Kinetic Mixing of U(1)s in Heterotic Orbifolds
aa r X i v : . [ h e p - t h ] O c t Preprint typeset in JHEP style - HYPER VERSION
DESY 11-184CERN-PH-TH/2011-253
Kinetic Mixing of U(1)s in HeteroticOrbifolds
Mark Goodsell , Sa´ul Ramos-S´anchez , Andreas Ringwald CERN, Theory Division, CH-1211 Geneva 23, Switzerland Department of Theoretical Physics, Physics Institute, UNAM, Mexico D.F. 04510,Mexico Deutsches Elektronen-Synchrotron DESY, Hamburg, GermanyE-mail: [email protected] , [email protected] , [email protected] Abstract:
We study kinetic mixing between massless U(1) gauge symmetries in thebosonic formulation of heterotic orbifold compactifications. For non-prime Z N factoris-able orbifolds, we find a simple expression of the mixing in terms of the properties of the N = 2 subsectors, which helps understand under what conditions mixing can occur. Withthis tool, we analyse Z -II heterotic orbifolds and find non-vanishing mixing even withoutincluding Wilson lines. We show that some semi-realistic models of the Mini-Landscapeadmit supersymmetric vacua with mixing between the hypercharge and an additional U(1),which can be broken at low energies. We finally discuss some phenomenologically appealingpossibilities that hidden photons in heterotic orbifolds allow. Keywords:
Heterotic strings, kinetic mixing, model building. ontents
1. Introduction 1
2. Kinetic mixing in heterotic orbifolds 3
3. Kinetic mixing in Z N heterotic orbifolds 7 Z –II orbifolds without Wilson lines 73.2 Semi-realistic Z –II orbifolds with kinetic mixing 93.2.1 A promising string realisation of kinetic mixing 10
4. Phenomenology of hidden photons from heterotic orbifolds 11
5. Discussion 17A. Further details on a promising model 18B. Other promising models with kinetic mixing 20
B.1 Example 1. Mixing in the observable E
1. Introduction
A good motivation for the existence of additional U(1) gauge symmetries is the “DarkForce” scenario. In this setting, dark matter arises from a standard model (SM) singletcharged under a hidden U(1), whose gauge boson has a GeV mass and mixes kinetically withthe observable photon [1–7]. Several features of such scenarios have been studied [8–10],including supersymmetry [4, 7, 11–13], in which some of the contemporary puzzles, such asthe data of PAMELA and Fermi, could find an explanation. Furthermore, the wealth of– 1 –xperiments [14–16] (see e.g. [17–19] for an overview) capable of probing hidden U(1)s overa very wide range of hidden gauge-boson mass and kinetic mixing values also motivatesthe study of the phenomenological potential of hidden sectors. On the other hand, stringtheory quite often offers a plethora of Abelian symmetries which might well help to embedthe Dark Force scenario in an ultra-violet complete and globally consistent theory. Thus,it seems natural to ask to what extent this idea is consistent with string constructions.As a starting point, we could well ask what hidden-sector models are generic and whatthe generic parameters for them are. In particular, we can explore the amount of kineticmixing that promising models exhibit.Predictions for kinetic mixing and its phenomenology were considered in type II stringsin [20–28]; in [24, 28] both masses and mixings were studied, and it was argued that theDark Force scenario could be accommodated provided that there is additional sequestering.However, it is to heterotic orbifolds that we shall turn in this paper. Orbifold models inthe fermionic formulation were the corner of the heterotic string landscape where kineticmixing was originally searched for [29,30] (it was also considered in heterotic models in thegeometric regime in [31, 32]), although the existence of interesting models (i.e. with non-zero mixing) was not established. We intend to resolve this issue, and also clarify underwhat conditions mixing may occur. To do so, we study kinetic mixing in heterotic orbifoldmodels in the bosonic formulation and focus particularly on those of the Z –II orbifoldMini-Landscape [33, 34], which are known to display many phenomenologically appealingfeatures [35]. Indeed, we hope that this paper paves the way for more exploration of therich phenomenology of kinetically mixed hidden sectors possible in heterotic models. In an unbroken supersymmetric theory, there is only one possible operator that can yieldkinetic mixing. It appears in the gauge kinetic part of the supergravity Lagrangian, and isthus a holomorphic function of other fields:
L ⊃ Z d θ (cid:26) g ha ) W a W a + 14( g hb ) W b W b − χ hab W a W b (cid:27) , (1.1)where W a , W b are the field strength superfields for the two U(1) gauge fields and χ hab , g ha , g hb are the holomorphic kinetic mixing parameter and gauge couplings that must run only atone loop. In the canonical basis, L canonical ⊃ Z d θ (cid:26) W a W a + 14 W b W b − χ ab W a W b (cid:27) , (1.2)where the canonical kinetic mixing is given in terms of the holomorphic quantity by theKaplunovsky-Louis type relation [24, 36, 37]: χ ab g a g b = ℜ ( χ hab ) + 18 π tr (cid:18) Q a Q b log Z (cid:19) − π X r n r Q a Q b ( r ) κ K. (1.3)Here, K is the full K¨ahler potential of the theory, and Z is the K¨ahler metric of matterfields formed by the second derivative of K . Q a,b are the charge operators of the two U(1)s.– 2 –he mixing can be read off from a one-loop string calculation of the S-matrix elementfor two gauge bosons. At one loop this is identical to the 1PI diagram, and so we obtainthe physical mixing. The calculation is analogous to that of gauge threshold corrections,and yields [29] χ ab g a g b = b ab π log M S µ + ∆ ab , (1.4)where M S is the string scale, and b ab and △ ab are mixed β function coefficients and stringthreshold contributions, respectively.The previous results have already been applied in several extensions of the SM, in-cluding string theory. For example, in [29] it was found that none of their models satisfied b ab = 0, condition that ensures no running of the kinetic mixing below the string scaleand corresponds to eliminating (chiral) light states charged under both U(1)s. They alsopointed out that the phenomenological problems associated with these states can be dodgedonce they are lifted out of the spectrum after certain SM singlets attain vacuum expec-tation values (VEVs) (although this would be very challenging to calculate in practise).Therefore, in their work ∆ ab was considered of prime interest as an unambiguous indicatorof kinetic mixing. We shall take the same approach: while it would be particularly pleasingto obtain models for which b ab = 0, we shall not restrict ourselves to that case. Unfortu-nately, ∆ ab was calculated for various models then available and exactly zero was found ineach case. This is in contrast to [38], which contained examples of Z × Z orbifolds withoutWilson lines for which b ab = 0 and ∆ ab = 0 but did not emphasize this result; however, theU(1)s descended from a higher-dimensional SU(2) in each case.Our discussion in the present work is structured as follows. In section 2, we ad-dress the general computation of kinetic mixing in heterotic orbifolds, and attempt toobtain a better understanding of when mixing is possible. Then in section 3, we presentexamples with and without Wilson lines where ∆ ab is non-zero and briefly study theirphenomenology. We show that the models of the Z –II heterotic Mini-Landscape canaccommodate mixing ∆ ab = 0 between the hypercharge and additional hidden forces inMSSM-like supersymmetry-preserving vacua. In section 4, we briefly explore how hiddenU(1) symmetries emerging from heterotic orbifolds could yield a Dark Force scenario andother interesting phenomenology. Finally, in section 5, we discuss our results and providean outlook. The appendices are devoted to the details of some sample models.A final remark is in order. In this work, we do not consider the case where b ab = 0 and∆ ab = 0 simultaneously. However, we note that in these cases, mixing can still be generatedby splitting the masses of multiplets charged under both U(1)s; in the case of light fieldsobtaining VEVs, this can generate sizable mixing, while splitting due to supersymmetrybreaking seems to generate only an incredibly small effect [24, 30].
2. Kinetic mixing in heterotic orbifolds
Let us start by introducing the main concepts that are important for computing the kineticmixing between U(1)s in the context of heterotic orbifolds (for reviews of these construc-– 3 –ions, see [39–42]). The Z N orbifolds in which we are interested are defined as quotients offactorisable six-dimensional tori T = T × T × T divided by a discrete set of isometries of T that form the so-called point group. In Z N heterotic orbifolds, the point group containsall different powers of a discrete rotation generator ϑ = diag( e π i v , e π i v , e π i v ), whereeach entry of the twist vector v encodes the action of the orbifold on each 2-torus and v + v + v = 0. Different powers k of ϑ (0 ≤ k < N ) define different twisted sectors . If theaction of ϑ k is trivial on only one T , i.e. if kv i = 0 mod 1 for an i , kv defines an N = 2subsector. As we shall see below, orbifold constructions that may allow for kinetic mixingrequire the existence of these N = 2 subsectors. Consequently, by inspecting all admissibletwist vectors (complete lists can be found in [43]), one finds that all Z N orbifolds withnon-prime N are candidates to provide kinetic mixing. We decompose U(1) gauge fields in terms of the Cartan generators of the initial E × E (or – but not here – Spin(32) / Z ), H I ( I = 1 , . . . , Q a = X I t Ia H I . (2.1)Then, a state with left-moving momenta p I will have U(1) charge t Ia p I . The vector t a willbe frequently called the generator of the U(1) a symmetry. The operators on the worldsheetcorresponding to the gauge bosons contain holomorphic currents j ( z ) satisfying the OPE j I ( z ) j J (0) ∼ k I δ IJ z + i f IJK z j K (0) , (2.2)where k I is the level of the algebra, which we shall take to be normalised to 2. For ourU(1)s, the currents appear in the combinations P I t Ia j I ( z ) and so the tree-level gaugekinetic function satisfies f ab = S X I k I t Ia t Ib , (2.3)where S is the (bosonic part of) the dilaton/axion chiral superfield. Hence the independentU(1)s satisfy P I t Ia t Ib = δ ab and have gauge kinetic function f ab = 2 Sδ ab . Note that this isnot the same as tr( Q a Q b ) = δ ab . Due to this orthogonality, there is no O ( z − ) term in theOPE of two different U(1)s.We are interested in the mixing between non-anomalous U(1)s, i.e. between Abeliansymmetries satisfying the universality condition [44, 45]12 | t b | tr( Q b Q a ) = 16 | t a | tr Q a = 124 tr Q a = 0 for all a, b , (2.4)where the traces run over all chiral-matter fields. It is well-known that the orbifold limitof the heterotic string has commonly (at most) one anomalous symmetry U(1) anom [46]. We note that also Z N × Z M orbifolds fall in this category, but we will not consider them here. Although we will mostly take | t a | = 1, as we mentioned before, we shall use the GUT-compatiblehypercharge normalisation | t Y | = 5 / – 4 –his anomaly is cancelled via the Green-Schwarz mechanism [47] which implies that, forthe anomalous case, (2.4) takes the form12 | t b | tr( Q b Q anom ) = 16 | t anom | tr Q = 124 tr Q anom = 8 π δ GS for all b , (2.5)where δ GS is the universal Green-Schwarz coefficient. The existence of U(1) anom inducesthe Fayet-Iliopoulos D -term [48] given (in string units) by ξ = g s δ GS = g s tr Q anom π , (2.6)where g s is the string coupling. It is precisely the appearance of this term what renders thisanomalous symmetry harmless for phenomenology. Since we do not expect supersymmetryto be broken at the compactification scale in realistic models, in an acceptable orbifoldvacuum ξ must be cancelled. This means that certain matter fields charged under U(1) anom need develop vacuum expectation values, breaking thereby this anomalous symmetry andavoiding dangerous mixing effects. Therefore, for practical purposes, we shall ignore theanomalous U(1) and focus on constructions with two or more non-anomalous U(1)s.Once the twist vector is set, different Z N orbifold models arise from the differentgauge embeddings that satisfy the modular invariance conditions [49–52] which ensure aneffective theory free of any anomalies. We are then interested in admissible shift vectors V and Wilson lines A i that comply with (no summation implied) N ( V − v ) = 0 mod 2 . (2.7a) N i A i · V = 0 mod 2 , (2.7b) N i A i = 0 mod 2 , (2.7c)gcd( N i , N j ) A i · A j = 0 mod 2 ( i = j ) , (2.7d)where the order of the Wilson lines N i is constrained by the geometry of T and gcd standsfor the greatest common denominator. We focus here on the ten-dimensional heteroticstring theory with gauge group E × E , which is broken down to the four-dimensionalgroup by the action of the shift and Wilson lines. For these models, the kinetic mixing at the string threshold is given by∆ ab ≡ π Z d τ Im τ [ B ab ( τ ) − b ab ]with B ab ( τ ) ≡ − Str( Q H Q a Q b e α ′ M R πiτ e α ′ M L πiτ ) , (2.8)where M R , M L are the masses of respectively right and left moving states in the theory(which are constrained to be equal), Q H denotes the helicity operator, and b ab ≡ − Str massless ( Q H Q a Q b ) (2.9)with Str massless being a supertrace over massless states.– 5 –ote that the result is entirely analogous to gauge threshold corrections, except thereis no moduli-independent piece proportional to the level: this is easily seen as being due tothe lack of an O ( z − ) term in the OPE of the currents. Following the reasoning in [38, 53],the result can be expressed as follows∆ ab = X i b iab | G i | π | G | (cid:20) log (cid:18) | η ( T i ) | Im( T i ) (cid:19) + log (cid:18) | η ( U i ) | Im( U i ) (cid:19)(cid:21) , (2.10)where the sum runs over all order | G i | N = 2 subsectors of the orbifold of order | G | , T i , U i are the untwisted K¨ahler and complex structure moduli (in Planck units) corresponding tothe torus fixed in the subsector i , and the β -like mixing coefficient associated to the i -thfixed torus b iab is given by b iab ≡ −
116 tr iV ( Q a Q b ) + 13 tr iF ( Q a Q b ) + 112 tr iS ( Q a Q b )= 14 (cid:20) − iV ( Q a Q b ) + tr iS ( Q a Q b ) (cid:21) . (2.11)Here, the traces in the first line are over the vectors, Weyl fermions and real scalars respec-tively, and in the second line we have simplified the expressions using supersymmetry –this is the form used in a brute force calculation of the mixing, by summing over all bosonicstates in the appropriate sector of the theory, identifying them as vectors or scalars, andweighting accordingly with P I,J ( t Ia p I )( t Jb p J ). Of course, the sum over states is equivalentto the more familiar sum over N = 2 vector and hypermultiplets b iab = 12 (cid:0) − iV, N =2 ( Q a Q b ) + tr iH, N =2 ( Q a Q b ) (cid:1) , where Q a,b are defined in eq. (2.1), i.e. they correspond to the N = 1 Abelian generators,whereas all summations in eq. (2.11) are over N = 2 states.We would also like now to understand when we can obtain kinetic mixing, and providea simple formula to explain its presence or absence. To this end, we shall rewrite b iab interms of the properties of the gauge subgroups b ′ in the different N = 2 sectors . In doingso, it is important to trace the origin of the U(1)s in the N = 2 sectors. For example, theremay be U(1)s in an N = 2 sector that may be broken or not in the full orbifold, and the N = 1 U(1)s may also arise from the non-Abelian groups in the N = 2 sector. The simpleroots of the N = 2 non-Abelian subgroups b ′ shall be denoted by ˆ α b ′ i with i running from1 to the sum of the ranks of the non-Abelian groups r ; the N = 2 Abelian generators aredenoted by t b ′ (clearly, the sum of ˆ α b ′ i s and t b ′ s is 16). For non-Abelian groups, the ˆ α b ′ i are normalised to h ˆ α b ′ i ˆ α c ′ j i = δ b ′ c ′ C b ′ ij , where C b ′ is the Cartan matrix of the group b ′ . Asfor N = 1, in the Abelian case we have h t b ′ t c ′ i = δ b ′ c ′ . We can then write the N = 1 U(1) a generators of interest t a as linear combinations: t a = r X i =1 m b ′ ,ia ˆ α b ′ i + X b ′ = r +1 n b ′ a t b ′ , m b ′ ,ia , n b ′ a ∈ R . (2.12) The indices a, b, . . . denote N = 1 gauge subgroups and a ′ , b ′ , . . . refer to the N = 2 theories. – 6 –efining additionally the matrix b N =2U(1) as (cid:16) b N =2U(1) (cid:17) a ′ b ′ = 12 ( − V, N =2 ( Q a ′ Q b ′ ) + tr H, N =2 ( Q a ′ Q b ′ )) (2.13)(with Q a ′ ,b ′ being the the analog of (2.1) for N = 2), we are ready to rewrite the mixingcoefficient b iab in terms of the previous matrix, the N = 2 β -function coefficients b N =2 b ′ ofthe non-Abelian groups, and the Cartan matrices C b ′ : b iab = (cid:16) m b ′ a C b ′ m b ′ b (cid:17) b N =2 b ′ + 2 (cid:16) n a b N =2U(1) n b (cid:17) . (2.14)This provides an explanation for when kinetic mixing can be present: the U(1)s musteither contain overlapping components in non-Abelian gauge groups in the N = 2 sector,or must derive from U(1)s that mix at that level. Note that clearly if the generators ofthe U(1)s lie entirely in separate E s, then only the second possibility is available providedthat the off-diagonal entries of b N =2U(1) are non-trivial.A final remark is in order. The actual value of kinetic mixing at low energies dependsnot only on the high energy contribution from the N = 2 subsectors, b iab (or ∆ ab ), but alsoimportantly on the N = 1 β function coefficient b ab , as in eq. (1.4). Since this dependencemight alter drastically any result coming from the high-energy string states, and wishingto have a hidden U(1), it is desirable to take b ab = 0. In heterotic orbifolds, this requiresthe spectrum to be vector-like w.r.t. the U(1)s, which imposes a strong constraint on themodels. This can be contrasted to D-brane models, where this constraint can be readilysatisfied by separating the branes supporting the U(1)s, but it is then necessary to ensurethe absence of mass mixing due to the St¨uckelberg mechanism (even though the U(1)smay not be anomalous) while preserving kinetic mixing [23]. This does not occur in theheterotic orbifold case since there are no axions that could generate the masses .
3. Kinetic mixing in Z N heterotic orbifolds Since only non-prime Z N orbifolds can produce non-trivial kinetic mixing, we shall considerbelow one promising candidate: Z –II, which has been found to lead to plenty of modelspossessing many phenomenologically appealing features [35]. Therefore, an interestingquestion is whether explicit models of this type exhibit kinetic mixing. Remarkably, even Z –II orbifold models without Wilson lines display non-trivial values of ∆ ab , as we nowdiscuss. Z –II orbifolds without Wilson lines The Z –II orbifold is defined by the twist vector v = 1 / , , −
3) acting on the T torusspanned by the root lattice of G × SU(3) × SO(4). The structure of these constructionsallows only for three K¨ahler moduli T , T , T and a complex-structure modulus U , wherethe subindexes refer to the three 2-tori of T . Without Wilson lines, there are 61 different In the geometric regime (or equivalently when the orbifold is blown up) [31, 32] it does happen. – 7 –auge embeddings V that fulfill (2.7a), but only roughly 1 / N = 2 subsectors of Z –II orbifolds: a Z subsector generated by 2 v that comprises the k = 2 , T invariant, and a Z subsector generated by 3 v which includes onlythe ϑ sector and leaves the second T untouched. Therefore, we have∆ ab = 116 π (cid:26) b ab (cid:18) | η ( T ) | Im( T ) (cid:19) + b ab (cid:20) log (cid:18) | η ( T ) | Im( T ) (cid:19) + log (cid:18) | η ( U ) | Im( U ) (cid:19)(cid:21)(cid:27) . (3.1)Using the methods described in section 2, we can compute b iab for all Z –II orbifoldswithout Wilson lines. We find that there are 10 models with3 . | b iab | . ⇒ − . | ∆ ab | . − , (3.2)where we have used moduli of order unity to estimate the values of ∆ ab . Remarkably, evenin this simple scenario, there is kinetic mixing. An example.
Consider now the model defined by the shift vector V = 112 (10 , , , , , , , , , , , , , , . (3.3)The unbroken gauge group is then SO(8) × SU(4) × SU(7) × U(1) anom × U(1) × U(1) . The(coefficients of the Cartan-expansion of the) U(1) generators are given by t anom = √ (0 , , , , , , , , , , , , , , , (3.4a) t = √ (0 , , , , , , , − , , , , , , , , (3.4b) t = ( − , , , , , , , , , , , , , , , (3.4c)(where we distinguish between the components of the first and second E s) which satisfythe orthogonality constraint t a · t b = δ ab . An interesting feature of this model is that therelevant U(1)s stem from different E groups; however, the N = 1 matter spectrum leadsto tr( Q Q ) = − √ = 0, implying that, although there is no kinetic mixing at tree level,at one-loop kinetic mixing may appear. In fact, in this case evaluating (3.1) results in∆ = b × π (cid:20) log (cid:18) | η ( T ) | Im( T ) (cid:19) + log (cid:18) | η ( U ) | Im( U ) (cid:19)(cid:21) = − √ π (cid:20) log (cid:18) | η ( T ) | Im( T ) (cid:19) + log (cid:18) | η ( U ) | Im( U ) (cid:19)(cid:21) , (3.5)which is about 10 − for order-one moduli.This result is interesting because it shows that, similarly to what happens in type IIscenarios with D3-branes at Z –II orbifold singularities [27], Z –II orbifold compactifica-tions of the heterotic string allow for kinetic mixing. The advantage of the latter is that the– 8 –(1)s are not located at singularities and, therefore, it is not necessary to build by hand asuitable pair of U(1)s that leads to this outcome. A shortcoming of the model presentedhere is of course that it has no chance of being a description of our universe, since it doesnot even exhibit the gauge group of the SM. Z –II orbifolds with kinetic mixing Introducing discrete Wilson lines in Z –II orbifolds leads to a large class of semi-realisticmodels with an observable sector displaying the exact spectrum of the MSSM and otherphenomenologically desirable properties [35]. In the hidden sector, there are typically someAbelian gauge symmetries, which can be broken spontaneously in explicit supersymmetricvacua [54, 55]. However, also supersymmetric vacua leading to two or more additionalmassless U(1)s exist [56], which may lead to observable kinetic mixing. In this section, weexplore this possibility.Before computing the kinetic mixing, a second effect of the presence of Wilson linebackgrounds must be considered. It is known that the original modular symmetry SL (2 , Z )is typically broken to its congruence subgroups Γ n ( N ) , Γ n ( N ) by the Wilson lines [57–59](for some n, N that depend on the chosen Wilson lines). As explained e.g. in [60, § η ( T i ) , η ( U i ) to be replaced by η ( p i T i ) , η ( q i U i ),where the factors p, q depend on the resulting modular subgroup. Therefore, includingthe effect of Wilson lines (3.1) takes the form∆
Y X = 116 π (cid:26) b Y X (cid:18) | η ( p T ) | Im( T ) (cid:19) + b Y X (cid:20) log (cid:18) | η ( p T ) | Im( T ) (cid:19) + log (cid:18) | η ( q U ) | Im( U ) (cid:19)(cid:21)(cid:27) . (3.6)Let us study now the subset of semi-realistic Z -II orbifold models obtained in theMini-Landscape [33, 34]. We find that almost all models (255) allow for mixing betweenthe hypercharge and one or more additional U(1) symmetries. The stringy contribution to χ in these constructions is approximately0 . . | b iY X | . ⇒ − . | ∆ Y X | . − , (3.7)where, as before, moduli are assumed to be order one. (See appendix B, for the detailsof some sample models of this kind.) This result contrasts with the one obtained in semi-realistic orbifolds in the fermionic formulation, where ∆ Y X was found to vanish [29].It is more challenging to find possibly realistic vacua in these scenarios. In particular,only a small fraction (11 out of 193) of the models found in [34] allow for supersymmetricvacua satisfying all the following constraints: i) both the hypercharge and an extra U(1) X remain massless, ii) the extra U(1) X is ‘hidden’, i.e. all SM-particles have no Q X charge, iii) all exotic particles are decoupled at a scale M d , close to the compactification scale, iv) at scales lower than M d there exist(s) some massless SM-singlet(s) with Q X = 0, E.g. p, q = N for the subgroup Γ ( N ). – 9 –hich can trigger the spontaneous breakdown of U(1) X at an intermediate scale.The subset of models leading to these properties share an additional feature: the unbrokengauge group after the action of the shift V contains an E factor, which is then broken downto the the SM gauge group times an extra gauge sector by the Wilson lines. This meansthat these scenarios are favoured in models with E local GUTs, in the jargon of [34]. Let us now inspect the details of one example. Consider the model defined by the shiftvector V = ( − , − , , , , , , , , , , , , ,
0) (3.8)and the Wilson lines A = (0 , , , − , − , , , , − , − , − , − , − , − , , (3.9a) A = ( − , , , − , , , , , , − , − , − , − , − , , (3.9b)satisfying eqs. (2.7). By itself, the shift V is known as the Z –II standard (gauge) embed-ding and leads to the breaking E × E → E × U(1) × E . After including both Wilson lines,the unbroken gauge group is SU(3) C × SU(2) L × U(1) Y × [SU(8) × U(1) X × U(1) anom × U(1) ].The 4D N = 1 matter spectrum is shown in Table 1. The modular group after compacti-fication is SL (2 , Z ) × Γ (3) T × Γ (2) T × Γ (2) U .The only relevant (as we shall shortly see) U(1) generators are given in the Cartanbasis of E × E by t Y = (0 , , , − / , − / , / , / , / , , , , , , , , (3.10a) t X = √ (0 , , , , , , , , − , − , − , − , , , , (3.10b)where we have taken the phenomenologically favoured normalisation for the hypercharge | t Y | = 5 /
6. Since tr( Q Y Q X ) = 4 √ = 0, there is a non-vanishing one-loop string contri-bution to the mixing between U(1) Y and U(1) X (see appendix A for further details):∆ Y X = 116 π √
23 log (cid:18) | η (3 T ) | Im( T ) (cid:19) , (3.11)with b Y X = 8 √ b Y X = 0. In this case, ∆
Y X is about 1 /
40 assuming that the moduluscan be stabilised at h T i ∼ { e s i } = { s , s , s , s , s , s , s , s , s , s , s , s , s , s , s } (3.12)develop non-zero VEVs, while the expectation values of all other fields vanish. The exis-tence of the holomorphic monomial ψ = s s s s ( s s s s s s s s ) ( s s s ) (3.13)– 10 – Irrep Label , ; ) (1 / , q i , ; ) ( − / , ¯ q i
45 ( , ; ) (0 , s i
13 ( , ; ) ( − / , ℓ i
10 ( , ; ) (1 / , ¯ ℓ i , ; ) (0 , √ / ξ + i , ; ) ( − / , ¯ u i , ; ) (2 / , u i , ; ) (0 , − √ / ξ − i , ; ) (1 , ¯ e i , ; ) ( − , e i , ; ) (0 , − / √ ¯ h i
10 ( , ; ) (1 / , ¯ d i , ; ) ( − / , d i , ; ) (0 , / √ h i , ; ) (1 / , √ / s ++ i , ; ) ( − / , −√ / s −− i , ; ) ( − / , √ / s − + i , ; ) (1 / , −√ / s + − i , ; ) (1 / , / √ σ + i , ; ) ( − / , − / √ ¯ σ − i , ; ) (1 / , / √ w + i , ; ) ( − / , − / √ ¯ w − i , ; ) (1 / , − / √ w − i , ; ) ( − / , / √ ¯ w + i , ; ) (0 , √ / m + i , ; ) (0 , −√ / m − i Table 1: Massless spectrum. Representations with respect to [SU(3) C × SU(2) L ] × [SU(8)]are given in bold face, the hypercharge and the U(1) X charge are indicated as subscript.ensures the cancellation of the Fayet–Iliopoulos term (2.6) and, thus, N = 1 supersymmetrybelow the compactification scale.Our choice of the vacuum (3.12) has further consequences. First, all vector–like exoticsattain large masses and decouple at M d ∼ . s i , and all h i , ¯ h i , ξ ± i but one pair of ( h i , ¯ h j ) and ( ξ + i , ξ − j ) acquire masses. Secondly,the gauge group is spontaneously broken down to G SM × [SU(8) × U(1) X ] hidden , (3.14)where SU(8) × U(1) X is hidden in the sense that no SM-field is charged under this group.Note that the only surviving Abelian symmetries correspond to the generators givenin (3.10). Therefore, the vacuum chosen contains only the spectrum of the MSSM plusthe two pairs of multiplets ( h i , ¯ h j ) and ( ξ + i , ξ − j ), both of which are charged under U(1) X .Let us call these SM singlets ( h + , h − ) and ( ξ + , ξ − ) respectively.It follows that b Y X = 0 below M d and, therefore, (1.4) becomes χ Y X g Y g X = 4 √ π log M S M d + ∆ Y X (3.15)and does not run. Consequently this has nearly the correct ingredients for an interestinghidden sector kinetically mixing with the hypercharge: the kinetic mixing is present and wehave some hidden vector-like matter. Such matter can cause higgsing of the hidden gaugegroup and may be interesting for dark-matter phenomenology or laboratory experimentsat the low energy, high intensity frontier, as emphasized in the introduction.
4. Phenomenology of hidden photons from heterotic orbifolds
Here we discuss the different hidden
U(1) phenomenology and the predictions from (orimplications for a discovery on) heterotic orbifolds.– 11 –he low energy limits of heterotic orbifolds provide consistent, complete and calculablerealisations of supersymmetric field theories. They therefore provide meaningful restric-tions on the phenomenology that we can obtain. Specifically with regard to supplementaryU(1) symmetries, in addition to the limit of the maximum total rank of all gauge groups,there are many further constraints. The most important features for phenomenology arewhether there is hidden matter (such as in the example above) and, if so, what its couplingsare; and whether the gauge boson has a mass. To obtain massive non-anomalous U(1)sin heterotic orbifolds we require a spontaneous breaking mechanism. This is because thetheory, in contrast to D-brane models, lacks the axions to give them St¨uckelberg masses.Moreover, only one anomalous U(1) is allowed, so there is no possibility of using a fermioncondensate to give masses to any others.In what follows, we shall assume that the hidden sectors play no role in supersymmetrybreaking, and we shall assume that moduli stabilisation and the integrating out of massivesinglets has taken place. Thus we shall treat the resulting theory as a softly broken globallysupersymmetric model, possibly with supersymmetry-breaking masses of similar order tothose in the visible sector (in the standard case of gravity mediation) or much smallermasses (for gauge-mediated scenarios, see for example [61]).
U(1) s A massless hidden U(1) can be interesting phenomenologically in a supersymmetric theorydue to its gaugino. The key issue in this case is that in heterotic orbifolds the order ofmagnitude of the hidden gauge coupling and the kinetic mixing (if present) are fixed, tobeing within roughly an order of magnitude or so of the standard model couplings and10 − respectively. As discussed in [30, 62], if the hidden gaugino is the LSP then it will beoverproduced, as the mixing cannot be reduced sufficiently to avoid this. This fate can beavoided in, for example, models with gauge mediation [61] where the hidden gaugino candecay to a gravitino. In that case, there could be signals due to displaced vertices at theLHC [63].Alternatively, in (the much more standard case of) models with gravity mediation, thedifficulty could potentially be avoided by allowing for hidden matter. However, this wouldthen possess millicharges under the hypercharge, and for mixing of order 10 − is constrainedto have masses above about 100 MeV [64]. Since the theory is supersymmetric, there wouldnecessarily be hidden fermions, and so there would need to be a hidden supersymmetricmass for these (either an explicit hidden µ -term or a form of hidden Higgs mechanismthat does not break the U(1)). Once this is allowed for, however, we could hope to detectthe hidden gauginos in collider experiments as above. However, the scenario would not beinteresting for dark matter experiments since we cannot substantially reduce the hiddengauge coupling; the self-interactions of a particle charged under the hidden U(1) would betoo strong, violating observations about the clustering of dark matter [65] (and a particlenot charged under the hidden U(1) would interact too weakly with the visible sector to bedetected). So then we should simply ensure that the relic density from the hidden sectoris small. – 12 – hidden Dirac fermion ψ with mass m D and hidden gauge coupling g h is thermalisedprovided that the rate of production is greater than the Hubble constant at some point.Assuming that the temperature is at some time above the mass of the hidden fermion, weobtain (roughly) 1 < Γ H ∼ ( g Y g h χ ) (cid:18) M P m D (cid:19) , (4.1)which implies that most such hidden sectors at experimentally accessible energies in het-erotic orbifolds are thermalised, since the gauge coupling and kinetic mixing (if presentand supersymmetric) cannot be substantially reduced in magnitude. For such a thermal species, the relic density is approximately given byΩ ψ h . ≈ − (cid:18) . g h (cid:19) (cid:16) m D GeV (cid:17) ≪ , (4.2)implying that the hidden matter cannot be too heavy. Including the constraints on mil-licharges and allowing for a hidden gauge coupling as large as 1, we then constrain thehidden matter to roughly lie in the range100 MeV < m D . GeV , (4.3)although the upper bound could be avoided if the reheating temperature is low. U(1) masses through supersymmetric breaking
Heterotic orbifold models typically begin with several U(1) factors that are broken super-symmetrically by the VEVs of standard-model singlets. These VEVs are induced by theeffective Fayet-Iliopoulos term corresponding to the one anomalous U(1), and their exactvalues depend on the details of moduli stabilisation. These expectation values are expectedto be of the order of 0 . M S , and the U(1) groups directly broken in this way will thus havevery large masses - so the number surviving at low energies is typically small. However,we could in principle obtain supersymmetric breaking of a hidden U(1) with a small mass:a prototypical example of such breaking would be a hidden-sector theory with three fields S h , H + and H − which have charges 0 , − W ⊃ λS h ( H + H − − µ ) . (4.4)This theory spontaneously generates a vacuum expectation value for H ± , giving a hiddenphoton mass 2 g h µ , and together with the D -term potential gives masses to all of thescalars and fermions. If λ ∼ g h then these are all of order the hidden photon mass. Ofcourse, we do not expect to obtain a tadpole term directly in the orbifold: µ should beregarded as effective, either arising from a term of the form M S s n S h , where s is somefield that obtains a string-scale VEV and n is suitably high ( ∼
32 for 10 GeV hidden Note that, even if the above bound is not met the hidden sector could become thermalised throughdecays of moduli etc, although this is somewhat model dependent. – 13 –auge bosons!) or more realistically arising from the effect of strong gauge dynamics; sincethe hidden sector of heterotic orbifolds typically includes a non-Abelian group, this can beused to induce supersymmetric breaking as above if there is light hidden matter. This does,however, potentially preclude the use of the non-Abelian group for moduli stabilisation - inorder to effect gauge symmetry breaking we assume that the moduli are stabilised already(there could be more than one non-Abelian hidden group, for example, with only one ofthem charged under some hidden light matter).
With the above assumption, an example of strong gauge dynamics breaking a hidden U(1)which could appear from a heterotic orbifold would be to have a hidden sector consistingof four fields ψ, ˜ ψ, φ, ˜ φ which are charged under SU( N ) × U(1). ψ, φ are fundamentalsof SU( N ) and have charges 0 , +1 respectively under the U(1); ˜ ψ, ˜ φ are antifundamentalsunder the SU( N ) and have charges 0 , − W pert = λM S ( φ ˜ ψ )( ψ ˜ φ ) , (4.5)where we have used brackets to show how the SU( N ) indices should be contracted. Belowthe strong coupling scale of the SU( N ), the fields condense into the (matrix of) mesons M ≡ ( φ, ψ ) ⊗ ( ˜ φ, ˜ ψ ) T . For SU( N ) with N f flavours the meson transforms under the adjointof SU( N f ) and we have the classic ADS 1PI superpotential (see e.g. [66] for a good review): W = ( N − N f ) (cid:18) Λ N − N f det M (cid:19) N − Nf + W pert , (4.6)where W pert is the perturbative potential, given in terms of the gauge invariants M . In thepresent example, we conveniently define U ≡ ( φ ˜ φ ) , V ≡ ( ψ ˜ ψ ) , H + ≡ ( φ ˜ ψ ) , H − ≡ ( ψ ˜ φ ) sothat det M = U V − H + H − , arriving at W = ( N − (cid:18) Λ N − U V − H + H − (cid:19) N − + λM S H + H − . (4.7)The D-term equations enforce H + = e iθ H − for some phase θ ; choosing the VEV of H + to be real fixes θ and we solve the F-term equations for the above to give (writing M S ≡ z Λ , z ≫ h H + i = Λ (cid:16) zλ (cid:17) N − N − . (4.8)Since now H + has dimension two, the canonical field is found by dividing by some scale oforder of the condensation scale. This produces the hidden-photon mass m γ ′ ∼ g h Λ (cid:16) zλ (cid:17) N − N − ∼ g h M S z − N N − λ − N − N − . (4.9)– 14 –ere g h is the coupling of the hidden U(1), and we shall write g N for the SU( N ) gaugecoupling with α N ≡ g N / (4 π ). The exponent of z will thus vary between − − /
2, sothe scale of breaking is set by the condensation scale, given by (using b ≡ N − N f )Λ = M D exp (cid:18) − π b g N ( M D ) (cid:19) z = M S M D exp (cid:18) πb α N ( M D ) (cid:19) , (4.10)where M D is the scale of the lightest heavy particle charged under the SU( N ) that isintegrated out (as mentioned previously, these are generically present in heterotic orbifolds).Interestingly, however, the hidden U(1) coupling will become weaker due to the extramatter; we have 1 g h (Λ) = 1 g h ( M D ) − N π log Λ /M D . (4.11)For some sample values, let us suppose that there is a large amount of matter above M D = 10 GeV so that g N ( M D ) is small. Taking N = 3 and α − N ( M D ) = 56 we find ahidden gauge boson mass of 1 GeV . Of course, such a value for the non-Abelian couplingis rather weak; however, this problem could be avoided by, for example, introducing anextra flavour that has an intermediate scale mass. In this way, the coupling would “walk”down from M D to this new scale, before becoming stronger - reducing the condensationscale. Alternatively, it would be interesting to consider an ISS-like model (by adding morelight flavours) which would potentially also break supersymmetry. However, the model ismodified, though, by invoking strong dynamics to break the Abelian gauge symmetry. Wewill find that the gauge boson mass depends exponentially on the scales and couplings,allowing a wide range of values and phenomenology. Most Dark Force models constructed in the literature break the hidden U(1) in the lowenergy theory after supersymmetry is broken. This can be induced by the electroweaksymmetry breaking in the visible sector, where the Higgs expectation values give a D-term to the hypercharge, and this is communicated to the hidden sector via the kineticmixing [11]. In the context of gauge mediation there are many possibilities for models, sincethe soft masses in the hidden sector can be naturally small, but in the context of gravitymediation this would require the hidden sector to be sequestered. Hence for heteroticorbifolds, as investigated in [13], an ideal scenario would involve radiative breaking ofthe hidden gauge group, as could be achieved in a simple hidden sector with three fields
S, H + , H − and superpotential W ⊃ λ S S H + H − . (4.12) We also take λ ∼ g h ∼ . This was first proposed as a Dark Force model in the context of gauge mediation in [4] and explored inmore detail in [11]. – 15 –his has the advantage of having no scales, and that we actually require λ S to be notsuppressed. Unfortunately we were so far not able to find examples in this class, and weleave this as a challenge for future work.The simplest supersymmetric Dark Force model would have superpotential W ⊃ µ ( ξ + ξ − ) , (4.13)where we require µ .
10 GeV and there to be a hidden gaugino mass (ideally of similarmagnitude). The kinetic mixing and hypercharge D-term would then break the hiddengauge group spontaneously [11].
The model of section 3.2.1 can in principle exactly realise a Dark Force scenario. This isbecause it contains a perturbative superpotential with terms W ⊃ s M S ( ξ + ξ − )( h + h − ) + s M S ( h + h − ) , (4.14)where s ∼ O (0 .
1) is the VEV of a G SM × U(1) X singlet expressed in string units, and h ± are SM singlets charged under both U(1) X and a hidden non-Abelian group. There arealso additional heavy fields h i , ¯ h i , ξ + i , ξ − i which will slow the running of the non-Abeliangroup above M D , and (if some of the h i , ¯ h i obtain VEVs) can break it down to SU(5). Sowe shall analyse an SU( N ) sector with superpotential W ⊃ s n M S ( ξ + ξ − )( h + h − ) + s m M S ( h + h − ) . (4.15)As in subsection 4.2.1, at the scale Λ there is a dynamically generated superpotential givenin terms of the gauge invariant M ≡ h + h − : W ⊃ ( N − (cid:18) Λ N − M (cid:19) / ( N − + s m M S M + s n M S ( ξ + ξ − ) M. (4.16) M then obtains a VEV (writing M S = z Λ, as before) of size M = Λ ( s m z ) − N − N . (4.17)We find thus an effective superpotential of the form W = µ ( ξ + ξ − ), with µ = s n Λ M S ( s m z ) − N − N = M S s n − m ( N − N )0 z − N − N . (4.18)Taking the model of section 3.2.1 (i.e. m = 8 , n = 5) we find that, in order to obtain a µ of 10 GeV, we need z ∼ , corresponding to a condensation scale of 10 GeV. If, asmentioned above, we break the gauge group down to SU(5) at a scale M D ∼ GeVthen we can take α − N ( M D ) = 24 to realise a Dark Force model.– 16 – . Discussion Hidden forces might play an important role in nature, most notably in the context of darkmatter. To probe the existence of Abelian hidden forces, the magnitude of their kineticmixing with the hypercharge in the proposed models must be contrasted to currentlyobserved bounds. On the other hand, string theory offers a remarkable playground forthese new interactions, since additional U(1)s generically appear in all kinds of stringcompactifications. Unfortunately, kinetic mixing has been studied only in some particularscenarios and in some cases only vanishing mixing has been found, which renders the newforces undetectable, thus irrelevant for nature.In this paper, we aimed at improving this situation. We have studied the kineticmixing between U(1)s in a special class of string constructions: heterotic orbifold models.We have noticed that the computation of kinetic mixing contribution ∆ ab here is akin tothe computation of threshold corrections. In particular, the result mostly depends on themassless modes of the N = 2 subsectors of the orbifold. This constrains the candidatesthat exhibit mixing to Z N with non-prime N and Z N × Z M orbifold models. Using theresulting expression for kinetic mixing we have been able to provide a simple formula thathelps one to recognize promising candidates with non-vanishing mixing. We also foundthat the resulting size of kinetic mixing depends on the size of the compact space, whichwe assumed to have been stabilised.As an application of our previous results, we have explored explicit Z –II orbifoldmodels with and without Wilson lines. In the former case, despite not having realistic gaugesymmetries, we find 10 out of a total of 61 orbifold models with Abelian kinetic mixing inthe interval 10 − . ∆ ab . − , values which are consistent with previous expectations [29].It is interesting to note that, although this result resembles the one obtained previouslyin the context of type II strings with D3-branes in orbifold backgrounds [27], the modelsstudied here are globally consistent constructions. This can also be contrasted to the toyexample with intersecting D6-branes in [23].The Z –II orbifold models with (2 and 3) Wilson lines that we have investigated arethose of the Mini-Landscape, which possess many properties of the (N)MSSM. We foundthat 255 (out of 267) models have non-trivial mixing between the hypercharge and at leastone additional U(1) X . The size of the stringy contribution to the kinetic mixing lies inthe range 10 − − − . This is in contrast to previous results [29], where semi-realisticorbifold models had been studied and no model with kinetic mixing was found. However,demanding the new Abelian symmetry to be truly hidden and to mix with the hyperchargein supersymmetric vacua turned out to be more challenging: only 11 models survive thesenew demands.We have also worked out explicitly the details of a sample model. We provided amodel with a supersymmetric vacuum, observable gauge group SU(3) C × SU(2) L × U(1) Y and hidden U(1) X , and the matter spectrum of the MSSM plus a couple of vectorlike SMsinglets that mediate interactions between both sectors. The mixing of the hyperchargeand the hidden U(1) occurs only in one N = 2 sector, rendering the result ∆ Y X ∼ stable against the running of the couplings.– 17 –inally, we have discussed some phenomenologically appealing possibilities that ourmodels allow. In particular, with the model that we have used as an example, we showedthat a Dark Force scenario arises due to the existence of a strongly interacting sector andadequate hidden matter couplings. This situation is generic, even though the precise detailsof the form of the couplings and the condensation scale depend on the particulars of eachmodel.Our study can be extended in several ways. First, we have focused on Z –II orbifoldsand ignored other promising models, such as Z –II and Z × Z , which have shown to be(at least) as promising as Z –II [67, 68]. In a fast test, we could verify that there areplenty of models similar to those of the Mini-Landscape with kinetic mixing. It wouldbe worth studying the details; in particular the phenomenology of the hidden sectors andhidden matter could be very interesting. Secondly, in this work we have concentrated ontruly hidden U(1)s, although Z ′ symmetries appear more frequently. A careful study ofthe kinetic mixing and other properties of these symmetries will be carried out elsewhere.It would also be interesting to perform our analysis avoiding the assumption of modulistabilisation, i.e. in heterotic orbifold models with stabilised moduli. Acknowledgements
We thank O. Lebedev and J. Erler for helpful discussions. M.G. was supported by SFBgrant 676 and ERC advanced grant 226371. S. R.-S. was partially supported by CONACyTproject 82291 and DGAPA project IA101811.
A. Further details on a promising model
We provide here the data of the model presented in section 3.2.1, which allows one tocompute the magnitude of kinetic mixing. Z N = 2 theory. The unbroken gauge group is SU(4) × SU(3) × U(1) × [SU(8) × U(1)],where we have used squared parenthesis in the subgroups arising from the second E of theoriginal heterotic string. The N = 2 U(1) generators are given by t ′ = √ ( − , , − , , , , , , , , , , , , , (A.1) t ′ = √ ( − , , , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , − , − , − , − , , , − . Listing complete hypermultiplets, the N = 2 matter spectrum contains the followinggauge representations , , ) ( − √ , √ , − √ , , , ) ( q , √ , √ , , , ) ( − q , √ , √ , , , ) (0 , , √ , − √ ) , , ) ( q , √ , , – 18 –onsequently, the β function coefficients are given by b N =2SU(4) = b N =2SU(3) = 4, b N =2SU(8) = − b N =2U(1) = √ q √ q q q
25 532 −
80 0 − . (A.2)In the notation of eq. (2.12), the overlap of the hypercharge and U(1) X is given by n Y = ( √ , √ , − √ , , m SU(4) Y = ( , , ) , m SU(3) Y = ( − , − ) , m SU(8) Y = 0 (A.3a) n X = (0 , , , , m b ′ X = 0 for all b ′ (A.3b)Thus, according to eq. (2.14), b Y X = 8 √ Z N = 2 theory. The unbroken gauge group is SU(4) × SU(2) a × SU(2) b × U(1) × [SU(8) × U(1)], where, as before, the parethesis refer to the subgroups arising from thesecond E of the original heterotic string. The N = 2 U(1) generators are t ′ = √ (11 , − , , , , , , , , , , , , , , (A.4) t ′ = √ (0 , , , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , − , − , − , − , , , − .t ′ = √ (0 , , − , , , , , , , , , , , , , Listing complete hypermultiplets, the N = 2 matter spectrum contains the followinggauge representations , , , ) ( √ , √ , , √ ) , , , ) (0 , , − √ , q ) , , , ) ( √ , q , , q ) , , , ) ( √ , − q , − √ , − √ ) , , , ) ( √ , − √ , , √ ) , , , ) ( − √ , − q , , q ) , , , ) ( − √ , q , , q ) , , , ) ( − √ , √ , , − √ ) , , , ) ( − √ , − √ , , − √ ) , , , ) ( − √ , − √ , , − √ ) , , , ) ( − √ , q , , − q ) , , , ) ( − √ , − √ , − √ , − √ ) , , , ) ( √ , − q , √ , q ) , , , ) ( − √ , q , √ , − √ ) , , , ) ( √ , √ , √ , √ ) Consequently, the β function coefficients are given by b N =2SU(4) = 12, b N =2SU(2) a = b N =2SU(2) b =16, b N =2SU(8) = − b N =2U(1) = − q − q
211 133 q − q
23 217199 283 √ − √ − q
211 283 √
33 103 43 √ q − √
11 43 √ . (A.5)In the notation of eq. (2.12), the overlap of the hypercharge and U(1) X is given by m SU(4) Y = ( , , ) , m SU(2) b Y = − , n Y = 0 , m b ′ Y = 0 for other b ′ , (A.6a) n X = (0 , , , , m b ′ X = 0 for all b ′ . (A.6b)Thus, according to eq. (2.14), b Y X = 0. – 19 – . Other promising models with kinetic mixing
The models presented here exhibit kinetic mixing. However, they do not satisfy all thequalities we demand in section 3.2.
B.1 Example 1. Mixing in the observable E Another interesting example of U(1) kinetic mixing in heterotic orbifolds arises from con-sidering the Z –II gauge embedding: V = (2 , − , − , , , , , , − , − , − , − , − , − , , (B.1a) A = (1 , , , , − , − , − , , − , − , − , − , − , , , (B.1b) A = (3 , − , − , − , − , − , − , , − , − , − , − , − , − , . (B.1c)The interesting U(1) generators are given in the Cartan basis of E × E by t Y = (0 , , , − / , − / , / , / , / , , , , , , , , (B.2a) t X = (0 , , , , , , , , , , , , , , . (B.2b)The matter spectrum can be obtained from [69] or by using the program orbifolder [70,71].Using the data below, the resulting kinetic mixing is∆ Y X = − π log (cid:18) | η (2 T ) η ( U / | Im( T )Im( U ) (cid:19) . (B.3) Z N = 2 theory. Using the notation of the previous appendix, the unbroken gaugegroup is SU(5) × U(1) × [SU(6) × SU(2) a × SU(2) b × U(1)], and the N = 2 U(1) generatorsare given by t ′ = √ (7 , , , , , − , − , − , , , , , , , , (B.4) t ′ = √ (0 , , − , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , , ,t ′ = (0 , , , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , , . The N = 2 matter spectrum contains the following hypermultiplet gauge representa-tions , , , ) ( √ , − √ , √ , − , , , , ) (0 , , , , , , , ) (0 , , , , q ) , , , ) ( q , − √ , √ , − , , , , ) ( q , − √ , − √ , − , , , , ) (0 , − √ , , , − √ ) , , , ) (0 , , − √ , , − √ ) – 20 –onsequently, the β function coefficients are given by b N =2SU(5) = − , b N =2SU(6) = b N =2SU(2) a =16, b N =2SU(2) b = − b N =2U(1) = − √ − q − √
107 927 q − √ √ − q q
27 312 −√ √ − √ −√ −√ √
42 4 √ −√ . (B.5)In the notation of eq. (2.12), the overlap of the hypercharge and U(1) X is given by n Y = ( − q , √ , −√ , , , m SU(5) Y = ( , , , ) , m b ′ Y = 0 for other b ′ (B.6a) n X = ( √ , √ , , , , m SU(5) X = ( − , − , − , − ) , m b ′ X = 0 for other b ′ (B.6b)Thus, according to eq. (2.14), b Y X = 0. Z N = 2 theory. The unbroken gauge group is SU(3) a × SU(3) b × U(1) × [SU(6) × U(1) ], and the N = 2 U(1) generators are t ′ = (1 , , , , , , , , , , , , , , , (B.7) t ′ = (0 , , , , , , , , , , , , , , ,t ′ = (0 , , , , , , , , , , , , , , ,t ′ = (0 , , , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , , , The N = 2 matter spectrum contains the following hypermultiplet gauge representa-tions , , ) (0 , , − , − , , , , , ) (0 , , − , , , , , , ) (0 , , , , , , − q ) , , ) (0 , , , , , , q ) , , ) (0 , , , − , , , , , ) ( , − , , , − √ , √ , , , ) ( , − , , , √ , − √ , , , ) ( , , , , − √ , − √ , , , ) ( , , , , , , , , ) ( − , , , , , − √ , , , ) ( , , , − , , , √ ) , , ) ( − , , , , , √ , , , ) ( , , , − , , , − √ ) , , ) ( , , − , − , , , , , ) ( , , , − , , , , , ) ( − , , − , , √ , √ , − √ ) , , ) ( − , , − , , , − √ , − √ ) , , ) ( , , − , , √ , − √ , − √ ) , , ) ( , − , − , , − √ , − √ , √ ) , , ) ( − , , − , , , √ , √ ) , , ) ( , − , , − , − √ , − √ , − √ ) , , ) ( , − , , , − √ , − √ , − q ) , , ) ( , − , , − , − √ , − √ , √ ) , , ) ( , − , , , − √ , − √ , , , ) ( , , − , , √ , − √ , √ ) – 21 –onsequently, the β function coefficients are given by b N =2SU(3) a = b N =2SU(3) b = b N =2SU(6) = 6and b N =2U(1) = − − −√ −√ − −
92 32 √ √ q − − − √ − − √ −√ √ − √ √ √ −√ √ √ q √ √ . (B.8)In the notation of eq. (2.12), the overlap of the hypercharge and U(1) X is given by n Y = (0 , , , , − √ , √ , , m SU(3) b Y = ( , ) , n Y = 0 , m b ′ Y = 0 for other b ′ , (B.9a) n X = (0 , , , , , , , m b ′ X = 0 for all b ′ . (B.9b)Thus, according to eq. (2.14), b Y X = − B.2 Example 2. Mixing in models with 3 Wilson lines
Let us consider now the gauge embedding with 3 wilson lines: V = ( − , − , , − , − , − , , , , , , , , , , (B.10a) A = (0 , , , − , − , − , , − , , , − , − , − , , , (B.10b) A ′ = ( − , − , , − , − , , , , , , − , , − , − , , (B.10c) A = ( − , , , − , − , − , − , − , , , , , , − , . (B.10d)The interesting U(1) generators are given in the Cartan basis of E × E by t Y = (0 , , , − / , − / , / , / , / , , , , , , , , (B.11a) t X = √ (0 , , − , , , , , , , , , , , , . (B.11b)The matter spectrum can be obtained from [69] or by using the program orbifolder [70,71].Using the data below, the resulting kinetic mixing is∆ Y X = − π r
32 log (cid:18) | η (3 T ) | Im( T ) (cid:19) . (B.12) Z N = 2 theory. The unbroken gauge group is SU(5) × U(1) × [SU(2) × SU(5) h × U(1) ],and the N = 2 U(1) generators are t ′ = (0 , , , , , , , , , , , , , , , (B.13) t ′ = √ ( − , , − , − , − , − , , , , , , , , , ,t ′ = √ ( − , , , − , − , − , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , , ,t ′ = ( − , , , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , − , . – 22 –he gauge representations of the N = 2 matter spectrum are: , , ) (0 , − √ , √ , √ , , , , , ) (0 , − q , √ , − √ , , , , , ) ( − , q , − √ , − √ , , , , , ) (0 , q , − √ , − √ , − √ , , , , ) ( − , q , √ , √ , √ , , √ ) , , ) ( − , − q , √ , − √ , √ , − , √ ) , , ) ( , q , √ , − √ , − √ , − , − √ ) , , ) ( − , q , √ , √ , − √ , − , − √ ) , , ) ( − , q , √ , √ , √ , − , √ ) , , ) (0 , √ , √ , √ , , , , , ) (0 , √ , √ , − √ , , , Hence, the β function coefficients are given by b N =2SU(5) = b N =2SU(2) = 0 , b N =2SU(5) h = − b N =2U(1) = − q − √ − √ − √ − − √ − q
514 16514 5728 q − q
57 154 q − √ − √ q
52 53756 114 q
72 414 √ − √ q − √ q
72 172 94 − √ q − √ − q
57 414 √
14 94 174 − √ q −
38 154 q − √ − √ − √ √ − √ − √
14 138 q
57 14 q
52 54 q
52 3 √
58 218 . (B.14)In the notation of eq. (2.12), the overlap of the hypercharge and U(1) X is given by n Y = (0 , √ , √ , √ , , − , , m SU(5) Y = ( − , − , − , − ) , m b ′ Y = 0 for other b ′ (B.15a) n X = ( √ , √ , , , , m SU(5) X = ( − , − , − , − ) , m b ′ X = 0 for other b ′ (B.15b)Thus, according to eq. (2.14), b Y X = − q . Z N = 2 theory. The unbroken gauge group is SU(3) a × SU(3) b × U(1) × [SU(2) × SU(5) × U(1) ], and the N = 2 U(1) generators are t ′ = (1 , , , , , , , , , , , , , , , (B.16) t ′ = (0 , , , , , , , , , , , , , , ,t ′ = (0 , , , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , , ,t ′ = √ (0 , , − , , , , , , , , , , , , ,t ′ = √ (0 , , , , , , , , , , , , , − , , The gauge representations of the N = 2 hypermultiplets are– 23 – Irrep U(1) charges , , , ) ( , , , , − √ , √ , , , , ) ( , , , , √ , − √ , , , , ) ( , , , , , , , , , ) ( , , , , − √ , √ , , , , ) ( , , , , √ , − √ , , , , ) ( , , , , √ , √ , , , , ) ( , , , , − √ , − √ , , , , ) ( − , , , − √ , √ , √ , , , , ) ( − , − , − , √ , − √ , − √ , , , , ) ( , , − , √ , − √ , √ , , , , ) ( − , − , , − √ , √ , √ , , , , ) ( , − , − , √ , − √ , − √ , , , , ) ( , , , − √ , − √ , √ , Consequently, the β function coefficients are given by b N =2SU(3) a = b N =2SU(3) b = 12 , b N =2SU(2) =2 , b N =2SU(5) = −
10 and b N =2U(1) = − √ − √ − √
03 6 1 − √ √ √ − − √ √ √ √ − √ − √ − q − q − √ √ √ − q − √ √ √ − q . (B.17)In the notation of eq. (2.12), the overlap of the hypercharge and U(1) X is given by n a = √ (0 , , , , − , , , m SU(3) b a = ( , ) , n a = 0 , m b ′ a = 0 for other b ′ , (B.18a) n b = (0 , , , , √ , √ , , m b ′ b = 0 for all b ′ . (B.18b)Thus, according to eq. (2.14), b Y X = 0.
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