CPTh/RR.349.0395
A way to break supersymmetry
C. Bachas ^{*}^{*}*email address:
Centre de Physique Théorique
Ecole Polytechnique
91128 Palaiseau, FRANCE
ABSTRACT
I study the spontaneous breakdown of supersymmetry when higherdimensional YangMills or the typeI string theory are compactified on magnetized tori. Because of the universal gyromagnetic ratio , the splittings of all multiplets are given by the product of charge times internal helicity operators. As a result such compactifications have two remarkable and robust features: (a) they can reconcile chirality with extended lowenergy supersymmetry in the limit of large tori, and (b) they can trigger gaugesymmetry breaking, via NielsenOlesen instabilities, at a scale tied classically to . I exhibit a compactification of the superstring, in which magnetic fields break spontaneously supersymmetry, produce the standardmodel gauge group with three chiral families of quarks and leptons, and trigger electroweak symmetry breaking. I discuss supertrace relations and the ensuing ultraviolet softness. As with other known mechanisms of supersymmetry breaking, the one proposed here faces two open problems: the threat to perturbative calculability in the decompactification limit, and the problem of gravitational stability and in particular of the cosmological constant. I explain, however, why a good classical description of the vacuum may require small tadpoles for the dilaton, moduli and metric.
March 1995
1. Introduction
Perhaps the main puzzle of superstring unification [1, 2] concerns the breaking of spacetime supersymmetry. Two proposals have so far been put forth in the literature: the nonperturbative scenario based on gaugino condensation in a hidden sector [3, 4, 5, 6, 7, 8], and the treelevel ScherkSchwarz mechanism [9, 10, 11, 12, 14, 13, 15, 16]. Both assume that the correct vacuum of the theory is an exact solution of the classical string equations, which leave undetermined the values of various continuous moduli. These should then hopefully be fixed by radiative or nonperturbative corrections. The mechanism of Scherk and Schwarz allows in particular for a a string realization of the noscale supergravity models [17, 18]. Supersymmetry breaks classically at a scale proportional to the values of one or more moduli, say , and the gauge hierarchy is presumably attributed to the logarithmic running of couplings which yields a very shallow deformation of the classicallyflat potential of . A merit of this scenario is that it is calculable at tree and oneloop order, at which point one encounters instabilities due among other things to a nonvanishing cosmological constant. Furthermore the embedding in the string, which makes oneloop gravitational corrections finite, raises a novel problem: within the oneloop approximation there is no hope to stabilize the dilaton and avoid among other things conflict with the principle of equivalence [19]. Gaugino condensation or other nonperturbative phenomena are conceptually more promising in this respect, but we lack the technology to study them directly at the string level. Most of the interesting attempts are therefore limited to guessing superpotential modifications, using as a guide field theory as well as the duality conjecture [7, 8]. Even with such guesswork a realistic vacuum without runaway dilaton and moduli and with vanishing cosmological constant has not yet been found. It is furthermore questionnable whether anything short of solving all of the above gravitational instabilities at once, would constitute real progress.
In view of this unsatisfactory situation, it is I believe fair to ask whether we have not taken too seriously the classical string equations of motion. What if the true string vacuum leads to small classical tadpoles which should be cancelled ultimately by higherloop and nonperturbative corrections? The ColemanWeinberg mechanism [20] serves to illustrate forcefully this point: one considers a complex scalar field with quartic potential, , coupled to a gauge field as well as to massless fermions through Yukawa couplings. The classical field equations tell us that , so that the photon and all fermions stay massless. Oneloop corrections on the other hand can change the shape of the potential leading to a nonzero vacuum expectation value for . Expanding around this nonzero vev gives thus a much more accurate description of the spectrum, even though classically there remains an uncancelled tadpole. Likewise, the price for getting a good description of the lowenergy world from the string may be to allow for small metric, dilaton and moduli tadpoles in the classical description of the ground state. To see how small is small, suppose that in what concerns the gauge sector, the vacuum of the heterotic string could be approximated well by compactification on a fourtorus times a two sphere with a magnetic monopole in its middle [21]. These backgrounds correspond [22, 23] to a model and free fields, so that apart from the function of the dilaton: , all other classical equations are satisfied [24]. Now we imagine that quantumgravity effects cancel the tadpoles of the dilaton and metric, but do not affect the gauge sector of the theory where supersymmetry is broken at a scale . For this scale to be we need . The classical prediction of string theory, that the effective spacetime dimension , would in this hypothetical case be a very good approximation of reality. But given our lack of control over the full quantum dynamics why should we expect it to be more accurate than one part in ten to the thirty?
Following this logic opens up a host of possibilities: any background deviating a little from a supersymmetric, classical solution of string theory could a priori be a good starting point to describe the vacuum. This nearby supersymmetric solution (NSS) must of course contain the right gross ingredients: a gauge group and enough chiral fermions to describe the families of quarks and leptons. The number of noncompact dimensions in the NSS may or may not be equal to four, since the decompactification theorems [25, 13] only apply when one restores supersymmetry along marginal directions. As for all other lowenergy phenomena, such as electroweak and supersymmetry breaking and the generation of mass, these are fine structure when viewed from the Planck scale and they therefore depend crucially on what kind of deviations one is willing to consider. To make some progress we need an ansatz for these deviations and I will here make the choice of turning on constant magnetic fields in torroidallycompactified dimensions. Though by no means compelling, this choice has the following attractive features:
(a) It corresponds, as I will explain below, to classicallystable solutions of higherdimensional gauge theory, as well as of typeI open string theory, if one arranges for NielsenOlesen instabilities [26] to be absent. Tadpoles only arise upon coupling the dilaton and metric, which makes it more plausible that after all the quantumgravity dust has settled, the classical spectrum is still a good approximation of reality.
(b) The pattern of supersymmetry breaking is elegant and simple, with all splittings being proportional to charge times internalhelicity operators. This is related to the fact that consistentlycoupled relativistic particles must have a gyromagnetic ratio [28, 29], and it implies powerful supertrace relations.
(c) Such compactifications can reconcile chirality with extended lowenergy supersymmetry in the limit of large tori. This is a consequence of the index theorem [27, 1], or put differently of the fact that the magnetic field shifts the masses of mirror fermions in opposite directions.
(d) The NielsenOlesen instability triggers gauge symmetry breaking when the magnetic field is embedded in nonabelian group factors. This provides a new mechanism to break electroweak symmetry and tie up classically to .
The use of open strings facilitates the analysis and postpones gravitational tadpoles to the oneloop (annulus or Möbius strip) level [30]. The above features should, however, continue to hold if one compactifies the heterotic string on magnetized spheres rather than tori. Furthermore, I expect the last two features to be robust and to characterize a much wider class of compactifications. Note that in contrast to the above mechanism, the breaking à la ScherkSchwarz is explicit rather than spontaneous at the global level, it can accomodate chiral fermions only for minimal lowenergy supersymmetry ( in four dimensions) [12] , and it cannot trigger electroweak breaking classically. Both mechanisms face, however, the same two serious difficulties: the problem of gravitational stability due in particular to a nonzero vacuum energy , and the problem of large internal dimensions [25, 13] which, though not in obvious conflict with experiment [10, 16], threaten weakcoupling calculability. This latter difficulty is due in our case to the Dirac quantization condition, which ties up the size of magnetic fields to the inverse area of tori. For chiral theories it poses, as we will see, a problem at oneloop order, while in the ScherkSchwarz scenario this problem can be postponed to two loops [15]. Likewise the vacuum energy has in our case a classical contribution, while the ScherkSchwarz breaking generates it only at oneloop order. The two mechanisms do not differ qualitatively in these respects, and any way out of these difficulties could a priori apply to either one or to both. A couple of points are nevertheless worth stressing: first, reconciling chirality with extended lowenergy supersymmetry raises the exciting possibility that the tractable dynamics of and supersymmetric gauge theories [31] could be of help in addressing these issues. Second, the classical energy of the magnetic fields, which is a drastic departure from the noscale idea, changes the nature of the gaugehierarchy puzzle ^{†}^{†}†Whether it can also help with the problem of the runaway dilaton [32] is unclear. Although this energy is multiplied by inverse powers of the coupling when one works with model backgrounds, this changes if one rescales fields so as to normalize the Einstein term in the action [24]. Note also that in WZW compactifications of the heterotic string the sign of this classical energy changes sign. : rather than explain why is so much smaller than we only have to explain why it is not exactly zero. Such a treelevel energy was in fact added by hand in recent phenomenological attempts to stabilize the mass of the gravitino [33].
Finally a note on references: there is of course a huge literature on compactifications of higherdimensional gauge theories, KaluzaKlein theories and supergravities in the presence of gaugefield backgrounds [34], and in particular on the twosphere with a magnetic monopole in its middle [21]. The role of gauge backgrounds in creating chirality through the index theorem is also a wellestablished fact of life [27, 34, 1], while monopole compactifications of the superstring were considered early on by Witten [35]. What was perhaps not appreciated in these earlier studies, was that magnetic fields can provide an elegant mechanism for spontaneous breaking of both supersymmetry and electroweak symmetry. These efforts were in any case silenced by the advent of string theory, as emphasis shifted to zeroenergy exact solutions of the classical equations of motion. The possibility of violating these equations was evoked by Rohm and Witten [22]. Their study of antisymmetrictensorfield backgrounds in relation to gaugino condensation in the heterotic string is closest in spirit to the present paper.
The plan of the paper is as follows: in section 2 I review some elementary facts about magnetic monopoles on the torus, and describe the splitting of hypermultiplets. In order to generalize the mass formula to vector and higherspin massive multiplets, it is technically more convenient to pass to the open superstring. This is explained in section 3, which contains some known material for completeness. In section 4 I discuss the NielsenOlesen instability, anomaly cancellation, as well as supertrace relations and the ensuing oneloop ultraviolet softness. I explain why despite this soft behaviour, and the absence of renormalization when the tori are demagnetized, the question of perturbative calculability in the decompactification limit remains open. In section 5 I exhibit a compactification of the superstring with a threechiralfamily standard model in the massless spectrum, broken supersymmetry and a negative mass square for the higgs doublets. Though this model has too much structure at the supersymmetry threshold to be considered at this point as phenomenologically viable, it is intriguing how close it comes to describing lowenergy physics. Finally in section 6 I give some concluding remarks.
2. Breaking SUSY with the magnetized torus
The simplest setting of interest is that of a sixdimensional gauge theory () compactified down to four dimensions on a twotorus. We assume for simplicity that the torus is generated by orthogonal vectors on the plane: and . Vacuum solutions, invariant under the fourdimensional Poincaré group, can be characterized by a constant magnetic field , and by the Wilson phases around a basis of noncontractible loops on the torus. Choosing a particular gauge we may write:
where and are constant. Charged quantum mechanical particles behave very differently according to whether or . In the former case their wavefunction is periodic, leading to a lattice of covariant momenta shifted from the origin by the charge () times the constant gauge field. Thus the spectrum of the twodimensional laplacian, which gives the mass shifts of towers of KaluzaKlein states, reads:
Here stands for the mass of the excitation minus the mass of the parent field in six dimensions. This spectrum changes continuously with the values of the (periodic) moduli and , which parametrize inequivalent vacua. For , all charged excitations are massive, including any charged (nonabelian) gauge bosons. This is the wellknown mechanism of gaugesymmetry breaking by Wilson lines. It respects all supersymmetries of the parent theory since the above mass formula is spinblind.
The situation changes drastically for a nonzero value of : the continuum of inequivalent vacua is replaced by a discrete set of states, and in the presence of charged fields all supersymmetries are spontaneously broken. To see why let us recall some elementary facts [36] about the gauge field, eq. (2.1). This is the field of a monopole, i.e. a nontrivial bundle over the torus with a transition function joining the to the regions:
Demanding that be singlevalued on the circle leads to the Dirac quantization condition:
if the unit of charge is set equal to one. Thus for fixed area of the torus, is a discrete modulus rather than a continuous parameter of compactification. Furthermore the spectra of KaluzaKlein excitations depend only on the commutator of covariant momenta,
but not on and . Indeed can be absorbed by a shift of , and by a change of gauge [36]. Thus, although from the point of view, and are scalar fields with a flat potential, their expectation values are physically irrelevant and do not label inequivalent vacuum states This is of course a common phenomenon: the Goldstone boson of a spontaneouslybroken global symmetry has also a physically irrelevant expectation value, as does the dilaton in a lineardilaton vacuum of string theory.
That the magnetic field breaks supersymmetry could be argued for on the basis of its positive contribution to vacuum energy. Strictly speaking this is incorrect, since in the absence of charged fields (and of gravity) all of the vacua would still be supersymmetric ^{‡}^{‡}‡This is of course possible because Poincaré invariance has been broken.. Let us assume therefore that the parent supersymmetric theory contains some charged chiral (and hence massless) hypermultiplet. This yields by trivial reduction two complex scalars and two Weyl spinors of opposite chirality in four dimensions [10]. Compactifying in the background of the magnetic field splits this hypermultiplet in an interesting way. First, the Laplace operator on the torus has now the spectrum of a harmonic oscillator, giving the following mass shifts for the scalar components of the multiplet:
Each Landau level in the above spectrum is, furthermore, times degenerate, with both and being integers by virtue of quantization of charge. A set of wavefunctions that span, for example, the lowest Landau level when are the following:
where is a normalization and . These have indeed the required periodicities: and . For higher Landau levels one must replace the first of the two exponentials above with higher excited harmonicoscillator eigenfunctions.
Consider next what happens to the Weyl spinor. Denoting by the Dirac matrices which obey , and using the commutator (2.5), one finds:
Each Landau level is again times degenerate, and is the spin operator projected along the magnetic field. Since the sixdimensional spinor was Weyl, we can identify the internal helicity with the fourdimensional chirality, so that for chiral or antichiral spinors. At the lowest Landau level we thus obtain massless chiral fermions, while their antichiral partners are shifted to a mass equal to . There they are joined by the chiral fermions of the first Landau level so as to form massive Dirac spinors, and the tower continues like that forever. All this is of course in agreement with the (twodimensional) index theorem:
which we could have used to predict the net number of massless chiral fermions surviving compactification on the magnetized torus [27, 34, 1].
It follows easily from the above expressions that for each Landau level separately
in accordance with the fact that the breaking of supersymmetry is spontaneous. Note that, in contrast, the ScherkSchwarz mechanism breaks global supersymmetry explicitly, by modifying the boundary conditions of fields as in the case of finite temperature, so that the above supertrace may [10] but need not vanish. Note also how chirality can be reconciled, as advertized, with lowenergy supersymmetry in four dimensions. In the stringy ScherkSchwarz scenario by contrast, chiral fermions live in twisted sectors of orbifolds, which are spectators of the symmetrybreaking process [12]. Equations (2.8) and (2.10) will stay valid for vector as well as massive higherspin multiplets. This can be shown easier in the context of the open superstring to which we now turn our attention.
3. TypeI superstring.
A constant electromagnetic background adds only quadratic boundary terms to the worldsheet action of an open string, so that the corresponding conformal field theory can be solved exactly [37, 38]. This has for instance been exploited to calculate the rate of stringpair creation in a constant electric field [39], and to show that the gyromagnetic ratio is for all higherspin string excitations [28]. The latter is the key observation which we want now to adapt in our context. We consider for definiteness some torroidal compactification of the superstring from ten down to four dimensions, and turn on for the time being a magnetic field in only one of the three planes of the hypertorus: ^{§}^{§}§ The magnetized torus was also considered in ref. [38]. This study was restricted to the bosonic string, so that the issue of supersymmetry breaking did not arise.. For later convenience we also add some Wilsonline breaking of the gauge group, by exploiting all six compact dimensions: for . We take all these backgrounds in the Cartan subalgebra of , and denote by the left(right) endpoint charges of the string in the direction of , and by the corresponding charges in the direction of . The worldsheet action on the strip reads:
Here is the Regge slope which we set equal to , the are real Majorana fermions, and () are the corresponding Dirac matrices in two dimensions with the conventions of ref. [1].
Since the gauge fields only couple at the boundary, all fermionic and bosonic coordinates satisfy freewave equations. For , the and have the usual Neumann, and Ramond (R) or NeveuSchwarz (NS) boundary conditions. The corresponding momenta lie on a lattice which is shifted by an amount from the origin, so that all charged states are generically massive. This is up to here a conventional torus compactification with Wilsonline breaking of the gauge group and maximal supersymmetry in six dimensions. We may also reduce the supersymmetry to , by orbifolding these extra compact dimensions [40, 41], without affecting the discussion that follows.
In contrast to the Wilsonline backgrounds, a nonvanishing magnetic field changes the boundary conditions of the remaining complex compact coordinate and of its superpartner . Recall that variations of the latter must be constrained at the boundary as follows [1]: at , and at , where is the left(right)moving component of the fermion and or in the NeveuSchwarz or Ramond sector. Extremizing the action leads therefore to the equations [38, 28, 39]:
where we have used the shorthand notation:
Since the boundary conditions are linear, we can expand the coordinates in orthonormal modes as usual:
with
and
with
The above expressions depend on the magnetic field through the nonlinear function [38]
which summarizes the effects of the nonminimal string coupling. Note that in the weakfield limit that interests us in this paper () we have
while for a field of the order of the string tension saturates to the values . Canonical quantization leads to the commutation relations:
and
All other commutators are zero.
The upshot of this tedious algebra is that the complex supercoordinate behaves like the coordinate of an orbifold [41] in a twisted sector with twist angle . There are, to be sure, some significant differences between the magnetized torus and an orbifold: the centerofmass position has in our case a nontrivial commutator, is not related to a discrete symmetry and can be arbitrarily small for a large torus, and we do not sum over twisted sectors. The orbifold analogy is nevertheless useful in deriving the spectrum of masses. To this end we note the following:
(i) The creation operators and () raise the helicity in the plane by one unit and have their worldsheet frequencies shifted by . Similarly and lower the helicity by one unit and have their frequencies shifted by .
(ii) The zero modes require special treatment: if is positive, annihilates the vacuum while creates the successive excited Landau levels. Likewise annihilates the lowestlying Ramond state of internal helicity , while the action of flips the helicity to and raises the square mass of the state by ^{¶}^{¶}¶With our conventions the square mass is given by twice the zerothmoment Virasoro generator.. If is negative one must reverse the roles of daggered and undaggered operators in the zeromode sector.
(iii) By virtue of worldsheet supersymmetry the magnetic field does not shift the position of the vacuum in the Ramond sector, while a straightforward calculation [41, 38, 39] gives a shift of for the square mass of the vacuum in the NeveuSchwarz sector.
Putting these observations together we arrive at the following remarquably simple expression for the mass shift of all string excitations when going down from six to four dimensions:
with the spin operator projected on the plane of the torus. The nontrivial commutator (3.9) ensures furthermore that each Landau level is degenerate times. The above expression reduces to eqs. (2.6) and (2.8) in the weakfield limit and for scalar or, respectively, spinor excitations, but generalizes these results to vector and higherspin multiplets.
Several remarks are in order here: first eq. (3.10) is of course only valid for , in which case the Wilson lines are irrelevant. For strings neutral with respect to the magnetic these Wilson lines shift the lattice of momenta as previously described. Secondly, the above analysis can be extended trivially to the case where all three tori are magnetized. Labelling these tori by a lowercase Latin index, and working from now on in the weakfield limit, eq. (3.7), we may write:
Here are the magnitudes of the three magnetic fields pointing in some directions inside the Cartan subalgebra of , and are the corresponding total charges. One can consider more general situations, such as magnetic fields not aligned with the planes of the tori, but we wont need these in the discussion that follows. When all the , a Weyl spinor is split by the three magnetic fields in such a way that only one chiral fermion remains massless. Indeed, one must fix all three internal helicities: , so as to cancel the positive mass shift common to all excitations at the lowest Landau level. This shows how chirality can be reconciled with maximal () lowenergy supersymmetry in the limit of three large tori. Note finally that the tracelessness of ensures that for any multiplet and every Landau level.
4. NielsenOlesen instability, anomalies and UV softness
Let us take now a closer look at vector multiplets. A charged gauge boson (and its conjugate) gives by trivial dimensional reduction two complex scalars with internal helicities , and a gauge boson with . The mass formula, eq. (3.10) implies that the lowest Landau excitation of one of the two scalars has a negative mass shift equal to . If the gauge boson was originally massless, the vacuum would therefore be unstable, as Nielsen and Olesen were the first to point out [26] ^{∥}^{∥}∥ Since we work with weak magnetic fields we wont worry about the extra instabilities which can occur when is of the order of [42].. We can of course eliminate the instability by rendering the charged gauge bosons massive. This can be achieved by turning on Wilson lines in the extra compact dimensions, so that the unbroken gauge group is of the form . Turning on several magnetic fields will also eliminate some of the instabilities, since only one internal helicity can be nonzero for lowlying scalars. A third option, whose consistency needs however to be checked in string theory, can a priori also be envisaged: since does not break the reflection symmetry of the worldsheet action, eq. (3.1), we could mod it out and convert the torus into a orbifold. This projects out of the spectrum the nonzero helicity components of all gauge bosons, thus eliminating the dangerous, potentially tachyonic states. Note that since the area of the orbifold is half that of the torus, the minimal magnetic charge must in this case be doubled. If despite all of the above cures there remain tachyonic scalars in the spectrum, they will acquire nonzero expectation values in the vacuum. In contrast to Wilsonline breaking, this mechanism can reduce the rank of the gauge group as the example in the following section will illustrate.
Setting magnetic instabilities aside for the moment, let me comment briefly on another important issue, i.e. the cancellation of anomalies. Suppose there are no pure gauge anomalies in six dimensions, so that the relevant box diagram is zero. Since the magnetic field gives mass to all charged gauge bosons, the unbroken gauge group after compactification is necessarily of the form . Let the chiral fermions transform in the representations of this gauge group. According to the index theorem, eq. (2.9), the net number of (left minus right) chiral fermions in the representation is . The triangle anomaly thus reads:
where the are generators of . Likewise one can show easily that all triangle anomalies involving vanish. This is a special case of the argument given for arbitary compactifications by Witten [35, 1]. The only subtle point is the fact that anomaly cancellation in ten dimensions makes use of the twoindex antisymmetrictensor , with modified field strength
where we neglect here gravitational backgrounds. Since contributes to the energy it must be globally welldefined, which means that [1, 35]
for any compact fourmanifold . It follows that consistent compactifications on several magnetized tori must obey ^{*}^{*}*We use to denote the Liealgebra valued magentic field, and its appropriately normalized magnitude.
i.e. the various magnetic fields should point in orthogonal directions in group space. This guarantees that in the theory any residual anomalies can still be cancelled by the mechanism of Green and Schwarz.
The reader may wonder why anomalies are treated differently from other ultraviolet divergences of the annulus or Möbiusstrip, which signal nonvanishing gravitational tadpoles. The reason in ten dimensions is that anomalies give rise to tadpoles of unphysical RamondRamond states [30], and cannot therefore, even in principle, be cured by shifting gravitational backgrounds. In four dimensions, on the other hand, they are a signal of an illdefined compactification as just noted. Another fact concerning eq. (4.3) is also worth pointing out: let the fourmanifold be the product of a torus with magnetic field on its surface, and of a sphere at spatial infinity. If all scalars go to constant values at infinity, the integral factorizes into the product of magnetic fluxes. In order to satisfy (4.3) the compactified theory should therefore have no monopoles with magnetic charge under . Does this mean that electricmagnetic duality is necessarily broken in this case? The answer is not obvious because the term in the Lagrangian gives a mass of order to the gauge boson, with furnishing its longitudinal component. This is a tiny mass indeed, since has gravitationalstrength couplings, but electric charges are all the same screened and duality could be possibly saved.
The last thing I would like to discuss in this section, is the issue of radiative corrections. These pose a threat to perturbative calculability, since above the supersymmetry threshold the theory is higherdimensional and hence nonrenormalizable, at least naively. Let me concentrate in particular on maximal supersymmetry, as this might offer the best hope of alleviating the problem. To keep the expressions simple I suppose that only one of the tori is magnetized, say , but the results will also hold in the more general situation. All particles belong to spin1 multiplets of , which contain one gauge boson, four Weyl fermions and six scalars. For massive multiplets one of the scalars is eaten by the longitudinal component of the vector. Out of the eight bosonic states two have internal helicity and all the others zero, while the eight fermionic states have in equal numbers. A simple counting then shows that at each Landau level separately
As a result the oneloop ultraviolet behaviour is indeed much more soft than naively expected. Consider for instance a multiplet of mass in six dimensions. Using the expansion
and the Schwinger propertime parametrization, one finds the following result for its contribution to the oneloop vacuum energy:
Since the double summation is absolutely convergent, the result is ultraviolet finite despite the infinite tower of Landau levels. Furthermore for massive multiplets, , the result vanishes like the sixth power of the supersymmetry scale:
Likewise one can show that the oneloop contribution of each multiplet to the effective gauge coupling is finite. A multiplet in a representation of some simple factor of the gauge group contributes indeed the following correction to :
where is the quadratic Casimir of the representation , and is here the helicity in four dimensions ^{†}^{†}†Note that can be obtained as the coefficient of the term in the oneloop vacuum energy, when one turns on a magnetic field in a threedimensional volume V. This can be used to check the relative normalizations of eqs. (4.7) and (4.9).. The double summation is once more convergent, and the result is finite even though we are dealing with a theory and supersymmetry is broken. Furthermore for particles much above the susy threshold we find
i.e. such particles do not even contribute a finite renormalization to the gauge couplings!
Does this mean that the couplings will stay small as we go up in energy towards the Planck scale? Things are, unfortunately, not so simple. First, this is a oneloop result, and there is certainly no guarantee that divergences will not show up at two or higherloop order. Second, if more than two compact dimensions become large, we must sum the above expressions over the masses () of new towers of KaluzaKlein states. The correction will thus grow logarithmically if there are two extra large radii, and it would grow like the square of energy if all six compact dimensions were large. But six large compact dimensions is precisely what we need in order to reconcile chirality with lowenergy supersymmetry, as previously noted. This example serves in fact to illustrate a crucial point: since for fixed torus area the magnetic field is a discrete modulus, the limit of supersymmetry restauration cannot be taken independently and need not commute with the sum over KaluzaKlein states. Put differently, although every multiplet makes a vanishinglysmall contribution to the running, the fact that there can be as many as of them can lead to a very large cummulative effect in the decompactification limit. The absence of renormalization in the supersymmetric compactification , does not in particular protect us agaist potential disaster. In chiral ScherkSchwarz compactifications powerlaw corrections can be suppressed at one loop [15], but this difficulty should show up at two and higherloop order.
5. A standard model with broken N=4
I will now describe a compactification of the superstring ^{‡}^{‡}‡See also ref. [35] for an earlier effort., that exhibits the two main features of magnetized tori: the reconciliation of chirality with extended lowenergy supersymmetry, and the triggering of electroweak breaking by the NielsenOlesen instability. I label the three magnetized tori by lowercase Latin indices, as in section 3, and consider the minimal fields allowed by the Dirac quantization condition:
Here is the area of the th torus, and is the corresponding generator of , normalized so that the elementary charge in the adjoint representation equals one. In order to define the we will use the following sequence of embeddings:
I assume a maximal embedding , and will denote by , , and the four generators in the order in which they appear above from left to right. Note that when these are normalized to unit charge, their trace in the adjoint of reads
where the refers to the subgroup in which the generator is embedded. Let us now choose the directions of the three magnetic fields as follows:
Using eq. (5.2) one can check easily that these are orthogonal generators of , so that the anomaly condition is satisfied.
Now recall that chiral fermions can only come from multiplets whose charges under all three magnetic ’s do not vanish. This is necessary in order to fix all internal helicities of the Weyl spinor, and can be also seen from the expression for the net chirality in four dimensions:
As a result there are no chiral fermions transforming under both the ”observable” gauge group , and the ”hidden” one . Simple inspection shows in fact that there are only three types of chiral fermions in the ”observable sector”, whose charges and multiplicities are listed in the table below:
Table 1. Chiral fermions transforming under the ”observable” gauge group, their charges under the three magnetic ’s and their multiplicities.
This is precisely the anomalyfree content of an grandunified model with three chiral families of quarks and leptons, and an extra horizontal symmetry under which transform the ’s as well as standardmodel singlets. To proceed further we would like to break , render massive other unwanted massless states and take care of NielsenOlesen instabilities. To these ends we still have Wilson lines, and the choice of the radii at our disposal. Rather than being systematic, let me make some choices and see where they lead us. Denoting by the hypercharge generator inside , we set
where and are constants. Recall that a Wilson line is relevant only when the charge under the magnetic field on the corresponding torus does not vanish. Chiral fermions and all their partners are not therefore affected by the above backgrounds. Gauge bosons of unbroken symmetries, on the other hand, have all three , so the and Wilson lines will break to the standard model group, and the horizontal symmetry to factors. Furthermore all particles charged under both the hidden and observable gauge groups have either or , so they will obtain a mass from either the or the Wilson line. By choosing moduli appropriately we can ensure that none of these states is tachyonic.
What about the breaking of electroweak symmetry? Candidate higgs doublets come from two different places: scalar partners of the chiral fermions in the representation of , and scalars in the multiplet, which does not contain chiral fermions since it has and . The scalars are preferable for two reasons: (i) they have nonvanishing (renormalizable) Yukawa couplings with the chiral fermions as can be seen from their transformation properties, and (ii) the Wilson line splits the colourtriplet from the doublet, and can thus ensure that remains unbroken. Using the formulae (3.11) and (2.2), and the fact that or for the doublet or (anti)triplet in the of , we find the following masses for the lowestlying scalars that transform nontrivially under the standard model gauge group:
and
Note that in what concerns eq. (5.7), the contribution of the Wilson line cannot exceed , the contribution of the background was omitted for simplicity, and there are excitations in the lowest Landau level. Now if we want only coloursinglet scalars to be tachyonic, we must demand that both and satisfy the triangle inequalities, and that
These constraints are mutually compatible, choosing for instance and will satisfy them all. In an appropriate region of parameter space the higgs doublets will thus acquire nonzero vevs, breaking electroweak symmetry and giving mass to the quarks and leptons ^{§}^{§}§By going to the corner of parameter space where the doublets are just barely tachyonic, we can be sure that their nonzero vevs will not have a big effect on the masses of all other scalars.. Furthermore, unless we fine tune parameters, the scale of electroweak breaking will be tied classically to .
Although this model has too much structure at the supersymmetry threshold to be considered at this point as realistic, it is surprising how close we come to a classical description of our lowenergy world with relatively little effort. Several features which have proven hard to achieve in previous string modelbuilding, come out rather easily here: adjoint scalars for breaking, three chiral families of quarks and leptons, and negative mass square for the higgs doublets. More conservative uses of magnetized tori can also be envisaged: since for example mass splittings are proportional to charge, one can restrict treelevel supersymmetry breaking to a hidden and possibly hypermassive sector. Alternatively, one may insist that there are only two large compact dimensions [10], so that the lowenergy world is supersymmetric. Finally, it could be desirable to have both magnetic fields and modified, ScherkSchwarz boundary conditions: the former would create chirality and trigger electroweak breaking, while the latter would give treelevel mass to all the standardmodel gauginos.
6. Outlook
No proposal for breaking supersymmetry avoids at present all gravitational tadpoles, for the dilaton, moduli and conformal factor of the metric. Allowing such tadpoles classically is thus a valid alternative, and may be necessary for getting a good approximation of our lowenergy world from the string. This logic opens up a host of possibilities of which perhaps the simplest one, compactification of string theory on magnetized tori, was studied in this paper in detail. Two remarkable features of such compactifications, namely the reconciliation of chirality with extended lowenergy supersymmetry, and the NielsenOlesentriggered electroweak breaking, should survive in more general settings. It should, in particular, be possible to extend the results of this paper to compactifications of the heterotic string on products of magnetized twospheres. As in the limiting field theory [21], the spectrum in the full string theory should also be calculable exactly, because string motion on a magnetized sphere can be described by a WZW model [22, 23]. The heterotic embedding can, however, be subtle. Whether the quantum string dynamics can stabilize such a vacuum, without going to negativelycurved supersymmetric spacetimes [43] is of course the big and open question. The other major difficulty, shared by the ScherkSchwarz scenario [13, 15] and other marginal deformations of classical string vacua [25], comes from the fact that is tied to the size of compact dimensions. In the compactification of section 5, for example, the entire string lies just beyond the supersymmetry threshold! Needless to say this would have dramatic consequences, such as infinite towers of mirror fermions shifted relative to each other by . Unfortunately, it also poses a threat to perturbative calculability, which as I explained above cannot be addressed by studying the exact supersymmetric limit.
Aknowledgements
I have benefited from discussions with I. Antoniadis, P. Fayet, I. Pavel and A. Zaffaroni. This research was supported in part by EEC grants SC1CT920792 and CHRXCT930340.
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