UFIFTHEP9819
The case for a
Standard Model with Anomalous
Pierre Ramond
Institute for Fundamental Theory,
Department of Physics, University of Florida
Gainesville FL 32611, USA
Invited Talk at Orbis Scientiae, Miami, FL, December 1977
Abstract
A gauged phase symmetry with its anomalies cancelled by a
GreenSchwarz mechanism, broken at a large
scale by an induced FayetIliopoulos term, is a generic feature of a large class of
superstring theories. It induces many desirable
phenomenological features: Yukawa coupling hierarchy, the emergence of a
small Cabibbolike expansion parameter, relating the Weinberg angle to
unification, and the linking of
Rparity conservation to neutrino masses. Some are discussed in the
context of a threefamily
model which reproduces all quark and lepton mass hierarchies as well
as the solar and atmospheric neutrino oscillations.
1 Introduction
The commonly accepted lore that string theories do not imply robust relations among measurable parameters is challenged in a large class of effective low energy theories derived from string models contain an anomalous with anomalies cancelled by the GreenSchwarz mechanism [1] at cutoff. As emphasized by ’t Hooft long ago, anomalies provide a link between infrared and ultraviolet physics. In these theories, this yields relations between the lowenergy parameters of the standard model and those of the underlying theory through anomaly coefficients. An equally important feature is that as the dilaton gets a vacuum value, it generates a FayetIliopoulos that triggers the breaking [2] of the anomalous gauged symmetry at a large computable scale.
Through the anomalous , the Weinberg angle at cutoff is related to anomaly coefficients [3]. A simple model [4] with one familydependent anomalous beyond the standard model was the first to exploit these features to produce Yukawa hierarchies and fix the Weinberg angle. It was soon realized that some features could be abstracted from the presence of the anomalous : expressing the ratio of downlike quarks to charged lepton masses in terms of the Weinberg angle [5, 6, 7], the suppression of the bottom to the top quark masses [8], relating [9] the uniqueness of the vacuum to Yukawa hierarchies and the presence of MSSM invariants in the superpotential, and finally relating the seesaw mechanism [10] to Rparity conservation.
These theories are expressed as effective lowenergy supersymmetric theories with a cutoff scale . The anomalous symmetry implies:

A Cabibbolike expansion parameter for the mass matrices.

Quark and charged lepton Yukawa hierarchies, and mixing, including the bottom to top Yukawa suppression.

The value of the Weinberg angle at unification.

Three massive neutrinos with mixings that give the smallangle MSW effect for the solar neutrino deficit, and the large angle mixing necessary for the atmospheric neutrino effect.

Natural Rparity conservation linked to massive neutrinos.

A hidden sector that contains strong gauge interactions with chiral matter.
An important theoretical requirement is that the vacuum, in which the anomalous symmetry is broken by stringy effects, be free of flat directions associated with the MSSM invariants, and preserve supersymmetry.
The anomalous also provides a possible explanation of supersymmetry breaking. Since the hidden sector contains a gauge theory with strong coupling, the GreenSchwarz mechanism requires that it have a mixed anomaly as well. This implies that the hidden matter is chiral with respect to the anomalous symmetry. As shown by Binétruy and Dudas [11], any strong coupling gauge theory with chiral fermions breaks supersymmetry (even QCD). A recent analysis [12] improves their mechanism by showing that the dilaton term does not vanish, providing for gaugino masses and possibly solving the FCNC problem.
In the following, we present the generic features of this type of model, and illustrate some in the context of a realistic threefamily model.
2 Applications to the standard model
We consider models which have a gauge structure broken in two sectors: a visible sector, and a hidden sector, linked by the anomalous symmetry and possibly other Abelian symmetries (as well as gravity).
(2.1) 
where is the hidden gauge group, and
(2.2) 
is the standard model. Of the extra symmetries,one which we call , is anomalous in the sense of GreenSchwarz.
The symmetries, , are spontaneously broken at a high scale by the FayetIliopoulos term generated by the dilaton vacuum. This DSW vacuum [2] is required by phenomenology to preserve both supersymmetry and the standard model symmetries.
We assume the smallest matter content needed to reproduce the observed quark and charged lepton hierarchy, cancel the anomalies associated with the extra gauge symmetries, and has a unique vacuum structure:

Three chiral families

One standardmodel vectorlike pair of Higgs weak doublets.

Three righthanded neutrinos ,

Standard model vectorlike pairs,

Chiral fields that are needed to break the three extra symmetries in the DSW vacuum. We denote these fields by .

Hidden sector gauge interactions and their matter, together with singlet fields, needed to cancel the remaining anomalies.
3 Anomalies
When viewed from the infrared, the anomaly constraints put strong restrictions on the low energy theory. In a fourdimensional theory, the GreenSchwarz anomaly compensation mechanism occurs through a dimensionfive term that couples an axion to all the gauge fields. As a result, any anomaly linear in the symmetry must satisfy the GreenSchwarz relations
(3.1) 
where is any gauge current. The anomalous symmetry must have a mixed gravitational anomaly, so that
(3.2) 
where is the energymomentum tensor. In addition, the anomalies compensated by the GreenSchwarz mechanism satisfy the universality conditions
(3.3) 
A similar relation holds for , the selfanomaly coefficient of the symmetry. These result in important numerical constraints, which restrict the matter content of the model. All other anomalies must vanish:
(3.4) 
In terms of the standard model, the vanishing anomalies are therefore of the following types:

The first involve only standardmodel gauge groups , with coefficients , which cancel for each chiral family and for vectorlike matter. Also the hypercharge mixed gravitational anomaly vanishes.

The second type is where the new symmetries appear linearly, of the type . If we assume that the are traceless over the three chiral families, these vanish over the three families of fermions with standardmodel charges. Hence they must vanish on the Higgs fields: with , it implies the Higgs pair is vectorlike with respect to the . It also follows that the mixed gravitational anomalies are zero over the fields with standard model quantum numbers.

The third type involve the new symmetries quadratically, of the form . These vanish by group theory except for those of the form . In general two types of fermions contribute: the three chiral families and standardmodel vectorlike pairs.

The remaining vanishing anomalies involve the anomalous charge .

With familyindependent, and familytraceless, the vanishing of the anomaly coefficients over the three families is assured: so they must also vanish over the Higgs pair. This means that is vectorlike on the Higgs pair. This is an important result, as it implies that the standardmodel invariant (the term) has zero and charges; it can appear by itself in the superpotential, but we are dealing with a string theory, where mass terms do not appear in the superpotential: it can appear only in the Kähler potential. This results, after supersymmetrybreaking in an induced term, of weak strength, as suggested by Giudice and Masiero [13].

The coefficients , . Since standardmodel singlets can contribute to these anomalies, we expect cancellation to come about through a combination of hidden sector and singlet fields.

The coefficient . This imposes an important constraint on the charges on the chiral families.

The coefficients ; with familytraceless symmetries, they vanish over the three families of fermions with standardmodel charges, but contributions are expected from other sectors of the theory.

The building of models in which these anomaly coefficients vanish is highly nontrivial. Finding a set of charges which satisfy all anomaly constraints, and reproduce phenomenology is highly constrained. In the threefamily model it will even prove predictive in the neutrino sector.
3.1 Standard Model Anomalies
In the standard model, we consider three anomalies associated with its three gauge groups,
(3.5) 
when stands for the trace. They can be expressed [8] in terms of the charges of the invariants of the MSSM
(3.6) 
(3.7) 
where is the charge of , that of , that of , and finally that of the term , where are the family indices. Also the mixed gravitational anomaly over the three chiral families is given by
(3.8) 
In theories derived from superstrings, the integer level numbers and are equal, resulting in the equality
(3.9) 
These imply that, as long as these anomaly coefficients do not vanish, the MSSM Yukawa invariants cannot all appear at tree level, as their charges are necessarily nonzero. This means that not all Yukawa couplings can be of the same order of magnitude, resulting in some sort of Yukawa hierarchy.
More specific conclusions can be reached by assuming that the charges are familyindependent and the are familytraceless. As we have seen, the term has vectorlike charges, .
By further assuming that the top quark Yukawa mass coupling occurs at treelevel, we have . This implies that the charge of the down quark Yukawa is proportional to the color anomaly, and thus cannot vanish: the down Yukawa is necessarily smaller than the top Yukawa, leading to the suppression of over , after electroweak breaking! The nonvanishing of the color anomaly implies the (observed) suppression of the bottom mass relative to the top mass.
The second anomaly equation simplifies to
(3.10) 
stating that the relative suppression of the down to the charged lepton sector is proportional to the difference of two anomaly coefficients. The data, extrapolated to near unification scales indicates that there is no relative suppression between the two sectors, suggesting that difference should vanish. Remarkably, the vanishing [3] of that combination fixes the value of the Weinberg angle through the string of relations
(3.11) 
This happens exactly at the phenomenologically preferred value of the Weinberg angle: the unification is related to the value of the Weinberg angle [5]!
The application of the GreenSchwarz structure to the standard model is consistent with many of its phenomenological patterns. However, more can be said through a careful study of the DSW vacuum.
4 The DSW vacuum
When the dilaton acquires its vacuum value, an anomalous Fayetiliopoulos term is generated through the gravitational anomaly. In the weak coupling limit of the string, it is given by
(4.1) 
where is the string coupling constant. This induces the breaking of and below the cutoff.
Phenomenology require that neither supersymmetry nor any of the standard model symmetries be broken at that scale. This puts severe restrictions on the form of the superpotential and the matter fields [9].
The analysis of the vacuum structure of supersymmetric theories is greatly facilitated by the fact that the solutions of the vacuum equations for the Dterms are in onetoone correspondance with holomorphic invariants. This analysis has been recently generalized to include an anomalous FayetIliopoulos term.
In order to get a unique determination of the DSW vacuum, we need as many singlet superfields, , as there are broken symmetries. Only they assume vev’s as a result of the FI term. They are standard model singlets, but not under and . If more fields than broken symmetries assume nonzero values in the DSW vacuum, we would have undetermined flat directions and hierarchies.
We assemble the charges in a matrix , whose rows are the , charges of the fields, respectively. Assuming the existence of a supersymmetric vacuum where only the fields have vacuum values, implies from the vanishing of the terms
(4.2) 
For this vacuum solution to exist, the matrix must have an inverse and the entries in the first row of its inverse must be positive. The solution to these equations naturally provide computably small expansion parameters . In the case when all expansion parameters are the same we can relate their value in terms of standard model quantities
(4.3) 
where is the unified gauge coupling at unification.
Another important consequence is that there is no holomorphic invariant polynomial involving the fields alone. Another is that the sector is necessarily anomalous. Indeed, let us assume that it has no mixed gravitational anomalies. This means that all the charges are traceless over the fields. now the fields form a representation of , and the tracelessness of the charges insures that they be members of . So we are looking for nonanomalous symmetries in , which is impossible except for . If two or more of the charges are the same on the ’s, we could have anomaly cancellation, but then the matrix would be singular, contrary to the assumption of the DSW vacuum. Hence this sector will in general be anomalous.
For a thorough analysis of the vacuum with FY term, we refer the reader to Ref. [14, 9]. Here, we simply note two striking generic facts of phenomenological import. Consider any invariant of the MSSM. It corresponds [15] to a possible flat direction of the nonanomalous supersymmetric vacuum. For that configuration, all its fields are aligned to the same vacuum value, as required by the vanishing of the nonanomalous terms of the standard model symmetries. It follows that the contribution of these terms of the anomalous will be proportional to its charge [16]. In order to forbid this flat direction to appear alone in the vacuum, it is therefore necessary to require that its charge be of the wrong sign to forbid a solution of . This implies a holomorphic invariant of the form , where is a holomorphic polynomial in the ’s. The term equations are not sufficient to forbid this flat direction together with fields. We have to rely on the terms associated with that invariant polynomial, and its presence is needed in the superpotential. Fortunately, phenomenology also requires such terms to appear in the superpotential. This is the first of several curious links between phenomenology and the vacuum structure near unification scales! One can see that the existence of this invariant is predicated on the invertibility of , the same condition for the DSW vacuum.
The second point addresses singlet fields that do not get vev’s in the DSW vacuum. To implement the seesaw mechanism, there must be righthanded neutrinos, . Since they have no vev, their Xcharge must also be of the wrong sign, which allows for holomorphic invariants of the form , where is a positive integer. The case is forbidden as it breaks supersymmetry. Thus . The case generates Majorana masses for these fields in the DSW vacuum. W scale. To single out we need to choose the charges of the to be a negative halfodd integers. To implement the seesaw, the righthanded neutrinos couple to the standard model invariants , which requires that is also a halfodd integer, while all other MSSM invariants have positive or zero integers charges.
5 A ThreeFamily Model
We can see how some of the features we have just discussed lead to phenomenological consequences in the context of a threefamily model [17, 18], with three Abelian symmetries broken in the DSW vacuum. The matter content of the theory is inspired by , which contains two Abelian symmetries outside of the standard model: the first , which we call , appears in the embedding
(5.1) 
The second , called , appears in
(5.2) 
The two nonanomalous symmetries are
(5.3) 
(5.4) 
The family matrices run over the three chiral families, so that are familytraceless. Since , there is no appreciable kinetic mixing between the nonanomalous s.
The charges on the three chiral families in the are of the form
(5.5) 
where are expressed in terms of the charges of (=3/2), that of (=3), and that of the vectorlike pair ,mass term (=3).
The matter content of this model is the smallest that reproduces the observed quark and lepton hierarchy while cancelling the anomalies associated with the extra gauge symmetries:

Three chiral families each with the quantum numbers of a of . This means three chiral families of the standard model, , , , , and , together with three righthanded neutrinos , three vectorlike pairs denoted by + and + , with the quantum numbers of the + of , and finally three real singlets .

One standardmodel vectorlike pair of Higgs weak doublets.

Chiral fields that are needed to break the three extra symmetries in the DSW vacuum. We denote these fields by . In our minimal model with three symmetries that break through the FI term, we just take . The sector is necessarily anomalous.

Other standard model singlet fields.

Hidden sector gauge interactions and their matter.
Finally, the charges of the fields is given in terms of the matrix
(5.6) 
Its inverse
(5.7) 
shows all three fields acquire the same vacuum value.
In the following, we will address only the features of the model which are of more direct phenomenological interest. For more details, the interested reader is referred to the original references [17, 18].
5.1 Quark and Charged Lepton Masses
The Yukawa interactions in the charge quark sector are generated by operators of the form
(5.8) 
in which the exponents must be positive integers or zero. Assuming that only the top quark Yukawa coupling appears at treelevel, a straighforward computation of their charges yields in the DSW vacuum the charge Yukawa matrix
(5.9) 
where is the common expansion parameter.
A similar computation is now applied to the charge Yukawa standard model invariants . The difference is that , its charge does not vanish. As long as , we deduce the charge Yukawa matrix
(5.10) 
Diagonalization of the two Yukawa matrices yields the CKM matrix
(5.11) 
This shows the expansion parameter to be of the same order of magnitude as the Cabibbo angle .
The eigenvalues of these matrices reproduce the geometric interfamily hierarchy for quarks of both charges
(5.12) 
(5.13) 
while the quark intrafamily hierarchy is given by
(5.14) 
implying the relative suppression of the bottom to top quark masses, without large . These quarksector results are the same as in a previously published model [17], but our present model is different in the lepton sector.
The analysis in the charged lepton sector proceeds in similar ways. No dimensionthree term appears and the standard model invariant have charge . For , there are supersymmetric zeros in the and position, yielding
(5.15) 
Its diagonalization yields the lepton interfamily hierarchy
(5.16) 
Our choice of insures that , which guarantees through the anomaly conditions the correct value of the Weinberg angle at cutoff, since
(5.17) 
it sets , so that
(5.18) 
It is a remarkable feature of this type of model that both inter and intrafamily hierarchies are linked not only with one another but with the value of the Weinberg angle as well. In addition, the model predicts a natural suppression of , which suggests that is of order one.
5.2 Neutrino Masses
Neutrino masses are naturally generated by the seesaw mechanism [10] if the three righthanded neutrinos acquire a Majorana mass in the DSW vacuum. The flat direction analysis indicates that their charges must be negative halfodd integers, with preferred by the vacuum analysis. Their standardmodel invariant masses are generated by terms of the form
(5.19) 
where is the cutoff of the theory. In the matrix element. The Majorana mass matrix is computed to be
(5.20) 
Its diagonalization yields three massive righthanded neutrinos with masses
(5.21) 
By definition, righthanded neutrinos are those that couple to the standardmodel invariant , and serve as Dirac partners to the chiral neutrinos. In our model,
(5.22) 
The superpotential contains the terms
(5.23) 
resulting, after electroweak symmetry breaking, in the orders of magnitude (we note )
(5.24) 
for the neutrino Dirac mass matrix. The actual neutrino mass matrix is generated by the seesaw mechanism. A careful calculation yields the orders of magnitude
(5.25) 
A characteristic of the seesaw mechanism is that the charges of the do not enter in the determination of these orders of magnitude as long as there are no massless righthanded neutrinos. Hence the structure of the neutrino mass matrix depends only on the charges of the invariants , already fixed by phenomenology and anomaly cancellation. In particular, the family structure is that carried by the lepton doublets . In our model, since and have the same charges, it ias not surprising that we have no flavor distinction between the neutrinos of the second and third family. In models with two nonanomalous flavor symmetries based on the matrix (5.25) is a very stable prediction of our model. Its diagonalization yields the neutrino mixing matrix [19]
(5.26) 
so that the mixing of the electron neutrino is small, of the order of , while the mixing between the and neutrinos is of order one. Remarkably enough, this mixing pattern is precisely the one suggested by the nonadiabatic MSW [20] explanation of the solar neutrino deficit and by the oscillation interpretation of the reported anomaly in atmospheric neutrino fluxes (which has been recently confirmed by the SuperKamiokande [21] and Soudan [22] collaborations).
Whether the present model actually fits both the experimental data on solar and atmospheric neutrinos or not depends on the eigenvalues of the mass matrix (5.25). A naive order of magnitude diagonalization gives a and neutrinos of comparable masses, and a much lighter electron neutrino:
(5.27) 
The overall neutrino mass scale depends on the cutoff . Thus the neutrino sector allows us, in principle, to measure it.
At first sight, this spectrum is not compatible with a simultaneous explanation of the solar and atmospheric neutrino problems, which requires a hierarchy between and . However, the estimates (5.27) are too crude: since the (2,2), (2,3) and (3,3) entries of the mass matrix all have the same order of magnitude, the prefactors that multiply the powers of in (5.25) can spoil the naive determination of the mass eigenvalues. A more careful analysis shows that even with factors of order one, it is possible to fit the atmospheric neutrino anomaly as well. A welcome byproduct of the analysis is that the mixing angle is actually driven to its maximum value. We refer the reader to Ref.[18] for more details. The main point of this analysis is that maximal mixing between the second and third family in the neutrino sector occurs naturally as it is determined from the structure of the quark and charged lepton hierarchies.
6 RParity
The invariants of the minimal standard model and their associated flat directions have been analyzed in detail in the literature [23]. In models with an anomalous , these invariants carry in general charges, which, as we have seen, determines their suppression in the effective Lagrangian. Just as there is a basis of invariants, proven long ago by Hilbert, the charges of these invariants are not all independent; they can in fact be expressed in terms of the charges of the lowest order invariants built out of the fields of the minimal standard model, and some anomaly coefficients.
The charges of the three types of cubic standard model invariants that violate parity as well as baryon and/or lepton numbers can be expressed in terms of the charges of the MSSM invariants and the Rparity violating invariant
(6.1) 
through the relations
Although they vanish in our model, we still display and , since these sum rules are more general.
In the analysis of the flat directions, we have seen how the seesaw mechanism forces the charge of to be halfodd integer. Also, the FroggattNielsen [24] suppression of the minimal standard model invariants, and the holomorphy of the superpotential require to be zero or negative integers, and the equality of the KácMoody levels of and forces , through the GreenSchwarz mechanism. Thus we conclude that the charges of these operators are halfodd integers, and thus they cannot appear in the superpotential unless multiplied by at least one . This reasoning can be applied to the higherorder / operators since their charges are given by
(6.2)  
(6.3)  
(6.4)  
(6.5) 
It follows that there are no parity violating operators, whatever their dimensions : through the righthanded neutrinos, parity is linked to halfodd integer charges, so that charge invariance results in parity invariance. Thus none of the operators that violate parity can appear in holomorphic invariants: even after breaking of the anomalous symmetry, the remaining interactions all respect parity, leading to an absolutely stable superpartner. This is a general result deduced from the uniqueness of the DSW vacuum, the GreenSchwarz anomaly cancellations, and the seesaw mechanisms.
7 Conclusion
The case for an anomalous extension to the standard model is particularly strong. We have presented many of its phenomenological consequences. In a very unique model, we detailed how the neutrino matrices are predicted. However much remains to be done: the nature of the hidden sector, and supersymmetry breaking. Our model only predicts orders of magnitude of Yukawa couplings. To calculate the prefactors, a specific theory is required. Many of the features we have discussed are found in the context of free fermion theories [25], which arise in the context of perturbative string theory. It is hoped that since they involve anomalies, these features can also be derived under more general assumptions. A particularly difficult problem is that of the cutoff scale. From the point of view of the low energy, there is only one scale of interest, that at which the couplings unify, and the GreenSchwarz mechanism, by fixing the weak and color anomalies, identifies the cutoff as the unification scale. On the other hand, another mass scale appears in the theory through the size of the anomalous FI term, and the two values do not coincide, the usual problem of string unification. It is hoped that the calculation of the FayetIliopoulos term in other regimes will throw some light on this problem.
Our simple model has too many desirable phenomenological features to be set aside, and we hope that a better understanding of fundamental theories will shed light on this problem.
Acknowledgements
I would like to thank Professor B. Kursunoglu for his kind hospitality and for giving me the opportunity to speak at this pleasant conference, as well as my collaborators, N. Irges and S. Lavignac, on whose work much of the above is based. This work was supported in part by the United States Department of Energy under grant DEFG0297ER41029.
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