the chiral generations is particular to the class
of quasi–realistic free fermionic models that has been explored to date, and other
possibilities may exist [15]. Additionally, the basis vectors that extend the NAHE–
set break the four dimensional gauge group. The SO(10) symmetry is broken to one
of the subgroups: SU(5)× U(1) [3]; SO(6)× SO(4) [5]; SU(3)× SU(2)× U(1)2 [6];
SU(3)×SU(2)2×U(1) [7]; or SU(4)×SU(2)×U(1) [8]. The three generations from
the sectors b1, b2 and b3 are decomposed under the final SO(10) subgroup. The flavour
SO(6)3 groups are broken to products of U(1)n with 3 ≤ n ≤ 9. The U(1)1,2,3 factors
arise from the three right–moving complex fermions η̄1,2,3. Additional U(1) currents
may arise from complexifications of right–moving fermions from the set {ȳ, ω̄}1,···,6.
The U(1) symmetry that will serve as the proton lifetime guard is a combination
of the three U(1) symmetries generated by the world–sheet complex fermions η̄1,2,3.
The states from each of the sectors b1, b2 and b3 are charged with respect to one of
these U(1) symmetries, i.e. with respect to U(1)1, U(1)2 and U(1)3, respectively.
Hence the U(1) combination
U(1)ζ = U1 + U2 + U3 (2)
is family universal. In the string derived models of ref. [3, 4, 5, 6] U(1)1,2,3 are
anomalous. Therefore, also U(1)ζ is anomalous and must be broken near the string
scale. In the string derived left–right symmetric models of ref [7] U(1)1,2,3 are anomaly
free, and hence also the combination U(1)ζ is anomaly free. It is this property of
these models which allows this U(1) combination to remain unbroken.
Subsequent to constructing the basis vectors and extracting the massless spectrum
the analysis of the free fermionic models proceeds by calculating the superpotential.
The cubic and higher-order terms in the superpotential are obtained by evaluating
the correlators
AN ∼ 〈V
3 · · ·VN〉, (3)
where V
i ) are the fermionic (scalar) components of the vertex operators, using
the rules given in [19]. Generically, correlators of the form (3) are of order O(gN−2),
and hence of progressively higher orders in the weak-coupling limit. Typically, one
of the U(1) factors in the free-fermion models is anomalous, and generates a Fayet–
Ilioupolos term which breaks supersymmetry at the Planck scale [20]. The anomalous
U(1) is broken, and supersymmetry is restored, by a non–trivial VEV for some scalar
field that is charged under the anomalous U(1). Since this field is in general also
charged with respect to the other anomaly-free U(1) factors, some non-trivial set of
other fields must also get non–vanishing VEVs V, in order to ensure that the vacuum
is supersymmetric. Some of these fields will appear in the nonrenormalizable terms
(3), leading to effective operators of lower dimension. Their coefficients contain fac-
tors of order V/M∼ 1/10. Typically the solution of the D– and F–flatness constraints
break most or all of the horizontal U(1) symmetries.
3 The proton lifeguard
In this section we discuss the characteristics of U(1)ζ in the left–right symmetric
string derived models [7], versus those of U(1)A in the string derived models of refs.
[3, 4, 5, 6]. We note that both U(1)ζ as well as U(1)A are obtained from the same
combination of complex right–moving world–sheet currents η̄1,2,3, i.e. both are given
by a combination of U1, U2, and U3. The distinction between the two cases, as we
describe in detail below, is due to the charges of the Standard Model states, arising
from the sectors b1, b2 and b3, under this combination. The key feature of U(1)ζ
in the models of ref. [7] is that it is anomaly free. To study the characteristics of
the proton protecting U(1) symmetry it is instructive to examine in combinatorial
notation the vacuum structure of the chiral generations from the sectors b1,2,3. The
vacuum of the sectors bj contains twelve periodic fermions. Each periodic fermion
gives rise to a two dimensional degenerate vacuum |+〉 and |−〉 with fermion numbers
0 and −1, respectively. The GSO operator, is a generalised parity operator, which
selects states with definite parity. After applying the GSO projections, we can write
the degenerate vacuum of the sector b1 in combinatorial form
)] {(
where 4 = {y3y4, y5y6, ȳ3ȳ4, ȳ5ȳ6}, 2 = {ψµ, χ12}, 5 = {ψ̄1,···,5} and 1 = {η̄1}. The
combinatorial factor counts the number of |−〉 in the degenerate vacuum of a given
state. The first term in square brackets counts the degeneracy of the multiplets,
being eight in this case. The two terms in the curly brackets correspond to the
two CPT conjugated components of a Weyl spinor. The first term among those
corresponds to the 16 spinorial representation of SO(10), and fixes the space–time
chirality properties of the representation, whereas the second corresponds to the CPT
conjugated anti–spinorial 16 representation. Similar vacuum structure is obtained
for b2 and b3. The periodic boundary conditions of the world–sheet fermions η̄
entails that the fermions from each sector bj are charged with respect to one of the
U(1)j symmetries. The charges, however, depend on the SO(10) symmetry breaking
pattern, induced by the basis vectors that extend the NAHE–set, and may, or may
not, differ in sign between different components of a given generation. In the models
of ref. [3, 6, 5] the charges of a given bj generation under U(1)j is of the same sign,
whereas in the models of ref. [7] they differ. In general, the distinction is by the
breaking of SO(10) to either SU(5) × U(1) or SO(6) × SO(4). In the former case
they will always have the same sign, whereas in the later they may differ. This
distinction fixes the charges of the Standard Model states under the U(1) symmetry
which safeguards the proton from decaying, while not obstructing the remaining
constraints listed above.
In the free fermionic standard–like models the SO(10) symmetry is broken to4
SU(3)× SU(2)× U(1)C × U(1)L. The weak hypercharge is given by
U(1)Y =
U(1)C +
U(1)L, (5)
and the orthogonal U(1)Z′ combination is given by
U(1)Z′ = U(1)C − U(1)L. (6)
The three twisted sectors b1, b2 and b3 produce three generations in the sixteen
representation of SO(10) decomposed under the final SO(10) subgroup. In terms of
4U(1)C = 3/2U(1)B−L ; U(1)L = 2U(1)T3
the SU(3)C × U(1)C × SU(2)L × U(1)L decomposition they take the values
E ≡ [(1, 3/2); (1, 1)];
U ≡ [(3̄,−1/2); (1,−1)];
Q ≡ [(3, 1/2); (2, 0)];
N ≡ [(1, 3/2); (1,−1)];
D ≡ [(3̄,−1/2); (1, 1)];
L ≡ [(1,−3/2); (2, 0)]. (7)
In terms of the SO(6)×SO(4) Pati–Salam decomposition [21] the Standard Model
fermion fields are embedded in the
FL ≡ (4, 2, 1) = Q + L ;
FR ≡ (4̄, 1, 2) = U +D + E +N , (8)
representations of SU(4) × SU(2)L × SU(2)R. In terms of the left–right symmetric
decomposition of ref. [7] the embedding is in the following representations:
QL = (3, 2, 1,
) , (9)
QR = (3̄, 1, 2,−
) = U +D , (10)
LL = (1, 2, 1,−
) , (11)
LR = (1, 1, 2,
) = E +N , (12)
of SU(3) × SU(2)L × SU(2)R × U(1)C . The Higgs fields in the later case are in a
bi–doublet representation
h = (1, 2, 2, 0) =
hu+ h
hu0 h
. (13)
Using the combinatorial notation introduced in eq. (4) the decomposition of the
16 representation of SO(10) in the Pati–Salam string models is
)] [(
)] [(
} (14)
The crucial point is that the Pati–Salam breaking pattern allows the first and second
terms in curly brackets to come with opposite charges under U(1)j . This results
from the operation of the GSO projection operator, which differentiates between
the two terms. Thus, in models that descend from SO(10) via the SU(5) × U(1)
breaking pattern the charges of a generation from a sector bj j = 1, 2, 3, under the
corresponding symmetry U(1)j are either +1/2, or −1/2, for all the states from
that sector. In contrast, in the left–right symmetric string models the corresponding
charges, up to a sign are,
Qj(QL;LL) = +1/2 ;Qj(QR;LR) = −1/2, (15)
i.e. the charges of the SU(2)L doublets have the opposite sign from those of the
SU(2)R doublets. This is in fact the reason that in the left–right symmetric string
models [7] it was found that, in contrast to the case of the FSU5 [3], Pati–Salam
[5] and standard–like [6], string models, the U(1)j symmetries are not part of the
anomalous U(1) symmetry [7].
It is therefore noted that the
U(1)ζ = U1 + U2 + U3 (16)
combination is a family–universal, anomaly free5, U(1) symmetry, and allows the
quark and lepton fermion mass terms
QLQRh and LLLRh . (17)
The two combinations of U(1)1, U(1)2 and U(1)3, that are orthogonal to U(1)ζ , are
family non–universal and may be broken at, or slightly below, the string scale.
The left–right symmetric heterotic string models of ref. [7] provide explicit quasi–
realistic string models, that realize the charge assignment of eq. (15). Furthermore,
the dimension four and five baryon number violating operators that arise from
QLQLQLLL → QQQL (18)
QRQRQRLR → {UDDN,UUDE} (19)
are forbidden, while the lepton number violating operators that arise from
QLQRLLLR → QDLN (20)
LLLLLRLR → LLEN (21)
are allowed.
The crucial observation is the opposite charge assignment of the left and right–
handed fields under U(1)ζ . This is available in models that descend from the Pati–
Salam symmetry breaking pattern of the underlying SO(10) GUT symmetry. In this
case the left– and right–moving fields carry opposite sign under the GSO projection
operator, induced by the basis vector that breaks SO(10) → SO(6) × SO(4). An
additional symmetry breaking stage of the Pati–Salam models [5], or left–right sym-
metric models [7], can be obtained at the string level or in the effective low–energy
5We note that there may exist string models in the classes of [3, 5, 6] in which U(1)ζ is anomaly
free. This may be the case in the so called self–dual vacua of ref. [15]. Such quasi–realistic string
models with an anomaly free U(1)ζ have not been constructed to date.
field theory by the Higgs fields in the representations {QH , Q̄H} = {(4̄, 1, 2), (4, 1, 2)}
or {LH , L̄H} = {(1̄, 1, 2,
), (1, 1, 2,−3
)}. The breaking can be achieved at the string
level, while preserving the desired charge assignment, as long as a basis vector of
the form 2γ of refs. [6], or b6 of ref. [5], are not introduced. The boundary condi-
tion assignments in these basis vectors entails that the N = 4 vacuum that we start
with factorizes the gauge degrees of freedom into E8 ×E8 or SO(16)× SO(16). The
consequence of this is that all the states from the twisted matter sectors bj carry
the same charge under U(1)j . Thus, this result is circumvented by not including the
vectors 2γ of [6], or b6 of [5] in the construction. In effect, such models are descending
from a different N = 4 underlying vacuum [7, 8]. Being SO(16) × E7 × E7 in the
models of ref. [7], which explicitly realize the desired breaking pattern in a class of
quasi–realistic string models. We assume below that the SU(2)R symmetry is bro-
ken directly at the string level in which case the remnant U(1)Z′ given in eq. (6)
has to be broken by the Higgs fields {NH , N̄H} = (1, 1, 0, 5/2), (1, 1, 0,−5/2) under
SU(3)× SU(2)× U(1)Y × U(1)Z′.
4 An effective string inspired Z ′ model
Inspired by the U(1) charge assignment in the left–right symmetric string derived
models [7], we present an effective field theory model incorporating these features.
At this stage our aim is to build an effective model that can be used in correspondence
with experimental data, rather than a complete effective field theory model below the
string scale, which is of further interest and will be discussed in future publications.
The charges of the fields in the low energy effective field theory of the string inspired
model are given by
Field U(1)Y U(1)Z′ U(1)ζ U(1)ζ′
Li −1
U i −2
Ei 1 1
N i 0 5
φi 0 0 0 0
φ0 0 0 0 0
−1 0 −1
HD −1
1 0 1
N̄H 0 −
ζH 0 0 1 1
ζ̄H 0 0 −1 −1
with i = 1, 2, 3. The U(1)ζ′ symmetry is the combination of U(1)Z′ and U(1)ζ left
unbroken by the vevs of NH and N̄H . The fields ζH and ζ̄H are needed to break the
residual U(1)ζ′ symmetry. States with the required quantum numbers in (22) exist in
the string models [7]. The fields φi are employed in an extended seesaw mechanism.
Using the superpotential terms
LiNjH
U , NiN̄Hφj , φiφjφk . (23)
The neutrino seesaw mass matrix takes the form
( νi Nk φm )
0 (kM
)ij 0
)ij 0 Mχ
0 Mχ O(Mφ)
, (24)
with Mχ ∼ 〈N̄H〉 and Mφ ∼ 〈φ0〉. The mass eigenstates are mainly νi, Nk and φm
with a small mixing and with the eigenvalues
mνj ∼Mφ
kM ju
mNj , mφ ∼Mχ .
A detailed fit to the neutrino data was discussed in ref [22]. We emphasize, however,
that our aim here is merely to demonstrate that the extra U(1)ζ′ , introduced below,
is not in conflict with the requirement of light neutrino masses. Alternatively, the
VEV of 〈N̄H〉 induces heavy Majorana mass terms for the right–handed neutrinos
from nonrenormalizable terms
NiNjN̄HN̄H . (25)
The effective Majorana mass scale of the right–handed neutrinos is then Mχ ∼
〈N̄H〉
2/M , which for 〈N̄H〉 ∼ 10
16GeV gives Mχ ∼ 10
14GeV. The VEV of 〈NH〉 may
induce unsuppressed dimension four baryon and lepton number violating interactions
η1QDL+ η2UDD (26)
from the nonrenormalizable terms given in eqs. (19) and (20). Therefore, if the VEV
of NH is of the order of the GUT, or intermediate, scale, as is required in the seesaw
mass matrix in eq. (24), then unsuppressed proton decay will ensue. However, this
VEV leaves the unbroken combination of U(1)Z′ and U(1)ζ given by
U(1)ζ′ =
U(1)Z′ − U(1)ζ . (27)
The induced dimension four lepton number violating operator that arises from eq.
(20) is invariant under U(1)ζ′ , whereas the induced dimension four baryon number
violating operator that arises from eq. (19) is not. Hence, to generate an unsup-
pressed dimension four baryon number violating operator we must break also U(1)ζ′.
Therefore, if U(1)ζ′ remains unbroken down to low energies, it suppresses proton de-
cay from dimension four operators. Similarly, the dimension five baryon and lepton
number violating operators given in eqs. (18) and (19) are not invariant under U(1)ζ′
and hence suppressed if U(1)ζ′ remains unbroken down to low energies.
5 Estimate of the U(1)ζ ′ mass scale
The dimension four and five proton decay mediating operators are forbidden by the
U(1)Z′ and U(1)ζ gauge symmetries. These symmetries are broken by some fields
and we can estimate the required symmetry breaking scale in order to ensure suffi-
cient suppression. In turn this will indicate the possible mass scale of the additional
Zζ′ vector boson, and whether it may exist in the range accessible to forthcoming
experiments. The dimension four operators that give rise to rapid proton decay,
η1UDD + η2QLD, are induced from the non–renormalizable terms of the form
η1(UDDN)Φ + η2(QLDN)Φ
′ (28)
where, Φ and Φ′ are combinations of fields that fix gauge invariance and the string
selection rules. The field NH can be the Standard Model singlet in the 16 represen-
tation of SO(10), or it can be a product of two fields, which effectively reproduces
the SO(10) charges of NH [12]. We take the VEV of NH , which breaks the B − L
symmetry, to be of the order of the GUT scale, i.e. 〈NH〉 ∼ 10
16GeV. This is the case
as the VEV of N̄H induces the seesaw mechanism, which suppresses the left–handed
neutrino masses. The VEVs of Φ and Φ′ then fixes the magnitude of the effective
proton decay mediating operators, with
η′1 ∼
; η′2 ∼
. (29)
We take M to be the heterotic string unification scale, M ∼ 1018GeV. Similarly,
the dimension five proton decay mediating operator QQQL can effectively be induced
from the nonrenormalizable terms
λ1QQQL(Φ
′′) (30)
The VEV of φ′′ then fixes the magnitude of the effective dimension five operator to
λ′1 ∼ λ1
〈φ′′〉
The experimental limits impose that the product (η′1η
2) ≤ 10
−24 and (λ′1/M) ≤ 10
Hence, for M ∼ Mstring ∼ 10
18GeV we must have λ′1 ≤ 10
−7, to guarantee that the
proton lifetime is within the experimental bounds. The induced dimension four lepton
number violating operator is invariant under U(1)ζ′. Hence, we can take n
′ = 0. The
dimension five baryon number violating operator is not invariant under U(1)ζ′ . Hence
we must have at least n′′ = 1. We assume that the dimension four baryon number
violating operator in eq. (26) is induced at the quintic order. The corresponding
nonrenormalizable term in eq. (28) contain one additional field that breaks the
proton protecting U(1)ζ′ at intermediate energy scale Λζ′. Hence, we have n = 1 in
eq. (29), and
(η′1η
Taking 〈N〉 ∼ 1016GeV and M ∼ 1018GeV, we obtain the estimate Λζ′ ≤ 10
−2GeV,
which is clearly too low. Taking 〈N〉 ∼ 1013GeV yields Λζ′ ≤ 10
4GeV. We also
have that in this case λ′1/M < 10
−14. Hence, the baryon and lepton number violat-
ing dimension five operator is adequately suppressed. On the other hand, we have
η′2 ∼ 10
−5. This may be too small to produce sizable effects in forthcoming col-
lider experiments, but may have interesting consequences for neutralino dark matter
searches.
6 Conclusions
The Standard Model gauge and matter spectrum clearly indicates the realization of
grand unification structures in nature. Most appealing in this respect is the struc-
ture of unification in the context of embedding the Standard Model chiral spectrum
into spinorial representations of SO(10). In this case each Standard Model gener-
ation together with the right–handed neutrino fits into a single SO(10) spinorial
representation. While this can be a mirage, it is the strongest hint from the avail-
able experimental data, accumulated over the past century. On the other hand,
grand unified theories, and many other extensions of the renormalizable Standard
Model, predict processes that lead to proton instability and decay. Proton longevity
is therefore another key ingredient in trying to understand the fundamental origin
of the Standard Model matter spectrum and interactions. A model that provides a
robust explanation for these two key observations, while not interfering with other
experimental and theoretical constraints, may indeed stand a good chance to pass
further experimental scrutiny.
String theory provides a viable framework for perturbative quantum gravity, while
at the same time giving rise to the gauge and matter structures that describe the in-
teractions of the Standard Model. In this respect string theory is unique and enables
the development of a phenomenological approach to the unification of the gauge and
gravitational interactions. Heterotic–string theory has the further distinction that by
giving rise to spinorial representations in the massless spectrum it also enables the
embedding of the Standard Model chiral spectrum in SO(10) spinorial representa-
tions. The free fermionic models provide examples of quasi–realistic three generation
heterotic–string models, in which the chiral spectrum arises from SO(10) spinorial
representations. These models therefore admit the SO(10) embedding of the Stan-
dard Model matter states. They satisfy the two pivotal criteria suggested by the
Standard Model data. These models are related to Z2×Z2 orbifolds at special points
in the moduli space. Other classes of quasi–realistic perturbative heterotic–string
models have also been studied on unrelated compactifications and using different
techniques [23].
Perhaps the most appealing explanation for the stability of the proton is the
existence of additional gauge symmetries that forbid the proton decay mediating
operators. However, such gauge symmetries should not interfere, or obstruct, the
other phenomenological requirements that must be imposed on any extension of the
Standard Model. Therefore, they should allow for generation of fermion masses and
suppression of neutrino masses. They should be anomaly free. Gauge symmetries that
may be observed in forthcoming collider experiments should also be family universal.
In this paper we examined the question of such an additional U(1) gauge symme-
try in the free fermionic models. While in most cases the additional gauge symmetries
that arise in the string models do not satisfy the needed requirements, we demon-
strated the existence of a U(1) symmetry in the class of models of ref. [7] that indeed
does pass all the criteria. The existence of this U(1) symmetry at low energies is
therefore motivated by the fact that it protects the proton from decaying, and it
may indeed exist in the range accessible to forthcoming experiments. It is noted
that although we investigated the additional U(1) in the context of the free fermionic
string models, the properties of the U(1) symmetry, and the charges of the Standard
Model state under it, rely solely on the weight charges of the string states under
the rank 16 gauge symmetry of the ten dimensional theory. A U(1) symmetry with
the properties that we extracted here may therefore arise in other classes of string
compactifications. We emphasize that the characteristics of the extra U(1) that we
extracted from a particular class of free fermionic models, do not depend on the
specific string compactification. It ought to be further noted that compactifications
that yielded the U(1) and the peculiar Standard Model charges under it, are not
decedent from the E8 ×E8 heterotic string in 10 dimensions. This is because a U(1)
symmetry which descends from the E8 × E8 (or SO(16) × SO(16)) will necessarily
have an embedding in E6 and as we demonstrated here the Standard Model U(1)
charges derived in this paper do not possess an E6 embedding, and do not descend
from E8. The properties of this U(1) symmetry therefore differ from those that have
been predominantly explored in the literature, which are inspired from compactifi-
cations of the E8 × E8 heterotic string. The investigation of the phenomenological
characteristics of this additional U(1) is therefore of further interest and we shall
return to it in future publications.
7 Acknowledgments
AEF would to thank the Oxford theory department for hospitality during the comple-
tion of this work. CC would like to thank the Liverpool theory division for hospitality.
This work was supported in part by PPARC (PP/D000416/1), by the Royal Society
and by the Marie Curie Training Research Network “UniverseNet” MRTN–CT–2006–
035863.
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