Neutrino Masses and Mixings from String Theory Instantons
aa r X i v : . [ h e p - ph ] J un IFT-UAM/CSIC-07-35
Neutrino Masses and Mixingsfrom String Theory Instantons
S. Antusch , , L.E. Ib´a˜nez and T. Macr`ı , Departamento de F´ısica Te´orica C-XI and Instituto de F´ısica Te´orica C-XVI,Universidad Aut´onoma de Madrid, Cantoblanco, 28049 Madrid, Spain Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut)F¨ohringer Ring 6, D-80805 M¨unchen, Germany Dipartimento di Fisica ’G. Galilei’, Universit`a di PadovaVia Marzolo 8, I-35131 Padua, Italy
Abstract
We study possible patterns of neutrino masses and mixings in string models in which Majorananeutrino masses are generated by a certain class of string theory instantons recently consideredin the literature. These instantons may generate either directly the dim=5 Weinberg operator orright-handed neutrino Majorana masses, both with a certain flavour-factorised form. A hierarchyof neutrino masses naturally appears from the exponentially suppressed contributions of differentinstantons. The flavour structure is controlled by string amplitudes involving neutrino fields andcharged instanton zero modes. For some simple choices for these amplitudes one finds neutrinomixing patterns consistent with experimental results. In particular, we find that a tri-bimaximalmixing pattern is obtained for simple symmetric values of the string correlators.
Introduction
Recently a new mechanism for the generation of neutrino Majorana masses in thecontext of string theory has been pointed out [1,2,3,4]. Certain string instanton effectscan generate right-handed neutrino masses from operators of the form e − U M ν R ν R M string . (1.1)Here M s is the string scale and U M is a complex scalar modulus field whose axion-likeimaginary part Im U M gets shifted under a gauged U (1) B − L symmetry in such a waythat the operator (1.1) is U (1) B − L gauge invariant. The size of these masses is oforder exp( − Re U M ) M s . Unlike ordinary, e.g., electroweak instanton effects which areof order exp( − /α ), these instantons need not be very much suppressed, Re U M is notthe inverse of any SM gauge coupling and may be relatively small. Thus right-handedneutrino masses may be large, as required phenomenologically. Furthermore it wasnoted [4] that analogous instantons can also generate a dimension 5 Weinberg operatorof the form e − U W M s HLHL . (1.2)This term gives rise directly to left-handed neutrino masses once the Higgs scalarsget a vev. Both these instanton effects only appear in a restricted class of stringcompactifications in which the SM gauge group is extended by a U (1) B − L gauge bosonwhich is massive through a Stuckelberg mass term. String compactifications in whichsuch instanton mechanism is operative have been recently discussed in [1, 2, 3, 4].In the present paper we make a first phenomenological exploration of the structureof neutrino masses and mixings obtained from this string instanton mechanism. In thisanalysis we will concentrate on a particular class of instantons, those with internal Sp (2)Chan-Paton (CP) symmetry which leads to the simplest structure and appear mostoften in available instanton searches [4]. For such instantons the flavour dependenceof both ν R -masses and the Weinberg operator factorises as product of flavour vectors(called d a and c a (a=1,2,3) in the main text for the ν R -masses and Weinberg operatorrespectively). These flavour vectors d a , c a may be in principle computable in terms ofthe specific underlying string compactification. This simple flavour structure and thefact that one expects several different instantons contributing to the amplitude makeit quite natural to obtain a hierarchy of neutrino masses [4].The structure of this paper is as follows. In the next section, section 2, we present abrief overview of the string instanton mechanism which is relevant for the generation ofneutrino Majorana masses. We discuss how the operators in Eqs. (1.1) and (1.2) may be1enerated as well as their flavour structure and the expected size of the neutrino masses.Turning to the phenomenological analysis in section 3, in section 3.1 we consider thecase in which the Weinberg operator is dominant compared to the see-saw contribution.We also assume in a first approximation that the large mixing in the leptonic sectororiginates in the neutrino mass matrix (and not in the charged leptons). In this case thephysical neutrino mass matrix is directly obtained from the discussed instanton effectsand the analysis is much easier. We show that, if there is a hierarchy of neutrinomasses (naturally induced by the above-named instanton effects), then one can obtaina neutrino mixing matrix consistent with experimental results for certain (not verystringent) constraints on the values of the flavour vectors c a . We also show that forflavour vectors c a along particular directions one can reproduce, e.g., tri-bimaximalmixing both for the normal and inverse hierarchy cases.We then consider the case in which the see-saw contribution to neutrino masses isdominant in section 3.2. In this case the final result for the physical neutrino massesdepends on the structure of the Dirac mass for the neutrinos. This makes the analysismore model-dependent. We consider a simple case in which the Dirac mass matrix isdiagonal. In this case one can obtain e.g. tri-bimaximal mixing if the flavour vector co-efficients d a of different contributing instantons align along certain directions in flavourspace. The case in which both the Weinberg operator and see-saw mechanism arerelevant is briefly discussed in section 3.3. Some final conclusions and some commentsare left to section 4. In large classes of string compactifications the gauge group of the SM fields includesan extra U (1) B − L gauge interaction. This is to be expected since U (1) B − L is theunique flavour-independent U (1) symmetry which is anomaly free (in the presence ofthree right-handed neutrinos ν aR , needed also for the cancellation of mixed U (1) B − L -gravitational anomalies). In fact practically all semi-realistic MSSM-like string com-pactifications constructed up to now have such an extra U (1) B − L interaction and threeright-handed neutrinos.We will focus on this class of string compactifications with an extra U (1) B − L . Ofcourse, such a gauge interaction forbids the presence of Majorana masses for neutrinos,since they would violate U (1) B − L gauge invariance. However, as pointed out in [1](see also [2]), string instanton effects may give rise to right-handed neutrino Majorana2asses under certain conditions. In particular a crucial point is that the U (1) B − L gauge boson should get a Stuckelberg mass from a B ∧ F type of coupling. Here B is a2-index antisymmetric field and F is the U (1) B − L field strength. This mechanism isubiquitous in string theory and it plays an important role in U (1) anomaly cancellationby the Green-Schwarz mechanism (for a simple discussion see e.g. [5]). Due to thepresence of the B ∧ F coupling, the pseudo-scalar η (dual to the B field) transformsunder a U (1) B − L gauge transformation of parameter Λ( x ) as: η ( x ) −→ η ( x ) + q Λ( x ) , (2.1)with q being some integer. The η scalar has then a Higgs-like behavior and gives a massof order the string scale M s to the U (1) B − L gauge boson. Thus, from the low-energypoint of view the gauge symmetry is just that of the SM (or possibly e.g. a SU (5)extension).As pointed out in [1, 2] in this class of models string instantons can give rise toterms of the form W ≃ e − U ins ν R ν R , (2.2)which give rise the right-handed neutrino Majorana masses. Here U ins is a complexmodulus scalar field characteristic of the instanton and the particular compactification.The point is that Im U ins is a linear combination of axion-like fields including η in sucha way that under a U (1) B − L gauge transformation transforms like Im U ins ( x ) −→ Im U ins ( x ) − x ) . (2.3)Then the operator exp( − U ins ) has effective B-L charge=2 and the operator (2.2) isgauge invariant, the gauge transformation of the neutrino bilinear is canceled by theexponential.This type of instanton contributions may appear in all 4-dimensional string con-structions but it is particularly intuitive in the case of Type IIA orientifolds with inter-secting D6-branes [6,7,8] (see e.g. [9] for reviews and references). These D6-branes havea 7-dimensional worldvolume including Minkowski space. The remaining 3-dimensionswrap a 3-cycle Π of volume V Π in the 6 compact dimensions. In these models quarksand leptons appear as string excitations localised at D6-brane intersections. In thesimplest configurations there are 4 different stacks of such D6-branes a,b,c,d associated These tensors come e.g. from the RR-sector of Type II string theory. In D = 4 they are dual topseudo-scalar fields η r which are the imaginary part of complex scalar moduli fields, either complexstructure moduli U r or Kahler moduli T r , depending on the specific compactification.
3o gauge groups U (3) a × SU (2) b × U (1) c × U (1) d . The U (1) a,d gauge symmetries corre-spond to baryon and lepton number respectively and U (1) c may be identified with thediagonal generator of right-handed weak isospin. Out of these 3 U (1)’s only the linearcombination Y = Q a / − Q c / − Q d / Q a + Q d has triangle anomalies and gets a Stuckelberg massas usual. The remaining orthogonal linear combination Y ′ = Q a / Q c / − Q d / U (1) B − L generator is given by Y + Y ′ .In these intersecting D6-brane models string instantons [10, 11] are D2-branes withtheir 3-dimensional volume wrapping a 3-cycle Π M on the 6 extra dimensions. Thisis just like D6-branes, the main difference being that these D2-branes are localised in D = 4 space and time (that is why they are identified with instantons). The action ofthese instantons is just the D2-brane classical action, which is given by the Born-Infeldaction which yields S D = V Π λ + i X r q M,r η r , (2.4)where V Π is the 3-volume wrapped by the D2-brane, λ is the string dilaton and theimaginary piece is a linear combination with integer coefficients of axion-like RR-fieldscharacteristic of the particular instanton M . For any given compactification and instan-tons S D may be written as a particular linear combination of moduli fields S D = U ins .In the particular case of Type IIA orientifolds with intersecting D6-branes, they arecomplex-structure moduli of the compact manifold.As described in [1, 2, 4] there is in fact an extra contribution to the instanton actionwhich comes from possible intersections of the D2-instanton and the relevant c andd D6-branes (see figure 1 for a pictorial view). Right-handed neutrinos come fromstring excitations around the intersections of d and c branes. On the other hand theD2-instanton may intersect with the d and c branes and at their intersections stringexcitations give rise to fermionic zero modes α i and γ i . We will soon see that for a ν R bilinear to be generated the multiplicity of these modes must be two, i.e., i, j = 1 , α i , γ i are not 3 + 1 dimensional particles, like the ν R ’s but rather 0+0 dimensional zeromodes. They will behave like Grassman variables over which one has to integrate.Indeed, in computing the contribution of instantons to a given amplitude (both withstandard YM instantons and with string instantons of the type here considered) one In the present discussion we will always work in string mass units with M s = ( α ′ ) − = 1,recovering the string mass dimensions for the final neutrino formulae. α γ R D6D6 c d D2 Figure 1: World-sheet disk amplitude inducing a cubic coupling on the D2-brane instanton action. Thecubic coupling involves the right-handed neutrinos lying at the intersection of the c and d D6-branes,and the instanton fermion zero modes α and γ from the D2-D6 intersections. has to integrate over the moduli of the instanton and these α i , γ i zero modes will bepart of it. Now, there are non-vanishing amplitudes among the right-handed neutrinos ν aR and the zero modes which contribute to the instanton action S ins ( α, γ ) = d ija ( α i ν a γ j ) , a = 1 , , . (2.5)Here d ija are coefficients which depend on the Kahler moduli of the compactification.In order to obtain the induced superpotential one has to integrate over the fermioniczero modes α i , γ i and one obtains a superpotential coupling proportional to [1, 2, 4] Z d θ Z d α d γ e − d ija ( α i ν a γ j ) = Z d θ ν a ν b ( ǫ ij ǫ kl d ika d jlb ) , (2.6)where we have made use of the Grassman integration rules R dα = 0, R dαα = 1 etc.Note that the fact that we have two α and γ zero modes is crucial in order to obtain abilinear. This expression is multiplied by the exponential of the classical action (2.4)so that the final expression for the right-handed neutrino Majorana mass has the form M Rab = M s ( ǫ ij ǫ kl d ika d jlb ) exp( − U ins ) , a, b = 1 , , , (2.7)where M s is the string scale and ǫ ij is the 2-index antisymmetric unit tensor. Note thatthe flavour information is encoded in the couplings d ija . As discussed in detail in [4],the relevant D2-instantons have a gauge symmetry which is realised only as a global There are additional factors of order one coming from the quantum fluctuations of massive modes(see e.g. [2, 12]) . We set those terms to one in the present analysis. D = 4 spectrum. The simplest and most frequent situationfound up to now is that the global symmetry is Sp (2) = SU (2), so that the α and γ zero modes are doublets of SU (2). In that situation one can write d ija = d a ǫ ij and theMajorana mass matrix takes a factorised form M Rab = 2 M s X r d ( r ) a d ( r ) b e − U r , (2.8)where the sum goes over the different instantons which may contribute to this Majoranamass term (in general there are several different instantons contributing). As notedin [4], this expression has an interesting flavour structure. Indeed one can write M R ∼ X r e − U r diag ( d ( r )1 , d ( r )2 , d ( r )3 ) · · diag ( d ( r )1 , d ( r )2 , d ( r )3 ) . (2.9)With this structure each instanton makes one particular linear combination of ν R ’ smassive, leaving two linear combinations massless. In particular one(two) instanton(s)contribution(s) would leave two(one) neutrinos massless. Thus with three or morecontributing instantons generically all three get a mass. Furthermore a hierarchy amongthe three eigenvalues may naturally appear taking into account that each instantonwill have in general a different suppression factor exp( − Re U r ). This will be one of thecrucial ingredients of our phenomenological analysis in the next sections.Once (large) right-handed neutrino masses are generated the standard see-sawmechanism [13] is expected to induce Majorana masses for the lightest eigenvaluesin the usual way, i.e. neutrino masses of the form M Lab (see-saw) = (cid:10) H (cid:11) M s h TD ( X r d ( r ) a d ( r ) b e − S r ) − h D , (2.10)where h D is the ordinary Yukawa coupling constant in h abD ( ν aR ¯ HL b ). The eigenvaluesof these matrices are the ones which should be compared with experiment.As we mentioned, there is a second lepton number violating operator which can berelevant for the structure of neutrino masses. This is the dim=5 Weinberg operator(in superpotential form) W W = λ ab M ( L a HL b H ) . (2.11)Once the Higgs fields get a vev, left-handed neutrino masses of order (cid:10) ¯ H (cid:11) λ ab /M aregenerated. One important aspect of this operator is that it does not involve any fieldbeyond those of the SM (not even ν R ’ s) and does not directly involve the see-saw6 bc d β δ LH − Figure 2: World-sheet disk amplitude inducing a quartic coupling on the D2-brane instanton action.The coupling involves the Higgs H and left-handed leptons L a lying at the intersection of the b , c and d D6-branes, and the instanton fermion zero modes β and δ from the D2-D6 intersections. mechanism. String instantons can again give rise to such a superpotential in the classof string models under consideration. A superpotential of the form W W = e − U ins λ ab M s ( L a HL b H ) . (2.12)may be generated in a way totally analogous to the one discussed above for the ν R Majorana masses. The only main difference is that this time the corresponding D2-instantons (which are different from the ones giving rise to ν R masses) have zero modes β i , δ i with quartic couplings S ins ( β, δ ) = c ija ( β i ( L a H ) δ j ) . (2.13)Again, in the simplest case with a SU (2) symmetry operating in the i, j indices one has c ija = c a ǫ ij and a flavour factorised expression is obtained for the left-handed neutrinoMajorana masses M Lab = 2 (cid:10) H (cid:11) M s X r c ( r ) a c ( r ) b e − U r , (2.14)where again the sum runs over possible different instantons contributing. Note thatthe flavour structure of this mass matrix is the same as that of Eq. (2.9) so that ahierarchy of neutrino masses naturally appears.A comment is in order concerning the relationship between the see-saw mecha-nism and the dim=5 Weinberg operator. For constant field-independent Majorana ν R masses, the exchange of the ν R fields gives rise to a see-saw superpotential contribu-tion to the Weinberg dim=5 term. On the other hand for field dependent masses like7hose generated from instantons, one cannot write down the see-saw contribution inthe form of a Weinberg holomorphic superpotential. So both contributions should beconsidered separately and in fact in the string models different instantons contributeto both effects [4].In a given string compactification both mechanisms (see-saw and direct dim=5 op-erator) may be present. Which is the dominant effect concerning the determination ofthe masses and mixings of the observed neutrinos will be model dependent. In partic-ular it will depend on the particular values of the instanton actions Re U ins , the valueof the string scale M s , the values of the coefficients c a , d a and of the neutrino Yukawacouplings h D . In particular, if the h D couplings are small and there is little suppressionfrom the exponential factors exp( − U r ), the Weinberg operator might be dominant. Aswe will see this case is particularly simple because then one can directly correlate theneutrino flavour structure to the string instanton mass generation formulae. In thesee-saw case the dependence on the string instanton effects is partially masked by thedependence on the flavour dependent h D Yukawa couplings.Looking at formulae (2.8) and (2.14) we see that the obtained neutrino massesdepend on three quantities, the string scale M s , the instanton actions Re U r and theinstanton couplings c a or d a . Before entering into the phenomenological analysis of thefollowing sections let us review what can be said about these quantities. Concerningthe string scale M s , it is in principle undetermined by present data and may be as smallas the TeV scale. On the other hand if we want to keep gauge coupling unificationand other simple features of MSSM-like scenarios, identifying M s with the GUT-scale(i.e. of order 10 GeV) is an attractive option. The Weinberg operator may induceneutrino masses of order 10 − − − eV as long as M s < ( c a ) e − Re U W GeV . (2.15)If this was the dominant source of neutrino masses, this would seem to favor values ofthe string scale below the unification scale 10 GeV. However, if there are a numberof different instantons contributing and
Re U W is small, it could still be computablewith M s of order the unification scale.If the dominant source of observed neutrino masses were the see-saw mechanism,one can obtain neutrino masses of order 10 − − − eV as long as M s < h d a e Re U M GeV , (2.16)where h is the size of the largest neutrino Yukawa coupling. In this case the size of thestring scale is essentially unconstrained. 8oncerning the values of the string instanton actions Re U r , it is important toremark that, unlike the standard YM instantons of electroweak interactions, stringinstanton effects are not particularly suppressed, since Re U r are unrelated to the (in-verse) gauge couplings of any SM interactions. This means that the exponential factorsexp( − Re U r ) appearing in the amplitudes may be in fact of order one or, say, O (1 / Re U r , are genericallydifferent. For example, in section 6.4 of [4] an example of compactification is shown inwhich there are three instantons contributing to right-handed neutrino masses whoseactions are on the ratios 1 : 3 . .
2. The overall normalization depends on the valueof the Type II string dilaton, which is a free parameter in perturbative compactifica-tions. This example illustrates how indeed a hierarchy of neutrino mass eigenvalues ispossible in the present context.Concerning the amplitudes c a and d a , a = 1 , ,
3, they are obtained from string cor-relators involving the chiral field operators ¯ HL a and ν aR respectively and the fermionicinstanton zero modes β i , δ i and α i , γ i , i = 1 ,
2. In the case of intersecting D6-branemodels they are in general functions of the Kahler moduli T k of the compactification.If the string compactification involves a known conformal field theory (CFT) like intoroidal (or orbifold) models or models obtained from Gepner orientifolds [14] (seealso [4] and references therein), c a , d a are computable in principle. In practice onlyfor toroidal models such computations are available at the moment (very much like ithappens with ordinary Yukawa couplings). However, although there have been con-structed Type IIA orbifold orientifolds with intersecting D6-branes and a MSSM-likespectrum (see [9] for reviews and references) none of them have a massive U (1) B − L as required for the present instanton mechanism to work. On the other hand thereare non supersymmetric intersecting D6-brane models in which such massive U (1) B − L gauge bosons occur. As pointed out in [1], for such models the d a ( T k ) amplitudesare analogous to those of ordinary Yukawa couplings [15] and they are typically pro-portional to (products of) Jacobi theta functions θ [ δ a , φ i , T k ). Here δ a are somefractional numbers which depend on the generation number a = 1 , , T k are Kahlermoduli and φ I are scalar moduli fields which parameterise the location of the D-branesin extra dimensions. However, the non-SUSY toroidal examples discussed in [1] needto be completed since they require the presence of further backgrounds in order to getthe adequate number of instanton zero modes for the neutrino mass operators to begenerated. Still this at least illustrates what type of functions could appear in more9ealistic computations. For example, we will see in the phenomenological applicationsin the next section that, e.g., in order to have a small θ in the neutrino mixing matrix,a suppressed c amplitude for the leading instanton would be required. So we mightwant to impose such a condition on candidate string compactifications. In this con-nection it is perhaps worth noticing that Jacobi theta functions do vanish in particularsymmetric points.Given our discussion above, our approach in the present paper concerning the am-plitudes c a , d a will be mostly phenomenological. We will address ourselves the question:under what circumstances are the present class of instanton induced neutrino massesconsistent with present experimental constraints? In the next section we will see thatunder very mild constraints on the c a coefficients the instanton generated Weinbergoperator will be consistent with present experimental data on neutrino masses andmixings. Furthermore we will see that if the c a amplitudes go along certain directionsin flavour space, e.g., tri-bimaximal neutrino mixing may be obtained. In general, the leptonic mixing matrix (PMNS matrix), is given in the standard PDGparameterisation as U PMNS = c c s c s e − iδ − c s − s s c e iδ c c − s s s e iδ s c s s − s c c e iδ − s c − s c s e iδ c c P Maj , (3.1)where we have used the abbreviations s ij = sin( θ ij ) and c ij = cos( θ ij ). Here δ isthe so-called Dirac CP violating phase which is in principle measurable in neutrinooscillation experiments, and P Maj = diag( e i α , e i α ,
0) contains the Majorana phases α , α . The PMNS mixing receives contributions from the matrix V e L diagonalizingthe mass matrix of the charged leptons and from V ν L diagonalizing the neutrino massmatrix, U PMNS = V e L V † ν L . (3.2)In the following, we assume that the large mixing in the lepton sector originates fromthe neutrino mass matrix. In fact such large mixings are generically expected in thepresent context, given the very different origins of the neutrino masses from stringinstantons, compared to the masses for charged leptons and quarks. Under this as-10umption, we may treat the small mixings of the charged lepton mass matrix as aperturbation. We first discuss the case that the contribution from the dimension 5 Weinberg operatordominates the neutrino mass matrix. As we said, in this case the left-handed neutrinomass matrix is directly related to the instanton contribution discussed in the previoussection.
When the Weinberg operator dominates, the instanton-induced neutrino mass matrixcan be written in the form M Lij = X r I r c ( r ) i c ( r ) j , (3.3)where we have defined I r = (cid:10) H (cid:11) M s e − S r . (3.4)Let us consider first the scenario in which we have three instanton contributions tothe neutrino mass matrix ( r = 1 , , | I c (3) i c (3) j | ≫ | I c (2) i c (2) j | ≫ | I c (1) i c (1) j | . (3.5)This may be motivated by a hierarchy of the instanton factors, e − Re ( S ) ≫ e − Re ( S ) ≫ e − Re ( S ) , (3.6)As we mentioned in the previous section, such modest hierarchies are likely in orien-tifold compactifications (see e.g. [4]). In this situation we can extract analytically theconditions under which the generated neutrino masses and mixing angles are compat-ible with the experimental results.In order to simplify the following discussion of the leptonic mixing angles and CPphases resulting from neutrino masses of the form in Eq. (3.3), we define c (3) i = p I c (3) i , c (2) i = p I c (2) i , c (1) i = p I c (1) i , (3.7)11nd furthermore φ (3) i = arg( c (3) i ) , φ (2) i = arg( c (2) i ) , φ (1) i = arg( c (1) i ) , (3.8)for i = 1 , ,
3. In the limit of Eq. (3.5), and using that the observed mixing angle θ is small, the mixing angles of the PMNS matrix (using the standard PDG parameter-isation [16]) are then given as tan θ ≈ | c (3)2 || c (3)3 | (3.9)tan θ ≈ | c (2)1 | c | c (2)2 | cos ˜ φ − s | c (2)3 | cos ˜ φ (3.10) θ ≈ e i ( ˜ φ + φ (2)1 − φ (3)2 ) | c (2)1 | ( c (3) ∗ c (2)2 + c (3) ∗ c (2)3 )[ | c (3)2 | + | c (3)3 | ] + e i ( ˜ φ + φ (3)1 − φ (3)2 ) | c (3)1 | q | c (3)2 | + | c (3)3 | , (3.11)where we have defined ˜ φ = φ (2)2 − φ (2)1 − ˜ φ + δ , (3.12)˜ φ = φ (2)3 − φ (2)2 + φ (3)2 − φ (3)3 − ˜ φ + δ . (3.13) δ and ˜ φ are determined from the condition/convention that tan θ and θ are real andpositive, respectively. Under the above conditions, the neutrino masses are given by m = ( | c (3)2 | + | c (3)3 | ) , (3.14) m = | c (2)1 | s , (3.15) m = O ( | c (1) | ) . (3.16)We note that from a technical point of view, the procedure which has been usedfor extracting the neutrino parameters is equivalent to the one for see-saw models ofneutrino masses with sequential right-handed neutrino dominance [17]. However, it isapplied here in a different physical context, namely that of neutrino masses from stringtheory instantons which generate the Weinberg operator.Let us now turn to the conditions for consistency with experiment. The presentexperimental status is summarised in Tab. 1. We see that under the “sequential dom-inance” assumptions of Eq. (3.5) the following general conditions are imposed on theparameters c ( r ) i : 12est-fit value Range C.L. θ [ ◦ ] 33 . . − . σ ) θ [ ◦ ] 45 . . − . σ ) θ [ ◦ ] − . − . σ )∆ m [eV ] 7 . · − . · − − . · −
99% (3 σ ) | ∆ m | [eV ] 2 . · − . · − − . · −
99% (3 σ ) Table 1: Experimental results for the neutrino mixing angles and mass squared differences, takenfrom the recent global fit of Ref. [18] to the present neutrino oscillation data. • Large, nearly maximal, mixing θ ≈ π/ | c (3)2 | ≃ | c (3)3 | . • Large (but non-maximal) θ implies that | c (2)1 | ≃ | c (2)2 | ≃ | c (2)3 | , or at least that | c (2)1 | and either | c (2)2 | or | c (2)3 | are of the same order. • Small θ requires that | c (3)1 | / | c (3)3 | is small.Generically, coefficients c ( r ) i of O (1) are a typical expectation in the present scheme.This means, large mixings are not only easy to accommodate, but are even expectedin the considered scenario. However, the explicit values depend on the details of themodel, and small (or even vanishing) values for the c ( r ) a amplitudes may emerge in par-ticular examples. The condition | c (3)1 | ≪ | c (3)3 | may thus give us information/constraintson which string constructions may be fully successful in describing the neutrino data.For a hierarchical neutrino spectrum, the conditions of Eq. (3.5) imply that theparticular parameters c (1) i (corresponding to the most suppressed instanton effect) donot play a significant role for the mixing angles. In fact, only two instantons arerequired to give masses m and m to two linear combination of neutrinos fields, whileone of the neutrinos could remain massless. The remaining constraint is that the twoinstanton contributions proportional to e − S and e − S have to generate neutrino masses m ≈ p ∆ m and m ≈ p | ∆ m | . One of the most popular proposed structures for neutrino mixing is that of tri-bimaximalmixing [19]. We would like now to study under what conditions the neutrino massmatrix from string theory instantons via the Weinberg operator, i.e. of the form ofEq. (3.3), can give rise to tri-bimaximal lepton mixing. Tri-bimaximal lepton mixing13s a pattern of neutrino mixing angles postulated by [19], where the PMNS matrix isgiven by U tri = p / / √ − / √ / √ − / √ − / √ / √ / √ . (3.17)In the standard PDG parameterisation [16], this corresponds to θ = arcsin(1 / √ ≈ . ◦ , θ = 0 and θ = arcsin(1 / √
2) = 45 ◦ in the lepton sector. The PMNS matrixis usually given in the basis where the so-called “unphysical phases” are eliminated byabsorbing a global phase factor in the definition of the lepton doublets. Since in ourcase neutrinos have Majorana masses, the PMNS matrix is multiplied by an additionalphase matrix P Maj = diag( e i α , e i α ,
0) from the right, which contains the Majoranaphases α , α . As stated earlier, we assume that the large mixing in the lepton sectororiginates from the neutrino mass matrix, such that we may treat the small mixingsof the charged lepton mass matrix as a perturbation. We will consider the minimalcase of two instantons, the minimal number required in order generate two massiveneutrinos.Let us try to find an example for tri-bimaximal mixing of the neutrino mass matrixfrom instantons. To start with, we may assume a normal hierarchy for the neutrinomasses, i.e. m ≪ m ≈ m , and set m to zero. Using the expression (3.17) oneobtains for the the neutrino mass matrix with tri-bimaximal mixing M tri = U tri diag(0 , m e iα , m ) U T tri = m e iα + m − − . (3.18)By comparing this form with Eq. (3.3), we see that a possibility to obtain this structureis to identify I = m , I = m and to choose the coefficients c (2) i , c (3) i as( c (2)1 , c (2)2 , c (2)3 ) = 1 √ e i α (1 , , , (3.19)( c (3)1 , c (3)2 , c (3)3 ) = 1 √ , − , . (3.20)We would like to remark that this is one particular possibility, not the most generalcase. However, with a hierarchy among the neutrino masses and a hierarchy amongthe instanton contributions, it is suggestive that one instanton generates m and the14ther one m . Note also that the “normalisation” of the “flavour vectors” c ( r ) i can bechanged to c ( r ) → N c ( r ) by choosing I r = N − m r instead of I r = m r .We see that obtaining precisely the structure of tri-bimaximal mixing requires theflavour vectors c (2) a , c (3) a to align along specific flavour directions . On the other handobtaining masses and mixings compatible with experiment require much milder con-straints on the flavour vectors, as we discussed in the previous sections. Experimentally, two possibilities for the ordering of the neutrino masses are allowed:The so-called normal ordering where m < m < m , and the so-called inverse orderingwhere m < m < m . If in the latter case m ≪ m . m , the neutrino spectrum iscalled inverse hierarchical. String theory instantons can in principle also give rise tothis scenario. However, we have to keep in mind that the splitting between m and m is very small, and that it would have to be explained why m ≈ m . Within the stringinstanton point of view, this would require the presence of two instantons D , D with approximately the same actions S (1) , S (2) but with very different flavour vectors c (1) a , c (2) a . Since the action is given essentially by the size of the wrapped 3-volumein extra dimensions and the latter are expected to be generically different, a certainamount of fine-tuning would be required. Different values for the different actions S ( r ) ,typically leading to some hierarchy seems more generic. Aiming at completeness, wewill nevertheless consider the inverse hierarchy case as well.Examples of patterns for the relevant coefficients c (1) i and c (2) i in this case can befound easily following the strategy used in the above subsection. Since general analyticformulae are rather lengthy, we will focus on a particular example here, noting thatmany variations and alternative patterns are possible and allowed by the experimentaldata. As above, we consider the example of tri-bimaximal mixing since approximatetri-bimaximality is well compatible with the present experimental data. For the inversehierarchy case, in principle tri-bimaximal mixing (and other patterns of neutrino mixingangles compatible with experiment) could be realised as well. Setting m = 0, the Note in particular that in order to exactly reproduce the tri-bimaximal mixing matrix there shouldbe three instantons with flavour vectors c ( r ) a aligning along the Cartan subalgebra generators of a U (3)group. M tri = U tri diag( m e iα , m e iα , U T tri = m e iα − − − − + m e iα , (3.21)and only two string instantons are required (in the most minimal case) to give neutrinomasses m and m . Again, comparing with Eq. (3.3) we see that a possible choice is I = m , I = m (with I ≃ I ) and( c (1)1 , c (1)2 , c (1)3 ) = 1 √ e i α ( − , , , (3.22)( c (2)1 , c (2)2 , c (2)3 ) = 1 √ e i α (1 , , . (3.23) We now turn to the general case, which includes the cases of quasi-degenerate (orpartially-degenerate) neutrino masses with m , m , m non-zero and with typically twoof them being nearly degenerate in mass. In our scheme one can in principle accom-modate this scenario as well. However, now the splitting between m , m and m are very small, and this almost degeneracy of the mass eigenvalues would have tobe explained. Explicitly, the masses have to satisfy the experimental constraints, i.e. m − m ≈ . × − eV , | m − m | ≈ . × − eV (c.f. Tab. 1), while m ≈ m ≈ m are much larger than the mass splitting. From the string instanton point of view, thiswould require again having three different instanton with almost identical action butvery different flavour vectors. Although possible such situation would require some finetuning and is generically unexpected.In order to give an example for a pattern of c ( r ) i compatible with the experimentallyfound mixing angles, let us consider again the concrete example of tri-bimaximal mix-ing. Tri-bimaximal mixing for quasi-degenerate neutrinos could be realised with threeinstantons with c (1) , c (2) , c (3) , chosen as( c (1)1 , c (1)2 , c (1)3 ) = 1 √ e i α ( − , , , (3.24)( c (2)1 , c (2)2 , c (2)3 ) = 1 √ e i α (1 , , , (3.25)( c (3)1 , c (3)2 , c (3)3 ) = 1 √ , − , . (3.26)and with I = m , I = m , I = m (and I ≃ I ≃ I )..16 .2 The see-saw case Up to now we have considered the case in which the leading contribution to the ob-served neutrino masses comes from the Weinberg operator. Let us consider now theinverse case in which the see-saw mechanism gives the leading contribution to neutrinomasses. In general, the see-saw contribution to the neutrino mass matrix can dependsignificantly on the structure of the neutrino Yukawa matrix h D , leading to a largevariety of possible patterns of h D and M Rab consistent with the experimental neutrinodata (assuming again small charged lepton mixing, as before).Obtaining analytic formulae for the most general see-saw case is difficult. In thefollowing, we discuss the special case in which only small mixing stems from the neu-trino Yukawa matrix h D . Explicitly, we will consider the limit that h D is diagonal,i.e. h D = diag( y ( ν ) e , y ( ν ) µ , y ( ν ) τ ) . (3.27)Small mixing induced by h D may be treated as a perturbation and can be includedin a straightforward way. We will furthermore assume that the see-saw contributiondominates the neutrino mass matrix.The neutrino mass matrix is then given by M Lab = X r d ( r ) a ( h TD ) aa d ( r ) b ( h D ) bb e I − r ! − , (3.28)where we have introduced e I r = (cid:10) ¯ H (cid:11) M s e − S r . (3.29)and which defines the quantity f M Lab := ( M Lab ) − = X r d ( r ) a ( h TD ) aa d ( r ) b ( h D ) bb e I − r . (3.30)The indices of the matrix f M Lab must be understood as those coming from the numbers d ( r ) a and d ( r ) b , while those of the matrix M Lab must be calculated as in the usual matrixcalculus.As an example, we now discuss how to choose the coefficients d ( r ) a in order to realisetri-bimaximal neutrino mixing. We note that this procedure can be readily generalisedto any other desired pattern of neutrino mixings. We first observe that if we find d ( r ) a such that U tri diagonalises f M Lab , then also M Lab has tri-bimaximal form, U T tri f M L U tri = diag( m , m , m ) ⇒ U T tri ( f M L ) − U tri = diag( m , m , m ) , (3.31)17ince U tri is orthogonal. The form of f M L required to realise tri-bimaximal mixing isthus given by f M L = U tri diag( m e iα , m e iα , m ) U T tri (3.32)16 m e iα − − − − + 13 m e iα + 12 m − − . A possible choice for the d ( r ) a is therefore (analogous to the Weinberg operator cases) e I = m , e I = m , e I = m and d (1)1 y ( ν ) e , d (1)2 y ( ν ) µ , d (1)3 y ( ν ) τ ! = 1 √ e − i α ( − , , , (3.33) d (2)1 y ( ν ) e , d (2)2 y ( ν ) µ , d (2)3 y ( ν ) τ ! = 1 √ e − i α (1 , , , (3.34) d (3)1 y ( ν ) e , d (3)2 y ( ν ) µ , d (3)3 y ( ν ) τ ! = 1 √ , − , . (3.35)As discussed for the Weinberg operator case, only two of the right-handed neutrinosare relevant in the limit of the normal and inverse hierarchy cases.We see that if ( h D ) aa = y ( ν ) a , a = e, µ, τ , are hierarchical, then also the d ( r ) a (for all r ) would have to have a very similar hierarchical structure, in order to generate largeneutrino mixing. Although possible in principle, this would be a significant constrainton models. On the other hand, with ( h D ) aa = y ( ν ) a being all of the same order, theconditions of Eq. (3.33) could be comparatively easier to satisfy and large neutrinomixing angles would be a generic expectation.A different possibility would of course be that only small mixing is induced by M R and that large mixing originates from h D . In this case, we recover the knownconditions on h D and M R discussed extensively in the literature on conventional see-saw models [20]. More generally, instantons may generate the Weinberg operator for neutrino masses,which provides a direct mass term for (some of) the three light neutrinos after EW sym-metry breaking, as well as the Majorana mass matrix for the right-handed neutrinos.The full neutrino mass matrix M has dimension 6 × M = M L vh TD vh D M R ! . (3.36)18eyond the discussion of the previous sections, there is the possibility that the contri-butions from the see-saw mechanism and from the Weinberg operator both contributewith similar strength to the mass matrix of the light neutrinos. For example, one mayhave the case that one of the contributions generates the dominant term in Eq. (3.18),while the other generates the sub-dominant one.Finally, it is also possible in principle that some of the right-handed neutrinos couldobtain rather small masses, such that there are more than three light neutrino masseigenstates (or right-handed neutrinos close to the EW scale). In specific string models,all ingredients of the neutrino mass matrix M are (in principle) computable. If suchmore unconventional scenarios should appear as predictions, a more careful analysis ofconstraints from oscillation experiments, electroweak decays and cosmological observa-tions would be required to test consistency of such a string model with respect to theneutrino sector data. In this paper we have explored the structure of neutrino masses originating from certainstring theory instanton effects recently pointed out in the literature [1, 2, 4]. Theyappear in string compactifications in which the SM group is extended by a U (1) B − L getting a Stuckelberg mass. Our analysis has concentrated in the simplest class of suchinstantons with a Sp (2) Chan-Paton symmetry. These instantons lead to a certainflavour-factorised form for both, the ν R mass matrix and the Weinberg operator. Ahierarchy of neutrino masses naturally appears from the different values of the actionsfor the different contributing instantons. For the case that the Weinberg operatorgives rise to the leading contribution to neutrino masses, we have shown how onecan reproduce the experimental patterns for neutrino masses and mixings under notvery restrictive conditions on the instanton amplitudes c ( r ) a . For particular directionsof these flavour vectors c ( r ) a one may reproduce, for example, the structure of tri-bimaximal mixing. This is true both for normal and inverted hierarchy cases, althoughthe latter seems more unlikely within the present scheme. In the opposite case in whichthe see-saw mechanism gives rise to the leading contribution, the structure of neutrinomasses depends strongly on the form of the Dirac neutrino mass matrix. In a simplifiedsituation with a diagonal Dirac mass matrix one can obtain, e.g., tri-bimaximal mixingif the flavour vector coefficients d ( r ) a align along certain flavour directions. The oftenassumed situation with a diagonal ν R mass matrix and mixing originated in the Dirac19ector is also possible.A number of extra possibilities should be explored. Other classes of string in-stantons [4] with Chan-Paton symmetries O (1) and U (1) in general do not lead toa factorised flavour dependence of both ν R masses and Weinberg operator. It wouldbe interesting to explore the phenomenological possibilities for these other classes ofinstantons. From the string model building point of view, it would be important tolearn more about the structure and flavour dependence of the flavour vectors c ( r ) a and d ( r ) a in particular string compactifications. To do that a search for models with anextra U (1) B − L gauge boson which becomes massive through a Stuckelberg is required.Getting a neutrino spectrum consistent with experimental constraints would be a newimportant test of string models.One assumption we have made is that the contribution to the leptonic mixingmatrix from the mass matrix of the charged leptons is small. This condition is satisfiedin many well motivated phenomenological models, where there is only small mixingin the mass matrices of quarks as well as charged leptons. We note however that ingeneral, large mixing can as well stem from the charged lepton sector (see e.g. [21]) orfrom a combination of both, neutrino and charged lepton contributions. The conditionsderived in this letter can be readily generalised to these scenarios as well. For the caseof small mixing from the charged lepton sector, the charged lepton contributions canbe treated as corrections to the neutrino mixing angles and CP phases (see e.g. [22]).The general conditions for consistency with neutrino data do not change due thesesmall corrections. In the case that the charged lepton mixing matrix is CKM-like,i.e., small and dominated by a 1-2 mixing, and for small 1-3 mixing in the neutrinomass matrix, the neutrino mixing sum rule θ − θ cos( δ ) ≈ θ ν [22, 23, 24] holdsbetween the measurable PMNS parameters θ , θ , δ and the theoretical prediction forthe 1-2 mixing angle θ ν from the diagonalisation of the neutrino mass matrix. Thus,the prediction for the neutrino mixing angle θ ν , which is directly connected to thestring instantons, can be tested by precisely measuring θ , θ and δ in future neutrinoexperiments [25].Regarding leptogenesis, there is one conceptually interesting fact: All of leptogenesiswould have its origin in instantons. ν R masses would come from string instantons, andthe transformation of lepton into baryon asymmetry would be due to SU (2) L gaugeinstantons. Leptogenesis would proceed via the out-of-equilibrium decay of the right-handed (s)neutrinos, and in the general case both, the other right-handed neutrinos aswell as the Weinberg operator would contribute to the decay asymmetries proportional20o their contribution to the neutrino mass matrix [26]. In this respect is worth notingthat the flavour vectors c ( r ) a , d ( r ) a are in general complex and so will be the generatedneutrino mass matrices. It would be interesting to explore in more detail whether(semi-)realistic string constructions consistent with the low energy neutrino data couldalso give rise to successful baryogenesis via leptogenesis. Acknowledgments
We thank F. Marchesano, B. Schellekens and A. Uranga for useful discussions.T.M. thanks the Instituto de F´ısica Te´orica IFT-UAM/CSIC for hospitality while thiswork was being carried out. This work has been partially supported by the EuropeanCommission under the RTN European Program MRTN-CT-2004-503369, the CICYT(Spain), and the Comunidad de Madrid under project HEPHACOS, P-ESP-00346.
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