Non-tachyonic semi-realistic non-supersymmetric

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Web End = Eur. Phys. J. C (2016) 76:208DOI 10.1140/epjc/s10052-016-4056-2

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Web End = Non-tachyonic semi-realistic non-supersymmetric heterotic-string vacua

Johar M. Ashfaquea, Panos Athanasopoulosb, Alon E. Faraggic, Hasan Sonmezd Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK

Received: 1 December 2015 / Accepted: 31 March 2016 / Published online: 15 April 2016 The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract The heterotic-string models in the free fermionic formulation gave rise to some of the most realistic-string models to date, which possess N = 1 spacetime supersym

metry. Lack of evidence for supersymmetry at the LHC instigated recent interest in non-supersymmetric heterotic-string vacua. We explore what may be learned in this context from the quasi-realistic free fermionic models. We show that constructions with a low number of families give rise to proliferation of a priori tachyon producing sectors, compared to the non-realistic examples, which typically may contain only one such sector. The reason being that in the realistic cases the internal six dimensional space is fragmented into smaller units. We present one example of a quasi-realistic, nonsupersymmetric, non-tachyonic, heterotic-string vacuum and compare the structure of its massless spectrum to the corresponding supersymmetric vacuum. While in some sectors supersymmetry is broken explicitly, i.e. the bosonic and fermionic sectors produce massless and massive states, other sectors, and in particular those leading to the chiral families, continue to exhibit FermiBose degeneracy. In these sectors the massless spectrum, as compared to the supersymmetric cases, will only differ in some local or global U(1) charges. We discuss the conditions for obtaining nb = n f at the mass

less level in these models. Our example model contains an anomalous U(1) symmetry, which generates a tadpole diagram at one-loop order in string perturbation theory. We speculate that this tadpole diagram may cancel the corresponding diagram generated by the one-loop non-vanishing vacuum energy and that in this respect the supersymmetric and non-supersymmetric vacua should be regarded on an equal footing. Finally we discuss vacua that contain two supersymmetry generating sectors.

a e-mail: mailto:[email protected]

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b e-mail: mailto:[email protected]

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c e-mail: mailto:[email protected]

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d e-mail: mailto:[email protected]

Web End [email protected]

1 Introduction

The discovery of the agent of electroweak symmetry breaking at the LHC [1,2] is a pivotal moment in particle physics. While conrmation of this agent as the Standard Model electroweak doublet representation will require experimental scrutiny in the decades to come, the data to date seems to vindicate this possibility. Substantiation of this interpretation of the data will reinforce the view that the electroweak symmetry breaking mechanism is intrinsically perturbative, and that the SM provides a viable perturbative parameterisation up to the Planck scale. Moreover, the large scale unication scenario is further motivated by the embedding of the SM matter states in the chiral SO(10) representation; by the logarithmic evolution of the SM parameters; by proton longevity; and by the suppression of left-handed neutrino masses. Gaining further insight into the fundamental origins of the SM parameters can then only be obtained by incorporating gravity into the picture.

String theory provides the most developed contemporary approach to study how the Standard Model parameters may arise from a unied theory of the gauge and gravitational interactions. For this purpose several models that reproduce the spectrum of the Minimal Supersymmetric Standard Model have been produced [38]. Amongst them the free fermionic models [35,923] are the most studied examples. The heterotic string in particular provides a compelling framework to study the gaugegravity synthesis in the large scale unication scenario, as it reproduces the embedding of the SM chiral spectrum in spinorial SO(10) representations.

The majority of semi-realistic heterotic-string models constructed to date possess N = 1 spacetime supersym

metry, while non-supersymmetric vacua were investigated sporadically [2431]. In the absence of evidence of super-symmetry at the LHC recent interest in non-supersymmetric heterotic-string vacua has emerged [3238]. It is therefore prudent to examine what may be learned in that context from the quasi-realistic free fermionic models. In this paper this

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question is considered. We discuss the different avenues that may be used to break supersymmetry directly at the string scale and how they compare with the recent analysis [34].

Our paper is organised as follows: in Sect. 2, we review the structure of the phenomenological free fermionic heterotic-string models. In Sect. 3, we discuss the phases that break supersymmetry in the string models and the different patterns that they induce. Further discussion of the existence of sectors producing tachyons in these models and the relation of the abundance of these sectors with the number of families is given. Moreover, in Sect. 4 a non-supersymmetric tachyon-free model is presented and its relation to the supersymmetric counterpart is inferred. In Sect.5 we discuss the construction of string vacua with split supersymmetry, in which super-symmetry is produced by two sectors. Section 6 contains our conclusions.

2 Phenomenological free fermionic models

In this section, we review the structure of the phenomenological free fermionic models. It should be stressed that these free fermionic models correspond to Z2 Z2 toroidal orbifolds

and their phenomenological characteristics are deeply rooted in the structure of the Z2 Z2 orbifolds. In this respect, the

free fermionic formalism merely provides an accessible set of tools to extract the spectra of the string vacua and their properties. Furthermore, the free fermionic machinery extends to the massive string spectrum via the analysis of the relevant partition function. This provides important insight into the symmetries that underly the string landscape and eventually may prove instrumental in understanding how the string vacuum is selected. However, one should not tie the cart before the horse. The fermionic and bosonic representations only provide complementary tools that are formally identical in two dimensions. The physically relevant properties of these free fermionic models are due to their underlying Z2 Z2

orbifold structure.

In the free fermionic formulation of the heterotic string in four dimensions, all the extra degrees of freedom needed to cancel the conformal anomaly are represented as free fermions propagating on the string worldsheet. It is important to note that the two dimensional fermions are free only at a special point in the moduli space [39]. However, the models can be deformed away from that point by incorporating the moduli as worldsheet Thirring interactions [4042]. Since the twisted matter spectrum of the Z2 Z2 orbifolds, which

gives rise to the Standard Model matter states, is independent of the moduli, working at the free fermionic point is just a convenient choice. In the light-cone gauge the supersymmetric left-moving sector includes the two transverse spacetime fermionic coordinates and 18 internal worldsheet real fermions I , whereas the right-moving bosonic sector con-

tains 44 real worldsheet fermions a. The worldsheet super-symmetry is realised non-linearly in the left-moving sector and the worldsheet supercurrent is given by

TF = X + fI J K I J K , (2.1)

where fI J K are the structure constants of the 18 dimensional semi-simple Lie group. The 18 left-moving world-sheet fermions I transform in the adjoint representation of the Lie group, which in the case of the fermionic Z2 Z2

orbifolds with N = 1 SUSY is SU(2)6. Such models pro

vide our starting point and we will discuss in later sections how supersymmetry is broken. The I therefore transform in the adjoint representation of SU(2)6, and they are denoted by I , yI , I with I = 1, . . . , 6. Under parallel transport

around a non-contractible loop of the one-loop vacuum to vacuum amplitude the worldsheet fermions pick up a phase

f ei( f ) f , (2.2)

with ( f ) (1, +1]. The phases for all worldsheet ferm

ions constitute the spin structure of the models and are given in the form of 64 dimensional boundary condition basis vectors. The partition function,

Z() =

, (2.3)

is a sum over all spin structures, where c are Generalised GSO (GGSO) projection coefcients and Tr

Tr(eiFeiH) with H being the hamiltonian, is the trace over the mode excitations of the worldsheet elds in the sector , subject to the GSO projections induced by the sector . Requiring invariance under modular transformations results in a set of constraints on the allowed spin structures and the GGSO projection coefcients. The Hilbert space of a given sector in the nite abelian additive group = knibi, where ni = 0, . . . , gzi 1, is obtained

by acting on the vacuum of the sector with bosonic, as well as fermionic oscillators with frequencies f , f , and subsequently imposing the GGSO projections

ei(bi F) c

|s = 0 (2.4)

with

(bi F)

real+complex

right

(bi( f )F( f )), (2.5)

where is the spacetime spin statistics index and F( f ) is a fermion number operator, counting each mode of f once (and if f is complex, f minus once). For Ramond fermions with ( f ) = 1 the vacuum is a doubly degenerate spinor | ,

annihilated by the zero modes f0 and f 0, and with fermion

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numbers F( f ) = 0, 1. The physical states in the string

spectrum satisfy the level matching condition

M2L =

2 +

2.1.1 The NAHE set

The NAHE set [51] is a set of ve boundary condition basis vectors {1, S, b1, b2, b3}. With 1 indicating Ramond

boundary conditions and 0 indicating NeveuSchwarz boundary conditions. The NAHE-set basis vectors are given by

12 34 56

1,...,5

1,...,8

L L

8 + NL = 1 +

R R

8 + NR = M2R

(2.6)

where = (L; R) is a sector in the additive group,

and

NL =

(L); NR =

(R); (2.7)

f =

1 ( f )

2 . (2.8)

The U(1) charges with respect to the Cartan generators of the gauge group in four dimensions are given by

Q( f ) =

1 1 1 1 1 1, . . . , 1 1 1 1 1, . . . , 1 S 1 1 1 1 0, . . . , 0 0 0 0 0, . . . , 0 b1 1 1 0 0 1, . . . , 1 1 0 0 0, . . . , 0 b2 1 0 1 0 1, . . . , 1 0 1 0 0, . . . , 0 b3 1 0 0 1 1, . . . , 1 0 0 1 0, . . . , 0

y3,...,6 y3,...,6 y1,2, 5,6 y1,2,

5,6 1,...,4

1,...,4

1 + ( f )

2 ; f =

1 1, . . . , 1 1, . . . , 1 1, . . . , 1 1, . . . , 1 1, . . . , 1 1, . . . , 1 S 0, . . . , 0 0, . . . , 0 0, . . . , 0 0, . . . , 0 0, . . . , 0 0, . . . , 0 b1 1, . . . , 1 1, . . . , 1 0, . . . , 0 0, . . . , 0 0, . . . , 0 0, . . . , 0 b2 0, . . . , 0 0, . . . , 0 1, . . . , 1 1, . . . , 1 0, . . . , 0 0, . . . , 0 b3 0, . . . , 0 0, . . . , 0 0, . . . , 0 0, . . . , 0 1, . . . , 1 1, . . . , 1(2.10)

A suitable choice of GGSO phases that preserves spacetime supersymmetry is given by

c bi

12( f ) + F( f ), (2.9) for each complex right-moving fermion f . In the usual notation the 64 worldsheet fermions in the light-cone gauge are denoted as , 1,...,6, y1,...,6, 1,...,6 and y1,...,6, 1,...,6,

1,...,5, 1,2,3, 1,...,8. Further details on the formalism and notation used in the free fermionic construction can be found in the literature [919,4350].

2.1 Construction of phenomenological models

Phenomenological free fermionic heterotic-string models were constructed following two main routes. The rst are the so-called NAHE-based models. This set of models utilise a set of eight or nine boundary condition basis vectors. The rst ve consist of the so-called NAHE set [51] and are common in all these models. The basis vectors underlying the NAHE-based models therefore differ by the additional three or four basis vectors that extend the NAHE set.

The second route follows from the classication methodology that was developed in [52] for the classication of type II free fermionic superstrings and adopted in [1821,23,46 48] for the classication of free fermionic heterotic-string vacua with SO(10) GUT symmetry and its PatiSalam [20,21] and ipped SU(5) [23] subgroups. The main difference between the two classes of models is that while the NAHE-based models allow for asymmetric boundary conditions with respect to the set of internal fermions {y, | y,

the classication method only utilises symmetric boundary conditions. This distinction affects the moduli spaces of the models [53], which can be entirely xed in the former case [54] but not in the later. On the other hand the classication method enables the systematic scan of spaces of the order of 1012 vacua, and led to the discovery of spinorvector duality [4648,5559] and exophobic heterotic-string vacua [20,21]. In this paper, for reasons that will be claried below, our discussion is focussed on the NAHE-based models.

S = c

1 1 = 1, (2.11)

where all other GGSO projection coefcients are determined by modular invariance. The basis vector S is the generator of spacetime supersymmetry. It merely corresponds to the Ramond vacuum of the worldsheet fermionic superpartners of the ten dimensional heterotic-string, and acts as a spectral ow operator that mixes between the spacetime fermionic and bosonic states.

The subset of basis vectors {1, S} produces a string vac

uum with N = 4 spacetime supersymmetry and SO(44)

gauge group. Adding the basis vectors b1 and b2 reduces the N = 4 spacetime supersymmetry to N = 1, where each of

these vectors on its own reduces N = 4 to N = 2, and their

combined action reduces N = 4 to N = 1. The characteristic

of these two vectors is that their overlap with the basis vector S yields S b1,2 = 2. Thus, any additional vector that satis

es this overlap with the basis vector S reduces the number of supersymmetries from N = 4 to N = 2. An example at

hand is the basis vector b3 of the NAHE set. However, the additional breaking induced by any additional basis vector with this property either preserves N = 1 supersymmetry or

reduces it further to N = 0.

This type of breaking therefore produces a type of non supersymmetric string vacua that follow the chain N = 4

N = 2 N = 1 N = 0. One characteristic of this type

of spacetime supersymmetry breaking is that the breaking will a priori not be family universal. The reason is that the

= c

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chiral families arise in the free fermionic models from the three sectors b1, b2 and b3, and this type of breaking necessarily violates the cyclic permutation symmetry among the three sectors b1, b2 and b3.

An example, where this cyclic permutation symmetry is instrumental in producing a family universal structure, is the supersymmetry breaking with a family universal anomalous U(1) of [60,61]. The basis vectors b1, b2 and b3 reduce the

SO(44) gauge symmetry to SO(10) SO(6)3 E8. The

gauge bosons that produce this gauge symmetry are obtained from the NeveuSchwarz (NS) sector and the sector Z =

1+b1 +b2 +b3, where the NS produces the vector bosons of

SO(10) SO(6)3 SO(16) and the sector Z complements

the SO(16) group factor to E8.

The NAHE set basis vectors b1, b2 and b3 correspond to the three twisted sectors of the Z2Z2 toroidal orbifold. Each

of these twisted sectors produces 16 multiplets in the spinorial 16 representation of SO(10) to give a total of 48 chiral generations. The correspondence of the quasi-realistic free fermionic models with Z2 Z2 orbifold has been amply dis

cussed in the literature [6269]. While the dictionary between specic models in the two approaches may be elusive, it is anticipated that for every model in one formalism there exists a representation in the alternative formalism and this should hold, at least for the Z2 Z2 orbifolds, and higher order

orbifolds may have a fermionic representation as well [70].

2.1.2 Beyond the NAHE set

The construction of the semi-realistic free fermionic models proceeds by adding three or four additional basis vectors to the NAHE set. The function of the additional basis vectors is to reduce the 48 spinorial 16 multiplets to three chiral generations, and at the same time to reduce the SO(10) GUT symmetry to one of its subgroups:

(i) SU(5) U(1) (FSU5) [9];

(ii) SU(3) SU(2) U(1)2 (SLM) [10,11];

(iii) SO(6) SO(4) (PS) [1214];

(iv) SU(3) U(1) SU(2)2 (LRS) [15,16];

(v) SU(4) SU(2) U(1) (SU421) [17].

The rst four cases produced viable three generation models, whereas in the last case it was shown that phenomenologically viable models cannot be constructed [17,72,73].The additional basis vectors may each preserve or break the SO(10) symmetry. Basis vectors that preserve the SO(10) symmetry are typically denoted by bi with (i = 4, 5, . . . ),

whereas those that break the SO(10) symmetry are denoted by {, , }. The overlap of the additional basis vectors with

the supersymmetry generator basis vector S determine the type of possible supersymmetry breaking. Thus, in the cases with S bi = 2 the pattern of supersymmetry breaking is

similar to the spacetime supersymmetry breaking discussed above, namely it follows the chain N = 4 N = 2

N = 1 N = 0.

An alternative is to use a basis vector with S ai = 0,

where ai may, or may not, break the SO(10) GUT symmetry. This type of supersymmetry breaking differs, however, from the one discussed above in that it induces the breaking pattern N = 4 N = 0. A general rule to construct

vacua that preserve N = 1 spacetime supersymmetry is to

impose c

S vi

= vi , where vi is any basis vector [71]. Thus,

relaxing this constraint would generically result in broken spacetime supersymmetry. Breaking spacetime supersymmetry with the additional basis vectors ai would generically also not affect the spectrum arising from the sectors bi that produce the chiral generations, but it may affect their super-partners as those are obtained from the sectors S + bi. As we

will show with an explicit example model in Sect. 4, in this case the effect of the projection is to select different components of the underlying N = 4 multiplets.

2.1.3 The classication set

In this paper our focus will be on non-supersymmetric NAHE-based free fermionic models. For completeness we discuss the construction of such models by using the classication methods of [1821,23,4648]. In this approach the set of basis vectors is xed and a large number of string models, of the order of 1012 vacua, is spanned by enumerating the independent GGSO projection coefcients. In this manner large spaces of string models with SO(10) [18,19], SO(6) SO(4) [20,21], SU(5) U(1) [23], and

SU(4)SU(2)U(1) [72,73], have been explored. A subset

of basis vectors that respect the SO(10) symmetry is given by the set of 12 basis vectors V = {v1, v2, . . . , v12}, where v1 = 1 = {, 1,...,6, y1,...,6, 1,...,6|

y1,...,6,

1,...,6,

1,2,3,

1,...,5,

1,...,8},

v2 = S = {, 1,...,6},

v2+i = ei = {yi, i| yi,

i}, i = 1, . . . , 6,

v9 = b1 = {34, 56, y34, y56| y34, y56,

1,...,5},

v10 = b2 = {12, 56, y12, y56| y12, y56,

1,...,5},

v11 = z1 = {

1,...,4}, v12 = z2 = {

5,...,8}. (2.12)

Additional vectors are added to the set given in (2.12) to construct vacua with SO(10) subgroups [20,21,23]. In the notation of Eq. (2.12) the worldsheet fermions appearing in the curly brackets have periodic boundary conditions, whereas all other worldsheet fermions are anti-periodic. The entries in the matrix of GGSO phases c vivj with i > j then span the space of string vacua. Additional constraints that are imposed on the string vacua, like the existence of spacetime

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supersymmetry leave 40 independent phases of the original66. One can then resort to a complete [4648] or statistical sampling1 of the total space [18,19], and classify the models by their twisted matter spectrum. The classication is facilitated by expressing the GGSO projections in algebraic form [18,19,4648]. The analysis of the entire spectrum of the string models is computerised and vacua with specic phenomenological characteristics can be shed our from the larger space of models.

In terms of spacetime supersymmetry breaking, as with the NAHE-set-based models the spacetime supersymmetry generator is the basis vector S. The subset {1, S} gives rise to

N = 4 spacetime supersymmetry, which is broken by b1 and

b2 to N = 2 spacetime supersymmetry and their combined

action breaks N = 4 N = 1. As with the NAHE-based

models imposing c

S vi

= vi ensures the preservation of

N = 1 supersymmetry. Projecting the remaining supersym

metry in this model is obtained by relaxing this condition. Furthermore, the basis vectors {ei, z1, z2} satisfy S ei = 0

and S zi = 0. These basis vectors therefore act as projectors

on the S sector. These basis vectors can be used to project all the states from the S sector and hence induce the breaking N = 4 N = 0 spacetime supersymmetry.

3 Tachyons in the free fermionic semi-realistic models

String models, in general, and heterotic-string models in particular, generically give rise to tachyonic states in their spectra. This can be seen from Eq. (2.6). Any sector that satises

M2L <

2 and M2R < 1 (3.1)

may produce tachyonic physical states. Tachyonic states can be obtained by acting on the vacuum with fermionic oscillators. They satisfy the level matching condition and survive all the GGSO projections. Their presence in the physical spectrum indicates the instability of the string vacuum. The existence of spacetime supersymmetry guarantees that all tachyonic states are projected out. The situation is altered if supersymmetry is broken to N = 0 spacetime supersymme

try by projecting all the states from the S sector. One then has to check in each model whether tachyonic states exist.

The existence of non-supersymmetric non-tachyonic string vacua has been known since the mid-1980s [24]. The gauge symmetry of this model is SO(16) SO(16), and its

non-perturbative extension was considered in [31]. In the free fermionic formalism the model is constructed by the set of boundary condition basis vectors {1, S, X, I} where

X = {

1,...,5,

1,...,8}. In ten dimensions

1 We note that analysis of large sets of string vacua has also been carried out by other groups See e.g.: [27,7485].

the choice of the GGSO phase c XI = 1 yields either

the supersymmetric E8 E8, or the non-supersymmetric

SO(16) SO(16), heterotic-string. This is necessarily the

case in ten dimensions because the supersymmetry generator is given by S = 1 + X + I and therefore the projec

tions on the three sectors are correlated. In the four dimensional models the same phase can be used to reduce the gauge symmetry from E8 E8 to SO(16) SO(16) with

out breaking supersymmetry. The same vacua can be constructed in the orbifold representation and can be connected by interpolations [25,26]. Hence, the supersymmetric and non-supersymmetric vacua exist on the boundary of the same moduli space.

Our interest in this paper is in the tachyonic states arising in the semi-realistic models. It is instructive to examine the case of the non-supersymmetric SO(16) SO(16) model

rst. In the four dimensional model supersymmetry may be broken from N = 4 N = 0 by the I or X projections.

The only sector that may produce tachyons in this model is the NS sector. The tachyonic states arising in this model are obtained by acting on the non-degenerate vacuum with a right-moving oscillator, and satisfy the level matching condition with M2L = M2R = 1/2. These tachyonic states are,

however, projected out by the S projection, which is given by

eiSFN S|t N S = S|t N S. (3.2)

As there are no oscillators acting on the left-moving vacuum in the tachyonic untwisted state, and the basis vector S is blind to the right-moving oscillators, the left-hand side of Eq. (3.2) is positive. On the other hand S = 1 because

the spacetime fermions are periodic in S. The mismatch between the two sides of Eq. (3.2) entails that the untwisted NS tachyons are projected out. This argument extends to any free fermionic model that contains the basis vector S. We conclude that in any non-supersymmetric free fermionic that includes the S sector the untwisted tachyons are always projected out, irrespective of the choice of the SUSY projecting phases.

The four dimensional SO(16) SO(16) non-supersym-

metric heterotic-string is therefore tachyon free. However, in this string vacuum the only sector that may give rise to tachyonic states is the untwisted sector. This is not the case in the semi-realistic free fermionic models. The three generation free fermionic models generically give rise to an abundance of sectors that may a priori give rise to tachyonic states. The reason is that the additional basis vectors that are required to reduce the number of families, break down the internal degrees of freedom into small units. This is exemplied by the set of vectors in Eq. (2.12), which is used as the basis for the classication of symmetric fermionic Z2 Z2 orbifolds. The basis vectors ei as well as their

ei + ej i = j and ei + ej + ek i = j = k combina

tions may produce physical tachyonic states by acting on the

1,2,3} and I = {

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non-degenerate vacuum with a right-moving NS fermionic

oscillator; similarly the sectors z , z +ei, z +ei +ej , i = j

z + ei + ej + ek, i = j = k may produce tachyonic states.

In total there are therefore 123 sectors, in addition to the NS sector, that may produce tachyons in these models.

This renders futile a systematic classication of nonsupersymmetric non-tachyonic semi-realistic vacua along the lines of [20,21,23]. The situation in NAHE-based models is similar. Typical models contain an abundance of sectors that may a priori produce tachyons. Furthermore, models that utilise fractional boundary conditions may contain additional tachyon producing sectors in which a fermionic oscillator with rational boundary conditions may act on the non-degenerate vacuum. A detailed sector by sector analysis is therefore required. A systematic procedure to extract the tachyon-free vacua is provided by performing a q-expansion of the partition function [35]. However, this method loses the detailed information on the structure of the string spectrum. The construction of non-supersymmetric vacua with quasi-realistic features therefore requires a detailed model by model analysis. One may then envision the existence of models in which the number of tachyonic producing sectors is restricted.

The best case scenario would be a model in which the only tachyon producing sector is the NS sector. In this case we are guaranteed that tachyons do not exist in the physical spectrum. However, a model with this property has not been found to date. The next best case scenario is a model that gives rise only to one type of tachyon producing sectors.Existence of a model with this characteristic may depend on further detailed phenomenological properties of the string vacua. For example, we were not able to nd such a model in the class of NAHE-based free fermionic models with reduced Higgs spectrum [54], whereas the class of leftright symmetric models [15,16] did produce a model with the desired property. The set of boundary condition basis vectors, beyond the NAHE set, generating the string vacuum is given by

12 34 56

1,...,5

The supersymmetric version of this model was presented in [15,16] with the set of GGSO phases given by

1 S b1 b2 b3

1 1 1 1 1 1 1 1 i

S 1 1 1 1 1 1 1 1

b1 1 1 1 1 1 1 1 i b2 1 1 1 1 1 1 1 i b3 1 1 1 1 1 1 1 i

1 1 1 1 1 1 1 1

(3.5)

The full mass spectrum of this model together with the cubic level superpotential was presented in [15,16]. The modication

c S

= 1 +1 and c

= 1 +1 (3.6)

projects the remaining gravitino and induces N = 1 N =

0. It can be checked that all the tachyonic states are projected out in this model. Furthermore, it can be veried that making the modication

c S

= 1 +1 and c

1,...,8

= 1 1 (3.7)

i.e. modifying only c S but not c S results in a model that

contains tachyonic states. The reason is that in this model all the sectors that may produce tachyons appear with the combination m( + ), where m = 0, 1. Hence, with the modi

cation given by Eq. (3.6) the S-projection on the tachyonic sectors is the same as in the corresponding supersymmetry preserving choice given in Eq. (3.5), whereas with the modication given by (3.7) the S-projection in some sectors is modied in comparison to the supersymmetric model and some tachyonic states are not projected out. We note that the construction of tachyonic-free semi-realistic vacua is highly nontrivial. In the next section we discuss the tachyon-free model in some detail.

4 An explicit tachyon-free model

We consider the model dened by the set of basis vectors

1 = {, 1,...,6, y1,...,6, 1,...,6| y1,...,6,

1,...,6,

1,2,3,

0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0

0 0 0 0 1

2 0 0

2 0 12 12 12 12 12 12 0

y3y6y4 y4y5 y5 y3 y6 y15y2 y26

6 y1

5 241

1 1 1 0 1 1 1 0 1 1 1 0

0 1 0 1 0 1 0 1 1 0 0 0

0 0 1 1 1 0 0 0 0 1 0 1

(3.3)

This model gives rise only to one type of tachyon producing sectors with

2L = 2 & 2R = 6 NR = 0. (3.4)

1,...,5, 1,...,8},

S = {, 1,...,6},b1 = {, 1,2, y3,...,6|y3,...,6, 1,...,5, 1},

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b2 = {, 3,4, y1,2, 5,6|y1,2, 5,6, 1,...,5, 2}, b3 = {, 5,6, 1,...,4|1,...,4, 1,...,5, 3},

b4 =

= {y1,...,6, 1,...,6|1, y2, 3, y4,5, 6, 1,2,3, 1,...,4}, b5 =

= {y2, 2, y4, 4|y1,...,4, 5, y6, 1,2,3, 1,...,4}, b6 =

= {y1, 1, y5, 5|1,2, y3, 4, y5,6, 1,2,3 =

2, 1,2,3

QC = Q

1 + Q

2 + Q

3 ,

Q8 = Q

2 + Q

3 + Q

4 ,

7 . (4.1)

To avoid writing fractional numbers all the charges in the table have been multiplied by 4. Finally, for every state the CPT conjugate is also understood to be in the spectrum and has not been written explicitly. Lastly, we comment that the states contain discrete charges corresponding to the action of real fermions that are not shown in the table. For example, the rst three states from the NeveuSchwarz sector are obtained by acting on the vacuum with two right-moving real fermions and are neutral under the gauge symmetry of the model. The weak hypercharge in the model is given by

U(1)Y =

13U(1)C + T3R,

where T3R is the diagonal generator of SU(2)R. The symmetry breaking to the Standard Model gauge group may be induced by a VEV for one of the Standard Model singlet scalar elds in the (1, 1, 2) representation from the sectors S+bj . This model exhibits many interesting features regard

ing supersymmetry. First of all, we observe that the model is manifestly non-supersymmetric. The gravitino and the gaugini are projected out and there is a clear mismatch between the number of states in the 0 and S sectors. Furthermore, there are eight sectors with only scalars and the sectors that contain the would-be superpartners are massive. These are

+ , + 3,

1 + b1 + b2 + b3 + + , 1 + b1 + b2 + b3 + + 3,

1 + b1 + b2 + b3 + + , 1 + b1 + b2 + b3 + + 3.

(4.2)

Such sectors would not remain in the spectrum in the supersymmetric choice of phases. The reason is that the spacetime supersymmetry generator in the supersymmetric model is the basis vector S, i.e. for a given sector , the

supersymmetric superpartners are obtained from the sector S + . All the sectors in Eq. (4.2) have ()2L = 4, whereas

(S + )2L = 8, i.e. in these sectors the would-be superpart

ners are massive. In the supersymmetric vacua the states from the sectors in Eq. (4.2) are necessarily projected out, as they break supersymmetry explicitly. However, once supersymmetry is broken they may appear in the spectrum, as is seen in our model. It is a highly non-trivial task to nd a model with three generations in which sectors of these type, which only appear when supersymmetry is broken, contain no tachyons, but this model provides exactly such an example. We collectively refer to all the sectors mentioned in this paragraph as sectors in which supersymmetry is badly broken.

On the other hand, there are (pairs of) sectors that are completely supersymmetric. This is due to the modication

Q9 = Q

5 + Q

6 + Q

2},

with the set of GGSO phases given by

1 S b1 b2 b3

1 1 1 1 1 1 1 1 i

S 1 1 1 1 1 1 1 1

b1 1 1 1 1 1 1 1 i b2 1 1 1 1 1 1 1 i b3 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

2, 2,...,7 =

This is a three generation model, with one generation appearing in each of the twisted sectors b1, b2 and b3. The full spectrum can be found in the tables of the appendix, with the exception of the gauge bosons which have been omitted in the interest of space. It is sufcient to state that the gauge group is

SU(3)C U(1)C SU(2)L SU(2)R

i=1

observable sector

SU(3)H1 SU(3)H2

j=7

hidden sector

The notation for the table is the following: The rst column describes if the states correspond to spacetime bosons or spacetime fermions and specically for bi the type of particle. The second column is the name of the sector. The third column gives the dimensionality of the states under SU(3)C SU(2)L SU(2)R and the fourth the charges of

the observable U(1)s. Columns 5 and 6 describe the hidden sector. The only charges appearing in the table that do not have a self-evident name are:

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208 Page 8 of 17 Eur. Phys. J. C (2016) 76 :208

(3.6) not affecting the GGSO projections in any sectors where none of the vectors S, or appear. Therefore such sectors will be identical to the corresponding sectors of the super-symmetric model. Nonetheless, for some of these sectors to remain supersymmetric as claimed above, the superpartners should be unchanged as well, or at least the effect must be (at most) a change in the R-charges of the superpartners. Sectors bi and 1 + bi + bj + 2 are of this type.

Finally, there are sectors that do not t any of the above categories. In these sectors the number of bosons and fermions is the same, but on the other hand some of the gauge charges of these states are different which in principle prevents us from grouping them together into supermultiplets. Most of the sectors are of this type. We use the term sectors in which super-symmetry is nicely broken when referring to this case.

Thus, while supersymmetry is broken, some segments of the string vacuum still respect the underlying supersymmetric degeneracy. This is in accordance with the ndings in [34], which showed that the partition function of string vacua with spontaneously broken supersymmetry can be divided into several orbits, some of which preserve the original super-symmetry.

Furthermore, we would like to comment in our model the fermionic states from the sectors b1, b2 and b3, as well as the bosonic states from the NS sector, are not affected by the GGSO phases that project the gravitino and gaugini from the S sector, and therefore break space-time supersymmetry. Therefore, the untwisted scalar states of our non-supersymmetric model as well as the fermionic states from the sectors b1, b2 and b3 are identical to those in the corresponding supersymmetric model. Consequently, the leading twistedtwisteduntwisted couplings in the nonsupersymmetric model, which are obtained by using the methods developed in [8688], are identical to those of the supersymmetric model. The model generated by Eqs. (3.3,3.5) contains electroweak doublet scalar representations from the twisted sectors that may be used as Higgs doublets.However, in this model the untwisted Higgs bi-doublets, which couple at leading order to the twisted sector states, are projected out and consequently the leading mass term which is identied with the top mass is absent. Other LRS [15,16] models, as well as the FSU5 [9], PS [1214] and SLM [35,10,11] models, do contain the untwisted Higgs doublets and in those cases a leading top mass term is obtained.

It is also worth noting that even for non-supersymmetric models the cosmological constant can be exponentially suppressed. As discussed in [35], this can be achieved if the massless spectrum has an equal number of bosons and fermions (irrespectively of their charges). Even though our model is not of this type and will therefore have an unsuppressed vacuum energy, our construction hints at how one might go about achieving such a goal. It is clear, for example, that we do not

have to worry about sectors that either respect supersymmetry or in which supersymmetry is nicely broken.

On the other hand, sectors that badly break supersymmetry will have to be carefully engineered. There are a few ways one might go about such a task. For example, one might entertain the idea that the addition of further basis vectors could project such sectors out of the spectrum. The biggest problem with this approach is that the removal of the gaugini from the S sector, even if some fermions transforming in a different from the adjoint representation are preserved, will create a mismatch of states in the S and NS sectors turning them into sectors that break supersymmetry badly; and it is impossible to project out the NS sector no matter what basis vectors are added. It is a priori possible that further basis vectors will remove exactly the correct number of bosons from the NS sector to match the remaining fermionic states in the S sector, but this method seems unnecessarily restricting.

An approach providing more freedom is to aim for an equality in the number of bosons and fermions not in each sector, but among different sectors. To cancel the surplus of bosons from the NS sector this would imply the existence of surviving fermionic states in different sectors, the bosonic counterpart of which has been projected out. The model presented in this section has an overall mismatch of bosonic and fermionic degrees of freedom and therefore does not satisfy this condition. Finding a semi-realistic model of suppressed cosmological constant appears to be very challenging, but it is of great interest as well and we hope to report on such constructions in a future publication.

5 Split SUSY models

In this section we briey discuss string models with a split supersymmetry structure. The basic idea is to use two basis vectors to generate spacetime supersymmetry. We recall from Sect. 2 that in the semi-realistic free fermionic models the supersymmetry generators arise from the basis vector S. The aim in split supersymmetry string models is to construct two basis vectors that produce supersymmetry generators. A particular aim is then to construct models in which gaugini are obtained from one generator, whereas those of the second generator are projected out, as well as the scalar superpartners of the twisted matter fermionic states. Our construction proceeds by keeping our previous basis vector S = {1,2, 1,...,6} S1. A second supersymmetry genera

tor is given by

S2 = {1,2, 1,2, 3,4, 5,6}. (5.1) The basis vectors b1 and b2 of the NAHE-set equation (2.10) are added as well as the basis vectors 1 and X, which is used to project the supersymmetric generators from S1, as discussed in Sect. 3. Shift basis vectors similar to the ei basis

123

Eur. Phys. J. C (2016) 76 :208 Page 9 of 17 208

vectors of Eq. 2.12 can be added, and variations that include the basis vector I of Sect. 2. We consider the set of six basis vectors given by

v1 = 1 = {, 1,...,6, y1,...,6, 1,...,6|y1,...,6, 1,...,6, 1,2,3, 1,...,5, 1,...,8},

v2 = S1 = {, 1,...,6},

v3 = S2 = {, 1,2, 3,...,6}, (5.2)

v4 = b1 = {34, 56, y34, y56| y34, y56,

1,...,5},

v5 = X = { 1,2,3,

1,...,5}, v6 = I = {

1,...,8}.with the set of GGSO phases given by

1 S1 S2 b1 X I

1 1 1 1 1 1 1

S1 1 1 1 1 1 1 S2 1 1 1 1 1 1 b1 1 1 1 1 1 1

X 1 1 1 1 1 1

I 1 1 1 1 1 1

. (5.3)

In this model the NS sector is the only sector that produces spacetime vector bosons. Hence the gauge symmetry in four dimensions is SO(8)SO(4)SO(4)SO(12)SO(16).

The sector b1 gives rise to spacetime fermions in the spinor and anti-spinor representations of SO(12). The supersymmetry generators of S1 are projected out, whereas the gaugini from S2 are retained. The model retains the scalar superpartners from the sector S2 + b1, and projects those from the

sector S1 + b1. Our general aim in the construction of mod

els with split supersymmetry is to construct models that retain the gaugini and spacetime fermions from S2 and b1, while projecting the gaugini (and hence the gravitini) from S1, as well as the superpartners from the sectors S1+b1 and S2+b1.

However, variations of the model in Eqs. (5.2, 5.3), including adding the ei projectors of Eq. (2.12) did not yield the desired result. The models in which supersymmetry is entirely broken, i.e. those in which the supersymmetry generators from S1 as well as S2 are projected out, typically contain tachyons.

We then face similar situation to the one discussed in Sect.2.1.3.

6 Conclusions

The observation of a scalar resonance compatible with the electroweak Higgs doublet reinforces the hypothesis that the Standard Model provides a viable parameterisation of all sub-atomic data up to the Planck scale. Synthesis of gravity with the sub-atomic interactions necessitates a departure from the local point particle idealisation of quantum

eld theories, which underly the Standard Model. The most developed framework to explore the gaugegravity augmentation is offered by string theory. Detailed phenomenological models that incorporate the salient feature of the Standard Model have been constructed. These detailed phenomenological constructions contain a new symmetry, i.e. N = 1

spacetime supersymmetry, which has not been observed to date in experiments.

A vital question is therefore to explore the consequences of breaking spacetime supersymmetry directly at the string scale. A generic feature of non-supersymmetric string vacua is the existence of tachyonic states in the physical spectrum. Non-supersymmetric string vacua, such as the SO(16)

SO(16) heterotic-string in ten dimensions, do not contain tachyonic states, but are typically connected in the moduli space to supersymmetric vacua, and tend to have large moduli states and group factors. More realistic constructions on the other hand, typically have reduced moduli spaces and contain more sectors that may a priori give rise to tachyons.

It is therefore important to examine the structure of nonsupersymmetric string vacua in more realistic setting. In this paper we undertook this task. We have shown that while generically the quasi-realistic non-supersymmetric vacua do contain tachyons, there also exist examples in which all the tachyonic states are projected out by the GGSO projections. Furthermore, given that the moduli spaces of the quasi-realistic constructions may be much reduced [53,54], one may entertain the possibility that the tachyon-free nonsupersymmetric quasi-realistic vacua may not be connected to supersymmetric solutions.

We have shown with a concrete example that the nonsupersymmetric quasi-realistic vacuum may retain some of the structure of the corresponding supersymmetric solution. This demonstrates that even though supersymmetry may be broken directly at the string level, the effective spectrum of the string vacuum, as well as its low energy effective eld theory, may still exhibit properties that are similar to those of the corresponding supersymmetric solutions, e.g. the existence of scalar replications of the chiral generations.

Another interesting point to note is the existence of an anomalous U(1) symmetry in the model. The anomalous U(1) is cancelled by the GreenSchwarzDineSeiberg Witten mechanism [89,90], but gives rise to a tadpole diagram at one-loop order in string perturbation theory [91], which reects the instability of the string vacuum. The mismatch between the fermionic and bosonic states at different mass levels gives rise to a non-vanishing vacuum energy, which similarly gives rise to a tadpole diagram, indicating the instability of the string vacuum. We may contemplate the possibility of employing one against the other so that they conspire to cancel. The anomalous U(1) contribution is proportional to the trace over the massless fermionic states and the sign can be altered by the GGSO projections [91,92]. It

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208 Page 10 of 17 Eur. Phys. J. C (2016) 76 :208

is proportional to the gauge coupling and consequently only depends on the dilaton moduli. On the other hand, the vacuum amplitude contribution depends on other moduli [35], and it may be tuned to obtain cancellation of the two contributions. In general, other background elds will be affected by the shift of the vacuum, and to demonstrate the existence of a stable vacuum one would need to solve the set of equations affecting those elds in the shifted vacuum. However, in this regard the same constraints would apply in the case of the supersymmetric vacua, where the FayetIliopoulos term [90,93], which is generated from the anomalousU(1) tadpole diagram [90,91], is cancelled by assigning VEVs to some massless scalar elds, along at supersymmetric directions.It would appear therefore that this shift of the vacuum is either legitimate, or illegitimate, in both cases. We therefore propose that the non-supersymmetric non-tachyonic string vacua should be considered on equal footing to the super-symmetric examples.

Acknowledgments We would like to thank Steve Abel, Ignatios Antoniadis, Costas Kounnas, Thomas Mohaupt, Marios Petropoulos and John Rizos for discussions. A.E.F. would like to thank the Institute for Advanced Study for hospitality. A.E.F. is supported in part by STFC under contract ST/L000431/1. P.A. acknowledges support from the Hellenic State Scholarship Foundation (IKY).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/

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Web End =ons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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A The spectrum of the model in Sect. 4

See Tables 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Table 1 The untwisted NeveuSchwarz sector matter states and charges

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

3,6 Q

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

b NS (1, 1, 1) 0 0 0 0 0 0 0 (1, 1) 0 0 0 0

(1, 1, 1) 0 0 0 0 0 0 0 (1, 1) 0 0 0 0

(1, 1, 1) 0 4 4 0 0 0 0 (1, 1) 0 0 0 0

(1, 1, 1) 0 0 4 4 0 0 0 (1, 1) 0 0 0 0

(1, 1, 1) 0 4 0 4 0 0 0 (1, 1) 0 0 0 0

Table 2 The untwisted S sector matter states and charges

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

y1 w

5 Q

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

f S (1, 1, 1) 0 0 0 0 0 0 0 (3, 3) 0 4 4 0

(1, 1, 1) 0 0 0 0 0 0 0 (3, 3) 0 4 4 0

(1, 1, 1) 0 0 0 0 0 0 0 (1, 1) 4 0 0 4

(3, 1, 1) 4 4 0 0 0 0 0 (1, 1) 0 0 0 0 (

3, 1, 1) 4 4 0 0 0 0 0 (1, 1) 0 0 0 0

(3, 1, 1) 4 0 4 0 0 0 0 (1, 1) 0 0 0 0 (

3, 1, 1) 4 0 4 0 0 0 0 (1, 1) 0 0 0 0

(3, 1, 1) 4 0 0 4 0 0 0 (1, 1) 0 0 0 0

3,6 Q

y1 w

5 Q

123

Eur. Phys. J. C (2016) 76 :208 Page 11 of 17 208

Table 2 continued

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

(

3, 1, 1) 4 0 0 4 0 0 0 (1, 1) 0 0 0 0

(1, 2, 2) 0 0 0 0 0 0 0 (1, 1) 0 0 0 0

(1, 1, 1) 0 0 0 0 4 0 0 (1, 1) 0 0 0 0

(1, 1, 1) 0 0 0 0 0 4 0 (1, 1) 0 0 0 0

(1, 1, 1) 0 0 0 0 4 0 0 (1, 1) 0 0 0 0

(1, 1, 1) 0 0 0 0 0 4 0 (1, 1) 0 0 0 0

Table 3 The observable matter sectors. All sectors, fermionic and bosonic, have CPT conjugates which are not displayed

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

3,6 Q

y1 w

5 Q

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

QL1 b1 (3, 2, 1) 2 2 0 0 2 0 0 (1, 1) 0 0 0 0 QR1 (3, 1, 2) 2 2 0 0 2 0 0 (1, 1) 0 0 0 0

LL1 (1, 2, 1) 6 2 0 0 2 0 0 (1, 1) 0 0 0 0 L R1 (1, 1, 2) 6 2 0 0 2 0 0 (1, 1) 0 0 0 0 b S + b1 (3, 1, 2) 2 2 0 0 2 0 0 (1, 1) 0 0 0 0

(3, 2, 1) 2 2 0 0 2 0 0 (1, 1) 0 0 0 0

(1, 2, 1) 6 2 0 0 2 0 0 (1, 1) 0 0 0 0

(1, 1, 2) 6 2 0 0 2 0 0 (1, 1) 0 0 0 0

QL2 b2 (3, 2, 1) 2 0 2 0 0 2 0 (1, 1) 0 0 0 0 QR2 (3, 1, 2) 2 0 2 0 0 2 0 (1, 1) 0 0 0 0

LL2 (1, 2, 1) 6 0 2 0 0 2 0 (1, 1) 0 0 0 0 L R2 (1, 1, 2) 6 0 2 0 0 2 0 (1, 1) 0 0 0 0 b S + b2 (3, 1, 2) 2 0 2 0 0 2 0 (1, 1) 0 0 0 0

(3, 2, 1) 2 0 2 0 0 2 0 (1, 1) 0 0 0 0

(1, 2, 1) 6 0 2 0 0 2 0 (1, 1) 0 0 0 0

(1, 1, 2) 6 0 2 0 0 2 0 (1, 1) 0 0 0 0

QL3 b3 (3, 2, 1) 2 0 0 2 0 0 2 (1, 1) 0 0 0 0 QR3 (3, 1, 2) 2 0 0 2 0 0 2 (1, 1) 0 0 0 0

LL3 (1, 2, 1) 6 0 0 2 0 0 2 (1, 1) 0 0 0 0 L R3 (1, 1, 2) 6 0 0 2 0 0 2 (1, 1) 0 0 0 0 b S + b3 (3, 1, 2) 2 0 0 2 0 0 2 (1, 1) 0 0 0 0

(3, 2, 1) 2 0 0 2 0 0 2 (1, 1) 0 0 0 0

(1, 2, 1) 6 0 0 2 0 0 2 (1, 1) 0 0 0 0

(1, 1, 2) 6 0 0 2 0 0 2 (1, 1) 0 0 0 0

3,6 Q

y1 w

5 Q

123

208 Page 12 of 17 Eur. Phys. J. C (2016) 76 :208

Table 4 Vector-like SO(10) singlet states. All sectors, fermionic and bosonic, have CPT conjugates which are not displayed

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

f S+ (1, 1, 1) 0 2 2 0 0 0 0 (1, 1) 0 0 0 4

b1 + b2 (1, 1, 1) 0 2 2 0 0 0 0 (1, 1) 0 0 0 4

+ + (1, 1, 1) 0 2 2 0 0 0 0 (1, 1) 0 0 0 4

(1, 1, 1) 0 2 2 0 0 0 0 (1, 1) 0 0 0 4

(1, 1, 1) 0 2 2 0 0 0 0 (3, 1) 0 4 0 0

(1, 1, 1) 0 2 2 0 0 0 0 (3, 1) 0 4 0 0 (1, 1, 1) 0 2 2 0 0 0 0 (3, 1) 0 4 0 0 (1, 1, 1) 0 2 2 0 0 0 0 (3, 1) 0 4 0 0

b b1 + b2 (1, 1, 1) 0 2 2 0 0 0 0 (1, 1) 4 0 0 0

+ + (1, 1, 1) 0 2 2 0 0 0 0 (1, 1) 4 0 0 0

(1, 1, 1) 0 2 2 0 0 0 0 (1, 1) 4 0 0 0

(1, 1, 1) 0 2 2 0 0 0 0 (1, 3) 0 0 4 0

(1, 1, 1) 0 2 2 0 0 0 0 (1, 3) 0 0 4 0 (1, 1, 1) 0 2 2 0 0 0 0 (1, 3) 0 0 4 0 (1, 1, 1) 0 2 2 0 0 0 0 (1, 3) 0 0 4 0

Table 5 Vector-like SO(10) singlet states. All sectors, fermionic and bosonic, have CPT conjugates which are not displayed

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

3,6 Q

y1 w

5 Q

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

f S+ (1, 1, 1) 0 2 0 0 0 2 2 (1, 1) 0 6 0 2

1 + b1 (1, 1, 1) 0 2 0 0 0 2 2 (1, 1) 0 6 0 2

+ + 2 (1, 1, 1) 0 2 0 0 0 2 2 (3, 1) 0 2 0 2

(1, 1, 1) 0 2 0 0 0 2 2 (3, 1) 0 2 0 2

b 1 + b1 (1, 1, 1) 0 2 0 0 0 2 2 (1, 1) 0 6 0 2

+ + 2 (1, 1, 1) 0 2 0 0 0 2 2 (1, 1) 0 6 0 2

(1, 1, 1) 0 2 0 0 0 2 2 (3, 1) 0 2 0 2 (1, 1, 1) 0 2 0 0 0 2 2 (3, 1) 0 2 0 2

f S+ (1, 1, 1) 0 0 2 0 2 0 2 (1, 1) 0 6 0 2

1 + b2 (1, 1, 1) 0 0 2 0 2 0 2 (1, 1) 0 6 0 2

+ + 2 (1, 1, 1) 0 0 2 0 2 0 2 (3, 1) 0 2 0 2

(1, 1, 1) 0 0 2 0 2 0 2 (3, 1) 0 2 0 2

b 1 + b2 (1, 1, 1) 0 0 2 0 2 0 2 (1, 1) 0 6 0 2

+ + 2 (1, 1, 1) 0 0 2 0 2 0 2 (1, 1) 0 6 0 2

(1, 1, 1) 0 0 2 0 2 0 2 (3, 1) 0 2 0 2 (1, 1, 1) 0 0 2 0 2 0 2 (3, 1) 0 2 0 2

f S+ (1, 1, 1) 0 0 2 0 2 0 2 (1, 1) 2 0 6 0

b1 + b3 (1, 1, 1) 0 0 2 0 2 0 2 (1, 1) 2 0 6 0

+ + 2 (1, 1, 1) 0 0 2 0 2 0 2 (1, 3) 2 0 2 0

(1, 1, 1) 0 0 2 0 2 0 2 (1, 3) 2 0 2 0

b b1 + b3 (1, 1, 1) 0 0 2 0 2 0 2 (1, 1) 2 0 6 0

+ + 2 (1, 1, 1) 0 0 2 0 2 0 2 (1, 1) 2 0 6 0

(1, 1, 1) 0 0 2 0 2 0 2 (1, 3) 2 0 2 0 (1, 1, 1) 0 0 2 0 2 0 2 (1, 3) 2 0 2 0

f S+ (1, 1, 1) 0 0 0 2 2 2 0 (1, 1) 2 0 6 0

3,6 Q

y1 w

5 Q

123

Eur. Phys. J. C (2016) 76 :208 Page 13 of 17 208

Table 5 continued

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

b1 + b2 (1, 1, 1) 0 0 0 2 2 2 0 (1, 1) 2 0 6 0

+ + 2 (1, 1, 1) 0 0 0 2 2 2 0 (1, 3) 2 0 2 0 (1, 1, 1) 0 0 0 2 2 2 0 (1, 3) 2 0 2 0

b b1 + b2 (1, 1, 1) 0 0 0 2 2 2 0 (1, 1) 2 0 6 0

+ + 2 (1, 1, 1) 0 0 0 2 2 2 0 (1, 1) 2 0 6 0

(1, 1, 1) 0 0 0 2 2 2 0 (1, 3) 2 0 2 0 (1, 1, 1) 0 0 0 2 2 2 0 (1, 3) 2 0 2 0

f S+ (1, 1, 1) 0 2 0 0 0 2 2 (1, 1) 2 0 6 0

b2 + b3 (1, 1, 1) 0 2 0 0 0 2 2 (1, 1) 2 0 6 0

+ + 2 (1, 1, 1) 0 2 0 0 0 2 2 (1, 3) 2 0 2 0 (1, 1, 1) 0 2 0 0 0 2 2 (1, 3) 2 0 2 0

b b2 + b3 (1, 1, 1) 0 2 0 0 0 2 2 (1, 1) 2 0 6 0

+ + 2 (1, 1, 1) 0 2 0 0 0 2 2 (1, 1) 2 0 6 0

(1, 1, 1) 0 2 0 0 0 2 2 (1, 3) 2 0 2 0 (1, 1, 1) 0 2 0 0 0 2 2 (1, 3) 2 0 2 0

f S+ (1, 1, 1) 0 0 0 2 2 2 0 (1, 1) 0 6 0 2

1 + b3 (1, 1, 1) 0 0 0 2 2 2 0 (1, 1) 0 6 0 2

+ + 2 (1, 1, 1) 0 0 0 2 2 2 0 (3, 1) 0 2 0 2 (1, 1, 1) 0 0 0 2 2 2 0 (3, 1) 0 2 0 2

b 1 + b3 (1, 1, 1) 0 0 0 2 2 2 0 (1, 1) 0 6 0 2

+ + 2 (1, 1, 1) 0 0 0 2 2 2 0 (1, 1) 0 6 0 2

(1, 1, 1) 0 0 0 2 2 2 0 (3, 1) 0 2 0 2 (1, 1, 1) 0 0 0 2 2 2 0 (3, 1) 0 2 0 2

Table 6 The table displays all the massless sectors for which the would-be superpartners are massive and do not form part of the massless spectrum. The would-be superpartners arise from the sectors that

3,6 Q

y1 w

5 Q

are obtained by adding the basis vector S to a given sector and are the fermionic counterparts

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

b (1, 1, 1) 3 1 1 1 2 0 2 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 1 1 2 0 2 (1, 1) 2 3 3 0

b (1, 1, 1) 3 1 1 1 0 2 2 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 1 1 0 2 2 (1, 1) 2 3 3 0

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

3,6 Q

y1 w

5 Q

123

208 Page 14 of 17 Eur. Phys. J. C (2016) 76 :208

Table 6 continued

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

b 1 + b1 (1, 1, 1) 3 1 1 1 0 2 2 (1, 1) 0 3 3 2

+b2 + b3 (1, 1, 1) 3 1 1 1 0 2 2 (1, 1) 0 3 3 2

+ (1, 1, 1) 3 1 1 1 0 2 2 (1, 1) 0 3 3 2

(1, 1, 1) 3 1 1 1 0 2 2 (1, 1) 0 3 3 2

b 1 + b1 (1, 1, 1) 3 1 1 1 2 0 2 (1, 1) 0 3 3 2

+b2 + b3 (1, 1, 1) 3 1 1 1 2 0 2 (1, 1) 0 3 3 2

+ (1, 1, 1) 3 1 1 1 2 0 2 (1, 1) 0 3 3 2

(1, 1, 1) 3 1 1 1 2 0 2 (1, 1) 0 3 3 2

Table 7 Vector-like exotic states. All sectors, fermionic and bosonic, have CPT conjugates which are not displayed

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

3,6 Q

y1 w

5 Q

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

f S+ (1, 1, 1) 3 3 1 1 0 0 0 (1, 1) 2 3 3 0

b2 + b3 (1, 1, 1) 3 1 3 1 0 0 0 (1, 1) 2 3 3 0

+ (1, 1, 1) 3 1 1 3 0 0 0 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 1 1 0 0 0 (1, 3) 2 3 1 0

(1, 1, 1) 3 3 1 1 0 0 0 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 3 1 0 0 0 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 1 3 0 0 0 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 1 1 0 0 0 (1, 3) 2 3 1 0 b b2 + b3 (3, 1, 1) 1 1 1 1 0 0 0 (1, 1) 2 3 3 0

+ (1, 1, 1) 3 1 1 1 0 0 0 (3, 1) 2 1 3 0

(3, 1, 1) 1 1 1 1 0 0 0 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 1 1 0 0 0 (3, 1) 2 1 3 0

f S+ (1, 1, 1) 3 3 1 1 0 0 0 (1, 1) 2 3 3 0

b1 + b3 (1, 1, 1) 3 1 3 1 0 0 0 (1, 1) 2 3 3 0

+ (1, 1, 1) 3 1 1 3 0 0 0 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 1 1 0 0 0 (1, 3) 2 3 1 0

(1, 1, 1) 3 3 1 1 0 0 0 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 3 1 0 0 0 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 1 3 0 0 0 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 1 1 0 0 0 (1, 3) 2 3 1 0 b b1 + b3 (3, 1, 1) 1 1 1 1 0 0 0 (1, 1) 2 3 3 0

+ (1, 1, 1) 3 1 1 1 0 0 0 (3, 1) 2 1 3 0

(3, 1, 1) 1 1 1 1 0 0 0 (1, 1) 2 3 3 0

(1, 1, 1) 3 1 1 1 0 0 0 (3, 1) 2 1 3 0

3,6 Q

y1 w

5 Q

123

Eur. Phys. J. C (2016) 76 :208 Page 15 of 17 208

Table 8 Vector-like exotic states. All sectors, fermionic and bosonic, have CPT conjugates which are not displayed

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

f S+ (3, 1, 1) 3 1 1 1 0 0 0 (1, 1) 0 3 3 2 1 + b2 (1, 1, 1) 3 1 1 1 0 0 0 (1, 3) 0 3 3 2

+ (3, 1, 1) 3 1 1 1 0 0 0 (1, 1) 0 3 3 2

(1, 1, 1) 3 1 1 1 0 0 0 (1, 3) 0 3 3 2

b 1 + b2 (1, 1, 1) 3 3 1 1 0 0 0 (1, 1) 0 3 3 2

+ (1, 1, 1) 3 1 3 1 0 0 0 (1, 1) 0 3 3 2

(1, 1, 1) 3 1 1 3 0 0 0 (1, 1) 0 3 3 2 (1, 1, 1) 3 1 1 1 0 0 0 (3, 1) 0 3 3 2

(1, 1, 1) 3 3 1 1 0 0 0 (1, 1) 0 3 3 2

(1, 1, 1) 3 1 3 1 0 0 0 (1, 1) 0 3 3 2

(1, 1, 1) 3 1 1 3 0 0 0 (1, 1) 0 3 3 2

(1, 1, 1) 3 1 1 1 0 0 0 (3, 1) 0 3 3 2

f S+ (1, 1, 1) 3 3 1 1 0 0 0 (1, 1) 0 3 3 2

1 + b1 (1, 1, 1) 3 1 3 1 0 0 0 (1, 1) 0 3 3 2

+ (1, 1, 1) 3 1 1 3 0 0 0 (1, 1) 0 3 3 2

(1, 1, 1) 3 1 1 1 0 0 0 (3, 1) 0 3 3 2

(1, 1, 1) 3 3 1 1 0 0 0 (1, 1) 0 3 3 2

(1, 1, 1) 3 1 3 1 0 0 0 (1, 1) 0 3 3 2

(1, 1, 1) 3 1 1 3 0 0 0 (1, 1) 0 3 3 2 (1, 1, 1) 3 1 1 1 0 0 0 (3, 1) 0 3 3 2

b 1 + b1 (3, 1, 1) 3 1 1 1 0 0 0 (1, 1) 0 3 3 2

+ (1, 1, 1) 3 1 1 1 0 0 0 (1, 3) 0 3 3 2 (3, 1, 1) 3 1 1 1 0 0 0 (1, 1) 0 3 3 2 (1, 1, 1) 3 1 1 1 0 0 0 (1, 3) 0 3 3 2

Table 9 Vector-like exotic states. All sectors, fermionic and bosonic, have CPT conjugates which are not displayed

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

3,6 Q

y1 w

5 Q

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

f 1 + b2 (1, 2, 1) 0 0 2 2 2 0 0 (1, 1) 2 0 0 2

+b3 + 2 (1, 2, 1) 0 0 2 2 2 0 0 (1, 1) 2 0 0 2

(1, 1, 2) 0 0 2 2 2 0 0 (1, 1) 2 0 0 2

b S+ (1, 2, 1) 0 0 2 2 2 0 0 (1, 1) 2 0 0 2

1 + b2 (1, 2, 1) 0 0 2 2 2 0 0 (1, 1) 2 0 0 2

+b3 + 2 (1, 1, 2) 0 0 2 2 2 0 0 (1, 1) 2 0 0 2

(1, 1, 2) 0 0 2 2 2 0 0 (1, 1) 2 0 0 2

f 1 + b1 (1, 2, 1) 0 2 0 2 0 2 0 (1, 1) 2 0 0 2

+b3 + 2 (1, 2, 1) 0 2 0 2 0 2 0 (1, 1) 2 0 0 2

(1, 1, 2) 0 2 0 2 0 2 0 (1, 1) 2 0 0 2

b S+ (1, 2, 1) 0 2 0 2 0 2 0 (1, 1) 2 0 0 2

1 + b1 (1, 2, 1) 0 2 0 2 0 2 0 (1, 1) 2 0 0 2

+b3 + 2 (1, 1, 2) 0 2 0 2 0 2 0 (1, 1) 2 0 0 2

(1, 1, 2) 0 2 0 2 0 2 0 (1, 1) 2 0 0 2

3,6 Q

y1 w

5 Q

123

208 Page 16 of 17 Eur. Phys. J. C (2016) 76 :208

Table 9 continued

F SEC (C; L; R) QC Q

1 Q

2 Q

3 Q y

3,6 Q

y1 w

5 Q

w2,4 SU(3)H1,2 Q 1 Q8 Q9 Q 8

f 1 + b1 (1, 2, 1) 0 2 2 0 0 0 2 (1, 1) 2 0 0 2

+b2 + 2 (1, 2, 1) 0 2 2 0 0 0 2 (1, 1) 2 0 0 2 (1, 1, 2) 0 2 2 0 0 0 2 (1, 1) 2 0 0 2

(1, 1, 2) 0 2 2 0 0 0 2 (1, 1) 2 0 0 2

b S+ (1, 2, 1) 0 2 2 0 0 0 2 (1, 1) 2 0 0 2

1 + b1 (1, 2, 1) 0 2 2 0 0 0 2 (1, 1) 2 0 0 2

+b2 + 2 (1, 1, 2) 0 2 2 0 0 0 2 (1, 1) 2 0 0 2 (1, 1, 2) 0 2 2 0 0 0 2 (1, 1) 2 0 0 2

f S+ (1, 1, 1) 6 0 0 2 0 0 0 (1, 1) 2 0 0 2 1 + b3 (3, 1, 1) 2 0 0 2 0 0 0 (1, 1) 2 0 0 2

+ + (1, 1, 1) 6 0 0 2 0 0 0 (1, 1) 2 0 0 2

+2 (3, 1, 1) 2 0 0 2 0 0 0 (1, 1) 2 0 0 2

b 1 + b3 (1, 1, 1) 6 0 0 2 0 0 0 (1, 1) 2 0 0 2

+ + (1, 1, 1) 6 0 0 2 0 0 0 (1, 1) 2 0 0 2

+2 (3, 1, 1) 2 0 0 2 0 0 0 (1, 1) 2 0 0 2 (3, 1, 1) 2 0 0 2 0 0 0 (1, 1) 2 0 0 2

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(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

The heterotic-string models in the free fermionic formulation gave rise to some of the most realistic-string models to date, which possess ...... spacetime supersymmetry. Lack of evidence for supersymmetry at the LHC instigated recent interest in non-supersymmetric heterotic-string vacua. We explore what may be learned in this context from the quasi-realistic free fermionic models. We show that constructions with a low number of families give rise to proliferation of a priori tachyon producing sectors, compared to the non-realistic examples, which typically may contain only one such sector. The reason being that in the realistic cases the internal six dimensional space is fragmented into smaller units. We present one example of a quasi-realistic, non-supersymmetric, non-tachyonic, heterotic-string vacuum and compare the structure of its massless spectrum to the corresponding supersymmetric vacuum. While in some sectors supersymmetry is broken explicitly, i.e. the bosonic and fermionic sectors produce massless and massive states, other sectors, and in particular those leading to the chiral families, continue to exhibit Fermi-Bose degeneracy. In these sectors the massless spectrum, as compared to the supersymmetric cases, will only differ in some local or global U(1) charges. We discuss the conditions for obtaining ...... at the massless level in these models. Our example model contains an anomalous U(1) symmetry, which generates a tadpole diagram at one-loop order in string perturbation theory. We speculate that this tadpole diagram may cancel the corresponding diagram generated by the one-loop non-vanishing vacuum energy and that in this respect the supersymmetric and non-supersymmetric vacua should be regarded on an equal footing. Finally we discuss vacua that contain two supersymmetry generating sectors.

Details

Title

Non-tachyonic semi-realistic non-supersymmetric heterotic-string vacua

Author

Ashfaque, Johar M; Athanasopoulos, Panos; Faraggi, Alon E; Sonmez, Hasan

Pages

1-17

Publication year

2016

Publication date

Apr 2016

Publisher

Springer Nature B.V.

ISSN

14346044

e-ISSN

14346052

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1140/epjc/s10052-016-4056-2

ProQuest document ID

1794086395

SIF and Springer-Verlag Berlin Heidelberg 2016

Non-tachyonic semi-realistic non-supersymmetric heterotic-string vacua

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