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Gauge Coupling Running in Minimal

arXiv:hep-th/9109053v1 27 Sep 1991CERN−TH.6241/91Gauge Coupling Running in MinimalSU(3) × SU(2) × U(1) Superstring UnificationLuis E. Ib´a˜nezDieter L¨ustCERN, 1211 Geneva 23, SwitzerlandandGraham G. RossDep. Theoretical Physics, 1 Keble Rd., Oxford, EnglandABSTRACTWe study the evolution of the gauge coupling constants in string unificationschemes in which the light spectrum below the compactification scale is exactlythat of the minimal supersymmetric standard model.

In the absence of stringthreshold corrections the predicted values sin2 θW = 0.218 and αs = 0.20 are ingross conflict with experiment, but these corrections are generically important.One can express the string threshold corrections to sin2 θW and αs in terms ofcertain modular weights of quark, lepton and Higgs superfields as well as themoduli of the string model. We find that in order to get agreement with theexperimental measurements within the context of this minimal scheme, certainconstraints on the modular weights of the quark, lepton and Higgs superfieldsshould be obeyed.

Our analysis indicates that this minimal string unificationscheme is a rather constrained scenario.CERN−TH.6241/91September 1991

−1 −Unification of coupling constants is a necessary phenomenon in string theory.Specifically, at tree level, the gauge couplings of a gauge group Ga have simplerelations [1] to the string coupling constant which is determined by the vacuumexpectation value of the dilaton field:1g2a =kag2string where ka is the level of thecorresponding Kac-Moody algebra. At higher loop levels this relation holds onlyat the typical string scale which is of the order of the Planck mass MP.

Below thisscale all couplings evolve according to their renormalization group equations inthe same way as in standard GUT theories as first discussed in [2]. This allowsa comparison of the coupling constants with the low energy data consideringa specific string model.

In addition, thresholds effects due the massive stringexcitations modify the above mentioned tree level relations.Let us recall briefly the exact definition of the string mass scale. It is givenin the MS scheme by [3]M 2string = 2 e(1−γ)√27πα′(1)where γ is the Euler constant and α′ = 16π/g2stringM 2P .

Numerically one finds [4]Mstring = 0.7 × gstring × 1018 GeV. (2)(Note that this value differs from the one found in [3].) This mass scale has tobe compared with low energy data using the field theory renormalization groupequations and taking into account also the model-dependent stringy thresholdcorrections.

A phenomenological very promising model is the minimal supersym-metric standard model with gauge group G = SU(3)×SU(2)×U(1). The relevantevolution of the electro-weak and strong coupling constants was considered sometime ago in [5], [6].

Recently this analysis was reconsidered [7] taking into ac-count the up-dated low energy data. The results for sin2 θW and αs are in verygood agreement with data for a value of the unification mass MX ≃1016 GeVand a susy threshold close to the weak scale.

On the other hand, as we showbelow, the large value for the string unification scale Mstring leads to rather em-barrassing results for the couplings sin2θ0W = 0.218 and α0s = 0.20. In this paperwe discuss the question whether one can make consistent the unification scale ofthe minimal supersymmetric standard model with the relevant string unification

−2 −scale Mstring. (String unification and threshold effects within the flipped SU(5)model were considered in [8].) At the first sight, this seems very unlikely sinceMstring is substantially larger than the minimal susy model scale MX.

However,one might hope that the effects of the string threshold contributions could makethe separation of these two scales consistent. Although the threshold effects arerather small in usual grand unified models [9] it is not obvious that the sameholds true in string unification since we have to remember that above the stringscale an infinite number of massive states contribute to the threshold.

This isobviously very different from field theory unification scenarios.The structure of the paper is the following. First we will collect some for-mulas about one-loop gauge coupling constants with special emphasis on stringthreshold corrections and their relation to target space duality.

Then we willapply these formulas to the case of the unification of the three physical couplingconstants g1, g2, g3. Our approach here will be mainly phenomenological.

We willconsider a possible situation in whicha) the massless particles with standard model gauge couplings are just those ofthe minimal supersymmetric standard modelb) there is no partial (field theoretical) unification scheme below the string scale.This is in principle the simplest string unification scheme that one can think ofand that is why we call it minimal string unification. Up to now no realisticstring model with this characteristics has been built but the model search doneup to now is extremely limited and by no means complete.

We would like toanswer the question whether such a minimal scenario can be made consistentwith the measured values of the low energy coupling constants.The one-loop running gauge coupling constant of a (simple) gauge group Gais of the following form:1g2a(µ) =kag2string+ba16π2 log M 2aµ2 + ∆a. (3)Here ba = −3C(Ga) + PRa hRaT(Ra) is the N = 1 β-function coefficient (hRa isthe number of chiral matter fields in a representation Ra).

Ma is the renormaliza-tion point below which the effective field theory running of the coupling constant

−3 −begins. (As we will discuss below, Ma will depend on the specific model andalso on the considered gauge group.) ∆a are the string threshold contributions[3] which arise due to the integration over the infinite number of massive stringstates, in particular momentum and winding states: ∆a ∝log det M, where Mis the mass matrix of the heavy modes.In the following we would like to give a brief description of how one derivesthe expressions for the field-dependent stringy threshold corrections and for therenormalization scale Ma which, in general, is also field dependent.Most di-rectly, these quantities can be obtained by world-sheet string computations ofstring amplitudes involving external gauge fields and moduli as done [10], [11]for the case of (2,2) symmetric orbifold compactifications [12].

These computa-tions are closely related to the calculation [13] of the target space free energiesof compactified strings.A second very useful approach to obtain information about the form of thestring threshold corrections is the use of the target space duality symmetries [14]present in many known string compactifications. Here, the main idea is relatedto the observation [15] that in string compactifications the scale Ma below whichthe effective field theory running of the gauge coupling constants starts becomesa moduli dependent quantity,M 2a = (2R2)αM 2string.

(4)R is a background parameter denoting (in Planck units) the overall radius of thecompact six-dimensional space, and the power α is a model- and gauge groupdependent parameter. (In naive field theory compactifications one expects α =−1.

However, as we will discuss in the following, for orbifold compactificationsα can also take different values. )Thus the running gauge coupling constanteq.

(3) generically depends on the background radius.To be specific considerorbifold type of compactifications. Here the radius is related to the real part ofa complex modulus field, T = R2 + iB, (B is an internal axion field) and thetarget space duality group is given [16] by the modular group PSL(2, Z), actingon T as T →aT −ibicT +d (a, b, c, d ∈Z, ad −bc = 1).

It follows that the effectiveaction involving the T-field must be target space modular invariant and is given

−4 −in terms of modular functions [17]. Now, requiring [18] the invariance of g2a(µ)under target space modular transformations enforces ∆a to be a non-trivial T-dependent function.

Specifically, as discussed in [18], [19] for the case α = −1,target space modular invariance, together with the requirement of having nopoles inside the fundamental region, implies∆a(T, ¯T) = αba16π2 log |η(T)|4,(5)where η(T) is the Dedekind function. Notice that for large T one recovers thelinear behavior found in ref.

[20] .The parameter α is intimately related to the modular weights of the chargedmatter fields which transform non-trivially under the gauge group Ga. To un-derstand this, consider a standard supergravity, Yang Mills field theory [21] withmassless gauge singlet chiral moduli fields Ti and massless⋆charged chiral mat-ter fields φRai(i = 1, . .

., hRa). The relevant part of the tree level supergravityLagrangian is specified by the following K¨ahler potential at lowest order in φRi :K(Ti, φRi ) = K(Ti, ¯Ti) + KRij(Ti, ¯Ti)φRi ¯φRj .

(6)In the following we assume that the K¨ahler metric for the charged fields is pro-portional to the K¨ahler metric of the moduli, which was shown [23] to be truefor (2,2) Calabi-Yau string compactifications [24], i.e. KRij ∝∂Ti∂¯TjK(Ti, ¯Ti).

Asdiscussed in [4], [25], [26], at the one loop level σ-model anomalies play a veryimportant role for the determination of the renormalized gauge coupling con-stant. Specifically, one has to consider two types of triangle diagrams with twogauge bosons and several moduli fields as external legs and massless gauginosand charged (fermionic) matter fields circulating inside the loop: First the cou-pling of the moduli to the charged fields contains a part described by a compositeK¨ahler connection, proportional to K(Ti, ¯Ti), which couples to gauginos as wellas to chiral matter fields φRai .

It reflects the (tree level) invariance of the theoryunder K¨ahler transformations. Second, there is a coupling between the moduli⋆If some of the matter fields become massive due to a trilinear superpotential one ends upwith the same results about the form of the threshold corrections [22].

−5 −and the φRai ’s by the composite curvature (holonomy) connection. It originatesfrom the non-canonical kinetic energy KRij of the matter fields φRi and shows the(tree level) invariance of the theory under general coordinate transformations onthe complex moduli space.

These two anomalous contributions lead, via super-symmetry, to the following one-loop modification of the gauge coupling constant[25],[4],[26]:1g2a=kag2string−116π2(C(Ga) −XRahRaT(Ra))K(Ti, ¯Ti)+ 2XRaT(Ra) log det KRij(Ti, ¯Ti). (7)Now assume that the string theory is invariant under target space dualitytransformations which are discrete reparametrizations of the moduli.

(The simpleR →1/R duality symmetry in bosonic string compactification was shown [27] tobe unbroken in each order of string perturbation.) These transformations do notleave invariant the K¨ahler potential K(Ti, ¯Ti) and also log det Kij.

Thus eq. (7) isnot invariant under duality transformations.

It follows that the duality anomaliesmust be cancelled by adding new terms to the effective action. Specifically, thereare two ways to cancel these anomalies.

First [4],[26], one can perform a modulidependent, but gauge group independent redefinition of the dilaton/axion field,the socalled S-field, such that S + ¯S transforms non-trivially under duality trans-formations and cancels in this way some part or all of the duality non-invarianceof eq.(7). This field redefinition of the S-field is analogous to Green-Schwarzmechanism [28] and leads to a mixing between the moduli and the S-field in theS-field K¨ahler potential.

Second, the duality anomaly can be cancelled by addingto eq. (7) a term which describes the threshold contribution due to the massivestring states.

(Only the specific knowledge about the massive string spectrumcan determine the exact coefficients for the Green-Schwarz and threshold termswhose combined variation cancels the total modular anomaly. However, as it willbecome clear in the following, the coefficient of Green-Schwarz term is irrelevantfor the determination of the unificaton mass scales.) In analogy to the Dedekindfunction the threshold contributions are given in terms of automorphic functionsof the target space duality group.

Specifically, as described in [13], for general

−6 −(2,2) Calabi-Yau compactifications there exist two types of automorphic func-tions: the first one provides a duality invariant completion of K(Ti, ¯Ti), wherethe second one is needed to cancel the duality anomaly coming from log det Kij.These two types of automorphic functions can be, at least formally, constructedfor all (2,2) Calabi-Yau compactifications [13].In the following we restrict ourselves to symmetric (but not necessarily (2,2)symmetric) ZN [12], [29] and ZN × ZM orbifolds [30]. Every orbifold of thistype has three complex planes corresponding to three two-dimensional subtori.For non-trivial examples each orbifold twist δm = (δ1, δ2, δ3) acts either simulta-neously on two or all three planes.

For simplicity we consider only the overallmodulus T = R2 + iB where the target space duality transformations are givenby the modular group PSL(2, Z). The K¨ahler potential for this overall moduluslooks like [31]K(T, ¯T) = −3 log(T + ¯T).

(8)The K¨ahler metric of the matter fields has the following generic form [23]:KRaij = δij(T + ¯T)nRa. (9)Target space modular invariance then implies that the matter fields transformunder PSL(2, Z) asφRai→φRai (icT + d)nRa.

(10)Thus we identify the integers nRa as the modular weights of φRai . (For the Z3orbifold see [32].) Specifically, for symmetric orbifold compactifications there arethree different types of matter fields:a) Untwisted matter with nR = −1.b) Twisted matter fields associated with an orbifold twist δm.

Here nR = −2 ifδm acts on all three planes. nR = −1 if the twist acts only on two of the threeplanes.

Thus the latter kind of twisted fields behave exactly like untwisted fieldsunder modular transformations.c) Twisted moduli with nR = −3 or nR = −2 if the corresponding twist acts onall three planes or only on two planes respectively.

−7 −Then, using eq. (7), the one-loop contribution to the gauge coupling constant dueto the anomalous triangle diagrams with massless charged fields has the followingform [25],[4]:1g2a=kag2string+116π2b′a log(T + ¯T),b′a = 3C(Ga) −XRahRaT(Ra)(3 + 2nRa) = −ba −2XRahRaT(Ra)(1 + nRa).

(11)As discussed already, the modular anomaly of this contribution to1g2a from themassless fields can be cancelled by by a universal Green-Schwarz term plus thethreshold contribution from the massive orbifold excitations. The orbifold thresh-old contribution takes the following form (up to a small T-independent term [3]):∆a(T, ¯T) =116π2(b′a −kabGS) log |η(T)|4.

(12)Here bGS is the universal coefficient of the Green-Schwarz term. Without goinginto any detail let us just state the main result concerning the coefficient b′a −kabGS [10],[11].

The threshold contribution of the massive fields, i.e. b′a −kabGS,is non-vanishing if at least one of the three complex planes is not rotated bysome of the orbifold twist δm.

Then, within this sector, the massive spectrumwith T-dependent masses is N = 2 space-time supersymmetric and b′a −kabGSis proportional to the N = 2 β-function coefficient. In this case b′a −kabGS is ingeneral non-zero for all gauge groups including the unbroken E8 in the hiddensector.

On the other hand, sectors corresponding to planes which are rotatedby all twists δi lead to a massive T-dependent spectrum with N = 4 space-timesupersymmetry and therefore do not contribute to the threshold corrections.Let us insert the threshold contribution eq. (12) into the one-loop runningcoupling constant eq.

(3):1g2a(µ) =kag2string+ba16π2 log M 2aµ2 +116π2(b′a −kabGS) log |η(T)|4,M 2a = (TR)αM 2string,α = b′a −kabGSba,(13)where TR = T + ¯T = 2R2. This expression is explicitly target space modularinvariant.

Here we have absorbed the piece from the massless fields in eq. (11)

−8 −which is not cancelled by the Green-Schwarz term into the definition of the renor-malization point Ma (the remainder is absorbed into1g2string [4]) since it is a fieldtheoretical, infrared effect and does not originate from the heavy string modes.Now we are finally ready to discuss the unification of the gauge coupling con-stants. The unification mass scale MX where two gauge group coupling constantsbecome equal, i.e.1kag2a(MX) =1kbg2b(MX), becomes using eq.

(13)MXMstring= [TR|η(T)|4]b′akb−b′bka2(bakb−bbka). (14)Note that since we are interested only in the difference of two gauge couplings, theuniversal Green-Schwarz term is irrelevant for MX.

Since the moduli-dependentfunction (T + ¯T)|η(T)|4 < 1 for all T it follows that MX/Mstring is smaller (bigger)than one if b′akb−b′bkabakb−bbka is bigger (smaller) than zero. Comparing with the definitionof b′a in eq.

(11) one recognizes that for ka = kb twisted states with nR < −1 arenecessarily required to have MX < Mstring.Let us now briefly discuss three known (2,2) orbifold examples (k = 1). Firstfor the Z3 and Z7 orbifolds, each of the three planes is simultaneously rotatedby all twists.

Thus b′a −bGS = 0 for all gauge groups. It trivially follows thatthe renormalization point is given as Ma = Mstring.

The radius independenceof the renormalization point is due to the fact that the spectrum of the massiveKaluza-Klein and winding states is N = 4 supersymmetric and has thereforeno effect in loop calculations.The absence of threshold corrections (in otherwords, only a universal gauge group independent piece contributes to the gaugecoupling constant at one loop) also trivially implies that the unification scaleMX, for example the unification point of E8 and E6, is given by Mstring.A second example, which is rather orthogonal to the previous case, is thesymmetric Z2 × Z2 orbifold. (Also many four-dimensional heterotic strings ob-tained by the fermionic [33], [34] or by the covariant lattice [35] construction fallinto his category.) Here each of the Z2 twists leaves invariant exactly one of thethree orbifold planes.

Then we obtain that bGS = 0, i.e. there is no one-loopS–T mixing in the K¨ahler potential of this model.

Furthermore, according to ourgeneral rules all twisted matter fields (27 of E6) have modular weight n27 = −1

−9 −and behave like untwisted fields. It follows that b′a = −ba (a = E8, E6).

Thus weobtain α = −1 and the radius dependence of the renormalization point agreeswith the naive field theoretical expectation: Ma = MstringT −1/2R. The unificationscale of E8 and E6 is given by MX = Mstring/(T 1/2R |η(T)|2) and is therefore largerthan the string scale for all values of the radius.Now let us apply the above discussion to the case of the unification of thegauge coupling constants within the minimal string unification.

We will make useof the threshold formulae of eqs. (13),(14) although they were originally derivedfor a general class of abelian ZN and ZN × ZM (2, 2) orbifolds.

In fact the gaugegroups in these cases is always E6×E8 and not anything looking like the standardmodel group. However we would like to argue for the validity of these formulaein the presence of Wilson lines and for (0, 2) type of gauge embeddings becausethe structure of the untwisted moduli is exactly the same as in the corresponding(2, 2) orbifold.

We will again only consider the string threshold effects dependenceon the overall modulus T. Then the T-field K¨ahler potential and the K¨ahlermetric of the matter fields are given by eqs. (8) and (9) respectively, and the lowenergy contribution to the gauge coupling constants is still described by eq.

(11).Thus, using the requirement of target space modular invariance, the thresholdformulae (13),(14) remain valid for generic symmetric orbifolds, and not onlyfor their standard embeddings. (For example one can check that for Z3 (0,2)orbifolds the b′ coefficients of all gauge groups again exactly agree.) These typeof models may in general yield strings with the gauge group of the standardmodel and appropriate matter fields as discussed e.g.in [29].In reality thethreshold effects will depend not only on the untwisted moduli but on othermarginal deformations like the twisted moduli and even on extra charged scalarswith flat potentials present in specific models.

We believe that considering justthe dependence on the overall (volume) modulus gives us an idea of the sizeand effects of the string threshold. Finally, as discussed above, we would expectto find similar results in more general (Calabi Yau) four dimensional strings inwhich the threshold effects have not been explicitly computed.

The low energyanomaly arguments should be valid for an arbitrary string and similar formulaeto those below should be found for those more general cases with the obviousreplacements due to the different duality groups involved.

−10 −Let us first consider the joining of the SU(2) and SU(3) gauge couplingconstants g2 and g3 at a field theory unification scale MX. If such a unificationtakes place eq.

(14) leads to the result (k2 = k3 = 1)M 2X = M 2string (TR |η(T)|4)b′2−b′3b2−b3(15)for the unification scale of the g2 and g3 coupling constants. Recalling that onealways has TR|η(T)|4 ≤1, one concludes that MX may be smaller or bigger thanMstring depending on the relative sign of (b′2 −b′3) versus (b2 −b3).

In general,the definition of b′i in eq. (11) shows that relative sign depends on the modularweights of the matter fields.

In some cases (e.g. when all matter fields havemodular weight n = −1) one has b′i = −bi and MX is necessarily bigger thanMstring.In these cases one cannot accommodate the difference between MXand Mstring we discussed above and the minimal string unification scheme issimply not viable.

This is the case of any model based on the Z2 × Z2 orbifold(or equivalent models constructed with free world-sheet fermions) since all matterfields have modular weight one.Let us now be a bit more quantitative and try to answer the following ques-tion: what are the values for modular weights nβ of quarks, leptons and Higgsas well as the corresponding values of TR which would allow for sin2 θW and αsvalues in reasonable agreement with data? Making use of equation (14) one getsfor the value of the electroweak angle θW after some standard algebrasin2 θW(µ) =k2k1 + k2−k1k1 + k2αe(µ)4πA log(M 2stringµ2) −A′ log(TR|η(T)|4)(16)where A is given byA ≡k2k1b1 −b2(17)and A′ has the same expression after replacing bi →b′i.

The standard grandunification values of the Kac-Moody levels correspond to the choice k2 = k3 = 1and k1 = 5/3. Finally, αe is the fine structure constant evaluated at a low energyscale µ (e.g.

µ = MZ). In an analogous way one can compute the low energy

−11 −value of the strong interactions fine structure constant αs1αs(µ) =k3(k1 + k2)1αe(µ) −14π B log(M 2stringµ2) −14π B′ log(TR|η(T)|4)(18)whereB ≡b1 + b2 −(k1 + k2)k3b3(19)and B′ has the same expression after replacing bi →b′i. Let us now defineδA ≡A′ + A = −2Xβ(nβ + 1)(k2k1Y 2(β) −T2(β))(20)where the sum runs over all the matter fields and nβ are the corresponding mod-ular weights.

Y (β) is the hypercharge of each field and T2(β) the correspondingSU(2) quadratic Casimir (T2 = 1/2 for a doublet). Analogously let us defineδB = B + B′ = −2Xβ(nβ + 1) (Y 2(β) + T2(β) −(k1 + k2)k3T3(β)).

(21)We can now write equations (16) and (18) as follows (k2 = k3 = 1, k1 = 5/3)sin2 θW = 38 −5αe32π A log(M 2Tµ2 ) −5αe32π δA log(TR|η(T)|4),(22)1αs(µ) =38αe−3B32π log(M 2Tµ2 ) −3δB32π log(TR|η(T)|4)(23)whereM 2T ≡M 2stringTR|η(T)|4. (24)All the model dependence (through the modular weights) is contained inδA, δB .

Denoting by niβ the modular weight of the i−th generation field of type

−12 −β = Q, U, D, L, E one can explicitly evaluate that dependence and findδA = 25NgenXi=1(7niQ + niL −4niU −niD −3niE) + 25 (2 + nH + n ¯H),(25)δB = 2NgenXi=1(niQ + niD −niL −niE) −2 (2 + nH + n ¯H)(26)where Ngen is the number of generations and nH, n ¯H are the modular weights ofthe Higgs fields.It is easy to see from equations (16) and (18) that both δA and δB have to bepositive in order to have any chance to obtain the correct values for sin2 θW andαs. In the minimal supersymmetric standard model one has b3 = −3, b2 = mHand b1 = 10 + mH, where mH is the number of pairs of Higgs doublets (mH = 1in the minimal case).

For the standard unification ki values one then finds A =6−25mH = 28/5 and B = 18+2mH = 20. As already explained, the requirementMX < Mstring implies A′/A > 0 and also B′/B > 0 (remember log(TR|η(T)|4) isnegative).

Then one has the conditionsδA > A = 285 ,(27)δB > B = 20(28)in the minimal model. Notice that these conditions are violated explicitly in theZ2 × Z2 orbifolds (in which case δA = δB = 0) and also in Z3 and Z7.

In thelatter cases one has δA = A and δB = B since A′ = B′ = 0. (Parenthetically,these latter equations can be used combined with eqs.

(25) and (26) in orderto get constraints on the number of SU(2)-doublets and SU(3)-triplets comingfrom untwisted, twisted and twisted moduli sectors in specific Z3 and Z7 (0, 2)orbifolds).Another point to remark is that if there are SU(5)-type boundary conditionsfor the matter kinetic terms (and, hence, for the modular weights) one has nU =nQ = nE and nD = nE. In this case δA = 25(2 + nH + n ¯H) and δB = −2(2 +

−13 −nH + n ¯H) = −5δA and both quantities cannot be simultaneously positive. ThusSU(5)-like boundary conditions for the modular weights cannot accommodatethe values of the measured low energy couplings in the context of minimal stringunification.

On the other hand there is no reason why those boundary conditionsshould hold since we are assuming that the gauge group is SU(3) × SU(2) ×U(1) up to the string scale. Furthermore, other unification schemes e.g.

insidesemisimple groups like SU(4)×SU(2)×SU(2) or SU(3)×SU(3)×SU(3) do notlead to those boundary conditions. All of these unification schemes are consistentwith k1 = 5/3.

Incidentally, let us recall that for the standard values k2 = k3 = 1and k1 = 5/3 the low energy symmetry is enlarged to SU(5) only in the casewe insist on the absence of massive fractionally charged states [36]. We do notinsist on that, we just assume the minimal low energy susy particle content butnothing specific about the massive sector.In principle, if the conditions (27) and (28) are met there may exist a value ofTR such that one can accommodate the measured low energy values of couplingconstants.

In the absence of string threshold effects (i.e. for δA = A and δB = B)one finds from equations (16),(18) sin2 θ0W(MZ) = 0.218 and α0s(MZ) = 0.205.The effect of non-vanishing threshold effects in the minimal scenario we are dis-cussing is displayed in figures 1 and 2.

The first shows the value of sin2 θW(MZ)as a function of ReT ≡TR/2 for different values of δA. A similar plot for αs(MZ)is shown in figure 2.

The shaded areas correspond to the experimental results.The bounds in eqs. (27) and (28) are apparent in the figures.

One also observesthat one can get results within the experimental constraints for sufficiently largevalues of δA, δB and TR. In fact one can eliminate the explicit dependence onTR by combining equations (22) and (23).

In this way one finds a linear equationrelating δA and δB:(δB −B) = γ (δA −A),(29)γ = 53αe ( 1αs0 −1αs(µ))(sin2θ0W −sin2θW(µ)!. (30)If δA/A = δB/B the string corrections may be entirely contained in a changein the scale in the original field theoretical analysis, and all three coupling

−14 −constants meet (at the one loop level) at the same energy scale.This corre-sponds to γ = B/A = 25/7. Allowing for the experimental errors in αs(µ) andsin2 θW(µ) one more generally finds 2.2 ≤γ ≤4.0.

One can then search forvalues for the modular weights nβ of the standard model particles compatiblewith eqs.(27),(28),(29). Assuming generation independence for the nβ as well−3 ≤nβ ≤−1 one finds, interestingly enough, a unique answer for the matterfields:nQ = nD = −1 ;nU = −2 ;nL = nE = −3(31)and a constraint nH + n ¯H = −5, −4.

For nH + n ¯H = −5 one obtains δA = 42/5,δB = 30, and the three coupling constants meet at a scale MX ∼2 × 1016GeVprovided that ReT is of order ReT ∼16 (see the two figures). For nH +n ¯H = −4one has δA = 44/5, δB = 28.

Now the three couplings only meet approximately(within the experimental errors of sin2 θW and αs) for similar values of ReT. Thuswe see that, in principle, a situation with a compactification scale below the stringscale may work within a minimal SU(3)×SU(2)×U(1) string provided, e.g., theabove modular weights are possible.

Notice that (twisted) moduli fields are notnecessarily gauge singlets (e.g. the SU(2) doublets in the Z4 orbifold).

(Allowingfor non-standard modular weights, i.e. n < −3, the minimal unification scenariowould be possible for smaller values of ReT, and in particular for ReT ∼1.

)We have just shown that the minimal string unification scenario is in princi-ple compatible with the measured low energy coupling constants for i) sufficientlylarge ReT and ii) restricted choices of standard particles modular weights. Thequestion now is wether these two conditions are easy to meet.

Concerning thefirst condition, we need to have an idea of the non-perturbative string dynamicswhich trigger the compactification process and fixes the value of ReT. In thecontext of duality-invariant effective actions, recent analysis [18],[19], [37] showsthat the preferred values of ReT are of order one.

This is expected since a dual-ity invariant potential will typically have its minima not very far away from theself-dual point. Thus large values of ReT are not expected within this philos-ophy.

However a deeper understanding about non-perturbative string effects isdefinitely needed to give a final answer to this question. Concerning the secondcondition, it would be interesting to investigate ZN and ZM ×ZN orbifold models

−15 −to see whether there are choices of modular weights leading to the appropriateδA, δB. It is certainly intriguing the degree of uniquenes in the possible choicesof modular weights (eq.

(31)) leading to adequate results and this point deservesfurther study. Notice, however that one may relax the condition of generationindependence which led to eq.(31).

In addition one can consider the separatecontribution of the three orbifold planes in terms of the three untwisted moduliTi. In summary, we believe that the analysis presented in this paper shows thatminimal string unification is a possible but rather constrained scenario.If no string model with the characteristics of the above minimal unificationscenario is found, it may still be possible to explain the success of the unificationof of the three g1, g2, g3 coupling constants in the context of strings providedone of the following possible alternatives is realized: i) There is an intermediategrand unification scale MX ∼1016 GeV at which a GUT simple group likeSU(5) or SO(10) is realized ; ii) There is instead some semi −simple group likeSU(4) × SU(2) × SU(2), SU(3)3 or non semi −simple group like SU(5) × U(1)beyond that scale.

Both of these possibilities has its shortcomings. The firstrequires that the Kac-Moody level of the gauge groups SU(5) or SO(10) be biggerthan one and the construction of higher level models is both complicated andphenomenologically problematic [38].

Alternative ii) has the problem that thereis further relative running of the coupling constants in the region MX −Mstringwhich will typically spoil the predictions of the minimal susy model. In any otherpossible alternative it would be difficult to understand why the couplings tendto join around 1016 GeV, it would just be a mere coincidence.For example,it is possible to consider extensions of the minimal particle content in such away that the low-energy gauge couplings directly meet around 1018 GeV.

Thispossibility was already considered in ref. [39] .

In any case it is clear that thepresent precision of the measurement of low energy gauge couplings has reacheda level which is sufficient to test some fine details of string models.We acknowledge useful discussions with M. Cvetic, S. Ferrara, C. KounnasC. Mu˜noz, F. Quevedo, A.N.

Schellekens and F. Zwirner.

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−19 −Figure CaptionsFigure 1: sin2 θW(MZ) as a function of the compactification radius2 ReT = R2for different values of δA. The shaded area corresponds to the experimentallyallowed range.Figure 2: αs(MZ) as a function of ReT for different values of δB.

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